diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzbfnf" "b/data_all_eng_slimpj/shuffled/split2/finalzzbfnf" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzbfnf" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\n\nCrystal structure prediction is increasingly becoming one of the most effective approaches for discovering new functional materials \\cite{oganov2019structure} due to the ease to obtain new compositions either by enumeration \\cite{davies2016computational}, heuristic knowledge, or the latest deep learning-based generative machine learning models \\cite{dan2020generative}. While the peer protein structure prediction problem has been recently almost solved by the deep learning-based AlphaFold and RossettaFold algorithms, the crystal structure prediction problem remains elusive for a majority of categories of compositions. There are mainly three types of crystal structure prediction approaches including the ab initio based global optimization \\cite{glass2006uspex, wang2015materials,liang2020cryspnet, demir2021ffcasp,falls2020xtalopt}, machine learning-based prediction \\cite{hu2021alphacrystal}, and template-based elemental substitution \\cite{hautier2011data}. The first approach instead depends on computationally expensive DFT calculations and is applicable to only small chemical systems. The second approach is inspired by the AlphaFold family of deep learning algorithms \\cite{senior2020improved,jumper2021highly}, but is only at the early stage of development. The last template-based CSP methods are the most widely used and easiest to implement. Even though this method cannot predict crystal structures of new prototypes, recent deep generative models can discover new prototype materials which can partially address this issue \\cite{zhao2021high}. In a pioneering work \\cite{hautier2011data}, Hautier et al. proposed a data mining-based approach to identify the probabilities for different pairs of ionic substitutions, which can be applied to any prototype structures to generate new structures or used to select templates for template-based crystal structure prediction. However, despite the wide usage of template-based CSP methods, there are many different ways to implement, and there is no working web app\/server that is user-friendly to make it accessible to all materials scientists (The structure predictor of the Materials Project \\cite{ceder2010materials} website was available before but is not functional now). \n\nHere we propose a fast and user-friendly template-based crystal structure prediction algorithm and related companion web server for broad adoption of crystal structure prediction in the daily life of materials science. Our algorithm TCSP is based on the careful selection of template structures based on chemical formula similarity and the match of oxidation states using an exhaustive enumeration strategy. Our predicted structures can be optimized by DFT or machine learning-based structure relaxation. By using seven case studies, we have shown that our user-friendly and fast crystal structure prediction web server has high prediction performance when appropriate templates are available. We also apply our TCSP algorithm to predict the structures of all 98,290 formulas using leave-one-out evaluation and have achieved good performance for a large portion of the targets: more than 13,145 target structures have been found with maximum RMSD less than 0.1. The good performance of this high-throughput crystal structure prediction shows that the template\/prototype-based element substitution CSP approach has big potential in exploratory materials discovery. With the development of large scale prototype databases \\cite{su2017construction,hicks2021aflow} and their applications in the generative design of new crystals \\cite{bushlanov2019topology,sorkun2020artificial}, the performance of our template-based crystal structure prediction algorithm can be further improved.\n\n\n\\section{Method}\n\\label{sec:headings}\n\n\n\\subsection{Template based crystal structure prediction algorithm}\n\nOur template-based crystal structure prediction algorithm is illustrated in Figure~\\ref{fig:predict_algorithm}. Given an input formula (e.g., SrTiO\\textsubscript{3}) with which can be specified by the user or predicted by algorithms \\cite{zhao2020machine,li2021composition} or without space group. We then search structure templates with the same prototype (sometimes called an anonymous formula) (e.g., ABC\\textsubscript{3}) and the same space group if specified. This step may retrieve too many matched templates, so we use Module A1, an Element's mover distance, to measure the composition similarity between the query formula and the compositions of all the template structures, which are then ranked by ascending order. We then pick the top K structures as template candidates with the smallest composition distances. For each of the candidate templates, we use the Pymatgen package to estimate its oxidation states and compare them to those of the query formula. If we find templates with identical oxidation states, we then add them to the final template list. If no such templates are found, we then neglect the oxidation match requirements and directly add them as final templates. The next step is to determine all the possible element substitution pairs between the query formula and the template formula using the algorithm described in Algorithm 2. Next, we will pick the template structure files and replace the elements according to the pair arrangements found by Algorithm 2. A replacement quality score is also calculated for each such element substitution arrangement using the procedure as described in Module 3. The resulting structures will then be subject to DFT or machine learning-based structure relaxation, which can be further used to calculate formation energy, e-above-hull energy, and phonon dispersion for validation. \n\n\n\n\n\\paragraph{Module A1: Element mover's distance for formula similarity calculation}\nWe use the Element's Mover Distance measure $ElMD$ \\cite{hargreaves2020earth} to select most similar template structures. ElMD is a metric that allows measuring the chemical similarity of two formulas in an explainable fashion. The EMD is computed between two compositions from the ratio of each of the elements and the absolute distance between the elements on the modified Pettifor scale (several other element similarities can also be used such as Mendeleev, Petti, Atomic, Mod\\_petti, Oliynyk, Oliynyk\\_sc, Jarvis, Jvarvis\\_sc, magpie,magpie\\_sc, CGCNN, Elemnet, mat2vec, Matscholar, megnet16, random). This metric shows clear strength at distinguishing compounds. It is shown that ElMD distances have greater alignment with chemical understanding than the Euclidean distance. The ElMD is defined in formula (1) below:\n\n\n\\begin{equation}\nElM D(X, Y)=\\min \\sum_{i=1}^{m} \\sum_{j=1}^{n} q_{i j}\\left|p_{i}-p_{j}\\right|\n\\textrm{, subject to} \\quad q_{i j} \\geq 0 \\quad \\textrm{for ${\\forall}$ } i, j\n\\end{equation}\n\n\n\\begin{equation}\n\\sum_{j=1}^{n} q_{i j} \\leq x_{i} \\textrm{, for ${\\forall}$ } 1 \\leq i \\leq m\n\\end{equation}\n\n\\begin{equation}\n\\sum_{i=1}^{m} q_{i j} \\leq y_{j} \\textrm{, for ${\\forall}$ } 1 \\leq j \\leq n\n\\end{equation}\n\n\\begin{equation}\n\\sum_{i=1}^{m} \\sum_{j=1}^{n} q_{i j}=1\n\\end{equation}\n\nwhere the distance is first calculated by matching and pairing each of the $m$ elements in a vector, $X$, to its most similar unmatched partner in the $n$ elements of a second vector $Y$, until all have been paired. The quantity matched, $q$, from the $i$-th element of $X$, to the $j$-th element of $Y$, is given by $q_{ij}$.\n\n\\paragraph{Algorithm A2: element replacement pair enumeration algorithm:} \n\nThis algorithm is used to enumerate all possible element replacement strategies between two pairs of formulas. \n\n\\begin{algorithm}[h]\n \\caption*{Algorithm A2: Element replacement pair enumeration algorithm}\n \\begin{algorithmic}[1]\n \\State Given two formula X,Y, calculate theirs oxidation states then get the statelist and elementlist.\n \n \\If{$stateList_x == stateList_y$}\n \n {Replace element pairs}\n \\Else{\\State Create elementGroupList to represent and distinguish elements of equal state\n \\For {$i = 0,1,\\ldots$}\n \\If{$i == 0$}\n \\State $elementGroup = elementList_y[i]$;\n \\ElsIf {($stateList_y[i] == stateList_y[i-1]$)}\n \\State append $elementList_y[i]$ to $elementGroup$;\n \\Else\n \\State append $elementList_y[i]$ to $elementGroup$;\n \\EndIf\n \\EndFor\n \\State Create $pre\\_patterns$ to represent permutation and combination of elements in $elementGroupList$\n \n \\For{$element =0,1,\\ldots,j$}\n \\If{$j = 0$}\n \\State $pre\\_patterns$ = permutations of $elementGroupList[j]$;\n \\Else\n \\State $patterns$ =permutations of $elementGroupList[j]$;\n \\For{$p1 = 0,1,\\ldots,m$} \n \\For{$p2 = 0,1,\\ldots,n$}\n \\State append $p1,p2$ to $new\\_patterns$;\n \\EndFor\n \\State $pre\\_patterns = new_patterns$;\n \\EndFor\n \\EndIf\n \\EndFor\n \n \\State If the element in $pre\\_patterns$ is different from the element in $elementList_x$, then replace the element\n \n \n \\EndIf\n}\n \\end{algorithmic} \n\\end{algorithm}\n\n\n\\paragraph{Module A3: element substitution scoring function:}\nTo differentiate the resulting structures from different element substitution arrangements between the query formula and the template structure and rank the final output structures from different templates, we use the ElMD distance in Module A1 to calculate the similarity score between each pair of substitution elements for a given query formula and the final structure. Then we sum up these similarity scores for all the element substitution pairs and use them to calculate the quality scores of final structures. \n\n\\begin{equation}\nS_{es}= ElMD(e1,e2), S_{r}= \\sum_{i=1}^{p} S^{i}_{es}\n\\end{equation}\n\n\nWhere $S_{es}$ measures the element matching quality between two substitution elements $e1$ and $e2$. $S_{r}$ is the replacement distance score for a given element substitution arrangement of a given query formula and template structure, which is equal to the sum of all the element similarity scores of all substitution element pairs in the element substitution arrangement.\n\nThe final quality of the generated structures is then measured by the $S_r$ scores, with lower scores corresponding to higher quality. \n\n\n\n\\paragraph{DFT or Genetic algorithm based structure relaxation:}\n\n\nAs with all predicted crystal structures, they usually need a fine-tuning or relaxation step to adjust the local atomic coordinates either using the DFT-based structural relaxation method or the recently developed machine learning and optimization-based relaxation approach \\cite{zuo2021accelerating} which is much faster than the DFT approach. In this study, we used the DFT approach for evaluation purposes.\n\n \n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=0.45\\linewidth]{predict_template.pdf}\n \\caption{Flowchart of our TCSP, a template based crystal structure prediction algorithm. The space group specification is optional. }\n \\label{fig:predict_algorithm}\n\\end{figure}\n\n\\FloatBarrier\n\n\\subsection{User interface of our web server}\n\nOur template-based crystal structure prediction server has a user-friendly web interface, as shown in Figure\\ref{fig:ui}. Each time, a user can just put in a formula\/composition, and then the target space group number from 1 to 230 can be set or just assigned to 0 to allow a template with any space group. Then the user types in their email for receiving the job completion notification email with a downloadable URL link for the predicted structures. After a few minutes, an email will be sent to the user with the download URL link for the predicted structures. After the zipped result file is downloaded and unzipped, the user can go into the folder and click the Name column to sort the files by filename. Then it shows several key files: (1) $results.txt$, which shows the template similarity scores, the templates with compatible oxidation states, and the element replace pairs for each template. (2) $similar\\_formulas.csv$ file shows the distance scores of all templates to the query formula; (3) $TemplateCandiates.csv$ shows the Materials Project IDs of the selected templates. (4) all the remaining $cif$ files are predicted, which are sorted by their replacement quality score (the number before \\_mp of the filename), which is better when the number is smaller. However, it is strongly suggested to validate a couple of top-scored candidate structures as the candidate structure with the top quality score is not always the best one.\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=0.75\\linewidth]{TCSP_UI.png}\n \\caption{User interface of our TCSP web app for crystal structure prediction.}\n \\label{fig:ui}\n\\end{figure}\n\n\n\\subsection{DFT validation of predicted structures}\n\nThe first principle calculations based on the density functional theory (DFT) are carried out using the Vienna \\textit{ab initio} simulation package (VASP) \\cite{Vasp1,Vasp2,Vasp3,Vasp4}. The projected augmented wave (PAW) pseudopotentials, where 520 eV plane-wave cutoff energy, were used to treat the electron-ion interactions \\cite{PAW1, PAW2}. The exchange-correlation functional was considered with the generalized gradient approximation (GGA) based on the Perdew-Burke-Ernzerhof (PBE) method \\cite{GGA1, GGA2}. The energy convergence criterion was set as 10$^{-5}$ eV, while the atomic positions were optimized with the force convergence criterion of 10$^{-2}$ eV\/{\\AA}. The Brillouin zone integration for the unit cells was computed using the $\\Gamma$-centered Monkhorst-Pack $k$-meshes. The Formation energies (in eV\/atom) of several materials were determined based on the expression in Eq.~\\ref{eq:form}, where $E[\\mathrm{Material}]$ is the total energy per unit formula of the considered structure, $E[\\textrm{A}_i]$ is the energy of $i^\\mathrm{th}$ element of the material, $x_i$ indicates the number of A$_i$ atoms in a unit formula, and $n$ is the total number of atoms in a unit formula($n=\\sum_i x_i$).\n\n\\begin{equation}\n E_{\\mathrm{form}} =\\frac{1}{n}(E[\\mathrm{Material}] - \\sum_i x_i E[\\textrm{A}_i])\n \\label{eq:form}\n\\end{equation}\n\n\n\n\n\n\\subsection{Evaluation criteria}\n\n\nTo evaluate the reconstruction performance of algorithm, we define the root mean square distance (RMSD) and mean absolute error (MAE) of two structures as below:\n\\begin{equation}\n \\begin{aligned}\n\\mathrm{RMSD}(\\mathbf{v}, \\mathbf{w}) &=\\sqrt{\\frac{1}{n} \\sum_{i=1}^{n}\\left\\|v_{i}-w_{i}\\right\\|^{2}} \\\\\n&=\\sqrt{\\frac{1}{n} \\sum_{i=1}^{n}\\left(\\left(v_{i x}-w_{i x}\\right)^{2}+\\left(v_{i y}-w_{i y}\\right)^{2}+\\left(v_{i z}-w_{i z}\\right)^{2}\\right)}\n\\end{aligned}\n\\end{equation}\n\n\n\n\\begin{equation}\n \\begin{aligned}\n\\mathrm{MAE}(\\mathbf{v}, \\mathbf{w}) &=\\frac{1}{n} \\sum_{i=1}^{n}\\left\\|v_{i}-w_{i}\\right\\| \\\\\n&=\\frac{1}{n} \\sum_{i=1}^{n}\\left(\\|v_{i x}-w_{i x}\\|+\\|v_{i y}-w_{i y}\\|+\\|v_{i z}-w_{i z}\\|\\right)\n\\end{aligned}\n\\end{equation}\n\nwhere $n$ is the number of independent atoms in the target crystal structure. For symmetrized CIF structures, $n$ is the number of independent atoms of the set of Wyckoff equivalent positions. For regular CIF structures, it is the total number of atoms in the compared structure. $v_i$ and $w_i$ are the corresponding atoms in the predicted crystal and the target crystal structure. \n\n\n\\section{Results}\n\\label{sec:others}\n\n\n\\subsection{Dataset}\n\nWe used more than 130,000 crystal structures deposited in the Materials Project database as our template sources. We picked seven test materials including SrTiO\\textsubscript{3} (mp-5229), Ni\\textsubscript{3}S\\textsubscript{4} (mp-1050), NiS\\textsubscript{2} (mp-849059), GaBN\\textsubscript{2} (mp-1007823), GaB\\textsubscript{3}N\\textsubscript{4} (mp-1019740), GaB\\textsubscript{2}N\\textsubscript{3} (mp-1245554), and Ga\\textsubscript{3}BN\\textsubscript{4} (mp-1019743). We then ran our algorithm and checked if it can predict the correct structures that match the target structures. \n\n\n\\subsection{ Performance of template based crystal structure prediction}\n\nWe selected seven formulas of target structures from materials project database for evaluating the capability of our TCSP algorithm for structure prediction. The first test target formula is SrTiO\\textsubscript{3}, which has three different phases corresponding to space groups of 140, 149, 221. Its most famous structure is the cubic perovskite structure as shown in Figure \\ref{fig:SrTiO3_target}. Our algorithm identified thousands of compatible templates and picked top 10 as templates including BaZrO\\textsubscript{3} (mp-3834), MgTiO\\textsubscript{3} (mp-1016830), CaZrO\\textsubscript{3} (mp-542112), MgZrO\\textsubscript{3} (mp-1017000), BaTiO\\textsubscript{3} (mp-504715), BaTiO\\textsubscript{3} (mp-5020), SrZrO\\textsubscript{3} (mp-613402), CaTiO\\textsubscript{3} (mp-5827), BaTiO\\textsubscript{3} (mp-2998), SrHfO\\textsubscript{3} (mp-4551), among which all are cubic templates except BaTiO\\textsubscript{3}(mp-5020), which is a trigonal structure with space group 160. The top four predicted structures all have a zero rmsd error compared to the perovskite target structure: they all have the identical fractional coordinates as the target structure except that the cubic lengths are different (the predicted cubic structure in Figure \\ref{fig:SrTiO3_predict} has a lattice length of 4.256 \u00c5 while the target structure has a lattice length of 3.945 \u00c5), which may be fine-tuned using DFT based relaxation.\n\nThe second test sample is Ni\\textsubscript{3}S\\textsubscript{4}, which only has one cubic phase with the space group of 227. The structure is shown in Figure \\ref{fig:Ni3S4_target}. Our algorithm found top four templates including Co\\textsubscript{3}S\\textsubscript{4} (mp-943), Co\\textsubscript{3}Se\\textsubscript{4} (mp-20456), Ni\\textsubscript{3}Se\\textsubscript{4} (mp-1120781), Co\\textsubscript{3}O\\textsubscript{4} (mp-18748), all of them are cubic structures with the space group of 227. The predicted structure with the lowest rmsd is 0.000714 which is predicted by our algorithm using Co\\textsubscript{3}S\\textsubscript{4} as the template. The rmsd errors of the structures from the other three templates are much larger, all around 0.288 . We can see that the predicted structure in Figure \\ref{fig:Ni3S4_predict} is very close to the target structure in Figure \\ref{fig:Ni3S4_target}. We also found that the predicted structure of NiS\\textsubscript{2} in Figure \\ref{fig:NiS2_predict} also matches well with the target structure in Figure \\ref{fig:NiS2_target}, which has the small rmsd error of 0.004918 .\n\nWe also tested four formulas of the chemical system Ga-B-N including GaBN\\textsubscript{2}, GaB\\textsubscript{3}N\\textsubscript{4}, Ga\\textsubscript{2}BN\\textsubscript{3}, and Ga\\textsubscript{3}BN\\textsubscript{4}. For the GaBN\\textsubscript{2} with the space group of 115, the best eight templates are found by our algorithm. Five of them including AlBN\\textsubscript{2} (mp-1008557) ,AlBN\\textsubscript{2} (mp-1008557), AlGaP\\textsubscript{2} (mp-1228888), AlGaN\\textsubscript{2} (mp-1008556), B\\textsubscript{2}AsP(mp-1008528) have the same tetragonal crystal system as the target structure with the space group of 115. The remaining are trigonal with the space group of 166. The predicted structures AlBN\\textsubscript{2} (mp-1008557) and AlBN\\textsubscript{2} (mp-1008557) have the same lowest rmsd error of 0.003889 . However they have different structure patterns as shown in Figure \\ref{fig:GaBN2_predict1} and Figure \\ref{fig:GaBN2_predict2}. For the same template, our algorithm suggests two element replacement strategies. In the first strategy, the element Ga in the test formula is used to replace the element Al in the template AlBN\\textsubscript{2}; in the second strategy, the element Ga in the test formula GaBN\\textsubscript{2} replaces the element B in the same template AlBN\\textsubscript{2}. For the formula GaB\\textsubscript{3}N\\textsubscript{4} with the space group of 215, TCSP finds top 9 most similar templates AlB\\textsubscript{3}N\\textsubscript{4} (mp-1019379), AlB\\textsubscript{3}N\\textsubscript{4} (mp-1019379), CrGa\\textsubscript{3}P\\textsubscript{4} (mp-985440), AlGa\\textsubscript{3}N\\textsubscript{4} (mp-1019508), Al\\textsubscript{3}GaN\\textsubscript{4} (mp-1019378), Al\\textsubscript{3}BN\\textsubscript{4} (mp-1019380), Al\\textsubscript{3}BN\\textsubscript{4} (mp-1019380), Ga\\textsubscript{3}BN\\textsubscript{4} (mp-1019743), Ga\\textsubscript{3}BN\\textsubscript{4} (mp-1019743) of the same space group 215 as well. The lowest rmsd error with different structure templates AlB\\textsubscript{3}N\\textsubscript{4} and AlB\\textsubscript{3}N\\textsubscript{4} is 0.002336 by using two element replacement strategies. The element Ga in the first strategy is used to replace Al in the template as shown in Figure \\ref{fig:GaB3N4_predict1} and\nas shown in Figure \\ref{fig:GaB3N4_predict2}, Ga replaces B in AlB\\textsubscript{3}N\\textsubscript{4} in the second strategy. For the formula GaB\\textsubscript{2}N\\textsubscript{3} (mp-1245554) in Figure \\ref{fig:GaB2N3_target} which has monoclinic structure with the space group 15, TCSP only find two templates AuC2N3 (mp-1245653) and AuC2N3 (mp-1245653) with the same space group. The lowest rmsd is 0.24746 for these two predicted structures. As shown in Figure \\ref{fig:GaB2N3_predict1}, our algorithm uses the first strategy, TCSP uses Ga in the test formula GaB\\textsubscript{2}N\\textsubscript{3} to replace Au in the first template AuC\\textsubscript{2}N\\textsubscript{3}. In the second strategy, Ga replaces B in the second template as shown in Figure \\ref{fig:GaB2N3_predict2}. For the formula Ga\\textsubscript{3}BN\\textsubscript{4}(mp-1019743) with the space group 215, TCSP finds 9 most similar templates Al\\textsubscript{3}BN\\textsubscript{4} (mp-1019380), Al\\textsubscript{3}BN\\textsubscript{4} (mp-1019380), Al\\textsubscript{3}GaN\\textsubscript{4} (mp-1019378), AlGa\\textsubscript{3}N\\textsubscript{4} (mp-1019508), CrGa\\textsubscript{3}P\\textsubscript{4} (mp-985440), AlB\\textsubscript{3}N\\textsubscript{4} (mp-1019379), AlB\\textsubscript{3}N\\textsubscript{4} (mp-1019379), GaB\\textsubscript{3}N\\textsubscript{4} (mp-1019740), GaB\\textsubscript{3}N\\textsubscript{4} (mp-1019740) with the same space group 215. The templates Al\\textsubscript{3}BN\\textsubscript{4} (mp-1019380) and Al\\textsubscript{3}BN\\textsubscript{4} (mp-1019380) with different structure patterns have the same lowest rmsd error of 0.004017781. For Al\\textsubscript{3}BN\\textsubscript{4}, AlB\\textsubscript{3}N\\textsubscript{4} and GaB\\textsubscript{3}N\\textsubscript{4}, they all have different structure patterns by using the two elements replacement strategies.\n\n\n\n\n\n\n\n\\begin{figure}[ht!] \n \\begin{subfigure}[t]{0.33\\textwidth}\n \\includegraphics[width=\\textwidth]{SrTiO3.png}\n \\caption{SrTiO\\textsubscript{3}(Target)}\n \\vspace{-3pt}\n \\label{fig:SrTiO3_target}\n \\end{subfigure}\n \\begin{subfigure}[t]{0.33\\textwidth}\n \\includegraphics[width=\\textwidth]{Ni3S4.png}\n \\caption{Ni\\textsubscript{3}S\\textsubscript{4}(Target)}\n \\vspace{-3pt}\n \\label{fig:Ni3S4_target}\n \\end{subfigure} \n \\begin{subfigure}[t]{0.33\\textwidth}\n \\includegraphics[width=\\textwidth]{NiS2.png}\n \\caption{NiS\\textsubscript{2}(Target)}\n \\vspace{-3pt}\n \\label{fig:NiS2_target}\n \\end{subfigure} \n \\begin{subfigure}[t]{0.33\\textwidth}\n \\includegraphics[width=\\textwidth]{2.0_mp-3834_SrTiO3_predict.png}\n \\caption{SrTiO\\textsubscript{3}(Predicted)}\n \\vspace{-3pt}\n \\label{fig:SrTiO3_predict}\n \\end{subfigure}\\hfill\n \\begin{subfigure}[t]{0.33\\textwidth}\n \\includegraphics[width=\\textwidth]{3.0_mp-943_Ni3S4_predict.png}\n \\caption{Ni\\textsubscript{3}S\\textsubscript{4}(Predicted)}\n \\vspace{-3pt}\n \\label{fig:Ni3S4_predict}\n \\end{subfigure}\n \\begin{subfigure}[t]{0.33\\textwidth}\n \\includegraphics[width=\\textwidth]{3.0_mp-849086_NiS2_predict.png}\n \\caption{NiS\\textsubscript{2}(Predicted)}\n \\vspace{-3pt}\n \\label{fig:NiS2_predict}\n \\end{subfigure}\\hfill\n \\begin{subfigure}[t]{0.33\\textwidth}\n \\includegraphics[width=\\textwidth]{GaBN2.png}\n \\caption{GaBN\\textsubscript{2}(Target)}\n \\vspace{-3pt}\n \\label{fig:GaBN2_target}\n \\end{subfigure}\n \\begin{subfigure}[t]{0.33\\textwidth}\n \\includegraphics[width=\\textwidth]{1.0_mp-1008557_GaBN2_predict.png}\n \\caption{GaBN\\textsubscript{2}(Predicted1)}\n \\vspace{-3pt}\n \\label{fig:GaBN2_predict1}\n \\end{subfigure}\\hfill \n \\begin{subfigure}[t]{0.33\\textwidth}\n \\includegraphics[width=0.9\\textwidth]{3.0_mp-1008557_GaBN2_predict.png}\n \\caption{GaBN\\textsubscript{2}(Predicted2)}\n \\vspace{-3pt}\n \\label{fig:GaBN2_predict2}\n \\end{subfigure}\\hfill \n \\begin{subfigure}[t]{0.33\\textwidth}\n \\includegraphics[width=\\textwidth]{GaB3N4.png}\n \\caption{GaB\\textsubscript{3}N\\textsubscript{4}(Target)}\n \\vspace{-3pt}\n \\label{fig:GaB3N4_target}\n \\end{subfigure} \n \\begin{subfigure}[t]{0.33\\textwidth}\n \\includegraphics[width=\\textwidth]{1.0_mp-1019379_GaB3N4_predict.png}\n \\caption{GaB\\textsubscript{3}N\\textsubscript{4}(Predicted1)}\n \\vspace{-3pt}\n \\label{fig:GaB3N4_predict1}\n \\end{subfigure}\\hfill\n \\begin{subfigure}[t]{0.33\\textwidth}\n \\includegraphics[width=\\textwidth]{3.0_mp-1019379_GaB3N4_predict.png}\n \\caption{GaB\\textsubscript{3}N\\textsubscript{4}(Predicted2)}\n \\vspace{-3pt}\n \\label{fig:GaB3N4_predict2}\n \\end{subfigure}\n \\begin{subfigure}[t]{0.33\\textwidth}\n \\includegraphics[width=\\textwidth]{GaB2N3.png}\n \\caption{GaB\\textsubscript{2}N\\textsubscript{3}(Target)}\n \\vspace{-3pt}\n \\label{fig:GaB2N3_target}\n \\end{subfigure} \n \\begin{subfigure}[t]{0.33\\textwidth}\n \\includegraphics[width=\\textwidth]{11.0_mp-1245653_GaB2N3_presict.png}\n \\caption{GaB\\textsubscript{2}N\\textsubscript{3}(Predicted1)}\n \\vspace{-3pt}\n \\label{fig:GaB2N3_predict1}\n \\end{subfigure} \n \\begin{subfigure}[t]{0.33\\textwidth}\n \\includegraphics[width=\\textwidth]{7.0_mp-1245653_GaB2N3_presict.png}\n \\caption{GaB\\textsubscript{2}N\\textsubscript{3}(Predicted2)}\n \\vspace{-3pt}\n \\label{fig:GaB2N3_predict2}\n \\end{subfigure} \n\n \\caption{predicted structures by TCSP compared to targets.}\n \\label{fig:predictedstructures}\n\\end{figure}\n\n\n\n\n\nOur TCSP algorithm can output multiple predictions using different templates. To understand this capability, Table~\\ref{tab:results_materials} shows the RMSD and quality scores of the top 10 predictions for each input formula. For SrTiO\\textsubscript{3}, the top 4 structures all have zero RMSD errors for their fractional coordinates with their replacement distance scores ranging from 2 to 3. The MAE errors of these four structures are also 0. We do find their lattice length is different from the target structures, which, however, can be tuned by DFT-based structure relaxation. For Ni\\textsubscript{3}S\\textsubscript{4}, only the top 1 result is very close to the target structure with three much worse results. For NiS\\textsubscript{2}, the top 2 predicted structures have RMSD of 0.0049 and 0.0124. Both are good compared to the target structures. For GaBN\\textsubscript{2}, the top four results all have small RMSD errors ranging from 0.0039 to 0.0209. The same is true for the predicted structures of GaB\\textsubscript{3}N\\textsubscript{4}. The worst prediction performance is on the formula GAB\\textsubscript{2}N\\textsubscript{3}, which can only find two compatible templates, both leading to very different structures from the target structure. Their MAE errors are 0.1853 . For GA\\textsubscript{3}BN\\textsubscript{4}, the top three predictions are all of high quality with RMSD of only 0.004\/0.004\/0.0206 respectively. In terms of the quality score distribution, we found that low replacement distance scores indicate predicted structures with good quality: for example, predictions with replacement distance scores greater than 5 are all low-quality results. However, low replacement distance scores do not always mean their structures are of high quality. For example, in the case of SrTiO3, the structures with high RMSD have lower replacement distance scores than those top 4 results. \n\n\\FloatBarrier\n\n\\begin{table}[th]\n\n\\centering\n\\caption{Prediction performance (rmsd error) of top 10 results for each sample in the benchmark set by TCSP\n}\n\\label{tab:results_materials}\n\\begin{tabular}{|llrrrrrrrrll|}\n\\hline\n\\rowcolor[HTML]{FFFFFF} \n\\textbf{Formula} & \\textbf{Metric} & \\multicolumn{1}{l}{\\cellcolor[HTML]{FFFFFF}\\textbf{Top1}} & \\multicolumn{1}{l}{\\cellcolor[HTML]{FFFFFF}\\textbf{Top2}} & \\multicolumn{1}{l}{\\cellcolor[HTML]{FFFFFF}\\textbf{Top3}} & \\multicolumn{1}{l}{\\cellcolor[HTML]{FFFFFF}\\textbf{Top4}} & \\multicolumn{1}{l}{\\cellcolor[HTML]{FFFFFF}\\textbf{Top5}} & \\multicolumn{1}{l}{\\cellcolor[HTML]{FFFFFF}\\textbf{Top6}} & \\multicolumn{1}{l}{\\cellcolor[HTML]{FFFFFF}\\textbf{Top7}} & \\multicolumn{1}{l}{\\cellcolor[HTML]{FFFFFF}\\textbf{Top8}} & \\textbf{Top9} & \\textbf{Top10} \\\\ \\hline\n\\multicolumn{1}{|l|}{\\cellcolor[HTML]{FFFFFF}SrTiO\\textsubscript{3}} & \\multicolumn{1}{l|}{rmsd} & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0} & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0} & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0} & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0} & \\multicolumn{1}{r|}{0.1667} & \\multicolumn{1}{r|}{0.2832} & \\multicolumn{1}{r|}{0.4082} & \\multicolumn{1}{r|}{0.4410} & \\multicolumn{1}{r|}{0.4410} & \\multicolumn{1}{r|}{0.4410} \\\\ \\hline\n\\multicolumn{1}{|l|}{} & \\multicolumn{1}{l|}{score} & \\multicolumn{1}{r|}{2} & \\multicolumn{1}{r|}{2} & \\multicolumn{1}{r|}{2} & \\multicolumn{1}{r|}{3} & \\multicolumn{1}{r|}{1} & \\multicolumn{1}{r|}{1} & \\multicolumn{1}{r|}{1} & \\multicolumn{1}{r|}{1} & \\multicolumn{1}{r|}{1} & \\multicolumn{1}{r|}{2} \\\\ \\hline\n\\multicolumn{1}{|l|}{\\cellcolor[HTML]{FFFFFF}Ni\\textsubscript{3}S\\textsubscript{4}} & \\multicolumn{1}{l|}{rmsd} & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0.0007} & \\multicolumn{1}{r|}{0.2885} & \\multicolumn{1}{r|}{0.2888} & \\multicolumn{1}{r|}{0.2897} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} \\\\ \\hline\n\\multicolumn{1}{|l|}{\\cellcolor[HTML]{FFFFFF}} & \\multicolumn{1}{l|}{score} & \\multicolumn{1}{r|}{3} & \\multicolumn{1}{r|}{4} & \\multicolumn{1}{r|}{1} & \\multicolumn{1}{r|}{4} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} \\\\ \\hline\n\\multicolumn{1}{|l|}{\\cellcolor[HTML]{FFFFFF}NiS\\textsubscript{2}} & \\multicolumn{1}{l|}{rmsd} & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0.0049} & \\multicolumn{1}{r|}{0.0124} & \\multicolumn{1}{r|}{0.0777} & \\multicolumn{1}{r|}{0.2282} & \\multicolumn{1}{r|}{0.2424} & \\multicolumn{1}{r|}{0.2846} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} \\\\ \\hline\n\\multicolumn{1}{|l|}{\\cellcolor[HTML]{FFFFFF}} & \\multicolumn{1}{l|}{score} & \\multicolumn{1}{r|}{3} & \\multicolumn{1}{r|}{1} & \\multicolumn{1}{r|}{3} & \\multicolumn{1}{r|}{2} & \\multicolumn{1}{r|}{1} & \\multicolumn{1}{r|}{2} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} \\\\ \\hline\n\\multicolumn{1}{|l|}{\\cellcolor[HTML]{FFFFFF}GaBN\\textsubscript{2}} & \\multicolumn{1}{l|}{rmsd} & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0.0039} & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0.0039} & \\multicolumn{1}{r|}{0.0174} & \\multicolumn{1}{r|}{0.0209} & \\multicolumn{1}{r|}{0.3410} & \\multicolumn{1}{r|}{0.3412} & \\multicolumn{1}{r|}{0.3412} & \\multicolumn{1}{r|}{0.3886} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} \\\\ \\hline\n\\multicolumn{1}{|l|}{\\cellcolor[HTML]{FFFFFF}} & \\multicolumn{1}{l|}{score} & \\multicolumn{1}{r|}{3} & \\multicolumn{1}{r|}{1} & \\multicolumn{1}{r|}{2} & \\multicolumn{1}{r|}{1} & \\multicolumn{1}{r|}{9} & \\multicolumn{1}{r|}{1} & \\multicolumn{1}{r|}{5} & \\multicolumn{1}{r|}{31} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} \\\\ \\hline\n\\multicolumn{1}{|l|}{GaB\\textsubscript{3}N\\textsubscript{4}} & \\multicolumn{1}{l|}{rmsd} & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0.0023} & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0.0023} & \\multicolumn{1}{r|}{0.0152} & \\multicolumn{1}{r|}{0.0162} & \\multicolumn{1}{r|}{0.3335} & \\multicolumn{1}{r|}{0.3344} & \\multicolumn{1}{r|}{0.3344} & \\multicolumn{1}{r|}{0.3347} & \\multicolumn{1}{r|}{0.3347} & \\multicolumn{1}{l|}{} \\\\ \\hline\n\\multicolumn{1}{|l|}{} & \\multicolumn{1}{l|}{score} & \\multicolumn{1}{r|}{1} & \\multicolumn{1}{r|}{3} & \\multicolumn{1}{r|}{22} & \\multicolumn{1}{r|}{1} & \\multicolumn{1}{r|}{1} & \\multicolumn{1}{r|}{1} & \\multicolumn{1}{r|}{3} & \\multicolumn{1}{r|}{4} & \\multicolumn{1}{r|}{0} & \\multicolumn{1}{l|}{} \\\\ \\hline\n\\multicolumn{1}{|l|}{\\cellcolor[HTML]{FFFFFF}GaB\\textsubscript{2}N\\textsubscript{3}} & \\multicolumn{1}{l|}{rmsd} & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0.2475} & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0.2475} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} \\\\ \\hline\n\\multicolumn{1}{|l|}{\\cellcolor[HTML]{FFFFFF}} & \\multicolumn{1}{l|}{score} & \\multicolumn{1}{r|}{11} & \\multicolumn{1}{r|}{7} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} & \\multicolumn{1}{l|}{} \\\\ \\hline\n\\multicolumn{1}{|l|}{\\cellcolor[HTML]{FFFFFF}Ga\\textsubscript{3}BN\\textsubscript{4}} & \\multicolumn{1}{l|}{rmsd} & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0.0040} & \\multicolumn{1}{r|}{\\cellcolor[HTML]{DDDDDD}0.0040} & \\multicolumn{1}{r|}{0.0206} & \\multicolumn{1}{r|}{0.3336} & \\multicolumn{1}{r|}{0.3337} & \\multicolumn{1}{r|}{0.3345} & \\multicolumn{1}{r|}{0.3345} & \\multicolumn{1}{r|}{0.3347} & \\multicolumn{1}{r|}{0.3347} & \\multicolumn{1}{l|}{} \\\\ \\hline\n\\multicolumn{1}{|l|}{\\cellcolor[HTML]{FFFFFF}} & \\multicolumn{1}{l|}{score} & \\multicolumn{1}{r|}{1} & \\multicolumn{1}{r|}{3} & \\multicolumn{1}{r|}{1} & \\multicolumn{1}{r|}{1} & \\multicolumn{1}{r|}{22} & \\multicolumn{1}{r|}{1} & \\multicolumn{1}{r|}{3} & \\multicolumn{1}{r|}{0} & \\multicolumn{1}{r|}{4} & \\multicolumn{1}{l|}{} \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\nTo further evaluate the performance of our TCSP algorithm, we conduct comprehensive predictions of all 98,290 formulas in the Materials Project database using the leave-one-out evaluation approach. For each formula, we predict its structure using existing templates that do not have the same formula. Here we use the strict mode for finding templates: only top 10 templates with the same prototype and compatible oxidation states are used to predict new structures. For each formula, we first identify all its corresponding mp-ids with corresponding structures and then for each of these target structures, we pick the structure with the lowest rmsd error out of all the predicted structures and we show the distribution of these rmsd errors to see how our template based TCSP algorithm can recover the structures in the MP database. The result is shown in Figure \\ref{fig:histogram_rmsd}. We find that for 34,569 MP structures, our algorithm has identified templates for structure prediction. Out of these target structures, TCSP has predicted hypothetical structures with a maximum RMSD less than 0.01 for 11,764 MP structures, or with RMDS less than 0.1 for 13,145 MP structures. We also find that for 10,433 structures, the algorithm could not find the correct templates that generates the same number of atoms in the unit cell for which we set the RMSD error at 1.0.\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=0.55\\linewidth]{histogram_tcsp.png}\n \\caption{Distribution of rmsd errors of all Materials Project structures as predicted by our TCSP using leave-one-out evaluation.}\n \\label{fig:histogram_rmsd}\n\\end{figure}\n\n\\FloatBarrier\n\n\\subsection{Discovery of new materials and DFT validation of the predict structures}\n\nWe are interested in how our TCSP algorithm can help to discover novel stable materials. We started with the Ga-B-N chemical system, for which we found four materials in the materials project database as shown in our test set discussed in the previous section. According to the materials project database, those available Ga-B-N structures have non-zero energy-above-hull values indicating those materials are thermodynamically unstable. We wonder if there exist thermodynamically and dynamically stable materials of this chemical system. We use the composition enumeration tool from our MaterialsAtlas.com toolbox to identify new Ga-B-N formulas prototypes and their predicted formation energy. We picked the top 41 formulas that do not exist in the MP database and used our TCSP to predict a set of candidate structures for each formula. We then conducted DFT relaxation and formation energy, e-above-hull energy, and phonon dispersion calculations to verify their thermodynamical and dynamical stability.\n\nOur calculations (Table 2) show that almost all candidate structures found by our TCSP have negative formation energies. This is immensely helpful for discovering new materials using the first-principles calculations. If most of the candidate structures of a selected composition have positive formation energies, we have to waste computational hours to find the structures with negative formation energies. However, our ML model is able to filter out the unsuitable candidate structures to reduce the computational burden.\n\nWe further calculated the e-above-hull to investigate the stability against the Ga-B-N competing phases. As given in the MP database, GaN, BN, Ga, B, and N2 are the stable competing phases available. Total energy calculations of competing phases were done with the same VASP settings used for Ga-B-N systems to determine the e-above-hull using the Pymatgen code. Our calculations suggest that 4 out of 41 materials exhibit zero e-bove-hull, indicating they are thermodynamically stable (Figure \\ref{fig:struct}). Those materials and their candidate structures are shown in Table~2. We further carried out phonon calculations for the candidate structures with the lowest formation energies for those four materials. It is clear from Fig.~\\ref{fig:phonons} that GaB$_2$N$_4$ material with mp-780282 template structure is dynamically stable at 0K temperature. It has an interesting layered honeycomb structure.\n\n\\begin{table}[]\n\\caption{Formation energy and corresponding templates of top 10 predictions for each of the four new materials}\n\\label{tab:my-table}\n\\begin{tabular}{|l|l|l|l|l|l|l|l|}\n\\hline\n\\multicolumn{2}{|c|}{GaB$_2$N$_4$} & \\multicolumn{2}{c|}{GaB$_4$N$_3$} & \\multicolumn{2}{c|}{Ga$_2$BN$_4$} & \\multicolumn{2}{c|}{GaBN$_4$} \\\\ \\hline\n\\multicolumn{1}{|c|}{mp-id} & \\multicolumn{1}{c|}{E$_\\textrm{form}$(eV)} & \\multicolumn{1}{c|}{mp-id} & \\multicolumn{1}{c|}{E$_\\textrm{form}$ (eV)} & \\multicolumn{1}{c|}{mp-id} & \\multicolumn{1}{c|}{E$_\\textrm{form}$(eV)} & \\multicolumn{1}{c|}{mp-id} & \\multicolumn{1}{c|}{E$_\\textrm{form}$ (eV)} \\\\ \\hline\nmp-780282 & -3.2957 & mp-1224009 & -2.4471 & mp-532446 & -2.9443 & mp-30979 & -3.29 \\\\ \\hline\nmp-778103 & -3.2414 & mp-1225800 & -2.2103 & mp-698589 & -2.8493 & mp-20790 & -2.9865 \\\\ \\hline\nmp-13335 & -3.2224 & mp-1228436 & -2.1092 & mp-761314 & -2.7696 & mp-1224951 & -2.7224 \\\\ \\hline\nmp-780395 & -3.1119 & mp-1120750 & -1.838 & mp-5712 & -2.5712 & mp-1224810 & -2.5462 \\\\ \\hline\nmp-30161 & -2.9124 & mp-29672 & -1.69 & mp-1212041 & -2.5703 & mp-555538 & -2.4165 \\\\ \\hline\nmp-1194477 & -2.4454 & mp-1019378 & -1.6865 & mp-1255006 & -2.5669 & mp-1071955 & -2.4012 \\\\ \\hline\nmp-756317 & -2.2645 & mp-1228943 & -0.3167 & mp-765466 & -2.5623 & mp-1102285 & -2.3984 \\\\ \\hline\nmp-36866 & -2.2639 & mp-29672 & -0.2741 & mp-753397 & -2.5622 & mp-1071955 & -2.398 \\\\ \\hline\nmp-1208866 & -2.1376 & mp-1019508 & -0.1409 & mp-756649 & -2.5621 & mp-27462 & -2.3971 \\\\ \\hline\nN\/A & N\/A & mp-1223879 & -0.0784 & mp-1178203 & -2.5621 & mp-1102285 & -2.3967 \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\n\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=0.8\\linewidth]{figures\/Structures.pdf}\n \\caption{Candidate new structures with zero e-above-hull for GaB$_2$N$_4$ (using template mp-780282), GaB$_4$N$_3$ (using template mp-1224009), GaBN$_4$ (using template mp-30979), and Ga$_2$BN$_4$ (using template mp-698589).}\n \\label{fig:struct}\n\\end{figure}\n\n\n\n\n\n\n\n\n\\section{Conclusion}\n\nCrystal structure prediction plays a key role in new materials discovery \\cite{oganov2019structure}. However, large-scale fast prediction of crystal structures is challenging, and user-friendly web apps are missing for such an important function despite the availability of a few public software that needs expensive high-performance computing (HPC) resources and expertise of computational materials. We believe such fast crystal structure prediction web apps are critical to the materials science community, as demonstrated by the bioinformatics field, which has more than 9000 web servers \\cite{hu2021materialsatlas}. Here we propose a template-based crystal structure prediction algorithm, TCSP, and its companion web server for fast and quick crystal structure prediction. Due to the widely observed structure similarity across many materials families such as perovskites in the materials database, TCSP achieves strong prediction performance as benchmarked on the whole Materials Project structure using leave-one-out evaluation due to its flexible template selection algorithm using prototype and oxidation information. To our knowledge, this is the largest experiments for crystal structure prediction. We believe our web-based TCSP algorithm will be of great interest to materials scientists for exploratory materials discovery. \n\n\\begin{figure}[htb]\n \\centering\n \\begin{subfigure}[t]{0.49\\textwidth}\n \\includegraphics[width=\\textwidth]{stableGaB2N4filling.png}\n \\caption{predicted structure of GaB$_2$N$_4$ }\n \\vspace{-3pt}\n \\label{fig:GaB2N3_predict1}\n \\end{subfigure} \n \\begin{subfigure}[t]{0.5\\textwidth}\n \\includegraphics[width=\\linewidth]{figures\/phonons.pdf}\n \\caption{Phonon dispersion of GaB$_2$N$_4$ with mp-780282 template structure.}\n \\label{fig:phonons}\n \\end{subfigure} \n \\caption{New material GaB$_2$N$_4$ discovered by our TCSP with zero e-above-hull energy, negative formation energy (-3.2957 eV) and dynamical stability}\n\\end{figure}\n\n\\section{Data Availability}\n\nDataset is downloaded from the Materials Project database website. \n\n\\section{Contribution}\nConceptualization, J.H.; methodology,J.H., L.W., W.Y., E.S., N.F., S.O., R.D.; software, J.H., L.W.,W.Y., N.F. ; resources, J.H.; writing--original draft preparation, J.H., L.W., E.S.,N.F.,R.X.; writing--review and editing, J.H, L.W.; visualization, J.H., L.W., E.S.; supervision, J.H.; funding acquisition, J.H.\n\n\\section*{Acknowledgement}\nThe research reported in this work was supported in part by National Science Foundation under the grant and 1940099 and 1905775. The views, perspectives, and content do not necessarily represent the official views of the NSF. We thank Andrew Hughes for his help in proofreading the manuscript. \n\n\\bibliographystyle{unsrt} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nRecently there was suggestion to extend the Universe beyond the Big Bang using the analytic continuation of the radiation-dominated epoch across the singularity\\cite{Turok2018,Turok2018b}. In this analytic continuation, at which the scale factor $a(\\tau)$ changes sign at $\\tau=0$, the gravitational tetrads also change sign giving rise to what is called the antispacetime. This means that the Universe on the other side of the Big Bang is the mirror image of the Universe on our side of the Big Bang.\n\nDifferent types of the antispacetime obtained by the space reversal $P$ and time reversal $T$ operations were considered ealier, including those where the determinant of the tetrads $e$ changes sign.\\cite{Diakonov2011,Diakonov2012,Rovelli2012b,Rovelli2012a,NissinenVolovik2018,Volovik2019b,Vergeles2019}\nLater the consideration has been extended to the thermal states, where the possible analytic continuation of the temperature across the transition from spacetime to antispacetime has been considered \\cite{Volovik2019}. Here we discuss two different scenarios of analytic contunuation across the Big Bang.\n\n\n\n\n\n\n\\section{CPT-symmetric Universe and negative temperature}\n\nThe CPT-symmetric Universe has been obtained in the conformal time frame\\cite{Turok2018}, where the metric in the spatially flat radiation-dominated era is:\n\\begin{equation}\ng_{\\mu\\nu}=a^2(\\tau)\\eta_{\\mu\\nu}\\,,\n\\label{metric}\n\\end{equation}\nwhere $\\eta_{\\mu\\nu}$ is the flat Minkowski metric; $a(\\tau)$ is the scale factor and $\\tau$ is the conformal time. \nSince the scale factor $a(\\tau)\\propto\\tau$, it was suggested that it can be analytically transformed to the region $\\tau<0$ before the Big Bang. Then the terad fields, which are proporional to $a(\\tau)$, also change sign under this analytic continuation,\n$e^a_\\mu(-\\tau)=-e^a_\\mu(\\tau)$. \n\nNow let us go further and extend the analytical continuation to the temperature of the system.\nIn the spatially flat radiation-dominated era one has\n\\begin{equation}\nT(\\tau)\\propto \\frac{1}{N(\\tau)}= \\frac{1}{e_{00}(\\tau)} \\,\\,, {\\rm or} \\,\\, \\beta(\\tau)\\propto\\tau\\,.\n\\label{TinCPT}\n\\end{equation}\nThe analytic continuation of $\\beta(\\tau)$ suggests that in the Universe at $\\tau<0$ the temperature is negative.\nThe transition between the states with positive and negative temperatures occurs via the infinite $T$ at $\\tau=0$ ($\\beta(0)=0$).\nThe states with negative temperature are typically unstable thermodynamically. This means that if the analytic continuation is really valid, then one of the two states (before or after Big Bang) is thermodynamically unstable.\nIt is more natural to assume, that the evolution of the system from $\\tau = -\\infty$ to $\\tau=0$ was equilibrium and corresponded to the positive temperature, $T(\\tau<0)>0$. This is because the system had enough time to equilibrate before the collapse. So we should have the following analytic time dependence of temperature:\n\\begin{equation}\nT(\\tau)\\propto -\\frac{1}{e_{00}(\\tau)} \\,\\,, {\\rm or} \\,\\, \\beta(\\tau)\\propto -\\tau\\,,\n\\label{TinCPTopposite}\n\\end{equation}\nand thus on our side of the Big Bang the temperature is negative, $T(\\tau>0)<0$. Of course, this may happen only at the first stage of the evolution of our Universe, i.e. immediately after the Big Bang. After some time the equilibration occurs, and the system returns again to the evolution with the equilibrium positive temperature, $T(\\tau>0)>0$.\nThe proper justification of this suggestion is beyond this comment.\n\n\\section{Euclidean signature Universe and spontaneously broken CPT symmetry}\n\n\n\\begin{figure}[t]\n\\includegraphics[width=\\linewidth]{Bifurcation.pdf}\n\\caption{ Bifurcation at Big Bang. Analytic continuation in physical proper time $t$: the scale factor\n$a(t)\\propto \\sqrt{t}$ changes sign around the Big Bang point, transforming spacetime into antispacetime.\nAt $t<0$ the scale factor is imaginary, which corresponds to the metric with Euclidean signature.\nCrossing the Big Bang from the $t<0$ semiaxis, the Universe approaches either spacetime or antispacetime. In this scenario, the Big Bang represents the bifurcation point at which the $Z_2$ symmetry \nbetween the spacetime and antispacetime is spontaneously broken. The quantum superposition of the two states is not allowed in the macroscopic system.\n}\n\\label{TwoRoads_Fig}\n\\end{figure}\n\n\nLet us consider the analytic continuation in terms of the physical proper time $t$. The metric for the radiation-dominated Universe is:\n\\begin{equation}\ng_{\\mu\\nu}=dt^2 -a^2(t)d{\\bf r}^2\\,\\,,\\, a^2(t) \\propto t \\,.\n\\label{metric2}\n\\end{equation}\nlet us now assume that the scale factor $a(t) \\sim \\sqrt{t}$ can be analytically extended around the singularity at $t=0$. Then the spacetime (with positive spatial components of tetrads) and antispacetime\n (with negative spatial components of tetrads) can be connected by analytic continuation: by $2\\pi$ rotation about $t=0$. As distinct from scenario in \\cite{Turok2018}, where the two Univereses live together,\nin the proper time scenario the spacetime and antispacetime represent two different realizations of the Universe at $t>0$, which exclude each other. \n\nThe Universe at $t>0$ can be obtained by analytic continuation from the negative $t$ region, where the metric has Euclidean signature. In this scenario, the Big Bang (at $t=0$) represents the point of \nthe phase transition from the Euclidean to Minkowski signature, which is similar to that, say, in \\cite{Steinacker2018,Stern2018,Sakharov1991,GibbonsHartle1990} (an example of such transition in condensed matter can be found e.g. in Ref. \\cite{NissinenVolovik2017b}). At the Big Bang, the state with Euclidean signature transforms either to the spacetime or to antispacetime, but not to the quantum superposition of the two Universes. The latter is not allowed for the macroscopic systems, which experience the spontaneous symmetry breaking.\\cite{Grady1994}\nIn principle, immediately after the Big Bang the Universe may evolve as the quantum superposition of these two states, but due to rapid decoherence only one of the two states survives. Thus in this analytic continuation the $Z_2$ ($CPT$) symmetry is spontaneously broken, as distinct from the scenario in \\cite{Turok2018}, which\ndescribes the creation of two Universes both evolving to the future with the conservation of the $CPT$ symmetry. \n\n\n\n\\section{Discussion}\n\nIn conclusion, we extended the analytic continuation across the Big Bang proposed in Ref.\\cite{Turok2018}in two different ways. The analytic continuation in the conformal time frame $\\tau$ was extended to the temperature of the system. This extension suggests that if the analytic continuation of the terad is valid, then it is possible that\nthe initial stage of the evolution of our Universe after the Big Bang was characterized by the negative temperature. The analytic continuation in the proper time $t$ suggests that the Big Bang is the bifurcation point of the second order quantum transition from the Euclidean to Minkowski signature, at which the symmetry between the spacetime and antispacetime is spontaneously broken.\n\n{\\bf Acknowledgements}. \n I thank A. Starobinsky for criticism. This work has been supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant Agreement No. 694248).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nHopf proved that the (singular) cohomology of a real connected compact Lie group $\\mathcal{G}$ is an exterior algebra on $\\mathrm{rank}(\\mathcal{G})$ generators of odd degree~\\cite{hopf1964topologie}. Its Poincar\\'e series is therefore given by \\[\\mathrm{Hilb}(H^*(\\mathcal{G});q) = \\prod_{i=1}^n (1+q^{2e_i+1}).\\]\nChevalley presented these $e_i$ for the exceptional simple Lie algebras in his 1950 address at the International Congress of Mathematicians~\\cite{chevalley1950betti}, and Coxeter recognized them from previous work with real reflection groups~\\cite{coxeter1951product}. This observation has led to deep relationships between the cohomology of $G$, and the invariant theory of the corresponding Weyl group $W=N_\\mathcal{G}(T)\/T$~\\cite{reeder1995cohomology,reiner2019invariant}---notably, $H^*(\\mathcal{G}) \\simeq \\left(H^*(\\mathcal{G}\/T) \\times H^*(T)\\right)^W \\simeq \udbff\udc01\\left(S\/S_+^W \\otimes \\bigwedge V^*\\right)^W.$\nFor more information, we refer the reader to the wonderful survey~\\cite{barcelo1994combinatorial}.\n\n\nIt turns out that one method to compute the $e_i$ is the generating function for the dimension of the fixed space ${\\mathrm{fix}}(w)=\\dim(\\ker(1-g))$ over the Weyl group:\n\\[\\sum_{w \\in W} q^{{\\mathrm{fix}}(w)} = \\prod_{i=1}^r (q+e_i).\\]\n\n\nShephard and Todd~\\cite{shephard1954finite} verified case-by-case that the same sum still factors when $W$ is replaced by a complex reflection group $G$. Let $G$ be a finite complex reflection group, acting by reflections on $V$. The $e_i$ are now determined by the degrees $d_i$ of the fundamental invariants of $G$ on $V$ as $e_i=d_i-1$. A case-free proof of this result was given by Solomon~\\cite{solomon1963invariants}, mirroring Hopf's result: writing $S=\\mathrm{Sym}(V^*)$ and $\\Lambda=\\bigwedge(V^*)$, $(\\left(S \\otimes \\Lambda\\right)^G$ is a free exterior algebra over the ring of $G$-invariant polynomials, which gives a factorization of the Poincar\\'e series \n\n\\begin{equation}\n\\label{eq:solomon}\n\\mathrm{Hilb}(\\left(S \\otimes \\Lambda \\right)^G;q,u) = \\prod_{i=1}^r \\frac{1+u q^{e_i}}{1-q^{d_i}}.\n\\end{equation}\nComputing the trace on $S \\otimes \\Lambda$ of the projection to the $G$-invariants $\\frac{1}{G}\\sum_{g\\in G} g$, specializing to $u=q(1-x)-1$, and taking the limit as $x \\to 1$ gives the Shephard-Todd result:\n\n\\begin{theorem}[{\\cite{shephard1954finite,solomon1963invariants}}]\\label{thm:OS-untwisted}\nFor any complex reflection group $G$,\n\\begin{equation}\n\\label{eq:shephard_todd}\n\\sum_{g \\in G} q^{{\\mathrm{fix}}(g)} = \\prod_{i=1}^r (q+e_i).\n\\end{equation}\n\\end{theorem}\n\nMore generally, define the \\defn{fake degree} of an $m$-dimensional (simple) $G$-module $M$ to be the polynomial encoding the degrees in which $M$ occurs in the coinvariant ring $\\mathbb{C}[V]_G$: $f_M(q)=\\sum_{i} (\\mathbb{C}[V]_G^i,M)q^i =\\sum_{i=1}^m q^{e_i(M)}.$\n\n\\medskip\nLet $G \\subset \\mathrm{GL}(V)$ be a complex reflection group. We say that $N \\triangleleft G$ is a \\defn{normal reflection subgroup} of $G$ if it is a normal subgroup of $G$ that is generated by reflections. For Weyl groups, nontrivial normal reflection subgroups can be constructed using root lengths. More generally, normal reflection subgroups are constructed by taking the union of conjugacy classes of reflections. We give the classification of normal reflection subgroups in~\\Cref{sec:classification}, and tie our work with previous work on their numerology in~\\Cref{sec:reflexponents}.\n\nThe following theorem is a special case of results in~\\cite{bessis2002quotients} (where the authors consider the more general notion of ``bon sous-groupe distingu\\'e'' in lieu of our normal reflection subgroup $N$ of $G$). We emphasize that our proof of this result in \\Cref{sec:quotients} follows the main ideas in~\\cite{bessis2002quotients}, specialized to our more restricted setting.\n\n\\begin{theorem}\n\\label{thm:reflection_action}\nLet $N$ be a normal reflection subgroup of a complex reflection group $G$ acting on $V$. Then the quotient group $H=G\/N$ acts as a reflection group on the vector space ${E}=V\/N$.\\end{theorem}\n\n\nIn \\Cref{sec:quotients} we will build on our proof of \\Cref{thm:reflection_action} to prove the following numerological identities.\n\n\\begin{theorem}\n\\label{thm:numbers}\nLet $G,N,H$ be as in~\\Cref{thm:reflection_action}. For a suitable choice of indexing of degrees and fake degrees, we have the following equalities:\n\\begin{align*}\ne_i^N(V) {+} e_i^G({E}) &= e_i^G(V) \\\\\nd_i^N \\cdot d_i^H & = d_i^G \\\\\nd_i^N \\cdot e_i^H({E}) & = e_i^G({E})\n\\end{align*}\n\\end{theorem}\n\n\\begin{example}\nTake $G=W(F_4)=G_{28}$ and $N$ to be the normal subgroup generated by the reflections corresponding to short roots. Then $N\\simeq W(D_4)$, $G\/N\\simeq W(A_2)=\\mathfrak{S}_3$ acting by reflections on $\\mathbb{C} \\oplus \\mathbb{C} \\oplus \\mathbb{C}^2$ (trivially on $\\mathbb{C}\\oplus\\mathbb{C}$), so that \\Cref{thm:numbers} gives the equations\n\\begin{align*}\n(1,5,3,3) {+} (0,0,4,8) &= (1,5,7,11) \\\\\n(2,6,4,4) \\cdot (1,1,2,3) & = (2,6,8,12) \\\\\n(2,6,4,4) \\cdot (0,0,1,2) & = (0,0,4,8).\n\\end{align*}\n\\end{example}\n\nThe following result simultaneously generalizes the results of~\\cite{williams2019reflexponents} and the Shephard-Todd formula~\\Cref{eq:shephard_todd} from \\Cref{thm:OS-untwisted}. We state a generalized version that incorporates Galois twists in \\Cref{sec:galois}.\n\n\\begin{theorem}\n\\label{thm:main_theorem}\nLet $N\\triangleleft \\ G$ be reflection groups acting by reflections on $V$, and let ${E}=V\/N$. Then\n\\[\\sum_{g \\in G} q^{{\\mathrm{fix}}_V g} t^{{\\mathrm{fix}}_{{E}} g} = \\prod_{i=1}^r \\left(qt+e_i^N(V) t + e_i^G({E})\\right).\\]\n\\end{theorem}\n\n\\begin{example}\n\\label{ex:c2}\nThe dihedral group $G=G(2,1,2)=\\left\\langle s,t | s^2=t^2=(st)^4=1\\right\\rangle$ acts as a reflection group on $V=\\mathbb{C}^2$ by $s=\\left[\\begin{smallmatrix} -1 & 0 \\\\ 0 & 1 \\end{smallmatrix} \\right] \\text{ and } t=\\left[\\begin{smallmatrix} 0 & 1 \\\\ 1 & 0 \\end{smallmatrix} \\right].$ Take $N$ to be the normal subgroup generated by the reflections conjugate to $s$. Then $N \\simeq C_2 \\times C_2$ is a normal reflection subgroup, isomorphic to the direct product of the cyclic group of order two with itself, with invariants $N_1=x_1^2$ and $N_2=x_2^2$, and $G$ acts on ${E}$ dual to ${E^*}=\\mathrm{span}_\\mathbb{C}\\{N_1,N_2\\}$ as the quotient group $G\/N\\simeq C_2$ by $s=\\left[\\begin{smallmatrix} 1 & 0 \\\\ 0 & 1 \\end{smallmatrix} \\right] \\text{ and } t=\\left[\\begin{smallmatrix} 0 & 1 \\\\ 1 & 0 \\end{smallmatrix} \\right].$ In this case, \\Cref{thm:main_theorem} expresses the equality\n\\[\\sum_{g \\in G} q^{{\\mathrm{fix}}_V g} t^{{\\mathrm{fix}}_{{E}} g} = q^2t^2+2qt^2+2qt+2t+t^2 =(qt+t)(qt+t+2).\\]\n\\end{example}\n\n\\noindent\n{\\bf Acknowledgements.} We thank Theo Douvropoulos for many helpful comments and suggestions. The first author was partially supported by NSF grant CCF-1815108. The second author was partially supported by Simons Foundation award number 585380. \n\n\\section{Quotients by Normal Reflection Subgroups}\\label{sec:quotients} \n\nLet $V$ be a complex vector space of dimension $r$. A \\defn{reflection} is an element of $\\mathrm{GL}(V)$ that fixes some hyperplane pointwise. A \\defn{complex reflection group} $G$ is a finite subgroup of $\\mathrm{GL}(V)$ that is generated by reflections. A complex reflection group $G$ is called \\defn{irreducible} if $V$ is a simple $G$-module; $V$ is then called the \\defn{reflection representation} of $G$. A \\defn{(normal) reflection subgroup} of $G$ is a (normal) subgroup of $G$ that is generated by reflections.\n\nLet $S(V^*)$ be the symmetric algebra on the dual vector space $V^*$, and write $S(V^*)^G$ for its $G$-invariant subring. By a classical theorem of Shephard-Todd~\\cite{shephard1954finite} and Chevalley~\\cite{chevalley1955invariants}, a subgroup $G$ of $\\mathrm{GL}(V)$ is a complex reflection group if and only if $S(V^*)^G$ is a polynomial ring, generated by $r$ algebraically independent homogeneous $G$-invariant polynomials---the \\defn{degrees} $d_1\\leq \\cdots \\leq d_r$ of these polynomials are invariants of $G$.\n\n\\begin{theorem}[\\cite{shephard1954finite,chevalley1955invariants}]\n\\label{thm:shephard_todd}\n Let $G \\leq \\mathrm{GL}(V)$ be finite. Then $G$ is a complex reflection group if and only if there exist $r$ homogeneous algebraically independent polynomials ${G}_1,\\ldots,{G}_r\\in S(V^*)^G$ such that $S(V^*)^G=\\mathbb{C}[{G}_1,\\ldots,{G}_r]$. In this case, $|G|=\\prod_{i=1}^r d_i$, where $d_i=\\mathrm{deg}({G}_i)$.\n\\end{theorem}\n\nAlthough \\Cref{thm:reflection_action} is a special case of results in~\\cite{bessis2002quotients} (where they consider the more general notion of ``bon sous-groupe distingu\\'e''), the proof is more straightforward in our restricted setting, and also leads directly to a proof of \\Cref{thm:numbers}.\n\n\n{\n\\renewcommand{\\thetheorem}{\\ref{thm:reflection_action}}\n\\begin{theorem}\nLet $N$ be a normal reflection subgroup of a complex reflection group $G$ acting on $V$. Then the quotient group $H=G\/N$ acts as a reflection group on the vector space ${E}=V\/N$.\n\\end{theorem}\n\\addtocounter{theorem}{-1}\n}\n\n\\begin{proof} We claim that there exist homogeneous generators ${N}_1,\\dots,{N}_r$ of $S(V^*)^N$ such that ${E}^*=\\mathrm{span}_\\mathbb{C}\\{{N}_1,\\ldots,{N}_r\\}$ is $H$-stable. By Theorem~\\ref{thm:shephard_todd}, $S(V^*)^N=\\mathbb{C}[\\tilde{{N}}_1,\\dots,\\tilde{{N}}_r]$ for some homogeneous algebraically independent $\\tilde{{N}}_i$. Let $I_+\\subset S(V^*)^N$ be the ideal generated by homogeneous elements of positive degree. Then both $I_+$ and $I_+^2$ are $H$-stable homogeneous ideals, and therefore the algebraic tangent space $I_+\/I_+^2$ to ${E}=V\/N$ at $0$ inherits a graded action of $H$ that is compatible with the (graded) quotient map $\\pi:I_+\\twoheadrightarrow I_+\/I_+^2$. Hence there exists a graded $H$-equivariant section $\\varphi:I_+\/I_+^2\\rightarrow I_+$. Letting ${N}_i=\\varphi\\circ\\pi(\\tilde{{N}}_i)$ we see that ${N}_1,\\dots,{N}_r$ are still homogeneous algebraically independent generators for $S(V^*)^N$ with $\\mathrm{deg}({N}_i)=\\mathrm{deg}(\\tilde{{N}}_i)$ and such that ${E}^*={\\mathrm{span}}_\\mathbb{C}\\{{N}_1,\\dots,{N}_r\\}$ is $H$-stable, as claimed.\n\n\nWrite ${G}_1, \\ldots, {G}_r$ for the homogeneous generators of $S(V^*)^G$, again as in Theorem~\\ref{thm:shephard_todd}. Consider the action of $H$ on ${E}^*$ defined by $(gN){N}_i\\coloneqq g{N}_i.$ Since $S(V^*)^G= (S(V^*)^N)^H=S(E^*)^H$, there exist polynomials ${H}_1,\\ldots,{H}_r \\in \\mathbb{C}[\\mathbf{{N}}]$ such that ${H}_i(\\mathbf{{N}})={G}_i(\\mathbf{x})$, where $\\mathbf{{N}}=\\{{N}_1,\\dots,{N}_r\\}$ and $\\mathbf{x}=\\{x_1,\\dots,x_r\\}$ denote dual bases for $E$ and $V$, respectively.\n\nSince any algebraic relation $f({H}_1,\\ldots,{H}_r)=0$ would result in an algebraic relation $f({G}_1,\\ldots,{G}_r)=0$, the ${H}_i$ must be algebraically independent. By \\Cref{thm:reflection_action}, $H$ is a complex reflection group.\\end{proof}\n\n\\begin{remark}\nNote that the quotient group $H=G\/N$ does not necessarily lift to a reflection subgroup of $G$ nor even a subgroup of $G$. A counterexample is given by $G(4,2,2)=N \\triangleleft G=G_8$, so that $G\/N \\simeq \\mathfrak{S}_3$.\n\\end{remark}\n\n\\begin{remark}\n\\label{rem:indexing}\nIn the course of the proof of \\Cref{thm:reflection_action} we showed that the vector space ${E}=V\/N$ on which $H$ acts by reflections is dual to ${E^*}:={\\mathrm{span}}_\\mathbb{C}\\{{N}_1,\\dots,{N}_r\\}$ for a certain choice of fundamental $N$-invariants ${N}_1,\\dots{N}_r\\in S(V^*)^N$ such that ${E^*}$ is $G$-stable. The resulting action of $H$ on ${E}$ respects the $\\mathbf{x}$-grading on the $N$-invariants ${N}_i(\\mathbf{x})$, and therefore there is a choice of fundamental $H$-invariants ${H}_1,\\dots,{H}_r\\in S({E^*})^H=S(V^*)^G$ such that each ${H}_i(\\mathbf{N})={G}_i(\\mathbf{x})$ is both $\\mathbf{x}$-homogeneous of $\\mathbf{x}$-degree $=:d_i^G$ and $\\mathbf{N}$-homogeneous of $\\mathbf{N}$-degree $=:d_i^H$. Since the action of $H$ on ${E^*}$ respects the homogeneous decomposition of ${E^*}$ according to $\\mathbf{x}$-degree, the ${H}_i(\\mathbf{N})$ may be chosen such that the $N$-invariants ${N}_j(\\mathbf{x})\\in\\mathbf{N}$ occurring non-trivially in ${H}_i(\\mathbf{N})$ are all of the same $\\mathbf{x}$-degree $=:d_i^N$. This relationship between the fundamental invariants for $N$, $G$, and $H$ acting on their corresponding reflection representations leads to interesting numerological identities.\n\\end{remark}\n\nThe following result motivates some of the theoretical ingredients in our proofs.\n\n\\begin{theorem}[{\\cite{solomon1963invariants}}]\n\\label{thm:solomon} If $G\\subset\\mathrm{GL}(V)$ is a complex reflection group, then the ring $\\left(S(V^*)\\otimes \\bigwedge V^*\\right)^G$ is a free exterior algebra over the ring of $G$-invariant polynomials:\n\\[\\left(S(V^*)\\otimes \\bigwedge V^*\\right)^G \\simeq S(V^*)^G \\otimes \\bigwedge U_G,\\] where $U_G={\\mathrm{span}}_\\mathbb{C}\\left\\{d{G}_1,\\ldots,d{G}_s\\right\\}$ and $d{G}_i=\\sum_{j=1}^r \\frac{\\partial {G}_i}{\\partial x_j} \\otimes x_j$.\\end{theorem}\n\n\\begin{comment}\n\\begin{proof}\nLet $u_i:=d{N}_i$. Consider the multiplication map $\\mu:{U^N}\\to {E}^*$ given by \\[\\mu(u_i)=\\sum_{j=1}^rx_j\\frac{\\partial {N}_i}{\\partial x_j},\\] and note that $\\mu$ is $H$-equivariant of degree $+1$ and $\\mu\\circ d{N}_i=d_i^N{N}_i$.\n\\end{proof}\n\\end{comment}\n\n{\n\\renewcommand{\\thetheorem}{\\ref{thm:numbers}}\n\\begin{theorem}\nLet $G,N,H$ be as in~\\Cref{thm:reflection_action}. For a suitable choice of indexing of degrees and fake degrees, we have the following equalities:\n\\begin{align*}\ne_i^N(V) {+} e_i^G({E}) &= e_i^G(V) \\\\\nd_i^N \\cdot d_i^H & = d_i^G \\\\\nd_i^N \\cdot e_i^H({E}) & = e_i^G({E})\n\\end{align*}\n\\end{theorem}\n\\addtocounter{theorem}{-1}\n}\n\n\\begin{proof} Having chosen fundamental $N$-invariants ${N}_1,\\dots,{N}_r\\in S(V^*)^N$ such that ${E^*}={\\mathrm{span}}_\\mathbb{C}\\{{N}_1,\\dots,{N}_r\\}$ is $G$-stable as in the proof of \\Cref{thm:reflection_action} and \\Cref{rem:indexing}, we have fundamental $G$-invariants ${G}_i(\\mathbf{x})={H}_i(\\mathbf{N})$ that are $\\mathbf{x}$-homogeneous of $\\mathbf{x}$-degree $d_i^G$ and $\\mathbf{N}$-homogeneous of $\\mathbf{N}$-degree $d_i^H$, and where the ${N}_j\\in\\mathbf{N}$ occurring non-trivially in ${H}_i(\\mathbf{N})$ are all of the same $\\mathbf{x}$-degree $d_i^N$. The equality $d_i^Nd_i^H=d_i^G$ is immediate.\n\nLet us show that this same choice of indexing of fundamental invariants for $N$, $G$, and $H$ results in the other two equalities. We begin by comparing $\\mathbf{x}$-degrees in \\[d{G}_i=\\sum_{j=1}^r\\frac{\\partial{G}_i}{\\partial x_j}\\otimes x_j=\\sum_{k=1}^r\\frac{\\partial {H}_i}{\\partial {N}_k}\\cdot d{N}_k=\\sum_{k=1}^r\\sum_{j=1}^r\\frac{\\partial {H}_i}{\\partial {N}_k}\\cdot\\frac{\\partial{N}_k}{\\partial x_j}\\otimes x_j.\\] Recall that $e_i^G(V)=d_i^G-1=\\mathrm{deg}_\\mathbf{x}(d{G}_i)$ and $e_i^N(V)=d_i^N-1=\\mathrm{deg}_\\mathbf{x}(d{N}_i)$. Similarly, $e_i^H(E)=d_i^H-1=\\mathrm{deg}_\\mathbf{N}(d{H}_i)$, where this time $d{H}_i=\\sum_{k=1}^r\\frac{\\partial {H}_i}{\\partial {N}_k}\\otimes{N}_k\\in (S({E^*})\\otimes {E^*})^H$. Since $\\frac{\\partial {H}_i}{\\partial {N}_k}=0$ whenever $\\mathrm{deg}_\\mathbf{x}({N}_k)\\neq d_i^N$, it follows that \\[e_i^G(V)=e_i^N(V)+d_i^N\\cdot (d_i^H-1)=e_i^N(V)+d_i^N\\cdot e_i^H(E).\\] It remains to show that $d_i^Ne_i^H(E)=e_i^G({E})$.\n\nThe $e_i^G({E})$ are known to coincide with the $\\mathbf{x}$-degrees of any set of homogeneous generators for $(S(V^*)\\otimes{E^*})^G$ as a free $S(V^*)^G$-module. Since ${E^*}$ consists of $N$-invariants, \\[(S(V^*)\\otimes{E^*})^G=((S(V^*)\\otimes{E^*})^N)^H=(S({E^*})\\otimes{E^*})^H\\simeq S({E^*})^H\\otimes U_H=S(V^*)^G\\otimes U_H,\\] where again $U_H:={\\mathrm{span}}_\\mathbb{C}\\{d{H}_1,\\dots,d{H}_r\\}$ and the non-trivial isomorphism comes from \\Cref{thm:solomon} applied to the reflection representation ${E}$ of $H$. Hence $(S(V^*)\\otimes{E^*})^G$ is generated by $d{H}_i$ as a free $S(V^*)^G$-module, whence $e_i^G({E})=\\mathrm{deg}_\\mathbf{x}(d{H}_i)=d_i^Ne_i^H({E})$.\\end{proof}\n\n\\begin{remark}\\label{rem:bigraded_solomon}\nThe same argument used in the proof of \\Cref{thm:numbers} shows more generally: \\[(S(V^*)\\otimes\\bigwedge {E^*})^G=((S(V^*)\\otimes \\bigwedge {E^*})^N)^H=(S({E^*})\\otimes\\bigwedge {E^*})^H\\simeq S(V^*)^G\\otimes\\bigwedge U_H.\\] \n\\end{remark}\n\n\\section{Poincar\\'e Series and Specializations}\\label{sec:poincare}\n\nOur goal in this section is to prove our main result:\n\n{\n\\renewcommand{\\thetheorem}{\\ref{thm:main_theorem}}\n\\begin{theorem}\nLet $N\\triangleleft \\ G$ be reflection groups acting by reflections on $V$, and let ${E}=V\/N$. Then\n\\[\\sum_{g \\in G} q^{{\\mathrm{fix}}_V g} t^{{\\mathrm{fix}}_{{E}} g} = \\prod_{i=1}^r \\left(qt+e_i^N(V) t + e_i^G({E})\\right).\\]\n\\end{theorem}\n\\addtocounter{theorem}{-1}\n}\n\nWe refer to the left-hand side of~\\Cref{thm:main_theorem} as the \\defn{sum side}, and to the right-hand side as the \\defn{product side}. \nWe prove \\Cref{thm:main_theorem} by computing the Poincar\\'e series for $(S(V^*)\\otimes\\bigwedge {E^*})^G$ in two different (and standard) ways, keeping track of the supplemental grading afforded by the $\\mathbf{x}$-degrees of $N$-invariants in ${E^*}={\\mathrm{span}}_\\mathbb{C}\\{{N}_1,\\dots,{N}_r\\}$: one way corresponds to the product side (\\Cref{sec:prod}), and the other to the sum side (\\Cref{sec:sum}). A subtlety arises when trying to compute the term-by-term specialization for the sum side, which is dealt with in~\\Cref{sec:coseti,sec:cosetii}.\n\nA more general version of~\\Cref{thm:main_theorem} that incorporates Galois twists is stated in~\\Cref{sec:galois}. For technical reasons that arise in that generalization, we will define the shifted homogeneous decomposition ${E}^*_m:={\\mathrm{span}}_\\mathbb{C}\\{{N}_i \\ | \\ \\mathrm{deg}_\\mathbf{x}({N}_i)=m+1\\}$, and similarly\n\\[({\\textstyle\\bigwedge^p}{E^*})_m:={\\mathrm{span}}_\\mathbb{C}\\{ {N}_{i_1}\\wedge\\dots\\wedge{N}_{i_p}\\in{\\textstyle\\bigwedge^p}{E^*} \\ | \\ {\\textstyle\\sum_{j=1}^p}\\mathrm{deg}_\\mathbf{x}({N}_{i_j})=m+p\\}.\\]\nWriting $S(V^*)_\\ell$ for the homogeneous component of $\\mathbf{x}$-degree $\\ell$, we define the Poincar\\'e series\n\\begin{equation}\\label{eq:p-series-defn}\n\\mathcal{P}(x,y,u):=\\sum_{\\ell,m,p\\geq 0}\\mathrm{dim}_\\mathbb{C}((S(V^*)_\\ell\\otimes ({\\textstyle\\bigwedge^p}{E^*})_m)^G)x^\\ell y^m u^p.\n\\end{equation}\nWe write ${\\mathcal{E}^N_\\sigma}=\\{e_1^N({V^\\sigma}),\\dots,e_r^N({V^\\sigma})\\}$ for the set of fake degrees of ${V^\\sigma}$ as an $N$-representation.\n\n\\subsection{Product Side}\n\\label{sec:prod}\nWe first obtain the following product formula for the Poincar\\'e series $\\mathcal{P}(x,y,u)$ defined in \\Cref{eq:p-series-defn} from \\Cref{thm:numbers} and \\Cref{rem:bigraded_solomon}, since $d{H}_i\\in S(V^*)_{e_i^G(E)}\\otimes {E}^*_{e_i^N(V)}$.\n\n\\begin{lemma}\n\\label{lem:product_side}\n$\\mathcal{P}(x,y,u)=\\displaystyle\\prod\\limits_{i=1}\\limits^r\\frac{1+x^{e_i^G({E})}y^{e_i^N(V)}u}{1-x_i^{d_i^G}}.$\n\\end{lemma}\n\n\\begin{corollary}\n\\label{cor:prod_side_specialization}\n$\\lim\\limits_{x\\to1}\\mathcal{P}\\left(x,x^t,qt(1-x)-1\\right)=\\displaystyle\\prod\\limits_{i=1}\\limits^r \\left(qt+e_i^N(V) t + e_i^G({E})\\right).$\n\\end{corollary}\n\n\n\\subsection{Sum Side}\n\\label{sec:sum}\nBy taking traces, we now compute a formula for the Poincar\\'e series $\\mathcal{P}(x,y,u)$ defined in \\Cref{eq:p-series-defn} as a sum over elements of $G$. To simplify notation, we denote by ${E}_m$ the homogeneous component of ${E}$ corresponding to the dual of ${E}^*_m$.\n\n\n\\begin{lemma} \\label{thm:5.2} \n$\n\\mathcal{P}(x,y,u)=\n\\frac{1}{|G|} \\displaystyle\\sum\\limits_{g \\in G} \\frac{\\prod_{m\\in{\\mathcal{E}^N_\\sigma}}\\mathrm{det}(1+uy^{m}g|_{{E}_m})}{\\det(1-xg|_{V})}.$\n\\end{lemma}\n\n\\begin{proof}\nSince ${(U^N_\\sigma)^*}\\simeq \\bigoplus_{m\\in{\\mathcal{E}^N_\\sigma}} {E}^*_m$, we have that\n$\\bigwedge {(U^N_\\sigma)^*}\\simeq \\bigotimes_{m\\in{\\mathcal{E}^N_\\sigma}}\\bigwedge {E}^*_m$ as $G$-modules. hence, for each $g\\in G$, \\[\\sum_{m,p\\geq 0} \\mathrm{tr}(g|({\\textstyle\\bigwedge^p}{(U^N_\\sigma)^*})_{m})y^mu^p= \\prod_{m\\in{\\mathcal{E}_N}}\\left(\\sum_{p\\geq 0}\\mathrm{tr}(g|{\\textstyle\\bigwedge^p} {E}^*_m )y^{pm}u^p\\right).\\]\n\nFor each $m\\in{\\mathcal{E}^N_\\sigma}$ we have $\\displaystyle\\sum\\limits_{p\\geq 0}\\left(\\mathrm{tr}(g|{\\textstyle\\bigwedge^p}{E}^*_m)y^{pm}u^p\\right)=\\mathrm{det}(1+y^{m}ug|_{{E}^*_{m}}).$\n Therefore,\n\\begin{align*}\n\\mathcal{P}(x,y,u)&=\\left(\\sum_{\\ell\\geq 0}\\mathrm{tr}(g|(S({V^*})_\\ell)x^\\ell\\right)\\left(\\sum_{m,p\\geq 0}\\mathrm{tr}(g|({\\textstyle \\bigwedge^p}{(U^N_\\sigma)^*})_m)y^mu^p\\right)\\\\\n&=\\frac{\\prod_{m\\in{\\mathcal{E}^N_\\sigma}}\\mathrm{det}(1+uy^{m}g^{-1}|_{{U^N_\\sigma}_m})}{\\det(1-xg^{-1}|_{V})}.\\end{align*}\nThe result follows after taking the average over $G$ on each side.\\qedhere\n\\end{proof}\n\nThe story is not quite so simple as just setting the sum over $G$ from~\\Cref{thm:5.2} equal to the product, and then specializing. The trouble is that in this specialized sum over $G$ from~\\Cref{thm:5.2}, each element of $G$ \\emph{does not} necessarily contribute the ``correct amount'' specified by the sum side of~\\Cref{thm:main_theorem}---in particular, $(ng)|_{{E}}$ often has larger fixed space than $(ng)|_{{V}}$, which causes many terms in the term-by-term limit to be zero. It turns out, as we will now show, that the contributions \\emph{are} correct when taken coset-by-coset.\n\n\n\n\\subsection{Sum Side, Coset-by-Coset I}\n\\label{sec:cosetii}\nFix some $g \\in G$. We find a product formula for \\Cref{thm:5.2} restricted to the coset $gN$. Define \\[\\mathcal{P}_{gN}(x,y,u):=\\frac{1}{|N|}\\sum_{n \\in N} \\frac{\\prod_{m\\in{\\mathcal{E}^N_\\sigma}} \\det\\left(1+u y^{m} (ng)|_{{E}^*_{m}}\\right) }{\\det(1-x(ng)|_{V^*})}.\\]\nGiven $g\\in G$, we can choose the fundamental $N$-invariants ${N}_1,\\dots,{N}_r\\in S(V^*)^G$ to also form a $g$-eigenbasis for ${E^*}={\\mathrm{span}}_\\mathbb{C}\\{{N}_1,\\dots,{N}_r\\}$, since this space is $G$-stable and $g$ has finite order. For $g \\in G$, let $\\epsilon_1^g,\\dots,\\epsilon_r^g$ denote the eigenvalues of $g$ on ${E^*}$, so that $g{N}_i=\\epsilon_i^g{N}_i$. \n\n\\begin{proposition}\n\\label{prop:twisted_graded}\n$\\mathcal{P}_{gN}(x,y,u) = \\displaystyle\\prod\\limits_{i=1}^r \\frac{1+ \\epsilon_i^g u y^{e_i^N(V)}}{1-\\epsilon_i^g x^{d_i^N}}.$\n\\end{proposition}\n\n\\begin{proof}\nFirst, $\\displaystyle\\prod\\limits_{m\\in{\\mathcal{E}^N_\\sigma}}\\mathrm{det}(1+uy^{m}(ng)|_{{E}^*_{m}})=\\displaystyle\\prod\\limits_{i=1}\\limits^r(1+\\epsilon_i^guy^{e_i^N({V^\\sigma})})$ uniformly for any $n\\in N$, since ${E}^*_m$ is $N$-invariant. It remains to show that \\[{\\frac{1}{|N|}\\sum_{n\\in N}\\frac{1}{\\mathrm{det}(1-x(ng)|_{{V^*}})}=\\prod_{i=1}^r\\frac{1}{1-\\epsilon_i^gx^{d_i^N}}}.\\]\n\nSince $S({E}^*)\\simeq\\bigotimes_{m\\in\\mathcal{E}_N}\\mathrm{Sym}({E}^*_m),$ where ${E}^*_m$ denotes the span of fundamental $N$-invariants having $\\mathbf{x}$-degree $m+1$ and $\\mathrm{Sym}({E}^*_m)$ denotes its symmetric algebra, we have\n\\[\\sum_{\\ell\\geq 0}\\mathrm{tr}\\bigl(g|S({E}^*)_\\ell\\bigr)x^\\ell = \\prod_{m\\in\\mathcal{E}_N}\\Bigl(\\sum_{\\ell\\geq 0}\\bigl(\\mathrm{tr}(g|\\mathrm{Sym}^\\ell({E}^*_m))(x^{m+1})^\\ell\\bigr), \\] where $S({E}^*)_\\ell:=S({E}^*)\\cap S(V^*)_\\ell$ and $S(V^*)_\\ell$ as before denotes the homogeneous subspace of polynomials of $\\mathbf{x}$-degree $\\ell$. On the other hand,\n\\[\\prod_{m\\in\\mathcal{E}_N}\\Bigl(\\sum_{\\ell\\geq 0}\\mathrm{tr}\\bigl(g|\\mathrm{Sym}^\\ell({E}^*_{m})\\bigr)(x^{m+1})^\\ell\\Bigr)=\\prod_{m\\in\\mathcal{E}_N}\\frac{1}{\\mathrm{det}(1-x^{m+1}(g|_{{E}^*_{m}}))}=\\prod_{i=1}^r\\frac{1}{1-\\epsilon_i^gx_i^{d_i^N}},\\]\nsince $d_i^N=e_i^N(V)$. Therefore, $\\displaystyle\\sum\\limits_{\\ell\\geq 0}\\mathrm{tr}\\bigl(g|S({E}^*)_\\ell\\bigr)x^\\ell=\\displaystyle\\prod\\limits_{i=1}^r\\frac{1}{1-\\epsilon_i^gx_i^{d_i^N}}.$ Since for each $n\\in N$ we have $\\displaystyle\\sum\\limits_{\\ell\\geq 0}\\mathrm{tr}\\bigl(ng|S(V^*)_\\ell\\bigr)x^\\ell = \\frac{1}{\\mathrm{det}(1-x(ng|_{V^*}))},$ it remains to show that, for each $\\ell\\geq 0$, \\[\\frac{1}{|N|}\\sum_{n\\in N}\\Bigl(\\mathrm{tr}\\bigl(ng|S(V^*)_\\ell\\bigr)\\Bigr)=\\mathrm{tr}\\bigl(g|S({E}^*)_\\ell\\bigr).\\]\nTo see this, note that the operator $\\frac{1}{|N|}\\displaystyle\\sum\\limits_{n\\in N}ng=g\\cdot\\left(\\frac{1}{|N|}\\displaystyle\\sum\\limits_{n\\in N}n\\right)=g\\circ\\mathrm{pr}^N_\\ell,$ where $\\mathrm{pr}^N_\\ell=\\frac{1}{|N|}\\sum_{n\\in N} n$ is the projection from $S(V^*)_\\ell$ onto its $g$-stable subspace $S(V^*)_\\ell^N=S(E^*)_\\ell$, whence $\\mathrm{tr}\\bigl((g\\circ\\mathrm{pr}^N_\\ell)|S(V^*)_\\ell\\bigr)=\\mathrm{tr}\\bigl(g|S(E^*)_\\ell\\bigr)$. \\end{proof}\n\n\\subsection{Sum Side, Coset-by-Coset II}\n\\label{sec:coseti}\nWe next specialize some results of \\cite{bonnafe2006twisted} to the case when $N$ is a normal reflection subgroup of a complex reflection group $G$. They consider the more general situation when $N$ is translated by an arbitrary element in the normalizer of $N$ in $\\mathrm{GL}(V)$. Continue to fix some $g \\in G$. \n\n\\begin{proposition}[{\\cite{bonnafe2006twisted}}]\n\\label{thm:twisted_os}\n\\[\\frac{1}{|N|}\\displaystyle\\sum\\limits_{n \\in N} \\frac{\\det(1+u (ng)|_{V^*})}{\\det(1-x (ng)|_{V^*})} = \\displaystyle\\prod\\limits_{i=1}^r \\frac{1+ \\epsilon_i^g u x^{e_i^N(V)}}{1-\\epsilon_i^g x^{d_i^N}} \\left( = \\mathcal{P}_{gN}(x,x,u)\\right).\\]\n\\end{proposition}\n\nSpecializing both sides of~\\Cref{thm:twisted_os} to $u=q(1-x)-1$ and then taking the limit $x \\to 1$ yields the following simple formula for sums over cosets.\n\n\\begin{corollary}[{\\cite{bonnafe2006twisted}}]\n\\label{cor:twisted_os} \n\\[\\displaystyle\\sum\\limits_{n \\in N} q^{{\\mathrm{fix}}_V(ng)} = \\left(\\displaystyle\\prod\\limits_{\\epsilon^g_i =1} q+e_i^N(V)\\right) \\left(\\displaystyle\\prod\\limits_{\\epsilon_i^g \\neq 1} d_i^N\\right).\\]\n\\end{corollary}\n\nUsing~\\Cref{cor:twisted_os}, we obtain the following crucial specialization of \\Cref{prop:twisted_graded}, exploiting the fact that \\Cref{prop:twisted_graded} gives the series $\\mathcal{P}_{gN}(x,y,u)$, while \\Cref{thm:twisted_os} gives the series $\\mathcal{P}_{gN}(x,x,u)$.\n\n\\begin{corollary}\n\\label{cor:twisted_graded}\n$\\lim_{x\\to 1} \\mathcal{P}_{gN}(x,x^t,qt(1-x)-1) = t^{{\\mathrm{fix}}_E g}\\sum_{n \\in N} q^{{\\mathrm{fix}}_V (ng)}.$\n\\end{corollary}\n\n\\subsection{Proof of Theorem \\ref{thm:main_theorem}} We now prove our main theorem.\n\\begin{proof}[Proof of~\\Cref{thm:main_theorem}] \n Equating the formulas from~\\Cref{lem:product_side,thm:5.2} gives\n\\begin{align}\\label{eq:sum_eq_product}\n\\mathcal{P}(x,y,u)=\\sum_{g \\in G} \\frac{\\prod_{m\\in{\\mathcal{E}^N_\\sigma}}\\mathrm{det}(1+uy^{m}g|_{{E}_m})}{\\det(1-xg|_{V})} = |G|\\prod_{i=1}^r\\frac{1+x^{e_i^G({U^N_\\sigma})}y^{e_i^N(V)}u}{1-x_i^{d_i^G}}. \\end{align}\n\nLet $\\{g_j\\}_{j=1}^{|H|}$ be a set of coset representatives for $N$ in $G$. By~\\Cref{cor:twisted_graded}, splitting the sum side of~\\Cref{eq:sum_eq_product} into a sum over the cosets of $N$ and specializing gives\n\\begin{align*}\n\\lim_{x\\to 1} \\mathcal{P}(x,x^t,qt(1-x)-1)&= \\sum_{j=1}^{|H|}\\lim_{x\\to 1} \\mathcal{P}_{gN}(x,x^t,qt(1-x)-1) \\\\ &= \\sum_{j=1}^{|H|} t^{{\\mathrm{fix}}_{E} g_j}\\sum_{n \\in N} q^{{\\mathrm{fix}}_V ng_j} = \\sum_{g \\in G} q^{{\\mathrm{fix}}_V g} t^{{\\mathrm{fix}}_{E} g}.\n\\end{align*}\n\nThe result now follows from~\\Cref{eq:sum_eq_product} by equating this specialization of the sum side with the same specialization of the product side from~\\Cref{cor:prod_side_specialization}.\n\\end{proof}\n\n\\section{Classification of Normal Reflection Subgroups}\n\\label{sec:classification}\nIn the interest of space, we restrict our classification of normal reflection subgroups to $\\mathrm{rank} \\geq 3$. In rank 2, there are two connected posets of imprimitive complex reflection groups ordered by normality: one has maximal element $G_{11}$ and minimal elements $G(4,2,2)$, $G_4$, and $G_{12}$, while the other has maximal element $G_{19}$ and minimal elements $G_{16}$, $G_{20}$, and $G_{22}$.\n\n\n\\begin{theorem}[{\\cite[Corollary 2.18]{lehrer2009unitary}}]\nFor $r\\geq 3$, the normal reflection subgroups of $G(ab,b,r)$ are $(C_d)^r$ and $G(ab,db,r)$ for $d|a$, giving quotients $G(ab,b,r)\/(C_d)^r = G((a\/d) b,b,r)$ and $G(ab,b,r)\/G(ab,db,r) = C_d$.\n\\end{theorem}\n\nAs $G_{26}$ and $G_{28}$ are the only exceptional reflection groups with more than a single orbit of reflections, there are three nontrivial exceptional (that is, not imprimitive) examples of normal reflection subgroups in rank greater than two: $G(3,3,3) \\triangleleft G_{26}$, with quotient $G_4$; $G_{25} \\triangleleft G_{26}$, with quotient $C_2$; and $G(2,2,4) \\triangleleft G_{28} \\simeq W(F_4)$, with quotient $\\mathfrak{S}_3$.\n\n\n\\section{Reflexponents}\n\\label{sec:reflexponents}\nFix $G$ a complex reflection group of rank $r$ with reflection representation $V$. Call an $r$-dimensional representation $M$ of $G$ \\defn{factorizing} if $M$ has dimension $r$ and\n\\[\\sum_{g \\in G} q^{\\mathrm{fix}_V(g)} t^{\\mathrm{fix}_M(g)} = \\prod_{i=1}^r \\Big(qt+(e_i^G(V)-m_i)t+ m_i\\Big),\\] for some nonnegative integers $m_1,\\ldots,m_r$. More generally, call a representation $M$ of $G$ of dimension $\\dim M \\leq r$ factorizing if it is factorizing in the above sense after adding in $r-\\dim M$ copies of the trivial representation.\n\nWe can now give a uniform explanation for certain ad-hoc identities from~\\cite{williams2019reflexponents}. Let $\\mathcal{H}$ be an orbit of reflecting hyperplanes, write $\\mathcal{R}_\\mathcal{H}$ for the set of reflections fixing some $L \\in \\mathcal{H}$, and let $N_\\mathcal{H} = \\left\\langle \\mathcal{R}_\\mathcal{H} \\right\\rangle$ be the subgroup generated by reflections around hyperplanes in $\\mathcal{H}$. Since these reflections form a conjugacy class in $G$, $N_\\mathcal{H}$ is a normal reflection subgroup of $G$. Furthermore: the quotient $G\/N_\\mathcal{H}$ acts as a reflection group on the $N_\\mathcal{H}$-invariants of $V$; and this action gives a $G$-representation $M_\\mathcal{H}$ that is factorizing by~\\Cref{thm:main_theorem}.\n\n\\section{Galois Twists}\\label{sec:galois}\n\n\nLet $V$ be an $r$-dimensional complex vector space and $G\\subset\\mathrm{GL}(V)$ be a complex reflection group. It is known that $G$ can be realized over $\\mathbb{Q}(\\zeta_G)$, where $\\zeta_G$ is a primitive $|G|$-th root of unity, in the sense that there is a basis for $V$ with respect to which $G\\subset \\mathrm{GL}_r(\\mathbb{Q}(\\zeta_G))$. For $\\sigma\\in\\mathrm{Gal}(\\mathbb{Q}(\\zeta_G)\/\\mathbb{Q})$, the \\defn{Galois twist} $V^\\sigma$ is the representation of $G$ on the same underlying vector space $V$ obtained by applying $\\sigma$ to the matrix entries of $g\\in\\mathrm{GL}(\\mathbb{Q}(\\zeta_G))$. Orlik and Solomon found a beautiful generalization of \\Cref{eq:shephard_todd} that takes into account these Galois twists. Below we write $\\lambda_1(g),\\dots,\\lambda_r(g)$ for the eigenvalues of $g\\in G$ on $V$.\n\n\\begin{theorem}[{\\cite{orlik1980unitary}}]\n\\label{thm:orlik_solomon}\nFix $G$ a reflection group and $\\sigma \\in \\mathrm{Gal}(\\mathbb{Q}(\\zeta_G)\/\\mathbb{Q})$. Then \n\\[\\sum_{g \\in G} \\left(\\prod_{\\lambda_i(g) \\neq 1} \\frac{1-\\lambda_i(g)^\\sigma}{1-\\lambda_i(g)}\\right) q^{{\\mathrm{fix}}_V g} = \\prod_{i=1}^r \\left(q+e_i(V^\\sigma) \\right).\\]\n\\end{theorem}\n\n\\begin{definition}\nLet $I^G_+\\subset S(V^*)$ be the ideal generated by homogeneous $G$-invariant polynomials of positive degree, and let $\\mathcal{C}_G$ be a $G$-stable homogeneous subspace of $S(V^*)$ such that $S(V^*)\\simeq I^G_+\\oplus \\mathcal{C}_G$ as $G$-modules. For $\\sigma\\in\\mathrm{Gal}(\\zeta_G)$, define the \\defn{Orlik-Solomon space} $U_G^\\sigma:=(\\mathcal{C}_G\\otimes ({V}^\\sigma)^*)^G={\\mathrm{span}}_\\mathbb{C}\\{u_1^G,\\dots,u_r^G\\}$, where the $u_i^G$ are homogeneous with $\\mathrm{deg}(u_i^G)=e_i^G(V^\\sigma)$ and such that $\\bigl(S(V^*)\\otimes (V^\\sigma)^*\\bigr)^G\\simeq S(V^*)^G\\otimes U_G^\\sigma$.\n\\label{def:osspace}\n\\end{definition}\n\n We can now state the general form of our main theorem.\n\n\\renewcommand{\\thetheorem}{\\ref{thm:main_theorem}}\n\\begin{theorem}\nLet $N\\triangleleft \\ G$ be reflection groups acting by reflections on $V$, and let ${E}=V\/N$. Let $\\sigma\\in\\mathrm{Gal}(\\mathbb{Q}(\\zeta_G)\/\\mathbb{Q})$ and define the Orlik-Solomon space $U_N^\\sigma$ as in \\Cref{def:osspace}. Then\n\\[\\sum_{g \\in G}\\left( \\prod_{\\lambda_i(g) \\neq 1} \\frac{1-\\lambda_i(g)^\\sigma}{1-\\lambda_i(g)}\\right) q^{{\\mathrm{fix}}_V g} t^{{\\mathrm{fix}}_{E} g} = \\prod_{i=1}^r \\left(qt+e_i^N(V^\\sigma) t + e_i^G((U_N^\\sigma)^*)\\right).\\]\n\\end{theorem}\n\\addtocounter{theorem}{-1}\n\n\\begin{remark}\nThe Orlik-Solomon space $U_N^\\sigma$ playes the role of ${E}^*$ in this generalization of~\\Cref{thm:main_theorem}. When $\\sigma=1$, a straightforward argument yields a graded $G$-module homomorphism (of degree $-1$) ${E^*}\\simeq U_N$. This is why we shifted the obvious $\\mathbf{x}$-grading of ${E}^*$ by $-1$ in \\Cref{sec:poincare}. However, it is not true in general that $({E}^\\sigma)^*\\simeq U_N^\\sigma$ when $\\sigma\\neq 1$.\n\\end{remark}\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\\IEEEPARstart{B}{ounds} on the performance of electrically small antennas provide meaningful estimates to the limiting trends in key metrics for wireless systems. Bounds on Q-factor\\footnote{Readers are directed to \\cite{Schab2017energy} for a detailed discussion of Q-factor and its various definitions for radiating systems.} (closely related to impedance bandwidth), and radiation efficiency are two of the most studied parameters, with mature methodologies available for their computation on a variety of objects. From analytic methods on spheres \\cite{Chu1948,Pfeiffer_FundamentalEfficiencyLimtisForESA} to advanced numerical procedures for arbitrary geometries \\cite{capek2017minimization,gustafsson2019trade}, the general trends are that both Q-factor and efficiency are dramatically constrained as the size of an electrically small antenna is reduced.\n\nOne possibility to circumvent these bounds is through the use of non-linear or time-varying (non-LTI) antennas, which are not necessarily constrained in the same ways as their linear time-invariant (LTI) counterparts. Several promising approaches have been reported using switched matching networks \\cite{Galejs1963,Vallese1972,Gamble1973,Xu2006,Zhu2014,manteghi2017,schab2019pulse,manteghi2019}, switched DC supplies \\cite{Merenda2006,Manteghi2016}, or time-varying tuning and matching elements on an antenna itself \\cite{Keller2010,Salehi2013} to successfully transmit broadband signals from electrically small antennas . However, lacking classically defined impedances or Q-factors, these time-varying systems cannot be directly compared to analogous LTI systems by the common practice of comparing impedance bandwidth and continuous wave (CW) efficiency. This presents a major obstacle in the fair, replicable evaluation of non-LTI devices and their relative performance tradeoffs compared to conventional systems. \n\nThe goal of this work is to reformulate well-known behavioral trends of LTI electrically small dipole antennas into forms compatible with time domain metrics applicable to non-LTI systems. These metrics are based heavily on those developed for assessing signal distortion (or equivalently, fidelity) in ultra-wideband (UWB) transmit systems \\cite{farr1992,Lamensdorf1994} which deal directly with the signals radiated by an arbitrary source. By moving to this methodology, we remove all dependence of assumptions on the LTI \/ non-LTI nature of a transmitter.\n\nWe begin by summarizing distortion, as used in this paper, in Section~\\ref{sec:dist-and-eff}. This is followed in Section~\\ref{sec:mappings} by development of field-circuit mappings necessary for the study of distortion due to both spherical TM$_{01}$ radiators (exact, specific) and single resonance electrically small dipole type antennas (approximate, general). Several numerical examples studying the accuracy of the approximate single resonance model and comparison of various simulated electrically small antennas are presented in Section~\\ref{sec:examples}. Finally we apply the developed methodology to assess the performance of measured conventional (LTI) and direct antenna modulation (DAM, non-LTI) systems reported in \\cite{schab2019pulse}.\n\n\n\n\n\\section{Distortion}\n\\label{sec:dist-and-eff}\nThe narrow impedance bandwidth of an LTI electrically small antenna distorts a transmitted or received signal, reducing the efficacy of information transfer. Typically, this distortion is not considered directly, but rather by proxy through the antenna's impedance bandwidth. In an effort to study electrically small antennas in ways applicable both to LTI (conventional) and non-LTI (time-varying systems for which impedance bandwidth is not defined), we revert to the study of signals produced by a radiating antenna directly in the time domain. \n\nLet the distortion measured between an ideal signal $x_\\mathrm{i}$ and a realized signal $x$ be defined as \\cite{Lamensdorf1994}\n\\begin{equation}\n\\label{eq:d-td}\nd(x_\\mathrm{i},x) = \\min_\\tau \\int_{-\\infty}^\\infty \\left| \\hat{x}_\\mathrm{i}(t) - \\hat{x}(t-\\tau) \\right|^2\\mathrm{d}t.\n\\end{equation}\nwhere $\\hat{\\cdot}$ denotes the normalization\n\\begin{equation}\n\t\\hat{f}(t) =\\frac{f}{\\left[\\int_{-\\infty}^{\\infty} |f|^2\\mathrm{d}t\\right]^{1\/2}}.\n\\end{equation}\nNote the optimization over time delay $\\tau$, with the true distortion defined by the minimum possible value. Parseval's Theorem allows this to be alternatively written in the frequency domain as \n\\begin{equation}\n\\label{eq:d-fd}\nd(X_\\mathrm{i},X) = \\min_\\tau \\int_{-\\infty}^\\infty \\left| \\hat{X}_\\mathrm{i}(\\omega) - \\hat{X}(\\omega)\\mathrm{e}^{-\\mathrm{j}\\omega\\tau} \\right|^2\\mathrm{d}\\omega.\n\\end{equation}\nFrom distortion, fidelity $F$ may also be calculated, as the two metrics are closely related by\n\\begin{equation}\n\\label{eq:fid}\n F= 1 -\\frac{d}{2}\n\\end{equation}\nwhere the minimization over time delay $\\tau$ is implicitly taken on whichever quantity is directly computed \\cite{Lamensdorf1994}. By these conventions, a linear system with a non-zero transfer function produces zero distortion (unity fidelity) under CW excitation. \n\nIf there exist multiple degrees of freedom in each observable, for example, two polarizations of a radiated field, the integrand in \\eqref{eq:d-td} and \\eqref{eq:d-fd} can be generalized to the norm of the difference between the ideal and realized signals in vector form, i.e.,\n\\begin{equation}\n\\label{eq:d-td-gen}\nd(\\boldsymbol{x}_\\mathrm{i},\\boldsymbol{x}) = \\min_\\tau \\int_{-\\infty}^\\infty \\left| \\boldsymbol{x}_\\mathrm{i}(t) - \\boldsymbol{x}(t-\\tau) \\right|^2\\mathrm{d}t.\n\\end{equation}\n\nIf measurement distortion (e.g., channel distortion, limited receiver bandwidth) or preconditioning (e.g., polarization selection) is known and representable by the linear operator $\\mathcal{L}$, the same formulation for distortion can be applied using \\cite{Lamensdorf1994}\n\\begin{equation}\n\\label{eq:op-transform}\n\tx = \\mathcal{L}y,\\quad x_\\mathrm{i} = \\mathcal{L}y_\\mathrm{i}\n\\end{equation}\nwhere $y$ and $y_\\mathrm{i}$ are the realized and ideal transmit signals, and now $x$ and $x_\\mathrm{i}$ are the corresponding observables. \n\nThough distortion is used extensively in the characterization of UWB transmitters, the impact of the bandwidth and efficiency limitations of small antennas on distortion bounds has not been previously described. Our goal here is to do so in order to provide a benchmark against which the distortion properties of non-LTI systems can be compared. We proceed by developing operational relationships between ideal signals (e.g., input voltage to an antenna) and some produced output (e.g., a normalized radiated field) in order to calculate distortion in terms of parameters bounded for LTI systems. In this way, we project the effect of an LTI system's Q-factor and efficiency onto the distortion metric which is compatible for comparison with non-LTI systems for which Q-factor, impedance bandwidth, and other frequency domain parameters are not defined. The rest of this paper uses distortion as the metric of interest, though all data and results can be recast in terms of fidelity using \\eqref{eq:fid}\n\n\\section{Radiated Fields from Electrically Small TM$_{01}$ Antennas}\n\\label{sec:mappings}\nThe goal of this section is to derive expressions for calculating the fields radiated by a small TM$_{01}$ antenna under specific (LTI) matching conditions. These in turn can be used to calculate distortion as defined in Sec.~\\ref{sec:dist-and-eff} under arbitrary excitation. A spherical radiator with a well-defined equivalent circuit is first studied before applying approximations to generalize the calculations to the study of any structure (antenna or current distribution) with a Q-factor and efficiency defined at a single frequency.\n\n\\subsection{Spherical Shell}\nIn a seminal paper studying the physical bounds of omnidirectional antennas, Chu established an equivalent circuit to model the impedance and radiation behavior of a dipole-like (TM$_{01}$) spherical antenna \\cite{Chu1948}. In terms of the input current $I$ to this equivalent circuit, the radiated electric field $\\boldsymbol{E}$ is\n\\begin{equation}\n \\boldsymbol{E} = \\hat{\\boldsymbol{\\theta}}\\frac{1}{\\rho\\mathrm{h}^{(2)}_1}\\sqrt{\\frac{3\\eta_0}{8\\pi}}\\sin\\theta \\frac{\\mathrm{e}^{-\\mathrm{j} kr}}{r} I.\n\\end{equation}\nHere $\\rho$ denotes the electrical size $ka = 2\\pi a\/\\lambda$ and $\\mathrm{h}^{(2)}_1$ is the first order spherical Hankel function of the second kind evaluated with argument $\\rho$. Without loss of generality, we consider only broadside ($\\theta=\\pi\/2$) radiated fields, drop the fixed constant $\\sqrt{3\\eta_0\/(8\\pi)}$, suppress the far field propagation term, and consider only the scalar magnitude of the $\\theta$-polarized field, i.e.,\n\\begin{equation}\n E = \\frac{I}{\\rho\\mathrm{h}^{(2)}_1}.\n \\label{eq:sphere-ei}\n\\end{equation}\nThe equivalent antenna input impedance of the TM$_{01}$ radiator is defined as\n\\begin{equation}\n Z_\\mathrm{a} = \\frac{\\mathrm{j}(\\rho\\mathrm{h}^{(2)}_1)'}{\\rho\\mathrm{h}^{(2)}_1} = \\frac{\\rho^4 - \\mathrm{j}\\rho}{\\rho^4+\\rho^2}\n\\label{eq:z1}\n\\end{equation}\nand can be readily implemented using a three element RLC circuit \\cite{Chu1948}. Here $(\\cdot)'$ denotes a derivative taken with respect to electrical size $\\rho$. Note that this impedance represents only an outward going wave impedance. Internal fields can easily be included using parallel networks \\cite{thal1978exact} with the effect of increased Q-factors in the calculation of electrically small spheres in Sec.~\\ref{sec:examples}.\n\nSuppose the antenna (equivalent circuit) is tuned to resonance at $\\rho_0$ by a series inductor $L$. Further, a series resistance $R_\\Omega$ is introduced to represent ohmic losses. This resistance can be determined based on material parameters and electrical size by various models. Because of the intricate nature of its definition and scaling (e.g., skin depth model or surface resistance) we leave this resistance as a free parameter to tune at will. With these alterations, the tuned, lossy impedance model is written as \n\\begin{equation}\n \\tilde{Z}_\\mathrm{a} = Z_\\mathrm{a} + \\mathrm{j}\\rho X_0\/\\rho_0 + R_\\Omega,\n\\end{equation}\nwhere the tuning inductance is scaled in terms of its reactance \\mbox{$X_0 = \\mathrm{Im}~\\tilde{Z}_\\mathrm{a}(\\rho_0)$} at the tuned frequency. A non-dispersive source with impedance \\mbox{$R_0 = \\mathrm{Re}~\\tilde{Z}_\\mathrm{a}(\\rho_0)$} is connected to the antenna circuit. For a given excitation $V_\\mathrm{s}$, the radiated field is then given by\n\\begin{equation}\n E = \\frac{V_\\mathrm{s}}{\\rho \\mathrm{h}^{(2)}_1(R_0+\\tilde{Z}_\\mathrm{a})}.\n \\label{eq:sphere-ev}\n\\end{equation}\nThe above expression is general in the sense that no approximations have been made regarding the electrical size of the spherical radiator, only that it radiates solely in the TM$_{10}$ mode. Using some form of this electric field as an observable quantity, the relation in \\eqref{eq:sphere-ev} can be applied to calculate distortion from a sphere of arbitrary size ($\\rho$), arbitrary loss parameters ($R_\\Omega$), and under arbitrary excitation ($V_\\mathrm{s}$); where the ideal observable $x_\\mathrm{i}$ may be the excitation itself.\n\n\n\\subsection{Arbitrary Electrically Small TM$_{01}$ Radiators}\nSevere Q-factor and efficiency limits arise in the electrically small limit ($\\rho\\rightarrow 0$). Here we simplify \\eqref{eq:sphere-ev} in this limit and make narrowband, stationary approximations to allow for the calculation of the radiated field $E$ with knowledge only of a radiator's tuned Q-factor and efficiency at a given operating frequency.\n\nThe current-field relation in \\eqref{eq:sphere-ei} simplifies to the expected time-derivative form in the electrically small limit, i.e.,\n\\begin{equation}\nE = -\\mathrm{j} \\rho I.\n\\end{equation}\nUsing this simplification to represent the general behavior of all small dipole-like radiators and the definition of the reflection coefficient \n\\begin{equation}\n \\varGamma = \\frac{\\tilde{Z}_\\mathrm{a} - R_0}{\\tilde{Z}_\\mathrm{a} + R_0},\n \\label{eq:gamma}\n\\end{equation}\nthe radiated field may be written as \n\\begin{equation}\n E = \\frac{\\mathrm{j} \\rho V_\\mathrm{s}}{2R_0}\\left(1-\\varGamma\\right).\n \\label{eq:esa-ev}\n\\end{equation}\nAssuming that the resonant system can be approximated locally in frequency as a series RLC circuit \\cite{Yaghjian2005}, the reflection coefficient can be rewritten using \\eqref{eq:gamma} as a function parameterized by the system's resonant radiation Q-factor $Q$ and efficiency $\\eta$\n\\begin{equation}\n \\varGamma = \\frac{\\mathrm{j} Q\\eta (\\rho-\\rho_0) \/ \\rho_0}{1+\\mathrm{j} Q\\eta (\\rho-\\rho_0) \/ \\rho_0}.\n \\label{eq:gamma}\n\\end{equation}\nHere $Q$ denotes only the radiation Q-factor where as $Q\\eta$ is the total system Q-factor including loss. In this RLC approximation, the ohmic and radiation losses are implicitly assumed to be stationary in frequency.\n\nThe expressions in \\eqref{eq:esa-ev} and \\eqref{eq:gamma} represent a framework for calculating the frequency domain radiated fields for an electrically small dipole-like antenna specified by a radiation Q-factor and efficiency at a single frequency. Arbitrary excitation $V_\\mathrm{s}$ may be applied and loss models may be parameterized by the total system efficiency $\\eta$. Given that closed form expressions are available for the radiation Q-factor and efficiency of a TM$_{01}$ spherical current distribution, the strength of the stationary narrowband approximations can be easily assessed. Similarly, many techniques allow for the calculation of optimal Q-factor and optimal efficiency currents on arbitrarily shaped objects (e.g., \\cite{capek2017minimization,gustafsson2019trade}), meaning that \\eqref{eq:esa-ev} and \\eqref{eq:gamma}, in conjunction with the metrics defined in Sec.~\\ref{sec:dist-and-eff}, can be applied to the study of the optimal distortion and efficiency achievable by arbitrarily shaped electrically small dipole-like radiators.\n\n\\subsection{Removal of Time Derivative Term}\nThe expressions \\eqref{eq:sphere-ev} and \\eqref{eq:esa-ev} suggest that the source voltage and radiated electric field may be candidates for the ideal and measured signals used in computing distortion via \\eqref{eq:d-td}. However, we note that even in the case of perfect broadband matching, there exists a time derivative term between these two quantities. In an effort to obtain a comparison that reduces to zero distortion in the ideally matched case, either the source voltage must be multiplied by this derivative term $\\mathrm{j}\\rho$ or the radiated field must be integrated to remove this term. We opt for the latter in this study to avoid issues related to the unbounded nature of the differentiation operator. Thus in applying equation \\eqref{eq:d-td} to calculate distortion, we take for the ideal signal $x_\\mathrm{i} = \\mathcal{F}^{-1}\\left[V_\\mathrm{s}\\right]$ and the measured signal $x= \\mathcal{F}^{-1}\\left[E\/(\\mathrm{j}\\rho)\\right]$ where $E$ is the radiated field calculated either by \\eqref{eq:sphere-ev} (exact sphere) or \\eqref{eq:esa-ev} (approximate single resonance model). This is similar to the treatment discussed in \\cite{farr1992}.\n\n\n\\subsection{Discussion of Parameter Dependencies}\nThe rigorous calculation of the electric field produced by a spherical shell \\eqref{eq:sphere-ev} depends on many parameters: input signal type, antenna materials, antenna efficiency $\\eta$, and electrical size $\\rho$. By contrast, it can be observed that, for signals classified by some nominal bandwidth $B$ (e.g., $99\\%$ power occupied bandwidth), all of these parameters can be collapsed into a single product $Q\\eta B$ using the single resonance model in \\eqref{eq:esa-ev}. Note that here $B$ represents the bandwidth of the input signal (not the antenna bandwidth). Thus, the complete term $Q\\eta B$ is proportional to the quotient of the signal and antenna impedance bandwidths \\cite{Yaghjian2005}. \n\nThis implies that all small antennas which fit the assumptions of the single resonance dipole model transmitting a signal of bandwidth $B$ can be classified by a single number $Q\\eta B$ regardless of their specific Q-factor, efficiency, or the bandwidth of the signal. Thus, the distortion for a specific signal type and bit sequence can be plotted as a function of this single composite variable. The resulting curve contains the distortion performance of any and all single resonance antennas transmitting that particular message. The shape of this curve depends on the exact message being transmitted for short signals but converges as the message length is increased and the power spectral density of the input signal approaches that of an infinite sequence of the chosen modulation type.\n\nAt this point it is necessary to highlight a few of the practical advantages of collapsing several antenna metrics into the combined parameter $Q\\eta B$. First, to predict realized distortion under a given signal bandwidth $B$, there is no need to evaluate an antenna's radiation Q-factor or efficiency, rather only the product $Q\\eta$ which can be easily estimated from an analytic, numerical, or measured antenna's input reflection coefficient \\cite{Yaghjian2005}. Second, the effect of broadbanding an antenna via resistive loading can quickly be assessed by calculating the associated shift in $Q\\eta B$ and moving along the single resonance model curve. Finally, if a non-LTI antenna is capable of operating in an LTI mode with no changes to hardware (e.g., with a dynamic switch held static as done in \\cite{schab2019pulse}), then its effective value of $Q\\eta B$ can be determined. The realized distortion in both LTI and non-LTI modes of such a transmitter can then be measured directly in the time domain and compared to the single resonance model presented here. Alternative methods for comparing non-LTI data using these metrics are discussed in Sec.~\\ref{sec:dam}.\n\n\n\n\\section{Example Calculations on LTI Systems}\n\\label{sec:examples}\nIn this section, we apply distortion calculations to assess LTI structures. In anticipating the analysis of datasets collected from non-LTI transmitters in \\cite{schab2019pulse}, we calculate distortion using pseudorandom bit sequence (PRBS) on-off-keyed (OOK) excitations where the data rate is defined by the number $N$ of carrier periods per symbol. The nominal bandwidth of these signals is defined here as $B = 1 \/ N$. Additional examples using differentiated Gaussian and modulated Gaussian pulses are also included. Throughout this section, $ka$ denotes the electrical size at the tuned frequency, i.e., $\\rho_0$.\n\n\\subsection{Assessment of Stationary, Narrowband Approximations}\nHere we compare the distortion resulting from the rigorous expressions derived for a spherical shell with those from the approximate single resonance model. A 128-bit PRBS sequence is used to excite spherical radiators of varying electrical size, with the radiated field calculated via \\eqref{eq:sphere-ev}. Frequency dependent losses were not considered, instead $R_\\Omega$ was selected to enforce a CW radiation efficiency of $\\eta = 10\\%$ at the tuned carrier frequency for each electrical size. At each frequency, the tuned factor $Q\\eta$ was calculated from the differential impedance bandwidth \\cite{Yaghjian2005}. Data from this calculation are overlaid with a curve of distortion values obtained by sweeping $Q\\eta$ in the single resonance model \\eqref{eq:esa-ev} in Fig.~\\ref{fig:sphere-comparison}. Excellent agreement is observed for all tested electrical sizes.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=3.25in]{figures\/1-sphere_points_processed.pdf}\n \\caption{Comparison of rigorous sphere distortion calculations (points) with those based on the single resonance model (solid line). Bracketed points are generated for a given electrical size $ka$, while opacity denotes the number of cycles $N$ per bit in the input 128-bit OOK PRBS.}\n \\label{fig:sphere-comparison}\n\\end{figure}\n\nMany time domain signals may yield the same distortion values. The level of tolerable distortion in a communications link would depend on many parameters related to the receiver architecture. Making judgements on what is or is not an acceptable level of distortion is beyond the scope of this work, however we do note that link resilience and capacity would nearly always improve with lowered distortion. For reference, in Fig.~\\ref{fig:sphere-td} we plot time domain signals produced by a small selection of the data points in Fig.~\\ref{fig:sphere-comparison} along with their corresponding values of distortion.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=3.25in]{figures\/2-sphere_signals_processed.pdf}\n \\caption{Normalized time-domain signals $\\hat{x}$ for points corresponding to \\mbox{$N = 20$} in Fig.~\\ref{fig:sphere-comparison}. The ideal signal $\\hat{x}_\\mathrm{i}$ is shown in the top panel. Time axis normalized to the symbol length $T_\\mathrm{s}$.}\n \\label{fig:sphere-td}\n\\end{figure}\n\nAdditionally, we can examine the validity of the single resonance model under extremely broadband conditions using traditional UWB pulses. Rather than the OOK PRBS signal used throughout the rest of this paper, Fig.~\\ref{fig:sphere-pulses} shows the same comparison as Fig.~\\ref{fig:sphere-comparison} using differentiated Gaussian \n\\begin{equation}\n v_\\mathrm{s} = -\\frac{t}{T_\\mathrm{p}}\\mathrm{e}^{-\\frac{1}{2}\\left(t\/T_\\mathrm{p}\\right)^2}\n\\end{equation}\nand modulated Gaussian \n\\begin{equation}\n v_\\mathrm{s} = \\sin\\left(2\\pi f_\\mathrm{c} t\\right)\\mathrm{e}^{-\\frac{1}{2}\\left(tf_\\mathrm{c}\/N\\right)^2}\n\\end{equation}\npulses \\cite{jin2015theory}. The width of the differentiated Gaussian pulses $T_\\mathrm{p}$ corresponds to a peak frequency component located at the tuned frequency of a spherical shell with swept electrical size. The pulse repetition period is $2048T_\\mathrm{p}$. In the case of the modulated Gaussian pulses, the pulse width is written in terms of the carrier period and is varied by the parameter $N$ and the pulse repetition rate is $2048\/f_\\mathrm{c}$, where $f_\\mathrm{c}$ corresponds to the modulation frequency and the tuned frequency of a spherical shell of size $ka = 0.05$. In each of these analyses, $\\eta = 10\\%$. Excellent agreement is observed. The nominal signal bandwidth is normalized to $B = 1$ for the differentiated Gaussian and $B = 1\/N$ for the modulated Gaussian pulses.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=3.25in]{figures\/3-sphere_pulses_points_processed.pdf}\n \\caption{Pulse distortion from a sphere via rigorous calculation (points) and single resonance model (curve). Differentiated Gaussian pulses for varying sphere sizes (top) and modulated Gaussian pulses for fixed sphere size and varying pulse width (bottom) are shown.}\n \\label{fig:sphere-pulses}\n\\end{figure}\n\n\\subsection{Analysis of Optimal Currents on Arbitrary Objects}\n\nWith the single resonance model, distortion may be predicted for any system with a Q-factor and efficiency defined at a single frequency. This includes non-driven current distributions on arbitrary structures, extending the general approach used for spheres to the analysis of any arbitrary geometry. Currents optimal in radiation Q-factor and efficiency are readily obtained using a method of moments formulation \\cite{capek2017minimization}. In many ways, these optimal, non-driven current distributions are exactly analogous to the spherical harmonics used by Chu, though their behavior is not obtained via analytic expansion, but rather by deterministic convex optimization methods.\n\nThe tradeoff between optimal radiation Q-factor ($Q$) and optimal efficiency may be calculated via a Pareto analysis~\\cite{gustafsson2019trade}. Here we conduct this analysis for an L-shaped plate\\footnote{Aspect ratio $2:1$, notch extends exactly halfway into each dimension.} with electrical size $ka = 0.5$, homogeneous surface resistivity \\mbox{$R_\\mathrm{s} = 0.1~\\Omega$}, and without the explicit constraint of self resonance (i.e., lossless tuning implied for non-resonant solutions). The results are plotted in the inset of Fig.~\\ref{fig:ell} as the tradeoff between $(Q\\eta)^{-1}$ (proportional to antenna bandwidth $B_\\mathrm{a}$) and efficiency $\\eta$. Three points are selected at various positions along the tradeoff curve and the properties of the associated optimal current distributions are used to calculate distortion of a 128-bit PRBS OOK signal with $N=10$. The resulting points are shown along side these current distributions in Fig.~\\ref{fig:ell}. Note that, unlike the preceding analysis of a sphere, these data points are computed directly from the single resonance model and lie exactly on the single resonance model curve, drawn in black.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=3.25in]{figures\/4-ell_points_processed.pdf}\n \\caption{Distortion calculated for Pareto-optimal currents on an L-shaped plate$^\\mathrm{1}$. Three points along the bandwidth-efficiency Pareto front (inset) are selected and the corresponding distortion values are calculated using the single resonance model.}\n \\label{fig:ell}\n\\end{figure}\n\nThis analysis demonstrates how distortion analysis may be combined with the study of optimal currents to predict \\textit{a priori} the ideal tradeoff between message distortion and radiator efficiency. This is particularly relevant for the analysis of non-LTI systems where it is prudent to assess how improvements in signal fidelity compare versus the gains that could have been achieved by resistively broadbanding a comparable LTI transmitter.\n\n\n\n\\subsection{Analysis of Driven Antennas}\n\nIn the preceding examples, non-driven optimal current distributions were assessed using hypothetical equivalent circuit models. By contrast, here we demonstrate the calculation of distortion from several driven antennas evaluated by full wave simulations and compare the obtained data to curves predicted by the single resonance model.\n\nWe begin with a thin wire dipole, depicted in Fig.~\\ref{fig:dipoles}. The dipole is simulated using NEC2++ \\cite{nec}, a method of moments solver, to obtain broadband input impedance and radiation data. These data are then used to numerically tune the dipole to a carrier frequency represented by an electrical size $ka$. In each case considered here, the dipole is tuned below its natural resonance using a series inductance. The tuned system is driven by a matched source using the same class of OOK signals as in the previous examples with bandwidths classified by the number of cycles per symbol $N$. For each signal considered, the broadside co-polarized field is computed, integrated, and compared to the driving voltage via calculation of distortion. Additionally, at each tuned frequency, the parameter $Q\\eta$ is estimated from simulated impedance data using the model in \\cite{Yaghjian2005}. This factor, combined with the nominal signal bandwidth yields the coordinate $Q\\eta B$ with which the simulated distortion can be compared to the single resonance distortion model. Fig.~\\ref{fig:dipoles} shows excellent agreement between the fullwave dipole simulation and single resonance distortion model over three electrical sizes and five data rates, with each parameter spanning at least one order of magnitude.\n\n\\begin{figure}\n \\centering \\includegraphics[width=3.25in]{figures\/5-dipole_points_processed.pdf}\n \\caption{Comparison of full wave distortion calculations on a copper dipole simulated using the method of moments (points) with those based on the single resonance model (solid line).}\n \\label{fig:dipoles}\n\\end{figure}\n\nSeveral self resonant structures\\footnote{Dimensions and materials are listed in Appendix A.} are subjected to the same simulation and distortion calculation procedure, with results shown in Fig.~\\ref{fig:sr-antennas}. A spherical helix, a quarterwave patch on a finite dielectric substrate, and a capacitively coupled circular patch \\cite{vanniekerk2013analysis} are examined. The latter two antennas were simulated over an infinite perfect electric conductor (PEC) ground plane using CST Microwave Studio. As in the previous example, the factor $Q\\eta$ was again estimated from simulated impedance data, though here each antenna is driven with a nominal $50~\\Omega$ source. All transmitted fields were calculated in the polarization and direction of maximum gain at the self resonance frequency. Despite the relatively high complexity of these antennas, we observe excellent agreement between the simulated distortion and the predicted single resonance curve. \n\n\\begin{figure}\n \\centering\n \\includegraphics[width=3.25in]{figures\/6-full_points_processed.pdf}\n \\caption{Comparison of full wave distortion calculations using three self-resonant antennas$^\\mathrm{2}$ (points) with those based on the single resonance model (solid line).}\n \\label{fig:sr-antennas}\n\\end{figure}\n\n\n\\section{Assessment of a non-LTI Transmitter}\n\\label{sec:dam}\nWith the accuracy of the single resonance model verified in previous examples, we move to our primary motivation for studying distortion: assessing the relative performance of LTI and non-LTI transmitters.\n\nAs stated in Sec.~\\ref{sec:intro}, while previous efforts to demonstrate non-LTI transmitters' ability to outperform comparable LTI transmitters have provided promising results, reporting of the realized benefits of non-LTI systems has been largely visual or qualitative. This is due in no small part to the fact that frequency domain quantities, such as impedance bandwidth, are not directly defined for non-LTI systems and thus cannot be used as performance metrics as is done with LTI systems. Here we use distortion, which can be measured directly in the time domain for either LTI or non-LTI systems, to compare the performance of the non-LTI OOK transmitter reported in \\cite{schab2019pulse}.\n\nThe transmitter studied in \\cite{schab2019pulse} is based on the switched matching network architecture described in \\cite{Galejs1963}. As such, it is capable of operating in conventional (LTI) and direct antenna modulation (DAM, non-LTI) modes without changes to the physical hardware. Thus, the transmitter can be characterized by a value of the parameter $Q\\eta$ measured when operating in LTI modes. Note that not all non-LTI systems may be characterized in this way, but as we shall point out later this does not prevent the use of distortion analysis.\n\nRaw over the air time domain data from the experiments conducted in \\cite{schab2019pulse} were used to calculate distortion for 64-bit PRBS OOK signals in conventional and DAM modes. Channel inversion was used to isolate distortion due to transmitter mismatch, see \\cite{schab2019pulse} for details. The resulting data are plotted in Fig.~\\ref{fig:dam}, where good agreement between the conventional transmitter mode and the single resonance model is observed. For all three transmissions using the DAM transmitter mode, the distortion is clearly reduced below that of the single resonance model, in line with the visual assessment of corresponding eye diagrams made in \\cite{schab2019pulse}.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=3.25in]{figures\/7-dam_points_processed.pdf}\n \\caption{Comparison of OOK LTI distortion bounds with measured non-LTI transmitter performance. Here $N\\sim 1\/B$ is a metric of symbol length. The same 64-bit OOK sequence was used for the analytic calculation and the measured transmissions. }\n \\label{fig:dam}\n\\end{figure}\n\nBeyond qualitative comparison, the use of distortion analysis enables the quantification of gains afforded by this non-LTI transmitter. Fig.~\\ref{fig:dam-detail} shows a detail view of the self resonance curve and measured distortion from the conventional and DAM transmitter modes with $N=3$. Starting from the indicated LTI measurement, we observe that the non-LTI system provides a four-fold reduction in distortion (from $0.36$ to $0.09$). To achieve equivalent distortion for the same transmission, the LTI system would require resistive broadbanding amounting to a reduction in efficiency (or gain, assuming constant directivity) by a multiplicative factor of $0.17$ or equivalently $-7.6~\\mathrm{dB}$. Conversely, since the LTI and non-LTI systems have the same efficiency and the signal bandwidths are identical in both measurements, the non-LTI system can be interpreted as having an ``effective Q-factor'' a multiplicative factor of $0.17$ lower than that of the LTI system, or equivalently an ``effective bandwidth'' that is a multiplicative factor of $5.7$ larger. \n\nNeither gain nor Q-factor is necessarily defined for the non-LTI transmitter. However, this example demonstrates how distortion may be used as an intermediate quantity to determine the non-LTI transmitter's potential benefits in terms of these critical system parameters. Additionally, while there is no non-trivial lower bound on distortion for non-LTI systems, the curve containing the distortion properties of all single resonance LTI antennas helps illustrate the relative gains and performance parameters afforded by the use of any particular non-LTI transmitter implementation.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=3.25in]{figures\/8-dam_detail.pdf}\n \\caption{Comparison of OOK LTI distortion bounds with measured non-LTI transmitter performance. The same 64-bit OOK sequence was used for the analytic calculation and the measured transmissions. }\n \\label{fig:dam-detail}\n\\end{figure}\n\n\n\\section{Conclusions}\n\nElectric field relations for a spherical shell and single resonance dipole-like antennas are cast in forms compatible with the calculation of distortion. We have provided several examples justifying the approximation that all LTI, electrically small, dipole-like radiators are well characterized by a single curve of distortion versus the composite Q-factor, efficiency, and signal bandwidth parameter $Q\\eta B$. This finding is used to compare the performance of a non-LTI transmitter by obtaining quantitative measures of its behavior relative to the bandwidth and \/ or efficiency of a comparable LTI system. \n\nGoing forward, further studies into quantifying the efficiency and directive nature (intentional or unintentional) of non-LTI radiators in order to complete a framework for assessing their integration into communications or sensing systems. Furthermore, the procedure presented in this paper will allow for the more rigorous analysis of previously proposed theoretical and experimental non-LTI transmitters.\n\n\\appendices\n\\section{Dimensions of Example Self-Resonant Antennas}\n\n\\subsection{Spherical Helix}\nThe five-turn spherical helix is simulated using NEC2++, a method of moments based solver. The simulated helix is made of a cylindrical wire of radius $r_\\mathrm{w} = 0.1~\\mathrm{mm}$ with conductivity $\\sigma = 5.8\\cdot10^{7}~\\mathrm{S\/m}$. The radius of the bounding sphere is \\mbox{$r = 1~\\mathrm{m}$} and the helix is described by the curve $(x(t),~y(t),~z(t))$ with coordinates\n\\begin{equation}\n z = -r + 2rt,\\quad\\quad\n \\rho = \\sqrt{r^2-z^2}\n\\end{equation}\n\\begin{equation}\n x = \\rho \\cos (10\\pi t),\\quad\\quad\n y = \\frac{z}{|z|}\\rho \\sin (10\\pi t)\n\\end{equation}\nwhere $t \\in [0,1]$. The structure is discretized into 175 segments and fed at the segment crossing $t=0.5~(z = 0~\\mathrm{m})$. The helix is resonant near $f = 6.6~\\mathrm{MHz}~(ka = 0.14)$.\n\n\\subsection{Quarterwave Patch}\nThe square quarterwave patch is simulated using the finite element method in CST. The ground plane and patch are made of PEC and the ground plane is of infinite extent. There is a shorting wall the entire extent of the patch on the side closest to the feed point. Parameters for this antenna are given in Table~\\ref{tab:diel}.\n\n\\subsection{Dual Reactively Loaded Monopole (DRLM)}\nThe DRLM is based on the horizontally meandered PIFA from \\cite{vanniekerk2013analysis}, where detailed dimensions may be obtained. The circumscribing sphere for this antenna (and its image below the infinite ground) has a radius of $22~\\mathrm{mm}$. Simulations of this antenna are carried out in CST with an infinite PEC ground plane. The self resonant frequency is near $810~\\mathrm{MHz}~(ka = 0.37)$. \n\n\\begin{table}[]\n \\centering\n \\begin{tabular}{l|c}\n \\textit{Parameter} & \\textit{Value}\\\\ \\hline\n Substrate dimensions & $9$~mm \\\\\n Patch dimensions & $8$~mm \\\\\n Shorting wall width & $8$~mm\\\\\n Feed distance from shorting wall & $3.4$~mm \\\\\n Substrate relative permittivity & $9.8$ \\\\\n Feed type & SMA coaxial \\\\ \n Self resonant frequency & $2.978~\\mathrm{GHz}$\\\\\n \\end{tabular}\n \\caption{Quarterwave patch parameters.}\n \\label{tab:diel}\n\\end{table}\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe cosmic infrared background (CIB or CIRB) contains a considerable fraction of the collective radiation emitted by stars through the cosmic time. Stars in the Epoch of Reionization (EoR) have the bulk of their radiation redshifted into the near-infrared band ($\\sim$0.7 - 5 $\\mu$m) and the CIB measured in this band is specifically named ``near-infrared background\" (NIRB). \nAs such, the NIRB offers\na unique opportunity to study faint high-$z$ galaxies that remain largely undetected in deep galaxy surveys (see e.g. \\citealt{2006MNRAS.367L..11S, 2006ApJ...646..703F,2010ApJ...710.1089F,2012ApJ...750...20F,2013ApJ...764...56F,2013MNRAS.433.2047F}). This is particularly important as these objects are commonly believed to provide most of the ionizing power that drives cosmic reionization \\citep{2007MNRAS.380L...6C,2011MNRAS.410..775R,2011MNRAS.414..847S}. Besides, the NIRB might also help characterizing the stellar populations of the first cosmic systems \n\\citep{2003MNRAS.339..973S,2006MNRAS.368L...6S,2002MNRAS.336.1082S,2004ApJ...608....1K,2005Natur.438...45K,2003MNRAS.342L..25M,2004MNRAS.351L..71C,2004ApJ...606..611C}. \n\nMost recent studies have converged on the prediction that on scales of $\\approx 1000''$ the fluctuation level from normal star-forming galaxies at $z \\lower.5ex\\hbox{\\gtsima} 5$ is $\\approx 10^{-3}$~nWm$^{-2}$sr$^{-1}$ at 3.6~$\\mu$m \\citep{2012ApJ...756...92C,2013MNRAS.431..383Y,2016MNRAS.455..282H}. However, extracting such signal from available data has been so far very challenging, as even when the deepest galaxy\nsubtraction from NIRB maps is applied, the remaining flux fluctuations\\footnote{NIRB studies usually concentrate on fluctuations rather than absolute flux, as the latter is difficult to measure due to the presence of an overwhelming foreground. However, as the foreground is rather smooth on scales at which the NIRB is measured, the fluctuations analysis is more robust -- see e.g. \\citealt{2007ApJ...657..669T,2011ApJ...742..124M,2012ApJ...753...63K,2012Natur.490..514C}.} still cannot be ascribed to the known high-$z$ galaxy population. On small angular scales, most of the signal arises from unresolved low-$z$ galaxies (see the first analysis by \\citealt{2002ApJ...579L..53K}).\nOn larger scales the measured power (see e.g. \\citealt{2004ApJ...608....1K,2005Natur.438...45K,2007ApJ...654L...1K,2007ApJ...654L...5K,2012ApJ...753...63K,2005ApJ...626...31M, 2011ApJ...742..124M,2015arXiv150405681S,2007ApJ...659L..91C,2012Natur.490..514C}) is $\\lower.5ex\\hbox{\\gtsima} 100$ times larger than the low-$z$ galaxies \\citep{2012ApJ...752..113H}, and $\\lower.5ex\\hbox{\\gtsima} 1000$ times larger than that expected from high-$z$ normal star-forming galaxies and first stars \\citep{2012ApJ...756...92C,2013MNRAS.431..383Y}. Therefore it must be attributed to some, yet unknown, alternative sources. Basically, two different explanations have been proposed for the origin of such large scale ($\\sim1000''$) ``power excess\". They involve either early accreting black holes \\citep{2013MNRAS.433.1556Y,2014MNRAS.440.1263Y} which could explain the detected NIRB-cosmic X-ray background coherence \\citep{2013ApJ...769...68C}, or ``intrahalo light'' that radiated by stars ejected from their parent galaxies during merger events \\citep{2012Natur.490..514C,2014Sci...346..732Z}. \n\nAt present it is unclear which hypothesis should be preferred. Besides, it is also possible that first stars or black holes are much more abundant than in the standard theoretical framework \\--- for example if the slope of the density fluctuations power spectrum sightly deviates from the standard one, the number of small halos (thus stars or black holes therein) could be boosted exponentially \\--- they can then provide sufficient radiation power. To be consistent with electron scattering reionization bounds, at the same time the ionizing photon escape probability must be rather low (for a detailed discussion see \\citealt{2016MNRAS.455..282H}). To further complicate the interpretation, recent observations \\citep{2014Sci...346..732Z} show that, on large scales and at least for the 1.1 and 1.6~$\\mu$m bands, diffuse Galactic light (DGL) might provide a non-negligible flux contribution. However \\citet{2007ApJ...654L...5K} (see \\citealt{2010ApJS..186...10A} as well) found that at the 3.6 and 4.5 $\\mu$m bands the DGL or the Galactic cirrus component is largely subdominant. \n\nFor the above reasons, it is urgent to devise new strategies that put our understanding on firmer grounds. In order to isolate the signal from increasingly high redshifts, with sufficient depth and angular resolution one can in principle remove foreground galaxies down to extremely faint levels. This is challenging for current instrumentation, and also not an easy task for the {\\tt JWST}, as the signal from high-$z$ galaxies is expected to be subdominant even at $\\sim$32 AB mag \\citep{2016MNRAS.455..282H}. \nAlternatively, cross-correlation studies seem promising. The NIRB-HI 21cm line cross-correlation \\citep{2014MNRAS.440..298F,2014ApJ...790..148M} has the advantage that it selectively picks up the signal from reionization sources. Also, the NIRB, if produced by sources in the EoR, would be cross-correlated with the CMB through the thermal Sunyaev-Zeldovich effect. This correlation could be seen in the forthcoming Euclid-based all-sky CIB maps \\citep{2014ApJ...797L..26A}. In addition to cross-correlation, a recent work \\citep{2015ApJ...804...99K} proposed the use of Lyman-break tomography to constrain the NIRB contribution from sources above a certain redshift.\n\n\\citet{2007ApJ...666L...1K} analyzed the cross-correlation between the ACS-detected faint sources and the source-subtracted NIRB. They found significant correlation on small scales, implying that the faint ACS galaxies do contribute to NIRB fluctuations. However, these sources (or objects associated with them) cannot account for the large scale clustering as the measured correlation is negligible. Their analysis did not pay particular attention to the high-$z$ sources, therefore the information on high-$z$ galaxies cannot be directly derived from there.\nIt is surprising that so far little attention has been devoted to the search for high-$z$ normal star-forming galaxy signatures in the NIRB, given that deep galaxy surveys have made tremendous progresses in obtaining the UV luminosity functions (LF) up to $z=10$ \\citep{2015ApJ...803...34B}, and the detection limits of Lyman break galaxy (LBG) surveys carried out by {\\tt HST} have already reached $H\\sim 29 - 31$ \\citep{2013ApJS..209....6I,2011ApJ...737...90B}. \nMeasuring their intensity fluctuations present (though tiny) in the NIRB is essentially one of the most exciting perspectives as it might convincingly show that NIRB fluctuation experiments can be used to study the first galaxies. \nSuch an experiment is also complementary to more traditional galaxy surveys that derive the LF of the brightest galaxies among early populations. \n\nIn summary, it is mandatory to show that the NIRB power spectrum signal of already known high-$z$ galaxies can be recovered robustly. Demonstrating a successful strategy will represent a major step forward in the methodology and allows to obtain, in addition to the clustering signal, other key quantities, as e.g. the colors of high-$z$ galaxies beyond the observed $H$ band. Colors, in turn, provide potentially information on stellar ages and initial mass function of the stars harbored by galaxies in the EoR. \n\nThe idea we propose here is to isolate the targeted LBG signal and show the feasibility of statistically detecting reionization sources by cross-correlating deep LBG surveys with NIRB maps. To this aim, we: (a) construct large scale mock maps of the source-subtracted NIRB and LBG catalogs using semi-numerical simulations; (b) perform a cross-correlation analysis between the two data sets to extract the contribution of high-$z$ galaxies in the NIRB. The paper is organized as follows. In Sec. \\ref{method} we describe the steps to construct the mock maps. In Sec. \\ref{results} we present the analysis about the correlation coefficient and the colors. Conclusions and discussions are presented in Sec. \\ref{conclusions}. Throughout this paper we use {\\tt Planck} cosmological parameters: $\\Omega_{\\rm m} = 0.31, \\Omega_\\Lambda = 0.69, \\Omega_{\\rm b} = 0.048, n_s = 0.96, \\sigma_8 = 0.82$ and $h = 0.68$ \\citep{2014A&A...571A..16P}. All magnitudes are in the AB-system \\citep{1983ApJ...266..713O}.\n\n\\section{Construction of mock maps}\\label{method}\n\n\\subsection{ High-$z$ galaxies }\nUsing the code {\\tt DexM}\\footnote{\\url{http:\/\/homepage.sns.it\/mesinger\/DexM___21cmFAST.html}} \\citep{2007ApJ...669..663M} we carry out semi-numerical simulations to get catalogs of halos with mass $M_{\\rm h}\\lower.5ex\\hbox{\\gtsima} 5\\times10^8~M_\\odot$ from $z = 5$ to 10 for every $\\Delta z = 0.1$. We adopt a 400~Mpc box size that corresponds to an angular size of $\\approx2.4$~deg at $z = 10$. We construct a cuboid by replicating the output boxes along the line-of-sight, adding random translations, rotations and reflections \\citep{2005MNRAS.360..159B}. This is our light-cone since we assume that all lines-of-sight are parallel \\--- an assumption that is convenient and safe enough when $z \\lower.5ex\\hbox{\\gtsima} 5$. \n\n \nTo construct flux maps from the light-cone, we link galaxy luminosities to halo mass $M_{\\rm h}$. The observed LFs could be reproduced exactly if we derive the $L_{\\rm UV} - M_{\\rm h}$ relation through abundance matching, i.e. we force the number density of galaxies with luminosity $>L_{\\rm UV}$ to match the number density of halos with mass $>M_{\\rm h}$. Formally, \n\\begin{equation}\n\\int_{M_{\\rm UV}} \\Phi(M^\\prime_{\\rm UV},z) dM^\\prime_{\\rm UV} = \\int_{M_h} \\frac{dn}{dM^\\prime_h}dM^\\prime_h,\n\\end{equation}\nwhere $\\Phi$ is the UV LF at 1600~\\AA~and we use the Schechter parameterization with the redshift-dependent fitting parameters given in \\citet{2015ApJ...803...34B}. As a reference, in the derived $L_{\\rm UV} - M_{\\rm h}$ relations \nour minimum mass $5\\times10^8~M_\\odot$ corresponds to an absolute magnitude $M_{\\rm UV} = -10.5, -12.1, -13.0$ at $z=5, 8, 10$ respectively. Luminosity at other UV wavelengths is obtained through the luminosity-dependent Spectral Energy Distribution (SED) slope $\\beta$ (i.e., $f_\\lambda \\propto \\lambda^{\\beta}$) in \\citet{2014ApJ...793..115B}\\footnote{$\\beta=\\beta_{-19.5}+\\frac{d\\beta}{dM_{\\rm UV}}(M_{\\rm UV}+19.5)$, the values of $\\beta_{-19.5}$ and $\\frac{d\\beta}{dM_{\\rm UV}}$ at $z= 4, 5, 6$ and 7 could be found in \\citet{2014ApJ...793..115B}. For convenience of using this form at in-between redshifts, we fit $z$-dependent forms \\citep{2015MNRAS.450.3829Y}: $\\beta_{-19.5}=-1.97-0.06(z-6)$ and $\\frac{d\\beta}{dM_{\\rm UV}}=-0.18-0.03(z-6)$. We use these fittings anyway when $5 < z < 10$.}. However, generally speaking this power-law only holds at $\\lambda \\lower.5ex\\hbox{\\ltsima} 2000 - 3000$~\\AA, while we need luminosities at least until $4.5\/(1+z)~\\mu$m, say 7500~\\AA~when $z=5$. Therefore at $\\lambda>$2000~\\AA~we use the SED template from {\\tt Starburst99}\\footnote{http:\/\/www.stsci.edu\/science\/starburst99\/docs\/default.htm} \\citep{1999ApJS..123....3L,2005ApJ...621..695V,2010ApJS..189..309L}, adopting a continuous star formation mode, with metallicity 0.1~$Z_\\odot$ and 200~Myr stellar age. The SB99 SEDs are normalized to match the power-law form at 2000~\\AA. \n\n\nThe flux received in each pixel in the map is the sum of radiation from all galaxies seen by the pixel,\n\\begin{equation}\nF(\\nu_0)=\\nu_0\\frac{1}{(\\theta_{\\rm pix})^2}\\sum_j \\frac{L^j(\\nu)(1+z_j)}{4\\pi r_j^2(1+z_j)^2},\n\\end{equation}\nwhere $\\nu=\\nu_0(1+z_j)$, $z_j$ is the redshift of the $j$-th halos\\footnote{We do not model photometric redshift uncertainties, because the redshift range considered here $z = 5-10$ is much larger than the uncertainties.} in the solid angle $(\\theta_{\\rm pix})^2$ (we adopt $\\theta_{\\rm pix}=3.6''$ for all mock maps in this work),\n $r_j$ is its comoving distance.\nIn Fig. \\ref{maps} we show the 3.6~$\\mu$m flux map (bottom left panel) from all galaxies between $z = 5 -10$ \n(flux from galaxies at $z > 10$ is negligible compared with galaxies at lower redshift, we ignore it here). \n \n\\begin{figure*}\n\\centering{\n\\subfigure{\\includegraphics[scale=0.4]{fig1a.eps}}\n\\subfigure{\\includegraphics[scale=0.4]{fig1b.eps}}\n\\subfigure{\\includegraphics[scale=0.4]{fig1c.eps}}\n\\subfigure{\\includegraphics[scale=0.4]{fig1d.eps}}\n \\caption{{\\it Upper:} The 1.6~$\\mu$m flux map constructed from resolved LBGs with $H_{\\rm lim} = 25$ (left) and $H_{\\rm lim} = 27$ (right) respectively. {\\it Lower:} Map of the 3.6~$\\mu$m flux from galaxies with $5 < z < 10$ (left) and map of contamination at the same wavelength (right). The mean flux of the latter is 1 nWm$^{-2}$sr$^{-1}$.}\n\\label{maps}\n}\n\\end{figure*}\n\nFinally, we construct the flux map of LBGs at $5 5$) galaxies, the observed NIRB also contains radiation from unresolved, low-$z$ galaxies, and an excess radiation from unknown sources \\citep{2013MNRAS.433.1556Y,2014MNRAS.440.1263Y,2012Natur.490..514C,2014Sci...346..732Z}. We collectively refer to these two components as {\\it contamination}, since in this work the targeted signal is the flux from high-$z$ galaxies. To model such contaminating signal we construct random maps with mean flux 1.0 (0.7) nWm$^{-2}$sr$^{-1}$ at 3.6 (4.5) $\\mu$m and reproduce the sum of (i) the angular power spectrum of the power {\\it excess} (see \\citealt{2013MNRAS.433.1556Y}) matching available observations \\citep{2012Natur.490..514C,2012ApJ...753...63K}, and (ii) the angular power spectrum of low-$z$ galaxies \\citep{2012ApJ...752..113H} producing shot noise level matching \n\\citealt{2012ApJ...753...63K} ($P_{\\rm SN} = 4.8\\times10^{-11}$nW$^{2}$m$^{-4}$sr$^{-1}$ at 3.6~$\\mu$m and $P_{\\rm SN} = 2.2\\times10^{-11}$nW$^{2}$m$^{-4}$sr$^{-1}$ at 4.5~$\\mu$m, the corresponding subtraction magnitude is $\\sim25$ mag). \nThe steps are:\n\\begin{itemize}\n\\item A white noise map, i.e. a Gaussian random field, is generated.\n\\item This map is then transformed into spatial-frequency space by FFT.\n\\item For each complex number in frequency space, its modulus is rescaled to be $\\sqrt{P(q)}$, where $P$ is the given power spectrum and $q$ is the spatial-frequency. The zero-frequency ($q = 0$) element is set to be the mean flux.\n\\item The above map is then transformed back into real space by inverse FFT, resulting in a synthetic image with the same 2-point clustering properties as the measured $P(q)$.\n\\end{itemize}\nIn Fig. \\ref{maps} we plot a single realization of the contamination map at 3.6~$\\mu$m as an example (bottom right). The contamination is not correlated with the high-$z$ galaxy component; however, it adds noise to the cross-correlation signal. To account for the statistical variance of the contamination, we make 30 independent realizations. In Fig. \\ref{APS} we show the angular power spectrum of bottom panels in Fig. \\ref{maps}. All maps are convolved with a circular symmetric {\\tt Spitzer} PSF before further analysis. \n\n\\begin{figure}\n\\centering{\n\\includegraphics[scale=0.4]{fig2.eps}\n \\caption{Angular power spectrum of the 3.6~$\\mu$m flux map for $5 < z < 10$ galaxies (triangles) and contamination (squares). Errorbars show uncertainties due to limited number of Fourier modes in each $q$ bin. As a comparison, in the same panel we also plot the \\citet{2012ApJ...752..113H} model for $z < 5$ galaxies (solid line), and the observational points in \\citet{2012ApJ...753...63K} (diamonds) and \\citet{2012Natur.490..514C} (hourglasses). \nOn large scales ($\\theta \\lower.5ex\\hbox{\\gtsima} 100''$) the angular power spectrum of our co-added map (quite similar to squares in the panel) is consistent with both observations. Compared with K12, at $\\theta \\lower.5ex\\hbox{\\ltsima} 100''$ our prediction slightly falls short, probably because the non-linear clustering of low-$z$ galaxies is not modeled here. The C12 observations have a shallower (i.e. $\\sim24$ mag) source subtraction, hence a higher shot-noise level.\n}\n\\label{APS}\n}\n\\end{figure}\n \n\\section{Cross-correlation of LBG\\lowercase{s} and NIRB}\\label{results}\n\n\\subsection{the correlation coefficient}\nWe first analyze the correlation coefficient between the LBG flux map and the NIRB map. It is defined as\n \\begin{equation}\n {\\cal R}=\\frac{\\mean{\\delta F_{1.6} \\delta F_{\\lambda_0}}}{\\sqrt{ \\mean{\\delta F^2_{1.6}} \\mean{ \\delta F^2_{\\lambda_0}} } },\n \\end{equation}\n where $\\lambda_0$ refers to either 3.6 $\\mu$m or 4.5 $\\mu$m, $\\delta F_{1.6}$ and $\\delta F_{\\lambda_0}$ are the flux fluctuations of the same pixel (zero lag) at those two wavelengths. The brackets refer to the pixel-averaged fluctuations. The correlation coefficient indicates the fraction of sources contributing to common signals. Before calculating the correlation coefficient we smooth both the NIRB maps and the LBG flux maps \nby a real space top-hat window function with diameter $\\theta=10''$. This is to suppress the instrumental noise since in the measured NIRB maps \\citep{2012ApJ...753...63K} the instrumental noise is negligible at $\\theta \\lower.5ex\\hbox{\\gtsima} 10''$.\nThe smoothing is not related to the LBG detection limits since it is performed on maps constructed from LBGs already present in catalogs.\nTo mimic surveys with different areas, we cut out sub-maps with different areas from the full map. We choose three areas:\n$(0.036)^2$, $(0.3)^2$ and $(1.2)^2$~deg$^2$, representing a survey region similar to HUDF\/XDF, UDS, and an hypothetical larger field, respectively. \n \nBefore calculating the correlation coefficient, in both maps we mask pixels containing galaxies brighter than mag $\\sim25$ at {\\it either} 3.6~$\\mu$m or 4.5~$\\mu$m. From this procedure we obtain the source-subtracted NIRB map. The correlation coefficient vs. LBG limiting magnitude is shown in Fig. \\ref{correlation_coefficient}. The filled regions contain 68.3\\% probability of all sub-map samples. \nNote that we have 30 random realizations for each full map ((2.4)$^2$ deg$^2$), so even for the $(1.2)^2$~deg$^2$ case there are 120 samples, sufficient to compute the correlation coefficient variance. \n\n\\begin{figure}\n\\centering{\n\\includegraphics[scale=0.4]{.\/fig3.eps}\n\\caption{\nCorrelation coefficient between the 3.6~$\\mu$m source-subtracted NIRB and LBG flux maps vs. $H$-band limiting magnitudes for three different map areas. Filled regions bracket the 68.3\\% probability ranges around the peaks.\nAll fields are smoothed on scale $\\theta=10''$. }\t\n\\label{correlation_coefficient}\n}\n\\end{figure}\n\nFig. \\ref{correlation_coefficient} shows that it is indeed feasible to detect the cross-correlation signal from the mock maps, even for a relatively shallow survey with $H_{\\rm lim}\\sim25$, for which ${\\cal R}\\approx 0.04$. By pushing the limiting magnitude to fainter values, the correlation coefficient rapidly increases by a factor $\\sim2$, and then slowly approaches $\\approx 0.09$ at $H_{\\rm lim}\\sim29$. It is worth noting that in small fields the measured correlation coefficient has a $\\sim 30 - 80\\%$ relative field-by-field scatter for $H_{\\rm lim} \\lower.5ex\\hbox{\\gtsima} 26$, and even larger scatter at $H_{\\rm lim} < 26$. In some extreme cases there would be no or negative cross-correlation detected, due to the small number of LBGs contained in the fields.\n\nTo elucidate the differential contribution of LBGs, for a $(0.3)^2$ deg$^2$ field, we further show in Fig. \\ref{correlation_coefficient_z} the correlation coefficient from LBGs at $>z$ for $H_{\\rm lim} = 25, 26$ and 27, respectively. For example, LBGs at $z>8$ contribute ${\\cal R}\\approx 0.01$ at $H\\lower.5ex\\hbox{\\ltsima} 27$.\n\nOur results are for NIRB maps with a fixed shot noise level matching \\citet{2012ApJ...753...63K} measurements. Since the shot noise is mainly from low-$z$ galaxies, an increased subtraction depth implies a relatively larger number of low-$z$ galaxies (with respect to the high-$z$ ones) are resolved and removed. As a result we expect a higher correlation coefficient. On the other hand, if we resolve more LBGs, the cross-power signal also becomes stronger. Thus, the strength of the cross-power signal vs. different LBGs detection depths can be used to determine the differential contribution of early galaxies. \n \n\\begin{figure}\n\\centering{\n\\includegraphics[scale=0.4]{fig4.eps}\n\\caption{ \nContribution to correlation coefficient from LBGs with $>z$. The 68.3\\% probability ranges are plotted by errorbars. For displaying purpose we slightly shift the x-positions of some curves.}\t\n\\label{correlation_coefficient_z}\n}\n\\end{figure}\n\n\\subsection{Detectability vs. scales}\n\nSo far we have investigated the behavior of the correlation coefficient by at a specific smoothing scale $\\theta=10''$. We now examine its dependence on angular scale. To this aim we re-define the correlation coefficient in the spatial-frequency domain via the power spectrum\n\\begin{equation} \\label{eqn_coherence}\n{\\cal R}_q(\\theta) = \\frac{P_{\\rm IR\\times G}(\\theta)}{ \\sqrt{P_{\\rm IR}(\\theta)\\times P_{\\rm G}(\\theta)} },\n\\end{equation}\nwhere $P_{\\rm 1\\times 2}(\\theta)$ is the cross-power spectrum and $P_{\\rm 1}$,$P_{\\rm 2}$ are the auto-power spectra. They are all calculated using the 2D FFT. Since $\\delta F^2 \\simeq q^2P\/2\\pi$, ${\\cal R}_q \\sim {\\cal R}$ for the same $\\theta$. \nFirst we derive the shot noise due to LBGs fainter than a given H-magnitude (Fig. \\ref{PSN}). It is the value of $P_{\\rm G}$ taken on a sufficiently small scale $\\theta \\approx 14''$ (a stability check of the results for scales up to $\\approx40''$ has been performed). As we will point out in the following, shot noise is the most important term contributing to the cross-correlation. The shot noise level shown here is a useful reference for possible follow-up work on real data. \n \n\\begin{figure}\n\\centering{\n\\includegraphics[scale=0.4]{fig5.eps}\n\\caption{Shot noise due to LBGs with magnitude $>H$ and at $5 < z < 10$.}\t\n\\label{PSN}\n}\n\\end{figure}\n \n \nThe ${\\cal R}_q(\\theta)$ between the source-subtracted NIRB at $3.6~\\mu$m and the 1.6~$\\mu$m flux map of LBGs with $H_{\\rm lim} = 25, 26$ and 27 is shown by Fig. \\ref{correlation_coefficient_q}. The errorbars are the r.m.s of 30 samples each one using different random contamination realizations. \n\\begin{figure}\n\\centering{\n\\includegraphics[scale=0.4]{fig6.eps}\n\\caption{The ${\\cal R}_q$ between the source-subtracted NIRB and the flux maps constructed from LBGs with different $H_{\\rm lim}$. The errorbars here show the r.m.s of 30 random realizations. For displaying purpose we slightly shift the x-positions of $H_{\\rm lim} = 26, 27$ curves.}\n\\label{correlation_coefficient_q}\n}\n\\end{figure}\n\nThe cross-power spectrum includes both the shot noise and clustering terms. Shot noise is from sources common to both NIRB and LBG flux, and it is dominant on scales $ \\theta \\lower.5ex\\hbox{\\ltsima} 100''$. On larger scales, the clustering term progressively takes over. This term arises from all sources sharing the same large scale structure, including galaxies even fainter than the LBG limiting magnitude. As a consequence, in principle the clustering term could allow the detection of fainter galaxies in the source-subtracted NIRB through the cross-correlation with relatively bright LBGs. However, as shown by Fig. \\ref{correlation_coefficient_q} the cross-correlation is more easily detected on small scales. Although the clustering term may contain more information, the main difficulty to be overcome in order to efficiently use this strategy is that even for a relatively large survey area of $(2.4)^2$~deg$^2$ (i.e. our full map), the signal-to-noise ratio never exceeds $\\sim 3$ at $\\theta \\gsim300''$ for $H_{\\rm lim} < 27$. While increasing the limiting magnitude to $>27$ does not help much to this aim, the noise could be reduced by using larger survey areas, as expected with, e.g. EUCLID and WFIRST. \n\n\n\\subsection{Colors}\n\nFrom our mock maps it is also possible to characterize the colors, i.e. the ratio between flux in two bands, of unresolved galaxies in NIRB observations. With the assumption \n\\begin{equation}\n\\frac{F_{4.5}}{F_{3.6}}\\approx \\frac{\\mean{ (\\delta F_{4.5} \\delta F_{1.6})_\\theta }} { \\mean{ (\\delta F_{3.6} \\delta F_{1.6})_\\theta } },\n\\end{equation} \nwe derive the magnitude difference as \n\\begin{align}\n[3.6]-[4.5] &= 2.5{\\rm log}\\left( \\frac{4.5F_{4.5}}{3.6F_{3.6}} \\right) \\nonumber \\\\\n&\\approx2.5{\\rm log}\\left( \\frac{4.5}{3.6} \\frac{\\mean{ (\\delta F_{4.5} \\delta F_{1.6} )_\\theta }} { \\mean{ (\\delta F_{3.6} \\delta F_{1.6} )_\\theta} } \\right) \n\\end{align} \nSuch magnitude difference, which might considerably vary for individual galaxies, represents the combined and weighted color of the unresolved galaxy population in these bands. In practice the PSF adds a bias to the measured flux ratio. Therefore before calculating the magnitude difference it would be necessary to first deconvolve the PSF from each map. We skip this step here and directly calculate the magnitude difference on maps without PSF convolution. Again we specify $\\theta=10''$. The predicted color as a function of $H_{\\rm lim}$ is reported in Fig. \\ref{color}, allowing us to conclude that the magnitude difference which is $\\sim$-0.13 mag, could be detected by cross-correlating survey maps with area $\\lower.5ex\\hbox{\\gtsima} (0.3)^2$ deg$^2$. The figure reiterates that smaller area fields would be affected by bias effects.\n\n\\begin{figure}\n\\centering{\n\\includegraphics[scale=0.4]{fig7.eps}\n\\caption{The magnitude difference, $[3.6]-[4.5]$, of fluctuations due to unresolved galaxies in the source-subtracted NIRB maps as a function of the survey limiting magnitude. Filled regions indicate 68.3\\% probability ranges.}\t\n\\label{color}\n}\n\\end{figure}\n \n \\section{Conclusions}\\label{conclusions} \n\nMotivated by the fact that LBG surveys carried out by the {\\tt HST} have reached detection limits much deeper than the NIRB measured by {\\tt Spitzer} at longer wavelengths, in this paper we investigated the feasibility to pick the resolved LBGs component out of the NIRB by a cross-correlation analysis. Our investigations were based on mock maps constructed from semi-numerical simulations of halo formation, and empirically determined $L_{\\rm UV} - M_{\\rm h}$ relations.\n \nWe found that in the NIRB observed at 3.6 and 4.5~$\\mu$m and with sources subtracted down to $\\sim$25 apparent magnitude, at smallest scales where the shot noise dominates, about 10\\% of the flux fluctuations arises from LBGs in $5 < z < 10$ and with $H \\lower.5ex\\hbox{\\ltsima} 29$. Such faint galaxies have already been resolved in the existing deep surveys. However, this fractional contribution, if measured from narrow fields with area $\\sim (0.036)^2$~deg$^2$ (as the HUDF\/XDF), could vary from $\\sim3\\%$ to $\\sim16\\%$. \nIf the limiting magnitude is decreased to $H \\sim 27$, the fractional contribution decreases to about 8\\%. If in this case we consider a larger field, e.g. with area similar to the UDS ($\\sim (0.3)^2$~deg$^2$), then the correlation coefficient varies in the narrower range $\\sim 6\\%-9\\%$. \nWe remind that the variance at hand here is due to both the large-scale inhomogeneity of the signal and the contamination: it is the field-to-field variance of the correlation itself. We do not model errors introduced by data analysis procedures, as for example mask effects. However, at least theoretically we have shown that the contribution from the faintest galaxies could be isolated from the NIRB through the cross-correlation analysis. \n\nWe pointed out that it is still challenging to use the cross-correlation arising from the clustering term of the cross-power spectrum to study galaxies unresolved not only in NIRB observations, but also in LBG surveys. This term is dominant at $\\lower.5ex\\hbox{\\gtsima} 200''$, but even for survey areas as large as $(2.4)^2$~deg$^2$, the signal has a small significance, with a S\/N ratio $\\lsim3$. Although very unlikely, if the NIRB clustering excess originates from LBGs we would detect a correlation coefficient close to 1 at scales where the clustering term dominates. Stated differently, a measured correlation coefficient $ {\\cal R} \\ll 1$ would actually rule out this scenario.\n\nOur results have interesting implications also for the colors of high-$z$ galaxy population. The colors of LBGs detected at wavelengths $< 1.6~\\mu$m but too faint to be individually resolved by existing telescopes at 3.6 and 4.5~$\\mu$m, can be obtained by using the cross-correlation with the NIRB in these two longer wavelength bands. \n\n \nIt is worth noting that the predictions presented in this paper assume a contaminating signal in the form of NIRB fluctuation excess which originates from direct collapse black holes at high $z$ \\citep{2013MNRAS.433.1556Y}. If however the excess is found to arise more locally, we might be able to model or subtract it more accurately, thereby making the signal calculated in this paper more easily detectable, i.e. ${\\mean {\\cal R}} \\lower.5ex\\hbox{\\gtsima} 0.1$. \n \nImportantly, the study proposed here can be further developed to infer the properties of even fainter high-$z$ galaxy populations that are currently inaccessible to direct telescopic detection. This could, for example, be accomplished by a Lyman-break tomography study designed to isolate high-$z$ populations via multi-band cross-correlations without requiring prior LBG detections \\citep{2015ApJ...804...99K}. \n\n\\section*{ACKNOWLEDGMENTS}\n\nWe thank A. Kashlinsky for valuable comments on the manuscript. \t\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}