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+{"text":"\\section{Abstract}\n\n\n\\section{Introduction}\n\n\nThe particle camera MX-10 (see Fig.~\\ref{fig:camera}) of the Medipix2 family~\\cite{HOLY2006254,VYKYDAL2006112} is a professional solid-state particle detector with the capability of energy and time measurements as well as particle identification based on the observed pattern of signaling pixels.\n\nThe device is an excellent educational equipment for demonstrating various properties of the ionizing radiation, able to show distinct patters for gamma (few-pixel dots), beta (visibly long and curved tracks), alpha (wider blobs due to the charge sharing between pixels), muons (long straight tracks) or more exotic wide tracks of heavily ionizing particles. We will describe the examples of all these in more detail in corresponding sections together with ideas on how to use the observations to explain key physics features of the conditions to obtain the patterns as well as the processes leading to observing the particles in the first place.\n\nOne of the device key features is the possibility to demonstrate the shielding effects of air or a sheet of paper to stop alpha particles, or various metals of different thickness to stop beta and gamma rays. The educational set provided by Czech company JABLOTRON ALARMS includes the shielding material as well as optionally also radiation sources like glass doped with uranium dioxide, metallic  electrode doped by a thorium dioxide, or americium sources modified to primary gamma or alpha sources. \n\n\n\\begin{figure}[!t]\n  \\centerline{\n    \\includegraphics[width=0.800\\textwidth]{fig1.JPG}  }\n  \\caption[]{The particle camera MX-10~\\cite{HOLY2006254,VYKYDAL2006112} based on the Medipix2 technology, with an the open cover showing the aluminum-coated sensor.}\n\\label{fig:camera} \n\\end{figure}\n\n\n\n\\section{Particle camera in the laboratory}\n\n\\subsection{Radiation background}\n\nLaboratory measurements offer many opportunities to analyze the shielding power of various materials like metals of different proton number. School and laboratory $\\alpha\/\\beta\/\\gamma$ radioactive sources can be used to demonstrate the shielding power in terms of the number as well as energy spectra of detected particles. One can also study several other accessible sources like potassium-rich fertilizers or uranium-oxide doped glass used for art and decoration purposes.\nOther sources include dust collected on a paper or cloth filter after vacuum-cleaning a room: it includes solid radioisotopes from the radon decay chain.\n\nBut the simplest one is the background itself, dominated by gamma and beta particles, occasionally spiced by alpha particles from the aforementioned and ubiquitous radon, but also more exotic particle species like cosmic muons or even heavy ionizing energetic particles, see~Fig.~\\ref{fig:bg_HI}.\n\n\\begin{figure}[!h]\n  \\includegraphics[width=0.450\\textwidth,height=0.450\\textwidth]{fig2.png}\n  \\includegraphics[width=0.450\\textwidth,height=0.450\\textwidth]{fig3.png} \\\\\n  \\caption{Example of a 10 min background exposure with a recorded alpha particle from natural background radiation (left) and a rare track of a highly-ionizing particle at the ground level (right).}\n\\label{fig:bg_HI}\n\\end{figure}\n\n\n\\subsection{Cosmic muons direction analysis}\n\nEducationally, a single image of a cosmic muon is a simultaneous proof of time dilatation by Einstein, proof of the existence of the second family of elementary particles, and a proof of radiation of extra-terrestrial origin.\n\nFor the analysis of the direction of incoming muons, data have been recorded in frames of 10~min each. During the total exposure of 662 hours, sometimes an accidental highly ionizing particle, perhaps from an extensive cosmic ray shower, was recorded, as seen in Fig.~\\ref{fig:bg_HI} (right), with a rate of approximately one per day. In total, 225 lines were found of length of at least 60 pixels, i.e. about one such a candidate per 20~min. Typical events are shown in Fig.~\\ref{fig:lab}.\n\nHough transformation, originally proposed for the analysis of bubble chamber pictures in 1959~\\cite{Hough:1959qva}, was used to search for line patters, candidates for the straight muon tracks. A simple, straightforward yet inefficient implementation of the Hough transformation was written in Python to analyze the recorded frames. Only the longest track from each frame was accepted to the analysis. As illustrated in Fig.~\\ref{fig:hough}, the algorithm transforms the 2D image from the $x-y$ plane to the $\\theta-r$ space, where $r$ is the closest approach of a line at angle $\\theta$ w.r.t. the origin. In essence, all lines are tried, pixels along the line are analyzed and the number of pixels with non-zero energy deposit is counted. This number is set as the value of the Hough-transformed histogram in the $\\theta-r$ space at coordinates corresponding to the line parameters. Searching numerically for the maxima leads to finding longest lines of given parameters.\n\nThe distribution of the azimuthal angle $\\theta_\\mu$ of the muon candidates in Fig.~\\ref{fig:zenith} is peaked around the direction of vertically-incoming muons. It is slightly shifted towards smaller angles, which could be attributed to shielding by the building (the experimental setup was kept close to window, i.e. close to a more open side of a building).\n\n\n\n\\begin{figure}[!h]\n  \\includegraphics[width=0.450\\textwidth]{fig4.png}\n  \\includegraphics[width=0.450\\textwidth]{fig5.png}\n  \\caption{Example of clean tracks of passing cosmic muons recorded in the laboratory during 10 min exposures, also with non-negligible gamma and beta background.}\n\\label{fig:lab}\n\\end{figure}\n\n\n\\begin{figure}[!h]\n  \\includegraphics[width=0.950\\textwidth]{fig6.pdf}\n \\caption{An example of an event with a straight track from a passing cosmic ray muon (left), the Hough-transformed image (middle) in the $\\theta-r$ space, and the reconstructed lines (right).}\n\\label{fig:hough}\n\\end{figure}\n\n\n\\begin{figure}[!h]\n  \\centerline{\n    \\includegraphics[width=0.750\\textwidth]{fig7.pdf}  }\n  \\caption{Distribution of the zenith angle of the incoming cosmic-ray muons, showing their dominant direction of coming from above.}\n\\label{fig:zenith}\n\\end{figure}\n\n\n\n\n\\section{Particle camera in caves}\n\nThe camera has been installed for a total of 2 days and 16 hours in a natural cave with an influx of carbon dioxide along a geological crack in limestone bedrock, expected to bring more radon from the depths of the Earth. A typical observed frame is shown in Fig.~\\ref{label:caves}, compared to the composition of particles recorded in a room on surface.\nInterestingly, in contrast to laboratory measurements, no muons were recorded due to the shielding of the rock and soil, and the frames are visually dominated by alpha particles, although the multiplicities of beta and gamma particles are higher (see Fig.~\\ref{fig:corr}).\n\nFurther, multiplicities of particles of different kinds were analyzed by a private code written by students, comparing very well to the original SW shipped with the camera, enabling to resurrect data on individual particle multiplicities lost in one set of measurements, where only the total multiplicity was saved to a data file. Correlations over frames between the observed numbers of particles of different kinds are shown in Fig.~\\ref{fig:corr}. Students have thus trained themselves in programming, pattern recognition, algorithm development and finally also in statistics, error treatment, and even covariance and correlation estimation.\n\nSome basic facts have been clarified, known perhaps to most of cave climatologists or radon specialists,\nbut it was found that the energy peaks of the observed alpha particle are consistent with those from polonium isotopes instead of the direct radon origin, i.e. the observed alpha particles seen come from solid decay products of radon, probably adhered in a form of aerosols directly on the chip. This is confirmed by the relatively sharp peaks at energies of about 6.4 and 8.2~MeV of alpha spectral particles as displayed in Fig.~\\ref{fig:alpha:spect}, corresponding probably to alpha particles from Po-218 and Po-214 decays, respectively, with an indication of the camera overestimating the energy by about 7\\% (the actual energies are about 6.0 and 7.7~MeV~\\cite{periodic}).\n\n\n\n\n\\begin{figure}[!h]\n\\centerline{ \n  \\includegraphics[width=0.450\\textwidth]{fig8.png}\n  \\includegraphics[width=0.450\\textwidth]{fig9.png}\n}\n  \\caption{Example of a frame of 10~min in a natural cave (left) and of 6~h of natural background (right).}\n\\label{label:caves}\n\\end{figure}\n\n\\begin{figure}[!p]\n  \\centerline{\n    \\includegraphics[width=0.750\\textwidth]{fig10.pdf}  }\n\\caption{Energy spectrum of particles as measured by the MX-10 camera, recorded in a natural cave. Two strong alpha peaks at energies of approximately 6.4 and 8.2 MeV are clearly seen, while their exact position is a subject to imperfect calibration of the particular device used.}  \n\\label{fig:alpha:spect}\n\\end{figure}\n\\begin{figure}[!p]\n\\centerline{ \n  \\includegraphics[width=0.950\\textwidth]{fig11.pdf}\n}\n  \\caption{Correlations over frames between the observed numbers of particles of different kinds, and their multiplicity over time in 5~min windows measured in a natural cave. Recorded using the MX-10 particle camera, with multiplicities analyzed by a private code.}\n\\label{fig:corr}\n\\end{figure}\n\n\n\n\\clearpage\n\\section{Particle camera aboard a commercial airplane}\n\nAs the camera works as a simple USB device, measurements with a laptop can be carried aboard a commercial airplane.\nFig.~\\ref{fig:plane} which shows examples of 15 min exposures at altitude of about 10 km.\nIn addition, pairing this information with data from an external GPS (in a digital camera) with a synchronized clock, the radiation level was studied as a function of the altitude~\\cite{SOC}.\nIn Fig.~\\ref{fig:dose} one can observe the effect of initially reduced radiation with altitude at about 1~km, followed by increase from cosmic rays, as observed originally also by V.~Hess and most notably by W.~Kolh\\\"{o}rster~\\cite{cosmic_wiki,FICK201450}. Error bars are statistical only, accounting for the fluctuation in the number of observed particles in frames included to the sum of the observed energy in given altitude range.\nThanks to the pixelized detector, one can clearly see many straight tracks of muons, as well as thick tracks of highly ionizing particles of energies up to 20 MeV, i.e. surpassing natural sources of radiation (alpha particles from natural radioisotopes have energies up to about 8~MeV).\nAll these are in sharp contrast to radiation levels and patters observed at the Earth surface. The dose recorded by the camera is about 15 times higher at altitude of 10~km, compared to the altitude of the laboratory of about 200~m.\n\n\n\\begin{figure}[!h]\n\\centerline{ \n  \\includegraphics[width=0.450\\textwidth,height=0.450\\textwidth]{fig12.png}\n  \\includegraphics[width=0.450\\textwidth,height=0.450\\textwidth]{fig13.png}\n}\n  \\caption{Example of 15 min exposures recorded aboard a commercial air plane over Europe at altitude of about 10~km.}\n\\label{fig:plane}\n\\end{figure}\n\\begin{figure}[!h]\n  \\centerline{\n    \\includegraphics[width=0.750\\textwidth]{fig14.pdf}  }\n  \\caption{The dose rate recorded by the particle camera silicon chip in mGy\/year as function of the altitude above the sea level in meters. The solid (dashed) line is a quadratic (power) fit to data. Error bars are statistical only, the standard $\\chi^2$ divided by the number of degrees of freedom (ndf) is indicated in plot legend.}\n\\label{fig:dose}\n\\end{figure}\n\n\nTracks of energies above 10~MeV were observed, with extreme cases of energies of~21 and even 40~MeV, the most energetic one carrying a sign of a Bragg peak, i.e. sharp increase of energy loss towards the end of the trajectory, followed by a rapid decrease, see Fig.~\\ref{fig:bragg}, which has an important application in medicine in hadron therapy. The observed range and losses pattern is similar to predicted losses of a deuteron in silicon with initial kinetic energy lower by 20\\% compared to the observed track. The reasonable agreement of the measured and predicted curves was reached by the change in the energy, motivated by the camera overcalibration, and also by the adopted model of losses below the application limit of the Bethe-Bloch formula~\\cite{pdg2} which was chosen to correspond to the particle velocity of $0.03c$, below which the energy losses were linearly interpolated to zero at rest.\n\nIn addition, one nuclear interaction was identified, with total of 77 MeV deposited into the device (see Fig.~\\ref{fig:candidates}).\n\n\\begin{figure}[!h]\n\\centerline{ \n  \\includegraphics[width=0.450\\textwidth,height=0.450\\textwidth]{fig15.pdf}\n  \\includegraphics[width=0.450\\textwidth,height=0.450\\textwidth]{fig16.pdf}\n}\n\\caption{Simulated Bragg peak (left), i.e. the enhancement of the energy loss per track length towards the end of a trajectory, of a 33~MeV deuteron in silicon, \n \n  and an example of a recorded 40 MeV track from flight data in MX-10 camera (right) exhibiting a similar pattern.}\n\\label{fig:bragg}\n\\end{figure}\n\\begin{figure}[!h]\n\\centerline{ \n  \\includegraphics[width=0.450\\textwidth,height=0.450\\textwidth]{fig17.png}\n  \\includegraphics[width=0.450\\textwidth,height=0.450\\textwidth]{fig18.png}\n}\n\\caption{The 40 MeV track candidate (left) exhibiting the Bragg peak and a nuclear interaction of total of 77 MeV deposited (right). The color coding of the pixels is as follows: 0--10 keV: green, 10--50 keV: yellow, 50--150 keV: orange, $\\geq 150$ keV: red.}\n\\label{fig:candidates}\n\\end{figure}\n\n\n\n\\section{Particle camera at an accelerator}\n\nIn Fig.~\\ref{fig:TB} one can see example frames recorded by the particle camera with the chip inserted parallel to a beam of muons (left) and charged pions (right) at the SPS test beam area at CERN. The figure clearly demonstrates the complexity of hadronic interactions of pions compared to muons.\nIn addition, it is not uncommon to see actually a break-up of a nucleus into several heavy-ionizing fragments, i.e. the alchemists' dream of changing a chemical element into another one.\n\n\n\\begin{figure}[!h]\n \\centerline{ \n \\includegraphics[width=0.450\\textwidth,height=0.450\\textwidth]{fig19.png}\n \\includegraphics[width=0.450\\textwidth,height=0.450\\textwidth]{fig20.png}\n}\n  \\caption{Example of frames recorded by the particle camera with the chip inserted parallel to a beam of muons (left) and charged pions (right) at the SPS test beam area at CERN.}\n\\label{fig:TB}\n\\end{figure}\n\n\\section{Discussion}\n\nStudents have performed and contributed in a major way to interesting experiments with various sources of the ionization radiation using the particle camera MX-10. Ranging from simple background measurements to the muon direction analysis, coming to radon survey in caves, the study of cosmic rays, dose variation with altitude, analysis of tracks of exotic heavily-ionizing particles, and finally to an example of the detector as a detection element in a fixed-target setup at an accelerator.\n\n\\subsection{Responsibility}\n\nAlready this high-school level research brings many questions on how to present and explain results to general public.\nThe issue is e.g. how to present (the interesting yet harmless for a mere visitor) elevated radiation background in the caves environment, in order not to disseminate confusion or even fear of ionization radiation levels on a plain or in caves, while bringing these exciting facts to interested students, readers, enthusiasts.\nIt is a sad truth that anything related to radon or ionizing radiation sparks often negative emotions in general public, although we are all being exposed to small harmless doses of radiation from many natural sources, including radon, medical imaging, food and airplane travel; and depending on the geographic location on Earth, not only as a function of the altitude above the see level but also as function of the Earth's crust local composition.\n\nMeasurements were first discussed in a local high-school journal in a form of an interview and then also in a local TV station, later, without letting authors and the supervisor properly know, appearing also in internet news, catching attention of the management of the caves, who were not particularly happy seeing the discussion of radon-related radiation in the caves (which is of course carefully being monitored in over long term periods). Thus, through a primarily science project, students actually found themselves be taught a lesson on the interaction with media and on the way how to communicate their results.\n\nNevertheless, we are responsible to the authorities and the public for their support and we have the social responsibility to spread knowledge and education. It is the duty of us, scientists, to present and explain facts in a way which is understandable to all. This can be particularly well achieved by attracting students to related projects already at the high-school level who can then present their findings to their schoolmates and who can then share knowledge in the most effective way, i.e. by themselves, directly within their age group. Last, we prove that high-school students can contribute to the process of writing a journal article containing their own work.\n\n\n\n\n\\section{Acknowledgments}\n\nThis educational research was performed using the MX-10 device by the company JABLOTRON ALARMS, equipped by the Medipix\/Timepix chip~\\cite{VYKYDAL2006112} and software Pixelman~\\cite{HOLY2006254} by IEAP, Czech Technical University, Prague, Czech Republic.\nData were analyzed by a private code in C++, Python and using libraries of the ROOT analysis framework~\\cite{Brun:1997pa}; and using Java, MS Excel and Visual Basic macros for the case of processing the flight data.\nWe thank T.~S\\'{y}kora for bringing our attention to the MX-10 camera and to the Hough transformation, and to L.~Chytka for recording the data with MX-10 in his spare time during the SPS test beam campaign for the time-of-flight detector for the ATLAS Forward Proton detector.\nWe also thank P. Baro\\v{n} for the interpretation of the alpha spectra.\nOur thanks belong to the Nature Conservation Agency of the Czech Republic and the staff of the Zbra\\v{s}ov Aragonite Caves, Teplice nad Be\\v{c}vou, Czech Republic, for providing us with the opportunity to perform measurements over several nights in the cave system.\nLast, the students would like to thank the pedagogical staff of the Grammar school in B\\'{\\i}lovec; and M.~Komsa, L.~Balc\\'{a}rek from the Grammar school in Uni\\v{c}ov and R.~D\\v{e}rda from the Technical and training school in Uni\\v{c}ov for their support and for allowing them to travel to Palack\\'{y} University in Olomouc in order to pursue their high school science projects. \nD.S. and J.P. took the third place in the physics category of the national competition of Czech high school science projects (SO\\v{C}) in 2018.\n\nJ.K. gratefully acknowledges the support by the Operational Programme Research, Development and Education -- European Regional Development Fund, project \\\\ no.~CZ.02.1.01\/0.0\/0.0\/16\\_019\/0000754 of the Ministry of Education, Youth and Sports (MSMT) of the Czech Republic; and support from the grant LTT17018 of MSMT, Czech Republic.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}   \n\n\nThe coincident detection of gravitational waves (GW) from a binary\nneutron star merger with aLIGO\/Virgo and short-lived gamma-ray (GR) emission\nwith Fermi-GBM (called GW 170817) in August 2017 is a milestone for the\nestablishment of multi-messenger astronomy \\citep{Abbott2017b},\ni.e. the measurement of electromagnetic radiation, gravitational waves\nand\/or particles or neutrinos from the same astrophysical source.\nMerging neutron stars (NS) represent the standard scenario \\citep{Eichler1989}\nfor short-duration ($<2$ s) gamma-ray bursts (sGRBs) which are produced in a\ncollimated, relativistically expanding jet with an opening angle of a\nfew degrees and a bulk Lorentz factor $\\Gamma$ of 300--1000.\nWhile the aLIGO detection\nis consistent with predictions, the measured faint gamma-ray emission from\nGW 170817A is about 1000x less luminous than known short-duration GRBs.\nHence, the presence of this sGRB in the local Universe is either a very rare\nevent, or points to a dramatic mis-understanding of the emission properties\nof sGRBs outside their narrow jets. By now we know that the jet in this\nGRB had an opening angle of $<$5\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi, but we observed it from\n$\\sim$20--30\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi\\ offaxis \\citep{Mooley+2018}. In all previous models,\nno emission was predicted to occur outside the opening angle.\n\n\nThus, also the previous estimates of the volume density of NS-NS mergers\nwas wrong, and needs to be corrected \\citep{Burgess+2020}.\nThis has important implications on our understanding of the chemical\nevolution of our Galaxy and the Universe, as NS-NS mergers are believed to be\nthe main source of heavy elements \\citep{Kasen+2017},\nso-called r-process elements (like gold and platinum). This material is\nexpelled both during the tidal disruption of the NSs and through winds\nduring the subsequent disk accretion onto the compact core.\nFurther progress in our understanding of NS-NS mergers will depend\non measurements in the electromagnetic regime, and these in turn\nwill only be possible if the localizations of these events can\nbe reduced to of order a few square degrees on the sky. While there exist\nseveral large-field-of-view optical sky surveys, covering\nup to several thousands square degrees, the challenge is to find the kilonova\namong the many other transient sources. Future NS-NS mergers will\nlikely all(!) be at larger distance than GW 170817, and thus their kilonova\nmuch fainter. Already for the only 3x more distant\nfour NS-NS merger events from 2019--2020, none detected in gamma-rays,\nthe optical emission would peak at 23rd mag\n(if at identical luminosity as GW 170817).\nExcept for one particularly poor localisation, the error regions of the\nother three events encompass 2300--14700 square degrees each \\citep{Wiki2020}.\nAt the expected optical brightness, there will be about 3--60 transient alerts\nper square degree down to 21 mag \\citep{Masci+2019}, or estimated\n5x more at 23rd mag, against which the kilonova will have to be identified.\nThus, a pre-requisite to identify the kilonova is the fast and precise\nlocalization of the GW\/GR event.\n\nExpectations for the fourth observing run O4 are 10$^{+52}_{-10}$ BNS\nmergers, with a median 33$^{+5}_{-5}$ square degrees localization.\nLikely not before 2026 \\citep{Abbott+2020},\nthe GW detector network of LIGO, Virgo and KAGRA is\nlooking forward to include LIGO-India, which promises a reduction of\nthe GW error regions to of order $<$10 square degrees. \nFurther reduction of the localization error is foreseen with the\nEinstein Telescope in Europe, or the Cosmic Explorer in the USA,\nboth not earlier than the mid 2030s.\n\n\nThus, accurate localization of the GW events should be sought elsewhere.\nGamma-rays provide an interesting\nalternative, at least for those NS-NS mergers for which the jet would\nbe broadly pointed towards us. With $\\gamma$-ray emission at large\noff-axis angles as in GRB 170817A, up to 30\\% of mergers will be\nsimultaneously detectable in $\\gamma$-rays \\cite[][]{Howell+2019, Burgess+2020}.\nObviously, accurate measurements of many GRBs will be beneficial\nfor other science questions beyond kilonova physics, such as\n(1) the structure of jets in GRBs \\citep[e.g.,][]{Janka+2006} and the\norigin of the off-axis emission which is distinctly different to on-axis\nemission \\citep{Begue+2017}, or\n(2) the potential emission of high-energy neutrinos as measured by\nIceCube \\citep{Aartsen+2013}, promising a potential \n'triple'-messenger, i.e. electromagnetic radiation, gravitational waves,\nand particles: \n\\cite{Kimura+2017} estimated that GRB 170817A could\nhave been detectable by IceCube if the jet had been viewed on-axis\ninstead of the $\\sim$30\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi\\ off-axis.\n\n\nHere, we propose adding a GRB detector on some of the next\ntwo dozen 2nd generation Galileo satellites (G2) in order to improve\nthe localization capability for short GRBs to the 1-degree level, \nreducing the error region by a factor of 100--1000.\n\n\n\\section{Prospects of accurate GRB localisation}\n\n\\subsection{Challenges of short GRBs}\n\nShort-duration GRBs (sGRB) have three properties which make their localisation\nin large numbers more difficult than that of long-duration GRBs:\\\\\n(1) Their short duration, of order 0.01--2 s, implies that their observable\nfluence is of order 5--50x smaller than in long-duration GRBs.\\\\\n(2) Their peak fluxes during their maximum spike is typically a factor 10\nsmaller than long-duration GRBs, making the discrepancy of (1) even larger.\\\\\n(3) sGRBs are also harder, with their spectral peak at\nhigher energies. This implies that the flux at soft gamma-rays (20--100 keV)\nis smaller than that in long-duration GRBs even if the energy-integrated\nflux is equal.\n\nThese factors together imply detection and localisation disadvantages\nin various detector types:\n(i) in coded-mask imagers like Swift\/BAT or INTEGRAL\/IBIS, the mask\nelements are getting increasingly more transparent at higher energy,\nleading to less ``sharp'' shadows, and thus detection sensitivity.\nThus, the ratio of long-to-short GRBs in Swift\/BAT\nis about 10:1, while it is 10:2.5 in Fermi\/GBM.\n(ii) in counting experiments like Fermi\/GBM, short spikes can more easily\nbe mistaken for noise spikes. Moreover, at the higher photon energies,\nthe cosine dependence of the\neffective area is much less pronounced in detectors with slab-like\nscintillators, due to the larger absorption probability \nat inclined incidence angles.\n\n\n\\begin{table*}[th]\n  \\smallskip\n  \\caption{Comparison of different $\\gamma$-localization methods in the\n    200--2000 keV band. The sensitivity column reports the peak flux threshold\n    over the 1--1000 keV band (for a Band function with $\\alpha$=--1,\n    E$_{\\rm peak}$=300 keV, $\\beta$=--2) of the listed detectors\n    \\citep{Band2003, Bosnjak+2014}. \\label{methods}}\n  \\vspace{-0.3cm}\n   \\begin{tabular}{llllclc}\n    \\hline\n     \\noalign{\\smallskip}\n     Method & Accuracy & Comments & E-range & GRBs & Example & Sensitivity \\\\\n            &          &          & (keV)   & (1\/yr) &       & (ph\/cm$^2$\/s) \\\\\n      \\noalign{\\smallskip}\n      \\hline\n      \\noalign{\\smallskip}\n      Triangulation    & arcsec  & cheap, all-sky & 10--1000  & 20-50 & IPN & 2.0 \\\\\n      Relative rates   & degrees & cheap, half sky & 8--500  &300 & BATSE, GBM & 1.0, 3.0 \\\\\n      Coded-mask       & arcmin  & small FOV & 10--200  & 10--100 & Swift\/BAT, ISGRI & 1.2, 0.6 \\\\\n      Photon-by-Photon$\\!\\!$ & degrees & heavy, big & 100--2000 & 10--30 & COMPTEL & 180 \\\\\n       \\noalign{\\smallskip}\n       \\hline\n    \\end{tabular}\n\\end{table*}\n\n\n\\subsection{Localisation methods}\n\nRather independent of the different $\\gamma$-ray detection technology\n(gas detectors, scintillation detectors or solid-state detectors)\nare the methods with which gamma-rays can be localized. \nThe four main methods with their advantages and disadvantages\n(Tab. \\ref{methods}) are described below. The summarizing statement\nis: large field-of-view (FOV) instruments with high GRB detection rates\nare operating at softer energies, not appropriate for short-duration GRBs,\nwhile detectors at higher energies are suffering from either bad\nlocalization capabilities or low detection rates. Over the last 20 years,\nall techniques except triangulation have been used in space applications\nwith the maximum possible capability.\n\n\\noindent{\\bf Triangulation:} \nAmong the first methods of localizing sources in gamma-ray astronomy\nwas triangulation, i.e. measuring the time difference\nof a signal arriving at different detectors. This was the method used\nby the Vela satellites in the 1960s to verify the Nuclear Arm Treaty\nbetween USA-Russia, which then led to the discovery of GRBs.\nThis method requires at least 3 detectors\/satellites, and accurate\nknowledge of time and the relative position of the detectors; it allows\nto cover all-sky, and provides localizations in the arcsec--arcmin range for\nwidely spaced satellites \\citep{Hurley+2017}.\nBut since GRB detectors on interplanetary\nspacecraft are auxiliary instruments, and thus small, triangulation\noffers substantial improvements.\n\n\\noindent{\\bf Orientation-dependent rate measurements:}\nMeasuring relative rates of orientation dependent $\\gamma$-ray\ndetectors, typically scintillation crystals, was used for\nGRB localizations with the BATSE instrument on the Compton\nObservatory, and is used presently with the GRB Monitor (GBM)\non Fermi. This method requires $>$4--6 detectors with different\norientations on the sky, and the localization accuracy is on the\ndegree-scale at best \\citep{Berlato+2019}.\n\n\\noindent{\\bf Coded-mask imaging:}\nCoded-mask imaging also allows a 2D reconstruction on the sky,\nand was frequently used over the last 30 years, such as Granat, Swift\nand INTEGRAL. It also requires large (m$^3$ scale) detector sizes, has\na restricted field-of-view, but allows localizations in the arcmin range.\n\n\\noindent{\\bf Photon-by-photon imaging:}\n Proper imaging (2D reconstruction) of individual photons on the\nsky. This method was used by the COMPTEL telescope in the\n1990s. Improved versions require electron tracking, and thus will\nbe large and heavy telescopes. However, localizations (degrees) and\nfield of view (up to 70\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi\\ radius) are advantageous.\n\n\\subsection{Missions with GRB capabilities}\n\nExcept for planned missions beyond 2027, \nthe near future can be summarized by the following three strategies:\n(i) new large(r) missions just represent a replication of existing missions,\n  such as GECAM (replicating Fermi\/GBM) and SVOM (Swift).\n(ii) new small(er) missions are mainly driven by enhancing the\n  sky coverage, not improving localizations\n(iii) an euphoric engagement in CubeSat swarms using triangulation\n  which due to their size and LEO will not provide accurate (degree)\n  localizations.\nAll of these strategies will not change\n the lack of well-localized short-duration GRBs.\nThe operational and planned (to our knowledge) missions \nare shortly sketched below.\n\nThe dedicated GRB mission {\\bf Swift} (USA) uses a coded-mask imager\n(BAT = Burst\nAlert Telescope) in the 15--150 keV range for GRB localization, to an\naccuracy of 3 arcmin radius \\citep{Barthelmy+2005}.\nIt has a 1.4 steradian field-of-view (half-coded), and detects about\n100 GRBs\/yr, predominantly as rate triggers (excess counts in the total rate\nof a detector module). Due to the soft energy band\nand the combined noise of the 32768 CdZnTe detector cells, the detection\nrate of short-duration GRBs is only $\\sim$10\\% (10 sGRBs\/yr).\n\nThe gamma-ray observatory Fermi (USA) features a Gamma-ray Burst Monitor\n({\\bf Fermi\/GBM})\naimed at localizing GRBs outside of the zenith-looking field-of-view of the\nprime instrument LAT (Large Area Telescope, 100 MeV -- 10 GeV). GBM consists\nof two sub-systems: (i) a collection of 12 NaI scintillation detectors\nfor the energy range 8--500 keV, and (ii) two thick BGO scintillation detectors\nfor the high-energy range up to 40 MeV \\citep{Meegan+2009}. It is presently\nthe most prolific GRB detector, with the detection and localization of\nabout 240 GRBs\/yr, among those about 40 short-duration GRBs\n\\citep{Kienlin+2020}. \nThe usual, 30-yr-long used localization method (based on orientation-dependent\nrates in different detectors) comes with large\nsystematic errors \\citep{Connaughton+2015}. The cause of these systematics\nhave recently been understood \\citep{Burgess+2018}, but even after\ncorrection the typical error regions have 5\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi--10\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi\\ radius\n\\citep{Berlato+2019}, with the 17\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi\\ error radius for GRB 170817A\ncompletely dominated by the statistical error.\n\nThe Interplanetary Network ({\\bf IPN}) is the logistic combination of different\nspacecrafts equipped with GRB detectors. The locations of GRBs are determined\nby the comparison of the arrival times of the event at the locations of the\nGRB detectors. The precision is proportional to the distance of spacecraft\nseparations, so that the localisation accuracy of a network with baselines\nof thousands of light-seconds can be equal or superior to that of any other\ntechnique \\citep{Hurley+2017}. \nA major disadvantage of the IPN method is the 1--2 day delay in the\ndownlink of the GRB data from the spacecraft.\nAt present, the main IPN contributors are Konus-WIND, Mars Odyssey,\nINTEGRAL, RHESSI, Swift, AGILE, BepiColombo, and Fermi\/GBM. \n\nThe European gamma-ray satellite {\\bf INTEGRAL} can detect GRBs with\nthree of its instruments, i.e. in the field-of-view of ISGRI (a 15-300 keV\ncoded-mask imager with few arcmin localization accuracy) or SPI (a\n200--8000 keV coded mask imager with degree localization accuracy but\nvery high energy resolution), and the SPI anti-coincidence system ACS\n(working at $>$80 keV). Due to the small field-of-view of ISGRI and SPI,\ntheir combined GRB detection rate is only $\\sim$10 GRBs\/yr\n\\citep{Bosnjak+2014}. The ACS detects about 150 GRBs\/yr, but has no\nlocalization capability \\citep{Savchenko+2012}.\n\n\\noindent\n{\\bf CALET} (Japan),  {\\bf Insight-HXMT} (China) and {\\bf AstroSat\/CZTI} (India)\nare operational satellite experiments with the capability of detecting GRBs\nin their particle detectors or shields, without localizations. Due to their\nlow-Earth orbit, they do not provide constraints via\ntriangulation, and thus are not (or very rarely) used in the IPN.\n\n\\noindent{\\bf GECAM} (China): The Gravitational Wave Electromagnetic\nCounterpart All-sky Monitor GECAM is a twin spacecraft mission to monitor GRBs\ncoincident with GW events \\citep{ZhengXiong2019}. With a dome-shaped\ndistribution of multiple scintillators it reaches an effective area\n(and energy range) similar to that of Fermi\/GBM.\nThe planned main advantage was the $\\approx$100\\% sky coverage due to the\n180 deg phasing of the two spacecrafts in their orbit. \nLaunched on Dec. 9th, 2020, only one of the spacecrafts\nreturns data.\n\n\\noindent{\\bf GRBAlpha} (Hungary\/Czech\/Slovakia\/Japan):\nGRBAlpha, launched on 2021 March 22, is a 1U CubeSat demonstration\nmission \\citep{Pal+2020} for a future CubeSat constellation\n\\citep{Werner+2018}. The detector consists of a 75 x 75 x 5 mm$^3$ CsI\nscintillator read out by a SiPM array, covering the energy\nrange 50--1000 keV.\n\n\\noindent{\\bf BurstCube} (USA) is a planned 6U CubeSat to be released into\nlow-Earth orbit from the ISS to detect GRBs. The instrument is\ncomposed of 4 CsI scintillator plates, each 9 cm diameter, read out by\narrays of silicon photo-multipliers \\citep{Smith2019}.\nIt reaches an effective area of 70\\% of Fermi\/GBM at 15\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi\\ incidence,\nbut the localisation accuracy is substantially worse, with 7\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi\\\nradius at best for the brightest GRBs\n(launch 2022).\n\n\\noindent{\\bf SVOM} (China\/France): The Space-based multi-band astronomical\nVariable Objects Monitor (SVOM) is a Swift-like mission with a wide field\nof view\n$\\gamma$-ray detector for GRB localization, and an X-ray and an optical\ntelescope for rapid follow-up of the GRB afterglow \\citep{Yu+2020}.\n60 GRBs\/yr will be localized to 10\\ifmmode ^{\\prime}\\else$^{\\prime}$\\fi\\ accuracy with a coded-mask telescope\nwith a 89\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi x89\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi\\ field of view, working in the 4--250 keV band.\nDue to this soft energy coverage the focus is on high-redshift GRBs\n(launch early 2023).\n\n\\noindent{\\bf HERMES} (Italy) is an Italian-led project to launch\n100 CubeSats with X-\/$\\gamma$-ray detectors to localize GRBs,\nand to derive limits\non Quantum Gravity \\citep{Fuschino+2019}. Presently, six 3U CubeSats\nare funded for a 2-year\npathfinder mission, with $\\sim$56 cm$^2$ effective area per CubeSat\nin the 3--1000 keV band (launch 2022). The anticipated\n  localisation accuracy for transients with ms variability is 3\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi\\ for\n the pathfinder and 10 arcsec for the full fleet in LEO\n\\citep{Fuschino+2019, Burderi+2020, Burderi+2021},\nthough this seems very optimistic given the detection of 0.05 ph\/ms\neven for the brightest bursts per CubeSat.\n\n\\noindent{\\bf Glowbug} (NASA) is a funded small (30x30x40 cm$^3$)\nsatellite to detect GRBs and  other transients in the 30 keV to 2 MeV\nband \\citep{Grove+2020}. With an effective area about 2.5x that of\nFermi\/GBM, about 70 short GRBs are expected per year. The localization\naccuracy is expected to be slightly better than GBM, \nin the 5\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi\\ (1$\\sigma$ radius) range. The nominal lifetime is 1 yr\n(launch 2023).\n\n\\noindent{\\bf POLAR-2} (China\/Switzerland) is a dedicated GRB polarimeter\nto be flown onboard China's space station. With a field of view\nof half the sky, the position determination will be a few degrees only.\nDetailed polarization measurements are expected for 50 GRBs\/yr, though more\nGRBs are expected to be detected \\citep{Kole2019} (launch 2024).\n\n\\noindent{\\bf COSI} (USA): The Compton Spectrometer and Imager is an\napproved NASA\/SMEX mission,\nworking in the 0.2--5 MeV band, and scheduled for launch in 2025.\nIts wide field of view of 3 sr will allow to detect 7--10 short GRBs per year, \nat sub-degree localisation \\citep{Tomsick+2021}.\n\n\\noindent{\\bf eXTP} (China\/ESA): The enhanced X-ray Timing and Polarimetry\nmission (eXTP) will study the X-ray\nsky with 4 different instruments, covering the 0.5--50 keV band.\nIt will likely be the first to simultaneously measure the\nspectral-timing-polarimetry properties of cosmic sources (launch 2027).\nRelevant for GRB detection is the Wide-field monitor (WFM): with a field of view\nof 1 sr (fully-coded) the detection of 16 GRBs\/day(!) is predicted\n\\citep{Zhang+2017}, the brighter ones with 1 arcmin localization accuracy.\nWhile this GRB rate is far ($\\sim$5x) above the predicted total number\nof GRBs in the Universe, the soft energy response of the WFM implies\na small fraction of short  GRBs (5\\%-10\\%).\n\n\\noindent{\\bf HSP} (USA):\nThe proposed High Resolution Energetic X-ray Imager SmallSat Pathfinder\n(HSP) is a wide-field hard X-ray (3-\u2013200 keV) coded aperture telescope\nwith 1024 cm$^2$ CdZnTe detectors and a Tungsten mask \\citep{Grindlay+2020a}.\nWith 4\\farcm7\nresolution covering 36\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi x 36\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi\\ (FWHM), HSP localizes transients\nand GRBs within $<$30\\ifmmode ^{\\prime\\prime}\\else$^{\\prime\\prime}$\\fi\\ in less than 10 min.\n(launch $>$2025).\n\nSummarizing, there is a need to better localize short-duration GRBs.\nWe propose that GRB triangulation with the Galileo satellite network\nprovides such an opportunity.\n\n\\section{The Galileo system as a perfect host for triangulation}\n\nThe Global Navigation Satellite System (GNSS) Galileo has five parameters\nwhich makes it a nearly ideal satellite constellation for triangulation:\n\\begin{itemize}\n \\vspace{-0.2cm}\\itemsep-.4ex\n\\item all satellites are synchronized with a very accurate atomic clock,\n  ensuring time-stamps for the GRB signal at the 10$^{-9}$\\,s level;\n\\item the satellites are distributed over three orbital planes, perpendicular\n  to each other, making triangulation positions close to round;\n\\item the position knowledge of all satellites is accurate to sub-meter\n  accuracy, and thus do not contribute to the error budget in\n  realistic GRB measurements, similar to the timing;\n\\item the knowledge of the orientation of each satellite is known to\n  better than 1\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi,\n  removing any ambiguity in the relative rate measurements for GRBs;\n\\item the orbital radius is large enough that one can realistically expect\n  sub-degree localizations, but small enough that light travel time\n  distances are short and communication (data transfer) is quick.\n  This differentiates it from the canonical Interplanetary Network (IPN),\n  where the baseline is much longer and thus allows arcmin-scale localizations,\n  but the triangulation can only be done 1-2 days after the GRB.\n\\vspace{-0.2cm}\n\\end{itemize}\n\n\\begin{figure}[th]\n  \\centering\n  \\vspace{-0.2cm}\n  \\includegraphics[width=0.5\\textwidth]{histogram_dist24.pdf}\n  \\vspace{-0.4cm}\n  \\caption[]{Histogram of pair-wise distances between Galileo satellites.\n    The peaks indicate constant distances between satellites in the same\n    orbital plane.\n  \\label{Fig_45histdist}}\n\\end{figure}\n\n\\noindent\nThe first satellites of the\npresent Galileo constellation were launched on October 21, 2011.\nToday, 26 satellites are in orbit, among them\ntwo unusable -\u2013 one with technical problems, one declared as spare due\nto issues with clocks \u2013- and two on non-nominal orbits due to launch\nfailure of the third rocket stage but otherwise fully operational and\nusable. In December 2016, Initial Service Declaration was announced.\nCurrently 12 satellites, the so-called Batch 3 satellites are under\nproduction, deployment shall start in 2021.\nAs the design lifetime of Galileo satellites is 12 years,\nthe constellation has to be replenished in the coming years.\n\nThe Galileo constellation is a Walker constellation \\citep{Walker1984}.\nThis constellation type is characterized by the three numbers 24\/3\/1:\n\\begin{itemize}\n \\vspace{-0.2cm}\\itemsep-.4ex\n\\item[~$n_1$:] Total number of satellites (i.e. eventually 24 satellites), equally distributed over the orbital planes.\n\\item[~$n_2$:] Number of equally spaced orbital planes (i.e. 3), with 8 satellites each.\n\\item[~$n_3$:] Relative spacing between satellites in adjacent planes.\nThe difference in argument of latitude (in degrees) for equivalent\nsatellites in neighboring planes is equal to $n_3*360\/n_1$.\n\\vspace{-0.2cm}\n\\end{itemize}\n\n\n\\noindent\nThe revolution period corresponds to 17 revolutions in 10\nsidereal days, i.e., 14h04m. With a semi-major axis of 29.600~km,\nthe orbital inclination is 56\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi.\nFor current Galileo satellites the eccentricity is below 0.0007 except\nfor the two satellites on non-nominal orbits that have an eccentricity\nof 0.166.\nFig.~\\ref{Fig_45histdist} shows the histogram of pair-wise distances between\nGalileo satellites in a Walker 24\/3\/1-constellation. Maximum distance is\n59.000 km, i.e., the orbit diameter, while the mean distance is 42.000 km.\nThe peaks in the figure indicate the constant distances between satellites\nin the same orbital plane.\n\nOrbits and clock corrections for Galileo satellites are available with\nhigh precision in real time. While precise orbits at a few cm level and\nclock corrections at below the ns level are available in post\nprocessing, sub-meter orbits and few ns clock corrections are available\nthrough the broadcast messages that are updated every 10 minutes.\nWhile the eccentric satellites show slightly larger broadcast\norbit errors, mainly in along-track, the rms of broadcast orbit errors\nfor the other satellites is at a level of 32~cm (12.5~cm radial,\n25~cm along-track, 15~cm cross-track).\n\nIn contrast, the broadcast clock quality of eccentric Galileo satellites \nis comparable to the other satellites. The rms of the difference is\n0.50~ns. For GRB triangulation it can thus be assumed that perfect\npositions and time tagging are known in real time at any given time.\n\n\nThe attitude of the Galileo satellites is a nominal yaw steering in order\nto point the navigation antenna (body-fixed z-axis) to the center of the\nEarth and the solar panel axis (y-axis) in perpendicular direction to the\ndirection of the Sun. The Sun is thus always located in the body-fixed\nx-z plane. While the positive x-surface is always illuminated by the Sun,\nthe negative x-surface constantly points to dark sky.\nThe nominal attitude is controlled by Earth- and Sun-sensors to below 0\\fdg1,\nexcept for non-nominal noon and midnight yaw maneuvers, if the Sun is\ncloser to the orbital plane than about 2\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi\\ for IOV and about 4\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi\\\nfor FOC, and the nominal yaw rate would exceed the maximum hardware\nrate of 0\\fdg2\/s. The negative z-axis is always pointing in zenith\ndirection with a precision below 0\\fdg1.\n\n\n\\section{Triangulation:  methods \\& prospects \\label{triangulation}}\n\n\nThe triangulation method uses measurements of differences of arrival times\nof the same signal (GRB) at different clocks (each on a different satellite).\nIn general, time differences between three independent satellite pairs \nare needed to derive a unique position on the sky, and not all these satellites\nshall be in the same plane.\n\nThe relation\ncos\\,$\\theta$ = $c{\\cdot}t \/ d$ holds for the time difference of a signal\nbetween two satellites. Under the simplifying\nassumption of perfect satellite clocks and perfectly known satellite\npositions, the width of the\nannulus $\\Delta\\theta$ is obtained as the derivative of the above equation,\ni.e. is just determined by the error $\\Delta t$ with which\nthe time delay $t$ between the two signals (light curves) can be measured:\n\n\\begin{equation}\n\\Delta\\theta = \\abs*{ \\frac{-1}{\\sqrt{1-(c\\cdot t \/ d)^2}}} \\Delta t \\cdot \\frac{c}{d} \\label{eq:1}\n\\end{equation}\n\n\\noindent\nThere are two possible approaches to triangulation: Firstly, one can\ncompare pairs of two light curves to each other to find the time lag, i.e.\nsimple cross-correlation\nof background subtracted time series \\citep{Hurley+2013, Palshin+2013}.\nEach satellite pair results in one time lag, and a\n corresponding triangulation ring. Combining multiple ($>$3) pairs then provides\n a unique sky position as the overlap of these triangulation rings.\n Cross-correlation is computationally fast, but\nsuffers from several draw-backs \\citep{Burgess+2021}:\n(i) they only work for binned light curves, at fixed binning;\n(ii) no mathematical method exists to estimate the proper error of the\ncross-correlation;\n(iii) the approximation of $\\chi^2$ rarely holds, in particular when\nsmall bin sizes are choosen in order to ``increase'' the temporal accuracy;\n(iv) the subtraction of two Poisson rates results in Skellam rather than\nPoisson distributed data, often leading to over-confidence;\n(v) it cannot take into account lightcurves taken at different energies.\nAs a second approach, one can forward-fold an\n  identical model  through the (different) response of each detector and fit\n  each observed light curve \\citep{Burgess+2021}.\n  This technique is computationally  expensive, but\n  offers the major advantage that\nit produces a complete posterior probability distribution allowing for a\nvery precise estimate of the uncertainty formally in $\\Delta\\theta$,\nbut due to the forward-folding directly in sky coordinates\n$\\Delta$RA, $\\Delta$Decl.\nThis nazgul code \\citep{Burgess+2021} has been made publicly\navailable\\footnote{\\url{https:\/\/github.com\/grburgess\/nazgul}}.\n\n\n\\section{Simulation set-up}\n\nIn order to test each of the localization methods and verify their\nperformance for different satellite configurations, we have developed\na simulation framework utilizing the Python package\nPyIPN\\footnote{\\url{https:\/\/github.com\/grburgess\/pyipn}}.\nThis package allows for the\ngeneration of synthetic GRB light curves as seen by detectors\ndistributed within the solar systems. We have added on top of this\nframe work a procedure to generate realistic light curve shapes and\ndetector configurations that mimic the orbit of the Galileo\nconstellation. Below we detail the setup and procedure for the\ngeneration of mock data sets which allow us to test our methods.\n\n\n\\subsection{Simulating GRB light curves}\n\nThe simulation of the triangulation capability of a network of\nGRB detectors requires to create mock GRB light curves which then\nhit differently-oriented detectors. These mock\nlight curves shall cover a peak flux range as bright as has\nbeen seen with previous experiments (CGRO\/BATSE, Swift\/BAT, Fermi\/GBM),\nand down to our proposed sensitivity limit of\n1x10$^{-7}$ erg\/cm$^2$\/s in the 25--150 keV band.\nWe pick the 256\\,ms timescale for the peak flux\nas a compromise between being short enough to cover spikes in short-duration\nGRBs, and being general enough also for long-duration GRBs.\n\nGRB light curves are generally very complex, and  unique  for  each  GRB.\nIn many cases the variability time-scale is significantly shorter than the\noverall burst duration. Only in a minority of GRB lightcurves there is only one\npeak, with no substructure. \nThe most straightforward way is to ``assemble'' GRB light curves by the\nsuperposition of different pulses.\nWe assume that candidate pulses can be modeled with the empirical\nfunctional pulse form of \\cite{Norris+1996, Norris+2005}: \n\\begin{equation}\n  I(t) = A \\lambda {\\rm e}^{-\\tau_1\/(t-t_s) - (t-t_s)\/\\tau_2} {\\rm cts\/s}\n  \\label{eq:lc}\n\\end{equation}\nwhere $t$ is time since trigger, $A$ is the pulse amplitude, $t_s$ is the\npulse start time, $\\tau_1$ and $\\tau_2$ are characteristics of the pulse\nrise and pulse decay, and the constant $\\lambda = {\\rm e}^{2(\\tau_1\/\\tau_2)^{1\/2}}$.\nThe pulse peak time occurs at time $\\tau_{peak} = t_s + \\sqrt(\\tau_1\\tau_2)$.\nTypically, the rise times in individual GRBs are very short (steep rise),\nand decay times substantially longer in most times. Thus, for single-pulse\nGRBs, the decay time $\\tau_2$ scales with the T90 duration of a GRB.\n\nIn order to implement the stochastic nature of the light emission\nprocess, and to incorporate unavoidable background at the measurement\nprocess, individual photon events are sampled according to an\ninhomogeneous-Poisson distribution following the intrinsic pulse shape\nspecified. The photon arrival times are sampled via an inverse\ncumulative distribution function rejection sampling scheme\n\\citep{Rubinstein+2016}. As the rate for the signal evolves with time,\na further rejection sampling step is implemented that thins the\narrival times according to the evolving light curve. This is done by\nfirst sampling a waiting time $t$ and computing the light curve\nintensity $I(t)$.  Another draw from $p \\in \\{0,1\\}$ is made and the\nsample is accepted if $p$ $\\mathrel{\\copy\\laxbox}$\\ $I(t)$ \\citep{Burgess+2021}.\n\nThe cross-correlation of two light curves depends crucially on the\nintensity of the GRB above background, and the structure of the light curves.\nWe therefore need a sample of different light curves.\nIn order to create a realistic sample, we need to make sure that\nwe reproduce\n\\begin{itemize}[leftmargin=12pt]\n  \\vspace{-0.2cm}\\itemsep-.3ex\n\\item a rough --3\/2 logN-logS intensity distribution\nbetween the brightest GRBs\nseen so far (2$\\times$10$^{-4}$ erg\/cm$^2$\/s)\n and our aimed-at limit of 1x10$^{-7}$ erg\/cm$^2$\/s:\nFor a canonical GRB spectrum below E$_{\\rm peak}$, i.e. a power law spectrum with\na slope in the range of -0.9...-1.1 (long) and 0.0...-0.2 (hard),\nthe following conversion holds for the 25\u2013-150 keV band\nwith an detector size of  3600 cm$^2$ (see below):\n1$\\times$10$^{-7}$ erg\/cm$^2$\/s = 0.65$\\pm$0.10 ph\/cm$^2$\/s.\nThus, we substitute the sampling over the\n2$\\times$10$^{-4}$ --- 1x10$^{-7}$ erg\/cm$^2$\/s range by that over\n1300 --- 0.65 ph\/cm$^2$\/s.\n\\item the observed T90 duration distribution of GRBs \\citep{Kouveliotou+1993}; and\n\\item some realistic distribution between single- and\n  multi-pulse light curve structure:\n  We assemble multi-pulse light curves by\n  overlapping multiple single pulses, each\n  with a shape a la \\cite{Norris+1996}, but with different parameters\n  and properly delayed to each other.\n  \\vspace{-0.2cm}\n\\end{itemize}\n\n\\noindent\nFor the latter, we implement a pulse avalanche, a linear Markov\nprocess, as proposed by \\cite{SternSvensson1996},\nand described in detail in appendix \\ref{avalanche}.\nExample light curves with this simulation set-up are shown in\nFig. \\ref{example-lc}.\n\n\\begin{figure}[ht]\n  \\hspace{-0.2cm}\\includegraphics[width=0.50\\textwidth]{lc_generation_example.pdf}\n  \\vspace{-0.4cm}\n  \\caption[]{Example mock light curves of the long-duration class\n    created with the pulse avalanche description of Norris-like pulses. \n    \\label{example-lc}}\n    \\vspace{-0.2cm}\n\\end{figure}\n\nEnergy dependent effects in GRB light curves are ignored.\nConceptually,\nwe treat our proposed energy band of 25--150 keV as a mono-energetic band.\n\n\n\\subsubsection{Lightcurve detectability by different detectors}\n\nThe previous steps provide theoretical light curves of GRBs (as emitted)\nwhich are representative in intensity distribution, duration distribution\nand pulse structure to the GRBs as measured over the last 30 years.\nThese light curves are now being measured by identical detectors\non a number of Galileo satellites. While the details of the Galileo\nsatellite network is described later, three effects combine together\nto establish the measured light curves: (1) since the detectors\nare oriented into different directions, each will detect photons\naccording to the cosine between the scintillator normal (we adopt\nthin, but large-area scintillator plates as baseline) and the GRB,\nand (2) the detector will measure quasi-isotropic $\\gamma$-ray background\nwhich has the effect of washing out\nlow-intensity features; (3) in the case of multiple detector plates\nper satellite, the sensitivity can be improved by co-adding the data.\nHowever, this\nhelps only for a certain incidence angle range, since the GRB signal\nvaries with the incidence angle, but the isotropic background does not.\n\n\\begin{figure}[th]\n  \\hspace{-0.2cm}\\includegraphics[width=0.50\\textwidth]{example_lcs3.pdf}\n  \\caption[]{Light curves of the same GRB as seen with\n    a flat detector plate from different incidence angles.\n    The labels in each plot\n    denote the angle under which the GRB impinges on the detector plane.\n    With increasing angle, the effective area shrinks with the\n    cosine of the angle, while the background remains the same.\n    The ``mother'' lightcurve\n    of this GRB has been generated with the above described\n    pulse avalanche scheme.\n  \\label{cosine-cts}}\n\\end{figure}\n\nAn example of such a set of 'measured' light curves for a given GRB\nand differently-oriented detectors is given in Fig. \\ref{cosine-cts}.\nThese are the final 'measured' light curves\n(in counts\/s) over a certain duration in the 25--150 keV band,\nwhich are then used for triangulation.\n\n\n\n\\subsubsection{Implementation of the discrete correlation function}\n\nA modified version of the discrete correlation function method\n\\citep{EdelsonKrolik1988} has been implemented in PyIPN\nwith the following three parts: First, the model is initialized by\nsetting a GRB position and define all detectors. The actual\nsimulation creates the GRB signal (light curve) and computes the\narrival times at the detectors as detailed above.\n\nThe final step performs the cross-correlation and computes the\ncenter point and opening angle of the circle for a \nspecified pair of detectors. \nRather than relying on mathematical covariance matrix estimation of\nuncertainty, cross-correlation methods heuristically derive uncertainties\nin one of two ways. First, the discretized time bins of the light curve\nyield discrete estimates of the time lag between each pair of light curves.\nFor each value, a pseudo-$\\chi^2$ statistics\\footnote{Note that the\npseudo-$\\chi^2$ values are incorrect in the first place due to the lack of\nfidelity in the low-count regime and the fact that count data are fundamentally\nPoisson distributed.} is derived yielding a grid of\nstatistics values hopefully in a parabolic shape. The minimum of these values\nis taken as the true time lag (best fit). The 1, 2, and 3 $\\sigma$ uncertainty\nregions are recovered by moving up the grid of statistics at the appropriate\nlevels and reading off the implied time lags. This has several drawbacks:\nfirst, the best-fit time lag can never be below the timing resolution of\nthe light curve. Additionally, the uncertainties are locked to the resolution\nof the grid and can thus easily be over- or under-estimated. To get around\nthis, another heuristic can be introduced. One can fit this grid of\nuncertainties with a parabolic shape to effectively interpolate to finer\ntiming resolution. While this alleviates the issues with discretization\nin the previous method, it introduces the problem that the interpolation\nhas an associated uncertainty which is not accounted for. Moreover, the\nchosen shape of this parabolic fit cannot incorporate asymmetries in the\ngrid and thus can easily over- or underestimate the true uncertainty. \nHowever, given the lack of a mathematically strict method, we use this\nprocedure, but keep the problems in mind.\n\n\n\n\\subsection{Gamma-ray background in the Galileo orbit\n\\label{sect8p1p1}}\n\nThe background which a $\\gamma$-ray detector (whether scintillation\ndetector or other type) experiences in space is composed of several\ndifferent components\n\\citep[e.g.][]{Weidenspointner+2003, Weidenspointner+2005, Wunderer+2006, Cumani+2019}.\nIn the 10--250 keV band, the most important\ncomponents are the extragalactic diffuse $\\gamma$-ray background, Earth albedo\nphotons (for LEO), Galactic cosmic-ray protons, and radioactive\ndecay of activated detector and spacecraft material due to cosmic-ray\nbombardment. For a satellite in MEO, the diffuse $\\gamma$-ray background\ndominates below 100 keV,\nwhile at $\\sim$200 keV the rising proton contribution has reached the level\nof the diffuse $\\gamma$-ray background.\nWe therefore just incorporate the extragalactic diffuse $\\gamma$-ray background\nin our simulations.\n\nWe adopt the following smoothly broken powerlaw for the diffuse\nbackground spectrum \\citep{Ajello+2008}:\n\\begin{equation}\n  E^2 {dN \\over dE} = E^2 \\times {C \\over (E\/E_{\\rm B})^{\\Gamma_1} + E\/E_{\\rm B})^{\\Gamma_2}}\n\\end{equation}\n\\noindent with the following constants:\nC = 0.102$\\pm$0.008 ${\\rm ph\\ cm^{-2}\\ s^{-1}\\ sr^{-1}\\ keV^{-1}}$,\n$\\Gamma_1 = -1.32 \\pm 0.02$, $\\Gamma_2 = -2.88 \\pm 0.02$ and a break at\n$E_{\\rm B} = 30.0 \\pm 1.1$ keV.\nIntegrating over the 25--150 keV energy range and the 2$\\pi$ sky coverage\nis consistent with both, the Konus-Wind \\citep{Aptekar+1995}\nas well as Fermi\/GBM \\citep{Burgess+2018} measurements,\nand leads to $\\approx$4 cts cm$^{-2}$ s$^{-1}$.\nThis background rate is then added to the scaled\nlight curve generated with\nthe pulse avalanche method (see previous subsection).\n\n\n\\subsection{Required detector timing}\n\nWith a dedicated GRB detector on the Galileo satellites, we can\ndramatically improve the localization accuracy.\nUsing the formal triangulation error (see  eq. \\eqref{eq:1}),\nit is easy\nto compute the required temporal resolution, usually the bin size in \nthe classical scheme, for a perfect system with satellites at known\ndistances. Fig. \\ref{accuracydelay} shows\nthat sub-ms accuracy  in the determination of the time delay is\nrequired to reach sub-degree localisation accuracy with two satellites at\na distance of 42000 km (which corresponds to the mean for\nGalileo's Walker constellation; see Fig. \\ref{Fig_45histdist}).\n\n\\begin{figure}[!bh]\n  \\vspace{-0.2cm}\n  \\includegraphics[width=0.50\\textwidth]{Accuracy_Delay.pdf}\n  \\vspace{-0.5cm}\n  \\caption[]{The 1$\\sigma$ half-width error $\\Delta\\theta$ of the\n    triangulation annulus is shown for a signal (GRB) arriving at a pair of\n    satellites 42000 km apart (corresponding to the mean of\n    Fig. \\ref{Fig_45histdist}),\n    with different delay times (bottom x-axis label) or angles relative\n    to the line connecting the two satellites (top x-axis) for different\n    accuracies $\\Delta t$ of 1\\,ms, 3\\,ms, 10\\,ms and 50\\,ms\n    with which the time delay can be measured. For comparison, the\n    Anti-Coincidence system of the INTEGRAL spectrometer (SPI-ACS) has\n    a time resolution of 50\\,ms, and the shortest binning of Fermi-GBM is\n    64\\,ms, with individual events time-tagged at 2\\,$\\mu$s.\n    \\label{accuracydelay}}\n    \\vspace{-0.1cm}\n\\end{figure}\n\nFig. \\ref{accuracydelay} only applies for a perfect system.\nAs discussed below, the classical triangulation method with its\nuse of a cross-correlation of binned data sets does not\nprovide a mathematically self-consistent error handling. In contrast, the\nalternative method by \\cite{Burgess+2020} does, but lacks the beauty\nof a simple equation. We therefore will show with simulations below\nhow close this new method gets to the estimate of eq. \\eqref{eq:1}.\n\n\\subsection{Required detector sensitivity}\n\nGood timing resolution provides a necessary but not yet sufficient condition.\nThe detector needs to\n(i) be large enough to detect a significant signal at these short\ntime scales, \n(ii) measure a significant signal independent on the arrival direction, and\n(iii) provide this high time-resolution data for analysis, either on-board\nor on the ground, rather than binning it up to save telemetry band width.\n\nA simple estimate of the minimum detector size can be made\nby recognizing that short-duration GRBs have structure, and do have\ndurations substantially longer than the 3\\,ms time scale which\nFig. \\ref{accuracydelay} implies as a requirement for sub-degree\nlocalization accuracy. \nAssuming a canonical shape of a short GRB prompt emission lightcurve,\nand knowing that for GRB 170817A a single Fermi\/GBM detector measured\n20--30 cts\/0.1\\,s in the 20--500 keV band at peak against $\\sim$30 cts rms\nfrom background fluctuations,\nwe estimate to need 2000 cts per short-duration (2\\,s) GRB\nor 10 cts\/1\\,ms at peak, so $\\sim$30x the effective area of a single\nGBM detector of 125 cm$^2$, that is 3500--4000 cm$^2$.\nIncorporating the correspondingly higher background rate\nat the Galileo orbit wrt. the LEO of Fermi will\nmodify this estimate, but for the simulations presented here,\nwe consider a detector of 60\\,cm x 60\\,cm geometrical area.\n\n\n\n\n\\begin{figure}[bh]\n  \\includegraphics[width=0.155\\textwidth]{detectorconf01.pdf}\n  \\includegraphics[width=0.155\\textwidth]{detectorconf02.pdf}\n  \\includegraphics[width=0.155\\textwidth]{detectorconf03.pdf}\\\\\n  \\includegraphics[width=0.155\\textwidth]{detectorconf04.pdf}\n  \\includegraphics[width=0.155\\textwidth]{detectorconf05.pdf}\n  \\includegraphics[width=0.155\\textwidth]{detectorconf06.pdf}\\\\\n  \\includegraphics[width=0.155\\textwidth]{detectorconf07.pdf}\n  \\includegraphics[width=0.155\\textwidth]{detectorconf08.pdf}\n  \\includegraphics[width=0.155\\textwidth]{detectorconf09.pdf}\n  \\caption[]{The different detector geometries simulated here,\n    detector numbers 1--3 (top row) to 7--9 (bottom row).\n    \\label{geometry}}\n\\end{figure}\n\n\\subsection{Detector geometry}\n\\label{sect:Detector-geometry}\n\n\\begin{figure}[ht]\n  \\includegraphics[width=0.48\\textwidth, clip]{effarea_vs_offaxis.jpg}\n  \\vspace{-0.3cm}\n  \\caption[]{Off-axis dependence of the different detector geometries.\n    The blue shaded range corresponds to the min-max range according\n    to azimuth angle for the cube detector.\n    Beyond 90\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi, the cube detector offers continued effective area,\n    but shadowing by the satellite bus leads to a rapid drop.\n    The green shaded area corresponds to equal-size detectors\n    on two neighboring sides, with the top boundary corresponding\n    to 45\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi\\ view onto both, and the lower to 90\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi\\\n    on only one of the two detectors.\n    \\label{Eff_area}}\n  \\vspace{-0.3cm}\n\\end{figure}\n\nAs we will show, the generally preferred and assumed zenith-looking\ndetector is not a good choice. Since the best localisation accuracy is\nreached at largest satellite separation and looking perpendicular to the\nsatellite-connecting line (see eq. \\eqref{eq:1} and Fig. \\ref{accuracydelay}),\ndetectors sensitive sidewards, i.e. 90\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi\\ of zenith, are preferred.\n\n\nSince the anticipated detector\nsize has $\\approx$60 cm sidelength, adding a cube of that size to the\nzenith-facing side of the Galileo satellite might be challenging in terms\nof satellite momentum balance or station keeping, thus\nwe also consider configurations, where one-dimensional\ndetector plates are mounted on different sides of the Galileo satellite\n(Fig. \\ref{geometry}).\nAny such 3D detector has several advantages:\n(i) multiple detector units provide independent measurements\n  to be used in the cross-correlation;\n(ii) for the same reason, a\n  coincidence veto against particle hits can be implemented, reducing\n  the rate of false triggers;\n(iii) since 3D detectors cover a field-of-view\n  (FOV) of more than\n  2$\\pi$ of the sky, detectors on some 'behind-the-Earth' Galileo satellites\n  will be able to detect the GRB, thus increasing not only the number\n  of measuring detectors, but more crucially extending the baseline\n  (maximum distance between detectors) for the time delay measurement.\n\nWe consider nine different detector geometries (Fig. \\ref{geometry}):\na single detector facing zenith (called detector 01),\na hollow cube detector with 30\\,cm height on the zenith-facing side (detector 02),\n4 detector plates looking sideways (03),\n2 neighboring sideways (+X, +Y) looking plates (04),\n4 sideways plus a zenith-looking detector (05),\n2 oppositely sideways and 1 zenith-looking (+X, -X; 06),\n1 sideway only (+X; 07),\n1 sideway plus 1 zenith-looking (+X; 08),\nand 2 oppositely sideways looking detectors (+X, -X; 09).\n\nAll the side-looking plates have also 60\\,cm $\\times$ 60\\,cm dimension\nand 1 cm thickness.\nThese configurations obviously change the zenith-angle dependent\nvariation of the effective area; see Fig. \\ref{Eff_area}:\nthe green area corresponds to two equal-size detectors on two neighboring\nsides (Fig. \\ref{geometry}). \nTwo-dimensional versions of theses dependencies (including azimuthal variation)\nare shown in Fig. \\ref{3Deffarea} further below.\n\n\n\\subsection{Set-up of GNSS configuration}\n\\label{sect:GNSS-setup}\n\nSince we use an existing satellite network, only one further\nconfiguration choice needs to be considered in the simulations,\nnamely the number of satellites per orbital plane that shall be\nequipped with GRB detectors to allow a $4\\pi$-coverage of the sky.\nWe use the notation of [1] or [0] if a GRB detector is\nplaced on a given satellite or not. With 8 satellites per orbital plane,\nand dealing with these planes consecutively, a configuration\nof every second satellite equipped with a GRB detector would read\n[10101010 10101010 10101010]. The set of simulated configurations is given\nin  Tab. \\ref{detconfigs}.\n\n\\begin{table}[th]\n    \\centering\n  \\caption{Overview on the simulated detector configurations \n    \\label{detconfigs}}\n  \\vspace{-0.3cm}\n  \\begin{tabular}{ccc}\n    \\hline\n    \\noalign{\\smallskip}\n    Sat   & Configuration & Detectors \\\\\n    \\noalign{\\smallskip}\n    \\hline\n    \\noalign{\\smallskip}\n    24 & 11111111 11111111 11111111 & 1,2,3,4,5,6,7,8,9 \\\\\n    12 & 10101010 10101010 10101010 & 1,2,3,4,5,6 \\\\\n     9 & 10010010 10010010 10010010 & 3,5 \\\\\n       & 10010010 01001001 10100100 & 3,5 \\\\\n     6 & 10001000 00100010 10001000 & 3,5 \\\\\n       & 10001000 10001000 10001000 & 3,5 \\\\\n       & 10010000 00010010 10010000 & 3,5 \\\\\n    \\noalign{\\smallskip}\n    \\hline\n   \\end{tabular}\n\\end{table}\n\n\nWe compute two maps:\none 'instantaneous' snapshot map, and one averaged over one\norbital period.\nSimulations are done in steps of 5\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi, which provides 72\nsubsequent snapshots for a full 14h04m orbital revolution of the Galileo\nsatellite network, i.e. the averaged map is the average of\nsuch 72 snapshots. GRBs are distributed on the sky on a 2\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi\\ grid,\nthus providing a full sky map for each snapshot.\n\nIn order to allow any arbitrary combination of Galileo satellites\nto be picked, a simulation tool has been set-up \\citep{Rott2020}\nwhich allows to switch on\/off single Galileo satellites\/detectors.\nThis is implemented as a\nMATLAB function {\\em galileo\\_skyCoverage.m},\nwhich computes the sky coverage of any Galileo constellation\nand a (or several) given off-axis detector response(s)\n(Fig. \\ref{skycov}).\n\n\\begin{figure}[bh]\n  \\includegraphics[width=0.47\\textwidth]{skycov_tensor.jpeg}\n  \\vspace{-0.2cm}\n  \\caption[]{Structure of the skycov tensor, i.e. sky maps for the\n    combinations of constellations and off-axis angle response.\n    For this example, a step size (time step) of 5\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi\\ is assumed,\n    resulting in 360\/5=72 sky maps.\n    \\label{skycov}}\n\\end{figure}\n\n\n\n\\begin{figure}[h]\n  \\includegraphics[width=0.48\\textwidth]{Coverage_2Sat_85d.jpeg}\n  \\vspace{-0.2cm}\n  \\caption[]{Example of an interactive plot for the sky coverage\n    achieved with two Galileo satellites in separate planes,\n    equipped with a flat GRB detector. The slide bar \n    moves through the 72 epochs at 5\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi\\ steps.\n    \\label{cov2sat85d}}\n\\end{figure}\n\n\\begin{figure}[h]\n  \\vspace{-0.2cm}\n  \\includegraphics[width=0.49\\textwidth, viewport=45 60 380 270, clip]{Rott_fig9.pdf}\n  \\vspace{-0.4cm}\n  \\caption[]{Mean sky coverage, i.e. the sum of 72 epochs similar to\n    Fig. \\ref{cov2sat85d}.\n    Only 50\\% coverage is achieved at the two poles of each of the two\n    orbital planes in which the detectors move.\n    \\label{cov2sat85d_72}}\n\\end{figure}\n\n\\begin{figure}[ht]\n  \\centering\n  \\vspace{-0.2cm}\n  \\includegraphics[width=0.39\\textwidth, viewport=45 60 385 270, clip]{Rott_fig10_a.pdf}\n  \\includegraphics[width=0.39\\textwidth, viewport=45 60 385 270, clip]{Rott_fig10_b.pdf}\n  \\includegraphics[width=0.39\\textwidth, viewport=45 60 385 270, clip]{Rott_fig10_c.pdf}\n  \\caption[]{An extreme example of the influence of the distribution of\n     GRB detectors over the orbital plane,\n     with {\\bf Top:} the identical mean sky coverage, ranging between 3.5 and\n    $\\sim$5.3;\n     {\\bf Middle:} all 12 detectors distributed over one pole,\n     and {\\bf Bottom:}\n    all 12 equally distributed. \n    The lower two panels show the standard deviation for each point on\n    the sky.\n    \\label{cov12satasym}}\n    \\vspace{-0.1cm}\n\\end{figure}\n\n\n\nAn example is shown in Fig. \\ref{cov2sat85d} for a flat\nGRB detector on two Galileo satellites in separate orbital planes,\nfor one single snapshot of the constellation.\nDark blue shows sky area not covered, light blue the sky covered\nby both detectors. Summing over one 14h04m period, i.e.\n72 snapshots with 5\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi\\ rotation steps, results in Fig. \\ref{cov2sat85d_72},\nshowing the mean sky coverage by two satellites.\n\nWe note that the distribution of detectors over the\nsatellites in a given orbital plane, or also between orbital planes,\nhas a substantial impact on the results. \nAs a demonstration of the effect\nFig. \\ref{cov12satasym} shows two different constellations with the\nsame number of satellites (with a GRB detector) per orbital plane,\nbut grossly different distribution: (1) one with equal distribution, i.e.\nevery second satellite has a GRB detector, and (2) all 4 satellites\n(with GRB detector) per orbital plane are centered over one pole\nof the Earth at the start configuration (epoch one out of the 72 epochs).\nThe top panel shows the total coverage  after 14h04m,\nwhich obviously is equal for both options, as it depends\njust on the number of satellites. The other two panels show the mean\nvariance of the coverage at any sky position.\nIn the equal distribution, there is no point on the sky which is covered\nby less than 5 and more than 6 satellites,\nwhile in the one-sphere constellation large regions of the\nsky are covered only with 1--2 satellites at a given time,\nwith the consequence that triangulation would not be possible\nwith this total number of satellites.\n\nSince the Earth is not infinitely small, there is a 5\\% chance that\none Galileo satellite is Earth-occulted at any given time. This\nis included in our computations.\n\n\n\\section{Simulations}\n\nThe simulations involve multiple steps:\n\n\\begin{enumerate}[leftmargin=14pt]\n  \\vspace{-0.28cm}\\itemsep-.4ex\n\\item define different detector geometries\n  (sect. \\ref{sect:Detector-geometry})\n  \\item define the number of satellites  and which satellites per\n    orbital plane are equipped (sect. \\ref{sect:GNSS-setup})\n  \\item compute the effective area per detector configuration\n      (sect. \\ref{sect:effarea})\n  \\item compute the accuracy of the localisation\n    via classical cross-correlation, depending on the effective\n    areas of the detectors on separate satellites\n    (sect. \\ref{sect:accuracymatrix})\n  \\item simulate the sky coverage and the GRB localisation\n    accuracy (sect. \\ref{sect:covloc})\n\\vspace{-0.22cm}\n\\end{enumerate}\n\n\\noindent\nIn order to cover the range of GRB peak intensities (amplitude $A$\nin eq. \\ref{eq:lc})\nfour intensity intervals are created such that \nfaint intensity levels can be differentiated,\nsee Tab. \\ref{tab:12Det_det1_acc}.\n\n\\begin{table}[th]\n  \\caption{Four GRB intensity intervals\n    \\label{tab:12Det_det1_acc}}\n  \\vspace{-0.3cm}\n  \\begin{tabular}{ccc}\n    \\hline\n    \\noalign{\\smallskip}\n    Intensity ID & Peak count rate & Peak flux bin  \\\\\n           & (ph\/cm$^2$\/s) & (10$^{-7}$ erg\/cm$^2$\/s)  \\\\\n    \\noalign{\\smallskip}\n    \\hline\n    \\noalign{\\smallskip}\n    1 & 1.5--2 & 2.3--3.0  \\\\\n    2 &  2--3  & 3.0--4.6  \\\\\n    3 &  3--6  & 4.6--9.2  \\\\\n    4 & 6--100 & 9.2--154  \\\\\n    \\noalign{\\smallskip}\n    \\hline\n   \\end{tabular}\n\\end{table}\n\n\\noindent\nWe fix the detector temporal resolution at 3\\,ms. Finally, we use the\npresent 24\/3\/1 walker configuration of the GNSS system, and\nassume that satellites in all three orbital planes are indeed equipped\nwith a detector.\n\n\\noindent These simulations return sky maps which are used to\n\\begin{itemize}[leftmargin=14pt]\n \\vspace{-0.2cm}\\itemsep-.2ex\n  \\item verify the extent to which 4$\\pi$ coverage is possible with a\n   homogeneous localisation accuracy over the sky\n  \\item verify the extent to which 4$\\pi$ coverage is possible with a\n   homogeneous flux sensitivity level\n \\item show the differences in sky coverage and localisation accuracy\n   as a function of different number of satellites to be equipped\n   with a detector and the detector geometry\n \\item provide absolute values of the GRB localisation accuracy\n   (distribution)  for both, single snapshots as well as time-averaged\n   over the GNSS orbital period of 14h04m.\n\\vspace{-0.22cm}\n\\end{itemize}\n\n\\noindent\nGiven the CPU-intensive forward-folding triangulation technique, the\nfull range of parameter testing in the simulation is done by using\nthe classical cross-correlation. Only one individual set-up is\ncomputed with the forward-folding triangulation technique\nin order to obtain proper error estimates and compare the absolute\nvalues of GRB location accuracy (distribution). Note that in this\ncase the above steps (3) and (4) are not necessary, since this\nis part of the model forward-folding.\n\n\n\\begin{figure}[!ht]\n  \\vspace{-0.2cm}\n  \\includegraphics[width=0.23\\textwidth]{Detector01_Area_signl.pdf}\n  \\includegraphics[width=0.23\\textwidth]{Detector01_Area_noise.pdf}\n  \\includegraphics[width=0.23\\textwidth]{Detector04_Area_signl.pdf}\n  \\includegraphics[width=0.23\\textwidth]{Detector04_Area_noise.pdf}\n  \\includegraphics[width=0.23\\textwidth]{Detector06_0_Area_signl.pdf}\n  \\includegraphics[width=0.23\\textwidth]{Detector06_0_Area_noise.pdf}\n  \\includegraphics[width=0.23\\textwidth]{Detector03_Area_signl.pdf}\n  \\includegraphics[width=0.23\\textwidth]{Detector03_Area_noise.pdf}\n  \\includegraphics[width=0.23\\textwidth]{Detector05_Area_signl.pdf}\n  \\includegraphics[width=0.23\\textwidth]{Detector05_Area_noise.pdf}\n  \\includegraphics[width=0.23\\textwidth]{Detector02_Area_signl.pdf}\n  \\hfill\\includegraphics[width=0.235\\textwidth]{Detector02_Area_noise.pdf}\n  \\vspace{-0.2cm}\n  \\caption[]{Two-dimensional variation of the effective area (left column)\n    and the illuminated area (right) for our\n    proposed geometries (each detector is 60cm x 60cm).\n    {\\bf Top row:} Flat detector, zenith looking.\n    {\\bf 2nd:} Two detectors on two neighboring satellite surfaces.\n    {\\bf 3rd:} Two detectors on opposite sides and zenith.\n    {\\bf 4th:}  Four detectors on each side of the satellite, none towards\n    zenith. The Earth shadow is included.\n    {\\bf 5th:} Five detectors, one on each side and one zenith pointing.\n    {\\bf Last row:} A zenith-looking cube\n     with 30\\,cm height; shadowing included.\n    \\label{3Deffarea}}\n  \\vspace{-0.8cm}\n\\end{figure}\n\n\n\\subsection{Classical scheme using cross-correlation}\n\n\\subsubsection{Look-up table for direction-dependent effective area\n\\label{sect:effarea}}\n\nDepending on the placement of single-plate  detectors on\ndifferent sides of a Galileo satellite, the three-dimensional distribution\nof the effective area is grossly different. For illustration, we show\nthese in Fig. \\ref{3Deffarea}: the left panels show the effective area\nas a function of azimuth and zenith angle between 0\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi\\ (figure center) to\n180\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi\\ (border of figure) of an illuminating source (GRB).\nThe right panel shows the corresponding area of the detectors that are\nilluminated, i.e., the area relevant for the noise. This is different\nfrom the left column figures, since the measured GRB counts per\ndetector scale with the cosine of the incidence angle, while the\nbackground (noise) is isotropic.\n\nThe corresponding 360x180 degree matrices are used as detector-lookup tables\nto identify the effective area for a given illumination direction. The\neffective areas obtained in this way from the two detectors of a baseline\nare used to access the lookup-table of the \naccuracy matrix, see next sub-section. With both together,\nthe different Galileo detector-equipment constellations are computed.\n\n\\begin{figure}[th]\n  \\centering\n  \\includegraphics[width=0.37\\textwidth, viewport=10 80 710 632, clip]{CrossCorr_median_delay.pdf}\n  \\includegraphics[width=0.37\\textwidth, viewport=10 80 710 632, clip]{Nazgul_median_delay.pdf}\n  \\vspace{-0.1cm}\n  \\caption[]{Distribution of the error of the time-delay (color coded)\n    for different angles of two detectors (x- and y-axis) at the same\n    position (=satellite), \n    (so the nominal time delay should be zero).\n    The matrices shown are the median of\n     all simulated GRBs in the bright fluence bin\n      (\\#4 in Tab. \\ref{tab:12Det_det1_acc}), separately computed\n    for the classical cross-correlation\n    method (top) and the forward-folding nazgul method (bottom),\n    and we have separate matrices for the other intensity bins.\n    For identical effective areas, i.e.\n    the diagonal, nazgul recovers the nominal time delay of zero, so it\n    was set to 0.3 to avoid division by zero in the follow-up steps.\n    The placement on the same satellite mimics (along the diagonal)\n      also the net effect on the accuracy of two identical detectors on\n      different satellites looking exactly towards the same sky location.\n    Each pixel is the median of the time-delay of many different GRB light\n    curves.\n    The time resolution of the detector is assumed to be 3\\,ms.\n    The effective area distribution\n    mimics a 1D detector with 3600 cm$^2$ seen at different off-axis angles.\n  \\label{delay3ms}}\n\\end{figure}\n\n\\subsubsection{Accuracy matrix \\label{sect:accuracymatrix}}\n\nFor the computation of the effects of the relative orientations\nof different detectors on different Galileo satellites according\nto the given satellite equipment scheme,\nwe need to map the effect of detector-related parameters on the\nlocalisation accuracy in  a way that they can be efficiently used.\nSince this localisation quality depends at least on\ntwo angles (the relative orientation of the detector normals\nof 2 detectors relative to the GRB direction) and the total\nintensity, this is a matrix rather than a factor.\nIt is straightforward to realize that\nthe cosine off-axis dependence of the detector sensitivity\nis a similar geometrical effect as different detector geometries.\nThus, instead of computing effective area matrices per angle pair,\nwe can incorporate the detector geometry (in terms of total\neffective area per direction) and compute the error of a\ndelay measurement per angle pair.\nSuch an ``accuracy matrix'' has been computed via\nboth methods\n  (Fig. \\ref{delay3ms}), and then  serves as input to the \n  Galileo satellite mapping simulation.\n  \n\\subsubsection{Results with different detectors\n  \\label{sect:covloc} }\n\n\\begin{figure*}[h]\n  \\includegraphics[width=0.49\\textwidth, viewport=0 90 825 515, clip]{24det_Detector01_11111111_11111111_11111111_SkyCov_first.pdf}\n  \\includegraphics[width=0.49\\textwidth, viewport=0 90 825 515, clip]{24det_Detector01_11111111_11111111_11111111_SkyCov_mean.pdf}\n  \\includegraphics[width=0.49\\textwidth, viewport=0 90 825 515, clip]{12det_Detector01_10101010_10101010_10101010_SkyCov_first.pdf}\n  \\hfill\\includegraphics[width=0.49\\textwidth, viewport=0 90 825 515, clip]{12det_Detector01_10101010_10101010_10101010_SkyCov_mean.pdf}\n  \\caption[]{Sky coverage for a zenith-looking detector each on 24 (top) and\n    12 satellites (bottom; every second along each orbital plane)\n    for an instantaneous moment (left) and averaged over one orbit (right).\n    The color-coding (note the different scales) gives the number of satellites\n    which see a GRB depending on where the GRB happens on the sky.\n    \\label{12sat_Det1_skycov}}\n\\end{figure*}\n\n\\paragraph{Detector 1: zenith-looking}\n\nWe start with a single detector plate, looking at zenith,\nwith every as well as every second\nof the 24 Galileo satellites equipped with one such detector.\nWe will use this constellation to show the different aspects of the\nsimulated data -- for the other detector geometries we will primarily\nshow example distributions and summarize the results in a table.\n\n\\begin{figure*}[h]\n  \\includegraphics[width=0.5\\textwidth, viewport=0 90 825 515, clip]{24det_Detector01_11111111_11111111_11111111_Bin4_first_sqrt2.pdf}\n  \\includegraphics[width=0.5\\textwidth, viewport=0 90 825 515, clip]{24det_Detector01_11111111_11111111_11111111_Bin4_mean_sqrt2.pdf}\n  \\includegraphics[width=0.5\\textwidth, viewport=0 90 825 515, clip]{24det_Detector01_11111111_11111111_11111111_Bin1_first_sqrt2.pdf}\n  \\includegraphics[width=0.5\\textwidth, viewport=0 90 825 515, clip]{24det_Detector01_11111111_11111111_11111111_Bin1_mean_sqrt2.pdf}\n  \\caption[]{Localisation accuracy for a zenith-looking detector each on\n    24 satellites \n    for an instantaneous moment (left) and averaged over one orbit (right),\n    for GRBs in the brightest peak flux bin of 6--100 ph\/cm$^2$\/s (top row)\n    and the faintest peak flux bin of 1.5--2 ph\/cm$^2$\/s (bottom row).\n    \\label{24sat_Det1}}\n\\end{figure*}\n\nFig. \\ref{12sat_Det1_skycov} shows the sky coverage for an instantaneous\nmoment (left) and averaged over the orbit (right) for the 24- and 12-satellite\noptions, and Figs. \\ref{24sat_Det1}, \\ref{12sat_Det1}\nshow the localisation accuracy for two different GRB flux intervals.\n\n\\begin{figure*}[th]\n   \\includegraphics[width=0.49\\textwidth, viewport=0 90 825 515, clip]{12det_Detector01_10101010_10101010_10101010_Bin4_first_sqrt2.pdf}\n   \\includegraphics[width=0.49\\textwidth, viewport=0 90 825 515, clip]{12det_Detector01_10101010_10101010_10101010_Bin4_mean_sqrt2.pdf}\n   \\includegraphics[width=0.49\\textwidth, viewport=0 90 825 515, clip]{12det_Detector01_10101010_10101010_10101010_Bin1_first_sqrt2.pdf}\n   \\hfill\\includegraphics[width=0.49\\textwidth, viewport=0 90 825 515, clip]{12det_Detector01_10101010_10101010_10101010_Bin1_mean_sqrt2.pdf}\n  \\caption[]{Same as Fig. \\ref{24sat_Det1} but for 12 satellites (every second along each orbital plane).\n    \\label{12sat_Det1}}\n\\end{figure*}\n\n\\begin{figure*}[h]\n  \\centering\n  \\hspace{-0.1cm}\n  \\includegraphics[width=0.28\\textwidth,height=0.265\\textwidth]{pointing.pdf}\\hspace{0.1cm}\n  \\includegraphics[width=0.34\\textwidth, viewport=0 0 390 310, clip]{zenzen-p2c_f2.pdf}\n  \\includegraphics[width=0.34\\textwidth, viewport=0 0 390 310, clip]{zenzen-p2c_f1.pdf}\n   \\caption[]{{\\bf Left:} Relation between distance between Galileo satellites\n     and zenith pointing difference.\n    {\\bf Middle:} Frequency of occurrence of satellite distances, averaged\n    over one orbital cycle for an isotropic GRB distribution, if all\n    Galileo satellites are equipped with a GRB detector.\n    {\\bf Right:} Same, but for every second Galileo satellite\n    equipped with a GRB detector.\n    \\label{1D-statistics}}\n\\end{figure*}\n\nFor the simple 1D detector plate facing to zenith,\nthe geometry of the satellite kinematics leads to a\nrelation between the distance of Galileo satellites and the difference\nof their zenith pointing direction, as shown in the left panel of\nFig. \\ref{1D-statistics}: \nThe larger the angle, the lower the area in\nthe sky that the GRB detectors on the two satellites jointly observe.\nMore importantly, since the sensitivity of triangulation is best\nfor GRBs occurring perpendicular to the connecting line of two\nsatellites, zenith-looking flat detectors\nwill not make use of the maximum baseline of the Galileo satellite system,\nbut use at most 2\/3 of it ($<$1.3 orbital radii).\nOur simulations over a full orbital cycle now return the frequency\nof occurrence of these distances between pairs of Galileo satellites\nfor an isotropic distribution of GRBs. This shows that for the maximum\nGRB-detector equipment rate on the GNSS, i.e.\na GRB detector on each of the 24 Galileo satellites, about half of the\ndetector pairs occur at satellite separations of $<$1.2 orbital radii\n(middle panel of Fig. \\ref{1D-statistics}). When reducing\nthe satellite equipment rate, this rate gets even worse (right panel of\nFig. \\ref{1D-statistics}). Thus, a single zenith-facing detector per\nsatellite is far from optimal.\n\n\n\\paragraph{Overview of all detector geometries}\n\nBefore elucidating the details of the other detector configurations,\nwe start with comparing the nine different GRB detector geometries\nby using the maximum equipment rate, i.e. all 24 Galileo satellites\ncarry a GRB detector, in Fig. \\ref{24sat_allDet}: \nthe sky coverage for any given moment (left column),\nthe average of the sky coverage over one Galileo orbital period (middle),\nand the mean localisation accuracy of the faintest GRB intensity bin (which\nis the one where our goal is to obtain a sub-degree localisation).\n\nOne interesting pattern (Fig. \\ref{24sat_allDet})\nare the green filled circles on brown sky background in the left\ncolumn for detectors \\#03, \\#05, \\#06, and \\#09. These detectors \nall cover the whole sky (ignoring the cosine\ndependence of the effective area), see right column of Fig. \\ref{3Deffarea}.\nThe green circles reduce the coverage by one, namely due to the\nshadowing of the Earth in nadir-direction, with a 12\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi\\ radius.\nDue to the three orbital planes being perpendicular to each other,\nthere are six positions in the sky where two satellites from two\norbital planes get close to each other, and their 12\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi\\ radii overlap\nto form a small region where the coverage is reduced by two satellites\n(blue regions).\n\nAnother aspect is symmetry:\nWhile a single detector looking towards zenith on all 24 satellites\nproduces a homogeneous sky coverage, this is not true anymore\nfor a single sidewards looking detector (\\#07, \\#08) or an\nasymmetrical detector (\\#04): given the Sun-pointing constraints\nof the Galileo system, their sky coverage is very asymmetrical.\n\nIn the following sub-sections, we describe most of these configurations\nin more detail.\n\n\\paragraph{Detector 2}\n\nAs a consequence of the average short baselines  for a flat, zenith-facing\ndetector (Fig. \\ref{1D-statistics}), we next test a cube detector on the\nzenith-facing side of the\nGalileo satellites: these have the same area as Detector 01 towards zenith,\nand half of this (due to the height of only 30\\,cm) towards all four sides.\nWith more satellites at large baselines and large effective area available\nfor large parts of the sky, this substantially improves the localisation accuracy\nof the zenith-looking flat detector (see Figs. \\ref{24sat_Det2}, \\ref{12sat_Det2}).\n\n\\begin{figure*}[!th]\n  \\centering\n  \\vspace{-0.1cm}\n  \\includegraphics[width=0.234\\textwidth, viewport=80 90 780 515, clip]{24det_Detector01_11111111_11111111_11111111_SkyCov_first.pdf}\n  \\includegraphics[width=0.234\\textwidth, viewport=80 90 780 515, clip]{24det_Detector01_11111111_11111111_11111111_SkyCov_mean.pdf}\n  \\includegraphics[width=0.234\\textwidth, viewport=80 90 780 515, clip]{24det_Detector01_11111111_11111111_11111111_Bin1_mean_sqrt2.pdf} \\\\\n  \\includegraphics[width=0.234\\textwidth, viewport=80 90 780 515, clip]{24det_Detector02_11111111_11111111_11111111_SkyCov_first.pdf}\n  \\includegraphics[width=0.234\\textwidth, viewport=80 90 780 515, clip]{24det_Detector02_11111111_11111111_11111111_SkyCov_mean.pdf}\n  \\includegraphics[width=0.234\\textwidth, viewport=80 90 780 515, clip]{24det_Detector02_11111111_11111111_11111111_Bin1_mean_sqrt2.pdf} \\\\\n  \\includegraphics[width=0.234\\textwidth, viewport=80 90 780 515, clip]{24det_Detector03_11111111_11111111_11111111_SkyCov_first.pdf}\n  \\includegraphics[width=0.234\\textwidth, viewport=80 90 780 515, clip]{24det_Detector03_11111111_11111111_11111111_SkyCov_mean.pdf}\n  \\includegraphics[width=0.234\\textwidth, viewport=80 90 780 515, clip]{24det_Detector03_11111111_11111111_11111111_Bin1_mean_sqrt2.pdf} \\\\\n  \\includegraphics[width=0.234\\textwidth, viewport=80 90 780 515, clip]{24det_Detector04_11111111_11111111_11111111_SkyCov_first.pdf}\n  \\includegraphics[width=0.234\\textwidth, viewport=80 90 780 515, clip]{24det_Detector04_11111111_11111111_11111111_SkyCov_mean.pdf}\n  \\includegraphics[width=0.234\\textwidth, viewport=80 90 780 515, clip]{24det_Detector04_11111111_11111111_11111111_Bin1_mean_sqrt2.pdf} \\\\\n  \\includegraphics[width=0.234\\textwidth, viewport=80 90 780 515, clip]{24det_Detector05_11111111_11111111_11111111_SkyCov_first.pdf}\n  \\includegraphics[width=0.234\\textwidth, viewport=80 90 780 515, clip]{24det_Detector05_11111111_11111111_11111111_SkyCov_mean.pdf}\n  \\includegraphics[width=0.234\\textwidth, viewport=80 90 780 515, clip]{24det_Detector05_11111111_11111111_11111111_Bin1_mean_sqrt2.pdf} \\\\\n  \\includegraphics[width=0.234\\textwidth, viewport=80 90 780 515, clip]{24det_Detector06_11111111_11111111_11111111_SkyCov_first.pdf}\n  \\includegraphics[width=0.234\\textwidth, viewport=80 90 780 515, clip]{24det_Detector06_11111111_11111111_11111111_SkyCov_mean.pdf}\n  \\includegraphics[width=0.234\\textwidth, viewport=80 90 780 515, clip]{24det_Detector06_11111111_11111111_11111111_Bin1_mean_sqrt2.pdf} \\\\\n  \\includegraphics[width=0.234\\textwidth, viewport=80 90 780 515, clip]{24det_Detector07_11111111_11111111_11111111_SkyCov_first.pdf}\n  \\includegraphics[width=0.234\\textwidth, viewport=80 90 780 515, clip]{24det_Detector07_11111111_11111111_11111111_SkyCov_mean.pdf}\n  \\includegraphics[width=0.234\\textwidth, viewport=80 90 780 515, clip]{24det_Detector07_11111111_11111111_11111111_Bin1_mean_sqrt2.pdf} \\\\\n  \\includegraphics[width=0.234\\textwidth, viewport=80 90 780 515, clip]{24det_Detector08_11111111_11111111_11111111_SkyCov_first.pdf}\n  \\includegraphics[width=0.234\\textwidth, viewport=80 90 780 515, clip]{24det_Detector08_11111111_11111111_11111111_SkyCov_mean.pdf}\n  \\includegraphics[width=0.234\\textwidth, viewport=80 90 780 515, clip]{24det_Detector08_11111111_11111111_11111111_Bin1_mean_sqrt2.pdf} \\\\\n  \\includegraphics[width=0.234\\textwidth, viewport=80 90 780 515, clip]{24det_Detector09_11111111_11111111_11111111_SkyCov_first.pdf}\n  \\includegraphics[width=0.234\\textwidth, viewport=80 90 780 515, clip]{24det_Detector09_11111111_11111111_11111111_SkyCov_mean.pdf}\n  \\includegraphics[width=0.234\\textwidth, viewport=80 90 780 515, clip]{24det_Detector09_11111111_11111111_11111111_Bin1_mean_sqrt2.pdf}\n  \\caption[]{Sky coverage for a single time slice (left), averaged over one\n    Galileo orbit (middle) and the averaged localisation accuracy \n     for the faintest\n    GRB intensity bin (right) for detector geometries 1--9\n    (from top to bottom) -- see also blue labels for each map.\n    \\label{24sat_allDet}}\n\\end{figure*}\n\n\n\\begin{figure*}[th]\n  \\centering\n  \\includegraphics[width=0.99\\columnwidth, viewport=0 90 825 515, clip]{24det_Detector02_11111111_11111111_11111111_Bin4_first_sqrt2.pdf}\n  \\includegraphics[width=0.99\\columnwidth, viewport=0 90 825 515, clip]{24det_Detector02_11111111_11111111_11111111_Bin4_mean_sqrt2.pdf}\n  \\includegraphics[width=0.99\\columnwidth, viewport=0 90 825 515, clip]{24det_Detector02_11111111_11111111_11111111_Bin1_first_sqrt2.pdf}\n  \\includegraphics[width=0.99\\columnwidth, viewport=0 90 825 515, clip]{24det_Detector02_11111111_11111111_11111111_Bin1_mean_sqrt2.pdf}\n  \\caption[]{Localisation accuracy for a zenith-looking cube detector each on\n    24 satellites \n    for an instantaneous moment (left) and averaged over one orbit (right),\n    for GRBs in the brightest peak flux bin of 6--100 ph\/cm$^2$\/s (top row)\n    and the faintest peak flux bin of 1.5--2 ph\/cm$^2$\/s (bottom row).\n    \\label{24sat_Det2}}\n\\end{figure*}\n\n\\begin{figure*}[h]\n   \\centering\n   \\includegraphics[width=0.99\\columnwidth, viewport=0 90 825 515, clip]{12det_Detector02_10101010_10101010_10101010_Bin4_first_sqrt2.pdf}\n   \\includegraphics[width=0.99\\columnwidth, viewport=0 90 825 515, clip]{12det_Detector02_10101010_10101010_10101010_Bin4_mean_sqrt2.pdf}\n   \\includegraphics[width=0.99\\columnwidth, viewport=0 90 825 515, clip]{12det_Detector02_10101010_10101010_10101010_Bin1_first_sqrt2.pdf}\n   \\includegraphics[width=0.99\\columnwidth, viewport=0 90 825 515, clip]{12det_Detector02_10101010_10101010_10101010_Bin1_mean_sqrt2.pdf}\n  \\caption[]{Same as Fig. \\ref{24sat_Det2} but for 12 satellites (every second along each orbital plane).\n    \\label{12sat_Det2}}\n\\end{figure*}\n\n\nFig. \\ref{aeff_Det0102} illustrates why the cube detector is so much better\nin performance: \nthe distribution of the mean baselines which are realized for given\npairs of detectors and their projected effective areas as\ndetermined by their viewing direction relative to a GRB shows that\nthe long baselines for higher effective areas dominate clearly.\nThus, we reach sub-degree localisation accuracy for the brightest\nGRB intensity bin (though for the faintest it is still of order 10\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi).\n\n\\paragraph{Detector 3}\n\n\\begin{figure*}[th]\n  \\centering\n \\includegraphics[width=0.65\\columnwidth]{Aeff_dist_24_detector01.pdf}\n \\includegraphics[width=0.65\\columnwidth]{Aeff_dist_24_detector02.pdf}\n \\includegraphics[width=0.65\\columnwidth]{Aeff_dist_24_detector03.pdf}\n   \\caption[]{Distribution of mean baseline lengths for combinations\n     of effective areas for pairs of detectors seeing\n     a GRB for the geometries of detector \\#01 (left) and \\#02 (middle),\n     and \\#03 (right).\n    \\label{aeff_Det0102}}\n\\end{figure*}\n\n\\begin{figure*}[h]\n  \\includegraphics[width=0.332\\textwidth, viewport=80 90 780 515, clip]{06det_Detector03_10001000_00100010_10001000_SkyCov_mean.pdf}\n  \\includegraphics[width=0.332\\textwidth, viewport=80 90 780 515, clip]{06det_Detector03_10001000_10001000_10001000_SkyCov_mean.pdf}\n   \\includegraphics[width=0.332\\textwidth, viewport=80 90 780 515, clip]{06det_Detector03_10010000_00010010_10010000_SkyCov_mean.pdf}\n\\includegraphics[width=0.332\\textwidth, viewport=80 90 780 515, clip]{09det_Detector03_10010010_01001001_10100100_SkyCov_mean.pdf}\n  \\includegraphics[width=0.332\\textwidth, viewport=80 90 780 515, clip]{09det_Detector03_10010010_10010010_10010010_SkyCov_mean.pdf}\n  \\includegraphics[width=0.332\\textwidth, viewport=80 90 780 515, clip]{12det_Detector03_10101010_10101010_10101010_SkyCov_mean.pdf}\n  \\caption[]{Sky coverage for 4-side detectors on 6 (top), 9 (left and middle of\n   bottom row) and 12 (lower right) satellites  averaged over one orbit (right).\n   Note the different distribution of satellites along each orbital plane,\n   with 3 different options for the 6 satellites, and two options for 9\n   satellites. The color-code provides the number of satellites which see\n   a given GRB at a given time, averaged over one orbital period.\n   \\label{allsat_Det3_skycov}}\n\\end{figure*}\n\n\\begin{figure*}[h]\n  \\includegraphics[width=0.33\\textwidth, viewport=80 90 780 515, clip]{06det_Detector03_10001000_10001000_10001000_Bin1_mean_sqrt2.pdf}\n  \\includegraphics[width=0.33\\textwidth, viewport=80 90 780 515, clip]{06det_Detector03_10010000_00010010_10010000_Bin1_mean_sqrt2.pdf}\\\\\n  \\includegraphics[width=0.33\\textwidth, viewport=80 90 780 515, clip]{09det_Detector03_10010010_01001001_10100100_Bin1_mean_sqrt2.pdf}\n  \\includegraphics[width=0.33\\textwidth, viewport=80 90 780 515, clip]{09det_Detector03_10010010_10010010_10010010_Bin1_mean_sqrt2.pdf}\n  \\includegraphics[width=0.33\\textwidth, viewport=80 90 780 515, clip]{12det_Detector03_10101010_10101010_10101010_Bin1_mean_sqrt2.pdf}\n  \\caption[]{Localisation accuracy for a 4-side detector each on\n    6 (top row), 9 (lower left and middle) and 12 satellites (lower right)\n    for the \n    faintest intensity interval. The 6- and 9-satellite options are shown for\n    two different configurations along the orbital plane.\n    \\label{comp_NumSat}}\n  \\vspace{-0.2cm}\n\\end{figure*}\n\nWith the zenith-looking detectors not being optimal, we now look at\nan arrangement where all 4 sides of a Galileo satellite are equipped\nwith a 60\\,cm x 60\\,cm detector, with the nadir- and zenith-looking\nsides without GRB detector. The localisation accuracy is very good,\nsee Fig. \\ref{comp_NumSat}, even for this faintest GRB intensity level.\nThe averaged\nsky coverage shows an identical sky pattern, independent of whether\nwe equip 6, 9 or 12 satellites with a GRB detector\n(see Fig. \\ref{allsat_Det3_skycov}). \nBut note the different color-code normalization: obviously, when\nmore satellites are equipped with a detector, then there is a larger number\nof satellites seeing a given GRB.\n\n\n\\begin{figure*}[h]\n  \\includegraphics[width=0.5\\textwidth, viewport=80 90 780 515, clip]{12det_Detector03_10101010_10101010_10101010_notmerged_Bin3_first_sqrt2.pdf}\n  \\includegraphics[width=0.5\\textwidth, viewport=80 90 780 515, clip]{12det_Detector03_10101010_10101010_10101010_merged_Bin3_first_sqrt2.pdf}\n  \\caption[]{Localisation accuracy for a 4-side detector each on\n    12 satellites for the first time slice, and for the second highest\n    intensity interval\n    for unbinned (left) and binned (right) ``accuracy matrix'', i.e.\n    when the 6\\,ms binned matrix performs better than the 3\\,ms matrix.\n    This happens for a certain intensity range where the increased S\/N-ratio\n    overcompensates the reduced temporal resolution of the detector.\n    For yet fainter intensity intervals, the same happens for 9\\,ms, and so on.\n    This holds for any number of satellites equipped with a GRB detector.\n    \\label{comp_merged}}\n\\end{figure*}\n\n\n\\begin{figure*}[h]\n   \\includegraphics[width=0.5\\textwidth, viewport=80 90 780 515, clip]{12det_Detector04_10101010_10101010_10101010_SkyCov_mean.pdf}\n   \\includegraphics[width=0.5\\textwidth, viewport=80 90 780 515, clip]{12det_Detector04_10101010_10101010_10101010_Bin3_mean_sqrt2.pdf}\n  \\caption[]{Sky coverage (left) and localisation accuracy (right; for the\n    second-brightest GRB intensity interval) of\n    12 satellites equipped with two detector plates on neighboring sides.\n    The sky coverage  is substantially\n    worse than any previous detector geometry.\n    \\label{Det04}}\n\\end{figure*}\n\nIn Fig. \\ref{comp_merged} we show the effect of the so-called\n\"merged\" configuration, i.e. the temporal re-binning to \n6\\,ms whenever the 3\\,ms sampling combined with the small baselines\nperforms worse. This is best shown with a single time slice, not\nthe orbit-averaged accuracy plot. The re-binning improves the\nbad localisation accuracy regions (red in the left panel of\nFig. \\ref{comp_merged}) by about 20\\% (from 1\\fdg1 to about 0\\fdg9).\n\n\\paragraph{Detector 4}\n\nWith the intention to minimize the number of detector plates\non a given Galileo satellite, we included this geometry with only\ntwo instead of 4 sides equipped with a 60\\,cm x 60\\,cm detector plate.\nThe simulations show that the sky coverage is substantially\nworse (Fig. \\ref{Det04}), which is a consequence of the ``eyes'' problem,\ni.e. that the detector on the y-side (together with a solar\npanel boom) will never look towards the Sun.\nIn addition, as a consequence of the yaw-steering attitude, the detector\nmounted on the +X surface always looks into the hemisphere containing the Sun,\ni.e., the direction towards the anti-Sun is not covered by any detector\non any satellite. The sky coverage and the localization precision thus\ndramatically degrade towards this direction.\n\n\\begin{figure*}[h]\n  \\includegraphics[width=0.5\\textwidth, viewport=80 90 780 515, clip]{12det_Detector05_10101010_10101010_10101010_Bin3_mean_sqrt2.pdf}\n   \\includegraphics[width=0.5\\textwidth, viewport=80 90 780 515, clip]{09det_Detector05_10010010_10010010_10010010_Bin3_mean_sqrt2.pdf}\n   \\caption[]{Localisation accuracy of 12 (left) and 9 (right)\n    satellites equipped with 4 lateral and a zenith-looking detector, \n    for the second brightest GRB intensity interval).\n    \\label{Det05}}\n\\end{figure*}\n\n\\begin{figure*}[h]\n   \\includegraphics[width=0.49\\textwidth, viewport=80 90 780 515, clip]{12det_Detector06_10101010_01010101_10101010_SkyCov_mean.pdf}\n   \\includegraphics[width=0.49\\textwidth, viewport=80 90 780 515, clip]{12det_Detector06_10101010_01010101_10101010_Bin3_mean_sqrt2.pdf}\n  \\caption[]{Sky coverage (left) and localisation accuracy (right; for the\n    second brightest GRB intensity interval) of 12 satellites\n    equipped with two opposite plus one zenith-looking detectors.\n    \\label{Det06}}\n\\end{figure*}\n\n\\paragraph{Detector 5}\n\nThis is a kind of 'maximum detector' concept per satellite,\nand unsurprisingly, the performance is very good (Fig. \\ref{Det05}).\nHowever, we see (Tab. \\ref{tab:Det_Sat_acc}) \nthat it performs slightly worse than the 4-lateral-only\ndetector geometry for fainter GRB intensity levels.\nThis is likely due to the fact that using detectors at large\ninclination angles towards the GRB does not help in improving\nthe S\/N-ratio, since co-adding the background noise of the\nsecond (or third) plate dominates over the gain in signal.\nFig. \\ref{comp_multi} shows the effect for a single plate,\nand the sum of two and three perpendicular-oriented plates:\nat large inclination angles, i.e. small effective area\ndue to the cosine effect, the S\/N after combining detectors does not improve.\nThis calls for an optimization of the co-adding of signals from\nmultiple detector plates: it should not be performed on the satellite,\nbut on the ground, as it depends on the actual noise level\nfor each satellite (which we expect to vary along the orbit).\nThen, cut-off angles can be applied\nabove which no co-addition is done.\nAlso for this detector we find 'eyes' in the spatial distribution of\nlocalization precision caused by the fact that maximum 2 instead of 3\nof the detector plates for any satellite can cover the Sun and anti-Sun\ndirections.\n\n\\paragraph{Detector 6}\n\nThis three-element option, which leaves the sides with the solar panels free,\neliminates the bad localization performance in Sun and anti-Sun directions\n(``eyes'' above). However, the  Sun-equator is less well covered\n(Fig. \\ref{Det06}).\nOtherwise, it provides a very uniform localisation capability over the sky,\nat substantially improved accuracy as compared to the case of\ntwo neighboring detectors.\n\n\n\n\\paragraph{Detectors 7--9}\n\nDetectors \\#07, \\#08 and \\#09 were included just for completeness\nand verification purposes, and the results are given in the overview\nplot of Fig. \\ref{24sat_allDet}.\n\n\\paragraph{Summary of detector geometries}\n\nTab. \\ref{tab:Det_Sat_acc} summarizes the different detector\ngeometries and satellite constellations considered, providing the\nall-sky averaged accuracy for each of the 4 GRB intensity intervals\nof Tab. \\ref{tab:12Det_det1_acc}.\n\nThus, and quite obviously, the localisation accuracy\nimproves with the number of satellites equipped, since among\nthe detectors seeing a GRB, there is a larger likelihood of\nhaving satellite pairs with a large distance (baseline):\nonly those are the ones\nwhich improve the localisation accuracy.\n\nThe placement of detectors on satellites positioned opposite to each other\nin the orbital plane causes moving patterns in the sky with reduced\nlocalization accuracy, and thus should be avoided; this applies primarily for\nlow equipment rates, e.g. the 6- and 9- satellite versions discussed above. \n\nAn interesting feature is seen in the case with \n6 satellites: when the GRB detectors are distributed isotropically,\ni.e. 2 per orbit at antipodal positions, there is a pattern on the\nsky at which the localisation is substantially worse (top left panel\nin Fig. \\ref{comp_NumSat}). This can be avoided by placing detectors\nnot in antipodal positions (next panel to the right in Fig. \\ref{comp_NumSat}).\n\nOne special effect to comment on are the two ``eyes'' in\nFig. \\ref{comp_NumSat} (lower row).\nThese are due to the position of the Sun in the simulation\n($\\alpha$ = 90\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi, $\\delta$ = 23\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi) and the anti-Sun direction,\nand in practice would move over the sky over the course of a year.\nThese are caused\nby the yaw-steering motion of the satellite guaranteeing pointing of the\nnavigation antenna continuously to the Earth and the solar panels to the Sun.\nAs a consequence of this attitude mode the +Y and-Y surfaces of the satellite\nwhere the solar panel booms are mounted never look towards the Sun.\nThe Sun and anti-Sun directions are thus covered only by one detector plate\nper satellite with varying orientation towards the Sun. The result is a\nreduced localization precision in these directions.\nFor the GRB\/GW application this is acceptable,\nsince optical follow-up of the GRB or neutron star merger close to the Sun\nis anyway not possible from the ground.\n\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.9\\columnwidth]{matrix_diagonal.pdf}\n  \\caption[]{Accuracy of different detector configurations: As one\n    adds perpendicular-oriented detector plates, and since the\n    background radiation is isotropic and does not scale with\n    the cosine of the incidence angle, the S\/N-ratio depends on\n    the relative inclination angles of the plates: it is best\n    for a single plate (green), and gets worse for two plates\n    inclined by 90 degrees (blue), and even worse for three\n    plates inclined by 90 degrees (red; corresponding to the\n    5-detector case for any given GRB).\n        \\label{comp_multi}}\n\\end{figure}\n\n\nFor obtaining the localisation accuracy in Tab. \\ref{tab:Det_Sat_acc},\nwe averaged over the full sky.\nBut looking at the various figures, it becomes obvious that\nthere are certain small (few to 10\\% of the sky) regions on the sky\nwhich are worse than the majority of the sky.\nWe therefore provide a more accurate accounting of the\nlocalisation accuracy for our best options in Tab. \\ref{tab:acc_final}.\nThis provides the worst accuracy for the best 50\\% and 90\\% of the sky\n(i.e. the accuracy is better than the specified value for that percentage of the sky),\nrespectively, as well as the best and the worst single GRB accuracy of the sky. \nFor the selected best detector and satellite configurations,\nwe also provide a graphical representation in Fig. \\ref{acc_persky}\nwhich allows to get the accuracy for any fraction of sky coverage.\n\n\\subsection{Bayesian scheme using nazgul}\n\nDue to the massive compute-time requirements, a simulation with\nnazgul was only done for one particular satellite constellation\n(9 satellites, 3 in each orbital plane, equally distributed)\nwith one detector (\\#03, looking towards 4 sides).\nWe use the same set-up as the one to reconstruct the time\nwith the cross-correlation algorithm.\nInstead of 1000 different GRB light curves, we use only one light curve shape,\nwith 5 different flux normalization.\nAlso, the triangulation was only computed at 134\nsky positions, instead of 10000.\nFrom each fit we obtain a distribution of the time delay which is used\nto compute both the ``best'' fit value (the median in this case)\nand the 68\\% probability uncertainties through the highest posterior\ndensity interval, i.e. the shortest possible interval, necessary\nto accumulate the chosen probability level.  \nWhile the source position distribution reconstructed by nazgul is\nnot, in general, an annulus, for the sake of a straightforward comparison\nwith the classical correlation method we compute\nan ``equivalent'' annulus from the fitted time delay. \nThe central ring of the annulus is computed from the median of\nthe time delay distribution, while the width is given by the uncertainties\nin an analogous way as what is done for the correlation method\n\\citep[see, e.g.,][]{Palshin+2013}.\nThis methodology, although to some extent simplistic, allows us to\ncompare the characteristic widths of the positional distributions\nfitted by nazgul and the correlation algorithm.\nThe corresponding 'map' is shown in Fig. \\ref{nazgulcomp}\nfor the faintest intensity interval, together\nwith the corresponding map from the cross-correlation method.\nThis shows, that the two methods are nicely compatible to each other.\n\n\\begin{figure}[h]\n  \\vspace{-0.22cm}\n  \\hspace{-0.05cm}\\includegraphics[width=0.52\\textwidth]{Accuracy_PercSky_A.pdf}\n  \\hspace{-0.18cm}\\includegraphics[width=0.52\\textwidth]{Accuracy_PercSky_B.pdf}\n  \\vspace{-0.22cm}\n  \\caption[]{Accuracy of different detector and satellite configurations\n    per sky fraction. Each curve shows the percentage of the sky for which the accuracy\n    is better than the corresponding  y-axis in degrees.\n    {\\bf Top:} for the faintest intensity bin for 24 satellite equipment rate;\n    {\\bf Bottom: } each color represents one configuration of\n    Tab. \\ref{tab:acc_final} for detector \\#3, with solid lines for the\n    brightest GRB interval, and dotted lines for the faintest. \n        \\label{acc_persky}}\n\\end{figure}\n\n\\begin{figure*}[h]\n  \\includegraphics[width=0.49\\textwidth, viewport=90 90 750 520, clip]{09det_Detector03_10010010_01001001_10100100_Bin1_first_sqrt2.pdf}\n  \\includegraphics[width=0.55\\textwidth]{annulus_half_width_nazgul_sim_example3.pdf}\n  \\caption[]{Localisation accuracy for detector \\#03 (4 sides) and 9 satellites\n    for the first of 72 snapshot per orbital phase and the faintest\n    GRB intensity bin, computed with\n    cross-correlation (left) and nazgul (right). Note that the\n    Sun is not included in the nazgul\n    simulation, and thus the ``eyes'' are missing.\n    \\label{nazgulcomp}}\n\\end{figure*}\n\n\\begin{figure}[h]\n\\includegraphics[width=0.5\\textwidth]{annulus_width_comparison_hist_2.pdf}\n\\caption[]{Distribution of the absolute value of the difference between\n    simulated and reconstructed time delays for nazgul (blue) and the\n    cross-correlation method (green), again for\n    detector \\#03 (4 sides) and 9 satellites.\n   \\label{nazgulcchist}}\n\\end{figure}\n  \n\nA more quantitative comparison of the localisation accuracy is\ngiven in Fig. \\ref{nazgulcchist}, showing the histogram\nof the 1$\\sigma$ localization errors of nazgul vs. the cross-correlation\nmethod. This shows,\nthat the nazgul distribution is a factor $\\sim$2 narrower\n(FWHM of about 4\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi\\ vs 8\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi),\nand has much less GRB reconstructions in the long tail.\nThus, the nazgul method leads to overall improvements, but is particularly\nsuperior at the faint end of the intensity distribution.\n\n\\subsection{Comparison to previous simulations}\n\nRecently, \\cite{Hurley2020} has combined a new localisation method\nwith the simulation of a near-Earth network of GRB detectors.\nThe basic concept of this method is similar to ours, namely avoiding\ncross-correlation and instead testing positions on the sky via a likelihood\nmethod.  While this method is a substantial\nimprovement over the classical cross-correlation, it still suffers\nfrom the above draw-backs (iii) and (iv) in sect. \\ref{triangulation} which\nis accounted for in our nazgul scheme \\citep{Burgess+2021}.\nIn his simulations, he uses individual detectors of 100 cm$^2$ effective\narea on a fleet of nine satellites, and derives localisation accuracies\nfor three different GRB peak intensities. His faintest and middle intensity\nintervals fall in our brightest interval. In terms of sky coverage,\n\\cite{Hurley2020} reaches only 40\\%, so a single-plate detector on each of\n9 satellites is by far too little to reach all-time, all-sky coverage.\nWhile we have not simulated such a constellation, this result is consistent\nwith our picture, i.e. the need to look to multiple sides at this small\nsatellite number. The 1$\\sigma$ average localisation for his faintest\nfluence GRB (16 ph\/cm$^2$\/s) is an ellipse with a\ndimension of 4\\fdg5$\\times$17\\fdg0, corresponding to an effective radius\n(same-area circle) of 4\\fdg5 (1$\\sigma$). Our closest constellation\nis a 1-side (zenith)\nlooking detector on 12 satellites, where our simulation for our brightest\nintensity interval (6--100 ph\/cm$^2$\/s) gives 2\\fdg9.\nThe difference between these two simulations is in effective area\n(100 vs. 3600 cm$^2$), orbit radius (7000 vs. 29.000 km), and\ntiming accuracy (0.1 vs. 3 ms). Assuming the typical square-root\ndependence on effective area, the combination of these three factors\nsuggests that our error should be $1 \/ \\sqrt{36} \/ 29 * 7 * 30 =  1.2\\times$\nthat of the \\cite{Hurley2020} simulation, pretty close to what we obtain\n(given that our intensity bin is very wide).\n\n\n\n\\subsection{Potential for improvements}\n\nGiven the comparison in the previous subsection, one could ask\nwhether or not reducing the time resolution in our simulations (fixed at 3\\,ms)\nwould substantially improve our localisation accuracy? The answer is:\nin theory yes, in practice likely not. The simple reason is that\nthe variability time scale in GRBs is at the level of a few milliseconds\n\\citep{MacLachlan+2013}, and not sub-milliseconds \\citep{Walker+2000}.\nThus, one needs to cross-correlate the rising edge of a pulse to better\nthen the rise time. This is complicated even further due to the fact that the\nslopes of the rise are energy dependent, i.e. detectors like the here preferred\nscintillator plates will see different slopes from the same GRB as soon as\nthe incidence angles on two detectors are not exactly the same. This is the\nreason why for instance co-adding of light curves of different GBM detectors\nis of no avail.\n\nFor configurations with more than one detector plate per satellite,\nwe have co-added multiple single-plate detectors per\nsatellite. As described earlier, this does not automatically\nprovide better S\/N-ratio, since the background radiation adds in full,\nnot diminished with the cosine law as the source counts. Therefore,\nsome optimization for adding two or three detector plates could\nbe implemented.\nThis affects mostly the faint end of the GRB intensity distribution,\nand the optimization is expected to improve the localization accuracy.\nOn a practical note, the data from different detector plates should thus\nnot be combined onboard, but send down to Earth separately.\nAs a side effect, the onboard triggering\nalgorithm can make use of the separate light curve measurements to\nfilter out particle hits, thus dramatically reducing false triggers.\n\nThere is yet another way to improve the accuracy distribution, computed\nvia the classical cross-correlation analysis, beyond what we presented: namely\nusing systematically a re-binning procedure (in an frequentist\napproach, and the corresponding $\\chi^2$ analysis). In practice,\nwhen moving from bright to fainter GRBs, there is a transition\nregion where re-binning from the original 3\\,ms time resolution\nof the detector towards, e.g., 6\\,ms provides a gain over the\nnoise fluctuations, and leads to an improved localisation.\nThis has been shown above with our 'merged' map for one case\n(Fig. \\ref{comp_merged}).\nBut moving further down in intensity, the same happens for\nfurther re-binning to 9\\,ms, or 12\\,ms, and so on.\nThe effect of this re-binning\nis to optimize between noise in the light curve vs. the best\naccuracy in the time delay measurement. \nObviously, it does not improve on the\nbest accuracy side, but improves the bad end by of order 15\\%-20\\%.\nThis, of course, does not apply to our forward-folding  nazgul\nlocalisation, since this is Bayesian, and the information in\n``low S\/N'' bins is properly accounted for.\n\n\\subsection{Inclusion of satellites beyond GNSS}\n\nThe inclusion of any satellite further out in space than the\nGalileo satellites would help reducing the localisation accuracy,\nas it shrinks linearly with the increase of the baseline.\nPotential options are a GRB detector on\n(i) the Gateway\\footnote{\\url{https:\/\/en.wikipedia.org\/wiki\/Lunar_Gateway}},\na multi-purpose space station in a highly elliptical (3000 km x 70.000 km)\nseven-day near-rectilinear halo orbit  around the Moon,\npresently planned to be assembled in the 2024--2028 timeframe, or\n(ii) the Moon LCS (Lunar navigation and communication system),\na network of 3--4 satellites that would provide communications\nand navigation services to support human and robotic exploration\non the Moon.\nA GRB detector in Lunar orbit would reduce the localisation error\nby a factor of $\\sim$6, if the GRB detector has the same size\nas discussed here for the GNSS.\nOf course, this improvement would only apply in one dimension of the\nerror box, for GRBs coming from a direction perpendicular to the Earth-Moon\nline.\n\n\n\n\\section{GNSS system requirements}\n\n\\subsection{Communication speed}\n\nThe GRB afterglow brightness fades by a\n  factor of three during the first 10 min. after the burst,\n  another factor of three during the next 50 min., and another\n  factor of three during the next 23 hrs. The kilonova emission\n  of short GRBs decays even faster. Moreover, clarifying\n  the presently hottest open astrophysics questions of merging\n  neutron stars such as distinguishing\n  the physical source of energy input (e.g. from the central remnant\n  or via radioactivity) or associated processes (e.g. internal\n  shock-reheating or heating of the outer ejecta by free neutrons)\n  requires ground-based optical\/near-infrared spectroscopy during\n  the first 12 hrs \\citep{Metzger2020}. Thus, rapid communication\n  at a timescale of minutes is required in order to support\n  the identification of the kilonova.\n\nGRBs occur at unpredictable time and sky position.\nFor the GRB position to be determined via triangulation,\nwe need the measured data of each of the $>$4 satellite detectors \non one computer. In order to be scientifically useful,\nthe data should be downlinked within of order a few minutes.\nThus, we require that at any time every\nGalileo satellite can send off its measured data, either\ndirectly or via another satellite to a ground-station.\nSince only\n6 TT\\&C stations around the world  are responsible for collecting\nand sending the telemetry data that was generated by the\nGalileo satellites,\nrelaying data between\ndifferent Galileo satellites to the one (or few) which\ndo have ground contact is a viable solution.\nThis should be done dynamically,\nwithout the need of commanding, i.e. each satellite (computer)\nshould know at any time its acting relay satellite.\n\nWe distinguish two data transmission rates:\n(1) full rate to be downlinked to Earth, within minutes: for a typical\nGRB, this implies sending 0.5--1 MB over a time period of a few minutes,\ne.g. 4--8 kB\/s over 2 minutes per satellite.\n(2) reduced rate for quick-look localisation.\nAs described above, this would reduce the data amount by a factor\nof 100--1000.\n\nThe inter-satellite data transmission rate is likely slower than\na satellite-ground contact rate. Assuming 4 Galileo satellites\nwithout ground-contact shall send data to one other satellite with \nground-contact leads to a required transfer rate of 4 MB\non a time-scale of minutes. In an ideal case, this can be done in parallel.\nIf not, the above 4--8 kB\/s are a minimum requirement.\n\nRapid up-link capability is not needed, since the GRB detectors should\nbe self-triggering.\n\n\\subsection{Ground segment}\n\nThe light curve data as measured by the multiple Galileo satellites\nshould be collected at one place on Earth, where the triangulation\n(and thus GRB localisation) can be computed.\nWe suggest that the final localisation is made publicly\navailable immediately -- the GRB community is using the GCN\n(Gamma-ray Burst Coordinate Network) for this since decades,\nwhich would guarantee distribution to every interested user\nin the world.\nThis would typically happen automatically, but oversight through\na, or several, (GRB) astronomer(s) is certainly not a bad idea.\nThis could be organized via forming a small group of interested\nscientists, similar to groups which collaborate in the follow-up\nobservations of GRBs at optical or radio observatories.\nIn parallel, also the raw data should be made publicly available\nat the shortest possible delay time, to allow other groups\nwith potential access to other long-baseline GRB data to\nuse those data. The high-energy mission archive at ESA would\nbe a logical place, but other satellite data centers\nin Europe might be alternative options.\n\n\\section{Conclusions}\n\nThe GNSS provides a close-to-perfect satellite system for\nthe localisation of gamma-ray bursts (GRBs) via triangulation.\nIt provides a very promising\ncompromise between satellite baselines (not too long to suffer\ndata transmission restrictions), number of satellites, and \nrequired size of GRB detectors to reach sub-degree localizations.\nIt is the combination of detector geometry (to how many of the\nsix directions of\nthe Galileo surfaces are the detectors facing) and the number\nof satellites to be equipped, which provides a scientifically\nuseful GRB triangulation network.\n\nSideways looking detectors are an extremely crucial ingredient.\nWe suggest to equip at least 12 satellites, four per orbital plane,\nwith a 4-side (excluding nadir and zenith) looking detector,\neach side with 3600 cm$^2$ and 1 cm thickness.\nThis will provide sub-degree localisation of GRBs, in particular\nfaint short-duration GRBs such as GRB 170817A,\nas expected from binary neutron star mergers\nto be routinely measured at a rate of dozens per year in the upcoming\nruns of the worldwide gravitational wave detectors.\nInstead, a flat, zenith-facing detector provides only 10-20\\ifmmode ^{\\circ}\\else$^{\\circ}$\\fi\\\nlocalizations. \nEquipping only 9 Galileo satellites with such a GRB detector leads\nto a 30\\% loss in localisation accuracy, while the 24-satellite solution\nimproves it by a factor 2.\n\nSuch a configuration should be feasible to implement given the moderate\nrequirements\nof mass ($<$20 kg) and power ($\\sim$20 W) of a single detector plate\n(i.e. $<$80 kg and $\\sim$80 W for the 4-side detector)\nas compared to the overall budget of a Galileo satellite,\nthough we note that this corresponds to about 10\\% of\nthe satellite mass.\nThe realization of such a large-format GRB detector\nplate is also technologically feasible: scintillators of the proposed type\nhave been flown since 40 years (TRL 9), and the Si detectors\nfor read-out have also seen their first space applications.\n\nEquipping second generation Galileo satellites with GRB detectors\nwould turn the navigation\nconstellation into an observatory supporting the research on\nfundamental astrophysical and cosmological problems.\n\n\n\\begin{acknowledgements}\n\nJMB acknowledges support from the Alexander von Humboldt foundation.\nWe are grateful to Dr Javier Ventura-Traveset, Dr Erik Kuulkers, Dr Luis \nMendes and Dr Francisco Amarillo for  their excellent scientific and \ntechnical support, as part of the European Space Agency supervision in the \nexecution of this research activity.\n\nThe work reported in this paper has been partly funded by the EU under a \ncontract of the European Space Agency in the frame of the EU Horizon 2020 \nFramework Programme for Research and Innovation in Satellite Navigation. \nThe view expressed herein can in no way be taken to reflect the official \nopinion of the European Union and\/or the European Space Agency. Neither \nthe European Union nor the European Space Agency shall be responsible for \nany use that may be made of the information it contains.\n\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:in}\n\nA \\emph{lattice point} is a point in ${\\mathbb Z}^D.$ Given any bounded set $S \\subseteq {\\mathbb R}^D,$ we let \\[\\mathrm{Lat}(S) := |S \\cap {\\mathbb Z}^D|\\] be the number of lattice points in $S.$\nGiven a polytope $P$ in ${\\mathbb R}^D,$ a natural enumerative problem is to compute $\\mathrm{Lat}(P)$. \nIn this paper, we will focus on \\emph{integral polytopes}, i.e., polytopes whose vertices are all lattice points, and generalized permutohedra -- a special family of polytopes.\n\n\\subsection{Motivation: Ehrhart positivity for generalized permutohedra}\nOne approach to study the question of computing $\\mathrm{Lat}(P)$ for an integral polytope $P$ is to consider a more general counting problem: For any nonnegative integer $t,$ let $tP := \\{ t \\mathbf{x} \\ | \\ \\mathbf{x} \\in P\\}$ be the \\emph{$t$-th dilation of $P$}, and then consider the function \n\\[\ni(P, t) := \\mathrm{Lat}(tP)\n\\]\nthat counts the number of lattice points in $tP.$\nIt is a classic result that $i(P,t)$ is a polynomial in $t$. More precisely:\n\\begin{thm}[Ehrhart \\cite{ehrhart}]\\label{thm:ehrhart0}\nThere exists a polynomial $f(x)$ such that $f(t) = i(P,t)$ for any $t \\in \\mathbb{Z}_{\\geq 0}$. \nMoreover, the degree of $f(x)$ is equal to the dimension of $P$.\n\\end{thm}\nWe call the function $i(P, t)$ the \\emph{Ehrhart polynomial} of $P.$\nA few coefficients of $i(P,t)$ are well understood: the leading coefficient is equal to the normalized volume of $P$, the second coefficient is one half of the sum of the normalized volumes of facets, and the constant term is always $1$. However, although formulas are derived for the other coefficients, they are quite complicated. One notices that the leading, second and last coefficients of the Ehrhart polynomial of any integral polytope are always positive; but it is not true for the rest of the coefficients for general polytopes. We say a polytope has \\emph{Ehrhart positivity} or is \\emph{Ehrhart positive} if it has positive Ehrhart coefficients.\n\\par\nThere are few families of polytopes known to be Ehrhart positive. Zonotopes, in particular the regular permutohedra, are Ehrhart positive \\cite[Theorem 2.2]{zonotopes}. Cyclic polytopes also have this property. Their Ehrhart coefficients are given by the volumes of certain projections of the original polytope \\cite{cyclic}. Stanley-Pitman polytopes are defined in \\cite{stanley-pitman} where a formula for the Ehrhart polynomial is given and from which Ehrhart positivity follows. Recently in \\cite{deloera} De Loera, Haws, and Koeppe study the case of matroid base polytopes and conjecture they are Ehrhart positive. Both Stanley-Pitman polytopes and matroid base polytopes fit into a bigger family: generalized permutohedra. \n\nIn \\cite{bible} Postnikov defines generalized permutohedra as polytopes obtained by moving the vertices of a usual permutohedron while keeping the same edge directions. That's what he calls a generalized permutohedron of type $z$. He also considers a strictly smaller family, type $y$, consisting of sums of dilated simplices. He describes the Ehrhart polynomial for the type $y$ family in \\cite[Theorem 11.3]{bible}, from which Ehrhart positivity follows. The type $y$ family includes the Stanley-Pitman polytopes, associahedra, cyclohedra, and more (see \\cite[Section 8]{bible}), but fails to contain matroid base polytopes, which are type $z$ generalized permutohedra \\cite[Proposition 2.4]{matroidpolytope}.\n\\par\nWe give the following conjecture:\n\\begin{conj}\\label{positivity}\nIntegral generalized permutohedra are Ehrhart positive.\n\\end{conj}\nNote that since generalized permutohedra contain the family of matroid base polytopes, our conjecture is a generalization of the conjecture on Ehrhart positivity of matroid base polytopes given in \\cite{deloera} by De Loera et al.\n\nInstead of studying the above conjecture directly, we will reduce it to another conjecture which only concerns regular permutohedra, a smaller family of polytopes. \n\n\\subsection{McMullen's formula and $\\alpha$-positivity}\nIn 1975 Danilov asked if it is possible to assign values $r_\\sigma$ to cones $\\sigma$ such that \n\\begin{equation}\\label{eqn:danilov0}\n\\operatorname{Td}(X(\\Delta)) = \\displaystyle \\sum_{\\sigma\\in\\Delta} r_\\sigma [V(\\sigma)],\n\\end{equation}\nwhere $\\Delta$ is a complete fan, $X(\\Delta)$ is the corresponding toric variety, and $V(\\sigma)$ is the closed subvariety associated with $\\sigma$.\nNotice the special feature of Formula \\eqref{eqn:danilov0}: the value of $r_\\sigma$ only depends on the cone $\\sigma$, but not on the fan $\\Delta.$ We call this the \\emph{Danilov condition}. \n\nIf such an expression exists, then the following \\emph{McMullen's formula} holds for any integral polytope $P$:\n\\begin{equation}\\label{equ:exterior}\n\\mathrm{Lat}(P) = \\displaystyle \\sum_{F: \\textrm{ a face of $P$}} \\alpha(F,P) \\ \\mathrm{nvol}(F) ,\n\\end{equation}\nwhere $\\alpha(F,P)$ is set to be $r_\\sigma$ where $\\sigma$ is the normal cone of $P$ at $F$, and $\\mathrm{nvol}(F)$ is the normalized volume of $F$. \n\nIn fact, one can asks directly the existence of McMullen's formula (independently from Danilov's question). More specifically, one can ask whether there are ways to assign values to cones such that McMullen's formula holds if $\\alpha(F,P)$ only depends on the normal cone of $P$ at $F.$\nWe will discuss the implication $\\eqref{eqn:danilov0} \\Rightarrow \\eqref{equ:exterior}$ in Section \\ref{sec:toric}. But for most the paper, we focus on McMullen's formula.\n\nMcMullen \\cite{mcmullen} was the first to confirm the exsitence of Formula \\eqref{equ:exterior} in a non constructive way (which was the reason we call this formula McMullen's formula). Morelli \\cite{morelli} supplied the first explicit way to choose $r_\\sigma$ answering Danilov's question. Pommersheim and Thomas \\cite{toddclass} gave a canonical construction of $r_\\sigma$ based on choices of flags.\nAs we discussed above, both of these two constructions naturally give a way to construct $\\alpha$ for McMullen's formula \\eqref{equ:exterior}\nBerline and Vergne \\cite{localformula} were able to construct such $\\alpha$ in a computable way. \nOne immediate consequence of the existence of Formula \\eqref{equ:exterior} is that if $\\alpha(F,P)$ is positive for each face $F$ of $P,$ then Ehrhart positivity follows. (See Theorem \\ref{thm:ehrhart} and Lemma \\ref{lem:red1}.)\nHence, it is natural to say that a polytope $P$ has \\emph{$\\alpha$-positivity} or is \\emph{$\\alpha$-positive} if all $\\alpha$'s associated to $P$ are positive.\n\nAlthough there are different constructions for $\\alpha(F,P)$, Berline-Vergne's construction has certain nice properties that are good for our purpose, and thus we will use their construction in our paper. We refer to their construction for $\\alpha(F, P)$ as the \\emph{BV-$\\alpha$-valuation}. To be more precise, we will use the terminologies \\emph{BV-$\\alpha$-positivity} and \\emph{BV-$\\alpha$-positive} to indicate the $\\alpha$'s we use are from the BV-$\\alpha$-valuation. \n\nAt present, the explicit computation of the BV-$\\alpha$-valuation is a recursive, complicated process, but we carry it out in the special example of regular permutohedra of small dimensions, whose symmetry simplifies the computations. Based on our empirical results, we conjecture the following:\n\\begin{conj}\\label{conj:alphas}\n  Every regular permutohedron is BV-$\\alpha$-positive.\n\\end{conj}\n\nOne important property of the BV-$\\alpha$-valuation enables us to reduce the problem of proving the Ehrhart positivity of all generalized permutohedra to proving the positivity of all the $\\alpha$'s arising from the regular permutohedra.\n\\begin{thm}\\label{thm:reduction}\n \n  \nConjecture \\ref{conj:alphas} implies Conjecture \\ref{positivity}.\n\\end{thm}\nTherefore, we focus on proving Conjecture \\ref{conj:alphas} instead. In this paper, we provide partial progress on proving Conjecture \\ref{conj:alphas} (and thus Conjecture \\ref{positivity}), as well as present equivalent statements to Conjecture \\ref{conj:alphas} in terms of mixed valuations and Todd class respectively. \n\n\\subsection*{Organization of the paper}\nIn Section \\ref{sec:background}, \nwe give basic definitions and review background results that are relevant to our paper. \nIn Section \\ref{sec:exterior}, we gives details of the BV-$\\alpha$-valuation, discuss consequences of the properties of this construction. In particular, we complete the proof of Theorem \\ref{thm:reduction}.\n\nIn Section \\ref{sec:generic}, assuming the $\\alpha$ function in McMullen's formula \\eqref{equ:exterior} is symmetric about the coordinates (a property of the BV-$\\alpha$-valuation), we derive a combinatorial formula for $\\mathrm{Lat}(\\operatorname{Perm}(\\bm{v}))$ indexed by subsets of $[n],$ where $\\operatorname{Perm}(\\bm{v})$ is a \\emph{generic permutohedron}, which belongs to a family of generalized permutohedra containing the regular permutohera.\nIn Section \\ref{sec:evidence}, we carry out direct computation to find values of BV-$\\alpha$-valuations for regular permutohedra, and verify that Conjecture \\ref{conj:alphas} is true for dimension up to $6.$ We are also able to verify positivity of $\\alpha(F,\\Pi_n)$ for faces $F$ of codimension $2$ or $3.$ There results provide evidences to Conjecture \\ref{conj:alphas}. \n\nIn addition to investigating positivity of the $\\alpha$'s, there are other questions one can ask about these constructions. Although there are different constructions for $\\alpha$, one still wonders whether under certain constraints, the construction is unique. In Section \\ref{sec:unique}, using the combinatorial formula we derived in Section \\ref{sec:generic} and theories of mixed valuations, we show that the $\\alpha$-values arising from the regular permutohedron is unique as long as we assume $\\alpha$ is symmetric about the coordinates. As a consequence of our techniques, we give an equivalent statement to Conjecture \\ref{conj:alphas} in terms of mixed valuations (Corollary \\ref{cor:equimixed}).\n\nIn Section \\ref{sec:toric}, we quickly review the connection with toric varieties as in \\cite[Section 5.3]{fulton}. Analogously to the treatment of mixed volumes in \\cite[Section 5.4]{fulton}, we can relate the more general mixed valuations to some intersection theoretic quantities. With these arguments we can give one more equivalence of our main Conjecture \\ref{conj:alphas}. It relates to the positivity of the Todd class of the permutohedral variety (in some) expression in terms of the torus invariant cycles.\n\n\n\n\n\n\n\\subsection*{Acknowledgement}\n\\label{sec:ack}\nThe authors thank Federico Ardila many helpful discussions. We also thank Nicole Berline and Michele Vergne for useful comments and for providing a code used in some of the computations.\n\n\n\\section{Background}\\label{sec:background}\n\nIn this section and the next section, we assume the ambient space is ${\\mathbb R}^D,$ and ${\\mathbb Z}^D$ is the \\emph{lattice} in ${\\mathbb R}^D.$\nFor any $D$-vector $\\bm{\\alpha}$,  $\\bm{\\alpha}_i( \\cdot)$ is the linear function that maps $\\mathbf{x} \\in {\\mathbb R}^D$ to the scalar product of $\\bm{\\alpha}$ and $\\mathbf{x}.$ Since we can consider $\\bm{\\alpha}( \\cdot)$ as a point in the dual space $({\\mathbb R}^D)^*$ of ${\\mathbb R}^D,$ we will use the notation $\\bm{\\alpha}$ (or any bold letter) to denote both the $D$-vector and the linear function.\n\n We assume familiarity with basic definitions of polyhedra and polytopes as presented in \\cite{grunbaum, zie}, and only review terminologies and setups that are relevant to us.  \n \n A \\emph{polyhedron} is the set of points defined by a linear system of equalities and inequalities \n\\begin{align}\n\\bm{\\alpha}_i(\\mathbf{x}) = w_i, &\\quad \\forall 1 \\le i \\le m_1,\\nonumber \\\\\n\\bm{\\beta}_i(\\mathbf{x}) \\le z_i, &\\quad \\forall 1 \\le i \\le m_2. \\label{ineq}\n\\end{align}\nFor simplicity, we let $A$ be the $m_1 \\times D$ matrix whose row vectors are $\\bm{\\alpha}_i$'s, $B$ the $m_2 \\times D$ matrix whose row vectors are $\\bm{\\beta}_i$'s, $\\bm{w} = (w_1,\\dots, w_{m_1})^T,$ and $\\bm{z} =(z_1, \\dots, z_{m_2})^T$, so the above linear system can be represented as\\[ A \\mathbf{x} = \\bm{w}, \\quad B \\mathbf{x} \\le \\bm{z}.\\] \n\n\nA \\emph{polytope} is a bounded polyhedron. (A polytope can also be defined as the convex hull of a finite set of points.)\n\nFor any polyhedron $P,$ we use $\\operatorname{vert}(P)$ to denote the vertex set of $P.$\nAn {\\it integral} polyhedron is a polyhedron whose vertices are all lattice points, i.e., points with integer coordinates.\n\nLet $V$ be a subspace of ${\\mathbb R}^D,$ and $\\Lambda := V \\cap {\\mathbb Z}^D$ the lattice in $V.$ \nFor any polytope $P$ that is lying in an affine space that is a translation of $V$, we define the \\emph{volume of $P$ normalized to the lattice $\\Lambda$} to be the integral\n\\[ \\mathrm{Vol}_\\Lambda(P) := \\int_P \\ 1 \\ d  \\Lambda,\\]\nwhere $d  \\Lambda$ is the canonical Lebesgue measure defined by the lattice $\\Lambda.$\nIn the case where $\\dim P = \\dim \\Lambda$, we get the \\emph{normalized volume} of $P$, denoted by $\\mathrm{nvol}(P).$\n\n\\subsection{Cones and fans} A \\emph{(polyhedral) cone} is the set of all nonegative linear combinations of a finite set of vectors. A \\emph{shifted cone} is a set of points in the form of $C + \\mathbf{x}$ where $C$ a cone and $\\mathbf{x}$ is a point. A shifted cone is \\emph{pointed} if it does not contain a line.\nA shifted cone $C + \\mathbf{x}$ is \\emph{rational} if the cone $C$ is generated by vectors with rational coordinates.\n\n\\begin{defn}\n  Suppose $P$ is a polyhedron and $F$ is a face. The \\emph{tangent cone} of $F$ at $P$ is:\n\\[\n{\\operatorname{tcone}(F,P)} = \\left\\{ F + \\bm{u}: F + \\delta \\bm{u} \\in P \\hspace{5pt}\\textrm{for sufficiently small $\\delta$}\\right\\}.\n\\]\nThe \\emph{feasible cone} of $F$ at $P$ is:\n\\[\n{\\operatorname{fcone}(F,P)} = \\left\\{ \\bm{u}: F + \\delta \\bm{u} \\in P \\hspace{5pt}\\textrm{for sufficiently small $\\delta$}\\right\\}\n\\]\nNote that $\\operatorname{tcone}(F,P)$ is a shifted cone, but not necessarily a cone, when $\\operatorname{fcone}(F,P)$ is always a cone.\n\nIn order to always work with pointed cones, we also define\n\\[{\\operatorname{tcone}}^p(F,P) = {\\operatorname{tcone}}(F,P)\/L \\quad \\text{ and } \\quad {\\operatorname{fcone}}^p(F,P) = {\\operatorname{fcone}}(F,P)\/L \\]\nwhere $L$ is the affine space spanned by $F$. Then $\\operatorname{tcone}^p(F,P)$ and $\\operatorname{fcone}^p(F,P)$ are pointed (shifted) cones with dimension $\\dim P - \\dim F$. \n\\end{defn}\n\n\n\n\n\\begin{defn} Suppose $V$ is a subspace of ${\\mathbb R}^D.$\n\tLet $K \\subseteq V$ be a cone. The \\emph{polar cone} of $K$ with respect to $V$ is the cone \n\t\\[ K_V^\\circ = \\{ \\bm{y} \\in V^* \\ | \\ \\bm{y}(\\mathbf{x}) \\le 0,  \\forall \\mathbf{x} \\in K\\}.\\]\nIn the situation where $K$ is full-dimensional in $V,$ we will omit the subscript $V$ and the words ``with respect to $V$''\n\\end{defn}\n\n\\begin{defn}\n  Suppose $V$ is a subspace of ${\\mathbb R}^D$ and $P \\subset V + \\bm{y}$ for some $\\bm{y} \\in {\\mathbb R}^D.$\nGiven any face $F$ of $P$, the \\emph{normal cone} of $P$ at $F$ with respect to $V$ is \n\\[\n\\operatorname{ncone}_V(F, P):=\\left\\{ \\bm{u} \\in V^*:\\quad \\bm{u}(\\bm{p}_1) \\geq \\bm{u}(\\bm{p}_2), \\quad \\forall \\bm{p}_1\\in F,\\quad \\forall \\bm{p}_2\\in P \\right\\}.\n\\]\nTherefore, $\\operatorname{ncone}_V(F,P)$ is the collection of linear functions $\\bm{u}$ in $V^*$ such that $\\bm{u}$ attains maximum value at $F$ over all points in $P.$\n\nThe \\emph{normal fan} $\\Sigma_V(P)$ of $P$ with respect to $V$ is the collection of all normal cones of $P$. \n\nIn the situation where the affine span of $P$ is $V + \\bm{y},$ i.e., $\\dim(P) = \\dim (V),$ we will omit the subscript $V$ and the words ``with respect to $V$''\n\\end{defn}\n\nWe have the following easy results for normal cones which will be useful for our paper.\n\\begin{lem}\\label{lem:ncone}\nLet $L$ be the shift of the affine span of $F$ to the origin. Then $\\operatorname{ncone}_V(F,P)$ spans the orthogonal complement of $L$ with respect to $V$. \nHence,\n\\begin{equation}\\label{equ:nconedim}\n  \\operatorname{ncone}_V(F, P) = \\dim V - \\dim F.\n  \\end{equation}\nFurthermore, the pointed feasible cone of $P$ at $F$ and the normal cone of $P$ at $F$ are polar to one another in the following sense:\n\\begin{equation}\\label{equ:fncone}\n  (\\operatorname{fcone}^p(F, P))_{V\/L}^\\circ = \\operatorname{ncone}_V(F, P) \\text{ and }  (\\operatorname{ncone}_V(F, P))_{V\/L}^\\circ = \\operatorname{fcone}^p(F, P).\n\\end{equation}\n\n\\end{lem}\n\n\n\\subsection{Generalized permutohedra} \nWe introduce \\emph{generalized permutohedra}, the main family of polytopes we study in this paper. In this part and any later part that is related to generalized permutohedra, we assume $D = n+1,$ i.e., the ambient is ${\\mathbb R}^{n+1}.$ First, we present the \\emph{usual permutohedron} as the convex hull of a finite number of points.\n\n\\begin{defn}\n  Given a point $\\bm{v} = (v_1,v_2,\\cdots,v_{n+1}) \\in {\\mathbb R}^{n+1}$, we construct the \\emph{usual permutohedron}\n\\[\\operatorname{Perm}(\\bm{v}) = \\operatorname{Perm} (v_1,v_2,\\cdots, v_{n+1}) := \\operatorname{conv}\\left(v_{\\sigma(1)},v_{\\sigma(2)},\\cdots, v_{\\sigma({n+1})}):\\quad \\sigma\\in {\\mathfrak S}_{n+1}\\right)\\]\nIn particular, if $\\mathbf{x} = (1, 2, \\dots, {n+1}),$ we obtain the \\emph{regular permutohedron}, denoted by $\\Pi_{n},$\n\\[ \\Pi_{n} := \\operatorname{Perm} (1, 2, \\dots, n+1).\\]\n\\end{defn}\nNote that as long as there are two different entries in $\\bm{v}$ we have $\\dim (\\operatorname{Perm}(\\bm{v})) = n$.\n\nThe \\emph{generalized permutohedra} is orginally introduced by Postnikov \\cite[Definition 6.1]{bible} as polytopes obtained from usual permutohedra by moving vertices while preserving all edge directions. \nIn \\cite{faces}, Postnikov, Reiner, and Williams give several equivalent definitions, one of which uses concepts of normal fans.\n\n\\begin{defn}\n  Let $V$ be the subspace of ${\\mathbb R}^{n+1}$ defined by $x_1 + x_2 + \\cdots + x_{n+1} = 0.$ The \\emph{braid arrangement fan}, denoted by ${\\mathfrak B}_n,$  is the complete fan in $V$ given by the hyperplanes\n\\[\n  x_i - x_j = 0 \\quad \\text{for all $i\\neq j$.}\n\\]\n\\end{defn}\n\n\\begin{prop}[Proposition 3.2 of \\cite{faces}] \\label{prop:coarser}\nLet $V$ be the subspace of ${\\mathbb R}^{n+1}$ defined by $x_1 + x_2 + \\cdots + \\mathbf{x}_{n+1} = 0.$ A polytope $P$ in $\\mathbb{R}^{n+1}$ is a generalized permutohedron if and only\nif its normal fan $\\Sigma_V(P)$ with respect to $V$ is refined by the braid arrangement fan ${\\mathfrak B}_n$.\n\\end{prop}\nIt follows from \\cite[Proposition 2.6]{bible} that as long as $\\bm{v}=(v_1,v_2,\\cdots,v_{n+1})$ has distinct coordinates, the associated usual permutohedron $\\operatorname{Perm}(\\bm{v})$ has the braid arrangement ${\\mathfrak B}_n$ as its normal fan. We call $\\operatorname{Perm}(\\bm{v})$ with $\\bm{v}$ of distinct coordinates a \\emph{generic permutohedron}.\nIn particular, the regular permutohedron $\\Pi_n$ is a generic permutohedron.\n\n\n\\subsection{Indicator functions and algebra of polyhedra}\n\nFor a set $S \\subseteq {\\mathbb R}^D$, the indicator function $[ S ]: {\\mathbb R}^D\n\\rightarrow {\\mathbb R}$ of $S$ is defined as \n\\[[ S ] (x) = \\left \\{\n\\begin{array}{ll} 1 \\mbox{ if }x \\in S, \\\\ 0 \\mbox{ if }x \\not \\in\nS.\\\\\n\\end{array}\\right .\\]\n\nLet $V$ be a subspace of ${\\mathbb R}^D.$ The \\emph{algebra of polyhedra}, denoted by $\\mathcal P(V)$, is the vector space defined as the span of the indicator functions of all polyhedra in $V.$ We similarly define $\\mathcal P_b(V)$ as the \\emph{algebra of polytopes}. For any $\\mathbf{x} \\in {\\mathbb R}^D,$ the \\emph{algebra of shifted cones at $\\mathbf{x}$}, denoted by $\\mathcal C_{\\mathbf{x}}(V)$, is the vector space defined as the span of the indicator functions of all shifted cones that are in the form of $C +\\mathbf{x}$ for some cone $C.$\n\n\nA linear transformation $\\phi: \\mathcal P(V), \\mathcal P_b(V), \\mathcal C_\\mathbf{x}(V) \\to W,$ where $W$ is a vector space, is a \\emph{valuation}.\n\nBoth volume $\\mathrm{Vol}_\\Lambda( \\cdot )$ and number of lattice points $\\mathrm{Lat}( \\cdot )$ are valuations on the algebra of polytopes. However, normalized volume $\\mathrm{nvol}( \\cdot )$ is not a valuation.\n\n\nBelow is an important result on indicator functions of feasible cones at vertices.\n\\begin{thm}[Theorem 6.6 of \\cite{barvinok}] \\label{thm:feasible}\nSuppose $P$ is a nonempty polytope. Then\n\\[\n  [0] \\equiv \\displaystyle \\sum_{\\bm{v}: \\textrm{ a vertex of $P$}} [{\\operatorname{fcone}}^p(\\bm{v},P)] \\quad \\textrm{modulo polyhedra with lines}\n\\]\n\\end{thm}\n\n\n\n\n\\subsection{Mixed valuations} \\label{subsec:mixed}\nLet $\\Lambda$ be a sublattice of ${\\mathbb Z}^D$ and $V$ is the span of $\\Lambda.$\n\nA valuation is a \\emph{$\\Lambda$-valuation} if it is invariant under $\\Lambda$-translation.\nWe say a valuation $\\phi$ is \\emph{homogeneous of degree $d$} if $\\phi([t P]) = t^d \\phi([P])$ for any integral polytope $P$ and $t \\in {\\mathbb Z}_{\\ge 0}.$ It's clear that $\\mathrm{Vol}_\\Lambda$ is homogeneous of degree $\\dim \\Lambda$, but $\\mathrm{Lat}$ is not homogenous. \n\n\n\n\n\nThe following important theorem by McMullen is a special case of \\cite[Theorem 6]{mcmullen}.\n\\begin{thm}\n\t\\label{thm:mixed}\nSuppose $\\phi$ is a homogeneous $\\Lambda$-valuation on $\\mathcal P_b(V)$ of degree $d.$\nThen there exists a function $\\mathcal M$ which takes $d$ integral polytopes as inputs such that\n\\begin{equation}\\label{equ:mixed0}\n  \\phi(t_1P_1+t_2P_2+\\cdots+t_kP_k) = \\sum_{j_1,\\cdots,j_d  \\in [d]} \\mathcal M(P_{j_1},P_{j_2},\\cdots,P_{j_d})t_{j_1}\\cdots t_{j_d},\n\\end{equation}\nfor any $k \\in {\\mathbb Z}_{>0},$ any integral polytopes $P_1, \\dots, P_k \\subset V$ and $t_1, \\dots, t_k \\in {\\mathbb Z}_{\\ge 0}.$\n\\end{thm}\n\nThe following definition and lemma are stated in \\cite[Section 3 of Chapter IV]{ewald} for the volume valuation (which is a homogeneous valuation). We give the general forms here.\n\\begin{defn}\n  Let $\\phi$ and $\\mathcal M$ be as in Theorem \\ref{thm:mixed}. We define another function $\\mathcal{M}\\phi$ that takes $d$ integral polytopes as inputs as an average of the function $\\mathcal M:$\n\\[ \\mathcal{M}\\phi(P_1, \\dots, P_d) := \\frac{1}{d!} \\sum_{\\sigma \\in {\\mathfrak S}_d} \\mathcal M(P_{\\sigma(1)}, \\dots, P_{\\sigma(d)}).\\]\nIt is easy to see that $\\mathcal{M}\\phi$ is uniquely chosen for each $\\phi$, and \\eqref{equ:mixed0} still holds for $\\mathcal{M}\\phi:$ \n\\begin{equation}\\label{equ:mixed}\n  \\phi(t_1P_1+t_2P_2+\\cdots+t_kP_k) = \\sum_{j_1,\\cdots,j_d \\in [d]} \\mathcal{M}\\phi(P_{j_1},P_{j_2},\\cdots,P_{j_d})t_{j_1}\\cdots t_{j_d},\n\\end{equation}\nWe call $\\mathcal{M}\\phi$ the \\emph{mixed valuation} of $\\phi.$ \n\\end{defn}\n\n\n\nThe lemma below gives two properties of the mixed valuation $\\mathcal{M}\\phi.$\n\\begin{lem}\\label{lem:mixval}\n  \\begin{ilist}\n  \\item For any integral polytopes $P_1, \\dots, P_d$, and any permutation $\\sigma \\in {\\mathfrak S}_d,$ we have\n\\begin{equation}\\label{equ:permprop}\n\\mathcal{M}\\phi(P_1, \\dots, P_d) = \\mathcal{M}\\phi(P_{\\sigma(1)}, \\dots, P_{\\sigma(d)}).\n\\end{equation}\n  \\item The function $\\mathcal{M}\\phi$ is a \\emph{multi-linear} function, that is, it is linear in each component.\n\\end{ilist}\n\\end{lem}\n\n\\begin{proof} (i) follows directly from the definition of $\\mathcal{M}\\phi.$ \n For (ii), we will just prove $\\mathcal{M}\\phi$ is linear in the first component, that is, to show for any integral polytopes $P_1, P_1', P_2, P_3, \\dots, P_n$ and nonnegative integers $s_1, s_1',$ we have\n\n\n\\begin{equation}\n  \\mathcal{M}\\phi(s_1 P_1 + s_1' P_1', P_2, \\dots, P_d) = s_1 \\mathcal{M}\\phi(P_1, P_2, \\dots, P_d) + s_1' \\mathcal{M}\\phi(P_1', P_2, \\dots, P_d), \\label{equ:multilinear}\n\\end{equation}\n\n\n\nWe apply \\eqref{equ:mixed} to both sides of the following equality:\n\\[\\phi( t_1 (s_1 P_1+ s_1' P_1') + t_2 P_2 + \\cdots + t_d P_d) = \\phi( (t_1 s_1) P_1 + (t_1 s_1') P_1' + t_2 P_2 + \\cdots + t_d P_d).\\]\nConsider $s_1$ and $s_1'$ as fixed numbers. Then each side gives a homogeneous polynomial in $t_1, t_2, \\dots, t_d.$ Since these two homogeneous polynomials agree on all $t_1, t_2, \\dots, t_n \\in {\\mathbb Z}_{\\ge 0},$ we conclude that they are exactly the same polynomials, and thus their coefficients agree. Then \\eqref{equ:multilinear} follows from \\eqref{equ:permprop} and comparing the coefficients of $t_1 t_2 \\dots t_n.$ \n\\end{proof}\n\nApply the above results to volume valuation, a homogeneous valuation, we obtain the following:\n\\begin{thm}[Theorem 3.2 of \\cite{ewald}] \\label{thm:mixedvol}\n  Suppose $P_1,\\cdots, P_k$ are integral polytopes with $\\dim(P_1+\\cdots +P_k)=d$. Let $\\Lambda$ be the $d$-dimensional lattice $\\mathrm{span}(P_1+\\cdots + P_k) \\cap {\\mathbb Z}^D.$ Then\n\\[\n\\mathrm{Vol}_\\Lambda(t_1P_1+t_2P_2+\\cdots+t_kP_k) = \\sum_{j_1,\\cdots,j_d =1}^d \\mathrm{\\mathcal{M}Vol}_\\Lambda(P_{j_1},P_{j_2},\\cdots,P_{j_d})t_{j_1}\\cdots t_{j_d}\n\\]\nwhere the sum is carried out independently over the $j_i$. The function $\\mathrm{\\mathcal{M}Vol}_\\Lambda(P_{j_1},P_{j_2},\\cdots,P_{j_d})$ is called the \\emph{mixed volume} of $P_{j_1},P_{j_2},\\cdots,P_{j_d}$.\n\\end{thm}\nFurthermore we have the following properties:\n\\begin{thm}[Theorem 4.13 of \\cite{ewald}] \\label{thm:mixvolprop}\nLet $P_1,\\cdots,P_d$ be integral polytopes. Then,\n\\begin{enumerate}\n\\item $\\mathrm{\\mathcal{M}Vol}_\\Lambda(P_1,\\cdots, P_d) \\geq 0 $\n\\item $\\mathrm{\\mathcal{M}Vol}_\\Lambda(P_1,\\cdots, P_d) > 0$ if and only if each $P_i$ contains a line segment $I_i=[a_i,b_i]$ such that $b_1-a_1,\\cdots,\nb_d-a_d$ are linearly independent.\n\\end{enumerate}\n\\end{thm}\n\nThe lattice point, or counting, valuation $\\mathrm{Lat}$ is not homogeneous. However it can be decomposed into homogeneous parts.\n\\begin{thm}[Theorem 5 of \\cite{mcmullen}] \\label{thm:decomposeLat}\nSuppose $V$ is $d$-dimensional. Then we can decompose the valuation $\\mathrm{Lat}$ as\n\\[\n\\mathrm{Lat} = \\mathrm{Lat}^d + \\cdots +\\mathrm{Lat}^1 + \\mathrm{Lat}^0\n\\]\nwhere $\\mathrm{Lat}^r$ is homogenenous of degree $r$.\n\\end{thm}\nThis decomposition corresponds to the coefficients of the Ehrhart polynomial, in particular $\\mathrm{Lat}^d$ corresponds to the volume valuation $\\mathrm{Vol}_\\Lambda,$ where $\\Lambda = V \\cap {\\mathbb Z}^D$.\nApplying Theorem \\ref{thm:mixed} and Lemma \\ref{lem:mixval} to each homogeneous function $\\mathrm{Lat}^r$ gives us the following result\n\n\\begin{thm}\\label{thm:mixedLat}\nSuppose $P_1,\\cdots, P_k$ are integral polytopes with $\\dim(P_1+\\cdots +P_k)=d$. Then \\[\n\\mathrm{Lat}(t_1P_1+t_2P_2+\\cdots+t_kP_k) = \\sum_{e=0}^d \\sum_{j_1,\\cdots,j_e =1}^e \\mathrm{\\mathcal{M}Lat}^e(P_{j_1},P_{j_2},\\cdots,P_{j_e})t_{j_1}\\cdots t_{j_e}.\n\\]\n\\end{thm}\n\nWe cannot expect $\\mathrm{\\mathcal{M}Lat}^r$ or any other mixed valuation $\\mathcal{M}\\phi$ to be nonnegative in general. However we have a way to compute them. \n\n\\begin{thm}\\label{thm:mobius}\nSuppose $\\phi$ is a homogeneous $\\Lambda$-valuation on $\\mathcal P_b(V)$ of degree $d.$For any integral polytopes $P_1,P_2,\\cdots, P_d \\subset V,$ we have\n\\[\nd! \\mathcal{M}\\phi(P_1,P_2,\\cdots, P_d) = \\sum_{J\\subseteq [d]} \\left(-1\\right)^{d-|J|} \\phi\\left(\\sum_{j\\in J}P_j\\right)\n\\]\n\\end{thm}\n\\begin{proof}\n  We define two functions $f$ and $g$ on the boolean algebra of order $d$. (See \\cite[Chapter 3]{enum}.) For any subset $T \\subseteq [d],$ let \n\\[\n  f(T) := \\phi\\left(\\sum_{i\\in T}P_i\\right) \\quad \\text{and} \\quad g(T) := \\sum_{\\substack{j_1, \\dots, j_d \\in [d] \\\\ \\cup_{i=1}^d \\{j_i\\} = S}} \\mathcal{M}\\phi \\left(P_{j_1}, P_{j_2},\\cdots, P_{j_d}\\right).\n\\]\nApply \\eqref{equ:mixed} with $t_i=1$ to $f(T),$ one sees that $f(T) = \\sum_{S \\subseteq T} g(S).$ Therefore, by Mobius inversion, we get\n\\[ g(T) = \\sum_{S \\subseteq T} (-1)^{|T|-|S|} f(S).\\]\nThen the theorem follows from evaluate the above equality at $T=[d].$\n\\end{proof}\n\n\n\\section{McMullen's formula and the BV-$\\alpha$-valuation}\\label{sec:exterior}\nRecall in the introduction, we've discussed the question of the existence of the following \\emph{McMullen's formula} \n\\begin{equation}\\label{equ:exterior1}\n\\mathrm{Lat}(P) = \\displaystyle \\sum_{F: \\textrm{ a face of $P$}} \\alpha(F,P)\n\\ \\mathrm{nvol}(F) \\end{equation}\nwhere $\\alpha(F,P)$ depends only on the normal cone of $P$ at $F$.\n\nOne immediate consequence of the existence of McMullen's formula \\eqref{equ:exterior1} is that it provides another way to prove Ehrhart's theorem. Moreover, it gives a description of each Ehrhart coefficient. We state the following modified version of Theorem \\ref{thm:ehrhart0}.\n\n\\begin{thm}\\label{thm:ehrhart}\n  For an integral polytope $P \\subset {\\mathbb Z}^D$ and any $t \\in {\\mathbb Z}_{\\ge 0}$, the function\n\\[i(P, t) =  \\mathrm{Lat}(t P) = | tP \\cap {\\mathbb Z}^n| \\]\nis a polynomial in $t$ of degree $\\dim P.$ Furthermore, the coefficient of $t^k$ in $i(P,t)$ is \n\\begin{equation}\\label{equ:coeff}\n  \\sum_{F: \\text{ a $k$-dim face of $P$}} \\alpha(F, P) \\ \\mathrm{nvol}(F).\n\\end{equation}\n\\end{thm}\n\n\\begin{proof}\nWhen we dilate the polytope $P$ by a factor of $t$, each face $F$ of $P$ becomes $tF,$ a face of $tP.$ It is clear that the normal cone does not change. Hence, applying McMullen's formula to $tP,$ we get\n\\[i(P,t) = \\sum_F \\alpha(tF,tP) \\ \\mathrm{nvol}(tF) = \\sum_F \\alpha(F,P) \\ \\mathrm{nvol}(F) \\ t^{\\dim F}.\\] \nThen our conclusion follows.\n\\end{proof}\n\nFormula \\eqref{equ:coeff} for the coefficients of the Ehrhart polynomial $i(P,t)$ gives a sufficient condition for Ehrhart positivity.\n\\begin{lem}\\label{lem:red1}\nLet $P$ be an integral polytope. For a fixed $k,$ if $\\alpha(F, P)$ is positive for any $k$-dim face of $P,$ then the coefficient of $t^k$ in the Ehrhart polynomial $i(P,t)$ of $P$ is positive.\n\nHence, if for every face $F$ of $P$, we have $\\alpha(F,P)>0$, then $P$ is Ehrhart positive.\n\\end{lem}\n\nAs discussed in the introduction, different constructions of $\\alpha(F, P)$ were given in the literature. In our paper, we will use Berline-Vergen's construction, which we refer to as the \\emph{BV-$\\alpha$-valuation}.\n\n\n\\subsection{Berline-Vergne's construction}\nBerline and Vergne construct in \\cite{localformula} a function $\\Psi([C], \\Lambda)$ on indicator functions of rational shifted cones $C$ with respect to a lattice $\\Lambda,$ where $\\Lambda$ is a quotient of the lattice $V \\cap {\\mathbb Z}^D$ and $C$ is inside the affine span of $\\Lambda.$\nThen they show $\\Psi$ has the following properties:\n\n\\begin{enumerate}[(P1)]\n  \\item Let $V$ be the affine span of a lattice $\\Lambda.$ Then $\\Psi( \\cdot, \\Lambda)$ is a valuation on $\\mathcal C_\\mathbf{x}(V)$ for any $\\mathbf{x} \\in V.$ (Recall that $\\mathcal C_\\mathbf{x}(V)$ is the algebra of shifted cones at $\\mathbf{x}.$)\n  \\item McMullen's formula \\eqref{equ:exterior1} holds for rational polytopes if we set \n  \\begin{equation}\\label{equ:defnalpha}\n    \\alpha(F,P) := \\Psi([\\operatorname{tcone}^p(F,P)], (V \\cap {\\mathbb Z}^D)\/L),\n\\end{equation}\nwhere $V$ nd $L$ are the affine spaces spanned by $P$ and $F$ respectively.\n\n\n\\item If a cone $C$ contains a line, then $\\Psi([C], \\Lambda)=0$.\n\\item $\\Psi$ is invariant under lattice translation, i.e., $\\Psi([C], \\Lambda) = \\Psi([C+\\mathbf{x}], \\Lambda+\\mathbf{x})$ for any lattice point $\\mathbf{x}$.\n\\item Its value on a lattice point is 1, i.e. $\\Psi([\\bm{0}], \\Lambda)=1$.\n\\item $\\Psi$ is invariant under orthogonal unimodular transformation; that is, if $T$ is an orthogonal unimodular transformation, for any cone $C,$ we have $\\Psi([C], \\Lambda) = \\Psi([T(C)], T(\\Lambda)).$ \n\\item Suppose $\\Lambda'$ is the lattice generated by a subset of a basis of the lattice $\\Lambda$ and $C$ is in the affine span of $\\Lambda'.$ Then $\\Psi([C], \\Lambda) = \\Psi([C], \\Lambda').$\n\\item It can be computed in polynomial time fixing the dimension.\n\\end{enumerate}\n\n\\begin{rem}\nNote that for integral polytopes, we have that $\\operatorname{tcone}^p(F,P)$ is a lattice translation of $\\operatorname{fcone}^p(F,P)$. Therefore, by Property (P5)\n\\begin{equation}\\label{equ:defnalpha1}\n  \\alpha(F, P) = \\Psi([\\operatorname{tcone}^p(F,P)], (V \\cap {\\mathbb Z}^D)\/L)= \\Psi([\\operatorname{fcone}^p(F,P)], (V' \\cap {\\mathbb Z}^D)\/L'),\n\\end{equation}\nwhere $V$ and $L$ are defined as for \\eqref{equ:defnalpha}, and $V'$ and $L'$ are obtained by shifting $V$ and $L$ to the origin.\nBecause $\\operatorname{fcone}^p(F, P)$ is determined by $\\operatorname{ncone}(F,P)$ as in \\eqref{equ:fncone}, one sees that $\\alpha(F, P)$ depends only on the normal cone of $P$ at $F$.\nTherefore, this construction of $\\alpha(F, P)$ does give McMullen's formula \\eqref{equ:exterior1}. \n\\end{rem}\n\nWe have an immediate corollary to Properties (P1) and (P3): \n\n\\begin{cor}\\label{cor:sum}\n Suppose $C_1, C_2, \\dots, C_k$ and $K$ are cones satisfying \n \\[\\left[C\\right] \\equiv \\displaystyle \\sum_{i=1}^k \\left[C_i\\right]\\quad\\textrm{modulo polyhedra with lines}.\\]\nThen\n\\[\\Psi\\left(\\left[C\\right], \\Lambda\\right) = \\displaystyle \\sum_{i=1}^k \\Psi\\left(\\left[C_i\\right], \\Lambda\\right). \\]\n\n\\end{cor}\n\n\\subsection{Reduction theorem}\nWe've already discused a consequence of the existence of McMullen's formula, which reduce the problem of proving Ehrhart positivity to proving $\\alpha$-positivity. Now we will discuss a very important consequence of the BV-$\\alpha$-valuation -- the reduction theorem -- using which we complete the proof of Theorem \\ref{thm:reduction}.\n\nFor the rest of the section, we assume $\\alpha(F,P)$ comes from the BV-$\\alpha$-valuation. Also, because we only deal with integral polytopes, we will take \\eqref{equ:defnalpha1} as the definition of $\\alpha(F,P).$\n\n\\begin{lem}\n Suppose $V$ is a subspace of ${\\mathbb R}^D,$ and $P \\subset V + \\mathbf{x}$ and $Q \\subset V + \\bm{y}$ are two integral polytopes for some points $\\mathbf{x}$ and $\\bm{y}.$ \n\n  Let $F$ be a face of $P.$ Suppose there exist faces $G_1, G_2, \\dots, G_r$ of $Q$ of the same dimension such that\n  \\begin{equation}\\label{equ:unionncone}\n    \\operatorname{ncone}_V(F, P) = \\cup_{i=1}^r \\operatorname{ncone}_V(G_i, Q).\n\\end{equation}\nThen $F$ is of the same dimension as $G_i$'s, and \n\\[ \\alpha(F, P) = \\sum_{i=1}^r \\alpha(G_i, Q).\\]\n\n\n\\end{lem}\n\n\\begin{proof}\n  The first consequence of Equation \\eqref{equ:unionncone} is that $\\operatorname{ncone}_V(F,P)$ and $\\operatorname{ncone}_V(G_i, Q)$'s all span the same subspace. Let $L$ be the orthogonal complement of this subspace with respect to $V.$ Then by Lemma \\ref{lem:ncone}, affine spans of $F$ and $G_i$'s are all shifts of $L.$ Hence, $F$ has the same dimension as $G_i$'s. Letting $\\Lambda := (V \\cap {\\mathbb Z}^D)\/L$, we have\n \\[ \\alpha(F,P) = \\Psi(\\operatorname{fcone}^p(F,P), \\Lambda), \\quad \\text{and} \\quad \\alpha(G_i,Q) = \\Psi(\\operatorname{fcone}^p(G_i,Q), \\Lambda) \\ \\forall i.\\] \n\nNext, since $\\operatorname{ncone}_V(G_i, Q) \\cap \\operatorname{ncone}_V(G_j, Q)$ is a lower dimensional cone for any $i \\neq j,$ we have \n  \\[ [ \\operatorname{ncone}(F, P) ] \\equiv \\sum_{i=1}^r [ \\operatorname{ncone}(G_i, Q)] \\quad \\text{modulo polyhedra contained in proper subspaces}.\\]\n  Taking the polar of the above identity and applying \\eqref{equ:fncone} yields\n \\[\\left[ \\operatorname{fcone}^p(F, P)\\right] \\equiv \\sum_{i=1}^r \\left[\\operatorname{fcone}^p(G_i,Q)\\right]\\quad\\text{modulo polyhedra with lines}.\\]\n  Applying Corollary \\ref{cor:sum}, we obtained the desired identity.\n\\end{proof}\n\n\\begin{thm}[Reduction Theorem] \\label{thm:reduction-gen}\nSuppose $V$ is a subspace of ${\\mathbb R}^D,$ and $P \\subset V + \\mathbf{x}$ and $Q \\subset V + \\bm{y}$ are two integral polytopes for some points $\\mathbf{x}$ and $\\bm{y}.$ \nAssume further the normal fan $\\Sigma_V(P)$ of $P$ with respect to $V$ is a refinement of the normal fan $\\Sigma_V(Q)$ of $Q$ with respect to $V$. \nThen for any fixed $k,$ if $\\alpha(F, P) >0 $ for every $k$-dimensional face $F$ of $P$, then $\\alpha(G, Q) >0$ for every $k$-dimensional face $G$ of $Q.$\n\nTherefore, BV-$\\alpha$-positivity of $P$ implies BV-$\\alpha$-positivity of $Q.$\n\\end{thm}\n\nThe above reduction theorem and Proposition \\ref{prop:coarser} immediately give the following result and complete the proof for Theorem \\ref{thm:reduction}\n\\begin{thm}[Reduction Theorem, special form]\\label{thm:reduction-spe}\nLet $Q \\subset {\\mathbb R}^{n+1}$ be a generalized permutohedron. \nThen for any fixed $k,$ if $\\alpha(F, \\Pi_{n}) >0 $ for every $k$-dimensional face $F$ of $\\Pi_{n}$, then $\\alpha(G, Q) >0$ for every $k$-dimensional face $G$ of $Q.$\n\nTherefore, BV-$\\alpha$-positivity of $\\Pi_{n}$ implies BV-$\\alpha$-positivity of $Q.$\\end{thm}\n\n\n\\begin{proof}[Proof of Theorem \\ref{thm:reduction}]\n The theorem follows from Theorem \\ref{thm:reduction-spe} and Lemma \\ref{lem:red1}.\n\\end{proof}\n\n\\begin{rem}\\label{rem:upton}\n  All permutohedra of dimension at most $n$ appear in ${\\mathbb R}^{n+1}$, and Proposition \\ref{prop:coarser} applies to all of these permutohera. Therefore, the polytope $Q$ in Theorem \\ref{thm:reduction-spe} could be any permutohedron of dimension up to $n.$\n\\end{rem}\n\n\n\\begin{rem}\\label{rem:generic}\n \n\tTheorem \\ref{thm:reduction-spe} stills holds if we replace $\\Pi_{n}$ with any generic permutohedron, that is, any $\\operatorname{Perm}(\\bm{v})$ where $\\bm{v} \\in {\\mathbb R}^{n+1}$ is a vector with distinct coordinates.\n \n\\end{rem}\n\n\\subsection{More applications}\\label{subsec:moreappl} \nTheorem \\ref{thm:reduction-gen} mainly follows from Properties (P1) and (P3) of the BV-$\\alpha$-valuation. In this subsection, we will give more applications of the BV-$\\alpha$-valuation as consequences of other properties, in particular Property (P6), showing reasons why this particular construction of $\\alpha$ is nice.\n\n\\begin{lem}\\label{lem:Psisum}\nSuppose $\\Lambda$ is a lattice, and $P$ is a nonempty polytope in the affine span of $\\lambda.$ Then\n\\begin{equation}\\label{equ:Psisum}\n\\sum_{\\bm{v} \\in \\operatorname{vert}(P)} \\Psi(\\operatorname{fcone}^p(\\bm{v},P), \\Lambda) = 1.\n\\end{equation}\nLet $C$ be a $1$-dimensional cone generated by a vector $\\bm{v} \\in \\Lambda.$ Then \n\\begin{equation}\\label{equ:1\/2}\n\\Psi(C, \\Lambda) = 1\/2.\n\\end{equation}\n\\end{lem}\n\n\\begin{proof}\n  Equation \\eqref{equ:Psisum} follows from Corollary \\ref{cor:sum} and the identity in Theorem \\ref{thm:feasible}. \n\nConsider the polytope $Q := \\operatorname{conv}\\{\\bm{0}, \\bm{v}\\},$ which has two vertices $\\bm{0}$ and $\\bm{v}$ and has pointed feasible cones:\n\\[ \\operatorname{fcone}^p(\\bm{0}, Q) = C, \\quad \\operatorname{fcone}^p(\\bm{v}, Q) = -C.\\]\nBy \\eqref{equ:Psisum}, we have\n\\[ \\Psi(C, \\Lambda) + \\Psi(-C, \\Lambda) = 1.\\]\nHowever, by Property (P6), we have\n\\[ \\Psi(C, \\Lambda) = \\Psi(-C, -\\Lambda) = \\Psi(-C, \\Lambda).\\]\nThen \\eqref{equ:1\/2} follows.\n\\end{proof}\n\nWe then apply the above lemma to obtain results on $\\alpha(F,P).$\n \\begin{lem}\\label{lem:sum}\nSuppose $P$ is a nonempty integral polytope. Then\n\\begin{eqnarray}\n  \\sum_{\\bm{v} \\in \\operatorname{vert}(P)} \\alpha(\\bm{v},P) &=& 1; \\label{equ:sum1} \\\\\n  \\alpha(F, P) &=& 1\/2, \\quad \\text{for any facet $F$ of $P$}; \\label{equ:facet} \\\\\n  \\alpha(P, P) &=& 1. \\label{equ:leading}\n\\end{eqnarray}\n\\end{lem}\n\n\\begin{proof}\n  Equalities \\eqref{equ:sum1} and \\eqref{equ:facet} follow from \\eqref{equ:defnalpha1}, Property (P7), and Lemma \\ref{lem:Psisum}.\n   Equality \\eqref{equ:leading} follows from the fact that $\\operatorname{fcone}^p(P, P) = \\bm{0}$.\n\\end{proof}\n\n\\begin{rem}\nFormulas \\eqref{equ:sum1} and \\eqref{equ:leading} hold for any constructions for $\\alpha(F,P)$; however, Formula \\eqref{equ:facet} does not hold for all the known constructions for $\\alpha(F,P)$, and thus is a special property of the BV-$\\alpha$-valuation.  \n\n\\end{rem}\n\n\n\n\nWe use the above lemma to give a quick proof for the following modified version of Pick's theorem, in which one sees that the $\\alpha$-values given above really corresponds to the coefficients appearing in Pick's theorem. \n\n\\begin{thm}[Pick's theorem]\nLet $P\\subset \\mathbb{Z}^2$ be an integral polygon. Then\n\\[\n\\mathrm{Lat}(P) = \\mathrm{area}(P) + \\frac{1}{2} \\  \\mathrm{Lat}(\\partial P) + 1,\n\\]\nwhere $\\partial P$ denotes the boundary of $P.$\n\\end{thm}\n\\begin{proof}\n  Applying Lemma \\ref{lem:sum}, we get\n  \\[ \\mathrm{Lat}(P) = 1 \\cdot \\mathrm{area}(P) + \\frac{1}{2} \\cdot \\sum_{E: \\text{ edge of $P$}} \\mathrm{nvol}(E) + 1.\\]\nIt is easy to see that $\\sum_{E: \\text{ edge of $P$}} \\mathrm{nvol}(E)$ is precisely the number of lattice points on the boundary of $P.$ Thus, the theorem follows.\n\\end{proof}\n\nWe remark that Lemma \\ref{lem:sum} also directly gives the results on the three coefficients of $i(P,t)$ with explicit simple descriptions: \nthe leading coefficient is equal to the normalized volume of $P$, the second coefficient is one half of the sum of the normalized volumes of facets, and the constant term is always $1$. This pattern extend naturally to \\emph{boxes}.\n\n\\begin{defn}\nAn $D$-dimensional \\emph{box} is a polytope defined by\n\\[\n\\left\\{ \\mathbf{x} \\in {\\mathbb R}^D:\\quad a_i \\leq x_i\\leq b_i \\quad \\forall i \\in [D] \\right\\}\n\\]\nfor some vectors $\\bm{a}, \\mathbf{b} \\in {\\mathbb R}^D.$\n\\end{defn}\n\n\\begin{ex}\\label{ex:boxes}\nLet $P$ be an $D$-dimensional box. The pointed feasible cone of $P$ at any vertex is equal to an orthant of ${\\mathbb R}^D$. By Property (P6),\n\\[ \\alpha(\\bm{v}, P) = \\Psi(\\bm{v}, {\\mathbb Z}^D) = \\Psi(\\bm{u}, {\\mathbb Z}^D) = \\alpha(\\bm{u}, P), \\quad \\forall \\bm{v}, \\bm{u} \\in \\operatorname{vert}(P).\\]\nSince $P$ has $2^D$ vertices and we have Equality \\eqref{equ:sum1}, we conclude\n\\begin{equation}\\label{equ:orthant} \\alpha(\\bm{v}, P) = \\frac{1}{2^D} \\quad \\forall \\bm{v} \\in \\operatorname{vert}(P), \\quad \\text{ and } \\quad \\Psi(C, {\\mathbb Z}^D) = \\frac{1}{2^D} \\quad \\text{for any $D$-dim orthant $C$}.\\end{equation}\nFurthermore, note that for any $k$-dimensional face of $P,$ the pointed feasible cone of $P$ at $F$ is an orthant of ${\\mathbb R}^{D-k}.$ So by \\eqref{equ:orthant},\n\\[ \\alpha(F, P) = \\Psi(\\operatorname{fcone}^p(F,P), {\\mathbb Z}^{D-k}) = \\frac{1}{2^{D-k}}.\\]\nTherefore, applying \\eqref{equ:coeff}, we get that the coefficient of $t^k$ in the Ehrhart polynomial $i(P,t)$ of the $n$-dimensional box $P$ is\n\\[ \\frac{1}{2^{D-k}} \\sum_{F: \\text{ a $k$-dim face of $P$}} \\mathrm{nvol}(F).\\]\nLet $v_k$ be the sum of the normalized volumes of all $k$-dimensional faces. Then \n\\[\n  i(P,t)=v_D t^D + \\frac{1}{2}v_{D-1}t^{D-1} + \\frac{1}{4}v_{D-2}t^{D-2} + \\cdots + \\frac{1}{2^{D-k}}v_k t^{k} + \\cdots + 1.\n\\]\n\\end{ex}\n\n\n\nFrom results we show above, one sees that Property (P6) is an important property for the BV-$\\alpha$-valuation. It provides us a way to obtain $\\alpha$-values for special situations without explicit computation for $\\Psi$ which could be quite complicated. (See examples in the next subsection.) In fact, the following lemma, which states a special case of this property, will be applied extensively when we compute $\\alpha$-vaules for regular permutohedra in Section \\ref{sec:generic}.\n\n\\begin{lem}\\label{lem:symmetric0}\n  The valuation $\\Psi$ is \\emph{symmetric about the coordinates}, i.e., for any cone $C \\in {\\mathbb R}^D$ and any permutation $\\sigma \\in {\\mathfrak S}_D,$ we have \n  \\[ \\Psi(C, \\Lambda) = \\Psi(\\sigma(C), \\sigma(\\Lambda)),\\]\n  where $\\sigma(T) = \\{ (x_{\\sigma(1)}, x_{\\sigma(2)}, \\dots, x_{\\sigma(D)}) \\ : \\ (x_1,\\dots, x_D)\\in T\\}$ for any set $T \\subseteq {\\mathbb R}^D.$\n\\end{lem}\n\n\\begin{proof}\nLet $M_\\sigma$ be the permutation matrix corresponding to $\\sigma.$ Then the lemma follows from the observation that $T$ is mapped to $\\sigma(T)$ under the linear transformation $M_\\sigma$ and any permutation matrix is orthonormal and unimodular.\n\\end{proof}\n\nThe above result motivates the following definition.\n\\begin{defn}\\label{defn:symmetric}\n  Suppose $\\alpha$ is a construction such that McMullen's formula \\eqref{equ:exterior1} holds. We say it is \\emph{symmetric about the coordinates} \n  \\[ \\alpha(F, P) = \\alpha(G, Q) \\]\n  whenever $ \\operatorname{fcone}^p(F,P) = \\sigma(\\operatorname{fcone}^p(G,Q))$ for some $\\sigma \\in {\\mathfrak S}_n.$\n\\end{defn}\n\nTherefore, we have the following:\n\\begin{lem}\\label{lem:symmetric}\nThe BV-$\\alpha$-valuation is symmetric about the coordinates.\n\\end{lem}\nProperty (P6), in particular Lemma \\ref{lem:symmetric}, does not hold for all the known $\\Psi$ or $\\alpha$-constructions. For example, the construction given by Pommersheim and Thomas in \\cite{toddclass} depends on an ordering of a basis for the vector space, which means their construction is not symmetric about coordinates. \nThis is one of the reasons why we work with the BV-$\\alpha$-valuations for this paper. \n\n\\subsection{Examples of computing $\\Psi$}\nBelow we will give examples of computing $\\Psi$ and using which to get $\\alpha(F,P)$. The computation of the function $\\Psi$ associated to Berline-Vergne's construction is carried out recursively. Hence, it is quicker to compute $\\Psi$ for lower dimensional cones. Since the dimension of $\\operatorname{fcone}^p(F, P)$ is equal to the codimension of $F$ with repect to $P,$ the value of $\\alpha(F,P)$ is easier to compute if $F$ is a higher dimensional face.\n\nIn general, the computation of $\\Psi(C, \\Lambda)$ is quite complicated. However, when $C$ is a unimodular cone (with respect to $\\Lambda$), that is, $C$ is generated by a basis of $\\Lambda$, computations are greatly simplified. In small dimensions we can even give a simple closed expression for $\\Psi$ of unimodular cones. The following result is given in \\cite[Example 19.3]{barvinok}.\n\n\\begin{lem}\\label{lem:2dim} \n  If $C = \\operatorname{Cone}(\\bm{u}_1, \\bm{u}_2),$ where $\\{\\bm{u}_1, \\bm{u}_2\\}$ is a basis for a lattice $\\Lambda$, then \n\\[\\Psi(C, \\Lambda)=\\displaystyle \\frac{1}{4}+\\frac{1}{12}\\left(\\frac{\\langle \\bm{u}_1,\\bm{u}_2\\rangle}{\\langle \\bm{u}_1,\\bm{u}_1\\rangle}+\\frac{\\langle \\bm{u}_1, \\bm{u}_2\\rangle}{\\langle \\bm{u}_2,\\bm{u}_2\\rangle}\\right).\\]\n\\end{lem}\n\n\n\\begin{figure}\n\\caption{The vertices add 1 in a nontrivial way}\n\\begin{center}\n\\begin{tikzpicture}\n\n \t\\draw[thick] (0,0) -- (0,2);\n \t\\draw[thick] (0,0) -- (4,0);\n \t\\draw[thick] (0,2) -- (4,0);\n\n    \n    \\node at (-0.25,-0.25) {(0, 0)};\n    \\node at (-0.5,2) {(0, 1)};\n    \\node at (4,-0.25) {(2, 0)};\n    \\node[above right] at (0,0) {$\\frac{1}{4}$};\n    \\node[above right] at (0,2) {$\\frac{3}{10}$};\n    \\node[above right] at (4,0) {$\\frac{9}{20}$};\n\n\\end{tikzpicture}\n\\end{center}\n\\end{figure}\n\\begin{ex} \n  Consider the polygon $P$ in ${\\mathbb R}^2$ with vertices $\\bm{v}_1 = (0,0),\\bm{v}_2=(2,0)$, and $\\bm{v}_3=(0,1)$. The pointed feasible cone of $P$ at $\\bm{v}_1$ is the first quadrant, with $\\Psi$ value $1\/4$ (see Example \\ref{ex:boxes}).\n  The pointed feasible cone of $P$ at $\\bm{v}_2$ is the unimodular cone $C_2 = \\operatorname{Cone}((-2,1), (-1,0))$. Applying Lemma \\ref{lem:2dim},\n  \\[ \\alpha(\\bm{v}_2, P) = \\Psi(C_2, {\\mathbb Z}^2) =\\displaystyle \\frac{1}{4}+\\frac{1}{12}\\left(\\frac{2}{5}+\\frac{2}{1}\\right) = \\frac{9}{20}.\\]\n  The pointed feasible cone of $P$ at $\\bm{v}_3$ is the cone $C_3 = \\operatorname{Cone}((0,-1), (2,-1))$, which is not unimodular, so we cannot directly apply Lemma \\ref{lem:2dim} to compute $\\Psi(C_3, {\\mathbb Z}^2)$. In order to compute it, we first decompose $C_3$ in the algebra of cones $\\mathcal C_\\bm{0}({\\mathbb R}^2):$\n\\[\n\\left[C_3\\right]  = \\left[\\operatorname{Cone}\\big((0,-1),(1,-1)\\big)\\right] + \\left[\\operatorname{Cone}\\big((1,-1),(2,-1)\\big)\\right] - \\left[\\operatorname{Cone}\\big((1,-1)\\big)\\right]\n\\]\n\n\n\\begin{figure}\n\\caption{Unimodular decomposition of $C_3$}\n\\begin{center}\n\\begin{tikzpicture}\n\n \t\\draw[thick] (0,0) -- (0,2);\n \t\\draw[thick] (0,2) -- (4,0);\n\t\\draw[thick] (0,2) -- (2,0);\n\t\n    \n    \\node at (-0.25,-0.25) {(0, 0)};\n    \\node at (-0.5,2) {(0, 1)};\n    \\node at (4,-0.25) {(2, 0)};\n\t\\node at (2,-0.25) {(1,0)};\n\t\\node at (0.5,0.7) {$\\frac{3}{8}$};\n\t\\node at (1.7,0.7) {$\\frac{17}{40}$};\n\n\\end{tikzpicture}\n\\end{center}\n\\end{figure}\nWe apply Lemma \\ref{lem:2dim} to the two first cones in the above decomposition and get $\\Psi$ values of $3\/8$ and $17\/40.$ Then applying Lemma \\ref{lem:Psisum}, we get $\\Psi$ value of the last cone is $1\/2.$ Finally, by Property (P1), we get\n\\[ \\alpha(\\bm{v}_3, P) = \\Psi(C_3, {\\mathbb Z}^2) = \\frac{3}{8} + \\frac{17}{40} - \\frac{1}{2} = \\frac{3}{10}.\\]\n\nWe can also verify Equality \\eqref{equ:sum1}:\n\\[\n  \\sum_{\\bm{v} \\in \\operatorname{vert}(P)} \\alpha(\\bm{v}, P) = \\frac{1}{4} + \\frac{9}{20} + \\frac{3}{10} = 1\n\\]\n\\end{ex}\n\nWe finish this part with a formula for computing $\\Psi$ of a $3$-dim unimodular cone, which was computed from Maple code.\n\\begin{lem}\\label{lem:3dim} \nIf $C = \\operatorname{Cone}(\\bm{u}_1,\\bm{u}_2,\\bm{u}_3)$ where $\\bm{u}_1, \\bm{u}_2, \\bm{u}_3$ is a basis for a lattice $\\Lambda,$ then\n\n\n\\[\n\\Psi(C, \\Lambda)= \\displaystyle \\frac{1}{8}+\\frac{1}{24}\\left( \\frac{\\langle \\bm{u}_1, \\bm{u}_2\\rangle}{\\langle \\bm{u}_1, \\bm{u}_1\\rangle}+\\frac{\\langle \\bm{u}_1, \\bm{u}_2\\rangle}{\\langle \\bm{u}_2,\\bm{u}_2\\rangle}+ \\frac{\\langle \\bm{u}_1, \\bm{u}_3\\rangle}{\\langle \\bm{u}_1,\\bm{u}_1\\rangle}+ \\frac{\\langle \\bm{u}_1, \\bm{u}_3\\rangle}{\\langle \\bm{u}_3, \\bm{u}_3\\rangle}+\\frac{\\langle \\bm{u}_3, \\bm{u}_2\\rangle}{\\langle \\bm{u}_2, \\bm{u}_2\\rangle}+ \\frac{\\langle \\bm{u}_3, \\bm{u}_2\\rangle}{\\langle \\bm{u}_3,\\bm{u}_3\\rangle} \\right). \n\\]\n\\end{lem}\n\n\\begin{rem}\nThe formulas for $2$-dim and $3$-dim unimodular cones appear to be simple. However, the apparent simplicity breaks down for dimension 4. The formula for $4$-dim unimodular cones include (way) more than 1000 terms.\n\\end{rem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Generic permutohedron}\\label{sec:generic} \n\nSince the proof of Theorem \\ref{thm:reduction} is completed in the last section, we only need to focus on the BV-$\\alpha$-valuation arising from the regular permutohedron $\\Pi_n,$ or any generic permutohedron. (See Remark \\ref{rem:generic}.) \n\nWe use the following setup. \n\\begin{setup} \\label{setup1}\n  \\begin{ilist}\n\\item Let $\\bm{v} = (v_1, v_2, \\dots, v_{n+1})$ be a vector with strictly increasing entries, and consider the generalized permutohedron \n\\[\\operatorname{Perm}(\\bm{v}) = \\operatorname{Perm}(v_1, v_2, \\dots, v_{n+1}).\\]\n\\item Suppose $\\alpha$ is a construction such that McMullen's formula \\eqref{equ:exterior1} holds and it is symmetric about the coordinates (see Definition \\ref{defn:symmetric}).\n\\end{ilist}\n\\end{setup}\nIt is clear that (i) covers all generic permutoheron, and the BV-$\\alpha$-valuation is a special case of (ii). Under this setup, we will analyze formula for computing $\\mathrm{Lat}(\\operatorname{Perm}(\\bm{v}))$ further, and derive a more combinatorial formula for computing $\\alpha$-values arising from $\\operatorname{Perm}(\\bm{v}).$ \n\nApplying McMullen's formula to $P = \\operatorname{Perm}(\\bm{v}),$ we get\n\\begin{equation}\\label{equ:exteriorpermv}\n\\mathrm{Lat}(\\operatorname{Perm}(\\bm{v})) = \\displaystyle \\sum_{F: \\textrm{ a face of $\\operatorname{Perm}(\\bm{v})$}} \\alpha(F, \\operatorname{Perm}(\\bm{v})) \\ \\mathrm{nvol}(F).\n\\end{equation}\nBecause of the symmetric properties of $\\operatorname{Perm}(\\bm{v})$ and $\\alpha,$ there are a lot of terms in the above summand coincident, and it is natural to group them together. In order to this, we need the following definition and proposition.\n\\begin{defn}\nThe symmetric group ${\\mathfrak S}_{n+1}$ acts linearly on ${\\mathbb R}^{n+1}$ by permuting the coordinates. Two subsets $A_1,A_2 \\subset {\\mathbb R}^{n+1}$ are said to be \\emph{symmetric} if they lie in the same orbit, i.e. if there exist $\\sigma\\in {\\mathfrak S}_{n+1}$ such that $\\sigma(A_1)=A_2$. Since the action is orthogonal, two symmetric sets are congruent, in particular, they have the same volume (if measurable).\n\\end{defn}\n\n\nThe following results are given in \\cite{??}.\n\\begin{prop}\\label{prop:faceposet}\nThere is a one-to-one correspondence between ordered set partitions of $[n+1]$ and faces of $\\operatorname{Perm}(\\bm{v})$ defined as follows:\n\nFor any ordered set partition ${\\mathcal P}= (P_1, P_2, \\cdots, P_l)$ of $[n+1]$, the corresponding face is obtained by maximizing any linear functional given by a vector $\\bm{c} \\in {\\mathbb R}^{n+1}$ with the property that \n\\begin{alist}\n\\item $c_i=c_j$ if $i$ and $j$ are both in $P_k$ for some $k$, and\n  \\item $c_i<c_j$ if $i\\in P_{k_1}$ and $j\\in P_{k_2}$ with $k_1< k_2$.\n\\end{alist}\nLet $m_i=|P_i|$. Then the corresponding face has dimension $n+1-l$ and it is congruent to\n\\[\n\\operatorname{Perm}(\\bm{v}_{M_1})\\times \\operatorname{Perm}(\\bm{v}_{M_2})\\times \\cdots \\operatorname{Perm}(\\bm{v}_{M_l}),\n\\]\nwhere $\\bm{v}_{M_i} = \\left(v_j:\\quad \\displaystyle\\sum_{k=1}^{i-1} m_k < j \\leq  \\sum_{k=1}^i m_k\\right)$. In other words, $\\bm{v}_{M_1}$ consists of the first $m_1$ entries of $\\bm{v}=(v_1,\\dots, v_{n+1}),$ $\\bm{v}_{M_2}$ consists of the next $m_2$ entries, and so on.\n\nWe call the ordered tuple $\\bm{m}:=(m_1,m_2,\\cdots,m_l)$ \\emph{the composition of $\\mathcal P$}. \n\\end{prop}\n\\begin{ex}\\label{ex:compositions} \n  Let $n=5$ and consider the ordered set partition $\\mathcal P = \\left(\\left\\{1,4,6\\right\\}, \\left\\{2,5\\right\\}, \\left\\{3\\right\\}\\right)$. Then the composition of $\\mathcal P$ is $(3,2,1).$ \n  \n  The face of $\\operatorname{Perm}(\\bm{v})$ corresponding to $\\mathcal P$ is the face which optimizes any linear functional $\\bm{c}=(c_1,c_2,c_3,c_4,c_5,c_6)$ with $c_1=c_4=c_6<c_2=c_5<c_3$. \n  In order to figure out this corresponding face, we look for vertices of $\\operatorname{Perm}(\\bm{v})$ optimizing such a functional $\\bm{c}.$ One sees that $v_1$,  $v_2$ and $v_3$ should be in positions 1, 4 and 6 of these vertices, $v_4$ and $v_5$ in positions 2 and 5, and $v_6$ in position 3. Therefore, the desired vertices are  \\[ \\left\\{ \\left(v_{\\mu(1)}, v_{\\tau(1)}, v_6, v_{\\mu(2)}, v_{\\tau(2)}, v_{\\mu(3)}\\right) \\ : \\ \\mu \\in {\\mathfrak S}_{\\{1,2,3\\}}, \\tau \\in {\\mathfrak S}_{\\{4,5\\}} \\right\\}.\n  \\]  \n\n  Hence, we conclude that the face that is corresponding to the ordered set partition $\\mathcal P= \\left(\\left\\{1,4,6\\right\\}, \\left\\{2,5\\right\\}, \\left\\{3\\right\\}\\right)$ is congruent to \n\\[\n\\operatorname{Perm}(v_1,v_2,v_3)\\times\\operatorname{Perm}(v_4,v_5)\\times \\operatorname{Perm}(v_6).\n\\]\n\\end{ex}\n\nBy Proposition \\ref{prop:faceposet}, two faces of $\\operatorname{Perm}(\\bm{v})$ are in the same orbit, i.e. they are symmetric, if and only if their corresponding ordered set partitions have the same composition. Therefore, the orbits of the ${\\mathfrak S}_{n+1}$-action on the faces of $\\operatorname{Perm}(\\bm{v})$ are indexed by compositions $\\bm{m}$ of $n+1$. \nWe denote the orbit corresponding to the composition $\\bm{m}$ by $\\mathcal O_n(\\bm{m}).$\n\n\nFurthermore, under Setup \\ref{setup1}, the construction $\\alpha$ is symmetric about the coordinates. Hence, for any fixed $\\bm{m},$ the value $\\alpha(F, \\operatorname{Perm}(\\bm{v}))$ is a constant on $\\mathcal O_n(\\bm{m}),$ and thus we can define $\\alpha_n(\\bm{m})$ to be this constant.\n\nFinally, a canonical representative of $\\mathcal O_n(\\bm{m})$ is chosen as below.\n\\begin{defn}\nLet $\\bm{m} = (m_1, m_2, \\dots, m_l)$ be a composition of $n+1$. Define an ordered set partition ${\\mathcal P}(\\bm{m})=\\big({\\mathcal P}(\\bm{m})_i\\big)$ where \n\\[\n{\\mathcal P}(\\bm{m})_i=\\left[\\displaystyle\\sum_{k=1}^{i-1} m_k +1, \\sum_{k=1}^i m_k\\right].\n\\]\nIn other words, the first subset $\\mathcal P(\\bm{m})_1$ consists of the first $m_1$ positive integers, the second subset $\\mathcal P(\\bm{m})_2$ consists of the next $m_2$ positive integers, and so on. \n\nThen we define $F_\\bm{m}$ to be the face corresponding to the ordered set partition ${\\mathcal P}(\\bm{m})$ under the bijection given in Proposition \\ref{prop:faceposet}.\n\\end{defn}\n\\begin{ex}\\label{ex:compositions2}\nLet $n=5$ and $\\bm{m}=(3,2,1)$. Then ${\\mathcal P}(\\bm{m})= \\left(\\left\\{1,2,3\\right\\}, \\left\\{4,5\\right\\}, \\left\\{6\\right\\}\\right),$ and \n\\begin{equation}\\label{eqn:exFm}\n  F_\\bm{m} = \\operatorname{conv}\\left\\{ \\left(v_{\\mu(1)}, v_{\\mu(2)}, v_{\\mu(3)}, v_{\\tau(1)}, v_{\\tau(2)}, v_6\\right) \\ : \\ \\mu \\in {\\mathfrak S}_{\\{1,2,3\\}}, \\tau \\in {\\mathfrak S}_{\\{4,5\\}} \\right\\}. \\end{equation}\n\\end{ex}\n\n\nApplying the above discussions to \\eqref{equ:exteriorpermv}, we get\n\\begin{equation}\\label{eqn:compositions}\n\\mathrm{Lat}(\\operatorname{Perm}(\\bm{v}))=\\displaystyle \\sum_{\\textrm{$\\bm{m}:$ composition of $n+1$}} |\\mathcal O_n(\\bm{m})| \\alpha_n(\\bm{m}) \\mathrm{nvol}(F_\\bm{m}).\n\\end{equation}\nNote that one of the terms in the above formula can be explicitly described: For a fixed $\\bm{m}=(m_1,\\cdots, m_l)$, the number of faces in $\\mathcal O_n(\\bm{m})$ is equal to the number of ordered set partitions whose compositions are $\\bm{m}.$ Thus,\n\\begin{equation}\\label{eqn:orbits}\n|\\mathcal O_n(\\bm{m})| = \\binom{n+1}{m_1,m_2,\\cdots, m_l}.\n\\end{equation}\n\nNext, we investigate properties of the face $F_\\bm{m}$. It is easy to see that $F_\\bm{m}$ is always adjacent to the vertex $\\bm{v} = (v_1, \\dots, v_n, v_{n+1}).$ Note that the (pointed) feasible cone of $\\operatorname{Perm}(\\bm{v})$ at $\\bm{v}$ is spanned by the following $n$ vectors:\n\\[ \\bm{e}_1 - \\bm{e}_2, \\bm{e}_2 - \\bm{e}_3, \\dots, \\bm{e}_n - \\bm{e}_{n+1}.\\]\nHence, subsets of these $n$-vectors are in one-to-one correspondence to faces of $\\operatorname{Perm}(\\bm{v})$ that are adjacent to $\\bm{v}.$ \nThis motivates the following definition.\n\n\\begin{defn}\\label{defn:comp2subset}\n  Let $\\bm{m}$ be a composition of $n+1,$ and $F_\\bm{m}$ the face of $\\operatorname{Perm}(\\bm{v})$ that is associated to $\\bm{m}.$ Define $S = \\mathcal S(\\bm{m})$ to be the subset of $[n]$ such that   \\[ \\textrm{affine span of }F_\\bm{m} = \\bm{v} + \\textrm{span}\\{ \\bm{e}_i - \\bm{e}_{i+1} \\ : \\ i \\in S\\}.\\]\n\\end{defn}\n\n\\begin{ex}\n  Let $n=5$ and $\\bm{m}=(3,2,1)$. Then $F_\\bm{m}$ is given by \\eqref{eqn:exFm}. One checks that \n\\[ \\textrm{affine span of }F_\\bm{m} = \\bm{v} + \\textrm{span}\\{ \\bm{e}_1 - \\bm{e}_2, \\bm{e}_2-\\bm{e}_3, \\bm{e}_4 - \\bm{e}_5\\}.\\]\nHence, $S = \\mathcal S(\\bm{m}) = \\{1, 2, 4\\}.$\n\\end{ex}\n\nNote that both compositions of $n+1$ and subsets of $[n]$ are counted by the same number, $2^n.$ So it is natural to ask whether the map $\\bm{m} \\mapsto \\mathcal S(\\bm{m})$ is a bijection between these two families of objects. Indeed, we have the following lemma.\n\n\\begin{lem}\\label{lem:bij}\n  The map $\\mathcal S$ defined in Definition \\ref{defn:comp2subset} is a bijection from compositions of $n+1$ to subsets of $[n].$\n\\end{lem}\n\n\\begin{proof}\nwe can define an inverse to $\\mathcal S$ in the following way: Suppose $S$ is a subset of $[n].$ Let $T = [n+1] \\setminus S.$ Suppose $T = \\{t_1 < t_2 < \\cdots < t_l\\}.$ Then one verifies that \n\\[ S \\mapsto \\bm{m} := (t_1, t_2-t_1, \\dots, t_l - t_{l-1})\\]\nis an inverse to $\\mathcal S,$ completing the proof. \n\\end{proof}\n\nThe above lemma tells us that the term $F_\\bm{m}$ appearing in the summand of Formula \\eqref{eqn:compositions} can also be understood as faces of $\\operatorname{Perm}(\\bm{v})$ that are adjacent to the vertex $\\bm{m}.$ Furthermore, abusing notation we can use subsets of $[n]$ to index Formula \\eqref{eqn:compositions}:\n\\begin{equation}\\label{eqn:subsets}\n\\mathrm{Lat}(\\operatorname{Perm}(\\bm{v})) = \\displaystyle \\sum_{S\\subseteq[n]} |\\mathcal O_n(S)| \\alpha_n(S) \\mathrm{nvol}(F_S).\n\\end{equation}\nNote that $\\dim(F_S)=|S|$.\nIn fact, for arguments we will carry out in both Section \\ref{sec:evidence} and Section \\ref{sec:unique}, it is more convenient to use subsets of $[n]$ as our indexing. \n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Evidence to conjectures}\\label{sec:evidence}\nWe will explore consequences of Formula \\eqref{eqn:subsets} further in Section \\ref{sec:unique}, and will only focus on computing $\\alpha_n(S)$ for some special cases in this section. \nThe main results of this section are the following two theorems, providing evidence to our conjectures. \n\n\\begin{thm}\\label{thm:truelowdim}\nFor all $n \\le 6,$ the regular permutohedron $\\Pi_n$ is BV-$\\alpha$-positive. \n\nTherefore, all the integral generalized permutohedra (including matroid base polytopes) of dimension at most $6$ are Ehrhart positive.\n\\end{thm}\n\n\n\\begin{thm}\\label{thm:34coeff}\nFor any $n,$ and any face $F$ of $\\Pi_n$ of codimension $2$ or $3,$ we have $\\alpha(F, \\Pi_n)$ is positive, where $\\alpha$ is the BV-$\\alpha$-valuation.\n\nHence, the third and fourth coefficients of the Ehrhart polynomial of any integral generalized permutohedra (including matroid base polytopes) are positive. \n\\end{thm}\n\nClearly, in order to prove the above two theorems, we just need to verify the following two statements respectively, assuming $\\alpha$ is the BV-$\\alpha$-valuation:\n\\begin{align}\n  \\alpha_n(S) > 0, \\quad &\\forall S \\subseteq [n], \\forall n \\le 6, \\label{equ:smalldim} \\\\\n  \\alpha_n(S) >0, \\quad &\\forall S \\subset [n], |S| = n-2, n-3. \\label{equ:34coeff}\n\\end{align}\n\n\n\\subsection{How to Compute $\\alpha_n(S)$} \\label{subsec:computealpha}\nWe first describe how to compute $\\alpha_n(S)$. Let $V$ be the $n$-dimensional subspace of ${\\mathbb R}^{n+1}$ determined by $x_1 + x_2 + \\cdots + x_{n+1} = 0.$ For any $S \\subseteq [n],$ let $L_S := \\textrm{span} \\{ \\bm{e}_i - \\bm{e}_{i+1} \\ : \\ i \\in S\\}.$ By Definition \\ref{defn:comp2subset}, \n\\[ \\textrm{affine span of }F_S = \\bm{v} + L_S.\\]\nTherefore, applying Formula \\eqref{equ:defnalpha1} to our situation, we get\n\\begin{equation}\\label{equ:alphaS}\n\\alpha_n(S) = \\Psi(\\operatorname{fcone}^p(F_S, \\Pi_n), \\Lambda_S),\n\\end{equation}\nwhere \n\\[ \\Lambda_S := (V \\cap {\\mathbb Z}^{n+1})\/L_S.\\]\nObserve that $\\{e_i - e_{i+1} \\ : \\ i \\in [n]\\}$ is a unimodular basis for $V.$ Hence, $\\Lambda_S$ is the lattice spanned by projections of vectors in \n\\[ \\{\\bm{e}_i - \\bm{e}_{i+1} \\ : \\ i \\in [n] \\setminus S\\} \\]\nto $L_S^\\perp,$ the orthogonal complement of $L_S$ in $V.$\nFurthermore, $\\operatorname{fcone}^p(F_S, \\Pi_n)$ is the span of these projection vectors\nWe then apply directly the Berline Vergne's recursive definition of $\\Psi$ to \\eqref{equ:alphaS} to find $\\alpha_n(S).$ \n\n\n\n\\subsection{Small dimensions}\nWe now prove \\eqref{equ:smalldim} by computing $\\alpha_n(S)$ directly, which implies Theorem \\ref{thm:truelowdim}.\n\nWhen $n=1,2$, $\\alpha_n(S)$ corresponds to $\\alpha(F, \\Pi_n)$ where $F$ is either $\\Pi_n$ or a facet of $\\Pi_n.$ Thus, the positivity of $\\alpha_n(S)$ follows from Lemma \\ref{lem:sum}. Hence, we only need to compute $\\alpha_n(S)$ for $3 \\le n \\le 6.$ We use the procedure described above to calculate the values of these $\\alpha_n(S)$ and summarize in the examples below.\n\n\\begin{ex}\nFor $n=3:$ \n  \n\\begin{tabular}{ | c || c | c | c | c | c | c | c | c | c | c | c | c | c | c | c | c | }\n    \\hline\n    $S$ & $\\emptyset$ & 1 & 2 & 3 & 12 & 13 & 23 & 123  \\\\ \\hline\n    $\\alpha_3(S)$  & 1\/24 & 11\/72 & 7\/36 & 11\/72 & 1\/2 & 1\/2 & 1\/2 & 1  \\\\ \\hline\n   \n  \\end{tabular}\n \n\n\\end{ex}\n\n\\begin{ex}\nFor $n=4:$\n\n  \\begin{tabular}{ | c || c | c | c | c | c | c | c | c | c | c | c | c | c | c | c | c | }\n    \\hline\n    $S$ & $\\emptyset$ & 1 & 2 & 3 & 4 & 12 & 13 & 14   \\\\ \\hline\n    $\\alpha_4(S)$  & 1\/120 & 5\/144 & 7\/144 & 7\/144 & 5\/144 & 7\/48 & 13\/72 & 5\/36 \\\\ \\hline\n  \\end{tabular}\n \n\n  \\begin{tabular}{| c || c | c | c | c | c | c | c | c | c |}\n  \\hline\n   $S$ & 23 & 24 & 34 & 123 & 124 & 134 & 234 & 1234 \\\\ \\hline\n   $\\alpha_4(S)$ & 5\/24 & 13\/72 & 7\/48 & 1\/2 & 1\/2 & 1\/2 & 1\/2 & 1 \\\\ \n   \\hline\n  \\end{tabular}\n\n\\end{ex}\n\n\\begin{ex}\\label{ex:pi5}\nFor $n=5:$\n\t\n  \\begin{tabular}{| c || c | c | c | c | c | c |}\n  \\hline\n  $S$ & $\\emptyset$ & 1 & 2 & 3 & 4 & 5 \\\\ \\hline\n  $\\alpha_5(S)$  & 1\/720 & 137\/21600 & 101\/10800 & 37\/3600 & 101\/10800 & 137\/21600 \\\\\n  \\hline\n  \\end{tabular}\n\n  \\begin{tabular}{| c || c | c | c | c | c | c | c | c | c | c |}\n  \\hline  \n  $S$ & 12 & 13 & 14 & 15 & 23 & 24 & 25 & 34 & 35 & 45 \\\\ \\hline\n  $\\alpha_5(S)$  & 1\/32 & 1\/24 & 1\/24 & 1\/36 & 5\/96 & 1\/18 & 1\/24 & 5\/96 & 1\/24 & 1\/32 \\\\\n  \\hline\n  \\end{tabular}\n  \n  \\begin{tabular}{| c || c | c | c | c | c | c | c | c | c | c |}\n  \\hline\n  $S$ & 123 & 124 & 125 & 134 & 135 & 145 & 234 \\\\ \\hline\n  $\\alpha_5(S)$  & 17\/120 & 31\/180 & 19\/144 & 47\/240 & 1\/6 & 19\/144 & 13\/60  \\\\\n  \\hline\n  \\end{tabular}\n  \n \\begin{tabular}{| c || c | c | c | c | c | c | c | c | c | c | c |}\n  \\hline  \n  $S$ & 235 & 245 & 345 & 1234 & 1235 & 1245 & 1345 & 2345 & 12345\\\\ \\hline\n  $\\alpha_5(S)$ & 47\/240 & 31\/180 & 17\/120  & 1\/2 & 1\/2 & 1\/2 & 1\/2 & 1\/2 & 1\\\\\n  \\hline\n  \\end{tabular} \n\\end{ex}\n\n\\begin{ex}\\label{ex:n6}\nFor $n=6:$\n\t\n  \\begin{tabular}{| c || c | c | c | c | c | c | c |}\n  \\hline\n  $S$ & $\\emptyset$ & 1 & 2 & 3 & 4 & 5 & 6 \\\\ \\hline\n  $\\alpha_6(S)$ & 1\/5040  & 7\/7200 & 1\/675 & 37\/21600 & 37\/21600 & 1\/675 & 7\/7200 \\\\\n  \\hline\n  \\end{tabular}\n\n  \\begin{tabular}{| c || c | c | c | c | c | c | c | c | c | c | c | c | c | c | c |}\n  \\hline  \n  $S$ & 12 & 13 & 14 & 15 & 16 & 23   \\\\ \\hline\n  $\\alpha_6(S)$  & 29\/5400 & 1\/135 & 541\/64800 & 149\/21600 & 151\/32400 & 211\/21600   \\\\\n  \\hline\n  \\end{tabular}\n  \n  \\begin{tabular}{| c || c | c | c | c | c | c | c | c |}\n  \\hline\n  $S$ & 24 & 25 & 26 & 34 & 35 & 36   \\\\ \\hline\n  $\\alpha_6(S)$ & 719\/64800 & 181\/16200 & 149\/21600 & 41\/3600 & 719\/64800 & 541\/64800   \\\\ \\hline \n\n  \\end{tabular}  \n  \n  \\begin{tabular}{| c || c | c | c | c | c | c | c | c | c | c | c | c | c | c | c | c | c | c | c | c | c | c | c | c |}\n  \\hline\n  $S$ & 45 & 46 & 56 & 123 & 124 & 125 & 126 & 134 & 135  \\\\ \\hline\n  $\\alpha_6(S)$ & 211\/21600 & 1\/135 & 29\/5400 & 7\/240 & 3\/80 & 11\/288 & 7\/288 & 11\/240  & 7\/144  \\\\\n  \\hline\n  \\end{tabular}\n  \n  \\begin{tabular}{| c || c | c | c | c | c | c | c | c | c | c | c | c | c | c | c | c |}\n  \\hline\n  $S$ & 136 & 145 & 146  & 156 & 234 & 235 & 236 \\\\ \\hline\n  $\\alpha_6(S)$ & 5\/144 & 13\/288 & 5\/144 & 7\/288 & 13\/240 & 17\/288 & 13\/288  \\\\ \\hline\n  \\end{tabular}\n\n  \\begin{tabular}{| c || c | c | c | c | c | c | c | c | c | c | c | c | c | c | c | c |}\n  \\hline\n  $S$ & 245 & 246 & 256 & 345 & 346 & 356 & 456\\\\ \\hline\n  $\\alpha_6(S)$ & 17\/288 & 7\/144 & 11\/288 & 13\/240 & 11\/240 & 3\/80 & 7\/240 \\\\ \\hline\n  \\end{tabular}  \n  \n \\begin{tabular}{| c || c | c | c | c | c | c | c | c | c | c | c | c | c | c | c |}\n  \\hline  \n  $S$ & 1234 & 1235 & 1236 & 1245 & 1246 & 1256 & 1345 & 1346 \\\\ \\hline\n  $\\alpha_6(S)$  & 5\/36 & 1\/6 & 23\/180 & 3\/16 & 19\/120 & 1\/8 & 37\/180 & 11\/60\\\\\n  \\hline\n  \\end{tabular} \n  \n  \\begin{tabular}{| c || c | c | c | c | c | c | c | c | c | c | c | c | c | c | c |}\n  \\hline\n  $S$ & 1356 & 1456 & 2345 & 2346 & 2356 & 2456 & 3456\\\\ \\hline\n  $\\alpha_6(S)$  & 19\/120 & 23\/180 & 2\/9 & 37\/180 & 3\/16 & 1\/6 & 5\/36\\\\\n  \\hline\n  \n  \\end{tabular}\n  \n  \\begin{tabular}{| c || c | c | c | c | c | c | c | c | c | c | c | c | c | c | c |}\n  \\hline\n  $S$ & 12345 & 12346 & 12356 & 12456 & 13456 & 23456 & 123456\\\\ \\hline\n  $\\alpha_6(S)$  & 1\/2 & 1\/2 & 1\/2 & 1\/2 & 1\/2 & 1\/2 & 1\\\\\n  \\hline\n  \n  \\end{tabular}\n  \n\\end{ex}\n\nOne sees in all the examples above that $\\alpha_n(S)$ is always positive for $n \\le 6.$ Hence, we complete our proof for Theorem \\ref{thm:truelowdim}.\n\n\\begin{rem}\n  By Remark \\ref{rem:upton},  if $\\alpha_6(S)$ is positive for all $S$, then we immediately have $\\alpha_n(S) >0$ for any $n < 6$ and any $S$. Hence, it is enough to use Example \\ref{ex:n6} to prove Theorem \\ref{thm:truelowdim}. We include all the other examples so that we have more data for $\\alpha_n(S),$ which might be helpful for future work. \n\\end{rem}\n\n\n\n\n\\subsection{Top coefficients}\n\n\nWe then prove \\eqref{equ:34coeff}, which implies Theorem \\ref{thm:34coeff}.\n\nWe will again apply the procedure described in Subsection \\ref{subsec:computealpha} to compute $\\alpha_n(S)$.\n\nWe start by considering the cases where $|S| = n-2.$ Suppose $[n] \\setminus S = \\{i,j\\}$ with $i < j.$ Then $L_S = \\textrm{span}(\\bm{e}_k - \\bm{e}_{k+1}: k \\neq i, j).$ For $k=i,j,$ let $\\bm{u}_k$ be the projection of $\\bm{e}_k - \\bm{e}_{k+1}$ to $L_S^\\perp$, the orthogonal complement of $L_S.$ Then \n\\begin{eqnarray*}\n  \\bm{u}_i &=& \\left( \\underbrace{\\frac{1}{i}, \\frac{1}{i}, \\cdots, \\frac{1}{i}}_i, \\underbrace{\\frac{-1}{j-i}, \\frac{-1}{j-i}, \\cdots, \\frac{-1}{j-i}}_{j-i}, \\underbrace{0,0, \\cdots, 0 }_{n+1-j} \\right), \\\\\n  \\bm{u}_j &=& \\left( \\underbrace{0, 0, \\cdots, 0}_i, \\underbrace{\\frac{1}{j-i}, \\frac{1}{j-i}, \\cdots, \\frac{1}{j-i}}_{j-i}, \\underbrace{\\frac{-1}{n+1-j},\\frac{-1}{n+1-j}, \\cdots, \\frac{-1}{n+1-j}}_{n+1-j} \\right).\n\\end{eqnarray*}\nNote that $\\{\\bm{u}_i, \\bm{u}_j\\}$ is a basis of $\\Lambda_S$ and also spans $\\operatorname{fcone}^p(F_S, \\Pi_n).$\nTherefore, applying Lemma \\ref{lem:2dim}, we obtain a precise formula for $\\alpha_n(S).$\n\n\\begin{lem}\\label{lem:3coeff}\n  Suppose $S \\subset [n]$ such that $[n] \\setminus S =\\{i,j\\}$ with $i < j.$ Then\n  \\begin{equation}\\label{equ:alphaS3}\n\\alpha_n(S) = \\frac{1}{4} - \\frac{1}{12}\\left(\\frac{i}{j} + \\frac{n+1-j}{n+1-i}\\right)\n\\end{equation}\n\\end{lem}\n \n\nWe repeat the same procedure for the cases where $|S| = n-3.$ Suppose $[n] \\setminus S = \\{i,j,k\\}$ with $i < j <k.$ Then $L_S = \\textrm{span}(\\bm{e}_l - \\bm{e}_{l+1}: l \\neq i, j).$ For $l=i,j,k,$ let $\\bm{u}_l$ be the projection of $\\bm{e}_l - \\bm{e}_{l+1}$ to $L_S^\\perp$, the orthogonal complement of $L_S.$ Then \n\\begin{eqnarray*}\n  \\bm{u}_i &=& \\left( \\underbrace{\\frac{1}{i}, \\frac{1}{i}, \\cdots, \\frac{1}{i}}_i, \\underbrace{\\frac{-1}{j-i}, \\frac{-1}{j-i}, \\cdots, \\frac{-1}{j-i}}_{j-i}, \\underbrace{0,0, \\cdots, 0 }_{n+1-j} \\right), \\\\\n  \\bm{u}_j &=& \\left( \\underbrace{0,\\cdots, 0}_i, \\underbrace{\\frac{1}{j-i},\\cdots, \\frac{1}{j-i}}_{j-i}, \\underbrace{\\frac{-1}{k-j}\\cdots, \\frac{-1}{k-j}}_{k-j}, \\underbrace{0,\\cdots ,0}_{n+1-k} \\right),\\\\\n  \\bm{u}_k&=& \\left( \\underbrace{0,\\cdots, 0}_j, \\underbrace{\\frac{1}{k-j},\\cdots, \\frac{1}{k-j}}_{k-j},\\underbrace{\\frac{-1}{n+1-k}\\cdots, \\frac{-1}{n+1-k}}_{n+1-k} \\right).\n\\end{eqnarray*}\nHere $\\{\\bm{u}_i, \\bm{u}_j, \\bm{u}_k\\}$ is a basis of $\\Lambda_S$ and also spans $\\operatorname{fcone}^p(F_S, \\Pi_n).$\nTherefore, applying Lemma \\ref{lem:3dim}, we obtain a precise formula for $\\alpha_n(S).$\n\\begin{lem}\\label{lem:4coeff}\nSuppose $S \\subset [n]$ such that $[n] \\setminus S =\\{i,j,k\\}$ with $i < j<k.$ Then\n\\begin{equation}\\label{equ:alphaS4}\n\\alpha_n(S) = \\frac{1}{8} - \\frac{1}{24}\\left(\\frac{i}{j} +1+ \\frac{n+1-k}{n+1-j}\\right)\n\\end{equation}\n\\end{lem}\n\n\n\n  It is easy to check that $\\alpha_n(S)$ are positive in both \\eqref{equ:alphaS3} and \\eqref{equ:alphaS4}. Hence, Theorem \\ref{thm:34coeff} follows.\n\n  \\begin{rem}\n     Similar as to the proof of Theorem \\ref{thm:truelowdim}, it is not necessary to prove $\\alpha_n(S)$ for both $|S|=n-2$ and $|S|=n-3.$ In fact, it follows from Remark \\ref{rem:upton} that if $\\alpha_n(S) >0$ for all $n$ and $S$ of size $n-3,$ then $\\alpha_n(S)>0$ for all $n$ and $S$ of size $n-2.$ We again include the cases of $|S|=n-2$ to provide more data of $\\alpha_n(S).$\n  \\end{rem}\n\n\n\\section{Uniqueness}\\label{sec:unique}\nIn this section, we take a different point of view and investigate the uniqueness of the $\\Psi$\/$\\alpha$ constructions for McMullen's formula. \nWe will apply the mixed valuation theories introduced in Subsection \\ref{subsec:mixed} to Minkowski sums of \\emph{hypersimplices}.\n\\begin{defn}\n  The \\emph{hypersimplex} $\\Delta_{k,n+1}$ is defined as\n  \\[ \\Delta_{k,n+1} = \\operatorname{Perm} ( \\underbrace{0,\\cdots,0}_{n+1-k}, \\underbrace{1,\\cdots,1}_{k} ).\\]\n\\end{defn}\nThe main result of this section is that the $\\alpha$-values of faces of $\\operatorname{Perm}(\\bm{v})$ are uniquely determined as a scalar of mixed valuation of hypersimplices if we require $\\alpha$ and $\\bm{v}$ to be given under Setup \\ref{setup1} (Theorem \\ref{thm:uniqueness}). Furthermore, as a consequence of this result, we give an equivalent statement of Conjecture \\ref{conj:alphas} in Corollary \\ref{cor:equimixed}.\n\n\nAs in Setup \\ref{setup1}, we consider the generalized permutohedron $\\operatorname{Perm}(\\bm{v}) = \\operatorname{Perm}(v_1,v_2,\\cdots, v_{n},v_{n+1})$ with $v_1<v_2<\\cdots<v_{n}<v_{n+1}$.\nWe have the following expression as Minkowski sum \\cite[Section 16]{bible}. \n\\[\n\\operatorname{Perm}(\\bm{v}) = w_1\\Delta_{1,n+1} + w_2\\Delta_{2,n+1} + \\cdots + w_{n}\\Delta_{n,n+1},\n\\]\nwhere \n\\begin{equation}\\label{equ:defnwi}\nw_i := v_{i+1}-v_i \\text{ for } i=1,2,\\dots,n.\n\\end{equation}\n(The $w_i$'s are actually edge lengths of edges of $\\operatorname{Perm}(\\bm{v}).$ But this is not relevant to our discussion.)\n\nUsing the results on mixed volumes -- Theorems \\ref{thm:mixedvol} and \\ref{thm:mixvolprop} -- we have the following: \n\\begin{lem}\\label{lem:voledges}\n  The normalized volume of $\\operatorname{Perm}(\\bm{v})$ is a homogeneous polynomial in $w_i$'s with strictly positive coefficients. \n\\end{lem}\n\n\nIn \\cite{bible}, the coefficients of the above homogeneous polynomial are called \\emph{mixed Eulerian numbers}, and some basic properties are established. One of the properties is the following: \n\\begin{lem}\\label{lem:squarefree}\n  The coefficient of $w_1w_2\\cdots w_n$, the unique squarefree monomial, in the homogeneous polynomial assumed in Lemma \\ref{lem:voledges} is $n!$\n\\end{lem}\n Note that Lemma \\ref{lem:squarefree} is equivalent to \n \\begin{eqnarray}\n\t & &\\sum_{\\sigma \\in {\\mathfrak S}_n} \\mathrm{\\mathcal{M}Lat}^n(\\Delta_{\\sigma(1), n+1}, \\Delta_{\\sigma(2),n+1}, \\dots, \\Delta_{\\sigma(n),n+1}) = n! \\nonumber \\\\\n &\\Longleftrightarrow&\n  \\mathrm{\\mathcal{M}Lat}^n(\\Delta_{1,n+1},\\Delta_{2,n+1},\\dots, \\Delta_{n,n+1}) = 1,\n\\label{equ:mixedhypersimplices}\n\\end{eqnarray}\nwhere the ``$\\Longleftrightarrow$'' follows from \\eqref{equ:permprop}.\n\nRecall in Section \\ref{sec:generic}, we associate a face $F_\\bm{m}$ of $\\operatorname{Perm}(\\bm{v})$ to any composition $\\bm{m}$ of $n+1,$ establish a bijection $\\mathcal S$ from $\\bm{m}$ to subsets $S$ of $[n]$, and rewrite $F_\\bm{m}$ as $F_S.$ We have the following result on the normalized volume of $F_S.$ \n\n\\begin{prop}\\label{prop:nvolumes}\n  Suppose $P=\\operatorname{Perm}(\\bm{v})$ and $S\\subseteq [n]$. Let $F_S$ be the corresponding face of $P$ as defined in Section \\ref{sec:generic}, and $\\bm{m}=(m_1,\\cdots, m_{l}) := \\mathcal S^{-1}(S)$ is the composition in bijection to $S$. Then $\\mathrm{nvol}(F_S)$ is a homogenenous polynomial in $\\{w_i \\ : \\ i \\in S\\},$ whose coefficient of $\\prod_{i \\in S} w_i$-- the unique squarefree monomial -- is \n  \\begin{equation}\\label{equ:CnS}\nC_n(S) :=(m_1-1)!(m_2-1)!\\cdots (m_{l}-1)!,\n\\end{equation}\n\\end{prop}\n\n\\begin{proof}\n  By Proposition \\ref{prop:faceposet}, the face $F_S$ is congruent to\n  \\[\n\\operatorname{Perm}(\\bm{v}_{M_1})\\times \\operatorname{Perm}(\\bm{v}_{M_2})\\times \\cdots \\operatorname{Perm}(\\bm{v}_{M_l}),\n\\]\nwhere $\\bm{v}_{M_i} = \\left(v_j:\\quad \\displaystyle\\sum_{k=1}^{i-1} m_k < j \\leq  \\sum_{k=1}^i m_k\\right)$.\n \nHence, $\\mathrm{nvol}(F_S) = \\prod_{i=1}^k \\mathrm{nvol}(\\operatorname{Perm}(\\bm{v}_{M_i})).$\nLet \n\\[ T_i := \\left\\{ j \\ : \\ \\sum_{k=1}^{i-1} m_k < j < \\sum_{k=1}^i m_k \\right\\}.\\]\nThen by Lemmas \\ref{lem:voledges} and \\ref{lem:squarefree}, the normalized volume of $\\operatorname{Perm}(\\bm{v}_{M_i})$ is a homogeneous polynomial in $\\{ w_j \\ : \\ j \\in T_i\\},$ and the coefficient of $\\prod_{j \\in T_i} w_j$ -- the unique squarefree monomial -- in this homogeneous polynomial is $(m_i-1)!.$ Therefore, the conclusion follows if we can show that $S = \\cup_{i=1}^l T_i.$ However, by the proof of Lemma \\ref{lem:bij}, the complement of $S$ with respect to $[n+1]$ is \n\\[ [n+1] \\setminus S = \\left\\{ \\sum_{k=1}^i m_i \\ : \\ i=1,2, \\dots, l\\right\\},\\]\nwhich is exactly $[n+1] \\setminus \\cup_{i=1}^l T_i$. Hence, we are done.  \n\\end{proof}\n\nThe following is the main result of this section.\n\\begin{thm}\\label{thm:uniqueness}\n  Suppose $\\alpha$ and $\\bm{v}$ are as in Setup \\ref{setup1}.  Then the $\\alpha$ values of faces of $\\operatorname{Perm}(\\bm{v})$ are uniquely determined. More specifically, if $S = \\{s_1, s_2, \\dots, s_k\\},$ we have\n  \\begin{equation}\\label{equ:alpha0}\n\\alpha_n(S) = \\frac{1}{C_n(S)|\\mathcal O_n(S)|}k!\\mathrm{\\mathcal{M}Lat}^{k}(\\Delta_{s_1,n+1},\\Delta_{s_2,n+1},\\cdots,\\Delta_{s_k,n+1}),\n\\end{equation}\nwhere $C_n(S)$ is defined in \\eqref{equ:CnS}.\n\nIn particular the above formula applies to the BV-$\\alpha$-valuation.\n\\end{thm}\n\nOne sees that the above theorem gives a connection between the $\\alpha$ arising from the regular permutohedron and the mixed lattice points counter function $\\mathrm{\\mathcal{M}Lat}^k$ on hypersimplices. Therefore, we have the following:\n\\begin{cor}\\label{cor:equimixed}\n  The following are equivalent:\n\\begin{enumerate}\n  \\item For any $S =\\{s_1, \\dots, s_k\\} \\subseteq [n],$ we have\n  \\[\\mathrm{\\mathcal{M}Lat}^{k}(\\Delta_{s_1,n+1},\\Delta_{s_2,n+1},\\cdots,\\Delta_{s_k,n+1}).\\]\n\\item The regular permutohedron $\\Pi_n$ is BV-$\\alpha$-positive.\n\\end{enumerate} \n  \\end{cor}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:uniqueness}]\n  Let $w_i$ be defined as in \\eqref{equ:defnwi}.\nTheorem \\ref{thm:mixedLat} or Theorems \\ref{thm:mixed} and \\ref{thm:decomposeLat} tell us that the number of lattice points in\n\\[\\operatorname{Perm}(\\bm{v}) =  w_1\\Delta_{1,n+1} + w_2\\Delta_{2,n+1} + \\cdots + w_{n}\\Delta_{n,n+1}\\] \nis a polynomial in the $w_i$ variables.\nWe denote this polynomial by $E = E(w_1, w_2,\\dots, w_n).$ We focus on the coefficient of squarefree monomials $w_S :=\\prod_{i\\in S}w_i$ in $E.$ On the one hand, by \\eqref{equ:mixed} and \\eqref{equ:permprop}, this coefficient is equal to\n\\begin{equation}\\label{eqn:coeff1}\n  k!\\mathrm{\\mathcal{M}Lat}^{k}(\\Delta_{s_1,n+1},\\Delta_{s_2,n+1},\\cdots, \\Delta_{s_k,n+1}).\n\\end{equation}\n\nNext by Equation \\eqref{eqn:subsets}, we have\n\\[\nE(w_1, \\dots, w_n) = \\displaystyle \\sum_{S\\subset[n]} |\\mathcal O_n(S)|\\alpha_n(S)\\mathrm{nvol}(F_S).\n\\]\n(Proposition \\ref{prop:nvolumes} guarantees that the right hand side of the above expression is indeed polynomial on the $w_i$ variables.)\nNote that according to Proposition \\ref{prop:nvolumes}, the only contribution to the monomial $w_S=\\prod_{i\\in S}w_i$ in the summand above is the term corresponding to $S$, and it is given by $C_n(S)$. Using these, we conclude that the coefficient of $w_S$ in $E(w_1,\\dots, w_n)$ is\n\\begin{equation}\\label{eqn:coeff2}\n\\alpha_n(S)|\\mathcal O_n(S)|C_n(S)\n\\end{equation}\nFinally, our two expressions, \\eqref{eqn:coeff1} and \\eqref{eqn:coeff2}, for the coefficient of $w_S$ in $E$ has to agree. Hence, the conclusion follows. \\end{proof}\n\n\n\n\n\n\nProposition \\ref{prop:nvolumes} and Equation \\eqref{eqn:orbits} allow us to give an explicit formula for $|\\mathcal O_n(S)|C_n(S)$ using the bijection of Definition \\ref{defn:comp2subset}.\n\\begin{equation}\\label{equ:OC}\n\\frac{1}{|\\mathcal O_n(S)|C_n(S)} = \\frac{m_1\\cdot m_2 \\cdots  m_{l}}{(n+1)!}.\n\\end{equation}\nHence, we rewrite Equation \\eqref{equ:alpha0} as\n  \\begin{equation}\\label{equ:alpha}\n\\alpha_n(S) = \\frac{m_1\\cdot m_2 \\cdots  m_{l}}{(n+1)!} k!\\mathrm{\\mathcal{M}Lat}^{k}(\\Delta_{s_1,n+1},\\Delta_{s_2,n+1},\\cdots,\\Delta_{s_k,n+1}).\n\\end{equation}\n\n\n\nFormula \\eqref{equ:alpha} allows us to compute some examples of $\\alpha_n(S)$.\n\n\\begin{ex}\n\tConsider $S=[n]$ as a subset of $[n+1]$. Its corresponding composition is $\\bm{m} = \\mathcal S^{-1}(S) = (n+1)$, and $F_S = \\operatorname{Perm}(\\bm{v}).$ By Equation \\eqref{equ:leading}, we have $\\alpha_n(S) = 1.$\n\tTherefore, Formula \\eqref{equ:alpha} gives: \n\t\\[ \\frac{n+1}{(n+1)!}n!\\mathrm{\\mathcal{M}Lat}^n(\\Delta_{1,n+1},\\Delta_{2,n+1},\\dots, \\Delta_{n,n+1}) = 1,\\]\n\tagreeing with \\eqref{equ:mixedhypersimplices}.\n\\end{ex}\n\\begin{ex}\n\tConsider $n=3$ and $S =\\{1,3\\} \\subseteq [3].$ The corresponding composition is $\\bm{m} = (2,2)$. Applying \\eqref{equ:alpha}, we get \n\t\\[ \\alpha_3(\\{1,3\\}) = \\frac{2\\cdot 2}{24} \\ 2! \\mathrm{\\mathcal{M}Lat}^2(\\Delta_{1,4}, \\Delta_{3,4}).\\]\n\tFurthermore, Theorem \\ref{thm:mobius} provides a way to compute mixed valuations:\n\t\\[ 2!\\mathrm{\\mathcal{M}Lat}^2(\\Delta_{1,4},\\Delta_{3,4}) = \\mathrm{Lat}^2(\\Delta_{1,4}+\\Delta_{3,4})-\\mathrm{Lat}^2(\\Delta_{1,4}) - \\mathrm{Lat}^2(\\Delta_{1,4}).\\]\n\tBy the comment after Theorem \\ref{thm:decomposeLat}, for any polytope $\\mathrm{Lat}^r(P)$ is the coefficient of $t^r$ in the Ehrhart polynomial $i(P,t).$ Hence, we can figure out the terms in the right hand side of the above equality by computing corresponding Ehrhart polynomials:\n\\begin{eqnarray*}\ni\\left(\\Delta_{14}+\\Delta_{34},t\\right) &=& \\frac{10}{3}t^3 + 5t^2 + \\frac{11}{3}t + 1,   \\\\ \ni\\left(\\Delta_{14},t\\right) &=& \\frac{1}{6}t^3 + t^2 + \\frac{11}{6}t +1, \\\\\ni\\left(\\Delta_{34},t\\right) &=& \\frac{1}{6}t^3 + t^2 + \\frac{11}{6}t +1.\n\\end{eqnarray*} \nTherefore,\n\\begin{align*}\n2!\\mathrm{\\mathcal{M}Lat}^2(\\Delta_{1,4},\\Delta_{3,4}) =&  5 - 1  - 1 =3, \n\\intertext{and} \n\\alpha_3(\\{1,3\\}) =& \\frac{2 \\cdot 2}{24} \\cdot 3 = \\frac{1}{2},\n\\end{align*}\nwhich agrees with \\eqref{equ:facet} since $F_{\\{1,3\\}}$ is a facet. \n\\end{ex}\n\n\\begin{ex}\n\tConsider $S =\\{i\\}$. Then $F_S$ is an edge, and the corresponding composition $\\bm{m}$ consists of one copy of $2$ and $n-1$ copies of $1$. Hence,\n\\[\n\\alpha_n(\\left\\{i\\right\\}) = \\frac{2}{(n+1)!}\\mathrm{Lat}^1(\\Delta_{i,n+1}).\n\\]\nWe now use the above formula to compute $\\alpha_5(\\{i\\})$ which are $\\alpha(E, \\Pi_5)$ for edges $E$ of $\\Pi_5.$\nAgain, $\\mathrm{Lat}^1(\\Delta_{i,6}$ is the linear coefficient of the Ehrhart polynomial of $\\Delta_{i,6}.$ So we compute these polynomials:\n\\begin{eqnarray*}\ni(\\Delta_{1,6},t) &=& \\frac{1}{120}t^5 + \\frac{1}{8}t^4 + \\frac{17}{24}t^3 + \\frac{15}{8}t^2 + \\frac{137}{60}t + 1,\\\\ \ni(\\Delta_{2,6},t) &=& \\frac{13}{60}t^5 + \\frac{3}{2}t^4 + \\frac{47}{12}t^3 + 5t^2 + \\frac{101}{30}t + 1,\\\\\ni(\\Delta_{3,6},t) &=& \\frac{11}{20}t^5 + \\frac{11}{4}t^4 + \\frac{23}{4}t^3 + \\frac{25}{4}t^2 + \\frac{37}{10}t + 1.\n\\end{eqnarray*}\nTherefore,\n\\begin{eqnarray*}\n\\alpha_5(\\left\\{1\\right\\}) = \\frac{2}{6!}\\mathrm{Lat}^1(\\Delta_{1,6}) &=& \\frac{2}{720}\\frac{137}{60} = \\frac{137}{21600}, \\\\\n\\alpha_5(\\left\\{2\\right\\}) = \\frac{2}{6!}\\mathrm{Lat}^1(\\Delta_{2,6}) &=& \\frac{2}{720}\\frac{101}{30} = \\frac{101}{10800},\\\\\n\\alpha_5(\\left\\{3\\right\\}) = \\frac{2}{6!}\\mathrm{Lat}^1(\\Delta_{3,6}) &=& \\frac{2}{720}\\frac{37}{10} = \\frac{37}{3600},\n\\end{eqnarray*}\nagreeing with Example \\ref{ex:pi5}. \n\\end{ex}\n\n\n\\section{Toric Perspective}\\label{sec:toric}\n\n\nMcMullen's originally showed that there are infinitely many solutions to McMullen's formula. In principle, even if Berline-Vergne's construction is not positive on the braid arrangement fan ${\\mathfrak B}_n$, there may be another construction that is positive. Our main purpose in this section is to prove Proposition \\ref{prop:toddequiv}, which says in particular that if any construction coming from Pommersheim-Thomas methods is positive on ${\\mathfrak B}_n$, then Berline-Vergne's construction is also positive. For this we use a standard argument in toric varieties \\cite[Section 5.3]{fulton}.\n\\begin{setup}\\label{set:toric}\nLet $M$ a lattice, $P\\subset \\operatorname{M}\\otimes_{\\mathbb Z}{\\mathbb R}$ be an $n$ dimensional lattice polytope, $\\Delta$ its normal fan, subdivided if necessary to obtain a unimodular fan, and $X=X(\\Delta)$ the corresponding toric variety. If $P$ has the facet description\n\\[\nP =\\{\\mathbf{x}\\in{\\mathbb R}^n:\\quad \\langle\\mathbf{x}, \\bm{n}_{F_i}\\rangle\\geq -a_i\\}\n\\]\nwhere $F_i$ are the facets of $P$ and $n_{F_i}$ are their normal vectors, then $P$ determines a divisor\n\\[\nD_P = \\displaystyle \\sum_{i} a_iD_{F_i}\n\\]\nwhere $D_{F_i}$ is the divisor associated to the facet $F_i$. The divisor $D$ is Cartier and it is generated by its global sections.\n\\end{setup}\n\nSince the torus invariant cycles generate the Chow ring, the Todd class of $X$ has a (non-unique) expression of the form\n\\begin{equation}\\label{eqn:danilov}\n\\operatorname{Td}(X) = \\displaystyle \\sum_{\\sigma\\in\\Delta} r_\\sigma [V(\\sigma)].\n\\end{equation}\n\nSuch an expression gives a solution to McMullen's formula\n\\begin{equation}\\label{eqn:mcmullen}\n\\mathrm{Lat}(P) = \\displaystyle \\sum_{F\\subset P} \\alpha(F,P)\\mathrm{nvol}(F)\n\\end{equation}\nwith $\\alpha(F,P)=r_\\sigma$, where $\\sigma = \\operatorname{ncone}(F,P)$. We briefly explain how.\n\nFirst of all, we have the following two key facts about the divisor $D_P$:\n\\begin{enumerate}\n\\item $\\dim H^0(X,D_P) = \\mathrm{Lat}(P)$.\n\\item $H^j(X,D_P)=0$ for $j>0$.\n\\end{enumerate}\nCombined together, we get $\\chi(X, D_P)=\\mathrm{Lat} (P)$. Now we can use the Riemann-Roch-Hirzebruch theorem to compute the euler characteristic.\n\\[\n\\chi(X,D_P) = \\textrm{deg }( \\textrm{ch}(D_P)\\cdot\\operatorname{Td}(X))_0.\n\\]\nIn the present case, where $D_P$ is a divisor, we have\n\\[\n\\textrm{ch}(D_P) = \\sum_{k=0}^n \\frac{D_P^k}{k!}.\n\\]\nThe zero sub index means that we only care about the zero dimensional part of the intersection, so we can write\n\\begin{equation}\\label{eqn:RRH}\n\\chi(X,D_P) = \\displaystyle \\sum_{k=0}^n \\sum_{\\dim \\sigma = k} r_\\sigma\\textrm{deg }\\left(\\frac{D_P^k}{k!}[V(\\sigma)]\\right).\n\\end{equation}\nOne of the pleasant connections between toric varieties and discrete geometry is the relation\n\\[\n\\textrm{deg }\\left(\\frac{D_P^k}{k!}[V(\\sigma)]\\right) = \\mathrm{nvol}(F_\\sigma),\n\\]\nwhere $F_\\sigma$ is the face of $P$ with normal cone $\\sigma$. Combined with $\\chi(X,D_P)=\\mathrm{Lat} (P)$ we get a solution for McMullen's formula\n\\[\n\\mathrm{Lat}(P) = \\displaystyle  \\sum_{F\\subseteq P} \\alpha(F,P) \\mathrm{nvol}(F)\n\\]\nwith $\\alpha(F,P)=r_\\sigma$, where $\\sigma = \\operatorname{ncone}(F,P)$.\n\n\nOriginally, Danilov asked if the coefficients in the Equation \\eqref{eqn:danilov} could be given depending just on $\\sigma$. Even though this is more general than McMullen's formula, Berline-Vergne's construction actually solve Danilov's question as well \\cite{bvtodd}.  \n\n\\begin{prop}\\label{prop:toddequiv}\nLet $\\Delta$ be the braid arrangement fan ${\\mathfrak B}_n$. The following are equivalent:\n\\begin{enumerate}\n\\item The Todd class is positive with respect to the torus invariant cycle, that is\n\\[\n\\operatorname{Td}(X(\\Delta)) = \\displaystyle \\sum_{\\sigma\\in\\Delta} r_\\sigma [V(\\sigma)],\n\\]\nwith $r_\\sigma>0$.\n\\item The regular permutohedron $\\Pi_n$ is BV-$\\alpha$-positive.\n\\end{enumerate} \n\\end{prop}\n\nNote that the condition in part (1) of Propsition \\ref{prop:toddequiv} is much weaker than Danilov's condition, since we only need to choose $r_\\sigma$ for $\\sigma$ in the braid arrangement fan $\\Delta$, and do not need to worry about assignments to cones in other fans or whether \\eqref{eqn:danilov0} holds for other fans.\n\n\\begin{proof}\n  Clearly we have $(2) \\Rightarrow (1)$ since Berline-Vergne's construction solves Danilov's question.\n  \n  Now assuming (1), we will prove (2). By Corollary \\ref{cor:equimixed}, we only need to show that the mixed lattice points valuation $\\mathrm{\\mathcal{M}Lat}^k$ on hypersimplices is always positive. We modify the Riemann-Roch-Hirzebruch argument to compute this mixed valuations.\n\nConsider $P_i=\\Delta_{i,n+1}$ for $1\\leq i\\leq n$. Each of them is compatible with the fan ${\\mathfrak B}_n$, so they each define a divisor $D_i$. Clearly, the divisor $D_P:=w_1D_1+\\cdots+w_nD_n$ corresponds to the polytope $P=w_1P_1+\\cdots+w_nP_n$. Now we use the Riemann-Roch-Hirzebruch argument as above. We have that:\n\\begin{equation}\\label{equ:Latpoly}\n\\mathrm{Lat}(P)=\\chi(X,D_P) = \\displaystyle \\sum_{k=0}^n \\sum_{\\dim \\sigma = k} r_\\sigma\\deg \\left(\\frac{D_P^k}{k!}[V(\\sigma)]\\right).\n\\end{equation}\nSince $D^k = (w_1D_1+\\cdots+w_nD_n)^k$, it follows naturally that $\\mathrm{Lat}(P)$ is a polynomial in the $w_i$'s. Let $S=\\{s_1,\\cdots,s_k\\}$, and consider coefficient of squarefree monomials $w_S :=\\prod_{i\\in S}w_i$ in this polynomial. On the one hand, it is clearly given by  \n\\[\n  \\sum_{\\dim \\sigma = k} r_\\sigma \\deg\\left(\\frac{D_S}{k!}[V(\\sigma)]\\right),\n\\]\nwhere $D_S = \\prod_{i\\in S}D_i$. \nOn the other hand, similar as in the proof of Theorem \\ref{thm:uniqueness}, this coefficient is given by \\eqref{eqn:coeff1}.\nTherefore, we get \n\\begin{equation}\\label{equ:mixedlatpos}\nk!\\mathrm{\\mathcal{M}Lat}^{k}(\\Delta_{s_1,n+1},\\Delta_{s_2,n+1},\\cdots\\Delta_{s_k,n+1}) = \\sum_{\\dim \\sigma = k} r_\\sigma \\deg\\left(\\frac{D_S}{k!}[V(\\sigma)]\\right),\n\\end{equation}\nwhere $D_S = \\prod_{i\\in S}D_i$. Since $D_i$ are generated by global sections, each intersection product $D_S[V(\\sigma)]$ of the right hand side of \\eqref{equ:mixedlatpos} has positive degree.\nTherefore we have that the value of \\eqref{equ:mixedlatpos} is nonnegative for $r_\\sigma >0,$ . It is left to show this value cannot be zero.\n\nWe assume to the contrary that the right hand side of \\eqref{equ:mixedlatpos} is equal to $0.$\nThen\n\\[\n  D_S[V(\\sigma)] = 0 \\qquad \\forall \\sigma \\textrm{ of codimension } k.\n\\]\nBut the torus invariant cycles generate the Chow groups \\cite[Section 5.1]{fulton}. so it follows that\n\\begin{equation}\\label{final}\nD_S\\cdot C = 0 \\qquad \\forall  C\\in A_k(X),\n\\end{equation}\nin the Chow ring. \nNote that the coefficient of $w_{[n]} = w_1 w_2 \\dots w_n$ of the polynomial \\eqref{equ:Latpoly} is given by $r_\\Delta \\deg (D_1D_2D_3\\cdots D_n)$ which is strictly posivity by Lemma \\ref{lem:squarefree}. Hence, we have \n\\[\\deg(D_1D_2D_3\\cdots D_n) = \\deg (D_{[n]}) >0.\\]\nSo $D_{[n]}$ is nonzero in the Chow ring. \n\nNow let $\\overline{S}$ be the set complement of $S$ in $[n]$. We have that $D_{\\overline{S}}\\in A_k(X)$ and $D_S\\cdot D_{\\overline{S}} = D_{[n]}\\neq 0$, contradicting Equation \\eqref{final}.\n\\end{proof}\n\nProposition \\ref{prop:toddequiv} provides us another way to attack Conjecture \\ref{conj:alphas} using theory of toric varieties. \nFor example, any construction that answers Danilov's question can potentially provide a way to prove our conjectures.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nQuantum systems tend to rapidly decohere due to the coupling to their\nenvironments, a process which is especially detrimental to quantum\ninformation processing and high resolution spectroscopy \\cite{Breuer:book}. \nOf the many methods which have been proposed in\nrecent years to overcome the damage caused by decoherence, we focus here on\ndynamical decoupling (DD), a method for suppressing decoherence whose\norigins can be traced to the Hahn spin echo \\cite{Hahn:50}. In DD, one\napplies a series of strong and frequent pulses to a system, designed to\ndecouple it from its environment\n\\cite{Viola:98,Ban:98,Zanardi:98b,Viola:99}. Recently, it was\ndiscovered how to optimally suppress decoherence of \na single qubit using DD, under the idealization of instantaneous\npulses \\cite{Uhrig:07,WFL:09,Pasini:09}. One of us found an optimal pulse\nsequence (later dubbed \nUhrig DD, or UDD) for suppressing pure dephasing (single-axis decoherence)\nof a qubit coupled to a boson bath with a hard frequency cut-off\n\\cite{Uhrig:07}. In UDD one applies a series of $N$ instantaneous\n$\\pi $ pulses at instants $t_{j}$ ($j\\in \\{1,2\\ldots N\\}$), with the instants \ngiven by $t_{j}=T\\delta _{j}$ where $T$ is the total time of the sequence and \n\\begin{equation}\n\\delta _{j}=\\sin ^{2}(j\\pi \/(2N+2)).  \n\\label{udd}\n\\end{equation}\nBy optimal it is meant that with each additional pulse the sequence\nsuppresses dephasing in one additional order in an expansion in $T$, i.e., \n$N$ pulses reduce dephasing to $\\mathcal{O}(T^{N+1})$. The existence and\nconvergence of an expansion in powers of $T$, at least as an asymptotic\nexpansion, is a necessary assumption \\cite{Yang:08,Pasini:09}.\n\nThe UDD sequence was first conjectured \\cite{lee:160505,Uhrig:08} and then\nproven \\cite{Yang:08} to be universal, in the sense that it applies to any\nbath causing pure dephasing of a qubit, not just bosonic baths. The\nperformance of the UDD sequence was tested, and its advantages over other\npulse sequences confirmed under appropriate circumstances, in a series of\nrecent experiments \\cite{Biercuk:09,biercuk:062324,Du:09}. Its limitations\nas a function of sharpness of the bath spectral density high frequency\ncut-off \\cite{Pasini:09a} and as a function of timing constraints\n\\cite{Hodgson:09} have also been discussed. \n\nIn order to suppress general (three-axis) decoherence on a qubit\nto all orders\nconcatenated sequences are needed \\cite{Khodjasteh:05,KhodjastehLidar:07},\nwhose efficiency can be\nimproved by using UDD building blocks \\cite{Uhrig:09b}.\nA near optimum suppression  is achieved by quadratic dynamic\ndecoupling (dubbed QDD). This scheme was proposed\nand numerically tested in Ref.\\ \\cite{WFL:09} and analytically\ncorroborated in Ref.\\ \\cite{Pasini:09}. \nIn QDD, a sequence of $(N+1)^{2}$\npulses comprising two nested UDD sequences suppresses general qubit\ndecoherence to $\\mathcal{O}(T^{N+1})$, which is known from brute-force\nsymbolic algebra solutions for small $N$ to be near-optimal \\cite{WFL:09}.\n\nWhile rigorous performance bounds have been derived previously for \nperiodic and concatenated DD pulse\nsequences \\cite{KhodjastehLidar:08,LZK:08,NLP:09}, no such\nperformance bounds have yet been derived for optimal decoupling pulse\nsequences, in particular UDD and QDD. In this\nwork we focus on UDD and obtain rigorous performance bounds.\nWe postpone the problem of finding rigorous QDD performance bounds to a\nfuture publication. Our main result here is an analytical upper bound for\nthe distance between UDD-protected states subject to pure\ndephasing and unperturbed states, as a function of the natural dimensionless \nparameters of the\nproblem, namely the total evolution time $T$ measured in units of the\nmaximal intra-bath energy $J_0$,\nand in units of the system-bath coupling strength $J_z$.\nThe bound shows that this distance\n(technically, the trace-norm distance), can be made arbitrarily small as a\nfunction of the number of pulses $N$, as \n$(1\/N!)(J_0 T+ J_z T)^{N}$.\nThis presumes that the bounds $J_\\alpha$ ($\\alpha\\in\\{0,z\\}$) are finite, an \nassumption which will fail for unbounded\nbaths, such as oscillator baths. In such cases, which includes the\nubiquituous spin-boson model, our analysis does not apply. Alternative\napproaches, such as those based on correlation function bounds \n\\cite{NLP:09}, are then required.\n\nWe begin by introducing the model of pure dephasing in the presence of\ninstantaneous DD pulses in Section \\ref{sec:model}. We then derive a\ngeneral time-evolution bound in Section \\ref{sec:t-bound}, without reference\nto any particular pulse sequence. In Section \\ref{sec:dep-bound} we\nspecialize this bound to the UDD sequence. Then, in Section\n\\ref{sec:D-bound}, we obtain our main result:\\ an upper bound on the\ntrace-norm distance between the UDD-protected and unperturbed\nstates. In Section \\ref{analysis} we analyze the implications of this\nbound in the more realistic setting when only a certain minimal interval\nbetween consecutive pulses can be attained.  Certain technical\ndetails are presented in the Appendix, including the first complete\nuniversality proof of the UDD sequence, which does not rely on the\ninteraction picture.\n\n\n\\section{Model}\n\\label{sec:model}\n\nWe start from the general, uncontrolled, time-independent system-bath\nHamiltonian for pure dephasing \n\\begin{equation}\nH_{\\mathrm{unc}}=I_{\\mathrm{S}}\\otimes B_0\n+\\sigma _{z}\\otimes B_z,\n\\label{hamil1}\n\\end{equation}\nwhere $B_0$ and \n$B_z$\nare bounded but otherwise arbitrary operators\nacting on the bath Hilbert space $\\mathcal{H}_{\\mathrm{B}}$, $I_{\\mathrm{S}}$\nis the identity operator on the system Hilbert space \n$\\mathcal{H}_{\\mathrm{S}}$, and $\\sigma _{z}$ is the diagonal Pauli matrix. \nThe bath operator  $B_z$ need not be traceless, i.e., \nwe allow for the possibility of a pure-system term \n$\\sigma _{z}\\otimes I_{\\mathrm{B}}$ in the system-bath interaction term \n$H_{\\mathrm{SB}}:= \\sigma _{z}\\otimes B_z$. \nSuch internal system dynamics will be\nremoved by the DD pulse sequence we shall add next, along with the coupling\nto the bath. However, the assumption of pure dephasing means that we assume\nthat the level splitting of the system, i.e., any term proportional to \n$\\sigma _{\\bot }$ (with $\\sigma _{\\bot }$ being $\\cos(\\varphi)\n\\sigma_x+\\sin(\\varphi)\\sigma_y$ for arbitrary $\\varphi$)\nacting on the system, is fully controllable. Otherwise the\nmodel is one of general decoherence, and our methods require a modification\nalong the lines of Refs.~\\cite{WFL:09} and \\cite{Pasini:09}. \nIf the system described by Eq.~\\eqref{hamil1} is subject to $N$ \ninstantaneous $\\pi $ pulses at the instants \n$\\{t_{j}:=T\\delta _{j}\\}_{j=1}^{N}$ about a\nspin axis perpendicular to the $z$-axis, i.e., if the Hamiltonian \n$H_{\\mathrm{DD}}(t)=\\frac{\\pi }{2}\\sum_{j=1}^{N}\\delta (t-t_{j})\\sigma _{\\bot\n}\\otimes I_{\\mathrm{B}}$ is added to $H_{\\mathrm{unc}}$, the interaction\npicture (\\textquotedblleft toggling-frame\\textquotedblright ) Hamiltonian \n$H_{\\mathrm{tog}}(t)=U_{\\mathrm{DD}}^{\\dag }(t)H_{\\mathrm{unc}}U_{\\mathrm{DD}\n}(t)$ reads \n\\begin{equation}\nH_{\\mathrm{tog}}(t)=I_{\\mathrm{S}}\\otimes B_0 + f(t) \\sigma_{z}\n\\otimes B_z,  \n\\label{hamil2}\n\\end{equation}\nwhere the unitary $U_{\\mathrm{DD}}(t)$ alternates between $I_{\\mathrm{S}\n}\\otimes I_{\\mathrm{B}}$ and $\\sigma _{\\bot }\\otimes I_{\\mathrm{B}}$ at the\ninstants $\\{t_{j}\\}_{j=1}^{N}$, and consequently the \\textquotedblleft\nswitching function\\textquotedblright\\ $f(t)=\\pm 1$ changes sign at the same\ninstants. \n\nWe shall also need the magnitudes of the two parts of the Hamiltonian \n\\begin{equation}\n\\label{Jbounds}\nJ_0 :=\\Vert B_0\\Vert <\\infty ,\\quad J_z:=\\Vert B_z \\Vert <\\infty ,\n\\end{equation}\nwhere $\\Vert \\cdot \\Vert $ is the sup-operator norm (see Appendix \n\\ref{app:norms}). There are certainly situations where either $J_0$ or $J_z$\ncan be divergent (e.g., $J_0$ in the case of oscillator baths). \nIn such cases our bounds will not apply, and other methods such as \ncorrelation function bounds are more appropriate (see, e.g., Ref.\\ \n\\cite{NLP:09} for such bounds applied to DD).\n\n\n\n\\section{Time evolution bounds}\n\n\\label{sec:t-bound}\n\nWe aim to bound certain parts of the time evolution\noperator induced by $H_\\mathrm{tog}(t)$\n\\begin{equation}\n\\label{Uintegral}\nU(T) = {\\cal T}\\exp\\left( -i\\int_0^T H_\\mathrm{tog}(t)dt \\right)\n\\end{equation}\nwhere ${\\cal T}$ is the time-ordering operator.\n\nStandard time dependent perturbation theory provides\nthe following Dyson series for $U(T)$\n\\begin{subequations}\n\\label{Useries}\n\\begin{eqnarray}\n\\label{vec-sum}\nU(T) &=& \\sum_{n=0}^\\infty (-iT)^n \n\\sum_{\\{\\vec{\\alpha};\\mathrm{dim}(\\vec{\\alpha})=n\\}} \nF_{\\vec{\\alpha}} \\; \\widehat Q_{\\vec{\\alpha}}\n\\\\\n\\label{f_def}\nF_{\\vec{\\alpha}} &:=& \n\\int_0^1 ds_n f_{\\alpha_n}(s_n) \\int_0^{s_n} ds_{n-1} f_{\\alpha_n}(s_n)\\ldots \n\\nonumber\\\\ \n&&\\int_0^{s_3} ds_2  f_{\\alpha_2}(s_{2}) \\int_0^{s_2} ds_1 f_{\\alpha_1}(s_{1})\n\\\\\n\\widehat \nQ_{\\vec{\\alpha}} &:=& \\sigma_{\\alpha_n} B_{\\alpha_n}   \\ldots \n\\sigma_{\\alpha_2}B_{\\alpha_2}   \\sigma_{\\alpha_1} B_{\\alpha_1},\n\\label{q_def}\n\\end{eqnarray}\n\\end{subequations}\nwhere $\\mathrm{dim}(\\vec{\\alpha})=n$ is\nthe dimension of the vector $\\vec{\\alpha}$.\nThe identity $I_{\\rm S}$\nin the Hilbert space of the qubit\/spin is denoted by $\\sigma_0$.\nIn all sums over \nthe vectors $\\vec{\\alpha}$ their components $\\alpha_j$\ntake the values $0$ or $z$. In this way, the summation includes\nall possible sequences of $B_0$ and $B_z$.\nThe function $f_0(s)$ is constant and equal to $1$ while $f_z(s):=f(s T)$\ntakes the valus $\\pm1$. We use the dimensionless relative time\n$s:=t\/T$ so that all dependence on $T$ appears as a power in the prefactor.\nNote that the coefficients $F_{\\vec{\\alpha}}$ do not depend on $T$.\n\n\nIn order to find an upper bound on \\emph{each} term \n$F_{\\vec{\\alpha}} \\; \\widehat Q_{\\vec{\\alpha}}$ separately we proceed\nin two steps. First, we use $|f_\\alpha|= 1$ to obtain\n\\begin{eqnarray}\n|F_{\\vec{\\alpha}}| &\\le&  \n\\int_0^1 ds_n  \\int_0^{s_n} ds_{n-1} \\ldots \n\\int_0^{s_3} ds_2  \\int_0^{s_2} ds_1\n\\notag\n\\\\\n&=& \\frac{1}{n!}.\n\\end{eqnarray}\nSecond, we use Eq.~\\eqref{Jbounds} and $\\Vert\\sigma_\\alpha\\Vert =1$ \nto arrive at\n\\begin{eqnarray}\n\\Vert \\widehat Q_{\\vec{\\alpha}} \\Vert &\\le& \\prod_{j=1}^n\nJ_{\\alpha_j} = J_0^{n-k(\\vec{\\alpha})} J_z^{k(\\vec{\\alpha})},\n\\end{eqnarray}\nwhere we used the submultiplicativity of the sup-operator norm\n(see Appendix~\\ref{app:norms}).\nThe number $k(\\vec{\\alpha})$ stands for the number\nof times that the factor $J_z$ occurs. Standard combinatorics of\nbinomial coefficients tells us that \nthe term $J_0^{n-k} J_z^{k}$ occurs $n!\/(k! (n-k)!)$ times\nin the sum over all the vectors $\\vec{\\alpha}$ of given \ndimensionality $n$ in \\eqref{vec-sum}. \nHence each term of the  time expansion of $U(T)$ is \nbounded by\n\\begin{equation}\n\\Big\\Vert \\sum_{\\{\\vec{\\alpha};\\mathrm{dim}(\\vec{\\alpha})=n\\}} \nF_{\\vec{\\alpha}} \\; \\widehat Q_{\\vec{\\alpha}} \\Big\\Vert\n\\le\n\\sum_{k=0}^n \\frac{1}{k!(n-k)!} J_0^{n-k}J_z^k .\n\\label{bound1}\n\\end{equation}\nWe therefore define the bounding series\n\\begin{subequations}\n\\begin{eqnarray}\n\\label{S_def0}\nS(J_0,J_z) &:=& \\sum_{n=0}^{\\infty} T^n \n\\sum_{k=0}^n \\frac{1}{k!(n-k)!} J_0^{n-k}J_z^k\n\\\\\n&=& \\exp((J_0+J_z)T).\n\\label{S_def}\n\\end{eqnarray}\n\\end{subequations}\nIt then follows from Eq.~(\\ref{bound1})\nthat each multinomial in $J_0$ and $J_z$ of the expansion of \n$S(J_0,J_z)$ is an upper bound on the norm of the sum of the corresponding\nmultinomial in the operators $B_0$ and $B_z$ of the expansion of $U(T)$\nin Eq.~(\\ref{vec-sum}). This is the property which we will use in the sequel.\n\n\n\\section{Bounds for dephasing}\n\n\\label{sec:dep-bound}\n\nFrom  $\\sigma_z^2=I_{\\rm S}$ it is obvious that only the odd powers in $B_z$\ncontribute to dephasing while the even ones do not. Hence we split \n$U(T)$ as \n\\begin{equation}\nU(T)=I_{\\rm S}\\otimes B_{+}(T)+\\sigma _{z}\\otimes B_{-}(T)  \n\\label{Usplit}\n\\end{equation}\nwhere the operators $B_{\\pm }$ act only on the bath while $I_{\\rm S}$ and \n$\\sigma_{z}$ act only on the qubit. The operator $B_+$ comprises\nall the terms with an even number $k$ of $\\sigma_z \\otimes B_z$, i.e., with\nan even number of $J_z$ in the bounding series $S(J_0,J_z)$.  \nThe operator $B_-$ comprises\nall the terms with odd number $k$ of $\\sigma_z\\otimes B_z$, i.e., with an odd\nnumber of $J_z$ in the bounding series $S(J_0,J_z)$. \nHence to bound the time series of $B_-(T)$ term by term we need the\nthe time series of the odd part of $S(J_0,J_z)$ in $J_z$. This, from\n\\eqref{S_def} is:\n\\begin{equation}\n\\label{Sminus_def}\nS_-(J_0,J_z) = \\exp(J_0 T)\n\\sinh(J_z T).\n\\end{equation}\nThe time series of $S_-(J_0,J_z)$ provides  a\nbounding series of $B_-(T)$ term by term.\nHence we define\n\\begin{equation}\nd_{k} := \\frac{1}{k!} \n\\frac{\\partial ^{k}}{\\partial T^{k}}S_{-}(J_0,J_z)\\Big|_{T=0},\n\\end{equation}\nsuch that $S_-(J_0,J_z)=\\sum_{k=0} ^\\infty d_k T^k$.\n\nWe know from the proof of Yang and Liu\n\\cite{Yang:08} that in the $B_0$-interaction picture \na UDD sequence with $N$ pulses (which we denote by UDD($N$))\nshould make the first $N$ powers in $T$ of $B_-(T)$ vanish, i.e.,\n$B_-(T)=O(T^{N+1})$.\nHowever, since the Yang-Liu proof does not directly apply to our\ndiscussion, we provide\na complete version of this proof which avoids the $B_0$-interaction picture\nin Appendix~\\ref{app:YL-proof}. The remaining powers\nare bounded by the corresponding coefficients\n$d_k$ of $S_-$. Thus the expression\n\\begin{equation}\n\\label{Delta_def}\n\\Delta_{N} := \\sum_{k=N+1}^{\\infty }d_{k}T^{k}\n\\end{equation}\nprovides an upper bound for $B_-(T)$ if UDD($N$) is applied:\n\\begin{equation}\n\\label{Bbound}\n\\Vert B_-(T) \\Vert \\le \\Delta_N.\n\\end{equation}\nDue to the obvious analyticity in the variable $T$\nof $S_-(J_0,J_z)$ as defined in \\eqref{Sminus_def}\nwe know that the residual term vanishes for\n$N\\to\\infty$, i.e.,\n\\begin{equation}\n\\lim_{N\\to \\infty} \\Delta_N = 0.\n\\end{equation}\nThis statement holds true irrespectively\nof the values of $J_0$ and $J_z$, as long as they are finite.\n\nWe can obtain a more explicit expression for $\\Delta _{N}$. Besides the\ndimensionless number of pulses $N$ the bound $\\Delta _{N}$ depends on \n$J_{0}T$ and on $J_{z}T$. It is convenient to introduce the dimensionless\nparameters \n\\begin{equation}\n\\varepsilon :=J_{0}T,\\quad \\eta := J_{z}\/J_{0} \n\\end{equation}\ninstead. In terms of these parameters we have\n\\begin{equation}\nS_{-}(\\eta ,\\varepsilon )=\\exp (\\varepsilon )\\sinh (\\varepsilon \\eta ).\n\\end{equation}\nFrom the series\n\\begin{subequations}\n\\begin{eqnarray}\n\\exp (\\varepsilon )\\sinh (\\varepsilon \\eta ) &=&\n\\frac{1}{2}[e^{\\varepsilon(1+\\eta)}-e^{\\varepsilon(1-\\eta)}]\n\\\\\n&=& \\sum_{l=0}^\\infty \\frac{\\varepsilon^l}{2l!}[(1+\\eta)^l-(1-\\eta)^l]\n\\qquad\n\\\\\n&=& \\sum_{l=0}^\\infty p_l(\\eta) \\varepsilon^l\n\\end{eqnarray}\n\\end{subequations}\nwith\n\\begin{equation}\np_{l}(\\eta ) := \\frac{1}{2 l!}\\left[(1+\\eta)^l - (1-\\eta)^l\\right].\n\\label{p_l}\n\\end{equation}\nwe obtain\n\\begin{subequations}\n\\begin{eqnarray}\n\\label{Del-simp}\n\\Delta _{N}(\\eta ,\\varepsilon ) &=& \\sum_{n=N+1}^{\\infty}\np_n(\\eta)\\varepsilon^{n} \n\\\\\n\\label{Del-series}\n&=& p_{N+1}(\\eta )\\varepsilon ^{N+1}+\\mathcal{O}(\\varepsilon ^{N+2}).\n\\end{eqnarray}\n\\end{subequations}\nThis, together with the bound \\eqref{Bbound}, is our key result: \nit captures how the ``error'' $\\Vert B_{-}(T)\\Vert$ is suppressed \nas a function of the relevant\ndimensionless parameters of the problem, $\\eta $, $\\varepsilon$, and $N$.\nNote  that convergence for $N\\to\\infty$ is always ensured\nby the factorial in the denominator, irrespectively\nof the values of $\\varepsilon$ and $\\eta$ as long as these\nare finite.\n\nFor practical purposes it is advantageous not to compute $\\Delta _{N}$\nby the infinite series in \\eqref{Del-simp}, but by \n\\begin{equation}\n\\Delta _{N}(\\eta ,\\varepsilon )=S_{-}(\\eta ,\\varepsilon\n)-\\sum_{n=0}^{N} p_n(\\eta)\\varepsilon ^{n} ,  \n\\label{Delta_pract}\n\\end{equation}\nwhich can easily be computed by computer\nalgebra programs. Figures~\\ref{fig1} and \\ref{fig2} depict the results of\nthis computation. Consider first Fig.~\\ref{fig1}. Each curve shows\n$\\Delta _{N}(\\eta ,\\varepsilon )$ as a function of $\\varepsilon$, at\nfixed $\\eta$ and $N$. The error $\\Vert B_{-}(T)\\Vert$ always\nlies under the corresponding curve. Clearly, the bound becomes tighter as\n$\\varepsilon $ decreases. Moreover, the more pulses are\napplied (the different panels) the higher the power in $\\varepsilon$ and \nthus the steeper the curve. Additionally, the curves are shifted to the \nright as $N$ increases. Clearly, then, a larger number of pulses improves \nthe error bound significantly, at fixed $\\varepsilon$ and $\\eta$.\nThis effect is even more conspicuous in Fig.~\\ref{fig2}, where $\\eta $ is \nfixed in each of the two panels, and the different curves correspond to \ndifferent values of \n$N$. The vertical line intersects the bounding function at progressively\nlower points as $N$ is increased, showing how the bound becomes tighter.\n\n\\begin{figure}[th]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{fig1}\n\\end{center}\n\\caption{(Color online) The bounding function $\\Delta _{N}$\nas a function of $\\varepsilon =J_{0}T$, as given in\nEq.~\\eqref{Delta_def} for various numbers of pulses $N$ and various \nvalues of the parameter $\\protect\\eta =J_z\/J_0\\in \\{0.01,0.1,1,10,100\\}$, \nwith $\\eta$ increasing from the rightmost curve to the leftmost curve in \neach panel.}\n\\label{fig1}\n\\end{figure}\n\n\\begin{figure}[th]\n\\begin{center}\n\\includegraphics[width=3.4in]{fig2}\n\\end{center}\n\\caption{(Color online) The bounding function $\\Delta _{N}$\nas a function of $\\varepsilon =J_{0}T$, as given in Eq.~\\eqref{Del-simp} for \nvarious numbers of pulses $N\\in \\{2,5,10,20\\}$, at fixed values of \n$\\protect\\eta$. In each panel the curves become steeper as $N$ increases.\n}\n\\label{fig2}\n\\end{figure}\n \n\n\\section{Distance bound}\n\n\\label{sec:D-bound}\n\nIntuitively, we expect the bound on $\\|B_{-}(T)\\|$ derived in the previous\nsection to be sufficient to bound the effect of dephasing. However, to make\nthis rigorous we need a bound on the trace-norm distance $D[\\rho _{\\mathrm{S}\n}(T),\\rho _{\\mathrm{S}}^{0}(T)]$ between the \\textquotedblleft\nactual qubit\\textquotedblright\\ state \n\\begin{equation}\n\\rho _{\\mathrm{S}}(T) := \\mathrm{tr}_{\\mathrm B}[\\rho_{\\mathrm SB}(T) ]\n\\end{equation}\n and the\n\\textquotedblleft ideal qubit\\textquotedblright\\ state \n\\begin{equation}\n\\rho _{\\mathrm{S}}^{0}(T) := \\mathrm{tr}_{\\mathrm B}[\\rho_{\\mathrm SB}^0(T) ]\n\\end{equation}\nwhere $\\rho_{\\mathrm SB}^0(T)$ is the time-evolved state without\ncoupling between qubit and bath.\nThe partial trace over the bath degrees of freedom is\na map from the joint system-bath Hilbert space to the \nsystem-only\nHilbert space (see Appendix~\\ref{app:norms}), and is\ndenoted by ${\\rm tr}_{\\mathrm B}$.\nAs we shall see, the term $I_{\\rm S}\\otimes B_{+}(T)$ \nin Eq.~(\\ref{Usplit}) indeed has a small, and in fact essentially\nnegligible effect.\n\nTo obtain the desired distance bound we consider a factorized initial state \n$\\rho _{\\mathrm{SB}}^{0}(0)=|\\psi \\rangle \\langle \\psi |\\otimes \\rho _{\n\\mathrm{B}}$, which evolves \nto $\\rho _{\\mathrm{SB}}(T)={U}(T)\\rho _{\\mathrm{SB}}^{0}(0){U}^{\\dag }(T)$ \nwhen the system-bath interaction is on (the\n\\textquotedblleft actual\\textquotedblright\\ state), \nor to $\\rho _{\\mathrm{SB}}^{0}(T)=\nI_{\\rm S}\\otimes U_{\\mathrm B}(T)\n\\rho _{\\mathrm{SB}}^{0}(0)\nI_{\\rm S}\\otimes U_{\\mathrm B}^\\dag(T)$ \nwhen the interaction is off\n(the \\textquotedblleft ideal\\textquotedblright\\ state).\nThe unitary time evolution operator without coupling reads\n\\begin{equation}\nU_{\\mathrm B}(T):=\\exp(-i T B_0),\n\\end{equation}\nwhere $B_0$ is the pure-bath term in Eq.~(\\ref{hamil1}).\nThe initial bath state \n$\\rho _{\\mathrm{B}}$ is arbitrary (e.g., a mixed thermal equilibrium state),\nwhile the initial system state is pure. Let us define the correlation functions\n\\begin{equation}\nb_{\\alpha \\beta }(T):=\\mathrm{tr}\n\\left[ B_{\\alpha }(T)\\rho _{\\mathrm{B}} B_{\\beta }^{\\dag }(T)\\right]\n\\label{b_ab}\n\\end{equation}\nwhere $\\alpha,\\beta\\in\\{+,-\\}$,\nand where all operators under the trace act only on the bath Hilbert space.\nExplicit computation (see Appendix \\ref{app:dist}) then yields: \n\\begin{align}\n& D[\\rho _\\mathrm{S}(T),\\rho _\\mathrm{S}^{0}(T)]  \\label{D-bound-final} \\\\\n& \\leq \\frac{1}{2}(|b_{++}(T)-1|+|b_{+-}(T)|+|b_{-+}(T)|+|b_{--}(T)|).\\notag\n\\end{align}\nWe will show that $b_{++}$ is very close to $1$ while the other $b_{\\alpha\n\\beta }$ quantities are small in the sense that they are bounded by\nEq.~\\eqref{Bbound}.\n\nFirst note from the unitarity of Eq.~\\eqref{Usplit} that\n\\begin{eqnarray}\nI &=& U^{\\dagger }U \n\\\\\n&=& I_{\\rm S}\\otimes (B_{+}^{\\dagger }B_{+}+B_{-}^{\\dagger }B_{-})\n+\\sigma _{z}\\otimes (B_{-}^{\\dagger }B_{+}+B_{+}^{\\dagger }B_{-})\\notag\n\\end{eqnarray}\nwhere we omitted the time dependence $T$ to lighten the notation. Hence we\nhave \n\\begin{subequations}\n\\begin{eqnarray}  \\label{Bunitary2}\nI &=&B_{+}^{\\dagger }B_{+}+B_{-}^{\\dagger }B_{-}  \\label{Bunitary21} \\\\\n0 &=&B_{+}^{\\dagger }B_{-}+B_{-}^{\\dagger }B_{+}.  \\label{Bunitary22}\n\\end{eqnarray}\n\\end{subequations}\nIt follows that $\\langle i|B_{+}^{\\dagger }B_{+}|i\\rangle =\\Vert\nB_{+}|i\\rangle \\Vert ^{2}\\leq 1$ for all normalized states $|i\\rangle $,\nbecause $\\langle i|B_{-}B_{-}^{\\dagger }|i\\rangle =\\Vert B_{-}|i\\rangle\n\\Vert ^{2}$ is non-negative. Thus in particular $\\max_{\\Vert |i\\rangle \\Vert\n=1}\\Vert B_{+}|i\\rangle \\Vert \\leq 1$, and we can conclude that \n\\begin{equation}\n  \\Vert B_{+}\\Vert \\leq 1.\n  \\label{Bunitary2a}\n\\end{equation}\nCyclic invariance of the trace in $b_{\\alpha\n\\beta }$ together with Eq.~\\eqref{Bunitary21} and the normalization \n$\\mathrm{tr} [\\rho _{\\mathrm{B}}]=1$ \nimmediately yields $b_{++}+b_{--}=1$, \nwhile the combination with Eq.~\\eqref{Bunitary22} implies $b_{+-}+b_{-+}=0$. \nHence Eq.~\\eqref{D-bound-final} can be simplified to \n\\begin{equation}\nD[\\rho _\\mathrm{S}(T),\\rho_\\mathrm{S}^{0}(T)]\\leq |b_{+-}(T)|+|b_{--}(T)|.\n\\label{D}\n\\end{equation}\n\n\nTo obtain a bound on the correlation functions $b_{\\alpha \\beta }$ we\nuse the following general correlation function inequality (for a proof\nsee Appendix~\\ref{app:cor-ineq}):\n\\begin{eqnarray}\n\\left| \\mathrm{tr}\n\\left[ Q\\rho_\\mathrm{B}Q^{\\prime}\\right]\\right| \\le\\|Q^{\\prime}\\|\\|Q\\|,\n\\label{Q}\n\\end{eqnarray}\nwhich holds for arbitrary bounded \nbath operators $Q,Q'$.\nApplying Eq.~(\\ref{Q}) to Eq.~\\eqref{b_ab} yields\n\\begin{subequations}\n\\begin{eqnarray}\n|b_{--}(T)| &\\leq &\\Vert B_{-}(T)\\Vert ^{2}, \\\\\n|b_{+-}(T)| &\\leq &\\Vert B_{+}(T)\\Vert \\Vert B_{-}(T)\\Vert\\\\\n& \\leq &\\Vert B_{-}(T)\\Vert ,\n\\end{eqnarray}\n\\end{subequations}\nwhere in the last inequality we used Eq.~\\eqref{Bunitary2a}.\n\nSummarizing, together with Eqs.~\\eqref{Bbound} and \\eqref{D} \nwe have obtained the following\nrigorous upper bound for the trace-norm distance\n\\begin{equation}\nD[\\rho _\\mathrm{S}(T),\\rho _\\mathrm{S}^{0}(T)]\\leq \\min \n[1,\\Delta _{N}(\\eta ,\\varepsilon)+\\Delta _{N}^{2}(\\eta ,\\varepsilon )].  \n\\label{Dbound}\n\\end{equation}\nThis upper bound completes our main result. Since as we saw\nin Eq.~\\eqref{Del-series} $\\Delta_{N}(\\eta ,\\varepsilon )=\np_{N+1} (\\eta )\\varepsilon ^{N+1}+\\mathcal{O} (\\varepsilon ^{N+2})$, the\nappearance of the squared term in Eq.~\\eqref{Dbound} \n(whose origin is $|b_{--}(T)|$) is not relevant in the sense that even in\nthe presence of this term the bound\n\\begin{equation}\nD[\\rho _\\mathrm{S}(T),\\rho _\\mathrm{S}^{0}(T)]\\leq p_{N+1}\n(\\eta )\\varepsilon ^{N+1}+\\mathcal{O}(\\varepsilon ^{N+2})\n\\end{equation}\nholds. Hence the result of Eq.~\\eqref{Del-simp}\ndepicted in Figs.\\ \\ref{fig1} and \\ref{fig2} provides the desired\nresult. Ignoring the $\\Delta _{N}^{2}$ term \nin Eq.~(\\ref{Dbound}), we note that Figs.\\ \\ref{fig1} and \\ref{fig2} also\nreveal the limitations of our bound when $\\varepsilon $ or $\\eta $ are \ntoo large for a given value of $N$:\nFor any pair of states it is always the case\nthat $D\\leq 1$, so that as soon as $\\Delta _{N}=1$ the bound no longer\nprovides any useful information.\n\nNote further that the results shown in Fig.\\ \\ref{fig1} are qualitatively\nsimilar to the results obtained for the analytically solvable\nspin-boson model for pure dephasing \\cite{Uhrig:07}.\nHeuristically, the necessary\nidentification is $J_0 =\\omega_{\\mathrm{D}}$ where $\\omega_{\\mathrm{D}}$\nis the hard cutoff of the spectral function and $\\eta \\propto \\alpha $\nwhere $\\alpha $ is the dimensionless coupling constant for Ohmic noise. We\nstress that the advantage of Eq.~\\eqref{Dbound} compared to the analytically\nexact results in Ref.\\ \\cite{Uhrig:07} is that it holds rigorously for a\nlarge class of pure dephasing models, namely those of bounded Hamiltonians.\n\n\n\\section{Analysis for finite minimum pulse interval}\n\n\\label{analysis}\n\nSo far we have essentially treated the total time $T$ and the number of\npulses $N$ as independent parameters. This is possible when there is no\nlower limit on the pulse intervals. However, in reality this is never the\ncase and in this section we analyze what happens when there is such a lower\nlimit. Note that it follows from Eq.~(\\ref{udd}) that the smallest pulse\ninterval is the first: $t_{1}=T\\sin ^{2}(\\pi \/(2N+2))$. Let us assume that \n$t_{1}$ is fixed, so that, given $t_{1}$ and $N$, the total time is\n\\begin{subequations}\n\\begin{eqnarray}\nT(N) &=&t_{1}q(N), \\\\\nq(N) &:=&\\csc ^{2}\\left( \\frac{\\pi }{2N+2}\\right) .\n\\end{eqnarray}\n\\end{subequations}\n\nFor large $N$ we can expand the $\\csc ^{2}$ function to first order in its\nsmall argument, yielding \n\\begin{equation}\nq(N)=\\left( \\frac{2N+2}{\\pi }\\right) ^{2}+\n\\frac{1}{3}+{\\cal O}\\left( N^{-2}\\right) ,\n\\end{equation}\nwhich shows how the total time grows as a function of $N$ at fixed minimum\npulse interval $t_{1}$. Along with $\\eta $, the relevant dimensionless\nparameter is now\n\\begin{equation}\n\\varepsilon_{1}:=J_0 t_{1},\n\\end{equation}\ninstead of $\\varepsilon = q(N)\\varepsilon _{1}$.\nWe can then rewrite the bounding function \\eqref{Del-simp} \nin terms of these quantities as\n\\begin{equation}\n\\label{Del1}\n\\Delta _{N}(\\eta ,\\varepsilon _{1}) =\n\\sum_{n=N+1}^\\infty p_n(\\eta)q^n(n)\\varepsilon_1^n .\n\\end{equation}\nConsidering now the large $N$ limit of the first term in this sum, we have\n\\begin{eqnarray}\np_{N+1}(\\eta)q^{N+1}(N+1)\\varepsilon_1^{N+1} &\\approx &\n\\frac{1}{2N!}(\\frac{2}{\\pi}N)^{2N}[(1+\\eta)\\varepsilon_1]^N \\notag \\\\\n\\label{Del1-1}\n&\\approx & (cN)^N\n\\end{eqnarray}\nwhere we kept only the leading order terms and neglected all additive \nconstants relative\nto $N$, and in Eq.~\\eqref{Del1-1} used Stirling's approximation\n$n! \\approx (n\/e)^n$. The constant $c$ is\n$\\frac{1}{2}(\\frac{2}{\\pi})^2 e (1+\\eta)\\varepsilon_1$.\nWe thus see clearly that for fixed $t_{1}$ it becomes\ncounterproductive to make $N$ too large, since\nno matter how small $c$ is, for large enough $N$ the factor $N^N$ will \neventually dominate.\nThis reflects the competition between the gains due to higher order\npulse sequences and the losses due to \nthe increased coupling time to the qubit allotted to the environment.\nSimilar conclusions, delineating regimes where increasingly long DD\nsequences become disadvantageous, have been reported for periodic \n\\cite{KhodjastehLidar:07,KhodjastehLidar:08} and concatenated \n\\cite{KhodjastehLidar:07,NLP:09,West:09} DD pulse sequences, as well as for the\nQDD sequence \\cite{WFL:09}.\n\nThese conclusions are further illustrated in Fig.~\\ref{fig3}, where we plot\nthe bound $\\Delta _{N}(\\eta ,\\varepsilon _{1})$ by replacing $\\varepsilon $\nwith $\\varepsilon _{1}q(N)$ in Eq.~\\eqref{Del-simp}. This figure should be\ncontrasted with Fig.~\\ref{fig2}. The most notable change is that increasing \n$N$ now no longer uniformly improves performance. Whereas in Fig.~\\ref{fig2}\nthe curves for different values of $N$ all tend to converge at high values\nof $\\varepsilon $, in Fig.~\\ref{fig3} a high $N$ value results in a steeper\nslope, but also moves the curve to the left. Thus, for a fixed value of \n$\\varepsilon _{1}$ it can be advantageous to use a small value of $N$ (e.g.,\nfor $\\eta =0.01$ and $\\varepsilon _{1}=0.1$ the $N=2$ curve provides the\ntightest bound).\n\n\\begin{figure}[th]\n\\begin{center}\n\\includegraphics[width=3.4in]{fig3}\n\\end{center}\n\\caption{\n(Color online) The bounding function $\\Delta _{N}$ \nas a function of $\\varepsilon_1=t_1 J_0$ (where $t_1$ is the smallest\npulse interval), at fixed values of $\\eta$.\nThe number of pulses $N\\in \\{2,5,10,20\\}$ is varied from curve \nto curve. In each panel the curves become steeper as $N$ increases.\n}\\label{fig3}\n\\end{figure}\n\n\\section{Conclusions}\n\n\\label{sec:conc}\n\nWe have derived rigorous performance bounds for the UDD sequence protecting\na qubit against pure dephasing. The derivation is based on\nthe existence of finite bounds for the relevant parts of the\nHamiltonian, captured in the dimensionless parameters $\\varepsilon$ and $\\eta$.\nUnder this assumption the bounds show rigorously that  \ndephasing is suppressed to leading order as $(1\/N!)[\\varepsilon (1+\\eta)]^{N}$\nWe consider it a vital step to know that irrespectively of any details of the\nbath, except for the existence of finite bounds, a large number $N$ of\npulses is always advantageous  at fixed $T$ -- at least under the\nidealized assumption of perfect and instantaneous pulses.\n\nAn immediate corollary of our results is that identical bounds apply\nfor the case of the UDD sequence protecting a qubit against\nlongitudinal relaxation. This is the case when the uncontrolled\nHamiltonian (\\ref{hamil1}) is replaced by $H_{\\mathrm{unc}}=I_{\\mathrm{S}}\n\\otimes H_{\\mathrm{B}}+\\sigma _{\\bot }\\otimes B$, and the UDD pulse sequence\nconsists of rotations about the spin-$z$ axis.\nA practical implication is that the bounds found here can be used\nto check numerical and approximate calculations.\nSuch calculations must obey our mathematically rigorous bounds,\nso that a testbed is provided.\n\nFurthermore, a number of interesting\ngeneralizations and extensions of our results readily suggest themselves.\nOne is to consider rigorous bounds for finite pulse-width UDD sequences. It\nis already known how to construct such sequences with pulse-width errors\nwhich appear only to third order in the value of the pulse width \n\\cite{Uhrig:09a}, but no rigorous bounds have been found. Another important\ngeneralization, as mentioned above, is to the QDD sequence for general\ndecoherence \\cite{WFL:09,Pasini:09}. \nWe expect that techniques similar to the ones we\nintroduced here will apply to both of these open problems. Yet another\ndirection, which will require different techniques, is to find rigorous UDD\nperformance bounds for unbounded baths, such as oscillator baths. It is\nlikely that a correlation function analysis similar to that performed in\nRef.~\\cite{NLP:09} for periodic and concatenated DD sequences will prove\nuseful in this case.\n\n\\begin{acknowledgments}\nG.S.U.  is supported under DFG grant UH 90\/5-1. D.A.L. is supported\nunder NSF grants CHE-924318 and PHY-802678,\nand by the United States Department of Defense. \n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nConsider a geometric morphism between toposes $\\mathcal{B}$ and $\\mathcal{A}$, i.e., a functor $\\mathcal{B} \\rightarrow \\mathcal{A}$ with a finite limit preserving left adjoint. If the right adjoint is fully faithful, we say that $\\mathcal{B}$ is a \\emph{subtopos} of $\\mathcal{A}$. If the left adjoint itself has a left adjoint, then we say $\\mathcal{B}$ is an \\emph{essential subtopos} of $\\mathcal{A}$, in which case we have a diagram:\n$$\\xymatrix{ \\mathcal{A} \\ar[r]|-{i^*}^-*+{\\perp}_-*+{\\perp} & \\mathcal{B} \\ar@\/^1.5pc\/[l]^{i_*} \\ar@\/_1.5pc\/[l]_{i_!}}$$ The right adjoint inclusion of $\\mathcal{B}$ into $\\mathcal{A}$ is a geometric morphism, which we think of as the sheaf inclusion of the essential subtopos. By contrast, the left adjoint inclusion, sometimes called ``essentiality,'' is not typically a geometric morphism, though in examples this is often the more natural way to think about objects of the subtopos in the context of the larger topos.\n\nKelly and Lawvere show that the essential subtoposes of a given topos form a complete lattice \\cite{kellylawverecompletelattice}. In light of this result, each such subtopos $\\mathcal{B}$ is referred to as a \\emph{level} of $\\mathcal{A}$. For each level $\\mathcal{B}$, $i_! i^*$ defines a comonad $\\mathrm{sk}_{\\mathcal{B}}$ and $i_*i^*$ defines a monad $\\mathrm{cosk}_{\\mathcal{B}}$ on $\\mathcal{A}$ such that $\\mathrm{sk}_{\\mathcal{B}}$ is left adjoint to $\\mathrm{cosk}_{\\mathcal{B}}$. \n\nFor example, suppose $\\mathcal{A}$ is the topos of presheaves on some small category $\\Delta$. Any fully faithful inclusion $i : \\Delta' \\hookrightarrow \\Delta$ induces functors $$\\xymatrix@R=50pt{\\mathrm{\\bf Set}^{\\Delta^{\\mathrm{op}}}\\ar[r]|-{i^*}_-*+{\\perp}^-*+{\\perp} & \\mathrm{\\bf Set}^{(\\Delta')^{\\mathrm{op}}} \\ar@\/_1.5pc\/[l]_{i_!} \\ar@\/^1.5pc\/[l]^{i_*}}$$ where $i^*$ is restriction and $i_!$ and $i_*$ are left and right Kan extension. These functors exhibit $\\mathrm{\\bf Set}^{\\Delta'^{\\mathrm{op}}}$ as an essential subtopos of $\\mathrm{\\bf Set}^{\\Delta^{\\mathrm{op}}}$. Up to isomorphism, the functor $i^*$ is a common retraction of $i_!$ and $i_*$, which are both fully faithful. This situation has been called \\emph{unity and identity of opposites} \\cite{lawvereuiop}, \\cite{lawveresomethoughts}.\n\nAn object $A$ of $\\mathcal{A}$ is $\\mathcal{B}$-\\emph{skeletal} if $A \\cong \\mathrm{sk}_{\\mathcal{B}}A$; likewise $A$ is $\\mathcal{B}$-\\emph{coskeletal} if $A \\cong \\mathrm{cosk}_{\\mathcal{B}}A$. A level $\\mathcal{B}'$ is \\emph{lower than} a level $\\mathcal{B}$ if the skeletal and coskeletal inclusions of $\\mathcal{B}'$ into $\\mathcal{A}$ factor through the skeletal and coskeletal inclusions, respectively, of $\\mathcal{B}$ in $\\mathcal{A}$. In the above example, the category of presheaves on a full subcategory $\\Delta'' \\hookrightarrow \\Delta'$ is lower than the category of presheaves on $\\Delta'$ . A level $\\mathcal{B}'$ is \\emph{way below} a level $\\mathcal{B}$ if in addition its skeletal inclusion into $\\mathcal{A}$ factors through the coskeletal inclusion of $\\mathcal{B}$ in $\\mathcal{A}$, i.e., if $\\mathcal{B}'$-skeletal implies $\\mathcal{B}$-coskeletal. The smallest level $\\mathcal{B}$ in the lattice of essential subtoposes of $\\mathcal{A}$ for which this condition holds, if such a level exists, is called the \\emph{Aufhebung} of $\\mathcal{B}'$, terminology introduced by Lawvere in deference to Hegel \\cite{lawveresomethoughts}. \n\nIn three toposes which have been important for the study of homotopy theory and higher category theory --- simplicial sets \\cite{gabrielzisman} \\cite{maysimplicial}, cubical sets \\cite{kanabstractI}, and reflexive globular sets \\cite{streetpetittopos} --- levels coincide with dimensions: the category of presheaves on a small category is equivalent to the presheaves on its Cauchy completion. Up to splitting of idempotents, the distinct full subcategories of, e.g., the simplicial category $\\Delta$ are the categories $\\Delta_n$ on objects $[0],\\ldots,[n]$ for each natural number. Thus, dimensions classify the essential subtoposes of the category of simplicial sets; a similar proof works for the other examples. For these toposes, a level $n$ is lower than a level $k$ precisely when $n \\leq k$, and the question of determining the Aufhebung of the level $n$ can be stated more colloquially: when does $n$-skeletal imply $k$-coskeletal?\n\nNaively, one might hope that $n$-skeletal implies $(n+1)$-coskeletal, and for reflexive globular sets this is indeed the case, as was first observed by Roy \\cite{roytopos}. A \\emph{reflexive globular set} is a presheaf on the globe category $\\mathbb{G}$, with the natural numbers as objects and maps of the form $\\sigma,\\tau : n \\rightarrow n+1$ such that $\\tau \\sigma = \\sigma \\sigma$ and $\\tau \\tau = \\sigma \\tau$ and  $\\iota : n+1 \\rightarrow n$ such that $\\iota \\sigma = \\mathrm{id} = \\iota\\tau$. For reflexive globular sets, $n$-skeletal implies $(n+1)$-coskeletal:\n\n\\begin{ex} A reflexive globular set is $n$-skeletal if and only if the only globs above level $n$ are identities and $(n+1)$-coskeletal if and only if there exists a unique filler for each parallel pair of $k$-globs, for $k > n$. Hence, the arrows of any parallel pair of $k$-globs for $k > n$ are both equal to the identity $k$-glob on (necessarily equal) domain and codomain. Such pairs are filled uniquely by their image under $\\iota$. This shows that $n$-skeletal implies $(n+1)$-coskeletal, and it is easy to construct examples to show this implication is as strong as possible.\n\\end{ex}\n\nHowever, for simplicial sets or cubical sets, the situation is rather more complicated. Some of this work was done over 20 years ago \\cite{zaksskeletal} but was never published. In light of continued interest in this problem \\cite{lawverefunctorialconcepts} \\cite{lawvereopenproblems}, the authors thought it was important that this work enter the literature in an easily accessible form.\n\nThe main goal of this paper is to determine the Aufhebung relation in two particular cases, that of simplicial sets and cubical sets. We will show in Theorems \\ref{rcubesetthm} and \\ref{ssetthm} that the Aufhebung relation for cubical sets is $2n$ and for simplicial sets is $2n-1$. The upper bound on the Aufhebung for simplicial sets is due to Zaks \\cite{zaksskeletal} and the upper bound for cubical sets is due to Kennett and Roy \\cite{kennettroyzaksanalysis}. The remaining author provided the examples which prove that these bounds are optimal and cleaned up the exposition.\n\nThe combinatorics involved in the proof for cubical sets is simpler, so we begin in \\S \\ref{cubesec} with this case, even though the proof for simplicial sets was discovered first. In \\S \\ref{simpsec}, we provide a complete proof for simplicial sets without reference to cubical sets, so that the reader who is only interested in that topos can skip directly there. Note that we have adopted similar notation for the face and degeneracy maps of simplicial and cubical sets to emphasize the analogy between the proofs for these toposes. As a result, notation introduced in \\S \\ref{cubesec} is redefined in \\S \\ref{simpsec}. Due to the logical independence of these sections, there should be no danger of confusion.\n\n\\begin{rmk} The apparent similarities in the arguments we present for the cubical and simplicial cases are related to the fact that the simplicial category $\\Delta$ and the cube category $\\mathbb{I}$ are both Reedy categories such that the degree-lowering arrows are uniquely determined by the set of their sections. (This last fact enables the proof of the Eilenberg-Zilber lemma.)  We expect that the combinatorial arguments presented in this paper could easily be adapted to similar situations, but without any other examples in mind, we were insufficiently motivated to do so ourselves.\n\\end{rmk}\n\nWe conclude both sections with a discussion of potential generalisations of these results that describes what seems to be possible as well as highlighting several pitfalls. In particular, we prove in \\S \\ref{simpsec} that the Aufhebung relation for cyclic sets, another topos of interest to homotopy theorists, is between $2n-1$ and $2n+1$. We hope that these remarks will aid future investigations relating to this problem.\n\n\\begin{thank}\nThe second author would like to thank the members of the Australian Category Seminar for being wonderful hosts, Ross Street for introducing her to this work, Richard Garner and Dominic Verity for stimulating conversations on this topic, and her advisor Peter May for his continued support. \n\\end{thank}\n\n\\section{Aufhebung of cubical sets}\\label{cubesec}\n\nThere are many variants in the notion of cubical sets \\cite{grandismauricubical}, which are defined to be presheaves on various cubical categories. We present the most elementary notion, popularized by Daniel Kan \\cite{kanabstractI}. Other versions of the cubical category contain the one described here, and for some of these variants, we expect that some results can be deduced from this one. See Remark \\ref{othercubermk}.\n\nWe write $I$ for the poset category $0 < 1$. Note that $I$ is an interval object: there are two maps $\\ensuremath{\\iota} : * \\rightarrow I$ with $\\ensuremath{\\iota} = 0,1$ from the terminal category to $I$ and a projection $I \\rightarrow *$ that is a common retraction of these maps. The \\emph{cube category} $\\mathbb{I} \\subset \\mathrm{\\bf Cat}$ is the free monoidal category containing an interval object. \n\nConcretely, its objects are the elementary cubes $I^n$ for each $n \\in \\mathbb{N}$. Its morphisms are generated by the elementary face and degeneracy maps, defined on coordinates by \n\\begin{align*} &\\al{\\ensuremath{\\iota}}{i} : I^{n-1} \\hookrightarrow I^n  &&\\mbox{where} \\quad \\al{\\ensuremath{\\iota}}{i} = \\langle \\pi_1,...,\\pi_{i-1},\\ensuremath{\\iota},\\pi_i,...,\\pi_{n-1}\\rangle, &&(i=1,...,n; \\ensuremath{\\iota}=0,1)\\\\ &\\be{i} : I^n \\twoheadrightarrow I^{n-1} &&\\mbox{where} \\quad \\be{i} = \\langle\\pi_1,...,\\pi_{i-1},\\pi_{i+1},...,\\pi_n\\rangle,&&(i=1,..,n),\\end{align*}\nwhere $\\pi_k$ denotes the $k$-th projection map for the product.  These maps satisfy the following relations \n\\begin{align} \n\\al{\\ensuremath{\\iota}}{j}\\al{\\upsilon}{i} &= \\al{\\upsilon}{i}\\al{\\ensuremath{\\iota}}{j-1}  \\quad \\quad i < j \\label{aleq} \\\\ \n\\be{j}\\be{i} &= \\be{i}\\be{j+1}\\; \\quad \\quad i \\leq j \\label{beeq}\\\\\n\\be{j}\\al{\\ensuremath{\\iota}}{i} &= \\begin{cases} \\al{\\ensuremath{\\iota}}{i}\\be{j-1} & i < j \\\\ \n\\text{id} & i = j \\\\\n\\al{\\ensuremath{\\iota}}{i-1}\\be{j} & i > j \n \\end{cases} \\label{mixeq2}\n\\end{align}\n\nThe category $\\mathbb{I}$ has many of the good properties of the simplicial category $\\Delta$.  Every morphism of $\\mathbb{I}$ can be expressed uniquely as a composite $\\mu\\ensuremath{\\epsilon}$ of a monomorphism $\\mu$ and an epimorphism $\\ensuremath{\\epsilon}$. Every epimorphism $\\ensuremath{\\epsilon} : I^n \\rightarrow I^m$ can be factorised uniquely as  $\\be{j_1}\\cdots \\be{j_t}$, where $1 \\leq j_1 < \\cdots < j_t \\leq n$. These are precisely the coordinates of the domain which are deleted. Every monomorphism $\\mu : I^m \\rightarrow I^n$ can be factorised uniquely as $\\al{\\ensuremath{\\iota}_1}{i_1}\\cdots \\al{\\ensuremath{\\iota}_s}{i_s}$, where $n \\geq i_1 >\\cdots > i_s \\geq 1$. The monomorphism $\\mu$ inserts the coordinate $\\ensuremath{\\iota}_k$ at position $i_k$.  \n\n\\begin{defn} The unique factorisation of a morphism of $\\mathbb{I}$ into a product of the form $$\\al{\\ensuremath{\\iota}_1}{i_1}\\cdots\\al{\\ensuremath{\\iota}_s}{i_s}\\be{j_1}\\cdots\\be{j_t}$$ as described above is called the \\emph{canonical factorisation} of the morphism.\n\\end{defn}\n\nThe cube category $\\mathbb{I}$ also has a strict monoidal structure inherited from the cartesian monoidal structure on $\\mathrm{\\bf Cat}$, which is perhaps the main advantage over $\\Delta$.\n\nA \\emph{cubical set} is a functor $X: \\mathbb{I}^{op}\\rightarrow \\mathrm{\\bf Set}$.  We will write $X_{n}$ for the image of the object $I^{n}$ under the functor $X$ and call elements of this set $n$-\\emph{cubes}.  Each arrow $\\tau:I^{m} \\rightarrow I^{n}$ in $\\mathbb{I}$ gives rise to a function $X_{n} \\rightarrow X_{m}$ whose value at $x \\in X_{n}$ is denoted $x\\tau$. An $n$-cube $x$ is \\emph{degenerate} if there exists an epimorphism $\\ensuremath{\\epsilon} : I^n \\rightarrow I^m$ with $m < n$ and an $m$-cube $y$ such that $x = y\\ensuremath{\\epsilon}$.\n\nWrite $\\mathbb{I}_n$ for the full subcategory of $\\mathbb{I}$ on the objects $I^1,\\ldots, I^n$. The essential subtopos $\\mathrm{\\bf Set}^{\\mathbb{I}_n^{\\mathrm{op}}}$ of the category $\\mathrm{\\bf Set}^{\\mathbb{I}^{\\mathrm{op}}}$ of cubical sets induces a pair of adjoint functors $\\mathrm{sk}_n \\dashv \\mathrm{cosk}_n$ on $\\mathrm{\\bf Set}^{\\mathbb{I}^{\\mathrm{op}}}$. Concretely, the $n$-\\emph{skeleton} $\\mathrm{sk}_{n}X$ of a cubical set $X$ consists of those cubes $x \\in X_{m}$ such that there exist $y \\in X_k$ with $k \\leq n$ and an epimorphism $ \\epsilon : I^m \\rightarrow I^k$ in $\\mathbb{I}$ such that $x=y \\epsilon$. As in the introduction, a cubical set $X$ is $n$-\\emph{skeletal} if it is isomorphic to its $n$-skeleton, i.e., when each $m$-simplex with $m >n$ is degenerate.\n\n\\begin{defn} An $k$-\\emph{sphere} or $k$-\\emph{cycle} $c$ in $X$ is a sequence of $(k-1)$-cubes $\\cc{0}{1}, \\cc{1}{1}, \\ldots ,\\cc{0}{k},\\cc{1}{k}$ satisfying the \\emph{cycle equations} \\begin{equation}\\label{ccycleeq} \\cc{\\ensuremath{\\iota}}{j}\\al{\\upsilon}{i} = \\cc{\\upsilon}{i}\\al{\\ensuremath{\\iota}}{j-1} \\hspace{.5cm} \\text{for}\\ i <j. \\end{equation} \n\\end{defn}\n\n\\begin{ex} Let $\\cc{\\ensuremath{\\iota}}{i} = x \\al{\\ensuremath{\\iota}}{i}$ for some $k$-cube $x$ in a cubical set $X$. Then $c$ is a $k$-sphere in $X$. \n\\end{ex}\n\nAs in the introduction, $X$ is $n$-\\emph{coskeletal} if it is isomorphic to $\\mathrm{cosk}_n X$. Concretely, this says that for any $k$-sphere in $X$ with $k >n$ there is a unique $k$-cube $y$ such that $y \\al{\\ensuremath{\\iota}}{i} = \\cc{\\ensuremath{\\iota}}{i}$ for all \n$0 \\leq i \\leq k$ and $\\ensuremath{\\iota}=0,1$.  \n\nImportantly, we have an Eilenberg-Zilber type lemma for cubes.\n\n\\begin{lem}\\label{cezlem}\nFor each $x \\in X_n$, there is a unique non-degenerate $y \\in X_k$ for some $k \\leq n$ together with a unique epimorphism $\\ensuremath{\\epsilon} : I^n \\rightarrow I^k$ such that $x = y\\ensuremath{\\epsilon}$.\n\\end{lem}\n\\begin{proof} \nExistence is obvious. For uniqueness, suppose $x=y\\ensuremath{\\epsilon}$ and $x=y'\\ensuremath{\\epsilon}'$ satisfy these conditions, where $y \\in X_k$ and $y' \\in X_{k'}$. Let $\\mu$ and $\\mu'$ be sections for $\\ensuremath{\\epsilon}$ and $\\ensuremath{\\epsilon}'$ respectively.  Then\n\\[y= y\\ensuremath{\\epsilon}\\mu = x \\mu = y' \\ensuremath{\\epsilon}' \\mu \\]\nSince $y$ is non-degenerate, the epimorphism portion of the canonical factorisation of $ \\ensuremath{\\epsilon}' \\mu$ must be trivial; thus $ \\ensuremath{\\epsilon}' \\mu :I^k \\rightarrow I^{k'}$ is a monomorphism. By a similar argument for $\\mu'$ and $\\ensuremath{\\epsilon}$ we have a monomorphism $\\ensuremath{\\epsilon}\\mu' : I^{k'} \\rightarrow I^k$. So $k=k'$, which means that these monomorphisms are both identities, and hence that $y=y'$. It follows that $\\mu$ is a section for both $\\ensuremath{\\epsilon}$ and $\\ensuremath{\\epsilon}'$. In the cube category $\\mathbb{I}$, a section uniquely determines its retraction; hence $\\ensuremath{\\epsilon}=\\ensuremath{\\epsilon}'$.\n\\end{proof}\n\nWhen $x = y\\ensuremath{\\epsilon}$ as in the lemma, we say that $x$ \\emph{has degeneracy} $n-k$ and write $\\mathrm{dgn}(x)=n-k$. Note, the canonical factorisation of $\\ensuremath{\\epsilon}$ will have the form $\\be{i_1}\\cdots\\be{i_{n-k}}$.\n\n\\begin{lem}\\label{cubedgnlem}\nLet $x$ be an $n$-cube. Then for $\\ensuremath{\\iota} = 0,1$ and all appropriate $i$\n\\begin{align*} \\mathrm{dgn}(x\\be{i}) & =  \\mathrm{dgn}(x) +1 \\\\ \n\\mathrm{dgn}(x\\al{\\ensuremath{\\iota}}{i}) & \\geq \\mathrm{dgn}(x) -1. \n\\end{align*}\n\\end{lem}\n\\begin{proof}\nObvious using Lemma \\ref{cezlem}.\n\\end{proof}\n\nThe degenerate cube $x\\be{i}$ has $x$ for its 0-th and 1-st faces, perpendicular to the $i$-th coordinate direction. All other faces are degenerate, even if $x$ is non-degenerate. We are interested in identifying which faces of a degenerate cube are least degenerate. Hence, the following definition.\n\n\\begin{defn}\nSay that $1 \\leq i \\leq n$ \\emph{reduces} an $n$-cube $x$ when $\\mathrm{dgn}(x\\al{\\ensuremath{\\iota}}{i}) = \\mathrm{dgn}(x)-1$ for some $\\ensuremath{\\iota}$. \n\\end{defn}\n\n\\begin{rmk}\nNote if $x$ is reduced by $i$ then $$x\\al{0}{i} = (x\\al{1}{i}\\be{i})\\al{0}{i} = x\\al{1}{i}(\\be{i}\\al{0}{i}) = x\\al{1}{i}.$$\n\\end{rmk}\n\nThere are several equivalent characterisations of this condition, as indicated by the following lemma, whose proof is an easy exercise.\n\n\\begin{lem}\\label{ctfaelem} The following are equivalent: \\\\ \\indent \\emph{(i)} $i$ reduces $x$. \\\\ \\indent \\emph{(ii)} $\\mathrm{dgn}(x\\al{0}{i}) = \\mathrm{dgn}(x\\al{1}{i})=\\mathrm{dgn}(x)-1$. \\\\ \\indent \\emph{(iii)} the epimorphism of the Eilenberg-Zilber decomposition of $x$ deletes the $i$-th \\\\ \\indent \\qquad coordinate.\\\\ \\indent \\emph{(iv)} $\\be{i}$ appears in the canonical factorisation of the epimorphism in the Eilenberg-\\indent \\qquad Zilber decomposition of $x$. \\\\ \\indent \\emph{(v)}  $x = x \\al{\\ensuremath{\\iota}}{i}\\be{i}$ for some $\\ensuremath{\\iota}$. \\\\ \\indent \\emph{(vi)} $x = x\\al{0}{i}\\be{i} = x\\al{1}{i}\\be{i}$.\n\\end{lem}\n\nNote the following obvious but useful consequence of these equivalent characterisations.\n\n\\begin{lem}\\label{tricklem}\nSuppose $x$ and $y$ are $n$-cubes which are both reduced by $i$. If $x\\al{\\ensuremath{\\iota}}{i}=y\\al{\\upsilon}{i}$ for some $\\ensuremath{\\iota}$ and $\\upsilon$ then $x=y$.\n\\end{lem}\n\nThe main technical tool in the computation of the Aufhebung relation for cubical sets is the following proposition, which we will use to show that spheres consisting of highly degenerate cubes can be filled by a cleverly chosen degenerate copy of one of the least degenerate constituent faces.\n\n\\begin{prop}\\label{techcubeprop}\nLet $X$ be a cubical set which is $n$-skeletal and let $c$ be a $k$-sphere with faces $\\cc{0}{1},\\cc{1}{1},\\ldots, \\cc{0}{k},\\cc{1}{k}$, all degenerate. Let $r$ be the minimal degeneracy of the faces $\\cc{\\ensuremath{\\iota}}{i}$ and let $m$ be the smallest ordinal with $\\mathrm{dgn}(\\cc{\\ensuremath{\\iota}}{m})=r$ for some $\\ensuremath{\\iota}$. If $k<2r+2$ then this sphere is filled by $\\cc{\\ensuremath{\\iota}}{m}\\be{m}$.\n\\end{prop}\n\\begin{proof}\nSuppose $\\mathrm{dgn}(\\cc{\\ensuremath{\\iota}}{m})=r$ with $m$ minimal. We will make repeated use of the set $M = \\{ j_1,\\ldots, j_r\\}$ of ordinals that reduce $\\cc{\\ensuremath{\\iota}}{m}$; i.e., write $M$ for the set of ordinals which appear in the canonical factorisation $\\be{j_1}\\cdots\\be{j_r}$ of the epimorphism part of the Eilenberg-Zilber decomposition of $\\cc{\\ensuremath{\\iota}}{m}$.\n\nBy a dimension argument, $M \\subset \\{1,\\ldots, k-1\\}$. In fact, because we chose $m$ to be minimal, each $j \\in M$ is such that $j \\geq m$: if $j<m$ reduces $\\cc{\\ensuremath{\\iota}}{m}$ then by the cycle equations,\n\\[\nr-1=\\mathrm{dgn}(\\cc{\\ensuremath{\\iota}}{m}\\al{\\upsilon}{j})=\\mathrm{dgn}(\\cc{\\upsilon}{j}\\al{\\ensuremath{\\iota}}{m})\\] which means $\\mathrm{dgn}(\\cc{\\upsilon}{j})=r$, contradicting our choice of $m$.\n\nFirst, we show that $\\cc{0}{m} = \\cc{1}{m}$. For any $j \\in M$ and some fixed $\\upsilon$, \\begin{equation}\\label{jinMeq}r-1 = \\mathrm{dgn} ( \\cc{\\ensuremath{\\iota}}{m} \\al{\\upsilon}{j} ) =\\mathrm{dgn}( \\cc{\\upsilon}{j+1} \\al{\\ensuremath{\\iota}}{m})\\end{equation} by the cycle equations. Because $r$ is the minimal degeneracy of the faces of $c$, this says that $m$ reduces $\\cc{\\upsilon}{j+1}$ which means that $\\cc{\\upsilon}{j+1} \\al{0}{m} = \\cc{\\upsilon}{j+1}\\al{1}{m}$ by Lemma \\ref{ctfaelem}. By applying the cycle equations to both sides of this equality, we see that the $\\al{\\upsilon}{j}$ faces of $\\cc{0}{m}$ and $\\cc{1}{m}$ are equal and Lemma \\ref{tricklem} implies that $\\cc{0}{m} = \\cc{1}{m}$. \n\nHenceforth, we write $c_m$ for $\\cc{0}{m}=\\cc{1}{m}$. We wish to show that $\\cc{\\ensuremath{\\iota}}{u} = c_m\\be{m}\\al{\\ensuremath{\\iota}}{u}$ for all faces of $c$. Immediately from (\\ref{mixeq2}), $\\cc{\\ensuremath{\\iota}}{m} = c_m\\be{m}\\al{\\ensuremath{\\iota}}{m}$ for $\\ensuremath{\\iota}=0,1$. We complete the proof by dividing the remaining faces into three cases.\n\nPart I: ($\\cc{\\ensuremath{\\iota}}{j+1} = c_m\\be{m}\\al{\\ensuremath{\\iota}}{j+1}$ for $j \\in M$). By (\\ref{jinMeq}), $m$ reduces $\\cc{\\ensuremath{\\iota}}{j+1}$ when $j \\in M$.  Hence, \n\\[ \\cc{\\ensuremath{\\iota}}{j+1} = \\cc{\\ensuremath{\\iota}}{j+1}\\al{\\ensuremath{\\iota}}{m}\\be{m} = c_m\\al{\\ensuremath{\\iota}}{j} \\be{m} = c_m\\be{m} \\al{\\ensuremath{\\iota}}{j+1}\\] by (\\ref{ccycleeq}) and then (\\ref{mixeq2}), recalling that $j\\geq m$. This is what we wanted to show.\n\nPart II: ($\\cc{\\ensuremath{\\iota}}{u} = c_m\\be{m}\\al{\\ensuremath{\\iota}}{u}$ for all $u<m$). If $u<m$ then $\\cc{\\ensuremath{\\iota}}{u}$ must be reduced by at least $r+1$ ordinals in $\\{1,\\ldots,k-1\\}$, by minimality of $m$.  If $k-1 < 2r+2$ then at least one of these lies in the $r+1$ element set $ \\{m-1 \\} \\cup M$.  Call this\nelement $p$.  Then we have\n\\begin{align*}\n\\cc{\\ensuremath{\\iota}}{u} & =  \\cc{\\ensuremath{\\iota}}{u}\\al{\\upsilon}{p}\\be{p} && \\mbox{$p$ reduces $ \\cc{\\ensuremath{\\iota}}{u}$} \\\\\n& =  \\cc{\\upsilon}{p+1}\\al{\\ensuremath{\\iota}}{u}\\be{p} && \\mbox{cycle equation ($u<p+1$)}  \\\\\n& =  \\cc{\\upsilon}{p+1}\\be{p+1}\\al{\\ensuremath{\\iota}}{u}&& \\mbox{cubical identity.} \\\\ \\intertext{If $p=m-1$ this is exactly what we want. Otherwise, $p \\in M$ and $\\cc{\\upsilon}{p+1} =c_m\\be{m}\\al{\\upsilon}{p+1}$ by Part I. By substitution}\n\\cc{\\ensuremath{\\iota}}{u} \n & =  c_m\\be{m}\\al{\\upsilon}{p+1}\\be{p+1}\\al{\\ensuremath{\\iota}}{u} \\\\\n & =  c_m\\al{\\upsilon}{p}\\be{m}\\be{p+1}\\al{\\ensuremath{\\iota}}{u} && \\mbox{cubical identity ($m<p+1$)} \\\\\n & =  c_m\\al{\\upsilon}{p}\\be{p}\\be{m}\\al{\\ensuremath{\\iota}}{u} && \n\\mbox{cubical identity} \\\\\n & =  c_m\\be{m}\\al{\\ensuremath{\\iota}}{u}  && \\mbox{$p$ reduces $c_m$.} \n\\end{align*}\nThis is what we wanted to show.\n\nPart III: ($\\cc{\\ensuremath{\\iota}}{u} = c_m\\be{m}\\al{\\ensuremath{\\iota}}{u}$ for $u>m$ and $u-1 \\notin M$). Let $$K = \\{ m\\} \\cup \\{ j+1 \\mid j \\in M, j+1< u \\} \\cup \\{ j \\mid j \\in M, j+1 > u \\}.$$ Because $u-1 \\notin M$ and $u>m$, $K$ has $r+1$ elements. Tautologically, $K\\subseteq\\{1,2,...,k-1\\}$. Because $\\cc{\\ensuremath{\\iota}}{u}$ is reduced by at least $r$ elements, if $k-1<2r+1$, there is a $p\\in K$ that reduces $\\cc{\\ensuremath{\\iota}}{u}$.\n\nCase 1: ($p<u$ so either $p=m$ or $p-1 \\in M$).\nIf $p=m$ then \\begin{align*} \\cc{\\ensuremath{\\iota}}{u} &= \\cc{\\ensuremath{\\iota}}{u}\\al{\\ensuremath{\\iota}}{m}\\be{m} && \\mbox{$m$ reduces $\\cc{\\ensuremath{\\iota}}{u}$} \\\\\n           & = \\cc{\\ensuremath{\\iota}}{m}\\al{\\ensuremath{\\iota}}{u-1}\\be{m} &&\n\\mbox{cycle equation ($m<u$)}  \\\\\n           & = \\cc{\\ensuremath{\\iota}}{m}\\be{m}\\al{\\ensuremath{\\iota}}{u} &&\n\\mbox{cubical identity}\\\\ \\intertext{as desired. If $p-1 \\in M$, then}\n\\cc{\\ensuremath{\\iota}}{u} &= \\cc{\\ensuremath{\\iota}}{u} \\al{\\ensuremath{\\iota}}{p}\\be{p} && \n\\mbox{$p$ reduces $\\cc{\\ensuremath{\\iota}}{u}$} \\\\\n & =  \\cc{\\ensuremath{\\iota}}{p}\\al{\\ensuremath{\\iota}}{u-1}\\be{p} &&\n\\mbox{cycle equation ($p<u$)}  \\\\\n & =  \\cc{\\ensuremath{\\iota}}{p}\\be{p}\\al{\\ensuremath{\\iota}}{u} &&\n\\mbox{cubical identity} \\\\\n             & = c_{m}\\be{m}\\al{\\ensuremath{\\iota}}{p}\n\\be{p}\\al{\\ensuremath{\\iota}}{u} && \\mbox{Part I}\\\\\n & = c_m\\al{\\ensuremath{\\iota}}{p-1}\\be{p-1}\n\\be{m}\\al{\\ensuremath{\\iota}}{u} && \\mbox{(\\ref{mixeq2}) then (\\ref{beeq}) ($m <p$)} \\\\\n             & = c_m\\be{m}\\al{\\ensuremath{\\iota}}{u} &&\n\\mbox{${p-1}$ reduces $c_m$}\n\\end{align*}\nwhich is the desired conclusion.\n\nCase 2: ($p\\geq u$ and hence $p \\in M$).\n\\begin{align*}\n\\cc{\\ensuremath{\\iota}}{u} & =  \\cc{\\ensuremath{\\iota}}{u}\\al{\\ensuremath{\\iota}}{p}\\be{p} &&\n\\mbox{$p$ reduces $\\cc{\\ensuremath{\\iota}}{u}$} \\\\\n\t\t & =  \\cc{\\ensuremath{\\iota}}{p+1}\\al{\\ensuremath{\\iota}}{u}\\be{p} && \\mbox{cycle equation $(u \\leq p)$} \\\\\n                 & =  \\cc{\\ensuremath{\\iota}}{p+1}\\be{p+1}\\al{\\ensuremath{\\iota}}{u} && \n\\mbox{cubical identity} \\\\\n             & =  c_m\\be{m}\\al{\\ensuremath{\\iota}}{p+1}\n\\be{p+1}\\al{\\ensuremath{\\iota}}{u} && \\mbox{Part I}\t \\\\\n        \t& = c_m\\al{\\ensuremath{\\iota}}{p}\\be{p}\\be{m}\\al{\\ensuremath{\\iota}}{u} && \\mbox{(\\ref{mixeq2}) then (\\ref{beeq}) ($m \\leq p$)}\n \\\\    & =  c_m\\be{m}\\al{\\ensuremath{\\iota}}{u} && \\mbox{$p$ reduces $c_m$} \n\\end{align*}\n\nCombining these cases, we have shown that if $k < 2r+2$ then $\\cc{\\ensuremath{\\iota}}{u} = c_m\\be{m}\\al{\\ensuremath{\\iota}}{u}$ for all $u=1,\\ldots,k$ and $\\ensuremath{\\iota}=0,1$. Hence, $c_m\\be{m}$ is a filler for the $k$-sphere $c$.\n\\end{proof}\n\nIn order to use Proposition \\ref{techcubeprop} to prove that $n$-skeletal implies $2n$-coskeletal, we must also show that the filler it constructs for high dimensional spheres is unique. This follows from the following lemma, which states that degenerate cubes are uniquely determined by their boundaries.\n\n\\begin{lem}\\label{cubeuniquelem}\nIf $x$ and $y$ are two degenerate $k$-cubes in $X$ with the same faces, i.e., if \n $x\\al{\\ensuremath{\\iota}}{i} = y\\al{\\ensuremath{\\iota}}{i}$ for all $1\\leq i\\leq k$ and \n$\\ensuremath{\\iota}=0,1$,  then $x=y$.\n\\end{lem}\n\\begin{proof} Because both $x$ and $y$ are degenerate there is some $i$ that reduces $x$ and some $j$ that reduces $y$. If $i=j$ we are done by Lemma \\ref{tricklem}, so we suppose without loss of generality that $i <j$. Then \n\\[ x = x\\al{\\ensuremath{\\iota}}{i}\\be{i} = y\\al{\\ensuremath{\\iota}}{i}\\be{i} = y\\al{\\ensuremath{\\iota}}{j}\\be{j}\\al{\\ensuremath{\\iota}}{i}\\be{i} = y \\al{\\ensuremath{\\iota}}{i}\\be{i}\\al{\\ensuremath{\\iota}}{j}\\be{j} = x\\al{\\ensuremath{\\iota}}{i}\\be{i}\\al{\\ensuremath{\\iota}}{j}\\be{j} = x\\al{\\ensuremath{\\iota}}{j}\\be{j}\\] using the hypothesis that $x$ and $y$ share the same faces, the definition of $i$ and $j$, and the cubical identities. This says that $j$ reduces $x$ as well as $y$ and the conclusion follows by Lemma \\ref{tricklem}.\n\\end{proof}\n\nIt is easy to check that a $0$-skeletal cubical set is $1$-coskeletal. For larger $n$, we use the preceding work to prove our main result.\n\n\\begin{thm}\\label{rcubeupperboundthm}\nIf a cubical set is $n$-skeletal, it is $2n$-coskeletal. Hence,\nthe Aufhebung relation for the topos of cubical sets is bounded above by $2n$. \n\\end{thm}\n\\begin{proof} We must show that any $k$-sphere in an $n$-skeletal cubical set $X$ with $k> 2n$ can be filled uniquely. The inequality $k > 2n$ can be rewritten as $k < 2(k-1-n)+2$. The faces of a $k$-sphere are $(k-1)$-cubes, which therefore have degeneracy at least $k-1-n$.  Applying Proposition \\ref{techcubeprop}, any $k$-sphere has a filler. By Lemma \\ref{cubeuniquelem}, it's unique.\n\\end{proof}\n \n\\begin{ex} Let $X$ be the $n$-skeletal cubical set generated by a single vertex $v$ and two $n$-cubes $x$ and $y$, with each face equal to the $(n-1)$-cube $v\\be{1}\\cdots\\be{n-1}$. We define a $2n$-sphere with faces $$\\cc{\\ensuremath{\\iota}}{i} = \\begin{cases} x\\be{1}\\cdots\\be{n-1} & 1 \\leq i \\leq n \\\\ y\\be{n+1}\\cdots\\be{2n-1} & n < i \\leq 2n. \\end{cases}$$ No cube of $X$ contains both $x$ and $y$ as faces, so this sphere has no filler.\\end{ex}\n\n\\begin{thm}\\label{rcubesetthm}\nThe Aufhebung relation for the topos of cubical sets is $2n$.\n\\end{thm}\n\\begin{proof} Immediate from Theorem \\ref{rcubeupperboundthm} and the preceding example, which shows that an $n$-skeletal cubical set is not necessarily $(2n-1)$-coskeletal.\n\\end{proof}\n\n\\begin{rmk}\\label{othercubermk}\nIn the literature, there are a plethora of definitions of a cubical category $\\mathbb{C}$: e.g., cubical categories with partial diagonals, conjunctions, connections, interchange, etc. These typically contain $\\mathbb{I}$ as a non-full subcategory. The categories $\\mathbb{C}$ are usually not Reedy categories, but are often generalized Reedy categories, in the sense of Berger and Moerdijk \\cite{bergermoerdijkextension}. For such categories, one may again describe canonical factorisations, which are typically only unique up to isomorphism.\n\nFor any of these examples, levels again coincide with dimensions. When the canonical factorisations in $\\mathbb{C}$ are particularly nice, restriction along the inclusion $\\mathbb{I} \\rightarrow \\mathbb{C}$ will be compatible with the skeletal and coskeletal inclusions of the levels, though this is by no means always the case. The example of cubical categories with partial diagonals has this nice property, and a straightforward argument due to Kennett and Roy \\cite{kennettroyzaksanalysis} can be used to prove that the Aufhebung relation is again $2n$. \n\nMore frequently, the restrictions are compatible with only one of the level inclusions. In particular, whenever some epimorphisms in $\\mathbb{C}$ cannot be factored as an epimorphism in $\\mathbb{I}$ followed by something else, $n$-skeletal presheaves on $\\mathbb{C}$ will not be $n$-skeletal as presheaves on $\\mathbb{I}$. Nonetheless, the above results at least provide a bound for the Aufhebung relation. This sort of situation is discussed in Corollary \\ref{cycliccor}.\n\\end{rmk}\n\n\\section{Aufhebung of simplicial sets}\\label{simpsec}\n\nLet $\\Delta$ be the category of finite non-empty ordinals and order preserving maps. The objects of $ \\Delta$ are the ordered sets $\\{0,1,2,\\ldots,n\\}$ denoted by $[n]$ for each non-negative integer $n$. The morphisms of $\\Delta$ are order preserving maps. These are generated by the elementary face and degeneracy maps: for each $n$ there are $n+1$ monics, \n\\[\n\\del{0} \\geq \\del{1} \\geq \\ldots \\geq \\del{n}:[n-1] \\rightarrow [n]\n\\]\nsuch that the image of $\\del{i}$ does not contain $i$ and there are $n$ epics \n\\[ \\sig{0} \\leq \\sig{1} \\leq \\ldots \\leq \\sig{n-1}:[n] \\rightarrow [n-1]\n\\]\nsuch that two elements map to $i \\in [n]$. Explicitly, \n\\begin{eqnarray*} \n\\sig{i}(j)=\\left \\{ \\begin{array}{lc}\n                      j & j \\leq i \\\\                      j-1 & j>i \\\\                     \n\\end{array} \n\\right. & \\quad \\del{i} (j)=\\left \\{ \\begin{array}{lc}\n                     j & j<i \\\\                      j+1 & j \\geq i. \\\\\n\\end{array} \\right. \\\\     \n\\end{eqnarray*}\nThese maps satisfy the following relations\n\\begin{eqnarray}\n\\del{j} \\del{i} & = & \\del{i}\\del{j -1} \\,\\,\\, \\qquad \ni < j \\label{deleq} \\\\\n\\sig{j} \\sig{i} & = & \\sig{i} \\sig{j +1} \\,\\, \\qquad \ni \\leq j \\label{sigeq} \\\\\n\\sig{j} \\del{i} & = & \\left \\{ \\begin{array}{ll}\n\t\t\\del{i} \\sig{j -1} & i < j \\\\\n\t\t\\mbox{id} & i = j \\,\\, \\mbox{or} \\,\\, i = j +1 \\\\\n\t\t\\del{i -1} \\sig{j} & i > j +1\\label{mixeq} \\\\ \n\\end{array} \\right.\n\\end{eqnarray}\n\nAny arrow of $\\Delta $ can be written uniquely as a composite\n$\\mu\\epsilon $ of a monic $\\mu $ and an epic $\\epsilon$. Each monic \n$\\mu:[m] \\rightarrow [n]$ can be factorised uniquely as $\\mu = \\del{i_{1}}\\del{i_{2}} \\ldots \\del{i_{s}}$ where $n \\geq i_{1}>i_{2}> \\cdots >i_{s}\\geq0$ are the elements of $[n]$ which are not in the image of $\\mu$.  Each epic $\\epsilon:[n] \\rightarrow [m]$ is uniquely of the form $\\epsilon = \\sig{j_{1}} \\cdots \\sig{j_{t}}$ where $0\\leq j_{1}<j_{2}< \\cdots <j_{t} \\leq n-1$ are the elements $j \\in [n]$ with $\\epsilon(j)=\\epsilon(j+1)$. \n\n\\begin{defn} The unique factorisation of a morphism of $\\Delta$ into a product of the form $$\\del{i_s}\\cdots \\del{i_1}\\sig{j_1}\\cdots \\sig{j_t}$$ as described above is called the \\emph{canonical factorisation} of the morphism.\n\\end{defn} \n\nA simplicial set is a functor $X:\\Delta^{\\!\\mathrm{op}} \\rightarrow \\mathrm{\\bf Set}$.  We will write $X_{n}$ for the image of the object $[n]$ under the functor $X$.  Elements of $X_n$ are called $n$-\\emph{simplices}. Each arrow $\\tau:[m] \\rightarrow [n]$ in $\\Delta$ gives rise to a function $X_{n} \\rightarrow X_{m}$ in $\\mathrm{\\bf Set}$ whose value at $x \\in X_{n}$ is denoted $x\\tau$. Alternatively, a simplicial set $X$ consists of sets $X_n$ for each $n$ together with right actions by the $\\del{i}$, which take $n$-simplices to $(n-1)$-simplices, and the $\\sig{j}$, which take $n$-simplices to $(n+1)$-simplices. An $n$-simplex $x$ is \\emph{degenerate} if there exists epimorphism $\\epsilon:[n] \\rightarrow [m]$ \nwith $m<n$ and an $m$-simplex $y$ such that  $x=y\\epsilon$.\n\nWrite $\\Delta_n$ for the full subcategory of $\\Delta$ on the objects $[0],\\ldots,[n]$. The essential subtopos $\\mathrm{\\bf Set}^{\\Delta_n^{\\mathrm{op}}}$ of the category $\\mathrm{\\bf Set}^{\\Delta^{\\mathrm{op}}}$ of simplicial sets induces pair of adjoint functors  $\\mathrm{sk}_n \\dashv \\mathrm{cosk}_n$ on $\\mathrm{\\bf Set}^{\\Delta^{\\mathrm{op}}}$. Concretely, the $n$-\\emph{skeleton} $\\mathrm{sk}_{n}X$ of a simplicial set $X$ is the subcomplex of $X$ that is formed by all $x \\in X_{k}$ such that there exist $y \\in X_m$ with $m \\leq n$ and an epimorphism $\\epsilon :[k] \\rightarrow [m]$ in $\\Delta$ such that $x=y \\epsilon$. As above, a simplicial set $X$ is $n$-\\emph{skeletal} if it is isomorphic to its $n$-skeleton, i.e., when each $k$-simplex with $k >n$ is degenerate.\n\n\\begin{defn} An $k$-\\emph{sphere} or $k$-\\emph{cycle} $c$ in $X$ is a sequence of $(k-1)$-simplices $c_{0}, \\ldots ,c_{k}$ satisfying the \\emph{cycle equations} \\begin{equation}\\label{cycleeq} c_{j}\\del{i} = c_{i}\\del{j-1} \\hspace{.5cm} \\text{for}\\ i <j. \\end{equation} \n\\end{defn}\n\n\\begin{ex} Let $c_i = x \\del{i}$ for some $k$-simplex $x$ in a simplicial set $X$. Then $c$ is a $k$-sphere in $X$. \n\\end{ex}\n\nAs in the introduction, $X$ is $n$-\\emph{coskeletal} if it is isomorphic to $\\mathrm{cosk}_n X$. Concretely, this says that for any $k$-sphere in $X$ with $k >n$ there is a unique $k$-simplex $y$ such that $y \\del{i} = c_{i}$ for all \n$0 \\leq i \\leq k$.  \n\nThe following lemma is very important.\n\n\\begin{lem}[Eilenberg-Zilber lemma]\\label{ezlem}\nFor each $x \\in X_{n}$, there is a unique non-degenerate $y \\in X_k$ for some $k\\leq n$ together with a unique epimorphism $\\ensuremath{\\epsilon} : [n] \\rightarrow [k]$ such that $x = y\\ensuremath{\\epsilon}$.\nis unique.\n\\end{lem}\n\\begin{proof} Similar to \\ref{cezlem} or see \\cite[pp 26-27]{gabrielzisman}.\n\\end{proof}\n\nWhen $x = y\\ensuremath{\\epsilon}$ as in the lemma, we say $x$ \\emph{has degeneracy} $n-k$ and write $\\mathrm{dgn}(x)=n-k$. Note, the canonical factorisation of $\\ensuremath{\\epsilon}$ will have the form $\\sig{j_1}\\cdots\\sig{j_{n-k}}$.\n\n\\begin{lem}\\label{dgnlem}\nLet $x$ be an $n$-simplex. Then for $0\\leq i \\leq n$\n\\begin{align*} \\mathrm{dgn}(x\\sig{i}) & =  \\mathrm{dgn}(x) +1 \\\\\n\\mathrm{dgn}(x\\del{i}) & \\geq \\mathrm{dgn}(x) -1.\n\\end{align*}\n\\end{lem}\n\\begin{proof}\nObvious using Lemma \\ref{ezlem}.\n\\end{proof}\n\nUsing (\\ref{mixeq}), the degenerate simplex $x\\sig{i}$ has $x$ as its $i$-th and $(i+1)$-th faces and degeneracies for all of the other faces, even if $x$ is non-degenerate. We will be interesting in identifying which faces of a degenerate simplex are least degenerate. Hence, the following definition. \n\n\\begin{defn}\nSay $i \\in [n]$ {\\em reduces} an $n$-simplex $x$ when $\\mathrm{dgn}(x\\del{i})=\\mathrm{dgn}(x)-1$. \n\\end{defn}\n\n\\begin{lem}\\label{tfaelem} Let $x$ be an $n$-simplex and suppose $0 \\leq i \\leq n$. The following are equivalent: \\\\ \\indent \\emph{(i)} $i$ reduces $x$ \\\\ \\indent \\emph{(ii)} $x=x\\del{i}\\sig{i}$ or $x=x\\del{i}\\sig{i-1}$ \n\\end{lem}\n\\begin{proof}\n(i) $ \\Rightarrow $ (ii). Write $x = y\\epsilon$ as in Lemma \\ref{ezlem}. If $\\epsilon\\del{i}$ is not epic, it factors through $[k-1]$, which contradicts the fact that $\\mathrm{dgn}(x\\del{i})=n-k-1$. Hence, $\\epsilon\\del{i}$ is epic, which means that $\\ensuremath{\\epsilon} (i) = \\ensuremath{\\epsilon}(i-1)$ or $\\ensuremath{\\epsilon}(i) = \\ensuremath{\\epsilon}(i+1)$. In the first case, $\\ensuremath{\\epsilon} = \\ensuremath{\\epsilon} \\del{i}\\sig{i}$ and in the second $\\ensuremath{\\epsilon} = \\ensuremath{\\epsilon}\\del{i}\\sig{i-1}$, which implies (ii).\n\n(ii) $ \\Rightarrow $ (i). Let $j=i-1$ or $j=i$, as appropriate. By (ii) and Lemma \\ref{dgnlem}, $$\\mathrm{dgn}(x)=\\mathrm{dgn}(x\\del{i}\\sig{j})=\\mathrm{dgn}(x\\del{i})+1 \\geq \\mathrm{dgn}(x).$$   So the inequality is an equality and $i$ reduces $x$.\n\\end{proof}\n\nBy the lemma, if $i$ reduces $x$ then $x$ is a degenerate copy of its $i$-th face. Either, this $i$-th face appears as the $i$-th and $(i+1)$-st faces of $x$, in which case $x = x \\del{i}\\sig{i}$; or it's the $(i-1)$-st and $i$-th faces, in which case $x=x\\del{i}\\sig{i-1}$. To enable subsequent accounting, we artificially choose to prefer the former and introduce terminology to distinguish these situations.\n\n\\begin{defn}\nSay $i\\in [n]$ \\emph{properly reduces} $x$ when $x=x\\del{i}\\sig{i}$.\n\\end{defn}\n\n\\begin{ex}\\label{prex} If $x = y\\sig{i}$, then \\[ x = y \\sig{i} = y (\\sig{i} \\del{i}) \\sig{i} = y \\sig{i} (\\del{i} \\sig{i}) = x \\del{i} \\sig{i} \\] by (\\ref{mixeq}). So $i$ properly reduces $x$.\n\\end{ex}\n\n\\begin{rmk}\\label{prrmk} It follows from the computation in \\ref{prex} that $i$ properly reduces $x$ if and only if $\\sig{i}$ appears in the canonical factorisation of the epimorphism of the Eilenberg-Zilber decomposition of $x$. In particular, $x$ is properly reduced by exactly $\\mathrm{dgn}(x)$ ordinals.\n\\end{rmk}\n\n\\begin{lem}\\label{oprlem} If $i$ properly reduces $x$, then $i+1$ reduces $x$ but not necessarily properly.\n\\end{lem}\n\\begin{proof} Assuming $i$ properly reduces $x$, then \\[ x \\del{i+1} \\sig{i} =  (x \\del{i} \\sig{i}) \\del{i+1} \\sig{i} = x \\del{i} (\\sig{i} \\del{i+1}) \\sig{i} = x \\del{i} \\sig{i} = x,\\] which says that $i+1$ reduces $x$ by Lemma \\ref{tfaelem}.\n\\end{proof}\n\n\\begin{lem}\\label{prlem} If $i$ reduces $x$ but $i$ does \\emph{not} properly reduce $x$, then $i-1$ properly reduces $x$.\n\\end{lem}\n\\begin{proof} By Lemma \\ref{tfaelem}, if $i$ reduces $x$ but not properly, then $x = x\\del{i}\\sig{i-1}$. By substitution and (\\ref{mixeq}), \\[ x \\del{i-1} \\sig{i-1} = x \\del{i} \\sig{i-1} \\del{i-1} \\sig{i-1} = x \\del{i} \\sig{i-1} = x, \\] which says that $i-1$ properly reduces $x$.\n\\end{proof}\n\nIt will be clear from the following lemma that the converse to Lemma \\ref{prlem} does not hold.\n\n\\begin{lem}Suppose $x=y\\ensuremath{\\epsilon}$ with $y$ non-degenerate and $\\ensuremath{\\epsilon}$ epic.  \nThen $i$ properly reduces $x$ precisely when $\\ensuremath{\\epsilon}(i) = \\ensuremath{\\epsilon}(i+1)$.\n\\end{lem}\n\\begin{proof}  If $\\ensuremath{\\epsilon}(i)= \\ensuremath{\\epsilon}(i+1)$ then $\\ensuremath{\\epsilon}\\del{i}\\sig{i}=\\ensuremath{\\epsilon}$, so $x \\del{i}\\sig{i} =x$, which says that $i$ properly reduces $x$. Conversely, if $y\\ensuremath{\\epsilon}\\del{i}\\sig{i} = y\\ensuremath{\\epsilon}$, we saw in the proof of Lemma \\ref{tfaelem} that $\\ensuremath{\\epsilon}\\del{i}$ is epi, so by uniqueness $\\ensuremath{\\epsilon} = \\ensuremath{\\epsilon}\\del{i}\\sig{i}$ and hence $\\ensuremath{\\epsilon}(i) =\\ensuremath{\\epsilon}(i+1)$.\n\\end{proof}\n\nThe main technical tool in the computation of the Aufhebung relation for simplicial sets is the following proposition, which we will use to show that spheres consisting of highly degenerate simplices can be filled by a cleverly chosen degenerate copy of one of the least degenerate constituent faces.\n\n\\begin{prop}\\label{techprop} Let $X$ be a simplicial set which is $n$-skeletal, and let $c$ be a $k$-sphere in $X$ whose faces $c_0,\\ldots, c_k$ all have degeneracy at least 2. Let $r$ be the minimal degeneracy the faces $c_i$  \nand let $m$ be the smallest ordinal with $\\mathrm{dgn}(c_{m})=r$.  If $k<2r+3$ then \nthis sphere is filled by $c_m\\sig{m}$.\n\\end{prop}\n\nTo aid the proof of this proposition, we use some technical lemmas.\n\n\\begin{lem}\\label{techlem1} Let $c$ be a $k$-sphere in $X$ with all faces degenerate. Let $r$ be the minimal degeneracy of the faces $c_0,\\ldots, c_k$, and let $m$ be the first ordinal with $\\mathrm{dgn}(c_m)=r$. Suppose $j$ reduces $c_m$. Then \\\\ \\indent \\emph{(a)} $j\\geq m$ and $m$ properly reduces $c_{j+1}$. \\\\ \\indent \\emph{(b)} Furthermore, $c_{j+1} = c_m \\sig{m} \\del{j+1}$. \\\\ \\indent \\emph{(c)} If $m$ reduces $c_m$, then $m$ properly reduces $c_m$ and $c_m = c_{m+1}$.\n\\end{lem}\n\\begin{proof} (a). If $j$ reduces $c_m$ and $j < m$, then $\\mathrm{dgn}(c_j \\del{m-1} ) = \\mathrm{dgn}(c_m \\del{j}) = r-1$. By minimality of $r$, $\\mathrm{dgn}(c_j) =r$, contradicting minimality of $m$. Hence $j \\geq m$. By the cycle equations,  $\\mathrm{dgn}(c_{j+1}\\del{m}) = \\mathrm{dgn}( c_m\\del{j})= r-1$; but $\\mathrm{dgn}(c_{j+1}) \\geq r$, which means that $m$ reduces $c_{j+1}$. If $m$ does not properly reduce $c_{j+1}$ then $m-1$ does, in which case $\\mathrm{dgn}(c_{m-1} \\del{j}) = \\mathrm{dgn}( c_{j+1} \\del{m-1}) = r-1$, contradicting minimality of $m$. So $m$ properly reduces $c_{j+1}$. \n\n(c). If $m$ reduces $c_m$, then $m$ properly reduces $c_m$ because $m-1$ cannot. Taking $j=m$ in part (a), we see that $m$ properly reduces $c_{m+1}$. By the cycle equations and the fact that $m$ properly reduces $c_m$ and $c_{m+1}$, we deduce that \\[ c_m = c_m \\del{m} \\sig{m} = c_{m+1} \\del{m} \\sig{m} = c_{m+1}.\\]\n\n(b). By part (a), $m$ properly reduces $c_{j+1}$. If $j>m$, \\[ c_{j+1} = c_{j+1} \\del{m}\\sig{m} = c_m \\del{j} \\sig{m} = c_m \\sig{m} \\del{j+1}\\] by (\\ref{cycleeq}) then (\\ref{mixeq}). If $j=m$, $\\sig{m}\\del{m+1}=\\text{id}$ and the conclusion follows immediately from (c).\n\\end{proof}\n\n\\begin{lem}\\label{techlem2} Let $c$, $X$, $r$, $m$ be as above. Then there are least $r+2$ faces $c_u$ of $c$ such that $\\mathrm{dgn}(c_u)=r$ and $c_u = c_m \\sig{m}\\del{u}$.\n\\end{lem}\n\\begin{proof} Let $\\ensuremath{\\epsilon}$ be the unique epimorphism the Eilenberg-Zilber decomposition of $c_m$, and let $$M = \\{ j \\mid \\ensuremath{\\epsilon}(j) = \\ensuremath{\\epsilon}(j+1)\\}$$ be the set of indices $m \\leq j \\leq k-2$ that properly reduce $c_m$; the lower bound is from Lemma \\ref{techlem1} and the upper bound is by a dimension argument. By Remark \\ref{prrmk}, $|M|=r$. By Lemma \\ref{oprlem}, if $l \\in [k-1]$ is the smallest ordinal not in $M$, then $l$ reduces $c_m$ also.\n\nLet $j \\in M \\cup \\{l\\}$. Then $$\\mathrm{dgn}(c_{j+1}\\del{m}) = \\mathrm{dgn}(c_m\\del{j}) = r-1,$$ which implies that $\\mathrm{dgn}(c_{j+1}) = r$. This gives us a set of $r+2$ elements of degeneracy $r$ \\[\\{c_m\\} \\cup \\{ c_{j+1} \\mid j \\in M\\} \\cup \\{c_{l+1}\\}.\\] Each of these faces also satisfies $c_u = c_m \\sig{m} \\del{u}$; the first one trivially by (\\ref{mixeq}) and the remaining $r+1$ by Lemma \\ref{techlem1} (b). \n\\end{proof}\n\nIn Proposition \\ref{techprop}, we will show that a sufficiently degenerate sphere $c$ is filled by $c_m\\sig{m}$ where $m$ is the smallest ordinal corresponding to a face of minimal degeneracy. In order for $c_m\\sig{m}$ to fill the sphere $c$, we must have $$c_{m+1} = c_m\\sig{m}\\del{m+1}=c_m.$$ The hardest part of the proof will be verifying this condition, so we tackle this first.\n\n\\begin{lem}\\label{mplus1lem} Let $X$ be a simplicial set which is $n$-skeletal, and let $c$ be a $k$-sphere in $X$ whose faces $c_0,\\ldots,c_k$ all have degeneracy at least $2$. Let $r$ be the minimal degeneracy of the faces $c_i$, and let $m$ be the smallest ordinal with $\\mathrm{dgn}(c_m)=r$. If any of the following hold, then $c_m=c_{m+1}$.\\\\ \\indent \\emph{(a)} $m$ reduces $c_m$. \\\\ \\indent \\emph{(b)} some $j$ properly reduces both $c_m$ and $c_{m+1}$. \\\\ \n\\indent \\emph{(c)} $m$ reduces $c_{m+1}$ and $\\mathrm{dgn}(c_{m+1})=r$. \\\\ \n\\indent \\emph{(d)} $k < 2r+3$.\\\\\nIn particular, if $k < 2r+3$, then $c_m = c_{m+1}$.\n\\end{lem}\n\\begin{proof}\n(a). This is shown in Lemma \\ref{techlem1} (c).\n\n(b). By Lemma \\ref{techlem1} (a), $j \\geq m$ and by part (a) just completed, it suffices to assume that $j > m$. Then \\begin{align*} c_{m+1} &= c_{m+1}\\del{j}\\sig{j} & & \\mbox{$j$ properly reduces $c_{m+1}$} \\\\ &= c_{j+1} \\del{m+1} \\sig{j} & & \\mbox{cycle equations $(m<j)$} \\\\ &= c_m\\sig{m}\\del{j+1}\\del{m+1}\\sig{j} & & \\mbox{Lemma \\ref{techlem1} (b)} \\\\&= c_m\\del{j}\\sig{j}\\sig{m}\\del{m+1} & & \\mbox{(\\ref{mixeq}) then (\\ref{mixeq}) then (\\ref{sigeq}) ($m < j$)} \\\\ &=c_m \\sig{m}\\del{m+1} & & \\mbox{$j$ properly reduces $c_m$} \\\\ &=c_m & & \\mbox{(\\ref{mixeq})} \n\\end{align*}\n\n(c). By the cycle equations $$\\mathrm{dgn}(c_m\\del{m}) = \\mathrm{dgn}(c_{m+1}\\del{m}) = r-1$$ so $m$ reduces $c_m$. Apply part (a).\n\n(d). In light of (a), we assume that $m$ does not reduce $c_m$. In light of (c), we assume that either $\\mathrm{dgn}(c_{m+1}) >r$ (which will eventually lead to a contradiction) or that $m$ does not reduce $c_{m+1}$. By Lemma \\ref{oprlem}, there are at least $r+1$ ordinals $0 \\leq j \\leq k-1$, $j \\neq m$, that reduce $c_m$. By the assumptions just made, there are likewise at least $r+1$ ordinals $0 \\leq j \\leq k-1$, $j \\neq m$, that reduce $c_{m+1}$. So if $k-1 < 2r+2$, then there is some $j$ that reduces both $c_m$ and $c_{m+1}$. Then\n\\begin{align*}  c_{m+1}\\del{j} &=c_{j+1}\\del{m+1} & & \\mbox{cycle equations ($m < j$)} \\\\\n&= c_m\\sig{m}\\del{j+1}\\del{m+1} & & \\mbox{Lemma \\ref{techlem1} (b)} \\\\\n&= c_m \\sig{m} \\del{m+1}\\del{j} & & \\mbox{simplicial identity}\\\\\n&= c_m \\del{j}\n\\end{align*}\n\nLet $y_m\\sig{j_1}\\ldots\\sig{j_r}$ be the Eilenberg-Zilber decomposition of $c_m$. Because $j$ reduces $c_m$, at least one of $\\sig{j}$ or $\\sig{j-1}$ appears in this sequence; let $s$ be the index such that $j_s=j$ if possible and $j_s=j-1$ otherwise. By repeated application of (\\ref{mixeq}), the Eilenberg-Zilber decomposition for $c_m\\del{j}$ is $$y_m\\sig{j_1}\\cdots \\sig{j_{s-1}}\\widehat{\\sig{j_s}} \\sig{j_{s+1}-1}\\cdots\\sig{j_r}.$$  Similarly, let $c_{m+1}= y_{m+1}\\sig{i_1}\\ldots\\sig{i_{r'}}$ and let $i_t=j$ if possible and take $i_t=j-1$ otherwise. The Eilenberg-Zilber decomposition of $c_{m+1}\\del{j}$ is $$y_{m+1}\\sig{i_1}\\cdots \\sig{i_{t-1}}\\widehat{\\sig{i_t}} \\sig{i_{t+1}-1}\\cdots \\sig{i_{r'}-1}$$ and by the above computation, these must be equal. Using the uniqueness statement, it follows that $r'=r$ (which means that $\\mathrm{dgn}(c_{m+1})=r$), $s=t$, $y_m=y_{m+1}$, and the sequences of elementary degeneracies (excluding $\\sig{j_s}$ and $\\sig{i_t}$) agree. In particular, because $r>1$, we can apply (b) to conclude that $c_m=c_{m+1}$.\n\\end{proof}\n\nFinally, we can prove Proposition \\ref{techprop}. Unfortunately, despite the lengthy preparation, this will still be considerably harder than it was to prove the analogous result for the topos of cubical sets.\n\n\\begin{proof}[Proof of Proposition \\ref{techprop}] We must show that $c_u = c_m \\sig{m} \\del{u}$ for all $0 \\leq u \\leq k$. Let $M$ be the set of indices which properly reduce $m$, and let $l$ be one greater than the largest element of $M$. As we saw in the proof of Lemma \\ref{techlem2}, each $j \\in M$ satisfies $m \\leq j \\leq k-2$ and $|M|=r$.\n\nIn Lemma \\ref{techlem2}, we showed that $c_u = c_m \\sig{m} \\del{u}$ for the $r+2$ element set $$\\{ m\\} \\cup \\{ j+1 \\mid j \\in M\\} \\cup \\{l+1\\}.$$ In Lemma \\ref{mplus1lem}, we proved the difficult case $u=m+1$. We divide the remaining proof into three parts, each with a few cases.\n\nPart I: ($c_u = c_m \\sig{m} \\del{u}$ for all $u < m$). If $m=0$ this case is vacuous, so we may assume $m>0$. If $u<m$ then $c_{u}$ must be properly reduced by at least $r+1$ ordinals in $[k-2]$ since $c_{m}$ is of \nminimal degeneracy and the first ordinal of such degeneracy.  If $k-1 < 2r+2$ then $c_{u}$ must be properly reduced by at least one ordinal in the set $\\{m-1\\} \\cup M$. Call this element $p$. Note that $u<m$ implies $u \\leq p$ also. By the hypothesis, (\\ref{cycleeq}), and (\\ref{mixeq}), \\begin{equation}\\label{quickthing}c_u = c_u \\del{p} \\sig{p} = c_{p+1} \\del{u} \\sig{p} = c_{p+1} \\sig{p+1} \\del{u}.\\end{equation}\n\nIf $p=m-1$, this is what we intended to show.\n\nIf $p=m$, then by part (c) of Lemma \\ref{techlem1}, $c_m=c_{m+1}$ and by hypothesis $m$ properly reduces $c_m$. We use these facts to compute \n\\begin{align*} \nc_u &= c_{m+1} \\sig{m+1} \\del{u} & & (\\ref{quickthing}) \\\\ &= c_m \\sig{m+1} \\del{u} & & \\mbox{Lemma \\ref{techlem1} (c)} \\\\ &= c_m \\del{m} \\sig{m} \\sig{m+1} \\del{u} & & \\mbox{$m$ properly reduces $c_m$} \\\\ &= c_m \\del{m} \\sig{m} \\sig{m} \\del{u} & & (\\ref{sigeq}) \\\\ &= c_m \\sig{m} \\del{u} & & \\mbox{$m$ properly reduces $c_m$} \\end{align*} as desired. \n\nIf $p>m$,\\begin{align*}\nc_{u} & =  c_{p+1}\\sig{p+1}\\del{u}  & &(\\ref{quickthing}) \\\\\n     & =  c_{m}\\sig{m}\\del{p+1}\\sig{p+1}\\del{u}  && \\mbox{Lemma \\ref{techlem1} (b)} \\\\   & =  c_{m}\\del{p}\\sig{p}\\sig{m}\\del{u} && \\mbox{(\\ref{mixeq}) then (\\ref{sigeq}) ($m<p$)} \\\\\n      & =  c_{m}\\sig{m}\\del{u}   && p\\ \\mbox{properly reduces}\\ c_m \\end{align*}\nas desired.\n\nPart II: ($c_u=c_m\\sig{m}\\del{u}$ for $u \\geq m$ with $\\mathrm{dgn}(u)>r$). By Lemmas \\ref{techlem2} and \\ref{mplus1lem}, $\\mathrm{dgn}(u)>r$ implies that $u \\not\\in \\{m,m+1 \\} \\cup \\{j+1 \\mid j \\in M \\} \\cup \\{l+1\\}$. Let \\[K= \\{m\\} \\cup \\{j+1 \\mid j \\in M, j+1<u \\} \\cup \\{j \\mid j \\in M, j+1 > u \\}.\\] Because $M \\subset \\{m, \\ldots, k-2\\}$, $K \\subset \\{m, \\ldots , k-1\\}$. In fact, we can deduce that $k-1 \\notin K$: $k-1 \\in K$ is only possible if $u = k$ is properly reduced by $k-2$, in which case $u=l+1$, contradicting the above.\n\nBecause $u-1 \\notin M$ and $u>m$, $|K|= r+1$.  Because $\\mathrm{dgn}(c_u) > r$, there are at least $r+1$ elements of $[k-2]$ that properly reduce $c_u$. If $k-1 < 2r+2$, then there is some $p \\in K$ that properly reduces $c_u$. We will use this $p$ to finish the proof for this case.\n\nCase 1: ($p=m$).  We have \\[ c_u = c_u \\del{m} \\sig{m} = c_m \\del{u-1} \\sig{m} = c_m \\sig{m} \\del{u} \\] by (\\ref{cycleeq}), (\\ref{mixeq}), and the fact that we may take $u > m+1$. This is what we wanted. \n\nCase 2: ($m < p$, $u=p+1$). Inspecting the definition of $K$, we see that $p-1$ properly reduces $c_m$. By Lemma \\ref{oprlem}, it follows that $p$ reduces $c_m$ so by Lemma \\ref{techlem1}, $c_u = c_m\\sig{m}\\del{u}$ as desired.\n\nCase 3: ($m < p$, $u>p+1$). As above, $u > p$ and $p \\in K$ implies that $p-1 \\in M$, which says that $p-1$ properly reduces $c_m$. We compute\n\\begin{align}\nc_{u} & =  c_{u}\\del{p}\\sig{p} & & \\makebox{$p$ properly reduces $c_{u}$}\\notag \\\\\n\t& = c_{p}\\del{u-1}\\sig{p} & &  \\makebox{cycle equations $(p<u)$} \\notag \\\\\n\t& = c_{m}\\sig{m}\\del{p}\\del{u-1}\\sig{p} & & \\makebox{Lemma \\ref{techlem1} (b)} \\notag \\\\\n\t& = c_{m}\\sig{m}\\del{p}\\sig{p}\\del{u} & & \\makebox{simplicial identity ($p+1 < u$)} \\label{thing} \\\\ \\intertext{\nIf $p>m+1$, it follows that} \nc_{u} & =  c_{m}\\del{p-1}\\sig{p-1}\\sig{m}\\del{u} && \\makebox{(\\ref{mixeq}) then (\\ref{sigeq})} \\notag\\\\\n\t& =  c_{m}\\sig{m}\\del{u} && \\makebox{$p-1$ properly reduces $c_{m}$} \\notag \\\\ \\intertext{as desired. If $p=m+1$, (\\ref{thing}) simplifies to $c_u = c_m \\sig{m+1} \\del{u}$. We saw above that $p-1=m$ properly reduces $c_m$. It follows that} c_u &= c_m \\sig{m+1} \\del{u} \\notag\\\\ &= c_m \\del{m} \\sig{m} \\sig{m+1} \\del{u} & & \\mbox{$m$ properly reduces $c_m$}\\notag \\\\ &= c_m \\del{m} \\sig{m} \\sig{m} \\del{u} & & \\mbox{simplicial identity (\\ref{sigeq})} \\notag\\\\ &= c_m \\sig{m} \\del{u} && \\mbox{$m$ properly reduces $c_m$}\\notag\\end{align} completing this case.\n\nCase 4: ($m< u \\leq p$). If $p \\in K$ and $p \\geq u$, then $p \\in M$. This says that $p$ properly reduces both $c_u$ and $c_m$. We compute\n\\begin{align*}\nc_{u} & = c_{u}\\del{p}\\sig{p} && \\mbox{$p$ properly reduces $c_{u}$} \\\\\n\t& =  c_{p+1}\\del{u}\\sig{p} && \\mbox{cycle equations ($u \\leq p$)} \\\\\n\t& =  c_{m}\\sig{m}\\del{p+1}\\del{u}\\sig{p} && \\mbox{Lemma \\ref{techlem1} (b)} \\\\\n\t& =  c_{m}\\del{p}\\sig{p}\\sig{m}\\del{u} && \\mbox{(\\ref{mixeq}) then (\\ref{mixeq}) then (\\ref{sigeq})} \\\\\n \t& =  c_{m}\\sig{m}\\del{u} && \\mbox{$p$ properly reduces $c_{m}$} \n\\end{align*}\n which is what we wanted to show.\n\nPart III: ($c_u = c_m \\sig{m} \\del{u}$ for $u > m+1$ with $\\mathrm{dgn}(u)=r$, not already covered by \\ref{techlem2}). It remains to consider $u \\not\\in \\{m,m+1 \\} \\cup \\{i+1 \\mid i \\in M \\} \\cup \\{l+1\\}$. We use the set $K$ defined in Part II.\n\nBecause $\\mathrm{dgn}(c_u) =r $, there are $r$ elements of $[k-2]$ that properly reduce $c_u$. Unless $c_u$ is properly reduced by the $r$ element set $\\{k-1-r,\\ldots, k-2\\}$, Lemma \\ref{oprlem} implies that there are $r+1$ elements of $[k-2]$ that reduce $c_u$. In this case, $k-1 < 2r+2$ implies that there is some $p \\in K$ that reduces $c_u$, though not necessarily properly.\n\nIf $c_u$ is properly reduced by the set $\\{k-1-r,\\ldots, k-2\\}$ and further if none of these ordinals lie in $K$, we must have $K \\subset \\{m,\\ldots, k-2-r\\}$, which necessitates $r \\leq k-2-r$. We've assumed $k < 2r+3$, so the first inequality is an equality and $K = \\{0,\\ldots, r\\}$, and hence $m=0$. The elements of $K$ all reduce $c_m$, so by Lemma \\ref{techlem1} (b), we may assume $u > r+1$. It follows that $M = \\{0,\\ldots,m-1\\}$ and $c_u$ is properly reduced by $r+1$, which we take for $p$ in this ``pathological'' case.\n\nWe will use the chosen $p$, however it was obtained, to complete the proof.\n\nCase 1: ($p < u$). By the above, either $p \\in K$ or $p$ is 2 greater than the maximal element of $M$. However we have chosen $p$, Lemma \\ref{techlem2} applies.  Using the cycle equations, $$\\mathrm{dgn}(c_u \\del{p} ) = \\mathrm{dgn}(c_p \\del{u-1}) = r-1.$$ If $p =m$, this says that $u-1$ reduces $c_m$, so we're done by Lemma \\ref{techlem1} (b). So we may suppose that $p > m$, in which case $u > m+1$. Then\n\\begin{align*} c_p\\del{u-1} &= c_m \\sig{m} \\del{p}\\del{u-1} & & \\mbox{Lemma \\ref{techlem2}} \\\\\n&= c_m \\del{u-1} \\sig{m} \\del{p} & & \\mbox{(\\ref{deleq}) then (\\ref{mixeq}) $(m+1< u)$}\n\\end{align*}\nThe degeneracy of the left hand side is $r-1$ by the above calculation; hence, $u-1$ reduces $c_m$ by Lemma \\ref{dgnlem}. We apply Lemma \\ref{techlem1} (b) to achieve the desired result.\n\nCase 2: ($p \\geq u$). Note that $u > m+1$, so $p > m$ in this case. The inequality $p \\geq u$ excludes the ``pathological'' case and implies that $p \\in M$. Because $u > m+1$\n\\begin{align*}  c_{u}\\del{p} &=c_{p+1}\\del{u} & & \\mbox{cycle equations} \\\\\n&= c_m\\sig{m}\\del{p+1}\\del{u} & & \\mbox{Lemma \\ref{techlem1} (b)} \\\\\n&= c_m \\del{u-1}\\sig{m}\\del{p} & & \\mbox{(\\ref{deleq}) then (\\ref{mixeq}) ($m+1 < u$)}\n\\end{align*}\nBy hypothesis, $\\mathrm{dgn}(c_u\\del{p})= r-1$, so by Lemma \\ref{dgnlem}, $u-1$ reduces $c_m$. Again apply Lemma \\ref{techlem1} (b) to achieve the desired result.\n\nCombining these (many) cases, we have shown that if $k<2r+3$ then $c_{u}=c_{m}\\sig{m}\\del{u}$ for all $u=0,1, \\ldots ,k$. Hence $\\sig{m}\\del{u}$ is a filler for the $k$-sphere $c$ in $X$.\n\\end{proof}\n\nIn order to use Proposition \\ref{techprop} to prove that $n$-skeletal implies $(2n-1)$-coskeletal, we must also show that the filler it constructs for high dimensional spheres is unique. This follows from the following lemma, which states that degenerate simplices are uniquely determined by their boundaries.\n\n\\begin{lem}\\label{dgnbdrylem}\nIf $x$ and $y$ are degenerate simplices with the same faces, i.e., if $x\\del{i}=y\\del{i}$ for all $i$, then $x=y$.\n\\end{lem}\n\\begin{proof}\nSince $x$ and $y$ are degenerate we can write them as $x=x'\\sig{m}$ and $y=y'\\sig{n}$. If $|m-n| \\leq 1$ then without loss of generality $m \\geq n$ and \n\\[\nx'=x'\\sig{m}\\del{m}=x\\del{m}=y\\del{m}=\ny'\\sig{n}\\del{m}=y'\n\\] by (\\ref{mixeq}). If $m=n$ it is clear that $x=y$. If $m = n+1$, \\[ y' = y \\del{n} = x \\del{n} = x' \\sig{n+1} \\del{n} = x' \\del{n} \\sig{n} \\] and \\[ x= x' \\sig{n+1} = x' \\del{n} \\sig{n} \\sig{n+1} = x' \\del{n} \\sig{n} \\sig{n} = y' \\sig{n} = y\\] by (\\ref{sigeq}).\n\nIf $|m-n|>1$, then \\begin{equation}\\label{mneq}(\\del{m}\\sig{m})(\\del{n}\\sig{n}) = (\\del{n}\\sig{n})(\\del{m}\\sig{m})\\end{equation} by the simplicial identities. By the computation in Example \\ref{prex}, $m$ properly reduces $x$ and $n$ properly reduces $y$. In fact, using (\\ref{mneq}), the same is true with $m$ and $n$ reversed: \\[ x = x \\del{m}\\sig{m} = y \\del{m}\\sig{m} = y \\del{n}\\sig{n}\\del{m}\\sig{m} = y \\del{m}\\sig{m} \\del{n} \\sig{n} = x \\del{m}\\sig{m} \\del{n} \\sig{n} = x \\del{n}\\sig{n}\\] and similarly $y = y \\del{m}\\sig{m}$. It follows that \\[x = x \\del{n} \\sig{n} = y \\del{n} \\sig{n} = y.\\quad\\qedhere\\]\n\\end{proof}\n\nAny $0$-skeletal simplicial set is $1$-coskeletal: there exists a path of 1-simplices connecting each pair of vertices in any sphere of dimension $k >1$. It follows that each of the vertices are the same and the sphere can be filled by the unique degenerate $k$-simplex on that vertex.  Any $1$-skeletal simplicial set is $2$-coskeletal: any sphere of dimension $k>2$ contains at most one non-degenerate edge. It follows that the initial $s$ vertices are the same and the final $k+1-s$ vertices are the same. From this point, it is easy to identify the unique non-degenerate filler.\n\nFor larger $n$, we use the preceding work to prove our main result.\n\n\\begin{thm}\\label{ssetupperboundthm}\nIf a simplicial set is $n$-skeletal with $n>1$, it is $(2n-1)$-coskeletal. Hence, the Aufhebung relation for the topos of simplicial sets is bounded above by $2n-1$. \n\\end{thm}\n\\begin{proof} \nLet $X$ be an $n$-skeletal simplicial set and $c$ be a $k$-sphere in $X$ with $k>2n-1$. The case $n=2$ and $k=4$ can be proven by considering which degenerate 3-simplices have faces which satisfy the cycle equations. Such an argument does not require the difficult combinatorics of Proposition \\ref{techprop}, and the details are left to the reader.\n\nIn general, the inequality $k>2n-1$ can be rewritten as \\[ k < 2k-2n+1 = 2(k-1-n)+3.\\] The faces of $c$ are $(k-1)$-simplices, which must have degeneracy at least $k-1-n$ which is greater than 1 in all cases which remain, so we may apply Proposition \\ref{techprop} to conclude that $c$ has a filler. The filler is necessarily degenerate, so by Lemma \\ref{dgnbdrylem} it's unique. This shows that $X$ is $(2n-1)$-coskeletal, as desired. \n\\end{proof}\n\n\\begin{ex}\\label{simpex} Let $X$ be the $n$-skeletal simplicial set, $n \\geq 3$, generated by a single vertex $v$, distinct $(n-1)$-simplices $x'$ and $y'$ whose faces are degeneracies at $v$, and two $n$-simplices $x$ and $y$ with $x\\del{0} = x'$, $y\\del{n} = y'$, and all other faces of $x$ and $y$ degeneracies at $v$. Let $c$ be the simplicial $(2n-1)$-sphere with $$c_0 = \\cdots = c_{n-1} = x\\sig{0}\\ldots \\sig{n-3} \\quad \\text{and} \\quad c_n = \\cdots = c_{2n-1} = y\\sig{n}\\cdots \\sig{2n-3}.$$ No simplex of $X$ contains both $x'$ and $y'$ as faces; hence, this sphere has no filler.\n\\end{ex}\n\n\\begin{thm}\\label{ssetthm}\nThe Aufhebung relation for the topos of simplicial sets is $2n-1$.\n\\end{thm}\n\\begin{proof} Immediate from Theorem \\ref{ssetupperboundthm} and the preceding example, which shows that an $n$-skeletal simplicial set is not necessarily $(2n-2)$-coskeletal.\n\\end{proof}\n\nThe results of this section can be used to compute a narrow bound on the Aufhebung relation for the topos of cyclic sets. We hope the details of this application will inspire others who are interested in comparing toposes which exhibit an analogous relationship.\n\nConnes' cyclic category $\\Lambda$ is a generalised Reedy category of interest to homotopy theorists \\cite{connescohomologiecyclique}, \\cite{dwyerhopkinskancyclic}, \\cite{lodaycyclic}. It bears the following close relationship to $\\Delta$: these categories have the same objects and a morphism $[n] \\rightarrow [m]$ of $\\Lambda$ can be written uniquely as a cyclic automorphism of $[n]$ followed by an arrow $[n] \\rightarrow [m]$ of $\\Delta$. Levels in the topos of \\emph{cyclic sets}, that is, presheaves on $\\Lambda$ again coincide with dimensions. However, restriction along the inclusion $\\Delta \\hookrightarrow \\Lambda$ only respects the coskeletal inclusions of the essential subtoposes, which complicates the comparison. Nonetheless, the results of this section have the following corollary.\n\n\\begin{cor}\\label{cycliccor} The Aufhebung relation for the topos of cyclic sets is between $2n-1$ and $2n+1$.\n\\end{cor}\n\\begin{proof}\nThe category $\\Lambda$ is generated by the face and degeneracy maps of $\\Delta$ together with cyclic automorphisms $\\tau_n : [n] \\rightarrow [n]$ of degree $n+1$ satisfying certain relations. See \\cite[Ch.~6]{lodaycyclic} for details. The underlying simplicial set of a cyclic set is its image under the restriction functor $\\mathrm{\\bf Set}^{\\Lambda^{\\mathrm{op}}} \\rightarrow \\mathrm{\\bf Set}^{\\Delta^{\\mathrm{op}}}$.\n\nA cyclic set $X$ is $k$-coskeletal if and only if its underlying simplicial set is $k$-coskeletal: a $k$-sphere in a cyclic set $X$ is a morphism from the $(k-1)$-skeleton of the cyclic set represented by the object $[k] \\in \\Lambda$ to $X$. Concretely, such a sphere consists of the usual faces $c_0,\\ldots, c_k$, together with rotations of these faces, satisfying certain relations. A simplicial sphere in a cyclic set determines a unique cyclic sphere of the same dimension: rotations of the faces will automatically satisfy the desired conditions. Furthermore, a filler for the simplicial sphere uniquely fills the cyclic sphere because the rotations of the simplicial filler will have the desired properties. Conversely, every filler for the cyclic sphere provides a filler for the underlying simplicial sphere in the underlying simplicial set. So a cyclic set is $k$-coskeletal as a cyclic set if and only if the underlying simplicial set is $k$-coskeletal.\n\nBy contrast, an $n$-skeletal cyclic set is  $(n+1)$-skeletal as a simplicial set. This follows most immediately from the presentation of the cyclic category $\\Lambda$ as the category generated by the simplicial face and degeneracy maps together with an extra degeneracy map $\\sig{n} : [n] \\rightarrow [n-1]$ for each $n$. This ``extra degeneracy'' satisfies the analogous relations, except that $\\sig{n}\\del{0}$ is an automorphism of $[n-1]$ of order $n$; this was denoted $\\tau_{n-1}$ above. An $n$-simplex in the image of $\\sig{n}$ is degenerate, when $X$ is regarded as a cyclic set, but not when $X$ is regarded as a simplicial set. However, any epimorphism in $\\Lambda$ can be expressed as a product $\\sig{j_0}\\cdots\\sig{j_t}$ where an ``extra degeneracy'' appears as $\\sig{j_0}$, if at all, and nowhere else. It follows that the dimension of a degenerate simplex changes at most by one when we regard the cyclic set as a simplicial set.\n\nWe may now compute a bound for the Aufhebung relation. Given an $n$-skeletal cyclic set, it is $(n+1)$-skeletal as a simplicial set, and so $(2n+1)$-coskeletal as a simplicial set, by Theorem \\ref{ssetupperboundthm}. By the above discussion, this implies that the cyclic set is $(2n+1)$-coskeletal. Hence, the Aufhebung relation is at most $2n+1$.\n\nFor the lower bound, let $X$ be the cyclic set that is generated by the simplices described in Example \\ref{simpex}. It is $n$-skeletal as a cyclic set. (Note however that its underlying simplicial set is $(n+1)$-skeletal and larger than the simplicial set described in the example.) The sphere described in the example cannot be filled for the reasons given above. So $X$ is an $n$-skeletal cyclic set which is not $(2n-2)$-coskeletal.\n\\end{proof}\n\nWe actually expect that the Aufhebung relation for cyclic sets is $2n-1$, based on the following intuition: the top dimensional non-degenerate simplices of an $n$-skeletal cyclic set, regarded as an $(n+1)$-skeletal simplicial set, are rotations of degenerate simplices, and we do not expect the process of rotation to substantially affect the combinatorics. We include Corollary \\ref{cycliccor} more as an illustration of potential extensions of our results than as a definitive analysis of the essential subtoposes of this topos.\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}