5$ kpc, the majority of disk stars are eliminated as shown in the lower panel of Figure REF .\",\n \"Finally, we build a sample of 13,554 halo K giants with 3D positions, 3D velocities, and metallicities.\",\n \"The spatial distribution of the halo K giants in $x-z$ plane is shown in Figure REF , and the distributions of distances and velocities are shown in Figure REF .\",\n \"The majority of halo K giants in our sample have Galactocentric distances in the range $\\\\mathrm {5-60~kpc}$ , with some stars up to $\\\\mathrm {120~kpc}$ .\",\n \"The errors of velocities and distances are shown in Figure REF .\",\n \"The typical errors are 13% in distance, 7 km s$^{-1}$ in line-of-sight velocity and 20 km s$^{-1}$ in tangential velocities.\",\n \"The sky coverage of the halo K giants with velocity color-coded in Figure REF shows that some K giants in the region of Sgr streams have similar velocities, so next we will detect and identify the substructures in position-velocity space from LAMOST halo K giants.\"\n ],\n [\n \"6Distance and Friends-of-friends Algorithm\",\n \"We now start quantifying the presence of any kinematic substructure and identifying members of the substructure in 6D phase-space using LAMOST halo K giants.\",\n \"The kinematically cold streams are not strongly phase-mixed, so the adjacent stars in stellar streams are supposed to have similar velocities.\",\n \"Here, we follow [79] and [41] and develop a statistic that focuses on the incidence of close pairs in $(l,b,d,V_{los},V_l,V_b)$ , and then we combine the friends-of-friends algorithm to group stars that are possible in structure.\"\n ],\n [\n \"6Distance\",\n \"We develop 6Distance from 4distance [79] to calculate a 6D separation of $(l,b,d,V_{los},V_{l},V_{b})$ of any two stars.\",\n \"$(l,b)$ are celestial position in the Galactic coordinate system, $d$ is distance to the Sun, $V_{los}$ is line-of-sight velocity, and $(V_{l},V_{b})$ are tangential velocities along $(l,b)$ .\",\n \"All velocities of $(V_{los},V_{l},V_{b})$ are in GSR frame (see Section REF ).\",\n \"6Distance between two stars $i$ and $j$ is defined as follows: $\\\\delta ^2_{6d}=\\\\omega _{\\\\theta }\\\\theta ^2_{ij}+\\\\omega _{\\\\Delta {d}}(d_i-d_j)^2+\\\\omega _{\\\\Delta {V_{l}}}(V_{l,i}-V_{l,j})^2+\\\\omega _{\\\\Delta {V_{b}}}(V_{b,i}-V_{b,j})^2+\\\\omega _{\\\\Delta {V_{\\\\rm {los}}}}(V_{\\\\rm {los},i}-V_{\\\\rm {los},j})^2,$ where $\\\\theta _{ij}$ is the great circle distance between two stars and calculated by: $\\\\cos {\\\\theta _{ij}}= \\\\cos {b_i}\\\\cos {b_j}\\\\cos {(l_i-l_j)}+\\\\sin {b_i}\\\\sin {b_j}.$ The five weights $\\\\omega _{\\\\theta }, \\\\omega _{\\\\Delta {d}}, w_{\\\\Delta V_{\\\\rm {los}}}, w_{\\\\Delta V_l},$ and $w_{\\\\Delta V_b}$ are used to normalize the corresponding components, and define as follows: $\\\\begin{array}{l}\\\\omega _{\\\\theta } = \\\\frac{1}{\\\\pi ^2},\\\\\\\\\\\\omega _{\\\\Delta {d}} = \\\\frac{1}{100^2} \\\\frac{(d_{\\\\rm {err}}(i)/d(i))^2+(d_{\\\\rm {err}}(j)/d(j))^2}{2^2},\\\\\\\\\\\\omega _{\\\\Delta {V_*}} = \\\\frac{1}{500^2}\\\\frac{V^2_{\\\\rm {*,err}}(i)+V^2_{\\\\rm {*,err}}(j)}{2^2},\\\\\\\\\\\\end{array}$ where $V_*$ stands for $V_l, V_b$ , or $V_{\\\\rm {los}}$ , $<...>$ denotes the average over all stars.\",\n \"The constants in the weights are the largest angular separation ($\\\\pi $ ), the largest heliocentric distance separation (100 kpc), and the largest velocity separation (500 km s$^{-1}$ ), for LAMOST halo K-giant sample.\",\n \"[79] and [88] had pointed out that 4Distance is insensitive to small changes in the weighting factors.\",\n \"We also tried weights ($w_\\\\theta =\\\\frac{1}{<\\\\theta ^2>}$ , $w_{\\\\Delta d}=\\\\frac{1}{<(\\\\Delta d)^2>}$ , $w_{\\\\Delta V_*}=\\\\frac{1}{<(\\\\Delta V_*)^2>}$ ) defined by [88] and find that different weights affect little to the substructure quantification and identification.\"\n ],\n [\n \"The diffuse halo system\",\n \"If position-velocity substructure is present, it is expected that the distribution of $\\\\delta _{6d}$ for the observed sample has more close pairs than the null hypothesis of a diffuse halo system where positions and velocities are uncorrelated.\",\n \"We construct the diffuse halo by only shuffling distances and velocities of our sample, but keeping the angular positions: $\\\\delta ^2_{6d_{r}}=\\\\omega _{\\\\theta }\\\\theta ^2_{ij}+\\\\omega _{\\\\Delta {d}}(d_{i_{r}}-d_{j_{r}})^2+\\\\omega _{\\\\Delta {V_{l}}}(V_{l,i_{r}}-V_{l,j_{r}})^2+\\\\omega _{\\\\Delta {V_{b}}}(V_{b,i_{r}}-V_{b,j_{r}})^2+\\\\omega _{\\\\Delta {V_{\\\\rm {los}}}}(V_{{\\\\rm {los}},i_{r}}-V_{{\\\\rm {los}},j_{r}})^2,$ where $\\\\omega _{\\\\theta }, \\\\omega _{\\\\Delta {d}}, w_{\\\\Delta V_{\\\\rm {los}}}, w_{\\\\Delta V_l}, w_{\\\\Delta V_b}$ , and the indices $(i,j)$ are exactly the same as in $\\\\delta _{6d}$ , but $(i_r,j_r)$ are shuffling indices.\",\n \"The selection function of LAMOST K giants varies with line-of-sight [52].\",\n \"However, it is a reasonable assumption that the distance of the stars in the same part of the sky are uncorrelated to the sample selection.\",\n \"Therefore, we do not shuffle the angular positions when constructing the diffuse halo system.\",\n \"Now, we can quantify the degree of substructure in LAMOST halo K giants by comparing the cumulative distribution of $\\\\delta _{6d}$ for halo K-giant sample, $N_{obs}(<\\\\delta _{6d})$ , with those of 100 null hypotheses of diffuse halo system $N_{null}(<\\\\delta _{6d})$ .\",\n \"Figure REF shows that $N_{obs}(<\\\\delta _{6d})$ exceeds $N_{null}(<\\\\delta _{6d})$ obviously for small values of $\\\\delta _{6d}$ , which means there are more close pairs in halo K giants.\",\n \"Since the null hypotheses of diffuse halo system have the same selection function with LAMOST halo K giants, more close pairs in the halo K giants is unlikely to be a result of selection function.\",\n \"Consequently, LAMOST halo K giants have substructure indeed.\"\n ],\n [\n \"Friends-of-Friends Algorithm\",\n \"The quantification of substructure is just the first step, and the identification of streams is of particular importance to understand the formation of the Milky Way, such as finding the progenitors of streams, exploring the chemical properties and mass of the progenitors, and constraining the dynamics of the Milky Way.\",\n \"FoF is a popular percolation algorithm of group finding.\",\n \"It defines groups that contain all stars separated by 6Distance less than a given linking length.\",\n \"[41] pointed out that FoF algorithm tends to find groups in the region of higher stellar density.\",\n \"As shown in Figure REF , LAMOST mainly observes northern Galactic hemisphere.\",\n \"[87] found the Galactic halo density profile traced by LAMOST halo K giants shows a single power law with index of $-4\\\\sim -5$ .\",\n \"Therefore, we adopt a sky-distance-dependent linking length (i.e., we divide our sample and allocate larger linking lengths for southern Galactic hemisphere and distant sub-samples).\",\n \"Sgr streams are the most prominent, coherent and widely studied tidal streams in the Milky Way, so its a good criterion to test our linking length.\",\n \"We choose linking length for each part by getting enough reliable members of Sgr streams.\",\n \"The reliability of Sgr members is evaluated by positions and velocities of Sgr stream in literature.\",\n \"We will show below (Section REF ) that the Sgr members obtained by our linking lengths are very consistent with simulation [48] and observations [7], [47].\",\n \"The details about the sub-samples and linking lengths will be discussed in Section .\",\n \"Obviously, the method employed here to identify stars in each substructure has an intrinsic uncertainty due to the choice of the linking length and the working coordinates (i.e.\",\n \"position-velocity space in this paper) itself.\",\n \"Therefore, the stellar samples associated to streams or overdensities in this paper suffer of contamination due to the mentioned reasons.\",\n \"However, it is not easy to quantify such contamination exactly.\",\n \"The Sgr streams are very coherent and dense in phase space, so the linking length suitable to identify Sgr streams should be a stringent choice.\",\n \"From the comparison with some known substructure properties in Section , we find the fraction of contamination is not high.\"\n ],\n [\n \"Results on Quantifying Substructure in LAMOST K Giants\",\n \"Both observations and simulations found that the Galactic halo is comprised of at least two overlapping components, an inner halo and an outer halo, with different metallcities, spatial distribution, and kinematics.\",\n \"The inner halo is the dominant component at galactocentric distance up to $\\\\sim $ 15-20 kpc and for metallicity $\\\\mathrm {[Fe/H]>-2.0}$ dex, while the outer halo dominates the region beyond 20 kpc and at metallicity $\\\\mathrm {[Fe/H]<-2.0}$ dex [14], [15], [23], [42], [3], [46], [1], [35], [44], [81], [80].\",\n \"The large sample size of LAMOST halo K giants enables us to quantify the substructure in inner halo and outer halo, as well as in different ranges of metallicity, and to test the contribution of Sgr streams.\"\n ],\n [\n \"Substructure in Inner and Outer Halo\",\n \"As predicted by the hierarchical galaxy formation model, substructures orbiting in outer halo are short after infall and very coherent in space, but substructures in inner halo are long after infall and spatially well-mixed [36].\",\n \"Recent studies used main-sequence turnoff stars (MSTO), blue horizontal branch stars (BHB), and K giants to quantify the degree of substructure and found the Galactic halo significantly more structured at larger radii $r_{\\\\rm {gc}} > $ 20 kpc [4], [88], [20], [41], [68], [17].\",\n \"Many cosmological simulations show a fully phase-mixed inner halo and increasing fraction of the substructure with distance [12], [59], [11], [24], [21], [65], [13].\",\n \"To test it, we divide LAMOST halo K giants into two sub-samples - one with $\\\\mathrm {5 ~kpc<}$ $r_{\\\\mathrm {gc}}$ $\\\\mathrm {<20~kpc}$ and the other with $r_{\\\\mathrm {gc}}$ $\\\\mathrm {>20~kpc}$ , and compare the substructure signals in them.\",\n \"Figure REF shows that the sub-sample beyond $\\\\mathrm {20~kpc}$ presents a stronger structure signal, $\\\\sim $ 3 times more than halo stars within $\\\\mathrm {20~kpc}$ at $lg(\\\\delta _{6d})=-1.0$ .\",\n \"It means the outer halo is more structured than the inner halo, which is consistent with the previous findings based on observations and simulations.\"\n ],\n [\n \"Substructure Dependence on Metallicity\",\n \"Covering large range of metallicities makes K giants good representative tracers of Galactic stellar halo.\",\n \"Metallicity of accreted stars can be used to infer the mass of their progenitor according to the mass-metallicity relation [49].\",\n \"The relation tells us that if a massive dwarf galaxy is accreted, its stellar populations are likely to be metal-rich.\",\n \"The meaningful statistics require large enough sample.\",\n \"We divide LAMOST halo K giants into three sub-samples with comparable sizes: one with [Fe/H] $<$ -1.6 dex, one with -1.6 dex $\\\\leqslant $ [Fe/H] $<$ -1.2 dex, and another with [Fe/H] $\\\\geqslant $ -1.2 dex.\",\n \"Figure REF shows the substructure measurements of the three sub-samples.\",\n \"The most metal-rich sub-sample has the strongest substructure signal, with $\\\\sim $ 9 times more than diffuse halo at $lg(\\\\delta _{6d})=-1.0$ .\",\n \"The sub-sample with intermediate metallicity has $\\\\sim $ 3 times more pairs than diffuse halo at $lg(\\\\delta _{6d})=-1.0$ , while the most metal-poor sub-sample shows the weakest substructure signal, with $\\\\sim $ 2.5 more pairs than diffuse halo at $lg(\\\\delta _{6d})=-1.0$ .\",\n \"These results suggest that the substructure signal is increasing with metallicity.\"\n ],\n [\n \"Contribution of Sgr Streams to the Substructure\",\n \"To study the contribution of the Sgr Stream to the substructure strength, we test the substructure-metallicity relation in two cases, with and without the Sgr Stream stars.\",\n \"In the non-Sgr Stream case, all the K giant stars with $|B|< 12^\\\\circ $ are removed following [53] and [5].\",\n \"Figure REF shows the distribution of the relation for the stars with different metallicity ranges in the two cases, solid lines represent the results for all K giant stars and dashed lines for those out of the plane.\",\n \"We can find a clear relation that the substructure strength is higher for the metal-richer sample in both cases, and a significant decrease of the substructure strength with $lg(\\\\delta _{6d})=-1.0$ between the two cases.\",\n \"What's more, the results of the metal-richer stars decrease more than that of the metal-poorer samples, e.g.\",\n \"the substructure strength of the most metal-rich sample decreases from 0.75 down to 0.53.\",\n \"While the strength of the most metal-poor samples decrease from 0.39 down to 0.33.\",\n \"The difference between the two cases indicates that the Sgr Stream significantly enhances this relation, which was also claimed by [41].\",\n \"All results above are suggesting that LAMOST is able to provide more help for further substructure investigation in the halo, including the Sgr Stream.\"\n ],\n [\n \"FoF Results\",\n \"As described in Section REF , we divide our sample and allocate different linking lengths for each sub-sample.\",\n \"Specifically, we divide our sample into 7 sub-samples according to the sky coverage and distance, as shown in Table REF .\",\n \"Note that there are some overlapped regions between the 7 sub-samples.\",\n \"The overlapped regions of sub-samples will produce common grouped members in the result of FoF groups.\",\n \"We will remove the common members from distant sub-sample groups to make sure the grouped members are unduplicated.\",\n \"By comparing our Sgr groups with the known properties of Sgr stream (e.g., distance, position, and velocities), we determine the linking length for each sub-sample.\",\n \"The specific physical sizes of each component corresponding to the linking lengths can be found in Table REF .\",\n \"The physical size is assuming two stars have 5 identical components of 6 phase-space, then calculating the difference component at a given linking lengths.\",\n \"For example, if two stars have identical values of $l,b,d,V_{l},V_{b}$ , a difference of 25 km s$^{-1}$ in $V_{\\\\rm {los}}$ would produce a $\\\\delta _{6d}$ of 0.05.\",\n \"Finally, we identify 25 groups (1517 K giants), associated to 5 known substructures: Sgr stream [39], [53], Monoceros Ring [62], Virgo Overdensity [62], Hercules-Aquila Cloud [6], and Orphan Stream [31], [6].\",\n \"Besides, 18 groups (350 K giants) can not be linked to any known substructure, so they may relate to some unknown substructures.\",\n \"In total, 1867 grouped stars are identified as shown in Figure REF , and the known substructures' sky distribution is shown in Figure REF .\",\n \"The corresponding properties of known substructures and unknown groups are listed in Table and Table , respectively.\"\n ],\n [\n \"Attributing Groups to Sgr Stream\",\n \"Sgr stream is the most prominent stellar stream, and it has become an important tool for studying the Milky Way halo.\",\n \"In Section , the spatial and velocity distributions have shown the exist of Sgr stream in the sample of LAMOST K giants (see Figure REF and Figure REF ).\",\n \"In this section, we link the FoF groups to Sgr streams by comparing them with models (LM10) and observations [7], [47].\",\n \"Comparing with the five most recent pericentric passages of LM10 model, we find 11 groups match well with LM10 model (see Figure REF ), of which 8 groups belong to Sgr leading arm (blue stars), and 3 groups belong to Sgr trailing arm (red stars).\",\n \"Figure REF shows Sgr streams traced by K giants have larger dispersion (even offset) in distance and tangential velocities than LM10 model.\",\n \"Larger dispersion may be caused by the errors of distances.\",\n \"Unlike “standard candles\\\" (e.g., BHB with distances good to 5%, RR Lyrae stars with distances good to 3% and red clump stars with distances better than 10% ), K giants have a typical error of about 15% because their intrinsic luminosities vary by two orders of magnitude with color and depend on metallicity and age.\",\n \"Given that tidal stripping generally eats away a satellite from the outside, so more recent pericentric passages means smaller mean internal (to the dwarf galaxy) orbital radii before they were unbound, and having higher metallicity [54], [34].\",\n \"Figure REF shows the Sgr trailing members are located in more recent pericentic passages than Sgr leading members, and the mean [Fe/H] value of the Sgr trailing members are indeed higher than that of the Sgr leading members.\",\n \"Besides the matched groups with LM10 model, there are 2 groups beyond distance range of LM10 model shown as the grey stars in Figure REF .\",\n \"However, they match well with Sgr debris found by [7].\",\n \"[47] and [7] traced the Sgr streams using red clump stars (RC), blue horizontal branch stars (BHB), main-sequence turn-off stars (MSTO), and red giants (RGB) drown from SDSS.\",\n \"Figure REF shows that the members associated with Sgr streams match well in line-of-sight velocity with tracks found by [7] using SDSS giant stars, but locate closer than the tracks traced by BHB stars and RC stars.\"\n ],\n [\n \"Attributing Groups to Monoceros Ring\",\n \"Monoceros Ring is a large overdensity firstly discovered by [62], and subsequent studies have shown it is a ring-like low latitude structure and could potentially encircle the entire galaxy [90], [40], [67].\",\n \"[90] traced the structure from $l =180^\\\\circ $ to $227^\\\\circ $ with SDSS faint turnoff stars ($(g-r)_0=0.2$ , $g_0$ =19.45).\",\n \"They found the substructure extends 5 kpc above and below the plane of the Galaxy, and stars at southern portion is about 2 kpc farther than those of northern portion.\",\n \"[67] used M giants from 2MASS, and they found the structure both in the north and south hemispheres and spanned at least 100$^\\\\circ $ .\",\n \"[40] detected the structure from colour-magnitude diagram in many lower latitude ($|b|<50^\\\\circ $ ) fields of Isaac Newton Telescope Wide Field Survey.\",\n \"Their structure from $(V-i)_0 \\\\sim 0.45$ , $V_0\\\\sim 0.9$ curved to $(V-i)_0 \\\\sim 1.0$ , $V_0\\\\sim 21.45$ in colour-magnitude diagram, which was also seen in the SDSS Monoceros fields [62].\",\n \"[77] found the structure stretching from 100$^\\\\circ $ to 230$^\\\\circ $ in Galactic longitude, and covering from -30$^\\\\circ $ to 35$^\\\\circ $ in Galactic latitude using Pan-STARRS1 survey.\",\n \"[50] identified the structure from SEGUE spectroscopy in northern Galactic hemisphere, and found a good match with the disrupting dwarf galaxy model by [64].\",\n \"At present, there is little consensus on the origin of the Monoceros Ring.\",\n \"Some studies attributed Monoceros Ring to the accretion debris from satellite [90], [55].\",\n \"While some studies argued that it may be parts of the flare or warp of the disk [57], [56], [18], [33].\",\n \"Rencently, [86] discovered an oscillating asymmetry in the disk in the direction of the anticenter, and associated the third oscillation line with the Monoceros Ring.\",\n \"In this work, we identify spectroscopic members of Monoceros Ring in both northern and southern Galactic hemisphere for the first time.\",\n \"There are four groups belonging to Monoceros Ring, of which one group is in the northern Galactic hemisphere and the other three groups are in the southern Galactic hemisphere.\",\n \"The members of Monoceros Ring mostly locate at 5-7 kpc from Galactic disk plane and show a mean rotation of $\\\\sim $ 185 km s$^{-1}$ , and mean metallicity is -0.66 dex (see Figure REF ).\",\n \"Comparing with [22], Gaia DR2 proper motions of Monoceros Ring show smaller dispersion and slighter gradients with Galactic latitude than SDSS-Gaia-DR1 shown as Figure REF .\",\n \"We compare the Monoceros members with the simulation by [64], which modeled the Monoceros Ring as the result of a disrupted dwarf galaxy.\",\n \"Figure REF shows the Monoceros members are consistent with the model in the northern Galactic hemisphere, but more distant and lower in Galactic latitude than the model in the southern Galactic hemisphere.\",\n \"The values of mean rotation velocity and metallicity for the Ring may reflect the contamination from other Galactic components, the thick disk in particular.\",\n \"Thus, the data-model inconsistency is likely caused by different origin mechanisms, or by contamination from other Galactic components.\"\n ],\n [\n \"Attributing Groups to Virgo Overdensity\",\n \"The substructure in Virgo constellation is very complex, and its nature is still uncertain.\",\n \"Because of a much higher stellar density exhibited by turnoff stars from SDSS, this region has become known as the “Virgo Overdensity\\\" [62].\",\n \"Virgo Overdensity is located 10$\\\\sim $ 20 kpc away from the Sun, over 1000 deg$^2$ [61], [43], [10], [28].\",\n \"We find four groups located in Virgo Overdensity.\",\n \"As shown in Figure REF , the Virgo Overdensity members has a mean metallicity of -1.31 dex, and heliocentric distance from 13 to 25 kpc.\"\n ],\n [\n \"Attributing Groups to Hercules-Aquila Cloud\",\n \"Hercules-Aquila Cloud was found as an overdensity using MSTO in SDSS DR5 by [6].\",\n \"They suggested that this cloud covers a huge area of sky, centered on Galactic longitude $l \\\\sim 40^\\\\circ $ , Galactic latitude $b$ from $- 50^\\\\circ $ to $+ 50^\\\\circ $ , and line-of-sight velocity $V_{\\\\rm {los}} \\\\sim 180\\\\ \\\\rm {km\\\\ s^{-1}}$ .\",\n \"Subsequently, [73] used RR Lyrae from SDSS found the Hercules-Aquila Cloud contained at least 1.6 times stellar density of the halo at heliocentric distance of 15 to 25 kpc, and its mean metallicity is similar to Galactic halo.\",\n \"[82] found the heliocentric distance of RR Lyrae stars in Hercules-Aquila Cloud is 21.9$\\\\pm $ 12.1 kpc, and metallicity is -1.43$\\\\pm $ 0.36 dex.\",\n \"Their study additionally presented a estimate of velocity of Hercules-Aquila Cloud.\",\n \"In their bottom panel of Figure 17, the mean velocity of MSTO stars in Hercules-Aquila Cloud is centered around $V_{\\\\rm {los}}=$ 25km s$^{-1}$ .\",\n \"[75] mapped the Hercules-Aquila Cloud using RR Lyrae from the Catalina Sky Survey [27].\",\n \"They found this substructure is more prominent in the southern Galactic hemisphere than in the north, peaking at a heliocentric distance of 18 kpc.\",\n \"We find three groups are asscoiated with Hercules-Aquila Could in the halo K giants.\",\n \"As shown in Figure REF , the Hercules-Aquila Cloud members has a mean metallicity of -1.31 dex, heliocentric distance from 12 to 21 kpc, and mean line-of-sight velocity $V_{\\\\rm {los}}=$ 33.37 km s$^{-1}$ .\"\n ],\n [\n \"Attributing Groups to Orphan Stream\",\n \"Orphan stream is a roughly $1 \\\\sim 2^\\\\circ $ wide stellar stream, and its progenitor has not been identified yet.\",\n \"It runs from ($165^\\\\circ ,-17^\\\\circ $ ) to ($143^\\\\circ ,48^\\\\circ $ ) in equatorial coordinate [31], [6].\",\n \"In addition, it was also traced with RR Lyrae stars by [74].\",\n \"They found that the most distant parts of Orphan stream is $40 \\\\sim 50$ kpc from the Sun, mean [Fe/H] value of -2.1 dex, and line-of-sight velocity $V_{\\\\rm {los}} \\\\sim 100$ km s$^{-1}$ .\",\n \"We find a group matches well with all these conditions.\",\n \"As shown in Figure REF , its mean [Fe/H] is -2.02 dex, and mean line-of-sight velocity $V_{\\\\rm {los}}=$ 118 km s$^{-1}$ .\"\n ],\n [\n \"Unknown Groups\",\n \"Besides the groups that can be attributed to known streams, there are 18 remaining groups likely relate to unknown substructure.\",\n \"The unknown groups and their velocity-position distribution are shown in Figure REF and Table .\",\n \"In addition, we plot the [Fe/H] distribution for groups with more than 20 members in the last two panels of Figure REF .\",\n \"The full catalog of K giants with more than 5 members are published online, and a sample is shown in Table .\",\n \"The stellar halo of our Milky Way are expected to be comprised largely of debris from disrupted satellite galaxies.\",\n \"The debris may appear as coherent streams for some time, but will phase-mix until they become difficult to recognize.\",\n \"Several prominent substructures in Galactic stellar halo have been found in the published literature, such as the famous Sgr streams.\",\n \"Some studies have attempted to quantify the position-velocity substructure of the stellar halo using K giants and BHB stars [79], [88], [20], [41].\",\n \"[41] even tried to identify members of substructures using SEGUE K giants.\",\n \"However, all previous studies used only 3D positions and 1D radial velocities because of the lack of proper motions at that time.\",\n \"Now, Gaia DR2 can provide useful proper motions for distant halo stars.\",\n \"LAMOST combining with Gaia enables to construct a large sample of $\\\\mathrm {13,554}$ halo K giants with distances up to 100 kpc, radial velocities, metallicities, and proper motions.\",\n \"This paper presents the first attempt to quantify and identify the substructure of the Milky Way’s stellar halo in 6D phase-space.\",\n \"Based on 4Distance used in previous studies [79], [88], [20], [41], we develop 6Distance to define the distance of two stars in phase-space.\",\n \"By comparing the number of close pairs between observed sample and the diffuse halo constructed by shuffling distances and velocities of the observed sample, we can quantify the amount of substructure in the sample.\",\n \"We find that the substructure increases from inner halo to outer halo, and from metal-poor population to metal-rich population, in agreement with the results of [88], [41].\",\n \"Besides quantifying substructures in stellar halo, identifying members of substructure is of particular importance to explore their the origin.\",\n \"We combine 6Distance with FoF algorithm, and manually assign a sky-distance-dependent linking length to identify the substructures.\",\n \"Finally, we find 43 FoF groups (1867 group members), in which 25 groups belong to 5 known substructures: Sgr stream (13 groups), Monoceros Ring (4 groups), Virgo Overdenstiy (4 groups), Hercules-Aquila Cloud (3 groups), and Orphan Stream (1 group); and 18 remaining groups are likely related to unknown substructures.\",\n \"It is worth to point out that for the first time we identify the spectroscopic members of Monoceros Ring both in northern and southern hemispheres, which demonstrates the advantage of LAMOST.\",\n \"The members of Sgr streams locate in distant halo, and are more metal-rich than other halo stars, we conclude that the Sgr stream dominates both trends of substructure versus metallicity and distance.\",\n \"In addition, we analyze the kinematics and metellicities of the Monoceros Ring, Hercules-Aqulia Cloud, Virgo Overdensity, and the unknown groups with more than 20 grouped members.\",\n \"Monoceros Ring shows more metal-rich than the typical halo stars, and its mean rotation velocity are closer to the thick disk, which may reflect the contamination from other Galactic components, the thick disk in particular.\",\n \"This study is supported by the National Natural Science Foundation of China undergrants (NSFC) Nos.\",\n \"11873052, 11890694, 11390371/2, 11573032 and 11773033.\",\n \"X.-X.X.\",\n \"thanks the “Recruitment Program of Global Youth Experts\\\" of China.\",\n \"J.L.\",\n \"acknowledges the NSFC under grants 11703019.\",\n \"L.Z.\",\n \"acknowledges supports from NSFC grants 11703038.\",\n \"This project was developed in part at the 2018 Gaia-LAMOST Sprint workshop supported by the NSFC under grants 11333003 and 11390372.\",\n \"Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope, LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences.\",\n \"Funding for the project has been provided by the National Development and Reform Commission.\",\n \"This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium).\",\n \"Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.\",\n \"Figure: Galactic longitude ll against the line-of-sight velocity V los V_{\\\\rm {los}} of LAMOST K giants.\",\n \"The top panel shows the entire sample, and the signature of disk rotation (sin-shaped) is clear here.\",\n \"In the bottom panel, after removing disk stars (|z|>5|z| > 5 kpc), the signature of disk largely disappeared.\",\n \"Some halo substructures are visible.\",\n \"Figure: The spatial distribution (x-zx-z plane) of 13,554\\\\mathrm {13,554} LAMOST halo K giants Figure: The Galactocentric distance distribution and velocity distributions along with Galactocentric distance of LAMOST halo K giants.\",\n \"Figure: The error distributions of distances and velocities along with distances.\",\n \"The distances have a typical error of 13%.\",\n \"A typical error of 7 km s -1 ^{-1} in line-of-sight velocity makes it the most accurate velocity component.\",\n \"The mean errors of the two tangential-velocity components are about 20 km s -1 ^{-1}, and can spread to ∼\\\\sim 100 km s -1 ^{-1}.\",\n \"Figure: Galactic sky coverage for LAMOST halo K giants.\",\n \"Stars are colored according to Galactocentric distance r gc r_{\\\\rm {gc}}, line-of-sight velocity V los V_{\\\\rm {los}}, and tangential velocities (V l V_{l}, V b V_{b}).\",\n \"In the region of Sgr streams (120 ∘ r_{\\\\rm {gc}} > 20 kpc (black solid line).\",\n \"The substructure signal exists in both regions, while it is stronger in r gc >r_{\\\\rm {gc}} > 20 kpc.\",\n \"Figure: The result of quantifying the halo K giants in different metallicity ranges.\",\n \"The substructure signal increases with metallicity.\",\n \"Figure: The comparison of quantifying the halo K giants before (solid lines) and after (dash lines) removing the Sgr stream (see more detals in the text).\",\n \"The colors represent the different metallicity ranges, which are marked in the legend.s Figure: The x-zx-z plane distribution of 43 groups (1867 stars) identified from LAMOST halo K giants.\",\n \"We combine different colors and symbols to represent different groups.\",\n \"Dashed lines are every 20 kpc in r gc r_{\\\\rm {gc}}.\",\n \"Because of the limited number of colors and symbols, some different groups are shown with same color and symbol.\",\n \"Figure: The sky coverage of five known substructures identified from LAMOST halo K giants: Sgr Stream (red stars), Monoceros Ring (blue triangles), Virgo Overdensity (green circles), Hercules-Aquila Cloud (orange pluses), and Orphan Stream (orange diamonds).\",\n \"The background (gray dots/filled circles) is literature sky coverage of these known substructure: Sgr leading arm and Sgr trailing arm , Monoceros , Virgo Overdensity , Hercules-Aquila Could , and Orphan stream .\",\n \"Figure: The comparison of Sgr members identified in LAMOST halo K giants with LM10 model.\",\n \"There are 8 FoF groups relate to Sgr leading arm (blue star symbols), 3 FoF groups attributed to Sgr trailing arm (red star symbols), and 2 FoF groups belonging to Sgr debris (grey star symbols).\",\n \"The color dots are from LM10 model, which is color-coded by the pericentric passage (values of -1 indicate debris which is still bound at the present day, values of 0 indicate debris stripped on the most recent pericentric passage of Sgr, and values of 1 indicate debris stripped on the previous pericentric passage, see LM10 for details).\",\n \"Comparing with LM10 model, Sgr streams traced by LAMOST K gaints are more diffuse, and locate closer.\",\n \"The mean [Fe/H] values of Sgr leading and trailing arms are -1.24 dex and -1.12 dex, and their dispersion are 0.47 dex and 0.49 dex, which consistent with the prediction of the model that the leading arm is composed by more metal-poor stars due to the origin from the periphery of Sgr dwarf galaxy.\",\n \"Figure: Comparison with literature Sgr streams traced by SDSS BHB and RC in r gc r_{\\\\rm {gc}}-Λ ˜ ⊙ \\\\widetilde{\\\\Lambda }_\\\\odot plane and V los V_{\\\\rm {los}}-Λ ˜ ⊙ \\\\widetilde{\\\\Lambda }_\\\\odot plane.\",\n \"Λ ˜ ⊙ \\\\widetilde{\\\\Lambda }_\\\\odot is same as the definition in .\",\n \"Blue, red, and grey star symbols represent the members of Sgr leading arm, trailing arm, and debris identified in LAMOST halo K giants.\",\n \"Black dots with error bars are from tables 1-5 of , and the red dots with error bars are from table 2 of , which has been increased by 0.35 mag to correct for the reddening towards the progenitor.\",\n \"The Sgr streams traced by LAMOST K giants locate closer than both the BHB stars of and the RC stars of .\",\n \"Figure: The position and velocity distributions of Virgo Overdensity (blue dots), Hercules-Aquila Cloud (red star symbols), and Orphan Stream (green triangles) identified in LAMOST K giants, and the metallicity distributions of the Virgo Overdensity, the Hercules-Aquila Cloud.\",\n \"There are 4 FoF groups relate to Virgo Overdensity, 3 FoF groups relate to Hercules-Aquila Cloud, and 1 FoF group relate to Orphan Stream.\",\n \"The mean [Fe/H] values of Hercules-Aquila Cloud and Virgo Overdensity are both -1.31 dex, and their dispersion are 0.42 dex and 0.40 dex, and the metallicity distribution of Hercules-Aquila Cloud shows three peaks at -1.6 dex, -1.2 dex, and -0.6 dex.\",\n \"Figure: The position and velocity distributions of 18 unknown groups (350 group members), and metallicity distributions of the groups with more than 20 members.\",\n \"The detail properties of these unknown groups are listed in Table .\",\n \"Table: Properties of Each sub-sample Table: Maximum Physical Component Size for Different Linking Length[h] Known Substructures in LAMOST Halo K Giants Table: NO_CAPTION[h] Unknown Groups in LAMOST Halo K Giants Table: NO_CAPTIONcccccccc Group Catalog: Groups with Five or More Members obsida R.A. Decl.\",\n \"$d$ $V_{\\\\rm {hel}}$ pmra pmdec GroupIDb (deg) (deg) (kpc) (km s$^{-1})$ (mas yr$^{-1}$ ) (mas yr$^{-1}$ ) 77201039 15.597092 5.310736 13.6997 -215.36 0.265 -4.238 0 20603144 19.739155 -0.7779 13.9554 -173.39 -1.366 -3.4597 0 185709128 39.33012 3.758879 12.3098 -179.17 -0.346 -2.451 0 187012195 23.141436 -0.383224 13.8292 -188.84 -0.805 -3.233 0 381006238 36.810744 -1.672415 12.8698 -148.22 -0.31 -2.639 0 21110160 38.951621 1.017566 12.9395 -156.68 -0.15 -2.698 0 381014061 33.997727 -0.539124 13.6571 -174.38 -0.4502 -3.003 0 203913110 39.6415 4.362719 13.0332 -199.16 -0.194 -2.99 0 187009063 22.400078 -1.022162 14.6656 -172.4 -0.861 -3.43 0 496704119 18.590844 8.725264 14.7694 -216.16 -1.173 -3.4581 0 aUnique identifier in LAMOST.\",\n \"b0-Sgr trailing arm; 1-Sgr leading arm; 2-Monoceros Ring; 3-Virgo Overdensity; 4-Hercules-Aquila; 5-Sgr Debris; 6-Orphan Stream; $>6$ -unknown.\",\n \"(This table is available in its entirety in a machine-readable form in the online journal.\",\n \"A portion is shown here for guidance regarding its form and content.)\"\n ]\n]"},"id":{"kind":"string","value":"1906.04373"}}},{"rowIdx":2589,"cells":{"text":{"kind":"list like","value":[["Classification of EEG Signals using Genetic Programming for Feature\n Construction"],["Abstract The analysis of electroencephalogram (EEG) waves is of critical importance for the diagnosis of sleep disorders, such as sleep apnea and insomnia, besides that, seizures, epilepsy, head injuries, dizziness, headaches and brain tumors.","In this context, one important task is the identification of visible structures in the EEG signal, such as sleep spindles and K-complexes.","The identification of these structures is usually performed by visual inspection from human experts, a process that can be error prone and susceptible to biases.","Therefore there is interest in developing technologies for the automated analysis of EEG.","In this paper, we propose a new Genetic Programming (GP) framework for feature construction and dimensionality reduction from EEG signals.","We use these features to automatically identify spindles and K-complexes on data from the DREAMS project.","Using 5 different classifiers, the set of attributes produced by GP obtained better AUC scores than those obtained from PCA or the full set of attributes.","Also, the results obtained from the proposed framework obtained a better balance of Specificity and Recall than other models recently proposed in the literature.","Analysis of the features most used by GP also suggested improvements for data acquisition protocols in future EEG examinations."],["Introduction","About 40% of the world's population suffers from some sleep disorder [41], [28].","Sleep quality directly affects the health and quality of life of the human being.","Poor sleep causes many people to seek out specialized clinics for an accurate diagnosis.","One of the most common techniques of analysis is done by observing brain activity, eye movement, muscle tension, and other body signals by polysomnography (PSG).","The examination consists of collecting data through a series of electrodes connected to the patient's skin and scalp during his or her usual nighttime sleep.","This examination allows the diagnosis of several disorders, such as obstructive sleep apnea, insomnia, narcolepsy, restless legs syndrome and bruxism.","It is also useful for the identification of visible waveforms like sleep spindles (SS) and K-complexes (KC) which, besides assisting in sleep staging, are related to the consolidation of memory and sensory systems.","Abnormalities in their forms may indicate neuropathologies or sleep disorders.","In patients with sleep or neurological disorders, the study of these waveforms helps in the understanding of the neurophysiological functioning and thus, allows to raise hypotheses about the problem.","In particular, sleep spindles have a number of theoretical and clinical implications in understanding how brain activity during sleep is affected and the development of the disorder [42].","Figure: Framework proposed in this work for the identification of structures in sleep EEGThe identification of waveforms on EEG signals is usually done by visual inspection by experts.","This is a time-consuming and tiring process, which may introduce biases and errors [34].","In consequence, specialists not always arrive in the same identification, as illustrated in Figure REF .","Because of this, there is an interest in the development of automated tools for waveform detection [42].","Figure: Example of Sleep Spindles and K-complexes identified from EEG data by two different experts.There are many challenges related to EEG signal analysis.","They have spatial and temporal co-variance, implying highly dependent samples.","They are also non-stationary, noisy and sensitive to external interference [19].","In order to describe these signals without losing information, a high number of features are necessary from the original signals, implying in high dimensionality samples.","One way to improve the automated classification of EEG signal structures is by using dimensionality reduction and feature construction techniques.","In this sense, Genetic Programming (GP) can be used to generate a function that generates a set of new, reduced features from the original ones.","Using GP for feature construction has two advantages: First, GP can generate non-linear combinations of features, making it more expressive than traditional feature reduction techniques.","Second, an analysis of the rules created by GP may allow insights about the importance of the different original features, as suggested by Ivert et al. [17].","Guo et al.","have previously proposed a GP framework for feature construction in the context of seizure detection in EEG signals using KNN [13].","In this paper, we build upon this work and apply it to the more difficult problem of detecting structures such as Sleep Spindles and K-complexes.","More specifically, we use short samples (2s vs 26s) to precisely identify the locations of the structure, we use AUC instead of Precision as the fitness function, and we explore five different classifiers instead of just KNN.","To test the proposed framework (Figure REF ) we perform the identification of Sleep Spindles and K-complexes on the DREAMS [9] dataset.","Starting from a set of 75 features per sample, the proposed GP finds a constructed set with a median of 12 features.","We show that the feature set found by GP achieved better AUC than using the full set of features, or even a set of 29 features selected by PCA.","Additionally, the proposed model achieve a better balance of Recall and Specificity when compared with other recently proposed models for the same problem.","Finally, and perhaps more interestingly, an analysis of the rules constructed by GP showed that we could use only one of the three EEG channels in the dataset to obtain the same quality of identification.","This result suggests that a simpler examination could be used, causing less discomfort for patients."],["The EEG Classification Problem","Electroencephalogram (EEG) is a typically noninvasive examination for the observation of electrical activity of the brain [36].","This information is obtained through electrodes attached to the scalp with a conductive paste.","Through the analysis of these data it is possible to detect diseases and psychiatric and neurological problems.","Usually the analysis of these signals done visually by experts (Figure REF ), which makes the process tiresome, tedious and susceptible to errors [34].","To assist specialists in this visual task, a number of methods of automatic processing and analysis of EEG signals has been proposed.","We emphasize the use of automatic methods for the study of apnea [38], epilepsy [13], drowsiness [4], sleep spindles [9], [8], [1], [21], [43], [5], K complexes [14], [31], [37], sleep stages [22] and schizophrenia [33]."],["Sleep Spindles and K-complexes","Sleep has two main phases: REM (Rapid Eye Movement) sleep and NREM (non-REM) sleep.","Occupying up to 80% of the sleep time, the NREM phase is divided into 4 stages, ranging from lightest to deepest sleep [30].","In particular, stage 2 of NREM sleep has as its main characteristic the appearance of specific waveforms, K-complexes and Sleep Spindles.","The beginning of this stage is defined by the occurrence of these signals.","Because of this well defined presence, they are very important for sleep staging.","Although Sleep Spindles mark entry into stage 2 of NREM sleep, they may also occur in stage 3 [2].","When a spindle occurs, the amplitude of the EEG signal increases and decreases progressively, having a minimum duration of 0.5 s with defined bandwidth between 12 and 14 Hz in the criterion of Rechtschafen and Kales [32] (some authors may consider from 11 until 16 Hz).","Peak-to-peak amplitude settings can also be found between 5 and 25 $\\mu V$ [9].","The occurrence of spindles contributes to memory consolidation, to continuous sleep [3] and in the study of sleep and neurological disorders.","The characteristics of the spindles change with the patient's age and sex [6].","The tendency is for it to occur less with advancing age [29].","As for gender, the phenomenon usually occurs twice more during the sleep of women, due to hormonal factors [24], [10].","The sleep spindles, in general, have a well-defined structure (occurrence, bandwidth, and amplitude).","However the advancement of the patient's age and pathologies cause inaccuracies in their shape.","Typically, the number of spindles decreases and their shape deteriorates [18].","Their shape may be distorted and are more subject to the occurrence of interference [8].","Patients with schizophrenia, for example, do not have normal patterns in the spindles [12].","Changes can also be observed due to fatal familial insomnia [27], autism and epilepsy [16].","This lack of standard is important for the diagnosis of neurological diseases, but it makes it more difficult to identify this phenomenon for specialists and automatic methods.","The K-complex is a negative acute wave immediately followed by a positive component that clearly arises in the EEG, having a minimum duration of 0.5s in the frequencies of 12 to 14 Hz [2].","In the identification, they can be easily confused with any waveform with high peaks [20], [11] (Figure REF ).","Abnormal activity of the K complex may be related to epilepsy, restless leg syndrome, and obstructive sleep apnea."],["DREAMS Data","We use the databases collected by the DREAMS project [9], which consist of a series of polysomnography (PSG) with expert annotations on phenomena or sleep disorders.","We use the sleep spindle and K-complexes datasets from this project.","Their purpose is to tune, train and test automatic detection algorithms.","The Spindles dataset consists of 30 minute stretches of the central EEG channel (extracted from full-night PSG records), independently annotated by two experts.","The data were acquired in a sleep laboratory of a Belgian hospital (BrainnetTM System of MEDATEC, Brussels) using a 32-channel digital polygraph.","It is important to highlight that all records on this dataset are from patients with various sleep pathologies: dyssonia, restless legs syndrome, insomnia, apnea syndrome or hypopnea.","EOG, EMG and EEG channels (channels FP1-A1, O1-A1 and C3-A1 or CZ-A1) were recorded, using the European standard data format (EDF) for storage.","Only EEG channels will be used in this work.","The sampling frequency varies between patients, having records of 200Hz, 100Hz or 50Hz of 30 minutes duration.","The recordings were given independently to two experts, who annotated their estimates for the locations of the sleep spindles.","The K-complex records were collected in the same hospital as the spindle registers, with the same equipment.","There are 10 polysomnographic recordings from healthy individuals.","Just like the previous base, we only use EEG channels.","The sampling frequency was 200 Hz for all patients with a 30 min duration.","In the same way, the excerpts were given independently to two experts.","To reduce bias, the experts did not have access to sleep staging of the records."],["Data preparation","The original EEG data was prepared for the automated identification process using the following procedure."],["Wavelets Transform","Wavelet transforms are mathematical tools capable of decomposing signals into several components that allow analysis at different time and frequency scales [7].","An input signal $x$ , passes through a low-pass filter $g$ and a high-pass filter $h$ (parallel, not sequentially), each with a cut-off frequency equal to one half of the sampling frequency of the input.","Then, the two generated sub-signals, that is, the output of the filters has their samples reduced in half (see Figure REF ).","Figure: Example of Digital Wavelet Transform in 3 levelsThis process can be repeated at several levels, causing the output of the low pass filter to be the input signal of a new pair of filters, followed by downsampling."],["Feature extraction","The EEG data used were sampled with different frequencies (50, 100 or 200 Hz).","For the application of the Digital Wavelet Transform (DWT) [23] with 5 decomposition levels, the data were resampled with an increased frequency of 256 Hz in all cases by means of interpolation through a cubic spline.","For each decomposition level ($D1$ to $D5$ ) in each EEG channel, the signal was separated into 2 second samples, and the following attributes were calculated for each sample: Signal amplitude average, Signal amplitude standard deviation (SD), Symmetry, Power Spectral Density (PSD), and Signal curve length.","Following this procedure, we obtain 900 samples (2s samples from 30 minutes of signal) with 75 real valued attributes (3 EEG channels, 5 DWT levels, and 5 attributes per level).","These attributes are summarized in tables REF , REF , and REF .","Here, the columns represent the coefficients of the DWT levels and the lines the operations performed.","Table: Attributes for the central EEG channel (CZ-A1 or C3-A1)Table: Attributes for the EEG channel FP1-A1Table: Attributes for the EEG channel O1-A1"],["Proposed Framework","Previously, Guo et al.","[13] proposed the use of Genetic Programming (GP) for the construction of features for EEG analysis in the classification of epileptic episodes.","Taking this framework as a base, we develop a framework for the identification of structures in sleep EEG, which we describe in this section.","There are many characteristics in the structure identification problem which differentiates it from the earlier classification work.","We must divide the EEG signals into multiple short samples in order to identify the position of the Spindles and K-complexes in the signal.","As a consequence, the data becomes highly unbalanced, complicating the training process.","Also, we work on three distinct EEG channels (as opposed to a single channel in the original work).","We tested several improvements on the original work to deal with this harder problem.","First, we use the Area Under the ROC Curve (AUC) of the classifiers instead of the accuracy as the fitness measure, since the AUC is more robust and discriminating [15].","Also, we compare several classifiers in addition to KNN, to explore the relationship between classifier choice and GP feature construction.","GP is widely applied in the construction and selection of features for its good performance.","In classification problems it is possible to evolve a tree for each problem class, selecting the best attributes and creating new features for each of them [25].","Even with unbalanced data, this approach with GP also gets good results [40].","In literature, there are also application studies in benchmarks [35] and in databases with few samples [26].","Finally, we publish the program of the proposed framework and experiments at our repositoryhttps://github.com/IcaroMarcelino/SleepEEG for reproducibility purposes."],["GP for Feature Construction","Our proposed framework uses Genetic Programming (GP) to generate the constructed set of features from the original features.","The GP tree is defined as follows: The input nodes are selected from the original attributes.","The intermediate nodes are selected from a set of arithmetic operators $\\lbrace +,-,\\times ,\\rbrace $ , as well as a set of protected operators $\\lbrace /,\\ln ,\\sqrt{\\rbrace }$ .","These protected operators have their definitions slightly modified to avoid errors such as division by zero, as follows: $\\text{protected division} (a, b) &=& \\left\\lbrace \\begin{array}{l l}1, & b = 0 \\\\\\frac{a}{b}, & b \\ne 0\\end{array} \\right.\\\\\\text{protected log}(a) &=& \\left\\lbrace \\begin{array}{l l}1, & b = 0 \\\\ln(|a|), & b \\ne 0\\end{array} \\right.\\\\\\text{protected square root} (a) &=& \\sqrt{|a|}$ Additionally, a special \"Feature Operator\" [13], $F$ , is used to indicate how to obtain the constructed features from the GP tree.","The feature operator returns the value of its input as its output, without any changes.","Its purpose is to mark a subtree as one of the constructed features.","Each $F$ operator will be the root of a subtree that expresses the function to calculate one attribute for the constructed set.","In this way, a GP tree containing ten nodes with the $F$ operator will generate a constructed attribute set with 10 attributes.","For example, the tree depicted in Figure REF shows a GP individual with two subtrees marked by the $F$ operator.","If we assume that the original attribute set has 26 attributes (a..z), this tree will generate a constructed attribute set with two attributes: $F_1 = a$ and $F_2 = b - 1$ .","The use of the $F$ operator allows a single GP tree to express multiple attribute constructing functions.","In this way, we avoid having to explicitly define how many attributes will be constructed beforehand, which would be necessary if each attribute were expressed by a separate tree [13].","Also, this allows GP trees in the population to exchange useful subtrees containing sets of constructed attributes that were successful.","We believe that this allows the GP to pass around the most relevant subtrees to the next generations and, with this, keep attributes and attribute subsets that facilitate classification.","Figure: Example of GP program marked with F operators.","This GP tree constructs two attributes: F 1 =aF_1 = a and F 2 =b-1F_2 = b-1To evaluate one GP tree using the structure described in the previous section, we use the following procedure.","First, we generate the set of constructed attributes for the GP tree using the $F$ operators.","Second, we train a classifier using this set of constructed attributes.","Finally, we use the AUC value of the classifier as the fitness value for the GP tree.","In this way, the product of the evolutionary process is both a set of constructed attributes, as well as a classifier trained on those attributes.","The parameters used for the evolutionary process in the current framework are listed in Table REF .","As the parameters provided good results in the initial runs, they were maintained for the following.","The \"Uniform mutation\" selects a random node from the individual and replaces subtree rooted in that node with a a randomly generated one.","Table: GP Model Parameterization"],["Classifiers","We compare the performance of the set of constructed attributes by GP by using five different classifiers: K Nearest Neighbors (KNN) Naive Bayes (NB) Support Vector Machines (SVM) Decision Tree (DT) Multilayer Perceptron (MLP) We perform an initial tuning procedure using the full set of 75 features to select the hyperparameters used by each classifier in the next experiments.","For each tuning classifier, we execute 100 runs for each parameter value tested in the following sets: KNN: $k \\in \\lbrace 3,5,7,9,11,13,15,17,19\\rbrace $ SVM: kernel $\\in \\lbrace $ Radial Basis Function (RBF), polynomial, sigmoid$\\rbrace $ MLP: activation function $\\in \\lbrace $ ReLU, logistic$\\rbrace $ , neurons in hidden layer $\\in \\lbrace 15,30,45,60,75\\rbrace $ The parameters that provide the best performance were selected.","KNN with $k = 5$ , SVM with RBF as kernel and MLP with a single hidden layer with 15 neurons and ReLU activation."],["Experiment","We perform several evaluations of the classifiers on the Sleep Spindles and K-Complexes datasets in order to analyse the performance of the propose framework.","The results of the classifiers are compared using the full set of 75 attributes, a reduced set of 29 attributes selected by PCA, and the set of attributes constructed by the GP framework.","The training dataset (used to train both the GP and the classifiers) was generated by simple random sampling 70% of the signal samples, and labelling them as positive samples (Sleep Spindles or K-complexes) if both specialists agreed on the label.","Additionally, because the dataset is highly unbalanced, we balance the training dataset by randomly removing samples from the majority class until both classes have the same number of signal samples.","The test dataset was generated by the remaining 30% of the samples, and each signal sample was labeled as positive if either specialist annotated it as positive.","Also, the balancing procedure is not performed on the testing data set.","This resulted in a slightly harder testing data set.","For each experiment, we repeat the training/testing procedure 10 times, and report the aggregate results of these 10 repetitions as described in the subsections below.","Figure: refF"],["Results","The first experiment was aimed at verifying the performance of the classifiers on the test dataset without the reduction of dimensionality by GP, i.e., using the 75 features.","The results are also useful to justify the application of feature construction.","The classifiers performance can be seen in Figure REF a.","Only MLP has good results, achieving an AUC greater than 0.7 with low SD.","Figure: Number of occurrences of the feature in the models generated by GP (Same models from Figure c).Figure: Number of dimensions in the models generated by GP (Same models from Figure c)Applying PCA on the data with a 95% threshold for variance, ensuring little loss of information, the initial set of 75 features can be represented with 29 attributes.","The performance of the trained classifiers with the feature set generated by PCA on the respective test set can be seen in Figure REF b.","For this problem, the PCA representation caused a performance reduction in all classifiers except NB.","The performance of applying the classifiers on the feature set generated by PCA can be seen in Figure REF b.","For this problem, the PCA representation caused a decrease in performance in all classifiers except NB.","Applying the GP feature reduction, the AUC of the classification increases for all classifiers except KNN (Figure REF c) and the SD reduces for all cases.","This method is capable to reduce the number of features from 75 to less than 29 (Figure REF ).","Using the same approach for training K-complexes classifiers, the models achieve high AUC scores too (Figure REF d).","Figure: Classifiers performance over the central EEG channel attributes for Sleep Spindles and K-complexes"],["Analysis of gender difference","As mentioned before, there are gender differences in sleep spindles.","To see if this difference affects the performance of the classifiers, the data were separated by gender.","Observing the Figures REF e and REF f, the classifier's performance for female patients is higher than the male patients.","As sleep spindles occur more often in female patients, in data there are more representative samples of the waveform (see Table REF ), which facilitates the training of more efficient classifiers."],["Constructed Features Analysis","In Figure REF , the frequency of occurrence of features in the models training shows that there are attributes more relevant than others in the dataset.","The greater occurrence of the features associated to the central EEG channel indicates that it has more important role to the identification of sleep spindles.","Using only this channel for training (i.e, only with the 25 first features of the dataset), the performance of all classifiers increase for sleep spindles and K-complexes identification (Figure REF ).","This attributes reduction contributes to a better understanding of the phenomenon, providing a more efficient approach by reducing the use of electrodes, consuming less resources and generating less discomfort to the patient.","Table: Comparison between the proposed model and literature models"],["Comparison to Literature Models","In the Table REF , the best generated model with the proposed approach (with NB classifier) was compared with the literature models which also used DREAMS data Further comparisons between sleep spindle identifiers can be seen in [21].","Tsanas at.","el [39] and Zhuang et al.","[44] proposed continuous wavelet transform (CWT) based approaches and the estimation of the probability of spindles occurrences.","Lachner-Piza et al.","[21] proposed a SVM approach with a feature selection method based on the label-feature and feature-feature correlations for determining the relevance and redundancy of each feature.","Observing the performance of the models, all obtained high specificity, indicating that the identification of samples where no spindles samples are present is reliable.","Moreover, there is a trade-off between sensitivity and precision.","In the context of applying automatic identifiers, false negatives are more unwanted than false positives.","That is, a highly accurate but not very sensitive classifier generates many false positives, indicating that it is not judicious.","In a semi-automatic application with low sensitivity, it is necessary for a specialist to inspect the markings performed by the classifier, eliminating the excess of false positives.","This is the case of the detector of Zhuang et al.","[44].","The detectors of Tsanas and Clifford [39] and Lachner-Piza et al.","[21] and the proposed model have achieved a better compromise between sensitivity (recall) and precision.","This implies that the identification of signal stretches as spindles is more reliable.","The proposed approach allows the generation of competitive models with the literature.","The Tsanas and Clifford model, although having a slightly higher sensitivity than the proposed model.","In contrast to the model of Lachner-Piza et al., our model has only minor precision, with 0.03 of difference."],["Discussion","The GP feature construction improves the performance of a classifier reducing the search space and generating more explicit relations between variables.","Observing the reduction in the number of attributes, the dimensionality of the problem is reduced by up to 7 times in most cases.","In addition, by analyzing the most frequent attributes, it is clear which ones are most relevant to the models.","With this information, it is possible to select the most important EEG channels.","In the case of K-complexes sleep spindles, only the central EEG channel is sufficient to perform the waveform identification.","The single channel approach already reduces search space by one-third.","Furthermore, fewer electrodes will be required for the examination, making it more comfortable for the patient, consequently, approaching the sleep in the laboratory of the daily sleep, avoiding biases."],["Conclusions","The use of automatic methods to identify sleep phenomena makes it possible to classify EEG signal segments with good performance, indicating whether or not a particular event occurs.","Excerpts of 30 minutes can have hundreds of events that you want to identify.","In this respect, the proposed model can be used to accelerate the process, and it is up to the expert to assign the classification.","The model was also useful for sleep staging, since the presence of spindles and K-complexes strongly characterize sleep stage 2.","It has also been shown that it is possible to significantly improve the performance of classifiers by selecting and constructing attributes.","In addition, the use of GP allows greater interpretability and mathematical analysis of the new attributes generated, which may help to better understand the model and the problem.","It is also possible to inspect the attributes generated through knowledge in the application domain.","The approach also does not require in-depth knowledge of the application domain.","In the first experiment, no assumptions about the data were performed.","The ease of defining the terms in which the solutions will be written, that is, the operators and the terminals, allow the creation of hybrid models with biomedical information.","It can also facilitate communication between specialists from different areas.","The automation of the selection and construction of attributes generates a dataset suitable for the desired classifier.","But, it is very simple to apply the attributes generated in another classifier.","The processing time is also reduced with the smaller number of dimensions.","The PSG generated signals used in sleep clinics are stored directly on computers.","Therefore, the application of the proposed technique can be easily applied in this context.","The generated models, once trained, make predictions quickly, facilitating a real-time approach.","Measurement of the micro-event activity on EEG signals in different populations can provide important information about abnormalities in brain signals and assist in the investigation and hypothesis assessment of observed phenomena or disturbances.","This underscores the importance of the study.","With the proposed models, the identification of the spindles or K-complexes is less costly for the specialist.","It may even replace its function in this task if the performance of the models is satisfactory for the requested analysis.","Therefore, the diagnosis can be faster and the return to the patient suffering from some disorder is more efficient.","Moreover, this methodology can easily be extended to other classification problems."]],"string":"[\n [\n \"Classification of EEG Signals using Genetic Programming for Feature\\n Construction\"\n ],\n [\n \"Abstract The analysis of electroencephalogram (EEG) waves is of critical importance for the diagnosis of sleep disorders, such as sleep apnea and insomnia, besides that, seizures, epilepsy, head injuries, dizziness, headaches and brain tumors.\",\n \"In this context, one important task is the identification of visible structures in the EEG signal, such as sleep spindles and K-complexes.\",\n \"The identification of these structures is usually performed by visual inspection from human experts, a process that can be error prone and susceptible to biases.\",\n \"Therefore there is interest in developing technologies for the automated analysis of EEG.\",\n \"In this paper, we propose a new Genetic Programming (GP) framework for feature construction and dimensionality reduction from EEG signals.\",\n \"We use these features to automatically identify spindles and K-complexes on data from the DREAMS project.\",\n \"Using 5 different classifiers, the set of attributes produced by GP obtained better AUC scores than those obtained from PCA or the full set of attributes.\",\n \"Also, the results obtained from the proposed framework obtained a better balance of Specificity and Recall than other models recently proposed in the literature.\",\n \"Analysis of the features most used by GP also suggested improvements for data acquisition protocols in future EEG examinations.\"\n ],\n [\n \"Introduction\",\n \"About 40% of the world's population suffers from some sleep disorder [41], [28].\",\n \"Sleep quality directly affects the health and quality of life of the human being.\",\n \"Poor sleep causes many people to seek out specialized clinics for an accurate diagnosis.\",\n \"One of the most common techniques of analysis is done by observing brain activity, eye movement, muscle tension, and other body signals by polysomnography (PSG).\",\n \"The examination consists of collecting data through a series of electrodes connected to the patient's skin and scalp during his or her usual nighttime sleep.\",\n \"This examination allows the diagnosis of several disorders, such as obstructive sleep apnea, insomnia, narcolepsy, restless legs syndrome and bruxism.\",\n \"It is also useful for the identification of visible waveforms like sleep spindles (SS) and K-complexes (KC) which, besides assisting in sleep staging, are related to the consolidation of memory and sensory systems.\",\n \"Abnormalities in their forms may indicate neuropathologies or sleep disorders.\",\n \"In patients with sleep or neurological disorders, the study of these waveforms helps in the understanding of the neurophysiological functioning and thus, allows to raise hypotheses about the problem.\",\n \"In particular, sleep spindles have a number of theoretical and clinical implications in understanding how brain activity during sleep is affected and the development of the disorder [42].\",\n \"Figure: Framework proposed in this work for the identification of structures in sleep EEGThe identification of waveforms on EEG signals is usually done by visual inspection by experts.\",\n \"This is a time-consuming and tiring process, which may introduce biases and errors [34].\",\n \"In consequence, specialists not always arrive in the same identification, as illustrated in Figure REF .\",\n \"Because of this, there is an interest in the development of automated tools for waveform detection [42].\",\n \"Figure: Example of Sleep Spindles and K-complexes identified from EEG data by two different experts.There are many challenges related to EEG signal analysis.\",\n \"They have spatial and temporal co-variance, implying highly dependent samples.\",\n \"They are also non-stationary, noisy and sensitive to external interference [19].\",\n \"In order to describe these signals without losing information, a high number of features are necessary from the original signals, implying in high dimensionality samples.\",\n \"One way to improve the automated classification of EEG signal structures is by using dimensionality reduction and feature construction techniques.\",\n \"In this sense, Genetic Programming (GP) can be used to generate a function that generates a set of new, reduced features from the original ones.\",\n \"Using GP for feature construction has two advantages: First, GP can generate non-linear combinations of features, making it more expressive than traditional feature reduction techniques.\",\n \"Second, an analysis of the rules created by GP may allow insights about the importance of the different original features, as suggested by Ivert et al. [17].\",\n \"Guo et al.\",\n \"have previously proposed a GP framework for feature construction in the context of seizure detection in EEG signals using KNN [13].\",\n \"In this paper, we build upon this work and apply it to the more difficult problem of detecting structures such as Sleep Spindles and K-complexes.\",\n \"More specifically, we use short samples (2s vs 26s) to precisely identify the locations of the structure, we use AUC instead of Precision as the fitness function, and we explore five different classifiers instead of just KNN.\",\n \"To test the proposed framework (Figure REF ) we perform the identification of Sleep Spindles and K-complexes on the DREAMS [9] dataset.\",\n \"Starting from a set of 75 features per sample, the proposed GP finds a constructed set with a median of 12 features.\",\n \"We show that the feature set found by GP achieved better AUC than using the full set of features, or even a set of 29 features selected by PCA.\",\n \"Additionally, the proposed model achieve a better balance of Recall and Specificity when compared with other recently proposed models for the same problem.\",\n \"Finally, and perhaps more interestingly, an analysis of the rules constructed by GP showed that we could use only one of the three EEG channels in the dataset to obtain the same quality of identification.\",\n \"This result suggests that a simpler examination could be used, causing less discomfort for patients.\"\n ],\n [\n \"The EEG Classification Problem\",\n \"Electroencephalogram (EEG) is a typically noninvasive examination for the observation of electrical activity of the brain [36].\",\n \"This information is obtained through electrodes attached to the scalp with a conductive paste.\",\n \"Through the analysis of these data it is possible to detect diseases and psychiatric and neurological problems.\",\n \"Usually the analysis of these signals done visually by experts (Figure REF ), which makes the process tiresome, tedious and susceptible to errors [34].\",\n \"To assist specialists in this visual task, a number of methods of automatic processing and analysis of EEG signals has been proposed.\",\n \"We emphasize the use of automatic methods for the study of apnea [38], epilepsy [13], drowsiness [4], sleep spindles [9], [8], [1], [21], [43], [5], K complexes [14], [31], [37], sleep stages [22] and schizophrenia [33].\"\n ],\n [\n \"Sleep Spindles and K-complexes\",\n \"Sleep has two main phases: REM (Rapid Eye Movement) sleep and NREM (non-REM) sleep.\",\n \"Occupying up to 80% of the sleep time, the NREM phase is divided into 4 stages, ranging from lightest to deepest sleep [30].\",\n \"In particular, stage 2 of NREM sleep has as its main characteristic the appearance of specific waveforms, K-complexes and Sleep Spindles.\",\n \"The beginning of this stage is defined by the occurrence of these signals.\",\n \"Because of this well defined presence, they are very important for sleep staging.\",\n \"Although Sleep Spindles mark entry into stage 2 of NREM sleep, they may also occur in stage 3 [2].\",\n \"When a spindle occurs, the amplitude of the EEG signal increases and decreases progressively, having a minimum duration of 0.5 s with defined bandwidth between 12 and 14 Hz in the criterion of Rechtschafen and Kales [32] (some authors may consider from 11 until 16 Hz).\",\n \"Peak-to-peak amplitude settings can also be found between 5 and 25 $\\\\mu V$ [9].\",\n \"The occurrence of spindles contributes to memory consolidation, to continuous sleep [3] and in the study of sleep and neurological disorders.\",\n \"The characteristics of the spindles change with the patient's age and sex [6].\",\n \"The tendency is for it to occur less with advancing age [29].\",\n \"As for gender, the phenomenon usually occurs twice more during the sleep of women, due to hormonal factors [24], [10].\",\n \"The sleep spindles, in general, have a well-defined structure (occurrence, bandwidth, and amplitude).\",\n \"However the advancement of the patient's age and pathologies cause inaccuracies in their shape.\",\n \"Typically, the number of spindles decreases and their shape deteriorates [18].\",\n \"Their shape may be distorted and are more subject to the occurrence of interference [8].\",\n \"Patients with schizophrenia, for example, do not have normal patterns in the spindles [12].\",\n \"Changes can also be observed due to fatal familial insomnia [27], autism and epilepsy [16].\",\n \"This lack of standard is important for the diagnosis of neurological diseases, but it makes it more difficult to identify this phenomenon for specialists and automatic methods.\",\n \"The K-complex is a negative acute wave immediately followed by a positive component that clearly arises in the EEG, having a minimum duration of 0.5s in the frequencies of 12 to 14 Hz [2].\",\n \"In the identification, they can be easily confused with any waveform with high peaks [20], [11] (Figure REF ).\",\n \"Abnormal activity of the K complex may be related to epilepsy, restless leg syndrome, and obstructive sleep apnea.\"\n ],\n [\n \"DREAMS Data\",\n \"We use the databases collected by the DREAMS project [9], which consist of a series of polysomnography (PSG) with expert annotations on phenomena or sleep disorders.\",\n \"We use the sleep spindle and K-complexes datasets from this project.\",\n \"Their purpose is to tune, train and test automatic detection algorithms.\",\n \"The Spindles dataset consists of 30 minute stretches of the central EEG channel (extracted from full-night PSG records), independently annotated by two experts.\",\n \"The data were acquired in a sleep laboratory of a Belgian hospital (BrainnetTM System of MEDATEC, Brussels) using a 32-channel digital polygraph.\",\n \"It is important to highlight that all records on this dataset are from patients with various sleep pathologies: dyssonia, restless legs syndrome, insomnia, apnea syndrome or hypopnea.\",\n \"EOG, EMG and EEG channels (channels FP1-A1, O1-A1 and C3-A1 or CZ-A1) were recorded, using the European standard data format (EDF) for storage.\",\n \"Only EEG channels will be used in this work.\",\n \"The sampling frequency varies between patients, having records of 200Hz, 100Hz or 50Hz of 30 minutes duration.\",\n \"The recordings were given independently to two experts, who annotated their estimates for the locations of the sleep spindles.\",\n \"The K-complex records were collected in the same hospital as the spindle registers, with the same equipment.\",\n \"There are 10 polysomnographic recordings from healthy individuals.\",\n \"Just like the previous base, we only use EEG channels.\",\n \"The sampling frequency was 200 Hz for all patients with a 30 min duration.\",\n \"In the same way, the excerpts were given independently to two experts.\",\n \"To reduce bias, the experts did not have access to sleep staging of the records.\"\n ],\n [\n \"Data preparation\",\n \"The original EEG data was prepared for the automated identification process using the following procedure.\"\n ],\n [\n \"Wavelets Transform\",\n \"Wavelet transforms are mathematical tools capable of decomposing signals into several components that allow analysis at different time and frequency scales [7].\",\n \"An input signal $x$ , passes through a low-pass filter $g$ and a high-pass filter $h$ (parallel, not sequentially), each with a cut-off frequency equal to one half of the sampling frequency of the input.\",\n \"Then, the two generated sub-signals, that is, the output of the filters has their samples reduced in half (see Figure REF ).\",\n \"Figure: Example of Digital Wavelet Transform in 3 levelsThis process can be repeated at several levels, causing the output of the low pass filter to be the input signal of a new pair of filters, followed by downsampling.\"\n ],\n [\n \"Feature extraction\",\n \"The EEG data used were sampled with different frequencies (50, 100 or 200 Hz).\",\n \"For the application of the Digital Wavelet Transform (DWT) [23] with 5 decomposition levels, the data were resampled with an increased frequency of 256 Hz in all cases by means of interpolation through a cubic spline.\",\n \"For each decomposition level ($D1$ to $D5$ ) in each EEG channel, the signal was separated into 2 second samples, and the following attributes were calculated for each sample: Signal amplitude average, Signal amplitude standard deviation (SD), Symmetry, Power Spectral Density (PSD), and Signal curve length.\",\n \"Following this procedure, we obtain 900 samples (2s samples from 30 minutes of signal) with 75 real valued attributes (3 EEG channels, 5 DWT levels, and 5 attributes per level).\",\n \"These attributes are summarized in tables REF , REF , and REF .\",\n \"Here, the columns represent the coefficients of the DWT levels and the lines the operations performed.\",\n \"Table: Attributes for the central EEG channel (CZ-A1 or C3-A1)Table: Attributes for the EEG channel FP1-A1Table: Attributes for the EEG channel O1-A1\"\n ],\n [\n \"Proposed Framework\",\n \"Previously, Guo et al.\",\n \"[13] proposed the use of Genetic Programming (GP) for the construction of features for EEG analysis in the classification of epileptic episodes.\",\n \"Taking this framework as a base, we develop a framework for the identification of structures in sleep EEG, which we describe in this section.\",\n \"There are many characteristics in the structure identification problem which differentiates it from the earlier classification work.\",\n \"We must divide the EEG signals into multiple short samples in order to identify the position of the Spindles and K-complexes in the signal.\",\n \"As a consequence, the data becomes highly unbalanced, complicating the training process.\",\n \"Also, we work on three distinct EEG channels (as opposed to a single channel in the original work).\",\n \"We tested several improvements on the original work to deal with this harder problem.\",\n \"First, we use the Area Under the ROC Curve (AUC) of the classifiers instead of the accuracy as the fitness measure, since the AUC is more robust and discriminating [15].\",\n \"Also, we compare several classifiers in addition to KNN, to explore the relationship between classifier choice and GP feature construction.\",\n \"GP is widely applied in the construction and selection of features for its good performance.\",\n \"In classification problems it is possible to evolve a tree for each problem class, selecting the best attributes and creating new features for each of them [25].\",\n \"Even with unbalanced data, this approach with GP also gets good results [40].\",\n \"In literature, there are also application studies in benchmarks [35] and in databases with few samples [26].\",\n \"Finally, we publish the program of the proposed framework and experiments at our repositoryhttps://github.com/IcaroMarcelino/SleepEEG for reproducibility purposes.\"\n ],\n [\n \"GP for Feature Construction\",\n \"Our proposed framework uses Genetic Programming (GP) to generate the constructed set of features from the original features.\",\n \"The GP tree is defined as follows: The input nodes are selected from the original attributes.\",\n \"The intermediate nodes are selected from a set of arithmetic operators $\\\\lbrace +,-,\\\\times ,\\\\rbrace $ , as well as a set of protected operators $\\\\lbrace /,\\\\ln ,\\\\sqrt{\\\\rbrace }$ .\",\n \"These protected operators have their definitions slightly modified to avoid errors such as division by zero, as follows: $\\\\text{protected division} (a, b) &=& \\\\left\\\\lbrace \\\\begin{array}{l l}1, & b = 0 \\\\\\\\\\\\frac{a}{b}, & b \\\\ne 0\\\\end{array} \\\\right.\\\\\\\\\\\\text{protected log}(a) &=& \\\\left\\\\lbrace \\\\begin{array}{l l}1, & b = 0 \\\\\\\\ln(|a|), & b \\\\ne 0\\\\end{array} \\\\right.\\\\\\\\\\\\text{protected square root} (a) &=& \\\\sqrt{|a|}$ Additionally, a special \\\"Feature Operator\\\" [13], $F$ , is used to indicate how to obtain the constructed features from the GP tree.\",\n \"The feature operator returns the value of its input as its output, without any changes.\",\n \"Its purpose is to mark a subtree as one of the constructed features.\",\n \"Each $F$ operator will be the root of a subtree that expresses the function to calculate one attribute for the constructed set.\",\n \"In this way, a GP tree containing ten nodes with the $F$ operator will generate a constructed attribute set with 10 attributes.\",\n \"For example, the tree depicted in Figure REF shows a GP individual with two subtrees marked by the $F$ operator.\",\n \"If we assume that the original attribute set has 26 attributes (a..z), this tree will generate a constructed attribute set with two attributes: $F_1 = a$ and $F_2 = b - 1$ .\",\n \"The use of the $F$ operator allows a single GP tree to express multiple attribute constructing functions.\",\n \"In this way, we avoid having to explicitly define how many attributes will be constructed beforehand, which would be necessary if each attribute were expressed by a separate tree [13].\",\n \"Also, this allows GP trees in the population to exchange useful subtrees containing sets of constructed attributes that were successful.\",\n \"We believe that this allows the GP to pass around the most relevant subtrees to the next generations and, with this, keep attributes and attribute subsets that facilitate classification.\",\n \"Figure: Example of GP program marked with F operators.\",\n \"This GP tree constructs two attributes: F 1 =aF_1 = a and F 2 =b-1F_2 = b-1To evaluate one GP tree using the structure described in the previous section, we use the following procedure.\",\n \"First, we generate the set of constructed attributes for the GP tree using the $F$ operators.\",\n \"Second, we train a classifier using this set of constructed attributes.\",\n \"Finally, we use the AUC value of the classifier as the fitness value for the GP tree.\",\n \"In this way, the product of the evolutionary process is both a set of constructed attributes, as well as a classifier trained on those attributes.\",\n \"The parameters used for the evolutionary process in the current framework are listed in Table REF .\",\n \"As the parameters provided good results in the initial runs, they were maintained for the following.\",\n \"The \\\"Uniform mutation\\\" selects a random node from the individual and replaces subtree rooted in that node with a a randomly generated one.\",\n \"Table: GP Model Parameterization\"\n ],\n [\n \"Classifiers\",\n \"We compare the performance of the set of constructed attributes by GP by using five different classifiers: K Nearest Neighbors (KNN) Naive Bayes (NB) Support Vector Machines (SVM) Decision Tree (DT) Multilayer Perceptron (MLP) We perform an initial tuning procedure using the full set of 75 features to select the hyperparameters used by each classifier in the next experiments.\",\n \"For each tuning classifier, we execute 100 runs for each parameter value tested in the following sets: KNN: $k \\\\in \\\\lbrace 3,5,7,9,11,13,15,17,19\\\\rbrace $ SVM: kernel $\\\\in \\\\lbrace $ Radial Basis Function (RBF), polynomial, sigmoid$\\\\rbrace $ MLP: activation function $\\\\in \\\\lbrace $ ReLU, logistic$\\\\rbrace $ , neurons in hidden layer $\\\\in \\\\lbrace 15,30,45,60,75\\\\rbrace $ The parameters that provide the best performance were selected.\",\n \"KNN with $k = 5$ , SVM with RBF as kernel and MLP with a single hidden layer with 15 neurons and ReLU activation.\"\n ],\n [\n \"Experiment\",\n \"We perform several evaluations of the classifiers on the Sleep Spindles and K-Complexes datasets in order to analyse the performance of the propose framework.\",\n \"The results of the classifiers are compared using the full set of 75 attributes, a reduced set of 29 attributes selected by PCA, and the set of attributes constructed by the GP framework.\",\n \"The training dataset (used to train both the GP and the classifiers) was generated by simple random sampling 70% of the signal samples, and labelling them as positive samples (Sleep Spindles or K-complexes) if both specialists agreed on the label.\",\n \"Additionally, because the dataset is highly unbalanced, we balance the training dataset by randomly removing samples from the majority class until both classes have the same number of signal samples.\",\n \"The test dataset was generated by the remaining 30% of the samples, and each signal sample was labeled as positive if either specialist annotated it as positive.\",\n \"Also, the balancing procedure is not performed on the testing data set.\",\n \"This resulted in a slightly harder testing data set.\",\n \"For each experiment, we repeat the training/testing procedure 10 times, and report the aggregate results of these 10 repetitions as described in the subsections below.\",\n \"Figure: refF\"\n ],\n [\n \"Results\",\n \"The first experiment was aimed at verifying the performance of the classifiers on the test dataset without the reduction of dimensionality by GP, i.e., using the 75 features.\",\n \"The results are also useful to justify the application of feature construction.\",\n \"The classifiers performance can be seen in Figure REF a.\",\n \"Only MLP has good results, achieving an AUC greater than 0.7 with low SD.\",\n \"Figure: Number of occurrences of the feature in the models generated by GP (Same models from Figure c).Figure: Number of dimensions in the models generated by GP (Same models from Figure c)Applying PCA on the data with a 95% threshold for variance, ensuring little loss of information, the initial set of 75 features can be represented with 29 attributes.\",\n \"The performance of the trained classifiers with the feature set generated by PCA on the respective test set can be seen in Figure REF b.\",\n \"For this problem, the PCA representation caused a performance reduction in all classifiers except NB.\",\n \"The performance of applying the classifiers on the feature set generated by PCA can be seen in Figure REF b.\",\n \"For this problem, the PCA representation caused a decrease in performance in all classifiers except NB.\",\n \"Applying the GP feature reduction, the AUC of the classification increases for all classifiers except KNN (Figure REF c) and the SD reduces for all cases.\",\n \"This method is capable to reduce the number of features from 75 to less than 29 (Figure REF ).\",\n \"Using the same approach for training K-complexes classifiers, the models achieve high AUC scores too (Figure REF d).\",\n \"Figure: Classifiers performance over the central EEG channel attributes for Sleep Spindles and K-complexes\"\n ],\n [\n \"Analysis of gender difference\",\n \"As mentioned before, there are gender differences in sleep spindles.\",\n \"To see if this difference affects the performance of the classifiers, the data were separated by gender.\",\n \"Observing the Figures REF e and REF f, the classifier's performance for female patients is higher than the male patients.\",\n \"As sleep spindles occur more often in female patients, in data there are more representative samples of the waveform (see Table REF ), which facilitates the training of more efficient classifiers.\"\n ],\n [\n \"Constructed Features Analysis\",\n \"In Figure REF , the frequency of occurrence of features in the models training shows that there are attributes more relevant than others in the dataset.\",\n \"The greater occurrence of the features associated to the central EEG channel indicates that it has more important role to the identification of sleep spindles.\",\n \"Using only this channel for training (i.e, only with the 25 first features of the dataset), the performance of all classifiers increase for sleep spindles and K-complexes identification (Figure REF ).\",\n \"This attributes reduction contributes to a better understanding of the phenomenon, providing a more efficient approach by reducing the use of electrodes, consuming less resources and generating less discomfort to the patient.\",\n \"Table: Comparison between the proposed model and literature models\"\n ],\n [\n \"Comparison to Literature Models\",\n \"In the Table REF , the best generated model with the proposed approach (with NB classifier) was compared with the literature models which also used DREAMS data Further comparisons between sleep spindle identifiers can be seen in [21].\",\n \"Tsanas at.\",\n \"el [39] and Zhuang et al.\",\n \"[44] proposed continuous wavelet transform (CWT) based approaches and the estimation of the probability of spindles occurrences.\",\n \"Lachner-Piza et al.\",\n \"[21] proposed a SVM approach with a feature selection method based on the label-feature and feature-feature correlations for determining the relevance and redundancy of each feature.\",\n \"Observing the performance of the models, all obtained high specificity, indicating that the identification of samples where no spindles samples are present is reliable.\",\n \"Moreover, there is a trade-off between sensitivity and precision.\",\n \"In the context of applying automatic identifiers, false negatives are more unwanted than false positives.\",\n \"That is, a highly accurate but not very sensitive classifier generates many false positives, indicating that it is not judicious.\",\n \"In a semi-automatic application with low sensitivity, it is necessary for a specialist to inspect the markings performed by the classifier, eliminating the excess of false positives.\",\n \"This is the case of the detector of Zhuang et al.\",\n \"[44].\",\n \"The detectors of Tsanas and Clifford [39] and Lachner-Piza et al.\",\n \"[21] and the proposed model have achieved a better compromise between sensitivity (recall) and precision.\",\n \"This implies that the identification of signal stretches as spindles is more reliable.\",\n \"The proposed approach allows the generation of competitive models with the literature.\",\n \"The Tsanas and Clifford model, although having a slightly higher sensitivity than the proposed model.\",\n \"In contrast to the model of Lachner-Piza et al., our model has only minor precision, with 0.03 of difference.\"\n ],\n [\n \"Discussion\",\n \"The GP feature construction improves the performance of a classifier reducing the search space and generating more explicit relations between variables.\",\n \"Observing the reduction in the number of attributes, the dimensionality of the problem is reduced by up to 7 times in most cases.\",\n \"In addition, by analyzing the most frequent attributes, it is clear which ones are most relevant to the models.\",\n \"With this information, it is possible to select the most important EEG channels.\",\n \"In the case of K-complexes sleep spindles, only the central EEG channel is sufficient to perform the waveform identification.\",\n \"The single channel approach already reduces search space by one-third.\",\n \"Furthermore, fewer electrodes will be required for the examination, making it more comfortable for the patient, consequently, approaching the sleep in the laboratory of the daily sleep, avoiding biases.\"\n ],\n [\n \"Conclusions\",\n \"The use of automatic methods to identify sleep phenomena makes it possible to classify EEG signal segments with good performance, indicating whether or not a particular event occurs.\",\n \"Excerpts of 30 minutes can have hundreds of events that you want to identify.\",\n \"In this respect, the proposed model can be used to accelerate the process, and it is up to the expert to assign the classification.\",\n \"The model was also useful for sleep staging, since the presence of spindles and K-complexes strongly characterize sleep stage 2.\",\n \"It has also been shown that it is possible to significantly improve the performance of classifiers by selecting and constructing attributes.\",\n \"In addition, the use of GP allows greater interpretability and mathematical analysis of the new attributes generated, which may help to better understand the model and the problem.\",\n \"It is also possible to inspect the attributes generated through knowledge in the application domain.\",\n \"The approach also does not require in-depth knowledge of the application domain.\",\n \"In the first experiment, no assumptions about the data were performed.\",\n \"The ease of defining the terms in which the solutions will be written, that is, the operators and the terminals, allow the creation of hybrid models with biomedical information.\",\n \"It can also facilitate communication between specialists from different areas.\",\n \"The automation of the selection and construction of attributes generates a dataset suitable for the desired classifier.\",\n \"But, it is very simple to apply the attributes generated in another classifier.\",\n \"The processing time is also reduced with the smaller number of dimensions.\",\n \"The PSG generated signals used in sleep clinics are stored directly on computers.\",\n \"Therefore, the application of the proposed technique can be easily applied in this context.\",\n \"The generated models, once trained, make predictions quickly, facilitating a real-time approach.\",\n \"Measurement of the micro-event activity on EEG signals in different populations can provide important information about abnormalities in brain signals and assist in the investigation and hypothesis assessment of observed phenomena or disturbances.\",\n \"This underscores the importance of the study.\",\n \"With the proposed models, the identification of the spindles or K-complexes is less costly for the specialist.\",\n \"It may even replace its function in this task if the performance of the models is satisfactory for the requested analysis.\",\n \"Therefore, the diagnosis can be faster and the return to the patient suffering from some disorder is more efficient.\",\n \"Moreover, this methodology can easily be extended to other classification problems.\"\n ]\n]"},"id":{"kind":"string","value":"1906.04403"}}},{"rowIdx":2590,"cells":{"text":{"kind":"list like","value":[["Spiral instabilities: Mechanism for recurrence"],["Abstract We argue that self-excited instabilities are the cause of spiral patterns in simulations of unperturbed stellar discs.","In previous papers, we have found that spiral patterns were caused by a few concurrent waves, which we claimed were modes.","The superposition of a few steadily rotating waves inevitably causes the appearance of the disc to change continuously, and creates the kind of shearing spiral patterns that have been widely reported.","Although we have found that individual modes last for relatively few rotations, spiral activity persists because fresh instabilities appear, which we suspected were excited by the changes to the disc caused by previous disturbances.","Here we confirm our suspicion by demonstrating that scattering at either of the Lindblad resonances seeds a new groove-type instability.","With this logical gap closed, our understanding of the behaviour in the simulations is almost complete.","We believe that our robust mechanism is a major cause of spiral patterns in the old stellar discs of galaxies, including the Milky Way where we have previously reported evidence for resonance scattering in the recently released Gaia data."],["Introduction","Theorists have long dreamed that the spiral patterns gracing most disc galaxies, and simulations thereof, reflect the normal modes of the disc.","That is, they are mildly non-linear manifestations of self-excited, linear instabilities of the stellar disc that are uniformly rotating and exponentially growing density waves.","Long ago, this grand program was set back by two early discoveries from normal mode analyses of apparently reasonable models of featureless discs: models where the rotation curve rose gently from the centre were dominated by bar-forming instabilities [25], [30], while models having a dense (bulge-like) centre had no modes whatsoever [61].","Here we begin by reviewing why we think spiral modes are in fact the most promising mechansim, despite this early setback, and add to our case by demonstrating how spiral modes recur.","A normal mode of any system is a self-sustaining, sinusoidal disturbance of fixed frequency and constant shape, save for a possible uniform rotation; the frequency would be complex if the mode were to grow or decay.","In the case of galaxy discs, the perturbed surface density of a mode is the real part of $\\delta \\Sigma (R,\\phi ,t) = A_m(R)e^{i(m\\phi - \\omega t)},$ where $m$ is the angular periodicity, $\\omega = m\\Omega _p + i\\beta $ , $\\Omega _p$ is the angular rate of rotation, usually called the pattern speed, and $\\beta $ is the growth rate.","The complex function $A_m(R)$ , which is independent of time, describes the radial variation of amplitude and phase of the mode.","Stability analysis of a system supposes small amplitude perturbations about the equilibrium state and is linearized when any terms that involve products of small amplitude terms are discarded – see [28] for a careful formulation.","The equilibrium is linearly unstable if any of the resulting normal modes have a positive growth rate, since its amplitude will exponentiate out of the noise until the neglected 2nd and higher order terms become no longer negligible.The word instability is sometimes used to imply a purely imaginary frequency, and a mode with a complex frequency is then described as an overstability.","Here we adopt the more usual convention of simply describing all frequencies with $\\beta >0$ as instabilities.","Normal modes can be standing wave oscillations of the system that exist between two reflecting barriers, as in organ pipes and guitar strings, which are generally described as cavity modes in galaxy discs.","The prime example in galaxies is the bar-forming mode, for which reflections take place at the centre and at corotation [61], [9].","An instability of this type is possible only if the disturbance has no inner Lindblad resonance (hereafter ILR), since linear theory [35] predicts that any disturbance that encounters an ILR will be absorbed, and therefore damped.","An ILR must be present for any reasonable pattern speed when the centre is dense, and therefore no small-amplitude cavity mode is possible in a featureless disc of this kind.","But $\\Omega _c-\\kappa /2$ has a maximum value in mass models having quasi-harmonic cores, and bar-forming instabilities avoid resonance damping by having pattern speeds that exceed this maximum.","Here $\\Omega _c(R)$ is the angular frequency of circular motion at radius $R$ in the disc mid-plane, and $\\kappa (R)$ is the usual frequency of small-amplitude radial oscillations about a circular orbit [9].","The dominant mode of several bar-unstable models has been identified in simulations, with excellent quantitative agreement of both the frequency and mode shape [49], [16].","Other types of cavity mode have been proposed by [36] and by [52] and are described below.","Galactic discs can also support another class of mode.","The best known examples are edge modes [61], [40] and groove modes [54], [53].","They are not standing waves, however, and maintain a fixed shape and frequency by other means.","In the case of the edge mode, a small non-axisymmetric distortion of the disc where the density decreases steeply, moves high density material out to places where the equilibrium density was lower, and conversely at other azimuthal phases.","On their own, such co-orbiting distortions would be neutrally stable and therefore of no interest.","But as [27] taught, a cool surrounding disc responds vigorously to a co-orbiting mass excess, creating a trailing wake that extends radially far into the shear flow on either side of the perturbing mass; this behaviour is a consequence of swing amplification [23], [27], [61], [8].","The distorted edge therefore excites a strong supporting response from the interior disc that is not balanced by the exterior response because the equilibrium density drops rapidly with radius at the edge.","The attraction of the interior wake on the original density excess increases its angular momentum, causing it to grow exponentially as it rotates.","The necessary conditions for instability were summarized by [62].","A groove in a disc is effectively two closely spaced edges, which however give rise to a single mode because the distortions on each edge are gravitationally coupled.","Note that it is a steep gradient in the angular momentum density that matters for both edge and groove modes; epicyclic blurring can mask the steepness in the actual surface density profile.","As for edge modes, it is the supporting response of the surrounding disc that causes the groove mode to have a substantial radial extent and to grow rapidly [53], and these authors were able to obtain good quantitative agreement between their analytic predictions and simulations.","Note that swing amplication, famously illustrated in the dust-to-ashes figure of [61], is not a mode both because the shape changes with time and the amplitude variation is not a simple exponential.","Also the wake response to an imposed co-orbiting mass clump [27] is not a mode because, to first order, it would disperse if the clump were removed, and it therefore is not self-sustaining.","Both are simply responses of the disc to externally imposed disturbances.","However, they are both very helpful concepts when trying to understand the mechanisms of self-sustaining modes."],["Noise, resonances, and heating","The collisionless fluid of stars invoked in these analytical treatments is an idealization that assumes the stars to be infinitely finely divided so that phase space is smooth.","The numbers of stars in galaxy discs is large enough that this assumption holds quite well [47], but galaxies contain mass clumps such as star clusters and giant molecular clouds, and the number of particles employed in a simulation is generally several orders of magnitude fewer than the number of stars.","Thus shot noise in both real galaxies and in simulations gives rise to significant inhomogeneities in the disc.","The spectrum of shot noise in a shearing distribution of randomly distributed gravitating masses inevitably contains leading wave components that will be strongly amplified as the shear carries them from leading to trailing.","This behaviour has two important consequences.","The first consequence of swing-amplified shot noise is that each heavy particle develops a wake [27], [8].","The two point correlation function of the particles becomes greater along the direction of the wake and lower in other directions, and the particle distribution is said to be polarized.","Since the distribution of particles is no longer perfectly random, the amplitude of all components of the noise spectrum is enhanced, causing subsequent noise-induced fluctuations to be stronger, although linear theory predicts this cycle should asymptote in a few epicycle periods to a mean steady excess over the level expected from uncorrelated noise [27], [64].","Second, the collective amplified response of any one component of the noise orbiting at the angular frequency $\\Omega _p$ , creates a coherent trailing wave in the disc that propagates away from corotation [59], [61] until it reaches a Lindblad resonance where it is absorbed [35].","For near circular orbits the Lindblad resonances occur where the Doppler shifted frequency at which the stars encounter the wave is equal to the epicyclic frequency of their small radial excursions, i.e., the radii at which $m\\left[\\Omega _p - \\Omega _c(R)\\right] = l\\kappa (R),$ with $l=\\pm 1$ at Lindblad resonances.","The negative sign is for the ILR, where stars overtake the wave, and the positive is for the OLR where the wave overtakes the stars, at the local epicycle frequency in both cases.","Wave-particle interactions at the resonance cause localized irreversible changes to the energy and angular momenta of stars.","Jacobi's invariant [9] implies that changes are related as $\\Delta E = \\Omega _p \\Delta L_z$ .","On average and to second order, particles lose $L_z$ at the ILR and gain at the OLR [34], [10], and this outward transfer of $L_z$ allows the wave to extract free energy from the galactic potential enabling the scattered particles to acquire additional random energy at both resonances.","The resulting depopulation of stars on near circular orbits over the narrow region of each resonance creates a “scratch” in the disc that may alter its stability properties.","It is important to realize that linear theory neglects this second order effect by assumption, i.e.","it does not allow for changes to the equilibrium state.","In fact, [46] found that the amplitudes of successive episodes of uncorrelated swing amplified noise rose slowly, but continuously, because the consequent scratches to the disc from each episode caused partial reflections of subsequent disturbances that allowed further amplification [52], [18].","This process continued until the the partial reflections became strong enough that the disc was able to support an unstable mode [46], [13], and coherent growth to large amplitude began.","We discuss this behaviour further in §REF .","Here we have used the word “scratch” to describe quite mild changes to the distribution function (hereafter DF) that can cause partial reflections of a wave propagating radially within the disc.","But scattering at a Lindblad resonance could also carve a similar feature that seeds a groove mode instead, and we will show below that this appears to be the more usual behaviour.","Note that the Lindblad resonances are closer to corotation, where $\\Omega _p = \\Omega _c(R_{\\rm CR})$ ($l=0$ in eq.","REF ), when $m$ is large than for waves of lower $m$ , implying that more free energy can be extracted from the potential, causing more rapid heating when the disc supports larger-scale waves.","The value of $m$ that is amplified most strongly [27], [61] is $m \\approx {R_{\\rm CR}\\kappa ^2 \\over 2\\pi X G\\Sigma },$ with $1 < X < 2$ in a flat rotation curve.","Thus the preferred $m$ varies inversely with the disc surface density $\\Sigma $ [51], [2].","Whatever the origin of disturbances, a sub-maximal disc will prefer higher $m$ , and therefore heat more slowly than would heavier discs.","Since spiral activity heats collisionless particles, or stars, an uncooled disc must become less able to support collective disturbances over time, as is well known.","[51] established that dynamical cooling by gas and star formation is needed to counter secular heating and to maintain spiral activity, a result that has been confirmed in many more recent simulations [43], [1], [3].","It nicely accounts for the observation [39] that almost all spiral patterns are seen in galaxies that contain gas and are forming stars."],["The spiral challenge","As already noted, a galaxy model having a dense centre and no sharp features, such as an edge or a groove, should not support any normal modes at all.","Since spirals are ubiquitous in disc galaxies containing a modest gas fraction, and also develop spontaneously in simulations of isolated discs, some mechanism is needed to excite them.","Currently, there are at least four proposed mechanisms: Following [36], [6] suggest that spirals result from a cavity type mode in a low-mass disc that is dynamically cool over most of the disc, but which also possess an inner “$Q$ barrier” to shield the ILR.","These authors imagine that most galaxies support a single, long-lived, mildly-unstable mode that persists for many tens of galactic rotations and becomes “quasi-steady” due to dissipative shocks in the gas, but allow that superposition of a second mode may be needed in some cases.","As given by eq.","(REF ), strong swing-amplification for $X\\sim 2$ sets a preference for multi-arm instabilities over 2-arm modes in low mass discs.","By considering only bi-symmetric disturbances in low-mass discs, [7] exploited the mild amplification when $X > 3$ in order to obtain slowly-growing spiral modes in their global stability analysis of many galaxy models.","Simulations by [45] of one of the cases presented by [7] confirmed that a single, slowly-growing mode was present when disturbance forces were restricted to $m=2$ .","Not surprisingly, however, he also found much more vigorous instabilities appeared when higher sectoral harmonics contributed to disturbance forces, and the contrived basic state of the disc that was designed to support the $m=2$ mode was rapidly changed by these more vigorous instabilities.","This evidence alone should have ruled out the theory, although [56] ignored it as he continued to advocate for their picture.","[63] and [64] abandoned the idea of spirals as normal modes, and advocated instead that a collection of massive clumps in the disc, each of which becomes dressed with its own wake, would create a “kaleidoscope” of shearing spiral patterns.","Their simulations of this process were all confined to a single shearing patch with a modest number of particles.","[15] conducted global simulations of a low-mass disc, embedded in a rigid halo, employing $10^8$ star particles to which they added a sprinkling of heavy particles that induced evolving multi-arm spiral patterns in the stars.","In a separate experiment they also tried a single perturber of mass $10^7\\;$ M$_\\odot $ , which they removed again after it had completed one orbit; the simulation continued to manifest spiral activity in response to the non-axisymmetric density distribution created by the original imposed mass, which they described as non-linear behaviour.","These authors suggest that “ragged” spiral activity in galaxies results from such responses to co-orbiting giant molecular clouds, massive star clusters, etc., and to the lingering disc responses should any disperse.","Although we do not doubt their numerical results, we remain unconvinced of their importance for spiral activity in galaxies.","First, the $10^7\\;$ M$_\\odot $ particle that [15] employed produced only a modest, and not very extensive wake in their low-mass disc.","Spirals in real galaxies generally have greater amplitude and radial extent than this, suggesting that yet more massive clumps would be needed.","Second, a collection of randomly placed heavy particles does not seem likely to produce a net response that is predominantly 2- or 3-armed, as are observed in the overwhelming majority of galaxies [11], [24], [65].","Third, clumps massive and numerous enough that their associated wakes produce large-amplitude and radially-extensive spiral patterns would scatter disc stars, and heat the disc rapidly so that the responses would fade quickly unless the disc were cooled aggressively, and the necessary cooling [63] seems rather extreme.","A fourth, and overriding, reason is that the idea is unnecessary, because discs readily support unstable spiral modes, as we discuss next.","[52] demonstrated that spiral activity in simulations resulted from superposition of a number of coherent, uniformly rotating waves.","They found that each wave grew and decayed, but was detectable over a period of some ten rotations at its corotation radius, and they presented substantial evidence that the waves were in fact modes.","The vigorous modes in their new picture differ substantially from those invoked by [6] because they work best where swing-amplication is strongest, do not last for nearly as long, and fresh instabilities develop to maintain spiral activity.","Note that the spiral appearance changes still more rapidly than do the modes in their simulations.","This is because the superposition of several modes, each having a different pattern speed and perhaps also angular periodicity, as well as time varying amplitude, causes the pattern of visible spiral arms to change radically in less than one orbit.An animation showing the time evolution of the net density when two notional patterns are superposed is at http://www.physics.rutgers.edu/$\\sim $ sellwood/spirals.html [52] also postulated a new cavity mode that relied upon partial reflection of travelling waves off an impedance variation created by a previous perturbation within the disc.","They suggested that ILR scattering by a past wave would have “scratched” an originally smooth distribution function (see §REF ) to create a deficiency of stars on near circular orbits at some radius.","A subsequent ingoing trailing wave encountering the scratch before reaching its own ILR, would find that the scratch in the otherwise smooth disc represents an abrupt change of impedance, and is therefore partially reflected into a outgoing leading wave.","A second reflection of the wave by swing amplification at corotation creates a cavity that will support standing waves having frequencies allowed by the usual phase closure condition.","They dubbed this new type of cavity mode a “mirror mode.” As for the bar mode that also includes swing amplification from leading to trailing, a reasonable reflected fraction at the partial mirror will cause a mirror mode to grow rapidly, and the instability must run its course within a modest number of pattern rotations.","Nevertheless, spiral activity can be maintained because other modes develop in rapid succession.","We present evidence in §REF below that the first real instability in the tests of the Mestel discs reported in [46] was almost certainly of this type.","Some mirror modes may also have been present in the simulations of [52], but we here (§) argue for a more probable recurrence cycle.","[20], [21], [5], [42], and others present simulations of mostly sub-maximal discs and report that spiral patterns are shearing structures, in which the density maxima wind as would material arms, or nearly so.","[20], [21], [31], and other authors have described spiral arm streaming motions that are consistent with expectations set out by [29]; however, such a flow pattern is required for any self-consistent spiral, irrespective of its nature or origin.","[5] and [37] find that spirals in their simulations behave as predicted by swing amplification theory [61].","[22] reported a correlation between the mean pitch angle of spiral arms in their simulations and the shear rate in the disc, which they also suggested was consistent with swing amplification theory.","[4] reported similar shearing and evolving spirals in the outer discs of barred simulations.","[32] excited regular spiral patterns that were particle wakes in the disc forced by a few heavy particles equally spaced around rings, finding additional swing amplification as the wakes lined up.","The earlier part of this body of work was summarized in the review by [14], and all these authors propose that shearing patterns are the fundamental character of spiral arms.","We discuss these findings in §REF .","The Gaia satellite has revealed the local phase space structure of the Milky Way in unprecedented detail [19].","[55] used these data to try to discriminate among the different theories just described.","They calculated the changes to a smooth distribution function that would be caused by a single episode of each of the last three spiral models and compared the predictions with the Gaia data.","They concluded that the features in action space seemed more consistent with the transient spiral mode model [52] than with either of the other two.","Note that [38] found evidence in the same data for scattering by the Milky Way bar, but some features remained that they could not attribute to the bar.","However, [26] concluded that other features in the same data were consistent with shearing spiral arms.","[55] also argued that the long-lived spiral mode model advocated by [6] would not cause any pronounced changes, because the ILR, where the largest changes generally occur, is shielded by the $Q$ barrier, no net change is expected at corotation, and the OLR would probably lie too far outside the solar circle to affect the local distribution.","Thus if their delicate mechanism for spiral generation were indeed to operate in the Milky Way, it would contribute little to the observd extensive sub-structure in phase space and would implausibly have to survive in the mild disequilibrium state of the disc revealed in the Gaia data."],["Modes or shearing patterns?","The shearing spiral behaviour that is apparent in almost all simulations of cool, isolated discs possesses many aspects of swing-amplification and wakes, as the above cited papers have reported.","We have also reported such behaviour: e.g., Fig.","3 of [51] indicated the time evolution of a single 3-arm spiral disturbance that apparently sheared and amplified to an open trailing pattern before decaying as it continued to wind and, in simulations of much improved numerical quality, [52] illustrated constantly changing patterns, which is qualitatively similar to the behaviour that all authors report.","However, we have also found [44], [45], [52] that the constantly changing appearance of spiral patterns results from the superposition of a modest number of longer lived waves [41].","Even though a mode (eq.","REF ) has a fixed pattern speed at all radii and a constant shape function, $A_m(R)$ , shearing patterns are the inevitable consequence of superposed modes.","All that is required to produce the appearance of a shearing transient spiral (see footnote 2) from two or more superposed patterns of fixed shape is that the modes closer to the centre have higher pattern speed, which is always true.","The proposition that the shearing patterns are the fundamental behaviour is not a satisfactory explanation for the origin of the spirals, because it is unable to answer a key question: how can the spiral amplitudes in simulations be largely independent of the number of particles?","No recent paper that argues for this interpretation addresses this issue, yet there is substantial evidence to support it: [45], [46], and [52] all reported similar final amplitudes in experiments in which the number of particles ranged over orders of magnitude.","As $N$ is increased, the amplitude of shot noise fluctuations must decrease as $N^{-1/2}$ , and some kind of growth mechanism is required to produce final amplitudes that are independent of $N$ .","Swing-amplification and/or wakes do not lead to indefinite growth, and therefore cannot account for final amplitudes that are independent of $N$ , at least in linear theory.","We would agree with an argument that non-linear scattering (§REF ) is important, but that plays into our case for unstable modes [52].","It is likely that most models adopted in simulations are unstable, because they allow some feedback through the centre and/or possess outer edges sharp enough to excite normal modes.","Mildly unstable modes will be seeded at low amplitude when $N$ is large and may not cause visible changes for several rotations.","Both [45] and [52] present cases in which the first visible features took longer and longer to appear as the number of particles was increased.Pronouncements of stability after a short period of evolution in large-$N$ models [15] are unlikely to hold in longer integrations.","Once the first instability has appeared, we find that activity takes off, as the particle distribution becomes more and more structured by the previous evolution; the new results presented in § below show in more detail how a recurrent cycle of modes can occur.","Figure: An old simulation of the half-mass Mestel disc.","The toppanel shows the amplitude evolution of m=2m=2 disturbances reproducedfrom the N=5×10 7 N=5\\times 10^7 case in Fig.","2 of .","The powerspectrum over the time range 100≤t≤400100\\le t \\le 400 in the middlepanel shows no coherent modes were present over this interval.","Thesolid curve indicates the radial variation of mΩ c m\\Omega _c while thedashed curves show mΩ c ±κm\\Omega _c \\pm \\kappa .","The bottom panel is forthe period 1300≤t≤16001300\\le t \\le 1600 and reveals that the more rapidamplitude growth from t=1000t=1000 is caused by more coherentinstabilities, dark horizontal streaks, that developed later in thismodel, as reported.","Note that the grey scaleamplitudes are logarithmic and differ in the two panels.However, it is not neccessary to start from a disc that possesses one or more global instabilities.","[61] claimed that the half-mass Mestel disc is linearly stable, and this case was studied in simulations by [46].","Fig.","REF illustrates the evolution of the $N = 5 \\times 10^7$ simulation from his paper; details of the model and the numerical method are given in § below.","The top panel reproduces the cyan curve from Fig.","2 of [46], which reported at each instant the greatest value of the ratio $\\Sigma _2(R)/\\Sigma _0(R)$ within the radius range $1.2 < R < 12$ , where $\\Sigma _2$ is the amplitude of the bi-symmetric disturbance density.","After an initial surge by a factor of a few as each particle created its own wake (§REF ), these $m=2$ features manifested slow secular growth.","During this phase, the amplitudes of successive swing-amplified episodes increased slowly because each created mild scratches in the previously smooth disc that enabled partial reflections.","The partial mirrors so created at first reflected small fractions of the incident waves, but as the disturbance amplitudes rose with each episode, the new scratches became more reflective.","No coherent modes were detectable during this secular growth phase, and the power spectrum, such as shown in the middle panel, was characterized by multiple uncorrelated frequencies arising from whichever noise features were strongest at the time; the corotation radius of each event determines the frequency of the wave that propagates to its ILR where it is absorbed.","These findings were consistent with Toomre's linear theory prediction of global stability, since the secular growth was caused by the non-linear scattering terms that his analysis neglected.","The largest amplitude disturbances reached an overdensity of $\\sim 2$ % by $t \\sim 1000$ , at which point the reflections became strong enough to cause true unstable modes in this originally stable disc.","The amplitude in the top panel of Fig.","REF began to rise more rapidly and the power spectrum, in the bottom panel, that has a different amplitude scale shows much stronger and more coherent waves, which are modes.","Figure: Fig.","6 of showed the shape of the mode fitted tothe data from his model 50c.","Here we show the logarithmic spiraltransform of that mode density.","Positive values of tanγ\\tan \\gamma aretrailing, negative are leading, where γ\\gamma is the angle betweenthe tangent to the spiral and the radius vector.","The amplitudescale is arbitrary.","[18] and [17] successfully developed a quasi-linear theoretical description of this behaviour.","[13] conducted a global mode analysis of this model at $t=1400$ , and estimated the frequency of the dominant instability to be $\\omega = 0.597 + 0.013i$ in excellent agreement with Sellwood's own estimate (his paper has factor 10 typo in the growth rate).","Although both analyses were of the model denoted 50c, in which the particles coordinates had been scrambled at $t=1400$ , the faster wave in the bottom panel of Fig.","REF confirms a mode of similar pattern speed when the evolution is uninterrupted.","Fig.","REF shows the logarithmic spiral decomposition of the mode [46] fitted to the data from his model 50c.","The leading/trailing bias reflects the trailing appearance of the mode, but the leading side has significant amplitude indicating that it was a mirror, or cavity, mode that operated by an inner reflection off the partial mirror to a leading wave, as [52] described.","[57] calculates the changes to the DF at the principal resonances caused by a transient disturbance, and attempts to show how they lead to further instabilities.","However, the correspondence between the modes he calculates and the groove and mirror modes that we have reported is not immediately apparent from his analysis.","Thus we consider that every simulation of a cool, unperturbed disc must inevitably possess, or develop, unstable modes.","The unbounded growth of instabilities, until second order terms become important, provides the only viable mechanism to give rise to large-amplitude spirals no matter how large a number of particles is employed.","With a recurrence mechanism, such as that described here (§), the simulations must support multiple unstable modes and their superposition naturally leads to the shearing transient activity that is almost universally reported.","Note that externally perturbed models, such as those presented by [15] and [32], may not manifest modes if the initial responses develop before any modes could grow to significant amplitude."],["Objective of this paper","The complicated mechanism just described would be required to account for the origin of spirals only in models that do not possess any linear instabilities.","The purpose of this paper is to show how a recurring cycle of instabilities can arise in models that do possess a single mild instability at the outset.","We argue that such an instability cycle is the origin of spiral activity in all simulations, and hopefully in real galaxies also.","[52] also reported that coherent waves appear and decay at increasing radii and lower frequency over time.","It is possible that the later instabilities are independent modes caused by uncorrelated noise fluctuations scratching the DF at larger radii.","However, [52] preferred the idea that the decay of one mode created conditions to seed a new instability, although they did not provide a detailed mechanism for recurrence.","Here we describe the recurrence mechanism in detail.","In fact, [54] presented a recurrence mechanism for self-gravitating spiral modes in low mass particle discs orbiting around a central mass.","The mechanism they proposed was that scattering at the OLR created a groove in phase space that excited a new instability.","We thought it unlikely that the same recurrence mechanism would work in heavy discs, for reasons that we give in §, but we here demonstrate that this expectation was wrong."],["Technique","Since the dynamical behaviour of fully self-consistent simulations can be very complicated, we find it fruitful to run simplified simulations that can capture the phenomena we wish to study without them being obscured by unrelated activity.","Once understood, it will naturally be important to show that the behaviour persists under more general conditions.","Accordingly, we adopt the razor-thin Mestel disc used in the studies by [66] and [61], which is characterized by a constant circular speed $V_0$ at all radii.","Note that since $\\Omega _c = V_0/R$ and $\\kappa = \\surd 2V_0/R$ , $\\Omega _c - \\kappa /m$ increases indefinitely towards the centre except for $m=1$ , when it is negative at all radii.","Thus an ILR will intervene to damp all disturbances having $\\Omega _p>0$ when $m\\ge 2$ , prohibiting possible cavity modes in a smooth disc, except for $m=1$ .","The axisymmetric surface density $\\Sigma _0(R) = V_0^2 / (2\\pi G R)$ would self-consistently yield the appropriate central attraction for centrifugal balance.","If the surface density is reduced to fraction $x$ ($0 < x \\le 1$ ), with the removed mass added to a rigid halo to maintain centrifugal balance, then eq.","(REF ) implies that the most vigorously amplified disturbances ($X\\simeq 1.5$ ) will have $m=1.33/x$ .","Since [66] had found that the full-mass disc was prone to $m=1$ cavity modes, [61] preferred a model with $x=0.5$ in order to avoid lop-sided modes.","Note that although swing-amplification will be near its most vigorous for $m=2$ disturbances for this choice of $x$ , bisymmetric cavity modes in a smooth disc are disallowed because an ILR would block the feedback loop.","Toomre employed the DF [60], [9] $f(E,L_z) = \\left\\lbrace \\begin{array}{ll}{ x F L_z^q e^{-E/\\sigma _R^2} & L_z>0 \\cr 0 & otherwise, \\cr }\\end{array}where \\right.q = V_0^2/\\sigma _R^2 - 1 and the normalization constant is\\begin{equation}F = {1 \\over G R_0(R_0V_0)^q} { (q/2 + 0.5)^{q/2+1} \\over \\pi ^{3/2}(q/2-0.5)!","}.\\end{equation}[\\cite {Se12} erroneously omitted the factor R_0^{-(q+1)}.","]Choosing q=11.44 yields a Gaussian distribution of velocities suchthat the x=0.5 disc has Q=1.5.","Toomre further multiplied the DFf by the double taper function\\begin{equation}T(L_z) =\\left[ 1 + \\left( {R_0V_0 \\over L_z} \\right)^\\nu \\right]^{-1}\\left[ 1 + \\left( {L_z \\over R_1V_0} \\right)^\\mu \\right]^{-1},\\end{equation}to create a central cut out centered at R_0 and an outer tapercentered at R_1, while maintaining the centripetal acceleration-V_0^2/R everywhere.","Setting the taper indices \\nu =4 and \\mu =5yielded an idealized, smooth disc model that Toomre claimed possessedno small amplitude unstable modes.","We choose R_1 = 11.5R_0, andlimit the radial extent of the disc by an energy cut-off thateliminates particles having sufficient energy to pass R=20R_0.","Herewe adopt units such that V_0 = R_0 = G = 1.$ The evolution of one of the simulations of this model reported by [46] was illustrated in Fig.","REF above.","The simulations in the present paper use the same code and disc model, have the same number of particles (see Table REF ), but are integrated to $t = 1000$ only and so are unaffected by the later rapid growth phase.","Table: Numerical parametersThe particles in our simulations are constrained to move in a plane over a 2D polar mesh at which the accelerations are calculated, and interpolated to the position of each particle.","In most cases, disturbance forces are restricted to a single sectoral harmonic, $m=2$ or $m=3$ .","Since the gravitational field is a convolution of the mass density with a Green function that is most efficiently computed by Fourier transforms, it is easy to restrict the sectoral hamonics that contribute to the field when using a polar grid.","A full description of our numerical procedures is given in the on-line manual [48] and the code itself is available for download.","Table REF gives the values of the numerical parameters for most simulations reported here, and we note when they are varied in a few cases.","[46] reported, and we have reconfirmed in this study, that all our results are insensitive to reasonable changes to grid resoution, time step and zones, and number of particles.","Changes to the softening length do affect the frequencies of the modes, but not the qualititative behaviour.","To set up each model we draw particles from the tapered DF (eqs.","REF and ) as described in the appendix of [12].","However, in some experiments we modified the distribution of selected particles in order to introduce a single additional feature into the DF, as set out in §REF .","We also restarted some simulations after “scrambling” a copy of the particle distribution.","By this we mean that we changed $(R,\\phi ,v_R,v_\\phi ) \\rightarrow (R,\\phi ^\\prime ,v_R,v_\\phi )$ for every particle, with $\\phi ^\\prime $ being chosen at random from a distribution that is uniform in 0 to $2\\pi $ .","Clearly scrambling resets the amplitudes of all non-axisymmetric disturbances back to the shot noise level of the initial disc, while preserving the radius and both velocity components (in polar coordinates), so that any features in the action distribution of the particles that had been introduced during prior evolution would be preserved.","This is therefore not equivalent to a fresh start with a different random seed.","Note that the radial action, $J_R \\equiv \\oint \\dot{R} dR/(2\\pi )$ [9], has dimensions of angular momentum, is zero for a circular orbit, and increases with the eccentricity of the orbit.","In an axisymmetric potential, the angular momentum, $L_z$ , is the other action for orbits confined to a plane."],["Results","In this section we study the effect of introducing two distinct features into the otherwise unperturbed disc, both designed to excite an initial mode of a specific type, namely: a groove mode [54], [53] and an outer edge mode [61], [40].","In both cases, we follow its evolution and study the further modes that are excited.","We show for each case that the second mode is a true instability of the modified disc by erasing all non-axisymmetric structure at the time the second mode began to grow, as just described, and demonstrating that the scrambled particle distribution possessed a very similar instability to that which developed in the continued original run.","Furthermore we identify the scattering feature in action space that was responsible for the second instability, as reported below in §REF .","Figure: The change in the densities of particles in action spacecreated by hand in order to insert a groove-like feature into theinitial model."],["Initial groove mode","We create a deficiency of low-$J_R$ particles over a narrow inclined range in $(L_z,J_R)$ space by giving some particles a larger $J_R$ , as shown in Fig.","REF .","The slope of this feature, $-1/2$ , approximately traces the locus of the ILR of a bisymmetric disturbance having a pattern speed $\\Omega _n = 0.1$ , although the disturbance was, in fact, purely hypothetical.","The ILR of a disturbance having this $\\Omega _n$ would lie at $L_z = 2.93$ for circular orbits, for which $J_R=0$ .Note that all the major resonances with any pattern speed and any angular periodicity in the self-similar Mestel disc lie on lines of slope $\\simeq -1/2$ in the space of these actions, for $J_R \\ll L_z$ .","We describe this feature as a groove, since it is a deficiency at low $J_R$ over a narrow range in $L_z$ .","Since orbits in this self-similar disc have typical epicyclic radii of $\\sigma _R/\\kappa \\simeq 0.2R_g$ about the guiding centre radius, this narrow groove in action space causes no noticeable change to the surface density profile.","Figure: The evolution of model G, which was seeded with the grooveshown in Fig. .","The red line in the top panelshows the amplitude evolution, the dotted black line indicates whatwould have happened without the groove, and explanations of thegreen and blue lines given in §§ & respectively.","The rapid, but small, variations in the amplitudereflect alternating constructive and destructive interference byuncorrelated episodes of swing-amplified noise, that have longerperiods at later times as this behaviour spreads to the outer partof the radial range.","The middle and bottom panels, for which againthe grey scales differ, present power spectra over the time ranges100≤t≤400100\\le t \\le 400 and 400≤t≤700400\\le t \\le 700 respectively, whicheach manifest a single mode.Table: Summary of the modes fitted to data from simulationsdescribed in §, including the fitted complexfrequencies, ω\\omega with β\\beta being the growth rate of themode, and the radii of the principal resonances for circular orbitsgiven by (m+l√2)/ℜ(ω)(m+l\\surd 2)/\\Re (\\omega ), although we leave uninterestingentries blank for clarity.Our, perhaps somewhat clumsy, procedure for shifting particles was motivated by our observations of resonance scattering in previous simulations.","We calculated the frequency distance from the resonance $\\delta \\omega = 2\\Omega _\\phi - \\Omega _R - 2\\Omega _n $ for each particle, where $\\Omega _\\phi $ and $\\Omega _R$ are respectively the angular frequencies of the guiding centre and radial oscillation for orbits of arbitrary eccentricity; note that these frequencies tend to the familiar quantities $\\Omega _\\phi \\rightarrow \\Omega _c$ and $\\Omega _R \\rightarrow \\kappa $ as the eccentricity $\\rightarrow 0$ .","We then select particles that have $g^2 = 1 - (\\delta \\omega / w_\\omega \\Omega _n)^2 > 0$ as candidates to be scattered.","The relative frequency width, $w_\\omega = 0.06$ , of this feature is deliberately rather narrow.","Of these, we further select only those particles whose energy of random motion $E_r = E - E_c(L_z) < g E_{r,{\\rm lim}}$ , where the groove extends to the limiting random energy $E_{r,{\\rm lim}} = 0.025$ at $g=1$ , i.e.","along the resonance locus.","The probability that these selected particles are given larger random energy decreases from $d$ to zero as $E_r$ increases from zero to this maximum, where the fractional depth of the groove $d = 0.25$ .","The new $E_r = E_{r,{\\rm lim}}(1+3p)$ where $p$ is the solution of $\\exp (-p^2)=s$ , with $s$ being randomly drawn from a distribution uniform in 0–1, and finally $\\Delta L_z = \\Delta E / \\Omega _n$ .","A simulation that employed this modified DF, model G, supported a groove instability at first.","The red curve in the top panel of Fig.","REF presents the time evolution of the ratio $\\langle \\Sigma _2(R)/\\Sigma _0(R)\\rangle $ averaged over the radius range $20$ as instabilities.\",\n \"Normal modes can be standing wave oscillations of the system that exist between two reflecting barriers, as in organ pipes and guitar strings, which are generally described as cavity modes in galaxy discs.\",\n \"The prime example in galaxies is the bar-forming mode, for which reflections take place at the centre and at corotation [61], [9].\",\n \"An instability of this type is possible only if the disturbance has no inner Lindblad resonance (hereafter ILR), since linear theory [35] predicts that any disturbance that encounters an ILR will be absorbed, and therefore damped.\",\n \"An ILR must be present for any reasonable pattern speed when the centre is dense, and therefore no small-amplitude cavity mode is possible in a featureless disc of this kind.\",\n \"But $\\\\Omega _c-\\\\kappa /2$ has a maximum value in mass models having quasi-harmonic cores, and bar-forming instabilities avoid resonance damping by having pattern speeds that exceed this maximum.\",\n \"Here $\\\\Omega _c(R)$ is the angular frequency of circular motion at radius $R$ in the disc mid-plane, and $\\\\kappa (R)$ is the usual frequency of small-amplitude radial oscillations about a circular orbit [9].\",\n \"The dominant mode of several bar-unstable models has been identified in simulations, with excellent quantitative agreement of both the frequency and mode shape [49], [16].\",\n \"Other types of cavity mode have been proposed by [36] and by [52] and are described below.\",\n \"Galactic discs can also support another class of mode.\",\n \"The best known examples are edge modes [61], [40] and groove modes [54], [53].\",\n \"They are not standing waves, however, and maintain a fixed shape and frequency by other means.\",\n \"In the case of the edge mode, a small non-axisymmetric distortion of the disc where the density decreases steeply, moves high density material out to places where the equilibrium density was lower, and conversely at other azimuthal phases.\",\n \"On their own, such co-orbiting distortions would be neutrally stable and therefore of no interest.\",\n \"But as [27] taught, a cool surrounding disc responds vigorously to a co-orbiting mass excess, creating a trailing wake that extends radially far into the shear flow on either side of the perturbing mass; this behaviour is a consequence of swing amplification [23], [27], [61], [8].\",\n \"The distorted edge therefore excites a strong supporting response from the interior disc that is not balanced by the exterior response because the equilibrium density drops rapidly with radius at the edge.\",\n \"The attraction of the interior wake on the original density excess increases its angular momentum, causing it to grow exponentially as it rotates.\",\n \"The necessary conditions for instability were summarized by [62].\",\n \"A groove in a disc is effectively two closely spaced edges, which however give rise to a single mode because the distortions on each edge are gravitationally coupled.\",\n \"Note that it is a steep gradient in the angular momentum density that matters for both edge and groove modes; epicyclic blurring can mask the steepness in the actual surface density profile.\",\n \"As for edge modes, it is the supporting response of the surrounding disc that causes the groove mode to have a substantial radial extent and to grow rapidly [53], and these authors were able to obtain good quantitative agreement between their analytic predictions and simulations.\",\n \"Note that swing amplication, famously illustrated in the dust-to-ashes figure of [61], is not a mode both because the shape changes with time and the amplitude variation is not a simple exponential.\",\n \"Also the wake response to an imposed co-orbiting mass clump [27] is not a mode because, to first order, it would disperse if the clump were removed, and it therefore is not self-sustaining.\",\n \"Both are simply responses of the disc to externally imposed disturbances.\",\n \"However, they are both very helpful concepts when trying to understand the mechanisms of self-sustaining modes.\"\n ],\n [\n \"Noise, resonances, and heating\",\n \"The collisionless fluid of stars invoked in these analytical treatments is an idealization that assumes the stars to be infinitely finely divided so that phase space is smooth.\",\n \"The numbers of stars in galaxy discs is large enough that this assumption holds quite well [47], but galaxies contain mass clumps such as star clusters and giant molecular clouds, and the number of particles employed in a simulation is generally several orders of magnitude fewer than the number of stars.\",\n \"Thus shot noise in both real galaxies and in simulations gives rise to significant inhomogeneities in the disc.\",\n \"The spectrum of shot noise in a shearing distribution of randomly distributed gravitating masses inevitably contains leading wave components that will be strongly amplified as the shear carries them from leading to trailing.\",\n \"This behaviour has two important consequences.\",\n \"The first consequence of swing-amplified shot noise is that each heavy particle develops a wake [27], [8].\",\n \"The two point correlation function of the particles becomes greater along the direction of the wake and lower in other directions, and the particle distribution is said to be polarized.\",\n \"Since the distribution of particles is no longer perfectly random, the amplitude of all components of the noise spectrum is enhanced, causing subsequent noise-induced fluctuations to be stronger, although linear theory predicts this cycle should asymptote in a few epicycle periods to a mean steady excess over the level expected from uncorrelated noise [27], [64].\",\n \"Second, the collective amplified response of any one component of the noise orbiting at the angular frequency $\\\\Omega _p$ , creates a coherent trailing wave in the disc that propagates away from corotation [59], [61] until it reaches a Lindblad resonance where it is absorbed [35].\",\n \"For near circular orbits the Lindblad resonances occur where the Doppler shifted frequency at which the stars encounter the wave is equal to the epicyclic frequency of their small radial excursions, i.e., the radii at which $m\\\\left[\\\\Omega _p - \\\\Omega _c(R)\\\\right] = l\\\\kappa (R),$ with $l=\\\\pm 1$ at Lindblad resonances.\",\n \"The negative sign is for the ILR, where stars overtake the wave, and the positive is for the OLR where the wave overtakes the stars, at the local epicycle frequency in both cases.\",\n \"Wave-particle interactions at the resonance cause localized irreversible changes to the energy and angular momenta of stars.\",\n \"Jacobi's invariant [9] implies that changes are related as $\\\\Delta E = \\\\Omega _p \\\\Delta L_z$ .\",\n \"On average and to second order, particles lose $L_z$ at the ILR and gain at the OLR [34], [10], and this outward transfer of $L_z$ allows the wave to extract free energy from the galactic potential enabling the scattered particles to acquire additional random energy at both resonances.\",\n \"The resulting depopulation of stars on near circular orbits over the narrow region of each resonance creates a “scratch” in the disc that may alter its stability properties.\",\n \"It is important to realize that linear theory neglects this second order effect by assumption, i.e.\",\n \"it does not allow for changes to the equilibrium state.\",\n \"In fact, [46] found that the amplitudes of successive episodes of uncorrelated swing amplified noise rose slowly, but continuously, because the consequent scratches to the disc from each episode caused partial reflections of subsequent disturbances that allowed further amplification [52], [18].\",\n \"This process continued until the the partial reflections became strong enough that the disc was able to support an unstable mode [46], [13], and coherent growth to large amplitude began.\",\n \"We discuss this behaviour further in §REF .\",\n \"Here we have used the word “scratch” to describe quite mild changes to the distribution function (hereafter DF) that can cause partial reflections of a wave propagating radially within the disc.\",\n \"But scattering at a Lindblad resonance could also carve a similar feature that seeds a groove mode instead, and we will show below that this appears to be the more usual behaviour.\",\n \"Note that the Lindblad resonances are closer to corotation, where $\\\\Omega _p = \\\\Omega _c(R_{\\\\rm CR})$ ($l=0$ in eq.\",\n \"REF ), when $m$ is large than for waves of lower $m$ , implying that more free energy can be extracted from the potential, causing more rapid heating when the disc supports larger-scale waves.\",\n \"The value of $m$ that is amplified most strongly [27], [61] is $m \\\\approx {R_{\\\\rm CR}\\\\kappa ^2 \\\\over 2\\\\pi X G\\\\Sigma },$ with $1 < X < 2$ in a flat rotation curve.\",\n \"Thus the preferred $m$ varies inversely with the disc surface density $\\\\Sigma $ [51], [2].\",\n \"Whatever the origin of disturbances, a sub-maximal disc will prefer higher $m$ , and therefore heat more slowly than would heavier discs.\",\n \"Since spiral activity heats collisionless particles, or stars, an uncooled disc must become less able to support collective disturbances over time, as is well known.\",\n \"[51] established that dynamical cooling by gas and star formation is needed to counter secular heating and to maintain spiral activity, a result that has been confirmed in many more recent simulations [43], [1], [3].\",\n \"It nicely accounts for the observation [39] that almost all spiral patterns are seen in galaxies that contain gas and are forming stars.\"\n ],\n [\n \"The spiral challenge\",\n \"As already noted, a galaxy model having a dense centre and no sharp features, such as an edge or a groove, should not support any normal modes at all.\",\n \"Since spirals are ubiquitous in disc galaxies containing a modest gas fraction, and also develop spontaneously in simulations of isolated discs, some mechanism is needed to excite them.\",\n \"Currently, there are at least four proposed mechanisms: Following [36], [6] suggest that spirals result from a cavity type mode in a low-mass disc that is dynamically cool over most of the disc, but which also possess an inner “$Q$ barrier” to shield the ILR.\",\n \"These authors imagine that most galaxies support a single, long-lived, mildly-unstable mode that persists for many tens of galactic rotations and becomes “quasi-steady” due to dissipative shocks in the gas, but allow that superposition of a second mode may be needed in some cases.\",\n \"As given by eq.\",\n \"(REF ), strong swing-amplification for $X\\\\sim 2$ sets a preference for multi-arm instabilities over 2-arm modes in low mass discs.\",\n \"By considering only bi-symmetric disturbances in low-mass discs, [7] exploited the mild amplification when $X > 3$ in order to obtain slowly-growing spiral modes in their global stability analysis of many galaxy models.\",\n \"Simulations by [45] of one of the cases presented by [7] confirmed that a single, slowly-growing mode was present when disturbance forces were restricted to $m=2$ .\",\n \"Not surprisingly, however, he also found much more vigorous instabilities appeared when higher sectoral harmonics contributed to disturbance forces, and the contrived basic state of the disc that was designed to support the $m=2$ mode was rapidly changed by these more vigorous instabilities.\",\n \"This evidence alone should have ruled out the theory, although [56] ignored it as he continued to advocate for their picture.\",\n \"[63] and [64] abandoned the idea of spirals as normal modes, and advocated instead that a collection of massive clumps in the disc, each of which becomes dressed with its own wake, would create a “kaleidoscope” of shearing spiral patterns.\",\n \"Their simulations of this process were all confined to a single shearing patch with a modest number of particles.\",\n \"[15] conducted global simulations of a low-mass disc, embedded in a rigid halo, employing $10^8$ star particles to which they added a sprinkling of heavy particles that induced evolving multi-arm spiral patterns in the stars.\",\n \"In a separate experiment they also tried a single perturber of mass $10^7\\\\;$ M$_\\\\odot $ , which they removed again after it had completed one orbit; the simulation continued to manifest spiral activity in response to the non-axisymmetric density distribution created by the original imposed mass, which they described as non-linear behaviour.\",\n \"These authors suggest that “ragged” spiral activity in galaxies results from such responses to co-orbiting giant molecular clouds, massive star clusters, etc., and to the lingering disc responses should any disperse.\",\n \"Although we do not doubt their numerical results, we remain unconvinced of their importance for spiral activity in galaxies.\",\n \"First, the $10^7\\\\;$ M$_\\\\odot $ particle that [15] employed produced only a modest, and not very extensive wake in their low-mass disc.\",\n \"Spirals in real galaxies generally have greater amplitude and radial extent than this, suggesting that yet more massive clumps would be needed.\",\n \"Second, a collection of randomly placed heavy particles does not seem likely to produce a net response that is predominantly 2- or 3-armed, as are observed in the overwhelming majority of galaxies [11], [24], [65].\",\n \"Third, clumps massive and numerous enough that their associated wakes produce large-amplitude and radially-extensive spiral patterns would scatter disc stars, and heat the disc rapidly so that the responses would fade quickly unless the disc were cooled aggressively, and the necessary cooling [63] seems rather extreme.\",\n \"A fourth, and overriding, reason is that the idea is unnecessary, because discs readily support unstable spiral modes, as we discuss next.\",\n \"[52] demonstrated that spiral activity in simulations resulted from superposition of a number of coherent, uniformly rotating waves.\",\n \"They found that each wave grew and decayed, but was detectable over a period of some ten rotations at its corotation radius, and they presented substantial evidence that the waves were in fact modes.\",\n \"The vigorous modes in their new picture differ substantially from those invoked by [6] because they work best where swing-amplication is strongest, do not last for nearly as long, and fresh instabilities develop to maintain spiral activity.\",\n \"Note that the spiral appearance changes still more rapidly than do the modes in their simulations.\",\n \"This is because the superposition of several modes, each having a different pattern speed and perhaps also angular periodicity, as well as time varying amplitude, causes the pattern of visible spiral arms to change radically in less than one orbit.An animation showing the time evolution of the net density when two notional patterns are superposed is at http://www.physics.rutgers.edu/$\\\\sim $ sellwood/spirals.html [52] also postulated a new cavity mode that relied upon partial reflection of travelling waves off an impedance variation created by a previous perturbation within the disc.\",\n \"They suggested that ILR scattering by a past wave would have “scratched” an originally smooth distribution function (see §REF ) to create a deficiency of stars on near circular orbits at some radius.\",\n \"A subsequent ingoing trailing wave encountering the scratch before reaching its own ILR, would find that the scratch in the otherwise smooth disc represents an abrupt change of impedance, and is therefore partially reflected into a outgoing leading wave.\",\n \"A second reflection of the wave by swing amplification at corotation creates a cavity that will support standing waves having frequencies allowed by the usual phase closure condition.\",\n \"They dubbed this new type of cavity mode a “mirror mode.” As for the bar mode that also includes swing amplification from leading to trailing, a reasonable reflected fraction at the partial mirror will cause a mirror mode to grow rapidly, and the instability must run its course within a modest number of pattern rotations.\",\n \"Nevertheless, spiral activity can be maintained because other modes develop in rapid succession.\",\n \"We present evidence in §REF below that the first real instability in the tests of the Mestel discs reported in [46] was almost certainly of this type.\",\n \"Some mirror modes may also have been present in the simulations of [52], but we here (§) argue for a more probable recurrence cycle.\",\n \"[20], [21], [5], [42], and others present simulations of mostly sub-maximal discs and report that spiral patterns are shearing structures, in which the density maxima wind as would material arms, or nearly so.\",\n \"[20], [21], [31], and other authors have described spiral arm streaming motions that are consistent with expectations set out by [29]; however, such a flow pattern is required for any self-consistent spiral, irrespective of its nature or origin.\",\n \"[5] and [37] find that spirals in their simulations behave as predicted by swing amplification theory [61].\",\n \"[22] reported a correlation between the mean pitch angle of spiral arms in their simulations and the shear rate in the disc, which they also suggested was consistent with swing amplification theory.\",\n \"[4] reported similar shearing and evolving spirals in the outer discs of barred simulations.\",\n \"[32] excited regular spiral patterns that were particle wakes in the disc forced by a few heavy particles equally spaced around rings, finding additional swing amplification as the wakes lined up.\",\n \"The earlier part of this body of work was summarized in the review by [14], and all these authors propose that shearing patterns are the fundamental character of spiral arms.\",\n \"We discuss these findings in §REF .\",\n \"The Gaia satellite has revealed the local phase space structure of the Milky Way in unprecedented detail [19].\",\n \"[55] used these data to try to discriminate among the different theories just described.\",\n \"They calculated the changes to a smooth distribution function that would be caused by a single episode of each of the last three spiral models and compared the predictions with the Gaia data.\",\n \"They concluded that the features in action space seemed more consistent with the transient spiral mode model [52] than with either of the other two.\",\n \"Note that [38] found evidence in the same data for scattering by the Milky Way bar, but some features remained that they could not attribute to the bar.\",\n \"However, [26] concluded that other features in the same data were consistent with shearing spiral arms.\",\n \"[55] also argued that the long-lived spiral mode model advocated by [6] would not cause any pronounced changes, because the ILR, where the largest changes generally occur, is shielded by the $Q$ barrier, no net change is expected at corotation, and the OLR would probably lie too far outside the solar circle to affect the local distribution.\",\n \"Thus if their delicate mechanism for spiral generation were indeed to operate in the Milky Way, it would contribute little to the observd extensive sub-structure in phase space and would implausibly have to survive in the mild disequilibrium state of the disc revealed in the Gaia data.\"\n ],\n [\n \"Modes or shearing patterns?\",\n \"The shearing spiral behaviour that is apparent in almost all simulations of cool, isolated discs possesses many aspects of swing-amplification and wakes, as the above cited papers have reported.\",\n \"We have also reported such behaviour: e.g., Fig.\",\n \"3 of [51] indicated the time evolution of a single 3-arm spiral disturbance that apparently sheared and amplified to an open trailing pattern before decaying as it continued to wind and, in simulations of much improved numerical quality, [52] illustrated constantly changing patterns, which is qualitatively similar to the behaviour that all authors report.\",\n \"However, we have also found [44], [45], [52] that the constantly changing appearance of spiral patterns results from the superposition of a modest number of longer lived waves [41].\",\n \"Even though a mode (eq.\",\n \"REF ) has a fixed pattern speed at all radii and a constant shape function, $A_m(R)$ , shearing patterns are the inevitable consequence of superposed modes.\",\n \"All that is required to produce the appearance of a shearing transient spiral (see footnote 2) from two or more superposed patterns of fixed shape is that the modes closer to the centre have higher pattern speed, which is always true.\",\n \"The proposition that the shearing patterns are the fundamental behaviour is not a satisfactory explanation for the origin of the spirals, because it is unable to answer a key question: how can the spiral amplitudes in simulations be largely independent of the number of particles?\",\n \"No recent paper that argues for this interpretation addresses this issue, yet there is substantial evidence to support it: [45], [46], and [52] all reported similar final amplitudes in experiments in which the number of particles ranged over orders of magnitude.\",\n \"As $N$ is increased, the amplitude of shot noise fluctuations must decrease as $N^{-1/2}$ , and some kind of growth mechanism is required to produce final amplitudes that are independent of $N$ .\",\n \"Swing-amplification and/or wakes do not lead to indefinite growth, and therefore cannot account for final amplitudes that are independent of $N$ , at least in linear theory.\",\n \"We would agree with an argument that non-linear scattering (§REF ) is important, but that plays into our case for unstable modes [52].\",\n \"It is likely that most models adopted in simulations are unstable, because they allow some feedback through the centre and/or possess outer edges sharp enough to excite normal modes.\",\n \"Mildly unstable modes will be seeded at low amplitude when $N$ is large and may not cause visible changes for several rotations.\",\n \"Both [45] and [52] present cases in which the first visible features took longer and longer to appear as the number of particles was increased.Pronouncements of stability after a short period of evolution in large-$N$ models [15] are unlikely to hold in longer integrations.\",\n \"Once the first instability has appeared, we find that activity takes off, as the particle distribution becomes more and more structured by the previous evolution; the new results presented in § below show in more detail how a recurrent cycle of modes can occur.\",\n \"Figure: An old simulation of the half-mass Mestel disc.\",\n \"The toppanel shows the amplitude evolution of m=2m=2 disturbances reproducedfrom the N=5×10 7 N=5\\\\times 10^7 case in Fig.\",\n \"2 of .\",\n \"The powerspectrum over the time range 100≤t≤400100\\\\le t \\\\le 400 in the middlepanel shows no coherent modes were present over this interval.\",\n \"Thesolid curve indicates the radial variation of mΩ c m\\\\Omega _c while thedashed curves show mΩ c ±κm\\\\Omega _c \\\\pm \\\\kappa .\",\n \"The bottom panel is forthe period 1300≤t≤16001300\\\\le t \\\\le 1600 and reveals that the more rapidamplitude growth from t=1000t=1000 is caused by more coherentinstabilities, dark horizontal streaks, that developed later in thismodel, as reported.\",\n \"Note that the grey scaleamplitudes are logarithmic and differ in the two panels.However, it is not neccessary to start from a disc that possesses one or more global instabilities.\",\n \"[61] claimed that the half-mass Mestel disc is linearly stable, and this case was studied in simulations by [46].\",\n \"Fig.\",\n \"REF illustrates the evolution of the $N = 5 \\\\times 10^7$ simulation from his paper; details of the model and the numerical method are given in § below.\",\n \"The top panel reproduces the cyan curve from Fig.\",\n \"2 of [46], which reported at each instant the greatest value of the ratio $\\\\Sigma _2(R)/\\\\Sigma _0(R)$ within the radius range $1.2 < R < 12$ , where $\\\\Sigma _2$ is the amplitude of the bi-symmetric disturbance density.\",\n \"After an initial surge by a factor of a few as each particle created its own wake (§REF ), these $m=2$ features manifested slow secular growth.\",\n \"During this phase, the amplitudes of successive swing-amplified episodes increased slowly because each created mild scratches in the previously smooth disc that enabled partial reflections.\",\n \"The partial mirrors so created at first reflected small fractions of the incident waves, but as the disturbance amplitudes rose with each episode, the new scratches became more reflective.\",\n \"No coherent modes were detectable during this secular growth phase, and the power spectrum, such as shown in the middle panel, was characterized by multiple uncorrelated frequencies arising from whichever noise features were strongest at the time; the corotation radius of each event determines the frequency of the wave that propagates to its ILR where it is absorbed.\",\n \"These findings were consistent with Toomre's linear theory prediction of global stability, since the secular growth was caused by the non-linear scattering terms that his analysis neglected.\",\n \"The largest amplitude disturbances reached an overdensity of $\\\\sim 2$ % by $t \\\\sim 1000$ , at which point the reflections became strong enough to cause true unstable modes in this originally stable disc.\",\n \"The amplitude in the top panel of Fig.\",\n \"REF began to rise more rapidly and the power spectrum, in the bottom panel, that has a different amplitude scale shows much stronger and more coherent waves, which are modes.\",\n \"Figure: Fig.\",\n \"6 of showed the shape of the mode fitted tothe data from his model 50c.\",\n \"Here we show the logarithmic spiraltransform of that mode density.\",\n \"Positive values of tanγ\\\\tan \\\\gamma aretrailing, negative are leading, where γ\\\\gamma is the angle betweenthe tangent to the spiral and the radius vector.\",\n \"The amplitudescale is arbitrary.\",\n \"[18] and [17] successfully developed a quasi-linear theoretical description of this behaviour.\",\n \"[13] conducted a global mode analysis of this model at $t=1400$ , and estimated the frequency of the dominant instability to be $\\\\omega = 0.597 + 0.013i$ in excellent agreement with Sellwood's own estimate (his paper has factor 10 typo in the growth rate).\",\n \"Although both analyses were of the model denoted 50c, in which the particles coordinates had been scrambled at $t=1400$ , the faster wave in the bottom panel of Fig.\",\n \"REF confirms a mode of similar pattern speed when the evolution is uninterrupted.\",\n \"Fig.\",\n \"REF shows the logarithmic spiral decomposition of the mode [46] fitted to the data from his model 50c.\",\n \"The leading/trailing bias reflects the trailing appearance of the mode, but the leading side has significant amplitude indicating that it was a mirror, or cavity, mode that operated by an inner reflection off the partial mirror to a leading wave, as [52] described.\",\n \"[57] calculates the changes to the DF at the principal resonances caused by a transient disturbance, and attempts to show how they lead to further instabilities.\",\n \"However, the correspondence between the modes he calculates and the groove and mirror modes that we have reported is not immediately apparent from his analysis.\",\n \"Thus we consider that every simulation of a cool, unperturbed disc must inevitably possess, or develop, unstable modes.\",\n \"The unbounded growth of instabilities, until second order terms become important, provides the only viable mechanism to give rise to large-amplitude spirals no matter how large a number of particles is employed.\",\n \"With a recurrence mechanism, such as that described here (§), the simulations must support multiple unstable modes and their superposition naturally leads to the shearing transient activity that is almost universally reported.\",\n \"Note that externally perturbed models, such as those presented by [15] and [32], may not manifest modes if the initial responses develop before any modes could grow to significant amplitude.\"\n ],\n [\n \"Objective of this paper\",\n \"The complicated mechanism just described would be required to account for the origin of spirals only in models that do not possess any linear instabilities.\",\n \"The purpose of this paper is to show how a recurring cycle of instabilities can arise in models that do possess a single mild instability at the outset.\",\n \"We argue that such an instability cycle is the origin of spiral activity in all simulations, and hopefully in real galaxies also.\",\n \"[52] also reported that coherent waves appear and decay at increasing radii and lower frequency over time.\",\n \"It is possible that the later instabilities are independent modes caused by uncorrelated noise fluctuations scratching the DF at larger radii.\",\n \"However, [52] preferred the idea that the decay of one mode created conditions to seed a new instability, although they did not provide a detailed mechanism for recurrence.\",\n \"Here we describe the recurrence mechanism in detail.\",\n \"In fact, [54] presented a recurrence mechanism for self-gravitating spiral modes in low mass particle discs orbiting around a central mass.\",\n \"The mechanism they proposed was that scattering at the OLR created a groove in phase space that excited a new instability.\",\n \"We thought it unlikely that the same recurrence mechanism would work in heavy discs, for reasons that we give in §, but we here demonstrate that this expectation was wrong.\"\n ],\n [\n \"Technique\",\n \"Since the dynamical behaviour of fully self-consistent simulations can be very complicated, we find it fruitful to run simplified simulations that can capture the phenomena we wish to study without them being obscured by unrelated activity.\",\n \"Once understood, it will naturally be important to show that the behaviour persists under more general conditions.\",\n \"Accordingly, we adopt the razor-thin Mestel disc used in the studies by [66] and [61], which is characterized by a constant circular speed $V_0$ at all radii.\",\n \"Note that since $\\\\Omega _c = V_0/R$ and $\\\\kappa = \\\\surd 2V_0/R$ , $\\\\Omega _c - \\\\kappa /m$ increases indefinitely towards the centre except for $m=1$ , when it is negative at all radii.\",\n \"Thus an ILR will intervene to damp all disturbances having $\\\\Omega _p>0$ when $m\\\\ge 2$ , prohibiting possible cavity modes in a smooth disc, except for $m=1$ .\",\n \"The axisymmetric surface density $\\\\Sigma _0(R) = V_0^2 / (2\\\\pi G R)$ would self-consistently yield the appropriate central attraction for centrifugal balance.\",\n \"If the surface density is reduced to fraction $x$ ($0 < x \\\\le 1$ ), with the removed mass added to a rigid halo to maintain centrifugal balance, then eq.\",\n \"(REF ) implies that the most vigorously amplified disturbances ($X\\\\simeq 1.5$ ) will have $m=1.33/x$ .\",\n \"Since [66] had found that the full-mass disc was prone to $m=1$ cavity modes, [61] preferred a model with $x=0.5$ in order to avoid lop-sided modes.\",\n \"Note that although swing-amplification will be near its most vigorous for $m=2$ disturbances for this choice of $x$ , bisymmetric cavity modes in a smooth disc are disallowed because an ILR would block the feedback loop.\",\n \"Toomre employed the DF [60], [9] $f(E,L_z) = \\\\left\\\\lbrace \\\\begin{array}{ll}{ x F L_z^q e^{-E/\\\\sigma _R^2} & L_z>0 \\\\cr 0 & otherwise, \\\\cr }\\\\end{array}where \\\\right.q = V_0^2/\\\\sigma _R^2 - 1 and the normalization constant is\\\\begin{equation}F = {1 \\\\over G R_0(R_0V_0)^q} { (q/2 + 0.5)^{q/2+1} \\\\over \\\\pi ^{3/2}(q/2-0.5)!\",\n \"}.\\\\end{equation}[\\\\cite {Se12} erroneously omitted the factor R_0^{-(q+1)}.\",\n \"]Choosing q=11.44 yields a Gaussian distribution of velocities suchthat the x=0.5 disc has Q=1.5.\",\n \"Toomre further multiplied the DFf by the double taper function\\\\begin{equation}T(L_z) =\\\\left[ 1 + \\\\left( {R_0V_0 \\\\over L_z} \\\\right)^\\\\nu \\\\right]^{-1}\\\\left[ 1 + \\\\left( {L_z \\\\over R_1V_0} \\\\right)^\\\\mu \\\\right]^{-1},\\\\end{equation}to create a central cut out centered at R_0 and an outer tapercentered at R_1, while maintaining the centripetal acceleration-V_0^2/R everywhere.\",\n \"Setting the taper indices \\\\nu =4 and \\\\mu =5yielded an idealized, smooth disc model that Toomre claimed possessedno small amplitude unstable modes.\",\n \"We choose R_1 = 11.5R_0, andlimit the radial extent of the disc by an energy cut-off thateliminates particles having sufficient energy to pass R=20R_0.\",\n \"Herewe adopt units such that V_0 = R_0 = G = 1.$ The evolution of one of the simulations of this model reported by [46] was illustrated in Fig.\",\n \"REF above.\",\n \"The simulations in the present paper use the same code and disc model, have the same number of particles (see Table REF ), but are integrated to $t = 1000$ only and so are unaffected by the later rapid growth phase.\",\n \"Table: Numerical parametersThe particles in our simulations are constrained to move in a plane over a 2D polar mesh at which the accelerations are calculated, and interpolated to the position of each particle.\",\n \"In most cases, disturbance forces are restricted to a single sectoral harmonic, $m=2$ or $m=3$ .\",\n \"Since the gravitational field is a convolution of the mass density with a Green function that is most efficiently computed by Fourier transforms, it is easy to restrict the sectoral hamonics that contribute to the field when using a polar grid.\",\n \"A full description of our numerical procedures is given in the on-line manual [48] and the code itself is available for download.\",\n \"Table REF gives the values of the numerical parameters for most simulations reported here, and we note when they are varied in a few cases.\",\n \"[46] reported, and we have reconfirmed in this study, that all our results are insensitive to reasonable changes to grid resoution, time step and zones, and number of particles.\",\n \"Changes to the softening length do affect the frequencies of the modes, but not the qualititative behaviour.\",\n \"To set up each model we draw particles from the tapered DF (eqs.\",\n \"REF and ) as described in the appendix of [12].\",\n \"However, in some experiments we modified the distribution of selected particles in order to introduce a single additional feature into the DF, as set out in §REF .\",\n \"We also restarted some simulations after “scrambling” a copy of the particle distribution.\",\n \"By this we mean that we changed $(R,\\\\phi ,v_R,v_\\\\phi ) \\\\rightarrow (R,\\\\phi ^\\\\prime ,v_R,v_\\\\phi )$ for every particle, with $\\\\phi ^\\\\prime $ being chosen at random from a distribution that is uniform in 0 to $2\\\\pi $ .\",\n \"Clearly scrambling resets the amplitudes of all non-axisymmetric disturbances back to the shot noise level of the initial disc, while preserving the radius and both velocity components (in polar coordinates), so that any features in the action distribution of the particles that had been introduced during prior evolution would be preserved.\",\n \"This is therefore not equivalent to a fresh start with a different random seed.\",\n \"Note that the radial action, $J_R \\\\equiv \\\\oint \\\\dot{R} dR/(2\\\\pi )$ [9], has dimensions of angular momentum, is zero for a circular orbit, and increases with the eccentricity of the orbit.\",\n \"In an axisymmetric potential, the angular momentum, $L_z$ , is the other action for orbits confined to a plane.\"\n ],\n [\n \"Results\",\n \"In this section we study the effect of introducing two distinct features into the otherwise unperturbed disc, both designed to excite an initial mode of a specific type, namely: a groove mode [54], [53] and an outer edge mode [61], [40].\",\n \"In both cases, we follow its evolution and study the further modes that are excited.\",\n \"We show for each case that the second mode is a true instability of the modified disc by erasing all non-axisymmetric structure at the time the second mode began to grow, as just described, and demonstrating that the scrambled particle distribution possessed a very similar instability to that which developed in the continued original run.\",\n \"Furthermore we identify the scattering feature in action space that was responsible for the second instability, as reported below in §REF .\",\n \"Figure: The change in the densities of particles in action spacecreated by hand in order to insert a groove-like feature into theinitial model.\"\n ],\n [\n \"Initial groove mode\",\n \"We create a deficiency of low-$J_R$ particles over a narrow inclined range in $(L_z,J_R)$ space by giving some particles a larger $J_R$ , as shown in Fig.\",\n \"REF .\",\n \"The slope of this feature, $-1/2$ , approximately traces the locus of the ILR of a bisymmetric disturbance having a pattern speed $\\\\Omega _n = 0.1$ , although the disturbance was, in fact, purely hypothetical.\",\n \"The ILR of a disturbance having this $\\\\Omega _n$ would lie at $L_z = 2.93$ for circular orbits, for which $J_R=0$ .Note that all the major resonances with any pattern speed and any angular periodicity in the self-similar Mestel disc lie on lines of slope $\\\\simeq -1/2$ in the space of these actions, for $J_R \\\\ll L_z$ .\",\n \"We describe this feature as a groove, since it is a deficiency at low $J_R$ over a narrow range in $L_z$ .\",\n \"Since orbits in this self-similar disc have typical epicyclic radii of $\\\\sigma _R/\\\\kappa \\\\simeq 0.2R_g$ about the guiding centre radius, this narrow groove in action space causes no noticeable change to the surface density profile.\",\n \"Figure: The evolution of model G, which was seeded with the grooveshown in Fig. .\",\n \"The red line in the top panelshows the amplitude evolution, the dotted black line indicates whatwould have happened without the groove, and explanations of thegreen and blue lines given in §§ & respectively.\",\n \"The rapid, but small, variations in the amplitudereflect alternating constructive and destructive interference byuncorrelated episodes of swing-amplified noise, that have longerperiods at later times as this behaviour spreads to the outer partof the radial range.\",\n \"The middle and bottom panels, for which againthe grey scales differ, present power spectra over the time ranges100≤t≤400100\\\\le t \\\\le 400 and 400≤t≤700400\\\\le t \\\\le 700 respectively, whicheach manifest a single mode.Table: Summary of the modes fitted to data from simulationsdescribed in §, including the fitted complexfrequencies, ω\\\\omega with β\\\\beta being the growth rate of themode, and the radii of the principal resonances for circular orbitsgiven by (m+l√2)/ℜ(ω)(m+l\\\\surd 2)/\\\\Re (\\\\omega ), although we leave uninterestingentries blank for clarity.Our, perhaps somewhat clumsy, procedure for shifting particles was motivated by our observations of resonance scattering in previous simulations.\",\n \"We calculated the frequency distance from the resonance $\\\\delta \\\\omega = 2\\\\Omega _\\\\phi - \\\\Omega _R - 2\\\\Omega _n $ for each particle, where $\\\\Omega _\\\\phi $ and $\\\\Omega _R$ are respectively the angular frequencies of the guiding centre and radial oscillation for orbits of arbitrary eccentricity; note that these frequencies tend to the familiar quantities $\\\\Omega _\\\\phi \\\\rightarrow \\\\Omega _c$ and $\\\\Omega _R \\\\rightarrow \\\\kappa $ as the eccentricity $\\\\rightarrow 0$ .\",\n \"We then select particles that have $g^2 = 1 - (\\\\delta \\\\omega / w_\\\\omega \\\\Omega _n)^2 > 0$ as candidates to be scattered.\",\n \"The relative frequency width, $w_\\\\omega = 0.06$ , of this feature is deliberately rather narrow.\",\n \"Of these, we further select only those particles whose energy of random motion $E_r = E - E_c(L_z) < g E_{r,{\\\\rm lim}}$ , where the groove extends to the limiting random energy $E_{r,{\\\\rm lim}} = 0.025$ at $g=1$ , i.e.\",\n \"along the resonance locus.\",\n \"The probability that these selected particles are given larger random energy decreases from $d$ to zero as $E_r$ increases from zero to this maximum, where the fractional depth of the groove $d = 0.25$ .\",\n \"The new $E_r = E_{r,{\\\\rm lim}}(1+3p)$ where $p$ is the solution of $\\\\exp (-p^2)=s$ , with $s$ being randomly drawn from a distribution uniform in 0–1, and finally $\\\\Delta L_z = \\\\Delta E / \\\\Omega _n$ .\",\n \"A simulation that employed this modified DF, model G, supported a groove instability at first.\",\n \"The red curve in the top panel of Fig.\",\n \"REF presents the time evolution of the ratio $\\\\langle \\\\Sigma _2(R)/\\\\Sigma _0(R)\\\\rangle $ averaged over the radius range $2\\mathrm {N}/2\\end{array}\\right.\\\\\\\\\\Pi _{qp}^{s}=-4n_{q}\\left(1-n_{q}\\right)n_{p}\\left(1-n_{p}\\right)\\quad \\;\\end{array}$ $\\mathcal {J}_{pq}$ , $\\mathcal {K}_{pq}$ , and $\\mathcal {L}_{pq}$ are the usual direct, exchange, and exchange-time-inversion two-electron integrals.","The first term of the energy in Eq.","(REF ) draws the system as independent N/2 electron pairs, whereas the second term contains the interactions between electrons belonging to different pairs.","PNOF7s provides the reference NOs to form $\\tilde{E}_{hf}$ in the NOF-MP2 method, Eq.","(REF ).","$E^{sta}$ is the sum of the static intrapair and interpair electron correlation energies: $\\begin{array}{c}E^{sta}=\\sum \\limits _{g=1}^{\\mathrm {N}/2}\\sum \\limits _{q\\ne p}\\sqrt{\\Lambda _{q}\\Lambda _{p}}\\,\\Pi _{qp}^{g}\\mathcal {\\,L}_{pq}+\\sum \\limits _{f\\ne g}^{\\mathrm {N}/2}\\sum \\limits _{p\\in \\Omega _{f}}\\sum \\limits _{q\\in \\Omega _{g}}\\Pi _{qp}^{s}\\mathcal {\\,L}_{pq}\\end{array}$ where $\\Lambda _{p}=1-\\left|1-2n_{p}\\right|$ is the amount of intra-pair static correlation in each orbital as a function of its occupancy.","$E^{dyn}$ is obtained from the second-order correction $E^{\\left(2\\right)}$ of the MP2 method.","The first-order wavefunction is a linear combination of all doubly excited configurations, and their amplitudes $T_{pq}^{fg}$ are obtained by solving the equations for the MP2 residuals [12].","The dynamic energy correction takes the form $E^{dyn}=\\sum \\limits _{g,f=1}^{\\mathrm {N}/2}\\sum \\limits _{p,q>N/2}^{M}\\left\\langle gf\\right|\\left.pq\\right\\rangle \\left[2T_{pq}^{gf}\\right.\\left.-T_{pq}^{fg}\\right]$ where $M$ is the number of basis functions, and $\\left\\langle gf\\right|\\left.pq\\right\\rangle $ are the matrix elements of the two-particle interaction.","In fact, $E^{dyn}$ is the modified $E^{\\left(2\\right)}$ in order to avoid double counting of the electron correlation.","It is divided into intra- and inter-pair contributions, and the amount of dynamic correlation in each orbital $p$ is defined by functions $C_{p}$ of its occupancy, namely, $\\begin{array}{c}C_{p}^{tra}={\\left\\lbrace \\begin{array}{ll}\\begin{array}{c}\\begin{array}{c}1-4h_{p}^{2}\\end{array}\\\\1-4n_{p}^{2}\\end{array} & \\begin{array}{c}p\\le \\mathrm {N}/2\\\\p>\\mathrm {N}/2\\end{array}\\end{array}\\right.","}\\\\\\:C_{p}^{ter}={\\left\\lbrace \\begin{array}{ll}\\begin{array}{c}\\begin{array}{c}1\\end{array}\\\\1-4h_{p}n_{p}\\end{array} & \\begin{array}{c}p\\le \\mathrm {N}/2\\\\p>\\mathrm {N}/2\\end{array}\\end{array}\\right.","}\\end{array}$ According to Eq.","(REF ), fully occupied and empty orbitals yield a maximal contribution to dynamic correlation, whereas orbitals with half occupancies contribute nothing.","Using these functions as the case may be (intra-pair or inter-pair), the modified off-diagonal elements of the Fock matrix ($\\tilde{\\mathcal {F}}$ ) are defined as $\\tilde{\\mathcal {F}}_{pq}={\\left\\lbrace \\begin{array}{ll}C_{p}^{tra}C_{q}^{tra}\\mathcal {F}_{pq}, & p,q\\in \\Omega _{g}\\\\C_{p}^{ter}C_{q}^{ter}\\mathcal {F}_{pq}, & otherwise\\end{array}\\right.","}$ as well as modified two-electron integrals: $\\widetilde{\\left\\langle pq\\right|\\left.rt\\right\\rangle }={\\left\\lbrace \\begin{array}{ll}C_{p}^{tra}C_{q}^{tra}C_{r}^{tra}C_{t}^{tra}\\left\\langle pq\\right|\\left.rt\\right\\rangle , & p,q,r,t\\in \\Omega _{g}\\\\C_{p}^{ter}C_{q}^{ter}C_{r}^{ter}C_{t}^{ter}\\left\\langle pq\\right|\\left.rt\\right\\rangle , & otherwise\\end{array}\\right.","}$ where the subspace index $g=1,...,\\mathrm {N}/2$ .","This leads to the following linear equation for the modified MP2 residuals: $\\widetilde{\\left\\langle ab\\right|\\left.ij\\right\\rangle }+\\left(\\mathcal {F}_{aa}\\right.+\\mathcal {F}_{bb}-\\mathcal {F}_{ii}-\\left.\\mathcal {F}_{jj}\\right)T_{ab}^{ij}\\:+$ ${\\displaystyle \\sum _{c\\ne a}\\mathcal {\\tilde{F}}_{ac}T_{cb}^{ij}}+{\\displaystyle \\sum _{c\\ne b}}T_{ac}^{ij}\\mathcal {\\tilde{F}}_{cb}-{\\displaystyle \\sum _{k\\ne i}}\\tilde{\\mathcal {F}}_{ik}T_{ab}^{kj}-{\\displaystyle \\sum _{k\\ne j}}T_{ab}^{ik}\\mathcal {\\tilde{F}}_{kj}=0$ where $i,j,k$ refer to the strong occupied NOs, and $a,b,c$ to weak occupied ones.","It should be noted that diagonal elements of the Fock matrix ($\\mathcal {F}$ ) are not modified.","By solving this linear system of equations the amplitudes $T_{pq}^{fg}$ are obtained, which are inserted into the Eq.","(REF ) to achieve $E^{dyn}$ .","All calculations have been carried out using the DoNOF code developed by M. Piris and coworkers.","The procedure is simple, showing a formal scaling of $M^{5}$ ($M$ : number of basis functions).","However, our implementation in the molecular basis set requires also four-index transformation of the electron repulsion integrals, which is a time-consuming step, though a parallel implementation of this part of the code has substantially improved its performance.","As a result, the possibility of addressing large systems opens up."],["Results and Discussion","Results are organized as follows.","First, the X-H bond dissociation energies for LiH, BH, CH$_{4}$ , NH$_{3}$ , H$_{2}$ O and HF molecules are studied using NOF-MP2.","Next, we analyze the performance of NOF-MP2 for describing C-H bond cleavage in a variety of 20 organic molecules.","Finally, radical stabilization energies are calculated based on the calculated C-H bond dissociation energies.","In all calculations, recall that the maximum value allowed by the basis set used is assumed for $\\mathrm {N}_{g}$ by default.","In NOF-MP2(3) calculations, only three orbitals ($\\mathrm {N}_{g}=3$ ) above the N/2 level in each electron pair are considered.","Geometries are taken from our previous publication [16], which were obtained at the M06-2X level of theory [37].","The dissociation limit is calculated by considering a frozen X-H distance of 5$\\textrm {Ã}$ , and optimizing the rest of internal coordinates.","At these geometries, single-point energies are evaluated at the NOF-MP2 level of theory.","The correlation-consistent valence double-$\\zeta $ (cc-pVDZ) or triple-$\\zeta $ (cc-pVTZ) basis sets developed by Dunning et al.","[38] are used.","The zero point vibrational energies (ZPVEs) were taken from the NITS Computational Chemistry Comparison and Benchmark Database (CCCBDB) [39], and corresponds to CCSD(T)/cc-pVTZ values.","We also provide the PNOF6 [40] and wavefunction-based calculations obtained in Ref.","[16].","For the latter, an active space was defined by the distribution of two electrons in two molecular orbitals, CASSCF(2,2) [41], [42].","The dynamic correlation effects were included through complete active space second-order perturbation theory calculations, CASPT2(2,2) [43].","MOLCAS 7.0 suite of programs [44] was used in Ref.","[16], for these wavefunction-based calculations."],["X-H homolytic bond cleavage ","X-H bond dissociation energies were calculated according to the following reaction: $\\mathrm {XH}\\rightarrow \\mathrm {^{.","}X}+\\mathrm {H}^{.","}$ with X = Li, B, CH$_{3}$ , NH$_{2}$ , OH, F. The results are presented in Figure REF and Table REF .","The different hydrides considered expand a wide range of dissociation energies, from 58.0 kcal/mol for LiH to 141.1 kcal/mol for FH.","The ordering in dissociation energies is $\\mathrm {LiH}<\\mathrm {BH}<\\mathrm {CH_{4}}<\\mathrm {NH_{3}}<\\mathrm {H_{2}O}<\\mathrm {FH}$ .","In general, NOF-MP2 reproduces satisfactorily these trends.","Figure: NOF-MP2 dissociation energies, in kcal/mol,for X-H bonds (X= Li ,B, CH 3 , NH 2 , OH ,F\\mathrm {X=Li,B,CH_{3},NH_{2},OH,F}) versus experimentalones.","Calculations carried out with the cc-pVTZ basis set.Let us focus our attention, for example, on the delicate case of the CH$_{4}$ /NH$_{3}$ ordering.","The difference in experimental dissociation energies for these two molecules is very small, only 2.9 kcal/mol with $\\mathrm {NH_{3}}$ having a higher dissociation energy.","NOF-MP2 is able to reproduce the correct ordering $\\mathrm {CH_{4}\\\\mathrm {N}/2\\\\end{array}\\\\right.\\\\\\\\\\\\\\\\\\\\Pi _{qp}^{s}=-4n_{q}\\\\left(1-n_{q}\\\\right)n_{p}\\\\left(1-n_{p}\\\\right)\\\\quad \\\\;\\\\end{array}$ $\\\\mathcal {J}_{pq}$ , $\\\\mathcal {K}_{pq}$ , and $\\\\mathcal {L}_{pq}$ are the usual direct, exchange, and exchange-time-inversion two-electron integrals.\",\n \"The first term of the energy in Eq.\",\n \"(REF ) draws the system as independent N/2 electron pairs, whereas the second term contains the interactions between electrons belonging to different pairs.\",\n \"PNOF7s provides the reference NOs to form $\\\\tilde{E}_{hf}$ in the NOF-MP2 method, Eq.\",\n \"(REF ).\",\n \"$E^{sta}$ is the sum of the static intrapair and interpair electron correlation energies: $\\\\begin{array}{c}E^{sta}=\\\\sum \\\\limits _{g=1}^{\\\\mathrm {N}/2}\\\\sum \\\\limits _{q\\\\ne p}\\\\sqrt{\\\\Lambda _{q}\\\\Lambda _{p}}\\\\,\\\\Pi _{qp}^{g}\\\\mathcal {\\\\,L}_{pq}+\\\\sum \\\\limits _{f\\\\ne g}^{\\\\mathrm {N}/2}\\\\sum \\\\limits _{p\\\\in \\\\Omega _{f}}\\\\sum \\\\limits _{q\\\\in \\\\Omega _{g}}\\\\Pi _{qp}^{s}\\\\mathcal {\\\\,L}_{pq}\\\\end{array}$ where $\\\\Lambda _{p}=1-\\\\left|1-2n_{p}\\\\right|$ is the amount of intra-pair static correlation in each orbital as a function of its occupancy.\",\n \"$E^{dyn}$ is obtained from the second-order correction $E^{\\\\left(2\\\\right)}$ of the MP2 method.\",\n \"The first-order wavefunction is a linear combination of all doubly excited configurations, and their amplitudes $T_{pq}^{fg}$ are obtained by solving the equations for the MP2 residuals [12].\",\n \"The dynamic energy correction takes the form $E^{dyn}=\\\\sum \\\\limits _{g,f=1}^{\\\\mathrm {N}/2}\\\\sum \\\\limits _{p,q>N/2}^{M}\\\\left\\\\langle gf\\\\right|\\\\left.pq\\\\right\\\\rangle \\\\left[2T_{pq}^{gf}\\\\right.\\\\left.-T_{pq}^{fg}\\\\right]$ where $M$ is the number of basis functions, and $\\\\left\\\\langle gf\\\\right|\\\\left.pq\\\\right\\\\rangle $ are the matrix elements of the two-particle interaction.\",\n \"In fact, $E^{dyn}$ is the modified $E^{\\\\left(2\\\\right)}$ in order to avoid double counting of the electron correlation.\",\n \"It is divided into intra- and inter-pair contributions, and the amount of dynamic correlation in each orbital $p$ is defined by functions $C_{p}$ of its occupancy, namely, $\\\\begin{array}{c}C_{p}^{tra}={\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\begin{array}{c}\\\\begin{array}{c}1-4h_{p}^{2}\\\\end{array}\\\\\\\\1-4n_{p}^{2}\\\\end{array} & \\\\begin{array}{c}p\\\\le \\\\mathrm {N}/2\\\\\\\\p>\\\\mathrm {N}/2\\\\end{array}\\\\end{array}\\\\right.\",\n \"}\\\\\\\\\\\\:C_{p}^{ter}={\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\begin{array}{c}\\\\begin{array}{c}1\\\\end{array}\\\\\\\\1-4h_{p}n_{p}\\\\end{array} & \\\\begin{array}{c}p\\\\le \\\\mathrm {N}/2\\\\\\\\p>\\\\mathrm {N}/2\\\\end{array}\\\\end{array}\\\\right.\",\n \"}\\\\end{array}$ According to Eq.\",\n \"(REF ), fully occupied and empty orbitals yield a maximal contribution to dynamic correlation, whereas orbitals with half occupancies contribute nothing.\",\n \"Using these functions as the case may be (intra-pair or inter-pair), the modified off-diagonal elements of the Fock matrix ($\\\\tilde{\\\\mathcal {F}}$ ) are defined as $\\\\tilde{\\\\mathcal {F}}_{pq}={\\\\left\\\\lbrace \\\\begin{array}{ll}C_{p}^{tra}C_{q}^{tra}\\\\mathcal {F}_{pq}, & p,q\\\\in \\\\Omega _{g}\\\\\\\\C_{p}^{ter}C_{q}^{ter}\\\\mathcal {F}_{pq}, & otherwise\\\\end{array}\\\\right.\",\n \"}$ as well as modified two-electron integrals: $\\\\widetilde{\\\\left\\\\langle pq\\\\right|\\\\left.rt\\\\right\\\\rangle }={\\\\left\\\\lbrace \\\\begin{array}{ll}C_{p}^{tra}C_{q}^{tra}C_{r}^{tra}C_{t}^{tra}\\\\left\\\\langle pq\\\\right|\\\\left.rt\\\\right\\\\rangle , & p,q,r,t\\\\in \\\\Omega _{g}\\\\\\\\C_{p}^{ter}C_{q}^{ter}C_{r}^{ter}C_{t}^{ter}\\\\left\\\\langle pq\\\\right|\\\\left.rt\\\\right\\\\rangle , & otherwise\\\\end{array}\\\\right.\",\n \"}$ where the subspace index $g=1,...,\\\\mathrm {N}/2$ .\",\n \"This leads to the following linear equation for the modified MP2 residuals: $\\\\widetilde{\\\\left\\\\langle ab\\\\right|\\\\left.ij\\\\right\\\\rangle }+\\\\left(\\\\mathcal {F}_{aa}\\\\right.+\\\\mathcal {F}_{bb}-\\\\mathcal {F}_{ii}-\\\\left.\\\\mathcal {F}_{jj}\\\\right)T_{ab}^{ij}\\\\:+$ ${\\\\displaystyle \\\\sum _{c\\\\ne a}\\\\mathcal {\\\\tilde{F}}_{ac}T_{cb}^{ij}}+{\\\\displaystyle \\\\sum _{c\\\\ne b}}T_{ac}^{ij}\\\\mathcal {\\\\tilde{F}}_{cb}-{\\\\displaystyle \\\\sum _{k\\\\ne i}}\\\\tilde{\\\\mathcal {F}}_{ik}T_{ab}^{kj}-{\\\\displaystyle \\\\sum _{k\\\\ne j}}T_{ab}^{ik}\\\\mathcal {\\\\tilde{F}}_{kj}=0$ where $i,j,k$ refer to the strong occupied NOs, and $a,b,c$ to weak occupied ones.\",\n \"It should be noted that diagonal elements of the Fock matrix ($\\\\mathcal {F}$ ) are not modified.\",\n \"By solving this linear system of equations the amplitudes $T_{pq}^{fg}$ are obtained, which are inserted into the Eq.\",\n \"(REF ) to achieve $E^{dyn}$ .\",\n \"All calculations have been carried out using the DoNOF code developed by M. Piris and coworkers.\",\n \"The procedure is simple, showing a formal scaling of $M^{5}$ ($M$ : number of basis functions).\",\n \"However, our implementation in the molecular basis set requires also four-index transformation of the electron repulsion integrals, which is a time-consuming step, though a parallel implementation of this part of the code has substantially improved its performance.\",\n \"As a result, the possibility of addressing large systems opens up.\"\n ],\n [\n \"Results and Discussion\",\n \"Results are organized as follows.\",\n \"First, the X-H bond dissociation energies for LiH, BH, CH$_{4}$ , NH$_{3}$ , H$_{2}$ O and HF molecules are studied using NOF-MP2.\",\n \"Next, we analyze the performance of NOF-MP2 for describing C-H bond cleavage in a variety of 20 organic molecules.\",\n \"Finally, radical stabilization energies are calculated based on the calculated C-H bond dissociation energies.\",\n \"In all calculations, recall that the maximum value allowed by the basis set used is assumed for $\\\\mathrm {N}_{g}$ by default.\",\n \"In NOF-MP2(3) calculations, only three orbitals ($\\\\mathrm {N}_{g}=3$ ) above the N/2 level in each electron pair are considered.\",\n \"Geometries are taken from our previous publication [16], which were obtained at the M06-2X level of theory [37].\",\n \"The dissociation limit is calculated by considering a frozen X-H distance of 5$\\\\textrm {Ã}$ , and optimizing the rest of internal coordinates.\",\n \"At these geometries, single-point energies are evaluated at the NOF-MP2 level of theory.\",\n \"The correlation-consistent valence double-$\\\\zeta $ (cc-pVDZ) or triple-$\\\\zeta $ (cc-pVTZ) basis sets developed by Dunning et al.\",\n \"[38] are used.\",\n \"The zero point vibrational energies (ZPVEs) were taken from the NITS Computational Chemistry Comparison and Benchmark Database (CCCBDB) [39], and corresponds to CCSD(T)/cc-pVTZ values.\",\n \"We also provide the PNOF6 [40] and wavefunction-based calculations obtained in Ref.\",\n \"[16].\",\n \"For the latter, an active space was defined by the distribution of two electrons in two molecular orbitals, CASSCF(2,2) [41], [42].\",\n \"The dynamic correlation effects were included through complete active space second-order perturbation theory calculations, CASPT2(2,2) [43].\",\n \"MOLCAS 7.0 suite of programs [44] was used in Ref.\",\n \"[16], for these wavefunction-based calculations.\"\n ],\n [\n \"X-H homolytic bond cleavage \",\n \"X-H bond dissociation energies were calculated according to the following reaction: $\\\\mathrm {XH}\\\\rightarrow \\\\mathrm {^{.\",\n \"}X}+\\\\mathrm {H}^{.\",\n \"}$ with X = Li, B, CH$_{3}$ , NH$_{2}$ , OH, F. The results are presented in Figure REF and Table REF .\",\n \"The different hydrides considered expand a wide range of dissociation energies, from 58.0 kcal/mol for LiH to 141.1 kcal/mol for FH.\",\n \"The ordering in dissociation energies is $\\\\mathrm {LiH}<\\\\mathrm {BH}<\\\\mathrm {CH_{4}}<\\\\mathrm {NH_{3}}<\\\\mathrm {H_{2}O}<\\\\mathrm {FH}$ .\",\n \"In general, NOF-MP2 reproduces satisfactorily these trends.\",\n \"Figure: NOF-MP2 dissociation energies, in kcal/mol,for X-H bonds (X= Li ,B, CH 3 , NH 2 , OH ,F\\\\mathrm {X=Li,B,CH_{3},NH_{2},OH,F}) versus experimentalones.\",\n \"Calculations carried out with the cc-pVTZ basis set.Let us focus our attention, for example, on the delicate case of the CH$_{4}$ /NH$_{3}$ ordering.\",\n \"The difference in experimental dissociation energies for these two molecules is very small, only 2.9 kcal/mol with $\\\\mathrm {NH_{3}}$ having a higher dissociation energy.\",\n \"NOF-MP2 is able to reproduce the correct ordering $\\\\mathrm {CH_{4} 0$ is the learning rate at iteration $t$ , $\\hat{\\mathbf {g}}_t = \\nabla f(\\mathbf {s}^{(t-1)})$ denotes the gradient of the attack loss $f$ evaluated at $\\mathbf {s}^{(t-1)}$ , and $\\Pi _{\\mathcal {S}} (\\mathbf {a}) \\mathrel {\\mathop :}=\\operatornamewithlimits{arg\\,min}_{\\mathbf {s} \\in \\mathcal {S}} \\Vert \\mathbf {s} - \\mathbf {a} \\Vert _2^2$ is the projection operator at $\\mathbf {a}$ over the constraint set $\\mathcal {S}$ .","In Proposition REF , we show that the projection operation yields the closed-form solution.","Proposition 1 Given $\\mathcal {S} = \\lbrace \\mathbf {s} \\, | \\, \\mathbf {1}^T s \\le \\epsilon , \\mathbf {s} \\in [0,1]^n\\rbrace $ , the projection operation at the point $\\mathbf {a}$ with respect to $\\mathcal {S}$ is $\\Pi _{\\mathcal {S}} (\\mathbf {a}) = \\left\\lbrace \\begin{array}{ll}P_{[\\mathbf {0}, \\mathbf {1}]} [\\mathbf {a} - \\mu \\mathbf {1}] &\\begin{array}{l}\\text{If $\\mu > 0$ and} \\\\\\mathbf {1}^T P_{[\\mathbf {0}, \\mathbf {1}]} [\\mathbf {a} - \\mu \\mathbf {1}] = \\epsilon ,\\end{array} \\\\& \\\\P_{[\\mathbf {0}, \\mathbf {1}]} [\\mathbf {a} ] & \\text{If $\\mathbf {1}^T P_{[\\mathbf {0}, \\mathbf {1}]} [\\mathbf {a} ] \\le \\epsilon $},\\end{array}\\right.$ where $P_{[0,1]} (x ) = x$ if $x \\in [0,1]$ , 0 if $x < 0$ , and 1 if $x > 1$ .","Proof: We express the projection problem as $\\begin{array}{ll}\\displaystyle \\operatornamewithlimits{\\text{minimize}}_{\\mathbf {s}} & \\frac{1}{2}\\Vert \\mathbf {s} - \\mathbf {a} \\Vert _2^2 + \\mathcal {I}_{[\\mathbf {0}, \\mathbf {1}]}(\\mathbf {s})\\\\\\operatornamewithlimits{\\text{subject to}}& \\mathbf {1}^\\top \\mathbf {s} \\le \\epsilon ,\\end{array}$ where $\\mathcal {I}_{[\\mathbf {0}, \\mathbf {1}]}(\\mathbf {s}) = 0$ if $\\mathbf {s} \\in [0,1]^{n}$ , and $\\infty $ otherwise.","The Lagrangian function of problem (REF ) is given by $& \\frac{1}{2} \\Vert \\mathbf {s} - \\mathbf {a} \\Vert _2^2 + \\mathcal {I}_{[\\mathbf {0}, \\mathbf {1}]}(\\mathbf {s}) + \\mu (\\mathbf {1}^\\top \\mathbf {s} - \\epsilon ) \\nonumber \\\\= & \\sum _{i} \\left( \\frac{1}{2} (s_i - a_i)^2 + \\mathcal {I}_{[0,1]}(s_i) + \\mu s_i \\right) - \\mu \\epsilon ,$ where $\\mu \\ge 0$ is the dual variable.","The minimizer to the above Lagrangian function (with respect to the variable $\\mathbf {s}$ ) is $\\mathbf {s} = P_{[\\mathbf {0}, \\mathbf {1}]} (\\mathbf {a} - \\mu \\mathbf {1} ),$ where $P_{[\\mathbf {0}, \\mathbf {1}]}$ is taken elementwise.","Besides the stationary condition (REF ), other KKT conditions for solving problem (REF ) are $& \\mu (\\mathbf {1}^\\top \\mathbf {s} - \\epsilon ) = 0 , \\\\& \\mu \\ge 0, \\\\& \\mathbf {1}^\\top \\mathbf {s} \\le \\epsilon .","$ If $\\mu > 0$ , then the solution to problem (REF ) is given by (REF ), where the dual variable $\\mu $ is determined by (REF ) and (REF ) $\\mathbf {1}^T P_{[\\mathbf {0}, \\mathbf {1}]} [\\mathbf {a} - \\mu \\mathbf {1}] = \\epsilon ,~\\text{and}~ \\mu > 0.$ If $\\mu = 0$ , then the solution to problem (REF ) is given by (REF ) and (), $\\mathbf {s} = P_{[\\mathbf {0}, \\mathbf {1}]} (\\mathbf {a}),~\\text{and}~ \\mathbf {1}^\\top \\mathbf {s} \\le \\epsilon ,$ The proof is complete.","$\\square $ In the projection operation (REF ), one might need to solve the scalar equation $\\mathbf {1}^T P_{[\\mathbf {0}, \\mathbf {1}]} [\\mathbf {a} - \\mu \\mathbf {1}] = \\epsilon $ with respect to the dual variable $\\mu $ .","This can be accomplished by applying the bisection method [2], [12] over $\\mu \\in [\\min (\\mathbf {a}-\\mathbf {1}), \\max (\\mathbf {a}) ]$ .","That is because $\\mathbf {1}^T P_{[\\mathbf {0}, \\mathbf {1}]} [\\mathbf {a} - \\max (\\mathbf {a}) \\mathbf {1}] \\le \\epsilon $ and $\\mathbf {1}^T P_{[\\mathbf {0}, \\mathbf {1}]} [\\mathbf {a} - \\min (\\mathbf {a} - \\mathbf {1}) \\mathbf {1} ] \\ge \\epsilon $ , where $\\max $ and $\\min $ return the largest and smallest entry of a vector.","We remark that the bisection method converges in the logarithmic rate given by $\\log _2 {[ (\\max (\\mathbf {a}) -\\min (\\mathbf {a}-\\mathbf {1}) )/\\xi ]}$ for the solution of $\\xi $ -error tolerance.","We summarize the PGD topology attack in Algorithm REF .","PGD topology attack on GNN [1] Input: $\\mathbf {s}^{(0)}$ , $\\epsilon > 0$ , learning rate $\\eta _t$ , and iterations $T$ $t = 1,2,\\ldots , T$ gradient descent: $\\mathbf {a}^{(t)} = \\mathbf {s}^{(t-1)} - \\eta _{t} \\nabla f(\\mathbf {s}^{(t-1)}) $ call projection operation in (REF ) call Algorithm REF to return $\\mathbf {s}^*$ , and the resulting $\\mathbf {A}^\\prime $ in (REF )."],["Min-max Topology Attack","We next solve the problem of min-max attack generation in (REF ).","By convex relaxation on the Boolean variables, we obtain the following continuous optimization problem $\\hspace*{-10.84006pt}\\begin{array}{cc}\\displaystyle \\operatornamewithlimits{\\text{minimize}}_{ \\mathbf {s}\\in \\mathcal {S} } \\displaystyle \\operatornamewithlimits{\\text{maximize}}_{\\mathbf {W}} \\, f(\\mathbf {s}, \\mathbf {W}) = \\sum _{i \\in \\mathcal {V}}f_i(\\mathbf {s} , \\mathbf {W} ; \\mathbf {A}, \\lbrace \\mathbf {x}_i\\rbrace , y_i ) ,\\end{array}$ where $\\mathcal {S}$ has been defined in (REF ).","We solve problem (REF ) by first-order alternating optimization [14], [15], where the inner maximization is solved by gradient ascent, and the outer minimization is handled by PGD same as (REF ).","We summarize the min-max topology attack in Algorithm REF .","We remark that one can perform multiple maximization steps within each iteration of alternating optimization.","This strikes a balance between the computation efficiency and the convergence accuracy [5], [18].","Min-max topology attack to solve (REF ) [1] Input: given $\\mathbf {W}^{(0)}$ , $\\mathbf {s}^{(0)}$ , learning rates $\\beta _t$ and $\\eta _t$ , and iteration numbers $T$ $t = 1,2,\\ldots , T$ inner maximization over $\\mathbf {W}$ : given $\\mathbf {s}^{(t-1)}$ , obtain $\\mathbf {W}^{t} = \\mathbf {W}^{t-1} + \\beta _t \\nabla _{\\mathbf {W}} f (\\mathbf {s}^{t-1},\\mathbf {W}^{t-1})$ outer minimization over $\\mathbf {s}$ : given $\\mathbf {W}^{(t)}$ , running PGD (REF ), where $\\hat{\\mathbf {g}}_t = \\nabla _{\\mathbf {s}} f (\\mathbf {s}^{t-1},\\mathbf {W}^{t})$ call Algorithm REF to return $\\mathbf {s}^*$ , and the resulting $\\mathbf {A}^\\prime $ in (REF )."],["Robust Training for GNNs","With the aid of first-order attack generation methods, we now introduce our adversarial training for GNNs via robust optimization.","Similar formulation is also used in [16].","In adversarial training, we solve a min-max problem for robust optimization: $\\begin{array}{cc}\\displaystyle \\operatornamewithlimits{\\text{minimize}}_{\\mathbf {W}} \\operatornamewithlimits{\\text{maximize}}_{ \\mathbf {s} \\in \\mathcal {S}} \\, -f(\\mathbf {s}, \\mathbf {W}) ,\\end{array}$ where $ f(\\mathbf {x}, \\mathbf {W})$ denotes the attack loss specified in (REF ).","Following the idea of adversarial training for image classifiers in [16], we restrict the loss function $f$ as the CE-type loss.","This formulation tries to minimize the training loss at the presence of topology perturbations.","We note that problems (REF ) and (REF ) share a very similar min-max form, however, they are not equivalent since the loss $f$ is neither convex with respect to $\\mathbf {s}$ nor concave with respect to $\\mathbf {W}$ , namely, lacking saddle point property [2].","However, there exists connection between (REF ) and (REF ); see Proposition REF .","Proposition 2 Given a general attack loss function $f$ , problem (REF ) is equivalent to $\\operatornamewithlimits{\\text{maximize}}_{\\mathbf {W}} \\operatornamewithlimits{\\text{minimize}}_{ \\mathbf {s} \\in \\mathcal {S}} \\, f(\\mathbf {s}, \\mathbf {W}),$ which further yields $(\\ref {eq: min_max_sol_same}) \\le (\\ref {eq: robust_attack_cont})$ .","Proof: By introducing epigraph variable $p$ [2], problem (REF ) can be rewritten as $\\begin{array}{ll}\\displaystyle \\operatornamewithlimits{\\text{minimize}}_{\\mathbf {W}, p} & p \\\\\\operatornamewithlimits{\\text{subject to}}& -f(\\mathbf {s}, \\mathbf {W}) \\le p, \\forall \\mathbf {s} \\in \\mathcal {S}.\\end{array}$ By changing variable $q \\mathrel {\\mathop :}=- p$ , problem (REF ) is equivalent to $\\begin{array}{ll}\\displaystyle \\operatornamewithlimits{\\text{maximize}}_{\\mathbf {W}, q} & q \\\\\\operatornamewithlimits{\\text{subject to}}&f(\\mathbf {s}, \\mathbf {W}) \\ge q, \\forall \\mathbf {s} \\in \\mathcal {S}.\\end{array}$ By eliminating the epigraph variable $q$ , problem (REF ) becomes (REF ).","By max-min inequality [2], we finally obtain that $\\operatornamewithlimits{\\text{maximize}}_{\\mathbf {W}} \\operatornamewithlimits{\\text{minimize}}_{ \\mathbf {s} \\in \\mathcal {S}} \\, f(\\mathbf {s}, \\mathbf {W}) \\le \\operatornamewithlimits{\\text{minimize}}_{ \\mathbf {s}\\in \\mathcal {S} } \\operatornamewithlimits{\\text{maximize}}_{\\mathbf {W}} \\, f(\\mathbf {s}, \\mathbf {W}).$ The proof is now complete.","$\\square $ We summarize the robust training algorithm in Algorithm for solving problem (REF ).","Similar to Algorithm REF , one usually performs multiple inner minimization steps (with respect to $\\mathbf {s}$ ) within each iteration $t$ to have a solution towards minimizer during alternating optimization.","This improves the stability of convergence in practice [18], [16].","Robust training for solving problem (REF ) [1] Input: given $\\mathbf {W}^{(0)}$ , $\\mathbf {s}^{(0)}$ , learning rates $\\beta _t$ and $\\eta _t$ , and iteration numbers $T$ $t = 1,2,\\ldots , T$ inner minimization over $\\mathbf {s}$ : given $\\mathbf {W}^{(t-1)}$ , running PGD (REF ), where $\\hat{\\mathbf {g}}_t = \\nabla _{\\mathbf {s}} f (\\mathbf {s}^{t-1},\\mathbf {W}^{t-1})$ outer maximization over $\\mathbf {W}$ : given $\\mathbf {s}^{(t)}$ , obtain $\\mathbf {W}^{t} = \\mathbf {W}^{t-1} + \\beta _t \\nabla _{\\mathbf {W}} f (\\mathbf {s}^{t},\\mathbf {W}^{t-1})$ return $\\mathbf {W}^{T}$ ."],["Experiments","In this section, we present our experimental results for both topology attack and defense methods on a graph convolutional networks (GCN) [10].","We demonstrate the misclassification rate and the convergence of the proposed 4 attack methods: negative cross-entropy loss via PGD attack (CE-PGD), CW loss via PGD attack (CW-PGD), negative cross-entropy loss via min-max attack (CE-min-max), CW loss via min-max attack (CW-min-max).","We then show the improved robustness of GCN by leveraging our proposed robust training against topology attacks."],["Experimental Setup","We evaluate our methods on two well-known datasets: Cora and Citeseer [19].","Both datasets contain unweighted edges which can be generated as symmetric adjacency matrix $\\mathbf {A} $ and sparse bag-of-words feature vectors which can be treated the input of GCN.","To train the model, all node feature vectors are fed into GCN but with only 140 and 120 labeled nodes for Cora and Citeseer, respectively.","The number of test labeled nodes is 1000 for both datasets.","At each experiment, we repeat 5 times based on different splits of training/testing nodes and report mean $\\pm $ standard deviation of misclassification rate (namely, 1 $-$ prediction accuracy) on testing nodes."],["Attack Performance","We compare our four attack methods (CE-PGD, CW-PGD, CE-min-max, CW-min-max) with DICE (`delete edges internally, connect externally’) [21], Meta-Self attack [26] and greedy attack, a variant of Meta-Self attack without weight re-training for GCN.","The greedy attack is considered as a fair comparison with our CE-PGD and CW-PGD attacks, which are generated on a fixed GCN without weight re-training.","In min-max attacks (CE-min-max and CW-min-max), we show misclassification rates against both natural and retrained models from Algorithm REF , and compare them with the state-of-the-art Meta-Self attack.","For a fair comparison, we use the same performance evaluation criterion in Meta-Self, testing nodes' predicted labels (not their ground-truth label) by an independent pre-trained model that can be used during the attack.","In the attack problems (REF ) and (REF ), unless specified otherwise the maximum number of perturbed edges is set to be $5\\%$ of the total number of existing edges in the original graph.","In Algorithm REF , we set the iteration number of random sampling as $K = 20$ and choose the perturbed topology with the highest misclassification rate which also satisfies the edge perturbation constraint.","In Table REF , we present the misclassification rate of different attack methods against both natural and retrained model from (REF ).","Here we recall that the retrained model arises due to the scenario of attacking an interactive GCN with re-trainable weights (Algorithm REF ).","For comparison, we also show the misclassification rate of a natural model with the true topology (denoted by `clean').","As we can see, to attack the natural model, our proposed attacks achieve better misclassification rate than the existing methods.","We also observe that compared to min-max attacks (CE-min-max and CW-min-max), CE-PGD and CW-PGD yield better attacking performance since it is easier to attack a pre-defined GCN.","To attack the model that allows retraining, we set 20 steps of inner maximization per iteration of Algorithm REF .","The results show that our proposed min-max attack achieves very competitive performance compared to Meta-Self attack.","Note that evaluating the attack performance on the retrained model obtained from (REF ) is not quite fair since the retrained weights could be sub-optimal and induce degradation in classification.","Table: Misclassification rates (%\\%) under 5%5\\% perturbed edgesIn Fig.","REF , we present the CE-loss and the CW-loss of the proposed topology attacks against the number of iterations in Algorithm REF .","Here we choose $T = 200$ and $\\eta _t = 200/\\sqrt{t}$ .","As we can see, the method of PGD converges gracefully against iterations.","This verifies the effectiveness of the first-order optimization based attack generation method.","Figure: CE-PGD and CW-PGD attack losses on Cora and Citeseer datasets."],["Defense Performance","In what follows, we invoke Algorithm to generate robust GCN via adversarial training.","We set $T = 1000$ , $\\beta _t = 0.01$ and $\\eta _t = 200/\\sqrt{t}$ .","We run 20 steps for inner minimization.","Inspired by [16], we increase the hidden units from 16 to 32 in order to create more capacity for this more complicated classifier.","Initially, we set the maximum number of edges we can modify as $5\\%$ of total existing edges.","In Figure REF , we present convergence of our robust training.","As we can see, the loss drops reasonably and the $1,000$ iterations are necessary for robust training rather than normal training process which only need 200 iterations.","We also observe that our robust training algorithm does not harm the test accuracy when $\\epsilon = 5\\%$ , but successfully improves the robustness as the attack success rate drops from $28.0\\%$ to $22.0\\%$ in Cora dataset as shown in Table REF , After showing the effectiveness of our algorithm, we explore deeper in adversarial training on GCN.","We aim to show how large $\\epsilon $ we can use in robust training.","So we set $\\epsilon $ from $5\\%$ to $20\\%$ and apply CE-PGD attack following the same $\\epsilon $ setting.","The results are presented in Table REF .","Note that when $\\epsilon =0$ , the first row shows misclassification rates of test nodes on natural graph as the baseline for lowest misclassification rate we can obtain; the first column shows the CE-PGD attack misclassification rates of natural model as the baseline for highest misclassification rate we can obtain.","We can conclude that when a robust model trained under an $\\epsilon $ constraint, the model will gain robustness under this $\\epsilon $ distinctly.","Considering its importance to keep the original graph test performance, we suggest generating robust model under $\\epsilon = 0.1$ .","Moreover, please refer to Figure REF that a) our robust trained model can provide universal defense to CE-PGD, CW-PGD and Greedy attacks; b) when increasing $\\epsilon $ , the difference between both test accuracy and CE-PGD attack accuracy increases substantially, which also implies the robust model under larger $\\epsilon $ is harder to obtain.","Table: Misclassification rates (%\\%) of robust training (smaller is better for defense task) with at most 5%5\\% of edge perturbations.","𝐀\\mathbf {A} means the natural graph, 𝐀 ' \\mathbf {A}^\\prime means the generated adversarial graph under ϵ=5%\\epsilon = 5\\%.","𝐗/M\\mathbf {X} / M means the misclassification rate of using model MM on graph 𝐗\\mathbf {X}.Table: Misclassification rates (%\\%) of CE-PGD attack against robust training model versus (smaller is better) different ϵ\\epsilon (%\\%) on Cora dataset.","Hereϵ=0\\epsilon =0 in training means natural model and ϵ=0\\epsilon =0 in attack means unperturbed topology.Figure: Robust training loss on Cora and Citeseer datasets.Figure: Test accuracy of robust model (no attack),CE-PGD attack against robust model, CW-PGD attack against robust model, Greedy attack against robust model and CE-PGD attack against natural model for different ϵ\\epsilon used in robust training and test on Cora dataset."],["Conclusion","In this paper, we first introduce an edge perturbation based topology attack framework that overcomes the difficulty of attacking discrete graph structure data from a first-order optimization perspective.","Our extensive experiments show that with only a fraction of edges changed, we are able to compromise state-of-the-art graph neural networks model noticeably.","Additionally, we propose an adversarial training framework to improve the robustness of GNN models based on our attack methods.","Experiments on different datasets show that our method is able to improve the GNN model's robustness against both gradient based and greedy search based attack methods without classification performance drop on original graph.","We believe that this paper provides potential means for theoretical study and improvement of the robustness of deep learning models on graph data."],["Acknowledgments","This work is supported by Air Force Research Laboratory FA8750-18-2-0058 and the MIT-IBM Watson AI Lab."]],"string":"[\n [\n \"Topology Attack and Defense for Graph Neural Networks: An Optimization\\n Perspective\"\n ],\n [\n \"Abstract Graph neural networks (GNNs) which apply the deep neural networks to graph data have achieved significant performance for the task of semi-supervised node classification.\",\n \"However, only few work has addressed the adversarial robustness of GNNs.\",\n \"In this paper, we first present a novel gradient-based attack method that facilitates the difficulty of tackling discrete graph data.\",\n \"When comparing to current adversarial attacks on GNNs, the results show that by only perturbing a small number of edge perturbations, including addition and deletion, our optimization-based attack can lead to a noticeable decrease in classification performance.\",\n \"Moreover, leveraging our gradient-based attack, we propose the first optimization-based adversarial training for GNNs.\",\n \"Our method yields higher robustness against both different gradient based and greedy attack methods without sacrificing classification accuracy on original graph.\"\n ],\n [\n \"Introduction\",\n \"Graph structured data plays a crucial role in many AI applications.\",\n \"It is an important and versatile representation to model a wide variety of datasets from many domains, such as molecules, social networks, or interlinked documents with citations.\",\n \"Graph neural networks (GNNs) on graph structured data have shown outstanding results in various applications [10], [20], [22].\",\n \"However, despite the great success on inferring from graph data, the inherent challenge of lacking adversarial robustness in deep learning models still carries over to security-related domains such as blockchain or communication networks.\",\n \"In this paper, we aim to evaluate the robustness of GNNs from a perspective of first-order optimization adversarial attacks.\",\n \"It is worth mentioning that first-order methods have achieved great success for generating adversarial attacks on audios or images [4], [23], [7], [24], [6].\",\n \"However, some recent works [8], [1] suggested that conventional (first-order) continuous optimization methods do not directly apply to attacks using edge manipulations (we call topology attack) due to the discrete nature of graphs.\",\n \"We close this gap by studying the problem of generating topology attacks via convex relaxation so that gradient-based adversarial attacks become plausible for GNNs.\",\n \"Benchmarking on node classification tasks using GNNs, our gradient-based topology attacks outperform current state-of-the-art attacks subject to the same topology perturbation budget.\",\n \"This demonstrates the effectiveness of our attack generation method through the lens of convex relaxation and first-order optimization.\",\n \"Moreover, by leveraging our proposed gradient-based attack, we propose the first optimization-based adversarial training technique for GNNs, yielding significantly improved robustness against gradient-based and greedy topology attacks.\",\n \"Our new attack generation and adversarial training methods for GNNs are built upon the theoretical foundation of spectral graph theory, first-order optimization, and robust (mini-max) optimization.\",\n \"We summarize our main contributions as follows: We propose a general first-order attack generation framework under two attacking scenarios: a) attacking a pre-defined GNN and b) attacking a re-trainable GNN.\",\n \"This yields two new topology attacks: projected gradient descent (PGD) topology attack and min-max topology attack.\",\n \"Experimental results show that the proposed attacks outperform current state-of-the-art attacks.\",\n \"With the aid of our first-order attack generation methods, we propose an adversarial training method for GNNs to improve their robustness.\",\n \"The effectiveness of our method is shown by the considerable improvement of robustness on GNNs against both optimization-based and greedy-search-based topology attacks.\"\n ],\n [\n \"Related Works\",\n \"Some recent attentions have been paid to the robustness of graph neural network.\",\n \"Both [25] and [8] studied adversarial attacks on neural networks for graph data.\",\n \"[8] studied test-time non-targeted adversarial attacks on both graph classification and node classification.\",\n \"Their work restricted the attacks to perform modifications on discrete structures, that is, an attacker is only allowed to add or delete edges from a graph to construct a new graph.\",\n \"White-box, practical black-box and restricted black-box graph adversarial attack scenarios were studied.\",\n \"Authors in [25] considered both test-time (evasion) and training-time (data poisoning) attacks on node classification task.\",\n \"In contrast to [8], besides adding or removing edges in the graph, attackers in [25] may modify node attributes.\",\n \"They designed adversarial attacks based on a static surrogate model and evaluated their impact by training a classifier on the data modified by the attack.\",\n \"The resulting attack algorithm is for targeted attacks on single nodes.\",\n \"It was shown that small perturbations on the graph structure and node features are able to achieve misclassification of a target node.\",\n \"A data poisoning attack on unsupervised node representation learning, or node embeddings, has been proposed in [1].\",\n \"This attack is based on perturbation theory to maximize the loss obtained from DeepWalk [17].\",\n \"In [26], training-time attacks on GNNs were also investigated for node classification by perturbing the graph structure.\",\n \"The authors solved a min-max problem in training-time attacks using meta-gradients and treated the graph topology as a hyper-parameter to optimize.\"\n ],\n [\n \"Problem Statement\",\n \"We begin by providing preliminaries on GNNs.\",\n \"We then formalize the attack threat model of GNNs in terms of edge perturbations, which we refer as `topology attack'.\"\n ],\n [\n \"Preliminaries on GNNs\",\n \"It has been recently shown in [10], [20], [22] that GNN is powerful in transductive learning, e.g., node classification under graph data.\",\n \"That is, given a single network topology with node features and a known subset of node labels, GNNs are efficient to infer the classes of unlabeled nodes.\",\n \"Prior to defining GNN, we first introduce the following graph notations.\",\n \"Let $\\\\mathcal {G} = (\\\\mathcal {V}, \\\\mathcal {E})$ denote an undirected and unweighted graph, where $\\\\mathcal {V}$ is the vertex (or node) set with cardinality $|\\\\mathcal {V}| = N$ , and $\\\\mathcal {E} \\\\in (\\\\mathcal {V} \\\\times \\\\mathcal {V})$ denotes the edge set with cardinality $|\\\\mathcal {E}| = M$ .\",\n \"Let $\\\\mathbf {A}$ represent a binary adjacency matrix.\",\n \"By definition, we have $A_{ij} = 0$ if $(i, j) \\\\notin \\\\mathcal {E}$ .\",\n \"In a GNN, we assume that each node $i$ is associated with a feature vector $\\\\mathbf {x}_i \\\\in \\\\mathbb {R}^{M_0}$ and a scalar label $y_i$ .\",\n \"The goal of GNN is to predict the class of an unlabeled node under the graph topology $\\\\mathbf {A}$ and the training data $\\\\lbrace (\\\\mathbf {x}_i, y_i )\\\\rbrace _{i=1}^{N_{\\\\text{train}}}$ .\",\n \"Here GNN uses input features of all nodes but only $N_{\\\\text{train}} < N$ nodes with labeled classes in the training phase.\",\n \"Formally, the $k$ th layer of a GNN model obeys the propagation rule of the generic form $\\\\mathbf {h}_i^{(k)} = g^{(k)} \\\\left( \\\\lbrace \\\\mathbf {W}^{(k-1)} \\\\mathbf {h}_j^{(k-1)} \\\\tilde{A}_{ij}, ~ \\\\forall j \\\\in \\\\mathcal {N}(i)\\\\rbrace \\\\right), ~ \\\\forall i\\\\in [N]$ where $\\\\mathbf {h}_i^{(k)} \\\\in \\\\mathbb {R}^{M_{k}}$ denotes the feature vector of node $i$ at layer $k$ , $\\\\mathbf {h}_i^{(0)} = \\\\mathbf {x}_i\\\\in \\\\mathbb {R}^{M_0}$ is the input feature vector of node $i$ , $ g^{(k)}$ is a possible composite mapping (activation) function, $\\\\mathbf {W}^{(k-1)} \\\\in \\\\mathbb {R}^{M_k \\\\times M_{k-1}}$ is the trainable weight matrix at layer $(k-1)$ , $\\\\tilde{ A}_{ij}$ is the $(i,j)$ th entry of $\\\\tilde{\\\\mathbf {A}}$ that denotes a linear mapping of $\\\\mathbf {A}$ but with the same sparsity pattern, and $\\\\mathcal {N}(i)$ denotes node $i$ 's neighbors together with itself, i.e., $\\\\mathcal {N}(i) = \\\\lbrace j | (i,j) \\\\in \\\\mathcal {E}, \\\\text{ or } j = i \\\\rbrace $ .\",\n \"A special form of GNN is graph convolutional networks (GCN) [10].\",\n \"This is a recent approach of learning on graph structures using convolution operations which is promising as an embedding methodology.\",\n \"In GCNs, the propagation rule (REF ) becomes [10] $\\\\mathbf {h}_i^{(k)} = \\\\sigma \\\\left( \\\\sum _{j \\\\in \\\\mathcal {N}_i} \\\\left( \\\\mathbf {W}^{(k-1)} \\\\mathbf {h}_j^{(k-1)} \\\\tilde{A}_{ij} \\\\right) \\\\right),$ where $\\\\sigma (\\\\cdot )$ is the ReLU function.\",\n \"Let $\\\\tilde{A}_{i,:}$ denote the $i$ th row of $\\\\tilde{\\\\mathbf {A}}$ and $\\\\mathbf {H}^{(k)} = \\\\left[ (\\\\mathbf {h}_1^{(k)})^\\\\top ; \\\\ldots ; (\\\\mathbf {h}_N^{(k)})^\\\\top \\\\right]$ , we then have the standard form of GCN, $\\\\mathbf {H}^{(k)} = \\\\sigma \\\\left( \\\\tilde{\\\\mathbf {A}} \\\\mathbf {H}^{(k-1)} ( \\\\mathbf {W}^{(k-1)} )^\\\\top \\\\right).$ Here $\\\\tilde{\\\\mathbf {A}}$ is given by a normalized adjacency matrix $\\\\tilde{\\\\mathbf {A}} = \\\\hat{\\\\mathbf {D}}^{-1/2} \\\\hat{\\\\mathbf {A}} \\\\hat{\\\\mathbf {D}}^{-1/2}$ , where $\\\\hat{\\\\mathbf {A}} = \\\\mathbf {A} + \\\\mathbf {I}$ , and $\\\\hat{\\\\mathbf {D}}_{ij} = 0$ if $i \\\\ne j$ and $\\\\hat{\\\\mathbf {D}}_{ii} = \\\\mathbf {1}^\\\\top \\\\hat{\\\\mathbf {A}}_{:,i}$ .\"\n ],\n [\n \"Topology Attack in Terms of Edge Perturbation\",\n \"We introduce a Boolean symmetric matrix $\\\\mathbf {S} \\\\in \\\\lbrace 0,1\\\\rbrace ^{N \\\\times N}$ to encode whether or not an edge in $\\\\mathcal {G}$ is modified.\",\n \"That is, the edge connecting nodes $i$ and $j$ is modified (added or removed) if and only if $S_{ij} = S_{ji}=1$ .\",\n \"Otherwise, $S_{ij } = 0$ if $i = j $ or the edge $(i,j)$ is not perturbed.\",\n \"Given the adjacency matrix $\\\\mathbf {A}$ , its supplement is given by $\\\\bar{\\\\mathbf {A}} = \\\\mathbf {1} \\\\mathbf {1}^T - \\\\mathbf {I} - \\\\mathbf {A} $ , where $\\\\mathbf {I}$ is an identity matrix, and $(\\\\mathbf {1} \\\\mathbf {1}^T - \\\\mathbf {I})$ corresponds to the fully-connected graph.\",\n \"With the aid of edge perturbation matrix $\\\\mathbf {S}$ and $\\\\bar{\\\\mathbf {A}}$ , a perturbed graph topology $\\\\mathbf {A}^\\\\prime $ against $\\\\mathbf {A}$ is given by $\\\\mathbf {A}^\\\\prime = \\\\mathbf {A} + \\\\mathbf {C} \\\\circ \\\\mathbf {S}, ~\\\\mathbf {C} = \\\\bar{\\\\mathbf {A}} - \\\\mathbf {A},$ where $\\\\circ $ denotes the element-wise product.\",\n \"In (REF ), the positive entry of $\\\\mathbf {C}$ denotes the edge that can be added to the graph $\\\\mathbf {A}$ , and the negative entry of $\\\\mathbf {C}$ denotes the edge that can be removed from $\\\\mathbf {A}$ .\",\n \"We then formalize the concept of topology attack to GNNs: Finding minimum edge perturbations encoded by $\\\\mathbf {S}$ in (REF ) to mislead GNNs.\",\n \"A more detailed attack formulation will be studied in the next section.\"\n ],\n [\n \"Topology Attack Generation: A First-Order Optimization Perspective\",\n \"In this section, we first define attack loss (beyond the conventional cross-entropy loss) under different attacking scenarios.\",\n \"We then develop two efficient attack generation methods by leveraging first-order optimization.\",\n \"We call the resulting attacks projected gradient descent (PGD) topology attack and min-max topology attack, respectively.\"\n ],\n [\n \"Attack Loss & Attack Generation\",\n \"Let $\\\\mathbf {Z} (\\\\mathbf {S}, \\\\mathbf {W}; \\\\mathbf {A}, \\\\lbrace \\\\mathbf {x}_i\\\\rbrace )$ denote the prediction probability of a GNN specified by $\\\\mathbf {A}^\\\\prime $ in (REF ) and $\\\\mathbf {W}$ under input features $ \\\\lbrace \\\\mathbf {x}_i\\\\rbrace $ .\",\n \"Then $Z_{i,c}$ denotes the probability of assigning node $i$ to class $c$ .\",\n \"It has been shown in existing works [9], [11] that the negative cross-entropy (CE) loss between the true labels ($ y_i$ ) and the predicted labels ($\\\\lbrace Z_{i,c} \\\\rbrace $ ) can be used as an attack loss at node $i$ , denoted by $f_i(\\\\mathbf {S}, \\\\mathbf {W};\\\\mathbf {A}, \\\\lbrace \\\\mathbf {x}_i\\\\rbrace , y_i )$ .\",\n \"We can also propose a CW-type loss similar to Carlili-Wagner (CW) attacks for attacking image classifiers [3], $& f_i(\\\\mathbf {S}, \\\\mathbf {W}; \\\\mathbf {A}, \\\\lbrace \\\\mathbf {x}_i\\\\rbrace , y_i ) =\\\\max \\\\left\\\\lbrace Z_{i,y_i} - \\\\max _{c \\\\ne y_i} Z_{i,c}, - \\\\kappa \\\\right\\\\rbrace ,$ where $\\\\kappa \\\\ge 0$ is a confidence level of making wrong decisions.\",\n \"To design topology attack, we seek $\\\\mathbf {S}$ in (REF ) to minimize the per-node attack loss (CE-type or CW-type) given a finite budge of edge perturbations.\",\n \"We consider two threat models: a) attacking a pre-defined GNN with known $\\\\mathbf {W}$ ; b) attacking an interactive GNN with re-trainable $\\\\mathbf {W}$ .\",\n \"In the case a) of fixed $\\\\mathbf {W}$ , the attack generation problem can be cast as $\\\\begin{array}{ll}\\\\displaystyle \\\\operatornamewithlimits{\\\\text{minimize}}_{\\\\mathbf {s} } & \\\\sum _{i \\\\in \\\\mathcal {V}}f_i(\\\\mathbf {s} ; \\\\mathbf {W}, \\\\mathbf {A}, \\\\lbrace \\\\mathbf {x}_i\\\\rbrace , y_i )\\\\\\\\\\\\operatornamewithlimits{\\\\text{subject to}}& \\\\mathbf {1}^\\\\top \\\\mathbf {s} \\\\le \\\\epsilon , ~ \\\\mathbf {s}\\\\in \\\\lbrace 0,1\\\\rbrace ^{n},\\\\end{array}$ where we replace the symmetric matrix variable $\\\\mathbf {S}$ with its vector form that consists of $n \\\\mathrel {\\\\mathop :}=N(N-1)/2$ unique perturbation variables in $\\\\mathbf {S}$ .\",\n \"We recall that $f_i$ could be either a CE-type or a CW-type per-node attack loss.\",\n \"In the case b) of re-trainable $\\\\mathbf {W}$ , the attack generation problem has the following min-max form $\\\\begin{array}{cc}\\\\displaystyle \\\\operatornamewithlimits{\\\\text{minimize}}_{ \\\\mathbf {1}^\\\\top \\\\mathbf {s} \\\\le \\\\epsilon , \\\\mathbf {s}\\\\in \\\\lbrace 0,1\\\\rbrace ^{n} } \\\\displaystyle \\\\operatornamewithlimits{\\\\text{maximize}}_{\\\\mathbf {W}} \\\\, \\\\sum _{i \\\\in \\\\mathcal {V}}f_i(\\\\mathbf {s} , \\\\mathbf {W} ; \\\\mathbf {A}, \\\\lbrace \\\\mathbf {x}_i\\\\rbrace , y_i ) ,\\\\end{array}$ where the inner maximization aims to constrain the attack loss by retraining $\\\\mathbf {W}$ so that attacking GNN is more difficult.\",\n \"Motivated by targeted adversarial attacks against image classifiers [3], we can define targeted topology attacks that are restricted to perturb edges of targeted nodes.\",\n \"In this case, we require to linearly constrain $\\\\mathbf {S}$ in (REF ) as $S_{i, \\\\cdot } = 0$ if $i$ is not a target node.\",\n \"As a result, both attack formulations (REF ) and (REF ) have extra linear constraints with respect to $s$ , which can be readily handled by the optimization solver introduced later.\",\n \"Without loss of generality, we focus on untargeted topology attacks in this paper.\"\n ],\n [\n \"PGD Topology Attack\",\n \"Problem (REF ) is a combinatorial optimization problem due to the presence of Boolean variables.\",\n \"For ease of optimization, we relax $\\\\mathbf {s} \\\\in \\\\lbrace 0,1\\\\rbrace ^{n}$ to its convex hull $s \\\\in [ 0,1]^{n}$ and solve the resulting continuous optimization problem, $\\\\begin{array}{ll}\\\\displaystyle \\\\operatornamewithlimits{\\\\text{minimize}}_{\\\\mathbf {s}} & f(\\\\mathbf {s}) \\\\mathrel {\\\\mathop :}=\\\\sum _{i \\\\in \\\\mathcal {V}}f_i(\\\\mathbf {s} ; \\\\mathbf {W}, \\\\mathbf {A}, \\\\lbrace \\\\mathbf {x}_i\\\\rbrace , y_i ) \\\\\\\\\\\\operatornamewithlimits{\\\\text{subject to}}& \\\\mathbf {s} \\\\in \\\\mathcal {S},\\\\end{array}$ where $\\\\mathcal {S} = \\\\lbrace \\\\mathbf {s} \\\\, | \\\\, \\\\mathbf {1}^T s \\\\le \\\\epsilon , \\\\mathbf {s} \\\\in [0,1]^n\\\\rbrace $ .\",\n \"Suppose that the solution of problem (REF ) is achievable, the remaining question is how to recover a binary solution from it.\",\n \"Since the variable $\\\\mathbf {s}$ in (REF ) can be interpreted as a probabilistic vector, a randomization sampling [13] is suited for generating a near-optimal binary topology perturbation; see details in Algorithm REF .\",\n \"Random sampling from probabilistic to binary topology perturbation [1] Input: probabilistic vector $\\\\mathbf {s}$ , $K$ is # of random trials $k = 1,2,\\\\ldots , K$ draw binary vector $\\\\mathbf {u}^{(k)}$ following $u^{(k)}_i = \\\\left\\\\lbrace \\\\begin{array}{ll}1 & \\\\text{with probability $s_{i}$}\\\\\\\\0 & \\\\text{with probability $1- s_{i}$}\\\\end{array} \\\\right.\",\n \", \\\\forall i$ choose a vector $\\\\mathbf {s}^*$ from $\\\\lbrace \\\\mathbf {u}^{(k)} \\\\rbrace $ which yields the smallest attack loss $f(\\\\mathbf {u}^{(k)} )$ under $\\\\mathbf {1}^T \\\\mathbf {s} \\\\le \\\\epsilon $ .\",\n \"We solve the continuous optimization problem (REF ) by projected gradient descent (PGD), $\\\\mathbf {s}^{(t)} = \\\\Pi _{\\\\mathcal {S}} \\\\left[ \\\\mathbf {s}^{(t-1)} - \\\\eta _{t} \\\\hat{\\\\mathbf {g}}_t \\\\right],$ where $t$ denotes the iteration index of PGD, $\\\\eta _t > 0$ is the learning rate at iteration $t$ , $\\\\hat{\\\\mathbf {g}}_t = \\\\nabla f(\\\\mathbf {s}^{(t-1)})$ denotes the gradient of the attack loss $f$ evaluated at $\\\\mathbf {s}^{(t-1)}$ , and $\\\\Pi _{\\\\mathcal {S}} (\\\\mathbf {a}) \\\\mathrel {\\\\mathop :}=\\\\operatornamewithlimits{arg\\\\,min}_{\\\\mathbf {s} \\\\in \\\\mathcal {S}} \\\\Vert \\\\mathbf {s} - \\\\mathbf {a} \\\\Vert _2^2$ is the projection operator at $\\\\mathbf {a}$ over the constraint set $\\\\mathcal {S}$ .\",\n \"In Proposition REF , we show that the projection operation yields the closed-form solution.\",\n \"Proposition 1 Given $\\\\mathcal {S} = \\\\lbrace \\\\mathbf {s} \\\\, | \\\\, \\\\mathbf {1}^T s \\\\le \\\\epsilon , \\\\mathbf {s} \\\\in [0,1]^n\\\\rbrace $ , the projection operation at the point $\\\\mathbf {a}$ with respect to $\\\\mathcal {S}$ is $\\\\Pi _{\\\\mathcal {S}} (\\\\mathbf {a}) = \\\\left\\\\lbrace \\\\begin{array}{ll}P_{[\\\\mathbf {0}, \\\\mathbf {1}]} [\\\\mathbf {a} - \\\\mu \\\\mathbf {1}] &\\\\begin{array}{l}\\\\text{If $\\\\mu > 0$ and} \\\\\\\\\\\\mathbf {1}^T P_{[\\\\mathbf {0}, \\\\mathbf {1}]} [\\\\mathbf {a} - \\\\mu \\\\mathbf {1}] = \\\\epsilon ,\\\\end{array} \\\\\\\\& \\\\\\\\P_{[\\\\mathbf {0}, \\\\mathbf {1}]} [\\\\mathbf {a} ] & \\\\text{If $\\\\mathbf {1}^T P_{[\\\\mathbf {0}, \\\\mathbf {1}]} [\\\\mathbf {a} ] \\\\le \\\\epsilon $},\\\\end{array}\\\\right.$ where $P_{[0,1]} (x ) = x$ if $x \\\\in [0,1]$ , 0 if $x < 0$ , and 1 if $x > 1$ .\",\n \"Proof: We express the projection problem as $\\\\begin{array}{ll}\\\\displaystyle \\\\operatornamewithlimits{\\\\text{minimize}}_{\\\\mathbf {s}} & \\\\frac{1}{2}\\\\Vert \\\\mathbf {s} - \\\\mathbf {a} \\\\Vert _2^2 + \\\\mathcal {I}_{[\\\\mathbf {0}, \\\\mathbf {1}]}(\\\\mathbf {s})\\\\\\\\\\\\operatornamewithlimits{\\\\text{subject to}}& \\\\mathbf {1}^\\\\top \\\\mathbf {s} \\\\le \\\\epsilon ,\\\\end{array}$ where $\\\\mathcal {I}_{[\\\\mathbf {0}, \\\\mathbf {1}]}(\\\\mathbf {s}) = 0$ if $\\\\mathbf {s} \\\\in [0,1]^{n}$ , and $\\\\infty $ otherwise.\",\n \"The Lagrangian function of problem (REF ) is given by $& \\\\frac{1}{2} \\\\Vert \\\\mathbf {s} - \\\\mathbf {a} \\\\Vert _2^2 + \\\\mathcal {I}_{[\\\\mathbf {0}, \\\\mathbf {1}]}(\\\\mathbf {s}) + \\\\mu (\\\\mathbf {1}^\\\\top \\\\mathbf {s} - \\\\epsilon ) \\\\nonumber \\\\\\\\= & \\\\sum _{i} \\\\left( \\\\frac{1}{2} (s_i - a_i)^2 + \\\\mathcal {I}_{[0,1]}(s_i) + \\\\mu s_i \\\\right) - \\\\mu \\\\epsilon ,$ where $\\\\mu \\\\ge 0$ is the dual variable.\",\n \"The minimizer to the above Lagrangian function (with respect to the variable $\\\\mathbf {s}$ ) is $\\\\mathbf {s} = P_{[\\\\mathbf {0}, \\\\mathbf {1}]} (\\\\mathbf {a} - \\\\mu \\\\mathbf {1} ),$ where $P_{[\\\\mathbf {0}, \\\\mathbf {1}]}$ is taken elementwise.\",\n \"Besides the stationary condition (REF ), other KKT conditions for solving problem (REF ) are $& \\\\mu (\\\\mathbf {1}^\\\\top \\\\mathbf {s} - \\\\epsilon ) = 0 , \\\\\\\\& \\\\mu \\\\ge 0, \\\\\\\\& \\\\mathbf {1}^\\\\top \\\\mathbf {s} \\\\le \\\\epsilon .\",\n \"$ If $\\\\mu > 0$ , then the solution to problem (REF ) is given by (REF ), where the dual variable $\\\\mu $ is determined by (REF ) and (REF ) $\\\\mathbf {1}^T P_{[\\\\mathbf {0}, \\\\mathbf {1}]} [\\\\mathbf {a} - \\\\mu \\\\mathbf {1}] = \\\\epsilon ,~\\\\text{and}~ \\\\mu > 0.$ If $\\\\mu = 0$ , then the solution to problem (REF ) is given by (REF ) and (), $\\\\mathbf {s} = P_{[\\\\mathbf {0}, \\\\mathbf {1}]} (\\\\mathbf {a}),~\\\\text{and}~ \\\\mathbf {1}^\\\\top \\\\mathbf {s} \\\\le \\\\epsilon ,$ The proof is complete.\",\n \"$\\\\square $ In the projection operation (REF ), one might need to solve the scalar equation $\\\\mathbf {1}^T P_{[\\\\mathbf {0}, \\\\mathbf {1}]} [\\\\mathbf {a} - \\\\mu \\\\mathbf {1}] = \\\\epsilon $ with respect to the dual variable $\\\\mu $ .\",\n \"This can be accomplished by applying the bisection method [2], [12] over $\\\\mu \\\\in [\\\\min (\\\\mathbf {a}-\\\\mathbf {1}), \\\\max (\\\\mathbf {a}) ]$ .\",\n \"That is because $\\\\mathbf {1}^T P_{[\\\\mathbf {0}, \\\\mathbf {1}]} [\\\\mathbf {a} - \\\\max (\\\\mathbf {a}) \\\\mathbf {1}] \\\\le \\\\epsilon $ and $\\\\mathbf {1}^T P_{[\\\\mathbf {0}, \\\\mathbf {1}]} [\\\\mathbf {a} - \\\\min (\\\\mathbf {a} - \\\\mathbf {1}) \\\\mathbf {1} ] \\\\ge \\\\epsilon $ , where $\\\\max $ and $\\\\min $ return the largest and smallest entry of a vector.\",\n \"We remark that the bisection method converges in the logarithmic rate given by $\\\\log _2 {[ (\\\\max (\\\\mathbf {a}) -\\\\min (\\\\mathbf {a}-\\\\mathbf {1}) )/\\\\xi ]}$ for the solution of $\\\\xi $ -error tolerance.\",\n \"We summarize the PGD topology attack in Algorithm REF .\",\n \"PGD topology attack on GNN [1] Input: $\\\\mathbf {s}^{(0)}$ , $\\\\epsilon > 0$ , learning rate $\\\\eta _t$ , and iterations $T$ $t = 1,2,\\\\ldots , T$ gradient descent: $\\\\mathbf {a}^{(t)} = \\\\mathbf {s}^{(t-1)} - \\\\eta _{t} \\\\nabla f(\\\\mathbf {s}^{(t-1)}) $ call projection operation in (REF ) call Algorithm REF to return $\\\\mathbf {s}^*$ , and the resulting $\\\\mathbf {A}^\\\\prime $ in (REF ).\"\n ],\n [\n \"Min-max Topology Attack\",\n \"We next solve the problem of min-max attack generation in (REF ).\",\n \"By convex relaxation on the Boolean variables, we obtain the following continuous optimization problem $\\\\hspace*{-10.84006pt}\\\\begin{array}{cc}\\\\displaystyle \\\\operatornamewithlimits{\\\\text{minimize}}_{ \\\\mathbf {s}\\\\in \\\\mathcal {S} } \\\\displaystyle \\\\operatornamewithlimits{\\\\text{maximize}}_{\\\\mathbf {W}} \\\\, f(\\\\mathbf {s}, \\\\mathbf {W}) = \\\\sum _{i \\\\in \\\\mathcal {V}}f_i(\\\\mathbf {s} , \\\\mathbf {W} ; \\\\mathbf {A}, \\\\lbrace \\\\mathbf {x}_i\\\\rbrace , y_i ) ,\\\\end{array}$ where $\\\\mathcal {S}$ has been defined in (REF ).\",\n \"We solve problem (REF ) by first-order alternating optimization [14], [15], where the inner maximization is solved by gradient ascent, and the outer minimization is handled by PGD same as (REF ).\",\n \"We summarize the min-max topology attack in Algorithm REF .\",\n \"We remark that one can perform multiple maximization steps within each iteration of alternating optimization.\",\n \"This strikes a balance between the computation efficiency and the convergence accuracy [5], [18].\",\n \"Min-max topology attack to solve (REF ) [1] Input: given $\\\\mathbf {W}^{(0)}$ , $\\\\mathbf {s}^{(0)}$ , learning rates $\\\\beta _t$ and $\\\\eta _t$ , and iteration numbers $T$ $t = 1,2,\\\\ldots , T$ inner maximization over $\\\\mathbf {W}$ : given $\\\\mathbf {s}^{(t-1)}$ , obtain $\\\\mathbf {W}^{t} = \\\\mathbf {W}^{t-1} + \\\\beta _t \\\\nabla _{\\\\mathbf {W}} f (\\\\mathbf {s}^{t-1},\\\\mathbf {W}^{t-1})$ outer minimization over $\\\\mathbf {s}$ : given $\\\\mathbf {W}^{(t)}$ , running PGD (REF ), where $\\\\hat{\\\\mathbf {g}}_t = \\\\nabla _{\\\\mathbf {s}} f (\\\\mathbf {s}^{t-1},\\\\mathbf {W}^{t})$ call Algorithm REF to return $\\\\mathbf {s}^*$ , and the resulting $\\\\mathbf {A}^\\\\prime $ in (REF ).\"\n ],\n [\n \"Robust Training for GNNs\",\n \"With the aid of first-order attack generation methods, we now introduce our adversarial training for GNNs via robust optimization.\",\n \"Similar formulation is also used in [16].\",\n \"In adversarial training, we solve a min-max problem for robust optimization: $\\\\begin{array}{cc}\\\\displaystyle \\\\operatornamewithlimits{\\\\text{minimize}}_{\\\\mathbf {W}} \\\\operatornamewithlimits{\\\\text{maximize}}_{ \\\\mathbf {s} \\\\in \\\\mathcal {S}} \\\\, -f(\\\\mathbf {s}, \\\\mathbf {W}) ,\\\\end{array}$ where $ f(\\\\mathbf {x}, \\\\mathbf {W})$ denotes the attack loss specified in (REF ).\",\n \"Following the idea of adversarial training for image classifiers in [16], we restrict the loss function $f$ as the CE-type loss.\",\n \"This formulation tries to minimize the training loss at the presence of topology perturbations.\",\n \"We note that problems (REF ) and (REF ) share a very similar min-max form, however, they are not equivalent since the loss $f$ is neither convex with respect to $\\\\mathbf {s}$ nor concave with respect to $\\\\mathbf {W}$ , namely, lacking saddle point property [2].\",\n \"However, there exists connection between (REF ) and (REF ); see Proposition REF .\",\n \"Proposition 2 Given a general attack loss function $f$ , problem (REF ) is equivalent to $\\\\operatornamewithlimits{\\\\text{maximize}}_{\\\\mathbf {W}} \\\\operatornamewithlimits{\\\\text{minimize}}_{ \\\\mathbf {s} \\\\in \\\\mathcal {S}} \\\\, f(\\\\mathbf {s}, \\\\mathbf {W}),$ which further yields $(\\\\ref {eq: min_max_sol_same}) \\\\le (\\\\ref {eq: robust_attack_cont})$ .\",\n \"Proof: By introducing epigraph variable $p$ [2], problem (REF ) can be rewritten as $\\\\begin{array}{ll}\\\\displaystyle \\\\operatornamewithlimits{\\\\text{minimize}}_{\\\\mathbf {W}, p} & p \\\\\\\\\\\\operatornamewithlimits{\\\\text{subject to}}& -f(\\\\mathbf {s}, \\\\mathbf {W}) \\\\le p, \\\\forall \\\\mathbf {s} \\\\in \\\\mathcal {S}.\\\\end{array}$ By changing variable $q \\\\mathrel {\\\\mathop :}=- p$ , problem (REF ) is equivalent to $\\\\begin{array}{ll}\\\\displaystyle \\\\operatornamewithlimits{\\\\text{maximize}}_{\\\\mathbf {W}, q} & q \\\\\\\\\\\\operatornamewithlimits{\\\\text{subject to}}&f(\\\\mathbf {s}, \\\\mathbf {W}) \\\\ge q, \\\\forall \\\\mathbf {s} \\\\in \\\\mathcal {S}.\\\\end{array}$ By eliminating the epigraph variable $q$ , problem (REF ) becomes (REF ).\",\n \"By max-min inequality [2], we finally obtain that $\\\\operatornamewithlimits{\\\\text{maximize}}_{\\\\mathbf {W}} \\\\operatornamewithlimits{\\\\text{minimize}}_{ \\\\mathbf {s} \\\\in \\\\mathcal {S}} \\\\, f(\\\\mathbf {s}, \\\\mathbf {W}) \\\\le \\\\operatornamewithlimits{\\\\text{minimize}}_{ \\\\mathbf {s}\\\\in \\\\mathcal {S} } \\\\operatornamewithlimits{\\\\text{maximize}}_{\\\\mathbf {W}} \\\\, f(\\\\mathbf {s}, \\\\mathbf {W}).$ The proof is now complete.\",\n \"$\\\\square $ We summarize the robust training algorithm in Algorithm for solving problem (REF ).\",\n \"Similar to Algorithm REF , one usually performs multiple inner minimization steps (with respect to $\\\\mathbf {s}$ ) within each iteration $t$ to have a solution towards minimizer during alternating optimization.\",\n \"This improves the stability of convergence in practice [18], [16].\",\n \"Robust training for solving problem (REF ) [1] Input: given $\\\\mathbf {W}^{(0)}$ , $\\\\mathbf {s}^{(0)}$ , learning rates $\\\\beta _t$ and $\\\\eta _t$ , and iteration numbers $T$ $t = 1,2,\\\\ldots , T$ inner minimization over $\\\\mathbf {s}$ : given $\\\\mathbf {W}^{(t-1)}$ , running PGD (REF ), where $\\\\hat{\\\\mathbf {g}}_t = \\\\nabla _{\\\\mathbf {s}} f (\\\\mathbf {s}^{t-1},\\\\mathbf {W}^{t-1})$ outer maximization over $\\\\mathbf {W}$ : given $\\\\mathbf {s}^{(t)}$ , obtain $\\\\mathbf {W}^{t} = \\\\mathbf {W}^{t-1} + \\\\beta _t \\\\nabla _{\\\\mathbf {W}} f (\\\\mathbf {s}^{t},\\\\mathbf {W}^{t-1})$ return $\\\\mathbf {W}^{T}$ .\"\n ],\n [\n \"Experiments\",\n \"In this section, we present our experimental results for both topology attack and defense methods on a graph convolutional networks (GCN) [10].\",\n \"We demonstrate the misclassification rate and the convergence of the proposed 4 attack methods: negative cross-entropy loss via PGD attack (CE-PGD), CW loss via PGD attack (CW-PGD), negative cross-entropy loss via min-max attack (CE-min-max), CW loss via min-max attack (CW-min-max).\",\n \"We then show the improved robustness of GCN by leveraging our proposed robust training against topology attacks.\"\n ],\n [\n \"Experimental Setup\",\n \"We evaluate our methods on two well-known datasets: Cora and Citeseer [19].\",\n \"Both datasets contain unweighted edges which can be generated as symmetric adjacency matrix $\\\\mathbf {A} $ and sparse bag-of-words feature vectors which can be treated the input of GCN.\",\n \"To train the model, all node feature vectors are fed into GCN but with only 140 and 120 labeled nodes for Cora and Citeseer, respectively.\",\n \"The number of test labeled nodes is 1000 for both datasets.\",\n \"At each experiment, we repeat 5 times based on different splits of training/testing nodes and report mean $\\\\pm $ standard deviation of misclassification rate (namely, 1 $-$ prediction accuracy) on testing nodes.\"\n ],\n [\n \"Attack Performance\",\n \"We compare our four attack methods (CE-PGD, CW-PGD, CE-min-max, CW-min-max) with DICE (`delete edges internally, connect externally’) [21], Meta-Self attack [26] and greedy attack, a variant of Meta-Self attack without weight re-training for GCN.\",\n \"The greedy attack is considered as a fair comparison with our CE-PGD and CW-PGD attacks, which are generated on a fixed GCN without weight re-training.\",\n \"In min-max attacks (CE-min-max and CW-min-max), we show misclassification rates against both natural and retrained models from Algorithm REF , and compare them with the state-of-the-art Meta-Self attack.\",\n \"For a fair comparison, we use the same performance evaluation criterion in Meta-Self, testing nodes' predicted labels (not their ground-truth label) by an independent pre-trained model that can be used during the attack.\",\n \"In the attack problems (REF ) and (REF ), unless specified otherwise the maximum number of perturbed edges is set to be $5\\\\%$ of the total number of existing edges in the original graph.\",\n \"In Algorithm REF , we set the iteration number of random sampling as $K = 20$ and choose the perturbed topology with the highest misclassification rate which also satisfies the edge perturbation constraint.\",\n \"In Table REF , we present the misclassification rate of different attack methods against both natural and retrained model from (REF ).\",\n \"Here we recall that the retrained model arises due to the scenario of attacking an interactive GCN with re-trainable weights (Algorithm REF ).\",\n \"For comparison, we also show the misclassification rate of a natural model with the true topology (denoted by `clean').\",\n \"As we can see, to attack the natural model, our proposed attacks achieve better misclassification rate than the existing methods.\",\n \"We also observe that compared to min-max attacks (CE-min-max and CW-min-max), CE-PGD and CW-PGD yield better attacking performance since it is easier to attack a pre-defined GCN.\",\n \"To attack the model that allows retraining, we set 20 steps of inner maximization per iteration of Algorithm REF .\",\n \"The results show that our proposed min-max attack achieves very competitive performance compared to Meta-Self attack.\",\n \"Note that evaluating the attack performance on the retrained model obtained from (REF ) is not quite fair since the retrained weights could be sub-optimal and induce degradation in classification.\",\n \"Table: Misclassification rates (%\\\\%) under 5%5\\\\% perturbed edgesIn Fig.\",\n \"REF , we present the CE-loss and the CW-loss of the proposed topology attacks against the number of iterations in Algorithm REF .\",\n \"Here we choose $T = 200$ and $\\\\eta _t = 200/\\\\sqrt{t}$ .\",\n \"As we can see, the method of PGD converges gracefully against iterations.\",\n \"This verifies the effectiveness of the first-order optimization based attack generation method.\",\n \"Figure: CE-PGD and CW-PGD attack losses on Cora and Citeseer datasets.\"\n ],\n [\n \"Defense Performance\",\n \"In what follows, we invoke Algorithm to generate robust GCN via adversarial training.\",\n \"We set $T = 1000$ , $\\\\beta _t = 0.01$ and $\\\\eta _t = 200/\\\\sqrt{t}$ .\",\n \"We run 20 steps for inner minimization.\",\n \"Inspired by [16], we increase the hidden units from 16 to 32 in order to create more capacity for this more complicated classifier.\",\n \"Initially, we set the maximum number of edges we can modify as $5\\\\%$ of total existing edges.\",\n \"In Figure REF , we present convergence of our robust training.\",\n \"As we can see, the loss drops reasonably and the $1,000$ iterations are necessary for robust training rather than normal training process which only need 200 iterations.\",\n \"We also observe that our robust training algorithm does not harm the test accuracy when $\\\\epsilon = 5\\\\%$ , but successfully improves the robustness as the attack success rate drops from $28.0\\\\%$ to $22.0\\\\%$ in Cora dataset as shown in Table REF , After showing the effectiveness of our algorithm, we explore deeper in adversarial training on GCN.\",\n \"We aim to show how large $\\\\epsilon $ we can use in robust training.\",\n \"So we set $\\\\epsilon $ from $5\\\\%$ to $20\\\\%$ and apply CE-PGD attack following the same $\\\\epsilon $ setting.\",\n \"The results are presented in Table REF .\",\n \"Note that when $\\\\epsilon =0$ , the first row shows misclassification rates of test nodes on natural graph as the baseline for lowest misclassification rate we can obtain; the first column shows the CE-PGD attack misclassification rates of natural model as the baseline for highest misclassification rate we can obtain.\",\n \"We can conclude that when a robust model trained under an $\\\\epsilon $ constraint, the model will gain robustness under this $\\\\epsilon $ distinctly.\",\n \"Considering its importance to keep the original graph test performance, we suggest generating robust model under $\\\\epsilon = 0.1$ .\",\n \"Moreover, please refer to Figure REF that a) our robust trained model can provide universal defense to CE-PGD, CW-PGD and Greedy attacks; b) when increasing $\\\\epsilon $ , the difference between both test accuracy and CE-PGD attack accuracy increases substantially, which also implies the robust model under larger $\\\\epsilon $ is harder to obtain.\",\n \"Table: Misclassification rates (%\\\\%) of robust training (smaller is better for defense task) with at most 5%5\\\\% of edge perturbations.\",\n \"𝐀\\\\mathbf {A} means the natural graph, 𝐀 ' \\\\mathbf {A}^\\\\prime means the generated adversarial graph under ϵ=5%\\\\epsilon = 5\\\\%.\",\n \"𝐗/M\\\\mathbf {X} / M means the misclassification rate of using model MM on graph 𝐗\\\\mathbf {X}.Table: Misclassification rates (%\\\\%) of CE-PGD attack against robust training model versus (smaller is better) different ϵ\\\\epsilon (%\\\\%) on Cora dataset.\",\n \"Hereϵ=0\\\\epsilon =0 in training means natural model and ϵ=0\\\\epsilon =0 in attack means unperturbed topology.Figure: Robust training loss on Cora and Citeseer datasets.Figure: Test accuracy of robust model (no attack),CE-PGD attack against robust model, CW-PGD attack against robust model, Greedy attack against robust model and CE-PGD attack against natural model for different ϵ\\\\epsilon used in robust training and test on Cora dataset.\"\n ],\n [\n \"Conclusion\",\n \"In this paper, we first introduce an edge perturbation based topology attack framework that overcomes the difficulty of attacking discrete graph structure data from a first-order optimization perspective.\",\n \"Our extensive experiments show that with only a fraction of edges changed, we are able to compromise state-of-the-art graph neural networks model noticeably.\",\n \"Additionally, we propose an adversarial training framework to improve the robustness of GNN models based on our attack methods.\",\n \"Experiments on different datasets show that our method is able to improve the GNN model's robustness against both gradient based and greedy search based attack methods without classification performance drop on original graph.\",\n \"We believe that this paper provides potential means for theoretical study and improvement of the robustness of deep learning models on graph data.\"\n ],\n [\n \"Acknowledgments\",\n \"This work is supported by Air Force Research Laboratory FA8750-18-2-0058 and the MIT-IBM Watson AI Lab.\"\n ]\n]"},"id":{"kind":"string","value":"1906.04214"}}},{"rowIdx":2594,"cells":{"text":{"kind":"list like","value":[["Introducing the Hearthstone-AI Competition"],["Abstract The Hearthstone AI framework and competition motivates the development of artificial intelligence agents that can play collectible card games.","A special feature of those games is the high variety of cards, which can be chosen by the players to create their own decks.","In contrast to simpler card games, the value of many cards is determined by their possible synergies.","The vast amount of possible decks, the randomness of the game, as well as the restricted information during the player's turn offer quite a hard challenge for the development of game-playing agents.","This short paper introduces the competition framework and goes into more detail on the problems and challenges that need to be faced during the development process."],["Introduction","The development of artificial intelligence (AI) was often guided by the plethora of available real world applications.","The recent success of AI agents such as AlphaGo, brought game AI back into the center of attention for media and researchers alike.","Games pose challenging and often well-balanced problems demanding the development of new agent architectures and allowing their evaluation based on their success in playing the game.","Due to their competitive nature, games offer an ideal test-bed for the comparison of multiple agents under differing conditions.","Game based competitions and benchmarks motivated researchers to create specialized agents in games of many different genres, such as Chess [1], Go [2], Poker [3], Pac-Man [4], Starcraft [5] and many more.","While these focused on the development of an agent for a single game, other competitions and frameworks tried to generalize the solutions to a broader scope.","The Arcade Learning Environment framework (ALE) [6] as well as the General Video Game AI framework (GVGAI) [7] test an agent's success on a wide range of games.","All of these benchmarks pose unique demands on the agent's planning and reasoning capabilities, while there accessibility ensures a simple generation and evaluation of new agents.","In this paper we introduce the Hearthstone AI competition, because we believe that it is an excellent addition to the set of currently available benchmarks.","The focus of this competition is the development of autonomously playing agents in the context of the online collectible card game Hearthstone.","Interesting features of collectible card games include, but are not limited to: Partial observable state space: Critical information is hidden from the player.","The agent typically does not know which cards it will draw and is unaware of the opponent's deck and hand cards.","This is especially relevant when estimating the risk of an action, since this often depends on the current options of our opponent.","High complexity: Hearthstone currently features more than 2000 different cards.","This high amount and the number of their unique effects drastically increases the game-tree complexity.","Randomness: In contrast to similar deck-building card-games, card effects in Hearthstone often involve randomness.","This makes it particularly difficult to plan ahead, such that the agent needs to continuously adapt its strategy according to the observed result.","Deck-building: A deck in Hearthstone consists of 30 cards, which do not necessarily need to be unique.","Card synergies mean that certain combinations of cards often have a stronger effect than playing the respective cards separately.","Exploiting these synergies is a very complex task but allows to play cards to their full potential.","Dynamic Meta-Game: The odds of winning the next game depend not only on the player's skill or the current deck, but also on the rate of other decks being played.","The analysis of this meta-game can be crucial for creating or choosing the next deck to be played.","Finally, a deep understanding of the meta-game can be used to predict future enemy moves based on previously seen cards and select appropriate actions based on them.","In the following we will give a short introduction to Hearthstone (hearthstone) and present the competition framework (competition).","Competition tracks of the Hearthstone AI'19 competition are highlighted in after which an outline of future competition tracks is presented (competition-tracks).","More information on the competition, results of the 2018's installment and additional resources for research on Hearthstone can be found at: ....... http://www.ci.ovgu.de/Research/HearthstoneAI.html We invite all developers to participate in this exciting research topic and hope to receive many interesting submissions.","Figure: Elements of the Hearthstone game board: (1) weapon slot (2) hero (bottom: player, top: opponent) (3) opponent's minions, (4) player's minions, (5) hero power, (6) hand cards, (7) mana, (8) decks, (9) history"],["Hearthstone: Heroes of Warcraft","Hearthstone is a turn-based digital collectible card game developed and published by Blizzard Entertainment [8].","Players compete in one versus one duels using self-constructed decks belonging to one hero out of nine available hero-classes.","In those matches players try to beat their opponents by reducing their starting health from 30 to 0.","This can be achieved by playing cards from the hand onto the game board at the cost of mana.","Played cards can be used to inflict damage to the opponent's hero or to destroy cards on his side of the game board.","The amount of mana available to the player increases every turn (up to a maximum of 10).","More mana gives access to increasingly powerful cards and increases the complexity of turn while the game progresses.","At the beginning of each turn the player draws a new card until his deck is empty, in which case he receives a step-wise increasing amount of fatigue-damage.","The standard game board is shown in fig:game-board.","Players need to construct their own decks to play the game.","Those consist of 30 cards, which can be chosen out of more than 2000 currently available cards.","Cards and game mechanics are added in regular updates.","Each card bears unique effects, which the players can use to their advantage.","Additionally, each player chooses a hero, which gives access to a class specific pool of cards and hero power.","Figure: General card types: cards include (1) mana cost, (2) attack damage, (3) health/durability, (4) and special effects.Cards can be of the type minion, spell, or weapon.","fig:cardtypes shows one example of each card type.","Minion cards assist and fight on behalf of the hero.","They usually have an attack, health, and mana cost-value, as well as a short ability text.","Furthermore, minions can belong to a special minion type, which is the basis for many synergy effects.","Once played, they can attack the enemies side of the board in every consecutive turn to inflict damage on either the opponent's minions or hero.","Attacking a target also reduces the attacker's health by the target's attack value.","In case any minion's health drops to zero, it is removed from the board and put into its player's graveyard.","Spell cards can be cast at the cost of mana to activate various abilities and are discarded after use.","They can have a wide range of effects, e.g.","raising a minion's attack or inflicting damage to a random minion.","Secrets, which are a special kind of spells, can be played without immediately activating their effect.","After a trigger condition was fulfilled, the secret will be activated.","Once activated, the secret is removed from the board.","Weapon cards are directly equipped to the player's hero and enable him to attack.","Their durability value limits the number of attacks till the weapon breaks.","Only one weapon can be equipped at the same time.","Hearthstone decks are often created around a common theme.","Multiple cards that positively influence each other can create strong synergies and increase the value of each card in context of its deck.","For this reason, the value of a single card highly depends on the player's hand, current elements on the board, and the deck in general.","Common examples are minion cards of the same type, e.g.","”Murloc“, which give each other additional advantages, e.g.","an attack boost.","Each of these minions is comparatively weak, but their value increases when they are played together.","Generated decks can be categorized into three major categories: aggro, mid-range, and control.","Aggro decks build on purely offensive strategies, which often include a lot of minions.","Control decks try to win in the long run by preventing the opponent's strategy and dominating the game situation.","The playing style of mid-range decks is between aggro and control.","They try to counter early attacks to dominate the game board with high-cost minions in the middle of the game.","Game length and branching factor can be dependent on the player's decks in the current game.","Some decks try to play single high-cost cards, whereas others build on versatile combinations.","The complexity of each turn and the uncertainty faced during the game makes Hearthstone a challenging problem for AI research."],["Hearthstone-AI Competition Framework","The Hearthstone-AI competition is based on the community driven simulator Sabberstone.","This framework is written in C# and the competition extends the original framework by multiple helper classes which provide an agent simple means of accessing the current game state limited to variables that would have been observable to a human player.","More specifically, each agent needs to inherit from the AbstractAgent class.","The included functions InitializeAgent and FinalizeAgent can be used to load and store information at the beginning and end of each session.","Additionally, InitializeGame and FinalizeGame are called at the beginning and end of each simulated game.","These functions can be used to setup a strategy or updating it based on the games' outcome.","During a game each time the agent needs to choose an action its GetMove function is called.","The agent is given a POGame object representing the partial observation of the current game-state.","It contains information about the visible part of the game board, a set of remaining cards in its deck, its hand cards, and the number of cards in their opponents' hand.","Furthermore, the opponent's deck as well as its hand cards are replaced by dummy-cards to ensure that this information remains hidden to the agent.","The agent is ensured 60s of computation time to step-wise return a set of actions and concluding its turn.","In case the turn was not ended by the agent, the returned action will be processed irreversibly and an updated game-state will be returned to the agent while asking for its next action.","Actions of both players are applied until a winner can be determined or a maximum number of turns (default = 50) is exceeded.","In the latter the game ends with a draw.","The POGameHandler class controls the simulation of multiple games and reports the result of these simulations in terms of a GameStats object.","The number of wins, draws, and loses as well as the total and average response times per agent are tracked and reported at the end of a simulation session."],["Competition Tracks","During the first years of the Hearthstone-AI competition two tracks will be open for entry.","We plan to extend this list in the following years to give users some time to accommodate with the framework and the game itself.","The following two tracks will be open for submission in the Hearthstone-AI'19: Premade Deck Playing”-track In the “Premade Deck Playing”-track participants will receive a list of six decks and play out all combinations against each other.","Only three of the six decks will be known to the developers before the final submission.","Determining and using the characteristics of player’s and the opponent’s deck to the player’s advantage will help in winning the game.","The long-term goal of this track will be the development of an agent that is capable of playing any deck.","User Created Deck Playing-track The “User Created Deck Playing”-track invites all participants to create their own decks or to choose from the vast amount of decks available online.","Finding combinations of decks and agents that can consistently beat others will play a key role in this competition track.","Additionally, it gives the participants the chance to optimize the agent’s strategy to the characteristics of their chosen deck.","A round robin tournament will be used to determine the average win-rate of each agent and rank them accordingly.","In case this process becomes unfeasible due to a large number of participants, we will split submissions into multiple sub-tournaments to determine the best performing agents among them and use a round robin tournament to determine the winner of the competition.","Matches will be repeated multiple times to accommodate for the randomness in the card draw.","In order to support incremental improvements of the agents' performance, we plan to make all submissions publicly available on the competition website after the competition evaluation has been completed."],["Conclusions and Future Plans","Previous work on collectible card games has been scattered on different games and frameworks.","With this competition we want to provide a unified way to develop and compare AI approaches on multiple collectible card game related tasks.","While current competition tracks focus on the agents' basic game playing capabilties, we like to cover various different tasks in future installments of this competition.","Specifically, we plan to implement the following future tracks: Deck-building: As soon as agents are reasonably skilled in playing a given deck we want to further explore the deck building process.","This is currently considered a highly creative task in which many game characteristics need to be considered to build outstanding decks.","Draft mode deck-building: A special form of deck building is the draft mode.","Here, an agent is presented 3 cards at a time of which it needs to choose one to include it in its deck.","This process is repeated until the deck is filled.","Strategic planning and estimating the value of each card are important to exploit synergies and building a competitive deck.","Game balancing/Card generation: During the development process of a collectible card game much of the time is put into the generation of new game mechanics and cards, since these need to fit and complement the current card pool.","A future track will aim to explore these balancing tasks in more detail.","We hope that this short introduction of our competition framework motivates further research in this very interesting topic.","More information on how to participate in the Hearthstone-AI'19 competition, results of the 2018's installment and additional resources for research on Hearthstone can be found at: ....... http://www.ci.ovgu.de/Research/HearthstoneAI.html"],["Acknowledgement","We would like to thank all who contributed to the Sabberstone framework on which this competition is based on.","Special thanks goes to darkfriend77 and Milva who are currently organizing the framework's ongoing development process.","This competition would not have been possible without them.","[Figure: NO_CAPTION [Figure: NO_CAPTION"]],"string":"[\n [\n \"Introducing the Hearthstone-AI Competition\"\n ],\n [\n \"Abstract The Hearthstone AI framework and competition motivates the development of artificial intelligence agents that can play collectible card games.\",\n \"A special feature of those games is the high variety of cards, which can be chosen by the players to create their own decks.\",\n \"In contrast to simpler card games, the value of many cards is determined by their possible synergies.\",\n \"The vast amount of possible decks, the randomness of the game, as well as the restricted information during the player's turn offer quite a hard challenge for the development of game-playing agents.\",\n \"This short paper introduces the competition framework and goes into more detail on the problems and challenges that need to be faced during the development process.\"\n ],\n [\n \"Introduction\",\n \"The development of artificial intelligence (AI) was often guided by the plethora of available real world applications.\",\n \"The recent success of AI agents such as AlphaGo, brought game AI back into the center of attention for media and researchers alike.\",\n \"Games pose challenging and often well-balanced problems demanding the development of new agent architectures and allowing their evaluation based on their success in playing the game.\",\n \"Due to their competitive nature, games offer an ideal test-bed for the comparison of multiple agents under differing conditions.\",\n \"Game based competitions and benchmarks motivated researchers to create specialized agents in games of many different genres, such as Chess [1], Go [2], Poker [3], Pac-Man [4], Starcraft [5] and many more.\",\n \"While these focused on the development of an agent for a single game, other competitions and frameworks tried to generalize the solutions to a broader scope.\",\n \"The Arcade Learning Environment framework (ALE) [6] as well as the General Video Game AI framework (GVGAI) [7] test an agent's success on a wide range of games.\",\n \"All of these benchmarks pose unique demands on the agent's planning and reasoning capabilities, while there accessibility ensures a simple generation and evaluation of new agents.\",\n \"In this paper we introduce the Hearthstone AI competition, because we believe that it is an excellent addition to the set of currently available benchmarks.\",\n \"The focus of this competition is the development of autonomously playing agents in the context of the online collectible card game Hearthstone.\",\n \"Interesting features of collectible card games include, but are not limited to: Partial observable state space: Critical information is hidden from the player.\",\n \"The agent typically does not know which cards it will draw and is unaware of the opponent's deck and hand cards.\",\n \"This is especially relevant when estimating the risk of an action, since this often depends on the current options of our opponent.\",\n \"High complexity: Hearthstone currently features more than 2000 different cards.\",\n \"This high amount and the number of their unique effects drastically increases the game-tree complexity.\",\n \"Randomness: In contrast to similar deck-building card-games, card effects in Hearthstone often involve randomness.\",\n \"This makes it particularly difficult to plan ahead, such that the agent needs to continuously adapt its strategy according to the observed result.\",\n \"Deck-building: A deck in Hearthstone consists of 30 cards, which do not necessarily need to be unique.\",\n \"Card synergies mean that certain combinations of cards often have a stronger effect than playing the respective cards separately.\",\n \"Exploiting these synergies is a very complex task but allows to play cards to their full potential.\",\n \"Dynamic Meta-Game: The odds of winning the next game depend not only on the player's skill or the current deck, but also on the rate of other decks being played.\",\n \"The analysis of this meta-game can be crucial for creating or choosing the next deck to be played.\",\n \"Finally, a deep understanding of the meta-game can be used to predict future enemy moves based on previously seen cards and select appropriate actions based on them.\",\n \"In the following we will give a short introduction to Hearthstone (hearthstone) and present the competition framework (competition).\",\n \"Competition tracks of the Hearthstone AI'19 competition are highlighted in after which an outline of future competition tracks is presented (competition-tracks).\",\n \"More information on the competition, results of the 2018's installment and additional resources for research on Hearthstone can be found at: ....... http://www.ci.ovgu.de/Research/HearthstoneAI.html We invite all developers to participate in this exciting research topic and hope to receive many interesting submissions.\",\n \"Figure: Elements of the Hearthstone game board: (1) weapon slot (2) hero (bottom: player, top: opponent) (3) opponent's minions, (4) player's minions, (5) hero power, (6) hand cards, (7) mana, (8) decks, (9) history\"\n ],\n [\n \"Hearthstone: Heroes of Warcraft\",\n \"Hearthstone is a turn-based digital collectible card game developed and published by Blizzard Entertainment [8].\",\n \"Players compete in one versus one duels using self-constructed decks belonging to one hero out of nine available hero-classes.\",\n \"In those matches players try to beat their opponents by reducing their starting health from 30 to 0.\",\n \"This can be achieved by playing cards from the hand onto the game board at the cost of mana.\",\n \"Played cards can be used to inflict damage to the opponent's hero or to destroy cards on his side of the game board.\",\n \"The amount of mana available to the player increases every turn (up to a maximum of 10).\",\n \"More mana gives access to increasingly powerful cards and increases the complexity of turn while the game progresses.\",\n \"At the beginning of each turn the player draws a new card until his deck is empty, in which case he receives a step-wise increasing amount of fatigue-damage.\",\n \"The standard game board is shown in fig:game-board.\",\n \"Players need to construct their own decks to play the game.\",\n \"Those consist of 30 cards, which can be chosen out of more than 2000 currently available cards.\",\n \"Cards and game mechanics are added in regular updates.\",\n \"Each card bears unique effects, which the players can use to their advantage.\",\n \"Additionally, each player chooses a hero, which gives access to a class specific pool of cards and hero power.\",\n \"Figure: General card types: cards include (1) mana cost, (2) attack damage, (3) health/durability, (4) and special effects.Cards can be of the type minion, spell, or weapon.\",\n \"fig:cardtypes shows one example of each card type.\",\n \"Minion cards assist and fight on behalf of the hero.\",\n \"They usually have an attack, health, and mana cost-value, as well as a short ability text.\",\n \"Furthermore, minions can belong to a special minion type, which is the basis for many synergy effects.\",\n \"Once played, they can attack the enemies side of the board in every consecutive turn to inflict damage on either the opponent's minions or hero.\",\n \"Attacking a target also reduces the attacker's health by the target's attack value.\",\n \"In case any minion's health drops to zero, it is removed from the board and put into its player's graveyard.\",\n \"Spell cards can be cast at the cost of mana to activate various abilities and are discarded after use.\",\n \"They can have a wide range of effects, e.g.\",\n \"raising a minion's attack or inflicting damage to a random minion.\",\n \"Secrets, which are a special kind of spells, can be played without immediately activating their effect.\",\n \"After a trigger condition was fulfilled, the secret will be activated.\",\n \"Once activated, the secret is removed from the board.\",\n \"Weapon cards are directly equipped to the player's hero and enable him to attack.\",\n \"Their durability value limits the number of attacks till the weapon breaks.\",\n \"Only one weapon can be equipped at the same time.\",\n \"Hearthstone decks are often created around a common theme.\",\n \"Multiple cards that positively influence each other can create strong synergies and increase the value of each card in context of its deck.\",\n \"For this reason, the value of a single card highly depends on the player's hand, current elements on the board, and the deck in general.\",\n \"Common examples are minion cards of the same type, e.g.\",\n \"”Murloc“, which give each other additional advantages, e.g.\",\n \"an attack boost.\",\n \"Each of these minions is comparatively weak, but their value increases when they are played together.\",\n \"Generated decks can be categorized into three major categories: aggro, mid-range, and control.\",\n \"Aggro decks build on purely offensive strategies, which often include a lot of minions.\",\n \"Control decks try to win in the long run by preventing the opponent's strategy and dominating the game situation.\",\n \"The playing style of mid-range decks is between aggro and control.\",\n \"They try to counter early attacks to dominate the game board with high-cost minions in the middle of the game.\",\n \"Game length and branching factor can be dependent on the player's decks in the current game.\",\n \"Some decks try to play single high-cost cards, whereas others build on versatile combinations.\",\n \"The complexity of each turn and the uncertainty faced during the game makes Hearthstone a challenging problem for AI research.\"\n ],\n [\n \"Hearthstone-AI Competition Framework\",\n \"The Hearthstone-AI competition is based on the community driven simulator Sabberstone.\",\n \"This framework is written in C# and the competition extends the original framework by multiple helper classes which provide an agent simple means of accessing the current game state limited to variables that would have been observable to a human player.\",\n \"More specifically, each agent needs to inherit from the AbstractAgent class.\",\n \"The included functions InitializeAgent and FinalizeAgent can be used to load and store information at the beginning and end of each session.\",\n \"Additionally, InitializeGame and FinalizeGame are called at the beginning and end of each simulated game.\",\n \"These functions can be used to setup a strategy or updating it based on the games' outcome.\",\n \"During a game each time the agent needs to choose an action its GetMove function is called.\",\n \"The agent is given a POGame object representing the partial observation of the current game-state.\",\n \"It contains information about the visible part of the game board, a set of remaining cards in its deck, its hand cards, and the number of cards in their opponents' hand.\",\n \"Furthermore, the opponent's deck as well as its hand cards are replaced by dummy-cards to ensure that this information remains hidden to the agent.\",\n \"The agent is ensured 60s of computation time to step-wise return a set of actions and concluding its turn.\",\n \"In case the turn was not ended by the agent, the returned action will be processed irreversibly and an updated game-state will be returned to the agent while asking for its next action.\",\n \"Actions of both players are applied until a winner can be determined or a maximum number of turns (default = 50) is exceeded.\",\n \"In the latter the game ends with a draw.\",\n \"The POGameHandler class controls the simulation of multiple games and reports the result of these simulations in terms of a GameStats object.\",\n \"The number of wins, draws, and loses as well as the total and average response times per agent are tracked and reported at the end of a simulation session.\"\n ],\n [\n \"Competition Tracks\",\n \"During the first years of the Hearthstone-AI competition two tracks will be open for entry.\",\n \"We plan to extend this list in the following years to give users some time to accommodate with the framework and the game itself.\",\n \"The following two tracks will be open for submission in the Hearthstone-AI'19: Premade Deck Playing”-track In the “Premade Deck Playing”-track participants will receive a list of six decks and play out all combinations against each other.\",\n \"Only three of the six decks will be known to the developers before the final submission.\",\n \"Determining and using the characteristics of player’s and the opponent’s deck to the player’s advantage will help in winning the game.\",\n \"The long-term goal of this track will be the development of an agent that is capable of playing any deck.\",\n \"User Created Deck Playing-track The “User Created Deck Playing”-track invites all participants to create their own decks or to choose from the vast amount of decks available online.\",\n \"Finding combinations of decks and agents that can consistently beat others will play a key role in this competition track.\",\n \"Additionally, it gives the participants the chance to optimize the agent’s strategy to the characteristics of their chosen deck.\",\n \"A round robin tournament will be used to determine the average win-rate of each agent and rank them accordingly.\",\n \"In case this process becomes unfeasible due to a large number of participants, we will split submissions into multiple sub-tournaments to determine the best performing agents among them and use a round robin tournament to determine the winner of the competition.\",\n \"Matches will be repeated multiple times to accommodate for the randomness in the card draw.\",\n \"In order to support incremental improvements of the agents' performance, we plan to make all submissions publicly available on the competition website after the competition evaluation has been completed.\"\n ],\n [\n \"Conclusions and Future Plans\",\n \"Previous work on collectible card games has been scattered on different games and frameworks.\",\n \"With this competition we want to provide a unified way to develop and compare AI approaches on multiple collectible card game related tasks.\",\n \"While current competition tracks focus on the agents' basic game playing capabilties, we like to cover various different tasks in future installments of this competition.\",\n \"Specifically, we plan to implement the following future tracks: Deck-building: As soon as agents are reasonably skilled in playing a given deck we want to further explore the deck building process.\",\n \"This is currently considered a highly creative task in which many game characteristics need to be considered to build outstanding decks.\",\n \"Draft mode deck-building: A special form of deck building is the draft mode.\",\n \"Here, an agent is presented 3 cards at a time of which it needs to choose one to include it in its deck.\",\n \"This process is repeated until the deck is filled.\",\n \"Strategic planning and estimating the value of each card are important to exploit synergies and building a competitive deck.\",\n \"Game balancing/Card generation: During the development process of a collectible card game much of the time is put into the generation of new game mechanics and cards, since these need to fit and complement the current card pool.\",\n \"A future track will aim to explore these balancing tasks in more detail.\",\n \"We hope that this short introduction of our competition framework motivates further research in this very interesting topic.\",\n \"More information on how to participate in the Hearthstone-AI'19 competition, results of the 2018's installment and additional resources for research on Hearthstone can be found at: ....... http://www.ci.ovgu.de/Research/HearthstoneAI.html\"\n ],\n [\n \"Acknowledgement\",\n \"We would like to thank all who contributed to the Sabberstone framework on which this competition is based on.\",\n \"Special thanks goes to darkfriend77 and Milva who are currently organizing the framework's ongoing development process.\",\n \"This competition would not have been possible without them.\",\n \"[Figure: NO_CAPTION [Figure: NO_CAPTION\"\n ]\n]"},"id":{"kind":"string","value":"1906.04238"}}},{"rowIdx":2595,"cells":{"text":{"kind":"list like","value":[["Metrics for Learning in Topological Persistence"],["Abstract Persistent homology analysis provides means to capture the connectivity structure of data sets in various dimensions.","On the mathematical level, by defining a metric between the objects that persistence attaches to data sets, we can stabilize invariants characterizing these objects.","We outline how so called contour functions induce relevant metrics for stabilizing the rank invariant.","On the practical level, the stable ranks are used as fingerprints for data.","Different choices of contour lead to different stable ranks and the topological learning is then the question of finding the optimal contour.","We outline our analysis pipeline and show how it can enhance classification of physical activities data.","As our main application we study how stable ranks and contours provide robust descriptors of spatial patterns of atmospheric cloud fields."],["Modelling data spaces","Topological data analysis (TDA) and particularly its subfield persistent homology, or persistence, aim at quantifying the global connectivity structure of data sets [1], [2], [3].","Given a set of data points it is often possible to endow it with some reasonable notion of relation between points, e.g.","distance measure or correlation.","Study of the connectivity is facilitated by first combining points into larger entities called simplices.","A $k$ -simplex is a declared subset of $k+1$ related points from the data set.","Collection of simplices makes up a simplicial complex $C$ , namely it is a collection of certain subsets of the data.","Requirements are that if $\\sigma $ is a simplex in $C$ then any subset of $\\sigma $ is also a simplex in $C$ and that the intersection of two simplices is a simplex or the empty set.","Above we have described an abstract simplicial complex.","Simplices can always be realized geometrically in some $\\mathbf {R}^n$ as convex hulls of their vertices: 0-simplices as points, 1-simplices as line segments, 2-simplices as filled triangles, 3-simplices as filled tetrahedra etc.","Simplicial complex is hence a model of the relational structure in the data.","Relational structure can be modelled by a graph but graphs only consider pairwise relations between points.","In many cases it makes sense to use higher-dimensional connectivity instead modelled with simplices.","As a justification consider the example explained in Fig.","REF .","More fundamental reason is that the simpicial approach views data as spaces spanned by their points and enables the use of powerful mathematical machinery of algebraic topology for the analysis of these spaces, as will be outlined in the following section.","Figure: Simplicial model of social relations.","To model relations between k+1k+1 points it is reasonable to use kk-simplex for the purpose.","Here the relations between {Arya, Bran, Jon} is depicted by the 2-simplex represented by the purple triangle.","From the point of view of TDA the prominent feature of this data is the single loop structure, whereas a graph would see two loops (the closed path (Arya, Bran, Jon) spanning the other loop in this case).For persistence analysis we define the relation to be a function $R$ on the data with values in $\\mathbf {R}=[0,\\infty )$ , i.e.","$R(x,y) \\mapsto t \\in \\mathbf {R}$ for data points $x$ and $y$ .","Concretely we say that $k+1$ data points $x_i$ create a $k$ -simplex at scale $t$ if the points satisfy pairwise $R(x_i,x_j) \\le t.$ This construction is called the Vietoris-Rips simplicial complex at scale $t$ .","At fixed scale we can then study the connectivity structure.","As a standard example, when data is endowed with a distance measure, clustering at some fixed scale corresponds to the 0-dimensional connectivity by only looking at the connected components of the simplicial complex.","Simplicial complexes can also contain 1-dimensional connectivity information in the form of loops and holes (see Fig.","REF ), 2-dimensional information in the form of voids or cavities, etc.","These are collectively called topological features.","Persistence aims to quantify the topological features in a data set and use this information for data analysis.","Loop structure might signal about a recurrent dynamics of the phenomenon behind the data.","Various dimensional voids can mark lack of information and connectivity or insufficient data collection.","Finding such voids in data sets has aroused interest in different areas of data analysis community, see for example [4] and references therein.","As noted in [4], voids can also indicate non-allowed combinations of feature values of data vectors.","One immediate difficulty arises in the simplicial modelling above: what is the appropriate scale of $R$ to capture the connectivity in various dimensions of an arbitrary set of points?","Persistence circumvents this by forming simplicial complexes at all scales $t \\ge 0$ and capturing the evolution of topological features.","If a simplex is generated at scale $t$ it is then present at any subsequent scale and the simplicial complexes are connected by inclusions: $\\cdots \\subseteq C_a \\subseteq C_b \\subseteq C_c \\subseteq \\cdots $ for $\\dots \\le a \\le b \\le c \\le \\dots $ The end result of the modelling step is then a mapping called filtration, $(D,R) \\times \\mathbf {R} \\rightarrow (C_t,\\subseteq _t)_{t \\in \\mathbf {R}}$ , where $(D,R)$ denotes a data set with real-valued relation and $(C_t,\\subseteq _t)_{t \\in \\mathbf {R}}$ denotes an $\\mathbf {R}$ -parameterized sequence of simplicial complexes and inclusions."],["Algebraic fingerprinting","Filtration contains all the information about the relations in the data set on various scales.","It is therefore very complicated object for infering the global structure of data and simplification is thus necessary.","TDA employs tools from mathematical field of algebraic topology, essentially it uses homology of simplicial complexes which transforms the geometric information into algebraic information.","We will outline the algorithm for computing homology to illustrate its very implementable nature and to gain intuition on why we are interested in homology in data analysis.","For details into homology and its computation see [5], [6], [7].","For simplicity we fix the field of coefficients to be $\\mathbf {F}_2$ , the field with two elements 0 and 1.","Let $C$ be a simplicial complex and denote by $C_k$ its set of $k$ -simplices.","Concretely, $C_0$ consists of the points of the original data set.","1) Choose an ordering (starting from zero) on $C_0$ and use it to order elements in any simplex.","If $\\lbrace \\text{Arya}, \\text{Bran}, \\text{Jon}\\rbrace $ is a 2-simplex in Fig.","REF , fix the order in which the points are listed and denote this ordered simplex by $[\\text{Arya}, \\text{Bran}, \\text{Jon}].$ 2) For natural numbers $k$ and $0\\le i\\le k$ and a simplex $\\sigma $ in $C_k$ , define a function $d_i \\colon C_k \\rightarrow C_{k-1}$ such that $d_i(\\sigma )$ is a simplex in $C_{k-1}$ formed by removing from $\\sigma $ its $i$ -th element.","The ordering on $C_0$ was needed to specify the $i$ -th element in a simplex.","For example, $d_1([\\text{Arya}, \\text{Bran}, \\text{Jon}]) = [\\text{Arya}, \\text{Jon}].$ 3) For any natural number $k$ , let $\\Delta (C)_k$ be the vector space over $\\mathbf {F}_2$ with a base given by all simplices in $C_k$ .","An element $\\tau $ in $\\Delta (C)_k$ is then given by a linear combination $\\tau = \\sum _{\\sigma \\in C_k} t_\\sigma \\sigma , \\ t_\\sigma \\in \\mathbf {F}_2.$ The base for $\\Delta (C)_2$ of the simplicial complex in Fig.","REF would be [Arya, Bran, Jon] whereas [Arya, Bran]+[Bran, Jon]+[Arya, Jon] would be linear combination of three basis elements in $\\Delta (C)_1$ .","4) Define $\\partial _k \\colon \\Delta (C)_k \\rightarrow \\Delta (C)_{k-1}$ to be the linear function assigning to a base element given by a simplex $\\sigma $ in $C_k$ the linear combination $\\sum _{i=0}^{k}d_i(\\sigma )$ of $k$ -1-simplices.","The map $\\partial _k$ is called the boundary operator.","Then $\\partial _k([\\text{Arya}, \\text{Bran}, \\text{Jon}]) = [\\text{Bran}, \\text{Jon}]+[\\text{Arya}, \\text{Jon}]+[\\text{Arya}, \\text{Bran}].$ The boundary operator thus formalizes the intuition that $[\\text{Bran}, \\text{Jon}]+[\\text{Arya}, \\text{Jon}]+[\\text{Arya}, \\text{Bran}]$ forms the boundary of $[\\text{Arya}, \\text{Bran}, \\text{Jon}].$ Define $\\Delta (C)_{-1} = 0$ and $\\Delta (C)_k = 0$ for $k>m$ , where 0 denotes the zero vector space.","5) The boundary operators connect the various simplices of a simplicial complex together.","Computationally the matrices of boundary operators store the global connectivity information in their elements, with coefficient field $\\mathbf {F}_2$ these are just binary matrices.","Homology on degree $k$ of a simplicial complex $C$ (over coefficients $\\mathbf {F}_2$ ) is then defined as a quotient vector space: $H_k(C)= \\frac{\\text{kernel of }\\partial _k\\colon \\Delta (C)_k \\rightarrow \\Delta (C)_{k-1}}{\\text{image of }\\partial _{k+1}\\colon \\Delta (C)_{k+1}\\rightarrow \\Delta (C)_{k}}, \\ \\ \\text{for } k \\ge 0.$ As noted in step 4) above, some 1-simplices might form the boundary of a 2-simplex.","Some 1-simplices on the other hand might form the boundary of an actual hole in the simplicial complex as in Fig.","REF .","Similarly some $k$ -1-simplices might form the boundary of a $k$ -simplex and some might form the boundary of a $k$ -dimensional hole.","By its definition homology quotients out linear combinations of simplices that are boundaries and we are left with those that actually represent linearly independent $k$ -dimensional holes in the complex.","For $k=0$ , $H_0$ measures the number of linearly independent points that make up boundaries of 1-simplices, effectively the number of connected components.","Homology thus gives us exactly the global connectivity information of the relational structure of data that we seek.","The full complexity of a filtration is now simplified by applying homology on degree $k$ .","Each simplicial complex is turned into a homology vector space and the inclusion functions are turned into linear maps.","The result is an $\\mathbf {R}$ -parameterized sequence of vector spaces and linear maps: $\\cdots \\rightarrow H_k(C_a) \\rightarrow H_k(C_b) \\rightarrow H_k(C_c) \\rightarrow \\cdots .$ We will abbreviate $H_k(C_a)$ as $H_{k,a}$ .","In this parameterized sequence the dimensions of homology vector spaces encode topological information: $H_{0,t}$ effectively measuring the number of connected components, $H_{1,t}$ measuring the number of one-dimensional holes and $H_{k,t}$ those of $k$ -dimensional voids at scale $t$ .","This algebraic step gives a mapping $(C_t,\\subseteq _t)_{t \\in \\mathbf {R}} \\rightarrow (H_{k,t},\\rightarrow _t)_{t \\in \\mathbf {R}}.$ The obtained result is not an arbitrary $\\mathbf {R}$ -parameterized vector space.","The vector spaces $H_{k,t}$ are finite dimensional and there are finitely many numbers $00w > 0; blue – w<0w < 0).b) Column liquid water content qlql (i. e. the maximum liquid water value in the vertical direction).c) Point representation of the cloud field by the local maxima of qlql(only connected components formed by at least 3 cells are considered),and 1-simplices of the Vietoris-Rips filtration using the distance relation between the points, at a distance scale of 1.5.An important issue in the study of cloud formation is the quantification of spatial organization, or lack thereof, in a given cloud field.","While methods to study spatial distributions exist in the statistical literature for objects which can be idealized as points, it is harder to work with objects that possess a spatial extent (i.e.","area or volume), as clouds do.","This leads to the necessity of computing a point representation for a cloud before being able to assess the spatial distribution of the cloud field.","Here we consider three different representations: assigning to each cloud its geometric centroid, its point with maximum $ql$ value, and a set of its points chosen at random.","A common metric in the assessment of spatial organization is the $I_\\text{org}$ index [21], defined as follows.","For a two-dimensional cloud field, such as the one shown in Fig.","REF (b), index the connected components (the individual clouds) as $c_i$ , and compute their geometric centroids, $\\bar{c}_i$ .","We are interested in how the spatial distribution of the $\\bar{c}_i$ compares to what we would expect under complete spatial randomness (CSR), that is, if the centroids represent a realization of a homogeneous Poisson point process.","To that end, we consider the nearest-neighbor distances $d_i$ , which are defined as $d_i = \\text{min}\\lbrace d(\\bar{c}_i, x) \\ | \\ x \\in \\bar{\\mathcal {C}} \\setminus \\lbrace \\bar{c}_i \\rbrace \\rbrace $ , where $\\bar{\\mathcal {C}}$ represents the set of all centroids.","The cumulative distribution function (CDF) of the $d_i$ is $G_{d_i}(r) = P[d_i \\le r],$ which in the case of a Poisson point process has the analytic expression $G_{CSR}(r) = 1 - \\exp {(-\\lambda \\, \\pi \\, r^2)},$ where $\\lambda $ is the Poisson intensity parameter.","The value of $I_\\text{org}$ is then defined to be the area under the graph $(G_{CSR}(r), \\hat{G}(r))$ , where $\\hat{G}(r) = \\frac{\\# \\lbrace \\bar{c}_i \\in \\bar{\\mathcal {C}} \\mid d_i \\le r \\rbrace }{\\# \\lbrace \\bar{c}_i \\in \\bar{\\mathcal {C}} \\rbrace }$ is the empirical estimator of $G(r)$ .","If $\\hat{G}$ matches well with $G_{CSR}$ , the value of $I_\\text{org}$ will be close to 0.5.","A value larger than this suggests spatial clustering, while a smaller one suggests dispersion or regularity.","Figure: Stable rank functions obtained from 100 realizations of a homogeneous Poisson point process with λ=100\\lambda = 100.","Left: S 0 * S_0^*.","Right: S 1 * S_1^*.Let $S^*_i$ denote the stable rank of $H_i$ with respect to the standard contour (Eq.","REF ), normalized by its value at 0.","If we define the function $G_{PH}^i(r) = 1 - S_i^*(r)$ , we note that it increases monotonically towards 1.","In fact, since the normalized stable rank at $r$ is an indication of the relative amount of homological features that persist beyond $r$ , the function $G_{PH}^i(r)$ can be understood as the empirical CDF of homological persistence.","For $n$ realizations of a Poisson point process with intensity parameter $\\lambda $ , we find that their normalized stable ranks $S^*_i$ , and therefore also $G_{PH}^i$ , oscillate within a narrow band (see Fig.","REF ).","At this point we do not have an analytic expression for the stable rank functions obtained from a Poisson point process, but we can define persistent homology analogues to the $I_\\text{org}$ index via a Monte Carlo procedure by taking the area under the curves defined by $(G_{PH,CSR}^{i}(r), G_{PH}^i(r))$ .","In the case of a point process in the plane we would then get two values $I_{PH,0}$ and $I_{PH,1}$ .","We define the index as their arithmetic mean, $I_{PH} = \\frac{I_{PH,0} + I_{PH,1}}{2}.$ Figure: Density histograms of the I org I_\\text{org} index and I PH I_{PH} (Eq. )","for 360 distinct cloud fields.","A: qlql max, B: qlql max removing cloud structures with size smaller than 3 cells, C: Geometric centroids, D: Geometric centroids removing cloud structures with size smaller than 3 cells.We tested the performance of the index $I_{PH}$ defined above, and compared it to the corresponding values of $I_\\text{org}$ in the dataset consisting of 360 distinct cloud fields (36 per simulation day).","The values of both indices are shown in Fig.","REF .","Each panel shows the 360 values of each index for all cloud fields, computed using 4 different point representations.","Panel A shows the values obtained from assigning to each connected component its point with maximum $ql$ value (local maxima); panel B shows the indices obtained when using the local maxima but only of those components with size at least 3 grid cells (all smaller components are ignored).","Panel C shows the results of using the geometric centroid of each connected component.","Finally, for panel D the geometric centroids were used after discarding the smaller components.","These small components can be attributed to numerical imprecision in the underlying model, and hence are not physically meaningful.","As discussed above, if these indices have a value close to 0.5, it would indicate that the point process that they are evaluated on is close to complete spatial randomness, or a Poisson point process.","In the simulations used here, we have cause to expect spatially random behavior: the domain size is too small to allow for deep convection and spatial organization to happen.","Moreover, the lack of land surface features or patterns means there are no forcings at different spatial scales.","Thus the spatial distribution of physical variables is dominated by the characteristic patterns present in atmospheric turbulence, itself an essentially random process.","The values of the persistent homology index $I_{PH}$ strongly support this hypothesis, while $I_\\text{org}$ exhibits values in general larger than 0.5.","This can be attributed to the fact that it is based on nearest-neighbor distances only, whereas the stable rank functions reflect the spatial relationships of the points throughout all spatial scales.","This is confirmed by the fact that removing the smaller structures in the fields (those less than 3 grid cells in size) brings the values of $I_\\text{org}$ closer to 0.5 on average, whereas the average for $I_{PH}$ is barely affected.","This highlights the fact that, by virtue of using all the spatial information available, the persistent homology based method is inherently more robust than any nearest-neighbor method.","Figure: Contour 1 (left) and contour 2 (right) used in the analysis of cloud fields.","Stem plot is from one sampling ofa cloud field at one time step.This result has been arrived at by using the standard contour only, which implies that spatial randomness in these cloud fields is obtained when all spatial scales present in the data are given the same weight.","It is possible to obtain different morphological classifications of the same fields by using alternative contours, which emphasize spatial features differently at varying scales, as presented with the classification in Section REF .","We used standard contour and contours visualized in Fig.","REF .","These contours are referred to as contour 1, denoted $C_1$ , and contour 2, denoted $C_2$ .","Figure: Cloud fields which are classified into different clusters, according to the methodology described in the text.We use the H 1 H_1 stable ranks and the interleaving metric to compute the distances between them.a) and b) are classified using contour C 1 C_1, and have I org I_\\text{org} values of 0.45 and 0.53 respectively.Cloud cover is similar at 14% for both.c) and d) are classified with C 2 C_2, and have I org I_\\text{org} values of 0.65 and 0.63 respectively, and cloud coverfor both is 9.2%.To reduce the effect of sampling, 10 random samples were drawn from each of the 360 cloud fields, with sample rate 5% of cloud size.","To each cloud field we assign the mean stable rank of these 10 samples.","Stable ranks were computed in $H_1$ with respect to standard contour, contour 1 and contour 2 and normalized to give $S^*_1$ function as explained above.","After removing those cloud fields without $H_1$ features, we have 254 normalized stable ranks $S^*_1$ for each class of contours.","Distance matrices using interleaving, $L_1$ - and $L_2$ -metrics (see Section REF ) were then computed for the three different classes of stable ranks.","Dendrograms from the distance matrices were visually analyzed to decide on a number of clusters of stable ranks.","From these computations the interleaving distance gave the clearest clustering results.","With respect to contours, $C_1$ and $C_2$ gave better clustering than standard contour.","An example of diverging morphological characteristics educed from the $C_{1,2}$ clustering schemes is shown in Fig.","REF : (a) and (b) are representatives of two different clusters obtained by using contour $C_1$ , while (c) and (d) stem from clusters in the $C_2$ classification.","As expected from the definition of the contours, the classifications they induce are influenced by different spatial scales.","Namely, despite the fact that cloud fields a) and b) have identical cloud cover, and their $I_\\text{org}$ values are very similar, the large-scale distribution of the individual clouds is significantly different for both.","In similar fashion, both c) and d) are indistinguishable in terms of cloud cover and $I_\\text{org}$ , yet are distinguished by the spatial pattern of smaller structures, even if the large-scale distribution is similar in both.","This study of cloud fields shows that the use of stable rank functions as descriptors for spatial distributions can reveal morphological properties which other methods cannot.","Crucially, the possibility of changing the contour enriches the scope for determining such properties.","Future investigation in this direction will address questions such as: what the optimal contour is for a given problem, what these methods can reveal about the temporal evolution of cloud formation, and how the homological properties thus discovered can be related to different physical variables in the system.","From general data analysis point of view, particularly the optimization of contours is crucial for making our pipeline a full scale machine learning approach."],["Acknowledgments","We gratefully acknowledge Roel Neggers for providing the DALES simulation data.","JLS acknowledges support by the DFG-funded transregional research collaborative TR32 on Patterns in Soil–Vegetation–Atmosphere Systems."]],"string":"[\n [\n \"Metrics for Learning in Topological Persistence\"\n ],\n [\n \"Abstract Persistent homology analysis provides means to capture the connectivity structure of data sets in various dimensions.\",\n \"On the mathematical level, by defining a metric between the objects that persistence attaches to data sets, we can stabilize invariants characterizing these objects.\",\n \"We outline how so called contour functions induce relevant metrics for stabilizing the rank invariant.\",\n \"On the practical level, the stable ranks are used as fingerprints for data.\",\n \"Different choices of contour lead to different stable ranks and the topological learning is then the question of finding the optimal contour.\",\n \"We outline our analysis pipeline and show how it can enhance classification of physical activities data.\",\n \"As our main application we study how stable ranks and contours provide robust descriptors of spatial patterns of atmospheric cloud fields.\"\n ],\n [\n \"Modelling data spaces\",\n \"Topological data analysis (TDA) and particularly its subfield persistent homology, or persistence, aim at quantifying the global connectivity structure of data sets [1], [2], [3].\",\n \"Given a set of data points it is often possible to endow it with some reasonable notion of relation between points, e.g.\",\n \"distance measure or correlation.\",\n \"Study of the connectivity is facilitated by first combining points into larger entities called simplices.\",\n \"A $k$ -simplex is a declared subset of $k+1$ related points from the data set.\",\n \"Collection of simplices makes up a simplicial complex $C$ , namely it is a collection of certain subsets of the data.\",\n \"Requirements are that if $\\\\sigma $ is a simplex in $C$ then any subset of $\\\\sigma $ is also a simplex in $C$ and that the intersection of two simplices is a simplex or the empty set.\",\n \"Above we have described an abstract simplicial complex.\",\n \"Simplices can always be realized geometrically in some $\\\\mathbf {R}^n$ as convex hulls of their vertices: 0-simplices as points, 1-simplices as line segments, 2-simplices as filled triangles, 3-simplices as filled tetrahedra etc.\",\n \"Simplicial complex is hence a model of the relational structure in the data.\",\n \"Relational structure can be modelled by a graph but graphs only consider pairwise relations between points.\",\n \"In many cases it makes sense to use higher-dimensional connectivity instead modelled with simplices.\",\n \"As a justification consider the example explained in Fig.\",\n \"REF .\",\n \"More fundamental reason is that the simpicial approach views data as spaces spanned by their points and enables the use of powerful mathematical machinery of algebraic topology for the analysis of these spaces, as will be outlined in the following section.\",\n \"Figure: Simplicial model of social relations.\",\n \"To model relations between k+1k+1 points it is reasonable to use kk-simplex for the purpose.\",\n \"Here the relations between {Arya, Bran, Jon} is depicted by the 2-simplex represented by the purple triangle.\",\n \"From the point of view of TDA the prominent feature of this data is the single loop structure, whereas a graph would see two loops (the closed path (Arya, Bran, Jon) spanning the other loop in this case).For persistence analysis we define the relation to be a function $R$ on the data with values in $\\\\mathbf {R}=[0,\\\\infty )$ , i.e.\",\n \"$R(x,y) \\\\mapsto t \\\\in \\\\mathbf {R}$ for data points $x$ and $y$ .\",\n \"Concretely we say that $k+1$ data points $x_i$ create a $k$ -simplex at scale $t$ if the points satisfy pairwise $R(x_i,x_j) \\\\le t.$ This construction is called the Vietoris-Rips simplicial complex at scale $t$ .\",\n \"At fixed scale we can then study the connectivity structure.\",\n \"As a standard example, when data is endowed with a distance measure, clustering at some fixed scale corresponds to the 0-dimensional connectivity by only looking at the connected components of the simplicial complex.\",\n \"Simplicial complexes can also contain 1-dimensional connectivity information in the form of loops and holes (see Fig.\",\n \"REF ), 2-dimensional information in the form of voids or cavities, etc.\",\n \"These are collectively called topological features.\",\n \"Persistence aims to quantify the topological features in a data set and use this information for data analysis.\",\n \"Loop structure might signal about a recurrent dynamics of the phenomenon behind the data.\",\n \"Various dimensional voids can mark lack of information and connectivity or insufficient data collection.\",\n \"Finding such voids in data sets has aroused interest in different areas of data analysis community, see for example [4] and references therein.\",\n \"As noted in [4], voids can also indicate non-allowed combinations of feature values of data vectors.\",\n \"One immediate difficulty arises in the simplicial modelling above: what is the appropriate scale of $R$ to capture the connectivity in various dimensions of an arbitrary set of points?\",\n \"Persistence circumvents this by forming simplicial complexes at all scales $t \\\\ge 0$ and capturing the evolution of topological features.\",\n \"If a simplex is generated at scale $t$ it is then present at any subsequent scale and the simplicial complexes are connected by inclusions: $\\\\cdots \\\\subseteq C_a \\\\subseteq C_b \\\\subseteq C_c \\\\subseteq \\\\cdots $ for $\\\\dots \\\\le a \\\\le b \\\\le c \\\\le \\\\dots $ The end result of the modelling step is then a mapping called filtration, $(D,R) \\\\times \\\\mathbf {R} \\\\rightarrow (C_t,\\\\subseteq _t)_{t \\\\in \\\\mathbf {R}}$ , where $(D,R)$ denotes a data set with real-valued relation and $(C_t,\\\\subseteq _t)_{t \\\\in \\\\mathbf {R}}$ denotes an $\\\\mathbf {R}$ -parameterized sequence of simplicial complexes and inclusions.\"\n ],\n [\n \"Algebraic fingerprinting\",\n \"Filtration contains all the information about the relations in the data set on various scales.\",\n \"It is therefore very complicated object for infering the global structure of data and simplification is thus necessary.\",\n \"TDA employs tools from mathematical field of algebraic topology, essentially it uses homology of simplicial complexes which transforms the geometric information into algebraic information.\",\n \"We will outline the algorithm for computing homology to illustrate its very implementable nature and to gain intuition on why we are interested in homology in data analysis.\",\n \"For details into homology and its computation see [5], [6], [7].\",\n \"For simplicity we fix the field of coefficients to be $\\\\mathbf {F}_2$ , the field with two elements 0 and 1.\",\n \"Let $C$ be a simplicial complex and denote by $C_k$ its set of $k$ -simplices.\",\n \"Concretely, $C_0$ consists of the points of the original data set.\",\n \"1) Choose an ordering (starting from zero) on $C_0$ and use it to order elements in any simplex.\",\n \"If $\\\\lbrace \\\\text{Arya}, \\\\text{Bran}, \\\\text{Jon}\\\\rbrace $ is a 2-simplex in Fig.\",\n \"REF , fix the order in which the points are listed and denote this ordered simplex by $[\\\\text{Arya}, \\\\text{Bran}, \\\\text{Jon}].$ 2) For natural numbers $k$ and $0\\\\le i\\\\le k$ and a simplex $\\\\sigma $ in $C_k$ , define a function $d_i \\\\colon C_k \\\\rightarrow C_{k-1}$ such that $d_i(\\\\sigma )$ is a simplex in $C_{k-1}$ formed by removing from $\\\\sigma $ its $i$ -th element.\",\n \"The ordering on $C_0$ was needed to specify the $i$ -th element in a simplex.\",\n \"For example, $d_1([\\\\text{Arya}, \\\\text{Bran}, \\\\text{Jon}]) = [\\\\text{Arya}, \\\\text{Jon}].$ 3) For any natural number $k$ , let $\\\\Delta (C)_k$ be the vector space over $\\\\mathbf {F}_2$ with a base given by all simplices in $C_k$ .\",\n \"An element $\\\\tau $ in $\\\\Delta (C)_k$ is then given by a linear combination $\\\\tau = \\\\sum _{\\\\sigma \\\\in C_k} t_\\\\sigma \\\\sigma , \\\\ t_\\\\sigma \\\\in \\\\mathbf {F}_2.$ The base for $\\\\Delta (C)_2$ of the simplicial complex in Fig.\",\n \"REF would be [Arya, Bran, Jon] whereas [Arya, Bran]+[Bran, Jon]+[Arya, Jon] would be linear combination of three basis elements in $\\\\Delta (C)_1$ .\",\n \"4) Define $\\\\partial _k \\\\colon \\\\Delta (C)_k \\\\rightarrow \\\\Delta (C)_{k-1}$ to be the linear function assigning to a base element given by a simplex $\\\\sigma $ in $C_k$ the linear combination $\\\\sum _{i=0}^{k}d_i(\\\\sigma )$ of $k$ -1-simplices.\",\n \"The map $\\\\partial _k$ is called the boundary operator.\",\n \"Then $\\\\partial _k([\\\\text{Arya}, \\\\text{Bran}, \\\\text{Jon}]) = [\\\\text{Bran}, \\\\text{Jon}]+[\\\\text{Arya}, \\\\text{Jon}]+[\\\\text{Arya}, \\\\text{Bran}].$ The boundary operator thus formalizes the intuition that $[\\\\text{Bran}, \\\\text{Jon}]+[\\\\text{Arya}, \\\\text{Jon}]+[\\\\text{Arya}, \\\\text{Bran}]$ forms the boundary of $[\\\\text{Arya}, \\\\text{Bran}, \\\\text{Jon}].$ Define $\\\\Delta (C)_{-1} = 0$ and $\\\\Delta (C)_k = 0$ for $k>m$ , where 0 denotes the zero vector space.\",\n \"5) The boundary operators connect the various simplices of a simplicial complex together.\",\n \"Computationally the matrices of boundary operators store the global connectivity information in their elements, with coefficient field $\\\\mathbf {F}_2$ these are just binary matrices.\",\n \"Homology on degree $k$ of a simplicial complex $C$ (over coefficients $\\\\mathbf {F}_2$ ) is then defined as a quotient vector space: $H_k(C)= \\\\frac{\\\\text{kernel of }\\\\partial _k\\\\colon \\\\Delta (C)_k \\\\rightarrow \\\\Delta (C)_{k-1}}{\\\\text{image of }\\\\partial _{k+1}\\\\colon \\\\Delta (C)_{k+1}\\\\rightarrow \\\\Delta (C)_{k}}, \\\\ \\\\ \\\\text{for } k \\\\ge 0.$ As noted in step 4) above, some 1-simplices might form the boundary of a 2-simplex.\",\n \"Some 1-simplices on the other hand might form the boundary of an actual hole in the simplicial complex as in Fig.\",\n \"REF .\",\n \"Similarly some $k$ -1-simplices might form the boundary of a $k$ -simplex and some might form the boundary of a $k$ -dimensional hole.\",\n \"By its definition homology quotients out linear combinations of simplices that are boundaries and we are left with those that actually represent linearly independent $k$ -dimensional holes in the complex.\",\n \"For $k=0$ , $H_0$ measures the number of linearly independent points that make up boundaries of 1-simplices, effectively the number of connected components.\",\n \"Homology thus gives us exactly the global connectivity information of the relational structure of data that we seek.\",\n \"The full complexity of a filtration is now simplified by applying homology on degree $k$ .\",\n \"Each simplicial complex is turned into a homology vector space and the inclusion functions are turned into linear maps.\",\n \"The result is an $\\\\mathbf {R}$ -parameterized sequence of vector spaces and linear maps: $\\\\cdots \\\\rightarrow H_k(C_a) \\\\rightarrow H_k(C_b) \\\\rightarrow H_k(C_c) \\\\rightarrow \\\\cdots .$ We will abbreviate $H_k(C_a)$ as $H_{k,a}$ .\",\n \"In this parameterized sequence the dimensions of homology vector spaces encode topological information: $H_{0,t}$ effectively measuring the number of connected components, $H_{1,t}$ measuring the number of one-dimensional holes and $H_{k,t}$ those of $k$ -dimensional voids at scale $t$ .\",\n \"This algebraic step gives a mapping $(C_t,\\\\subseteq _t)_{t \\\\in \\\\mathbf {R}} \\\\rightarrow (H_{k,t},\\\\rightarrow _t)_{t \\\\in \\\\mathbf {R}}.$ The obtained result is not an arbitrary $\\\\mathbf {R}$ -parameterized vector space.\",\n \"The vector spaces $H_{k,t}$ are finite dimensional and there are finitely many numbers $00w > 0; blue – w<0w < 0).b) Column liquid water content qlql (i. e. the maximum liquid water value in the vertical direction).c) Point representation of the cloud field by the local maxima of qlql(only connected components formed by at least 3 cells are considered),and 1-simplices of the Vietoris-Rips filtration using the distance relation between the points, at a distance scale of 1.5.An important issue in the study of cloud formation is the quantification of spatial organization, or lack thereof, in a given cloud field.\",\n \"While methods to study spatial distributions exist in the statistical literature for objects which can be idealized as points, it is harder to work with objects that possess a spatial extent (i.e.\",\n \"area or volume), as clouds do.\",\n \"This leads to the necessity of computing a point representation for a cloud before being able to assess the spatial distribution of the cloud field.\",\n \"Here we consider three different representations: assigning to each cloud its geometric centroid, its point with maximum $ql$ value, and a set of its points chosen at random.\",\n \"A common metric in the assessment of spatial organization is the $I_\\\\text{org}$ index [21], defined as follows.\",\n \"For a two-dimensional cloud field, such as the one shown in Fig.\",\n \"REF (b), index the connected components (the individual clouds) as $c_i$ , and compute their geometric centroids, $\\\\bar{c}_i$ .\",\n \"We are interested in how the spatial distribution of the $\\\\bar{c}_i$ compares to what we would expect under complete spatial randomness (CSR), that is, if the centroids represent a realization of a homogeneous Poisson point process.\",\n \"To that end, we consider the nearest-neighbor distances $d_i$ , which are defined as $d_i = \\\\text{min}\\\\lbrace d(\\\\bar{c}_i, x) \\\\ | \\\\ x \\\\in \\\\bar{\\\\mathcal {C}} \\\\setminus \\\\lbrace \\\\bar{c}_i \\\\rbrace \\\\rbrace $ , where $\\\\bar{\\\\mathcal {C}}$ represents the set of all centroids.\",\n \"The cumulative distribution function (CDF) of the $d_i$ is $G_{d_i}(r) = P[d_i \\\\le r],$ which in the case of a Poisson point process has the analytic expression $G_{CSR}(r) = 1 - \\\\exp {(-\\\\lambda \\\\, \\\\pi \\\\, r^2)},$ where $\\\\lambda $ is the Poisson intensity parameter.\",\n \"The value of $I_\\\\text{org}$ is then defined to be the area under the graph $(G_{CSR}(r), \\\\hat{G}(r))$ , where $\\\\hat{G}(r) = \\\\frac{\\\\# \\\\lbrace \\\\bar{c}_i \\\\in \\\\bar{\\\\mathcal {C}} \\\\mid d_i \\\\le r \\\\rbrace }{\\\\# \\\\lbrace \\\\bar{c}_i \\\\in \\\\bar{\\\\mathcal {C}} \\\\rbrace }$ is the empirical estimator of $G(r)$ .\",\n \"If $\\\\hat{G}$ matches well with $G_{CSR}$ , the value of $I_\\\\text{org}$ will be close to 0.5.\",\n \"A value larger than this suggests spatial clustering, while a smaller one suggests dispersion or regularity.\",\n \"Figure: Stable rank functions obtained from 100 realizations of a homogeneous Poisson point process with λ=100\\\\lambda = 100.\",\n \"Left: S 0 * S_0^*.\",\n \"Right: S 1 * S_1^*.Let $S^*_i$ denote the stable rank of $H_i$ with respect to the standard contour (Eq.\",\n \"REF ), normalized by its value at 0.\",\n \"If we define the function $G_{PH}^i(r) = 1 - S_i^*(r)$ , we note that it increases monotonically towards 1.\",\n \"In fact, since the normalized stable rank at $r$ is an indication of the relative amount of homological features that persist beyond $r$ , the function $G_{PH}^i(r)$ can be understood as the empirical CDF of homological persistence.\",\n \"For $n$ realizations of a Poisson point process with intensity parameter $\\\\lambda $ , we find that their normalized stable ranks $S^*_i$ , and therefore also $G_{PH}^i$ , oscillate within a narrow band (see Fig.\",\n \"REF ).\",\n \"At this point we do not have an analytic expression for the stable rank functions obtained from a Poisson point process, but we can define persistent homology analogues to the $I_\\\\text{org}$ index via a Monte Carlo procedure by taking the area under the curves defined by $(G_{PH,CSR}^{i}(r), G_{PH}^i(r))$ .\",\n \"In the case of a point process in the plane we would then get two values $I_{PH,0}$ and $I_{PH,1}$ .\",\n \"We define the index as their arithmetic mean, $I_{PH} = \\\\frac{I_{PH,0} + I_{PH,1}}{2}.$ Figure: Density histograms of the I org I_\\\\text{org} index and I PH I_{PH} (Eq. )\",\n \"for 360 distinct cloud fields.\",\n \"A: qlql max, B: qlql max removing cloud structures with size smaller than 3 cells, C: Geometric centroids, D: Geometric centroids removing cloud structures with size smaller than 3 cells.We tested the performance of the index $I_{PH}$ defined above, and compared it to the corresponding values of $I_\\\\text{org}$ in the dataset consisting of 360 distinct cloud fields (36 per simulation day).\",\n \"The values of both indices are shown in Fig.\",\n \"REF .\",\n \"Each panel shows the 360 values of each index for all cloud fields, computed using 4 different point representations.\",\n \"Panel A shows the values obtained from assigning to each connected component its point with maximum $ql$ value (local maxima); panel B shows the indices obtained when using the local maxima but only of those components with size at least 3 grid cells (all smaller components are ignored).\",\n \"Panel C shows the results of using the geometric centroid of each connected component.\",\n \"Finally, for panel D the geometric centroids were used after discarding the smaller components.\",\n \"These small components can be attributed to numerical imprecision in the underlying model, and hence are not physically meaningful.\",\n \"As discussed above, if these indices have a value close to 0.5, it would indicate that the point process that they are evaluated on is close to complete spatial randomness, or a Poisson point process.\",\n \"In the simulations used here, we have cause to expect spatially random behavior: the domain size is too small to allow for deep convection and spatial organization to happen.\",\n \"Moreover, the lack of land surface features or patterns means there are no forcings at different spatial scales.\",\n \"Thus the spatial distribution of physical variables is dominated by the characteristic patterns present in atmospheric turbulence, itself an essentially random process.\",\n \"The values of the persistent homology index $I_{PH}$ strongly support this hypothesis, while $I_\\\\text{org}$ exhibits values in general larger than 0.5.\",\n \"This can be attributed to the fact that it is based on nearest-neighbor distances only, whereas the stable rank functions reflect the spatial relationships of the points throughout all spatial scales.\",\n \"This is confirmed by the fact that removing the smaller structures in the fields (those less than 3 grid cells in size) brings the values of $I_\\\\text{org}$ closer to 0.5 on average, whereas the average for $I_{PH}$ is barely affected.\",\n \"This highlights the fact that, by virtue of using all the spatial information available, the persistent homology based method is inherently more robust than any nearest-neighbor method.\",\n \"Figure: Contour 1 (left) and contour 2 (right) used in the analysis of cloud fields.\",\n \"Stem plot is from one sampling ofa cloud field at one time step.This result has been arrived at by using the standard contour only, which implies that spatial randomness in these cloud fields is obtained when all spatial scales present in the data are given the same weight.\",\n \"It is possible to obtain different morphological classifications of the same fields by using alternative contours, which emphasize spatial features differently at varying scales, as presented with the classification in Section REF .\",\n \"We used standard contour and contours visualized in Fig.\",\n \"REF .\",\n \"These contours are referred to as contour 1, denoted $C_1$ , and contour 2, denoted $C_2$ .\",\n \"Figure: Cloud fields which are classified into different clusters, according to the methodology described in the text.We use the H 1 H_1 stable ranks and the interleaving metric to compute the distances between them.a) and b) are classified using contour C 1 C_1, and have I org I_\\\\text{org} values of 0.45 and 0.53 respectively.Cloud cover is similar at 14% for both.c) and d) are classified with C 2 C_2, and have I org I_\\\\text{org} values of 0.65 and 0.63 respectively, and cloud coverfor both is 9.2%.To reduce the effect of sampling, 10 random samples were drawn from each of the 360 cloud fields, with sample rate 5% of cloud size.\",\n \"To each cloud field we assign the mean stable rank of these 10 samples.\",\n \"Stable ranks were computed in $H_1$ with respect to standard contour, contour 1 and contour 2 and normalized to give $S^*_1$ function as explained above.\",\n \"After removing those cloud fields without $H_1$ features, we have 254 normalized stable ranks $S^*_1$ for each class of contours.\",\n \"Distance matrices using interleaving, $L_1$ - and $L_2$ -metrics (see Section REF ) were then computed for the three different classes of stable ranks.\",\n \"Dendrograms from the distance matrices were visually analyzed to decide on a number of clusters of stable ranks.\",\n \"From these computations the interleaving distance gave the clearest clustering results.\",\n \"With respect to contours, $C_1$ and $C_2$ gave better clustering than standard contour.\",\n \"An example of diverging morphological characteristics educed from the $C_{1,2}$ clustering schemes is shown in Fig.\",\n \"REF : (a) and (b) are representatives of two different clusters obtained by using contour $C_1$ , while (c) and (d) stem from clusters in the $C_2$ classification.\",\n \"As expected from the definition of the contours, the classifications they induce are influenced by different spatial scales.\",\n \"Namely, despite the fact that cloud fields a) and b) have identical cloud cover, and their $I_\\\\text{org}$ values are very similar, the large-scale distribution of the individual clouds is significantly different for both.\",\n \"In similar fashion, both c) and d) are indistinguishable in terms of cloud cover and $I_\\\\text{org}$ , yet are distinguished by the spatial pattern of smaller structures, even if the large-scale distribution is similar in both.\",\n \"This study of cloud fields shows that the use of stable rank functions as descriptors for spatial distributions can reveal morphological properties which other methods cannot.\",\n \"Crucially, the possibility of changing the contour enriches the scope for determining such properties.\",\n \"Future investigation in this direction will address questions such as: what the optimal contour is for a given problem, what these methods can reveal about the temporal evolution of cloud formation, and how the homological properties thus discovered can be related to different physical variables in the system.\",\n \"From general data analysis point of view, particularly the optimization of contours is crucial for making our pipeline a full scale machine learning approach.\"\n ],\n [\n \"Acknowledgments\",\n \"We gratefully acknowledge Roel Neggers for providing the DALES simulation data.\",\n \"JLS acknowledges support by the DFG-funded transregional research collaborative TR32 on Patterns in Soil–Vegetation–Atmosphere Systems.\"\n ]\n]"},"id":{"kind":"string","value":"1906.04436"}}},{"rowIdx":2596,"cells":{"text":{"kind":"list like","value":[["A Graph-theoretic Method to Define any Boolean Operation on Partitions"],["Abstract The lattice operations of join and meet were defined for set partitions in the nineteenth century, but no new logical operations on partitions were defined and studied during the twentieth century.","Yet there is a simple and natural graph-theoretic method presented here to define any n-ary Boolean operation on partitions.","An equivalent closure-theoretic method is also defined.","In closing, the question is addressed of why it took so long for all Boolean operations to be defined for partitions."],["Introduction","The lattice operations of join and meet were defined on set partitions during the late nineteenth century, and the lattice of partitions on a set was used as an example of a non-distributive lattice.","But during the entire twentieth century, no new logical operations were defined on partitions.","Equivalence relations are so ubiquitous in everyday life that we often forget about their proactive existence.","Much is still unknown about equivalence relations.","Were this situation remedied, the theory of equivalence relations could initiate a chain reaction generating new insights and discoveries in many fields dependent upon it.","This paper springs from a simple acknowledgement: the only operations on the family of equivalence relations fully studied, understood and deployed are the binary join $\\vee $ and meet $\\wedge $ operations.","[3] Papers on the \"logic\" of equivalence relations [7] or partitions only involved the join and meet, and not the crucial logical operation of implication.","Yet, there is a general graph-theoretic methodThe method is, strictly speaking, an algorithm only when $U$ is finite.","by which any $n$ -ary Boolean (or truth-functional) operation $f:\\left\\lbrace T,F\\right\\rbrace ^{n}\\rightarrow \\left\\lbrace T,F\\right\\rbrace $ can be used to define the corresponding $n$ -ary operation $f:\\prod \\left( U\\right) ^{n}\\rightarrow \\prod \\left(U\\right) $ where $\\prod \\left( U\\right) $ is the set of partitions on a set $U$ .","A partition $\\pi =\\left\\lbrace B,B^{\\prime },...\\right\\rbrace $ on a set $U=\\left\\lbrace u,u^{\\prime },...\\right\\rbrace $ is a set of disjoint non-empty subsets $B,B^{\\prime },...$ of $U$ , called blocks, whose union is $U$ .","The corresponding equivalence relation, denoted $\\operatornamewithlimits{indit}\\left(\\pi \\right) $ , is the set of ordered pairs of elements of $U$ that are in the same block of $\\pi $ , and are called the indistinctions or indits of $\\pi $ , i.e., $\\operatornamewithlimits{indit}\\left( \\pi \\right) =\\left\\lbrace \\left( u,u^{\\prime }\\right) \\in U\\times U:\\exists B\\in \\pi ,u,u^{\\prime }\\in B\\right\\rbrace $ .","The complement $\\operatornamewithlimits{dit}\\left( \\pi \\right) =U\\times U-\\operatornamewithlimits{indit}\\left( \\pi \\right) $ is the set of distinctions or dits of $\\pi $ , i.e., ordered pairs of elements in different blocks.","As binary relations, the sets of distinctions or ditsets $\\operatornamewithlimits{dit}\\left( \\pi \\right) $ of some partition $\\pi $ on $U$ are called partition (or apartness) relations.","Given partitions $\\pi =\\left\\lbrace B,B^{\\prime },...\\right\\rbrace $ and $\\sigma =\\left\\lbrace C,C^{\\prime },...\\right\\rbrace $ on $U$ , the refinement relation is the partial order defined by: $\\sigma \\preceq \\pi $ if $\\forall B\\in \\pi ,\\exists C\\in \\sigma ,B\\subseteq C $ .","At the top of the refinement partial order is the discrete partition $\\mathbf {1}=\\left\\lbrace \\left\\lbrace u\\right\\rbrace :u\\in U\\right\\rbrace $ of all singletons and at the bottom is the indiscrete partition $\\mathbf {0}=\\left\\lbrace U\\right\\rbrace $ with only one block consisting of $U$ .","In terms of binary relations, the refinement partial order is just the inclusion partial order on ditsets, i.e., $\\sigma \\preceq \\pi $ iff $\\operatornamewithlimits{dit}\\left( \\sigma \\right) \\subseteq \\operatornamewithlimits{dit}\\left( \\pi \\right) $ .","It should be noted that most of the previous literature on partitions (e.g., [1]) uses the opposite partial order of `unrefinement' corresponding to the inclusion relation on equivalence relations–which reverses the definitions of the join and meet of partitions."],["The Join Operation on Partitions","The join $\\pi \\vee \\sigma $ of partitions $\\pi $ and $\\sigma $ (least upper bound using the refinement partial order) is the partition whose blocks are the non-empty intersections $B\\cap C$ of the blocks of $\\pi $ and $\\sigma $ (under the unrefinement ordering, it is the meet).","In terms of ditsets, $\\operatornamewithlimits{dit}\\left( \\pi \\vee \\sigma \\right) =\\operatornamewithlimits{dit}\\left(\\pi \\right) \\cup \\operatornamewithlimits{dit}\\left( \\sigma \\right) $ .","The general method for defining Boolean operations on partitions will be first illustrated with the join operation whose corresponding Boolean operation is disjunction with the truth table.","Table: NO_CAPTIONTruth table for disjunction.","Let $K\\left( U\\right) $ be the complete undirected graph on $U$ .","The links $u-u^{\\prime }$ corresponding to dits, i.e., $\\left( u,u^{\\prime }\\right)\\in \\operatornamewithlimits{dit}\\left( \\pi \\right) $ , of a partition are labelled with the `truth value' $T_{\\pi }$ and corresponding to indits $\\left( u,u^{\\prime }\\right) \\in \\operatornamewithlimits{indit}\\left( \\pi \\right) $ are labelled with the `truth value' $F_{\\pi }$ .","Given the two partitions $\\pi $ and $\\sigma $ , each link in the complete graph $K\\left( U\\right) $ is labelled with a pair of truth values.","The graph $G\\left( \\pi \\vee \\sigma \\right) $ of the join is obtained by putting a link $u-u^{\\prime }$ where the truth function applied to the pair of truth values on the link in $K\\left(U\\right) $ gives an $F$ .","Thus in the case at hand, the only links in $G\\left( \\pi \\vee \\sigma \\right) $ are for the $u-u^{\\prime }$ labelled with $F_{\\pi }$ and $F_{\\sigma }$ in $K\\left( U\\right) $ .","Then the partition $\\pi \\vee \\sigma $ is obtained as the connected components of its graph $G\\left(\\pi \\vee \\sigma \\right) $ .","Thus $u$ and $u^{\\prime }$ are in the same block (connected component of $G\\left( \\pi \\vee \\sigma \\right) $ ) if and only if the link $u-u^{\\prime }$ was labelled $F_{\\pi }$ and $F_{\\sigma }$ , i.e., $u$ and $u^{\\prime }$ were in the same block of $\\pi $ and in the same block of $\\sigma $ .","Thus the graph-theoretic definition of the join reproduces the set-of-blocks definition of the join defined as having its blocks the non-empty intersections of the blocks of $\\pi $ and $\\sigma $ ."],["The Meet Operation on Partitions","On the combined set of blocks $\\pi \\cup \\sigma $ of $\\pi $ and $\\sigma $ , define the overlap relation $B\\between C$ on two blocks if they have a non-empty intersection or overlap (see [8]).","The reflexive-symmetric-transitive closure of this relation is an equivalence relation, and the union of the blocks in each equivalence class gives the blocks of the meet $\\pi \\wedge \\sigma $ .","The corresponding truth-functional operation is conjunction with the following truth table.","Table: NO_CAPTIONTruth table for conjunction.","The same method is applied except that the links of the graph $G\\left(\\pi \\wedge \\sigma \\right) $ are the ones for which the conjunction truth table gives an $F$ when applied to the truth values on each link $u-u^{\\prime }$ .","Thus $G\\left( \\pi \\wedge \\sigma \\right) $ contains a link $u-u^{\\prime }$ if $\\left( u,u^{\\prime }\\right) \\in \\operatornamewithlimits{indit}\\left( \\pi \\right) $ , $\\left( u,u^{\\prime }\\right) \\in \\operatornamewithlimits{indit}\\left( \\sigma \\right)$ , or both.","Then the blocks of the partition $\\pi \\wedge \\sigma $ are the connected components of the graph $G\\left( \\pi \\wedge \\sigma \\right) $ .","The proof that the graph-theoretic definition of the meet gives the usual set-of-blocks definition of the meet boils down to showing that: $B\\in \\pi $ and $C\\in \\sigma $ are contained in the same block of the usual meet $\\pi \\wedge \\sigma $ (i.e., there is a chain of overlaps $B\\between C^{\\prime }\\between ...\\between B^{\\prime }\\between C$ connecting $B$ and $C$ ) if and only for any $u\\in B$ and $u^{\\prime }\\in C$ , $u$ and $u^{\\prime }$ are in the same connected component of $G\\left( \\pi \\wedge \\sigma \\right) $ .","If any two blocks $B^{\\prime }\\between C^{\\prime }$ overlap in the overlap chain, then there is an element $u^{\\prime \\prime }\\in B^{\\prime }\\cap C^{\\prime }$ such any $u\\in B^{\\prime }$ had a link $u-u^{\\prime \\prime }$ in $G\\left( \\pi \\wedge \\sigma \\right) $ and similarly any $u^{\\prime }\\in C^{\\prime }$ has a link $u^{\\prime \\prime }-u^{\\prime }$ in $G\\left( \\pi \\wedge \\sigma \\right) $ .","Hence the existence of an overlap chain connecting $B$ and $C$ implies that any $u\\in B$ and $u^{\\prime }\\in C$ are in the same connected component of $G\\left( \\pi \\wedge \\sigma \\right) $ .","Conversely, if $u\\in B$ and $u^{\\prime }\\in C$ are in the same connected component of $G\\left( \\pi \\wedge \\sigma \\right) $ , then there is some chain of links $u=u_{0}-u_{1}-...-u_{n-1}-u_{n}=u^{\\prime }$ where each link $u_{i}-u_{i+1}$ for $i=0,...,n-1$ has either $\\left( u_{i},u_{i+1}\\right) \\in \\operatornamewithlimits{indit}\\left( \\pi \\right) $ , $\\left( u_{i},u_{i+1}\\right)\\in \\operatornamewithlimits{indit}\\left( \\sigma \\right) $ , or both.","Every link $u_{i}-u_{i+1}$ that is in one indit set but not the other, say, $\\left(u_{i},u_{i+1}\\right) \\in \\operatornamewithlimits{indit}\\left( \\pi \\right) $ and $\\left( u_{i},u_{i+1}\\right) \\notin \\operatornamewithlimits{indit}\\left(\\sigma \\right) $ , establishes an overlap between the block of $\\pi $ containing $u_{i},u_{i+1}$ and the block of $\\sigma $ containing $u_{i}$ as well as the different block of $\\sigma $ containing $u_{i+1}$ .","Thus the chain of links connecting $u\\in B$ and $u^{\\prime }\\in C$ establishes a chain of overlapping blocks connecting $B$ and $C$ ."],["The Implication Operation on Partitions","The real beginning of the logic of partitions, as opposed to the lattice theory of partitions, was the discovery of the set-of-blocks definition of the implication operation $\\sigma \\Rightarrow \\pi $ for partitions ([5], [6]).","The intuitive idea is that $\\sigma \\Rightarrow \\pi $ functions like an indicator or characteristic function to indicate which blocks $B$ of $\\pi $ are contained in a block of $\\sigma $ .","View the discretized version of $B\\in \\pi $ , i.e., $B$ replaced by the set of singletons of the elements of $B$ , as the local version $\\mathbf {1}_{B}$ of the discrete partition $\\mathbf {1}$ , and view the block $B$ remaining whole as the local version $\\mathbf {0}_{B}$ of the indiscrete partition $\\mathbf {0}$ .","Then the partition implication as the inclusion indicator function is: the blocks of $\\sigma \\Rightarrow \\pi $ are for any $B\\in \\pi $ : $\\left\\lbrace \\begin{array}[c]{l}\\mathbf {1}_{B}\\text{ if }\\exists C\\in \\sigma ,B\\subseteq C\\\\\\mathbf {0}_{B}=B\\text{ otherwise.}\\end{array}\\right.",".$ In the case of the Boolean logic of subsets, for any subsets $S,T\\subseteq U$ , the conditional $S\\supset T=S^{c}\\cup T$ has the property: $S\\supset T=U$ iff $S\\subseteq T$ , i.e., the conditional $S\\supset T$ equals the top of the lattice of subsets of $U$ iff the inclusion relation $S\\subseteq T$ holds.","Similarly, it is immediate that the corresponding relation holds in the partition case: $\\sigma \\Rightarrow \\pi =\\mathbf {1}$ iff $\\sigma \\preceq \\pi $ .","This set-of-blocks definition of the partition implication operation accounts for the important new non-lattice-theoretic properties revealed in the algebra of partitions $\\prod \\left( U\\right) $ on $U $ (defined with the join, meet, and implication as partition operations).","A logical formula in the language of join, meet, and implication is a subset tautology if for any non-empty universe $U$ and any subsets of $U$ substituted for the variables, the whole formula evaluates by the set-theoretic operations of join, meet, and implication (conditional) to the top $U$ .","Similarly, a formula in the same language is a partition tautology if for any universe $U$ with $\\left|U\\right|>1$ and for any partitions on $U$ substituted for the variables, the whole formula evaluates by the partition operations of join, meet, and implication to the top $\\mathbf {1}$ (the discrete partition).","All partition tautologies are subset tautologies but not vice-versa.","Modus ponens $\\left(\\sigma \\wedge \\left( \\sigma \\Rightarrow \\pi \\right) \\right) \\Rightarrow \\pi $ is both a subset and partition tautology but Peirce's law, $\\left( \\left(\\sigma \\Rightarrow \\pi \\right) \\Rightarrow \\sigma \\right) \\Rightarrow \\sigma $ , accumulation, $\\sigma \\Rightarrow \\left( \\pi \\Rightarrow \\left( \\sigma \\wedge \\pi \\right) \\right) $ , and distributivity, $\\left( \\left( \\pi \\vee \\sigma \\right) \\wedge \\left( \\pi \\vee \\tau \\right) \\right) \\Rightarrow \\left(\\pi \\vee \\left( \\sigma \\wedge \\tau \\right) \\right) $ , are examples of subset tautologies that are not partition tautologies.","The importance of the implication for partition logic is emphasized by the fact that the only partition tautologies using only the lattice operations, e.g., $\\pi \\vee \\mathbf {1}$ , correspond to general lattice-theoretic identities, i.e., $\\pi \\vee \\mathbf {1}=\\mathbf {1}$ (see [9]).","The graph-theoretic method automatically gives a partition operation corresponding to the Boolean conditional or implication with the truth table: Table: NO_CAPTIONTruth table for conditional and it is not trivial that the two definitions are the same.","It may be helpful to restate the truth table in terms of the partitions.","Table: NO_CAPTIONImplication truth table for partition `truth values'.","For the graph-theoretic definition of $\\sigma \\Rightarrow \\pi $ , we again label the links $u-u^{\\prime }$ in the complete graph $K\\left( U\\right) $ with $T_{\\pi }$ if $\\left( u,u^{\\prime }\\right) \\in \\operatornamewithlimits{dit}\\left(\\pi \\right) $ and $F_{\\pi }$ otherwise, and similarly for $\\sigma $ .","Then we construct the graph $G\\left( \\sigma \\Rightarrow \\pi \\right) $ by putting in a link $u-u^{\\prime }$ only in the case the link is labeled $T_{\\sigma }$ and $F_{\\pi }$ , i.e., $F_{\\sigma \\Rightarrow \\pi }$ .","Then the partition $\\sigma \\Rightarrow \\pi $ is the partition of connected components in the graph $G\\left( \\sigma \\Rightarrow \\pi \\right) $ .","To prove the graph-theoretic and set-of-blocks definitions equivalent, we might first note that if $\\left( u,u^{\\prime }\\right) \\in \\operatornamewithlimits{dit}\\left( \\pi \\right) $ , then $T_{\\pi }$ is assigned to that link in $K\\left(U\\right) $ so there is no link $u-u^{\\prime }$ in $G\\left( \\sigma \\Rightarrow \\pi \\right) $ .","And if $\\left( u,u^{\\prime }\\right) \\in \\operatornamewithlimits{indit}\\left( \\pi \\right) $ but also $\\left( u,u^{\\prime }\\right) \\in \\operatornamewithlimits{indit}\\left( \\sigma \\right) $ , then $T_{\\sigma \\Rightarrow \\pi }$ is assigned to the link in $K\\left( U\\right) $ so again there is no link $u-u^{\\prime }$ in $G\\left( \\sigma \\Rightarrow \\pi \\right) $ .","There is a link $u-u^{\\prime }$ in $G\\left( \\sigma \\Rightarrow \\pi \\right) $ in and only in the following situation where $\\left(u,u^{\\prime }\\right) \\in \\operatornamewithlimits{indit}\\left( \\pi \\right) $ and $\\left( u,u^{\\prime }\\right) \\in \\operatornamewithlimits{dit}\\left( \\sigma \\right)$ –which is exactly the situation when $B$ is not contained in any block $C$ of $\\sigma $ : Figure: NO_CAPTION Figure 1: Links $u-u^{\\prime }$ in $G\\left( \\sigma \\Rightarrow \\pi \\right) $ .","Then for any other element $u^{\\prime \\prime }\\in B$ so that $\\left(u,u^{\\prime \\prime }\\right) $ and $\\left( u^{\\prime },u^{\\prime \\prime }\\right)\\in \\operatornamewithlimits{indit}\\left( \\pi \\right) $ , we must have either $\\left(u,u^{\\prime \\prime }\\right) \\in \\operatornamewithlimits{dit}\\left( \\sigma \\right) $ or $\\left( u^{\\prime },u^{\\prime \\prime }\\right) \\in \\operatornamewithlimits{dit}\\left(\\sigma \\right) $ so $u^{\\prime \\prime }$ is linked in $G\\left( \\sigma \\Rightarrow \\pi \\right) $ to either $u$ or to $u^{\\prime }$ .","Thus all the elements of $B$ are in the same connected component of the graph $G\\left(\\sigma \\Rightarrow \\pi \\right) $ whenever $B$ is not contained in any block of $\\sigma $ .","If, on the other hand, $B$ is contained in some block $C$ of $\\sigma $ , then any $u\\in B$ cannot be linked to any other $u^{\\prime }$ .","In order to that $F_{\\pi }$ assigned to the link $u-u^{\\prime }$ , the two elements have to both belong to $B$ and thus since $B\\subseteq C$ , they both belong to $C$ so $F_{\\sigma }$ and thus $T_{\\sigma \\Rightarrow \\pi }$ is also assigned to that link.","Thus when $B$ is contained in a block $C\\in \\sigma $ , then any point $u\\in B$ is a disconnected component to itself in $G\\left( \\sigma \\Rightarrow \\pi \\right) $ so $B$ is discretized in the graph-theoretic construction of $\\sigma \\Rightarrow \\pi $ .","Thus the graph-theoretic and set-of-blocks definitions of the partition implication are equivalent.","Figure: NO_CAPTION Figure 2: Example of graph for partition implication Example 1 Let $U=\\left\\lbrace a,b,c,d\\right\\rbrace $ so that $K(U)=K_{4}$ is the complete graph on four points.","Let $\\sigma =\\left\\lbrace \\left\\lbrace a\\right\\rbrace ,\\left\\lbrace b,c,d\\right\\rbrace \\right\\rbrace $ and $\\pi =\\left\\lbrace \\left\\lbrace a,b\\right\\rbrace ,\\left\\lbrace c,d\\right\\rbrace \\right\\rbrace $ so we see immediately from the set-of-blocks definition, that the $\\pi $ -block of $\\left\\lbrace c,d\\right\\rbrace $ will be discretized while the $\\pi $ -block of $\\left\\lbrace a,b\\right\\rbrace $ will remain whole so the partition implication is $\\sigma \\Rightarrow \\pi =\\left\\lbrace \\left\\lbrace a,b\\right\\rbrace ,\\left\\lbrace c\\right\\rbrace ,\\left\\lbrace d\\right\\rbrace \\right\\rbrace $ .","After labelling the links in $K\\left( U\\right) $ , we see that only the $a-b$ link has the $F_{\\sigma \\Rightarrow \\pi }$ `truth value' so the graph $G\\left(\\sigma \\Rightarrow \\pi \\right) $ has only that $a-b$ link (thickened in Figure 2).","Then the connected components of $G\\left( \\sigma \\Rightarrow \\pi \\right) $ give the same partition implication $\\sigma \\Rightarrow \\pi =\\left\\lbrace \\left\\lbrace a,b\\right\\rbrace ,\\left\\lbrace c\\right\\rbrace ,\\left\\lbrace d\\right\\rbrace \\right\\rbrace $ .","The partition implication is quite rich in defining new structures in the algebra of partitions (i.e., the lattice of partitions extended with other partition operations such as the implication).","For instance, for a fixed partition $\\pi $ on $U$ , all the partitions of the form $\\sigma \\Rightarrow \\pi $ (for any partitions $\\sigma $ on $U$ ) form a Boolean algebra under the partition operations of implication, join, and meet, e.g., $\\left(\\sigma \\Rightarrow \\pi \\right) \\Rightarrow \\pi $ is the negation of $\\sigma \\Rightarrow \\pi $ , called the Boolean core of the upper segment $\\left[ \\pi ,\\mathbf {1}\\right] $ in the partition algebra $\\prod \\left(U\\right) $ .","A relation is a subset of a product, and, dually, a corelation is a partition on a coproduct.","Any partition $\\pi $ on $U$ can be canonically represented as a relation: $\\operatornamewithlimits{dit}\\left(\\pi \\right) \\subseteq U\\times U$ .","Dually any subset $S\\subseteq U$ can be canonically represented as a corelation, namely the partition $\\pi \\left(S\\right) $ on the coproduct (disjoint union) $U\\uplus U$ where the only nonsingleton blocks in $\\pi \\left( S\\right) $ are the pairs $\\left\\lbrace u,u^{\\ast }\\right\\rbrace $ of $u$ and its copy $u^{\\ast }$ for $u\\notin S$ .","Using this corelation construction, any powerset Boolean algebra $\\wp \\left(U\\right) $ can be canonically represented as the Boolean core of the upper segment $\\left[ \\pi ,\\mathbf {1}\\right] $ in the partition algebra $\\prod \\left( U\\uplus U\\right) $ where $\\pi =\\pi \\left( \\emptyset \\right) $ is the partition on the disjoint union $U\\uplus U$ whose blocks are all the pairs $\\left\\lbrace u,u^{\\ast }\\right\\rbrace $ for each element $u\\in U$ and its copy $u^{\\ast }$ .","Each partition of the form $\\sigma \\Rightarrow \\pi $ on $U\\uplus U$ is $\\pi \\left( S\\right) $ for some $S\\subseteq U$ since $\\sigma \\Rightarrow \\pi $ is essentially the characteristic function of some subset $S$ of $U$ with $\\mathbf {1}\\Rightarrow \\pi =\\pi \\left( \\emptyset \\right) $ playing the role of the empty set $\\emptyset $ and $\\pi \\Rightarrow \\pi =\\mathbf {1}_{U\\uplus U}$ playing the role of $U$ ."],["The General Graph-Theoretic Method","Let $f:\\left\\lbrace T,F\\right\\rbrace ^{n}\\rightarrow \\left\\lbrace T,F\\right\\rbrace $ be an $n$ -ary Boolean function and let $\\pi _{1},...,\\pi _{n}$ be $n$ partitions on $U$ .","In order to define the corresponding $n$ -ary partition operation $f\\left( \\pi _{1},...,\\pi _{n}\\right) $ , we again consider the complete graph $K\\left( U\\right) $ and then use each partition $\\pi _{i}$ to label each link $u-u^{\\prime }$ with $T_{\\pi _{i}}$ if $\\left( u,u^{\\prime }\\right)\\in \\operatornamewithlimits{dit}\\left( \\pi _{i}\\right) $ and $F_{\\pi _{i}}$ if $\\left(u,u^{\\prime }\\right) \\in \\operatornamewithlimits{indit}\\left( \\pi _{i}\\right) $ .","Then on each link we may apply $f$ to the $n$ `truth values' on the link and retain the link in $G\\left( f\\left( \\pi _{1},...,\\pi _{n}\\right) \\right) $ if the result was $F_{f\\left( \\pi _{1},...,\\pi _{n}\\right) }$ .","The partition $f\\left( \\pi _{1},...,\\pi _{n}\\right) $ is obtained as the connected components of the graph $G\\left( f\\left( \\pi _{1},...,\\pi _{n}\\right)\\right) $ ."],["An Equivalent Closure-theoretic Method","Given any subset $S\\subseteq U\\times U$ , the reflexive-symmetric-transitive (RST) closure $\\overline{S}$ is the intersection of all equivalence relations on $U$ containing $S$ .","The `topological' terminology of calling a subset closed if $S=\\overline{S}$ is used even though the RST closure operator is not a topological closure operator since the union of two closed sets is not necessarily closed.","The closed sets in $U\\times U$ are the equivalence relations (or indit sets of partitions), and their complements, the open sets, are the partition relations (or ditsets of partitions).","As usual, the interior operator $\\operatornamewithlimits{int}\\left( S\\right) =\\left(\\overline{S^{c}}\\right) ^{c}$ is the complement of the closure of the complement, and the open sets are the ones equalling their interiors.","The closure-theoretic method of defining Boolean operations on partitions will be illustrated using the symmetric difference or inequivalence operation $\\pi \\oplus \\sigma $ .","Every $n$ -ary Boolean operation can be defined by a truth table such as the one for symmetric difference in this case: Table: NO_CAPTIONTruth table for symmetric difference.","The disjunctive normal form (DNF) for the formula $P\\oplus Q$ is given by the rows where the formula evaluates as $T$ , i.e., $P\\oplus Q=\\left( P\\wedge \\lnot Q\\right) \\vee \\left( \\lnot P\\wedge Q\\right) $ , while the DNF for the negation of the formula is given by the other rows where the formula evaluates as $F$ , i.e., $\\lnot \\left( P\\oplus Q\\right) =\\left( P\\wedge Q\\right)\\vee \\left( \\lnot P\\wedge \\lnot Q\\right) $ .","Given two partitions $\\pi $ and $\\sigma $ on $U$ , the closure-theoretic method of obtaining the partition $\\pi \\oplus \\sigma $ is to start with the DNF for the negated Boolean formula and replace each unnegated variable by the corresponding ditset and each negated variable by the corresponding indit set–as well as replacing the disjunctions and conjunctions by the corresponding subset operations of union and intersection.","Applied to $\\lnot \\left( P\\oplus Q\\right) =\\left( P\\wedge Q\\right) \\vee \\left( \\lnot P\\wedge \\lnot Q\\right) $ , this procedure would yield $\\left( \\operatornamewithlimits{dit}\\left( \\pi \\right) \\cap \\operatornamewithlimits{dit}\\left( \\sigma \\right) \\right) \\cup \\left( \\operatornamewithlimits{indit}\\left(\\pi \\right) \\cap \\operatornamewithlimits{indit}\\left( \\sigma \\right) \\right) \\subseteq U\\times U$ .","Then the indit set of $\\pi \\oplus \\sigma $ is obtained as the RST closure: $\\operatornamewithlimits{indit}\\left( \\pi \\oplus \\sigma \\right) =\\overline{\\left(\\operatornamewithlimits{dit}\\left( \\pi \\right) \\cap \\operatornamewithlimits{dit}\\left(\\sigma \\right) \\right) \\cup \\left( \\operatornamewithlimits{indit}\\left( \\pi \\right)\\cap \\operatornamewithlimits{indit}\\left( \\sigma \\right) \\right) }$ and the partition $\\pi \\oplus \\sigma $ is the set of equivalence classes of this equivalence relation.","The graph-theoretic method of obtaining the partition $\\pi \\oplus \\sigma $ would label each link $u-u^{\\prime }$ in $K\\left( U\\right) $ by the two `truth values' given by $\\pi $ and $\\sigma $ , and then retain in the graph $G\\left(\\pi \\oplus \\sigma \\right) $ the links where the truth values evaluated to $F_{\\pi \\oplus \\sigma }$ , namely the ones labelled with $T_{\\pi },T_{\\sigma }$ and $F_{\\pi },F_{\\sigma }$ .","Then the partition $\\pi \\oplus \\sigma $ is obtained as the connected components of the graph $G\\left( \\pi \\oplus \\sigma \\right) $ .","To see the equivalence between the two methods, note first that the links retained in $G\\left( \\pi \\oplus \\sigma \\right) $ are precisely the pairs $\\left( u,u^{\\prime }\\right) $ in $\\left( \\operatornamewithlimits{dit}\\left(\\pi \\right) \\cap \\operatornamewithlimits{dit}\\left( \\sigma \\right) \\right) \\cup \\left(\\operatornamewithlimits{indit}\\left( \\pi \\right) \\cap \\operatornamewithlimits{indit}\\left(\\sigma \\right) \\right) $ .","The equivalence proof is completed by showing that taking connected components in the graph $G\\left( \\pi \\oplus \\sigma \\right) $ is equivalent to taking the RST closure of $\\left( \\operatornamewithlimits{dit}\\left(\\pi \\right) \\cap \\operatornamewithlimits{dit}\\left( \\sigma \\right) \\right) \\cup \\left(\\operatornamewithlimits{indit}\\left( \\pi \\right) \\cap \\operatornamewithlimits{indit}\\left(\\sigma \\right) \\right) $ .","The elements $u$ and $u^{\\prime }$ are in the same connected component of $G\\left( \\pi \\oplus \\sigma \\right) $ iff there is a chain of links $u=u_{0}-u_{1}-...-u_{n-1}-u_{n}=u^{\\prime }$ in the graph $G\\left( \\pi \\oplus \\sigma \\right) $ so each link has to be originally labelled $T_{\\pi },T_{\\sigma }$ or $F_{\\pi },F_{\\sigma }$ in the graph on $K\\left(U\\right) $ .","But the condition for $\\left( u,u^{\\prime }\\right) $ to be included in the RST closure $\\overline{\\left( \\operatornamewithlimits{dit}\\left(\\pi \\right) \\cap \\operatornamewithlimits{dit}\\left( \\sigma \\right) \\right) \\cup \\left(\\operatornamewithlimits{indit}\\left( \\pi \\right) \\cap \\operatornamewithlimits{indit}\\left(\\sigma \\right) \\right) }$ is that there is a chain of pairs $\\left(u,u_{1}\\right) ,\\left( u_{1},u_{2}\\right) ,...,\\left( u_{n-1},u^{\\prime }\\right) $ such that each pair is either in $\\operatornamewithlimits{dit}\\left(\\pi \\right) \\cap \\operatornamewithlimits{dit}\\left( \\sigma \\right) $ or in $\\operatornamewithlimits{indit}\\left( \\pi \\right) \\cap \\operatornamewithlimits{indit}\\left(\\sigma \\right) $ .","Hence the two methods give the same result.","The example suffices to illustrate the general closure-theoretic method and its equivalence to the graph-theoretic method of defining Boolean operations on partitions."],["Relationships between Boolean operations on partitions","For two subset variables, there are $2^{4}=16$ binary Boolean operations on subsets–corresponding to the sixteen ways to fill in the truth table for a binary Boolean operation.","Any compound Boolean function of two variables will be truth-table equivalent to one of the sixteen binary Boolean operations.","For instance, the Pierce's Law formula $\\left( \\left( Q\\Rightarrow P\\right)\\Rightarrow Q\\right) \\Rightarrow Q$ defines a compound binary operation that is equivalent to the constant function $T$ since it is a subset tautology.","Certain subsets of the sixteen binary operations suffice to define all the binary operations, e.g., $\\lnot $ and $\\vee $ .","Matters are rather different for the Boolean operations on partitions.","Using the graph-theoretic or the closure-theoretic method, partition versions of sixteen binary Boolean operations are easily defined.","And certain combinations of the sixteen operations suffice to define all sixteen, e.g., $\\vee $ , $\\wedge $ , $\\Rightarrow $ , and $\\oplus $ [5].","But when the sixteen operations are compounded, still keeping to two variables, then the resulting binary partition operations does not necessarily reduce to one of the sixteen–due to the complicated compounding of the closure operations.","For instance, the Pierce's Law formula $\\left( \\left(\\sigma \\Rightarrow \\pi \\right) \\Rightarrow \\sigma \\right) \\Rightarrow \\sigma $ for partitions is not equivalent to the constant function $\\mathbf {1}$ since it is not a partition tautology.","The topic of the total number of binary operations on partitions obtained by compounding the sixteen basic binary Boolean operations is one of many topics in partition logic that awaits future research."],["Concluding Remarks","In conclusion, perhaps some remarks are in order as to why it took so long to extend the Boolean operations to partitions.","The Boolean operations are normally associated with subsets of a set or, more specifically, with propositions.","Boole originally defined his logic as the logic of subsets [2] of a universe set.","It is then a theorem that the same set of subset tautologies is obtained as the truth-table tautologies.","Perhaps because “logic” has been historically associated with propositions, the texts in mathematical logic throughout the twentieth century (to the author's knowledge) ignored the Boolean logic of subsets and started with the special case of the logic of propositions and then took the truth-table characterization as the definition of a tautology.","By the middle of the twentieth century, category theory was defined [4] and the category-theoretic duality was established between subobjects and quotient objects, e.g., between subsets of $U$ and quotient sets (or equivalently equivalence relations or partitions) of $U$ .","The conceptual cost of restricting subset logic to the special case of propositional logic is that subsets have the category-theoretic dual concept of partitions while propositions have no such dual concept.","Hence the focus on “propositional logic” did not lead to the search for the dual logic of partitions ([5], [6]) or to the simple and natural application of Boolean operations to partitions as well as subsets–which has been our topic here."]],"string":"[\n [\n \"A Graph-theoretic Method to Define any Boolean Operation on Partitions\"\n ],\n [\n \"Abstract The lattice operations of join and meet were defined for set partitions in the nineteenth century, but no new logical operations on partitions were defined and studied during the twentieth century.\",\n \"Yet there is a simple and natural graph-theoretic method presented here to define any n-ary Boolean operation on partitions.\",\n \"An equivalent closure-theoretic method is also defined.\",\n \"In closing, the question is addressed of why it took so long for all Boolean operations to be defined for partitions.\"\n ],\n [\n \"Introduction\",\n \"The lattice operations of join and meet were defined on set partitions during the late nineteenth century, and the lattice of partitions on a set was used as an example of a non-distributive lattice.\",\n \"But during the entire twentieth century, no new logical operations were defined on partitions.\",\n \"Equivalence relations are so ubiquitous in everyday life that we often forget about their proactive existence.\",\n \"Much is still unknown about equivalence relations.\",\n \"Were this situation remedied, the theory of equivalence relations could initiate a chain reaction generating new insights and discoveries in many fields dependent upon it.\",\n \"This paper springs from a simple acknowledgement: the only operations on the family of equivalence relations fully studied, understood and deployed are the binary join $\\\\vee $ and meet $\\\\wedge $ operations.\",\n \"[3] Papers on the \\\"logic\\\" of equivalence relations [7] or partitions only involved the join and meet, and not the crucial logical operation of implication.\",\n \"Yet, there is a general graph-theoretic methodThe method is, strictly speaking, an algorithm only when $U$ is finite.\",\n \"by which any $n$ -ary Boolean (or truth-functional) operation $f:\\\\left\\\\lbrace T,F\\\\right\\\\rbrace ^{n}\\\\rightarrow \\\\left\\\\lbrace T,F\\\\right\\\\rbrace $ can be used to define the corresponding $n$ -ary operation $f:\\\\prod \\\\left( U\\\\right) ^{n}\\\\rightarrow \\\\prod \\\\left(U\\\\right) $ where $\\\\prod \\\\left( U\\\\right) $ is the set of partitions on a set $U$ .\",\n \"A partition $\\\\pi =\\\\left\\\\lbrace B,B^{\\\\prime },...\\\\right\\\\rbrace $ on a set $U=\\\\left\\\\lbrace u,u^{\\\\prime },...\\\\right\\\\rbrace $ is a set of disjoint non-empty subsets $B,B^{\\\\prime },...$ of $U$ , called blocks, whose union is $U$ .\",\n \"The corresponding equivalence relation, denoted $\\\\operatornamewithlimits{indit}\\\\left(\\\\pi \\\\right) $ , is the set of ordered pairs of elements of $U$ that are in the same block of $\\\\pi $ , and are called the indistinctions or indits of $\\\\pi $ , i.e., $\\\\operatornamewithlimits{indit}\\\\left( \\\\pi \\\\right) =\\\\left\\\\lbrace \\\\left( u,u^{\\\\prime }\\\\right) \\\\in U\\\\times U:\\\\exists B\\\\in \\\\pi ,u,u^{\\\\prime }\\\\in B\\\\right\\\\rbrace $ .\",\n \"The complement $\\\\operatornamewithlimits{dit}\\\\left( \\\\pi \\\\right) =U\\\\times U-\\\\operatornamewithlimits{indit}\\\\left( \\\\pi \\\\right) $ is the set of distinctions or dits of $\\\\pi $ , i.e., ordered pairs of elements in different blocks.\",\n \"As binary relations, the sets of distinctions or ditsets $\\\\operatornamewithlimits{dit}\\\\left( \\\\pi \\\\right) $ of some partition $\\\\pi $ on $U$ are called partition (or apartness) relations.\",\n \"Given partitions $\\\\pi =\\\\left\\\\lbrace B,B^{\\\\prime },...\\\\right\\\\rbrace $ and $\\\\sigma =\\\\left\\\\lbrace C,C^{\\\\prime },...\\\\right\\\\rbrace $ on $U$ , the refinement relation is the partial order defined by: $\\\\sigma \\\\preceq \\\\pi $ if $\\\\forall B\\\\in \\\\pi ,\\\\exists C\\\\in \\\\sigma ,B\\\\subseteq C $ .\",\n \"At the top of the refinement partial order is the discrete partition $\\\\mathbf {1}=\\\\left\\\\lbrace \\\\left\\\\lbrace u\\\\right\\\\rbrace :u\\\\in U\\\\right\\\\rbrace $ of all singletons and at the bottom is the indiscrete partition $\\\\mathbf {0}=\\\\left\\\\lbrace U\\\\right\\\\rbrace $ with only one block consisting of $U$ .\",\n \"In terms of binary relations, the refinement partial order is just the inclusion partial order on ditsets, i.e., $\\\\sigma \\\\preceq \\\\pi $ iff $\\\\operatornamewithlimits{dit}\\\\left( \\\\sigma \\\\right) \\\\subseteq \\\\operatornamewithlimits{dit}\\\\left( \\\\pi \\\\right) $ .\",\n \"It should be noted that most of the previous literature on partitions (e.g., [1]) uses the opposite partial order of `unrefinement' corresponding to the inclusion relation on equivalence relations–which reverses the definitions of the join and meet of partitions.\"\n ],\n [\n \"The Join Operation on Partitions\",\n \"The join $\\\\pi \\\\vee \\\\sigma $ of partitions $\\\\pi $ and $\\\\sigma $ (least upper bound using the refinement partial order) is the partition whose blocks are the non-empty intersections $B\\\\cap C$ of the blocks of $\\\\pi $ and $\\\\sigma $ (under the unrefinement ordering, it is the meet).\",\n \"In terms of ditsets, $\\\\operatornamewithlimits{dit}\\\\left( \\\\pi \\\\vee \\\\sigma \\\\right) =\\\\operatornamewithlimits{dit}\\\\left(\\\\pi \\\\right) \\\\cup \\\\operatornamewithlimits{dit}\\\\left( \\\\sigma \\\\right) $ .\",\n \"The general method for defining Boolean operations on partitions will be first illustrated with the join operation whose corresponding Boolean operation is disjunction with the truth table.\",\n \"Table: NO_CAPTIONTruth table for disjunction.\",\n \"Let $K\\\\left( U\\\\right) $ be the complete undirected graph on $U$ .\",\n \"The links $u-u^{\\\\prime }$ corresponding to dits, i.e., $\\\\left( u,u^{\\\\prime }\\\\right)\\\\in \\\\operatornamewithlimits{dit}\\\\left( \\\\pi \\\\right) $ , of a partition are labelled with the `truth value' $T_{\\\\pi }$ and corresponding to indits $\\\\left( u,u^{\\\\prime }\\\\right) \\\\in \\\\operatornamewithlimits{indit}\\\\left( \\\\pi \\\\right) $ are labelled with the `truth value' $F_{\\\\pi }$ .\",\n \"Given the two partitions $\\\\pi $ and $\\\\sigma $ , each link in the complete graph $K\\\\left( U\\\\right) $ is labelled with a pair of truth values.\",\n \"The graph $G\\\\left( \\\\pi \\\\vee \\\\sigma \\\\right) $ of the join is obtained by putting a link $u-u^{\\\\prime }$ where the truth function applied to the pair of truth values on the link in $K\\\\left(U\\\\right) $ gives an $F$ .\",\n \"Thus in the case at hand, the only links in $G\\\\left( \\\\pi \\\\vee \\\\sigma \\\\right) $ are for the $u-u^{\\\\prime }$ labelled with $F_{\\\\pi }$ and $F_{\\\\sigma }$ in $K\\\\left( U\\\\right) $ .\",\n \"Then the partition $\\\\pi \\\\vee \\\\sigma $ is obtained as the connected components of its graph $G\\\\left(\\\\pi \\\\vee \\\\sigma \\\\right) $ .\",\n \"Thus $u$ and $u^{\\\\prime }$ are in the same block (connected component of $G\\\\left( \\\\pi \\\\vee \\\\sigma \\\\right) $ ) if and only if the link $u-u^{\\\\prime }$ was labelled $F_{\\\\pi }$ and $F_{\\\\sigma }$ , i.e., $u$ and $u^{\\\\prime }$ were in the same block of $\\\\pi $ and in the same block of $\\\\sigma $ .\",\n \"Thus the graph-theoretic definition of the join reproduces the set-of-blocks definition of the join defined as having its blocks the non-empty intersections of the blocks of $\\\\pi $ and $\\\\sigma $ .\"\n ],\n [\n \"The Meet Operation on Partitions\",\n \"On the combined set of blocks $\\\\pi \\\\cup \\\\sigma $ of $\\\\pi $ and $\\\\sigma $ , define the overlap relation $B\\\\between C$ on two blocks if they have a non-empty intersection or overlap (see [8]).\",\n \"The reflexive-symmetric-transitive closure of this relation is an equivalence relation, and the union of the blocks in each equivalence class gives the blocks of the meet $\\\\pi \\\\wedge \\\\sigma $ .\",\n \"The corresponding truth-functional operation is conjunction with the following truth table.\",\n \"Table: NO_CAPTIONTruth table for conjunction.\",\n \"The same method is applied except that the links of the graph $G\\\\left(\\\\pi \\\\wedge \\\\sigma \\\\right) $ are the ones for which the conjunction truth table gives an $F$ when applied to the truth values on each link $u-u^{\\\\prime }$ .\",\n \"Thus $G\\\\left( \\\\pi \\\\wedge \\\\sigma \\\\right) $ contains a link $u-u^{\\\\prime }$ if $\\\\left( u,u^{\\\\prime }\\\\right) \\\\in \\\\operatornamewithlimits{indit}\\\\left( \\\\pi \\\\right) $ , $\\\\left( u,u^{\\\\prime }\\\\right) \\\\in \\\\operatornamewithlimits{indit}\\\\left( \\\\sigma \\\\right)$ , or both.\",\n \"Then the blocks of the partition $\\\\pi \\\\wedge \\\\sigma $ are the connected components of the graph $G\\\\left( \\\\pi \\\\wedge \\\\sigma \\\\right) $ .\",\n \"The proof that the graph-theoretic definition of the meet gives the usual set-of-blocks definition of the meet boils down to showing that: $B\\\\in \\\\pi $ and $C\\\\in \\\\sigma $ are contained in the same block of the usual meet $\\\\pi \\\\wedge \\\\sigma $ (i.e., there is a chain of overlaps $B\\\\between C^{\\\\prime }\\\\between ...\\\\between B^{\\\\prime }\\\\between C$ connecting $B$ and $C$ ) if and only for any $u\\\\in B$ and $u^{\\\\prime }\\\\in C$ , $u$ and $u^{\\\\prime }$ are in the same connected component of $G\\\\left( \\\\pi \\\\wedge \\\\sigma \\\\right) $ .\",\n \"If any two blocks $B^{\\\\prime }\\\\between C^{\\\\prime }$ overlap in the overlap chain, then there is an element $u^{\\\\prime \\\\prime }\\\\in B^{\\\\prime }\\\\cap C^{\\\\prime }$ such any $u\\\\in B^{\\\\prime }$ had a link $u-u^{\\\\prime \\\\prime }$ in $G\\\\left( \\\\pi \\\\wedge \\\\sigma \\\\right) $ and similarly any $u^{\\\\prime }\\\\in C^{\\\\prime }$ has a link $u^{\\\\prime \\\\prime }-u^{\\\\prime }$ in $G\\\\left( \\\\pi \\\\wedge \\\\sigma \\\\right) $ .\",\n \"Hence the existence of an overlap chain connecting $B$ and $C$ implies that any $u\\\\in B$ and $u^{\\\\prime }\\\\in C$ are in the same connected component of $G\\\\left( \\\\pi \\\\wedge \\\\sigma \\\\right) $ .\",\n \"Conversely, if $u\\\\in B$ and $u^{\\\\prime }\\\\in C$ are in the same connected component of $G\\\\left( \\\\pi \\\\wedge \\\\sigma \\\\right) $ , then there is some chain of links $u=u_{0}-u_{1}-...-u_{n-1}-u_{n}=u^{\\\\prime }$ where each link $u_{i}-u_{i+1}$ for $i=0,...,n-1$ has either $\\\\left( u_{i},u_{i+1}\\\\right) \\\\in \\\\operatornamewithlimits{indit}\\\\left( \\\\pi \\\\right) $ , $\\\\left( u_{i},u_{i+1}\\\\right)\\\\in \\\\operatornamewithlimits{indit}\\\\left( \\\\sigma \\\\right) $ , or both.\",\n \"Every link $u_{i}-u_{i+1}$ that is in one indit set but not the other, say, $\\\\left(u_{i},u_{i+1}\\\\right) \\\\in \\\\operatornamewithlimits{indit}\\\\left( \\\\pi \\\\right) $ and $\\\\left( u_{i},u_{i+1}\\\\right) \\\\notin \\\\operatornamewithlimits{indit}\\\\left(\\\\sigma \\\\right) $ , establishes an overlap between the block of $\\\\pi $ containing $u_{i},u_{i+1}$ and the block of $\\\\sigma $ containing $u_{i}$ as well as the different block of $\\\\sigma $ containing $u_{i+1}$ .\",\n \"Thus the chain of links connecting $u\\\\in B$ and $u^{\\\\prime }\\\\in C$ establishes a chain of overlapping blocks connecting $B$ and $C$ .\"\n ],\n [\n \"The Implication Operation on Partitions\",\n \"The real beginning of the logic of partitions, as opposed to the lattice theory of partitions, was the discovery of the set-of-blocks definition of the implication operation $\\\\sigma \\\\Rightarrow \\\\pi $ for partitions ([5], [6]).\",\n \"The intuitive idea is that $\\\\sigma \\\\Rightarrow \\\\pi $ functions like an indicator or characteristic function to indicate which blocks $B$ of $\\\\pi $ are contained in a block of $\\\\sigma $ .\",\n \"View the discretized version of $B\\\\in \\\\pi $ , i.e., $B$ replaced by the set of singletons of the elements of $B$ , as the local version $\\\\mathbf {1}_{B}$ of the discrete partition $\\\\mathbf {1}$ , and view the block $B$ remaining whole as the local version $\\\\mathbf {0}_{B}$ of the indiscrete partition $\\\\mathbf {0}$ .\",\n \"Then the partition implication as the inclusion indicator function is: the blocks of $\\\\sigma \\\\Rightarrow \\\\pi $ are for any $B\\\\in \\\\pi $ : $\\\\left\\\\lbrace \\\\begin{array}[c]{l}\\\\mathbf {1}_{B}\\\\text{ if }\\\\exists C\\\\in \\\\sigma ,B\\\\subseteq C\\\\\\\\\\\\mathbf {0}_{B}=B\\\\text{ otherwise.}\\\\end{array}\\\\right.\",\n \".$ In the case of the Boolean logic of subsets, for any subsets $S,T\\\\subseteq U$ , the conditional $S\\\\supset T=S^{c}\\\\cup T$ has the property: $S\\\\supset T=U$ iff $S\\\\subseteq T$ , i.e., the conditional $S\\\\supset T$ equals the top of the lattice of subsets of $U$ iff the inclusion relation $S\\\\subseteq T$ holds.\",\n \"Similarly, it is immediate that the corresponding relation holds in the partition case: $\\\\sigma \\\\Rightarrow \\\\pi =\\\\mathbf {1}$ iff $\\\\sigma \\\\preceq \\\\pi $ .\",\n \"This set-of-blocks definition of the partition implication operation accounts for the important new non-lattice-theoretic properties revealed in the algebra of partitions $\\\\prod \\\\left( U\\\\right) $ on $U $ (defined with the join, meet, and implication as partition operations).\",\n \"A logical formula in the language of join, meet, and implication is a subset tautology if for any non-empty universe $U$ and any subsets of $U$ substituted for the variables, the whole formula evaluates by the set-theoretic operations of join, meet, and implication (conditional) to the top $U$ .\",\n \"Similarly, a formula in the same language is a partition tautology if for any universe $U$ with $\\\\left|U\\\\right|>1$ and for any partitions on $U$ substituted for the variables, the whole formula evaluates by the partition operations of join, meet, and implication to the top $\\\\mathbf {1}$ (the discrete partition).\",\n \"All partition tautologies are subset tautologies but not vice-versa.\",\n \"Modus ponens $\\\\left(\\\\sigma \\\\wedge \\\\left( \\\\sigma \\\\Rightarrow \\\\pi \\\\right) \\\\right) \\\\Rightarrow \\\\pi $ is both a subset and partition tautology but Peirce's law, $\\\\left( \\\\left(\\\\sigma \\\\Rightarrow \\\\pi \\\\right) \\\\Rightarrow \\\\sigma \\\\right) \\\\Rightarrow \\\\sigma $ , accumulation, $\\\\sigma \\\\Rightarrow \\\\left( \\\\pi \\\\Rightarrow \\\\left( \\\\sigma \\\\wedge \\\\pi \\\\right) \\\\right) $ , and distributivity, $\\\\left( \\\\left( \\\\pi \\\\vee \\\\sigma \\\\right) \\\\wedge \\\\left( \\\\pi \\\\vee \\\\tau \\\\right) \\\\right) \\\\Rightarrow \\\\left(\\\\pi \\\\vee \\\\left( \\\\sigma \\\\wedge \\\\tau \\\\right) \\\\right) $ , are examples of subset tautologies that are not partition tautologies.\",\n \"The importance of the implication for partition logic is emphasized by the fact that the only partition tautologies using only the lattice operations, e.g., $\\\\pi \\\\vee \\\\mathbf {1}$ , correspond to general lattice-theoretic identities, i.e., $\\\\pi \\\\vee \\\\mathbf {1}=\\\\mathbf {1}$ (see [9]).\",\n \"The graph-theoretic method automatically gives a partition operation corresponding to the Boolean conditional or implication with the truth table: Table: NO_CAPTIONTruth table for conditional and it is not trivial that the two definitions are the same.\",\n \"It may be helpful to restate the truth table in terms of the partitions.\",\n \"Table: NO_CAPTIONImplication truth table for partition `truth values'.\",\n \"For the graph-theoretic definition of $\\\\sigma \\\\Rightarrow \\\\pi $ , we again label the links $u-u^{\\\\prime }$ in the complete graph $K\\\\left( U\\\\right) $ with $T_{\\\\pi }$ if $\\\\left( u,u^{\\\\prime }\\\\right) \\\\in \\\\operatornamewithlimits{dit}\\\\left(\\\\pi \\\\right) $ and $F_{\\\\pi }$ otherwise, and similarly for $\\\\sigma $ .\",\n \"Then we construct the graph $G\\\\left( \\\\sigma \\\\Rightarrow \\\\pi \\\\right) $ by putting in a link $u-u^{\\\\prime }$ only in the case the link is labeled $T_{\\\\sigma }$ and $F_{\\\\pi }$ , i.e., $F_{\\\\sigma \\\\Rightarrow \\\\pi }$ .\",\n \"Then the partition $\\\\sigma \\\\Rightarrow \\\\pi $ is the partition of connected components in the graph $G\\\\left( \\\\sigma \\\\Rightarrow \\\\pi \\\\right) $ .\",\n \"To prove the graph-theoretic and set-of-blocks definitions equivalent, we might first note that if $\\\\left( u,u^{\\\\prime }\\\\right) \\\\in \\\\operatornamewithlimits{dit}\\\\left( \\\\pi \\\\right) $ , then $T_{\\\\pi }$ is assigned to that link in $K\\\\left(U\\\\right) $ so there is no link $u-u^{\\\\prime }$ in $G\\\\left( \\\\sigma \\\\Rightarrow \\\\pi \\\\right) $ .\",\n \"And if $\\\\left( u,u^{\\\\prime }\\\\right) \\\\in \\\\operatornamewithlimits{indit}\\\\left( \\\\pi \\\\right) $ but also $\\\\left( u,u^{\\\\prime }\\\\right) \\\\in \\\\operatornamewithlimits{indit}\\\\left( \\\\sigma \\\\right) $ , then $T_{\\\\sigma \\\\Rightarrow \\\\pi }$ is assigned to the link in $K\\\\left( U\\\\right) $ so again there is no link $u-u^{\\\\prime }$ in $G\\\\left( \\\\sigma \\\\Rightarrow \\\\pi \\\\right) $ .\",\n \"There is a link $u-u^{\\\\prime }$ in $G\\\\left( \\\\sigma \\\\Rightarrow \\\\pi \\\\right) $ in and only in the following situation where $\\\\left(u,u^{\\\\prime }\\\\right) \\\\in \\\\operatornamewithlimits{indit}\\\\left( \\\\pi \\\\right) $ and $\\\\left( u,u^{\\\\prime }\\\\right) \\\\in \\\\operatornamewithlimits{dit}\\\\left( \\\\sigma \\\\right)$ –which is exactly the situation when $B$ is not contained in any block $C$ of $\\\\sigma $ : Figure: NO_CAPTION Figure 1: Links $u-u^{\\\\prime }$ in $G\\\\left( \\\\sigma \\\\Rightarrow \\\\pi \\\\right) $ .\",\n \"Then for any other element $u^{\\\\prime \\\\prime }\\\\in B$ so that $\\\\left(u,u^{\\\\prime \\\\prime }\\\\right) $ and $\\\\left( u^{\\\\prime },u^{\\\\prime \\\\prime }\\\\right)\\\\in \\\\operatornamewithlimits{indit}\\\\left( \\\\pi \\\\right) $ , we must have either $\\\\left(u,u^{\\\\prime \\\\prime }\\\\right) \\\\in \\\\operatornamewithlimits{dit}\\\\left( \\\\sigma \\\\right) $ or $\\\\left( u^{\\\\prime },u^{\\\\prime \\\\prime }\\\\right) \\\\in \\\\operatornamewithlimits{dit}\\\\left(\\\\sigma \\\\right) $ so $u^{\\\\prime \\\\prime }$ is linked in $G\\\\left( \\\\sigma \\\\Rightarrow \\\\pi \\\\right) $ to either $u$ or to $u^{\\\\prime }$ .\",\n \"Thus all the elements of $B$ are in the same connected component of the graph $G\\\\left(\\\\sigma \\\\Rightarrow \\\\pi \\\\right) $ whenever $B$ is not contained in any block of $\\\\sigma $ .\",\n \"If, on the other hand, $B$ is contained in some block $C$ of $\\\\sigma $ , then any $u\\\\in B$ cannot be linked to any other $u^{\\\\prime }$ .\",\n \"In order to that $F_{\\\\pi }$ assigned to the link $u-u^{\\\\prime }$ , the two elements have to both belong to $B$ and thus since $B\\\\subseteq C$ , they both belong to $C$ so $F_{\\\\sigma }$ and thus $T_{\\\\sigma \\\\Rightarrow \\\\pi }$ is also assigned to that link.\",\n \"Thus when $B$ is contained in a block $C\\\\in \\\\sigma $ , then any point $u\\\\in B$ is a disconnected component to itself in $G\\\\left( \\\\sigma \\\\Rightarrow \\\\pi \\\\right) $ so $B$ is discretized in the graph-theoretic construction of $\\\\sigma \\\\Rightarrow \\\\pi $ .\",\n \"Thus the graph-theoretic and set-of-blocks definitions of the partition implication are equivalent.\",\n \"Figure: NO_CAPTION Figure 2: Example of graph for partition implication Example 1 Let $U=\\\\left\\\\lbrace a,b,c,d\\\\right\\\\rbrace $ so that $K(U)=K_{4}$ is the complete graph on four points.\",\n \"Let $\\\\sigma =\\\\left\\\\lbrace \\\\left\\\\lbrace a\\\\right\\\\rbrace ,\\\\left\\\\lbrace b,c,d\\\\right\\\\rbrace \\\\right\\\\rbrace $ and $\\\\pi =\\\\left\\\\lbrace \\\\left\\\\lbrace a,b\\\\right\\\\rbrace ,\\\\left\\\\lbrace c,d\\\\right\\\\rbrace \\\\right\\\\rbrace $ so we see immediately from the set-of-blocks definition, that the $\\\\pi $ -block of $\\\\left\\\\lbrace c,d\\\\right\\\\rbrace $ will be discretized while the $\\\\pi $ -block of $\\\\left\\\\lbrace a,b\\\\right\\\\rbrace $ will remain whole so the partition implication is $\\\\sigma \\\\Rightarrow \\\\pi =\\\\left\\\\lbrace \\\\left\\\\lbrace a,b\\\\right\\\\rbrace ,\\\\left\\\\lbrace c\\\\right\\\\rbrace ,\\\\left\\\\lbrace d\\\\right\\\\rbrace \\\\right\\\\rbrace $ .\",\n \"After labelling the links in $K\\\\left( U\\\\right) $ , we see that only the $a-b$ link has the $F_{\\\\sigma \\\\Rightarrow \\\\pi }$ `truth value' so the graph $G\\\\left(\\\\sigma \\\\Rightarrow \\\\pi \\\\right) $ has only that $a-b$ link (thickened in Figure 2).\",\n \"Then the connected components of $G\\\\left( \\\\sigma \\\\Rightarrow \\\\pi \\\\right) $ give the same partition implication $\\\\sigma \\\\Rightarrow \\\\pi =\\\\left\\\\lbrace \\\\left\\\\lbrace a,b\\\\right\\\\rbrace ,\\\\left\\\\lbrace c\\\\right\\\\rbrace ,\\\\left\\\\lbrace d\\\\right\\\\rbrace \\\\right\\\\rbrace $ .\",\n \"The partition implication is quite rich in defining new structures in the algebra of partitions (i.e., the lattice of partitions extended with other partition operations such as the implication).\",\n \"For instance, for a fixed partition $\\\\pi $ on $U$ , all the partitions of the form $\\\\sigma \\\\Rightarrow \\\\pi $ (for any partitions $\\\\sigma $ on $U$ ) form a Boolean algebra under the partition operations of implication, join, and meet, e.g., $\\\\left(\\\\sigma \\\\Rightarrow \\\\pi \\\\right) \\\\Rightarrow \\\\pi $ is the negation of $\\\\sigma \\\\Rightarrow \\\\pi $ , called the Boolean core of the upper segment $\\\\left[ \\\\pi ,\\\\mathbf {1}\\\\right] $ in the partition algebra $\\\\prod \\\\left(U\\\\right) $ .\",\n \"A relation is a subset of a product, and, dually, a corelation is a partition on a coproduct.\",\n \"Any partition $\\\\pi $ on $U$ can be canonically represented as a relation: $\\\\operatornamewithlimits{dit}\\\\left(\\\\pi \\\\right) \\\\subseteq U\\\\times U$ .\",\n \"Dually any subset $S\\\\subseteq U$ can be canonically represented as a corelation, namely the partition $\\\\pi \\\\left(S\\\\right) $ on the coproduct (disjoint union) $U\\\\uplus U$ where the only nonsingleton blocks in $\\\\pi \\\\left( S\\\\right) $ are the pairs $\\\\left\\\\lbrace u,u^{\\\\ast }\\\\right\\\\rbrace $ of $u$ and its copy $u^{\\\\ast }$ for $u\\\\notin S$ .\",\n \"Using this corelation construction, any powerset Boolean algebra $\\\\wp \\\\left(U\\\\right) $ can be canonically represented as the Boolean core of the upper segment $\\\\left[ \\\\pi ,\\\\mathbf {1}\\\\right] $ in the partition algebra $\\\\prod \\\\left( U\\\\uplus U\\\\right) $ where $\\\\pi =\\\\pi \\\\left( \\\\emptyset \\\\right) $ is the partition on the disjoint union $U\\\\uplus U$ whose blocks are all the pairs $\\\\left\\\\lbrace u,u^{\\\\ast }\\\\right\\\\rbrace $ for each element $u\\\\in U$ and its copy $u^{\\\\ast }$ .\",\n \"Each partition of the form $\\\\sigma \\\\Rightarrow \\\\pi $ on $U\\\\uplus U$ is $\\\\pi \\\\left( S\\\\right) $ for some $S\\\\subseteq U$ since $\\\\sigma \\\\Rightarrow \\\\pi $ is essentially the characteristic function of some subset $S$ of $U$ with $\\\\mathbf {1}\\\\Rightarrow \\\\pi =\\\\pi \\\\left( \\\\emptyset \\\\right) $ playing the role of the empty set $\\\\emptyset $ and $\\\\pi \\\\Rightarrow \\\\pi =\\\\mathbf {1}_{U\\\\uplus U}$ playing the role of $U$ .\"\n ],\n [\n \"The General Graph-Theoretic Method\",\n \"Let $f:\\\\left\\\\lbrace T,F\\\\right\\\\rbrace ^{n}\\\\rightarrow \\\\left\\\\lbrace T,F\\\\right\\\\rbrace $ be an $n$ -ary Boolean function and let $\\\\pi _{1},...,\\\\pi _{n}$ be $n$ partitions on $U$ .\",\n \"In order to define the corresponding $n$ -ary partition operation $f\\\\left( \\\\pi _{1},...,\\\\pi _{n}\\\\right) $ , we again consider the complete graph $K\\\\left( U\\\\right) $ and then use each partition $\\\\pi _{i}$ to label each link $u-u^{\\\\prime }$ with $T_{\\\\pi _{i}}$ if $\\\\left( u,u^{\\\\prime }\\\\right)\\\\in \\\\operatornamewithlimits{dit}\\\\left( \\\\pi _{i}\\\\right) $ and $F_{\\\\pi _{i}}$ if $\\\\left(u,u^{\\\\prime }\\\\right) \\\\in \\\\operatornamewithlimits{indit}\\\\left( \\\\pi _{i}\\\\right) $ .\",\n \"Then on each link we may apply $f$ to the $n$ `truth values' on the link and retain the link in $G\\\\left( f\\\\left( \\\\pi _{1},...,\\\\pi _{n}\\\\right) \\\\right) $ if the result was $F_{f\\\\left( \\\\pi _{1},...,\\\\pi _{n}\\\\right) }$ .\",\n \"The partition $f\\\\left( \\\\pi _{1},...,\\\\pi _{n}\\\\right) $ is obtained as the connected components of the graph $G\\\\left( f\\\\left( \\\\pi _{1},...,\\\\pi _{n}\\\\right)\\\\right) $ .\"\n ],\n [\n \"An Equivalent Closure-theoretic Method\",\n \"Given any subset $S\\\\subseteq U\\\\times U$ , the reflexive-symmetric-transitive (RST) closure $\\\\overline{S}$ is the intersection of all equivalence relations on $U$ containing $S$ .\",\n \"The `topological' terminology of calling a subset closed if $S=\\\\overline{S}$ is used even though the RST closure operator is not a topological closure operator since the union of two closed sets is not necessarily closed.\",\n \"The closed sets in $U\\\\times U$ are the equivalence relations (or indit sets of partitions), and their complements, the open sets, are the partition relations (or ditsets of partitions).\",\n \"As usual, the interior operator $\\\\operatornamewithlimits{int}\\\\left( S\\\\right) =\\\\left(\\\\overline{S^{c}}\\\\right) ^{c}$ is the complement of the closure of the complement, and the open sets are the ones equalling their interiors.\",\n \"The closure-theoretic method of defining Boolean operations on partitions will be illustrated using the symmetric difference or inequivalence operation $\\\\pi \\\\oplus \\\\sigma $ .\",\n \"Every $n$ -ary Boolean operation can be defined by a truth table such as the one for symmetric difference in this case: Table: NO_CAPTIONTruth table for symmetric difference.\",\n \"The disjunctive normal form (DNF) for the formula $P\\\\oplus Q$ is given by the rows where the formula evaluates as $T$ , i.e., $P\\\\oplus Q=\\\\left( P\\\\wedge \\\\lnot Q\\\\right) \\\\vee \\\\left( \\\\lnot P\\\\wedge Q\\\\right) $ , while the DNF for the negation of the formula is given by the other rows where the formula evaluates as $F$ , i.e., $\\\\lnot \\\\left( P\\\\oplus Q\\\\right) =\\\\left( P\\\\wedge Q\\\\right)\\\\vee \\\\left( \\\\lnot P\\\\wedge \\\\lnot Q\\\\right) $ .\",\n \"Given two partitions $\\\\pi $ and $\\\\sigma $ on $U$ , the closure-theoretic method of obtaining the partition $\\\\pi \\\\oplus \\\\sigma $ is to start with the DNF for the negated Boolean formula and replace each unnegated variable by the corresponding ditset and each negated variable by the corresponding indit set–as well as replacing the disjunctions and conjunctions by the corresponding subset operations of union and intersection.\",\n \"Applied to $\\\\lnot \\\\left( P\\\\oplus Q\\\\right) =\\\\left( P\\\\wedge Q\\\\right) \\\\vee \\\\left( \\\\lnot P\\\\wedge \\\\lnot Q\\\\right) $ , this procedure would yield $\\\\left( \\\\operatornamewithlimits{dit}\\\\left( \\\\pi \\\\right) \\\\cap \\\\operatornamewithlimits{dit}\\\\left( \\\\sigma \\\\right) \\\\right) \\\\cup \\\\left( \\\\operatornamewithlimits{indit}\\\\left(\\\\pi \\\\right) \\\\cap \\\\operatornamewithlimits{indit}\\\\left( \\\\sigma \\\\right) \\\\right) \\\\subseteq U\\\\times U$ .\",\n \"Then the indit set of $\\\\pi \\\\oplus \\\\sigma $ is obtained as the RST closure: $\\\\operatornamewithlimits{indit}\\\\left( \\\\pi \\\\oplus \\\\sigma \\\\right) =\\\\overline{\\\\left(\\\\operatornamewithlimits{dit}\\\\left( \\\\pi \\\\right) \\\\cap \\\\operatornamewithlimits{dit}\\\\left(\\\\sigma \\\\right) \\\\right) \\\\cup \\\\left( \\\\operatornamewithlimits{indit}\\\\left( \\\\pi \\\\right)\\\\cap \\\\operatornamewithlimits{indit}\\\\left( \\\\sigma \\\\right) \\\\right) }$ and the partition $\\\\pi \\\\oplus \\\\sigma $ is the set of equivalence classes of this equivalence relation.\",\n \"The graph-theoretic method of obtaining the partition $\\\\pi \\\\oplus \\\\sigma $ would label each link $u-u^{\\\\prime }$ in $K\\\\left( U\\\\right) $ by the two `truth values' given by $\\\\pi $ and $\\\\sigma $ , and then retain in the graph $G\\\\left(\\\\pi \\\\oplus \\\\sigma \\\\right) $ the links where the truth values evaluated to $F_{\\\\pi \\\\oplus \\\\sigma }$ , namely the ones labelled with $T_{\\\\pi },T_{\\\\sigma }$ and $F_{\\\\pi },F_{\\\\sigma }$ .\",\n \"Then the partition $\\\\pi \\\\oplus \\\\sigma $ is obtained as the connected components of the graph $G\\\\left( \\\\pi \\\\oplus \\\\sigma \\\\right) $ .\",\n \"To see the equivalence between the two methods, note first that the links retained in $G\\\\left( \\\\pi \\\\oplus \\\\sigma \\\\right) $ are precisely the pairs $\\\\left( u,u^{\\\\prime }\\\\right) $ in $\\\\left( \\\\operatornamewithlimits{dit}\\\\left(\\\\pi \\\\right) \\\\cap \\\\operatornamewithlimits{dit}\\\\left( \\\\sigma \\\\right) \\\\right) \\\\cup \\\\left(\\\\operatornamewithlimits{indit}\\\\left( \\\\pi \\\\right) \\\\cap \\\\operatornamewithlimits{indit}\\\\left(\\\\sigma \\\\right) \\\\right) $ .\",\n \"The equivalence proof is completed by showing that taking connected components in the graph $G\\\\left( \\\\pi \\\\oplus \\\\sigma \\\\right) $ is equivalent to taking the RST closure of $\\\\left( \\\\operatornamewithlimits{dit}\\\\left(\\\\pi \\\\right) \\\\cap \\\\operatornamewithlimits{dit}\\\\left( \\\\sigma \\\\right) \\\\right) \\\\cup \\\\left(\\\\operatornamewithlimits{indit}\\\\left( \\\\pi \\\\right) \\\\cap \\\\operatornamewithlimits{indit}\\\\left(\\\\sigma \\\\right) \\\\right) $ .\",\n \"The elements $u$ and $u^{\\\\prime }$ are in the same connected component of $G\\\\left( \\\\pi \\\\oplus \\\\sigma \\\\right) $ iff there is a chain of links $u=u_{0}-u_{1}-...-u_{n-1}-u_{n}=u^{\\\\prime }$ in the graph $G\\\\left( \\\\pi \\\\oplus \\\\sigma \\\\right) $ so each link has to be originally labelled $T_{\\\\pi },T_{\\\\sigma }$ or $F_{\\\\pi },F_{\\\\sigma }$ in the graph on $K\\\\left(U\\\\right) $ .\",\n \"But the condition for $\\\\left( u,u^{\\\\prime }\\\\right) $ to be included in the RST closure $\\\\overline{\\\\left( \\\\operatornamewithlimits{dit}\\\\left(\\\\pi \\\\right) \\\\cap \\\\operatornamewithlimits{dit}\\\\left( \\\\sigma \\\\right) \\\\right) \\\\cup \\\\left(\\\\operatornamewithlimits{indit}\\\\left( \\\\pi \\\\right) \\\\cap \\\\operatornamewithlimits{indit}\\\\left(\\\\sigma \\\\right) \\\\right) }$ is that there is a chain of pairs $\\\\left(u,u_{1}\\\\right) ,\\\\left( u_{1},u_{2}\\\\right) ,...,\\\\left( u_{n-1},u^{\\\\prime }\\\\right) $ such that each pair is either in $\\\\operatornamewithlimits{dit}\\\\left(\\\\pi \\\\right) \\\\cap \\\\operatornamewithlimits{dit}\\\\left( \\\\sigma \\\\right) $ or in $\\\\operatornamewithlimits{indit}\\\\left( \\\\pi \\\\right) \\\\cap \\\\operatornamewithlimits{indit}\\\\left(\\\\sigma \\\\right) $ .\",\n \"Hence the two methods give the same result.\",\n \"The example suffices to illustrate the general closure-theoretic method and its equivalence to the graph-theoretic method of defining Boolean operations on partitions.\"\n ],\n [\n \"Relationships between Boolean operations on partitions\",\n \"For two subset variables, there are $2^{4}=16$ binary Boolean operations on subsets–corresponding to the sixteen ways to fill in the truth table for a binary Boolean operation.\",\n \"Any compound Boolean function of two variables will be truth-table equivalent to one of the sixteen binary Boolean operations.\",\n \"For instance, the Pierce's Law formula $\\\\left( \\\\left( Q\\\\Rightarrow P\\\\right)\\\\Rightarrow Q\\\\right) \\\\Rightarrow Q$ defines a compound binary operation that is equivalent to the constant function $T$ since it is a subset tautology.\",\n \"Certain subsets of the sixteen binary operations suffice to define all the binary operations, e.g., $\\\\lnot $ and $\\\\vee $ .\",\n \"Matters are rather different for the Boolean operations on partitions.\",\n \"Using the graph-theoretic or the closure-theoretic method, partition versions of sixteen binary Boolean operations are easily defined.\",\n \"And certain combinations of the sixteen operations suffice to define all sixteen, e.g., $\\\\vee $ , $\\\\wedge $ , $\\\\Rightarrow $ , and $\\\\oplus $ [5].\",\n \"But when the sixteen operations are compounded, still keeping to two variables, then the resulting binary partition operations does not necessarily reduce to one of the sixteen–due to the complicated compounding of the closure operations.\",\n \"For instance, the Pierce's Law formula $\\\\left( \\\\left(\\\\sigma \\\\Rightarrow \\\\pi \\\\right) \\\\Rightarrow \\\\sigma \\\\right) \\\\Rightarrow \\\\sigma $ for partitions is not equivalent to the constant function $\\\\mathbf {1}$ since it is not a partition tautology.\",\n \"The topic of the total number of binary operations on partitions obtained by compounding the sixteen basic binary Boolean operations is one of many topics in partition logic that awaits future research.\"\n ],\n [\n \"Concluding Remarks\",\n \"In conclusion, perhaps some remarks are in order as to why it took so long to extend the Boolean operations to partitions.\",\n \"The Boolean operations are normally associated with subsets of a set or, more specifically, with propositions.\",\n \"Boole originally defined his logic as the logic of subsets [2] of a universe set.\",\n \"It is then a theorem that the same set of subset tautologies is obtained as the truth-table tautologies.\",\n \"Perhaps because “logic” has been historically associated with propositions, the texts in mathematical logic throughout the twentieth century (to the author's knowledge) ignored the Boolean logic of subsets and started with the special case of the logic of propositions and then took the truth-table characterization as the definition of a tautology.\",\n \"By the middle of the twentieth century, category theory was defined [4] and the category-theoretic duality was established between subobjects and quotient objects, e.g., between subsets of $U$ and quotient sets (or equivalently equivalence relations or partitions) of $U$ .\",\n \"The conceptual cost of restricting subset logic to the special case of propositional logic is that subsets have the category-theoretic dual concept of partitions while propositions have no such dual concept.\",\n \"Hence the focus on “propositional logic” did not lead to the search for the dual logic of partitions ([5], [6]) or to the simple and natural application of Boolean operations to partitions as well as subsets–which has been our topic here.\"\n ]\n]"},"id":{"kind":"string","value":"1906.04539"}}},{"rowIdx":2597,"cells":{"text":{"kind":"list like","value":[["Almost Optimal Semi-streaming Maximization for k-Extendible Systems"],["Abstract In this paper we consider the problem of finding a maximum weight set subject to a $k$-extendible constraint in the data stream model.","The only non-trivial algorithm known for this problem to date---to the best of our knowledge---is a semi-streaming $k^2(1 + \\varepsilon)$-approximation algorithm (Crouch and Stubbs, 2014), but semi-streaming $O(k)$-approximation algorithms are known for many restricted cases of this general problem.","In this paper, we close most of this gap by presenting a semi-streaming $O(k \\log k)$-approximation algorithm for the general problem, which is almost the best possible even in the offline setting (Feldman et al., 2017)."],["positioning arrows backgrounds calc,arrows.meta every node=[font=] [ framed, align=center, node distance=0.25cm, roundnode/.style=circle, draw=black!60, minimum width=width(\"-2L+2\"),, blanknode/.style=circle, draw=white!60, minimum width=width(\"-2L+2\"),font=, ] blanknode] (-2L) $\\vdots $ ; blanknode] (-2L+1) [right=of -2L] $\\vdots $ ; blanknode] (-2L+2) [right=of -2L+1] $\\vdots $ ; blanknode] (qdots0) [right=of -2L+2] ; roundnode, fill=green] (-L) [below=of -2L] -$\\ell $ ; roundnode, fill=orange] (-L+1) [right=of -L] -$\\ell $ +1; roundnode, fill=yellow] (-L+2) [right=of -L+1] -$\\ell $ +2; blanknode] (qdots1) [right=of -L+2] $\\cdots $ ; roundnode, fill=red] (-1) [right=of qdots1] -1; roundnode, fill=green] (0) [below=of -L] 0; roundnode, fill=orange] (1) [right=of 0] 1; roundnode, fill=yellow] (2) [right=of 1] 2; blanknode] (qdots2) [right=of 2] $\\cdots $ ; roundnode, fill=red] (L-1) [right=of qdots2] $\\ell $ -1; roundnode, fill=green] (L) [below=of 0] $\\ell $ ; roundnode, fill=orange] (L+1) [right=of L] $\\ell $ +1; roundnode, fill=yellow] (L+2) [right=of L+1] $\\ell $ +2; blanknode] (qdots3) [right=of L+2] $\\cdots $ ; roundnode, fill=red] (2L-1) [right=of qdots3] 2$\\ell $ -1; blanknode] (2L) [below=of L] $\\vdots $ ; blanknode] (2L+1) [right=of 2L] $\\vdots $ ; blanknode] (2L+2) [right=of 2L+1] $\\vdots $ ; blanknode] (qdots4) [right=of 2L+2] ; blanknode] (-L-1) [above=of -1] $\\vdots $ ; blanknode] (3L-1) [below=of 2L-1] $\\vdots $ ;"]],"string":"[\n [\n \"Almost Optimal Semi-streaming Maximization for k-Extendible Systems\"\n ],\n [\n \"Abstract In this paper we consider the problem of finding a maximum weight set subject to a $k$-extendible constraint in the data stream model.\",\n \"The only non-trivial algorithm known for this problem to date---to the best of our knowledge---is a semi-streaming $k^2(1 + \\\\varepsilon)$-approximation algorithm (Crouch and Stubbs, 2014), but semi-streaming $O(k)$-approximation algorithms are known for many restricted cases of this general problem.\",\n \"In this paper, we close most of this gap by presenting a semi-streaming $O(k \\\\log k)$-approximation algorithm for the general problem, which is almost the best possible even in the offline setting (Feldman et al., 2017).\"\n ],\n [\n \"positioning arrows backgrounds calc,arrows.meta every node=[font=] [ framed, align=center, node distance=0.25cm, roundnode/.style=circle, draw=black!60, minimum width=width(\\\"-2L+2\\\"),, blanknode/.style=circle, draw=white!60, minimum width=width(\\\"-2L+2\\\"),font=, ] blanknode] (-2L) $\\\\vdots $ ; blanknode] (-2L+1) [right=of -2L] $\\\\vdots $ ; blanknode] (-2L+2) [right=of -2L+1] $\\\\vdots $ ; blanknode] (qdots0) [right=of -2L+2] ; roundnode, fill=green] (-L) [below=of -2L] -$\\\\ell $ ; roundnode, fill=orange] (-L+1) [right=of -L] -$\\\\ell $ +1; roundnode, fill=yellow] (-L+2) [right=of -L+1] -$\\\\ell $ +2; blanknode] (qdots1) [right=of -L+2] $\\\\cdots $ ; roundnode, fill=red] (-1) [right=of qdots1] -1; roundnode, fill=green] (0) [below=of -L] 0; roundnode, fill=orange] (1) [right=of 0] 1; roundnode, fill=yellow] (2) [right=of 1] 2; blanknode] (qdots2) [right=of 2] $\\\\cdots $ ; roundnode, fill=red] (L-1) [right=of qdots2] $\\\\ell $ -1; roundnode, fill=green] (L) [below=of 0] $\\\\ell $ ; roundnode, fill=orange] (L+1) [right=of L] $\\\\ell $ +1; roundnode, fill=yellow] (L+2) [right=of L+1] $\\\\ell $ +2; blanknode] (qdots3) [right=of L+2] $\\\\cdots $ ; roundnode, fill=red] (2L-1) [right=of qdots3] 2$\\\\ell $ -1; blanknode] (2L) [below=of L] $\\\\vdots $ ; blanknode] (2L+1) [right=of 2L] $\\\\vdots $ ; blanknode] (2L+2) [right=of 2L+1] $\\\\vdots $ ; blanknode] (qdots4) [right=of 2L+2] ; blanknode] (-L-1) [above=of -1] $\\\\vdots $ ; blanknode] (3L-1) [below=of 2L-1] $\\\\vdots $ ;\"\n ]\n]"},"id":{"kind":"string","value":"1906.04449"}}},{"rowIdx":2598,"cells":{"text":{"kind":"list like","value":[["Pykg2vec: A Python Library for Knowledge Graph Embedding"],["Abstract Pykg2vec is an open-source Python library for learning the representations of the entities and relations in knowledge graphs.","Pykg2vec's flexible and modular software architecture currently implements 16 state-of-the-art knowledge graph embedding algorithms, and is designed to easily incorporate new algorithms.","The goal of pykg2vec is to provide a practical and educational platform to accelerate research in knowledge graph representation learning.","Pykg2vec is built on top of TensorFlow and Python's multiprocessing framework and provides modules for batch generation, Bayesian hyperparameter optimization, mean rank evaluation, embedding, and result visualization.","Pykg2vec is released under the MIT License and is also available in the Python Package Index (PyPI).","The source code of pykg2vec is available at https://github.com/Sujit-O/pykg2vec."],["Introduction","In recent years, Knowledge Graph Embedding (KGE) methods have been applied in benchmark datasets including Wikidata ([5]), Freebase ([2]), DBpedia ([1]), and YAGO ([8]).","Applications of KGE methods include fact prediction, question answering, and recommender systems.","KGE is an active area of research and many authors have provided reference software implementations.","However, most of these are standalone reference implementations and therefore it is difficult and time-consuming to: (i) find the source code; (ii) adapt the source code to new datasets; (iii) correctly parameterize the models; and (iv) compare against other methods.","Recently, this problem has been partially addressed by libraries such as OpenKE [6] and AmpliGraph [4] that provide a framework common to several KGE methods.","However, these frameworks take different perspectives, make specific assumptions, and thus the resulting implementations diverge substantially from the original architectures.","Furthermore, these libraries often force the user to use preset hyperparameters, or make implicit use of golden hyperparameters, and thus make it tedious and time-consuming to adapt the models to new datasets.","This paper presents pykg2vec, a single Python library with 16 state-of-the-art KGE methods.","The goals of pykg2vec are to be practical and educational.","The practical value is achieved through: (a) proper use of GPUs and CPUs; (b) a set of tools to automate the discovery of golden hyperparameters; and (c) a set of visualization tools for the training and results of the embeddings.","The educational value is achieved through: (d) a modular and flexible software architecture and KGE pipeline; and (e) access to a large number of state-of-the-art KGE models."],["Knowledge Graph Embedding Methods","A knowledge graph contains a set of entities $\\mathbb {E}$ and relations $\\mathbb {R}$ between entities.","The set of facts $\\mathbb {D}^+$ in the knowledge graph are represented in the form of triples $(h, r, t)$ , where $h,t\\in \\mathbb {E}$ are referred to as the head (or subject) and the tail (or object) entities, and $r\\in \\mathbb {R}$ is referred to as the relationship (or predicate).","The problem of KGE is in finding a function that learns the embeddings of triples using low dimensional vectors such that it preserves structural information, $f:\\mathbb {D}^+\\rightarrow \\mathbb {R}^d$ .","To accomplish this, the general principle is to enforce the learning of entities and relationships to be compatible with the information in $\\mathbb {D}^+$ .","The representation choices include deterministic point ([3]), multivariate Gaussian distribution ([7]), or complex number ([9]).","Under the Open World Assumption (OWA), a set of unseen negative triplets, $\\mathbb {D}^-$ , are sampled from positive triples $\\mathbb {D}^+$ by either corrupting the head or tail entity.","Then, a scoring function, $f_r(h, t)$ is defined to reward the positive triples and penalize the negative triples.","Finally, an optimization algorithm is used to minimize or maximize the scoring function.","KGE methods are often evaluated in terms of their capability of predicting the missing entities in negative triples $(?, r, t)$ or $(h, r, ?",")$ , or predicting whether an unseen fact is true or not.","The evaluation metrics include the rank of the answer in the predicted list (mean rank), and the ratio of answers ranked top-k in the list (hit-k ratio)."],["Software Architecture","The pykg2vec library is built using Python and TensorFlow.","TensorFlow allows the computations to be assigned on both GPU and CPU.","In addition to the main model training process, pykg2vec utilizes multi-processing for generating mini-batches and performing an evaluation to reduce the total execution time.","The various components of the library (see Figure REF ) are as follows: Figure: Pykg2vec software architecture[align=left,leftmargin=*] KG Controller: handles all the low-level parsing tasks such as finding the total unique set of entities and relations; creating ordinal encoding maps; generating training, testing and validation triples; and caching the dataset data on disk to optimize tasks that involve repetitive model testing.","Batch Generator: consists of multiple concurrent processes that manipulate and create mini-batches of data.","These mini-batches are pushed to a queue to be processed by the models implemented in TensorFlow.","The batch generator runs independently so that there is a low latency for feeding the data to the training module running on the GPU.","Core Models: consists of 16 KGE algorithms implemented as Python modules in TensorFlow.","Each module consists of a modular description of the inputs, outputs, loss function, and embedding operations.","Each model is provided with configuration files that define its hyperparameters.","Configuration: provides the necessary configuration to parse the datasets and also consists of the baseline hyperparameters for the KGE algorithms as presented in the original research papers.","Trainer and Evaluator: the Trainer module is responsible for taking an instance of the KGE model, the respective hyperparameter configuration, and input from the batch generator to train the algorithms.","The Evaluator module performs link prediction and provides the respective accuracy in terms of mean ranks and filtered mean ranks.","Visualization: plots training loss and common metrics used in KGE tasks.","To facilitate model analysis, it also visualizes the latent representations of entities and relations on the 2D plane using t-SNE based dimensionality reduction.","Bayesian Optimizer: pykg2vec uses a Bayesian hyperparameter optimizer to find a golden hyperparameter set.","This feature is more efficient than brute-force based approaches."],["Usage Examples","Pykg2vec provides users with two utilization examples ($train.py$ and $tune\\_model.py$ ) available in the pykg2vec/example folder.","Training is performed with the following script: $ python train.py -h # Check the manual for input arguments$ python train.py -mn transe Train TransE To apply the best setting described in the paper, the following script can be invoked.","$ python train.py -mn transe -ghp True # Train using the golden setting$ Some of the results plotted after training $TransE$ and $TransH$ are shown in Figure REF .","Figure: Visualization examples from Freebase15KTo tune the model, the following script can be invoked.","$ python tune_model.py -mn transe # Tune hyperparametersFound Golden Setting: # dummy tuning result{^{\\prime }L1_flag^{\\prime }: False, ^{\\prime }batch_size^{\\prime }: 256, ^{\\prime }epochs^{\\prime }: 5, ^{\\prime }hidden_size^{\\prime }: 32,^{\\prime }learning_rate^{\\prime }: 0.001, ^{\\prime }margin^{\\prime }: 0.4, ^{\\prime }opt^{\\prime }: ^{\\prime }sgd^{\\prime },^{\\prime }samp^{\\prime }: ^{\\prime }bern^{\\prime }}$"],["Conclusion","Pykg2vec is a Python library with extensive documentation that includes the implementations of a variety of state-of-the-art Knowledge Graph Embedding methods and modular building blocks of the embedding pipeline.","This library aims to help researchers and developers to quickly test algorithms against their custom knowledge base or utilize the modular blocks to adapt the library for their custom algorithms."]],"string":"[\n [\n \"Pykg2vec: A Python Library for Knowledge Graph Embedding\"\n ],\n [\n \"Abstract Pykg2vec is an open-source Python library for learning the representations of the entities and relations in knowledge graphs.\",\n \"Pykg2vec's flexible and modular software architecture currently implements 16 state-of-the-art knowledge graph embedding algorithms, and is designed to easily incorporate new algorithms.\",\n \"The goal of pykg2vec is to provide a practical and educational platform to accelerate research in knowledge graph representation learning.\",\n \"Pykg2vec is built on top of TensorFlow and Python's multiprocessing framework and provides modules for batch generation, Bayesian hyperparameter optimization, mean rank evaluation, embedding, and result visualization.\",\n \"Pykg2vec is released under the MIT License and is also available in the Python Package Index (PyPI).\",\n \"The source code of pykg2vec is available at https://github.com/Sujit-O/pykg2vec.\"\n ],\n [\n \"Introduction\",\n \"In recent years, Knowledge Graph Embedding (KGE) methods have been applied in benchmark datasets including Wikidata ([5]), Freebase ([2]), DBpedia ([1]), and YAGO ([8]).\",\n \"Applications of KGE methods include fact prediction, question answering, and recommender systems.\",\n \"KGE is an active area of research and many authors have provided reference software implementations.\",\n \"However, most of these are standalone reference implementations and therefore it is difficult and time-consuming to: (i) find the source code; (ii) adapt the source code to new datasets; (iii) correctly parameterize the models; and (iv) compare against other methods.\",\n \"Recently, this problem has been partially addressed by libraries such as OpenKE [6] and AmpliGraph [4] that provide a framework common to several KGE methods.\",\n \"However, these frameworks take different perspectives, make specific assumptions, and thus the resulting implementations diverge substantially from the original architectures.\",\n \"Furthermore, these libraries often force the user to use preset hyperparameters, or make implicit use of golden hyperparameters, and thus make it tedious and time-consuming to adapt the models to new datasets.\",\n \"This paper presents pykg2vec, a single Python library with 16 state-of-the-art KGE methods.\",\n \"The goals of pykg2vec are to be practical and educational.\",\n \"The practical value is achieved through: (a) proper use of GPUs and CPUs; (b) a set of tools to automate the discovery of golden hyperparameters; and (c) a set of visualization tools for the training and results of the embeddings.\",\n \"The educational value is achieved through: (d) a modular and flexible software architecture and KGE pipeline; and (e) access to a large number of state-of-the-art KGE models.\"\n ],\n [\n \"Knowledge Graph Embedding Methods\",\n \"A knowledge graph contains a set of entities $\\\\mathbb {E}$ and relations $\\\\mathbb {R}$ between entities.\",\n \"The set of facts $\\\\mathbb {D}^+$ in the knowledge graph are represented in the form of triples $(h, r, t)$ , where $h,t\\\\in \\\\mathbb {E}$ are referred to as the head (or subject) and the tail (or object) entities, and $r\\\\in \\\\mathbb {R}$ is referred to as the relationship (or predicate).\",\n \"The problem of KGE is in finding a function that learns the embeddings of triples using low dimensional vectors such that it preserves structural information, $f:\\\\mathbb {D}^+\\\\rightarrow \\\\mathbb {R}^d$ .\",\n \"To accomplish this, the general principle is to enforce the learning of entities and relationships to be compatible with the information in $\\\\mathbb {D}^+$ .\",\n \"The representation choices include deterministic point ([3]), multivariate Gaussian distribution ([7]), or complex number ([9]).\",\n \"Under the Open World Assumption (OWA), a set of unseen negative triplets, $\\\\mathbb {D}^-$ , are sampled from positive triples $\\\\mathbb {D}^+$ by either corrupting the head or tail entity.\",\n \"Then, a scoring function, $f_r(h, t)$ is defined to reward the positive triples and penalize the negative triples.\",\n \"Finally, an optimization algorithm is used to minimize or maximize the scoring function.\",\n \"KGE methods are often evaluated in terms of their capability of predicting the missing entities in negative triples $(?, r, t)$ or $(h, r, ?\",\n \")$ , or predicting whether an unseen fact is true or not.\",\n \"The evaluation metrics include the rank of the answer in the predicted list (mean rank), and the ratio of answers ranked top-k in the list (hit-k ratio).\"\n ],\n [\n \"Software Architecture\",\n \"The pykg2vec library is built using Python and TensorFlow.\",\n \"TensorFlow allows the computations to be assigned on both GPU and CPU.\",\n \"In addition to the main model training process, pykg2vec utilizes multi-processing for generating mini-batches and performing an evaluation to reduce the total execution time.\",\n \"The various components of the library (see Figure REF ) are as follows: Figure: Pykg2vec software architecture[align=left,leftmargin=*] KG Controller: handles all the low-level parsing tasks such as finding the total unique set of entities and relations; creating ordinal encoding maps; generating training, testing and validation triples; and caching the dataset data on disk to optimize tasks that involve repetitive model testing.\",\n \"Batch Generator: consists of multiple concurrent processes that manipulate and create mini-batches of data.\",\n \"These mini-batches are pushed to a queue to be processed by the models implemented in TensorFlow.\",\n \"The batch generator runs independently so that there is a low latency for feeding the data to the training module running on the GPU.\",\n \"Core Models: consists of 16 KGE algorithms implemented as Python modules in TensorFlow.\",\n \"Each module consists of a modular description of the inputs, outputs, loss function, and embedding operations.\",\n \"Each model is provided with configuration files that define its hyperparameters.\",\n \"Configuration: provides the necessary configuration to parse the datasets and also consists of the baseline hyperparameters for the KGE algorithms as presented in the original research papers.\",\n \"Trainer and Evaluator: the Trainer module is responsible for taking an instance of the KGE model, the respective hyperparameter configuration, and input from the batch generator to train the algorithms.\",\n \"The Evaluator module performs link prediction and provides the respective accuracy in terms of mean ranks and filtered mean ranks.\",\n \"Visualization: plots training loss and common metrics used in KGE tasks.\",\n \"To facilitate model analysis, it also visualizes the latent representations of entities and relations on the 2D plane using t-SNE based dimensionality reduction.\",\n \"Bayesian Optimizer: pykg2vec uses a Bayesian hyperparameter optimizer to find a golden hyperparameter set.\",\n \"This feature is more efficient than brute-force based approaches.\"\n ],\n [\n \"Usage Examples\",\n \"Pykg2vec provides users with two utilization examples ($train.py$ and $tune\\\\_model.py$ ) available in the pykg2vec/example folder.\",\n \"Training is performed with the following script: $ python train.py -h # Check the manual for input arguments$ python train.py -mn transe Train TransE To apply the best setting described in the paper, the following script can be invoked.\",\n \"$ python train.py -mn transe -ghp True # Train using the golden setting$ Some of the results plotted after training $TransE$ and $TransH$ are shown in Figure REF .\",\n \"Figure: Visualization examples from Freebase15KTo tune the model, the following script can be invoked.\",\n \"$ python tune_model.py -mn transe # Tune hyperparametersFound Golden Setting: # dummy tuning result{^{\\\\prime }L1_flag^{\\\\prime }: False, ^{\\\\prime }batch_size^{\\\\prime }: 256, ^{\\\\prime }epochs^{\\\\prime }: 5, ^{\\\\prime }hidden_size^{\\\\prime }: 32,^{\\\\prime }learning_rate^{\\\\prime }: 0.001, ^{\\\\prime }margin^{\\\\prime }: 0.4, ^{\\\\prime }opt^{\\\\prime }: ^{\\\\prime }sgd^{\\\\prime },^{\\\\prime }samp^{\\\\prime }: ^{\\\\prime }bern^{\\\\prime }}$\"\n ],\n [\n \"Conclusion\",\n \"Pykg2vec is a Python library with extensive documentation that includes the implementations of a variety of state-of-the-art Knowledge Graph Embedding methods and modular building blocks of the embedding pipeline.\",\n \"This library aims to help researchers and developers to quickly test algorithms against their custom knowledge base or utilize the modular blocks to adapt the library for their custom algorithms.\"\n ]\n]"},"id":{"kind":"string","value":"1906.04239"}}},{"rowIdx":2599,"cells":{"text":{"kind":"list like","value":[["N\\'{e}el and stripe ordering from spin-orbital entanglement in\n $\\alpha$-Sr$_2$CrO$_4$"],["Abstract The rich phenomenology engendered by the coupling between the spin and orbital degrees of freedom has become appreciated as a key feature of many strongly-correlated electron systems.","The resulting emergent physics is particularly prominent in a number of materials, from Fe-based unconventional superconductors to transition metal oxides, including manganites and vanadates.","Here, we investigate the electronic ground states of $\\alpha$-Sr$_2$CrO$_4$, a compound that is a rare embodiment of the spin-1 Kugel-Khomskii model on the square lattice -- a paradigmatic platform to capture the physics of coupled magnetic and orbital electronic orders.","We have used resonant X-ray diffraction at the Cr-$K$ edge to reveal N\\'{e}el magnetic order at the in-plane wavevector $\\mathbf{Q}_N = (1/2, 1/2)$ below $T_N = 112$ K, as well as an additional electronic order at the 'stripe' wavevector $\\mathbf{Q}_s = (1/2, 0)$ below T$_s$ $ \\sim 50$ K. These findings are examined within the framework of the Kugel-Khomskii model by a combination of mean-field and Monte-Carlo approaches, which supports the stability of the spin N\\'{e}el phase with subsequent lower-temperature stripe orbital ordering, revealing a candidate mechanism for the experimentally observed peak at $\\mathbf{Q}_s$.","On the basis of these findings, we propose that $\\alpha$-Sr$_2$CrO$_4$ serves as a new platform in which to investigate multi-orbital physics and its role in the low-temperature phases of Mott insulators."],[" Néel and stripe ordering from spin-orbital entanglement in $\\alpha $ -Sr$_2$ CrO$_4$ Z. H. Zhu$^1$ zzh@mit.edu W. Hu$^2$ C. A. Occhialini$^1$ J. Li$^1$ J. Pelliciari$^1$ C. S. Nelson$^3$ M. R. Norman$^4$ Q. Si$^2$ R. Comin$^1$ rcomin@mit.edu $^1$ Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA $^2$ Department of Physics and Astronomy, Rice Center for Quantum Materials, Rice University, Houston, TX 77005, USA $^3$ National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, NY 11973, USA $^4$ Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA The rich phenomenology engendered by the coupling between the spin and orbital degrees of freedom has become appreciated as a key feature of many strongly-correlated electron systems.","The resulting emergent physics is particularly prominent in a number of materials, from Fe-based unconventional superconductors to transition metal oxides, including manganites and vanadates.","Here, we investigate the electronic ground states of $\\alpha $ -Sr$_2$ CrO$_4$ , a compound that is a rare embodiment of the spin-1 Kugel-Khomskii model on the square lattice – a paradigmatic platform to capture the physics of coupled magnetic and orbital electronic orders.","We have used resonant X-ray diffraction at the Cr-$K$ edge to reveal Néel magnetic order at the in-plane wavevector $\\mathbf {Q}_N = (1/2, 1/2)$ below $T_N = 112$ K, as well as an additional electronic order at the “stripe” wavevector $\\mathbf {Q}_s = (1/2, 0)$ below T$_s$ $ \\sim 50$ K. These findings are examined within the framework of the Kugel-Khomskii model by a combination of mean-field and Monte-Carlo approaches, which supports the stability of the spin Néel phase with subsequent lower-temperature stripe orbital ordering, revealing a candidate mechanism for the experimentally observed peak at $\\mathbf {Q}_s$ .","On the basis of these findings, we propose that $\\alpha $ -Sr$_2$ CrO$_4$ serves as a new platform in which to investigate multi-orbital physics and its role in the low-temperature phases of Mott insulators.","Multi-orbital physics plays an important role for several collective electronic phenomena in strongly-correlated systems, including high-$T_c$ superconductivity and exotic magnetism [1], [2].","Some model materials, such as LaMnO$_3$ [3], KCuF$_3$ [4], and YVO$_3$ [5], have enabled the exploration of the active role played by the orbital degrees of freedom in determining the symmetry of the electronic ground state.","In particular, the essential ingredients of unconventional superconductivity in the iron-based superconductors (FeSCs) are purported to involve multi-orbital electronic states and their interplay with magnetism [6], [7], [8], in remarkable contrast to the effective single-orbital nature of cuprate superconductors [9].","A few chromate perovskites ACrO$_3$ (A =Ca, Sr, and Pb) with a Cr$^{4+}$ valence state were synthesized by several groups four decades ago [10], [11], [12].","Recent studies on CaCrO$_3$ and SrCrO$_3$ have revealed novel physics, drawing more attention to these chromate compounds [13], [14], [15].","For example, both compounds are antiferromagnetic conductors, but the exact nature of the ground state has remained elusive.","An interesting orbital ordering transition with electronic phase coexistence has been discovered in SrCrO$_3$ [14], though it is unclear if the orbital degree of freedom is essential to carrier itinerancy within the antiferromagnetic ground state.","Moreover, despite the isovalency of Ba and Sr, the sister compound BaCrO$_3$ was found to be an antiferromagnetic insulator [16], [17], with theoretical studies proposing that the mechanism for the insulating state involves, again, some form of orbital ordering [18], [19].","A recent study of bilayered Sr$_3$ Cr$_2$ O$_7$ revealed the presence of an exotic ordered phase of orbital singlets arising from the interplay between the spin and orbital degrees of freedom [20].","It is thus not surprising that similar phenomena at the nexus between magnetism and orbital physics were proposed also for the single-layer ($n=1$ ) member of the Ruddlesden-Popper series of chromates, Sr$_2$ CrO$_4$ [21].","However, the microscopic nature of the ground state of this compound remains unknown.","In the present work, we focus on $\\alpha $ -Sr$_2$ CrO$_4$ , which crystallizes in the K$_2$ NiF$_4$ -type structure that makes it isostructural to La$_2$ CuO$_4$ .","With its rare Cr$^{4+}$ oxidation state, it realizes a unique $3d^2$ electronic configuration among layered oxides, resulting in active $t_{2g}$ orbitals with total spin $S=1$ on a square lattice.","Despite several studies characterizing this unique material [22], [23], [24], a clear picture of its physical properties has remained elusive, partly owing to known challenges with the synthesis of bulk single crystals.","Successful growth of polycrystalline samples of $\\alpha $ -Sr$_2$ CrO$_4$ by means of high-pressure ($> 5$ GPa) and high-temperature (1500 $^{\\circ }$ C) synthesis has been reported by several groups[22], [25], [26].","Magnetization characterization of these polycrystalline samples reveal a sharp antiferromagnetic transition at T$_N$ $\\approx 112$ K[22], which was further confirmed by muon spin rotation ($\\mu ^+$ SR) measurements [23], [24].","The resistivity of sintered pellets of $\\alpha $ -Sr$_2$ CrO$_4$ reveal insulating behavior [22].","In addition, single-crystalline samples of $\\alpha $ -Sr$_2$ CrO$_4$ have been synthesized as thin films using pulsed laser deposition (PLD), which are determined by optical conductivity to be Mott-Hubbard type insulators with a charge gap $\\Delta $ $\\approx $ 0.3 eV [27].","While previous studies provide evidence for an antiferromagnetic insulating ground state in $\\alpha $ -Sr$_2$ CrO$_4$ , no direct reciprocal space studies of the long-range electronic order in these compounds have been reported to date.","Figure: Resonant X-ray scattering measurements of α\\alpha -Sr 2 _2CrO 4 _4 at the Cr-KK edge.","A: unit cell of α\\alpha -Sr 2 _2CrO 4 _4, which is tetragonal with I4/mmm space group at room temperature.","B: Reciprocal space cartography of scattering scans in the H,K\\left( H, K \\right) plane.","C1-C4: individual RXS scans across the symmetry-equivalent ordering vectors ±0.5,±0.5,4.84\\left(\\pm 0.5, \\pm 0.5, 4.84 \\right) (see insets).To address this open question, we have synthesized single-crystalline films of $\\alpha $ -Sr$_2$ CrO$_4$ with the aim of resolving its magnetic ground state and to search for other ordered phases within the electronic degrees of freedom.","We have sought reciprocal space signatures of electronic long-range ordering using resonant x-ray scattering (RXS) measurements at the Cr-$K$ edge (with photon energy $E_{ph} \\approx 6.0$ keV).","At the Cr-$K$ edge, photons that are tuned to the Cr-$1s \\rightarrow 4p$ transition become resonantly sensitive to the charge, spin, and orbital degrees of freedom at the Cr site, a capability that has made RXS a valuable probe to study electronic ordering in transition metal oxides [28], [29].","Thin film samples of $c$ -oriented Sr$_2$ CrO$_4$ have been grown on (001) SrTiO$_3$ substrates using pulsed laser deposition (PLD) from a KrF excimer laser ($\\lambda = 248$ nm).","Deposition was performed under high vacuum with a laser pulse energy and repetition rate of 400 mJ and 4 Hz, respectively, with the substrate heated to 900 $^{\\circ }$ C. The laser target contained a stoichiometric mixture of SrCO$_3$ and Cr$_2$ O$_3$ prepared by a solid state reaction at 920 $^{\\circ }$ C under a constant N$_2$ flow for 24 hours.","The products of this reaction were ground and pressed into a pellet, then sintered under an Ar flow at 1200 $^{\\circ }$ C for 24 hours.","The main phase of the target was determined to be $\\beta $ -Sr$_2$ CrO$_4$ , however the final $\\sim 100$ nm film was confirmed by X-ray diffraction to be pure $\\alpha $ -Sr$_2$ CrO$_4$ , with no traces of the sister phase (see Supplementary Information, in particular Fig.","S1, for additional details).","Resonant X-ray measurements were performed at the Integrated In-Situ and Resonant X-ray Studies (ISR) beamline- 4-ID of NSLS-II at Brookhaven National Laboratory.","Figure: Nature and characteristics of the Néel-ordered state.","A: energy-integrated sample rocking scans across the resonant reflection 𝐐 N =1/2,1/2,L{\\mathbf {Q}}_{N} = \\left( 1/2, 1/2, L \\right) at various temperatures.","B: temperature dependence of the integrated scattering intensity around 𝐐 N {\\mathbf {Q}}_{N} (markers) and of the magnetic susceptibility χT\\chi \\left( T \\right), from (gray line).","C: energy dependence of the integrated RXS intensity around 𝐐 N {\\mathbf {Q}}_{N} (purple diamonds) and near-edge X-ray absorption scan (XANES, orange trace), both acquired at 20 K; the dash-dotted gray and teal lines represent, respectively, simulations of the XANES spectrum and the RXS signals from magnetic scattering using FDMNES program (see text and Supplementary Information).","D: azimuthal angle (ψ\\psi ) dependence of the integrated intensity (green squares) of the 𝐐 N {\\mathbf {Q}}_{N} reflection, with the blue curve representing the simulated azimuthal dependence expected for the magnetic model described in the text.Figure REF A shows the unit cell of $\\alpha $ -Sr$_2$ CrO$_4$ , which is tetragonal with space group I4/mmm at room temperature.","Throughout this paper, reciprocal lattice vectors (H,K,L) are scaled to reciprocal lattice units (r.l.u) for the room temperature tetragonal structure (that is, in units of $2\\pi / a$ , $2\\pi / a$ , $2\\pi / c$ , where $a = b = 3.87$ Å, $c = 12.55$ Å).","Resonant X-ray scattering scans were performed in the reciprocal $\\left( H, K \\right)$ plane parallel to the crystallographic CrO$_2$ layers, as illustrated in Figure REF B.","We have surveyed the high-symmetry directions $\\left[ H, 0 \\right]$ and $\\left[ H, H \\right]$ using incident X-rays tuned to the Cr-$K$ edge resonance E = 6.009 keV to resonantly enhance Bragg reflections of electronic nature.","Under these experimental conditions, we have observed four well-defined, symmetry-equivalent diffraction peaks at the Néel vectors ${\\mathbf {Q}}_{N} = \\left( \\pm 1/2, \\pm 1/2, L \\right)$ (r.l.u), with $L \\sim 4.84$ , as shown in Fig.","REF C1-C4.","These diffraction signatures reveal the presence of a $\\sqrt{2} \\times \\sqrt{2}$ electronic superlattice which is forbidden by the high-symmetry structure.","The disappearance of the superlattice reflection away from the Cr-$K$ resonance suggests that it results from a reordering of the electronic subspace.","The in-plane correlation length $\\xi _{ab}$ = a$(\\pi FWHM)^{-1}$ $\\sim $ 4000 Å($\\sim $ 1000 unit cells) is much greater than the out-of-plane correlation length $\\xi _{c}$ $\\sim $ 250 Å($\\sim $ 20 unit cells), though we note that the latter could be limited by the approximate 1000 Åthickness of the films in the $c$ direction.","The non-integer out-of-plane projection of the ordering vector ($L \\approx 4.84$ ) reveals an incommensurate nature of the ordering pattern along the c axis.","To uncover the nature of these reflections, we have studied their evolution as a function of temperature, photon energy, and azimuthal angle.","Figure REF A shows a series of representative rocking curves at ${\\mathbf {Q}}_{N} = \\left( 1/2, 1/2, L \\right)$ for various temperatures, with the integrated intensity extracted from Gaussian fits and plotted in Fig.","REF B.","A clear transition occurs at $T \\approx 110$ K, which coincides with the reported Néel temperature for polycrystalline samples revealed by bulk magnetometry (see susceptibility trace from Ref.","[22] in Fig.","REF B), as well as muon spin rotation measurements [23], [24].","In Fig.","REF C, we plot the integrated intensity of the rocking curve for the $ {\\mathbf {Q}}_{N} $ reflection as a function of the photon energy at $T = 20$ K, showing a clear resonant enhancement around $E = 6.009$ keV, at the main Cr-$K$ edge, measured using near edge X-ray absorption (XANES) spectrum measured in fluorescence mode.","Given the close correspondence of the transition temperature of this peak with the reported Néel temperature, we performed ab initio calculations using the FDMNES program to model the RXS intensity resulting from Néel-type magnetic order .","The RXS intensity and XANES curves are overlaid onto the experimental data points in Fig.","REF C, confirming the resonant enhancement of the magnetic peak at the Cr-$1s \\rightarrow 4p$ transition (for more details, see Supplementary Information).","To further examine the symmetry of the ordered phase, we also study the azimuthal angle dependence of the integrated intensity at $ {\\mathbf {Q}}_{N} $ .","RXS scans across $ {\\mathbf {Q}}_{N} $ were collected for different values of the azimuthal $\\psi $ , which corresponds to a rotation around the axis parallel to $ {\\mathbf {Q}}_{N} $ .","Here, an azimuthal angle $\\psi = 0$ coincides with a sample orientation that has the $\\left( 1, 1, L \\right)$ crystallographic direction lying in the scattering plane.","The clear 2-fold modulation of the RXS intensity as a function of the azimuthal angle, shown in Fig.","REF D, is consistent with a magnetic structure with moments oriented along $\\left( 1, -1, 0 \\right)$ , i.e.","perpendicular to the in-plane projection of the Néel ordering vector.","The collective information, including the resonance energy profile, azimuthal symmetry and polarization restrictions (from the incoming photons) are most consistent with magnetic scattering.","This interpretation is further bolstered by the close correspondence between the observed onset temperature and previous reports of the Néel transition [22], [23], [24].","We also note that, among the possible high-symmetry magnetic space subgroups, the one determined to be most consistent with the azimuthal symmetry is equivalent to that found in other isostructural compounds, such as La$_2$ CuO$_4$ .","Figure: Stripe-like electronic ordering along H,0\\left[ H, 0 \\right].","A: representative momentum scans across 𝐐 s =1/2,0,L {\\mathbf {Q}}_{s} = \\left( 1/2, 0, L \\right) at a photon energy E=6.009E = 6.009 keV.","B: integrated RXS intensity at 𝐐 s {\\mathbf {Q}}_{s} vs. temperature; the inset shows a series of RXS rocking curves across 𝐐 s {\\mathbf {Q}}_{s} for a few representative temperatures.Having identified the nature and microscopic traits of the ordered state below $T_N \\sim 110$ K, we have focused on the $\\left[ H, 0 \\right]$ reciprocal space direction, to search for additional ordering instabilities.","This further exploration was motivated by previous heat capacity data hinting at a second ordering transition around 140 K, whose nature has remained undisclosed [22].","Consistently with a dual-instability scenario, we have observed a resonant reflection at the wavevector for period-2 stripe order $ {\\mathbf {Q}}_{s} = \\left( 1/2, 0, L \\right)$ (with $L = 4.84$ here, as well) as shown in Figure REF A.","Similar to the reflection at the Néel vector $ {\\mathbf {Q}}_{N}$ , the diffraction peak at the stripe ordering vector also exhibits a resonant enhancement as the photon energy is tuned to $E = 6.009$ keV (see Supplementary Information).","The integrated intensity of the rocking curves as a function of temperature is shown in Fig.","REF B (RXS scans at representative temperatures are reported in the inset).","Surprisingly, the transition was seen to occur around $T_s \\sim 50$ K, namely at a significantly lower temperature than the proposed orbital ordering transition.","Here, we should note that a remnant scattering signal persists above the transition and up to the highest measured temperature (200 K), yet shows no variation around 140 K. This unusual temperature dependence has been noted in other transition metal compounds with coupled spin and orbital order parameters (OPs), in particular KCuF$_3$ , though we cannot rule out additional OPs of possibly different symmetry contributing to the remnant signal above the clear transition at $T_s$ .","We reconcile the puzzle of this second transition at $T_s$ by noting that the ordering temperature is close to that of the appearance of a second oscillatory component near 45 K as recorded by zero-field muon spin rotation (ZF-$\\mu ^+$ SR) measurements on polycrystalline samples [24].","Figure: Phase diagrams at zero and nonzero temperatures.","A: Thephase diagram at zero temperature for the Kugel-Khomskii model with J 1 =J 4 =1.0J_1=J_4=1.0, based on a site-factorized wavefunction approach.","Three stable phases appear in the phase diagram, including S Ne ´el T CAFO S_{N\\acute{e}el}T_{CAFO}, S Ne ´el T AFO S_{N\\acute{e}el}T_{AFO}, and S CAFM T AFO S_{CAFM}T_{AFO}.","The inset is a schematic illustration of the S Ne ´el T CAFO S_{N\\acute{e}el}T_{CAFO} ordered state.","The blue and red ovals represent |xz〉\\vert xz \\rangle and |yz〉\\vert yz \\rangle orbitals, respectively.","The black arrows denote the spin state on the |xz〉\\vert xz \\rangle or |yz〉\\vert yz \\rangle orbital.","S Ne ´el _{N\\acute{e}el} and S CAFM _{CAFM} stand for Néel order and stripe antiferromagnetic order, respectively.","T CAFO T_{CAFO} marks a stripe-like orbital order with momentum (π,0)/(0,π)(\\pi ,0)/(0,\\pi ),and T AFO _{AFO} represents Néel-like orbital order with momentum (π,π)(\\pi ,\\pi ).","B: The temperature-dependence of the magnetic (m S m_S) and orbital (m T m_T) order parameters from classical Monte Carlo simulations of the phase S Ne ´el T CAFO S_{N\\acute{e}el}T_{CAFO} with J 1 =J 4 =1.0,J 6 =0.4,J 8 =0.0J_1=J_4=1.0,J_6=0.4,J_8=0.0 on L=24,32,40L=24,32,40 clusters (marked by the black star in panel A).","The model parameters J i J_i are set as J i =t i 2 /UJ_i=t^2_i/U, where UU is the on-site interaction and t i t_i represent hopping amplitudes between orbitals up to next-nearest neighbors.","J 1 J_1 and J 4 J_4 are the nearest-neighbor couplings, and J 6 J_6 and J 8 J_8 the next-nearest-neighbor couplings.","Further details are given in the Supplementary Information.To understand the microscopic physics behind the observed scattering peaks, we examine an effective spin-1 Kugel-Khomskii model for $\\alpha $ -Sr$_2$ CrO$_4$ (see Supplementary Information for additional details).","In this compound, the necessarily high-spin $3d^2$ electronic configuration form an active $t_{2g}$ orbital subspace.","Of particular interest are results from density functional theory (DFT) calculations showing a reversed crystal field splitting compared to typical tetragonal systems with an elongated c-axis lattice parameter [21].","Within this scenario, two electrons occupy the $\\left\\lbrace \\vert xy \\rangle , \\vert xz \\rangle , \\vert yz \\rangle \\right\\rbrace $ $t_{2g}$ orbital manifold, with a proposed negative energy level splitting $\\Delta \\sim -0.6eV$ between the low-lying $\\vert xy \\rangle $ and $\\vert xz \\rangle $ /$ \\vert yz \\rangle $ orbitals.","This reversed crystal field leaves a single electron in the two-fold degenerate $\\left\\lbrace \\vert xz \\rangle , \\vert yz \\rangle \\right\\rbrace $ submanifold, setting the stage for active orbital physics as encoded in a pseudospin-1/2 degree of freedom.","In order to elucidate the essential physics arising from the interplay of the spin and orbital degrees of freedom in $\\alpha $ -Sr$_2$ CrO$_4$ , , , we consider the spin-1 Kugel-Khomskii Hamiltonian as the effective model: $\\small H_{KK}^{(i,j)} = -\\frac{1}{3}({\\bf S}_i\\cdot {\\bf S}_j+2)Q^{(1)}({\\bf T}_i,{\\bf T}_j)+ \\frac{1}{3}({\\bf S}_i\\cdot {\\bf S}_j-1) Q^{(2)}({\\bf T}_i,{\\bf T}_j),$ with $&&Q^{(n)}({\\bf T}_i,{\\bf T}_j)=\\nonumber \\\\&&f_{zz}^{(n)}T_{i}^{z}T_{j}^{z}+\\frac{1}{2} f_{+-}^{(n)}(T_{i}^{+}T_{j}^{-}+T_{i}^{-}T_{j}^{+})+f_{z}^{(n)}(T_{i}^{z}+T_{j}^{z})\\nonumber \\\\&&+\\frac{1}{2} f_{++}^{(n)}(T_{i}^{+}T_{j}^{+}+T_{i}^{-}T_{j}^{-})+f_{zx}^{(n)}(T_{i}^{z}T_{j}^{x}+T_{i}^{x}T_{j}^{z})\\nonumber \\\\&&+f_{x}^{(n)}(T_{i}^{x}+T_{j}^{x}) +f_{0}^{(n)},$ where ${\\bf S}_i$ and ${\\bf T}_i$ are the spin-1 and pseudospin-$1/2$ operators, respectively.","$f^{(n)}$ with $n=1,2$ are functions of microscopic parameters that include the hopping amplitudes between orbitals up to next-nearest neighbor and the onsite Hubbard interactions and Hund's coupling (see Supplementary Information for details).","We have performed calculations using a combination of mean-field and classical Monte Carlo methods.","As shown in Fig.","REF A, we find S$_{N\\acute{e}el}$ T$_{CAFO}$ , a Néel order accompanied by a stripe-like orbital order, over an extended parameter regime in the phase diagram.","This phase corresponds to the same ordering vector as found in RXS experiments.","In Fig.REF B, we show the results of classical Monte Carlo simulations for the evolution of the magnetic and orbital order parameters vs. temperature for the phase S$_{N\\acute{e}el}$ T$_{CAFO}$ .","The numerical results clearly show that, as temperature is lowered, the magnetic order parameter develops first, which is followed by the emergence of the stripe-like orbital order.","The proposed theoretical description captures the salient features of the experimental observations, including the ordering wavevectors ($\\mathbf {Q}_s$ and $\\mathbf {Q}_N$ ) and their respective transition temperatures.","The Néel spin order setting in at the higher-temperature transition is consistent with our RXS measurements (see Supplementary Information) as well as the magnetic measurements in polycrystalline samples [22], [24].","In addition, because the orbital order below the lower-temperature transition develops in a time-reversal-broken background, it is expected to develop an additional orbital moment at the same wavevector, in the presence of spin-orbit coupling.","Given that the involved orbitals are $\\vert xz \\rangle $ and $\\vert yz \\rangle $ , the orbital moment is likely to dominantly point along the $c$ -axis, and this appears to be the most consistent with the existing RXS azimuthal dependence (see Supplementary Information) although further spectroscopic studies will be needed in order to reach a firm conclusion.","Finally, while our model study has focused on the largest part of the energetics, a complete understanding will require the addition of couplings with smaller magnitude, such as the interlayer couplings.","In summary, our experimental and theoretical results uncover the microscopic nature of the antiferromagnetic state in $\\alpha $ -Sr$_2$ CrO$_4$ , which is here shown using resonant X-ray scattering to arise from collinear Néel-type order at ${\\mathbf {Q}}_{N} = \\left( \\pm 1/2, \\pm 1/2, L \\right)$ with the spin axis in the CrO$_2$ plane, in close analogy with the parent state of cuprate superconductors.","In addition to Néel order, we have detected coexisting stripe-like order setting in at a lower temperature $T_s \\sim 50$ K, with ordering vector $ {\\mathbf {Q}}_{s} = \\left( 1/2, 0, L \\right)$ .","A Monte Carlo study of the spin-1 Kugel-Khomskii Hamiltonian captures the observed experimental signatures of Néel and stripe phases and their ordering sequence, thereby pointing to the physical realization of this model for a system of $d^2$ ions on a square lattice.","Our study singles out $\\alpha $ -Sr$_2$ CrO$_4$ as a unique platform to explore the rich electronic phases in a Kugel-Khomskii-like spin-orbital system, and underscores the great scientific potential of a unique transition metal oxide, whose electronic phase diagram as a function of carrier doping is yet to be explored.","We are grateful to A. Chubukov, D. Puggioni, J.M.","Tranquada, and P.A.","Lee for insightful discussions.","Work at MIT (Z.H.Z., C.A.O., J.L., J.P., and R.C.)","has been supported by the Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award Number DE-SC0019126.","Part of this research concerning sample synthesis (Z.H.Z.)","has been supported by NSF through the Massachusetts Institute of Technology Materials Research Science and Engineering Center DMR – 1419807.","Work at Rice (W.H.","and Q.S.)","has in part been supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award No.","DE-SC0018197 and by the Robert A. Welch Foundation Grant No.","C-1411.","R.C.","acknowledges support from the Alfred P. Sloan Foundation.","M.N.","was supported by the Materials Sciences and Engineering Division, Basic Energy Sciences, Office of Science, US DOE.","J. P. acknowledges financial support by the Swiss National Science Foundation Early Postdoc Mobility Fellowship Project No.","P2FRP2_171824 and P400P2_180744.","This research used beamline 4-ID of the National Synchrotron Light Source II, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Brookhaven National Laboratory under Contract No.","DE-SC0012704.","D.~Vaknin, S.~K.","Sinha, D.~E.","Moncton, D.~C.","Johnston, J.~M.","Newsam, C.~R.","Safinya, and H.~E.","King~Jr., Physical Review Letters 58, 2802 (1987).","L.~Paolasini, R.~Caciuffo, A.~Sollier, P.~Ghigna, and M.~Altarelli, Physical Review Letters 88, 106403 (2002).","A.~M.","Oleś, G.~Khaliullin, P.~Horsch, and L.~F.","Feiner, Physical Review B 72, 214431 (2005), URL https://link.aps.org/doi/10.1103/PhysRevB.72.214431.","A.~M.","Oleś, P.~Horsch, and G.~Khaliullin, Physical Review B 75, 184434 (2007), URL https://link.aps.org/doi/10.1103/PhysRevB.75.184434.","F.~Krüger, S.~Kumar, J.~Zaanen, and J.~van~den Brink, Physical Review B 79, 054504 (2009), URL https://link.aps.org/doi/10.1103/PhysRevB.79.054504.","authorD.~Vaknin, authorS.~K.","Sinha, authorD.~E.","Moncton, authorD.~C.","Johnston, authorJ.~M.","Newsam, authorC.~R.","Safinya, and authorH.~E.","King~Jr., journalPhysical Review Letters volume58, pages2802 (year1987).","authorL.~Paolasini, authorR.~Caciuffo, authorA.~Sollier, authorP.~Ghigna, and authorM.~Altarelli, journalPhysical Review Letters volume88, pages106403 (year2002).","authorA.~M.","Oleś, authorG.~Khaliullin, authorP.~Horsch, and authorL.~F.","Feiner, journalPhysical Review B volume72, pages214431 (year2005), https://link.aps.org/doi/10.1103/PhysRevB.72.214431.","authorA.~M.","Oleś, authorP.~Horsch, and authorG.~Khaliullin, journalPhysical Review B volume75, pages184434 (year2007), https://link.aps.org/doi/10.1103/PhysRevB.75.184434.","authorF.~Krüger, authorS.~Kumar, authorJ.~Zaanen, and authorJ.~van~den Brink, journalPhysical Review B volume79, pages054504 (year2009), https://link.aps.org/doi/10.1103/PhysRevB.79.054504.","authorY.~Joly, authorO.~Bunǎu, authorJ.~E.","Lorenzo, authorR.~M.","Galéra, authorS.~Grenier, and authorB.~Thompson, journalJournal of Physics: Conference Series volume190, pages012007 (year2009), ISSN issn1742-6596,"]],"string":"[\n [\n \"N\\\\'{e}el and stripe ordering from spin-orbital entanglement in\\n $\\\\alpha$-Sr$_2$CrO$_4$\"\n ],\n [\n \"Abstract The rich phenomenology engendered by the coupling between the spin and orbital degrees of freedom has become appreciated as a key feature of many strongly-correlated electron systems.\",\n \"The resulting emergent physics is particularly prominent in a number of materials, from Fe-based unconventional superconductors to transition metal oxides, including manganites and vanadates.\",\n \"Here, we investigate the electronic ground states of $\\\\alpha$-Sr$_2$CrO$_4$, a compound that is a rare embodiment of the spin-1 Kugel-Khomskii model on the square lattice -- a paradigmatic platform to capture the physics of coupled magnetic and orbital electronic orders.\",\n \"We have used resonant X-ray diffraction at the Cr-$K$ edge to reveal N\\\\'{e}el magnetic order at the in-plane wavevector $\\\\mathbf{Q}_N = (1/2, 1/2)$ below $T_N = 112$ K, as well as an additional electronic order at the 'stripe' wavevector $\\\\mathbf{Q}_s = (1/2, 0)$ below T$_s$ $ \\\\sim 50$ K. These findings are examined within the framework of the Kugel-Khomskii model by a combination of mean-field and Monte-Carlo approaches, which supports the stability of the spin N\\\\'{e}el phase with subsequent lower-temperature stripe orbital ordering, revealing a candidate mechanism for the experimentally observed peak at $\\\\mathbf{Q}_s$.\",\n \"On the basis of these findings, we propose that $\\\\alpha$-Sr$_2$CrO$_4$ serves as a new platform in which to investigate multi-orbital physics and its role in the low-temperature phases of Mott insulators.\"\n ],\n [\n \" Néel and stripe ordering from spin-orbital entanglement in $\\\\alpha $ -Sr$_2$ CrO$_4$ Z. H. Zhu$^1$ zzh@mit.edu W. Hu$^2$ C. A. Occhialini$^1$ J. Li$^1$ J. Pelliciari$^1$ C. S. Nelson$^3$ M. R. Norman$^4$ Q. Si$^2$ R. Comin$^1$ rcomin@mit.edu $^1$ Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA $^2$ Department of Physics and Astronomy, Rice Center for Quantum Materials, Rice University, Houston, TX 77005, USA $^3$ National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, NY 11973, USA $^4$ Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA The rich phenomenology engendered by the coupling between the spin and orbital degrees of freedom has become appreciated as a key feature of many strongly-correlated electron systems.\",\n \"The resulting emergent physics is particularly prominent in a number of materials, from Fe-based unconventional superconductors to transition metal oxides, including manganites and vanadates.\",\n \"Here, we investigate the electronic ground states of $\\\\alpha $ -Sr$_2$ CrO$_4$ , a compound that is a rare embodiment of the spin-1 Kugel-Khomskii model on the square lattice – a paradigmatic platform to capture the physics of coupled magnetic and orbital electronic orders.\",\n \"We have used resonant X-ray diffraction at the Cr-$K$ edge to reveal Néel magnetic order at the in-plane wavevector $\\\\mathbf {Q}_N = (1/2, 1/2)$ below $T_N = 112$ K, as well as an additional electronic order at the “stripe” wavevector $\\\\mathbf {Q}_s = (1/2, 0)$ below T$_s$ $ \\\\sim 50$ K. These findings are examined within the framework of the Kugel-Khomskii model by a combination of mean-field and Monte-Carlo approaches, which supports the stability of the spin Néel phase with subsequent lower-temperature stripe orbital ordering, revealing a candidate mechanism for the experimentally observed peak at $\\\\mathbf {Q}_s$ .\",\n \"On the basis of these findings, we propose that $\\\\alpha $ -Sr$_2$ CrO$_4$ serves as a new platform in which to investigate multi-orbital physics and its role in the low-temperature phases of Mott insulators.\",\n \"Multi-orbital physics plays an important role for several collective electronic phenomena in strongly-correlated systems, including high-$T_c$ superconductivity and exotic magnetism [1], [2].\",\n \"Some model materials, such as LaMnO$_3$ [3], KCuF$_3$ [4], and YVO$_3$ [5], have enabled the exploration of the active role played by the orbital degrees of freedom in determining the symmetry of the electronic ground state.\",\n \"In particular, the essential ingredients of unconventional superconductivity in the iron-based superconductors (FeSCs) are purported to involve multi-orbital electronic states and their interplay with magnetism [6], [7], [8], in remarkable contrast to the effective single-orbital nature of cuprate superconductors [9].\",\n \"A few chromate perovskites ACrO$_3$ (A =Ca, Sr, and Pb) with a Cr$^{4+}$ valence state were synthesized by several groups four decades ago [10], [11], [12].\",\n \"Recent studies on CaCrO$_3$ and SrCrO$_3$ have revealed novel physics, drawing more attention to these chromate compounds [13], [14], [15].\",\n \"For example, both compounds are antiferromagnetic conductors, but the exact nature of the ground state has remained elusive.\",\n \"An interesting orbital ordering transition with electronic phase coexistence has been discovered in SrCrO$_3$ [14], though it is unclear if the orbital degree of freedom is essential to carrier itinerancy within the antiferromagnetic ground state.\",\n \"Moreover, despite the isovalency of Ba and Sr, the sister compound BaCrO$_3$ was found to be an antiferromagnetic insulator [16], [17], with theoretical studies proposing that the mechanism for the insulating state involves, again, some form of orbital ordering [18], [19].\",\n \"A recent study of bilayered Sr$_3$ Cr$_2$ O$_7$ revealed the presence of an exotic ordered phase of orbital singlets arising from the interplay between the spin and orbital degrees of freedom [20].\",\n \"It is thus not surprising that similar phenomena at the nexus between magnetism and orbital physics were proposed also for the single-layer ($n=1$ ) member of the Ruddlesden-Popper series of chromates, Sr$_2$ CrO$_4$ [21].\",\n \"However, the microscopic nature of the ground state of this compound remains unknown.\",\n \"In the present work, we focus on $\\\\alpha $ -Sr$_2$ CrO$_4$ , which crystallizes in the K$_2$ NiF$_4$ -type structure that makes it isostructural to La$_2$ CuO$_4$ .\",\n \"With its rare Cr$^{4+}$ oxidation state, it realizes a unique $3d^2$ electronic configuration among layered oxides, resulting in active $t_{2g}$ orbitals with total spin $S=1$ on a square lattice.\",\n \"Despite several studies characterizing this unique material [22], [23], [24], a clear picture of its physical properties has remained elusive, partly owing to known challenges with the synthesis of bulk single crystals.\",\n \"Successful growth of polycrystalline samples of $\\\\alpha $ -Sr$_2$ CrO$_4$ by means of high-pressure ($> 5$ GPa) and high-temperature (1500 $^{\\\\circ }$ C) synthesis has been reported by several groups[22], [25], [26].\",\n \"Magnetization characterization of these polycrystalline samples reveal a sharp antiferromagnetic transition at T$_N$ $\\\\approx 112$ K[22], which was further confirmed by muon spin rotation ($\\\\mu ^+$ SR) measurements [23], [24].\",\n \"The resistivity of sintered pellets of $\\\\alpha $ -Sr$_2$ CrO$_4$ reveal insulating behavior [22].\",\n \"In addition, single-crystalline samples of $\\\\alpha $ -Sr$_2$ CrO$_4$ have been synthesized as thin films using pulsed laser deposition (PLD), which are determined by optical conductivity to be Mott-Hubbard type insulators with a charge gap $\\\\Delta $ $\\\\approx $ 0.3 eV [27].\",\n \"While previous studies provide evidence for an antiferromagnetic insulating ground state in $\\\\alpha $ -Sr$_2$ CrO$_4$ , no direct reciprocal space studies of the long-range electronic order in these compounds have been reported to date.\",\n \"Figure: Resonant X-ray scattering measurements of α\\\\alpha -Sr 2 _2CrO 4 _4 at the Cr-KK edge.\",\n \"A: unit cell of α\\\\alpha -Sr 2 _2CrO 4 _4, which is tetragonal with I4/mmm space group at room temperature.\",\n \"B: Reciprocal space cartography of scattering scans in the H,K\\\\left( H, K \\\\right) plane.\",\n \"C1-C4: individual RXS scans across the symmetry-equivalent ordering vectors ±0.5,±0.5,4.84\\\\left(\\\\pm 0.5, \\\\pm 0.5, 4.84 \\\\right) (see insets).To address this open question, we have synthesized single-crystalline films of $\\\\alpha $ -Sr$_2$ CrO$_4$ with the aim of resolving its magnetic ground state and to search for other ordered phases within the electronic degrees of freedom.\",\n \"We have sought reciprocal space signatures of electronic long-range ordering using resonant x-ray scattering (RXS) measurements at the Cr-$K$ edge (with photon energy $E_{ph} \\\\approx 6.0$ keV).\",\n \"At the Cr-$K$ edge, photons that are tuned to the Cr-$1s \\\\rightarrow 4p$ transition become resonantly sensitive to the charge, spin, and orbital degrees of freedom at the Cr site, a capability that has made RXS a valuable probe to study electronic ordering in transition metal oxides [28], [29].\",\n \"Thin film samples of $c$ -oriented Sr$_2$ CrO$_4$ have been grown on (001) SrTiO$_3$ substrates using pulsed laser deposition (PLD) from a KrF excimer laser ($\\\\lambda = 248$ nm).\",\n \"Deposition was performed under high vacuum with a laser pulse energy and repetition rate of 400 mJ and 4 Hz, respectively, with the substrate heated to 900 $^{\\\\circ }$ C. The laser target contained a stoichiometric mixture of SrCO$_3$ and Cr$_2$ O$_3$ prepared by a solid state reaction at 920 $^{\\\\circ }$ C under a constant N$_2$ flow for 24 hours.\",\n \"The products of this reaction were ground and pressed into a pellet, then sintered under an Ar flow at 1200 $^{\\\\circ }$ C for 24 hours.\",\n \"The main phase of the target was determined to be $\\\\beta $ -Sr$_2$ CrO$_4$ , however the final $\\\\sim 100$ nm film was confirmed by X-ray diffraction to be pure $\\\\alpha $ -Sr$_2$ CrO$_4$ , with no traces of the sister phase (see Supplementary Information, in particular Fig.\",\n \"S1, for additional details).\",\n \"Resonant X-ray measurements were performed at the Integrated In-Situ and Resonant X-ray Studies (ISR) beamline- 4-ID of NSLS-II at Brookhaven National Laboratory.\",\n \"Figure: Nature and characteristics of the Néel-ordered state.\",\n \"A: energy-integrated sample rocking scans across the resonant reflection 𝐐 N =1/2,1/2,L{\\\\mathbf {Q}}_{N} = \\\\left( 1/2, 1/2, L \\\\right) at various temperatures.\",\n \"B: temperature dependence of the integrated scattering intensity around 𝐐 N {\\\\mathbf {Q}}_{N} (markers) and of the magnetic susceptibility χT\\\\chi \\\\left( T \\\\right), from (gray line).\",\n \"C: energy dependence of the integrated RXS intensity around 𝐐 N {\\\\mathbf {Q}}_{N} (purple diamonds) and near-edge X-ray absorption scan (XANES, orange trace), both acquired at 20 K; the dash-dotted gray and teal lines represent, respectively, simulations of the XANES spectrum and the RXS signals from magnetic scattering using FDMNES program (see text and Supplementary Information).\",\n \"D: azimuthal angle (ψ\\\\psi ) dependence of the integrated intensity (green squares) of the 𝐐 N {\\\\mathbf {Q}}_{N} reflection, with the blue curve representing the simulated azimuthal dependence expected for the magnetic model described in the text.Figure REF A shows the unit cell of $\\\\alpha $ -Sr$_2$ CrO$_4$ , which is tetragonal with space group I4/mmm at room temperature.\",\n \"Throughout this paper, reciprocal lattice vectors (H,K,L) are scaled to reciprocal lattice units (r.l.u) for the room temperature tetragonal structure (that is, in units of $2\\\\pi / a$ , $2\\\\pi / a$ , $2\\\\pi / c$ , where $a = b = 3.87$ Å, $c = 12.55$ Å).\",\n \"Resonant X-ray scattering scans were performed in the reciprocal $\\\\left( H, K \\\\right)$ plane parallel to the crystallographic CrO$_2$ layers, as illustrated in Figure REF B.\",\n \"We have surveyed the high-symmetry directions $\\\\left[ H, 0 \\\\right]$ and $\\\\left[ H, H \\\\right]$ using incident X-rays tuned to the Cr-$K$ edge resonance E = 6.009 keV to resonantly enhance Bragg reflections of electronic nature.\",\n \"Under these experimental conditions, we have observed four well-defined, symmetry-equivalent diffraction peaks at the Néel vectors ${\\\\mathbf {Q}}_{N} = \\\\left( \\\\pm 1/2, \\\\pm 1/2, L \\\\right)$ (r.l.u), with $L \\\\sim 4.84$ , as shown in Fig.\",\n \"REF C1-C4.\",\n \"These diffraction signatures reveal the presence of a $\\\\sqrt{2} \\\\times \\\\sqrt{2}$ electronic superlattice which is forbidden by the high-symmetry structure.\",\n \"The disappearance of the superlattice reflection away from the Cr-$K$ resonance suggests that it results from a reordering of the electronic subspace.\",\n \"The in-plane correlation length $\\\\xi _{ab}$ = a$(\\\\pi FWHM)^{-1}$ $\\\\sim $ 4000 Å($\\\\sim $ 1000 unit cells) is much greater than the out-of-plane correlation length $\\\\xi _{c}$ $\\\\sim $ 250 Å($\\\\sim $ 20 unit cells), though we note that the latter could be limited by the approximate 1000 Åthickness of the films in the $c$ direction.\",\n \"The non-integer out-of-plane projection of the ordering vector ($L \\\\approx 4.84$ ) reveals an incommensurate nature of the ordering pattern along the c axis.\",\n \"To uncover the nature of these reflections, we have studied their evolution as a function of temperature, photon energy, and azimuthal angle.\",\n \"Figure REF A shows a series of representative rocking curves at ${\\\\mathbf {Q}}_{N} = \\\\left( 1/2, 1/2, L \\\\right)$ for various temperatures, with the integrated intensity extracted from Gaussian fits and plotted in Fig.\",\n \"REF B.\",\n \"A clear transition occurs at $T \\\\approx 110$ K, which coincides with the reported Néel temperature for polycrystalline samples revealed by bulk magnetometry (see susceptibility trace from Ref.\",\n \"[22] in Fig.\",\n \"REF B), as well as muon spin rotation measurements [23], [24].\",\n \"In Fig.\",\n \"REF C, we plot the integrated intensity of the rocking curve for the $ {\\\\mathbf {Q}}_{N} $ reflection as a function of the photon energy at $T = 20$ K, showing a clear resonant enhancement around $E = 6.009$ keV, at the main Cr-$K$ edge, measured using near edge X-ray absorption (XANES) spectrum measured in fluorescence mode.\",\n \"Given the close correspondence of the transition temperature of this peak with the reported Néel temperature, we performed ab initio calculations using the FDMNES program to model the RXS intensity resulting from Néel-type magnetic order .\",\n \"The RXS intensity and XANES curves are overlaid onto the experimental data points in Fig.\",\n \"REF C, confirming the resonant enhancement of the magnetic peak at the Cr-$1s \\\\rightarrow 4p$ transition (for more details, see Supplementary Information).\",\n \"To further examine the symmetry of the ordered phase, we also study the azimuthal angle dependence of the integrated intensity at $ {\\\\mathbf {Q}}_{N} $ .\",\n \"RXS scans across $ {\\\\mathbf {Q}}_{N} $ were collected for different values of the azimuthal $\\\\psi $ , which corresponds to a rotation around the axis parallel to $ {\\\\mathbf {Q}}_{N} $ .\",\n \"Here, an azimuthal angle $\\\\psi = 0$ coincides with a sample orientation that has the $\\\\left( 1, 1, L \\\\right)$ crystallographic direction lying in the scattering plane.\",\n \"The clear 2-fold modulation of the RXS intensity as a function of the azimuthal angle, shown in Fig.\",\n \"REF D, is consistent with a magnetic structure with moments oriented along $\\\\left( 1, -1, 0 \\\\right)$ , i.e.\",\n \"perpendicular to the in-plane projection of the Néel ordering vector.\",\n \"The collective information, including the resonance energy profile, azimuthal symmetry and polarization restrictions (from the incoming photons) are most consistent with magnetic scattering.\",\n \"This interpretation is further bolstered by the close correspondence between the observed onset temperature and previous reports of the Néel transition [22], [23], [24].\",\n \"We also note that, among the possible high-symmetry magnetic space subgroups, the one determined to be most consistent with the azimuthal symmetry is equivalent to that found in other isostructural compounds, such as La$_2$ CuO$_4$ .\",\n \"Figure: Stripe-like electronic ordering along H,0\\\\left[ H, 0 \\\\right].\",\n \"A: representative momentum scans across 𝐐 s =1/2,0,L {\\\\mathbf {Q}}_{s} = \\\\left( 1/2, 0, L \\\\right) at a photon energy E=6.009E = 6.009 keV.\",\n \"B: integrated RXS intensity at 𝐐 s {\\\\mathbf {Q}}_{s} vs. temperature; the inset shows a series of RXS rocking curves across 𝐐 s {\\\\mathbf {Q}}_{s} for a few representative temperatures.Having identified the nature and microscopic traits of the ordered state below $T_N \\\\sim 110$ K, we have focused on the $\\\\left[ H, 0 \\\\right]$ reciprocal space direction, to search for additional ordering instabilities.\",\n \"This further exploration was motivated by previous heat capacity data hinting at a second ordering transition around 140 K, whose nature has remained undisclosed [22].\",\n \"Consistently with a dual-instability scenario, we have observed a resonant reflection at the wavevector for period-2 stripe order $ {\\\\mathbf {Q}}_{s} = \\\\left( 1/2, 0, L \\\\right)$ (with $L = 4.84$ here, as well) as shown in Figure REF A.\",\n \"Similar to the reflection at the Néel vector $ {\\\\mathbf {Q}}_{N}$ , the diffraction peak at the stripe ordering vector also exhibits a resonant enhancement as the photon energy is tuned to $E = 6.009$ keV (see Supplementary Information).\",\n \"The integrated intensity of the rocking curves as a function of temperature is shown in Fig.\",\n \"REF B (RXS scans at representative temperatures are reported in the inset).\",\n \"Surprisingly, the transition was seen to occur around $T_s \\\\sim 50$ K, namely at a significantly lower temperature than the proposed orbital ordering transition.\",\n \"Here, we should note that a remnant scattering signal persists above the transition and up to the highest measured temperature (200 K), yet shows no variation around 140 K. This unusual temperature dependence has been noted in other transition metal compounds with coupled spin and orbital order parameters (OPs), in particular KCuF$_3$ , though we cannot rule out additional OPs of possibly different symmetry contributing to the remnant signal above the clear transition at $T_s$ .\",\n \"We reconcile the puzzle of this second transition at $T_s$ by noting that the ordering temperature is close to that of the appearance of a second oscillatory component near 45 K as recorded by zero-field muon spin rotation (ZF-$\\\\mu ^+$ SR) measurements on polycrystalline samples [24].\",\n \"Figure: Phase diagrams at zero and nonzero temperatures.\",\n \"A: Thephase diagram at zero temperature for the Kugel-Khomskii model with J 1 =J 4 =1.0J_1=J_4=1.0, based on a site-factorized wavefunction approach.\",\n \"Three stable phases appear in the phase diagram, including S Ne ´el T CAFO S_{N\\\\acute{e}el}T_{CAFO}, S Ne ´el T AFO S_{N\\\\acute{e}el}T_{AFO}, and S CAFM T AFO S_{CAFM}T_{AFO}.\",\n \"The inset is a schematic illustration of the S Ne ´el T CAFO S_{N\\\\acute{e}el}T_{CAFO} ordered state.\",\n \"The blue and red ovals represent |xz〉\\\\vert xz \\\\rangle and |yz〉\\\\vert yz \\\\rangle orbitals, respectively.\",\n \"The black arrows denote the spin state on the |xz〉\\\\vert xz \\\\rangle or |yz〉\\\\vert yz \\\\rangle orbital.\",\n \"S Ne ´el _{N\\\\acute{e}el} and S CAFM _{CAFM} stand for Néel order and stripe antiferromagnetic order, respectively.\",\n \"T CAFO T_{CAFO} marks a stripe-like orbital order with momentum (π,0)/(0,π)(\\\\pi ,0)/(0,\\\\pi ),and T AFO _{AFO} represents Néel-like orbital order with momentum (π,π)(\\\\pi ,\\\\pi ).\",\n \"B: The temperature-dependence of the magnetic (m S m_S) and orbital (m T m_T) order parameters from classical Monte Carlo simulations of the phase S Ne ´el T CAFO S_{N\\\\acute{e}el}T_{CAFO} with J 1 =J 4 =1.0,J 6 =0.4,J 8 =0.0J_1=J_4=1.0,J_6=0.4,J_8=0.0 on L=24,32,40L=24,32,40 clusters (marked by the black star in panel A).\",\n \"The model parameters J i J_i are set as J i =t i 2 /UJ_i=t^2_i/U, where UU is the on-site interaction and t i t_i represent hopping amplitudes between orbitals up to next-nearest neighbors.\",\n \"J 1 J_1 and J 4 J_4 are the nearest-neighbor couplings, and J 6 J_6 and J 8 J_8 the next-nearest-neighbor couplings.\",\n \"Further details are given in the Supplementary Information.To understand the microscopic physics behind the observed scattering peaks, we examine an effective spin-1 Kugel-Khomskii model for $\\\\alpha $ -Sr$_2$ CrO$_4$ (see Supplementary Information for additional details).\",\n \"In this compound, the necessarily high-spin $3d^2$ electronic configuration form an active $t_{2g}$ orbital subspace.\",\n \"Of particular interest are results from density functional theory (DFT) calculations showing a reversed crystal field splitting compared to typical tetragonal systems with an elongated c-axis lattice parameter [21].\",\n \"Within this scenario, two electrons occupy the $\\\\left\\\\lbrace \\\\vert xy \\\\rangle , \\\\vert xz \\\\rangle , \\\\vert yz \\\\rangle \\\\right\\\\rbrace $ $t_{2g}$ orbital manifold, with a proposed negative energy level splitting $\\\\Delta \\\\sim -0.6eV$ between the low-lying $\\\\vert xy \\\\rangle $ and $\\\\vert xz \\\\rangle $ /$ \\\\vert yz \\\\rangle $ orbitals.\",\n \"This reversed crystal field leaves a single electron in the two-fold degenerate $\\\\left\\\\lbrace \\\\vert xz \\\\rangle , \\\\vert yz \\\\rangle \\\\right\\\\rbrace $ submanifold, setting the stage for active orbital physics as encoded in a pseudospin-1/2 degree of freedom.\",\n \"In order to elucidate the essential physics arising from the interplay of the spin and orbital degrees of freedom in $\\\\alpha $ -Sr$_2$ CrO$_4$ , , , we consider the spin-1 Kugel-Khomskii Hamiltonian as the effective model: $\\\\small H_{KK}^{(i,j)} = -\\\\frac{1}{3}({\\\\bf S}_i\\\\cdot {\\\\bf S}_j+2)Q^{(1)}({\\\\bf T}_i,{\\\\bf T}_j)+ \\\\frac{1}{3}({\\\\bf S}_i\\\\cdot {\\\\bf S}_j-1) Q^{(2)}({\\\\bf T}_i,{\\\\bf T}_j),$ with $&&Q^{(n)}({\\\\bf T}_i,{\\\\bf T}_j)=\\\\nonumber \\\\\\\\&&f_{zz}^{(n)}T_{i}^{z}T_{j}^{z}+\\\\frac{1}{2} f_{+-}^{(n)}(T_{i}^{+}T_{j}^{-}+T_{i}^{-}T_{j}^{+})+f_{z}^{(n)}(T_{i}^{z}+T_{j}^{z})\\\\nonumber \\\\\\\\&&+\\\\frac{1}{2} f_{++}^{(n)}(T_{i}^{+}T_{j}^{+}+T_{i}^{-}T_{j}^{-})+f_{zx}^{(n)}(T_{i}^{z}T_{j}^{x}+T_{i}^{x}T_{j}^{z})\\\\nonumber \\\\\\\\&&+f_{x}^{(n)}(T_{i}^{x}+T_{j}^{x}) +f_{0}^{(n)},$ where ${\\\\bf S}_i$ and ${\\\\bf T}_i$ are the spin-1 and pseudospin-$1/2$ operators, respectively.\",\n \"$f^{(n)}$ with $n=1,2$ are functions of microscopic parameters that include the hopping amplitudes between orbitals up to next-nearest neighbor and the onsite Hubbard interactions and Hund's coupling (see Supplementary Information for details).\",\n \"We have performed calculations using a combination of mean-field and classical Monte Carlo methods.\",\n \"As shown in Fig.\",\n \"REF A, we find S$_{N\\\\acute{e}el}$ T$_{CAFO}$ , a Néel order accompanied by a stripe-like orbital order, over an extended parameter regime in the phase diagram.\",\n \"This phase corresponds to the same ordering vector as found in RXS experiments.\",\n \"In Fig.REF B, we show the results of classical Monte Carlo simulations for the evolution of the magnetic and orbital order parameters vs. temperature for the phase S$_{N\\\\acute{e}el}$ T$_{CAFO}$ .\",\n \"The numerical results clearly show that, as temperature is lowered, the magnetic order parameter develops first, which is followed by the emergence of the stripe-like orbital order.\",\n \"The proposed theoretical description captures the salient features of the experimental observations, including the ordering wavevectors ($\\\\mathbf {Q}_s$ and $\\\\mathbf {Q}_N$ ) and their respective transition temperatures.\",\n \"The Néel spin order setting in at the higher-temperature transition is consistent with our RXS measurements (see Supplementary Information) as well as the magnetic measurements in polycrystalline samples [22], [24].\",\n \"In addition, because the orbital order below the lower-temperature transition develops in a time-reversal-broken background, it is expected to develop an additional orbital moment at the same wavevector, in the presence of spin-orbit coupling.\",\n \"Given that the involved orbitals are $\\\\vert xz \\\\rangle $ and $\\\\vert yz \\\\rangle $ , the orbital moment is likely to dominantly point along the $c$ -axis, and this appears to be the most consistent with the existing RXS azimuthal dependence (see Supplementary Information) although further spectroscopic studies will be needed in order to reach a firm conclusion.\",\n \"Finally, while our model study has focused on the largest part of the energetics, a complete understanding will require the addition of couplings with smaller magnitude, such as the interlayer couplings.\",\n \"In summary, our experimental and theoretical results uncover the microscopic nature of the antiferromagnetic state in $\\\\alpha $ -Sr$_2$ CrO$_4$ , which is here shown using resonant X-ray scattering to arise from collinear Néel-type order at ${\\\\mathbf {Q}}_{N} = \\\\left( \\\\pm 1/2, \\\\pm 1/2, L \\\\right)$ with the spin axis in the CrO$_2$ plane, in close analogy with the parent state of cuprate superconductors.\",\n \"In addition to Néel order, we have detected coexisting stripe-like order setting in at a lower temperature $T_s \\\\sim 50$ K, with ordering vector $ {\\\\mathbf {Q}}_{s} = \\\\left( 1/2, 0, L \\\\right)$ .\",\n \"A Monte Carlo study of the spin-1 Kugel-Khomskii Hamiltonian captures the observed experimental signatures of Néel and stripe phases and their ordering sequence, thereby pointing to the physical realization of this model for a system of $d^2$ ions on a square lattice.\",\n \"Our study singles out $\\\\alpha $ -Sr$_2$ CrO$_4$ as a unique platform to explore the rich electronic phases in a Kugel-Khomskii-like spin-orbital system, and underscores the great scientific potential of a unique transition metal oxide, whose electronic phase diagram as a function of carrier doping is yet to be explored.\",\n \"We are grateful to A. Chubukov, D. Puggioni, J.M.\",\n \"Tranquada, and P.A.\",\n \"Lee for insightful discussions.\",\n \"Work at MIT (Z.H.Z., C.A.O., J.L., J.P., and R.C.)\",\n \"has been supported by the Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award Number DE-SC0019126.\",\n \"Part of this research concerning sample synthesis (Z.H.Z.)\",\n \"has been supported by NSF through the Massachusetts Institute of Technology Materials Research Science and Engineering Center DMR – 1419807.\",\n \"Work at Rice (W.H.\",\n \"and Q.S.)\",\n \"has in part been supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award No.\",\n \"DE-SC0018197 and by the Robert A. Welch Foundation Grant No.\",\n \"C-1411.\",\n \"R.C.\",\n \"acknowledges support from the Alfred P. Sloan Foundation.\",\n \"M.N.\",\n \"was supported by the Materials Sciences and Engineering Division, Basic Energy Sciences, Office of Science, US DOE.\",\n \"J. P. acknowledges financial support by the Swiss National Science Foundation Early Postdoc Mobility Fellowship Project No.\",\n \"P2FRP2_171824 and P400P2_180744.\",\n \"This research used beamline 4-ID of the National Synchrotron Light Source II, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Brookhaven National Laboratory under Contract No.\",\n \"DE-SC0012704.\",\n \"D.~Vaknin, S.~K.\",\n \"Sinha, D.~E.\",\n \"Moncton, D.~C.\",\n \"Johnston, J.~M.\",\n \"Newsam, C.~R.\",\n \"Safinya, and H.~E.\",\n \"King~Jr., Physical Review Letters 58, 2802 (1987).\",\n \"L.~Paolasini, R.~Caciuffo, A.~Sollier, P.~Ghigna, and M.~Altarelli, Physical Review Letters 88, 106403 (2002).\",\n \"A.~M.\",\n \"Oleś, G.~Khaliullin, P.~Horsch, and L.~F.\",\n \"Feiner, Physical Review B 72, 214431 (2005), URL https://link.aps.org/doi/10.1103/PhysRevB.72.214431.\",\n \"A.~M.\",\n \"Oleś, P.~Horsch, and G.~Khaliullin, Physical Review B 75, 184434 (2007), URL https://link.aps.org/doi/10.1103/PhysRevB.75.184434.\",\n \"F.~Krüger, S.~Kumar, J.~Zaanen, and J.~van~den Brink, Physical Review B 79, 054504 (2009), URL https://link.aps.org/doi/10.1103/PhysRevB.79.054504.\",\n \"authorD.~Vaknin, authorS.~K.\",\n \"Sinha, authorD.~E.\",\n \"Moncton, authorD.~C.\",\n \"Johnston, authorJ.~M.\",\n \"Newsam, authorC.~R.\",\n \"Safinya, and authorH.~E.\",\n \"King~Jr., journalPhysical Review Letters volume58, pages2802 (year1987).\",\n \"authorL.~Paolasini, authorR.~Caciuffo, authorA.~Sollier, authorP.~Ghigna, and authorM.~Altarelli, journalPhysical Review Letters volume88, pages106403 (year2002).\",\n \"authorA.~M.\",\n \"Oleś, authorG.~Khaliullin, authorP.~Horsch, and authorL.~F.\",\n \"Feiner, journalPhysical Review B volume72, pages214431 (year2005), https://link.aps.org/doi/10.1103/PhysRevB.72.214431.\",\n \"authorA.~M.\",\n \"Oleś, authorP.~Horsch, and authorG.~Khaliullin, journalPhysical Review B volume75, pages184434 (year2007), https://link.aps.org/doi/10.1103/PhysRevB.75.184434.\",\n \"authorF.~Krüger, authorS.~Kumar, authorJ.~Zaanen, and authorJ.~van~den Brink, journalPhysical Review B volume79, pages054504 (year2009), https://link.aps.org/doi/10.1103/PhysRevB.79.054504.\",\n \"authorY.~Joly, authorO.~Bunǎu, authorJ.~E.\",\n \"Lorenzo, authorR.~M.\",\n \"Galéra, authorS.~Grenier, and authorB.~Thompson, journalJournal of Physics: Conference Series volume190, pages012007 (year2009), ISSN issn1742-6596,\"\n ]\n]"},"id":{"kind":"string","value":"1906.04194"}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":25,"numItemsPerPage":100,"numTotalItems":1867602,"offset":2500,"length":100}},"jwt":"eyJhbGciOiJFZERTQSJ9.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc1MDcwMDM0Niwic3ViIjoiL2RhdGFzZXRzL2hvd2V5L3VuYXJYaXZlIiwiZXhwIjoxNzUwNzAzOTQ2LCJpc3MiOiJodHRwczovL2h1Z2dpbmdmYWNlLmNvIn0.VRS460ADZs--5rQxjFr1aloBcRZpcHgTf3TdAVQdoeXBpPlGcnL49az_2gIejcFJPaV2cCc2SHorAT4IhLuoBA","displayUrls":true},"discussionsStats":{"closed":0,"open":1,"total":1},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
text
sequencelengths 2
2.54k
| id
stringlengths 9
16
|
|---|---|
[
[
"Inflation in the general Poincar\\'e gauge cosmology"
],
[
"Abstract The general Poincar\\'e gauge cosmology given by a nine-parameter gravitational Lagrangian with ghost- and tachyon-free conditions is studied from the perspective of field theory.",
"By introducing new variables for replacing two (pseudo-) scalar torsions, the Poincar\\'e gauge cosmological system can be recast into a gravitational system coupled to two-scalar fields with a potential up to quartic-order.",
"We discussed the possibility of this system producing two types of inflation without any extra inflatons.",
"The hybrid inflation with a first-order phase transition can be ruled out, while the slow rollover can be achieved.",
"The numerical analysis shows that the two-scalar fields system evolved in a potential well processes spontaneously four stages: \"pre-inflation\", slow-roll inflation with large enough e-folds, \"pre-reheating\" and reheating.",
"We also studied the stableness of this system by setting large values of initial kinetic energies.",
"The results show that even if the system evolves past the highest point of the potential well, the scalar fields can still return to the potential well and cause inflation.",
"The general Poincar\\'e gauge cosmology provides us with a self-consistent candidate of inflation."
],
[
"Introduction",
"The standard model (SM) framework of cosmology based on Einstein's general relativity (GR) is quite successful in describing the evolution of the Universe on large enough scales [1].",
"The SM infers that the Universe experienced a significant accelerated expansion in the very early period, which is called the inflation.",
"By introducing the inflation, problems such as the horizon, the flatness, the origin of perturbations, and the monopoles, that have plagued cosmologists before 1980s, can be solved naturally [2].",
"After years of development, some inflationary models, such as the standard single-field (inflaton) and Starobinsky's inflation, can match the current observations in very high precision [3], [4].",
"Unfortunately, those models lack a more essential mechanism for the origin of the inflaton(s), namely the source(s) of inflation.",
"The classical theory of inflation requires the Universe to experience an exponentially accelerated expansion from about $10^{-36}s$ to $10^{-32}s$ in the cosmic chronology [5].",
"This expansion drove the spatial curvature of the Universe towards extreme flatness, and established the causal correlations on the uniformity of the cosmic microwave background (CMB), and generated the seeds of large-scale structure [6].",
"At the end of this stage, the expansion decelerated spontaneously and the Universe exited from the adiabatic process.",
"The subsequent reheating led to various particles to be generated [7], [8].",
"According to the way of exiting the expansion, the inflationary models can be classified into the slow rollover and the first-order phase transition [9].",
"In addition, an effective correction from the loop quantum cosmology (LQC) enables to push the beginning of the whole process back to the Planck scale, where the big bang singularity have been replaced by the big bounce [10], [11].",
"The mechanism of inflation from big bounce to reheating is clear phenomenologically.",
"As a single-field model, Starobinsky's inflation given by a Lagrangian $\\tilde{R}+\\tilde{R}^2/6M^2$ plus some small non-local terms (which are crucial for reheating after inflation) is an internally self-consistent cosmological model, which possess a (quasi-)de Sitter stage in the early Universe with slow-roll decay, and a graceful exit to the subsequent radiation-dominated Friedmann-Lemaître-Robertson-Walker (FLRW) stage [12], [13], [14].",
"This is one of the most appealing from both theoretical and observational perspectives among different models of inflation [15].",
"Besides adding directly higher order curvature invariants or scalar fields to the Einstein-Hilbert (EH) action, another more fundamental way to generalize GR from the geometric and gauge perspectives has been introduced systematically since 1970's [16], [17], which is called the Poincaré gauge gravity (PGG).",
"As the maximum group of Minkowski spacetime isometrics, Poincaré group possesses both translations and rotations, which totally has 10 degree of freedom.",
"If constructing a gauge field theory based on the local invariance of the Poincaré group, the gravity will be represented by two independent gauge fields: tetrads $e$ and spin-connections $\\omega $ , corresponding to the translations and rotations, respectively.",
"Analogous to the Yang-Mills theory, one can verify that torsion $T$ and curvature $R$ are just their gauge field strengths.",
"According to the Noether's theorems, the symmetries of translation and rotation lead to two conservation objects: energy-momentum and spin spin-angular momentum.",
"Further more, the energy-momentum can be connected through Einstein's equation with curvature, and the spin-angular momentum with torsion through Cartan's equation, which mean that the sources of spacetime curvature and torsion are energy-momentum and spin of matter, respectively.",
"The above are fundamental ideas of PGG, which follows the schemes of standard Yang-Mills theory.",
"From the geometrical perspective, the spacetime extends from Riemann's to Riemann-Cartan's, where curvature measures the difference of a vector after parallel transporting along a infinitesimal loop, and torsion for the failure of closure of the parallelogram made of the infinitesimal displacement.",
"In order to show the extension of PGG to GR, we plot the following diagram: Einstein = [rectangle, rounded corners, minimum width = 1.cm, minimum height=0.5cm,text centered,text width=2cm, draw = blue] Cartan = [rectangle, rounded corners, minimum width = 1.cm, minimum height=0.5cm,text centered,text width=2.5cm, draw = red] EinLor = [rectangle, minimum width=0.5cm, minimum height=0.5cm, text centered,text width=2cm, draw=blue] CarPoi = [rectangle, minimum width=0.5cm, minimum height=0.5cm, text centered,text width=2cm, draw=red] arrow = [thin,<->,>=stealth] arrow0 = [] arrow1 = [thin,->,>=stealth] arrow2 = [thin,->,>=stealth] [node distance=2cm] Einstein](cur)Curvature; Einstein, right of = cur, xshift = 1.0cm, yshift = -2.7cm](con)Connection; Einstein, left of = cur, xshift = -1.0cm, yshift = -2.7cm](EMT)Energy-Momentum; Cartan, right of = EMT, xshift = 0.01cm, yshift = -2.7cm, inner sep=0.001pt](tra)Translation; Einstein, left of = con, xshift = -0.01cm, yshift = -2.7cm, inner sep=0.001pt](rot)Rotation; Cartan, left of = tra, xshift = -0.01cm, yshift = -2.7cm](can)Canonical 1-form; Cartan, right of = rot, xshift = 0.01cm, yshift = -2.7cm](SAM)Spin Angular Momentum; Cartan, right of = can, xshift = 1.0cm, yshift = -2.7cm](tor)torsion; CarPoi, left of = tra, xshift = -0.5cm, inner sep=0.001pt](PG)Poincaré; EinLor, right of = rot, xshift = 0.5cm, inner sep=0.001pt](LG)Lorentz; CarPoi, above of = tor, yshift = 0.5cm](Car)Cartan; EinLor, below of = cur, yshift = -0.5cm](Ein)Einstein; (point1) at (-3cm, -6cm); [arrow] (cur) – node [above,rotate=-35] 2nd structure node [below,rotate=-35] $R=\\mathcal {D}_\\omega \\omega $ (con.north); [arrow] (cur) – node [above,rotate=35] Einstein eq.",
"(EMT.north); [arrow] (EMT.south) – node [above,rotate=-45] Noether's thm.",
"(tra.north); [arrow] (rot.north) – node [above,rotate=45] Gauge potential (con.south); [arrow] (tra.south) – node [below,rotate=45] Gauge potential (can.north); [arrow] (can.south) – node [above,rotate=-35] $T=\\mathcal {D}_\\omega \\theta $ node [below,rotate=-35] 1st structure (tor); [arrow0] (rot) – node [] $\\oplus $ (tra); [arrow] (tor) – node [below,rotate=35] Cartan eq.",
"(SAM.south); [arrow] (SAM.north) – node [below,rotate=-45] Noether's thm.",
"(rot.south); [arrow1] (tra) – (PG); [arrow2] (LG) – (rot); General speaking, the crucial different between PGG and GR-based theories, such as $f(R)$ gravity, is that the former removed the restriction of torsion-free.",
"However, the direct generalization from the EH action will be back to Einstein's theory, when the spin tensor of matter vanishes because of the algebraic Cartan equation, i.e.",
"torsion can not propagate.",
"This reminds us that in order to obtain the propagating torsion in the vacuum, the action should be also generalized.",
"The standard PGG Lagrangian has a quadratic field strength form [18]: $\\mathcal {L}_G\\sim \\Lambda +curvature+torsion^2+\\frac{1}{\\varrho }curvature^2,$ where $\\Lambda $ is the cosmological constant, and $\\varrho $ the parameter with certain dimension.",
"The additional quadratic terms are naturally at most second derivative if one regards tetrads and spin-connections as the fundamental variables.",
"It is likely that such terms introduce ghost degrees of freedom, when one considers the particle substance of the gravity.",
"That would be something troublesome even for a simple modified gravity theory.",
"The existence of the ghost is closely related to the fact that the modified equation of motion has orders of time-derivative higher than two, for example, scale factor $a$ will be fourth-order over time in the general quadratic curvature case in FLRW cosmology.",
"Due to Ostrogradsky's theorem [19], a system is not (kinematically) stable if it is described by a non-degenerate higher time-derivative Lagrangian.",
"To avoid the ghosts, a bunch of scalar-tensor theories of gravity was introduced, such as the Horndeski theory and beyond [20], [21].",
"Another way to evade Ostrogradsky's theorem is to break Lorentz invariance in the ultraviolet and include only high-order spatial derivative terms in the Lagrangian, while still keeping the time derivative terms to the second order.",
"This is exactly what Hořava did recently [22], [23].",
"In addition, another recipe to treat the ghosts is not removing them from the action, while focusing on the higher-order instability in the equations of motion [24].",
"For the general second-order Lagrangian with propagating torsion, a systematical way to remove the ghosts and tachyons was introduced in [25], [26] using spin projection operators.",
"The gauge fields $(e,\\omega )$ can be decomposed irreducibly by $su(2)$ group into different spin modes by means of the weak-field approximation.",
"In addition to the graviton, three classes spin-$0^{\\pm },1^{\\pm },2^{\\pm }$ modes of torsion were introduced.",
"[26] studied the general quadratic Lagrangian with nine-parameter and obtained the conditions on the parameters for not having ghosts and tachyons at the massive and massless sectors, respectively.",
"In this work, to develop a good cosmology based on PGG, we will adopt their nine-parameter Lagrangian with ghost- and tachyons-free conditions on parameters.",
"The Hamiltonian analysis of PGG for different modes can be found in [27], [28], which tell us that the only safe modes of torsion are spin-$0^{\\pm }$ , corresponding to the scalar and pseudo-scalar components of torsion, respectively.",
"It's natural to apply the corresponding Poincaré gague cosmology (PGC) on understanding the evolution of the Universe.",
"The last decade, a series of work [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39] (from both analytical and numerical approach) proved that it is possible to reproduce the late-time acceleration in PGC without “dark energy”.",
"In Ref.",
"[40], the authors discussed the early-time behaviors of the expanding solution of PGC with a scalar field (inflaton), while in Ref.",
"[41], a power-law inflation was studied in a $R+R^2$ model of PGC without inflaton.",
"The current work is a continuation of our previous one: Late-time acceleration and inflation in a Poincaré gauge cosmological model [42].",
"In our previous work, we proposed several fundamental assumptions to define the PGC on FLRW level.",
"Then we studied the general nine-parameter PGC Lagrangian with ghost- and tachyon-free constraints on parameters.",
"With specific choice of parameters, we obtained two Friedmann-like analytical solutions by varying the Lagrangian, where the scalar torsion $h$ -determined solution is consistent with the Starobinsky cosmology in the early time and the $f$ -determined solution contains naturally a constant geometric “dark energy” density, which cover the $\\Lambda $ CDM model in the late-time.",
"We further constrained the magnitudes of parameters using the latest observations.",
"However, we left a problem unsolved that two solutions are mutually exclusive even they are derived from a same Lagrangian.",
"The reason comes probably from that the restraint on $B_1$ (vanishing) is too strong, so that the high-order terms of $f$ are removed.",
"Therefore, in current work, we will investigate the general case at least ghost-free, and use the new results for leading to the slow-roll inflation without any extra fields.",
"This series of work aims to build self-consistent cosmology to solve the problem of SM in describing the evolution of the Universe, where “self-consistent” means without extra hypothesis of inflaton and “dark energy”.",
"This paper is organized as follows.",
"In Sec.",
", we start from the nine-parameter Lagrangian with the ghost- and tachyon-free conditions on parameters.",
"Then replacing the scalar and pseudo-scalar torsion by two new variables, we rewrite the cosmological equations obtained in [42] into new forms.",
"In Sec.",
", we discuss the possibility of the hybrid inflation with a first-order phase transition generated in this gravitational system.",
"In Sec.",
", we study the slow-roll inflation of this system.",
"We conclude and discuss our work in Sec.",
".",
"To learn more about what the current work based on, please see [42] and references therein."
],
[
"Cosmological equations",
"We consider the nine-parameter gravitational Lagrangian $\\mathcal {L}_G$ , which reads: 2 I=d4xg[12LG+LM], LG=R+LT+LR, LT a1TT+a2TT+a3TT, LR b1RR+b2RR+b3RR +b4RR+b5RR, where $\\kappa \\equiv 8\\pi G=8\\pi m_{PI}^{-2}$ with $m_{PI}$ the Planck mass, and $\\alpha $ , $a_1\\sim a_3$ are freely dimensionless Lagrangian parameters, while $b_1\\sim b_5$ are free Lagrangian parameters with dimension $m_{PI}^{-2}$ .",
"The $R^2$ term need not be included due to the use of the Chern-Gauss-Bonnet theorem [43]: $\\int d^4x\\sqrt{\\vert g\\vert }(R_{\\mu \\nu \\rho \\sigma }R^{\\mu \\nu \\rho \\sigma }-4R_{\\mu \\nu }R^{\\mu \\nu }+R^2)=0,$ for spacetime topologically equivalent to flat space.",
"For the pair of gauge field ($e,\\omega $ ) as the dynamical variables, the field equations are up to 2nd-order.",
"However, the gauge fields ($e,\\omega $ ) can be decomposed irreducibly by $su(2)$ group into different spin modes by means of the weak-field approximation.",
"In addition to the graviton, three classes spin-$0^{\\pm },1^{\\pm },2^{\\pm }$ modes of torsion were introduced.",
"It is obvious that in such a general quadratic, the ghosts and tachyons are inevitable for certain modes.",
"Fortunately, the authors studied this Lagrangian in [26] using the spin projection operators and obtained the conditions on parameters for not having ghosts and tachyons at the massive and massless sectors, respectively.",
"According to [26], we summarize the ghost- and tachyon-free conditions on parameters for action () in TABLE I: Table: The ghost- and tachyon-free conditions on parameters for six spin modes, respectively.For the massless sector, the ghost-free condition is just: $\\alpha >0$ .",
"We will still focus on the FLRW cosmology, where the spatial curvature free metric and non-vanishing components of torsion are, respectively $ds^{2} = -dt^{2}+a^{2}d\\mathbf {x}^{2},$ $T_{ij0} = a^2 h \\delta _{ij}, ~~ T_{ijk} = a^3 f \\epsilon _{ijk}, ~~ i,j,k=1,2,3.$ Where $a$ , $h$ , $f$ are scale factor, scalar torsion, and pseudo-scalar torsion respectively, and are functions of cosmic time $t$ .",
"The general cosmological equations corresponding to the action (), on the background can be found in our former work [42], which are cumbersome and the physical meaning lost.",
"While, we noticed that the equations (23) and (24) in [42] can be regarded as the dynamic evolutions of scalar torsion $h$ and pseudo-scalar torsion $f$ , and they are second-order of $h$ and $f$ , respectively.",
"We would like to recast them into the Klein-Gordon-like form by introducing the following new variables: $h&\\mapsto &\\phi _h=a(h-H),\\nonumber \\\\f&\\mapsto &\\phi _f=af,$ (the scale factors $a$ in transformations are for absorbing the Hubble rate $H$ would occur in the potential to avoid tricky recursions) then, the cosmological equations (14)$\\sim $ (17) in [42] can be rewritten as: 4 H2=13+13, 2H+3H2=-p-p, =-3H(+p), =1212B0a2h2-1212B1a2f2+V(h,f)-3A1-22H2, 12B0ah+12B0aHh+12(2B0-2B1-B2)fa2f+aV(h,f)h+3(A1-2)H=0, -12B1af-12B1aHf-12(2B0-2B1-B2)fa2h+aV(h,f)f=0, V(h,f)=3A1-22h2a2-3(4A0-)f2a2-6B0a4(f4-2B0+2B1B0f2h2+h4), where the combinations of parameters are (corresponding to (25) in [42]): 2 A0a1-a2, A12a1+a2+3a3, B0b1+b2+b3+b4+12b5, B1b1-12b5, B24b2+b3+b4+b5.",
"The degeneracy among these Lagrangian parameters on background makes the inequalities can not be solved completely.",
"However, it is obvious that ghost- and tachyon-free spin-$0^{\\pm }$ “particles” require: $B_0>0,~~B_1<0,~~\\alpha -4A_0>0,~~\\alpha A_1(A_1-\\alpha )>0.$ Now, the physical picture is quite clear, that the nine-parameter PGC system is equivalent to a gravitational system coupled two-scalar fields $(\\phi _h,\\phi _f)$ , with a potential up to quartic-order, $V(\\phi _h,\\phi _f)$ .",
"() and () are the equations of motion for $\\phi _h$ and $\\phi _f$ , respectively.",
"They look very symmetrical except the last term in $(\\ref {eq_phi_h})$ .",
"If $B_0$ , $B_1$ and $(2B_0-2B_1-B_2)$ don't vanish, $|(2B_0-2B_1-B_2)\\phi _f/a|$ represents the strength of interaction between two scalar fields.",
"We conclude that $B_0$ and $B_1$ must have the opposite sign so that the interaction terms in the equations of motion have opposite sign too.",
"The different between $A_1$ and $\\alpha $ measures the weight of $\\phi _h^2$ in the potential $V(\\phi _h,\\phi _f)$ , and analogously, $A_0$ and $\\alpha $ for $\\phi _f^2$ .",
"It will be convenient to overlook the $1/a$ factor in front of field $\\phi _h$ or $\\phi _f$ because of the inverse factor occurred in (REF ).",
"The ghost- and tachyon-free conditions for spin-$0^{\\pm }$ ensure that the kinetic energy terms are positive in (), as well as a potential well can be formed from () when one require $\\alpha >0$ and $A_1>0$ .",
"In addition, we notice that when setting $A_1=0$ , the last term in () will offset $H^2$ in the left hand side of Friedmann equation (), which degrades the entire system into the trivial situations as we studied in our former work [42].",
"If $A_1\\ne 0$ , to remove the possible recursion in (), we should set exactly $A_1=2$ .",
"The above system is general because we didn't set any additional assumptions on the parameters yet except the ghost- and tachyon-free conditions (REF ).",
"In the rest of this work, we will focus on the inflationary period of this system, thus the energy densities $\\rho $ and pressures $p$ of the matters (with equation of state parameters, i.e.",
"EOS: $w=0,1/3$ ) will be neglected.",
"According to the choosing of parameters, the potential () can be classified into several types of inflation."
],
[
"Hybrid inflation with first-order phase transition",
"We start from considering two types of hybrid inflation by means of the features of two scalar fields, $\\text{Case I:}\\\\V_I(\\phi _h,\\phi _f)&=&[12(B_0+2B_1)\\frac{\\phi _h^2}{a^2}-3(4A_0-\\alpha )]\\frac{\\phi _f^2}{a^2}-6B_0\\frac{\\phi _f^4}{a^4}\\nonumber \\\\&+&V_I^{eff}(\\phi _h),\\\\V_I^{eff}(\\phi _h)&\\equiv &3(1-\\alpha )\\frac{\\phi _h^2}{a^2}-6B_0\\frac{\\phi _h^4}{a^4},$ $\\text{Case II:}\\\\V_{II}(\\phi _h,\\phi _f)&=&[12(B_0+2B_1)\\frac{\\phi _f^2}{a^2}+3(1-\\alpha )]\\frac{\\phi _h^2}{a^2}-6B_0\\frac{\\phi _h^4}{a^4}\\nonumber \\\\&+&V_{II}^{eff}(\\phi _f),\\\\V_{II}^{eff}(\\phi _f)&\\equiv &-3(4A_0-\\alpha )\\frac{\\phi _f^2}{a^2}-6B_0\\frac{\\phi _f^4}{a^4},$ where we regard $\\phi _h$ as the inflaton caused the slow-roll inflationary phase, and $\\phi _f$ as an auxiliary field which can trigger a phase transition, occurring either before or just after the breaking of slow-roll conditions in Case I, and interchange the positions of $\\phi _h$ and $\\phi _f$ in Case II.",
"In the standard hybrid inflation, the curvature of potential in the auxiliary direction should be much greater than in the inflaton direction, so that the slow-roll inflation could happen when the auxiliary field rolled down to its minimum value, whereas the inflaton could remain large for a much longer time [9].",
"In Case I, the effective mass squared of the field $\\phi _f$ is equal to $2[12(B_0+2B_1)\\frac{\\phi _h^2}{a^2}-3(4A_0-\\alpha )]$ .",
"Therefore for $\\phi _h>\\phi ^c_{h}\\equiv \\frac{a}{2}\\sqrt{\\frac{4A_0-\\alpha }{B_0+2B_1}}$ , the only minimum of the potential is at $\\phi _f=0$ .",
"For this reason, we will consider the stage of inflation at large $\\phi _h$ with $\\phi _f=0$ .",
"However, we notice that in the potential, both $\\phi _h$ and $\\phi _f$ have the same orders and the same coefficient $-6B_0$ in the quartic term.",
"$\\phi _h$ and $\\phi _f$ are highly symmetrical, especially when $A_0$ approximates to $1/4$ , so it's difficult to distinguish them from their speed of slow-rolling.",
"Fortunately, the extra term $6(1-\\alpha )H$ in the equation of motion of $\\phi _h$ () may help us to break the symmetry in potential.",
"During the inflation, the Hubble rate $H$ is approximated as a constant, so it can be absorbed into the potential, and the effective potential of inflton $\\phi _h$ in Case I can be modified as $V_I^{eff^{\\prime }}(\\phi _h)= 6(1-\\alpha )H\\frac{\\phi _h}{a}+3(1-\\alpha )\\frac{\\phi _h^2}{a^2}-6B_0\\frac{\\phi _h^4}{a^4}.$ When $\\phi _h$ falls down to the critical point $\\phi ^c_h$ , the phase transition with another symmetry breaking occurs.",
"To fulfill the condition of the standard hybrid inflation with small quartic term [44], we assume the 1st-order term dominates the effective potential of $\\phi _h$ in (REF ) at the moment of phase transition (also throughout the inflation), i.e.",
"(substituting $\\phi ^c_h$ into (REF )) $&&\\frac{1-\\alpha }{4}\\frac{4A_0-\\alpha }{B_0+2B_1}-\\frac{B_0}{8}(\\frac{4A_0-\\alpha }{B_0+2B_1})^2\\nonumber \\\\&\\ll & (1-\\alpha )H\\sqrt{\\frac{4A_0-\\alpha }{B_0+2B_1}},$ which requires that $4A_0-\\alpha \\ll B_0+2B_1$ .",
"Then, the Hubble rate $H$ can be estimated as: $H\\cong \\kappa (1-\\alpha )\\sqrt{\\frac{4A_0-\\alpha }{B_0+2B_1}},$ which is a constant as we expected.",
"However, it's easy to check that by substituting (REF ) into (REF ), the magnitudes on both sides of “$\\ll $ ” are at the same order.",
"This contradicts our assumption of the hybrid inflation with small quartic term, which means there is no a minimum effective potential with a non-zero vacuum energy.",
"Theoretically speaking, the approach to hybrid inflation with a first-order phase transition failed for Case I.",
"The same conclusion can be get for Case II."
],
[
"Slow-roll inflation and numerical analysis",
"According to our presupposition that $B_0>0$ , if the coefficients of the quadratic terms in potential () are positive with the same magnitude, and $|B_1|$ has the same magnitude with $B_0$ , we can get a potential well with a effective radius $r_\\phi \\lesssim \\sqrt{\\frac{A_1-2\\alpha }{4B_0}}$ .",
"By choosing special values of parameters, the slow-roll inflation can be obtained in this potential well.",
"In this scenario, the numerical analysis is more clear and convincing than the theoretical analysis.",
"The dimensions for parameters and quantities read: 2 A0A11, B0B1B2mPI-2, tmPI-1, HhfmPI, hfmPI2.",
"In the unit of $m_{PI}=1$ , and by considering that $a$ can be rescaled, we set the initial data (labeled with “B”) as: 2 tB=0, aB=1, h(tB)=f(tB)=0, h(tB)=f(tB)=1.",
"To investigate the effect of each parameter on this system, we choose a set of fiducial values: 2 1-=0.99, A0=-0.245, B0=1, B1=-1, Bi2B0-2B1-B2=0, then vary a parameter and plot while keeping other parameters to maintain the fiducial value.",
"FIG.",
"REF , REF , REF , REF , REF are the evolution curves of Hubble rate $H$ for every choice of parameter, while FIG.",
"REF , REF , REF , REF , REF for the EOS of scalar fields $w_{\\phi }$ , where $w_{\\phi }\\equiv p_{\\phi }/\\rho _{\\phi }$ .",
"These evolution curves show that the system starts from a “pre-inflation” stage, then enters into the slow-roll inflation, meanwhile the EOS changes from positive to negative.",
"After a nearly constant stage, the system decay rapidly, which we call it “pre-reheating”.",
"Then the subsequent oscillation indicates that the system has entered a stage of reheating.",
"To match the current observations, the stage of inflation should last long enough, which can be quantified by e-folds $N_{inf}$ : $N_{inf}:=\\ln \\frac{a_f}{a_i},$ where $a_i$ is the scale factor at the moment $t_i$ of the inflationary onset, which is defined by the time when the Universe begins to accelerate $\\ddot{a}(t_i)=0$ , i.e.",
"$\\ddot{a}(t)$ first changes its sign right after the bouncing phase [45].",
"The end of the inflation is defined by the time $t_f$ when the accelerating expansion of the Universe stops, i.e.",
"$w_{\\phi }(t_f)=-1/3$ .",
"The current observations require that $N_{inf}>60$ .",
"Figure: Evolution of Hubble rate HH over time.",
"The large dashed (blue), the solid (orange) and the small dashed (red) lines correspond to β=0.9,0.99,0.999\\beta =0.9,0.99,0.999, respectively.",
"To compare with the fiducial value, the smaller β\\beta makes the decay advanced.",
"The e-folds for three scenarios read N inf =18.2,160.8,33.1N_{inf}=18.2,160.8,33.1.Figure: Evolution of EOS w φ w_{\\phi } over time.",
"The large dashed (blue), the solid (orange) and the small dashed (red) lines correspond to β=0.9,0.99,0.999\\beta =0.9,0.99,0.999, respectively.",
"w φ <-1/3w_{\\phi }<-1/3 during the whole inflationary stage, and w φ w_{\\phi } is approximately equal to -1-1 in the deep inflationary stage.",
"The inflation ends when w φ =-1/3w_{\\phi }=-1/3 again, then reheating starts.",
"The e-folds for three scenarios read N inf =18.2,160.8,33.1N_{inf}=18.2,160.8,33.1.Figure: Evolution of Hubble rate HH over time.",
"The large dashed (blue), the solid (orange) and the small dashed (red) lines correspond to A 0 =-0.645,-0.245,-1.045A_0=-0.645,-0.245,-1.045, respectively.",
"We can see that this parameter has little effect on the Hubble rate.",
"The e-folds for three scenarios read N inf =151.7,160.8,151.1N_{inf}=151.7,160.8,151.1.Figure: Evolution of EOS w φ w_{\\phi } over time.",
"The large dashed (blue), the solid (orange) and the small dashed (red) lines correspond to A 0 =-0.645,-0.245,-1.045A_0=-0.645,-0.245,-1.045, respectively.",
"w φ <-1/3w_{\\phi }<-1/3 during the whole inflationary stage, and w φ w_{\\phi } is approximately equal to -1-1 in the deep inflationary stage.",
"The inflation ends when w φ =-1/3w_{\\phi }=-1/3 again, then reheating starts.",
"The smaller value of A 0 A_0 don't influence the decay and reheating but leads to oscillation before inflation.",
"The e-folds for three scenarios read N inf =151.7,160.8,151.1N_{inf}=151.7,160.8,151.1.Figure: Evolution of Hubble rate HH over time.",
"The large dashed (blue), the solid (orange) and the small dashed (red) lines correspond to B 0 =0.5,1.0,2.0B_0=0.5,1.0,2.0, respectively.",
"To compare with the fiducial value, the smaller B 0 B_0 makes the decay advanced.",
"The e-folds for three scenarios read N inf =139.2,160.8,82.2N_{inf}=139.2,160.8,82.2.Figure: Evolution of EOS w φ w_{\\phi } over time.",
"The large dashed (blue), the solid (orange) and the small dashed (red) lines correspond to B 0 =0.5,1.0,2.0B_0=0.5,1.0,2.0, respectively.",
"w φ <-1/3w_{\\phi }<-1/3 during the whole inflationary stage, and w φ w_{\\phi } is approximately equal to -1-1 in the deep inflationary stage.",
"The inflation ends when w φ =-1/3w_{\\phi }=-1/3 again, then reheating starts.",
"To compare with the fiducial value, the smaller B 0 B_0 makes the curve overall left shift, while right shift for larger B 0 B_0.",
"The e-folds for three scenarios read N inf =139.2,160.8,82.2N_{inf}=139.2,160.8,82.2.Figure: Evolution of Hubble rate HH over time.",
"The large dashed (blue), the solid (orange) and the small dashed (red) lines correspond to B 1 =-0.1,-1.0,-1.5B_1=-0.1,-1.0,-1.5, respectively.",
"To compare with the fiducial value, the larger absolute value of B 1 B_1 makes the decay advanced.",
"The e-folds for three scenarios read N inf =157.2,160.8,55.6N_{inf}=157.2,160.8,55.6.Figure: Evolution of EOS w φ w_{\\phi } over time.",
"The large dashed (blue), the solid (orange) and the small dashed (red) lines correspond to B 1 =-0.1,-1.0,-1.5B_1=-0.1,-1.0,-1.5, respectively.",
"w φ <-1/3w_{\\phi }<-1/3 during the whole inflationary stage, and w φ w_{\\phi } is approximately equal to -1-1 in the deep inflationary stage.",
"The inflation ends when w φ =-1/3w_{\\phi }=-1/3 again, then reheating starts.",
"The larger absolute value of B 1 B_1 makes the decay advanced, while the smaller absolute value of B 1 B_1 makes w φ w_{\\phi } sinking before inflation.",
"The e-folds for three scenarios read N inf =157.2,160.8,55.6N_{inf}=157.2,160.8,55.6.Figure: Evolution of Hubble rate HH over time.",
"The large dashed (blue), the solid (orange) and the small dashed (red) lines correspond to B i =-50,0,50B_i=-50,0,50, respectively.",
"To compare with the fiducial value, any interaction between two-scalar fields makes the decay advanced.",
"The e-folds for three scenarios read N inf =12.6,160.8,105.6N_{inf}=12.6,160.8,105.6.Figure: Evolution of EOS w φ w_{\\phi } over time.",
"The large dashed (blue), the solid (orange) and the small dashed (red) lines correspond to B i =-50,0,50B_i=-50,0,50, respectively.",
"w φ <-1/3w_{\\phi }<-1/3 during the whole inflationary stage, and w φ w_{\\phi } is approximately equal to -1-1 in the deep inflationary stage.",
"The inflation ends when w φ =-1/3w_{\\phi }=-1/3 again, then reheating starts.",
"The interaction between two-scalar fields leads to oscillation on w φ w_{\\phi } before inflation.",
"The e-folds for three scenarios read N inf =12.6,160.8,105.6N_{inf}=12.6,160.8,105.6.According to the numerical analysis, we summarize the influences of every parameter comparing with the fiducial value as following: 1) smaller $\\beta $ can lead to the decay advanced; 2) $A_0$ doesn't influence the decay but causes oscillation before inflation; 3) smaller $B_0$ leads to the curve overall left shift; 4) smaller $|B_1|$ makes $w_{\\phi }$ before inflation; 5) the interaction between two-scalar fields makes the decay advanced and leads to oscillation on $w_{\\phi }$ before inflation.",
"Figure: The 3D phase diagram of two-scalar fields φ h \\phi _h and φ f \\phi _f (over aa) evolute in the potential well in the fiducial scenario.",
"The shape of potential well is determined by the values of parameters given by ().",
"The initial data is given by () where the initial kinetic energies (speeds) read φ h ˙(t B )=1\\dot{\\phi _h}(t_B)=1, φ f ˙(t B )=1\\dot{\\phi _f}(t_B)=1.Figure: The 3D phase diagram of two-scalar fields φ h \\phi _h and φ f \\phi _f (over aa) evolute in the potential well in the fiducial scenario (), but start with very large initial kinetic energies (speeds): φ h ˙(t B )=1000\\dot{\\phi _h}(t_B)=1000, φ f ˙(t B )=500\\dot{\\phi _f}(t_B)=500 corresponding to the thick trajectory and φ h ˙(t B )=1000\\dot{\\phi _h}(t_B)=1000, φ f ˙(t B )=-500\\dot{\\phi _f}(t_B)=-500 to the thin one.",
"The potential well looks shallower than the previous one in FIG.",
", not because we changed the fiducial parameters, but instead expanded the ranges of φ h \\phi _h and φ f \\phi _f (over aa).The 3D phase diagram FIG.",
"REF visualizes the evolutions of two-scalar fields $\\phi _h$ and $\\phi _f$ (over $a$ ) on the potential well in the fiducial scenario.",
"The system starts from the on set point where we set the initial data (), where the non-trivial values are the kinetic energies of two fields.",
"Then the system is driven by the kinetic energies and climbs to the high level of the potential well, and prepares for the slow-roll inflation.",
"The slow-roll inflation occurs on the wall of the potential well, where the deep inflation happened after the inflection point, where $\\phi _f$ is approximate to 0, and $\\phi _h$ is almost constant.",
"At last, the system decays and drops down from the wall of the potential well, towards the on set point of the phase space, then leads to the reheating.",
"It seems that if the initial kinetic energies are large enough, the system may cross the highest point of the potential well, causing it to collapse.",
"To test the stableness of this system, we keep the fiducial values of parameters () unchanged, but increase the initial kinetic energies (speeds) of two-scalar fields.",
"FIG.",
"REF is the 3D phase diagram of this case, where we set two pairs of very large initial kinetic energies (speeds): $\\dot{\\phi _h}(t_B)=1000$ , $\\dot{\\phi _f}(t_B)=500$ and $\\dot{\\phi _h}(t_B)=1000$ , $\\dot{\\phi _f}(t_B)=-500$ .",
"Both trajectories can cross the highest point of the potential well, but can return to the potential well and cause inflation anyway.",
"Numerical analysis shows that the system has good stability."
],
[
"Conclusion and discussion",
"PGG as a gauge field gravitational theory is a natural extension of Einstein's GR to the Poincaré group.",
"It is worth looking forward using PGC, the cosmology of PGG, to solve the problems in the cosmological SM, especially the mechanisms of inflation and late-time acceleration.",
"In this work, we started from the general nine-parameter gravitational Lagrangian of PGC, and introduced the ghost- and tachyon-free conditions for this Lagrangian.",
"By introducing new variables $\\lbrace \\phi _h,\\phi _f\\rbrace $ for replacing the scalar and pseudo-scalar torsion $\\lbrace h,f\\rbrace $ , we found the general PGC on background is equivalent to a gravitational system coupled to two-scalar fields with a potential up to quartic-order.",
"We analyzed the possibility of this system producing the hybrid inflation with first-order phase transition, and concluded that it is not feasible.",
"Then by choosing appropriate parameters, we constructed a potential well from the quartic-order potential, and studied the slow-roll inflation numerically.",
"We chose a set of fiducial values for parameters, and investigated the effects of each parameter on this system.",
"All the evolution curves show that this system experiences four different stages: “pre-inflation” (on set), slow-roll inflation, “pre-reheating” (decay) and reheating.",
"Most scenarios possess large enough e-folds which is required by the current theories and observations.",
"The 3D phase diagram of two-scalar fields shows clearly four stages of the evolution in the potential well.",
"At last, we studied the stableness of this system by setting large values of initial kinetic energies (speeds).",
"We found that even if the system evolves past the highest point of the potential well, the scalar fields can still return to the potential well and cause inflation.",
"In short, the numerical analysis for this general PGC system on background indicated that it is a good self-consistent candidate for the slow-roll inflation.",
"Further studies on the aspect of perturbation will be our next work, especially the primordial power spectrum from this system and it's effects on CMB.",
"It is also worth looking forward to unify the inflation and the late-time acceleration under PGC in the future.",
"We thank Prof. Abhay Ashtekar for helpful comments.",
"Lixin Xu is supported in part by National Natural Science Foundation of China under Grant No.",
"11675032.",
"Hongchao Zhang is supported from the program of China Scholarships Council No.",
"201706060084."
]
] | 1906.04340 |
[
[
"Abelian tropical covers"
],
[
"Abstract The goal of this article is to classify unramified covers of a fixed tropical base curve $\\Gamma$ with an action of a finite abelian group G that preserves and acts transitively on the fibers of the cover.",
"We introduce the notion of dilated cohomology groups for a tropical curve $\\Gamma$, which generalize simplicial cohomology groups of $\\Gamma$ with coefficients in G by allowing nontrivial stabilizers at vertices and edges.",
"We show that G-covers of $\\Gamma$ with a given collection of stabilizers are in natural bijection with the elements of the corresponding first dilated cohomology group of $\\Gamma$."
],
[
"Introduction",
"Class field theory is a pillar of algebraic number theory; it is mostly concerned with classifying finite abelian extensions of a fixed local or global field $K$ .",
"Similarly, abelian covers of a fixed Riemann surface $X$ can be classified in terms of its first homology group $H_1(X,\\mathbb {Z})$ , or in terms of its Jacobian $J(X)\\simeq H_1(X,\\mathbb {R}/\\mathbb {Z})$ .",
"André Weil, in a letter to his sister from 1940 (see [41]), pointed out the analogy between these two situations, as well as a potential bridge: the theory of abelian extensions of function fields over finite fields, an area that is now known as geometric class field theory (see [36]).",
"More recently, another analogy has entered the mathematical stage: between a Riemann surface $X$ and a metric graph $\\Gamma $ , or more generally a tropical curve $\\scalebox {0.8}[1.3]{\\sqsubset }$ .",
"Many classical geometric constructions for Riemann surfaces, such as the theory of divisors, linear equivalence, Jacobians, theta functions, and moduli spaces, have natural analogues for tropical curves, as beautifully illustrated in [31].",
"The success of this analogy is, of course, not a coincidence.",
"A tropical curve naturally arises as the dual graph $\\Gamma _X$ of a semistable degeneration $\\mathcal {X}$ of an algebraic curve $X$ (with the metric encoding the deformation parameters at the nodes of $\\mathcal {X}$ ).",
"Geometric constructions on $\\Gamma _X$ then naturally arise as combinatorial specializations of their classical counterparts on $X$ .",
"We refer the reader for example to [10] for a survey of this story in the case of linear series.",
"In this article, we develop a theory of $G$ -covers of a tropical curve $\\scalebox {0.8}[1.3]{\\sqsubset }$ , where $G$ is a finite abelian group.",
"A $G$ -cover of $\\scalebox {0.8}[1.3]{\\sqsubset }$ is an unramified harmonic morphism $\\scalebox {0.8}[1.3]{\\sqsubset }^{\\prime }\\rightarrow \\scalebox {0.8}[1.3]{\\sqsubset }$ (such morphisms were studied in [18] under the name of tropical admissible covers), together with an action of $G$ on $\\scalebox {0.8}[1.3]{\\sqsubset }^{\\prime }$ that preserves and acts transitively on the fibers.",
"We show that such covers are classified by two objects.",
"The first is a dilation stratification $\\mathcal {S}$ of $\\scalebox {0.8}[1.3]{\\sqsubset }$ , indexed by the subgroups of $G$ , that encodes the local stabilizer subgroups (see Def.",
"REF ).",
"The second is an element of a dilated cohomology group $H^1(\\scalebox {0.8}[1.3]{\\sqsubset },S)$ associated to $\\scalebox {0.8}[1.3]{\\sqsubset }$ and a dilation stratification $\\mathcal {S}$ of $\\scalebox {0.8}[1.3]{\\sqsubset }$ (see Def.",
"REF ).",
"In the spirit of the above analogies, one may think of this work as the starting point for a tropical version of class field theory.",
"Our principal result is the following (see Thm.",
"REF ): Theorem A Let $\\scalebox {0.8}[1.3]{\\sqsubset }$ be a tropical curve, let $G$ be a finite abelian group, and let $\\mathcal {S}$ be an admissible dilation stratification of $\\scalebox {0.8}[1.3]{\\sqsubset }$ .",
"Then there is a natural bijection between the set of unramified $G$ -covers of $\\scalebox {0.8}[1.3]{\\sqsubset }$ having dilation stratification $\\mathcal {S}$ and the dilated cohomology group $H^1(\\scalebox {0.8}[1.3]{\\sqsubset },\\mathcal {S})$ .",
"The main technical ingredient in the classification of $G$ -covers of a tropical curve $\\scalebox {0.8}[1.3]{\\sqsubset }$ is a theory of dilated cohomology groups of a graph marked by subgroups of $G$ .",
"This theory generalizes simplicial cohomology with coefficients in $G$ and satisfies a number of natural properties such as functoriality and pullback, and admits a long exact sequence.",
"It seems natural to generalize dilated cohomology to arbitrary simplicial complexes, but this is beyond the scope of our paper.",
"Since our methods are cohomological, they do not readily generalize to non-abelian groups.",
"In a future paper, we plan to treat the non-abelian case by relating dilated cohomology to Bass–Serre theory [35], [7] and developing a Galois theory for non-abelian unramified covers of tropical curves."
],
[
"Earlier and related works",
"A number of authors study graphs and tropical curves with a group action.",
"The simplest example is the case of tropical hyperelliptic curves, which are $\\mathbb {Z}/2\\mathbb {Z}$ -covers of a tree ([11], [16], [13], [2], [32], [8], [28]).",
"Brandt and Helminck [9] consider arbitrary cyclic covers of a tree, while Helminck [24] looks at the tropicalization of arbitrary abelian covers of algebraic curves from a non-Archimedean perspective.",
"Jensen and Len [26] classify unramified $\\mathbb {Z}/2\\mathbb {Z}$ -covers of arbitrary tropical curves in terms of dilation cycles, which is a special case of our dilation stratification; with this article we aim to generalize this aspect of their work.",
"While we do not pursue this direction here, $G$ -covers of curves may be used to produce interesting loci of special divisors and linear series.",
"For instance, Jensen and Len [26] and Len and Ulirsch [29] develop a theory of tropical Prym varieties associated to $\\mathbb {Z}/2\\mathbb {Z}$ -covers of tropical curves, with applications to algebraic Prym–Brill–Noether theory.",
"In a similar vein, Song [37] considers $G$ -invariant linear systems with the goal of studying their descent properties to the quotient.",
"From a moduli-theoretic perspective, studying degenerations of $G$ -covers of algebraic curves is equivalent to studying the compactification of the moduli space of $G$ -covers in terms of the moduli space of $G$ -admissible covers, as constructed in [5] and [12].",
"In [12] the authors have already introduced a graph-theoretic gadget to understand the boundary strata of this moduli space: so-called modular graphs with an action of a finite (not necessarily abelian) group $G$ .",
"This idea seems to have appeared independently in other works as well: Chiodo and Farkas [15] study the boundary of the moduli space of level curves, which is equivalent to a component of the moduli space of $G$ -admissible covers for a cyclic group $G$ , and look at cyclic covers of an arbitrary graph.",
"Their work has been extended to an arbitrary finite group $G$ by Galeotti in [22], [23].",
"Finally, in [38], Schmitt and van Zelm apply a graph-theoretic approach to the boundary of the moduli space of $G$ -admissible covers (for an arbitrary finite group $G$ ) to study their pushforward classes in the tautological ring of $\\overline{\\mathcal {M}}_{g,n}$ .",
"In [18] Cavalieri, Markwig, and Ranganathan develop a moduli-theoretic approach to the tropicalization of the moduli space of admissible covers (without a fixed group operation).",
"We extend this aspect of their article to the moduli space of $G$ -admissible covers in Section below.",
"In [17], Caporaso, Melo, and Pacini study the tropicalization of the moduli space of spin curves, which, in view of the results in [26], is closely related to to our story in the case $G=\\mathbb {Z}/2\\mathbb {Z}$ .",
"The problem of classifying covers of a graph with an action of a given group (not necessarily abelian) was studied by Corry in [19], [20], [21].",
"However, Corry considered a different category of graph morphisms, allowing edge contraction but not dilation.",
"To the best of our knowledge, no author has considered the problem of classifying all unramified covers of a given graph with an action of a fixed group.",
"It is instructive to recall the theory of abelian covers in two categories, both directly related to tropical geometry: topological covering spaces and algebraic étale covers.",
"Let $X$ be a path-connected, locally path-connected and semi-locally simply connected topological space, let $x_0\\in X$ be a base point, and let $G$ be a group.",
"A regular $G$ -cover of $(X,x_0)$ is a based covering space $(Y,y_0)\\rightarrow (X,x_0)$ together with an $G$ -action on $Y$ such that $G$ acts freely and transitively on fibers.",
"Based regular $G$ -covers of $(X,x_0)$ are classified by monodromy homomorphisms $\\pi _1(X,x_0)\\rightarrow G$ (the cover is connected if and only if the homomorphism is surjective).",
"If $G$ is a finite abelian group, then we can identify the set of such homomorphisms, canonically and independently of $x_0$ , with the cohomology group $H^1(X,G)$ .",
"We note that a $G$ -cover is rigidified by the $G$ -action: for example, if $p$ is a prime number, there is a single connected degree $p$ covering space $S^1\\rightarrow S^1$ , but there are $p-1$ connected $\\mathbb {Z}/p\\mathbb {Z}$ -covers of $S^1$ corresponding to the non-trivial elements of $H^1(S^1,\\mathbb {Z}/p\\mathbb {Z})\\simeq \\mathbb {Z}/p\\mathbb {Z}$ .",
"If $X$ is the underlying topological space of a tropical curve $\\scalebox {0.8}[1.3]{\\sqsubset }$ , then any regular $G$ -cover $X^{\\prime }\\rightarrow X$ can be given the structure of an unramified $G$ -cover $\\scalebox {0.8}[1.3]{\\sqsubset }^{\\prime }\\rightarrow \\scalebox {0.8}[1.3]{\\sqsubset }$ of tropical curves by pulling back the genus function from $\\scalebox {0.8}[1.3]{\\sqsubset }$ to $\\scalebox {0.8}[1.3]{\\sqsubset }^{\\prime }$ .",
"These $G$ -covers, which we call topological, have the property that $G$ -action on the fibers is free (see Ex.",
"REF and Ex.",
"REF ).",
"The corresponding dilation stratification $\\mathcal {S}$ on $\\scalebox {0.8}[1.3]{\\sqsubset }$ is trivial, and the dilated cohomology group $H^1(\\scalebox {0.8}[1.3]{\\sqsubset },\\mathcal {S})$ reduces to $H^1(X,G)$ .",
"Let $X$ be an algebraic variety over a field $k$ and $x_0$ a geometric base point of $X$ .",
"Like its topological counterpart, the étale fundamental group $\\pi _1^{ét}(X,x_0)$ of $X$ classifies finite étale covers of $X$ .",
"For a finite abelian group $G$ the set of continuous homomorphisms $\\operatorname{Hom}(\\pi _1^{ét}(X,x_0),G)$ is equal to the set of Galois coverings of $X$ with Galois group $G$ .",
"If $X$ is a smooth projective curve, the abelian coverings of $X$ naturally arise as pullbacks of (always abelian) coverings of its Jacobian $J$ along the Abel-Jacobi map $X\\rightarrow J$ .",
"In particular, we have an induced isomorphism $\\pi _1^{ét}(X,x_0)^{ab}\\simeq \\pi _1^{ét}(J,x_0)$ .",
"In Section we will see how our a priori purely combinatorial construction can be thought of a tropical limit of this well-known story.",
"Our paper is organized as follows.",
"In Sec.",
", we review the necessary definitions from graph theory and tropical geometry.",
"In Sec.",
", we introduce $G$ -covers, $G$ -dilation data, and dilated cohomology groups.",
"We are primarily interested in classifying abelian covers of tropical curves, however, our constructions are purely graph-theoretic in nature and may be of interest to specialists in graph theory and topology.",
"For this reason, we first develop the theory of $G$ -covers for unweighted graphs.",
"In Sec.",
"we prove our main classification results, and then extend them to weighted graphs, weighted metric graphs, and tropical curves.",
"Finally, in Sec.",
"we relate our constructions to the tropicalization of the moduli space of admissible $G$ -covers.",
"The authors would like to thank Matthew Baker, Madeline Brandt, Renzo Cavalieri, Gavril Farkas, Paul Helminck, David Jensen, Andrew Obus, Sam Payne, Matthew Satriano, Johannes Schmitt, and Jason van Zelm for useful discussions.",
"This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie-Skłodowska-Curie Grant Agreement No.",
"793039.",
"Figure: NO_CAPTION"
],
[
"Definitions and notation",
"We develop the theory of $G$ -covers of graphs on several levels successively: graphs, weighted graphs, metric graphs, and tropical curves.",
"In this section, we recall the necessary definitions from graph theory."
],
[
"Graphs",
"We first consider unweighted graphs without a metric.",
"Definition 2.1 A graph with legs $\\Gamma $ , or simply a graph, consists of the following: A finite set $X(\\Gamma )$ .",
"An idempotent root map $r:X(\\Gamma )\\rightarrow X(\\Gamma )$ .",
"An involution $\\iota :X(\\Gamma )\\rightarrow X(\\Gamma )$ whose fixed set contains the image of $r$ .",
"The image $V(\\Gamma )$ of $r$ is the set of vertices of $\\Gamma $ , and its complement $H(\\Gamma )=X(\\Gamma )\\backslash V(\\Gamma )$ is the set of half-edges of $\\Gamma $ .",
"The involution $\\iota $ preserves $H(\\Gamma )$ and partitions it into orbits of size 1 and 2; we call these respectively the legs and edges of $\\Gamma $ and denote the corresponding sets by $L(\\Gamma )$ and $E(\\Gamma )$ .",
"The root map assigns one root vertex to each leg and two root vertices to each edge.",
"A loop is an edge whose root vertices coincide.",
"We note that, from a graph-theoretic point of view, there is essentially no difference between a leg and an extremal edge.",
"This distinction is important, however, from a tropical viewpoint: legs are the tropicalizations of marked points, while an extremal edge represents a rational tail.",
"Note that, unlike an extremal edge, a leg does not have a vertex at its free end.",
"The tangent space $T_v \\Gamma $ and valency $\\operatorname{val}(v)$ of a vertex $v\\in V(\\Gamma )$ are defined by $T_v\\Gamma =\\big \\lbrace h\\in H(\\Gamma )\\big |r(h)=v\\big \\rbrace \\textrm { and } \\operatorname{val}(v)=\\#(T_v\\Gamma ).$ Definition 2.2 Let $\\Gamma $ be a graph.",
"A subgraph $\\Delta $ of $\\Gamma $ is a subset of $X(\\Gamma )$ closed under the root and involution maps.",
"Given a subgraph $\\Delta \\subset \\Gamma $ and a vertex $v\\in V(\\Delta )$ , we denote $\\operatorname{val}_{\\Delta }(v)$ the valency of $v$ viewed as a vertex of $\\Delta $ .",
"A subgraph $\\Delta \\subset \\Gamma $ is called a cycle if $\\operatorname{val}_{\\Delta }(v)$ is even for every $v\\in V(\\Delta )$ .",
"A subgraph $\\Delta \\subset \\Gamma $ is called edge-maximal if every edge $e\\in E(\\Gamma )$ having both root vertices in $\\Gamma $ lies in $\\Delta $ .",
"It is clear that a subgraph of $\\Gamma $ is edge-maximal if and only if it is the largest subgraph of $\\Gamma $ with a given set of vertices.",
"Definition 2.3 Let $\\Gamma $ be a graph.",
"An orientation on $\\Gamma $ is a choice of order $(h,h^{\\prime })$ on each edge $e=\\lbrace h,h^{\\prime }\\rbrace \\in E(\\Gamma )$ .",
"We call $s(e)=r(h)$ and $t(e)=r(h^{\\prime })$ the source and target vertices of $e$ .",
"Definition 2.4 A finite morphism of graphs $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ , or simply a morphism, is a map of sets $\\varphi :X(\\Gamma ^{\\prime })\\rightarrow X(\\Gamma )$ which commutes with the root and involution maps, such that edges map to edges and legs map to legs.",
"An automorphism $\\varphi :\\Gamma \\rightarrow \\Gamma $ of a graph $\\Gamma $ is a morphism with an inverse.",
"We denote the group of automorphisms of $\\Gamma $ by $\\operatorname{Aut}(\\Gamma )$ .",
"We remark that a nontrivial graph automorphism may act trivially on the vertex and edge sets.",
"For example, the graph $\\Gamma $ consisting of one vertex $v$ and one loop $e=\\lbrace h,h^{\\prime }\\rbrace $ has a nontrivial automorphism fixing $v$ and exchanging $h$ and $h^{\\prime }$ .",
"To form quotients of graphs by group actions, we need to exclude such automorphisms from consideration.",
"Definition 2.5 Let $\\Gamma $ be a graph, and let $G$ be a group.",
"A $G$ -action on $\\Gamma $ is a homomorphism of $G$ to the automorphism group $\\operatorname{Aut}(\\Gamma )$ such that for every $g\\in G$ , the corresponding automorphism does not flip edges.",
"In other words, for every edge $e=\\lbrace h,h^{\\prime }\\rbrace \\in E(\\Gamma )$ either $\\varphi (e)\\ne e$ , or $\\varphi (h)=h$ and $\\varphi (h^{\\prime })=h^{\\prime }$ .",
"Given a $G$ -action on $\\Gamma $ , we define the quotient graph $\\Gamma /G$ by setting $X(\\Gamma /G)=X(\\Gamma )/G$ .",
"The root and involution maps on $\\Gamma $ are $G$ -invariant and descend to $\\Gamma /G$ .",
"It is clear that $V(\\Gamma /G)=V(\\Gamma )/G$ and $H(\\Gamma /G)=H(\\Gamma )/G$ , and the no-flipping assumption implies that the $G$ -action does not identify the two half-edges of any edge of $\\Gamma $ .",
"Therefore $E(\\Gamma /G)=E(\\Gamma )/G$ and $L(\\Gamma /G)=L(\\Gamma )/G$ , and the quotient map $\\pi :\\Gamma \\rightarrow \\Gamma /G$ is a finite morphism."
],
[
"Weighted graphs, harmonic morphisms, and ramification",
"We now consider graphs with vertex weights.",
"Heuristically, one may think of a vertex of weight $g$ as an infinitesimally small graph with $g$ loops (cf.",
"[3]).",
"Definition 2.6 A weighted graph $(\\Gamma ,g)$ is a pair consisting of a graph $\\Gamma $ and a vertex weight function $g:V(\\Gamma )\\rightarrow \\mathbb {Z}_{\\ge 0}$ .",
"We will usually suppress $g$ and denote weighted graphs by $\\Gamma $ .",
"We define the Euler characteristic $\\chi (v)$ of a vertex $v\\in V(\\Gamma )$ on a weighted graph $\\Gamma $ as $\\chi (v)=2-2g(v)-\\operatorname{val}(v).$ The genus of a connected graph $\\Gamma $ is defined to be $g(\\Gamma )=\\#(E(\\Gamma ))-\\#(V(\\Gamma ))+1+\\sum _{v\\in V(\\Gamma )}g(v).$ We define the Euler characteristic $\\chi (\\Gamma )$ of a graph $\\Gamma $ by $\\chi (\\Gamma )=\\sum _{v\\in V(\\Gamma )} \\chi (v);$ this is not to be confused with the topological Euler characteristic of $\\Gamma $ .",
"An easy calculation shows that, if $\\Gamma $ is connected, then $\\chi (\\Gamma )=2-2g(\\Gamma )-\\#(L(V)).$ A subgraph $\\Delta \\subset \\Gamma $ of a weighted graph $\\Gamma $ is naturally given the structure of a weighted graph by restricting the weight function $g$ .",
"In this case, denote $\\chi _{\\Delta }(v)=2-2g(v)-\\operatorname{val}_{\\Delta }(v)$ the Euler characteristic of a vertex $v$ of $\\Delta $ .",
"We say that a vertex $v\\in \\Gamma $ of a weighted graph $\\Gamma $ is unstable if $\\chi (v)\\ge 1$ , semistable if $\\chi (v)\\le 0$ , and stable if $\\chi (v)\\le -1$ .",
"An unstable vertex has genus zero and is either isolated or extremal.",
"A semistable vertex that is not unstable is either an isolated vertex of genus one or a valency two vertex of genus zero, in which case we call it simple.",
"We say that a graph $\\Gamma $ is semistable if all of its vertices are semistable, and stable if all of its vertices are stable.",
"Let $\\Gamma $ be a connected weighted graph with $\\chi (\\Gamma )<0$ .",
"Following [4], we construct a stable graph $\\Gamma _{st}$ , called the stabilization of $\\Gamma $ , as follows.",
"First, we construct the semistabilization $\\Gamma _{sst}$ of $\\Gamma $ by inductively removing all extremal edges ending at an extremal vertex of genus zero (but not the legs).",
"The graph $\\Gamma _{sst}$ is a semistable subgraph of $\\Gamma $ , and it is clear that $\\chi (\\Gamma _{sst})=\\chi (\\Gamma )$ , and that any vertices of $\\Gamma _{sst}$ that are not stable are simple.",
"We then construct $\\Gamma _{st}$ by gluing together the two half-edges at each simple vertex $v$ of $\\Gamma _{sst}$ .",
"Specifically, if $v$ is an endpoint of two edges $e_1$ and $e_2$ , we replace $v$ , $e_1$ , and $e_2$ with a new edge connecting the other endpoints of $e_1$ and $e_2$ .",
"If $v$ is an endpoint of an edge $e$ and a leg $l$ , we replace $v$ , $e$ , and $l$ with a new leg rooted at the other endpoint of $e$ .",
"The result is a stable graph $\\Gamma _{st}$ with $\\chi (\\Gamma _{st})=\\chi (\\Gamma _{sst})=\\chi (\\Gamma )$ .",
"Definition 2.7 Let $\\Gamma $ and $\\Gamma ^{\\prime }$ be graphs.",
"A finite harmonic morphism $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ , or simply a harmonic morphism, consists of a finite morphism $\\Gamma ^{\\prime }\\rightarrow \\Gamma $ and a map $d_{\\varphi }:X(\\Gamma ^{\\prime })\\rightarrow \\mathbb {Z}_{>0}$ , called the degree of $\\varphi $ , such that the following properties are satisfied: If $e^{\\prime }=\\lbrace h^{\\prime }_1,h^{\\prime }_2\\rbrace \\in E(\\Gamma ^{\\prime })$ is an edge then $d_{\\varphi }(h^{\\prime }_1)=d_{\\varphi }(h^{\\prime }_2)$ .",
"We call this number the degree of $\\varphi $ along $e^{\\prime }$ and denote it $d_{\\varphi }(e^{\\prime })$ .",
"For every vertex $v^{\\prime }\\in V(\\Gamma ^{\\prime })$ and every tangent direction $h\\in T_{\\varphi (v)}\\Gamma $ , we have $d_{\\varphi }(v^{\\prime })=\\sum _{\\begin{array}{c}h^{\\prime }\\in T_{v^{\\prime }}\\Gamma ^{\\prime }, \\\\ \\varphi (h^{\\prime })=h\\end{array}}d_{\\varphi }(h^{\\prime }).$ In particular, this sum does not depend on the choice of $h$ .",
"Let $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ be a harmonic morphism of graphs, where $\\Gamma $ is connected.",
"The sum $\\deg (\\varphi )=\\sum _{\\begin{array}{c}v^{\\prime }\\in V(\\Gamma ^{\\prime }),\\\\ \\varphi (v^{\\prime })=v\\end{array}}d_{\\varphi }(v^{\\prime })=\\sum _{\\begin{array}{c}e^{\\prime }\\in E(\\Gamma ^{\\prime }), \\\\ \\varphi (e^{\\prime })=e\\end{array}}d_{\\varphi }(e^{\\prime })=\\sum _{\\begin{array}{c}l^{\\prime }\\in L(\\Gamma ^{\\prime }),\\\\ \\varphi (l^{\\prime })=l\\end{array}}d_{\\varphi }(l^{\\prime })$ does not depend on the choice of $v\\in V(\\Gamma )$ , $e\\in E(\\Gamma )$ or $l\\in L(\\Gamma )$ and is called the degree of $\\varphi $ (see Section 2 of [1]).",
"Definition 2.8 Let $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ be a harmonic morphism of weighted graphs.",
"The ramification degree $\\operatorname{Ram}_{\\varphi }(v^{\\prime })$ of $\\varphi $ at a vertex $v^{\\prime }\\in V(\\Gamma ^{\\prime })$ is equal to $\\operatorname{Ram}_{\\varphi }(v^{\\prime })=d_{\\varphi }(v^{\\prime })\\chi (\\varphi (v^{\\prime }))-\\chi (v^{\\prime }).$ We say that $\\varphi $ is effective if $\\operatorname{Ram}_{\\varphi }(v^{\\prime })\\ge 0$ for all $v^{\\prime }\\in V(\\Gamma ^{\\prime })$ , and unramified if $\\operatorname{Ram}_{\\varphi }(v^{\\prime })=0$ for all $v^{\\prime }\\in V(\\Gamma ^{\\prime })$ .",
"Remark 2.9 Unramified morphisms were studied extensively in [18], where they were called tropical admissible covers.",
"We partly preserve this terminology: for example, we call a dilation stratification admissible if it corresponds to an unramified cover.",
"A simple calculation shows that our definition of ramification degree agrees with the standard one in the literature (see, for example, Sec.",
"2.2 in [2] or Def.",
"16 in [18]): $\\operatorname{Ram}_{\\varphi }(v^{\\prime })=d_{\\varphi }(v^{\\prime })\\left(2-2g(\\varphi (v^{\\prime }))\\right)-\\left(2-2g(v^{\\prime })\\right)-\\sum _{h^{\\prime }\\in T_{v^{\\prime }}\\Gamma ^{\\prime }} \\left(d_{\\varphi }(h^{\\prime })-1\\right).$ For an unramified harmonic morphism $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ , we call equation (REF ) the local Riemann–Hurwitz condition at $v^{\\prime }\\in V(\\Gamma ^{\\prime })$ .",
"Adding together these conditions at all $v^{\\prime }\\in V(\\Gamma ^{\\prime })$ , we obtain the global Riemann–Hurwitz condition $\\chi (\\Gamma ^{\\prime })=\\deg (\\varphi )\\chi (\\Gamma ).$ Example 2.10 Let $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ be an unramified harmonic morphism, and suppose that $d_{\\varphi }(v^{\\prime })=1$ for some $v^{\\prime }$ .",
"By the harmonicity condition, each $h\\in T_{\\varphi (v^{\\prime })}\\Gamma $ has a unique preimage in $T_{v^{\\prime }}\\Gamma ^{\\prime }$ , hence $\\operatorname{val}(v^{\\prime })=\\operatorname{val}(\\varphi (v^{\\prime }))$ .",
"Furthermore, we have $\\chi (\\varphi (v^{\\prime }))=\\chi (v^{\\prime })$ , which implies that $g(v^{\\prime })=g(\\varphi (v^{\\prime }))$ .",
"It follows that $\\varphi $ is a local isomorphism of weighted graphs in a neighborhood of $v^{\\prime }$ .",
"In particular, an unramified harmonic morphism of degree one is a graph isomorphism, and vice versa.",
"We observe that if $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ is an effective harmonic morphism and $\\Delta \\subset \\Gamma $ is a subgraph with preimage $\\Delta ^{\\prime }=\\varphi ^{-1}(\\Delta )$ , then the induced map $\\varphi |_{\\Delta ^{\\prime }}:\\Delta ^{\\prime }\\rightarrow \\Delta $ is also an effective harmonic morphism, since the ramification degree does not decrease when a half-edge and its preimages are removed.",
"However, if $\\varphi $ is unramified, then $\\varphi |_{\\Delta ^{\\prime }}$ is not necessarily unramified.",
"We now show that unramified morphisms naturally restrict to stabilizations.",
"Let $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ be an unramified harmonic morphism of connected graphs, and assume that $\\chi (\\Gamma )<0$ (or, equivalently by (REF ), that $\\chi (\\Gamma ^{\\prime })<0$ ).",
"For any two vertices $v^{\\prime }\\in V(\\Gamma ^{\\prime })$ and $v=\\varphi (v^{\\prime })\\in V(\\Gamma )$ , Eq.",
"(REF ) implies that $v^{\\prime }$ is unstable if and only if $v$ is unstable, in which case $\\chi (v^{\\prime })=\\chi (v)=1$ and $d_{\\varphi }(v^{\\prime })=1$ .",
"Let $v\\in V(\\Gamma )$ be an extremal vertex of genus 0, let $e\\in E(\\Gamma )$ be the unique edge rooted at $v$ , and let $u\\in V(\\Gamma )$ be the other root vertex of $v$ .",
"By the above, we see that $v\\in V(\\Gamma )$ has $\\deg (\\varphi )$ preimages $v^{\\prime }_i$ in $\\Gamma ^{\\prime }$ , each of which is a root vertex of a unique extremal edge $e^{\\prime }_i$ mapping to $e$ with local degree 1.",
"For any $u^{\\prime }\\in \\varphi ^{-1}(u)$ , $d_{\\varphi }(u^{\\prime })$ of the edges $e^{\\prime }_i$ are rooted at $u^{\\prime }$ .",
"Therefore, removing $v^{\\prime }_i$ , $e^{\\prime }_i$ , $v$ , and $e$ increases $\\chi (u)$ by 1 and increases each $\\chi (u^{\\prime })$ by $d_{\\varphi }(u^{\\prime })$ , hence does not change the local Riemann–Hurwitz condition at $u^{\\prime }$ .",
"Proceeding in this way, we remove all unstable vertices of $\\Gamma ^{\\prime }$ and $\\Gamma $ and obtain an unramified harmonic morphism $\\varphi _{sst}:\\Gamma ^{\\prime }_{sst}\\rightarrow \\Gamma _{sst}$ .",
"Similarly, we see that for two vertices $v^{\\prime }\\in V(\\Gamma ^{\\prime }_{sst})$ and $v=\\varphi (v^{\\prime })\\in V(\\Gamma _{sst})$ , Eq.",
"(REF ) implies that one is simple if and only if the other is.",
"Furthermore, by the harmonicity condition, the degrees of $\\varphi $ at the two half-edges at $v^{\\prime }$ are equal, hence we can remove $v^{\\prime }$ and $v$ , glue together the free half-edges, and extend $\\varphi $ ; this does not change the local Riemann–Hurwitz condition at any remaining vertex of $\\Gamma ^{\\prime }_{sst}$ .",
"Proceeding in this way, we obtain an unramified morphism $\\varphi _{st}:\\Gamma ^{\\prime }_{st}\\rightarrow \\Gamma _{st}$ .",
"Definition 2.11 Let $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ be an unramified harmonic morphism of connected weighted graphs, such that $\\chi (\\Gamma )<0$ (or, equivalently, $\\chi (\\Gamma ^{\\prime })<0$ ).",
"The unramified morphism $\\varphi _{st}:\\Gamma ^{\\prime }_{st}\\rightarrow \\Gamma _{st}$ constructed above is called the stabilization of $\\varphi $ .",
"Finally, we define the contraction of a graph along a subset of its edges; this can be viewed as a non-finite harmonic morphism of degree one.",
"Definition 2.12 Let $\\Gamma $ be a weighted graph, and let $S\\subset E(\\Gamma )$ be a set of edges of $\\Gamma $ .",
"We define the weighted edge contraction $\\Gamma /S$ of $\\Gamma $ along $S$ as follows.",
"Let $\\Delta $ be the minimal subgraph of $\\Gamma $ whose edge set contains $S$ , and let $\\Delta _1,\\ldots ,\\Delta _k$ be the connected components of $\\Delta $ .",
"We obtain $\\Gamma /S$ from $\\Gamma $ by contracting each $\\Delta _i$ to a vertex $v_i$ of genus $g(\\Delta _i)$ .",
"Given a harmonic morphism $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ of weighted graphs, we can contract a subset of edges $S\\subset E(\\Gamma )$ of $\\Gamma $ , and their preimages in $\\Gamma ^{\\prime }$ .",
"Connected components of graphs map to connected components, and degree is constant when restricted to a connected component, so there is a natural harmonic morphism $\\varphi _{S}:\\Gamma ^{\\prime }/\\varphi ^{-1}(S)\\rightarrow \\Gamma /S$ .",
"A simple calculation shows that if $\\varphi $ is unramified, then so is $\\varphi _S$ : Proposition 2.13 (Proposition 19 in [18]) Let $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ be an unramified harmonic morphism of unweighted graphs, let $S\\subset E(\\Gamma )$ be a subset of the edges of $\\Gamma $ , and let $\\Gamma ^{\\prime }/\\varphi ^{-1}(S)$ and $\\Gamma /S$ be the weighted edge contractions.",
"Then $\\varphi _{S}:\\Gamma ^{\\prime }/\\varphi ^{-1}(S)\\rightarrow \\Gamma /S$ is unramified."
],
[
"Metric graphs and tropical curves",
"Finally, we consider weighted graphs with a metric, as well as tropical curves.",
"Definition 2.14 A weighted metric graph consists of a weighted graph $(\\Gamma ,g)$ and a function $\\ell :E(\\Gamma )\\rightarrow \\mathbb {R}_{>0}$ .",
"A finite harmonic morphism of weighted metric graphs $\\varphi :(\\Gamma ^{\\prime },\\ell ^{\\prime })\\rightarrow (\\Gamma ,\\ell )$ , or simply a harmonic morphism, is a finite harmonic morphism $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ of the underlying weighted graphs such that for every edge $e^{\\prime }\\in E(\\Gamma ^{\\prime })$ we have $\\ell (\\varphi (e^{\\prime }))=d_{\\varphi }(e^{\\prime })\\ell ^{\\prime }(e^{\\prime }).$ In other words, $\\varphi $ dilates each edge $e^{\\prime }\\in E(\\Gamma ^{\\prime })$ by a factor of $d_{\\varphi }(e^{\\prime })$ .",
"A harmonic morphism $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ of weighted metric graphs is called effective or unramified if it is so as a map of weighted graphs.",
"Remark 2.15 Given a finite harmonic morphism $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ of weighted graphs and a length function $\\ell $ on $\\Gamma $ , there is a unique length function $\\ell ^{\\prime }$ on $\\Gamma ^{\\prime }$ satisfying the dilation condition (REF ).",
"Similarly, a length function on $\\Gamma ^{\\prime }$ uniquely induces a length function on $\\Gamma $ .",
"It follows that the classification of unramified covers of weighted metric graphs, in particular abelian covers, is independent of the choice of metric.",
"For this reason, in this paper we mostly work with graphs and weighted graphs without metrics.",
"Given a connected weighted metric graph $\\Gamma $ with $\\chi (\\Gamma )<0$ , we give $\\Gamma _{st}$ the structure of a weighted metric graph in the obvious way, by setting $\\ell (e)=\\ell (e_1)+\\ell (e_2)$ whenever we replace two edges $e_1$ and $e_2$ with a new edge $e$ .",
"It is clear that an unramified morphism of weighted metric graphs $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ induces an unramified morphism $\\varphi _{st}:\\Gamma ^{\\prime }_{st}\\rightarrow \\Gamma _{st}$ .",
"Definition 2.16 Let $(\\Gamma ,\\ell )$ be a weighted metric graph.",
"We define a metric space $|\\Gamma |$ , called the metric realization of $(\\Gamma ,l)$ , as follows.",
"Consider a closed interval $I_e\\subset \\mathbb {R}$ of length $\\ell (e)$ for each edge $e\\in E(\\Gamma )$ , and a half-open interval $I_l=[0,\\infty )$ for each leg $l\\in L(\\Gamma )$ .",
"We obtain $|\\Gamma |$ from the $I_e$ and the $I_l$ by treating their endpoints as the root vertices and gluing accordingly.",
"We then give $|\\Gamma |$ the path metric.",
"A harmonic morphism $\\varphi :(\\Gamma ^{\\prime },\\ell ^{\\prime })\\rightarrow (\\Gamma ,\\ell )$ of weighted metric graphs naturally induces a continuous map $|\\varphi |:|\\Gamma ^{\\prime }|\\rightarrow |\\Gamma |$ where, for a pair of edges $e=\\varphi (e^{\\prime })$ , the map is given by dilation by a factor of $d_{\\varphi }(e^{\\prime })$ , and similarly for a pair of legs $l=\\varphi (l^{\\prime })$ .",
"The map is piecewise-linear with integer slope with respect to the metric structure.",
"A basic inconvenience of tropical geometry is that different weighted metric graphs may have the same metric realizations.",
"This motivates the following definition.",
"Definition 2.17 A tropical curve $(\\scalebox {0.8}[1.3]{\\sqsubset },g)$ is a pair consisting of a metric space $\\scalebox {0.8}[1.3]{\\sqsubset }$ and a weight function $g:\\scalebox {0.8}[1.3]{\\sqsubset }\\rightarrow \\mathbb {Z}_{\\ge 0}$ such that there exists a weighted metric graph $(\\Gamma ,g,\\ell )$ and an isometry $m:|\\Gamma |\\rightarrow \\scalebox {0.8}[1.3]{\\sqsubset }$ of its metric realization with $\\Gamma $ such that the weight functions agree: $g(x)=\\left\\lbrace \\begin{array}{cc}g(v)& \\textrm { if }x=m(v) \\textrm { for a } v\\in V(\\Gamma ), \\\\ 0 & \\mbox{otherwise}.\\end{array}\\right.$ We call a quadruple $(\\Gamma ,g,\\ell ,m)$ satisfying these properties a model for $\\scalebox {0.8}[1.3]{\\sqsubset }$ .",
"The genus of a connected tropical curve $\\scalebox {0.8}[1.3]{\\sqsubset }$ is given by $g(\\scalebox {0.8}[1.3]{\\sqsubset })=b_1(\\scalebox {0.8}[1.3]{\\sqsubset })+\\sum _{x\\in \\scalebox {0.8}[1.3]{\\sqsubset }} g(x)$ and is equal to the genus of any model of $\\scalebox {0.8}[1.3]{\\sqsubset }$ .",
"For a point $x\\in \\scalebox {0.8}[1.3]{\\sqsubset }$ on a tropical curve $\\scalebox {0.8}[1.3]{\\sqsubset }$ with model $(\\Gamma ,g,l,m)$ , we define its valency $\\operatorname{val}(x)$ to be $\\operatorname{val}(v)$ if $x=m(v)$ for some $v\\in V(\\Gamma )$ and 2 otherwise.",
"We similarly define the Euler characteristic as $\\chi (x)=2-2g(x)-\\operatorname{val}(x)$ ; these numbers do not depend on the choice of model.",
"We define the Euler characteristic of a tropical curve $\\scalebox {0.8}[1.3]{\\sqsubset }$ to be $\\chi (\\scalebox {0.8}[1.3]{\\sqsubset })=\\chi (\\Gamma )$ for any model $\\Gamma $ .",
"It is clear that $\\chi (\\scalebox {0.8}[1.3]{\\sqsubset })=\\sum _{x\\in \\scalebox {0.8}[1.3]{\\sqsubset }}\\chi (x),$ where $\\chi (x)=0$ for all but finitely many $x\\in \\scalebox {0.8}[1.3]{\\sqsubset }$ .",
"Remark 2.18 Our definition differs from Def.",
"2.14 in [2], where a tropical curve was defined as an equivalence class of weighted metric graphs up to tropical modifications.",
"Given a tropical curve $\\scalebox {0.8}[1.3]{\\sqsubset }$ with model $\\Gamma $ , we can form another model $\\Gamma ^{\\prime }$ by splitting any edge or leg of $\\Gamma $ at a new vertex.",
"Conversely, any tropical curve $\\scalebox {0.8}[1.3]{\\sqsubset }$ (other than $\\mathbb {R}$ and $S^1$ ) has a unique minimal model $\\Gamma _{min}$ having no simple vertices.",
"We say that a connected tropical curve $\\scalebox {0.8}[1.3]{\\sqsubset }$ is stable if $\\chi (x)\\le 0$ for all $x\\in \\scalebox {0.8}[1.3]{\\sqsubset }$ , or, equivalently, if its minimal model is a stable graph.",
"We define the stabilization of a connected tropical curve $\\scalebox {0.8}[1.3]{\\sqsubset }$ with $\\chi (\\scalebox {0.8}[1.3]{\\sqsubset })<0$ by removing all trees of edges having no vertices of positive genus, or, equivalently, as the geometric realization of the stabilization of any model of $\\scalebox {0.8}[1.3]{\\sqsubset }$ .",
"Any tropical curve other than the real line has a well-defined set of maximal legs.",
"A morphism of tropical curves is a continuous, piecewise-linear map that sends legs to legs and is eventually linear on each leg.",
"Definition 2.19 A morphism $\\tau :\\scalebox {0.8}[1.3]{\\sqsubset }^{\\prime }\\rightarrow \\scalebox {0.8}[1.3]{\\sqsubset }$ of tropical curves is a continuous, piecewise-linear map with integer slopes such that for any leg $l^{\\prime }\\subset \\scalebox {0.8}[1.3]{\\sqsubset }^{\\prime }$ , there exists a leg $l\\subset \\scalebox {0.8}[1.3]{\\sqsubset }$ and numbers $a\\in \\mathbb {Z}_{>0}$ and $b\\in \\mathbb {R}$ such that, identifying $l^{\\prime }$ and $l$ with $[0,+\\infty )$ , we have $\\tau (x)=ax+b\\in l$ for $x\\in l^{\\prime }$ sufficiently large.",
"We note that $\\tau $ may map a finite section of $l^{\\prime }$ to $\\scalebox {0.8}[1.3]{\\sqsubset }\\backslash l$ .",
"Let $\\tau :\\scalebox {0.8}[1.3]{\\sqsubset }^{\\prime }\\rightarrow \\scalebox {0.8}[1.3]{\\sqsubset }$ be a morphism of tropical curves.",
"A model for $\\tau $ is a pair of models $(\\Gamma ^{\\prime },g^{\\prime },\\ell ^{\\prime },m^{\\prime })$ and $(\\Gamma ,g,\\ell ,m)$ for $\\scalebox {0.8}[1.3]{\\sqsubset }^{\\prime }$ and $\\scalebox {0.8}[1.3]{\\sqsubset }$ , respectively, and a morphism $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ of weighted metric graphs such that $m\\circ |\\varphi |=\\tau \\circ m^{\\prime }$ .",
"We say that $\\tau $ is harmonic, effective or unramified if $\\varphi $ has the corresponding property.",
"Given a morphism $\\tau :\\scalebox {0.8}[1.3]{\\sqsubset }^{\\prime }\\rightarrow \\scalebox {0.8}[1.3]{\\sqsubset }$ of tropical curves, we construct a model $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ by choosing the vertex set $V(\\Gamma ^{\\prime })$ to contain the finite set of points where $\\tau $ changes slope, and then enlarging $V(\\Gamma ^{\\prime })$ and $V(\\Gamma )$ to ensure that the image and the preimage of a vertex is a vertex.",
"We let the degree of $\\varphi $ on each edge and leg be the slope of $\\tau $ .",
"Given a model $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ of $\\tau $ , we can produce another model by adding more vertices to $\\Gamma ^{\\prime }$ and $\\Gamma $ .",
"Conversely, any morphism $\\tau :\\scalebox {0.8}[1.3]{\\sqsubset }^{\\prime }\\rightarrow \\scalebox {0.8}[1.3]{\\sqsubset }$ to a tropical curve $\\scalebox {0.8}[1.3]{\\sqsubset }$ with $\\chi (\\scalebox {0.8}[1.3]{\\sqsubset })<0$ has a unique minimal model $\\varphi _{min}:\\Gamma ^{\\prime }_{min}\\rightarrow \\Gamma _{min}$ with the property that every simple vertex $v\\in V(\\Gamma _{min})$ has at least one preimage that is not simple.",
"Example 2.20 Let $\\tau :\\scalebox {0.8}[1.3]{\\sqsubset }^{\\prime }\\rightarrow \\scalebox {0.8}[1.3]{\\sqsubset }$ be an unramified morphism of tropical curves of local degree one.",
"Then $\\varphi $ is a topological covering space of degree $\\deg \\tau $ .",
"Conversely, if $\\scalebox {0.8}[1.3]{\\sqsubset }$ is a tropical curve and $f:\\scalebox {0.8}[1.3]{\\sqsubset }^{\\prime }\\rightarrow \\scalebox {0.8}[1.3]{\\sqsubset }$ is a covering space of finite degree, then there is a unique way to give $\\scalebox {0.8}[1.3]{\\sqsubset }^{\\prime }$ the structure of a tropical curve such that $f$ is unramified: we define the genus function on $\\scalebox {0.8}[1.3]{\\sqsubset }^{\\prime }$ as the pullback of the genus function on $\\scalebox {0.8}[1.3]{\\sqsubset }$ ."
],
[
"Dilated cohomology",
"In the following two sections, we fix a finite abelian group $G$ and classify the $G$ -covers of a given unweighted graph $\\Gamma $ .",
"These are defined as surjective finite morphisms $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ together with an $G$ -action on $\\Gamma ^{\\prime }$ that preserves and acts transitively on the fibers.",
"We will see that a $G$ -cover of $\\Gamma $ is uniquely determined by two objects.",
"The first is a $G$ -dilation datum $D$ on $\\Gamma $ (equivalently, a $G$ -stratification $\\mathcal {S}$ of $\\Gamma $ ), recording the fibers of $\\varphi $ in terms of local stabilizer subgroups of $G$ .",
"The second is an element of a dilated cohomology group $H^1(\\Gamma ,D)$ (or $H^1(\\Gamma ,\\mathcal {S})$ ), which generalizes the first simplicial cohomology group $H^1(\\Gamma ,G)$ by taking the local stabilizers into account.",
"We introduce $G$ -covers, $G$ -dilation data and $G$ -stratifications in Sec.",
"REF .",
"In Sec.",
"REF , we introduce the dilated cohomology groups $H^i(\\Gamma ,D)$ of a pair $(\\Gamma ,D)$ , where $\\Gamma $ is a graph and $D$ is a $G$ -dilation datum on $\\Gamma $ .",
"In Sec.",
"REF we introduce the long exact sequence in dilated cohomology and study the cohomology groups of a subgraph $\\Delta \\subset \\Gamma $ .",
"Once all the relevant definitions have been established, we reach Sec.",
", which is mostly dedicated to proving our classification results."
],
[
"$G$ -covers, dilation data, and stratifications",
"Throughout this section, we only consider unweighted graphs with legs.",
"We now give the main definition of our paper.",
"Definition 3.1 Let $\\Gamma $ be a graph.",
"A $G$ -cover of $\\Gamma $ is a finite surjective morphism $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ together with an action of $G$ on $\\Gamma ^{\\prime }$ , such that the following properties are satisfied: The action is invariant with respect to $\\varphi $ .",
"For each $x\\in X(\\Gamma )$ , the group $G$ acts transitively on the fiber $\\varphi ^{-1}(x)$ .",
"Example 3.2 Let $\\Gamma $ be a graph with a $G$ -action (see Def.",
"REF ), then the quotient map $\\pi :\\Gamma \\rightarrow \\Gamma /G$ is a $G$ -cover.",
"Example 3.3 Let $\\Gamma $ be a graph.",
"Viewing $\\Gamma $ as a topological space, an element of $H^1(\\Gamma ,G)$ determines a covering space $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ with a $G$ -action.",
"It is clear that we can equip $\\Gamma ^{\\prime }$ with the structure of a graph such that $\\varphi $ is a $G$ -cover of graphs.",
"Such $G$ -covers, which we call topological $G$ -covers, are distinguished by the property that $G$ acts freely on each fiber $\\varphi ^{-1}(x)$ .",
"For such covers, the $G$ -dilation datum is trivial, while the the dilated cohomology group is $H^1(\\Gamma ,G)$ .",
"An example with $G$ the Klein group is given below in Fig.",
"REF .",
"Our goal is to describe all $G$ -covers $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ of a given graph $\\Gamma $ .",
"We begin our description by considering the local stabilizer subgroups.",
"Definition 3.4 Let $\\Gamma $ be a graph.",
"A $G$ -dilation datum $D$ on $\\Gamma $ is a choice of a subgroup $D(x)\\subset G$ for every $x\\in X(\\Gamma )$ , such that $D(h)\\subset D(r(h))$ for every half-edge $h\\in H(\\Gamma )$ , and such that $D(h)=D(h^{\\prime })$ for each edge $e=\\lbrace h,h^{\\prime }\\rbrace \\in E(\\Gamma )$ .",
"Given $G$ -dilation data $D$ and $D^{\\prime }$ on $\\Gamma $ , we say that $D$ is a refinement of $D^{\\prime }$ if $D(x)\\subset D^{\\prime }(x)$ for all $x\\in X(\\Gamma )$ .",
"A $G$ -dilated graph is a pair $(\\Gamma ,D)$ consisting of a graph $\\Gamma $ and a $G$ -dilation datum $D$ on $\\Gamma $ .",
"We call $D(x)$ the dilation group of $x\\in X(\\Gamma )$ , and for an edge $e=\\lbrace h,h^{\\prime }\\rbrace \\in E(\\Gamma )$ we call $D(e)=D(h)=D(h^{\\prime })$ the dilation group of $e$ .",
"If $e$ is an edge with root vertices $u$ and $v$ (which may be the same), then $D(e)\\subset D(u)\\cap D(v)$ .",
"We call $C(e)=D(u)+D(v)$ the vertex dilation group of the edge $e$ .",
"Remark 3.5 A $G$ -dilation datum on a graph $\\Gamma $ is an example of a graph of groups, as defined by Bass (see Def.",
"1.4 in [7]).",
"In a future paper, we plan to explore the relationship between the cohomology groups $H^i(\\Gamma ,D)$ and the fundamental group of the graph of groups defined by $D$ , with the goal of extending our theory to the non-abelian case.",
"Definition 3.6 Let $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ be a $G$ -cover.",
"We define the $G$ -dilation datum $D_{\\varphi }$ of $\\varphi $ by setting $D_{\\varphi }(x)$ for $x\\in X(\\Gamma )$ to be the stabilizer group of any $x^{\\prime }\\in \\varphi ^{-1}(x)$ .",
"The group $G$ is assumed to be abelian, therefore the stabilizer group of $x^{\\prime }\\in \\varphi ^{-1}(x)$ does not depend on the choice of $x^{\\prime }$ .",
"Remark 3.7 If $D_{\\varphi }$ is the $G$ -dilation datum of a $G$ -cover $\\varphi $ that is the tropicalization of a $G$ -cover of algebraic curves, then the dilation subgroup of every half-edge is cyclic (this follows, for instance, from [38]).",
"As a result, many of the covers described throughout this paper are not algebraically realizable, e.g.",
"the cover REF below.",
"Our approach is to develop, as far as possible, an independent theory of $G$ -covers of graphs, so we do not impose this condition from the start.",
"In any case, as we shall see, the dilation groups of the half-edges play a secondary role in the classification of $G$ -covers.",
"For any $x\\in X(\\Gamma )$ , the fiber $\\varphi ^{-1}(x)$ of a $G$ -cover $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ is a $G/D_{\\varphi }(x)$ -torsor.",
"If $h\\in T_v(\\Gamma )$ is a half-edge rooted at $v\\in V(\\Gamma )$ , then the root map $r:\\varphi ^{-1}(h)\\rightarrow \\varphi ^{-1}(v)$ is an equivariant map of transitive $G$ -sets, which implies that $D_{\\varphi }(h)\\subset D_{\\varphi }(v)$ .",
"Furthermore, it is clear that $D_{\\varphi }(h)=D_{\\varphi }(h^{\\prime })$ for any edge $e=\\lbrace h,h^{\\prime }\\rbrace \\in E(\\Gamma )$ .",
"Therefore, $D_{\\varphi }$ is a $G$ -dilation datum.",
"The cardinality of each fiber $\\varphi ^{-1}(x)$ equals the index of $D_{\\varphi }(x)$ in $G$ : $\\#(\\varphi ^{-1}(x))=[G:D_{\\varphi }(x)].$ Furthermore, for a half-edge $h\\in H(\\Gamma )$ rooted at $r(h)=v\\in V(\\Gamma )$ , the $[G:D_{\\varphi }(h)]$ half-edges in the fiber $\\varphi ^{-1}(h)$ are partitioned by their root vertices into $\\#(\\varphi ^{-1}(v))=[G:D_{\\varphi }(v)]$ subsets, each containing $[D_{\\varphi }(v):D_{\\varphi }(h)]$ elements.",
"Example 3.8 (Klein covers) We now give several of examples of $G$ -covers in the simplest non-cyclic case, when $G=\\mathbb {Z}/{2}\\mathbb {Z}\\oplus \\mathbb {Z}/{2}\\mathbb {Z}$ is the Klein group.",
"The base graph $\\Gamma $ consists of two vertices $u$ and $v$ joined by two edges $e$ and $f$ .",
"We use the following notation to describe a $G$ -cover $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ .",
"We denote the elements of $G$ by 00, 10, 01, and 11, and denote the subgroups generated by 10, 01, and 11 by respectively $H_1$ , $H_2$ , and $H_3$ .",
"The vertices of $\\Gamma ^{\\prime }$ lying above $u$ and $v$ are labeled (non-uniquely if the corresponding stabilizer is non-trivial) $u_{ij}$ and $v_{ij}$ for $ij\\in G$ , and the action of $G$ on $\\varphi ^{-1}(u)$ and $\\varphi ^{-1}(v)$ is the natural additive action on the indices.",
"We color the edges $\\varphi ^{-1}(e)$ and $\\varphi ^{-1}(f)$ red and blue, respectively, and label them with indices $ij$ in such a way that $e_{ij}$ and $f_{ij}$ are attached to $u_{ij}$ .",
"The sizes of the vertices and the thickness of the edges of $\\Gamma ^{\\prime }$ denote the size of the dilation subgroup.",
"In the caption, we indicate the nontrivial dilation groups.",
"In Ex.",
"REF , we will enumerate all Klein covers of $\\Gamma $ .",
"Figure: Klein covers of a genus 1 graphWe now give an alternative way to record a $G$ -dilation datum on $\\Gamma $ , by means of a stratification of $\\Gamma $ indexed by the subgroups of $G$ .",
"This description is often easier to visualize, and generalizes more naturally to tropical curves.",
"Definition 3.9 Let $\\Gamma $ be a graph.",
"A $G$ -stratification $\\mathcal {S}=\\big \\lbrace \\Gamma _H\\big \\vert H\\in S(G)\\big \\rbrace $ on $\\Gamma $ is a collection of subgraphs $\\Gamma _H\\subset \\Gamma $ indexed by the set $S(G)$ of subgroups of $G$ , such that $\\begin{split}\\Gamma _0&=\\Gamma ,\\\\\\Gamma _K&\\subset \\Gamma _H\\mbox{ if }H\\subset K, \\textrm { and} \\\\\\Gamma _H\\cap \\Gamma _K&=\\Gamma _{H+K}\\mbox{ for all }H,K\\in S(G).\\end{split}$ We allow the $\\Gamma _H$ to be empty or disconnected for $H\\ne 0$ .",
"The union of the $\\Gamma _H$ for $H\\ne 0$ is called the dilated subgraph of $\\Gamma $ and is denoted $\\Gamma _{dil}$ .",
"We can associate a $G$ -stratification of $\\Gamma $ to a $G$ -dilation datum $D$ , and vice versa.",
"Definition 3.10 Let $\\Gamma $ be a graph, and let $D$ be a $G$ -dilation datum on $\\Gamma $ .",
"We define the $G$ -stratification $\\mathcal {S}(D)=\\lbrace \\Gamma _H:H\\in S(G)\\rbrace $ associated to $D$ as follows: $\\Gamma _H=\\big \\lbrace x\\in X(\\Gamma )\\big \\vert H\\subset D(x)\\big \\rbrace .$ We observe that for any half-edge $h\\in H(\\Gamma )$ we have $D(h)\\subset D(r(h))$ , therefore each $\\Gamma _H$ is indeed a subgraph of $\\Gamma $ .",
"Remark 3.11 Let $D_{\\varphi }$ be the $G$ -dilation datum associated to a $G$ -cover $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ .",
"Then for any $H\\in S(G)$ , $\\Gamma _H$ is the image under $\\varphi $ of the subgraph of $\\Gamma ^{\\prime }$ fixed under the action of $H$ .",
"A $G$ -dilation datum $D$ can be uniquely recovered from a $G$ -stratification $\\mathcal {S}$ as follows.",
"Condition (REF ) implies that the set $X(\\Gamma )$ is partitioned into disjoint subsets (which are not subgraphs in general) $X(\\Gamma )=\\coprod _{H\\in S(G)}\\Gamma _H\\backslash \\Gamma ^0_H,\\mbox{ where } \\Gamma _{H}^0=\\bigcup _{H\\subsetneq K} \\Gamma _K.$ For any $x\\in X(\\Gamma )$ we set $D(x)=H$ , where $H$ is the unique subgroup of $G$ such that $x\\in \\Gamma _H\\backslash \\Gamma ^0_H$ .",
"We also define a dual stratification associated to a $G$ -dilation datum.",
"Definition 3.12 Let $D$ be a $G$ -dilation datum on $\\Gamma $ .",
"The dual stratification $\\mathcal {S}^*(D)=\\lbrace \\Gamma ^H:H\\in S(G)\\rbrace $ of $\\mathcal {S}$ is defined as follows.",
"For $H\\in S(G)$ , we define $\\Gamma ^H$ to be the edge-maximal subgraph of $\\Gamma $ whose vertex set is $V(\\Gamma ^H)=\\bigcup _{K\\subset H}V(\\Gamma _K\\backslash \\Gamma ^0_K)=\\big \\lbrace v\\in V(\\Gamma )\\big \\vert D(v)\\subset H\\big \\rbrace .$ In other words, a leg of $\\Gamma $ with root vertex $v$ lies in $\\Gamma ^H$ if and only if $D(v)\\subset H$ , and an edge $e\\in E(\\Gamma )$ with root vertices $u$ and $v$ lies in $\\Gamma ^H$ if and only if $C(e)=D(u)+D(v)\\subset H$ .",
"The dual stratification satisfies the following properties: $\\begin{split}\\Gamma ^G&=\\Gamma ,\\\\\\Gamma ^H&\\subset \\Gamma ^K\\mbox{ if }H\\subset K, \\textrm { and }\\\\ \\Gamma _H\\cap \\Gamma _K&=\\Gamma _{H\\cap K}\\mbox{ for all }H,K\\in S(G).\\end{split}$ Remark 3.13 Unlike $\\mathcal {S}(D)$ , the dual stratification $\\mathcal {S}^*(D)$ of a $G$ -dilation datum does not uniquely determine $D$ .",
"For a vertex $v\\in V(\\Gamma )$ , we can recover $D(v)$ as the smallest subgroup $H\\subset G$ such that $v\\in V(\\Gamma ^H)$ , but the dilation groups $D(h)$ of the edges cannot be determined.",
"For example, let $\\Gamma $ be the graph consisting of a vertex $v$ and a loop $e$ , let $D(v)=H$ be a subgroup of $G$ , and let $D(e)$ be any subgroup of $H$ .",
"The dual stratification is $\\Gamma ^K=\\left\\lbrace \\begin{array}{cc} \\Gamma & \\textrm { if } \\ H\\subseteq K,\\\\ \\emptyset & \\textrm { if } \\ H\\subsetneq K,\\end{array}\\right.$ so we can recover $H$ but not $D(e)$ .",
"Finally, we define morphisms of $G$ -covers of $\\Gamma $ .",
"Definition 3.14 Let $\\varphi _1:\\Gamma _1^{\\prime }\\rightarrow \\Gamma $ and $\\varphi _2:\\Gamma _2^{\\prime }\\rightarrow \\Gamma $ be $G$ -covers.",
"A morphism of $G$ -covers from $\\varphi _1$ to $\\varphi _2$ is a $G$ -equivariant morphism $\\psi :\\Gamma _1^{\\prime }\\rightarrow \\Gamma _2^{\\prime }$ such that $\\varphi _1=\\varphi _2\\circ \\psi $ .",
"We observe that if $\\psi :\\Gamma _1^{\\prime }\\rightarrow \\Gamma _2^{\\prime }$ is a morphism of $G$ -covers from $\\varphi _1:\\Gamma _1^{\\prime }\\rightarrow \\Gamma $ to $\\varphi _2:\\Gamma _2^{\\prime }\\rightarrow \\Gamma $ , then for any $x\\in X(\\Gamma )$ the restriction of $\\tau $ to the fiber $\\varphi _1^{-1}(x)$ is a $G$ -equivariant surjective map onto $\\varphi _2^{-1}(x)$ , which implies that $D_{\\varphi _1}(x)\\subset D_{\\varphi _2}(x)$ , in other words $D_{\\varphi _1}$ is a refinement of $D_{\\varphi _2}$ .",
"Remark 3.15 In this paper, we only consider $G$ -covers of a fixed base graph $\\Gamma $ (except that we do consider restrictions of covers to a subgraph).",
"It is also possible to define morphisms of $G$ -covers of graphs that are related by a morphism.",
"For example, given a $G$ -cover $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ and a morphism $\\psi :\\Delta \\rightarrow \\Gamma $ , we define the pullback $G$ -cover $\\varphi ^{\\prime }:\\Delta ^{\\prime }\\rightarrow \\Delta $ by taking $\\Delta ^{\\prime }$ to be the fiber product $\\Gamma ^{\\prime }\\times _{\\Gamma } \\Delta ^{\\prime }$ (defined by $X(\\Delta ^{\\prime })=X(\\Gamma ^{\\prime })\\times _{X(\\Gamma )}X(\\Delta ^{\\prime })$ with coordinatewise involution and root maps), and letting $G$ act on the first factor.",
"All of the constructions of this chapter are functorial with respect to such operators, so for example the $G$ -dilation datum $D_{\\varphi ^{\\prime }}$ on $\\Delta $ is equal to the pullback $G$ -dilation datum $\\psi ^*D_{\\varphi }=D_{\\varphi }\\circ \\psi $ ."
],
[
"Cohomology of $G$ -data",
"In this subsection, we define the cohomology groups $H^0(\\Gamma ,D)$ and $H^1(\\Gamma ,D)$ of a $G$ -dilated graph $(\\Gamma ,D)$ .",
"These groups generalize the simplicial cohomology groups $H^i(\\Gamma ,G)$ of $\\Gamma $ with coefficients in $G$ .",
"The groups $H^i(\\Gamma ,D)$ do not depend on the legs of $\\Gamma $ , so we assume for simplicity that $\\Gamma $ has no legs.",
"The legs of $\\Gamma $ will again play a role in Sec.",
"REF , when we classify unramified $G$ -covers of weighted graphs.",
"Rather than only considering $G$ -dilation data on a graph $\\Gamma $ , we work in a larger category of $G$ -data on $\\Gamma $ , a $G$ -datum being simply a choice of a $G$ -group at every vertex and every edge of $\\Gamma $ that is consistent with the root maps (see Definition REF ).",
"A $G$ -datum $D_\\varphi $ arising from a $G$ -cover $\\varphi $ is always a $G$ -dilation datum.",
"However, cohomology groups of the more general $G$ -data appear in the long exact sequence (REF ) that relates the cohomology groups $H^i(\\Gamma ,D)$ of a $G$ -dilation datum $D$ on $\\Gamma $ to the cohomology groups $H^i(\\Delta ,D|_{\\Delta })$ of the restriction of $D$ to a subgraph $\\Delta \\subset \\Gamma $ .",
"We begin by recalling the simplicial cohomology groups of a graph $\\Gamma $ with coefficients in $G$ .",
"Choose an orientation on the edges, and let $s,t:E(\\Gamma )\\rightarrow V(\\Gamma )$ be the source and target maps.",
"The simplicial chain complex of $\\Gamma $ is $\\begin{tikzcd}0[r] & \\mathbb {Z}^{E(\\Gamma )}[r,\"\\delta \"]& \\mathbb {Z}^{V(\\Gamma )} [r]& 0,\\end{tikzcd}$ with the boundary map defined on the generators of $\\mathbb {Z}^{E(\\Gamma )}$ by $\\delta (e)=t(e)-s(e)$ .",
"Applying the functor $\\operatorname{Hom}(-, G)$ and identifying $\\operatorname{Hom}(\\mathbb {Z}^{V(\\Gamma )},G)=G^{V(\\Gamma )}\\quad \\textrm { and } \\quad \\operatorname{Hom}(\\mathbb {Z}^{E(\\Gamma )},G)=G^{E(\\Gamma )},$ we obtain the simplicial cochain complex of $\\Gamma $ with coefficients in $G$ : $\\begin{tikzcd}0 [r] & G^{V(\\Gamma )} [r,\"\\delta ^*\"] & G^{E(\\Gamma )} [r] & 0.\\end{tikzcd}$ We identify elements of $G^{V(\\Gamma )}$ and $G^{E(\\Gamma )}$ with functions $\\xi :V(\\Gamma )\\rightarrow G$ and $\\eta :E(\\Gamma )\\rightarrow G$ , respectively.",
"Under this identification, the duals $s^*,t^*:G^{V(\\Gamma )}\\rightarrow G^{E(\\Gamma )}$ of the maps $s$ and $t$ are $s^*(\\xi )(e)=\\xi (s(e))\\quad \\textrm { and }\\quad t^*(\\xi )(e)=\\xi (t(e)),$ and the coboundary map is equal to $\\delta ^*=t^*-s^*.$ The simplicial cohomology groups of $\\Gamma $ with coefficients in $G$ are $H^0(\\Gamma ,G)=\\operatorname{Ker}\\delta ^* \\quad \\textrm { and }\\quad H^1(\\Gamma ,G)= \\operatorname{Coker}\\delta ^*.$ We now generalize this construction by replacing every copy of $G$ in the cochain complex (REF ) with an arbitrary $G$ -group.",
"We recall that a $G$ -group is a map of abelian groups $f:G\\rightarrow H$ , and a morphism of $G$ -groups from $f_1:G\\rightarrow H_1$ to $f_2:G\\rightarrow H_2$ is a group homomorphism $g:H_1\\rightarrow H_2$ such that $f_2=g\\circ f_1$ .",
"Definition 3.16 A $G$ -datum $A$ on an oriented graph $\\Gamma $ consists of the following: For every vertex $v\\in V(\\Gamma )$ , a $G$ -group $f_v:G\\rightarrow A(v)$ .",
"For every edge $e\\in E(\\Gamma )$ , a $G$ -group $f_e:G\\rightarrow A(e)$ and morphisms of $G$ -groups $s_e:A\\big (s(e)\\big )\\rightarrow A(e)$ and $t_e:A\\big (t(e)\\big )\\rightarrow A(e)$ such that $s_e\\circ f_{s(e)}=t_e\\circ f_{t(e)}=f_e$ , i.e.",
"for which the diagram $\\begin{tikzcd}& G [dl,\"f_{s(e)}\"^{\\prime }] [d,\"f_e\"] [dr,\"f_{t(e)}\"] & \\\\ A(s(e)) [r,\"s_e\"^{\\prime }] & A(e) & A(t(e)) [l,\"t_e\"] \\\\\\end{tikzcd}$ commutes.",
"In other words, a $G$ -datum on $\\Gamma $ is a functor to the category of $G$ -groups from the category whose objects are $V(\\Gamma )\\cup E(\\Gamma )$ , and whose non-trivial morphisms are the source and target maps.",
"In contrast with Remark REF , a $G$ -datum is not necessarily a graph of groups in the sense of [7], since the maps $f_{s(e)}$ and $f_{t(e)}$ are not required to be injective.",
"To verify that $G$ -data, in fact, generalize the notion of $G$ -dilation data, we associate a $G$ -datum $A^D$ to each $G$ -dilation datum $D$ .",
"First, let $H_1$ and $H_2$ be subgroups of $G$ , let $f_i:G\\rightarrow G/H_i$ be the projections, and let $\\iota _i:G/H_i\\rightarrow G/H_1\\oplus G/H_2$ be the embeddings.",
"The coproduct of $f_1$ and $f_2$ is the $G$ -group $G/H_1\\sqcup _G G/H_2=(G/H_1\\oplus G/H_2)/(\\operatorname{Im}f_1\\oplus -f_2).$ The natural map $f_1\\sqcup f_2:G\\rightarrow G/H_1\\sqcup _G G/H_2$ is equal to $\\pi \\circ \\iota _1\\circ f_1=\\pi \\circ \\iota _2\\circ f_2$ , where $\\pi :G/H_1\\oplus G/H_2\\rightarrow G/H_1\\sqcup _G G/H_2$ is the projection.",
"It is clear that $f_1\\sqcup f_2$ is surjective and that $\\operatorname{Ker}f_1\\sqcup f_2=H_1+H_2$ , hence the $G$ -group $G\\rightarrow G/H_1\\sqcup _G G/H_2$ can be identified with the quotient $G\\rightarrow G/(H_1+H_2)$ .",
"Definition 3.17 Let $\\Gamma $ be an oriented graph, and let $D$ be a $G$ -dilation datum on $\\Gamma $ .",
"We define the associated $G$ -datum $A^D$ as follows.",
"For each $v\\in V(\\Gamma )$ , we set $A^D(v)=G/D(v)$ , and let $f_v$ be the natural projection map: $f_v:G\\rightarrow A^D(v)=G/D(v).$ For an edge $e\\in E(\\Gamma )$ , we let $f_e=f_{s(e)}\\sqcup f_{t(e)}$ be the coproduct.",
"In other words, we let $A^D(e)=[G/D(s(e))\\oplus G/D(t(e))]/(\\operatorname{Im}f_{s(e)}\\oplus -f_{t(e)})\\simeq G/C(e),$ where $C(e)=D(s(e))+D(t(e))$ is the edge dilation group.",
"We let $f_e:G\\rightarrow A^D(e)\\simeq G/C(e)$ be the quotient map, and we let $s_e:A^D(s(e))\\rightarrow A^D(e) \\quad \\textrm { and } \\quad t_e:A^D(t(e))\\rightarrow A^D(e)$ be the natural quotient maps $G/D(s(e))\\rightarrow G/C(e)$ and $G/D(t(e))\\rightarrow G/C(e)$ .",
"We now define the cochain complex and cohomology groups of a $G$ -datum $A$ on an oriented graph $\\Gamma $ .",
"Definition 3.18 Let $G$ be an oriented graph, and let $A$ be a $G$ -datum on $\\Gamma $ .",
"We define the cochain groups of the pair $(\\Gamma ,A)$ as follows: $C^0(\\Gamma ,A)=\\prod _{v\\in V(\\Gamma )}A(v)=\\Big \\lbrace \\xi :V(\\Gamma )\\rightarrow \\coprod _{v\\in V(\\Gamma )} A(v):\\xi (v)\\in A(v)\\Big \\rbrace ,$ $C^1(\\Gamma ,A)=\\prod _{e\\in E(\\Gamma )}A(e)=\\Big \\lbrace \\eta :E(\\Gamma )\\rightarrow \\coprod _{e\\in E(\\Gamma )} A(e):\\eta (e)\\in A(e)\\Big \\rbrace .$ We define the morphisms $s^*,t^*:C^0(\\Gamma ,A)\\rightarrow C^1(\\Gamma ,A)$ by $s^*(\\xi )(e)=s_e(\\xi (s(e)))\\quad \\textrm { and } \\quad t^*(\\xi )(e)=t_e(\\xi (t(e))).$ We define the cochain complex of the pair $(\\Gamma ,A)$ as $\\begin{tikzcd}0 [r] & C^0(\\Gamma ,A)[r,\"\\delta _{\\Gamma ,A}^*\"] & C^1(\\Gamma ,A) [r] & 0,\\end{tikzcd}$ where the coboundary map $\\delta _{\\Gamma ,A}^*$ is $\\delta _{\\Gamma ,A}^* =t^*-s^*.$ We define the cohomology groups of the pair $(\\Gamma ,A)$ as $H^0(\\Gamma ,A)=\\operatorname{Ker}\\delta _{\\Gamma ,A}^*\\quad \\textrm { and } \\quad H^1(\\Gamma ,A)=\\operatorname{Coker}\\delta _{\\Gamma ,A}^*.$ Specializing to $G$ -dilation data, we obtain the main definition of this section.",
"Definition 3.19 Let $(\\Gamma ,D)$ be a $G$ -dilated graph, and let $A^D$ be the $G$ -datum associated to $D$ .",
"The cochain complex of $(\\Gamma ,D)$ is the cochain complex of the pair $(\\Gamma ,A^D)$ : $\\begin{tikzcd}0 [r] & C^0(\\Gamma ,D)[r,\"\\delta _{\\Gamma ,D}^*\"] & C^1(\\Gamma ,D) [r] & 0,\\end{tikzcd}$ where $C^i(\\Gamma ,D)=C^i(\\Gamma ,A^D) \\quad \\textrm { and } \\quad \\delta _{\\Gamma ,D}^*=\\delta _{\\Gamma ,A^D}^*.$ The dilated cohomology groups $H^i(\\Gamma ,D)$ are the cohomology groups of $(\\Gamma ,A^D)$ : $H^0(\\Gamma ,D)=\\operatorname{Ker}\\delta _{\\Gamma ,D}^*=H^0(\\Gamma ,A^D)\\quad \\textrm { and }\\quad H^1(\\Gamma ,D)=\\operatorname{Coker}\\delta _{\\Gamma ,D}^*=H^1(\\Gamma ,A^D).$ For the sake of clarity, and for future use, we give an explicit description of $H^1(\\Gamma ,D)$ as a quotient.",
"The cochain group $C^1(\\Gamma ,D)$ is the direct product of $A^D(e)$ over all $e\\in E(\\Gamma )$ , where each $A^D(e)$ is the coproduct $G/C(e)$ of $G\\rightarrow G/D(s(e))$ and $G\\rightarrow G/D(t(e))$ .",
"In other words, each $\\eta \\in C^1(\\Gamma ,D)$ is given by choosing a pair of elements $(\\eta _s(e),\\eta _t(e))\\in G/D(s(e))\\oplus G/D(t(e))$ for each $e\\in E(\\Gamma )$ .",
"A tuple $(\\eta _s(e),\\eta _t(e))_{e\\in E(\\Gamma )}$ is equivalent to $(\\widetilde{\\eta }_s(e),\\widetilde{\\eta }_t(e))_{e\\in E(\\Gamma )}$ if and only if there exist elements $\\omega (e)\\in G$ for all $e\\in E(\\Gamma )$ such that $\\begin{split}\\eta _s(e)&=\\widetilde{\\eta }_s(e)+\\omega (e)\\operatorname{mod}D(s(e))\\\\ \\eta _t(e)&=\\widetilde{\\eta }_t(e)-\\omega (e)\\operatorname{mod}D(t(e)).\\end{split}$ Note that, instead of assuming that $\\omega (e)\\in G$ , we may assume that $\\omega (e)$ lies in any quotient group between $G$ and $G/(D(s(e))\\cap D(t(e)))$ , and it is natural to assume that in fact $\\omega (e)\\in G/D(e)$ .",
"An element of $C^0(\\Gamma ,D)$ is given by choosing $\\xi (v)\\in G/D(v)$ for each $v\\in V(\\Gamma )$ .",
"Putting everything together, we see that an element $[\\eta ]\\in H^1(\\Gamma ,D)$ is given by choosing a pair of elements $(\\eta _s(e),\\eta _t(e))\\in G/D(s(e))\\oplus G/D(t(e))$ for each $e\\in E(\\Gamma )$ , and that two choices $(\\eta _s(e),\\eta _t(e))_{e\\in E(\\Gamma )}$ and $(\\widetilde{\\eta }_s(e),\\widetilde{\\eta }_t(e))_{e\\in E(\\Gamma )}$ represent the same element of $H^1(\\Gamma ,D)$ if and only if there exist elements $\\omega (e)\\in G/D(e)$ for all $e\\in E(\\Gamma )$ and elements $\\xi (v)$ for all $v\\in V(\\Gamma )$ such that $\\begin{split}\\eta _s(e)&=\\widetilde{\\eta }_s(e)-\\xi (s(e))+\\omega (e)\\operatorname{mod}D(s(e))\\\\ \\eta _t(e)&=\\widetilde{\\eta }_t(e)+\\xi (t(e))-\\omega (e)\\operatorname{mod}D(t(e))\\end{split}$ for all $e\\in E(\\Gamma )$ .",
"Remark 3.20 The dilated cochain complex of $(\\Gamma ,D)$ , and hence the cohomology groups $H^i(\\Gamma ,D)$ , depend only on the dilation groups $D(v)$ of the vertices $v\\in V(\\Gamma )$ , and do not depend on the edge groups $D(e)$ .",
"Specifically, given a graph $\\Gamma $ , we can choose the dilation groups $D(v)$ of the vertices $v\\in V(\\Gamma )$ arbitrarily, and for each edge $e\\in E(\\Gamma )$ choose $D(e)$ to be any subgroup of $D(s(e))\\cap D(t(e))$ .",
"The resulting groups $H^0(\\Gamma ,D)$ and $H^1(\\Gamma ,D)$ are independent of the choice of the $D(e)$ .",
"In other words, the dilated cohomology groups of $(\\Gamma ,D)$ only depend on the dual stratification $\\mathcal {S}^*(D)$ .",
"For the remainder of the paper, with the exception of Sec.",
"REF below, we restrict our attention to $G$ -dilation data and their cohomology groups.",
"Before we proceed, we calculate our first example, showing that we have in fact generalized simplicial cohomology.",
"Example 3.21 Let $\\Gamma $ be a graph, and let $A_G$ be the trivial $G$ -datum, namely $A_G(v)=G$ and $A_G(e)=G$ for all $v\\in V(\\Gamma )$ and all $e\\in E(\\Gamma )$ , with all structure maps being the identity.",
"Alternatively, $A_G$ is the $G$ -datum associated to the trivial $G$ -dilation datum $D_0$ given by $D_0(x)=0$ for all $x\\in X(\\Gamma )$ .",
"It is clear that $C^i(\\Gamma ,A_G)=C^i(\\Gamma ,G)$ , and that the coboundary map $\\delta ^*_{\\Gamma ,A_G}$ given by (REF ) is equal to $\\delta ^*$ given by (REF ).",
"Hence $H^i(\\Gamma ,A_G)=H^i(\\Gamma ,D_0)=H^i(\\Gamma ,G)$ .",
"We now work out several explicit examples of the cohomology groups $H^i(\\Gamma ,D)$ of $G$ -dilated graphs $(\\Gamma ,D)$ .",
"In the previous example, we saw that the cohomology of the trivial $G$ -dilation datum on $\\Gamma $ is the simplicial cohomology of $\\Gamma $ with coefficients in $G$ .",
"In particular, $H^1(\\Delta ,G)$ is trivial for any tree $\\Delta $ .",
"We now show that $H^1(\\Delta ,D)=0$ for any $G$ -dilation datum $D$ on a tree $\\Delta $ .",
"Proposition 3.22 Let $\\Delta $ be a tree.",
"Then $H^1(\\Delta ,D)=0$ for any $G$ -dilation datum $D$ on $\\Delta $ .",
"Let $\\Gamma $ be an arbitrary graph, and suppose that $D$ and $D^{\\prime }$ are two $G$ -dilation data on $\\Gamma $ , such that $D$ is a refinement of $D^{\\prime }$ .",
"In this case, we can define natural surjective maps $\\pi ^i:C^i(\\Gamma ,D)\\rightarrow C^i(\\Gamma ,D^{\\prime })$ by taking coordinatewise quotients.",
"These maps commute with the coboundary maps and induce maps $\\pi ^i:H^i(\\Gamma ,D)\\rightarrow H^i(\\Gamma ,D^{\\prime })$ , and furthermore the map $\\pi ^1$ is surjective.",
"Now suppose that $\\Delta $ is a tree, and $D$ is a $G$ -dilation datum on $\\Delta $ .",
"Let $D_0$ be the trivial $G$ -dilation datum on $\\Delta $ .",
"Then $D_0$ is a refinement of $D$ , so there is a surjective map $H^1(\\Delta ,D_0)\\rightarrow H^1(\\Delta ,D)$ .",
"But by Ex.",
"REF we know that $H^1(\\Delta ,D_0)=H^1(\\Delta ,G)=0$ , hence $H^1(\\Delta ,D)=0$ .",
"We now work out an example of $H^i(\\Gamma ,D)$ for a topologically non-trivial graph $\\Gamma $ .",
"Example 3.23 Let $\\Gamma $ be the graph consisting of two vertices $v_1$ and $v_2$ joined by $n$ edges $e_1,\\ldots ,e_n$ , oriented such that $s(e_i)=v_1$ and $t(e_i)=v_2$ .",
"Let $H_1$ and $H_2$ be two subgroups of $G$ , and consider the following $G$ -dilation datum on $\\Gamma $ : $D(v_1)=H_1,\\quad D(v_2)=H_2\\quad \\textrm { and }\\quad D(e_i)\\mbox{ are arbitrary subgroups of }H_1\\cap H_2.$ We see that $C(e_i)=H_1+ H_2$ for all $i$ , therefore $C^0(\\Gamma ,D)=G/H_1\\oplus G/H_2\\quad \\textrm { and }\\quad C^1(\\Gamma ,D)=[G/(H_1+ H_2)]^n.$ The coboundary map $\\delta ^*_{\\Gamma ,D}$ is the composition of the projection $\\pi :G/H_1\\oplus G/H_2\\rightarrow G/H_1\\sqcup G/H_2\\simeq G/(H_1+H_2)$ and the diagonal map.",
"Therefore $H^0(\\Gamma ,D)=\\operatorname{Ker}\\pi \\simeq G/(H_1\\cap H_2)\\quad \\textrm { and }\\quad H^1(\\Gamma ,D)\\simeq [G/(H_1+ H_2)]^{n-1}.$ We also show that cohomology of $G$ -dilation data can be used to compute simplicial cohomology of edge-maximal subgraphs of $\\Gamma $ , with coefficients in any quotient group of $G$ .",
"Example 3.24 Let $\\Gamma $ be a graph, let $\\Delta \\subset \\Gamma $ be an edge-maximal subgraph, and let $H\\subset G$ be a subgroup.",
"Consider the following $G$ -dilation datum on $\\Gamma $ : $D_{\\Delta ,H}(x)=\\left\\lbrace \\begin{array}{cc} H,& x\\in X(\\Delta ), \\\\ G, & x\\notin X(\\Delta ).\\end{array}\\right.$ By definition, an edge $e\\in E(\\Gamma )$ lies in $\\Delta $ if and only if both of its root vertices do.",
"It follows that the dilated cochain complex $C^*(\\Gamma ,D_{\\Delta ,H})$ is equal to the simplicial cochain complex $C^*(\\Delta ,G/H)$ , and hence $H^i(\\Gamma ,D_{\\Delta ,H})=H^i(\\Delta ,G/H)\\mbox{ for }i=0,1.$ Remark 3.25 We will show in Sec.",
"REF that the group $H^1(\\Gamma ,D)$ classifies $G$ -covers of $\\Gamma $ with dilation datum $D$ .",
"We do not know of a similar geometric interpretation of the group $H^0(\\Gamma ,D)$ .",
"For the trivial dilation datum $D=0$ , the group $H^0(\\Gamma ,D)=H^0(\\Gamma ,G)$ is equal to $G$ for any connected graph $\\Gamma $ .",
"In general, the group $H^0(\\Gamma ,D)$ can be quite large, even on a connected graph.",
"For example, let $\\Gamma $ be a chain of $2n$ vertices, let $p$ and $q$ be distinct prime numbers, let $G=\\mathbb {Z}/pq\\mathbb {Z}$ , and let $H_1=\\mathbb {Z}/p\\mathbb {Z}$ and $H_2=\\mathbb {Z}/q\\mathbb {Z}$ be the nontrivial subgroups of $G$ .",
"Label the vertices of $\\Gamma $ by $H_1$ and $H_2$ in an alternating fashion.",
"Then $C(e)=G/(H_1+H_2)=0$ for any edge of $e$ , hence $C^1(\\Gamma ,D)=0$ and therefore $H^0(\\Gamma ,D)=C^0(\\Gamma ,D)=H_1^n\\oplus H_2^n=G^n$ .",
"We have already noted (see Rem.",
"REF ) that we restrict our attention to a fixed base graph $\\Gamma $ .",
"It is possible to define morphisms between pairs consisting of a graph and a $G$ -datum on it (it is necessary to require that the graph morphism be finite).",
"Such morphisms define natural pullback maps on the cochain and cohomology groups.",
"In the next section, we work out these pullback maps for a single example, namely the relationship (Prop.",
"REF ) between the cohomology groups $H^i(\\Gamma ,A)$ of a $G$ -datum $A$ on $\\Gamma $ , and the cohomology groups $H^i(\\Delta ,A|_{\\Delta })$ of the restriction of $A$ to a subgraph $\\Delta \\subset \\Gamma $ ."
],
[
"Relative cohomology and reduced cohomology",
"This section is somewhat technical in nature, and deals with a single question: how to relate the cohomology groups $H^i(\\Gamma ,D)$ of a $G$ -dilated graph $(\\Gamma ,D)$ to the cohomology groups $H^i(\\Delta ,D|_{\\Delta })$ of the restriction of $D$ to a subgraph $\\Delta \\subset \\Gamma $ .",
"This question is natural from the point of view of tropical geometry: we often study tropical curves by contracting edges and forming simpler graphs, and hence we may need to understand the classification of $G$ -covers of a graph in terms of $G$ -covers of its contractions.",
"We fix a graph $\\Gamma $ , a subgraph $\\Delta \\subset \\Gamma $ , and a $G$ -datum $A$ on $\\Gamma $ .",
"We will see that the cohomology groups $H^i(\\Gamma ,A)$ and $H^i(\\Delta , A|_{\\Delta })$ fit into an exact sequence, which is the analogue of the long exact sequence of the cohomology groups of a pair of topological spaces.",
"The relative cohomology groups occurring in this sequence can be computed as reduced cohomology groups of an induced $G$ -datum $A_{\\Gamma /\\Delta }$ on the quotient graph $\\Gamma /\\Delta $ .",
"Unfortunately, the $G$ -datum $A_{\\Gamma /\\Delta }$ is not in general the $G$ -datum associated to a $G$ -dilation datum on $\\Gamma /\\Delta $ , even when $A$ is associated to a $G$ -dilation datum on $\\Gamma $ .",
"Definition 3.26 Let $\\Gamma $ be an oriented graph, let $\\Delta \\subset \\Gamma $ be a subgraph, and let $A$ be a $G$ -datum on $\\Gamma $ .",
"Then $A|_{\\Delta }$ is a $G$ -datum on $\\Delta $ .",
"Viewing $C^0(\\Gamma ,A)$ and $C^0(\\Delta ,A|_{\\Delta })$ as sets of $A(v)$ -valued maps from $V(\\Gamma )$ and $V(\\Delta )$ , respectively, we define a surjective map $\\iota ^0:C^0(\\Gamma ,A)\\rightarrow C^0(\\Delta ,A|_{\\Delta })$ by restricting from $V(\\Gamma )$ to $V(\\Delta )$ .",
"We similarly define a surjective restriction map $\\iota ^1:C^1(\\Gamma ,A)\\rightarrow C^1(\\Delta ,A|_{\\Delta })$ , and define the relative cochain complex of the triple $(\\Gamma ,\\Delta ,A)$ : $\\begin{tikzcd}0[r] & C^0(\\Gamma ,\\Delta ,A) [r,\"\\delta ^*_{\\Gamma ,\\Delta ,A}\"] & C^1(\\Gamma ,\\Delta ,A) [r] & 0,\\end{tikzcd}$ by setting $C^i(\\Gamma ,\\Delta ,A)=\\operatorname{Ker}\\iota ^i$ for $i=0,1$ and $\\delta ^*_{\\Gamma ,\\Delta ,A}$ to be the restriction of $\\delta ^*_{\\Gamma ,A}$ to $C^0(\\Gamma ,\\Delta ,A)$ .",
"The relative cohomology groups of the triple $(\\Gamma ,\\Delta ,A)$ are $H^0(\\Gamma ,\\Delta ,A)=\\operatorname{Ker}\\delta ^*_{\\Gamma ,\\Delta ,A}\\quad \\textrm { and } \\quad H^1(\\Gamma ,\\Delta ,A)=\\operatorname{Coker}\\delta ^*_{\\Gamma ,\\Delta ,A}.$ We note that $\\delta _{\\Gamma ,A}^*\\circ \\iota ^0=\\iota ^1\\circ \\delta ^*_{\\Delta ,A|_{\\Delta }}$ , in other words the $\\iota ^i$ form a cochain map.",
"Hence we have a short exact sequence of cochain complexes: $\\begin{tikzcd}& 0 [d]& 0 [d]& \\\\0[r] & C^0(\\Gamma ,\\Delta ,A) [r,\"\\delta ^*_{\\Gamma ,\\Delta ,A}\"] [d] & C^1(\\Gamma ,\\Delta ,A) [r] [d] & 0 \\\\0[r] & C^0(\\Gamma ,A) [r,\"\\delta ^*_{\\Gamma ,A}\"] [d,\"\\iota ^0\"] & C^1(\\Gamma ,A) [r] [d,\"\\iota ^1\"] & 0 \\\\0[r] & C^0(\\Delta ,A|_{\\Delta }) [r,\"\\delta ^*_{\\Delta ,A|_{\\Delta }}\"] [d] & C^1(\\Delta ,A|_{\\Delta }) [r] [d] & 0 \\\\& 0 & 0 &\\end{tikzcd}$ By the snake lemma, the cohomology groups of $(\\Gamma ,A)$ , $(\\Delta ,A|_{\\Delta })$ and the triple $(\\Gamma ,\\Delta ,A)$ fit into an exact sequence $\\begin{tikzcd} 0 [r]& H^0(\\Gamma ,\\Delta ,A)[r] & H^0(\\Gamma ,A)[r] & H^0(\\Delta ,A|_{\\Delta }) [r] & \\, \\\\\\,[r] &H^1(\\Gamma ,\\Delta ,A) [r] & H^1(\\Gamma ,A)[r] & H^1(\\Delta ,A|_{\\Delta })[r] &0.\\end{tikzcd} $ We now show that the relative cohomology groups of the triple $(\\Gamma ,\\Delta ,A)$ are equal to the reduced cohomology groups of the contracted graph $\\Gamma /\\Delta $ with a certain induced $G$ -datum.",
"First, we define the reduced cohomology groups of a pair $(\\Gamma ,A)$ .",
"Definition 3.27 Let $\\Gamma $ be an oriented graph and let $A$ be a $G$ -datum on $\\Gamma $ .",
"Let $d:G\\rightarrow C^0(\\Gamma ,A)$ be the diagonal morphism given by $d(g)=\\xi _g\\in C^0(\\Gamma ,A)$ , where $\\xi _g(v)=f_v(g)\\in A(v)\\mbox{ for all }v\\in V(\\Gamma )\\mbox{ and all }g\\in G.$ For any $g\\in G$ and any $e\\in E(\\Gamma )$ we have $\\delta _{\\Gamma ,A}^*(d(g))(e)=t^*(\\xi _g)(e)-s^*(\\xi _g)(e)=t_e(f_{t(e)}(g))-s_e(f_{s(e)}(g))=f_e(g)-f_e(g)=0,$ hence $\\operatorname{Im}d\\subset \\operatorname{Ker}\\delta _{\\Gamma ,A}^*$ .",
"Therefore we can define the reduced cochain complex of the pair $(\\Gamma ,A)$ $\\begin{tikzcd}0 [r] & \\widetilde{C}^0(\\Gamma ,A)[r,\"\\widetilde{\\delta }_{\\Gamma ,A}^*\"] & \\widetilde{C}^1(\\Gamma ,A) [r] & 0\\end{tikzcd}$ by $ \\widetilde{C}^0(\\Gamma ,A)=C^0(\\Gamma ,A)/\\operatorname{Im}d\\quad \\textrm { and } \\quad \\widetilde{C}^1(\\Gamma ,A)=C^1(\\Gamma ,A), $ and the reduced cohomology groups of $(\\Gamma ,A)$ : $\\widetilde{H}^0(\\Gamma ,A)=\\operatorname{Ker}\\widetilde{\\delta }_{\\Gamma ,A}^*=H^0(\\Gamma ,A)\\operatorname{mod}G\\quad \\textrm { and }\\quad \\widetilde{H}^1(\\Gamma ,A)=\\operatorname{Coker}\\widetilde{\\delta }_{\\Gamma ,A}^*=H^1(\\Gamma ,A).$ We define the quotient of a graph $\\Gamma $ by a subgraph $\\Delta \\subset \\Gamma $ by contracting $\\Delta $ to a single vertex.",
"Note that this definition comes from topology, and differs from weighted edge contraction (see Def.",
"REF ), wherein each connected component of $\\Delta $ is contracted to a separate vertex.",
"Definition 3.28 Let $\\Gamma $ be an oriented graph and let $\\Delta $ be a subgraph.",
"We define the graph $\\Gamma /\\Delta $ as follows: $V(\\Gamma /\\Delta )=V(\\Gamma )\\backslash V(\\Delta )\\cup \\lbrace w\\rbrace \\quad \\textrm { and }\\quad E(\\Gamma /\\Delta )=E(\\Gamma )\\backslash E(\\Delta ),$ as well as $s(e)=\\left\\lbrace \\begin{array}{cc} w & \\textrm { if }s(e)\\in V(\\Delta ), \\\\ s(e) & \\textrm { if }s(e)\\notin V(\\Delta ), \\end{array}\\right.\\quad t(e)=\\left\\lbrace \\begin{array}{cc} w & \\textrm { if } t(e)\\in V(\\Delta ), \\\\ t(e) & \\textrm { if }t(e)\\notin V(\\Delta ).",
"\\end{array}\\right.$ Now let $\\Gamma $ be an oriented graph, let $\\Delta $ be a subgraph, and let $A$ be a $G$ -datum on $\\Gamma $ .",
"We define the $G$ -datum $A_{\\Gamma /\\Delta }$ on $\\Gamma /\\Delta $ by restricting $A$ to all vertices except $w$ and all edges, and by placing the trivial $G$ -datum at $w$ .",
"Specifically, the $G$ -groups $f^{\\prime }_v:G\\rightarrow A_{\\Gamma /\\Delta }(v)$ corresponding to the vertices $v\\in V(\\Gamma /\\Delta )$ are $f^{\\prime }_v:G\\rightarrow A_{\\Gamma /\\Delta }(v)=\\left\\lbrace \\begin{array}{ll} f_v:G\\rightarrow A(v)&\\textrm { if } v\\in V(\\Gamma )\\backslash V(\\Delta ), \\\\ \\operatorname{Id}:G\\rightarrow G& \\textrm { if } v=w.\\end{array}\\right.$ The $G$ -groups $f^{\\prime }_e:G\\rightarrow A_{\\Gamma /\\Delta }(e)$ corresponding to $e\\in E(\\Gamma /\\Delta )$ are the same as $f_e:G\\rightarrow A(e)$ .",
"Finally, the source and target maps $s^{\\prime }_e:A_{\\Gamma /\\Delta }\\big (s(e)\\big )\\rightarrow A_{\\Gamma /\\Delta }(e)$ and $t^{\\prime }_e:A_{\\Gamma /\\Delta }\\big (t(e)\\big )\\rightarrow A_{\\Gamma /\\Delta }(e)$ are $s^{\\prime }_e=\\left\\lbrace \\begin{array}{ll} s_e:A\\big (s(e)\\big )\\rightarrow A(e) & \\textrm { if } s(e)\\ne w, \\\\f_e:G\\rightarrow A(e) & \\textrm { if } s(e)=w,\\end{array}\\right.$ and $t^{\\prime }_e=\\left\\lbrace \\begin{array}{ll} t_e:A\\big (t(e)\\big )\\rightarrow A(e) & \\textrm { if } t(e)\\ne w, \\\\f_e:G\\rightarrow A(e) & \\textrm { if } t(e)=w.\\end{array}\\right.$ Remark 3.29 If $A=A^D$ is the $G$ -datum associated to a $G$ -dilation datum $D$ , then so is $A|_{\\Delta }$ , but not, in general, $A_{\\Gamma /\\Delta }$ .",
"Specifically, the edge groups of $A^D$ are the coproducts of the vertex groups, which is no longer the case for $A_{\\Gamma /\\Delta }$ .",
"In other words, the relationship between the dilated cohomology groups $H^i(\\Gamma ,D)$ and $H^i(\\Delta ,D|_{\\Delta })$ cannot be expressed without using the more general framework of $G$ -data and their cohomology.",
"The edge groups of $A_{\\Gamma /\\Delta }$ retain a record of the dilation groups $D(v)$ of the edges $v\\in V(\\Delta )$ that are contracted in $\\Gamma /\\Delta $ .",
"Proposition 3.30 Let $\\Gamma $ be an oriented graph, let $\\Delta $ be a subgraph, and let $A$ be a $G$ -datum on $\\Gamma $ .",
"The relative cohomology groups of the triple $(\\Gamma ,\\Delta ,A)$ are equal to the reduced cohomology groups of $(\\Gamma /\\Delta ,A_{\\Gamma /\\Delta })$ : $H^i(\\Gamma ,\\Delta ,A)=\\widetilde{H}^i(\\Gamma /\\Delta ,A_{\\Gamma /\\Delta }).$ The group $G$ acts diagonally on $C^0(\\Gamma /\\Delta ,A_{\\Gamma /\\Delta })$ , and the action is free and transitive on the $w$ -coordinate, since by definition $A_{\\Gamma /\\Delta }(w)=G$ .",
"Therefore any element $[\\xi ]$ in the quotient group $\\widetilde{C}^0(\\Gamma /\\Delta ,A_{\\Gamma /\\Delta })$ has a unique representative $\\xi \\in C^0(\\Gamma /\\Delta ,A_{\\Gamma /\\Delta })$ satisfying $\\xi (w)=0$ .",
"Since $A_{\\Gamma /\\Delta }(v)=A(v)$ for $v\\in V(\\Gamma \\backslash \\Delta )$ , we can define an extension by zero map $j^0:\\widetilde{C}^0(\\Gamma /\\Delta ,A_{\\Gamma /\\Delta })\\rightarrow C^0(\\Gamma ,A)$ by $j^0([\\xi ])(v)=\\left\\lbrace \\begin{array}{ll}\\xi (v) & \\textrm { if } v\\in V(\\Gamma )\\backslash V(\\Delta ), \\\\ 0 & \\textrm { if }v\\in V(\\Delta ).\\end{array}\\right.$ Similarly, since $A_{\\Gamma /\\Delta }(e)=A(e)$ for all $e\\in E(\\Gamma /\\Delta )$ , we can define an extension by zero map $j^1:\\widetilde{C}^1(\\Gamma /\\Delta ,A_{\\Gamma /\\Delta })=C^1(\\Gamma /\\Delta ,A_{\\Gamma /\\Delta })\\rightarrow C^1(\\Gamma ,A)$ by $j^1(\\eta )(e)=\\left\\lbrace \\begin{array}{ll}\\eta (e) & \\textrm { if }e\\in E(\\Gamma )\\backslash E(\\Delta ), \\\\ 0 & \\textrm { if }e\\in E(\\Delta ).\\end{array}\\right.$ We claim that the $j^i$ form a chain map.",
"Denote for simplicity $\\delta ^*=t^*-s^*=\\delta ^*_{\\Gamma ,\\Delta }$ and $\\widetilde{\\delta }^*=\\widetilde{\\delta }^*_{\\Gamma /\\Delta ,\\alpha _{\\Gamma /\\Delta }}$ .",
"For $[\\xi ]\\in \\widetilde{C}^0(\\Gamma /\\Delta ,A_{\\Gamma /\\Delta })$ let $\\xi \\in C^0(\\Gamma /\\Delta ,A_{\\Gamma /\\Delta })$ be the representative satisfying $\\xi (w)=0$ .",
"Let $e\\in E(\\Gamma )$ be an edge.",
"If $e\\in E(\\Delta )$ then $(j^1\\circ \\widetilde{\\delta }^*)\\big ([\\xi ]\\big )(e)=0$ .",
"If $e\\in E(\\Gamma )\\backslash E(\\Delta )$ has root vertices $u=s(e)$ and $v=t(e)$ , then, using $(j^1\\circ \\widetilde{\\delta }^*)\\big ([\\xi ]\\big )(e)=t^{\\prime }_e\\big (\\xi (v)\\big )-s^{\\prime }_e\\big (\\xi (u)\\big )$ we find $(j^1\\circ \\widetilde{\\delta }^*)\\big ([\\xi ]\\big )(e)=\\left\\lbrace \\begin{array}{ll} t_e(\\xi (v))-s_e(\\xi (u)) & \\textrm { if } v\\in V(\\Gamma )\\backslash V(\\Delta ) \\textrm { and } u\\in V(\\Gamma )\\backslash V(\\Delta ) , \\\\ t_e\\big (\\xi (v)\\big ) & \\textrm { if } v\\in V(\\Gamma )\\backslash V(\\Delta ) \\textrm { and } u\\in V(\\Delta ), \\\\ -s_e\\big (\\xi (u)\\big ) & \\textrm { if } v\\in V(\\Delta ) \\textrm { and } u\\in V(\\Gamma )\\backslash V(\\Delta ), \\\\ 0 &\\textrm { if } v\\in V(\\Delta ) \\textrm {and} u\\in V(\\Delta ),\\end{array}\\right.$ because $\\xi (w)=0$ .",
"On the other hand, $j^0([\\xi ])$ is the element of $C^0(\\Gamma ,\\Delta )$ obtained by setting $j^0([\\xi ])(v)=\\xi (v)$ for all vertices $v\\in V(\\Gamma )\\backslash V(\\Delta )$ and $\\xi (v)=0$ for all $v\\in V(\\Delta )$ .",
"It is clear that $(\\delta ^*\\circ j^0)([\\xi ])(e)$ is given by (REF ) for any $e\\in E(\\Gamma )\\backslash E(\\Delta )$ , and $(\\delta ^*\\circ j^0)([\\xi ])(e)=0$ for any $e\\in E(\\Delta )$ , because any such edge has root vertices in $\\Delta $ and $\\xi $ vanishes at those vertices.",
"It follows that $\\delta ^*\\circ j^0=j^1\\circ \\widetilde{\\delta }^*$ , hence the $j^i$ form a chain map.",
"We now consider the diagram (REF ).",
"By definition, $C^0(\\Gamma ,\\Delta ,A)=\\left\\lbrace \\xi \\in C^0(\\Gamma ,A):\\xi (v)=0\\mbox{ for all }v\\in V(\\Delta )\\right\\rbrace $ and $C^1(\\Gamma ,\\Delta ,A)=\\left\\lbrace \\eta \\in C^1(\\Gamma ,A):\\chi (e)=0\\mbox{ for all }e\\in E(\\Delta )\\right\\rbrace .$ It is clear that $j^i$ maps $\\widetilde{C}^i(\\Gamma /\\Delta ,A_{\\Gamma /\\Delta })$ bijectively onto $C^i(\\Gamma ,\\Delta ,A)$ for $i=0,1$ .",
"It follows that $j^i$ is a chain isomorphism from $\\widetilde{C}^i(\\Gamma /\\Delta ,A_{\\Gamma /\\Delta })$ to $C^i(\\Gamma ,\\Delta ,A)$ , which completes the proof."
],
[
"Classification of $G$ -covers of graphs and tropical curves",
"In this section, we use dilated cohomology groups, defined in the previous section, to classify $G$ -covers of graphs and tropical curves.",
"In Sec.",
"REF we give the main classification result for unweighted graphs, Thm.",
"REF , which identifies the set of $G$ -covers of $\\Gamma $ with a given dilation datum $D$ with the group $H^1(\\Gamma ,D)$ .",
"Among these covers, we characterize the connected ones in Prop.",
"REF , and give examples.",
"The case of unramified $G$ -covers of a weighted graph is treated in Sec.",
"REF , the only novelty being a numerical restriction on the dilation datum $D$ imposed by the local Riemann–Hurwitz condition (REF ).",
"Finally, the case of weighted metric graphs and tropical curves is summarized in Sec.",
"REF ."
],
[
"$G$ -covers of graphs",
"In this section, we determine all $G$ -covers of an unweighed graph $\\Gamma $ with a given $G$ -dilation datum $D$ .",
"Our theorem generalizes the standard result that the set of topological $G$ -covers of $\\Gamma $ (i.e.",
"with trivial stabilizers) is identified with $H^1(\\Gamma ,G)$ (see Ex.",
"REF and Ex.",
"REF ).",
"Theorem 4.1 Let $\\Gamma $ be a graph, let $G$ be a finite abelian group, and let $D$ be a $G$ -dilation datum on $\\Gamma $ .",
"Then there is a natural bijection between $H^1(\\Gamma ,D)$ and the set of $G$ -covers having dilation datum $D$ .",
"We first explain how to associate an element $[\\eta _{\\varphi }]\\in H^1(\\Gamma ,D)$ to a $G$ -cover $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ .",
"Pick an orientation on $E(\\Gamma )$ and a consistent orientation on $E(\\Gamma ^{\\prime })$ , and denote $s,t:E(\\Gamma ^{\\prime })\\rightarrow V(\\Gamma ^{\\prime })$ and $s,t:E(\\Gamma )\\rightarrow V(\\Gamma )$ the source and target maps.",
"For each $x\\in X(\\Gamma )$ , the preimage $\\varphi ^{-1}(x)$ is a $G/D(x)$ -torsor, so pick a $G$ -equivariant bijection $f_x:\\varphi ^{-1}(x)\\rightarrow G/D(x)$ .",
"Namely, for every $x^{\\prime }\\in \\varphi ^{-1}(x)$ and every $g\\in G$ we have $f_x(gx^{\\prime })=f_x(x^{\\prime })+g\\ \\operatorname{mod}D(x).$ We require that if $e=(h_1,h_2)\\in E(\\Gamma )$ , then, under the identification $\\varphi ^{-1}(h_1)=\\varphi ^{-1}(h_2)=\\varphi ^{-1}(e)$ , the two maps $f_{h_1}$ and $f_{h_2}$ are equal, in which case we denote them by $f_e$ .",
"If $l\\in L(\\Gamma )$ is a leg, then $D(l)\\subset D(r(l))$ and we have the following diagram of $G$ -sets: -1(l)[r,\"r\"] [d,\"fl\"] -1(r(l)) [d,\"fr(l)\"] G/D(l)[r] G/D(r(l)) The vertical maps are bijections, while the horizontal maps are surjections.",
"Adding a constant to $f_l$ if necessary, we can assume that the lower horizontal map is reduction modulo $D({r(l)})/D(l)$ .",
"Now let $e\\in E(\\Gamma )$ be an edge, then $D(s(e))$ and $D(t(e))$ are subgroups of $G$ containing $D(e)$ .",
"The source and target maps restrict to $G$ -equivariant surjections $s:\\varphi ^{-1}(e)\\rightarrow \\varphi ^{-1}(s(e))$ and $t:\\varphi ^{-1}(e)\\rightarrow \\varphi ^{-1}(t(e))$ , and we have a commutative diagram of $G$ -sets -1(s(e))[d,\"fs(e)\"] -1(e)[l,\"s\"'][r,\"t\"] [d,\"fe\"] -1(t(e)) [d,\"ft(e)\"] G/D(s(e)) G/D(e) [l,\"+s(e)\"'] [r,\"+t(e)\"] G/D(t(e)) where the vertical arrows are bijections.",
"The lower horizontal arrows are surjections, and are therefore given by adding certain elements $\\eta _t(e)\\in G/D({t(e)})$ and $\\eta _s(e)\\in G/D(s(e))$ , and then reducing modulo $D({t(e)})$ and $D(s(e))$ , respectively.",
"Hence the cover $\\varphi $ determines an element $\\eta _{\\varphi }=\\big (\\eta _s(e),-\\eta _t(e)\\big )_{e\\in E(\\Gamma )}\\in \\prod _{e\\in E(\\Gamma )} G/D\\big (s(e)\\big )\\oplus G/D\\big (t(e)\\big ).$ Denote $[\\eta _{\\varphi }]$ its class in $H^1(\\Gamma ,D)$ .",
"We need to verify that the association $\\varphi \\mapsto [\\eta _{\\varphi }]$ is independent of all choices.",
"Suppose that we chose different bijections $\\widetilde{f}_x:\\varphi ^{-1}(x)\\rightarrow G/D(x)$ .",
"For any leg $l\\in L(\\Gamma )$ we can assume, as above, that the induced map $G/D(l)\\rightarrow G/D(r(l))$ is reduction modulo $D(r(l))/D(l)$ .",
"Now let $e\\in E(\\Gamma )$ be an edge.",
"We have a diagram of $G$ -sets G/D(s(e)) G/D(e) [r,\"+t(e)\"][l,\"+s(e)\"'] G/D(t(e)) -1(s(e))[d,\"fs(e)\"] [u,\"fs(e)\"'] -1(e)[l,\"s\"'][r,\"t\"] [d,\"fe\"] [u,\"fe\"'] -1(t(e)) [d,\"ft(e)\"] [u,\"ft(e)\"'] G/D(s(e)) G/D(e) [r,\"+t(e)\"][l,\"+s(e)\"'] G/D(t(e)) The top horizontal maps define an element $\\widetilde{\\eta }_{\\varphi }=(\\widetilde{\\eta }_s(e), -\\widetilde{\\eta }_t(e))_{e\\in E(\\Gamma )}$ and a corresponding class $[\\widetilde{\\eta }_{\\varphi }]$ in $H^1(\\Gamma ,D)$ .",
"The middle column consists of isomorphisms of $G$ -sets, hence the map $\\widetilde{f}_e\\circ f_e^{-1}:G/D(e)\\rightarrow G/D(e)$ is the addition of an element $\\omega (e)\\in G/D(e)$ .",
"Similarly, the isomorphisms $\\widetilde{f}_{s(e)}\\circ f_{s(e)}^{-1}:G/D\\big (s(e)\\big )\\rightarrow G/D\\big (s(e)\\big )$ and $\\widetilde{f}_{t(e)}\\circ f_{t(e)}^{-1}:G/D\\big ({t(e)}\\big )\\rightarrow G/D\\big ({t(e)}\\big )$ are given by adding certain elements $\\xi \\big (s(e)\\big )\\in G/D\\big (s(e)\\big )$ and $\\xi \\big ({t(e)}\\big )\\in G/D\\big ({t(e)}\\big )$ .",
"The two vertical rectangles give the following relations on all of these elements: $\\begin{split}\\eta _s(e)+\\xi (s(e))&=\\omega (e)+\\widetilde{\\eta }_s(e)\\operatorname{mod}D(s(e))\\\\\\eta _t(e)+\\xi ({t(e)})&=\\omega (e)+\\widetilde{\\eta }_t(e) \\operatorname{mod}D({t(e)}).\\end{split}$ Comparing this with Eq.",
"(REF ), we see that $[\\eta _{\\varphi }]=[\\widetilde{\\eta }_{\\varphi }]$ , so the $G$ -cover $\\varphi $ determines a well-defined element of $H^1(\\Gamma ,D)$ .",
"Conversely, let $D$ be a $G$ -dilation datum on $\\Gamma $ , let $[\\eta ]\\in H^1(\\Gamma ,D)$ be an element, and let $(\\eta _t(e),\\eta _s(e))_{e\\in E(\\Gamma )}$ be a lift of $[\\eta ]$ .",
"Running the above construction in reverse, we obtain a $G$ -cover $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ with associated $G$ -dilation datum $D$ .",
"Specifically, let $V(\\Gamma ^{\\prime })=\\coprod _{v\\in V(\\Gamma )} G/D(v)$ , $E(\\Gamma ^{\\prime })=\\coprod _{e\\in E(\\Gamma )} G/D(e)$ , and $L(\\Gamma ^{\\prime })=\\coprod _{l\\in L(\\Gamma )} G/D(l)$ .",
"We define $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ by sending each $G/D(x)$ to the corresponding $x\\in X(\\Gamma )$ .",
"For a leg $l\\in L(\\Gamma )$ , we define the lifting $r:G/D(l)\\rightarrow G/D\\big (r(l)\\big )$ of the root map to $\\Gamma ^{\\prime }$ as reduction modulo $D(r(l))/D(l)$ .",
"Finally, for an edge $e\\in L(\\Gamma )$ , we define the liftings $s:G/D(e)\\rightarrow G/D(s(e))$ and $t:G/D(e)\\rightarrow G/D(t(e))$ of the source and target maps as $s(g)=g+\\eta _s(e)\\operatorname{mod}D(s(e))/D(e) \\quad \\textrm { and }\\quad t(g)=g-\\eta _t(e)\\operatorname{mod}D(t(e))/D(e).$ We observe that there is at least one $G$ -cover associated to any $G$ -dilation datum $D$ on $\\Gamma $ , namely the trivial $G$ -cover with dilation datum $D$ , corresponding to the identity element of $H^1(\\Gamma ,D)$ .",
"Explicitly, the source graph $\\Gamma ^{\\prime }$ is the union of the sets $G/D(x)$ for all $x\\in X(\\Gamma )$ , and the root maps $G/D(x)\\rightarrow G/D\\big (r(x)\\big )$ are the quotient maps corresponding to the injections $D(x)\\subset D\\big (r(x)\\big )$ .",
"Note also that the set of $G$ -covers with dilation datum $D$ depends only on the vertex groups $D(v)$ for $v\\in V(\\Gamma )$ , or, alternatively, on the dual stratification $\\mathcal {S}^*(D)$ .",
"Remark 4.2 (Functoriality) The correspondence $\\varphi \\mapsto \\eta _{\\varphi }$ between $G$ -covers of $\\Gamma $ with dilation datum $D$ and elements of $H^1(\\Gamma ,D)$ given in Thm.",
"REF is functorial, in the following sense.",
"Let $\\varphi _1:\\Gamma ^{\\prime }_1\\rightarrow \\Gamma $ and $\\varphi _2:\\Gamma ^{\\prime }_2\\rightarrow \\Gamma $ be $G$ -covers, and let $\\tau :\\Gamma ^{\\prime }_1\\rightarrow \\Gamma ^{\\prime }_2$ be a morphism of $G$ -covers (in the sense of Def.",
"REF ).",
"Then the dilation datum $D_{\\varphi _1}$ is a refinement of $D_{\\varphi _2}$ , and by the proof of Prop.",
"REF there is a surjective map $\\pi :H^1(\\Gamma ,D_{\\varphi _1})\\rightarrow H^1(\\Gamma ,D_{\\varphi _2})$ .",
"It is easy to check that $\\pi (\\eta _{\\varphi _1})=\\eta _{\\varphi _2}$ .",
"More generally, the correspondence $\\varphi \\mapsto \\eta _{\\varphi }$ is functorial with respect to pullback maps induced by finite harmonic morphisms $\\Delta \\rightarrow \\Gamma $ , which, as we have already remarked, are beyond the scope of our paper.",
"Remark 4.3 (Trivialization along a tree) We saw in Prop.",
"REF that $H^1(\\Delta ,D)=0$ for any $G$ -dilation datum on a tree $\\Delta $ .",
"In other words, any $G$ -cover of a tree is isomorphic to the trivial $G$ -cover associated to some dilation datum $D$ .",
"This statement allows us to give a somewhat explicit description of $G$ -covers of an arbitrary graph $\\Gamma $ .",
"Let $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ be a $G$ -cover with dilation datum $D$ .",
"Pick a spanning tree $\\Delta \\subset \\Gamma $ , and let $\\lbrace e_1,\\ldots ,e_n\\rbrace =E(\\Gamma )\\backslash E(\\Delta )$ be the remaining edges.",
"The restricted $G$ -cover $\\varphi |_{\\Delta }$ is isomorphic to the trivial $G$ -cover of $\\Delta $ with dilation datum $D|_{\\Delta }$ , in other words there is a $G$ -equivariant bijection $\\tau :\\varphi ^{-1}(\\Delta )\\rightarrow \\coprod _{x\\in X(\\Delta )} G/D(x),\\quad \\tau \\big (\\varphi ^{-1}(x)\\big )=G/D(x).$ The cover $\\varphi $ is then completely determined by the way that the fibers $G/D(e_i)$ are attached to the fibers $G/D(s(e_i))$ and $G/D(t(e_i))$ .",
"As we saw in the proof above, this attachment datum can be recorded (in general, non-uniquely) by an $n$ -tuple of elements of $\\eta _i\\in A^D(e_i)=G/C(e_i)$ .",
"In terms of the dilated cohomology group, we have shown that any element $[\\eta ]\\in H^1(\\Gamma ,D)$ can be represented by a cochain $\\eta \\in C^1(\\Gamma ,D)$ such that $\\eta (e)=0$ unless $e=e_i$ for some $i=1,\\ldots ,n$ (cf.",
"Lemma 2.3.4 in [29])."
],
[
"Connected covers",
"Given a connected graph $\\Gamma $ , it is natural to ask which of its $G$ -covers constructed above are connected.",
"To answer this question, we first consider the following construction.",
"Let $H$ be a proper subgroup of $G$ , and let $D$ be an $H$ -dilation datum on a connected graph $\\Gamma $ .",
"We can then view $D$ as a $G$ -dilation datum, which we denote by $D^G$ to prevent confusion.",
"There are natural injective chain maps $\\iota ^i:C^i(\\Gamma ,D)\\rightarrow C^i(\\Gamma ,D^G)$ that induce maps $\\iota ^i:H^i(\\Gamma ,D)\\rightarrow H^i(\\Gamma ,D^G)$ .",
"Lemma 4.4 The maps $\\iota ^i:H^i(\\Gamma ,D)\\rightarrow H^i(\\Gamma ,D^G)$ are injective.",
"The cochain groups $C^0(\\Gamma ,D)$ and $C^0(\\Gamma ,D^G)$ are the products of $H/D(v)$ and $G/D(v)$ , respectively, over all $v\\in V(\\Gamma )$ .",
"It follows that $\\operatorname{Coker}\\iota ^0$ can be identified with the cochain group $C^0(\\Gamma ,G/H)$ , and similarly $\\operatorname{Coker}\\iota ^1=C^1(\\Gamma ,G/H)$ .",
"By the snake lemma, we have a long exact sequence of cohomology groups: $\\begin{tikzcd} 0 [r]& H^0(\\Gamma ,D)[r,\"\\iota ^0\"] & H^0(\\Gamma ,D^G)[r] & H^0(\\Gamma ,G/H) [r] & H^1(\\Gamma ,D) [r,\"\\iota ^1\"] & H^1(\\Gamma ,D^G).\\end{tikzcd}$ Therefore $\\iota ^0$ is injective.",
"To prove that $\\iota ^1$ is injective, we show that the map $\\pi :H^0(\\Gamma ,D^G)\\rightarrow H^0(\\Gamma ,G/H)$ is surjective.",
"All of our chain complexes split into direct sums over the connected components of $\\Gamma $ , so we assume that $\\Gamma $ is connected.",
"In this case $H^0(\\Gamma ,G/H)=G/H$ , and moreover any $[\\xi ]\\in H^0(\\Gamma ,G/H)$ is represented by a constant cochain $\\xi (v)=\\overline{g}$ for some $\\overline{g}\\in G/H$ .",
"Pick $g\\in G$ representing $\\overline{g}$ , then the constant cochain $\\xi ^{\\prime }(v)=g\\operatorname{mod}G/D(v)$ in $C^0(\\Gamma ,D^G)$ lies in $\\operatorname{Ker}\\delta ^0_{\\Gamma ,D^G}$ , hence represents a class $[\\xi ^{\\prime }]\\in H^0(\\Gamma ,D^G)$ , and $\\pi \\big ([\\xi ^{\\prime }]\\big )=[\\xi ]$ .",
"Therefore $\\pi $ is surjective, so $\\iota ^0$ is injective.",
"There is a natural way to associate a $G$ -cover of $\\Gamma $ to an $H$ -cover of $\\Gamma $ that corresponds, under the bijection of Thm.",
"REF , to the injective map $\\iota ^1:H^1(\\Gamma ,D)\\rightarrow H^1(\\Gamma ,D^G)$ .",
"Definition 4.5 Let $\\Gamma $ be a graph, let $H\\subset G$ be abelian groups, and let $\\varphi \\colon \\Gamma ^{\\prime }\\rightarrow \\Gamma $ be an $H$ -cover with $H$ -dilation datum $D$ .",
"We define the $G$ -cover $\\varphi ^G:\\Gamma ^{\\prime G}\\rightarrow \\Gamma $ with $G$ -dilation datum $D^G$ , called the extension of $\\varphi $ by $G$, as follows.",
"For each $x\\in X(\\Gamma )$ , pick an identification of $H$ -sets, as in the proof of Thm.",
"REF , of $\\varphi ^{-1}(x)$ with $H/D(x)$ , and for every edge $e\\in E(\\Gamma )$ let $\\eta _t(e)\\in H/D\\big (t(e)\\big )$ and $\\eta _s(e)\\in H/D\\big (s(e)\\big )$ be the elements that determine the root maps $t:\\varphi ^{-1}(e)\\rightarrow \\varphi ^{-1}\\big (t(e)\\big )$ and $s:\\varphi ^{-1}(e)\\rightarrow \\varphi ^{-1}\\big (s(e)\\big )$ .",
"We define $\\varphi ^G$ by identifying each fiber $(\\varphi ^G)^{-1}(x)$ with the $G$ -set $G/D(x)$ , and rooting $(\\varphi ^G)^{-1}(e)$ to $(\\varphi ^G)^{-1}\\big (t(e)\\big )$ and $(\\varphi ^G)^{-1}\\big (s(e)\\big )$ using $\\eta _t(e)$ and $\\eta _s(e)$ , viewed, respectively, as elements of $G/D\\big (t(e)\\big )$ and $G/D\\big (s(e)\\big )$ .",
"Looking at the proof of REF , it is clear that $\\iota ^1(\\eta _{\\varphi })=\\eta _{\\varphi ^G}$ .",
"Furthermore, the cover $\\varphi ^G$ is disconnected (unless $H=G$ ), since the root maps $t:G/D(e)\\rightarrow G/D(t(e))$ and $s:G/D(e)\\rightarrow G/D(s(e))$ preserve the decomposition into $H$ -cosets.",
"We now show that all disconnected $G$ -covers of a connected graph $\\Gamma $ arise in this way.",
"Indeed, let $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ be a $G$ -cover of a connected graph with $G$ -dilation datum $D$ , and let $\\Gamma ^{\\prime }=\\Gamma ^{\\prime }_1\\sqcup \\cdots \\sqcup \\Gamma ^{\\prime }_n$ be the connected components of $\\Gamma ^{\\prime }$ .",
"The group $G$ acts on the connected components by permutation.",
"Let $H=\\big \\lbrace g\\in G\\big \\vert g(\\Gamma ^{\\prime }_1)=\\Gamma ^{\\prime }_1\\big \\rbrace $ , then $D(v)\\in H$ for all $v\\in V(\\Gamma )$ .",
"We view $D$ as an $H$ -dilation datum, which we denote $D_H$ .",
"It is clear that the restriction $\\varphi _{\\Gamma ^{\\prime }_1}\\colon \\Gamma ^{\\prime }_1\\rightarrow \\Gamma $ is a connected $H$ -cover with $H$ -dilation datum $D_H$ , and that $\\varphi $ is isomorphic to the $G$ -extension of $\\varphi |_{\\Gamma ^{\\prime }_1}$ by $G$ .",
"In other words, every disconnected $G$ -cover of $\\Gamma $ is the extension of an $H$ -cover, where $H\\subset G$ is some proper subgroup.",
"We have proved the following result, which classifies connected $G$ -covers of a connected graph $\\Gamma $ .",
"Proposition 4.6 Let $\\Gamma $ be a connected graph, and let $D$ be a $G$ -dilation datum on $\\Gamma $ .",
"If the groups $D(v)$ span $G$ , then every $G$ -cover with dilation datum $D$ is connected.",
"If not, then the set of disconnected $G$ -covers with dilation datum $D$ is the union of the images of the maps $H^1(\\Gamma ,D_H)\\rightarrow H^1(\\Gamma ,D)$ over all proper subgroups $H\\subset G$ such that $D(v)\\subset H$ for all $v\\in V(\\Gamma )$ , where for each such $H$ , $D_H$ denotes $D$ viewed as an $H$ -dilation datum.",
"Example 4.7 (Klein covers continued) We now apply the results of this section to enumerate all $G$ -covers of the graph $\\Gamma $ consisting of two vertices $u$ and $v$ joined by two edges $e$ and $f$ , when $G=\\mathbb {Z}/{2}\\mathbb {Z}\\oplus \\mathbb {Z}/{2}\\mathbb {Z}$ is the Klein group.",
"In particular, we describe the covers of Ex.",
"REF in terms of dilated cohomology.",
"We recall that we denote 00, 10, 01, and 11 the elements of $G$ , and $H_1$ , $H_2$ , and $H_3$ the subgroups of $G$ generated respectively by 10, 01, and 11.",
"We enumerate the covers in the following way: first, we enumerate the choices for $D(u)$ and $D(v)$ , then, for each choice, we consider the possible $D(e), D(f)\\subset D(u)\\cap D(v)$ , and finally $\\#H^1(\\Gamma ,D)$ counts the $G$ -covers with such $G$ -dilation data (note that the last two steps are independent, since $H^1(\\Gamma ,D)$ does not depend on the edge dilation groups).",
"We saw in Ex.",
"REF that $H^1(\\Gamma ,D)=G/(D(u)+D(v))$ for any $G$ -dilation datum on $\\Gamma $ .",
"We now make this identification more explicit.",
"Orient $\\Gamma $ so that $s(e)=s(f)=u$ and $t(e)=t(f)=v$ .",
"An element $[\\eta ]\\in H^1(\\Gamma ,D)$ is represented by two pairs of elements $\\big (\\eta _s(e),\\eta _t(e)\\big ), \\big (\\eta _s(f),\\eta _t(f)\\big )\\in G/D(u)\\oplus G/D(v),$ modulo the relations (REF ).",
"It is clear that for any $[\\eta ]$ we can pick a representative with $\\eta _s(e)=0$ , $\\eta _s(f)=0$ , and $\\eta _t(f)=0$ (in other words, we trivialize $[\\eta ]$ along the spanning tree $\\lbrace u,v,f\\rbrace $ ), so we can represent $[\\eta ]$ with a single element $\\eta _t(e)\\in G/D(v)$ .",
"Furthermore, the class of this $\\eta _t(e)$ in $G/(D(u)+D(v))$ is equal to $[\\eta ]$ under the isomorphism $G\\big /\\big (D(u)+D(v)\\big )=H^1(\\Gamma ,D)$ .",
"Explicitly, the cover corresponding to $[\\eta ]$ is constructed as follows: define the sets $\\lbrace u_{ij}\\rbrace =G/D(u)$ , $\\lbrace v_{ij}\\rbrace =G/D(v)$ , $\\lbrace e_{ij}\\rbrace =G/D(e)$ , and $\\lbrace f_{ij}\\rbrace = G/D(f)$ (where the labeling is non-unique for a nontrivial dilation group), attach $f_{ij}$ to $u_{ij}$ and $v_{ij}$ , and attach $e_{ij}$ to $u_{ij}$ and $v_{ij+\\eta _t(e)}$ .",
"$D(u)=D(v)=0$ .",
"This is the topological case, with trivial dilation.",
"Here $D(e)=D(f)=0$ , $H^1(\\Gamma ,D)=H^1(\\Gamma ,G)=G$ , and there are four covers, three of them non-trivial.",
"All of these covers are disconnected, since there are no surjective maps $\\pi _1(\\Gamma )=\\mathbb {Z}\\rightarrow G$ .",
"The cover corresponding to $\\eta _t(e)=10$ is given in Fig.",
"REF .",
"$D(u)=0$ , $D(v)=H_i$ for $i=1,2,3$ .",
"In this case $D(e)=D(f)=0$ , $H^1(\\Gamma ,D)=G/H_i$ , so for each $i$ there is one trivial and one nontrivial cover.",
"For example, Fig.",
"REF shows the non-trivial cover with $D(v)=H_1$ and $\\eta _t(e)=01$ .",
"There are a total of six covers of this type: three trivial disconnected covers and three non-trivial connected covers.",
"$D(u)=H_i$ for $i=1,2,3$ , $D(v)=0$ .",
"This case is symmetric to the one above, with three connected and three disconnected covers.",
"$D(u)=D(v)=H_i$ for $i=1,2,3$ .",
"Each of the groups $D(e)$ and $D(f)$ can be chosen to be 0 or $H_i$ .",
"Since $H^1(\\Gamma ,D)=G/H_i$ , there is one trivial and one non-trivial cover for each choice.",
"For example, when $D(u)=D(v)=D(e)=H_2$ and $D(f)=0$ , we obtain the non-trivial cover of Fig.",
"REF by choosing $\\eta _t(e)=10$ , and the trivial cover of Fig.",
"REF by choosing $\\eta _t(e)=00$ .",
"There are a total of 24 such covers, 12 connected and 12 disconnected.",
"$D(u)=H_i$ , $D(v)=H_j$ , $i\\ne j$ .",
"The only possibility is $D(e)=D(f)=0$ , and $H^1(\\Gamma ,D)=0$ , so for each $i\\ne j$ there is a unique trivial cover, for a total of six covers, all connected.",
"If one or both of the groups $D(u)$ and $D(v)$ are equal to $G$ , then $H^1(\\Gamma ,D)=0$ .",
"Picking $D(e)$ and $D(f)$ to be arbitrary subgroups of $D(u)\\cap D(v)$ , we obtain 51 connected trivial covers.",
"Two such covers are given in Figs.",
"REF and REF .",
"We note that 9 of these covers, including the one on REF , have non-cyclic edge dilation groups, and are therefore not algebraically realizable.",
"In total, there are 97 Klein covers of $\\Gamma $ , including 75 connected covers."
],
[
"Weighted graphs and unramified $G$ -covers",
"We now consider the category of weighted graphs and finite harmonic morphisms between them.",
"Given a weighted graph $\\Gamma $ and a $G$ -cover $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ (where we view $\\Gamma $ as an unweighted graph and $\\varphi $ as a morphism), there is a natural way to promote $\\varphi $ to a harmonic morphism of degree equal to $\\#(G)$ .",
"Since the action of $G$ is transitive on the fibers, the genera of all vertices of $\\Gamma ^{\\prime }$ lying in a single fiber are equal.",
"Therefore a $G$ -cover of $\\Gamma $ with a given dilation datum $D$ is uniquely specified by an element of $H^1(\\Gamma ,D)$ and a weight function $g^{\\prime }:V(\\Gamma )\\rightarrow \\mathbb {Z}_{\\ge 0}$ (which we lift to $\\Gamma ^{\\prime }$ ).",
"There is a natural way to specify this weight: require $\\varphi $ to be unramified.",
"This condition imposes a numerical restriction on the $G$ -dilation datum $D$ .",
"Definition 4.8 Let $\\Gamma $ be a weighted graph, and let $G$ be a finite abelian group.",
"A $G$ -cover of $\\Gamma $ is a finite harmonic morphism $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ together with an action of $G$ on $\\Gamma ^{\\prime }$ , such that the following properties are satisfied: The action is invariant with respect to $\\varphi $ .",
"For each $x\\in X(\\Gamma )$ , the group $G$ acts transitively on the fiber $\\varphi ^{-1}(x)$ .",
"$\\#(G)=\\deg \\varphi $ .",
"We say that a $G$ -cover $\\varphi $ is effective or unramified if it is so as a harmonic morphism.",
"Remark 4.9 This definition is similar to Definition 7.1.2 in [12].",
"Example 4.10 Let $\\Gamma $ be a weighted graph.",
"In Example REF , we saw that an element $\\eta \\in H^1(\\Gamma ,G)$ determines a topological $G$ -cover $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ .",
"We now weight $\\Gamma ^{\\prime }$ by setting $g(v^{\\prime })=g(v)$ for all $v\\in V(\\Gamma )$ and all $v^{\\prime }\\in \\varphi ^{-1}(v)$ .",
"Setting $\\deg _{\\varphi }(x)=1$ for all $x\\in X(\\Gamma ^{\\prime })$ , we see that $\\varphi $ is an unramified $G$ -cover.",
"Conversely, it is clear that a $G$ -cover $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ is a topological $G$ -cover if and only if $\\deg _{\\varphi }(x)=1$ for all $x\\in X(\\Gamma ^{\\prime })$ .",
"We now classify all $G$ -covers and unramified $G$ -covers of a given weighted graph $\\Gamma $ .",
"We first note that there is no difference between studying $G$ -covers of a weighted graph and $G$ -covers of the underlying unweighted graph.",
"Indeed, let $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ be a $G$ -cover of a weighted graph $\\Gamma $ , and let $D_{\\varphi }$ be the associated $G$ -dilation datum.",
"An element $g\\in G$ determines an automorphism of $\\Gamma ^{\\prime }$ , which in particular is an unramified cover of degree one.",
"Therefore, for any $x\\in X(\\Gamma )$ , the harmonic morphism $\\varphi $ has the same degree at $x$ and at $g(x)$ .",
"Since $G$ acts transitively on $\\varphi ^{-1}(x)$ , we see that $d_{\\varphi }(x^{\\prime })$ is the same for all $x^{\\prime }\\in \\varphi ^{-1}(x)$ .",
"Since $\\deg \\varphi =\\sum _{x^{\\prime }\\in \\varphi ^{-1}(x)} d_{\\varphi }(x^{\\prime })=d_{\\varphi }(x^{\\prime }) \\#\\big (\\varphi ^{-1}(x)\\big )=d_{\\varphi }(x^{\\prime })\\big [G:D_{\\varphi }(x)\\big ],$ we see that $d_{\\varphi }(x^{\\prime })=\\#\\big (D_{\\varphi }(x)\\big )$ for all $x^{\\prime }\\in \\varphi ^{-1}(x)$ .",
"Therefore, the local degrees of $\\varphi $ are uniquely defined by the associated dilation datum.",
"Conversely, if $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ is a $G$ -cover of $\\Gamma $ viewed as an unweighted graph, then Eq.",
"(REF ) gives the unique way to promote $\\varphi $ to a harmonic morphism of degree $\\#(G)$ .",
"As a result, the classification of $G$ -covers of weighted graphs reduces trivially to the unweighted case, except that we need to manually specify the weights on the cover.",
"Theorem 4.11 Let $\\Gamma $ be a weighted graph, let $G$ be a finite abelian group, let $D$ be a $G$ -dilation datum on $\\Gamma $ , and let $g^{\\prime }:V(\\Gamma )\\rightarrow \\mathbb {Z}_{\\ge 0}$ be a function.",
"Then the there is a natural bijection between $H^1(\\Gamma ,D)$ and the set of $G$ -covers $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ having dilation datum $D$ , such that $g(v^{\\prime })=g^{\\prime }(\\varphi (v^{\\prime }))$ for all $v^{\\prime }\\in V(\\Gamma ^{\\prime })$ .",
"This follows immediately from Thm.",
"REF , since $G$ acts transitively on each fiber $\\varphi ^{-1}(v)$ and therefore the numbers $g(v^{\\prime })$ for $v^{\\prime }\\in \\varphi ^{-1}(v)$ are all equal to some $g^{\\prime }(v)$ .",
"For the remainder of this section, we restrict our attention to unramified $G$ -covers, which are the graph-theoretic analogues of étale maps.",
"Given such a cover $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ , we consider the Riemann–Hurwitz condition (REF ) at all vertices $v^{\\prime }\\in V(\\Gamma ^{\\prime })$ .",
"This condition uniquely specifies the genera of the vertices of $\\Gamma ^{\\prime }$ .",
"However, these genera may fail to be non-negative integers, which imposes a numerical constraint on the $G$ -dilation data on $\\Gamma $ that are associated to unramified $G$ -covers.",
"Definition 4.12 Let $(\\Gamma ,D)$ be a $G$ -dilated graph.",
"We define the index function $a_{\\Gamma ,D}\\colon V(\\Gamma )\\times S(G)\\rightarrow \\mathbb {Z}_{\\ge 0}$ of $(\\Gamma ,D)$ by $a_{\\Gamma ,D}(v;H)=\\#\\big \\lbrace h\\in T_v\\Gamma \\big \\vert D(h)=H\\big \\rbrace .$ Proposition 4.13 Let $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ be an unramified $G$ -cover, let $D_{\\varphi }$ be the associated $G$ -dilation datum, and let $a_{\\Gamma ,D}$ be the index function of $(\\Gamma ,D_{\\varphi })$ .",
"Let $v\\in V(\\Gamma )$ be a vertex with dilation group $D(v)$ , and let $S(D(v))$ be the set of subgroups of $D(v)$ .",
"Then $2-2g^{\\prime }(v)-\\sum _{K\\in S(D(v))} a_{\\Gamma ,D}(v;K)\\big [D(v):K\\big ]=\\#\\big (D(v)\\big )\\Big [2-2g(v)-\\sum _{K\\in S(D(v))} a_{\\Gamma ,D}(v;K)\\Big ]$ where $g^{\\prime }(v)$ is the genus of any vertex $v^{\\prime }\\in \\varphi ^{-1}(v)$ .",
"For any half-edge $h\\in T_v\\Gamma $ , the dilation group $D(h)$ is a subgroup of $D(v)$ .",
"Hence $\\operatorname{val}(v)=\\sum _{K\\in S(D(v))}a_{\\Gamma ,D}(v;K).$ As noted above, each $h\\in T_v\\Gamma $ has $[D(v):D(h)]$ preimages in $\\Gamma ^{\\prime }$ attached to $v^{\\prime }$ .",
"Therefore $\\operatorname{val}(v^{\\prime })=\\sum _{K\\in S(D(v))} a_{\\Gamma ,D}(v;K)\\big [D(v):K\\big ].$ Plugging this into (REF ), we obtain (REF ).",
"Definition 4.14 Let $\\Gamma $ be a weighted graph.",
"A $G$ -dilation datum $D$ on $\\Gamma $ is called admissible if for every $v\\in V(\\Gamma )$ the number $g^{\\prime }(v)=\\#\\big (D(v)\\big )\\big [g(v)-1\\big ]+1+\\frac{1}{2}\\sum _{K\\in S(D(v))}a_{\\Gamma ,D}(v;K)\\big (\\#(D(v))-[D(v):K]\\big )$ determined by (REF ) is a non-negative integer.",
"A $G$ -stratification $\\mathcal {S}$ is called admissible if the associated $G$ -stratification $D$ is admissible.",
"It is clear that the $G$ -dilation datum associated to an unramified $G$ -cover is admissible.",
"Conversely, if $D$ is an admissible $G$ -dilation datum, then (REF ) uniquely specifies the weight function on any $G$ -cover of $\\Gamma $ with dilation datum $D$ .",
"Hence we obtain the following result.",
"Theorem 4.15 Let $(\\Gamma ,D)$ be $G$ -dilated weighted graph.",
"If $D$ is admissible, then there is a natural bijection between the set of unramified $G$ -covers of $\\Gamma $ having dilation datum $D$ and $H^1(\\Gamma ,D)$ .",
"Otherwise, there are no unramified covers of $\\Gamma $ having dilation datum $D$ .",
"The result follows immediately from Thm.",
"REF and Prop.",
"REF .",
"Condition (REF ) imposes two restrictions on a $G$ -stratification $\\mathcal {S}$ (equivalently, on a $G$ -dilation datum $D$ ): a stability condition ($g^{\\prime }(v)$ is non-negative) and a parity condition ($g^{\\prime }(v)$ is an integer).",
"We make a number of general observations.",
"First, we note that the admissibility condition is trivially satisfied at each undilated vertex $v\\in V(\\Gamma )\\backslash V(\\Gamma _{dil})$ .",
"Indeed, if $D(v)=0$ , then equation (REF ) reduces to $g^{\\prime }(v)=g(v)$ .",
"We also observe that $g^{\\prime }(v)$ is positive, and hence the stability condition is satisfied, if $g(v)\\ge 1$ .",
"We also note that $g^{\\prime }(v)$ is an integer if $\\#(D(v))$ is odd, so (REF ) does not impose a parity condition if the order of $G$ is odd.",
"Remark 4.16 Equation (REF ) is the only role that the weight function on $\\Gamma $ plays in the classification of unramified $G$ -covers of $\\Gamma $ .",
"Furthermore, when checking the admissibility condition at a vertex $v\\in V(\\Gamma )$ , we only need to know whether $g(v)$ is positive or not, the actual value is not important.",
"Therefore, for example, two weighted graphs having the same underlying unweighted graph, and having the same set of genus zero vertices with respect to the two weight functions, will have the same set of unramified $G$ -covers.",
"We now show that an admissible $G$ -stratification has the following semistability properties.",
"Proposition 4.17 Let $\\mathcal {S}$ be an admissible $G$ -stratification of a graph $\\Gamma $ .",
"For every simple vertex $v\\in V(\\Gamma )$ and for each $H\\in S(G)$ , either $v\\in V(\\Gamma _H)$ and $\\operatorname{val}_{\\Gamma _H}(v)=2$ , or $v\\notin V(\\Gamma _H)$ .",
"Suppose that $\\mathcal {S}$ is admissible.",
"Let $v\\in V(\\Gamma )$ be a vertex with $g(v)=0$ and two tangent directions $h_1$ and $h_2$ .",
"We write condition (REF ) at $v$ : $g^{\\prime }(v)=1-\\frac{1}{2}\\left([D(v):D(h_1)]+[D(v):D(h_2)]\\right).$ The only way that this number can be a non-negative integer is $D(v)=D(h_1)=D(h_2)$ , hence $v\\in V(\\Gamma _H)$ and $\\operatorname{val}_{\\Gamma _H}(v)=2$ if $H\\subset D(v)$ and $v\\notin V(\\Gamma _H)$ otherwise.",
"Proposition 4.18 Let $\\mathcal {S}$ be an admissible $G$ -stratification of a graph $\\Gamma $ .",
"Then the dilated subgraph $\\Gamma _{dil}\\subset \\Gamma $ is semistable.",
"We recall that $\\Gamma _{dil}$ is the union of the $\\Gamma _H$ for all subgroups $H\\subset G$ except $H=0$ .",
"Let $D$ be the $G$ -dilation datum associated to $\\mathcal {S}$ , and let $v\\in V(\\Gamma _{dil})$ be a vertex, so that $D(v)\\ne 0$ , and assume that $g(v)=0$ .",
"If $v$ is an isolated vertex of $\\Gamma _{dil}$ , then $a_{\\Gamma ,D}(v;K)=0$ for all subgroups $K\\subset D(v)$ such that $K\\ne 0$ .",
"It follows that the sum in the right hand side of (REF ) vanishes, hence $g^{\\prime }(v)=-\\#\\big (D(v)\\big )+1<0$ .",
"Similarly, suppose that $v$ is an extremal vertex of $\\Gamma _{dil}$ , so that there exists a unique edge $h\\in T_v \\Gamma _{dil}$ with $H=D(h)\\ne 0$ .",
"It follows that $a_{\\Gamma ,D}(v;H)=1$ and $a_{\\Gamma ,D}(v;K)=0$ for all $K\\ne 0,H$ , hence $g^{\\prime }(v)= -\\#\\big (D(v)\\big )+1+\\frac{1}{2}\\big (\\#(D(v))-[D(v):H]\\big )=1-\\frac{\\#\\big (D(v)\\big )}{2}\\left(1+\\frac{1}{\\#(H)}\\right)<0,$ since $\\#\\big (D(v)\\big )\\ge \\#(H)\\ge 2$ .",
"Therefore $\\operatorname{val}_{\\Gamma _{dil}}(v)\\ge 2$ and $\\Gamma _{dil}$ is semistable."
],
[
"Unramified $G$ -covers and stability",
"Let $\\Gamma $ be a weighted graph, and let $\\Gamma _{st}$ be its stabilization.",
"We have seen in Def.",
"REF that any unramified cover $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ descends to an unramified cover $\\varphi _{st}:\\Gamma ^{\\prime }_{st}\\rightarrow \\Gamma _{st}$ .",
"It follows that we can restrict unramified $G$ -covers of $\\Gamma $ to its stabilization, and vice versa.",
"Proposition 4.19 Let $\\Gamma $ be a weighted graph.",
"Then there is a natural bijection between the unramified $G$ -covers of $\\Gamma $ and the unramified $G$ -covers of $\\Gamma _{st}$ .",
"Let $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ be an unramified $G$ -cover.",
"The $G$ -action descends to the subgraph $\\Gamma ^{\\prime }_{sst}\\subset \\Gamma ^{\\prime }$ , hence $\\varphi _{sst}:\\Gamma ^{\\prime }_{sst}\\rightarrow \\Gamma _{sst}$ is a $G$ -cover.",
"We note that the supporting arguments for Def.",
"REF show that $\\varphi $ is undilated on $\\Gamma ^{\\prime }\\backslash \\Gamma _{sst}$ ; alternatively, this follows from Prop.",
"REF , since any semistable subgraph of $\\Gamma $ is contained in $\\Gamma _{sst}$ .",
"Therefore, for any vertex $v\\in V(\\Gamma _{sst})$ , any adjacent half-edge $h\\in H(\\Gamma )\\backslash H(\\Gamma _{sst})$ has $\\deg \\varphi $ preimages in $H(\\Gamma ^{\\prime })$ , evenly split among the preimages of $v$ .",
"It follows that $\\varphi _{sst}$ is an unramified $G$ -cover.",
"It is then clear how to descend the $G$ -action to $\\varphi _{st}:\\Gamma ^{\\prime }_{st}\\rightarrow \\Gamma _{st}$ : for any $g\\in G$ and any simple vertex $v^{\\prime }\\in V(\\Gamma ^{\\prime }_{sst})$ that is replaced by an edge or a leg, $g$ maps that edge or leg to the edge or leg that replaces $g(v)$ .",
"Conversely, let $\\Gamma $ be a weighted graph, and let $\\varphi _{st}:\\Gamma ^{\\prime }_{st}\\rightarrow \\Gamma _{st}$ be an unramified $G$ -cover, where $\\Gamma ^{\\prime }_{st}$ is a stable weighted graph.",
"The semistabilization $\\Gamma _{sst}$ is obtained from $\\Gamma _{st}$ by splitting edges and legs at new vertices of genus 0.",
"Performing the same operation on the preimages of these vertices in $\\Gamma ^{\\prime }_{st}$ , we obtain an unramified $G$ -cover $\\varphi _{sst}:\\Gamma ^{\\prime }_{sst}\\rightarrow \\Gamma _{sst}$ .",
"The graph $\\Gamma $ is obtained from $\\Gamma _{sst}$ by attaching trees having no vertices of positive genus.",
"For each such tree $T$ attached at $v\\in V(\\Gamma _{sst})$ , we attach $\\#(G)$ copies of $T$ to $\\Gamma ^{\\prime }_{sst}$ at the fiber $\\varphi ^{-1}(v)$ , and extend the $G$ -action in the obvious way.",
"We obtain an unramified $G$ -cover $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ whose stabilization is $\\varphi _{st}$ ."
],
[
"$G$ -covers of weighted metric graphs and tropical curves",
"In this final section, we reformulate our classification results for weighted metric graphs and tropical curves.",
"There is essentially no new mathematical content obtained by adding metrics to graphs, so this section is essentially a restatement and a summary of the results of the previous sections, and is included for the reader's convenience.",
"First, we introduce $G$ -covers of weighted metric graphs: Definition 4.20 Let $(\\Gamma ,\\ell )$ be a weighted metric graph.",
"A $G$ -cover of $(\\Gamma ,\\ell )$ is a finite harmonic morphism $\\varphi :(\\Gamma ^{\\prime },\\ell ^{\\prime })\\rightarrow (\\Gamma ,\\ell )$ together with an action of $G$ on $(\\Gamma ^{\\prime },\\ell ^{\\prime })$ , such that the following properties are satisfied: The action is invariant with respect to $\\varphi $ .",
"For each $x\\in X(\\Gamma )$ , $G$ acts transitively on the fiber $\\varphi ^{-1}(x)$ .",
"$\\#(G)=\\deg \\varphi $ .",
"We say that a $G$ -cover $\\varphi $ is unramified if it is an unramified harmonic morphism.",
"In other words, a $G$ -cover $\\varphi :(\\Gamma ^{\\prime },\\ell ^{\\prime })\\rightarrow (\\Gamma ,\\ell )$ is a $G$ -cover $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ of the underlying weighted graph $\\Gamma $ that satisfies the dilation condition (REF ).",
"Given $\\varphi $ and $\\ell $ , there is a unique way to choose $\\ell ^{\\prime }$ such that the dilation condition is satisfied (see Rem.",
"REF ).",
"It follows that the classification of $G$ -covers of $(\\Gamma ,\\ell )$ is identical to the classification of $G$ -covers of $\\Gamma $ .",
"Specifically, such a cover is uniquely determined by choosing the dilation subgroups, an element of the corresponding dilated cohomology group, and a genus assignment on $\\Gamma $ which is then lifted to $\\Gamma ^{\\prime }$ .",
"To obtain unramified $G$ -covers, we require the dilation data to be admissible, and pick the genus using Eq.",
"(REF ): Theorem 4.21 Let $(\\Gamma ,\\ell )$ be a weighted metric graph.",
"There is a natural bijection between the set of $G$ -covers of $(\\Gamma ^{\\prime },\\ell ^{\\prime })$ and the set of triples $(D,\\eta ,g^{\\prime })$ , where $D$ is a $G$ -dilation datum on the underlying weighted graph $\\Gamma $ , $\\eta $ is an element of $H^1(\\Gamma ,D)$ , $g^{\\prime }$ is a map from $V(\\Gamma )$ to $\\mathbb {Z}_{\\ge 0}$ .",
"The set of unramified $G$ -covers is obtained by choosing $D$ to be an admissible $G$ -dilation datum, and defining $g^{\\prime }$ by Eq.",
"(REF ).",
"This follows immediately from Thms.",
"REF and REF , and Rem.",
"REF .",
"Finally, we describe $G$ -covers of tropical curves.",
"Definition 4.22 Let $\\scalebox {0.8}[1.3]{\\sqsubset }$ be a tropical curve.",
"A $G$ -cover of $\\scalebox {0.8}[1.3]{\\sqsubset }$ is a finite harmonic morphism $\\tau :\\scalebox {0.8}[1.3]{\\sqsubset }^{\\prime }\\rightarrow \\scalebox {0.8}[1.3]{\\sqsubset }$ together with an action of $G$ on $\\scalebox {0.8}[1.3]{\\sqsubset }^{\\prime }$ such that the following properties are satisfied: The action is invariant with respect to $\\tau $ .",
"For each $x\\in \\scalebox {0.8}[1.3]{\\sqsubset }$ , $G$ acts transitively on the fiber $\\tau ^{-1}(x)$ .",
"$\\#(G)=\\deg \\tau $ .",
"We say that a $G$ -cover $\\tau $ is unramified if it is an unramified harmonic morphism.",
"To describe $G$ -covers of a tropical curve $\\scalebox {0.8}[1.3]{\\sqsubset }$ , we need to define $G$ -dilation data on $\\scalebox {0.8}[1.3]{\\sqsubset }$ .",
"We can define this to be a $G$ -dilation datum on some model of $\\scalebox {0.8}[1.3]{\\sqsubset }$ .",
"It is more convenient to define dilation in terms of the associated stratification, which does not involve choosing a model.",
"The following definition generalizes Defs.",
"REF , REF , and REF to the case of tropical curves.",
"Definition 4.23 Let $\\scalebox {0.8}[1.3]{\\sqsubset }$ be a tropical curve.",
"A $G$ -stratification $\\mathcal {S}=\\lbrace \\scalebox {0.8}[1.3]{\\sqsubset }_H:H\\in S(G)\\rbrace $ on $\\scalebox {0.8}[1.3]{\\sqsubset }$ is a collection of subcurves $\\scalebox {0.8}[1.3]{\\sqsubset }_H\\subset \\scalebox {0.8}[1.3]{\\sqsubset }$ , indexed by the set $S(G)$ of subgroups of $G$ , such that $\\scalebox {0.8}[1.3]{\\sqsubset }_0=\\scalebox {0.8}[1.3]{\\sqsubset }$ , $\\scalebox {0.8}[1.3]{\\sqsubset }_K\\subset \\scalebox {0.8}[1.3]{\\sqsubset }_H$ if $H\\subset K$ , $\\scalebox {0.8}[1.3]{\\sqsubset }_H\\cap \\scalebox {0.8}[1.3]{\\sqsubset }_K=\\scalebox {0.8}[1.3]{\\sqsubset }_{H+K}$ for all $H,K\\in S(G)$ .",
"We allow $\\scalebox {0.8}[1.3]{\\sqsubset }_H$ to be empty or disconnected for $H\\ne 0$ .",
"A $G$ -stratification $\\mathcal {S}$ partitions $\\scalebox {0.8}[1.3]{\\sqsubset }$ into disjoint subsets $\\scalebox {0.8}[1.3]{\\sqsubset }=\\coprod _{H\\in S(G)}\\scalebox {0.8}[1.3]{\\sqsubset }_H\\backslash \\scalebox {0.8}[1.3]{\\sqsubset }^0_H \\quad \\textrm { and }\\quad \\scalebox {0.8}[1.3]{\\sqsubset }_{H}^0=\\bigcup _{H\\subsetneq K} \\scalebox {0.8}[1.3]{\\sqsubset }_K.$ For $x\\in \\scalebox {0.8}[1.3]{\\sqsubset }$ we define the dilation subgroup $D(x)$ to be the unique subgroup $H\\subset G$ such that $x\\in \\scalebox {0.8}[1.3]{\\sqsubset }_H\\backslash \\scalebox {0.8}[1.3]{\\sqsubset }^0_H$ .",
"We define the index function $a_{\\mathcal {S}}:\\scalebox {0.8}[1.3]{\\sqsubset }\\times S(G)\\rightarrow \\mathbb {Z}_{\\ge 0}$ of $\\mathcal {S}$ by setting $a_{\\mathcal {S}}(x,H)$ to be the number of connected components of the intersection of $\\scalebox {0.8}[1.3]{\\sqsubset }_H\\backslash \\scalebox {0.8}[1.3]{\\sqsubset }_H^0$ with a sufficiently small punctured neighborhood of $x$ .",
"We say that $\\mathcal {S}$ is admissible if for every $x\\in \\scalebox {0.8}[1.3]{\\sqsubset }$ the number $g^{\\prime }(x)=\\#(D(x))\\big [g(x)-1\\big ]+1+\\frac{1}{2}\\sum _{K\\in S(D(x))}a_{\\mathcal {S}}(x;K)\\big (\\#(D(x))-[D(x):K]\\big )$ is a non-negative integer.",
"Finally, we define the dual stratification $\\mathcal {S}^*$ of a stratification $\\mathcal {S}$ of $\\scalebox {0.8}[1.3]{\\sqsubset }$ as follows.",
"Choose a model $\\Gamma $ of $\\scalebox {0.8}[1.3]{\\sqsubset }$ minimal with respect to the property that each element $\\scalebox {0.8}[1.3]{\\sqsubset }_H$ of $\\mathcal {S}$ corresponds to a subgraph $\\Gamma _H$ of $\\Gamma $ .",
"Then the $\\Gamma _H$ form a $G$ -stratification of the weighted metric graph $\\Gamma $ , and we let $\\mathcal {S}^*$ be the dual of this stratification.",
"We note that choosing a larger model $\\Gamma ^{\\prime }$ will result in a larger dual stratification, which will, however, retract to $\\mathcal {S}^*$ .",
"Similarly, we can define the dilated cohomology groups of a tropical curve with a $G$ -stratification: Definition 4.24 Let $\\mathcal {S}$ be a stratification of a tropical curve $\\scalebox {0.8}[1.3]{\\sqsubset }$ .",
"Pick a model $(\\Gamma ,\\ell )$ for $\\scalebox {0.8}[1.3]{\\sqsubset }$ such that each $\\scalebox {0.8}[1.3]{\\sqsubset }_H$ corresponds to a subgraph $\\Gamma _H$ of $\\Gamma $ , then $\\mathcal {S}$ is a $G$ -stratification of $\\Gamma $ and induces a $G$ -dilation datum $D$ on $\\Gamma $ .",
"We define the dilated cohomology group $H^1(\\scalebox {0.8}[1.3]{\\sqsubset },\\mathcal {S})$ as the cohomology group $H^1(\\Gamma ,D)$ ; it is clear that this group does not depend on the choice of model.",
"We can now state our main classification result for $G$ -covers of tropical curves, which is simply a restatement of Thm.",
"REF using the equivalent description of dilation by means of a stratification: Theorem 4.25 Let $\\scalebox {0.8}[1.3]{\\sqsubset }$ be a tropical curve.",
"There is a natural bijection between the set of $G$ -covers of $\\scalebox {0.8}[1.3]{\\sqsubset }$ and the set of triples $(\\mathcal {S},\\eta ,g^{\\prime })$ , where $\\mathcal {S}$ is a $G$ -stratification of $\\scalebox {0.8}[1.3]{\\sqsubset }$ , $\\eta $ is an element of $H^1(\\scalebox {0.8}[1.3]{\\sqsubset },\\mathcal {S})$ , $g^{\\prime }$ is a function from $\\scalebox {0.8}[1.3]{\\sqsubset }$ to $\\mathbb {Z}_{\\ge 0}$ .",
"The set of unramified $G$ -covers of $\\scalebox {0.8}[1.3]{\\sqsubset }$ is obtained by requiring $\\mathcal {S}$ to be an admissible $G$ -stratification, and defining $g^{\\prime }$ by (REF ).",
"We also restate Prop.",
"REF for tropical curves.",
"Proposition 4.26 Let $\\scalebox {0.8}[1.3]{\\sqsubset }$ be a tropical curve.",
"Then there is a natural bijection between the unramified $G$ -covers of $\\scalebox {0.8}[1.3]{\\sqsubset }$ and the unramified $G$ -covers of $\\scalebox {0.8}[1.3]{\\sqsubset }^{st}$ .",
"Remark 4.27 Any tropical curve $\\scalebox {0.8}[1.3]{\\sqsubset }$ has infinitely many $G$ -covers for any nontrivial group $G$ , since we can choose a dilation stratification with arbitrarily many connected components.",
"However, the number of unramified $G$ -covers of $\\scalebox {0.8}[1.3]{\\sqsubset }$ is finite.",
"Indeed, Prop.",
"REF shows that if $\\mathcal {S}$ is an admissible stratification of $\\scalebox {0.8}[1.3]{\\sqsubset }$ , then no $\\scalebox {0.8}[1.3]{\\sqsubset }_H$ can contain any simple point $x\\in \\scalebox {0.8}[1.3]{\\sqsubset }$ as an unstable extremal point.",
"Since any tropical curve has only finitely many non-simple points, it follows that the number of admissible stratifications of a tropical curve is finite, and hence so is the number of unramified $G$ -covers."
],
[
"Cyclic covers of prime order",
"We now classify the unramified $G$ -covers of a tropical curve $\\scalebox {0.8}[1.3]{\\sqsubset }$ in the case when $G=\\mathbb {Z}/p\\mathbb {Z}$ , where $p$ is prime.",
"These covers were studied in [26] and [8] for $p=2$ , and in [9] for arbitrary $p$ in the case when $\\scalebox {0.8}[1.3]{\\sqsubset }$ is a tree.",
"Let $\\scalebox {0.8}[1.3]{\\sqsubset }$ be a tropical curve, let $p$ be a prime number, and let $G=\\mathbb {Z}/p\\mathbb {Z}$ .",
"A $G$ -stratification $\\mathcal {S}=\\lbrace \\scalebox {0.8}[1.3]{\\sqsubset }_0,\\scalebox {0.8}[1.3]{\\sqsubset }_G\\rbrace $ of $\\scalebox {0.8}[1.3]{\\sqsubset }$ has a single nontrivial element $\\scalebox {0.8}[1.3]{\\sqsubset }_G=\\scalebox {0.8}[1.3]{\\sqsubset }_{dil}$ , the dilated subcurve.",
"Condition (REF ) is trivially satisfied at any non-dilated point.",
"If $x\\in \\scalebox {0.8}[1.3]{\\sqsubset }_G$ , then $D(x)=\\mathbb {Z}/p\\mathbb {Z}$ and $a_{\\mathcal {S}}(x;\\mathbb {Z}/p\\mathbb {Z})=\\operatorname{val}_{\\scalebox {0.8}[1.3]{\\sqsubset }_G}(x)$ , and condition (REF ) is $g^{\\prime }(x)=\\big [g(x)-1\\big ]p+1+\\frac{p-1}{2}\\operatorname{val}_{\\scalebox {0.8}[1.3]{\\sqsubset }_G}(x).$ We see that $g^{\\prime }(x)$ is non-negative if $g(x)>0$ , or if $g(x)=0$ and $\\operatorname{val}_{\\scalebox {0.8}[1.3]{\\sqsubset }_G}(x)\\ge 2$ .",
"Similarly, $g^{\\prime }(x)$ is an integer if $p\\ge 3$ , or if $p=2$ and $\\operatorname{val}_{\\scalebox {0.8}[1.3]{\\sqsubset }_G}(x)$ is even.",
"We therefore have the following result: For $p\\ge 3$ , a $\\mathbb {Z}/p\\mathbb {Z}$ -stratification $\\scalebox {0.8}[1.3]{\\sqsubset }$ is admissible if and only if the dilated subcurve $\\scalebox {0.8}[1.3]{\\sqsubset }_G$ is semistable.",
"For $p=2$ , a $\\mathbb {Z}/p\\mathbb {Z}$ -stratification $\\scalebox {0.8}[1.3]{\\sqsubset }$ is admissible if and only if the dilated subcurve $\\scalebox {0.8}[1.3]{\\sqsubset }_G$ is a semistable cycle.",
"This was observed in [26] (see Corollary 5.5).",
"If $\\mathcal {S}$ is an admissible $\\mathbb {Z}/p\\mathbb {Z}$ -stratification of $\\scalebox {0.8}[1.3]{\\sqsubset }$ , Ex.",
"REF and Thm.",
"REF shows that the set of unramified $G$ -covers of $\\scalebox {0.8}[1.3]{\\sqsubset }$ having dilation stratification $\\mathcal {S}$ is equal to $H^1(\\scalebox {0.8}[1.3]{\\sqsubset }^0,\\mathbb {Z}/p\\mathbb {Z})$ , where $\\scalebox {0.8}[1.3]{\\sqsubset }^0$ is the nontrivial element of the dual stratification $\\mathcal {S}^*(D)$ .",
"Specifically, $\\scalebox {0.8}[1.3]{\\sqsubset }^0$ is obtained from $\\scalebox {0.8}[1.3]{\\sqsubset }$ by removing $\\scalebox {0.8}[1.3]{\\sqsubset }_G$ , and then removing any edges or legs that are missing an endpoint.",
"In other words, an unramified $\\mathbb {Z}/p\\mathbb {Z}$ -cover of $\\scalebox {0.8}[1.3]{\\sqsubset }$ is uniquely specified by choosing a (possibly empty) semistable subcurve $\\scalebox {0.8}[1.3]{\\sqsubset }_G\\subset \\scalebox {0.8}[1.3]{\\sqsubset }$ , which is required to be a cycle when $p=2$ , and an element of $H^1(\\scalebox {0.8}[1.3]{\\sqsubset }^0,\\mathbb {Z}/p\\mathbb {Z})$ .",
"As an example, we count the number of unramified $\\mathbb {Z}/p\\mathbb {Z}$ -covers of the following genus two tropical curve $\\scalebox {0.8}[1.3]{\\sqsubset }$ with one leg (the edge lengths are arbitrary and irrelevant): [ultra thick] (0.5,0) circle(.5); [ultra thick] (0.5,0.5) – (0.5,1.0); [ultra thick] (0.5,1.0) – (0.5,1.5); [ultra thick] (0.5,1.0) – (1.0,1.0); [ultra thick] (0.5,2.0) circle(.5); [fill] (0.5,0.5)circle(.10); [fill] (0.5,1.0)circle(.10); [fill] (0.5,1.5)circle(.10); This curve has eight admissible $\\mathbb {Z}/p\\mathbb {Z}$ -stratifications, listed below, with the last one being admissible only when $p$ is odd.",
"We draw the semistable dilated subgraph $\\scalebox {0.8}[1.3]{\\sqsubset }_G$ in blue, and the corresponding element $\\scalebox {0.8}[1.3]{\\sqsubset }^0$ of the dual stratification in red.",
"Below each stratification we list the number $p^{b_1\\left(\\scalebox {0.8}[1.3]{\\sqsubset }^0\\right)}$ of $\\mathbb {Z}/p\\mathbb {Z}$ -covers with the given stratification.",
"Table: NO_CAPTIONHence there are a total of $p^2+4p+3$ covers when $p$ is odd, and 14 covers when $p=2$ ."
],
[
"Tropicalizing the moduli space of admissible $G$ -covers",
"In this section we explain how unramified tropical $G$ -covers naturally arise as tropicalizations of algebraic $G$ -covers from a moduli-theoretic perspective, expanding on [4] and [18] (recall that tropical unramified covers are called tropical admissible covers in [18]).",
"Throughout this section we assume that the genus $g\\ge 2$ and we work over an algebraically closed field $k$ endowed with the trivial absolute value.",
"In this section, we do not need to assume that $G$ is abelian."
],
[
"Compactifying the moduli space of $G$ -covers",
"Let $G$ be a finite group and let $X\\rightarrow S$ be a family of smooth projective curves of genus $g$ .",
"A $G$ -cover of $X$ is a finite unramified Galois morphism $f\\colon X^{\\prime }\\rightarrow X$ together with an isomorphism $\\operatorname{Aut}(X^{\\prime }/X)\\simeq G$ .",
"Denote by $\\mathcal {H}_{g,G}$ the moduli space of connected $G$ -covers of smooth curves of genus $g$ (see e.g.",
"[34] for a construction).",
"There is a good notion of a limit object as $X$ degenerates to a stable curve, as introduced in [5].",
"The definition below generalizes this construction, by allowing a fixed ramification profile along marked points.",
"Definition 5.1 Let $G$ be a finite group and let $X\\rightarrow S$ be a family of stable curves of genus $g$ with $n$ marked disjoint sections $s_1,\\ldots , s_n$ .",
"Let $\\mu =(r_1,\\ldots , r_n)$ be a $n$ -tuple of natural numbers that divide $\\#(G)$ , and denote $k_i=\\#(G)/r_i$ for $i=1,\\ldots , n$ .",
"An admissible $G$ -cover of $X$ consists of a finite morphism $f\\colon X^{\\prime }\\rightarrow X$ from a family of stable curves $X^{\\prime }\\rightarrow S$ that is Galois and unramified away from the sections of $X$ , an action of $G$ on $X^{\\prime }$ , and disjoint sections $s^{\\prime }_{ij}$ of $X^{\\prime }$ over $S$ for $i=1,\\ldots ,n$ and $j=1,\\ldots ,k_i$ , subject to the following conditions: The map $f:X^{\\prime }\\rightarrow X$ is a principal $G$ -bundle away from the nodes and sections of $X$ .",
"The preimage of the set of nodes in $X$ is precisely the set of nodes of $X^{\\prime }$ .",
"The preimage of a section $s_i$ is precisely given by the sections $s^{\\prime }_{i1},\\ldots , s^{\\prime }_{ik_i}$ .",
"Let $p$ be a node in $X$ and $p^{\\prime }$ a node of $X^{\\prime }$ above $p$ .",
"Then étale-locally $p^{\\prime }$ is given by $x^{\\prime }y^{\\prime }=t$ for $t\\in \\mathcal {O}_S$ and $p$ is étale-locally given by $xy=t^r$ for some integer $r\\ge 1$ with $x^{\\prime }=x^r$ and $y^{\\prime }=y^r$ , and the stabilizer of $G$ at $p^{\\prime }$ is cyclic of order $r$ and operates via $(x^{\\prime },y^{\\prime })\\longmapsto (\\zeta x^{\\prime },\\zeta ^{-1} y^{\\prime })$ for an $r$ -th root of unity $\\zeta \\in \\mu _r$ .",
"Étale-locally near the sections $s_i$ and $s^{\\prime }_{ij}$ respectively, the morphism $f$ is given by $\\mathcal {O}_S[t_i]\\rightarrow \\mathcal {O}_S[t_{ij}^{\\prime }]$ with $(t_{ij}^{\\prime })^{r_i}=t_i$ , and the stabilizer of $G$ along $s_{ij}$ is cyclic or order $r_i$ and operates via $t^{\\prime }_{ij}\\mapsto \\zeta t$ , for an $r_i$ -th root of unity $\\zeta \\in \\mu _{r_i}$ .",
"We emphasize that the $G$ -action is part of the data, in particular, an isomorphism between two admissible $G$ -covers has to be a $G$ -equivariant isomorphism.",
"As explained in [5], the moduli space $\\overline{\\mathcal {H}}_{g,G}(\\mu )$ of $G$ -admissible covers of stable $n$ -marked curves of genus $g$ is a smooth and proper Deligne-Mumford stack that contains the locus $\\mathcal {H}_{g,G}(\\mu )$ of $G$ -covers of smooth curves of ramification type $\\mu $ as an open substack.",
"The complement of $\\mathcal {H}_{g,G}(\\mu )$ is a normal crossing divisor.",
"Remark 5.2 Although closely related, the moduli space $\\overline{\\mathcal {H}}_{g, G}(\\mu )$ is not quite the same as the one constructed in [5].",
"The quotient $\\big [\\overline{\\mathcal {H}}_{g, G}(\\mu )/S_{k_1}\\times \\ldots \\times S_{k_{n}}\\big ]$ which forgets about the order of the marked sections on $s^{\\prime }_{ij}$ of $X^{\\prime }$ over $S$ for $i=1,\\ldots ,n$ and $j=1,\\ldots ,k_i$ , is equivalent to a connected component of the moduli space of twisted stable maps to $\\mathbf {B}G$ in the sense of [6], [5], indexed by ramification profile and decomposition into connected components.",
"Our variant of this moduli space $\\overline{\\mathcal {H}}_{g, G}(\\mu )$ , with ordered sections on $X^{\\prime }$ , has also appeared in [38] and in [25] (the latter permitting admissible covers with possibly disconnected domains).",
"An object in this stack is technically not a admissible $G$ -cover $X^{\\prime }\\rightarrow X$ but rather a $G$ -cover $X^{\\prime }\\rightarrow \\mathcal {X}$ of a twisted stable curve $\\mathcal {X}$ .",
"A twisted stable curve $\\mathcal {X}\\rightarrow S$ is a Deligne-Mumford stack $\\mathcal {X}$ with sections $s_1,\\ldots , s_n\\colon S\\rightarrow \\mathcal {X}$ whose coarse moduli space $X\\rightarrow S$ is a family of stable curves over $S$ with $n$ marked sections (also denoted by $s_1,\\ldots , s_n$ ) such that The smooth locus of $\\mathcal {X}$ is representable by a scheme, The singularities are étale-locally given by $\\big [\\lbrace x^{\\prime }y^{\\prime }=t\\rbrace /\\mu _r\\big ]$ for $t\\in \\mathcal {O}_S$ , where $\\zeta \\in \\mu _r$ acts by $\\zeta \\cdot (x^{\\prime },y^{\\prime })=(\\zeta x^{\\prime },\\zeta ^{-1}y^{\\prime })$ .",
"In this case the singularity in $X^{\\prime }$ is locally given by $xy=t^{r}$ .",
"The stack $\\mathcal {X}$ is a root stack $\\big [\\@root r_i \\of {s_i/X}\\big ]$ along the section $s_i$ for all $i=1,\\ldots n$ .",
"Both notions are naturally equivalent: given a $G$ -admissible cover $X^{\\prime }\\rightarrow X$ the associated twisted $G$ -cover is given by $X^{\\prime }\\rightarrow [X^{\\prime }/G]$ .",
"Conversely, given a twisted $G$ -cover $X^{\\prime }\\rightarrow \\mathcal {X}$ in the corresponding connected component, the composition $X^{\\prime }\\rightarrow \\mathcal {X}\\rightarrow X$ with the morphism to the coarse moduli space $X$ is a $G$ -admissible cover.",
"We refer the interested reader to [12] for an alternative approach to this construction."
],
[
"The moduli space of unramified tropical $G$ -covers",
"We now construct a moduli space $H_{g,G}^{trop}(\\mu )$ of unramified $G$ -covers of stable tropical curves of genus $g$ with $n$ marked points and ramification profile $\\mu =(r_1,\\ldots ,r_n)$ , where each $r_i$ divides $\\#(G)$ .",
"Denote $k_i=\\#(G)/r_i$ , as well as $k=k_1+\\cdots +k_n$ , and assume that $n\\cdot \\#(G)-k$ is even.",
"A point $\\big [\\varphi ,l,l^{\\prime }\\big ]$ of $H_{g,G}^{trop}(\\mu )$ consists of the following data: A $G$ -equivariant isomorphism class of an unramified $G$ -cover $\\varphi \\colon \\scalebox {0.8}[1.3]{\\sqsubset }^{\\prime }\\rightarrow \\scalebox {0.8}[1.3]{\\sqsubset }$ of a stable tropical curve $\\scalebox {0.8}[1.3]{\\sqsubset }$ of genus $g$ with $n$ legs, by a stable tropical curve $\\scalebox {0.8}[1.3]{\\sqsubset }^{\\prime }$ of genus $g^{\\prime }=(g-1)\\cdot \\#(G)+1+(n\\cdot \\#(G)-k)/2$ with $k$ legs.",
"A marking $l:\\lbrace 1,\\ldots ,n\\rbrace \\simeq L(\\scalebox {0.8}[1.3]{\\sqsubset })$ of the legs of $\\scalebox {0.8}[1.3]{\\sqsubset }$ .",
"A marking $l^{\\prime }\\colon \\big \\lbrace (1,\\ldots , k_1),\\ldots ,(1,\\ldots ,k_n)\\big \\rbrace \\simeq L(\\scalebox {0.8}[1.3]{\\sqsubset }^{\\prime })$ of the legs of $\\scalebox {0.8}[1.3]{\\sqsubset }^{\\prime }$ such that $\\varphi (l^{\\prime }_{ij})=l_i$ , where we denote $l_i=l(i)$ and $l^{\\prime }_{ij}=l(i,j)$ .",
"Proposition 5.3 The moduli space $H_{g,G}^{trop}(\\mu )$ naturally carries the structure of a generalized cone complex.",
"We need to show that $H_{g,G}^{trop}(\\mu )$ is naturally the colimit of a diagram of rational polyhedral cones connected by (not necessarily proper) face morphisms.",
"We first construct an index category $J_{g,G}(\\mu )$ as follows: The objects are tuples $\\big (\\varphi \\colon \\Gamma ^{\\prime }\\rightarrow \\Gamma ,l,l^{\\prime }\\big )$ , where $\\Gamma ^{\\prime }$ and $\\Gamma $ are stable weighted graphs of genera $g^{\\prime }$ and $g$ having respectively $k$ and $n$ legs, $\\varphi $ is an unramified $G$ -cover, and $l^{\\prime }$ and $l$ are markings of the legs of $\\Gamma ^{\\prime }$ and $\\Gamma $ , respectively, such that $\\varphi (l^{\\prime }_{ij})=l_i$ .",
"The morphisms are generated by the automorphisms of $\\Gamma ^{\\prime }\\rightarrow \\Gamma $ that preserve the markings on both $\\Gamma $ and $\\Gamma ^{\\prime }$ , and weighted edge contractions (see Def.",
"REF ) of the target graph $\\Gamma $ .",
"We recall that a weighted edge contraction of the target graph $\\Gamma $ induces a weighted edge contraction of the source graph $\\Gamma ^{\\prime }$ along the preimages of the contracted edges.",
"Moreover, the $G$ -action on $\\Gamma ^{\\prime }$ induces a $G$ -action on its weighted edge contraction, which is an unramified $G$ -cover by Prop.",
"REF .",
"We then consider a functor $\\Sigma _{g,G}(\\mu )\\colon J_{g,G}(\\mu )\\rightarrow \\mathbf {RPC}_{face}$ to the category $\\mathbf {RPC}_{face}$ of rational polyhedral cones with (not necessarily proper) face morphisms defined as follows: An object $\\big (\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma ,l,l^{\\prime }\\big )$ is sent to the rational polyhedral cone $\\sigma _{\\varphi }=\\mathbb {R}_{\\ge 0}^{E(\\Gamma )}$ .",
"An automorphism of $\\big (\\Gamma ^{\\prime }\\rightarrow \\Gamma ,l,l^{\\prime }\\big )$ induces an automorphism of $\\sigma _{\\varphi }$ that permutes the entries according to the induced permutation of the edges of $\\Gamma $ ; for a set of edges $S\\subset E(\\Gamma )$ , a weighted edge contraction $\\varphi _S:\\Gamma ^{\\prime }/\\varphi ^{-1}(S)\\rightarrow \\Gamma /S$ of $\\varphi :\\Gamma ^{\\prime }\\rightarrow \\Gamma $ induces a morphism $\\sigma _{\\varphi _S}\\hookrightarrow \\sigma _{\\varphi }$ that sends $\\sigma _{\\varphi _S}$ to the face of $\\sigma _{\\varphi }$ given by setting all entries of the contracted edges equal to zero.",
"The natural maps $\\sigma _{\\varphi }\\rightarrow H_{g,G}^{trop}(\\mu )$ are given by associating to a point $(a_e)_{e\\in E(\\Gamma )}\\in \\mathbb {R}_{\\ge 0}^{E(\\Gamma )}$ an unramified $G$ -cover $\\big [\\scalebox {0.8}[1.3]{\\sqsubset }^{\\prime }\\rightarrow \\scalebox {0.8}[1.3]{\\sqsubset }\\big ]$ defined as follows: In the special case that $a_e\\ne 0$ for all $e\\in E(\\Gamma )$ , the tropical curve $\\scalebox {0.8}[1.3]{\\sqsubset }$ is given by the graph $\\Gamma $ with the metric $\\ell (e)=a_e$ .",
"In general, the tropical curve $\\scalebox {0.8}[1.3]{\\sqsubset }$ is given by contracting those edges $e\\in E(\\Gamma )$ for which $a_e=0$ and then by endowing the contracted weighted graph with the induced edge length given by the $a_e\\ne 0$ .",
"The tropical curve $\\scalebox {0.8}[1.3]{\\sqsubset }^{\\prime }$ is defined accordingly: we first contract all edges that map to an edge $e$ with $a_e=0$ , and then endow $\\scalebox {0.8}[1.3]{\\sqsubset }^{\\prime }$ with the edge length $\\ell ^{\\prime }(e^{\\prime })= \\ell (\\varphi (e^{\\prime }))/d_\\varphi (e)$ so that the induced map $\\scalebox {0.8}[1.3]{\\sqsubset }^{\\prime }\\rightarrow \\scalebox {0.8}[1.3]{\\sqsubset }$ is an unramified $G$ -cover of tropical curves.",
"The maps $\\sigma _{\\varphi }\\rightarrow H_{g,G}^{trop}(\\mu )$ naturally commute with the morphisms induced by $J_{g,G}(\\mu )$ , and therefore descend to a map $\\varinjlim _{(\\varphi ,l,l^{\\prime })\\in J_{g,G}(\\mu )} \\sigma _{\\varphi } \\simeq H_{g,G}^{trop}(\\mu )$ that is easily checked to be a bijection.",
"This realizes $H_{g,G}^{trop}(\\mu )$ as a colimit of a diagram of (not necessarily proper) face morphisms and therefore endows it with the structure of a generalized cone complex.",
"There are natural source and target morphisms $\\operatorname{src}_{g,G}^{trop}(\\mu )\\colon H_{g,G}^{trop}(\\mu )\\longrightarrow M_{g^{\\prime },k}^{trop}\\\\\\qquad \\textrm { and } \\qquad \\operatorname{tar}_{g,G}^{trop}(\\mu )\\colon H_{g,G}^{trop}(\\mu )\\longrightarrow M_{g,n}^{trop}\\\\$ that are given by the associations $\\big [\\scalebox {0.8}[1.3]{\\sqsubset }^{\\prime }\\rightarrow \\scalebox {0.8}[1.3]{\\sqsubset },l,l^{\\prime }\\big ]\\longmapsto \\big [\\scalebox {0.8}[1.3]{\\sqsubset }^{\\prime },l^{\\prime }\\big ] \\qquad \\textrm { and } \\qquad \\big [\\scalebox {0.8}[1.3]{\\sqsubset }^{\\prime }\\rightarrow \\scalebox {0.8}[1.3]{\\sqsubset },l,l^{\\prime }\\big ]\\longmapsto \\big [\\scalebox {0.8}[1.3]{\\sqsubset },l\\big ]$ respectively.",
"By Rem.",
"REF , the map $\\operatorname{tar}_{g,G}^{trop}(\\mu )$ has finite fibers.",
"Remark 5.4 The functor $J_{g,G}(\\mu )\\rightarrow \\mathbf {RPC}_{face}$ in the proof of Proposition REF defines a category fibered in groupoids over $\\mathbf {RPC}_{face}$ , i.e.",
"a combinatorial cone stack in the sense of [14].",
"So we may think of $H_{g,G}^{trop}(\\mu )$ as a \"coarse moduli space\" of a cone stack $\\mathcal {H}_{g,G}^{trop}(\\mu )$ , a geometric stack over the category of rational polyhedral cones (see [14] for details), that parametrizes families of unramified tropical $G$ -covers over rational polyhedral cones."
],
[
"A modular perspective on tropicalization",
"Denote by $\\mathcal {H}_{g,G}^{an}(\\mu )$ the Berkovich analytic spaceWe implicitly work with the underlying topological space of the Berkovich analytic stack $\\mathcal {H}_{g,G}^{an}(\\mu )$ , as introduced in [40].",
"associated to $\\mathcal {H}_{g,G}(\\mu )$ .",
"We define a natural tropicalization map $\\begin{split}\\operatorname{trop}_{g,G}(\\mu )\\colon \\mathcal {H}_{g,G}^{an}(\\mu )&\\longrightarrow H_{g,G}^{trop}(\\mu )\\\\[X^{\\prime }\\rightarrow X,s_i,s^{\\prime }_{ij}]& \\longmapsto \\big [\\scalebox {0.8}[1.3]{\\sqsubset }_{X^{\\prime }}\\rightarrow \\scalebox {0.8}[1.3]{\\sqsubset }_X,l,l^{\\prime }\\big ]\\end{split}$ that associates to an admissible $G$ -cover $X^{\\prime }\\rightarrow X$ of smooth curves over a non-Archimedean extension $K$ of $k$ an unramified tropical $G$ -cover $\\scalebox {0.8}[1.3]{\\sqsubset }_{X^{\\prime }}\\rightarrow \\scalebox {0.8}[1.3]{\\sqsubset }_X$ of the dual tropical curve $\\scalebox {0.8}[1.3]{\\sqsubset }_X$ of $X$ that is defined in the following way.",
"Let $X$ be a smooth projective curve of genus $g$ over a non-Archimedean extension $K$ of $k$ with $n$ marked sections $s_1,\\ldots , s_n$ over $K$ .",
"Let $(X^{\\prime }\\rightarrow X,s^{\\prime }_{ij})$ be a $G$ -cover of $X$ , where $i=1,\\ldots ,n$ and $j=1,\\ldots ,k_i$ .",
"By the valuative criterion for properness, applied to the stack $\\overline{\\mathcal {H}}_{g,G}(\\mu )$ , there is a finite extension $L$ of $K$ such that $X^{\\prime }_L\\rightarrow X_L$ extends to a family of admissible $G$ -covers $f:\\mathcal {X}^{\\prime }\\rightarrow \\mathcal {X}$ defined over the valuation ring $R$ of $L$ (with marked sections also denoted by $s_i$ and $s^{\\prime }_{ij}$ ).",
"The dual tropical curve $(\\scalebox {0.8}[1.3]{\\sqsubset }_X,l)$ of $\\mathcal {X}$ (and similarly $(\\scalebox {0.8}[1.3]{\\sqsubset }_{X^{\\prime }},l^{\\prime })$ of $\\mathcal {X}^{\\prime }$ ) is given by the following data: The dual graph $\\Gamma _{\\mathcal {X}_0}$ of the special fiber $\\mathcal {X}_0$ of $\\mathcal {X}$ : the components of $\\mathcal {X}_0$ correspond to vertices, nodes correspond to edges, and the sections correspond to legs.",
"A vertex weight $V(\\Gamma _{\\mathcal {X}_0})\\rightarrow \\mathbb {Z}_{\\ge 0}$ that associates to a vertex $v$ the genus of the normalization of the corresponding component of $\\mathcal {X}_0$ .",
"A marking $l\\colon \\lbrace 1,\\ldots n\\rbrace \\simeq L(\\Gamma _{\\mathcal {X}_0})$ of the legs of $\\Gamma _{\\mathcal {X}_0}$ according to the full order of $s_1,\\ldots , s_n$ .",
"An edge length $\\ell \\colon E(\\Gamma _{\\mathcal {X}_0})\\rightarrow \\mathbb {R}_{>0}$ that associates to an edge $e$ the positive real number $r\\cdot \\operatorname{val}(t)$ , where the corresponding node is étale-locally given by an equation $xy=t^r$ for $t\\in R$ .",
"The map $f:\\mathcal {X}^{\\prime }\\rightarrow \\mathcal {X}$ induces a map $\\varphi :\\Gamma _{\\mathcal {X}^{\\prime }_0}\\rightarrow \\Gamma _{\\mathcal {X}_0}$ : Every component $X^{\\prime }_{v^{\\prime }}$ of $\\mathcal {X}^{\\prime }_0$ is mapped to exactly one component $X_{v}$ of $\\mathcal {X}_0$ .",
"Every node $p_{e^{\\prime }}$ of $\\mathcal {X}^{\\prime }_0$ , over a node $p_e$ of $\\mathcal {X}_0$ given by $xy=t^r$ for $t\\in R$ on the base, has a local equation $x^{\\prime }y^{\\prime }=t$ which determines the dilation factor $r=d_\\varphi (e^{\\prime })$ .",
"Étale-locally around the sections $s_i$ and $s^{\\prime }_{ij}$ respectively, the morphism $f$ is given by $\\mathcal {O}_S[t_i]\\rightarrow \\mathcal {O}[t^{\\prime }_{ij}]$ with $(t^{\\prime }_{ij})^{r_i}=t_i$ , so the dilation factor $d_\\varphi (l^{\\prime }_{ij})$ is given by $r_i$ .",
"The map $f:\\Gamma _{\\mathcal {X}^{\\prime }_0}\\rightarrow \\Gamma _{\\mathcal {X}_0}$ is harmonic by [1] (identifying both $\\scalebox {0.8}[1.3]{\\sqsubset }_{X^{\\prime }}$ and $\\scalebox {0.8}[1.3]{\\sqsubset }_X$ with the non-Archimedean skeletons of $(X^{\\prime })^{an}$ and $X^{an}$ respectively).",
"Applying the Riemann–Hurwitz formula to $X_{v^{\\prime }}\\rightarrow X_v$ shows that it is unramified.",
"The operation of $G$ on $\\mathcal {X}^{\\prime }_0$ induces an operation of $G$ on $\\Gamma _{\\mathcal {X}^{\\prime }_0}$ for which the map $\\scalebox {0.8}[1.3]{\\sqsubset }_{X^{\\prime }}\\rightarrow \\scalebox {0.8}[1.3]{\\sqsubset }_X$ is $G$ -invariant.",
"The stabilizer at every edge $e^{\\prime }_i$ and of every leg $l^{\\prime }_{ij}$ is a cyclic group of order $r_i$ and $r_{ij}$ respectively by Definition REF (iii) and (iv).",
"Since $\\mathcal {X}^{\\prime }_0\\rightarrow \\mathcal {X}_0$ is a principal $G$ -bundle away from the nodes, the operation of $G$ on the fiber over each point in $\\scalebox {0.8}[1.3]{\\sqsubset }_{X}$ is transitive and so $\\scalebox {0.8}[1.3]{\\sqsubset }_{X^{\\prime }}\\rightarrow \\scalebox {0.8}[1.3]{\\sqsubset }_X$ is a $G$ -cover.",
"Remark 5.5 In the language of twisted stable curves, the stabilizers of the $G$ -operation on the nodes and legs of $\\mathcal {X}_0^{\\prime }$ give rise to the dilation datum on the dual graph.",
"So one may think of a dilation datum as a stack-theoretic enhancement of a tropical curve.",
"Since the boundary of $\\overline{\\mathcal {H}}_{g,G}(\\mu )$ has normal crossings, the open immersion $\\mathcal {H}_{g,G}(\\mu )\\hookrightarrow \\overline{\\mathcal {H}}_{g,G}(\\mu )$ is a toroidal embedding in the sense of [27].",
"Therefore, as explained in [39], [4], there is a natural strong deformation retraction $\\rho _{g,G}\\colon \\mathcal {H}_{g,G}^{an}(\\mu )\\rightarrow \\mathcal {H}_{g,G}^{an}(\\mu )$ onto a closed subset of $\\mathcal {H}_{g,G}^{an}(\\mu )$ that carries the structure of a generalized cone complex, the non-Archimedean skeleton $\\Sigma _{g,G}(\\mu )$ of $\\mathcal {H}_{g,G}^{an}(\\mu )$ .",
"Expanding on [18], we have: Theorem 5.6 The tropicalization map $\\operatorname{trop}_{g,G}(\\mu )\\colon \\mathcal {H}_{g,G}^{an}(\\mu )\\longrightarrow H_{g,G}^{trop}(\\mu )$ factors through the retraction to the non-Archimedean skeleton $\\Sigma _{g,G}(\\mu )$ of $\\mathcal {H}_{g,G}^{an}(\\mu )$ , so that the restriction $\\operatorname{trop}_{g,G}(\\mu )\\colon \\Sigma _{g,G}(\\mu )\\longrightarrow H^{trop}_{g,G}(\\mu )$ to the skeleton is a finite strict morphism of generalized cone complexes.",
"Moreover, the diagram $\\begin{tikzcd}\\mathcal {H}_{g,G}^{an}(\\mu )[rr,\"\\operatorname{src}_{g,G}^{an}(\\mu )\"] [dd,\"\\operatorname{tar}_{g,G}^{an}(\\mu )\"^{\\prime }][rd,\"\\operatorname{trop}_{g,G}(\\mu )\"]& & \\mathcal {M}_{g^{\\prime },k}^{an}[d,\"\\operatorname{trop}_{g^{\\prime },k}\"]\\\\& H_{g,G}^{trop}(\\mu ) [r,\"\\operatorname{src}_{g,G}^{trop}(\\mu )\"][d,\"\\operatorname{tar}_{g,G}^{trop}(\\mu )\"^{\\prime }]& M_{g^{\\prime },k}^{trop}\\\\\\mathcal {M}_{g,n}^{an}[r,\"\\operatorname{trop}_{g,n}\"^{\\prime }]& M_{g,n}^{trop} &\\end{tikzcd}$ commutes.",
"In other words, the restriction of $\\operatorname{trop}_{g,G}(\\mu )$ onto a cone in $\\Sigma _{g,G}(\\mu )$ is an isomorphism onto a cone in $H^{trop}_{g,G}(\\mu )$ and every cone in $H_{g,G}^{trop}(\\mu )$ has at most finitely many preimages in $\\Sigma _{g,G}(\\mu )$ .",
"[Proof of Theorem REF ] Let $x$ be a closed point in $\\overline{\\mathcal {H}}_{g,G}(\\mu )$ , which corresponds to an admissible $G$ -cover $X^{\\prime }\\rightarrow X$ over $k$ .",
"Denote by $\\varphi :\\Gamma _{X^{\\prime }}\\rightarrow \\Gamma _X$ the corresponding unramified $G$ -cover of the dual graphs.",
"Denote by $\\mathfrak {o}_k$ either $k$ when $\\operatorname{char}k=0$ or the unique complete local ring with residue field $k$ when $\\operatorname{char}k=p>0$ (using Cohen's structure theorem).",
"The complete local ring at $x$ is given by $\\widehat{\\mathcal {O}}_{\\overline{\\mathcal {H}}_{g,G}(\\mu ),x}\\simeq \\mathfrak {o}_k\\big \\llbracket t_1,\\ldots , t_{3g-3+n}\\big \\rrbracket $ where $t_i=0$ for $i=1,\\ldots ,r$ cuts out the locus where the corresponding node $q_i$ of $X$ remains a node.",
"The retraction to the skeleton is locally given by $\\begin{split}\\big (\\operatorname{Spec}\\widehat{\\mathcal {O}}_{\\overline{\\mathcal {H}}_{g,G}(\\mu ),x}\\big )^\\beth &\\longrightarrow \\overline{\\mathbb {R}}_{\\ge 0}^r\\\\x&\\longmapsto \\big ( -\\log \\vert t_1\\vert _x, \\ldots , -\\log \\vert t_r\\vert _x\\big ) \\ ,\\end{split}$ where $(.",
")^\\beth $ denotes the generic fiber functor constructed in [39] and $\\overline{\\mathbb {R}}=\\mathbb {R}\\cup \\lbrace \\infty \\rbrace $ .",
"We find that under the isomorphism $\\sigma _{\\varphi }\\simeq \\mathbb {R}_{\\ge 0}^r$ the restriction of (REF ) to the preimage of $\\mathbb {R}_{\\ge 0}^r$ is nothing but the tropicalization map $\\operatorname{trop}_{g,G}(\\mu )$ defined above.",
"We observe the following: A degeneration of $X^{\\prime }\\rightarrow X$ in $\\overline{\\mathcal {H}}_{g,G}(\\mu )$ to another admissible $G$ -cover $X_0^{\\prime }\\rightarrow X_0$ with corresponding $\\varphi _0:\\Gamma _{X^{\\prime }_0}\\rightarrow \\Gamma _{X_0}$ may be described by additional coordinates $t_{r+1}, \\ldots , t_{r_0}$ that encode the new nodes $q_{r+1},\\ldots , q_{r_0}$ in the degeneration.",
"The induced map $\\mathbb {R}_{\\ge 0}^r\\hookrightarrow \\mathbb {R}_{\\ge 0}^{r_0}$ describes $\\sigma _{\\varphi }$ as the face of $\\sigma _{\\varphi _0}$ that corresponds to letting the edges $e_{r+1},\\ldots , e_{r_0}$ have length zero.",
"Denote by $E\\subset \\overline{\\mathcal {H}}_{g,G}(\\mu )$ the toroidal stratum containing $x$ and by $\\widetilde{E}$ and $\\widetilde{x}$ respectively their images in $\\overline{\\mathcal {M}}_{g,n}$ .",
"The operation of the fundamental group $\\pi _1(E,x)$ of $E$ on $\\mathbb {R}_{\\ge 0}^r\\simeq \\operatorname{Hom}(\\Lambda ^+_E,\\mathbb {R}_{\\ge 0})$ , where $\\Lambda ^+_E$ denotes the monoid of effective divisors supported on the closure of $E$ , naturally factors through the operation of $\\pi _1(\\widetilde{E},\\widetilde{x})$ on $\\mathbb {R}_{\\ge 0}^r\\simeq \\operatorname{Hom}(\\Lambda ^+_{\\widetilde{E}},\\mathbb {R}_{\\ge 0})$ .",
"Analogously, the operation of the automorphisms of $\\Gamma _{X^{\\prime }}\\rightarrow \\Gamma _X$ on $\\mathbb {R}_{\\ge 0}^r$ naturally factors through the operation of the automorphisms of $\\Gamma _X$ on $\\mathbb {R}_{\\ge 0}^r=\\mathbb {R}_{\\ge 0}^{E(\\Gamma _X)}$ .",
"Therefore, by [4], the images of the automorphism groups of both $\\pi _1(E,x)$ and $\\operatorname{Aut}(\\Gamma _{X^{\\prime }}\\rightarrow \\Gamma _X)$ in the permutation group of the entries of $\\mathbb {R}_{\\ge 0}^r$ are equal.",
"This shows that the isomorphisms $\\mathbb {R}_{\\ge 0}^r\\simeq \\sigma _{\\varphi }$ induce a necessarily strict morphism of generalized cone complexes $\\Sigma _{g,G}(\\mu )\\rightarrow H_{g,G}^{trop}(\\mu )$ that factors the tropicalization map as $\\mathcal {H}_{g,G}^{an}(\\mu )\\rightarrow \\Sigma _{g,G}(\\mu )\\rightarrow H_{g,G}^{trop}(\\mu )$ .",
"Its fibers are finite, since above every toroidal stratum of $\\overline{\\mathcal {M}}_{g,n}$ there are only finitely many toroidal strata of $\\overline{\\mathcal {H}}_{g,G}(\\mu )$ .",
"Finally, the commutativity of (REF ) is an immediate consequence of the definition of $\\operatorname{trop}_{g,G}(\\mu )$ .",
"Remark 5.7 In general, not every unramified tropical cover is realizable.",
"We may, for example, consider an unramified cover for which the local ramification profile is not of Hurwitz type (e.g.",
"when $d=4$ and the ramification profile at a vertex is given by $\\big \\lbrace (3,1),(2,2),(2,2)\\big \\rbrace $ ).",
"This explains why the tropicalization map on the moduli space of admissible covers (without the $G$ -action), as considered in [18], is not surjective.",
"We refer the reader to [13] for a discussion of this issue in the context of comparing algebraic and tropical gonality and to [33] for a survey of the underlying widely open problem, the so-called Hurwitz existence problem.",
"We do not know whether, for a general finite abelian group $G$ , every unramified tropical $G$ -cover (with cyclic stablizers at the nodes) is realizable.",
"When $G$ itself is cyclic and there are no marked legs (along which ramification is possible), we do expect every $G$ -admissible cover to be realizable, since by [26] the tropicalization map on the level of $n$ -torsion points on Jacobians is surjective.",
"We will return to this topic in its proper setting in the upcoming [30].",
"tocsectionReferences Yoav Len School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA Email address: [email protected] Martin Ulirsch Institut für Mathematik, Goethe-Universität Frankfurt, 60325 Frankfurt am Main, Germany E-mail address: [email protected] Dmitry Zakharov Department of Mathematics, Central Michigan University, Mount Pleasant, MI 48859, USA E-mail address: [email protected]"
]
] | 1906.04215 |
[
[
"What Does BERT Look At? An Analysis of BERT's Attention"
],
[
"Abstract Large pre-trained neural networks such as BERT have had great recent success in NLP, motivating a growing body of research investigating what aspects of language they are able to learn from unlabeled data.",
"Most recent analysis has focused on model outputs (e.g., language model surprisal) or internal vector representations (e.g., probing classifiers).",
"Complementary to these works, we propose methods for analyzing the attention mechanisms of pre-trained models and apply them to BERT.",
"BERT's attention heads exhibit patterns such as attending to delimiter tokens, specific positional offsets, or broadly attending over the whole sentence, with heads in the same layer often exhibiting similar behaviors.",
"We further show that certain attention heads correspond well to linguistic notions of syntax and coreference.",
"For example, we find heads that attend to the direct objects of verbs, determiners of nouns, objects of prepositions, and coreferent mentions with remarkably high accuracy.",
"Lastly, we propose an attention-based probing classifier and use it to further demonstrate that substantial syntactic information is captured in BERT's attention."
],
[
"Introduction",
"Large pre-trained language models achieve very high accuracy when fine-tuned on supervised tasks [8], [26], [28], but it is not fully understood why.",
"The strong results suggest pre-training teaches the models about the structure of language, but what specific linguistic features do they learn?",
"Recent work has investigated this question by examining the outputs of language models on carefully chosen input sentences [19] or examining the internal vector representations of the model through methods such as probing classifiers [1], [3].",
"Complementary to these approaches, we studyCode will be released at https://github.com/clarkkev/attention-analysis.",
"the attention maps of a pre-trained model.",
"Attention [2] has been a highly successful neural network component.",
"It is naturally interpretable because an attention weight has a clear meaning: how much a particular word will be weighted when computing the next representation for the current word.",
"Our analysis focuses on the 144 attention heads in BERTWe use the English base-sized model.",
"[9], a large pre-trained Transformer [37] network that has demonstrated excellent performance on many tasks.",
"We first explore generally how BERT's attention heads behave.",
"We find that there are common patterns in their behavior, such as attending to fixed positional offsets or attending broadly over the whole sentence.",
"A surprisingly large amount of BERT's attention focuses on the deliminator token [SEP], which we argue is used by the model as a sort of no-op.",
"Generally we find that attention heads in the same layer tend to behave similarly.",
"We next probe each attention head for linguistic phenomena.",
"In particular, we treat each head as a simple no-training-required classifier that, given a word as input, outputs the most-attended-to other word.",
"We then evaluate the ability of the heads to classify various syntactic relations.",
"While no single head performs well at many relations, we find that particular heads correspond remarkably well to particular relations.",
"For example, we find heads that find direct objects of verbs, determiners of nouns, objects of prepositions, and objects of possessive pronouns with $>$ 75% accuracy.",
"We perform a similar analysis for coreference resolution, also finding a BERT head that performs quite well.",
"These results are intriguing because the behavior of the attention heads emerges purely from self-supervised training on unlabeled data, without explicit supervision for syntax or coreference.",
"Figure: Examples of heads exhibiting the patterns discussed in Section .",
"The darkness of a line indicates the strength of the attention weight (some attention weights are so low they are invisible).Our findings show that particular heads specialize to specific aspects of syntax.",
"To get a more overall measure of the attention heads' syntactic ability, we propose an attention-based probing classifier that takes attention maps as input.",
"The classifier achieves 77 UAS at dependency parsing, showing BERT's attention captures a substantial amount about syntax.",
"Several recent works have proposed incorporating syntactic information to improve attention [10], [6], [32].",
"Our work suggests that to an extent this kind of syntax-aware attention already exists in BERT, which may be one of the reason for its success."
],
[
"Background: Transformers and BERT",
"Although our analysis methods are applicable to any model that uses an attention mechanism, in this paper we analyze BERT [9], a large Transformer [37] network.",
"Transformers consist of multiple layers where each layer contains multiple attention heads.",
"An attention head takes as input a sequence of vectors $h = [h_1, ..., h_n]$ corresponding to the $n$ tokens of the input sentence.",
"Each vector $h_i$ is transformed into query, key, and value vectors $q_i, k_i, v_i$ through separate linear transformations.",
"The head computes attention weights $\\alpha $ between all pairs of words as softmax-normalized dot products between the query and key vectors.",
"The output $o$ of the attention head is a weighted sum of the value vectors.",
"$ \\alpha _{ij} = \\frac{\\exp {(q_i^T k_j)}}{\\sum _{l=1}^n \\exp {(q_i^T k_l)}} \\quad \\phantom{aa} o_i = \\sum _{j=1}^n \\alpha _{ij} v_j $ Attention weights can be viewed as governing how “important\" every other token is when producing the next representation for the current token.",
"BERT is pre-trained on 3.3 billion tokens of English text to perform two tasks.",
"In the “masked language modeling\" task, the model predicts the identities of words that have been masked-out of the input text.",
"In the “next sentence prediction\" task, the model predicts whether the second half of the input follows the first half of the input in the corpus, or is a random paragraph.",
"Further training the model on supervised data results in impressive performance across a variety of tasks ranging from sentiment analysis to question answering.",
"An important detail of BERT is the preprocessing used for the input text.",
"A special token [CLS] is added to the beginning of the text and another token [SEP] is added to the end.",
"If the input consists of multiple separate texts (e.g., a reading comprehension example consists of a separate question and context), [SEP] tokens are also used to separate them.",
"As we show in the next section, these special tokens play an important role in BERT's attention.",
"We use the “base\" sized BERT model, which has 12 layers containing 12 attention heads each.",
"We use $<$ layer$>$ -$<$ head_number$>$ to denote a particular attention head."
],
[
"Surface-Level Patterns in Attention",
"Before looking at specific linguistic phenomena, we first perform an analysis of surface-level patterns in how BERT's attention heads behave.",
"Examples of heads exhibiting these patterns are shown in Figure REF .",
"Setup.",
"We extract the attention maps from BERT-base over 1000 random Wikipedia segments.",
"We follow the setup used for pre-training BERT where each segment consists of at most 128 tokens corresponding to two consecutive paragraphs of Wikipedia (although we do not mask out input words or as in BERT's training).",
"The input presented to the model is [CLS]$<$ paragraph-1$>$ [SEP]$<$ paragraph-2$>$ [SEP].",
"Figure: Each point corresponds to the average attention a particular BERT attention head puts toward a token type.",
"Above: heads often attend to “special\" tokens.",
"Early heads attend to [CLS], middle heads attend to [SEP], and deep heads attend to periods and commas.",
"Often more than half of a head's total attention is to these tokens.",
"Below: heads attend to [SEP] tokens even more when the current token is [SEP] itself.Figure: Gradient-based feature importance estimates for attention to [SEP], periods/commas, and other tokens.Figure: Entropies of attention distributions.",
"In the first layer there are particularly high-entropy heads that produce bag-of-vector-like representations."
],
[
"Relative Position",
"First, we compute how often BERT's attention heads attend to the current token, the previous token, or the next token.",
"We find that most heads put little attention on the current token.",
"However, there are heads that specialize to attending heavily on the next or previous token, especially in earlier layers of the network.",
"In particular four attention heads (in layers 2, 4, 7, and 8) on average put $>$ 50% of their attention on the previous token and five attention heads (in layers 1, 2, 2, 3, and 6) put $>$ 50% of their attention on the next token."
],
[
"Attending to Separator Tokens",
"Interestingly, we found that a substantial amount of BERT's attention focuses on a few tokens (see Figure REF ).",
"For example, over half of BERT's attention in layers 6-10 focuses on [SEP].",
"To put this in context, since most of our segments are 128 tokens long, the average attention for a token occurring twice in a segments like [SEP] would normally be around 1/64.",
"[SEP] and [CLS] are guaranteed to be present and are never masked out, while periods and commas are the most common tokens in the data excluding “the,\" which might be why the model treats these tokens differently.",
"A similar pattern occurs for the uncased BERT model, suggesting there is a systematic reason for the attention to special tokens rather than it being an artifact of stochastic training.",
"One possible explanation is that [SEP] is used to aggregate segment-level information which can then be read by other heads.",
"However, further analysis makes us doubtful this is the case.",
"If this explanation were true, we would expect attention heads processing [SEP] to attend broadly over the whole segment to build up these representations.",
"However, they instead almost entirely (more than 90%; see bottom of Figure REF ) attend to themselves and the other [SEP] token.",
"Furthermore, qualitative analysis (see Figure REF ) shows that heads with specific functions attend to [SEP] when the function is not called for.",
"For example, in head 8-10 direct objects attend to their verbs.",
"For this head, non-nouns mostly attend to [SEP].",
"Therefore, we speculate that attention over these special tokens might be used as a sort of “no-op\" when the attention head's function is not applicable.",
"To further investigate this hypothesis, we apply gradient-based measures of feature importance [33].",
"In particular, we compute the magnitude of the gradient of the loss from BERT's masked language modeling task with respect to each attention weight.",
"Intuitively, this value measures how much changing the attention to a token will change BERT's outputs.",
"Results are shown in Figure REF .",
"Starting in layer 5 – the same layer where attention to [SEP] becomes high – the gradients for attention to [SEP] become very small.",
"This indicates that attending more or less to [SEP] does not substantially change BERT's outputs, supporting the theory that attention to [SEP] is used as a no-op for attention heads."
],
[
"Focused vs Broad Attention",
"Lastly, we measure whether attention heads focus on a few words or attend broadly over many words.",
"To do this, we compute the average entropy of each head's attention distribution (see Figure REF ).",
"We find that some attention heads, especially in lower layers, have very broad attention.",
"These high-entropy attention heads typically spend at most 10% of their attention mass on any single word.",
"The output of these heads is roughly a bag-of-vectors representation of the sentence.",
"We also measured entropies for all attention heads from only the [CLS] token.",
"While the average entropies from [CLS] for most layers are very close to the ones shown in Figure REF , the last layer has a high entropy from [CLS] of 3.89 nats, indicating very broad attention.",
"This finding makes sense given that the representation for the [CLS] token is used as input for the “next sentence prediction\" task during pre-training, so it attends broadly to aggregate a representation for the whole input in the last layer."
],
[
"Probing Individual Attention Heads",
"Next, we investigate individual attention heads to probe what aspects of language they have learned.",
"In particular, we evaluate attention heads on labeled datasets for tasks like dependency parsing.",
"An overview of our results is shown in Figure REF ."
],
[
"Method",
"We wish to evaluate attention heads at word-level tasks, but BERT uses byte-pair tokenization [30], which means some words ($\\sim $ 8% in our data) are split up into multiple tokens.",
"We therefore convert token-token attention maps to word-word attention maps.",
"For attention to a split-up word, we sum up the attention weights over its tokens.",
"For attention from a split-up word, we take the mean of the attention weights over its tokens.",
"These transformations preserve the property that the attention from each word sums to 1.",
"For a given attention head and word, we take whichever other word receives the most attention weight as that model's predictionWe ignore [SEP] and [CLS], although in practice this does not significantly change the accuracies for most heads."
],
[
"Dependency Syntax",
"Setup.",
"We extract attention maps from BERT on the Wall Street Journal portion of the Penn Treebank [21] annotated with Stanford Dependencies.",
"We evaluate both “directions\" of prediction for each attention head: the head word attending to the dependent and the dependent attending to the head word.",
"Some dependency relations are simpler to predict than others: for example a noun's determiner is often the immediately preceding word.",
"Therefore as a point of comparison, we show predictions from a simple fixed-offset baseline.",
"For example, a fixed offset of -2 means the word two positions to the left of the dependent is always considered to be the head.",
"Table: The best performing attentions heads of BERT on WSJ dependency parsing by dependency type.Numbers after baseline accuracies show the best offset found (e.g., (1) means the word to the right is predicted as the head).",
"We show the 10 most common relations as well as 5 other ones attention heads do well on.",
"Bold highlights particularly effective heads.Figure: BERT attention heads that correspond to linguistic phenomena.",
"In the example attention maps, the darkness of a line indicates the strength of the attention weight.",
"All attention to/from red words is colored red; these colors are there to highlight certain parts of the attention heads' behaviors.",
"For Head 9-6, we don't show attention to [SEP] for clarity.",
"Despite not being explicitly trained on these tasks, BERT's attention heads perform remarkably well, illustrating how syntax-sensitive behavior can emerge from self-supervised training alone.Results.",
"Table REF shows that there is no single attention head that does well at syntax “overall\"; the best head gets 34.5 UAS, which is not much better than the right-branching baseline, which gets 26.3 UAS.",
"This finding is similar to the one reported by [29], who also evaluate individual attention heads for syntax.",
"However, we do find that certain attention heads specialize to specific dependency relations, sometimes achieving high accuracy and substantially outperforming the fixed-offset baseline.",
"We find that for all relations in Table REF except pobj, the dependent attends to the head word rather than the other way around, likely because each dependent has exactly one head but heads have multiple dependents.",
"We also note heads can disagree with standard annotation conventions while still performing syntactic behavior.",
"For example, head 7-6 marks 's as the dependent for the poss relation, while gold-standard labels mark the complement of an 's as the dependent (the accuracy in Table REF counts 's as correct).",
"Such disagreements highlight how these syntactic behaviors in BERT are learned as a by-product of self-supervised training, not by copying a human design.",
"Figure REF shows some examples of the attention behavior.",
"While the similarity between machine-learned attention weights and human-defined syntactic relations are striking, we note these are relations for which attention heads do particularly well on.",
"There are many relations for which BERT only slightly improves over the simple baseline, so we would not say individual attention heads capture dependency structure as a whole.",
"We think it would be interesting future work to extend our analysis to see if the relations well-captured by attention are similar or different for other languages."
],
[
"Coreference Resolution",
"Having shown BERT attention heads reflect certain aspects of syntax, we now explore using attention heads for the more challenging semantic task of coreference resolution.",
"Coreference links are usually longer than syntactic dependencies and state-of-the-art systems generally perform much worse at coreference compared to parsing.",
"Setup.",
"We evaluate the attention heads on coreference resolution using the CoNLL-2012 datasetWe truncate documents to 128 tokens long to keep memory usage manageable.",
"[27].",
"In particular, we compute antecedent selection accuracy: what percent of the time does the head word of a coreferent mention most attend to the head of one of that mention's antecedents.",
"We compare against three baselines for selecting an antecedent: Picking the nearest other mention.",
"Picking the nearest other mention with the same head word as the current mention.",
"A simple rule-based system inspired by [18].",
"It proceeds through 4 sieves: (1) full string match, (2) head word match, (3) number/gender/person match, (4) all other mentions.",
"The nearest mention satisfying the earliest sieve is returned.",
"We also show the performance of a recent neural coreference system from [41].",
"Results.",
"Results are shown in Table REF .",
"We find that one of BERT's attention heads achieves decent coreference resolution performance, improving by over 10 accuracy points on the string-matching baseline and performing close to the rule-based system.",
"It is particularly good with nominal mentions, perhaps because it is capable of fuzzy matching between synonyms as seen in the bottom right of Figure REF .",
"Table: Accuracies (%) for systems at selecting a correct antecedent given a coreferent mention in the CoNLL-2012 data.",
"One of BERT's attention heads performs fairly well at coreference."
],
[
"Probing Attention Head Combinations",
"Since individual attention heads specialize to particular aspects of syntax, the model's overall “knowledge\" about syntax is distributed across multiple attention heads.",
"We now measure this overall ability by proposing a novel family of attention-based probing classifiers and applying them to dependency parsing.",
"For these classifiers we treat the BERT attention outputs as fixed, i.e., we do not back-propagate into BERT and only train a small number of parameters.",
"The probing classifiers are basically graph-based dependency parsers.",
"Given an input word, the classifier produces a probability distribution over other words in the sentence indicating how likely each other word is to be the syntactic head of the current one.",
"Attention-Only Probe.",
"Our first probe learns a simple linear combination of attention weights.",
"$ p(i|j) \\propto \\exp {\\bigg (\\sum _{k=1}^{n} w_k\\alpha ^k_{ij} + u_k \\alpha ^k_{ji}\\bigg )} $ where $p(i|j)$ is the probability of word $i$ being word $j$ 's syntactic head, $\\alpha ^k_{ij}$ is the attention weight from word $i$ to word $j$ produced by head $k$ , and $n$ is the number of attention heads.",
"We include both directions of attention: candidate head to dependent as well as dependent to candidate head.",
"The weight vectors $w$ and $u$ are trained using standard supervised learning on the train set.",
"Attention-and-Words Probe.",
"Given our finding that heads specialize to particular syntactic relations, we believe probing classifiers should benefit from having information about the input words.",
"In particular, we build a model that sets the weights of the attention heads based on the GloVe [25] embeddings for the input words.",
"Intuitively, if the dependent and candidate head are “the\" and “cat,\" the probing classifier should learn to assign most of the weight to the head 8-11, which achieves excellent performance at the determiner relation.",
"The attention-and-words probing classifier assigns the probability of word $i$ being word $j$ 's head as $ p(i|j) \\propto \\exp \\bigg ( \\sum _{k=1}^{n} &W_{k,:}(v_i \\oplus v_j)\\alpha ^k_{ij} +\\\\ &U_{k,:}(v_i \\oplus v_j) \\alpha ^k_{ji} \\bigg ) $ Where $v$ denotes GloVe embeddings and $\\oplus $ denotes concatenation.",
"The GloVe embeddings are held fixed in training, so only the two weight matrices $W$ and $U$ are learned.",
"The dot product $W_{k,:}(v_i \\oplus v_j)$ produces a word-sensitive weight for the particular attention head.",
"Results.",
"We evaluate our methods on the Penn Treebank dev set annotated with Stanford dependencies.",
"We compare against three baselines: A right-branching baseline that always predicts the head is to the dependent's right.",
"A simple one-hidden-layer network that takes as input the GloVe embeddings for the dependent and candidate head as well as distance features between the two words.Indicator features for short distances as well as continuous distance features, with distance ahead/behind treated separately to capture word order Our attention-and-words probe, but with attention maps from a BERT network with pre-trained word/positional embeddings but randomly initialized other weights.",
"This kind of baseline is surprisingly strong at other probing tasks [7].",
"Results are shown in Table REF .",
"We find the Attn + GloVe probing classifier substantially outperforms our baselines and achieves a decent UAS of 77, suggesting BERT's attention maps have a fairly thorough representation of English syntax.",
"As a rough comparison, we also report results from the structural probe from [14], which builds a probing classifier on top of BERT's vector representations rather than attention.",
"The scores are not directly comparable because the structural probe only uses a single layer of BERT, produces undirected rather than directed parse trees, and is trained to produce the syntactic distance between words rather than directly predicting the tree structure.",
"Nevertheless, the similarity in score to our Attn + Glove probing classifier suggests there is not much more syntactic information in BERT's vector representations compared to its attention maps.",
"Overall, our results from probing both individual and combinations of attention heads suggest that BERT learns some aspects syntax purely as a by-product of self-supervised training.",
"Other work has drawn a similar conclusions from examining BERT's predictions on agreement tasks [12] or internal vector representations [14], [20].",
"Traditionally, syntax-aware models have been developed through architecture design (e.g., recursive neural networks) or from direct supervision from human-curated treebanks.",
"Our findings are part of a growing body of work indicating that indirect supervision from rich pre-training tasks like language modeling can also produce models sensitive to language's hierarchical structure.",
"Table: Results of attention-based probing classifiers on dependency parsing.",
"A simple model taking BERT attention maps and GloVe embeddings as input performs quite well.",
"*Not directly comparable to our numbers; see text.Figure: BERT attention heads embedded in two-dimensional space.Distance between points approximately matches the average Jensen-Shannon divergences between the outputs of the corresponding heads.Heads in the same layer tend to be close together.Attention head “behavior\" was found through the analysis methods discussed throughout this paper."
],
[
"Clustering Attention Heads",
"Are attention heads in the same layer similar to each other or different?",
"Can attention heads be clearly grouped by behavior?",
"We investigate these questions by computing the distances between all pairs of attention heads.",
"Formally, we measure the distance between two heads $\\textsc {H}_i$ and $\\textsc {H}_j$ as: $ \\sum _{\\text{token} \\in \\text{data}} JS(\\textsc {H}_i(\\text{token}), \\textsc {H}_j(\\text{token})) $ Where $JS$ is the Jensen-Shannon Divergence between attention distributions.",
"Using these distances, we visualize the attention heads by applying multidimensional scaling [17] to embed each head in two dimensions such that the Euclidean distance between embeddings reflects the Jensen-Shannon distance between the corresponding heads as closely as possible.",
"Results are shown in Figure REF .",
"We find that there are several clear clusters of heads that behave similarly, often corresponding to behaviors we have already discussed in this paper.",
"Heads within the same layer are often fairly close to each other, meaning that heads within the layer have similar attention distributions.",
"This finding is a bit surprising given that [36] show that encouraging attention heads to have different behaviors can improve Transformer performance at machine translation.",
"One possibility for the apparent redundancy in BERT's attention heads is the use of attention dropout, which causes some attention weights to be zeroed-out during training."
],
[
"Related Work",
"There has been substantial recent work performing analysis to better understand what neural networks learn, especially from language model pre-training.",
"One line of research examines the outputs of language models on carefully chosen input sentences [19], [16], [13], [23].",
"For example, the model's performance at subject-verb agreement (generating the correct number of a verb far away from its subject) provides a measure of the model's syntactic ability, although it does not reveal how that ability is captured by the network.",
"Another line of work investigates the internal vector representations of the model [1], [11], [42], often using probing classifiers.",
"Probing classifiers are simple neural networks that take the vector representations of a pre-trained model as input and are trained to do a supervised task (e.g., part-of-speech tagging).",
"If a probing classifier achieves high accuracy, it suggests that the input representations reflect the corresponding aspect of language (e.g., low-level syntax).",
"Like our work, some of these studies have also demonstrated models capturing aspects of syntax [31], [4] or coreference [35], [34], [20] without explicitly being trained for the tasks.",
"With regards to analyzing attention, [38] builds a visualization tool for the BERT's attention and reports observations about the attention behavior, but does not perform quantitative analysis.",
"[5] analyze the attention of memory networks to understand model performance on a question answering dataset.",
"There has also been some initial work in correlating attention with syntax.",
"[29] evaluate the attention heads of a machine translation model on dependency parsing, but only report overall UAS scores instead of investigating heads for specific syntactic relations or using probing classifiers.",
"[22] propose heuristic ways of converting attention scores to syntactic trees, but do not quantitatively evaluate their approach.",
"For coreference, [39] show that the attention of a context-aware neural machine translation system captures anaphora, similar to our finding for BERT.",
"Concurrently with our work [40] identify syntactic, positional, and rare-word-sensitive attention heads in machine translation models.",
"They also demonstrate that many attention heads can be pruned away without substantially hurting model performance.",
"Interestingly, the important attention heads that remain after pruning tend to be ones with identified behaviors.",
"[24] similarly show that many of BERT's attention heads can be pruned.",
"Although our analysis in this paper only found interpretable behaviors in a subset of BERT's attention heads, these recent works suggest that there might not be much to explain for some attention heads because they have little effect on model perfomance.",
"[15] argue that attention often does not “explain\" model predictions.",
"They show that attention weights frequently do not correlate with other measures of feature importance.",
"Furthermore, attention weights can often be substantially changed without altering model predictions.",
"However, our motivation for looking at attention is different: rather than explaining model predictions, we are seeking to understand information learned by the models.",
"For example, if a particular attention head learns a syntactic relation, we consider that an important finding from an analysis perspective even if that head is not always used when making predictions for some downstream task."
],
[
"Conclusion",
"We have proposed a series of analysis methods for understanding the attention mechanisms of models and applied them to BERT.",
"While most recent work on model analysis for NLP has focused on probing vector representations or model outputs, we have shown that a substantial amount of linguistic knowledge can be found not only in the hidden states, but also in the attention maps.",
"We think probing attention maps complements these other model analysis techniques, and should be part of the toolkit used by researchers to understand what neural networks learn about language."
],
[
"Acknowledgements",
"We thank the anonymous reviews for their thoughtful comments and suggestions.",
"Kevin is supported by a Google PhD Fellowship."
]
] | 1906.04341 |
[
[
"Evolution of primordial black hole spin due to Hawking radiation"
],
[
"Abstract Near extremal Kerr black holes are subject to the Thorne limit $a<a^*_{\\rm lim}=0.998$ in the case of thin disc accretion, or some generalized version of this in other disc geometries.",
"However any limit that differs from the thermodynamics limit $a^* < 1$ can in principle be evaded in other astrophysical configurations, and in particular if the near extremal black holes are primordial and subject to evaporation by Hawking radiation only.",
"We derive the lower mass limit above which Hawking radiation is slow enough so that a primordial black hole with a spin initially above some generalized Thorne limit can still be above this limit today.",
"Thus, we point out that the observation of Kerr black holes with extremely high spin should be a hint of either exotic astrophysical mechanisms or primordial origin."
],
[
"Introduction",
"Primordial Black Holes (PBHs) are appealing candidates for solving the long-standing issue of dark matter [1], [2], [3], [4].",
"PBHs could have been created at the end of the inflationary stage of the early Universe, when relatively high density fluctuations $\\Delta \\rho /\\rho \\gtrsim 1$ re-entered the Hubble horizon.",
"The mass collapsing into a PBH through this mechanism is not subject to the lower Chandrasekhar limit of $\\sim 1.4\\,M_\\odot $ [5], as this limit is a consequence of the stellar origin of Black Holes (BHs).",
"Thus, the detection of a sub-solar BH in the merger events of forthcoming gravitational waves detectors such as LISA [6] would certainly point to a primordial origin [7].",
"Most excitingly, there are powerful observational constraints, primarily from gravitational microlensing in the subsolar mass regime, but a substantial window remains open for PBHs as dark matter in the mass range that extends from asteroid mass scales down to the mass set by evaporation limits [8].",
"In principle, depending on their mechanism of production at the end of inflation, there is no restriction on the initial spin of a PBH up to near extremal values $a^* \\lesssim 1$ .",
"On the other hand, for BHs with astrophysical origin, Thorne has shown that in the case of thin disc accretion, there is a limit to the reduced spin $a^*_{\\rm lim} \\approx 0.998$ .",
"This limit comes from accretion of the surrounding gas on a BH, and its balance with superradiance effects [9].",
"Surprisingly, the same $a^*_{\\rm lim} \\approx 0.998$ is found in [10] for BH mergers.",
"In the case of accretion, this limit has been recently generalized to other accretion regimes and disc geometries in [11], based on earlier work [12]; reaching somewhat higher values depending on the disc parameters.",
"The overall state-of-art is that, except for really specific accretion environments, the spin of a BH of astrophysical origin should not exceed a generalized version of the Thorne limit [13].",
"However, the superior limit $a^* < 1$ holds in any case due to the third law of thermodynamics [14].",
"Indeed, a BH with $a^* = 1$ would have $T = 0$ , which is classically forbidden for any statistical system.",
"Moreover, its horizon would have disappeared, thus revealing a naked space-time singularity and violating the Cosmic Censorship Conjecture (a comprehensive discussion of extremal BHs is given in e.g.",
"[15], [16]).",
"We would like to emphasize the fact that, even if the thermodynamics limit $a^* < 1$ remains true, nothing prevents a priori astrophysical BHs from reaching a near extremal spin value $a^* = 0.999999...$ , yet the mechanisms are still to be proposed.",
"Thus, the detection of BHs with a reduced spin higher than a generalized Thorne limit $a^* > a^*_{\\rm lim}$ could point either towards primordial origin [17], [18] or astrophysical origin with an exotic accretion history.",
"In this paper, we focus on the mechanisms allowing a PBH to have today a spin higher than a generalized Thorne limit.",
"For this purpose, we compute the mass and spin loss through Hawking radiation of a Kerr PBH and evaluate the minimum initial mass a PBH should have in order to experience a current spin value above a generalized Thorne limit.",
"Hawking showed that BHs are not as black as was first supposed [19].",
"Throughout, we use a natural system of units where $G = c = k_{\\rm B} = \\hbar = 1$ .",
"Hawking used a semi-classical treatment, that is to say the general relativity Kerr (or Schwarzschild) metric for space-time ds2 = ( 1 - 2Mr2 )dt2 + 4aMr()22dtd- 2 dr2 - 2d2 - ( r2 + a2 + 2a2Mr()22 )()2d2 , where $M$ is the BH mass, $a \\equiv J/M$ is the BH spin parameter ($J$ is the BH angular momentum), $\\Sigma \\equiv r^2 + a^2\\cos (\\theta )^2$ and $\\Delta \\equiv r^2 -2Mr + a^2$ , and a quantum mechanics treatment of Standard Model (SM) particles through a wave function $\\psi $ satisfying the Dirac equation for fermions (spin $1/2$ ) $(i{\\partial } - \\mu )\\psi = 0\\,, $ where ${\\partial } \\equiv \\gamma _\\nu \\partial ^\\nu $ is the standard Feynman notation, and the Proca equation for bosons (spin 0, 1 or 2) $(\\square + \\mu ^2)\\psi = 0\\,, $ where $\\mu $ is the particle rest mass.",
"Setting $\\mu = 0$ in these equations of motion also allows to compute the propagation of the massless fields (in the following neutrinos, photons, gravitons).",
"In these equations, we neglect the couplings between the fields, since they do not affect the probability of emission of (primary) SM particles via Hawking radiation, but we consider them to obtain the abundance of the final (secondary) particles at infinity, which come from the hadronization or decay of the primary particles.",
"Solving these equations shows that there is a net emission of particles of type $i$ called the Hawking radiation (HR).",
"The number of particles emitted per unit time and energy is $\\dfrac{{\\rm d}^2N_i}{{\\rm d}t{\\rm d}E} = \\dfrac{1}{2\\pi }\\sum _{\\rm dof.",
"}\\dfrac{\\Gamma ^{lm}_i(E,M,a^*)}{e^{E^\\prime /T}\\pm 1}\\,, $ where $T$ is the Kerr BH Hawking temperature $T \\equiv \\dfrac{1}{2\\pi }\\left( \\dfrac{r_+ - M}{r_+^2 + a^2} \\right)\\,,$ and $r_\\pm \\equiv M\\left(1 \\pm \\sqrt{1 - (a^{*})^2}\\right)$ are the Kerr horizons radii; $a^* \\equiv a/M$ is the Kerr dimensionless spin parameter, it is 0 for a Schwarzschild – non rotating – BH and 1 for a Kerr extremal BH; $E^\\prime \\equiv E - m\\Omega $ is the energy of the particle that takes into account the horizon rotation with angular velocity $\\Omega \\equiv a^*/(2r_+)$ on top of the total energy $E \\equiv E_{\\rm kin.}",
"+ \\mu $ ; $m$ is the particle angular momentum projection $m \\in [-l,+l]$ .",
"The sum of Eq.",
"(REF ) is on the degrees of freedom (dof.)",
"of the particle considered, that is to say the color and helicity multiplicity as well as the angular momentum $l$ and its projection $m$ .",
"The quantity $\\Gamma _i^{lm}(E,M,a^*)$ is called the greybody factor and has been extensively studied in the literature (see below).",
"It encodes the probability that a particle of type $i$ with angular momentum $l$ and projection $m$ generated at the horizon of a BH escapes its gravitational well and reaches space infinity."
],
[
"Evolution of BHs",
"After computing the greybody factors $\\Gamma _i^{lm}$ , it is possible to compute the mass and spin loss rates by integrating Eq.",
"(REF ) over all energies and summing over all – massive and massless – SM particles $i$ (6 quarks + 6 antiquarks, 3 neutrinos + 3 antineutrinos, 3 charged leptons + 3 charged antileptons, 8 gluons, weak $W^+$ , $W^-$ and $Z^0$ bosons, the photon and the Higgs boson), plus the graviton.",
"We define the (positive) $f$ and $g$ factors following [20], [21] f(M,a*) -M2 dMdt = M20+ idof.",
"E2ilm(E,M,a*)eE/T1 dE , g(M,a*) -Ma* dJdt = Ma*0+ idof.m2 ilm(E,M,a*)eE/T1dE .",
"Inverting these equations and using the definition of $a^*$ , we obtain the differential equations governing the mass and spin of a Kerr BH $\\dfrac{{\\rm d}M}{{\\rm d}t} = -\\dfrac{f(M,a^*)}{M^2}\\,, $ and $\\dfrac{{\\rm d}a^*}{{\\rm d}t} = \\dfrac{a^*(2f(M,a^*) - g(M,a^*))}{M^3}\\,.",
"$ We solve Eqs.",
"(REF ) and (REF ) numerically, using a new code entitled BlackHawk [22]https://blackhawk.hepforge.org/.",
"This code contains tabulated values of $f(M,a^*)$ and $g(M,a^*)$ obtained through Eqs.",
"(REF ) and (REF ).",
"Within BlackHawk, efforts have been made to compute the greybody factors $\\Gamma ^{lm}_i(E,M,a^*)$ numerically.",
"Teukolsky and Press [23], [24] have shown that the Dirac and Proca equations (REF ) and (REF ), once developed in the Kerr metric (REF ), can be separated into a radial and an angular part with the wave function written as $\\psi (t,r,\\theta ,\\phi ) = R(r) \\, S_{lm}(\\theta ) \\, e^{-iEt} \\,e^{im\\phi }$ .",
"In the following, we make the approximation that the particles are massless for the greybody factor computations, since the main effect of the non-zero particle rest mass is to induce a cut in the emission spectra at energies $E < \\mu $ [25].",
"This cut in the energy spectra is applied as a post-process and included in the computation of the integrals (REF ) and (REF ).",
"We checked that the approximated spectra only slightly differ from the ones with a full massive computation, and the small differences are smoothed out by the integration in Eqs.",
"(REF ) and (REF ).",
"The radial equation on $R(r)$ for a massless field of spin $s$ reads 1sddr( s+1dRdr ) + ( K2+2is(r-M)K - 4is E r - slm)R = 0 , where $\\lambda _{slm}$ is the eigenvalue of the angular part (we use the polynomial expansion of [21] to compute $\\lambda _{slm}$ ) and $K\\equiv (r^2 + a^{*2}M^2)E + a^*Mm$ .",
"Then, Chandrasekhar and Detweiler [26], [27], [28], [29] have shown that through suitable function transformation $R\\rightarrow Z$ and a change of variable from the Boyet-Lindquist radial coordinate to a generalized tortoise coordinate $r\\rightarrow r^*$ , one can transform Eq.",
"(REF ) into a wave equation with a short-range potential $\\dfrac{{\\rm d}^2 Z}{{\\rm d}r^{*2}} + \\left(E^2 - V(r^*)\\right)Z = 0\\,.",
"$ For details about this transformation, we refer the reader to Appendix .",
"We solve this wave equation numerically with Mathematica, starting from an ingoing plane wave at the horizon $Z_{\\rm hor} \\underset{r^*\\rightarrow -\\infty }{\\rightarrow } e^{iEr^*}\\,,$ and integrating to space infinity where the solution is $Z_\\infty \\underset{r^*\\rightarrow +\\infty }{\\rightarrow } Ae^{iEr^*} + Be^{-iEr^*}\\,,$ we identify the transmission coefficient (greybody factor) $\\Gamma \\equiv |A|^2\\,.$ This allows us to perform the integrals (REF ) and (REF ).",
"BlackHawk uses an Euler-based adaptive time step method to compute accurately the last stages of the BH life, when its mass goes down to the Planck mass $M_{\\rm P}$ very quickly.",
"In practice, the time step is decreased whenever the relative changes in mass or spin are too large ($|\\Delta M|/M$ or $|\\Delta a^*|/a^*$ $>0.1$ ), leading to a precision of a few percent.",
"When $M\\sim M_{\\rm P}$ , we consider that Hawking evaporation terminates."
],
[
"Evolution of Kerr BHs",
"The main difference between Kerr ($a^* \\ne 0$ ) and Schwarzschild ($a^* = 0$ ) BHs is that Kerr BHs are axially symmetric and not spherically symmetric.",
"This gives a favored axial direction when computing the Hawking radiation.",
"The emission of particles with an angular momentum spinning in the same direction as that of the BH is enhanced when $a^*$ increases.",
"Moreover, for sufficiently small energies and high angular momentum $E < E_{\\rm SR} \\equiv \\dfrac{a^*m}{2r_+}\\,,$ we enter the regime of superradiance (SR), with enhanced emission.",
"This asymmetry in the Hawking radiation causes a net spin loss by the BH (hence the positivity of the $g$ factor defined in Eq.",
"(REF )) through the emission of high angular momentum particles.",
"This enhanced radiation also causes a mass loss larger than in the Schwarzschild case.",
"Thus, Kerr BHs have a shorter lifetime than Schwarzschild BHs, and it gets shorter and shorter as the initial spin $a^*_i$ gets close to 1.",
"Figure: Kerr BH lifetimes t BH t_{\\rm BH} as functions of the BH mass M BH M_{\\rm BH} for different initial spins a i * ={0,0.9,0.9999}a^*_i = \\lbrace 0, 0.9, 0.9999\\rbrace (blue, green and red curves respectively).",
"The age of the universe is indicated as a grey horizontal line.Fig.",
"REF shows the lifetime of Kerr BHs as a function of their mass for different initial spins $a^*_i$ .",
"The Hawking radiation computed with BlackHawk for the evaporation includes all the SM particles (both massive and massless) as well as one massless graviton.",
"We see that the spin indeed reduces the lifetime but the difference is small compared to the enormous time range spanned by the BH lifetimes.",
"We verify that the lifetime is approximately given by $t_{\\rm BH} \\sim M^3\\,,$ which can be derived from Eq.",
"(REF ) by considering that $f(M,a^*)$ is a constant.",
"This approximate relation still holds in presence of angular momentum $a^* \\ne 0$ .",
"Figure: Kerr BH mass MM (plain curve, normalized over the initial mass M i =10 16 M_i = 10^{16}\\,g) and reduced spin a * a^* (dashed curve, starting from an initial spin a i * =0.9a^*_i = 0.9) as functions of time tt (normalized to the BH lifetime t BH t_{\\rm BH}).Fig.",
"REF shows an example of the evolution of the Kerr BH mass and spin through time.",
"We see that the reduced spin $a^*$ has a slightly shorter timescale than the mass $M$ .",
"This is easy to understand when looking at Eqs.",
"(REF ) and (REF ).",
"The first stage of the evolution is a strong decrease of both mass and spin, corresponding to the Kerr regime when the Hawking radiation is enhanced.",
"When we leave the high-spin region ($a^*\\lesssim 0.2$ ), the emission becomes similar to that of a Schwarzschild BH and the mass evolution is more monotonic.",
"At the end of the BH life (the last $10\\%$ ), a final stage of very fast evaporation occurs, during which the BH loses the major part of its mass ($\\sim 50\\%$ ).",
"This is in agreement with the results of [17].",
"When reaching the Planck mass, Hawking's theory does not tell what happens of the BH.",
"Figure: Comparison of Kerr BH mass MM (plain curves, normalized to the initial mass M i =10 16 M_i = 10^{16}\\,g which is the same for all curves) and spin a * a^* (dashed curves) evolutions as functions of time tt (normalized to the Schwarzschild BH lifetime t S t_{\\rm S}), for different values of the initial spin a i * a^*_i ranging (right to left) from a i * =0a_i^* = 0 (Schwarzschild case) to a i * =0.9999a_i^* = 0.9999 (near extremal case).Fig.",
"REF shows the mass and spin evolutions for the same initial mass $M_i = 10^{16}\\,$ g but different initial spins $a_i^* = \\lbrace 0, 0.9, 0.9999\\rbrace $ .",
"We see that the lifetime of a Kerr BH can be reduced by almost $\\sim 60\\%$ when going from the Schwarzschild case $a^*_i = 0$ to the near extremal case $a^*_i = 0.9999$ .",
"This is compatible with the results of [21].",
"The higher the initial spin is, the stronger the initial mass loss will be, so the shorter the BH lifetime.",
"We can see that after most of the spin is radiated away, all curves share the same shape as the Schwarzschild one.",
"Figure: Kerr BH lifetimes t BH t_{\\rm BH} (normalized to the Schwarzschild case t S t_{\\rm S}) for different initial masses M i ={10 9 ,10 13 ,10 18 }M_i = \\lbrace 10^{9}, 10^{13}, 10^{18}\\rbrace \\,g (blue, green and red lines, respectively) as functions of the initial spin a i * a^*_i.",
"The xx-axis has been reversed to show 1-a i * 1-a^*_i in a logarithmic scale.Fig.",
"REF shows the evolution of the lifetime of a Kerr BH as a function of the initial spin $a^*_i$ , for different values of the initial mass $M_i = \\lbrace 10^{9}, 10^{13}, 10^{18}\\rbrace \\,$ g. We have reversed the $x$ -axis to focus on the near extremal region $a^*_i \\lesssim 1$ .",
"We see that the lifetime decreases as the initial spin increases, but this saturates as we come closer to the extremal Kerr case $a^*_i \\lesssim 1$ .",
"The decrease of the lifetime relative to the Schwarzschild case is not the same for all initial masses since they have a different Hawking emission history: lighter BHs can emit massive particles at the beginning of their evaporation (in the Kerr regime) while heavier BHs can only emit them at the end of their evaporation (in the Schwarzschild regime).",
"The difference in the evolution of the lifetimes remains small."
],
[
"Maximum spin",
"Using these data on the Kerr BH evolution, which is a function of both mass and spin, we can estimate the maximum spin a BH can still have today, starting from some initial spin, and depending on its initial mass.",
"We know that some generalized Thorne limit prevents BHs with a disc from having a spin higher than $a^*_{\\rm lim} \\gtrsim 0.998$ , due to accretion and superradiance effects [9], [12], [11].",
"We also know that the same limit applies to the outcome of BH mergers due to general relativistic dynamics [10].",
"One possibility of overcoming $a^*_{\\rm lim}$ , while remaining below the thermodynamics limit $a^* < 1$ , may be to form a Kerr PBH with an initial spin $a_i^* > a^*_{\\rm lim}$ and to maintain this spin over time until today.",
"As mentioned in the Introduction, the precise value of the Thorne limit can depend on the disc geometry and parameters (accretion regime, viscosity) [11], thus the numerical results presented in this Section have to be adapted to somewhat higher generalized Thorne limits.",
"Figure: Minimum initial mass M i M_i needed to have a relative spin loss today ϵ≡Δa * /a i * \\epsilon \\equiv \\Delta a^*/a^*_i for different values of ϵ\\epsilon .",
"The xx-axis has been reversed to show 1-a i * 1-a^*_i in a logarithmic scale.We have seen that the spin decrease time-scale corresponds roughly to that of the mass decrease $t_{\\rm BH} \\sim M_i^3$ .",
"That means that in order to maintain a spin value really close to the extremal Kerr case, the BH initial mass must be sufficiently high.",
"Fig.",
"REF shows the minimum initial mass needed as a function of the initial spin, for different values of the desired relative spin change $\\epsilon \\equiv \\Delta a^* / a^*_i$ .",
"As expected, the more we want to have a spin today close to the initial one ($\\epsilon \\rightarrow 0$ ) the more massive the BH has to have been originally.",
"As $\\epsilon \\rightarrow 1$ (all initial spin is lost), the minimum mass, for all initial spins, goes to $M_{\\rm lim}(a^*_i) \\sim 10^{15}\\,$ g the mass of the BHs just evaporating today.",
"Figure: Value of the spin today a today * a^*_{\\rm today} as a function of the initial mass M i M_i for different initial spins a i * ={0.9922,0.9994,0.9999}a^*_i = \\lbrace 0.9922, 0.9994, 0.9999\\rbrace (blue, green and red lines, respectively).",
"The Thorne limit a lim * ≈0.998a^*_{\\rm lim} \\approx 0.998 is shown as an horizontal dashed line.",
"The yy-axis has been reversed to show 1-a today * 1-a^*_{\\rm today} on a logarithmic scale.Fig.",
"REF gives a reversed view of Fig.",
"REF : starting from an initial spin $a^*_i = \\lbrace 0.9922, 0.9994, 0.9999\\rbrace $ below or above the Thorne limit, it shows the value of the spin today $a^*_0$ as a function of the initial mass.",
"We see that for sufficiently high initial masses $M_i \\gtrsim 10^{16}-10^{17}\\,$ g, the value of the spin has barely changed, as could be already guessed from Fig.",
"REF .",
"For initial masses $M_i > M_i^{\\rm lim} \\approx 2-3\\times 10^{16}\\,$ g, the spin value today is still higher than the Thorne limit for the two cases where it was higher at the beginning.",
"That means that a Kerr PBH of initial spin $a^*_i > a^*_{\\rm lim}$ could still have a spin $a^*_0 > a^*_{\\rm lim}$ today if it were sufficiently heavy.",
"The same picture could be drawn for even higher initial spins $a^*_i = 0.999999...$ with a decrease of $M_i^{\\rm lim}$ when $a^*_i$ increases.",
"Indeed, starting from a higher spin, a smaller initial mass is necessary to reach the Thorne limit today through Hawking radiation."
],
[
"Accretion and mergers",
"The above computation of the PBH spin evolution is relevant only if the mechanisms leading to the establishment of the generalized Thorne limit are avoided, that is to say accretion of material surrounding the PBH and mergers with other PBHs.",
"The accretion part is clearly not a problem as accretion is dominated by Hawking radiation for sufficiently light PBHs during the radiation-dominated era.",
"During the matter-dominated era, PBHs do not necessarily evolve in a matter-rich environment as they do not come from the collapse of a star.",
"Thus, the spin loss is only given by the Hawking radiation, as computed with BlackHawk.",
"The merging part should not be bothersome if the PBH merging rate is sufficiently small, which should be the case if PBHs do not contribute too much to the dark matter fraction (thus preventing the formation of binaries).",
"At least, some of them should have been isolated until today.",
"Thus, the generalized Thorne limit does not apply to sufficiently rare and light PBHs."
],
[
"Formation",
"The question on how to generate such high-spin PBHs can be answered by a profusion of models of inflation and early Universe cosmology.",
"Every model involving PBH creation should generate high spin PBHs at least as a tail in the distribution [30], [31], [32].",
"In these cases, the observation of high spin BHs should remain a rare event not necessarily incompatible with an unexpected astrophysical origin.",
"However, some models predict a domination of high spin values for PBHs.",
"We refer to one recent example, that of PBHs formed by scalar field fragmentation during the matter-dominated period that precedes reheating in an inflationary universe [33], [34].",
"Precise spin measurements could be accomplished with further LIGO-Virgo data [18].",
"We nevertheless point out that a more realistic scenario in which the transient matter-domination is not complete, taking into account the increasing proportion of radiation when this phase ends, predicts somewhat less extreme PBHs, and in fewer quantities [35].",
"One last remark concerns the possible destruction of such extreme PBHs by external perturbations.",
"It has been shown in [36] that falling matter can not overspin a near extremal Kerr-Newman BH, and a fortiori a near extremal Kerr one, thus the horizon should persist and the Hawking radiation paradigm should hold during the PBH evolution.",
"In this paper, we have computed the evolution of BH spin through Hawking radiation using our new code BlackHawk.",
"We have seen that a way to presently have BHs with a spin near the Kerr extremal value $a^*\\lesssim 1$ and above some generalized Thorne limit $a^*_{\\rm lim} \\gtrsim 0.998$ is to generate it in the primordial Universe through post-inflationary mechanisms with an initial spin $a^*_i > a^*_{\\rm lim}$ .",
"Then, if its mass is sufficiently high, Hawking radiation is too slow to drive its spin below the generalized Thorne limit.",
"One interesting result is that the initial PBH mass needed to retain such a high spin until today is well below the mass of the Sun.",
"We conclude that near extremal Kerr black holes may exist in nature, if primordial black holes constitute all of the dark matter in the observationally allowed window, or at least some of it in the higher mass range.",
"Moreover when such near extremal black holes enter the galactic environment, accretion of order 0.001 of their rest mass would render them sub-extremal and induce Hawking evaporation.",
"Such potential black hole \"bombs\" may make primordial black holes directly detectable via X-ray or gamma ray emission."
],
[
"From the Teukolsky equation to a Schrödinger-like wave equation",
"In this Appendix we will briefly present the analytical method used by Chandrasekhar and Detweiler [26], [27], [28], [29] to transform the Teukolsky radial equation Eq.",
"(REF ) into the Schrödinger-like wave equation Eq.",
"(REF ).",
"The first change consists in moving from the Boyet-Lindquist coordinate $r$ to a generalized tortoise coordinate $r^*$ related to $r$ through $\\dfrac{{\\rm d}r^*}{{\\rm d}r} = \\dfrac{\\rho ^2}{\\Delta }\\,, $ where $\\Delta \\equiv r^2 - 2Mr + a^2$ , $\\rho \\equiv r^2 + \\alpha ^2$ and $\\alpha ^2 \\equiv a^2 + am/E$ .",
"This differential change of variable can be solved to give r*(r) = r + rSr+ + am/Er+ - r-( rr+ - 1 ) - rSr- + am/Er+ - r-( rr- - 1 ) , where $r_{\\rm S} \\equiv 2M$ is the Schwarzschild radius.",
"The inverse relation has no analytical expression and must be computed numerically by solving the differential equation (REF ).",
"On top of this change, one changes the function from $R(r)$ to $Z(r^*)$ by imposing that the final result is a Schrödinger-like wave equation.",
"Surprisingly, this is always something one can do for the values of the spin $s = 0,1,2,1/2$ and for both Schwarzschild ($a^* = 0$ ) and Kerr ($a^*\\ne 0$ ) BHs.",
"The precise transformations are given in the papers by Chandrasekhar and Detweiler and are of the form Z(r*) = A(r*)R(r(r*)) + B(r*)dRdr* , dZdr* = C(r*)R(r(r*)) + D(r*)dRdr* .",
"Imposing that the equation governing $Z$ is Eq.",
"(REF ) while $R$ satisfies Eq.",
"(REF ) gives a system of equations that the functions $A$ , $B$ , $C$ and $D$ must fulfill.",
"Solutions of this system give the form of the potential $V_s(r(r^*))$ .",
"These potentials are, for a field of spin $s$ $V_0(r) =\\dfrac{\\Delta }{\\rho ^4}\\left( \\lambda _{0\\,lm} + \\dfrac{\\Delta + 2r(r-M)}{\\rho ^2} \\dfrac{3r^2\\Delta }{\\rho ^4} \\right)\\,, $ V1/2,(r) = (1/2 lm+1)4 (1/2,l,m+1)4 ( (r-M) - 2r2 ) , $V_{1,\\pm }(r) = \\dfrac{\\Delta }{\\rho ^4}\\left( (\\lambda _{1\\,lm}+2)-\\alpha ^2\\dfrac{\\Delta }{\\rho ^4} \\mp i\\alpha \\rho ^2 \\dfrac{{\\rm d}}{{\\rm d}r}\\left( \\dfrac{\\Delta }{\\rho ^4} \\right) \\right)\\,, $ V2(r) = 8( q - 2(q-)2( (q-)( 2q - 22q - 2r(q- q)) + 2(2 - q + )(q- q) ) ) .",
"The different potentials for a given spin lead to the same results.",
"In the potential for spin 2 particles, the following quantities appear q(r) = 4 + 32(r2-a2) - 3r2 , q(r) = r( (4+ 6)2 - 6(r2 - 3Mr + 2a2) ) , q(r) = (4+6)2 + 8r2 - 6r2 + 36Mr - 12a2 , q- q= -2(r-M)4 + 22(2r-3M(r2 + a2) + 6ra2) + 12r(Mr - a2) , $\\beta _\\pm = \\pm 3\\alpha ^2\\,,$ $\\kappa _\\pm = \\pm \\sqrt{36M^2 -2\\nu (\\alpha ^2(5\\nu +6)-12a^2) + 2\\beta \\nu (\\nu +2)}\\,,$ $q-\\beta _+\\Delta = \\rho ^2(\\nu \\rho ^2 + 6Mr - 6a^2)\\,,$ $q-\\beta _-\\Delta = \\nu \\rho ^4 + 6r^2(\\alpha ^2-a^2) + 6Mr(r^2-\\alpha ^2)\\,,$ where $\\nu \\equiv \\lambda _{2\\,lm} + 4$ .",
"More details on how we numerically solve the Schrödinger-like wave equation (REF ) with the potentials Eqs.",
"(REF ) to () are presented in the BlackHawk manual [22]."
]
] | 1906.04196 |
[
[
"EdMot: An Edge Enhancement Approach for Motif-aware Community Detection"
],
[
"Abstract Network community detection is a hot research topic in network analysis.",
"Although many methods have been proposed for community detection, most of them only take into consideration the lower-order structure of the network at the level of individual nodes and edges.",
"Thus, they fail to capture the higher-order characteristics at the level of small dense subgraph patterns, e.g., motifs.",
"Recently, some higher-order methods have been developed but they typically focus on the motif-based hypergraph which is assumed to be a connected graph.",
"However, such assumption cannot be ensured in some real-world networks.",
"In particular, the hypergraph may become fragmented.",
"That is, it may consist of a large number of connected components and isolated nodes, despite the fact that the original network is a connected graph.",
"Therefore, the existing higher-order methods would suffer seriously from the above fragmentation issue, since in these approaches, nodes without connection in hypergraph can't be grouped together even if they belong to the same community.",
"To address the above fragmentation issue, we propose an Edge enhancement approach for Motif-aware community detection (EdMot).",
"The main idea is as follows.",
"Firstly, a motif-based hypergraph is constructed and the top K largest connected components in the hypergraph are partitioned into modules.",
"Afterwards, the connectivity structure within each module is strengthened by constructing an edge set to derive a clique from each module.",
"Based on the new edge set, the original connectivity structure of the input network is enhanced to generate a rewired network, whereby the motif-based higher-order structure is leveraged and the hypergraph fragmentation issue is well addressed.",
"Finally, the rewired network is partitioned to obtain the higher-order community structure."
],
[
"Introduction",
"Recent years have witnessed a growing trend in many different disciplines that model and interpret structured data as networks [1], [19].",
"As ubiquitous abstractions depicting relationships among entities, networks have become a focus of data science and network analysis has attracted an increasing amount of attention from different fields such as physics, biology, mathematics and computer science [23], [24], [14].",
"Community detection is an important task in network analysis that aims to partition the network into communities of strongly connected nodes.",
"Although many community detection methods have been proposed [31], [10], [5], most of them only take into consideration the lower-order structure of the network at the level of individual nodes and edges, which fails to unravel the higher-order organization of the network.",
"Recently, to go beyond the lower-order connectivity patterns, some motif-based higher-order community detection methods have been proposed [34], [3], [20], [13], [21], [15], which can capture the motif-based characteristics and gain new insights into the organization of the network.",
"However, they typically focus on only the higher-order connections that are encoded in the motif-based hypergraph but underestimate and even violate the original lower-order topological structure.",
"In particular, the hypergraph is assumed to be a connected graph in which the consequent partitioning procedure can be applied.",
"However, in some real-world networks, such assumption cannot be ensured.",
"That is, the hypergraph may be fragmented to a large number of connected components (with various sizes) and isolated nodes.",
"This is because the higher-order connections among nodes are built upon motifs.",
"Two nodes are said to have higher-order connection if they have involved in at least one common motif and vise versa.",
"It's worth noting that two nodes can be separated in the hypergraph of higher-order connections despite that they are connected in the original network.",
"In this way, the number of connected components would increase and more isolated nodes would appear.",
"These isolated nodes (from the perspective of higher-order connections) will render the community structure with instability.",
"Therefore, the existing higher-order methods would suffer seriously from the above fragmentation issue since nodes without higher-order connections can't be grouped together even if they belong to the same community.",
"To address the above fragmentation issue, we propose an Edge enhancement approach for Motif-aware community detection (EdMot).",
"The main idea is as follows.",
"Firstly, a motif-based hypergraph is constructed and the top $K$ largest connected components (measured by the number of contained nodes) in the hypergraph are partitioned into modules.",
"Afterwards, the connectivity structure within each module is strengthened by constructing an edge set to derive a clique from each module.",
"Based on the new edge set, the original connectivity structure of the input network is enhanced to generate a rewired network, whereby both the higher-order structure and lower-order structure are integrated.",
"Finally, by partitioning the rewired network, the higher-order community structure can be discovered.",
"In this study, we focus on the triangle motif for its ubiquitousness in social networks [12], [26], [29], but our technique can be extended to other motifs as well.",
"Extensive experiments have been conducted on eight real-world datasets and the results show the effectiveness of the proposed method in improving the community detection performance of state-of-the-art methods.",
"We summarize the main contributions as follows: We present and formalize the hypergraph fragmentation issue suffered by the existing motif-based community detection methods, where higher-order connections are preserved but the original lower-order structure may be underestimated and even violated.",
"We propose an Edge enhancement approach for Motif-aware community detection (EdMot), which can not only leverage higher-order connections of the network but also overcome the hypergraph fragmentation issue.",
"Extensive experiments are conducted on eight real-world datasets to show the effectiveness of the proposed method."
],
[
"Background and Problem Statement",
"Assume that we are given a network $\\mathcal {G}=\\lbrace \\mathcal {V},\\mathcal {E}\\rbrace $ , where $\\mathcal {V} = \\lbrace v_{i}|i=1,\\dots ,n\\rbrace $ represents the node set consisting of $n$ nodes and $\\mathcal {E}=\\lbrace e_{i}|i=1,\\dots ,m\\rbrace $ represents the edge set consisting of $m$ undirected and unweighted edges.",
"A node adjacency matrix $A\\in \\mathbb {R}^{n \\times n}$ is used to encode the node-wise connection of the network, which is also known as the lower-order structure of the network [3].",
"In this paper, the largest connected component of the original network will be extracted and utilized if it contains isolated nodes.",
"We aim to infer the corresponding higher-order structure, which may characterize the building block of the network.",
"In particular, the typical higher-order connectivity patterns, i.e., motifs, are identified to gain new insights into the organization of the network.",
"Formally, a network motif with $p$ nodes and $q$ edges can be denoted as: $\\mathbf {M}_{p}^{q} = \\lbrace \\mathcal {V}_\\mathbf {M},\\mathcal {E}_\\mathbf {M}\\rbrace $ where ${\\mathcal {V}_\\mathbf {M}}\\subseteq {\\mathcal {V}}$ represents the set of $p$ nodes and ${\\mathcal {E}_\\mathbf {M}}\\subseteq {\\mathcal {E}}$ represents the set of $q$ edges.",
"Following the convention of the literature [12], [26], [29], we focus on $\\mathbf {M}_{3}^{3}$ , i.e., the triangle motif (as shown in the middle part of REF ).",
"However, our technique can be well extended to other motifs.",
"Conceptually, given the original node adjacency matrix $A$ , the motif adjacency matrix constructed from the triangle motif, denoted as $W_{\\mathbf {M}} \\in \\mathbb {R}^{n \\times n}$ , can be defined: (WM)ij = number of motif instances containing nodes $i$ and $j$ .",
"Note that $W_{\\mathbf {M}}$ encodes the motif-based higher-order connections of the network, where edges correspond to co-occurrences in motifs and $(W_{\\mathbf {M}})_{ij}$ can be 0 even though $A_{ij}>0$ .",
"Several methods have been developed in terms of motif-based community detection [3], [34].",
"The common denominator of these methods is the dedication to leverage higher-order network structures effectively for community detection.",
"However, they rely on the construction of a hypergraph whose edges correspond to motifs [3].",
"In this way, the motif-based higher-order structure of the network is highlighted and well preserved but the original lower-order structure is underestimated and even violated.",
"That is, edges that do not involve in any motifs would be eliminated.",
"This leads to the serious issue called hypergraph fragmentation.",
"By “fragmentation”, it means that the hypergraph or the motif adjacency matrix constructed from the original single connected component is usually fragmented into a large number of connected components with various sizes and isolated nodes due to the lack of higher-order connection, i.e., lack of involvement in the motif [37].",
"As shown in REF , by constructing the motif-based hypergraph from the original network structure of the Cora dataset, a large number of connected components with various sizes and isolated nodes having no connection to any other nodes will be generated from the original single large connected component.",
"As a consequence, most of the existing higher-order community detection methods that utilize the hypergraph would suffer from the hypergraph fragmentation issue [34], [37], [3].",
"In particular, by definition of communities (nodes should be densely connected within communities but sparsely connected between communities), the existing methods assume that the intermediate hypergraph is a single connected component [3], [34], from which the eventual higher-order community structure can be discovered.",
"However, as discussed above, the hypergraph is usually fragmented with a large number of connected components and isolated nodes.",
"Therefore, the performance of the existing higher-order community detection methods can not be guaranteed.",
"For example, there will be a large number of dead nodes in the random walk process.",
"Such isolated nodes would present confusing or rather, generate uncertain community memberships if the random label assignment strategy or the exclusion strategy is applied [16], leading to the degenerate performance.",
"A naive solution is directly grouping the connected components or isolated nodes together in the hypergraph to form the communities.",
"Unfortunately, these components may be of different sizes and nodes with different ground-truth community labels may reside in the same component.",
"For example, in REF , the largest connected component in the hypergraph consists of nodes from 7 different ground-truth communities.",
"Furthermore, serious randomness would be introduced by the isolated nodes due to the lack of higher-order connections because there is no such community definition customized for isolated nodes.",
"In this case, community memberships will be rendered with randomness if the isolated nodes are grouped randomly.",
"To address the above hypergraph fragmentation issue, we propose a novel method termed EdMot for motif-aware community detection.",
"On the one hand, the motif-based higher-order structure is leveraged.",
"On the other hand, the hypergraph fragmentation issue suffered by the traditional higher-order community detection methods can be well addressed by the newly designed edge enhancement strategy."
],
[
"Connected Component Identification",
"We commence by constructing a motif adjacency matrix, (a.k.a.",
"hypergraph) $W_{\\mathbf {M}}$ through the original network structure to encode the higher-order connections.",
"It can also be represented in the set form as follows: $\\mathcal {G}^{\\mathbf {M}} = \\lbrace \\mathcal {V},\\mathcal {E}^{\\mathbf {M}} \\rbrace $ In the above equation, $\\mathcal {G}^{\\mathbf {M}}$ represents the motif-based hypergraph, $\\mathcal {V}$ represents the node set that is the same as the original network, and $\\mathcal {E}^{\\mathbf {M}}$ represents the edge set consisting of $m_{w}$ weighted edges generated based on triangle motifs: $\\mathcal {E}^{\\mathbf {M}} = \\lbrace (a,b,\\tau )_{i}\\rbrace $ where $a,b\\in \\mathcal {V}$ are two end nodes of the $i$ -th edge ($i \\in \\lbrace 1,\\cdots , m_{w}\\rbrace $ ) and $\\tau $ is the edge weight, i.e., the number of motif instances that contain node $a$ and node $b$ together.",
"Accordingly, a set of $c_{\\Phi }$ connected components of the hypergraph can be identified: $\\Phi = \\lbrace \\phi _{i}\\rbrace $ where $\\phi _{i}$ is the $i$ -th connected component.",
"That is, $\\phi _{i} = \\lbrace \\mathcal {V}^{\\phi }_{i}, \\mathcal {E}^{\\mathbf {M}}_{i} \\rbrace $ , $i \\in \\lbrace 1, \\cdots ,c_{\\Phi }\\rbrace $ , where $\\mathcal {V}^{\\phi }_{i}\\subseteq \\mathcal {V}$ is the set of nodes involved in the $i$ -th connected component and $\\mathcal {E}^{\\mathbf {M}}_{i}\\subseteq \\mathcal {E}^{\\mathbf {M}}$ is the weighted edge set of the $i$ -th connected component respectively.",
"Apart from $c_{\\Phi }$ connected components, there is a set of isolated nodes in the hypergraph, denoted as $\\mathcal {V}_{iso} = \\mathcal {V} - \\bigcup _{i \\in \\lbrace 1, \\cdots , c_{\\Phi }\\rbrace } \\mathcal {V}^{\\phi }_{i}$ Actually, motif structures are well preserved in these components, i.e., triangle motifs can certainly be found in the components containing 3 or more nodes.",
"The top $K$ largest connected components (measured by the number of contained nodes), i.e., $ \\Phi _{K} \\subseteq \\Phi $ ($K<c_{\\Phi }$ ) will be obtained.",
"And each connected component in $\\Phi _{K}$ will be further partitioned into modulesTo avoid confusing the partitioning results in each connected component with the eventual partitioning results, the “module” is used to name the partitioning results in each connected component.",
"by using some traditional graph partitioning methods.",
"The influence of the parameter $K$ will be analyzed in later section."
],
[
"Connected Component Partitioning",
"As an example, we adopt the Louvain [5] method that heuristically maximizes the well-known community structure evaluation measure called modularity [24], [25] to partition each connected component $\\phi _{l}\\in \\Phi _{K}$ into modules.",
"In particular, by taking each connected component $\\phi _{l}$ as the input network, the modularity $Q$ can be given as [24]: $Q = \\frac{1}{4\\mu }\\sum _{ij}(A_{ij}-\\frac{k_{i}k_{j}}{2\\mu })(s_{i}s_{j}+1) = \\frac{1}{4\\mu }\\sum _{ij}(A_{ij}-\\frac{k_{i}k_{j}}{2\\mu })s_{i}s_{j}$ where $k_{i}$ and $k_{j}$ are the degrees of nodes $i$ and $j$ respectively and $\\mu = \\frac{1}{2}\\sum _{i}k_{i}$ is the total number of edges in the network.",
"$s_{i}$ is the community label of node $i$ .",
"$\\frac{k_{i}k_{j}}{2\\mu }$ is the expected number of edges between node $i$ and node $j$ in the randomly rewired graph preserving the same degree distribution.",
"$s_{i}s_{j}$ is equal to 1 if node $i$ and node $j$ belong to the same community and -1 otherwise.",
"The output of the above modularity-based community detection procedure is the partitions (modules) in the $l$ -th connected component $\\phi _{l}\\in \\Phi _{K}$ .",
"By putting all the partitions (modules) of all the top $K$ largest connected components together, we can obtain a module set, denoted as $\\lbrace \\mathcal {M}_{1},\\cdots , \\mathcal {M}_{\\bar{m}}\\rbrace $ , where $\\bar{m}$ is the number of modules obtained by partitioning all the top $K$ largest connected components.",
"It is worthy noting that many other graph partitioning schemes can also be applied."
],
[
"Network Rewiring via Edge Enhancement",
"The module set $\\lbrace \\mathcal {M}_{1},\\cdots , \\mathcal {M}_{\\bar{m}}\\rbrace $ obtained in the previous subsection partially encodes the higher-order community structure.",
"The reason for using “partially” is as follows.",
"Property I: If two nodes belong to the same module, it is likely that they have the same ground-truth community label as shown by the existing higher-order community detection approaches [2], [3], i.e., discovering higher-order communities from each connected component in the hypergraph.",
"However, it is also possible that two different modules may belong to the same ground-truth community.",
"Property II: The module set $\\lbrace \\mathcal {M}_{1},\\cdots , \\mathcal {M}_{\\bar{m}}\\rbrace $ obtained so far involves only part of nodes in the network since only the top $K$ largest connected components in the hypergraph are processed in the connected component partitioning.",
"To completely reveal the higher-order community structure, the lower-order structure should be taken into account as a complement to the higher-order connections.",
"To this end, an edge enhancement approach is developed to rewire the connectivity structure of the original network, whereby both the connectivity structure within each module in $\\lbrace \\mathcal {M}_{1},\\cdots , \\mathcal {M}_{\\bar{m}}\\rbrace $ is strengthened and the original lower-order structure is considered.",
"In this way, not only the higher-order connections of the network can be leveraged but also the hypergraph fragmentation issue can be well addressed.",
"First of all, the connectivity structure within each module $\\lbrace \\mathcal {M}_{1},\\cdots , \\mathcal {M}_{\\bar{m}}\\rbrace $ is strengthened as follows.",
"For the nodes that have already been partitioned into the same module, e.g., $\\mathcal {M}_{i} \\in \\lbrace \\mathcal {M}_{1},\\cdots , \\mathcal {M}_{\\bar{m}}\\rbrace (i \\in \\lbrace 1,\\cdots , \\bar{m}\\rbrace )$ , their connectivity is strengthened in line with the assumption that motif-based connection should have higher priority compared with the lower-order connection.",
"This is because motif-based connection reflects the social transitivity and may encode more impressive characteristics of the network [35], [33].",
"To this end, a clique is obtained for each module $\\mathcal {M}_{i}\\in \\lbrace \\mathcal {M}_{1},\\cdots , \\mathcal {M}_{\\bar{m}}\\rbrace $ by constructing an edge to each node pair in $\\mathcal {M}_{i}$ .",
"In this way, a new set of edges is constructed, denoted as $\\mathcal {E}^{*}_{mod}$ , $\\mathcal {E}^{*}_{mod}=\\lbrace (a,b)|\\forall a,b\\in \\mathcal {M}_{i}, \\forall i=1,\\cdots ,\\bar{m}\\rbrace $ Notice that, the nodes within each module $\\mathcal {M}_{i}$ are interconnected with each other by the strongest connectivity pattern, i.e.",
"a clique structure, which is almost impossible to be destroyed in the consequent partitioning procedure.",
"According to Property I, it is rational to establish such strong connectivity.",
"However, according to Property II, it is also necessary to take into account the nodes residing out of the module set as well as the original lower-order connectivity pattern to overcome the fragmentation issue.",
"Therefore, based on the new edge set $\\mathcal {E}^{*}_{mod}$ , the original connectivity structure of the input network is enhanced to generate a rewired network $\\mathcal {G}_{A}^{\\mathbf {M}} = \\lbrace \\mathcal {V},\\mathcal {E}_{A}^{\\mathbf {M}}\\rbrace \\text{ with~~} \\mathcal {E}_{A}^{\\mathbf {M}} = \\mathcal {E} \\cup \\mathcal {E}^{*}_{mod}$ In this way, the edge enhanced network contains the same node set as the original network but encodes the connectivity patterns from both the original lower-order network structure in terms of $\\mathcal {E}$ and the higher-order connections in terms of $\\mathcal {E}^{*}_{mod}$ ."
],
[
"Method Summary and Computational Complexity",
"The rewired network, i.e., $\\mathcal {G}_{A}^{\\mathbf {M}}$ , is fed into some graph partitioning methods to obtain the final community structure, i.e., $\\mathcal {\\widehat{C}} = \\lbrace \\mathcal {C}_{1}, \\cdots , \\mathcal {C}_{c}\\rbrace $ , where $c$ is the number of communities in the final partitioning.",
"For clarification, Algorithm 1 summarizes the main procedure of the proposed EdMot method.",
"In addition, REF exhibits the whole process intuitively.",
"[htb] The proposed EdMot approach.",
"Input: Original network $A\\in \\mathbb {R}^{n \\times n}$ , parameter $K$ , a graph partitioning method S. [1] Construct motif adjacency matrix $W_{\\mathbf {M}}$ from $A$ via ().",
"Obtain connected components $\\Phi $ via (REF ) and select top $K$ largest connected components $\\Phi _{K} \\subseteq \\Phi $ .",
"Apply S to partition each $\\phi _{l} \\in \\Phi _{K}$ , $\\forall l \\in \\lbrace 1,\\cdots , K\\rbrace $ and aggregate the results to form the module set $\\lbrace \\mathcal {M}_{1},\\cdots , \\mathcal {M}_{\\bar{m}}\\rbrace $ .",
"Construct a new edge set $\\mathcal {E}^{*}_{mod}$ via (REF ).",
"Rewire the original network to obtain $\\mathcal {G}_{A}^{\\mathbf {M}}$ via (REF ).",
"Feed $\\mathcal {G}_{A}^{\\mathbf {M}}$ into S to obtain final community structure $\\mathcal {\\widehat{C}}$ .",
"Output: Community structure $\\mathcal {\\widehat{C}}$ .",
"Figure: Illustration of the proposed EdMot algorithm.",
"A synthetic network is designed to serve as the original network, where nodes and edges in three communities are denoted with different colors and the black edges represent the inter-community edges.",
"Specifically, by constructing the motif-based hypergraph in step 1, the hypergraph fragmentation issue arises, where two connected components and three isolated nodes are generated in the hypergraph.",
"By partitioning the largest connected component into modules in step 2, two modules can be identified and a new edge set is constructed to derive a clique from each module, as shown as the dashed line.",
"By rewiring the network in step 3, a rewired network can be obtained by substituting the new edge set into the original network.",
"Finally, by partitioning the rewired network into communities, the community structure can be discovered.We now analyze the computational complexity of the proposed EdMot method.",
"Overall, the complexity of the method is governed by the computations of the motif adjacency matrix $W_{\\mathbf {M}}$ and the graph partitioning in both the connected components and rewired network.",
"For simplicity, we assume that we can access the edges in a graph in $O(1)$ time and can access and modify the elements in the matrix in $O(1)$ time.",
"The computational time of constructing $W_{\\mathbf {M}}$ is bounded by the time to find all the motif instances in the graph.",
"Theoretically, we can compute $W_{\\mathbf {M}}$ in $O(n^{p})$ time for a motif with $p$ nodes.",
"However, most real-world networks are sparse, we can instead focus on the computational complexity in terms of the number of edges in the network.",
"In particular, for the triangle motifs discussed in this paper, the motif instances can be found in $O(m^{1.5})$ time [18], [4], [3].",
"As for the graph partitioning, in this paper, we adopt the heuristic modularity maximization method as an example, which can be finished in $O(n\\log n)$ on average for its ability to find the hierarchical community structure [5].",
"Therefore, the overall computational complexity is $O(m^{1.5}+n\\log n)$ .",
"In this section, extensive experiments are conducted to confirm the effectiveness of the proposed method.",
"The Matlab code is available in https://github.com/lipzh5/EdMot_pro.git."
],
[
"Datasets and Evaluation Measures",
"In our experiments, eight real-world datasets are used (the edges are treated to be undirected for all the datasets).",
"On the first four datasets, the ground-truth community labels are provided for evaluation purpose and on the rest four datasets, there is no ground-truth community information where the internal evaluation measures such as modularity are used for evaluation.",
"Datasets with ground-truth community labels: polbooks[1].",
"A network of books about US politics, where edges between books represent frequent copurchasing of books by the same buyers.",
"It consists of 105 nodes and 441 edges.",
"email-Eu-core[2].",
"A email network of communication between institution members, which is generated using email data from a large European research institution.",
"It consists of 1005 nodes and 25571 edges.",
"polblogs[1].",
"A network of hyperlinks between weblogs on US politics, which is recorded in 2005.",
"It consists of 1490 nodes and 19090 edges.",
"Cora[3].",
"A citation network of scientific publications, which consists of maching learning papers that can be classified into seven classes.",
"It consists of 2708 nodes and 5429 edges.",
"Datasets without ground-truth community labels: power[4].",
"A network that represents the topology of the Western States Power Grid of the United States.",
"It consists of 4941 nodes and 6594 edges.",
"ca-GrQc[2].",
"A collaboration network of scientific collaborations between authors with papers submitted to General Relativity and Quantum Cosmology category.",
"It consists of 5242 nodes and 14496 edges.",
"as-22july06[4].",
"A symmetric snapshot of the structure of the Internet at the level of autonomous systems.",
"It consists of 22963 nodes and 48436 edges.",
"email-Enron[4].",
"A email network that covers all the email communication within a dataset of around half million emails.",
"It consists of 36692 nodes and 367662 edges.",
"[1]http://www-personal.umich.edu/~mejn/netdata/ [2]http://snap.stanford.edu/data/ [3]http://linqs.cs.umd.edu/projects/projects/lbc/ [4]https://graph-tool.skewed.de/static/doc/collection.html Three commonly used evaluation measures are adopted for evaluating the quality of the discovered community structure, namely Normalized Mutual Information (NMI), F-score and Modularity (Q) [8].",
"The first two measures require the ground-truth community labels for evaluation purpose and the values are in the range between 0 and 1.",
"The last measure $Q$ ranges from -1 to 1.",
"For all the three measures, a higher value indicates better performance."
],
[
"Graph Partitioning Methods",
"We adopt the following four graph partitioning methods in our experiments: Louvain [5]: It is a greedy community detection method that can reveal the hierarchical community structure.",
"Spectral Clustering (SC) [27]: It performs a spectral clustering of the node adjacency matrix into $c$ clusters.",
"Affinity Propagation (AP) [9]: It is a fast clustering method based on similarities between pairs of data points.",
"Nonnegative Matrix Factorization (NMF) [6]: It obtains the new property representation by factorizing the node adjacency matrix into two nonnegative matrices.",
"Each of the above four graph partitioning methods is taken as the input “graph partitioning method S” of Algorithm REF , by which different variants of EdMot are obtained, namely EdMot-Louvain, EdMot-SC, EdMot-AP and EdMot-NMF.",
"Similarly, as the conventional higher-order community detection methods, each of the above four graph partitioning methods takes as input the hypergraph of higher-order connectivity, i.e.",
"the motif adjacency matrix $W_{\\mathbf {M}}$ , by which different variants of the higher-order community detection methods are obtained, namely Motif-Louvain, Motif-SC, Motif-AP and Motif-NMF.",
"And the original graph partitioning method, the corresponding higher-order variant, and the corresponding EdMot variant are compared and analyzed.",
"For instance, Louvain, Motif-Louvain and EdMot-Louvain are compared."
],
[
"Comparison Results",
"Comparison results are reported from Table REF to Table REF , where the scores are averaged over 20 runs for every method and the standard deviations are also reported.",
"Table REF to Table REF report the results on datasets with ground-truth community labels while Table REF is for two datasets without ground-truth community labels.",
"Additionally, the average rank is also provided in the last column, which is computed by averaging the ranking positions of each method across the testing datasets.",
"Specifically, Table REF reports the comparison results among Louvain, Motif-Louvain and EdMot-Louvain.",
"As can be seen, the best scores are achieved by EdMot-Louvain in terms of NMI, F-score and modularity on polbooks, polblogs and Cora.",
"On average, about 17% and 14% improvements (in terms of NMI) have been achieved by EdMot-Louvain over Louvain and Motif-Louvain respectively.",
"This may be due to the utilization of edge enhancement and the original network structure, which plays an important role in avoiding hypergraph fragmentation and preserving the structural information of the network.",
"On the email-Eu-core dataset, the proposed method performs worse than Motif-Louvain in terms of NMI.",
"The reason is that this dataset has a relatively denser linkage structure (i.e., 1005 nodes and 25571 edges) and the hypergraph fragmentation issue does not appear on this dataset.",
"That is, the motif-based hypergraph is a single connected component.",
"While the polblogs and Cora datasets suffer from the hypergraph fragmentation issue, which accounts for the better performance achieved by the proposed method.",
"As for the polbooks dataset, even though the hypergraph fragmentation does not appear, the proposed method still performs well.",
"Similar analysis can be made in Table REF to Table REF .",
"In summary, the proposed method can perform better than the original graph partitioning methods and the traditional higher-order methods with hypergraph fragmentation issue.",
"What's more, it may also improve the performance of some state-of-the-art graph partitioning methods despite there is no hypergraph fragmentation issue.",
"Table: Comparison results in terms of modularity on the two datasets without ground-truth community labels."
],
[
"Parameter Analysis",
"In this section, parameter analysis is conducted to investigate the effect of the parameter $K$ on the performance of our EdMot method.",
"Due to the space limit, we will take the Louvain method as the input graph partitioning method.",
"However, similar analysis can be conducted by using any other graph partitioning methods.",
"The effect of $K$ on the performance of EdMot-Louvain is shown in REF and REF .",
"Specifically, the effect of $K$ on polblogs and Cora is presented in REF .",
"Since the motif-based hypergraphs of polbooks and email-Eu-core contain only one connected component, the discussion for these two datasets is omitted here.",
"As can be seen, the scores of the evaluation measures hardly change as $K$ increases.",
"This is because the motif-based hypergraph often contains a largest connected component that may consist of nodes from different communities and a large number of peripheral nodes that are isolated (see REF for illustration).",
"Therefore, it is beneficial to partition the largest connected component in the hypergraph ($K=1$ ) into modules.",
"Similar phenomenon can also be observed from the power and ca-GrQc datasets as shown in REF .",
"As for the as-22july06 ($n=22963$ , $m=48436$ ) and the email-Enron ($n=36692$ , $m=367662$ ) datasets, which are relatively larger than other datasets, the best performance is achieved when $K=3$ and $K=2$ respectively.",
"Thus, we set $K=3$ for the as-22july06 dataset, $K=2$ for the email-Enron dataset and $K=1$ for the other datasets in the experiments."
],
[
"Lower-Order Community Detection",
"Lower-order community detection methods discover communities by mainly leveraging the lower-order connectivity patterns of the network at the level of individual nodes and edges.",
"For example, the Louvain method was proposed to reveal the hierarchical community structure by heuristically optimizing modularity [5].",
"The Nonnegative Matrix Factorization (NMF) method factorizes the node adjacency matrix into two nonnegative matrices and yields a new parts-based data representation [7], [6].",
"From the perspective of clustering nodes in the network into clusters, the Affinity Propagation (AP) clustering method was proposed to cluster nodes based on pair-wise similarities [9].",
"Similarly, the Spectral Clustering (SC) method was proposed to cluster nodes using eigenvectors of matrix [27].",
"Besides, a generative model termed Stochastic BlockModel (SBM) was proposed to detect communities by fitting blockmodels to the network data [17], [28].",
"And the permanence based method was also proposed to provide a more fine-grained view of the modular structure of the network [8].",
"In addition, the label propagation based methods were developed, which possess some advantages such as the simplicity and nearly linear time complexity [30]."
],
[
"Motif-based Higher-Order Community Detection",
"Network motifs are defined as patterns of interconnections occurring in networks at numbers that are significantly higher than those in the random networks [22], [3].",
"As the building blocks of the network, motifs are widely applied to unravel the design principles of gene regulation networks [32] and the underlying mechanisms of social networks [11], [35].",
"Different from the lower-order community detection methods, higher-order community detection methods discover communities by leveraging the higher-order connectivity patterns of the network at the level of small network subgraphs, e.g., motifs.",
"For example, motif was used to define communities by extending the mathematical expression of Newman Girvan modularity in [2].",
"A graph sparsification principle based on graph motifs was proposed so as to improve the efficiency and quality for graph partitioning [37].",
"The motif conductance based framework was also proposed to reveal higher-order organization of complex networks [3].",
"Besides, a highly effective heuristic method was proposed based on motifs, where a random walk interpretation of the graph reweighing scheme was developed [34].",
"In addition, motifs have been leveraged in local higher-order graph partitioning [36].",
"Despite the success in preserving the building blocks, the existing higher-order community detection methods may violate the original lower-order structure of the network.",
"Specifically, the motif-based hypergraph may consist of a large number of connected components with various sizes and isolated nodes having no connections to any other nodes.",
"In this case, they may suffer seriously from the hypergraph fragmentation issue.",
"To address this issue, we propose an edge enhancement approach for motif-aware community detection (EdMot), which can not only leverage higher-order connections of the network but also overcome the hypergraph fragmentation issue."
],
[
"Conclusion",
"In this paper, we for the first time propose a novel motif-aware community detection method termed EdMot for addressing the hypergraph fragmentation issue.",
"Different from the existing higher-order community detection methods that directly operate on the possibly fragmented hypergraph, EdMot partitions the top $K$ largest connected components in the hypergraph into modules.",
"And then, an edge enhancement approach is designed for enhancing the connectivity structure of the original network as follows.",
"1) First, a new edge set is constructed to derive a clique from each module.",
"2) Based on the new edge set, the original connectivity structure of the input network is enhanced to generate a rewired network, whereby the motif-based higher-order structure is leveraged and the hypergraph fragmentation issue is well addressed.",
"After the edge enhancement, the rewired network is partitioned to obtain the higher-order community structure.",
"Extensive experiments have been conducted to show the effectiveness of the proposed method."
],
[
"Acknowledgments",
"This work was supported by NSFC (61876193), National Key Research and Development Program of China (2016YFB1001003), Guangdong Natural Science Funds for Distinguished Young Scholar (2016A030306014), and Key Areas Research and Development Program of Guangdong (2018B010109007)."
]
] | 1906.04560 |
[
[
"Computation of the dynamic critical exponent of the three-dimensional\n Heisenberg model"
],
[
"Abstract Working in and out of equilibrium and using state-of-the-art techniques we have computed the dynamic critical exponent of the three dimensional Heisenberg model.",
"By computing the integrated autocorrelation time at equilibrium, for lattice sizes $L\\le 64$, we have obtained $z=2.033(5)$.",
"In the out of equilibrium regime we have run very large lattices ($L\\le 250$) obtaining $z=2.04(2)$ from the growth of the correlation length.",
"We compare our values with that previously computed at equilibrium with relatively small lattices ($L\\le 24$), with that provided by means a three-loops calculation using perturbation theory and with experiments.",
"Finally we have checked previous estimates of the static critical exponents, $\\eta$ and $\\nu$, in the out of equilibrium regime."
],
[
"Introduction",
"The study of the dynamics in and out of equilibrium in a critical phase is of paramount importance since it permits to extract the critical exponents of the system, hence, to characterize its universality class.",
"In the last decades a great amount of work, analytical, numerical and experimental, has been devoted to study these issues.",
"One of the studied systems has been the three dimensional (isotropic) classical Heisenberg model.",
"The dynamic critical exponent, $z$ , has been computed using field theory by studying its Model A dynamics (pure relaxational dynamics in which the order parameter is not conserved).",
"[1], [2], [3] A three-loop computation reported in Ref.",
"[4] provided $z=2.02$ .",
"$ z =2 + c \\eta \\,\\,,\\,\\, c = 0.726-0.137 \\epsilon + O(\\epsilon ^2)\\,, $ where $\\eta $ is the anomalous dimension of the field (from static) and $\\epsilon =4-d$ , $d$ being the dimensionality of the model.",
"The equilibrium dynamics of this model was studied by means of numerical simulations in Ref.",
"[5] and $z=1.96(6)$ was reported.",
"The authors were aware that this exponent was slightly below the analytical computation of Ref.",
"[4] and discuss in the paper different systematic bias.",
"For example, a relatively narrow range of the lattice sizes and despite the accuracy of their values for the correlation times, a more precise determination of these times were needed to study the corrections to the scaling presented in the model.",
"From the experimental side, the situation is complicated due to the crossover from the Heisenberg universality class to the dipolar one which induces a change from $z\\sim 2.5$ (Heisenberg with conserved magnetization and reversible forces, model J [1], [2], [3]) to $z\\sim 2$ (dipolar).",
"[6], [7] In particular using PAC Perturbed Angular Correlations of $\\gamma $ ray spectroscopy.",
"Hohenemser et al.",
"found $z=2.06(4)$ [7] for Ni and $z\\simeq 2$ for Fe; Dunlap et al.",
"[8] using ESR Electron Spin Resonance.",
"found $z=2.04(7)$ for EuO; and $z=2.09(6)$ was found by Bohn et al for EuS [9] using inelastic neutron scattering.",
"It seems that the interplay of spin dipoles with orbital angular momentum or dipolar interactions breaks the conservation of the magnetization on these materials, producing a crossover between Heisenberg model J ($z\\sim 2.5$ ) and Heisenberg model A ($z\\sim 2$ ).",
"[6], [7] Recently, Pelissetto and Vicari [10] have used the value provided by field theory in the scaling analysis of their numerical data to study the off-equilibrium behavior of three-dimensional $O(N)$ models driven by time-dependent external fields and assigned it an error of 0.01, so $z=2.02(1)$ , to take into account the uncertainty on the extrapolation to $\\epsilon =1$ of the three-loop-expansion result.",
"Consequently, it is of paramount importance to obtain an accurate value for this dynamic critical exponent, in order to be used in future numerical analysis and experiments, and also to check the accuracy of the three-loops analytical computation.",
"The main goal of our study is to improve the value of $z$ using numerical simulations by studying the dependence of the integrated correlation time in the equilibrium regime and the behavior of the correlation length, susceptibility and energy with the simulation time in the out of equilibrium region.",
"In this way, we can compare the performance of equilibrium and out-of-equilibrium methods in the computation of the dynamic critical exponent.",
"Nowadays, a great amount of work has been devoted to study numerically the dynamics of disordered systems, see for example Refs.",
"[11], [12], [13], [14], [15], [16], [17].",
"In general a sudden quenched is performed to work in the off-equilibrium, yet, in other studies the models have been simulated at equilibrium.",
"For example in the three dimensional diluted Ising model both approaches gave the same dynamic critical exponent.",
"[12] In the equilibrium part of the paper we compute the integrated correlation time, avoiding some of the problems which appear in the computation of the exponential one (e.g.",
"assume that the autocorrelation function is a single exponential function [18], [19]).",
"We also study the correlation length in the out-of-equilibrium regime.",
"In the last two decades, this observable has played an important role both in numerical simulations [13], [14] and experiments out of equilibrium [20] in spin glasses.",
"Due to this, powerful numerical techniques has been developed to compute this observable with high accuracy which has allowed a precise determination of the dynamic critical exponent just at the critical point as well as inside the critical spin glass phase.",
"[16] We apply these techniques to the three dimensional (non disordered) Heisenberg model.",
"In addition to the computation of the dynamic critical exponent, we have checked the consistency of previous and very accurate determinations of the static critical exponents ($\\nu $ and $\\eta $ ) in the out-of-equilibrium regime.",
"Our starting point will be the (very precise) critical temperature computed in Ref.",
"[21] and the static critical exponents reported in Refs.",
"[22], [23].",
"We have also measured the dynamic critical exponent from the decay of the energy at criticality.",
"This decay has also been studied in the past in finite dimensional spin glass [13] and recently has played a central role together with the behavior of the correlation length in the analysis of the Mpemba effect, a striking memory effect.",
"[17] The structure of the paper is the following.",
"In the next section we introduce the model and the observables.",
"In section we describe our numerical results: in Sec.",
"REF we report the equilibrium determination of $z$ via the integrated autocorrelation time; in Sec.",
"REF we study the dependence of the correlation length with time and the computation of $z$ out of equilibrium; in Sec.",
"REF and Sec.",
"REF the correlation function and the energy have been studied (respectively).",
"Sec.",
"is devoted to the conclusions.",
"Two appendices close the paper, one to describe our implementation of GPU and the last one to describe how we have computed the statistical error of the exponents with highly correlated data."
],
[
"The model and Observables",
"The Hamiltonian of the three dimensional Heisenberg model is ${\\mathcal {H}} = - \\sum _{<{\\mathitbf r},{\\mathitbf r^{\\prime }}>}\\mathitbf {S}_{\\mathitbf r}\\cdot \\mathitbf {S}_{\\mathitbf r^{\\prime }} \\, .$ $\\mathitbf {S}_{\\mathitbf r}$ is a classical three component spin on the site $\\mathitbf {r}$ of a three dimensional cubic lattice with volume $V=L^3$ and periodic boundary conditions.",
"Without loss of generality we will assume that the spins are unit vectors.",
"The sum runs over all pairs of nearest neighbors spins.",
"We have simulated this model with the standard Metropolis algorithm In the equilibrium simulations, Sec.",
"REF , we have used the Metropolis algorithm proposing a random spin in the unit sphere.",
"In the out of equilibrium simulations we have used the Metropolis algorithm in the standard way [24]: we modify the original spin by adding a random vector and normalizing the final vector to the unit sphere.",
"The magnitude of the random vector is selected in order to maintain an acceptance between 40% and 60%.",
"Both versions of the Metropolis algorithm belong to the same dynamic universality class.",
"[18] and we have run in CPU (smaller time simulations) and GPU (for larger time simulations).",
"Details on the simulations can be found in the appendix A."
],
[
"Equilibrium",
"We address the problem of the computation of the dynamic critical exponent in the equilibrium regime by means the computation and further analysis of the integrated aucorrelation time as a function of the lattice size.",
"We compute for a given observable ${\\cal O}(t)$ , the autocorrelation function (we follow Refs.",
"[25], [18], [24]): $C_{\\cal O}(t)\\equiv \\langle {\\cal O}(s) {\\cal O}(t+s) \\rangle -\\langle {\\cal O}(t) \\rangle ^2\\,,$ and its normalized version $\\rho _{\\cal O}(t)\\equiv C_{\\cal O}(t)/C_{\\cal O}(0)\\,.$ The integrated autocorrelation time is given by $\\tau _{\\mathrm {int},{\\cal O}}= \\frac{1}{2}+ \\sum _{t=0}^\\infty \\rho _{\\cal O}(t)\\,.$ In a run with $N$ measurements, the number of independent measurements of the observable ${\\cal O}$ is just $N/(2 \\tau _{\\mathrm {int},{\\cal O}})$ .",
"[25], [18], [24] If the number of measurements is finite, for large times $t$ the noise will dominate the signal in $\\rho _{\\cal O}(t)$ and to bypass this problem we use the following self-consistent method to compute the integrated time $\\tau _{\\mathrm {int},{\\cal O}}= \\frac{1}{2}+ \\sum _{t=0}^{ c \\tau _{\\mathrm {int},{\\cal O}}} \\rho _{\\cal O}(t)\\,,$ where $c$ is usually taken to be 6 or bigger.",
"[25], [18], [24] At the critical point the integrated autocorrelation time of a long distance observable diverges with the size of the system [12] $\\tau _{\\mathrm {int},{\\cal O}}\\sim L^z \\large (1+ O(L^{-\\omega })\\large )\\,.$ where $z$ is the dynamic critical exponent and $\\omega $ is the leading correction-to-scaling exponent (the leading irrelevant eigenvalue of the theory).",
"Another time, the exponential correlation time, is defined as $\\tau _\\mathrm {exp,{\\cal O}} \\equiv \\limsup _{t\\rightarrow \\infty } \\frac{-t}{\\log \\rho _{\\cal O}(t)} \\,,$ which also depends on the observable used to define $\\rho _{\\cal O}(t)$ .",
"The exponential autocorrelation time controls the approach to the equilibrium.",
"[18] Once we have defined the exponential correlation function we can write the general scaling form for the correlation function [18] $\\rho _{\\cal O}(t)= t^{-p_{\\cal {O}}} f_{\\cal {O}}\\Big (\\frac{t}{\\tau _{\\mathrm {exp},{\\cal O}}}, \\frac{\\xi (L)}{L}\\Big )\\,,$ where $\\xi (L)$ is the equilibrium correlation length computed on a system of size $L$ .",
"Integrating Eq.",
"(REF ) in time, we obtain the integrated correlation time and that $\\tau _\\mathrm {int}\\sim \\tau _\\mathrm {exp} ^{1-p_{\\cal {O}}}$ .",
"Both times are proportional if and only if $p_{\\cal {O}}=0$ and in this situation $z$ is the same for both times.",
"Otherwise, $p_{\\cal {O}}\\ne 0$ , and $\\tau _\\mathrm {exp}$ and $\\tau _\\mathrm {int}$ will provide with different dynamic critical exponents.",
"See also the discussion of Ref.",
"[12].",
"In this paper we will use the slowest mode provided by the non local operator ${\\cal O}=\\mathitbf {M}^2$ , where the magnetization ${\\mathitbf M}$ is defined as ${\\mathitbf M} = \\sum _{\\mathitbf x} {\\mathitbf S}_{\\mathitbf x}\\,.$"
],
[
"Out of equilibrium",
"We have focused on only one local observable, the energy, defined as $e(t)=\\frac{\\langle {\\mathcal {H}} \\rangle _t}{V}\\,.$ We denote the average over different initial conditions at the Monte Carlo time $t$ by $\\langle (\\cdots )\\rangle _t$ .",
"The renormalization group predicts [24], [3], at the critical point, the following behavior for this observable: $e(t)=e_\\infty + C t^{(d-1/\\nu )/z} \\left(1+ A t^{-\\omega /z} \\right)\\,,$ where $d$ is the dimensionality of the space (three in this study) and $\\nu $ is the critical exponent which controls the divergence of the equilibrium correlation length.",
"The other two exponents $z$ and $\\omega $ has been defined in the previous subsection.",
"One of the main observables on this paper is the correlation function defined as: $C(r,t)=\\frac{1}{V} \\sum _{\\mathitbf x}\\langle {\\mathitbf S}_{\\mathitbf x} {\\mathitbf S}_{\\mathitbf {r+x}} \\rangle _t\\,,$ satisfying, at criticality, the following scaling law [3] $C(r,t)=\\frac{1}{r^a} f\\left(\\frac{r}{\\xi (t)}\\right)$ which defines the dynamic correlation length, $\\xi (t)$ .",
"As we approach the equilibrium regime, $\\xi (t)$ reaches its equilibrium value.",
"At the $d=3$ critical point and in equilibrium, one should expect $C(r,t)\\sim \\frac{1}{r^{d-2+\\eta }} = \\frac{1}{r^{1+\\eta }} \\,,$ $\\eta $ being the anomalous dimension of the field.",
"The correlation length $\\xi (t)$ can be estimated by computing [13], [14] $I_k(t)=\\int _{0}^{L/2} dr~ r^k C(r,t)\\,,$ by means of $\\xi _{k,k+1}(t)\\equiv \\frac{I_{k+1}(t)}{I_k(t)}\\,.$ We focus in this work on $\\xi _{2,3}$ .",
"On spin glasses was measured $\\xi _{1,2}$ with a correlation function decaying like $1/r^{0.5}$ .",
"[13], [14] In our case, to decrease the weight of the smallest distances we have resorted to compute higher values of $I_k$ .",
"In the appendix B we describe the detailed procedure we have used to compute the integrals and how we have estimate the statistical error associated with $\\xi _{k,k+1}(t)$ .",
"The dependence of the dynamic correlation length with time is $\\xi _{k,k+1}(t)\\sim t^{1/z} \\left(1+ A_k t^{-\\omega /z} \\right)\\,.$ The magnetic susceptibility is given by $\\chi (t)=\\frac{1}{V} \\langle {\\mathitbf M}^2 \\rangle _t\\,,$ or equivalently by $\\chi (t)=\\int d^3 x~ C(|\\mathitbf {x}|,t) \\,.$ In the regime of large $\\xi (t)$ we recover rotational invariance and we obtain $\\chi (t)=4 \\pi I_2(t)\\,.$ The temporal dependence of $\\chi (t)$ is $\\chi (t)\\sim t^{(2-\\eta )/z} \\left(1+ A t^{-\\omega /z} \\right)\\,\\,,$ which can be rewritten as $\\chi (t)\\sim \\xi _{k,k+1}(t)^{2-\\eta } \\left(1+ C_k \\xi (t)^{-\\omega } \\right)\\,\\,.$"
],
[
"Numerical results",
"In this section we report the computation of the integrated correlation time at equilibrium.",
"After this analysis, we describe our results in the out of equilibrium regime.",
"In particular, we consider the short and long time behavior of correlation length and the long time behavior of the correlation function and that of the energy.",
"The data are obtained after a sudden quench from $T=\\infty $ to $T=1/\\beta _c$ .",
"All the numerical simulations were performed at $\\beta _c=0.693001$ .",
"[21]"
],
[
"Equilibrium",
"To obtain the dynamic critical exponent in the equilibrium regime, we compute the integrated correlation time of $\\mathitbf {M}^2$ when the numerical simulation has reached the equilibrium.",
"We follow the methodology described in Sec.",
"REF , using the self consistent windowm algorithm with a window size given by $c\\tau _\\mathrm {int}$ .",
"We have analyzed the correlation functions with $c=6$ , 8, 10 and 12 and we have checked that the $c=10$ data are fully compatible with that of $c=8$ and 12.",
"We report in the following $c=10$ integrated autocorrelation times.",
"In Table REF we report the values of $\\tau _{\\mathrm {int},\\mathitbf {M}^2}$ and other parameters of the performed runs.",
"In order to improve the statistics on $\\tau _{\\mathrm {int},\\mathitbf {M}^2}$ we have performed 50 independent runs (initial conditions).",
"We have computed the statistical error on the integrated autocorrelation times by using the jackknife method over the independent runs.",
"[26], [27] Table: Integrated correlation time of M 2 \\mathitbf {M}^2,τ int ,M 2 \\tau _{\\mathrm {int},\\mathitbf {M}^2} for c=10c=10, as a function of the latticesize, LL.",
"We also report the length of the run at equilibrium,n sweep n_\\mathrm {sweep}, in units ofτ int ,M 2 \\tau _{\\mathrm {int},\\mathitbf {M}^2}.",
"For each lattice size wehave performed 50 intial conditions.",
"Notice that all the reportedruns satisfyn sweep >10000τ int ,M 2 n_\\mathrm {sweep}>10000~\\tau _{\\mathrm {int},\\mathitbf {M}^2}.We have fitted $\\tau _{\\mathrm {int},\\mathitbf {M}^2}$ to Eq.",
"(REF ) using $8\\le L \\le 64$ obtaining $z=2.033(5)$ and $\\omega =2.7(3)$ with a $\\chi ^2/\\mathrm {d.o.f.",
"}=0.36/3$ .In this case, the data for different lattice sizes are not correlated and we can safely use the diagonal covariance matrix.",
"We report this fit and the numerical data in Fig.",
"REF .",
"Fitting the data using only a power law (i.e.",
"neglecting the correction-to-scaling term) we obtain a good fit only for $L\\ge 24$ obtaining $z=2.026(4)$ with $\\chi ^2/\\mathrm {d.o.f.",
"}=0.28/2$ .",
"Both reported values are fully compatible.",
"Figure: (color online) Behavior of the integrated correlation time,τ int ,M 2 \\tau _{\\mathrm {int},\\mathitbf {M}^2}, as a function of the lattice size, LL.",
"Wehave also shown our best fit taking into account corrections to thescaling (see the text).Finally we present a scaling analysis of the $\\rho (t)$ function at equilibrium and at the critical point to show that $\\tau _\\mathrm {exp}$ and $\\tau _\\mathrm {int}$ are proportional and therefore both times diverge with the same dynamic critical exponent $z$ .",
"Fig.",
"REF shows the scaling law of the correlation function $\\rho (t)$ as a function of $t/\\tau _\\mathrm {int}(L)$ instead of $t/\\tau _\\mathrm {exp}(L)$ , as stated in Eq.",
"(REF ).",
"Scaling in the new variable holds if and only if $\\tau _\\mathrm {exp} \\propto \\tau _\\mathrm {int}$ and this is the case apart from small scaling corrections on the $L\\le 16$ data induced by the term $\\xi (L)/L$ in the scaling function $f_{\\cal {O}}$ of Eq.",
"(REF ), see also the inset of this figure for a detailed and more quantitative view of this effect.",
"Figure: (color online) Behavior of the integrated correlationfunction, ρ(t)\\rho (t) for the squared magnetization as a function oft/τ int (L)t/\\tau _\\mathrm {int}(L).",
"Inset: we show a zoom of the long time region ofthe main plot, drawing only a small number of points, so the readercan see in a better way the differences among the different latticesizes.",
"Notice that all the data with L≥16L\\ge 16 collapse in thescaling formula and this fact provides a numerical verification of theproportionality of the integrated and exponential correlationtimes.",
"We have also plotted a single pure exponential, exp(-x)\\exp (-x),to show that the correlation function is not a single exponential.In Ref.",
"[5] the (biggest) exponential correlation time was computed for the magnetization $\\sqrt{\\mathitbf {M}^2}$ .",
"However, the computation of this exponential time is very involved in the case the autocorrelation function $\\rho (t)$ does not show a single exponential decay [19], [10]: in Fig.",
"REF we have plotted a single exponential decay and the correlation function clearly departs from this behavior."
],
[
"Correlation length: Shorter times",
"We report in Fig.",
"REF the behavior of $\\xi _{23}$ as a function of time for different lattice sizes that we have been able to thermalize.",
"The $\\xi _{23}$ -plateaus obtained for the largest times are a clear evidence that the numerical simulations have reached the equilibrium.",
"In order to extract the dynamic critical exponent by using Eq.",
"(REF ), we need to work in the out of equilibrium regime, avoiding the transient regime and the equilibrium domain.",
"Therefore, we need to check the following points: We need to avoid the transient regime between the power law behavior (pure out of equilibrium regime) and the plateau one (equilibrium one).",
"Eq.",
"(REF ) holds after the initial transient of the dynamics.",
"Therefore, we need to fix a minimum time $t_\\mathrm {min}$ .",
"Finally, in order to avoid finite size effects, we need to compare the correlation lengths for different lattice sizes.",
"Figure: (color online) Behavior of the dynamic correlation length forL=16L=16, 24, 32, 48, 64 and L=128L=128.",
"We have only plotted simulationsin which the equilibrium regime has been reached: notice the clearplateau of the different correlation length curves.Figure: (color online) We show the difference of dynamic correlationlengths, Δξ 23 (t)\\Delta \\xi _{23}(t) for three pairs of lattice sizes as afunction of time: ξ L=250 -ξ L=200 \\xi _{L=250}-\\xi _{L=200} andξ L=250 -ξ L=128 \\xi _{L=250}-\\xi _{L=128}.",
"The zero value has been marked with ahorizontal line.",
"Notice that the L=250L=250 data are asymptotic (ascompared with those of L=200L=200) for t<8100t<8100 (the data lie, at most, at onestandard deviation of the zero value).In this section we have performed numerical simulations with $t<10240$ (in order to avoid the transient and equilibrium regimes).",
"We have simulated 4000 random initial conditions for $L=128$ and $L=200$ , and 5325 initial conditions for $L=250$ .",
"In Fig.",
"REF we have plotted, to check finite size effects, the differences among the correlation lengths of $L=128$ and 200 and that of 250.",
"In this figure we can see that the data of the $L=200$ and $L=250$ lattices are compatible in the statistical error for $t<8100$ .",
"From the previous discussion we must fit the data for $\\xi _{23}$ in the time interval given by $t\\in [t_\\mathrm {min},8100]$ using the $L=250$ data: $t_\\mathrm {min}$ being the smallest value of the Monte Carlo time that provides a good $\\chi ^2/\\mathrm {d.o.f.",
"}$ (e.g.",
"$\\sim 1$ ) by fitting the data to Eq.",
"(REF ).",
"In Fig.",
"REF we show the behavior of $\\xi _{23}$ for the largest lattice size simulated in this time regime, $L=250$ .",
"By fitting $L=250$ data in the interval $t \\in [100,8100)$ we have obtained $z=2.04(2)$ and $\\omega =2.2(4)$ with $\\chi ^2/\\mathrm {d.o.f.",
"}=287/796$ .",
"Furthermore, we can report that a fit neglecting the contribution of the correction-to-scaling term provides $z=2.012(13)$ with $t \\in [400,8100)$ and $\\chi ^2/\\mathrm {d.o.f.",
"}=361/768$ .",
"All two reported values are statistically compatible.",
"We have computed the statistical error on the $z$ -exponent by means of the jackknife method.",
"[26], [27] As described in the appendix , we compute the $\\chi ^2$ using a diagonal covariance matrix (neglecting the correlations of the data), but we use a jackknife procedure to take into account the (important) different correlations among the data.",
"Hence, in the following all $\\chi ^2$ are computed assuming a diagonal covariance matrix; we refer the reader to the appendix for a discussion of the interpretation of this diagonal $\\chi ^2$ and for more details on the procedure we have followed to take into account the correlation among the data (in time or in distance, see below) and the way we have computed the statistical errors on the values of the critical exponents.",
"The same fit performed with the help of Gnuplot [37] (with a diagonal covariance matrix) provides a $z= 2.012$ with an asymptotic error of $0.00075$ .",
"In order to obtain the right statistical error, we need to divide this asymptotic error by $\\sqrt{\\chi ^2/\\mathrm {d.o.f.",
"}}=0.687$ [27], obtaining the final value of $z=2.012(1)$ .",
"Notice that the computed error discarding correlations among the different times is a factor 13 times smaller.",
"Figure: (color online) Behavior of the dynamic correlation length forL=250L=250 in the out of equilibrium regime.",
"The fit is only for the L=250L=250 data inthe region 100≤t<8100100\\le t<8100.Having computed $\\xi _{23}$ and $I_2 \\propto \\xi _{23}^{2-\\eta }$ we can, as a check, estimate the $\\eta $ exponent.",
"Fig.",
"REF shows $I_2$ as a function of $\\xi _{23}$ .",
"We can compute $\\eta $ using the time interval $t\\in [100,8100)$ obtaining $\\eta =0.029(20)$ and $\\omega =0.8(4)$ with $\\chi ^2/\\mathrm {d.o.f.",
"}=342/796$ .",
"In this case the $\\omega $ -exponent is similar to the equilibrium value $\\omega \\simeq 0.78$ .",
"[28], [29] We can improve the value of $\\eta $ by fixing the correction-to-scaling exponent $\\omega $ to the equilibrium value $\\omega =0.78$ , obtaining $\\eta =0.044(7)$ with $\\chi ^2/\\mathrm {d.o.f.",
"}=666/757 $ ($t\\in [50,8100)$ ).",
"Our value compares very well (but with 20 times more error) with that computed at equilibrium: $\\eta =0.0378(3)$ .",
"[22], [23] Finally, neglecting the correction-to-scaling term, we have found $\\eta =0.043(6)$ with $\\chi ^2/\\mathrm {d.o.f.",
"}=676/768$ ($t\\in [400,8100)$ ).",
"All three reported values are statistically compatible.",
"Figure: (color online) Behavior of I 2 (ξ 23 )∝χI_2(\\xi _{23})\\propto \\chi forL=250L=250.",
"We also plot the fit taking into account scalingcorrections but with the ω\\omega -exponent fixed to theequilibrium value as described in the text."
],
[
"Correlation function for larger times",
"In Fig.",
"REF we plot $C(r,t)$ for different times using $L=128$ data (200 initial conditions) and very long times.",
"One can see the crossover of the dynamic correlation function between the off-equilibrium regime and the equilibrium one.",
"In appendix we provide more details about the functional form of $C(r,t)$ in the out of equilibrium regime.",
"We can also check that we have reached the equilibrium regime by plotting the behavior of $\\xi _{23}(t)$ (see $L=128$ curve of Fig.",
"REF ).",
"This non-local observable has clearly reached its equilibrium (plateau) value.",
"We can safely assume that for $t>4\\times 10^5$ we have thermalized the $L=128$ lattice and we can try to extract the value of the the anomalous dimension by averaging the correlation function above this time.",
"The analytical behavior at the critical point in this regime (large $L$ ) is given by Eq.",
"(REF ).",
"Having in mind that we are using periodic boundary conditions, we can write the following improved equation to fit our numerical data $C(r,L) =\\frac{A}{r^{1+\\eta }}+ \\frac{A}{(L-r)^{1+\\eta }}\\,.$ By fitting the data of Fig.",
"REF to this functional form, we obtain $\\eta =0.026(4)$ (by using only $t>4\\times 10^5$ , $r\\ge 16$ and $\\chi ^2/\\mathrm {d.o.f}=44/48$ ) in a good statistical agreement with the value drawn from equilibrium studies $\\eta =0.0378(3)$ .",
"We have followed the method described in appendix in order to obtain the error in the $\\eta $ exponent.",
"The same fit, assuming no correlation among the different values of the correlation function, provides an error of 0.0013, three times smaller than that obtained in our procedure.",
"Figure: (color online) Correlation function at criticality for aL=128L=128 lattice.",
"We have drawn different times in order to show thecrossover between the out of equilibrium region and the equilibriumone.",
"Notice the bad signal-noise ratio in the tail of C(r,t)C(r,t) forlarge rr and shorter times tt, and how this ratio improves withtime, generating a plateau (due the periodic boundary conditions)with small error.Figure: (color online) Equilibrium correlation function atcriticality for a L=128L=128 lattice.",
"The continuous line is a fit toEq.",
"() with η=0.026\\eta =0.026."
],
[
"Energy for larger times",
"We have analyzed the behavior of the energy at criticality in order to compute the ratio of critical exponents $(d-1/\\nu )/z$ .",
"To analyze this behavior, we have run $L=128$ (153 initial conditions, i.c.",
"in the following), $L=160$ (600 i.c.",
"), $L=200$ (684 i.c.)",
"and $L=250$ (684 i.c.)",
"for longer times $t<102400$ .",
"Firstly, we study in Fig.",
"REF the effect of a finite size lattice on the values of energy as a function of time.",
"From this figure one can see that it is safe to take fits only in the range $t<48000$ in order to avoid finite size effects (at least in the precision of our simulation).",
"In Fig.",
"REF we show the results for the largest lattice $L=250$ .",
"We have fitted the $L=250$ data to a power law, in the time interval $t\\in [1000, 48000)$ obtaining $z=2.034(22)$ and $e_\\infty =-0.989505(17)$ , with a diagonal $\\chi ^2/\\mathrm {d.o.f.",
"}=985/939$ .",
"We have fixed in the fit the value $\\nu =0.7117(5)$ .",
"[22], [23] The really small error bar of the $\\nu $ exponent has not a measurable effect in the final error bar of $z$ .",
"To finish the analysis of the energy, we have also checked corrections to scaling for this observable and we have found that the exponent $\\omega _\\mathrm {eff}=2\\times 0.78$ describes very well the numerical data obtaining $z=2.13(7)$ and $e_\\infty =-0.989525(22)$ with $\\chi ^2/\\mathrm {d.o.f}=980/947$ .",
"Figure: (color online) Behavior of the energy e(t)e(t) at the criticalpoint for the L=250L=250 run.",
"We draw also the fit in order to extractthe ratio (d-1/ν)/z(d-1/\\nu )/z with d=3d=3, ν=0.7117(5)\\nu =0.7117(5) fixed gettingz=2.034(22)z=2.034(22).By performing in and out equilibrium numerical simulations we have computed the dynamic critical exponent $z$ .",
"The most accurate value has been computed in the equilibrium regime by studying the integrated correlation time as a function of the lattice size: $z=2.033(5)$ .",
"We have found a correction-to-scaling exponent $\\omega =2.7(3)$ .",
"In addition we have provided strong numerical evidences about the proportionality of the integrated and exponential correlation times.",
"We have also computed the $z$ exponent in the out-of-equilibrium regime obtaining $z=2.04(2)$ and $\\omega =2.2(4)$ which is similar to the $\\omega $ -exponent computed at equilibrium.",
"Moreover, we have checked the consistency of the computed critical exponents at equilibrium with the out of equilibrium ones with and without considering corrections to scaling.",
"The (equilibrium) value of $\\nu $ provides us, by monitoring the energy, with a another dynamic exponent estimate ($z=2.034(22)$ ) fully compatible with the previous ones.",
"Furthermore, our value of $z$ has improved the statistical precision of that computed in numerical simulations performed at equilibrium in relatively small lattices ($L\\le 24$ ).",
"[5] Our computed values match very well with that obtained in experiments and with the exponent computed using field theoretical techniques [4] (although in this framework it is very difficult to assign an uncertainty to this estimate).",
"We thank L. A. Fernandez, M. Lulli, A. Pelissetto, V. Martin-Mayor, J. Salas, J.",
"A. del Toro and D. Yllanes for discussions.",
"This work was partially supported by Ministerio de Economía y Competitividad (Spain) through Grant No.",
"FIS2016-76359-P, by Junta de Extremadura (Spain) through Grant No.",
"GRU10158 and IB16013 (partially funded by FEDER).",
"We have run the simulations in the computing facilities of the Instituto de Computación Científica Avanzada (ICCAEx) and in the CETA-Ciemat thanking Dr. A. Paz for his support."
],
[
"Details of the numerical simulations and GPU parallel implementation",
"We have simulated the Heisenberg model using the Metropolis Algorithm on CPUs and GPUs (see Table REF ).",
"We have simulated $L=128$ , 160, 200 and 250 for more than 10000 random initial conditions.",
"The GPU code has been programmed in CUDA C. [31] The original C code which simulates the Heisenberg model has been parallelized in three parts: Computation of the nearest neighbors of each spin: the C code has a loop which goes sequentially through all the spins one by one.",
"However, in the GPU code each spin has associated an execution thread and all the nearest neighbors of every spin are computed at once.",
"Metropolis Algorithm: in the sequential C code we can find several loops in the Metropolis part.",
"So, the parallel GPU code reduces meaningfully the execution time especially in large systems ($L \\sim 200$ ).",
"Moreover, the lattice has been divided using a checkboard scheme (Fig.",
"REF ).",
"[32] In this way, the Metropolis algorithm has been executed first of all in the “white” spins and after that in the “black” ones.",
"Random numbers: to have high quality random numbers is mandatory in Computational Physics.",
"Initially, we have used the CURAND random numbers which are part of the CUDA C distribution.",
"[31] The problems with the CURAND random numbers have appeared when we have performed long simulations using a huge quantity of random numbers.",
"To avoid these problems we have used Congruential Random Numbers.",
"[36] Table: Hardware features of the CPUs and GPUs.Figure: (color online) Division of the three dimensional latticeusing a checkboard scheme.Making use of the GPU Tesla K80 we have achieved a speedup of 22 which represents an important reduction of the execution time."
],
[
"Details of the analysis of the computation of the correlation length",
"We describe the different steps we have followed in order to compute $\\xi (t)$ and its associated exponent $z$ .",
"[13], [14], [33], [34] The important point of this approach is to avoid the use of the full covariance matrix since this matrix is frequently singular (see for example [33], [35]).",
"Thus, our procedure is the following: We compute using the jackknife method over the set of the initial conditions, the statistical error of $C(r,t)$ , denoted as $\\sigma [C(\\Lambda ,t)]$ .",
"To compute $I_k$ we introduce a cutoff to have a good control of the signal to noise ratio of $C(r,t)$ for large values of $r$ (see also Fig.",
"REF ).",
"We compute the cutoff $\\Lambda $ using the condition $\\sigma [C(\\Lambda ,t)]=4 C(\\Lambda ,t)$ .",
"For a fixed $t$ and $r_\\mathrm {min}< r < \\Lambda $ we fit the correlation function to the functional form given by $C(r,t)=\\frac{a_1}{r^{a_2}} \\exp (-a_{3} r^{a_4})\\,.$ with $r_\\mathrm {min}$ is the minimum value of $r$ which provided, for $C(r,t)$ , a good fit (e.g.",
"$\\chi ^2/\\mathrm {d.o.f.",
"}\\sim 1$ ) to Eq.",
"(REF ).",
"In Fig.",
"REF we report the dependence of the exponents $a_2$ and $a_4$ with the Monte Carlo time.",
"Notice that $a_2$ converges to the equilibrium value (see Eq.",
"(REF )) given by $1+\\eta =1.0378$ and $a_4\\simeq 1.8$ .",
"Figure: (color online) Behavior of the exponents a 2 a_2 and a 4 a_4 as afunction of time for L=200L=200.",
"The horizontal line is the equilibrium theoreticalexpectation for a 2 a_2, namely 1+η=1.03781+\\eta =1.0378.",
"We compute the integral in Eq.",
"(REF ) using the numerical values of the correlation $C(r,t)$ for $r<\\Lambda $ and using the values provided by the fit (Eq.",
"(REF )) for $\\Lambda < r <L/2$ .",
"Using the previous procedure, we compute the statistical error of $\\xi (t)$ using again the jackknife method over the set of the initial conditions.",
"The time interval for the fit is decided by imposing a diagonal $\\chi ^2/\\mathrm {d.o.f}\\sim 1$ .",
"The jackknifed $\\xi $ 's are used to compute the jackknifed values of $z$ and this allows us to compute the statistical error of the dynamic critical exponent using the standard deviation in the jackknife method.",
"Notice that for extracting $z$ on each jackknife block, we use the diagonal covariance matrix.",
"However, the jackknife procedure reproduces with high accuracy the effect of the correlations among the different times.",
"Notice that the diagonal $\\chi ^2/\\mathrm {d.o.f.",
"}$ has not a rigorous interpretation as that of the full (non diagonal) one.",
"One can show (see the detailed analysis of this procedure carried out in section B.3.3.1 of Ref.",
"[33]) that the diagonal $\\chi ^2/\\mathrm {d.o.f.",
"}$ behaves as if there were a small number of degrees of freedom, hence, one can not compute confident limits as usual.",
"Finally, in Ref.",
"[15] was shown that the error bars are essentially equal (using this jackknife procedure, neglecting the correlations among the data) to those obtained taking into account all the statistical correlations among the data."
]
] | 1906.04518 |
[
[
"New loop expansion for the Random Magnetic Field Ising Ferromagnets at\n zero temperature"
],
[
"Abstract We apply to the Random Field Ising Model at zero temperature (T= 0) the perturbative loop expansion around the Bethe solution.",
"A comparison with the standard epsilon-expansion is made, highlighting the key differences that make the new expansion much more appropriate to correctly describe strongly disordered systems, especially those controlled by a T = 0 RG fixed point.",
"This new loop expansion produces an effective theory with cubic vertices.",
"We compute the one-loop corrections due to cubic vertices, finding new terms that are absent in the epsilon-expansion.",
"However, these new terms are subdominant with respect to the standard, supersymmetric ones, therefore dimensional reduction is still valid at this order of the loop expansion."
],
[
"Expanding around the Bethe solution",
"The expansion proposed in ref.",
"[22] is an expansion around the Bethe solution: by replicating the model $M$ times, and rewiring the $M$ copies, one can show that the limit $M\\rightarrow \\infty $ gives the Bethe approximation, while the original model is recovered for $M=1$ .",
"A diagrammatic loop expansion can then be constructed expanding in powers of $1/M$ , similarly to the standard perturbative expansion.",
"We call this framework the $M$ -layer or the BL approach.",
"At leading order, the correlation functions on the $M$ -layer are strictly related to the ones found on a BL.",
"The latter are easy to compute since in the thermodynamic limit the BL contains no loops of finite length.",
"Using the $M$ -layer expansion, one finds that a generic correlation $G$ (either $C$ or $R$ ) between the lattice origin and a point $x\\in Z^D$ on the lattice can be written, at the leading order, as $G(x)=\\sum _{L=1,\\infty }{\\cal N}(x,L)\\, G^{\\text{\\tiny {BL}}}(L)$ where ${\\cal N}(x,L)$ is the number of non-backtracking paths going from the origin to $x$ of length $L$ on the original lattice and $G^{\\text{\\tiny {BL}}}(L)$ is the correlation function computed on the BL between two spins at distance $L$ .",
"The BL has the same connectivity $z=2D$ of the finite dimensional lattice and the same probability distribution for the random fields.",
"In the region of large $x$ and $L$ we find that in $D$ dimensions [25] ${\\cal N}(x,L) \\propto (2D-1)^L \\exp \\left(-x^2/(4L)\\right) L^{-D/2},$ where $x^2\\equiv \\Vert x\\Vert ^2$ , and $\\propto $ denote equality up to a constant.",
"and we obtain for the Fourier transform of eq.",
"[REF ] in the small momentum region $\\widetilde{G}(p)\\propto \\sum _{L=1,\\infty } (2D-1)^L \\exp (-L \\, p^2) G^{\\text{\\tiny {BL}}}(L)\\ .$ So we just need to compute how correlations behave on a BL at large distances.",
"As shown in the SI, at $T=0$ the crucial quantity that encodes all the information about two spins $\\sigma _1$ and $\\sigma _2$ at positions $x_1$ and $x_2$ is the dependence of the ground state energy on their values, after that we minimize over all the other spins.",
"For Ising spins the energy must be of the form $\\mathcal {H} [\\sigma _1,\\sigma _2] = -h_1 \\sigma _1 -h_2 \\sigma _2 -J \\sigma _1\\sigma _2 + E.$ On the BL, we can derive a recursive equation for the joint probability distribution $P_L(h_1,h_2,J)$ for two spins at distance $L$ , and obtain an explicit expression for large $L$ by imposing some consistency condition.",
"The computation is presented in the SI, and from here on we discuss the large $L$ behaviour, which is the relevant one at criticality.",
"The result can be written in the form: $P_L(h_1,h_2,J)=Q_L(h_1,h_2)\\delta (J)+a L\\lambda ^{L} \\hat{g}(h_1) \\hat{g}(h_2)F_L(J)$ where $\\lambda , \\hat{g}, F_L$ and $Q_L$ depend implicitly on the external random field distribution and $F_L(J)=\\rho L \\exp (-\\rho LJ)$ at the leading order in $L$ .",
"In spin-glass jargon, $\\lambda $ is called the anomalous eigenvalue and governs the decay of ferromagnetic susceptibilities along a chain in the BL [32], [30].",
"The expression for the marginal probability $P_L(J)\\equiv \\int dh_1 dh_2\\, P_L(h_1,h_2,J) $ is thus given by: $P_L(J)=(1-aL\\lambda ^ L)\\delta (J) +a\\rho L^2\\lambda ^ L \\exp (-\\rho LJ)$ The coefficient $\\rho $ can be computed exactly, as shown in the SI.",
"The result [REF ] is quite surprising: the quantity $J$ is either exactly 0 or is of order $1/L$ with a probability of order $L \\lambda ^ L$ .",
"Denoting with $\\overline{\\bullet }^L$ averages over $P_L$ , we obtain $\\overline{J}^L\\propto \\lambda ^ L$ It can be shown that the response function $R_{12}$ receives contributions only from the event $J>|h_1|$ , and can be easily computed.",
"We obtain that the average response function on a BL behaves as the average effective coupling: $R^{\\text{\\tiny {BL}}}(L)\\propto \\lambda ^ L$ In a similar way, if we look to $P_L(h_1,h_2)=\\int dJ P_L(h_1,h_2,J)$ we find that for large $L$ $P_L(h_1,h_2)=P(h_1)P(h_2)+L \\lambda ^L f(h_1)f(h_2),$ where the function $f(h)$ is even and $P(h)$ is the Bethe distribution of cavity fields.",
"One can see that the dominant contribution to the disconnected correlation on the BL at distance $L$ comes from the correlated fields $h_1$ and $h_2$ .",
"For this reason we find: $C^{\\text{\\tiny {BL}}}(L)\\propto L \\lambda ^ L$ The behaviors for $R^{\\text{\\tiny {BL}}}$ and $C^{\\text{\\tiny {BL}}}$ correspond to the known ones on a line [29], [30].",
"We have found that for large $L$ , $G^{\\text{\\tiny {BL}}}(L)\\approx \\mathcal {G}^{\\text{\\tiny {BL}}}(L) \\lambda ^{L}$ , where $\\cal {G}^{\\text{\\tiny {BL}}}$ is a polynomial in $L$ .",
"On the BL, the correlation functions decrease exponentially and the critical point is located where their exponential decrease matches the exponential increase of the number of paths (resulting in a diverging susceptibility).",
"In both the BL and in finite dimensional model at the zeroth-order of the $M$ -layer construction, the critical point is located at $\\lambda _c=\\frac{1}{2D-1}\\,.$ Near the critical point, starting from eq.",
"[REF ] we can write $\\widetilde{G}(p)\\propto \\int _0^\\infty dL\\ \\exp \\left(-L (p^2 +\\tau )\\right) \\mathcal {G}^{\\text{\\tiny {BL}}}(L) $ where the sum over $L$ has been replaced by an integral and $\\tau $ is the inverse of the correlation length $\\xi $ and it is given by $\\tau \\equiv -\\log (\\lambda (2D-1))$ As usual, $\\tau =0$ at the critical point.",
"The representation in eq.",
"[REF ] is the equivalent of the proper time representation in a field theory context.",
"If we now put eqs.",
"[REF ,REF ] in eq.",
"[REF ], we obtain the leading order of the expansion of the correlation functions of a $D$ -dimensional model around the BL: $\\left.\\begin{aligned}\\widetilde{C}(p)\\propto & \\int _0^\\infty dL\\ \\exp \\left(-L (p^2 +\\tau )\\right) L = \\frac{1}{(p^2+\\tau )^2}\\\\\\widetilde{R}(p)\\propto & \\int _0^\\infty dL\\ \\exp \\left(-L (p^2 +\\tau )\\right) = \\frac{1}{p^2+\\tau }\\end{aligned}\\right.$ The formulae in eq.",
"[REF ] are the same of those coming from the LG effective Hamiltonian approach, with a few crucial differences though.",
"In the LG approach, near the critical point and in the zero-loop approximation, the equations for the stationary points are linear: the unique solution is $\\phi (x)\\!=\\!\\int \\!dy R(x-y) h(y)$ , where $R(x)$ is the Fourier transform of $\\widetilde{R}(p)$ .",
"In the BL approach, near the critical point and in the zero-loop approximation, there is an infinite number of local minima (i.e.",
"configurations whose energy does not decrease if a finite number of spins are flipped), but the only thermodynamically relevant configuration is the global minimum [33].",
"In the LG approach, the response function $R(x)$ does not depend on the field and it does not fluctuate: more precisely, all possible paths give the same contribution.",
"In the case of the BL approach, only an exponentially small number of paths gives a contribution to the response function: for large $L$ the probability of a given path to have a non zero $J$ and consequently to contribute to the response is exponentially small [29].",
"The above differences become strikingly evident if we consider avalanches, a well-studied phenomenon in the RFIM.",
"We are interested in seeing the change in the magnetizations when we change by a finite amount the magnetic field in a given point.",
"Denoting the original field at position $x$ by $h^*$ , we define $A(y,x;h)\\equiv \\langle \\sigma (y)\\rangle _{h^*+h} -\\langle \\sigma (y)\\rangle _{h^*-h}$ where the label denotes the field at $x$ .",
"The quantity $A(y,x;h)$ is the variation of the magnetization at $y$ when we change the magnetic field at $x$ adding or subtracting a term $h$ .",
"In the limit of small field avalanches are related to an alternative choice of the response function that we call $\\hat{R}(y-x)$ .",
"It amounts to consider only (the density of) excitations with strictly zero energy cost.",
"More precisely, we have: $\\hat{R}(y-x)=\\lim _{h\\rightarrow 0}\\overline{\\frac{A(y,x;h)}{2 h}}.$ One can show that $\\hat{R}(y-x)$ is proportional to the zero temperature limit of the two-point connected correlation function, with a factor proportional to the inverse temperature.",
"The avalanche size $S(x;h)$ is given by $S(x;h)=\\frac{1}{2}\\int d^Dy\\, A(y,x;h)\\,.$ For small $h$ we have $\\overline{S(x;h)}\\approx h\\, \\chi \\equiv h \\int dy\\, \\hat{R}(y-x)\\,,$ with $\\chi $ the susceptibility associated to the response function $\\hat{R}(y-x)$ .",
"The susceptibility diverges as $1/\\tau $ at the critical point both in the LG approach and on the Bethe lattice, but new features arise when we consider the probability distribution $P(S)$ of $S(x;h)$ .",
"Let us compare what happens in the two approaches at the zeroth-order of the loop expansion.",
"In the LG approach, since $\\hat{R}$ is related to connected correlation functions we easily find that $S$ does not fluctuates: $P(S)=\\delta \\left(S-\\frac{c}{\\tau }\\right)\\,$ for some constant $c$ .",
"Therefore, we find: $\\overline{S}\\propto \\tau ^{-1} ,\\quad \\overline{S^2}\\propto \\tau ^{-2}$ .",
"The median value of $S$ is divergent.",
"In the BL approach, following [26], one can argue that: $P(S)\\propto S^{-3/2} \\exp (-S\\tau ^2 ).$ We thus find: $\\overline{S}\\propto \\tau ^{-1}, \\quad \\overline{S^2}\\propto \\tau ^{-3}$ .",
"The median value of $S$ is finite and the divergence of $\\overline{S}$ and $\\overline{S^2}$ stems from rare events in the tail of the distribution.",
"The power law divergence of $\\overline{S^2}$ is quite different in the two approaches: $\\tau ^{-2}$ in LG and $\\tau ^{-3}$ on the BL (more details in the SI).",
"We have seen that at the zeroth-order the physical behavior is quite different in the BL and in the LG approaches, although the critical behavior of the correlation function is superficially similar: in the LG approach, anomalous large fluctuations do not exist, while on the BL everything is dominated by rare large fluctuations.",
"The superficial similarity for the average two-point correlations disappears if we look to high-order correlation functions (responsible for avalanches).",
"At this point, it is not clear what happens when we consider the loop expansion in the BL approach.",
"The natural question is whether this loop expansion produces the same results as in LG.",
"We have two alternative scenarios: The difference in the high-order correlations that we have seen at the tree level (zeroth-order BL) contaminates the two-points correlations when the leading contributions coming from the loop are considered.",
"In this case, we would have additional terms at $T=0$ that are ignored in the LG approach.",
"This would lead to the appearance of extra terms in the $\\epsilon $ expansion in $6-\\epsilon $ and DR should fail already in the $\\epsilon $ expansion.",
"The difference in the high-order correlations do not produce leading discrepancies on the two-points correlation functions and the contribution of the loops is the same as in the LG approach.",
"As a consequence, in an unexpected way, we would recover perturbative dimensional reduction in $6-\\epsilon $ .",
"One can present many hand-waving arguments in favor of the first or the second scenario.",
"However, the proof of the pudding is in the eating.",
"In the following we prove that at one loop the results of the BL and LG approaches are the same.",
"This will be done presenting a computation (down to the metal) of the one loop correction in the case of the BL.",
"Roughly speaking the idea at the basis of the loop expansion around the BL is to start approximating, at least locally, the $D$ -dimensional lattice with loopless (acyclical) graphs: these are Caley trees with self-consistent conditions at the boundary or Bethe lattices.",
"Of course, loops are present in the $D$ -dimensional lattice and their effect is introduced perturbatively, by considering a sequence of BL with a finite number of loops.",
"The expansion is similar to the virial expansion, where the complex interaction among infinitely many particles is decomposed in terms of simpler interactions between a finite number of particles.",
"Figure: One loop topological diagrams important for the first order expansion around the Bethe lattice: they can have vertices with four lines (left), or vertices with three lines (right).The loop expansion around the BL is an expansion in topological diagrams.",
"The contribution of a given topological diagram can be written as the probability of finding such a topological diagram embedded in the $D$ -dimensional lattice times the averaged value that the observable takes on that given structure when inserted in a loop-less and infinite Bethe lattice.",
"Only the topologically connected part of the average of the observable has to be consider.",
"As it is shown in ref.",
"[22], this connectivization procedure practically corresponds to adding the value of the observable evaluated on each of the subgraphs that are obtained from the original structure by sequentially removing its lines times a factor $-1$ for each line removed.",
"The one loop contribution comes from the two diagrams shown in Fig.",
"REF .",
"They look similar to standard Feynman diagrams, however their physical interpretation is quite different.",
"In standard Feynman diagrams the loops do not have a special meaning, here instead they have a geometrical meaning.",
"Generalizing eq.",
"[REF ], the one loop contribution, in Fourier space and as a function of the incoming moment, can be written as $\\widetilde{G}^{loop}(p)=\\int _0^\\infty d\\vec{L} \\ \\tilde{\\cal {N}}(p,{\\vec{L}})\\,\\lambda ^{\\Sigma (\\vec{L})}\\,{\\cal G}^{\\text{\\tiny {BL}}}(\\vec{L})$ where $\\vec{L}$ is a vector containing the lengths of each line in the topological diagram, the factor $\\tilde{\\cal {N}}(p,\\vec{L})$ accounts for the number of such topological diagrams, while $\\Sigma (\\vec{L})$ is the sum of all $L$ 's, and $\\lambda $ is the same eigenvalue on the BL as in the previous discussion.",
"The term ${\\cal G}^{\\text{\\tiny {BL}}}(\\vec{L})$ is the generalization of $ {\\cal G}^{\\text{\\tiny {BL}}}(L)$ at the zero-th order: it is the only term depending on the model and has to be carefully computed on the BL.",
"In the case of the two diagrams in Fig.",
"REF , we find: for the left diagram N(p,L)(2D-1)(L)D(L)D/2 (-(LI +LO)p2) D(L)=LA, (L)=LI+LO+LA; for the right diagram N(p,L)(2D-1)(L)D(L)D/2(-(LI +LO+LALBD(L))p2), D(L)=LA+LB, (L)=LI+LO+LA+LB.",
"Setting $\\mathcal {G}(\\vec{L})=1$ we recover the conventional diagrams of the field theory approach in the cases of a $\\phi ^4$ interaction (left diagram) or $\\phi ^3$ interaction (right diagram) written in the Feynman proper time representation: $\\left.\\begin{aligned}\\widetilde{G}^{loop}_{\\phi ^4}(p)&=\\frac{1}{(p^2+\\tau )^2}\\int d^Dq \\:\\frac{1}{q^2+\\tau } \\\\\\widetilde{G}^{loop}_{\\phi ^3}(p)&=\\frac{1}{(p^2+\\tau )^2}\\int d^Dq \\:\\frac{1}{q^2+\\tau }\\:\\frac{1}{(p-q)^2+\\tau }\\end{aligned}\\right.$ where $\\tau =-\\log (\\lambda (2D-1))$ as usual.",
"In fact, we can go backward from last expression containing integrals in momentum space to the previous one: for each line we have to use the representation $\\frac{1}{p^2+\\tau }=\\int _0^{\\infty } dL\\ \\exp \\left(-L(p^2+\\tau )\\right)$ In this way, the integral over the loop momentum $q$ becomes Gaussian: it can be readily done and we obtain the previous results, eqs.",
"[REF ,REF ].",
"It is clear that the crucial point is the computation of the function ${\\cal G}^{\\text{\\tiny {BL}}}(\\vec{L})$ , since it contains all the information related to the theory we are considering.",
"In the case of the standard LG approach, only the left diagram is present: a standard computation gives for the disconnected and the connected correlation functions ${\\cal G}_C^{\\text{\\tiny {LG}}}(\\vec{L})=L_A (L_I+L_O) \\qquad {\\cal G}_R^{\\text{\\tiny {LG}}}(\\vec{L})=L_A$ and, using the representation $\\frac{1}{(p^2+\\tau )^2}=\\int _0^{\\infty } dL \\ L\\,\\exp \\left(-L(p^2+\\tau )\\right)\\,,$ we obtain the standard result where some lines have a single pole, $(p^2+\\tau )^{-1}$ , while others have a double pole $(p^2+\\tau )^{-2}$ .",
"This representation can be derived also for higher orders of the perturbative expansion.",
"The first perturbative proof [9] of DR was based on the use of the identity for the diagrams: ${\\cal G}_R^{\\text{\\tiny {LG}}}(\\vec{L})={\\cal D}(\\vec{L}).$ In this way, the denominator in $\\tilde{\\cal N}$ becomes ${\\cal D}({\\vec{L}})^{D/2-1}$ and the final expression for the diagrams is the same of a vanilla $\\phi ^4$ theory in dimensions $D-2$ .",
"How to compute the factors ${\\cal G}^{\\text{\\tiny {BL}}}(\\vec{L})$ in the BL approach?",
"We have to compute the average connected and disconnected correlations on a BL where we have the same local geometry ($z=2D$ ) plus a manually injected topological diagram.",
"The final results can be summarized as follows The left diagram gives the same type of contribution of the diagrams of LG reproducing DR.",
"In the region where either $L_A$ or $L_B$ is small the right diagram has a behavior quite similar to the left diagram.",
"Nothing new comes from this diagram in this region.",
"The real interesting region for the right diagram is when all the $L$ 's are large: this gives the relevant contribution at large distances (small momentum) discussed below.",
"In order to compute factors ${\\cal G}^{\\text{\\tiny {BL}}}(\\vec{L})$ on a BL, we have to go through a sequence of simple steps.",
"Some of them are rather lengthy yet straightforward.",
"We consider a BL where we add a loop of the type of (Fig.",
"REF , right).",
"Apart from the loop, the rest of the lattice is a standard BL with fixed connectivity $z=2D$ : this means that variables $\\sigma _I$ and $\\sigma _O$ are the root of $z-1$ infinite tree-like branches, variables $\\tau _I$ and $\\tau _O$ are the root of $z-3$ tree-like branches, while the spins along the topological lines are the root of $z-2$ tree-like branches.",
"We are interested in computing the probability distribution of the random restricted Hamiltonian $\\mathcal {H} [\\sigma _I,\\sigma _O]$ , i.e.",
"the one we obtain after minimizing with respect to all the other variables.",
"This 2-spins Hamiltonian is obtained from the 4-spins Hamiltonian $\\mathcal {H} [\\sigma _I,\\sigma _O,\\tau _I,\\tau _O]$ , where the distances among are fixed to the values $L_I,L_O,L_A,L_B$ shown in Fig.",
"REF , by $\\mathcal {H} [\\sigma _I,\\sigma _O]=\\min _{\\tau _I,\\tau _O} \\mathcal {H} [\\sigma _I,\\sigma _O,\\tau _I,\\tau _O]$ The 4-spins Hamiltonian can be computed by summing four statistically independent 2-spins Hamiltonian and the cavity fields on $\\sigma _I,\\sigma _O,\\tau _I,\\tau _O$ coming from the infinite trees.",
"A two-spin Hamiltonian for a line of length $L$ is described by two fields and a coupling, $(u_1, u_2, J)$ , whose joint law for large $L$ can be written in the form $P_L(u_1,u_2,J)=P(u_1)P(u_2)\\delta (J)+\\\\+Q^D_L(u_1,u_2)\\delta (J) +Q^C_L(u_1,u_2,J)$ Last equations differs from eq.",
"[REF ] only in the fact that here we do not include the contribution by the external fields and the cavity fields for the spin at the extremities of the line.",
"When we compute the probability distribution of the quantity $\\mathcal {H} [\\sigma _I,\\sigma _O,\\tau _I,\\tau _O]$ it factorizes into the product of four terms coming from each of the lines.",
"Connectivization of the diagram, as prescribed by ref.",
"[22], corresponds to removing the term $P(u_1)P(u_2)\\delta (J)$ on each line.",
"Therefore, on each line we can decide if we take the contribution $Q^D$ or $Q^C$ : in the first case we have a disconnected term that can be represented with a line bearing a cross, in the second case we have a connected term that can be represented with a line without a cross.",
"In this way the diagrammatics becomes graphically equivalent to the one of the LG approach, with the addiction of extra diagmas containing cubic vertices.",
"We note that in this computation it is not obvious that the most divergent diagrams will be the ones with the maximal number of possible crosses, as in the standard LG expansion that leads to DR.",
"In fact, this is not the case for diagrams with cubic vertices.",
"We obtain the following results: the connected correlation (response) function is not renormalized at one loop, since ${\\cal G}^{\\text{\\tiny {BL}}}_R(\\vec{L})=0$ ; a factor ${\\cal G}^{\\text{\\tiny {BL}}}_C(\\vec{L})=\\text{const}\\ne 0$ appears for the disconnected correlation function when $L_I=L_O=L_A=L_B=L$ .",
"Numerically, when the four lengths are all different, the result is consistent with the behaviour ${\\cal G}^{\\text{\\tiny {BL}}}_C(\\vec{L})= a+b\\left(\\frac{L_A}{L_B}+\\frac{L_B}{L_A}\\right)$ .",
"The detailed derivation of this result is presented in the SI, together with a numerical consistency check.",
"At this point one should compare this new contribution to the one obtained from the diagrams coming from the standard expansion around the LG theory, looking at the power-law divergence when $\\tau \\rightarrow 0$ in the limit $p\\rightarrow 0$ .",
"Let's focus on $C$ .",
"Within the LG approach, the divergence of the diagram is of order $\\tau ^{-5}$ .",
"Noticing that $\\int _1^{\\infty } dL \\frac{\\exp \\left(-L(p^2+\\tau )\\right)}{L}\\,=\\Gamma \\left[0,(p^2+\\tau )\\right]$ and that the Incomplete Euler Gamma function behaves as $\\Gamma \\left[0,(p^2+\\tau )\\right]\\simeq -\\log [p^2+\\tau ]$ for ${(p^2+\\tau )\\rightarrow 0}$ , and using eqs.",
"[REF ,REF ], we find that the divergence of the new diagram is at most $-\\tau ^{-4}\\log (\\tau )<\\tau ^{-5}$ .",
"The new diagrams coming from cubic vertices are thus sub-dominant with respect to the standard ones in the one loop expansion of two point correlation functions.",
"In this work, we have applied the new topological expansion around the Bethe solution proposed in ref.",
"[22] to the RFIM at $T=0$ , numerically and semi-analytically, obtaining consistent results.",
"It is crucial that we expand around the Bethe solution because it is deeply different from the one found in a standard Landau-Ginzburg approach: while in the latter fluctuations do not play any role, the Bethe solution is dominated by rare fluctuations, especially at $T=0$ ; this is of primary importance given that the RFIM critical behavior is controlled by a $T=0$ fixed point.",
"A direct consequence of the fluctuations-dominated behavior at $T=0$ is that higher order correlations do not decay faster than the average correlation, $\\overline{G(x)^p}\\approx \\overline{G(x)}$ , and this produces an effective theory with vertices of all degrees, including cubic vertices, essentially because diagrams with multiple lines between the same vertices are allowed.",
"We have analyzed the first two one-loop corrections to the correlation functions due to cubic vertices, finding that they give a contribution that is divergent at $D<6$ , as also happens for the standard quartic diagrams.",
"We also found that they give an extra contribution.",
"However, this contribution is sub-dominant with respect to the one given by the usual one-loop diagram coming from the standard LG theory.",
"This means that, within our framework and at the 1-loop order, Dimensional Reduction is still valid at $6-\\epsilon $ dimensions because the most divergent diagrams remain the super-symmetric ones Let us finally remark that the analyzed cubic vertices are really important already at the mean-field level (zero-th order).",
"A peculiarity of the RFIM at $T=0$ on finite connectivity lattices is the existence of avalanches: this collective $T=0$ phenomenon cannot be described within the standard field theoretical treatment, while it appears naturally if $\\phi ^3$ vertices are introduced.",
"At the critical point, the avalanches size distribution follows a power law with a nontrivial exponent $\\tau $ .",
"In our framework, we easily enough find the correct mean-field value $\\tau =3/2$ that cannot be computed within the standard LG approach.",
"Avalanches have a fractal dimension $d_f$ that is connected to fluctuations in the $T=0$ integrated response $\\chi $ via $\\overline{\\chi ^2}/\\overline{\\chi }\\propto L^{d_f}$ .",
"We plan to compute the one-loop correction for $\\chi ^2$ , i.e.",
"for three-point functions, and obtain in this way the $\\epsilon $ -expansion for $d_f$ .",
"This research has been supported by the European Research Council under the European Unions Horizon 2020 research and innovation programme (grant No.",
"694925 – Lotglassy, G Parisi) and by the Simons Foundation (grant No.",
"454949, G Parisi).",
"Supporting Information for “New loop expansion for the Random Magnetic Field Ising Ferromagnets at zero temperature”"
],
[
"Standard diagrammatic rules for the random field Ising model",
"The standard way to compute the loop expansion for the Random Field Ising Model (RFIM) is to introduce an effective replicated $\\phi _4$ model once the disorder has been integrated out [31].",
"In practice one is left with few operating rules to construct Feynmann diagrams, that we briefly recall here.",
"The main difference with a standard $\\phi _4$ theory is that the bare propagator is composed of two parts: a connected part, that is commonly indicated with a line, going as $\\widetilde{R}(p)\\propto \\frac{1}{p^2+\\tau }$ , that will contribute to the connected correlation function $\\overline{\\langle \\sigma _i \\sigma _j\\rangle _c}=\\overline{\\langle \\sigma _i \\sigma _j\\rangle -\\langle \\sigma _i\\rangle \\langle \\sigma _j\\rangle }$ and a disconnected part, indicated with a line plus a cross, $\\widetilde{C}(p)\\propto \\frac{1}{(p^2+\\tau )^2}$ , that will be the dominant contribution to the disconnected correlation function $\\overline{\\langle \\sigma _i\\rangle \\langle \\sigma _j\\rangle }$ .",
"In practice, to build Feynmann diagrams, one should put vertices with 4 lines, that could be connected or disconnected.",
"The only rule in the construction of the diagrams for the expansion of the two-point connected correlation function is that at least one connected path between the two points should be present.",
"Selecting only the most divergent diagrams at each order, one discovers that they correspond to the diagrams with the highest number of allowed crosses at each order in the perturbative expansion.",
"They are shown in Fig.",
"REF for the connected correlation function up to second order."
],
[
"Solution of the Random Field Ising Model on the Bethe Lattice at $T=0$",
"The solution presented in this Section is the 0th order of the loop expansion presented in the main text.",
"It has been presented in some detail in Refs.",
"[29], [30], but we find useful to reported here again for completeness.",
"We consider a model with Hamiltonian $H = -J \\sum _{(ij)\\in E} \\sigma _i\\sigma _j - \\sum _i h_i^R \\sigma _i\\;,$ where $J>0$ and $h_i^R$ are i.i.d.",
"random variables extracted from a Gaussian probability distribution with zero mean and standard deviation $\\sigma _h$ .",
"The edge set $E$ defines a Bethe lattice (BL) of finite connectivity $z$ (mathematically speaking it is a random regular graph of constant degree $z$ ).",
"Following the standard cavity method, we consider cavity fields $h_{i \\rightarrow j}$ and $u_{i\\rightarrow j}$ defined on each edge of the graph.",
"They parametrize, respectively, the marginal probability distribution on $\\sigma _i$ in the cavity graph where edge $(ij)$ has been removed, and the marginal probability distribution on $\\sigma _j$ in the cavity graph where all edges involving vertex $j$ , but $(ij)$ , have been removed.",
"At $T=0$ the self-consistency equations among cavity fields read $h_{i\\rightarrow j} &=& h_i^R + \\sum _{k \\in \\partial i \\setminus j} u_{k\\rightarrow i}\\\\u_{i\\rightarrow j} &=& \\operatornamewithlimits{sign}(h_{i\\rightarrow j})\\;\\min (|h_{i\\rightarrow j}|,J)$ where $\\partial i$ is the set of neighbours of $i$ , i.e.",
"spins linked to $i$ via an edge of the graph.",
"Within the cavity method one is interested, rather than in the specific solution on a given graph, in the solution averaged over the ensemble of random graphs and random fields.",
"To this end it is enough to solve Eqs.",
"[REF ,] in distribution sense and compute the probability distributions of cavity fields $h$ and $u$ .",
"We call $P(u)$ the latter.",
"Willing to compute the correlations between spins $\\sigma _0$ and $\\sigma _L$ that are connected by a line of length $L$ (the path linking $\\sigma _0$ and $\\sigma _L$ is unique on a BL in the thermodynamic limit) we need to integrate out all the spins along the line and compute the triplet $(u_0, u_L, J_L)$ , where $J_L$ is the effective coupling between $\\sigma _0$ and $\\sigma _L$ , while $u_0$ and $u_L$ are the effective fields on $\\sigma _0$ and $\\sigma _L$ coming from the line.",
"Such a triplet can be computed in a recursive way.",
"Let us join two chains, the first one between $\\sigma _1$ and $\\tau $ , characterized by the triplet $(u_1,u_{\\tau , 1},J_1)$ , and the second one between $\\tau $ and $\\sigma _2$ identified by $(u_{\\tau , 2},u_2,J_2)$ .",
"In order to compute the triplet describing the effective Hamiltonian between $\\sigma _1$ and $\\sigma _2$ we need to sum over $\\tau $ and keep only the lowest energy term (we are working at $T=0$ ) $\\begin{aligned}\\mathcal {H}(\\sigma _1,\\sigma _2) &= -\\sigma _1u_1-\\sigma _2u_2+\\min _\\tau \\left[-J_1\\sigma _1 \\tau - h \\tau -J_2 \\tau \\sigma _2 \\right]\\\\& \\equiv E -(u_1+u_1^{\\prime })\\sigma _1 -J_{12} \\sigma _1\\sigma _2-(u_2+u_2^{\\prime })\\sigma _2\\end{aligned}$ with $h=h_{\\tau }^R+u_{\\tau ,1}+u_{\\tau ,2}+\\sum _{k\\in \\partial \\tau \\setminus 1,2}u_{k\\rightarrow \\tau }$ , and $u_{k\\rightarrow \\tau }$ are independent random variables extracted from $P(u)$ .",
"Explicit expressions for $u_1^{\\prime }$ , $u_2^{\\prime }$ and $J_{12}$ , assuming $J_2\\ge J_1\\ge 0$ , are given in Table REF , where $h_{\\pm } = \\big (h \\pm \\operatornamewithlimits{sign}(h) (J_2-J_1)\\big )/2$ .",
"We went from the two initial triplets $(u_1,u_{\\tau , 1},J_1)$ , and $(u_{\\tau , 2},u_2,J_2)$ to the new one $(u_1+u_1^{\\prime },u_2+u_2^{\\prime }, J_{12})$ , with the insertion of $z-2$ cavity fields acting on the central spin $\\tau $ .",
"Table: Rules for evolving cavity fields and effective coupling in the computation of correlations at T=0T=0.In practice we start from a population $P_{L=1}(u_0,u_1,J_1)$ of triplets all equal to $(0,0,J)$ .",
"To evolve the population $P_{L-1}$ into population $P_{L}$ we follow the rules summarized in Table REF , where each triplet of the population $P_{L-1}$ is joined to a triplet $(0,0,J)$ and $z-2$ cavity fields $u_{k\\rightarrow \\tau }$ extracted from $P(u)$ are added on the central spin.",
"Unfortunately this procedure is very ineffective, because at each step a constant fraction of the population (the one satisfying the condition $|h|>J_{L-1}+J$ ) produces a new triplet with $J_L=0$ .",
"Given that $J_L=0$ is a fixed point of the iteration, the part of the population keeping information about branches with non-zero effective couplings shrinks exponentially fast during the iteration.",
"To amplify this signal, we evolve two populations of the same size: one population keeps the pairs $(u_0,u_L)$ along branches with $J_L=0$ , while the second one stores the triplets along branches with $J_L\\ne 0$ .",
"At the same time we measure the probability $p_L=\\mathbb {P}[J_L\\ne 0]$ , that is the relative weight of the second population to the first one, which is found to decay exponentially fast with $L$ : $p_L=a L \\lambda ^L+b\\lambda ^L+o(\\lambda ^L)$ as shown in Fig.",
"REF .",
"$\\lambda $ is the largest eigenvalue associated to the linearization of the BP eqs.",
"[REF ,] around the fixed point.",
"At the critical point, $\\sigma _{h}=\\sigma _{h,c}$ , $\\lambda (\\sigma _{h,c})=1/(z-1)$ holds.",
"The average coupling on the second population decays as $\\overline{J_L}=\\frac{a}{L}+\\frac{b}{L^2}+o(L^{-2})$ as shown in Fig.",
"REF .",
"We see that $\\overline{u_0 u_L}\\propto \\frac{1}{L^2}$ on the population with $J\\ne 0$ while $\\overline{u_0 u_L}\\propto L\\lambda ^L$ on the population with $J=0$ .",
"Moreover $\\overline{u_0^2 J}-\\overline{u_0^2}\\,\\overline{J}\\propto \\frac{1}{L^2}$ on the population with $J\\ne 0$ .",
"Figure: Exponential decay of the probability to have J L ≠0J_L\\ne 0 on a chain of length LL in a BL, data computed at the critical field σ h,c =1.037\\sigma _{h,c}= 1.037 for z=3z= 3.",
"Errors are smaller than points.Figure: Average coupling computed on the population of triplets (u 0 ,u L ,J L )(u_0,u_L,J_L) with J L ≠0J_L\\ne 0.",
"Its decay follows the law J L =a L+b L 2 +o(L -2 )J_L=\\frac{a}{L}+\\frac{b}{L^2}+o(L^{-2}).",
"Data are computed at the critical field σ h,c =1.037\\sigma _{h,c}= 1.037 for z=3z= 3.",
"Errors are smaller than points.Once we have $p_L$ and the two populations at each length $L$ , it is quite simple to compute correlation functions.",
"Indeed, given a triplet $(u_1,u_2,J_{12})$ associated to a path, where the internal spins have been integrated out, the effective two spin Hamiltonian reads $\\mathcal {H} [\\sigma _1,\\sigma _2] = -h_1 \\sigma _1 -J_{12} \\sigma _1\\sigma _2 -h_2 \\sigma _2.$ with $h_1=h^R_1+u_1+\\sum _{k1=1}^{z-1}u_{k1}$ , $h_2=h^R_2+u_2+\\sum _{k2=1}^{z-1}u_{k2}$ , and $u_{k1}$ , $u_{k2}$ extracted from $P(u)$ .",
"At zero temperature the Gibbs measure is concentrated on the ground state $(\\sigma ^*_1,\\sigma ^*_2)$ of the Hamiltonian, that can be easily computed using the rules listed in Table REF .",
"Table: Rules for computing the ground state configuration given the triplet of cavity messages (h 1 ,h 2 ,J 12 )(h_1,h_2,J_{12}).Since we are at $T=0$ , the disconnected correlation function is given by $C_{ij} \\equiv \\overline{\\langle \\sigma _i \\rangle \\langle \\sigma _j\\rangle } \\equiv \\overline{\\sigma _i^*\\sigma _j^*}$ where $\\sigma ^*$ is the ground state configuration, computed according to the rules listed in Table REF .",
"The connected correlation function $C^{con}_{i j} = \\overline{\\langle \\sigma _i \\sigma _j\\rangle _c}$ is ill defined since it is identically equal to zero at $T=0$ , therefore we work with the response $R_{ij} = \\frac{1}{2}\\ \\overline{1 - \\sigma ^*_i\\,\\langle \\sigma _i\\rangle _j}$ where $\\langle \\cdot \\rangle _j$ denotes the expectation over the ground state of the system conditioned to the flipping of the spin $\\sigma _j$ , i.e.",
"$\\langle \\sigma _j\\rangle _j = -\\sigma ^*_j$ .",
"This can be achieved adding a field $h_j=-\\sigma ^*_j \\cdot \\infty $ on the spin $\\sigma _j$ .",
"An alternative and more general definition of response would be $R_{ij} = \\frac{1}{2}\\ \\overline{\\sigma ^*_j(\\sigma ^*_i - \\,\\langle \\sigma _i\\rangle _j)}$ .",
"Since in the RFIM the couplings are ferromagnetic, the spin $\\sigma _i$ can only flip in the same direction of the flip of $\\sigma _j$ , therefore the two definitions are equivalent and $0\\le R_{ij}\\le 1$ .",
"It can be shown that the response can be expressed as the zero temperature limit of an opportunely normalized connected correlation function, that is $R_{ij}=\\lim _{\\beta \\rightarrow \\infty } \\overline{\\left[\\frac{\\langle \\sigma _i \\sigma _j\\rangle _c}{1-\\langle \\sigma _i\\rangle ^2}\\right]}.$ Figure: Connected and disconnected correlation functions as a function of the distance LL on the Bethe lattice.",
"Data are computed at the critical field σ h,c =1.037\\sigma _{h,c}= 1.037 for z=3z= 3.In terms of the probability law of random triplets $(h_1,h_2,J_{12})$ the correlation functions can be written as $C_{12}&=&\\mathbb {P}\\left[J_{12}>\\min (|h_1|,|h_2|)\\right]+\\nonumber \\\\&&\\mathbb {E}\\left[\\operatornamewithlimits{sign}(h_1)\\operatornamewithlimits{sign}(h_2)\\, |\\, J_{12}<\\min (|h_1|,|h_2|)\\right]\\\\R_{12}&=&\\mathbb {P}\\left[J_{12} > |h_1|\\right]=\\mathbb {P}\\left[J_{12} > |h_2|\\right].$ On chains, $h_1$ and $h_2$ are positively correlated,therefore $C_L \\ge R_L$ .",
"Notice that only events with a non-zero effective couplings contribute to the response function: this is the reason why amplifying the population of cavity messages with $J_L\\ne 0$ is mandatory to have a precise measurement of correlations in the $T=0$ limit.",
"In Fig.",
"REF we show the connected and disconnected correlation functions at distance $L$ , averaged over the population of the triplets generated as explained before, in a BL with fixed connectivity $z=3$ , at zero temperature and critical standard deviation $\\sigma _{h,c}=1.037$ for the external field.",
"We find the the connected correlation function decays as $R_L \\propto \\lambda ^L$ , with $\\lambda =\\frac{1}{z-1}$ , while the disconnected correlation function is larger and decays as $C_L = (a L + b) \\lambda ^L$ , as already found analytically in Refs.",
"[29], [30].",
"The corresponding susceptibilities can be computed by summing over all the vertices of the graph $\\chi _{disc}=\\sum _j C_{0j}=\\sum _L n_L C_L$ where $n_L$ is the number of spins at distance $L$ that in a BL is $n_L=\\frac{z}{z-1}\\cdot (z-1)^{L}$ .",
"Substituting $n_L$ and $C_L$ in the equation for $\\chi $ one gets: disc=zz-1L ((z-1))L (a L+b)= =zz-1[a (z-1) (1-(z-1))2+b1-(z-1)] At the critical point $\\lambda (\\sigma _{h,c})=\\frac{1}{z-1}$ and the susceptibility diverges.",
"How can we relate this computation of the susceptibility on the Bethe lattice to the perturbative expansion for a finite dimensional model in dimension $D$ around the Bethe theory?",
"Following ref.",
"[22], the zeroth order expansion for the susceptibility is just eq.",
"[REF ] where $n_L$ is replaced with the number of non-backtracking paths of length $L$ starting from a point in a $D$ dimensional lattice: $n_L\\propto (2D-1)^L$ .",
"Therefore at zeroth order the expansion predicts a divergence located at the same critical point $\\sigma _{h,c}$ of a Bethe lattice with connectivity $z=2D$ ."
],
[
"One loop BL correction",
"The first order in the BL expansion considers the presence of structures with one spatial loop, as reported in Fig.",
"REF .",
"In this section, we compute the one-loop correction to the connected and disconnected correlation function, coming from the topological structure in the right part of Fig.",
"REF , that is the one that gives an additional term with respect to the standard Landau-Ginsburg (LG) expansion.",
"Following the prescription of ref.",
"[22], we should compute the correlation on a BL in which such structure has been manually injected and subtracting the values of the correlation computed on the two paths $L_I+L_A+L_O$ , $L_I+L_B+L_O$ , supposed as independent.",
"Operatively we build the loop putting together four triplets or couples extracted independently from the populations of single Bethe lines obtained as explained in the previous section: two of length $L_I$ and $L_O$ for the external lines, and two of length $L_A$ and $L_B$ for the internal lines of the loop.",
"The internal lines of the loop, with triplets $(u_{\\tau _I}^A,u_{\\tau _O}^A, J_A)$ and $(u_{\\tau _I}^B,u_{\\tau _O}^B, J_B)$ , will just result in a new triplet whose coupling is the sum of the couplings: $J_T=J_A+J_B$ and whose fields are the sum of the fields $u_{\\tau _I}^T=u_{\\tau _I}^A+u_{\\tau _I}^B$ .",
"Then the new triplet is attached to the external legs, as in eq.",
"(REF ), with the only difference being that in $\\tau _I$ and $\\tau _O$ there are just $z-3$ additionally cavity fields (instead of $z-2$ ones) extracted from $P(u)$ .",
"We then end up with a new triplet describing the loop.",
"We compute the correlations implied by this triplet and subtract the correlation implied by the paths $L_I+L_A+L_O$ and $L_I+L_B+L_O$ considered as independent: This is the one-loop contribution to the correlation function, that we will indicate with $G_{loop}(\\vec{L})=G_{loop}({L_I,L_O,L_A,L_B})$ .",
"In the following we report the one-loop results for $G=C,R$ , that we obtain looking numerically to the behaviors when $L_A$ or $L_B$ are small or large.",
"Let us first analyze the response.",
"We know that the contribution to $R$ given by the external legs should, in any case, be proportional to $\\lambda ^{(L_I+L_O)}$ , because the diagram should be connected to contribute to the connected correlation function.",
"Thus we concentrate on the internal legs.",
"First of all, we fix also $L_B$ to a finite value $L_B=3$ and we measure the contribution of the loop to the response function as a function of $L_A=L$ .",
"We measure the behavior $R_{loop}({L_I,L_O,L,3})\\propto L\\lambda ^{L+L_I+L_O},$ as shown in Fig.",
"REF .",
"As expected, the behaviour is the same of the one-loop diagram coming from the standard theory, that is the left diagram of Fig.",
"REF .",
"In fact, two $\\phi _3$ vertices reduces to a tadpole $\\phi _4$ vertex once one internal line is fixed to a finite length.",
"For a tadpole $\\phi _4$ diagram, LG theory predicts that the maximal divergent contribution comes from the second diagram of Fig.",
"REF , that has indeed the same behavior of as in eq.",
"[REF ].",
"Figure: Absolute value of the one-loop BL contribution to the connected correlation function divided by λ L \\lambda ^L,when L I =L B =L O =3L_I=L_B=L_O=3 and L A =LL_A=L, as function of distance LL.",
"The behaviour R loop (3,3,L,3)∝λ L LR_{loop}(3,3,L,3)\\propto \\lambda ^L L is evident.",
"The sign of the contribution is negative.",
"Data is computed at the critical field σ h,c =1.037\\sigma _{h,c}= 1.037 for z=3z= 3.Next, we fix $L_A=L_B$ and we measure $R_{loop}(L_I,L_O,L,L)$ .",
"It receives contributions from two different diagrams: we call “Contribution A” the one coming from the loop with both $J_A$ and $J_B$ different from zero, that is, in the language of eq.",
"[36] of the main text, taking the contribution coming from $Q^C_L(u_1,u_2,J)$ on both lines; “Contribution B” instead, is the one from a loop with just one coupling different from zero, that is, taking a contribution coming from $Q^C_L(u_1,u_2,J)$ on one line, and a contribution coming from $Q^D_L(u_1,u_2)\\delta (J)$ on the other line.",
"(Please note that it is not possible to take contribution from $Q^D_L(u_1,u_2)\\delta (J)$ on both internal lines because it will result in a disconnected loop that gives zero contribution to connected correlation functions).",
"Separately, contribution A and B, multiplied by their occurrence probabilities, have a dominant behaviour in $L$ of the type $(a+bL)\\lambda ^{L_I+L_O+2L}$ , but they have opposite sign.",
"When summing the two contributions, the dominant term in $L$ is exactly cancelled, and the total contribution is left with the subdominant part $R_{loop}({L_I,L_O,L,L})\\propto \\lambda ^{L_I+L_O}\\lambda ^{4L},$ as shown in fig.",
"REF .",
"As described in the main text, we can write $R_{loop}(\\vec{L})=\\lambda ^{\\Sigma (\\vec{L})}{\\cal G}_R(\\vec{L})+o(\\lambda ^{\\Sigma (\\vec{L})})$ , with $\\Sigma (\\vec{L})=L_I+L_O+L_A+L_B$ .",
"Eq.",
"[REF ] corresponds to ${\\cal G}_R(\\vec{L})=0$ .",
"Figure: One-loop BL contribution to the connected correlation functionwhen L I =L O =3L_I=L_O=3 and L A =L B =LL_A=L_B=L, as function of the length of internal lines LL.The total contribution is the sum of Contribution A and B.",
"They separately go like (a+bL)λ 2L (a+bL)\\lambda ^{2L}, but they have opposite sign.Once they are summed, the result decays as λ 4L \\lambda ^{4L}.",
"Data are computed at the critical field σ h,c =1.037\\sigma _{h,c}= 1.037 for z=3z= 3.Things are different for the disconnected correlation function.",
"Also in this case, first of all we fix $L_B$ to a finite value and we measure the behaviour of the loop as a function of $L_A=L$ .",
"We measure the behavior $C_{loop}({L_I,L_O,L,3})\\propto L\\lambda ^{L+L_I+L_O},$ that again is the same contribution of the tadpole diagram from the usual $\\phi _4$ LG theory.",
"Then we put $L_A=L_B=L_I=L_O=L$ , and we observe $C_{loop}({L,L,L,L})\\propto \\lambda ^{4L},$ that corresponds to ${\\cal G}_C(L,L,L,L)=const$ .",
"Figure: Comparison between the one-loop BL contribution to the connected and disconnected correlation functionwhen L I =L O =3L_I=L_O=3 and L A =L B =LL_A=L_B=L, as function of the length of internal lines LL.",
"Data are computed at the critical field σ h,c =1.037\\sigma _{h,c}= 1.037 for z=3z= 3.We now want to compute the one-loop correction to the susceptibilities.",
"In an analogous way to eq.",
"[REF ], we should account for all the subgraphs of the type of the right dyagram in Fig.",
"REF that are presents in a finite-dimensional lattice.",
"The computation can be done exactly using the number of non-backtracking paths, however, the large $L$ behavior of this counting factor is also captured if we assume that the number of paths from 0 to $x$ of length $L$ is given by the random walk probability of reaching $x$ in time $L$ in $D$ dimensions multiplied by the number of generic non-backtracking paths of length $L$ starting from 0: $n_L(x)\\propto \\frac{(z-1)^L}{L^D} e^{-x^2/(2L)}$ .",
"In the same way we can compute $n_{L_A,L_B}(x,y)$ , defined as the number of paths of length $L_A$ and $L_B$ that have the same starting and ending point $x$ and $y$ : $n_L(x,y)\\propto \\frac{(z-1)^{L_A+L_B}}{(L_AL_B)^D} \\left(e^{-\\frac{(x-y)^2}{2}(\\frac{1}{L_A}+\\frac{1}{L_B})}\\right)$ .",
"The one loop correction to the susceptibility associated to a generic correlation function $G$ is thus: loopLI,LA,LB,LO x,y,dnLI(x)nLA,LB(y-x) nLO(d-y)Gloop(L).",
"Replacing the sums over $x,y,d$ with integrals, and performing the integrals we obtain: loop L Gloop(L)(z-1)(L)1(LA+LB)D/2.",
"It is now clear that the contribution to the connected susceptibility is always finite at the critical point, because $R_{loop}(\\vec{L})$ decays more rapidly than $(z-1)^{-\\Sigma (\\vec{L})}$ .",
"Things are different for the disconnected susceptibility.",
"In this case, we can apply the Ginzburg criterion to identify the upper critical dimension: we look at $\\frac{d\\chi }{d \\lambda }$ , that is divergent at the critical point in $D\\le 6$ .",
"At this point, one could think that Dimensional Reduction is broken at $D=6$ .",
"In fact for $D<6$ , we have cubic diagrams, not of the type of the super-symmetric ones, that are important.",
"However, as we explained in the main text, once we compare their divergence with the divergence of the standard one-loop $\\phi _4$ diagrams, we discover that their contribution is sub-dominant with respect to the usual $\\phi _4$ term.",
"This implies that DR is still valid at $6-\\epsilon $ dimensions.",
"To conclude, we just mention that until now we do not know the exact behavior of $C_{loop}$ as a function of the length of the legs.",
"Having measured $C_{loop}(L,L,L,L)\\propto \\lambda ^{4L}$ we can think to different cases: A) $C_{loop}(\\vec{L})\\propto \\lambda ^{L_I+L_A+L_B+L_O}$ B) $C_{loop}(\\vec{L})\\propto \\lambda ^{L_I+L_A+L_B+L_O}(\\frac{L_A}{L_B}+\\frac{L_B}{L_A})$ C) $C_{loop}(\\vec{L})\\propto \\lambda ^{L_I+L_A+L_B+L_O}(\\frac{L_I}{L_B}+\\frac{L_I}{L_A}+\\frac{L_O}{L_B}+\\frac{L_O}{L_A})$ We expect that the presence of terms of the type $\\lambda ^L/L$ should signal the presence of squared disconnected correlation function We somehow expect the presence of important square correlations at zero temperature, see the Conclusions.",
"In fact we measured numerically that $\\overline{(u_0 u_L)^2}\\propto \\lambda ^L/L$ , and we expect that $\\overline{(\\langle \\sigma _0\\rangle \\langle \\sigma _L\\rangle )^2}\\propto \\overline{(u_0 u_L)^2}$ , given that $\\overline{\\langle \\sigma _0\\rangle \\langle \\sigma _L\\rangle }\\propto \\overline{u_0 u_L}$ .",
"To understand which terms are present, we measure $C_0\\equiv C_{loop}(L_I,L_O,L,L)$ and $C_1\\equiv C_{loop}(L_I,L_O,2L,L)$ , at fixed, finite values of $L_I$ and $L_O$ , and we look at them as a function of $L$ .",
"We obtain the behaviours: $C_0\\propto \\lambda ^{2L}$ , $C_1\\propto \\lambda ^{3L}$ .",
"The ratio $Q(L_I,L_O)\\equiv \\frac{C_0/ \\lambda ^{2L}}{C_1 /\\lambda ^{3L}}$ is independent from $L_I$ and $L_O$ .",
"This result tells us that the case C) is not present.",
"Indeed this is what we expected: the case C) corresponds to a connected loop, but we know that the connected correlation, that can receive contribution only by a connected loop, is not renormalized at one loop.",
"We thus expect that the connected loop gives no contribution to $C$ , as found.",
"We numerically find that $Q=0.96$ : if the situation A) were the only present, $Q=1$ , while in the case B) $Q=0.8$ : to recover the measured $Q=0.96$ we need a linear combination of the two cases.",
"From the numerical computation, we thus expect the one-loop contribution to the disconnected correlation function to have the form $C_{loop}(\\vec{L})\\propto \\lambda ^{\\Sigma (\\vec{L})}\\left[a+b\\left(\\frac{L_A}{L_B}+\\frac{L_B}{L_A}\\right)\\right],$ with $a=1$ , $b=0.1$ ."
],
[
"BL results for the distribution of Avalanches",
"The distribution of the size of the avalanches $s$ at the critical point is expected to be $P(s)=\\frac{1}{s^{\\rho }}$ with $\\rho $ the critical exponent for the avalanches whose value on the BL is $\\rho _{MF}=\\frac{3}{2}$ .",
"This distribution can be obtained in the framework of percolation on the BL (see [26] and refs.",
"therein).",
"We explained in the main text that $s$ is proportional to the susceptibility associated to the connected correlation function: $\\chi =\\sum _x \\langle \\sigma _0\\sigma _x\\rangle _c$ .",
"Given that $\\overline{\\chi }\\propto \\tau ^{-1}$ , the distribution [REF ] in the MF region implies that $\\overline{\\chi ^2}\\propto \\tau ^{-3}\\ne \\overline{\\chi }^2$ .",
"This result cannot be recovered from the LG theory.",
"In this case, in fact, the global magnetization is the only important variable, there are no fluctuations in the magnetization nor in the susceptibility, for which therefore we can write $\\overline{\\chi ^2}=\\overline{\\chi }^2\\propto \\tau ^{-2}$ .",
"Let us now look in detail to what are the field-theoretical predictions on $\\chi ^2=\\sum _{x,y} \\langle \\sigma _0\\sigma _x\\rangle _c\\langle \\sigma _0\\sigma _y\\rangle _c$ , for which we have to look to three point functions.",
"If we admit that there are only $\\phi _4$ vertices, as in the MF FC model, the diagram with no loop is the left one in Fig.",
"REF .",
"Giving that each line corresponds to a connected propagator and thus bring a factor $\\tau ^{-1}$ , the left diagram will be associated at the critical point to a divergence of the type $\\overline{\\chi ^2}\\propto \\tau ^{-2}$ , recovering the FC MF result.",
"If now we imagine that the associated field theory includes also $\\phi _3$ vertices, the situation will change: the right diagram in Fig.",
"REF is possible, leading to a critical behaviour: $\\overline{\\chi ^2}\\propto \\tau ^{-3}$ .",
"Figure: Important topological diagrams for the calculation of the leading term for χ 2 \\chi ^2 in the standard φ 4 \\phi _4 theory (left) and in a theory with φ 3 \\phi _3 vertices (right).We have announced that, following ref.",
"[22], diagrams with $\\phi _3$ vertices should be present in the field-theoretical description of the RFIM at $T=0$ when expanding around the finite connectivity Bethe solution: in this section, we have shown that their presence is perfectly compatible with the MF description of the avalanches, for which we can recover the critical exponent $\\rho ^{MF}=\\frac{3}{2}$ , in contrast to the standard FC $\\phi _4$ theory that cannot justify the probability distribution of the avalanches.",
"In ref.",
"[27], the following connection between avalanches and DR is stated: DR breaks down due to avalanches if they are “big enough”, more precisely if the fractal dimension $d_f$ of the largest typical critical avalanches satisfies the condition $d_f=D-d_{\\phi }$ , with $D$ the spatial dimension and $d_{\\phi }$ the scaling dimension of the field near the relevant zero-temperature fixed point.",
"In a way analogous to Ref.",
"[27], we have seen that the $\\phi _3$ diagrams will not automatically destroy DR: in particular, at one loop they are sub-dominant with respect to the standard $\\phi _4$ ones implying that at $D=6-\\epsilon $ , DR is preserved."
],
[
"Ansatz for coupling and fields at distance L",
"In this section, we verify our previous numerical results using a different method.",
"We introduce an Ansatz $P_L(u_0,u_L,J)$ for the joint distribution of the effective coupling and fields between two spins at distance $L$ in a BL, which should capture the leading behviour at large $L$ .",
"We assume the form $\\begin{aligned}P_L(u_0,u_L,J)=& \\delta (J)\\bigg [ P(u_0)P(u_L)- c_0 L \\lambda ^L g(u_0)g(u_L)\\\\& - c_1 L \\lambda ^L g^{\\prime }(u_0)g^{\\prime }(u_L)- c_2 L \\lambda ^L g^{\\prime \\prime }(u_0)g^{\\prime \\prime }(u_L) \\bigg ]\\\\&+ a L^2 \\lambda ^L \\rho \\, e^{-\\rho J L}g(u_0)g(u_L) +o(L\\lambda ^L)\\end{aligned}$ and check it's consistency.",
"$P(u)$ is the already mentioned Bethe distribution of cavity fields, while $g(u)$ is the eigenfunction associated to the largest eigenvalue $\\lambda $ with respect to a perturbation of $P(u)$ [34].",
"$g(u)$ is symmetric, therefore $g^{\\prime }(u)$ is anti-symmetric.",
"We impose $\\int _{-\\infty }^{\\infty } g(u)du = 1$ .",
"We impose normalization: $1=\\int \\text{d}u_0\\text{d}u_L\\text{d}J\\ P_L(u_0,u_L,J) = 1- c_0 L\\lambda ^L + a L \\lambda ^L,$ obtaining the relation $c_0= a $ .",
"The functional form REF has to reproduce itself when attaching two chains to create a new one of length $L_1+L_2=L$ , according do the rules of Table REF .",
"Using the symbol $\\otimes $ to denote the iteration in distribution of two chains according to these rules, or the addition of a field to an extremity of a chain, we have to check that $P_L = P_{L_{1}}\\otimes Q_{z-2} \\otimes P_{L_{2}}$ where $Q_{z-2}$ is the distribution of the sum of $z-2$ cavity fields extracted from $P(u)$ plus the random external field.",
"More explicitely, we have $\\begin{aligned}P_L(u_0,u_L,J)=&\\int dP_{L_{1}}(v_0,v_{\\tau ,1},J_1) \\,dQ_{z-2}(h)\\,dP_{L_{2}}(v_{\\tau ,2},v_L,J_2)\\\\\\times &\\delta (u_0 - v_0 - f_1(v_{\\tau ,1}+v_{\\tau ,2}+h,J_1,J_2))\\\\\\times &\\delta (u_L - v_L - f_2(v_{\\tau ,1}+v_{\\tau ,2}+h,J_1,J_2))\\\\\\times &\\delta (J - f_J(v_{\\tau ,1} + v_{\\tau ,2} + h,J_1 , J_2)),\\end{aligned}$ where the functions $f_1$ , $f_2$ and $f_3$ can be deduced from Table REF .",
"A careful computation of the leading order terms in the right hand side, shows that the Ansatz (REF ) holds, provided that $a = \\frac{\\rho }{2 \\hat{P}(0)} , \\quad c_2=0,$ where $\\hat{P}(h) =\\int \\text{d} u\\,\\text{d} v\\,\\text{d} h \\ g(u)g(v)Q_{z-2}(h) \\ \\delta (h-(u+v+h))$ It turns out that $\\rho $ and $c_1$ are left undetermined.",
"The final form for the Ansatz is thus given by: $\\begin{aligned}P_L(u_0,u_L,J)=& \\delta (J)\\bigg [ P(u_0)P(u_L)- a\\,L \\lambda ^L g(u_0)g(u_L)+ \\\\& - c_1 L \\lambda ^L g^{\\prime }(u_0)g^{\\prime }(u_L) \\bigg ]+\\\\&+ a L^2 \\lambda ^L \\rho \\, e^{-\\rho J L}g(u_0)g(u_L)\\end{aligned}$ In this form, the Ansatz is normalized and stable under the merging of two chains up to $o(L\\lambda ^L)$ terms.",
"The coefficient $\\rho $ can be derived using a few additional arguments, see next Section.",
"This form for the Ansatz is compatible with what we know from the previous sections: the coupling $J$ either is exactly 0 or, with a probability of order $L\\lambda ^L$ , is of order $1/L$ .",
"The two effective fields and the coupling are independently distributed when conditioning on the event $J_L>0$ .",
"The two fields have correlation of order $O(L\\lambda ^L)$ when conditioning on $J_L=0$ instead.",
"Moreover, the Ansatz reproduces the numerical behavior of the correlation functions at length $L$ .",
"Now we want to reproduce the numerical results for the loop contribution to the correlation functions using the Ansatz.",
"In order to obtain the joint distribution $P_{\\text{loop}}(u_{I},u_{O}, J_{\\text{loop}})$ of the effective fields and coupling for two spin at the extremities of a loop as in the right part of Fig.",
"REF , we convolve the two internal branches $L_A$ and $L_B$ , yielding a distribution that we call $(P_{L_A}* P_{L_B})$ on the two internal spins, and attaching the external legs $L_I$ and $L_O$ : $P_{\\text{loop}}=P_{L_{I}}\\otimes Q_{z-3}\\otimes \\left(P_{L_{A}}*P_{L_{B}}\\right)\\otimes Q_{z-3}\\otimes P_{L_{O}}$ We have already said that the loop contribution to the observable is given by the value of the observable computed on the loop minus the observable computed on the two paths $L_I+L_A+L_O$ and $L_I+L_B+L_O$ considered as independent.",
"We can easily obtain this loop correction defining the “topologically connected” loop distribution $\\tilde{P}_{\\text{loop}}(u_I,u_O,J_{\\text{loop}})$ as in eq.",
"(REF ) but substituting to $P_L$ the (improper) distribution $\\widetilde{P}_L$ given by $\\begin{aligned}\\widetilde{P}_L(u_0,u_L,J)=& \\delta (J)\\bigg [- a\\,L \\lambda ^L g(u_0)g(u_L)+ \\\\& - c_1 L \\lambda ^L g^{\\prime }(u_0)g^{\\prime }(u_L) \\bigg ]+\\\\&+ a L^2 \\lambda ^L \\rho \\, e^{-\\rho J L}g(u_0)g(u_L),\\end{aligned}$ that is, the same as $P_L$ but without its asymptotic term.",
"In this way, the loop correction is just the mean value of the observable on $\\tilde{P}_{\\text{loop}}(u_I,u_O,J_{\\text{loop}})$ .",
"The loop correction for both the connected and disconnected correlation function computed on $\\tilde{P}_{\\text{loop}}(u_I,u_O,J_{\\text{loop}})$ gives 0.",
"While this is in agreement with the numerical computation for $R$ , we had been expecting a non-zero contribution for $C$ .",
"However, being the Ansatz consistent up to order $O(L\\lambda ^L)$ , it could only give a contribution to $C_{loop}(L,L,L,L)=O(L\\lambda ^{\\Sigma (\\vec{L})})$ that in fact is not present from the numerical analysis (Please notice that higher contributions are prohibited for symmetry reasons).",
"Thus the analytical Ansatz predictions are fully compatible with the numerical results up to the chosen order.",
"To go to next order, we should introduce terms $O(\\lambda ^L)$ in the ansatz.",
"Unfortunately, the addition of new terms in the original Ansatz makes the computation much more involved, and we did not perform it entirely.",
"In particular these new terms should take into account correlations between fields and coupling in the $J\\ne 0$ part, as found from the numerical analysis.",
"However terms in the $J=0$ part can be added without much effort, in particular we added the terms $-b_0\\lambda ^L g(u_0)g(u_L)-b_1\\lambda ^L g^{\\prime }(u_0)g^{\\prime }(u_L)-b_2\\lambda ^L g^{\\prime \\prime }(u_0)g^{\\prime \\prime }(u_L)$ and checked how they behave under iteration.",
"Imposing normalization and self-consistency we do find that $b_0=b_2=0$ and $b_1=\\frac{1}{(2\\hat{P}(h=0))^2}$ .",
"The addition of this new term gives no contribution to $R$ while it gives a contribution $C_{loop}(L,L,L,L)=O(\\lambda ^{\\Sigma (\\vec{L})})$ for the disconnected correlation function, as found from the numerical computation.",
"We stress however that we lack some terms coming from the correction of the $J\\ne 0$ part of the Ansatz to order $O(\\lambda ^L)$ that do not allow us to compute exactly $C_{loop}(L,L,L,L)$ at order $O(\\lambda ^{\\Sigma (\\vec{L})})$ ."
],
[
"Computation of the mean coupling decay",
"The analytical Ansatz presented in the previous Section requires the knowledge of 2 parameters: $c_1$ and $\\rho $ .",
"Here we show how to compute the latter in a very effective way.",
"We follow the ideas of Ref.",
"[29], but correcting an error made in that work.",
"In practice we are interested in computing the mean value of the effective coupling at distance $L$ along the branches of the BL, where the coupling is non zero $\\langle J \\rangle _{J>0} = \\frac{1}{\\rho L}$ Without loss of generality and to make analytical expressions more compact we fix the single link coupling to $J=1$ hereafter.",
"A possible numerical method has been already discussed in the previous sections and consists in evolving a population of triplets $(u_1,u_2,J)_L$ reweighted in a such a way that triplets with $J\\ne 0$ do not decrease exponentially fast but remain constant in number: this trick allows to follow triplets with non-zero effective coupling for a long enough time to measure accurately the exponent $\\rho $ .",
"As an example we show in Figure REF the inverse of the mean effective coupling as a function of $L$ , measured at the critical point $\\sigma _{h,c}\\simeq 1.037$ for $z=3$ .",
"Figure: The inverse of the mean effective coupling 〈J L 〉 J>0 \\langle J_L \\rangle _{J>0} scales linearly with the distance LL on the BL.",
"The fit 〈J〉 J>0 -1 =1.035(5)+0.755(1)L\\langle J \\rangle _{J>0}^{-1} = 1.035(5) + 0.755(1) L interpolates well the data points, even at relatively small values of LL, as can be appreciated in the inset, where we plot 〈J〉 J>0 -1 -0.755L\\langle J \\rangle _{J>0}^{-1} - 0.755 L versus LL.Although the fit shown in Figure REF is very good and provides an estimate to $\\rho = 0.755(1)$ we have to remind that the reported uncertainty only represent the statistical error given the fitting function.",
"It is much more difficult to estimate the systematic error, that would depend — among others — on the corrections to the asymptotic scaling.",
"For this reason we would be much more confident if we could derive an analytical expression for $\\rho $ .",
"Given that the effective couplings becomes very small even on the BL branches where they are non-zero, we would like to exploit this observation to better study the asymptotic distribution of cavity messages.",
"Let us consider the equations for updating the triplets reported in Table REF and let us rewrite it in a more explicit form, concentrating on the messages acting on the spin at distance $L$ (we ignore the messages arriving on the spin at the root).",
"Schematically we have that, adding one new link, the messages change according to the following rules ($*$ messages are irrelevant in the present computation) (*,uL,JL) + (0,0,1) (*,uL+1,JL+1) uL+1 = u(hz-2+uL,JL) JL+1 = J(hz-2+uL,JL) where $h_{z-2}\\sim Q_{z-2}$ , i.e.",
"$h_{z-2} = h^R+ \\sum _{i=1}^{z-2} u_i$ in distribution, and the functions are defined as follows u(h,J) = *sign(h) { ll |h| if |h|<1-J |h|+1-J2 if 1-J<|h|<1+J 1 if 1+J<|h| .",
"J(h,J) = { ll J if |h|<1-J 1+J-|h|2 if 1-J<|h|<1+J 0 if 1+J<|h| .",
"From the above expressions we understand that during the evolution with probability $\\mathbb {P}[1+J<|h|]$ the effective coupling becomes null, but we are interested in the complementary events, when the effective coupling remains non-zero.",
"With probability $\\mathbb {P}[|h|<1-J]$ the coupling remains unaltered and with probability $\\mathbb {P}[1-J<|h|<1+J]$ it decreases.",
"We notice that the last event becomes very rare in the limit of small $J$ , because the random variable $h$ has a continuous probability density function $P_h$ with no Dirac deltas in 1 or -1, so $\\mathbb {P}[1-J<|h|<1+J]\\simeq (P_h(1)+P_h(-1))2J$ .",
"In practice in the large $L$ limit, when all effective couplings are very small, $J_L\\ll 1$ , the evolution proceeds essentially by keeping the $J_L$ constant until it jumps directly to $J_L=0$ .",
"Let us move now to the analysis of the cavity messages $u_L$ .",
"Assuming we are in the large $L$ limit and all effective couplings are very small, we can work under the above hypothesis that the effective coupling stays constant in $L$ until it becomes null.",
"So hereafter we fix $J_L=J$ , where $J\\ll 1$ is a small constant.",
"We call $Q_J$ the probability distribution of the cavity messages on the branches where the effective coupling is fixed to $J$ .",
"From Eq.",
"[] it is easy to derive that asymptotically $Q_J$ has support in $(-1+J,1-J)$ and satisfies the following equation (J) QJ(u') = =E du QJ(u) (u'-(hz-2+u)) I(|u'|<1-J) , where $\\mathbb {I}$ is the indicator function and the normalizing factor $\\lambda (J)$ is given by $\\lambda (J) = \\int du Q_J(u) \\mathbb {I}(|h_{z-2}+u|<1-J) =\\\\= \\mathbb {P}[|h_{z-2} + u^{(J)}|<1-J]\\,,$ where $u^{(J)}\\sim Q_J$ .",
"In practice $\\lambda (J)$ is the rate of survival of a non-zero effective coupling equal to $J$ .",
"From this we can obtain the probability distribution of couplings in the $J\\ll 1$ limit $\\mathbb {P}[J_L=J] \\propto \\lambda (J)^L \\simeq \\big [\\lambda (0)+\\lambda ^{\\prime }(0)J\\big ]^L \\propto \\left(1+\\frac{\\lambda ^{\\prime }(0)}{\\lambda (0)}J\\right)^L \\propto \\\\\\propto \\exp \\left(\\frac{\\lambda ^{\\prime }(0)}{\\lambda (0)} J L\\right) \\propto \\exp (-\\rho J L) \\Rightarrow \\rho = -\\frac{\\lambda ^{\\prime }(0)}{\\lambda (0)}\\,.$ Given that we are mostly interested in studying the decay at the critical point it is worth reminding that at criticality $\\lambda (0)=1/(z-1)$ holds and thus we have $(z-1) \\lambda (J) \\simeq 1-\\rho J \\;\\text{ for }\\; J\\ll 1\\,.$ Figure: λ(J)\\lambda (J) computed at the critical field σ h,c =1.037\\sigma _{h,c}=1.037 for z=3z=3.",
"The curve is the best fit to the function λ(J)=λ(0)+λ ' (0)J+aJ 2 \\lambda (J)=\\lambda (0)+\\lambda ^{\\prime }(0)J+a J^2.In Figure REF we show data for $\\lambda (J)$ computed at criticality for $z=3$ together with best interpolation via the following function (J)=(0)+'(0)J+a J2 The curve interpolates perfectly the data within the statistical uncertainties and it returns an estimate $\\rho =0.754(1)$ , compatible with the numerical estimate coming from the triplets evolution described in previous sections."
]
] | 1906.04437 |
[
[
"On the quest for generalized Hamiltonian descriptions of $3D$-flows\n generated by curl of a vector potential"
],
[
"Abstract We study Hamiltonian analysis of three-dimensional advection flow $\\mathbf{\\dot{x}}=\\mathbf{v}({\\bf x})$ of incompressible nature $\\nabla \\cdot {\\bf v} ={\\bf 0}$ assuming that dynamics is generated by the curl of a vector potential $\\mathbf{v} = \\nabla \\times \\mathbf{A}$.",
"More concretely, we elaborate Nambu-Hamiltonian and bi-Hamiltonian characters of such systems under the light of vanishing or non-vanishing of the quantity $\\mathbf{A} \\cdot \\nabla \\times \\mathbf{A}$.",
"We present an example (satisfying $\\mathbf{A} \\cdot \\nabla \\times \\mathbf{A} \\neq 0$) which can be written as in the form of Nambu-Hamiltonian and bi-Hamiltonian formulations.",
"We present another example (satisfying $\\mathbf{A} \\cdot \\nabla \\times \\mathbf{A} = 0$) which we cannot able to write it in the form of a Nambu-Hamiltonian or bi-Hamiltonian system.",
"On the hand, this second example can be manifested in terms of Hamiltonian one-form and yields generalized or vector Hamiltonian equations $\\dot{x}_i = - \\epsilon_{ijk}{\\partial \\eta_j}/{\\partial x_k}$."
],
[
"Introduction",
"Hamiltonian analysis of finite dimensional systems has a huge literature and still attracts deep attention of many autors.",
"There are several types of 3D flows which you can interpret in terms of Nambu-Poisson Hamiltonian dynamics.",
"As it is well known, the generalized force field can be decomposed into the sum of two vector fields.",
"One which is equal to minus of the gradient of some potential function whereas the other does not come from a potential.",
"This decomposition resembles the decomposition of a vector field into a curl-free (irrotational) component and a solenoidal (divergence-free) component arising from the celebrated Helmholtz theorem.",
"A particular instance of this Helmholtz-like decomposition is the one where the irrotational and the divergence-free parts are orthogoal [35].",
"We start with a three dimensional system in form $\\mathbf {\\dot{x}}=\\mathbf {v}({\\bf x},t),$ where we denote a three-tuple by $\\mathbf {x}=(x,y,z)$ , describing the evolution with time $t$ of the spatial position ${\\bf x}$ of a fluid particle under the action of the fluid velocity field $\\mathbf {v}({\\bf x},t)$ , known as the advection equation, which can be described briefly as follows [33], [34].",
"The transport of a passive scalar $\\psi $ embedded in a fluid flow governed by the ideal advection equation $ \\frac{\\partial \\psi }{\\partial t} + {\\bf v}({\\bf x},t) = 0 \\qquad \\Leftrightarrow \\qquad \\frac{d\\psi }{dt} = 0.$ Since the passive scalar is frozen into the fluid element, the distribution function $\\psi $ at arbitrary time can be found to be $\\frac{d{\\bf x}(\\xi ,t)}{dt} = {\\bf v}({\\bf x},t)$ with the initial condition ${\\bf x}(\\xi , t=0) = \\xi $ .",
"In general, (REF ) is a nonlinear dynamical system capable of exhibiting chaotic dynamics, then this flow is a mixing flow over some region of Lagrangian topology.",
"The flow is mixing means whether the flow trajectory is globally ergodic, i.e., trajectory visits every point in a closed domain.",
"If the flow is incompressible $\\nabla \\cdot {\\bf v} = 0$ , then the dynamical system (REF ) is conservative.",
"The representation of divergence-free vector fields as curls in two and three dimensions has been studied in [2].",
"In such a situation ${\\bf v}$ can be represented as ${\\bf v}({\\bf x}) = \\nabla \\gamma _1 \\times \\nabla \\gamma _2$ , which can be manifested in terms of Nambu-Hamiltonian form [7], [25].",
"In this paper we show an example involving an additional condition on the dynamics of the flow ${\\bf v}({\\bf x}) = \\nabla \\times {\\bf A}$ , i.e., null helicity condition etc, then Hamiltonian formalism can not be given in terms of Nambu-Hamiltonian.",
"This can be restored if we break the symmetry of the null helicity.",
"Accordingly, we notice one can grossly divide these into two categories.",
"In the first category, there are 3D flows satisfying both the divergence free condition and the integrability condition, that is $\\nabla \\cdot \\mathbf {v}=0,\\qquad \\mathbf {v} \\cdot \\nabla \\times \\mathbf {v} = \\mathbf {0}.$ Here, the integrability condition $\\mathbf {v} \\cdot \\nabla \\times \\mathbf {v} = 0$ of the vector field $\\mathbf {v}$ is defined up to a multiplier $\\mu $ .",
"This can be shown as follows.",
"Let us consider a system of equation $ \\dot{x} = P(x,y,z), \\qquad \\dot{y} = Q(x,y,z),\\qquad \\dot{z} = R(x,y,z).",
"$ Then a direct calculation follows $&&\\mu \\mathbf {v} \\cdot \\big ( \\nabla \\times \\mu \\mathbf {v}) \\\\ &&\\qquad = \\mu P\\big ( \\frac{\\partial }{\\partial y}(\\mu R) -\\frac{\\partial }{\\partial z}(\\mu Q) \\big ) + \\mu Q\\big ( \\frac{\\partial }{\\partial z}(\\mu P) -\\frac{\\partial }{\\partial x}(\\mu R) \\big ) + \\mu R\\big ( \\frac{\\partial }{\\partial x}(\\mu Q) -\\frac{\\partial }{\\partial y}(\\mu P) \\big )\\\\&&\\qquad = \\mu ^2 \\Big ( P \\big (\\frac{\\partial R}{\\partial y} - \\frac{\\partial Q}{\\partial z}\\big ) +Q \\big (\\frac{\\partial P}{\\partial z} - \\frac{\\partial R}{\\partial x}\\big ) + R \\big (\\frac{\\partial Q}{\\partial x} -\\frac{\\partial P}{\\partial y}\\big ) \\Big ) \\\\&&\\qquad = \\mu ^2 ( \\mathbf {v} \\cdot \\nabla \\times \\mathbf {v} ) = 0.$ Note that, in the second line the other terms are identically vanishing.",
"It is immediate to see that, if $\\nabla \\cdot \\mathbf {v}=0$ we have that $\\mathbf {v} = K\\nabla H, \\qquad \\text{or} \\qquad \\nabla \\times \\mathbf {v} = \\nabla K \\times \\nabla H,$ for two real valued functions $K$ and $H$ .",
"As an example for this case, consider the Euler top equation $ \\dot{x} = Ayz, \\qquad \\dot{y} = Bxz, \\qquad \\dot{z} = Cxy,$ where $A$ , $B$ and $C$ being constants.",
"The divergence free condition generates Liouville’s theorem of Nambu mechanics, therefore the state space can be regarded as an incompressible fluid.",
"It is interesting to notice that if the coefficients $A$ , $B$ and $C$ in the system (REF ) are all equal to each other, then $\\nabla \\times \\mathbf {v}$ vanishes identically.",
"This motivates the following special case of such flows $ \\nabla \\cdot \\mathbf {v} = 0, \\qquad \\nabla \\times \\mathbf {v} = \\mathbf {0}.$ As an example, consider the Lagrange system, the case $A=B=C=1$ , $\\dot{x} = yz, \\qquad \\dot{y} = xz, \\qquad \\dot{z} = xy,$ and 3D circle map equation $\\mathbf {\\dot{x}} = x^2$ .",
"These flows are the usual (conservative) potential flows $\\mathbf {v} = - \\nabla V$ generating by a potential function $V$ .",
"As an another category, let the generator vector field $\\mathbf {v}$ in a $3D$ system is curl of a potential field $\\mathbf {A}$ .",
"In general, the integrability condition is not fulfilled.",
"In this case, we have $\\mathbf {v}=\\nabla \\times \\mathbf {A},\\qquad \\nabla \\times \\mathbf {v}\\ne \\mathbf {0},$ where $\\mathbf {A}$ is the potential vector field.",
"Motivating from the Hamiltonian curl forces in two dimensions discussed in [3], [4], [5], our focus in this study is to elaborate Hamiltonian analysis of three dimensional differential system satisfying (REF ).",
"In this present work, we distinguish two subcategories according to the satisfaction of the potential vector field $\\mathbf {A}$ of the integrability condition.",
"In the first subcategory, we study $ \\mathbf {A} \\cdot \\nabla \\times \\mathbf {A} \\ne 0.$ It is known from the standard text book that for a solenoidal and continuously differentiable vector function such as a vector field $\\mathbf {v}$ can be expressed as a cross-product of two gradients, $\\mathbf {v} = \\nabla H \\times \\nabla K,$ where $H$ and $K$ are assumed to smooth functions [2].",
"The standard proof assumes that there exist two families of surfaces, $H(x, y, z) = c_1$ and $K(x, y, z) = c_2$ , for which all the flow lines are confined to the surface of intersections.",
"This is the essential ingredient of the Nambu mechanics too.",
"It is worth noting that if we start with $ \\mathbf {A} = H\\nabla K$ or $\\mathbf {v} = \\nabla H \\times \\nabla K$ , the scalar product $\\mathbf {A}\\cdot \\mathbf {v} = 0$ .",
"It is known that the vector potential corresponding to vector field $\\mathbf {v}$ can be presented in the Clebsch representation [6] form $\\mathbf {A} = - K \\nabla H + \\nabla M,$ where the function $M$ must be multi-valued.",
"This implies that the function $M$ has a surface $\\Sigma $ inside the volume $\\Omega $ where it has a jump, then the contribution from the jump surface $\\Sigma $ is added to the integral over $\\partial \\Omega $ , which results in the nonzero helicity.",
"It is a challenging question to find the function $M$ and to understand why it has to be multi-valued [6], [30].",
"In three dimensions, the geometric aspects can be discussed in terms of bi-Hamiltonian and Nambu-Hamiltonian formulations apart from the two dimensional ones.",
"Accordingly, we shall discuss the Hamiltonian characters of the systems (REF ) in terms of the bi-Hamiltonian and the Nambu-Hamiltonian frameworks.",
"We give two different examples for this subcategory, one possesses the Nambu structure whereas the other one has the almost Nambu structure involving a multiplier.",
"The second subcategory covers a divergence free flow satisfying integrability condition $ \\mathbf {A} \\cdot \\nabla \\times \\mathbf {A} = 0.$ We should expect if these conditions are satisfied then $\\mathbf {A} = H\\nabla K$ , then it would yield Nambu Hamiltonian system, unfortunately we fail to get it.",
"The closed two form $\\omega $ associated to dynamics is the product of two one-forms $\\omega = J_1 \\wedge J_2$ , unlike the previous cases in this $J_1$ and $J_2$ are not exact.",
"The Euler potentials could be discontinuous, although the vector potential and $\\mathbf {v} = \\nabla \\times \\mathbf {A}$ might not have discontinuities on the intersection of surfaces.",
"In this paper, we demonstrate if we deform the system to another which satisfies divergence free condition and $\\mathbf {A} \\cdot \\nabla \\times \\mathbf {A} \\ne 0$ , then it becomes Nambu-Hamiltonian.",
"Our motivation to study such flows coming from the work of Berry and Shukla on curl forces we have studied in this paper curl velocities on 3D spaces.",
"We have studied 3D vector fields generated by a curl of a vector potential.",
"In general, the theory for 3D systems is less well-developed [36], there are many unanswered questions regarding the nature of dynamics in 3D flows.",
"The flows which have been investigated in this paper are also divergence free vector fields like Berry-Shukla, and these could be either Hamiltonian or non-Hamiltonian.",
"In this point, we want to make a final remark that 3D flows possess extremely diverse characters.",
"Consider, for example, the following 3D flow, called as SIR equation, given by $\\dot{S} = -rSI, \\qquad \\dot{I} = rSI - aI, \\qquad \\dot{R} = aI,$ which neither satisfies $\\nabla \\cdot \\mathbf {v} = 0$ nor $\\mathbf {v}=\\nabla \\times \\mathbf {A}$ , but still one can express it in terms of Nambu Hamiltonian form.",
"Organization of the paper.",
"In Section 2, we present basics of Hamiltonian realizations of $3D$ systems.",
"In Section 3 we exhibit some illustrations that posses Nambu-Hamiltonian and bi-Hamiltonian character.",
"In Section 4, we examine a counter example which cannot in the Nambu-Hamiltonian and bi-Hamiltonian formalism."
],
[
"Three Dimensional Hamiltonian Systems",
"Let $(\\mathcal {P},\\lbrace \\bullet ,\\bullet \\rbrace )$ be a 3-dimensional Poisson manifold equipped with Poisson bracket $\\lbrace \\bullet ,\\bullet \\rbrace $ satisfying the Jacobi identity.",
"The Hamilton's equation generated by a Hamiltonian function is $\\mathbf {\\dot{x}}=\\lbrace \\mathbf {x},H\\rbrace ,$ for local coordinates $(\\mathbf {x})$ on $\\mathcal {P}$ .",
"In 3-dimensions, we can replace the role of a Poisson bracket with a Poisson vector $\\mathbf {J}$ , [10], [17], [18].",
"In this case, the Jacobi identity turns out to be the following equation $ \\mathbf {J}\\cdot (\\nabla \\times \\mathbf {J})=0,$ whereas the Hamilton's equation takes the particular form $ \\mathbf {\\dot{x}}=\\mathbf {J}\\times \\nabla H.$ Here, $H$ is a Hamiltonian function defined on $\\mathcal {P}$ , and $\\nabla H$ is the gradient of $H$ .",
"The following theorem is exhibiting all possible solutions of Jacobi identity given in (REF ) so that characterizes all Poisson Poisson structures in 3-dimensions, [1], [19], [20], [21].",
"Theorem 2.1 The general solution of the vector equation (REF ) is $\\mathbf {J}= \\left({1}/{M} \\right) \\nabla F$ for arbitrary functions $M$ and $F$ .",
"The existence of scalar multiple ${1}/{M}$ in the solution is a manifestation of conformal invariance of the identity (REF ).",
"In the literature, $M$ is called as the Jacobi's last multiplier [22], [23].",
"In this picture, a Hamiltonian system has the following generic form $ \\mathbf {\\dot{x}}= \\frac{1}{M} \\nabla F \\times \\nabla H.$ A dynamical system is bi-Hamiltonian if it admits two different Hamiltonian structures $\\mathbf {\\dot{x}}= \\lbrace \\mathbf {x},H_2\\rbrace _{1}=\\lbrace \\mathbf {x},H_1\\rbrace _{2},$ with the requirement that the Poisson brackets $\\lbrace \\bullet ,\\bullet \\rbrace _{1}$ and $\\lbrace \\bullet ,\\bullet \\rbrace _{2}$ be compatible [13], [28].",
"Recalling the system (REF ), we arrive at that a Hamiltonian system in the form (REF ) is bi-Hamiltonian $ \\mathbf {\\dot{x}}= \\frac{1}{M} \\nabla F\\times \\nabla H=\\mathbf {J}_{1}\\times \\nabla H=\\mathbf {J}_{2}\\times \\nabla F,$ where, the first Poisson vector $\\mathbf {J}_{1}$ is given by $(1/M)\\nabla F$ whereas the second Poisson vector $\\mathbf {J}_{2}$ is given by $-(1/M)\\nabla H$ .",
"The following theorem determine the Hamiltonian picture of three dimensional dynamical systems admitting an integral invariant.",
"For the proof, we refer [10], [11].",
"Theorem 2.2 A three dimensional dynamical system $\\mathbf {\\dot{x}}=\\mathbf {v}(\\bf {x})$ having a time independent first integral is bi-Hamiltonian if and only if there exist a Jacobi's last multiplier $M$ which makes $M\\mathbf {v}$ divergence free."
],
[
"$3D$ Nambu-Poisson manifolds",
"Let $\\mathcal {P}$ be a three dimensional manifold.",
"A Nambu-Poisson bracket of order 3 is a ternary operation, denoted by $\\lbrace \\bullet ,\\bullet ,\\bullet \\rbrace $ , on the space of smooth functions, satisfying both the generalized Leibniz identity $ \\left\\lbrace F_{1},F_{2},FH\\right\\rbrace =\\left\\lbrace F_{1},F_{2},F\\right\\rbrace H+F\\left\\lbrace F_{1},F_{2},H\\right\\rbrace $ and the fundamental (or Takhtajan) identity $\\left\\lbrace F_{1},F_{2},\\lbrace H_{1},H_{2},H_{3}\\rbrace \\right\\rbrace =\\sum _{k=1}^{3}\\lbrace H_{1},...,H_{k-1},\\lbrace F_{1},F_{2},H_{k}\\rbrace ,H_{k+1},...,H_{3}\\rbrace ,$ for arbitrary functions $F,F_{1},F_{2},H,H_{1},H_{2}$ , see [26], [32].",
"Assume that $(\\mathcal {P},\\lbrace \\bullet ,\\bullet ,\\bullet \\rbrace )$ be a Nambu-Poisson manifold.",
"For a pair $(H_1,H_2)$ of Hamiltonian functions, the associated Nambu-Hamiltonian vector field $X_{H_1,H_2}$ is defined through $X_{H_1,H_2}(F)=\\lbrace F,H_1,H_2\\rbrace .$ The distribution of the Nambu-Hamiltonian vector fields are in involution and defines a foliation of the manifold $\\mathcal {P}$ .",
"A dynamical system is called Nambu-Hamiltonian with a pair $(H_1,H_2)$ of Hamiltonian functions if it can be recasted as $\\mathbf {\\dot{x}}=\\left\\lbrace \\mathbf {x} ,H_{1},H_{2}\\right\\rbrace .",
"$ If a system is in the Nambu-Hamiltonian form (REF ) then by fixing one of the Hamiltonian functions in the pair $(H_1,H_2)$ , we can write it in the bi-Hamiltonian form as well $\\mathbf {\\dot{x}}=\\left\\lbrace \\mathbf {x},H_{1}\\right\\rbrace ^{H_{2}}=\\left\\lbrace \\mathbf {x},H_{2}\\right\\rbrace ^{H_{1}} $ where the brackets $\\lbrace \\bullet ,\\bullet \\rbrace ^{H_{2}}$ and $\\lbrace \\bullet ,\\bullet \\rbrace ^{H_{1}}$ are compatible Poisson structures defined by $\\left\\lbrace F,H\\right\\rbrace ^{{H}_{2}}=\\left\\lbrace F,H,H_{2}\\right\\rbrace \\text{, \\ \\ \\ }\\left\\lbrace F,H\\right\\rbrace ^{{H}_{1}}=\\left\\lbrace F,H_{1},H\\right\\rbrace , $ respectively.",
"Let $\\mathcal {P}$ be a 3 dimensional manifold equipped with a non-vanishing volume manifold $\\mu $ .",
"Then the following identity $\\lbrace F_1,F_2,F_3\\rbrace \\mu =dF_1\\wedge dF_2 \\wedge dF_3$ defines a Nambu-Poisson bracket on $\\mathcal {P}$ [15], [16].",
"In this case, the equation (REF ) relating a Hamiltonian pair $(H_1,H_2)$ and a Nambu-Hamiltonian vector field $X_{H_1,H_2}$ can be written, in a covariant formulation, as $ \\iota _{X_{H_1,H_2}}\\mu = dH_1 \\wedge dH_2,$ where $\\iota $ is the interior derivative.",
"We call (REF ) as the Nambu-Hamilton's equations [12].",
"Note that, by taking the exterior derivative of both hand side of (REF ), we arrive at the preservation of the volume form by the Nambu-Hamiltonian vector field, that is $ \\mathcal {L}_{X_{H_1,H_2}}\\mu =0.$ Integration of this conservation law gives that the flow of a Nambu-Hamiltonian vector field is a volume preserving diffeomorphism.",
"Consider a local frame (called as the standard basis) given by a three-tuple $(u,v,w)$ such that the volume form is $\\mu = du \\wedge dv \\wedge dw.$ In this picture the Nambu-Poisson three-vector takes the particular form $N= \\frac{\\partial }{\\partial u} \\wedge \\frac{\\partial }{\\partial v} \\wedge \\frac{\\partial }{\\partial w}.$ Locally, the Nambu-Hamiltonian vector field $X_{H_1,H_2}$ defined in (REF ) for a pair $(H_1,H_2) $ of Hamiltonian functions can be computed as $ X_{H_1,H_2} = \\lbrace H_1,H_2\\rbrace _{u,v} {\\frac{\\partial }{\\partial w}} + \\lbrace H_1,H_2\\rbrace _{v,w} {\\frac{\\partial }{\\partial u}} +\\lbrace H_1,H_2\\rbrace _{w,u} {\\frac{\\partial }{\\partial v}},$ where the coefficients are, for example, $ \\lbrace H_1,H_2\\rbrace _{a,b} = {\\frac{\\partial H_1}{\\partial a}}{\\frac{\\partial H_2}{\\partial b}} - {\\frac{\\partial H_1}{\\partial b}}{\\frac{\\partial H_2}{\\partial a}}.$ For the particular case of a three dimensional Euclidean space, the present discussion reduces to the following form.",
"Let $F_1$ , $F_{2}$ and $F_{3}$ be three real valued functions, and consider the triple product $\\left\\lbrace F_{1},F_{2},F_{3}\\right\\rbrace = \\nabla F_1 \\cdot \\nabla F_{2}\\times \\nabla F_{3} $ of the gradients of these functions.",
"It is evident that the bracket (REF ) is a Nambu-Poisson bracket with corresponding Nambu–Poisson three–vector field in the standard form (REF ).",
"The Nambu-Hamiltonian vector field presented in (REF ) takes the particular form $X_{H_1,H_2} = \\nabla H_{1}\\times \\nabla H_{2}.$ It follows that, the Nambu-Hamilton's equations (REF ) turn out to be $ \\mathbf {\\dot{x}}=\\left\\lbrace \\mathbf {x} ,H_{1},H_{2}\\right\\rbrace = \\nabla H_{1}\\times \\nabla H_{2}.$ The bi-Hamiltonian character of this system can easily be observed by employing (REF ).",
"The divergence of (REF ) generates Liouville’s theorem of Nambu mechanics: $ \\nabla \\cdot \\mathbf {\\dot{x}} = \\nabla \\cdot \\big (\\nabla H_{1}\\times \\nabla H_{2}\\big ) = 0.$ The main advantage of using the two conserved quantities (Hamiltonians) in Nambu formulation, is the representation of the phase space trajectory as intersection line of two surfaces based on the conserved quantities.",
"This geometric application illustrates the kind of motion without explicitly solving the equations of motion, this is the key feature of the (maximal) superintegrability."
],
[
"Three Dimensional Systems",
"Our motivation stems from the equation of the chaotic advection of dye $\\dot{\\bf x} = {\\bf v}, \\qquad \\nabla \\times \\mathbf {v}\\ne \\mathbf {0}$ where the velocity field is assumed to have been determined a priori and satisfies the incompressibility condition $\\nabla \\cdot {\\bf v} = 0$ .",
"All the flows we consider in this section satisfy the Frobenius integrability condition.",
"An handy way to describe Frobenius integrability condition has been prescribed by Ollagnier and Strelcyn [27].",
"Consider a smooth one form $J$ corresponding to a first integral $I$ of a smooth dynamical vector field $\\mathbf {v}$ defined on ${R}^3$ .",
"It is tautological that the volume form $\\Omega $ and $J$ satisfy $\\Omega \\wedge J = 0$ .",
"If we take interior product with respect to the vector field $\\mathbf {v}$ , we get $i_{\\mathbf {v}}\\Omega \\wedge J = 0$ .",
"We now state a general remark that a smooth function $I$ is a first integral of a smooth vector field $\\mathbf {v}$ defined on ${R}^3$ if and only if $dI \\wedge i_{\\mathbf {v}}\\Omega = 0$ .",
"The advection equations are non-integrable in general.",
"A large class of conservative systems, which exhibit chaotic behavior, has a Hamiltonian representation.",
"Two of the well-known examples are the magnetic field $B$ and the velocity field $v$ of a divergence free fluid.",
"Let us start by considering a vector potential $\\mathbf {A}=(A_x,A_y,A_z)$ , and the following system of equations ${\\bf \\dot{x}} = \\mathbf {v}(x)= \\nabla \\times \\mathbf {A}(x).$ It is easy to observe now that the vector potential $\\mathbf {A}$ is far from being unique since $\\nabla \\times (\\mathbf {A} + \\nabla \\phi ) = \\nabla \\times \\mathbf {A} + \\nabla \\times \\nabla \\phi = \\nabla \\times \\mathbf {A} $ for an arbitrary real valued function $\\phi $ .",
"This is the gauge invariance of the dynamics.",
"It is immediate to observe that $\\nabla \\times \\mathbf {v}$ of the velocity field does not necessarily zero for arbitrary vector potential $\\mathbf {A}$ .",
"In terms of the differential forms, the picture is as follows.",
"Consider a differential one-form $ \\alpha =\\mathbf {A}(x)\\cdot d\\mathbf {x},$ so that we have $*d \\alpha =\\nabla \\times \\mathbf {A}\\cdot d\\mathbf {x}=\\mathbf {v}(\\mathbf {x})\\cdot d\\mathbf {x},$ where $*$ is the Hodge star operator with respect to the Euclidean norm.",
"Alternatively, start with a volume form $\\Omega = dx \\wedge dy \\wedge dz$ .",
"If we contract with the dynamical vector field $\\hat{\\mathbf {v}} = v_i {\\partial }/{\\partial x_i}$ we obtain the two form $d\\alpha $ .",
"Therefore, the equations of motion presented in (REF ) can also be expressed in the following form $\\mathbf {\\dot{x}}=\\ast (d\\alpha \\wedge d\\mathbf {x}).$ In 3-dimensions, any two-from is decomposable.",
"Accordingly, we write the exact two-form $d\\alpha $ as the wedge product of two one-forms $J_1$ and $J_2$ that is $d\\alpha = dv_x \\wedge dx + dv_y \\wedge dy + dv_z \\wedge dz = J_1 \\wedge J_2.$ Here, $J_1$ and $J_2$ are two integral invariants of the system.",
"If both $J_1$ and $J_2$ are closed then by Poincaré lemma, we arrive at two first integrals $I_1$ and $I_2$ satisfying $J_1 = dI_1$ and $J_2 = dI_2$ respectively.",
"If this is the case, then 3-dimensional phase flow can be described by means of the first integrals.",
"From geometric point of view, this gives that a solution to the system, that is an integral curve, can be realized as the intersection of two level surfaces defined by the first integrals $I_1$ and $I_2$ .",
"So that we can write the system (REF ) as follows $\\dot{\\bf x} = \\nabla I_1 \\times \\nabla I_2.",
"$ In this case, we can write $\\mathbf {A}=I_1\\nabla I_2$ .",
"Note that this representation of the dynamics coincide with the standard form of the Nambu-Hamilton equations exhibited in (REF ).",
"Further, by employing (REF ), we see that this description is bi-Hamiltonian as well.",
"We will illustrate two types of flows; one kind of 3D flows yield Nambu-Hamiltonian mechanics and they satisfy $\\mathbf {A}\\cdot \\nabla \\times \\mathbf {A} \\ne 0$ , where $\\mathbf {v} = \\nabla \\times \\mathbf {A}$ .",
"Other type flow does not possess Hamiltonian framework and it satisfies $\\mathbf {A}\\cdot \\nabla \\times \\mathbf {A}= 0$ ."
],
[
"Illustrations",
"In this section we give two examples, one is a relatively simple one and the other one is more complicated."
],
[
"A superintegrable system ",
"We now consider an example of 3D system [9] generated by the vector potential $\\mathbf {A}= \\frac{1}{4}\\left(z^2 - xy^2,x^2y - 2yz,y^2 - 2xz \\right).",
"$ The equations of motion (REF ) can be written as $\\dot{x} = y, \\qquad \\dot{y} = z, \\qquad \\dot{z} = xy.$ One can check trivially $\\mathbf {A}\\cdot (\\nabla \\times \\mathbf {A}) \\ne 0$ .",
"It is immediate to see that $\\nabla \\times \\mathbf {v}$ does not vanish for $\\mathbf {v}=(y,x,xy)$ .",
"The two-forms in (REF ) turns out to be $zdz \\wedge dx + ydy \\wedge dz + xy dx\\wedge dy =J_1 \\wedge J_2$ where the integral one-forms $J_1$ and $J_2$ can be computed to be $J_1 = zdx + xdz - ydy - x^2dx, \\qquad J_2 = xdx - dz$ respectively.",
"It is immediate to check the invariance of the one-forms by $\\iota _{\\mathbf {v}} J_i = 0$ .",
"Note that, $J_1 = dI_1$ and $J_2 = dI_2$ are exact with the potential functions $I_1 = xz - \\frac{y^2}{2} - \\frac{x^3}{3}, \\qquad I_2 = \\frac{x^2}{2} - z,$ See that $I_1$ and $I_1$ are smooth first integrals of the dynamics (REF ).",
"Thus (REF ) is a maximal superintegrable 3D dynamics generated by the curl of the vector potential $\\mathbf {A}$ .",
"Using these two first integrals, we can express equation (REF ) in the Nambu-Hamiltonian form (REF ) as follows $\\mathbf {\\dot{x}} = \\lbrace \\mathbf {x},I_1,I_2\\rbrace =\\nabla I_1 \\times \\nabla I_2$ where $I_1$ and $I_1$ are the ones in (REF ).",
"According to (REF ), we can represent the maximal superintegrable system (REF ) in the bi-Hamiltonian formulation $\\left(\\begin{array}{c}\\dot{x}\\\\\\dot{y}\\\\\\dot{z}\\\\\\end{array}\\right) = \\left(\\begin{array}{ccc}0 & -1 & 0\\\\1 & 0 & x \\\\0 & -x & 0\\\\\\end{array}\\right)\\left(\\begin{array}{c}\\frac{\\partial I_1}{\\partial x}\\\\\\frac{\\partial I_1}{\\partial y}\\\\\\frac{\\partial I_1}{\\partial z}\\\\\\end{array}\\right) = \\left(\\begin{array}{ccc}0 & x & -y\\\\-x & 0 & -z+x^2 \\\\y & -x & 0\\\\\\end{array}\\right)\\left(\\begin{array}{c}\\frac{\\partial I_2}{\\partial x}\\\\\\frac{\\partial I_2}{\\partial y}\\\\\\frac{\\partial I_2}{\\partial z}\\\\\\end{array}\\right).$"
],
[
"Lotka-Volterra equation",
"The generalized Lotka-Volterra equation is given by $\\dot{x_1} = x_1(a_3x_2 + x_3 + l_1), \\,\\,\\,\\, \\dot{x_2} = x_2(a_1x_3 + x_1 + l_2) \\,\\,\\,\\,\\dot{x_3} = x_3(a_2x_1 + x_2 + l_3).$ The divergence free condition ${\\partial v_i}/{\\partial x_i} = 0$ imposes conditions on $a_i$ and $l_i$ , such that $a_i = -1$ and $l_1 + l_2 + l_3 = 0$ .",
"Then the reduced set of equations becomes $\\dot{x_1} = x_1(-x_2 + x_3 + l_1), \\qquad \\dot{x_2} = x_2(-x_3 + x_1 + l_2), \\qquad \\dot{x_3} = x_3(-x_1 + x_2 + l_3).$ One can check directly that $ \\nabla \\times \\mathbf {v} \\ne 0$ .",
"Recast this in the vector potential form $\\mathbf {A}$ with $ A_i = x_1x_2x_3 + \\tilde{l_i},$ where the constants are satisfying $l_1 = \\tilde{l_3} - \\tilde{l_2},\\,\\,\\,\\, l_2 = \\tilde{l_1} - \\tilde{l_3} \\,\\,\\,\\, l_3 = \\tilde{l_2} - \\tilde{l_1}.$ One can also check that $\\mathbf {A}\\cdot (\\nabla \\times \\mathbf {A}) \\ne 0$ .",
"Just like the previous example we can express $d\\ast \\alpha $ in terms of $J_1$ and $J_2$ using a multiplier $m = x_1x_2x_3$ : $x_1(-x_2 + x_3 + l_1)dx_2 \\wedge dx_3 + x_2(-x_3 + x_1 + l_2)dx_3 \\wedge dx_1 + x_3(a_2x_1 + x_2 + l_3)dx_1 \\wedge dx_2 = m (J_1 \\wedge J_2),$ where $ J_1 = \\frac{dx_1}{x_1} + \\frac{dx_2}{x_2} + \\frac{dx_3}{x_3}, \\qquad J_2 = dx_1 + dx_2 + dx_3 + \\frac{l_3dx_2}{x_2} - \\frac{l_2dx_3}{x_3}.",
"$ This immediately shows $dJ_1 = dJ_2 = 0$ and the two Hamiltonians are $H_1 = \\ln x_1 + \\ln x_2 + \\ln x_3, \\qquad H_2 = x_1 + x_2 + x_3 + l_3\\ln x_2 - l_2 \\ln x_3.",
"$ Thus equations can be obtained from the standard Nambu-Hamiltonian formalism."
],
[
"Generalized Hamiltonian case and vector equations",
"Consider the following vector potential, see for example [8], $\\mathbf {A} = \\frac{1}{2}\\left(y^3 - x^2z, z^3 - xy^2,x^3 - yz^2\\right).",
"$ The dynamics generated by the curl of $\\mathbf {A}$ is computed to be $\\dot{x} = z^2, \\qquad \\dot{y} = x^2, \\qquad \\dot{z} = y^2.$ See that $\\nabla \\times \\mathbf {v}$ is not vanishing for $\\mathbf {v}=(z^2,x^2,y^2)$ .",
"The vector potential satisfies $ \\mathbf {A}\\cdot \\nabla \\times \\mathbf {A} = z^2(y^3 - x^2z) + x^2(z^3 - xy^2) + y^2(x^3 - yz^2) = 0.",
"$ Thus the helicity of the local flow is identically zero, due to this identity Sposito's result shows that the flow streamlines are confined to flat 2D manifolds.",
"Hence we can not get the Nambu-Hamiltonian structure here.",
"The identity (REF ) becomes $y^2dx\\wedge dy + x^2 dz\\wedge dx +z^2 dy \\wedge dz = J_1 \\wedge J_2,$ where $ J_1 = (z^2 dy - x^2 dx), \\qquad J_2 = (dz - \\frac{y^2}{z^2}dx).",
"$ It is easy to check that $J_1$ and $J_2$ are invariant one-forms that is $\\iota _{\\mathbf {v}}J_i = 0$ .",
"The integrals one-forms $J_1$ and $J_2$ presented in (REF ) are not closed, i.e., $ dJ_1 = 2z dz \\wedge dy \\ne 0, \\qquad dJ_2 = 2\\frac{y^2}{z^3}dz \\wedge dx - 2\\frac{y}{z^2} dy \\wedge dx \\ne 0.$ Hence we can not express the equation (REF ) neither in the Nambu-Hamiltonian nor the bi-Hamiltonian forms.",
"We can find two more such pairs like $J_1$ and $J_2$ which also satisfy $ y^2dx\\wedge dy + x^2 dz\\wedge dx + z^2 dy \\wedge dz = K_1 \\wedge K_2 = L_1 \\wedge L_2, $ where $(K_1,K_2) = (y^2 dx - z^2dz, dy - \\frac{x^2}{z^2}dx)$ and $(L_1,L_2) = ( z^2 dy - x^2 dx, dz- \\frac{y^2}{x^2}dy)$ but none of them are closed."
],
[
"Homotopy operator and closed form",
"Let $M$ be a manifold and let $I = [0,1]$ Suppose $\\omega \\in \\Omega ^k(M \\times I)$ , every $\\omega $ can be uniquely decomposed to $\\omega = \\alpha _1 + \\alpha _2 \\wedge dt$ with $\\alpha _1(0,t) \\in \\Omega ^k(M)$ and $\\alpha _2(0,t) \\in \\Omega ^{k-1}(M)$ .",
"We define the following mapping $D_k : \\Omega ^k(M \\times I) \\rightarrow \\Omega ^{k-1}(M), \\qquad (D_k\\Omega )(m) : = (-1)^{k-1}\\int _{0}^{1}\\alpha _2(m,t) dt,$ where the integral is to be understood as an integral of a function on the interval $I$ with values in the vector space $\\wedge ^{k-1}T_{m}^{\\ast }M$ , [29].",
"Notice that, this satisfies $d(D_k\\omega ) + D_{k+1}(d\\omega ) = \\omega |_{t=1} - \\omega |_{t=0}.$ Let $\\alpha \\in \\Omega ^k(N)$ , let $ \\phi _{0}, \\phi _1 : M \\rightarrow N$ be smooth mapping, Then we set $\\Omega = F^{\\ast }\\alpha $ and obtain $d(D_kF^{\\ast }\\alpha ) + D_{k+1}(dF^{\\ast }\\alpha ) = \\phi _{1}^{\\ast }\\alpha - \\phi _{0}^{\\ast }\\alpha .$ Here, the operator $H_k = D_k \\circ F^{\\ast }$ is called the homotopy operator.",
"If $\\alpha $ belongs to a cohomology class in $H^k(N)$ , then $dD_kF^{\\ast }\\alpha = \\phi _{1}^{\\ast }\\alpha - \\phi _{0}^{\\ast }\\alpha $ .",
"So $\\big (\\phi _{1}^{\\ast }\\alpha - \\phi _{0}^{\\ast }\\alpha \\big )$ differ by an exact form, hence they define the same cohomology class.",
"The operator $D_k$ defined in (REF ) yields an explicit potential form of a closed form on a contractible manifold.",
"We recall that if $M$ is contractible then by Poincaré lemma $H^k(M) = {R}$ for $k =0$ otherwise it is zero for all other $k > 0$ .",
"Let $F : M \\times I \\rightarrow N$ be a homotopy fulfilling $F(m,0) = \\phi _0(m)$ and $F(m,1) = \\phi _1(m)$ , where $I =[0,1]$ .",
"From (REF ), we obtain $ dD_kF^{\\ast }\\alpha + D_{k+1}dF^{\\ast }\\alpha = -\\alpha , $ for all $\\alpha $ , $d\\alpha = 0$ , then, $\\beta = -D_kF^{\\ast }\\alpha .$ Claim 4.1 Let $\\alpha = y^2 dx \\wedge dy + x^2 dz \\wedge dx + z^2 dy \\wedge dz$ be a two form and $F((x,y,z),t) = (tx,ty,tz)$ be a homotopy mapping.",
"Then the exact one form $\\eta = -D_2F^{\\ast }\\alpha $ is given by $\\eta = - \\frac{1}{4}\\big ( (y^3 - x^2z)dx + (z^3 - xy^2)dy + (x^3 - z^2y)dz \\big ),$ such that $\\alpha = d\\eta $ .",
"Notice that $\\eta = \\eta _1 dx + \\eta _2 dy + \\eta _3 dz$ plays the role of Hamiltonian (one) form.",
"Let us consider first order Hamiltonian equations in 2D, we can express it $\\dot{x}\\Omega = dx \\wedge dH, \\qquad \\dot{p}\\Omega = dp\\wedge dH, \\,\\,\\,\\,\\hbox{ where }\\,\\,\\,\\,\\Omega = dx \\wedge dp,$ which yields Hamiltonian equations in the standard form $\\dot{x} = \\frac{\\partial H}{\\partial p}$ and $\\dot{x} = -\\frac{\\partial H}{\\partial x}$ .",
"Similarly, we can express 3D equations of motion as $\\dot{x}_i{\\tilde{\\Omega }} = dx_i \\wedge d\\eta , \\qquad \\hbox{ where } \\,\\,\\,\\, {\\tilde{\\Omega }} = dx \\wedge dy \\wedge dz,$ where $x_i = x,y,z$ .",
"Expanding in components we obtain $\\dot{x} = -\\frac{\\partial \\eta _2}{\\partial z} + \\frac{\\partial \\eta _3}{\\partial y}, \\qquad \\dot{y} = -\\frac{\\partial \\eta _3}{\\partial x} + \\frac{\\partial \\eta _1}{\\partial z}, \\qquad \\dot{z} = -\\frac{\\partial \\eta _1}{\\partial y} + \\frac{\\partial \\eta _2}{\\partial x}.$ This set of equations are also obtained by Dumachev [8], [9] and he called as vector Hamiltonian equation."
],
[
"Deformation and Hamiltonization",
"In this section we deform the two form (REF ) in such a way that the dynamics involved in the modified system also yields velocity as a curl of potential vector field and during this process we will get rid of null condition, i.e.",
"$\\mathbf {A}\\cdot \\nabla \\times \\mathbf {A} = 0$ .",
"Let us assume one form $\\tilde{J}_1 = xdx + ydy + zdz$ and demand another form $\\tilde{J}_2 = A_1 dx + A_2 dy + A_3 dz$ in such a way that it would yield the original form (REF ) and at the same time it should be exact.",
"We arrive at the following set of (deformed) equations $\\dot{x} = z^2 - y^2 + xz - xy, \\,\\,\\,\\,\\, \\dot{y} = x^2 - z^2 + xy - yz, \\,\\,\\,\\,\\, \\dot{z} = y^2 - x^2 + yz - xz.$ It is easy to check that (REF ) yields a divergence free vector field $\\mathbf {v}$ and it satisfies $\\mathbf {v} = \\nabla \\times \\mathbf {A}$ , where the vector potential $\\mathbf {A}$ is given by $\\mathbf {A} = \\big ( (y^2 + z^2)x + xyz, (x^2 + z^2)y + xyz, (x^2 + y^2)z + xyz \\big ).$ It is easy to check that the vector potential $\\mathbf {A}$ for the deformed equation satisfies $\\mathbf {A}\\cdot \\nabla \\times \\mathbf {A} \\ne 0$ .",
"The identity (REF ) becomes $\\big (y^2 - x^2 + yz - xz \\big ) dx\\wedge dy + \\big ( x^2 - z^2 + xy - yz \\big ) dz\\wedge dx+ \\big (z^2 - y^2 + xz - xy \\big ) dy \\wedge dz = \\hat{J}_1 \\wedge \\hat{J}_2,$ where $\\hat{J}_1$ and $\\hat{J}_2$ are exact one forms, $\\hat{J}_1 = dI_1$ and $\\hat{J}_{2} = dI_2$ with the potential functions $I_1 = xy + yz + zx, \\qquad I_2 = \\frac{1}{2}( x^2 + y^2 + z^2).$ Using these two first integrals, we can express equation (REF ) in the Nambu-Hamiltonian form (REF ) as follows $\\mathbf {\\dot{x}} = \\lbrace \\mathbf {x},I_1,I_2\\rbrace =\\nabla I_1 \\times \\nabla I_2$ where $I_1$ and $I_1$ are the ones in (REF )."
],
[
"Outlook",
"We studied Hamiltonian aspects of divergence-free vector fields in dimension 3, chaotic aspects of these kind of equations have been studied in [33], [34], in general handful of papers are known in the literature for three-dimensional divergence-free vector fields.",
"In particular, we have studied $\\dot{\\bf x} = \\nabla \\times \\mathbf {A}$ type flows, and all these flows satisfy Frobenius integrability condition in a sense that $i_{\\mathbf {v}}\\Omega \\wedge J = 0$ , or in other words, $i_{\\mathbf {v}}\\Omega = J \\wedge K$ , where $J$ and $K$ are 1-forms.",
"We explored that not all the flows yield (Nambu) Hamiltonian framework, it depends on the nature of $J$ and $K$ , whether they are closed or not.",
"We have demonstrated that when we deform the second class of system to get rid of null condition $\\mathbf {A} \\cdot \\nabla \\times \\mathbf {A} = 0$ , the system possesses the Hamiltonian realization.",
"In this paper we could not prove the existence theorem for Hamiltonian framework but instead of that we have demonstrated the existence of both Hamiltonian and non-Hamiltonian type flows for $\\mathbf {v} = \\nabla \\times \\mathbf {A}$ type 3D flows.",
"Also, very little is known about divergence-free vector fields in dimension $n \\ge 4$ , so we will focus on this problem in our next project.",
"This piece of work also raised several questions regarding the applicability of the Euler theorem of potential."
],
[
"Acknowledgements",
"We would like to express our sincere appreciation to Professors Sir Michael Berry, Tony Bloch, Larry Bates and Jean-Luc Thiffeault for their interest and valuable comments.",
"PG is also grateful to Vishal Vasan for enlighting discussion.",
"This work has been done while PG is visiting Gebze Technical University, Department of Mathematics, under TUBITAK 2221 Fellowships for Visiting Scientists and Scientists on Sabbatical Leave program.",
"He would like to express his sincerest gratitude to all members of department for their warm hospitality, especially to the chairman Mansur Hoca."
]
] | 1906.04476 |
[
[
"Tests of Acoustic Scale Shifts in Halo-based Mock Galaxy Catalogues"
],
[
"Abstract We utilise mock catalogues from high-accuracy cosmological $N$-body simulations to quantify shifts in the recovery of the acoustic scale that could potentially result from galaxy clustering bias.",
"The relationship between galaxies and dark matter halos presents a complicated source of systematic errors in modern redshift surveys, particularly when aiming to make cosmological measurements to sub-percent precision.",
"Apart from a scalar, linear bias parameter accounting for the density contrast ratio between matter tracers and the true matter distribution, other types of galaxy bias, such as assembly and velocity biases, may also significantly alter clustering signals from small to large scales.",
"We create mocks based on generalised halo occupation populations of 36 periodic boxes from the \\abacuscosmos release with.",
"In a total volume of $48 \\, h^{-3} \\mathrm{Gpc}^3$, we test various biased models along with an unbiased base case.",
"Two reconstruction methods are applied to galaxy samples and the apparent acoustic scale is derived by fitting the two-point correlation function multipoles.",
"With respect to the baseline, we find a $0.3\\%$ shift in the line-of-sight acoustic scale for one variation in the satellite galaxy population, and we find an $0.7\\%$ shift for an extreme level of velocity bias of the central galaxies.",
"All other bias models are consistent with zero shift at the $0.2\\%$ level after reconstruction.",
"We note that the bias models explored are relatively large variations, producing sizeable and likely distinguishable changes in small-scale clustering, the modelling of which would further calibrate the BAO standard ruler."
],
[
"Introduction",
"The standard ruler provided by baryon acoustic oscillations (BAO) has become a powerful probe in the past decade for studying the large-scale structure of the universe and constraining properties of dark energy.",
"The next generation of dark energy experiments, such as the Dark Energy Spectroscopic Instrument (DESI) [9], Euclid [23], and WFIRST [39], are designed and built with BAO as a primary method to measure the expansion history of the universe to unprecedented precision.",
"The acoustic scale around $100\\,h^{-1}{} $ is much larger than the scales relevant for nonlinear gravitational evolution, and galaxy formation and remains in the linear regime today.",
"However, nonlinear effects still have small but important consequences, despite the acoustic scale being a robust standard ruler for measuring cosmological distances.",
"Systematic errors on the distance scales inferred from BAO measurements are indeed dominated by nonlinear structure growth as well as redshift-space distortions.",
"As reconstruction has been shown to substantially reduce these major contributors to systematics if not reverse them entirely [34], the next subleading source of systematics, galaxy bias, is increasingly relevant with the precision of BAO measurements reaching the $0.1\\%$ level in future surveys [25].",
"Density field reconstruction uses Lagrangian perturbation theory to reduce anisotropies in the clustering and reverse the large-scale gravitational bulk flows in the observed galaxy sample [11].",
"This procedure is very effective in undoing the smoothing of the BAO feature and the shift of the BAO scale due to nonlinear structure growth, thereby improving the precision of BAO measurements.",
"The one-step version, known as standard reconstruction, has been field tested extensively with on-sky data [27], [1], [2], [3], [20], [30] and become part of the standard analysis procedure in modern galaxy redshift surveys.",
"There has been much recent development in reconstruction [32], [31], [35], [50], [17], [42].",
"While standard reconstruction has worked well to restore and enhance the BAO signature, whether newer algorithms can perform better is a question.",
"To find out, we employ one of the new methods, the iterative reconstruction in particular [17], which provides better correlation to the matter density field and closer-to-truth statistics than standard reconstruction when applied to galaxy mocks [18].",
"Galaxies as tracers of the underlying matter density field are far from perfect and tend to overweigh overdense regions.",
"It is well known that galaxy bias can be scale-dependent and, even with a linear bias correction, shift the acoustic scale measurement [26], [25], increasing systematic errors in cosmic distances and cosmological parameters.",
"Many mechanisms in the halo model that give rise to galaxy bias and induce sub-percent level shifts of the acoustic scale can be studied through simulations.",
"[25] showed that linear bias can be removed by reconstruction for $b\\leqslant 3.29$ and no significant shift of the BAO scale is detected in a volume of $44 \\, h^{-3}{{3}}$ .",
"For example, galaxy or halo assembly bias is a significant source of systematic error in the galaxy-halo relationship at small scales [46]; the CMASS BOSS galaxy sample was found to have significant velocity bias for the central galaxies, and between velocity and spatial distributions of satellites, at least one is biased [15].",
"[44] found significant changes of the power spectrum caused by non-Poisson distribution and velocity bias of satellite galaxies.",
"This study seeks to assess the sensitivity of the acoustic scale in the two-point correlation functions (2PCF) to the effects of galaxy bias.",
"Our analysis packages together a number of improvements over existing literature.",
"The anisotropic acoustic scale is measured in both radial and transverse directions.",
"Several types of galaxy bias are considered, including satellite distribution bias, assembly bias for central and satellite galaxies, velocity bias for centrals and satellites, and more sub-halo-scale effects.",
"The cosmological $N$ -body simulations are generated by the high-accuracy Abacus code [12], [13], and the total simulation volume of $48 \\, h^{-3}{{3}}$ is the largest as of the writing of this paper.",
"A newer iterative reconstruction method is applied in addition to standard reconstruction [17].",
"Halo catalogues are populated with mock galaxies in a generalised halo occupation distribution (HOD) model akin to [45], which can be implemented in a deterministic way such that clustering statistics are differentiable with respect to variations of the input HOD model parameters.",
"The layout of this paper is as follows.",
"In § methods, §REF describes the $N$ -body simulations and halo catalogues, §REF covers generation of mock galaxy catalogues with biased HOD models and reconstruction parameters, §REF –REF goes through the correlation and covariance calculation, and §REF motivates the BAO fitting test for iterative reconstruction with alternative models.",
"In §, we examine the differential change in 2PCF resulting from all bias models in §REF , and then presents the shift of the BAO scale measurement for of each type of bias in §REF .",
"Finally, the conclusions section § highlights the findings.",
"Our study uses cosmological N-body simulations produced by the Abacus code [13].",
"Specifically, 36 simulation boxes assuming the Planck cosmology [29] are used.",
"They all have particle mass about $4\\times 10^{10} \\, h^{-1}M_\\odot $ , particle count $1440^3$ , box size $1100 \\,h^{-1}{} $ , periodic boundary conditions, and initial conditions created with the same input linear power spectrum and independent initial phases [12].",
"The total volume adds up to $48 \\, h^{-3}{{3}}$ .",
"The only difference among the 36 boxes is that 16 were run with Plummer force softening and the other 20 with spline softening.",
"Detailed descriptions of the force softening can be found in the Abacus Cosmos public data release paper [14].",
"These two sets are first analysed separately from applying halo mass cuts all the way through to the BAO scale measurements, where we find all clustering statistics within $1\\sigma $ range of each other when analysing matter density field and galaxy samples created by any given HOD model.",
"There is no difference between two simulations at any confidence level, despite their slightly different halo mass functions.",
"With fixed halo finder and HOD parameters, spline softening typically results in more massive and large halos, meaning more halos pass the mass cut and more galaxies are generated.",
"But as we have found, the BAO scale evolves slowly with respect to the halo mass function, and it is safe to combined two simulations for a larger total volume.",
"All results hereafter have two sets combined and treated as one simulation of 36 boxes.",
"Several time slices after $z=1$ are saved with data products available, and we take the $z=0.5$ snapshot to mimic luminous red galaxy (LRG) samples and stay close to the redshifts of the mocks used in galaxy surveys [41].",
"The halo catalogues are created by the Rockstar halo finder [5].",
"To incorporate galaxy assembly bias later in the HOD, the halo NFW concentration, defined as $c_\\text{NFW} \\equiv R_\\text{virial}/R_\\text{s, Klypin}$ , is added to the halo properties, where the virial radius of the halo is chosen as $r_\\text{virial} = r_\\text{200}$ and the scale radius is given by [21], which is more stable than the traditional scale radius for small halos [5].",
"A mass cut at 70 dark matter (DM) particles, or approximately $M_\\text{halo} = 4\\times 10^{12} M_\\odot $ , was applied, as small halos have essentially zero chance of hosting galaxies and slow down the computation.",
"Subhalos are not reliable indicators of of in-halo galaxy distribution and are removed as well.",
"Instead, we use the position and velocity of dark matter particles to generate satellite galaxies.",
"Each halo catalogue is accompanied by a DM particle catalogue, which is a 10% subsample of the particles enclosed by halo boundaries.",
"Particles within subhalos are associated only to their host subhalos in Rockstar catalogues, not the higher level host halos.",
"This becomes a problem after subhalos are removed, and the particle associations have to be rebuilt before satellite generation to ensure that all particles are associated to their highest-level host halos.",
"In addition, a $10\\%$ uniform subsample of all $1440^3$ particles in any given simulation box was also used to validate our BAO fitter, as well as to quantify how much cosmic variance there exists in the BAO measurements and are cancelled out when the difference is taken.",
"The HOD model used for populating halos with galaxies is based on the classic 5-parameter model by [49] with decorations accounting for assembly bias, velocity bias, satellite distribution bias, and perihelion distance bias [45].",
"Abacus provides direct access to DM particles from which halos have been found in the simulation.",
"Although halo finders only produce spherical halo boundaries and therefore spherical DM particle distributions, the matter distribution is still a much better representation of the actual density profile within a halo than NFW profiles.",
"Satellite galaxies generated with DM particles more realistically trace the matter distribution of the halo.",
"A key feature of this HOD model is the deterministically seeded random numbers used for populating each simulation box.",
"The seed is chosen such that all halos and DM particles always receive the same random number assignment completely irrespective of the HOD parameters specified.",
"This means any infinitesimal change in the input HOD parameters would correspond to an infinitesimal change in the clustering results.",
"Our revamped HOD implementationhttps://github.com/duanyutong/abacus_baofit largely shares the same formalism and equations as GRAND-HOD [45], so we focus on only the differences or advantages it offers in terms of science and software implementation.",
"The mean halo occupations for central and satellite galaxies take the following forms, $\\left< N_\\text{cen} (M) \\right> &= \\frac{1}{2} \\left[ 1 + \\mathrm {erf} \\left( \\frac{\\ln M - \\ln M_\\text{cut}}{\\sqrt{2}\\sigma ^\\prime } \\right) \\right] \\nonumber \\\\&= \\frac{1}{2} \\mathrm {erfc} \\left( \\frac{\\ln (M_\\text{cut} / M)}{\\sqrt{2}\\sigma ^\\prime } \\right) \\\\\\left< N_\\text{sat} (M) \\right> &= \\left< N_\\text{cen} (M) \\right> \\left( \\frac{M - \\kappa M _\\text{cut} }{M_1} \\right) ^\\alpha \\, .$ This parametrisation is consistent with the original HOD prescription by [49], up to a rescaling of the dispersion $\\sigma ^\\prime \\equiv (\\ln 10\\sqrt{2}) \\sigma $ for a more natural interpretation [43] and $\\kappa \\equiv M_0/M_\\text{cut}$ , and is a popular choice in recent literature [40], [48], [24], [16], [47], [37], [6].",
"The explicit coupling between $\\left< N_\\text{sat} \\right>$ to $\\left< N_\\text{cen} \\right>$ by multiplication maintains the reasonably physical assumption that the central and satellite galaxies in the same halo are correlated to some extent, and that a high-mass halo hosting already hosting a central galaxy has a higher chance of also hosting one or more satellites.",
"Accordingly, we make satellite occupation terminate at a higher halo mass scale than the central occupation cutoff.",
"The dependence of $\\left< N_\\text{sat} (M) \\right>$ on $\\left< N_\\text{cen} (M) \\right>$ introduces complications in the fitting procedures when one tries to determine the coupled central and satellite parameters simultaneously in a given HOD model.",
"If the goal is not to constrain HOD parameters and this fitting difficulty is not of concern, then there is no advantage or motivation for dropping this assumption and insisting on no correlation between centrals and satellites [7].",
"Assembly bias is implemented as comparing the halo concentration to the median concentration of all halos of that mass.",
"The more concentrated halo may have a more favourable assembly history and a higher probability of hosting central or satellite galaxies, or vice versa.",
"The median halo concentration as a function of halo mass $c_\\text{med}(M)$ is obtained by putting all halos into mass bins and fitting a polynomial to $c_\\text{med}(M)$ .",
"Instead of taking the halos from a single simulation box as the sample and repeating the fit for all boxes, we take the entire halo population from all boxes in a given simulation and perform the fitting once for all.",
"This has a better theoretical motivation because the function $c_\\text{med}(M)$ , in principle, is independent from the phase of initial conditions and does not vary across boxes.",
"Even though the entire halo population is now an order of magnitude larger than that of a single box, if a smooth fit is desired and the bin specification is fine, there are still some mass bins with few halos and thereby, small variances in halo concentration.",
"The usual weight definition for all mass bins is $w=1/\\sigma _c = \\sqrt{(N-1)/\\Sigma _i (c_i-\\mu _c)}$ up to a normalisation constant, where $N$ is the number of halos in the mass bin, $c_i$ is the concentration of each halo in the bin, and $\\mu _c$ is the mean concentration of the bin.",
"With $\\sqrt{N}$ in the numerator, this gives disproportionally large weights for those least populated bins, resulting in poor fits.",
"We corrected for this pathology by adjusting the bin weight definition, multiplying the canonical weight by $(N-1)$ in powers of $1/2$ in an attempt to increase the weight for the more populated bins.",
"Both $w=\\sqrt{N-1}/\\sigma _c$ and $w=(N-1)/\\sigma _c$ produced quality fits which were almost identical.",
"We chose $w \\equiv \\sqrt{N-1}/\\sigma _c = (N-1)/\\sqrt{\\Sigma _i (c_i-\\mu _c)}$ as the weight definition.",
"To optimise I/O performance and the size of data products, Abacus saves each halo catalogue for a single box into many HDF5 files, each being a subset of the halo catalogue.",
"When populating halos, GRAND-HOD proceeds on a subset-by-subset basis as it goes through the HDF5 files sequentially.",
"As a result, all halo ranking operations involved in the decorations, e.g.",
"pseudomass calculations, are limited to the current HDF5 subset of halos only.",
"Our code utilises the halo reader built-in to Abacus and loads the complete halo catalogue at once in the standard Halotools [19] format together with the DM particle subsample with corrected host IDs as part of the mock.",
"The new code fully conforms to the Astropy [4] and Halotools standards, supports the Halotools prebuilt HOD models, and is compatible with both the latest Python 3 and 2 builds.",
"It provides flexibility in customising HOD parameters and preset models, and boosts performance with parallelism."
],
[
"Biased HOD Models",
"The generalised HOD framework easily allows any aforementioned source of bias, or combination of sources, to be introduced into the model with flexibility.",
"The biased HOD models tested in this paper are summarised in Table REF .",
"All models, except the baseline (denoted Base 1), have only a single mechanism of galaxy bias applied in order to explore the “unit vector” directions in the space of variations.",
"Although this HOD parametrisation does not exactly preserve the number density of galaxies, the resulting number density of all biased models only differ from the baseline by about $0.1\\%$ , which is negligible.",
"By default, central galaxies inherit the velocity of the host halos, and satellite galaxies generated with DM particles assume the DM particle velocities, unless velocity bias is applied.",
"To understand how each bias mechanism changes clustering statistics and the acoustic scale, we first establish a base case, free of any decoration for reference.",
"The baseline is a simple 5-parameter model; its parameters are unimportant as we are only interested in the differential change with respect to the baseline.",
"We reiterate that our HOD implementation is differentiable, which is precisely what enables us to set up a baseline and subtract it from the bias model results.",
"For each simulation box, the random numbers are strictly model-independent with both the order and quantity being fixed.",
"Besides the baseline two more undecorated models are defined, which vary only the vanilla HOD parameters.",
"The models named Base 2 and 3 differ from the baseline only in the satellite parameters in Eqn.",
": the exponent $\\alpha $ is changed by $\\pm 25\\%$ , and the denominator in the power law term, $M_1$ , is tuned accordingly.",
"The specific combinations of $M_1$ and $\\alpha $ values are chosen to match the number density in the baseline model, $4\\times 10^{-4} h^{3}{{-3}}$ , which agrees with realistic LRG sample densities.",
"Next, there are a number of single-decoration models that come in pairs.",
"Single-decoration means that each model has only one origin of bias present with respect to the baseline model.",
"And each pair of models have opposite changes in the parameter(s) of interest.",
"Two models have assembly bias only for central galaxies with opposite assembly bias parameter $A_\\text{cen}$ , and similarly two assembly bias models for satellites only.",
"Then there are two central velocity bias models, one assumes a more realistic, $20\\cdot v_\\text{rms}$ velocity dispersion for the central galaxy relative to the DM halo, and the other assuming a more extreme, $100\\cdot v_\\text{rms}$ dispersion.",
"The last six models investigate three bias effects arising from sub-halo-scale astrophysics, by giving preferential treatment to DM particles based on the particle's speed, halo centric distance, or total mechanical energy (quantified by perihelion distance) when assigning satellite galaxies.",
"Below is a brief review of the definitions of the parameters, and readers are referred to [45] for a more detailed discussion.",
"In Table REF , the first five columns are the standard 5 parameters as defined in Eqn.",
"REF and which govern the mean halo occupation for central and satellite galaxies.",
"$A_\\text{cen}$ and $A_\\text{sat}$ are assembly bias parameters for centrals and satellites, implemented as comparing the halo concentration to its peers of similar masses and re-assigning $\\log M_\\text{pseudo} = \\log M + A_\\text{cen/sat} \\left[ 2 \\Theta (c-c_\\text{med}) - 1 \\right],$ a pseudomass to the halo as input for $\\langle N_\\text{cen}(M) \\rangle $ or $\\langle N_\\text{sat}(M) \\rangle $ .",
"Here $c_\\text{med}$ is the median concentration in the halo mass bin to which the halo belongs, and $\\Theta (c-c_\\text{med})$ is the Heaviside step function.",
"While [8] showed that the formation time (redshift) and halo concentration do not capture the assembly history of halos as far as small-scale two-halo terms in the correlation function is concerned, we still use halo concentration as a proxy for assembly bias, as our mock central and satellite galaxies and do not exactly preserve the 1-halo terms and bias effects may manifest at large scales.",
"The velocity bias for central galaxies draws randomly from a normal distribution scaled by $\\alpha _\\text{cen}$ , and adds that peculiar velocity relative to the halo to the line-of-sight component of the halo velocity, $v_{\\text{pec}} &\\sim N(0, \\frac{v_\\text{rms}}{\\sqrt{3}} \\alpha _\\text{cen}) \\\\v_^\\prime &= v_+ v_{\\text{pec}}$ where $v_\\text{rms}$ is the RMS velocity dispersion of all DM particles within the halo.",
"The 20% case is realistic while 100% is rather extreme, as central galaxies do not move as fast as DM particles relative to the halos in which they reside.",
"The last three $s$ parameters control in-halo satellite generation by modifying the probability of each particle hosting a satellite as $p_i = \\overline{p} \\left[ 1 + s_{\\_,v,p} (1 - \\frac{2 r_i}{N_\\text{part} - 1}) \\right],$ where $\\overline{p} \\equiv \\langle N_\\text{sat} (M) \\rangle / N_\\text{part}$ is the uniform probability for each particle to begin with, the three ranking parameters need to satisfy $s_{\\_,v,p} \\in (-1, 1)$ to conserve the total probability, $N_\\text{part}$ is the total number of particles within the halo, and $r_i=0, 1, 2, \\ldots , N_\\text{part}-1$ is $i$ th particle's ranking by halo centric distance, speed, or perihelion (total mechanical energy), all at the snapshot taken at $z=0.5$ .",
"We stress that the parameters for the bias models listed in Table REF are intentionally chosen to be quite extreme.",
"For a common 20-particle halo and $s=0.9$ , for example, the innermost particle (rank $r=0$ ) has 19 times the probability of the outermost particle ($r=19$ ) to match to a satellite galaxy.",
"The purpose is to increase the chance of detecting a shift in the acoustic peak location and to explore the worse-case scenarios."
],
[
"Redshift-Space Distortion and Reconstruction",
"By adding decorations to the base HOD class, we are modifying the line-of-sight velocity of galaxies and assuming an alternative truth velocity.",
"As apparent RSD depends on the true peculiar velocity in addition to the Hubble flow, we artificially apply RSD to the line-of-sight coordinate as the last step of mock galaxy generation by modifying $x^\\prime _= x_+ v_/ \\left[a H(a) \\right]$ in the line-of-sight direction, after all decorations are completed.",
"In practice, this is implemented as $x^\\prime _= x_+ v_/ \\left[a H_0 E(z) \\right]$ where $E(z)$ is the Astropy efunc defined as $H(z) \\equiv H_0 E(z)$ .",
"Now the complete mock galaxy catalogue is ready to be treated as observed data.",
"Two reconstruction methods are applied and compared side by side: the standard reconstruction [11] and a recent iterative reconstruction method [17].",
"For both reconstruction methods, the optimal smoothing scale $\\Sigma = 15 \\,h^{-1}{} $ is used.",
"Additional parameters for iterative reconstruction used are grid size $N_\\text{grid} = 480^3$ , galaxy bias $b = 2.23$ , initial smoothing scale $\\Sigma _\\text{ini} = 15 \\,h^{-1}{} $ , annealing parameter $\\mathcal {D} = 1.2$ , weight $w = 0.7$ , and number of iterations $n_\\text{iter} = 6$ .",
"These parameters are chosen based on [17], [18] where variations in the input galaxy bias $b$ up to 20 was found to hardly impact the iterative reconstruction result.",
"The rest of the procedures including 2PCF, covariance, and fitting is performed on all three types of galaxy catalogues: pre-reconstruction, post-reconstruction (standard), and post-reconstruction (iterative).",
"To suppress the shot noise in galaxy generation, for the same simulation box and same biased HOD model, 12 realisations of the galaxy catalogue are generated repeatedly with varied initial seed.",
"The initial condition phase for a box $p$ is an integer index labelling the simulation box.",
"The realisation index $r$ is an integer from 0 to 11.",
"The random number generator seed is reset as $s = 100p + r$ before every galaxy generation, which guarantees that the random numbers are always deterministic and model-independent, as long as we never exceed 100 realisations.",
"Tn the beginning of each galaxy catalogue generation, a fixed quantity of random numbers ($N_\\text{halos} + N_\\text{particles}$ ) are thrown for centrals and satellites before any other operation takes place which may involve throwing more random numbers (e.g.",
"central velocity bias which draws randomly from a Gaussian distribution).",
"As $N_\\text{halos}$ and $N_\\text{particles}$ are both constant for a given simulation box, this ensures that the same random numbers are always generated, regardless of the HOD model imposed, and assigned to every halo or DM particle in a fixed order.",
"For each realisation of the galaxy sample, the 2PCF and their Legendre multipole decompositions are calculated using a Fast Fourier Transform algorithm on the $k$ -grid [38], as well as using the pair-counting method for cross-checking when applicable.",
"All pair-counting is done in fine $(s, \\mu )$ and $(r_p, \\pi )$ bins using a highly efficient pair-counting code Corrfunc [36]: $s$ bin edges from 0 to $150 \\,h^{-1}{} $ at $1 \\,h^{-1}{} $ steps and $\\mu $ bin edges from 0 to 1 at 0.01 steps.",
"The pair-counts are re-binned with optimal bin size $\\Delta s = 5 \\,h^{-1}{} $ found in [30].",
"2PCF functions are calculated from raw pair-counts using a generalised form of the [22] estimator, which works for both auto- and cross-correlations.",
"Given data samples $D_1, D_2$ , random samples $R_1, R_2$ in the same respective volumes, and sample sizes $N_{D1}, N_{D2}, N_{R1}, N_{R2}$ (number of data or random galaxies in the sample), the correlation as a function of pair-counts is $\\xi _\\text{LS}&= \\frac{\\frac{D_1D_2}{N_{D1}N_{D2}} - \\frac{D_1R_2}{N_{D1}N_{R2}} - \\frac{R_1D_2}{N_{R1}N_{D2}} + \\frac{R_1R_2}{N_{R1}N_{R2}}}{\\frac{R_1R_2}{N_{R1}N_{R2}}} \\nonumber \\\\&= \\frac{\\overline{D_1D_2} - \\overline{D_1R_2} - \\overline{R_1D_2} + \\overline{R_1R_2}}{\\overline{R_1R_2}}$ where pair-counts with bars denotes normalised pair-counts, i.e.",
"raw pair-counts weighted by sample population sizes $1/(N_1N_2)$ .",
"For the auto-correlation of galaxy samples, we may simply set $D_1 = D_2$ and $R_1 = R_2$ .",
"No FKP weighting is included as the galaxy distribution is homogeneous in one redshift bin in our simulations.",
"Figure: (Colour online) The first two multipoles of the galaxy 2PCF for the baseline HOD model showing fluctuations among simulation boxes.",
"The first row shows N box =36N_\\text{box}=36 multipole samples derived from co-adding 12 realisations of each box; the curves are the mean monopole or quadrupole, and the shaded regions are ±1σ\\pm 1\\sigma intervals around the mean.",
"The second row shows the re-sampled, delete-1 jackknife multipoles derived from co-adding N box -1N_\\text{box}-1 boxes at a time; the shaded ±1σ\\pm 1\\sigma regions are in fact the standard deviation rescaled by N box -1=35\\sqrt{N_\\text{box}-1}=\\sqrt{35} to account for the jackknife re-sampling, though difficult to see.",
"Three columns are pre-reconstruction, post-reconstruction (standard), and post-reconstruction (iterative).",
"From left to right, the monopole BAO peak is sharpened and the quadrupole getting closer to zero, indicating less anisotropy and better restored spherical BAO shell.",
"Although different boxes in a given simulation only differ by the initial condition phase and over ten realisations are generated and co-added, there are still considerable fluctuations in the 2PCF among boxes.",
"Jackknife re-sampling yields much smoother and stabler samples, and greatly reduces the uncertainty in the mean correlations.The multipoles from all realisations are co-added into one correlation sample for a given simulation box.",
"To further increase signal-to-noise ratio in the clustering statistics, we take the delete-1 jackknife samples of the multipoles by ignoring one box and co-adding all the other boxes at a time.",
"A comparison between individual box samples and jackknife samples is shown in Fig.",
"REF .",
"There are large enough fluctuations across different boxes that jackknife re-sampling is a necessary step before BAO fitting and significantly reduces sample variance.",
"The $N_\\text{box}=36$ jackknife samples are then passed on to the fitter."
],
[
"Covariance Estimation",
"The covariance between all multipoles and $(s, \\mu )$ bins in the 2PCF must be estimated before fitting for any physical parameter.",
"Since we are interested in the acoustic scale around $100 \\,h^{-1}{} $ , an order of magnitude smaller than the simulation box size $1100 \\,h^{-1}{} $ , we opt to divide the box into subvolumes, increasing the number of correlation samples while still retaining the BAO signal, and bootstrap the covariance.",
"We choose $N_\\text{sub} = 3$ along each dimension, so that each subvolume has a side length of over 3 times the BAO scale of interest.",
"For each realisation, the full box galaxy sample is divided into $N_\\text{sub}^3 = 27$ subvolumes.",
"The galaxies in each subvolume are cross-correlated with the full box volume using Eqn.",
"REF , with index 1 being the full box and index 2 being the subvolume.",
"Every subvolume is treated independently, and all its realisations are co-added to prevent shot noise from entering the covariance matrix.",
"By taking cross-correlations between the full box and the subvolume, we obtain $N_\\text{sub}^3=27$ times the number of auto-correlation samples in $1/N_\\text{sub}^3=1/27$ of the box volume.",
"In the end, the joint monopole-quadrupole covariance matrix is derived from $N_\\text{box}N_\\text{sub}^3=36\\times 3^3=972$ correlation samples.",
"As covariance scales inversely with the spatial volume in which it is calculated, and the auto-correlations used in the fitting are for the full box, this covariance matrix is re-scaled by a factor of $1/N_\\text{sub}^3$ to account for the subvolume division.",
"An additional factor of $1/(N_\\text{box}-1)$ is needed if fitting to jackknife multipoles averaged over $N_\\text{box}-1$ samples.",
"It is worth noting that for every bias model and every type of correlation there is a different covariance matrix.",
"Pre-reconstruction galaxy samples are given uniform, analytic randoms to calculate the correlations and covariance.",
"Standard reconstruction produces a shifted galaxy catalogue as well as shifted numerical randoms, which can then be both subdivided to estimate the covariance.",
"Our standard reconstruction implementation produces a shifted random set 200 times the size of the data set.",
"For the purpose of estimating covariance, it is computationally expensive and unnecessary to use all of it.",
"We opt to speed up the pair-counting by randomly downsampling the the random set to a level of 10 times the data.",
"To further balance the pair-counting workload between the $DR$ and $RR$ terms in the correlation estimator for covariance bootstrap, while $D_1R_2$ and $R_1D_2$ are counted using the $10\\times $ subsample of shifted randoms, for $R_1R_2$ the $10\\times $ subsample is split into 10 copies, each of size $1\\times $ the galaxy sample, and $R_1R_2$ is counted 10 times using the $1\\times $ split samples and then the pair-counts averaged.",
"Iterative reconstruction does not provide any shifted sample after it completes, only the auto-correlations.",
"We assume it shares the same covariance matrix as standard reconstruction, given how similar their post-reconstruction correlations are.",
"For BAO fitting, which involves about 10 degree of freedoms, this estimate of the covariance matrix is acceptable but of course not perfect [28].",
"We emphasise that the purpose is to determine the shifts in the acoustic scale, not the confidence level of the chi-square fit in the $(\\alpha _\\perp , \\alpha _)$ space.",
"The covariance matrices only weight the fit overall and still give the correct acoustic scale."
],
[
"Fitting 2PCF for the BAO Scale",
"Following the tried-and-true fitting methods described in previous BAO analyses of galaxy redshift surveys [2], [3], [30], we fit to the monopole and quadrupole jackknife samples and determine the anisotropic BAO scale in transverse and radial directions .",
"Our new BAO fitter supports any arbitrary input linear power spectrum and transforms it with flexible choices of parameters into correlation multipole templates.",
"Starting with the input linear power spectrum of the simulation (e.g.",
"from Camb) and the no-wiggle power spectrum [10], $P(k, \\mu ) = C^2(k, \\mu , \\Sigma _s)\\left[ (P_\\text{lin} - P_\\text{nw}) e^{-k^2\\sigma _v^2} + P_\\text{nw} \\right] $ where $\\sigma _v^2 &=\\frac{(1-\\mu ^2)\\Sigma _\\perp ^2}{2} + \\frac{\\mu ^2 \\Sigma _^2}{2}\\\\C(k, \\mu , \\Sigma _s) &= \\frac{1+\\mu ^2\\beta \\left[ 1-S(k)\\right]}{1+\\frac{k^2\\mu ^2\\Sigma _s^2}{2}}\\\\S(k) &= e^{-\\frac{k^2\\Sigma _r^2}{2}}\\\\\\Sigma _&= \\frac{\\Sigma _\\perp }{1-\\beta }.$ Here the reconstruction smoothing scale $\\Sigma _r = 15 \\,h^{-1}{} $ and the streaming scale $\\Sigma _s = 4 \\,h^{-1}{} $ are fixed.",
"The last equation is a convenient approximation such that the user only needs to specify $\\Sigma _\\perp $ , and in the isotropic case, $\\beta =0$ enforces $\\Sigma _\\perp = \\Sigma _$ .",
"We have experimented with various choices of the other parameters and checked which one(s) best recovered the truth acoustic scale in the input power spectrum.",
"For pre-reconstruction matter density field, $\\Sigma _\\perp = 1.5 \\,h^{-1}{} $ , and for pre-reconstruction galaxy catalogue, $\\Sigma _\\perp = 5 \\,h^{-1}{} $ provide the appropriate smoothing of the BAO peak.",
"For all post-reconstruction samples, even with the lowest choice $\\Sigma _\\perp = 0$ and least smoothing, the monopole peak of the template is still slightly wider than that of the data, so $\\Sigma _\\perp = 0$ is used.",
"When varying $\\beta \\in [0, 0.5]$ , we find that for pre-reconstruction samples, the resulting $\\alpha $ scale is relatively stable and insensitive to $\\beta $ while $\\chi ^2$ would increase by up to $50\\%$ as $\\beta $ increases; for post-reconstruction samples, $\\beta =0$ best recovers the truth acoustic scale.",
"Therefore in all cases we set $\\beta =0$ .",
"The power spectrum in Eqn.",
"REF is decomposed into power multipoles and then Fourier transformed to correlation multipoles in the usual manner $P_\\ell (k) &= \\frac{2\\ell + 1}{2} \\int _{-1}^{1} P(k, \\mu ) L_\\ell (\\mu ) \\mu \\\\\\xi _\\ell (r) &= \\frac{i^\\ell }{2\\pi ^2} \\int _{k_\\text{min}}^{k_\\text{max}} k^2 P_\\ell (k)j_\\ell (kr) e^{-(ka)^2 r} k$ where $L_\\ell (\\mu )$ is the Legendre polynomial, $j_\\ell (kr)$ is the spherical Bessel function of the first kind, $k_\\text{min}$ and $k_\\text{max}$ are the limits in the input linear power, and $a = 0.35 \\,h^{-1}{} $ controls the exponential damping term which suppresses high-$k$ oscillations and is necessarily needed to produce the correct shape of correlations multipoles.",
"The rest of the fitting procedure is the same as in [30].",
"The fiducial fitting model assumes fitting range $r \\in (50 \\,h^{-1}{}, 150 \\,h^{-1}{})$ , bin size $5 \\,h^{-1}{} $ , and the poly3 nuisance form (third-order inverse polynomial in power $k$ ), $A_\\ell (r) = A_{2}/r^2 + A_{1}/r + A_{0}$ .",
"As a validation of the fitter, we fit to a $0.2\\%$ uniform subsample of the matter field ($6\\times 10^6$ of $1440^3$ DM particles in a box) without RSD applied in addition to mock galaxy samples.",
"We find the fitted matter field acoustic scale in excellent agreement with the theoretical template at the $0.3\\%$ level up to nonlinear corrections after standard reconstruction.",
"We expect this level of shift due to nonlinear evolutions when fitting with a linear power spectrum as input [33], and that is observed in the first two panels of Fig.",
"REF .",
"The uniform matter field and baseline galaxy results have a mean BAO shift of $0.3\\%$ to $0.4\\%$ .",
"We also find that $N_\\text{box}=36$ jackknife fitting results in the $(\\alpha _, \\alpha _\\perp )$ plane are distributed like an ellipse, indicating that the errors are Gaussian and we may estimate the confidence region using the elliptical distribution of points.",
"Ultimately we are concerned with the differential change in the BAO scale $\\alpha $ when a certain source of galaxy bias is introduced, and the sample variance in the fitted $\\alpha $ values should largely cancel out.",
"In the last panel of Fig.",
"REF showing the $\\alpha _\\text{gal} - \\alpha _\\text{mat}$ subtraction, the uncertainty regions indeed shrink and the data points are much more tightly bound together compared to the first panel without subtraction.",
"For a given simulation box, this subtraction does cancel out a significant amount of sample variance.",
"The post-reconstruction point being at the origin also indicates that galaxy bias in the base model has introduced no statistically significant shift relative to the matter.",
"Figure: (Colour online) An overplotted comparison between pre- and post-reconstruction multipoles for the baseline HOD model, averaged over all simulation boxes.",
"In the top monopole panel, two reconstruction methods both sharpen and narrow the BAO peak, and essentially overlap with each other from 60 to 150h -1 150 \\,h^{-1}{} , only differing on small-scales.",
"In the bottom quadrupole plot, iterative reconstruction does a significantly better job reducing anisotropy and appears nearly flat down to 20h -1 20 \\,h^{-1}{} , whereas standard reconstruction has large residual anisotropies below 80h -1 80 \\,h^{-1}{} .",
"Two reconstruction methods also produce slightly different monopole on small scales below 50h -1 50 \\,h^{-1}{} , with iterative reconstruction supposedly be more accurate, but this range is usually discarded in BAO fitting to better isolate the BAO signal.Although the poly3 fitting form has been shown to be robust for samples with and without standard reconstruction in previous analyses [27], [2], alternative choices are worth considering again for our Abacus mocks with the new iterative reconstruction method applied.",
"Iterative reconstruction does significantly better than standard reconstruction in reducing the anisotropies in the quadrupole, especially on intermediate to small scales, as shown in Fig.",
"REF .",
"Using a simpler $A(r)$ form or a wider $r$ range in the fitting model might yield better fits.",
"Polynomials with simple Fourier transformation properties are motivated by the need to marginalise over the broadband shape of the galaxy correlation functions and to isolate the BAO feature.",
"Having a nonzero $A(r)$ is important for ameliorating inaccuracies of the assumed fiducial cosmology and keeping the fit robust against variations in the input, and polynomials of degrees as high as 4 risks over-fitting the data with too much freedom.",
"This means that $A_\\ell (r) = 0$ (poly0) and $A_\\ell (r) = A_1/r^2 + A_2/r + A_3 + A_4 r$ (poly4) are both disfavoured.",
"The broadband correlation contains unwanted information such as scale-dependent bias, uneven galaxy number densities, and redshift-space distortions among other observational effects, which might not be present in our simulation mocks in the first place.",
"We experimented with several other nuisance forms, $A_\\ell (r) &= \\frac{A_1}{r^2} && (\\textit {poly}1)\\\\A_\\ell (r) &= \\frac{A_1}{r^2} + \\frac{A_2}{r} && (\\textit {poly}2) \\\\A_\\ell (r) &= \\frac{A_1}{r^2} + \\frac{A_2}{r} + A_3 && (\\textit {poly}3) \\\\A_\\ell (r) &= \\frac{A_2}{r} + A_3 \\, .",
"&& (\\textit {poly}3^\\prime )$ along with increased fitting ranges and found no improvement over the fiducial fitting model in any case, in terms of recovering the true BAO $\\alpha $ and reducing $\\chi ^2$ of the fit.",
"The HOD bias results in § are all obtained with the fiducial fitting model."
],
[
"Two-Point Correlation Functions of Bias Models",
"Before presenting BAO fitting results, we first examine the 2PCF resulting from different biased HOD models and reconstruction methods.",
"Fig.",
"REF shows the $\\xi (r_p, \\pi )$ 2PCF for the baseline model before and after standard reconstruction, averaged over all boxes up to $120 \\,h^{-1}{} $ (heatmap colour is rescaled by $r^2 = r_p^2 + \\pi ^2$ or $r^2 = r_\\perp ^2 + r_^2$ ).",
"Standard reconstruction does a fine job restoring the isotropy of the spherical BAO shell in the redshift-space 2PCF around $r=100 \\,h^{-1}{} $ .",
"On intermediate to large scales, all bias models look very similar in this $\\xi (r_p, \\pi )$ plot, with differences being obvious only on the small scales.",
"Figure: (Colour online) Changes in 2D 2PCF of each biased HOD model with respect to the baseline model, averaged over all boxes.",
"Each row contains a pair of two comparable models; each model has a pre-reconstruction panel and a post-reconstruction (standard) one.",
"Symmetrical positive and negative changes in the HOD parameters in all pairs of models but assembly bias ones induce symmetrical change in the correlation Δξ\\Delta \\xi in rows 1, 5, 6 and 7.Zooming in to the small-scale correlations around the origin in Fig.",
"REF , we takes a closer look at the effects of galaxy bias on clustering in Fig.",
"REF by plotting the correlation difference with respect to the baseline model, $\\Delta \\xi =\\xi _\\text{model} - \\xi _\\text{base}$ , again rescaled by $r^2 = r_p^2 + \\pi ^2$ or $r^2 = r_\\perp ^2 + r_^2$ .",
"Each row is a side-by-side comparison between two HOD models with symmetric changes in the bias parameter, as defined in Table.",
"REF .",
"There are substantial changes on small scales in may cases, and the finger-of-god effect is especially exacerbated.",
"For assembly bias in rows 2 and 3, because re-assigning halo masses in our implementation essentially changes the mass distribution of galaxies, reversing the sign of the assembly bias parameter does not simply result in the opposite change in the correlation.",
"With the exception of assembly bias models, symmetrical positive and negative changes in the HOD parameters in all the other bias models induce symmetrical $\\Delta \\xi $ when comparing columns 1 to 3, or columns 2 to 4.",
"All single-variation models tested produce distinct $\\Delta \\xi $ patterns (up to normalisation by $r^2$ ) on small scales and can be easily distinguishable from each other.",
"When more than source of galaxy bias is present, the resulting 2D correlation will be a combination of all the contributions, making the pattern difficult to parse and likely creating degeneracies.",
"Comparing the pre- and post-reconstruction columns, we see that the peripheral regions become noisy after reconstruction.",
"This means that although the decorations imposed may cause nontrivial changes in 2PCF on intermediate scales around 40, these changes are largely removed by reconstruction.",
"Figure: (Colour online) Phase-matched difference of jackknife multipoles for each biased HOD model with respect to the baseline; monopole in blue solid line, quadrupole in red dashed line.",
"Every row is a biased HOD model with and without reconstruction applied.",
"The shaded regions are ten times the size of the actual 1σ1\\sigma error bands, which would have been hardly visible.",
"All axes share the same scaling.Fig.",
"REF shows the phase-matched differences of the first two jackknife multipoles for each biased HOD model with respect to the baseline model, i.e.",
"$\\Delta \\xi _\\ell = \\xi _{\\ell ,\\text{model}} - \\xi _{\\ell ,\\text{base}}$ rescaled by $r^2$ .",
"The sample variance across simulation boxes is already very small after jackknife resampling, and the $1\\sigma $ errors are too small when plotted, so the error regions in Fig.",
"REF are artificially enlarged 10 times to improve visibility and demonstrate the reduction of sample variance by two reconstruction methods.",
"The correlation multipoles show the same trend as does the 2D redshift-space correlation function.",
"Satellite variations in the first two rows change small-scale clustering significantly, indicating the re-distribution of satellite galaxies mostly changing the 1-halo term; on larger scales the changes get noisier but one can barely see that there exist rises and drops around the BAO scale, and monopole and quadrupoles change in opposite directions.",
"The assembly bias models in rows 3-6 re-assign halo masses by ranking halo pseudomasses, and the mass distribution of halos selected with the mass cut varies across simulation boxes, so these plots are the noisiest ones; the changes in 2PCF are not symmetrical as the assembly bias parameters flips signs.",
"From the first six rows, one notices that reconstruction shrinks the error bands considerably, reducing the sample variance.",
"It also reduces the net change, bringing the multipoles closer to the unbiased zero point.",
"As velocity dispersion of central galaxies is increased from 20% to 100%, the multipole differences grow drastically, and a decrease in BAO peak amplitude is seen on large scales after reconstruction right around $100 \\,h^{-1}{} $ , as central velocity bias smears the central-central contribution in the 2-halo term.",
"Models in the last six rows incorporate sub-halo-scale physics and only significantly affect small-scale clustering up to about $50 \\,h^{-1}{} $ as expected.",
"These models are extremely stable across random realisations, with very small scatter in spite of highly exaggerated error ranges.",
"One can see that for each pair of models with symmetric changes in the HOD parameters, the $\\Delta \\xi _\\ell $ plots are essentially symmetric in the same way as in the redshift-space 2PCF (Fig.",
"REF ).",
"Again, the difference between standard and iterative reconstruction mostly lies in small to intermediate scales, and on large scales they are visually the same."
],
[
"Effects of Galaxy Bias on the Acoustic Scale Measurement",
"Having computed the correlation functions of our various models, we fit the acoustic signal to yield transverse and radial scale measurements.",
"Then the $\\alpha $ values are subtracted by the baseline model value for each simulation box in a phased-matched manner.",
"The subtraction reveals the differential change in BAO $\\alpha $ for each biased model relative to the baseline.",
"Phase-matching is critical when taking the difference, because every box has its own sample variance shared by the realisations of all models and this sample variance can be cancelled out effectively as demonstrated in §REF .",
"Table REF summarises the shifts of the acoustic scale for all biased HOD models with respect to the baseline model.",
"Two reconstruction methods are listed separately.",
"The same numerical results are plotted in Fig.",
"REF , where each row shows a pair of comparable models and each column is a pre- or post-reconstruction sample type.",
"The first two models in the first row with opposite variations in satellite mass cutoff and power law exponent see asymmetric shifts.",
"It is encouraging to see that iterative reconstruction manages to completely eliminate the bias of the Base 3 model (higher $M_1$ and $\\alpha $ ), bringing the red cross back right onto the origin, and slightly reduces the bias of the Base 2 model (lower $M_1$ and $\\alpha $ ).",
"This is one of the only two cases where iterative reconstruction noticeably performs better than standard reconstruction at restoring the unbiased BAO scale.",
"By lowering $M_1$ and $\\alpha $ , Base 2 effectively moves satellite galaxies from high mass halos to lower mass halos.",
"There are many more possible locations to put them at a lower mass cut, and low mass halos do not mark the highest initial overdensities as much as high mass halos do, so correcting for the bias for Base 2 is naturally more difficult than for Base 3, resulting in a residual shift of 0.3% post-reconstruction primarily in the radial direction.",
"For the assembly bias models in rows 2 and 3 of Fig.",
"REF , the shifts induced by opposite bias parameters are almost symmetrical in the horizontal, or traverse, direction, but in the radial direction all models are slightly biased toward a higher $\\alpha _$ regardless of the sign of the bias parameter.",
"The previous plots (Fig.",
"REF and REF ) do not clearly separate the two directions and show this difference.",
"The second and third rows have similar mean values for the same coloured models post-reconstruction, only the uncertainty is smaller for satellite assembly bias.",
"This resemblance implies that, whether assembly bias is present in the central or satellite galaxies, the end effect on the acoustic scale is virtually the same if reconstruction is applied.",
"In other words, assembly bias does not distinguish between centrals and satellites.",
"Comparing the models with negative signs only (red markers and orange ellipses), however, one sees that the satellite assembly bias results are better constrained than central assembly bias ones, making the same 0.2% shift more significant for the satellite negative model.",
"But it is only about $2\\sigma $ , and at the 0.2% level all four models are consistent with zero shift.",
"For the velocity bias models in the fourth row, a realistic dispersion at the 20% level hardly causes any shift, whereas the more extreme 100% case resulted in large shifts of 0.7%.",
"This is the other one of the two cases where iterative reconstruction performs noticeably better than standard reconstruction.",
"In fact, there is even a qualitative difference here—the orange error region of standard reconstruction seems to be consistent with zero shift at the $2\\sigma $ level, but the iterative reconstruction uncertainty is much more confined and certainly rules out the no shift possibility.",
"The last three rows are six models with sub-halo-scale astrophysics.",
"Galaxy bias originating from halo centric distances of DM particles clearly does not impact the BAO measurement.",
"On the other hand, satellite velocity bias and perihelion bias (or dependence on the total mechanical energy of DM particle) do make small differences of 0.1% to 0.2%.",
"Satellite velocity bias results in almost $0.2\\%$ shifts with just above $2\\sigma $ significance.",
"The perihelion bias models have smaller shifts of 0.1%, which are insignificant.",
"The symmetric pattern of shifts can again be seen in the last two rows for models of opposite signs.",
"Again, at the 0.2% level all six models are consistent with zero shift.",
"Iterative reconstruction offers better BAO fitting results than standard reconstruction in general.",
"In rows 1 and 4 of Fig.",
"REF , it made a real difference by significantly reducing the bias and uncertainty in the acoustic scale measurement, as mentioned above.",
"For other models, it is not drastically better and the mean $\\alpha $ values are extremely close to the standard reconstruction ones, but still it yields slightly less bias in the acoustic scale and tighter constraints.",
"Lastly, we also notice a reliable inverse correlation between the sign of $\\Delta \\alpha _$ and the sign of $\\Delta \\xi _2$ , when comparing Fig.",
"REF with REF .",
"This is obeyed by every model tested, even if the changes in 2PCF and acoustic scale are not very significant.",
"A positive $\\Delta \\xi _2$ relative to the baseline corresponds to a negative $\\Delta \\alpha _$ , and vice versa.",
"While we used the fiducial BAO fitting model with a fitting range of $r \\in (50 \\,h^{-1}{}, 150 \\,h^{-1}{})$ and the $\\textit {poly}3^\\prime $ nuisance form, improved fitting methods may exploit this inverse correlation by extending the range to smaller scales."
],
[
"Conclusions",
"In this work, we test the effect of galaxy bias on the acoustic scale by considering several bias mechanisms to their extremes.",
"With accurate N-body simulations in a total volume of $48 \\, h^{-3}{{3}}$ and a generalised HOD approach, every biased model can be compared to the baseline to derive the differential shift in the acoustic scale measurement that precisely corresponds to the change in the input HOD parameters.",
"We find a 0.3% shift in the line-of-sight acoustic scale for one variation in the satellite galaxy population, the Base 2 model.",
"The model with an extreme level of velocity bias of the central galaxies produces the largest shift in the BAO measurement, $0.7\\%$ relative to the unbiased scale.",
"All the other bias models result in either small ($0.2\\%$ or less) or statistically insignificant ($2\\sigma $ or less) shifts.",
"Except for the highly unlikely event that the central galaxies have very large velocity dispersions relative to the halo bulk (close to the typical speed of DM particles), we find the shifts caused by single-variation models to be below 0.3%.",
"However, this is by no means a claim that the theoretical systematic error in the acoustic scale measurement due to galaxy bias originating from the halo-galaxy connection in the halo model is only $0.3\\%$ .",
"Observed galaxy samples from redshift surveys may well be subject to not one, but many processes which can introduce galaxy bias.",
"Combinations of bias mechanisms at play may potentially compound the uncertainties we found and push the BAO shift above 0.3%.",
"That said, the biggest shift of the acoustic scale at 0.7% comes from increased velocity bias for central galaxies, which seems readily detectable—there would be sizeable changes in the velocity dispersion of clusters compared to their weak lensing masses.",
"In addition, these bias models also create substantial changes on small scales, which may in fact allow one to detect these effects and thereby improve the modelling of the acoustic scale.",
"In Fig.",
"REF , many bias models have greatly increased finger-of-god effect in the correlation function, both pre- and post-reconstruction.",
"Given the inverse correlation we found between the sign of $\\Delta \\alpha _$ and the sign of $\\Delta \\xi _2$ , small-scale clustering data offer a promising opportunity to correct for galaxy bias and further calibrate the acoustic peak ruler with future development in the 2PCF fitting formalism.",
"In regards to reconstruction, both standard and iterative reconstruction methods show similar efficacy in reducing the imposed bias and recovering the unbiased 2PCF and acoustic scale.",
"In terms of the performance for BAO fitting, iterative reconstruction is more robust against galaxy bias, bringing the BAO measurements closer to the true, unbiased acoustic scale; it is also more precise, being less prone to sample variance and producing less uncertainty.",
"It is a promising new method with the potential to benefit from further development and optimisation in the future.",
"Although it currently requires an order of magnitude more resources in CPU time and memory allocation than standard reconstruction does, reconstruction and BAO fitting comprise a minor fraction of the computation time in comparison to the time needed for genuine $N$ -body simulations.",
"The extra time needed for iterative reconstruction is worthwhile If one is concerned with minimising the bias and uncertainty in the BAO analysis.",
"While current galaxy surveys, such as the Sloan Digital Sky Survey (SDSS) III Baryon Oscillations Spectroscopic Survey (BOSS), measure the acoustic scale to about 1% precision and are insensitive to the galaxy bias effects shown, upcoming dark energy experiments, including DESI, Euclid, and WFIRST, will make use of the BAO standard ruler to sub-percent level precision and these effects can no longer be overlooked.",
"Our ability to account for or even correct for galaxy bias in the modelling of the acoustic scale directly impacts the measurement precision of the BAO ruler and the success of upcoming surveys.",
"Having shown that the systematic effects of galaxy bias alone could amount to 0.3%, we find the majority of the shifts values in Table REF encouraging and note that these bias effects may be detected.",
"Accurate modelling of the galaxy-halo connection, in conjunction with the bias that comes with it, is a need of growing urgency as the statistical uncertainties of larger surveys approach the level of cosmic variance limit and theoretical systematics.",
"Future analysis of the BAO signal may benefit from the inclusion of velocity dispersion and small-scale clustering to mitigate the non-negligible systematic effects of galaxy bias."
],
[
"Acknowledgements",
"The authors would like to thank Lehman Garrison for producing the Abacus simulation products and managing the computing cluster.",
"DYT thanks Lehman Garrison for helpful discussions on Corrfunc and Rockstar halo and particle catalogue products, Ryuichiro Hada for providing the reconstruction code, Ashley Ross for sharing his BAO fitter used in BOSS DR12, and Prof. Steve Ahlen for his continued support and mentorship.",
"DJE is supported by U.S. Department of Energy grant DE-SC0013718 and as a Simons Foundation Investigator.",
"DYT is supported by U.S. Department of Energy grant DE-SC0015628."
]
] | 1906.04262 |
[
[
"Hybrid Function Sparse Representation towards Image Super Resolution"
],
[
"Abstract Sparse representation with training-based dictionary has been shown successful on super resolution(SR) but still have some limitations.",
"Based on the idea of making the magnification of function curve without losing its fidelity, we proposed a function based dictionary on sparse representation for super resolution, called hybrid function sparse representation (HFSR).",
"The dictionary we designed is directly generated by preset hybrid functions without additional training, which can be scaled to any size as is required due to its scalable property.",
"We mixed approximated Heaviside function (AHF), sine function and DCT function as the dictionary.",
"Multi-scale refinement is then proposed to utilize the scalable property of the dictionary to improve the results.",
"In addition, a reconstruct strategy is adopted to deal with the overlaps.",
"The experiments on Set14 SR dataset show that our method has an excellent performance particularly with regards to images containing rich details and contexts compared with non-learning based state-of-the art methods."
],
[
"Introduction",
"The target of the single image super resolution (SISR) is to produce a high resolution (HR) image from an observed low resolution (LR) image.",
"This is meaningful due to its valuable applications in many real-world scenarios such as remoting sensing, magnetic resonance and surveillance camera.",
"However, SR is an ill-posed issue which is very difficult to obtain the optimal solution.",
"The approaches for SISR can be divided into three categories including interpolation-based methods [22], [4], [11], [16], [3], reconstruction-based methods [2], [18], [6], [7], [9], and learning-based methods[10], [12], [20], [5], [17], [15].",
"Reconstruction-based methods and learning-based methods often yield more accurate results than interpolation-based algorithm, since those methods can acquire more information from statistical prior and external data even though more time-consuming.",
"Algorithm can be integrated from multiple categories mentioned above, which is not strictly independent.",
"As classic approaches in SR, interpolation-based methods follow a basic strategy to construct the unknown pixel of HR image such as bicubic, bilinear and nearest-neighbor(NN).",
"However, nearest-neighbor interpolation always causes a jag effect.",
"In this aspect, bicubic and bilinear perform better than NN and are widely used as zooming algorithms but easier to generate a blur effect on images.",
"Reconstruction-based methods usually model an objective function with reconstruction constraints from image priors and regularization terms to lead better solutions for the inverse problem.",
"The image priors for reconstructions include the gradient priors [18], sparsity priors [6], [7], [14] and edge priors [9].",
"Those methods usually restore the images with sharp edges and less noises, but will be invalid when the upscaling factor becomes larger.",
"Learning-based algorithms mainly explore the relationship between LR and HR image patches.",
"In the early time, Freeman et al.",
"[10] predict the HR image from LR input by Markov Random Field trained by belief propagation.",
"Sparse-representation for SISR is firstly proposed by Yang et al.",
"[20] and achieves state-of-the-art performance.",
"Zhang et al.",
"[24] propose a novel classification method based on sparse coding autoextractor.",
"Zeyde et al.",
"[21] improve the algorithm by using K-SVD [1] based dictionary and orthogonal matching pursuit (OMP) optimization method.",
"Recently, deep neural network has been extensively used in super-resolution task.",
"Dong et al.",
"[5] firstly design an end-to-end mapping from LR to HR called SRCNN.",
"Generative adversarial network [13] is introduced to SR by [17], [15] to encourage the network to favor solutions that look more like natural images.",
"The learning-based algorithms are not generally promoted, since most of the algorithms only focus on the specific upscaling factor and need to be retrained on external large dataset once the upscaling factor is changed.",
"In this paper, we propose hybrid function sparse representation (HFSR), which uses function-based dictionary to replace conventional training-based dictionary.",
"The special property of such dictionary are utilized to reinforce the SR algorithm.",
"Primary idea of the proposed method is that the upscaling of the function curve will keep it's fidelity, and to represent the patches from LR image by linear combination of functions.",
"We select approximate Heaviside function (AHF), discrete cosine transformation (DCT) function, and sine function to form the dictionary.",
"Unlike the work of [4], we adopt diverse functions instead of single AHF, and do not mix the sparsity and intensity priors together for representation which needs ADMM to solve the objective loss.",
"As a consequence, approach applying with sparse representation performs much faster than the algorithm in [4].",
"Different from the conventional sparse representation method, our algorithm requires no additional training for dictionary on high resolution images, and there is no scale constraint since the dictionary can be scaled up to any size.",
"Once being represented by hybrid functions, dictionary can be stored by parameters rather than matrixs with large size.",
"The algorithm we design is related to the interpolation-based method and reconstruction-based method.",
"In view of this, we compare our results with two interpolation-based algorithms and another reconstruction-based algorithm [12].",
"The remaining part of the paper is organized as follows: Section briefly introduces the sparse representation algorithm for single image super resolution.",
"Section describes the proposed HFSR approach from dictionary designing to multi-scale refinement and then to the construction of whole image.",
"Section discusses the results of the proposed method and the paper is concluded in Section ."
],
[
"Super Resolution via sparse Representation Preliminaries",
"The goal of single image super resolution is to recover the HR image $\\mathbf {x}$ by giving an observed LR image $\\mathbf {y}$ .",
"Relationship between LR images and HR images is denoted as: $ \\mathbf {y} = D \\mathbf {x} + v$ where $D$ is the degradation matrix combining with down sampling operator and blur operator, and $v$ is the independent Gaussian noise.",
"Sparse representation method processes the image in patch level, which means a LR image is divided into several small patches of the same size.",
"Then each patch is approximately represented as linear combination of the dictionary atoms.",
"A HR patch $p_H$ could be reconstructed by the corresponding HR dictionary with the shared representation $\\alpha $ obtained from LR patch $p_L$ .",
"$ p_L \\approx _1 \\alpha ,\\ \\ \\ \\ p_H = _s \\alpha $ We denote $_s \\in \\mathcal {R}^{m\\cdot s^2 \\times N}$ as the dictionary, where $s$ is the upscaling factor.",
"If the value of $s$ is 1, $_s$ exactly becomes $_1 \\in \\mathcal {R}^{m \\times N}$ for the LR patches.",
"$N$ is the size of the whole dictionary, and $m = w \\cdot w$ is the vectorization size of square patch with $w$ length.",
"Therefore, the size of $p_L$ is $m$ , while the size of $p_H$ for HR image is $m\\cdot s^2$ .",
"The SR task is ill-posed which means there are infinite $\\mathbf {x}$ satisfying the Eq.",
"(REF ).",
"The sparse representation methods provide sparse prior for the representation of patch $p_L$ to regularize the task.",
"Sparse prior assumes $\\Vert \\alpha \\Vert _0$ , the coefficients of representation, should be as small as possible, while still contains the low reconstruction error.",
"With the constraint above, SR task can be formulated as an optimization problem: $ \\mathop {\\min } \\Vert \\alpha \\Vert _0 \\ \\ \\ \\ s.t.",
"\\ \\ \\Vert p_L - _1 \\alpha \\Vert _2^2 \\le \\varepsilon $ The aforementioned problem is still non-convex and NP-hard.",
"However, work on [8] shows that (REF ) can be converted by replacing the objective with the $l^1$ -norm: $ \\mathop {\\min } \\Vert \\alpha \\Vert _1 \\ \\ \\ \\ s.t.",
"\\ \\ \\Vert p_L - _1 \\alpha \\Vert _2^2 \\le \\varepsilon $ Applied with Langrange multipliers, (REF ) becomes a convex problem which can be efficiently solved by LASSO [19].",
"$ \\alpha = \\mathop {\\arg } \\mathop {\\min }_{\\alpha } \\left\\lbrace \\Vert p_L - _1 \\alpha \\Vert _2^2 + \\lambda \\Vert \\alpha \\Vert _1 \\right\\rbrace $ $\\lambda $ denotes sparse regularization to balance the sparsity of the representation and the reconstruction error.",
"In conventional sparse representation method [20], we need external images to train the dictionary $_1$ and $_s$ .",
"However, since it is not required in the proposed algorithm, the part of describing dictionary from training has been excluded."
],
[
"Function-based dictionary",
"It is critically important to choose dictionary for sparse coding.",
"Despite the training-based algorithm, another type of dictionary uses wavelets and curvelets which shares some similarity with proposed function-based dictionary.",
"However, the ability of adapting different types of data for this dictionary is limited and do not perform well under sparse prior [23].",
"There are some advantages of using the function to generate the dictionary, in which we can leverage the property of the function to fine tune parameters or enlarge the dictionary to any scale as is required.",
"The relation between dictionary and functions is connected as: $ _1^k(i, j | \\Theta ^k) = f^k (i, j, \\Theta ^k)$ Here, $\\Theta =\\lbrace \\Theta ^1,\\Theta ^2,....,\\Theta ^N\\rbrace $ is the set of parameters for the whole dictionary.",
"$_1^k \\in \\mathcal {R}^m$ represents the $k^{th}$ element in the LR dictionary, which is the vectorized result from the patch with size $\\mathcal {R}^{w \\times w}$ , so the integer pair $(i, j)$ maps the pixel value by coordinate from the raw patch to the $(i*w + j)^{th}$ position in $_1^k$ , $f^k$ and $\\Theta ^k$ is the $k^{th}$ function in dictionary and its corresponding parameter, respectively.",
"$ f^k (i, j, \\Theta ^k) \\in \\left\\lbrace \\begin{array}{lll}\\mathop {\\arctan } \\left( (i \\cdot \\cos \\theta ^k + j \\cdot \\sin \\theta ^k + b^k) / \\xi ^k \\right) / \\pi \\\\\\sin ( (i\\cdot \\cos \\theta ^k + j\\cdot \\sin \\theta ^k + b^k) \\cdot a^k ) \\\\\\cos \\left( \\frac{\\pi a^k}{w} (i + \\frac{1}{2}) \\right) \\cdot \\cos \\left( \\frac{\\pi b^k}{h} (j + \\frac{1}{2}) \\right) \\end{array} \\right.$ Figure: The visualization of LR patches and corresponding HR patches sampled from the hybrid of(a) AHF, (b) DCT function and (c) Sin function.The first row are the LR patches from 1 _1, the second row arethe HR patches from 2 _2.As is shown in Eq.",
"(REF ), three categories of function are combined here.",
"The approximate Heaviside function can be used to fit the edge in patch, where one-dimensional AHF is $\\psi (x) = \\frac{1}{2} + \\frac{1}{\\pi } \\arctan (\\frac{x}{\\xi })$ , and the parameters $\\xi $ control the smoothness of margin between 1 and 0.",
"Then the function is extended to two-dimensional case for our task, thus $f_{AHF}(\\mathbf {z}) = \\psi (\\mathbf {z} \\cdot \\mathbf {d}^T + b)$ , where $\\mathbf {d} = (\\cos \\theta , \\sin \\theta )$ and $\\mathbf {z} = (i, j)$ is the coordinate in patch.",
"Since sine is a simple periodic function and is suitable for representing stripes, we adopt it as one of the basic functions and extend it to two-dimensional case as what we did for AHF.",
"$f_{Sin}(\\mathbf {z}) = \\sin (a \\mathbf {z} \\cdot \\mathbf {d}^T + b)$ , where parameter $a$ and $\\theta $ for 2D-sine function controls the intensity and direction of the stripes, respectively.",
"The idea of using over-complete discrete cosine transformation (DCT) as one of the functions is inspired by [1], the pixel value in DCT patch is generated by $\\cos \\left( \\frac{\\pi a}{w} (i + \\frac{1}{2}) \\right) \\cdot \\cos \\left( \\frac{\\pi b}{h} (j + \\frac{1}{2}) \\right)$ .",
"We have designed several functions and tried various mixed strategies, then we find the hybrid of these three functions work better than single functions or other mixed forms.",
"$ \\Theta ^k \\in \\left\\lbrace \\begin{array}{lll} \\lbrace \\theta _k, b_k, \\xi _k \\rbrace \\\\\\lbrace \\theta _k, a_k, b_k \\rbrace \\\\ \\lbrace a_k, b_k \\rbrace \\end{array} \\right.$ The parameter selections of three functions are listed in the Eq.",
"(REF ).",
"Fig.",
"REF depict upscaling process using three functions above, respectively.",
"One of the advantages of HFSR approach is that the dictionary for reconstructing the HR image with a specific size could be directly obtained by functions.",
"Generating the HR dictionary and LR Dictionary can be connected in a similar way.",
"$_s(i, j | \\Theta ) = _1(i/s, j/s | \\Theta ) = f (i/s, j/s, \\Theta )$"
],
[
"Multi-scale refinement",
"The patches are sampled in raster-scan order from the raw LR image and processed independently.",
"The full representation for each patch is computed by two processes.",
"Sparse coding is firstly applied to get coarse representation $\\alpha _{coarse}$ , and then the proposed multi-scale refinement is used to generate fine representation $\\alpha _{fine}$ .",
"Suppose $\\alpha $ is the optimal solution, then the difference of $p_L - D _s \\alpha $ should equal to zero, where $D$ is the downsampling operator.",
"However, the condition is typically unsatisfied with $\\alpha _{coarse}$ , and the difference usually contains some residual edges.",
"To address this issue, we pick $p_L - D _s \\alpha $ as a new LR patch to fit.",
"Then recompute a representation $\\Delta \\alpha $ for the new residual input and add $\\Delta \\alpha $ to $\\alpha $ for updating.",
"When the values in residual image are small enough, refinement ends with a better representation $\\alpha _{fine}$ .",
"Generic refinement method downsample the patch from HR patch with specific upscaling factor $s$ .",
"To utilize the scalable property of our function-based dictionary, we also put forward a new refinement algorithm called multi-scale refinement.",
"Scaling up an image with $s$ by interpolation-based algorithm can be viewed as a process of iteratively expanding an image by multi-steps.",
"During each step, images are enlarged slightly compared with the ones in the previous step.",
"As such, our proposed algorithm could perform this procedure due to the scalable property.",
"Here, it is necessry to make an assumption that during the iterative process of expanding an image, if representation for each patch is well-obtained, the reconstruction could be of high quality at each zooming step rather than perform well only in a certain scale but distort at other optional scaling factors.",
"This assumption can be viewed as a regularization that saves the ill-posed problem from massive solutions.",
"Therefore, we make the refinement for each scale to enforce the stability of its reconstruction results.",
"The difference of $k^{th}$ scale refinement with scaling factor $s_k$ hence becomes $p_L - D _{s_k} \\alpha $ .",
"Experiments demonstrate that multi-scale refinement improved the results and is less time consuming.",
"[H] {$p_L$ : low resolution patch } {$$ : dictionary} {$\\alpha _{fine}$ : representation} $\\alpha _{coarse} = \\mathop {\\arg } \\mathop {\\min }_{\\alpha } \\Vert p_L - _1 \\alpha \\Vert _2^2 + \\lambda _1 \\Vert \\alpha \\Vert _1 $ $k = 1$ $p_L^0 = p_L$ $\\alpha ^0 = \\alpha _{coarse}$ $\\bar{s}\\in scales$ $iter$ = 1: $iters_{\\bar{s}}$ $p_L^k = p_L^{k-1} - D _{\\bar{s}} \\alpha ^{k-1}$ $\\alpha ^{k} = \\mathop {\\arg } \\mathop {\\min }_{\\alpha } \\Vert p_L^k - _{1} \\alpha \\Vert _2^2 + \\lambda _2 \\Vert \\alpha \\Vert _1 $ $k = k + 1$ $\\alpha _{fine} = \\alpha _{coarse} + \\sum _{i=1}^k \\alpha ^i$ HFSR with multi-scale refinement for patch approximation"
],
[
"Image reconstruction from patches",
"The whole image can be reconstructed after calculating the representation for each patch in raster-scan order.",
"However the borders of adjacent patches are not compatible which may cause patchy result.",
"To deal with this issue, the adjacent patches are sampled with overlaps which would assign multiple values to each pixel.",
"General method is to average these pixel values for reconstruction.",
"Based on the aforementioned analysis, objective loss represents the reconstruction quality of the patch and can be involved in reconstruction process for more precise result.",
"Following this idea, we make weighted summation according to the reconstruction objective loss to recover the pixel value.",
"$HR(x,y) = \\frac{1}{\\sum _{j=1}^{|p|}loss^j} \\sum _i^{|p|} loss^i \\cdot p_H^i(x^i,y^i)$ $HR(x,y)$ represents the pixel in HR image with coordinate $(x,y)$ .",
"Suppose the pixel is covered by $|p|$ patches, and $p_H^i(x^i,y^i)$ is the corresponding value in $i^{th}$ constructed patch while $loss^i$ is the objective loss.",
"The strategy works better than generic average approach especially when the objective loss vary dramatically among different patches.",
"Fig.",
"REF demonstrates the concrete procedures of HFSR algorithm."
],
[
"Experimental Results",
"In this section, the proposed algorithm HFSR is evaluated with two classical interpolation methods and one of the state-of-the art methods proposed by glasner et al.",
"[12].",
"The hyperparameter and some implementation details are also analyzed and discussed here.",
"ScSR [20] is not compared here because it requires HR images for training.",
"AHF [4] approach is not sparse coding method and needs the inverse of matrix to solve ADMM problem which runs extremely slowly, the comparison is hence excluded either.",
"All experiments are implemented in Python 3.6 on Macbook Pro of 8Gb RAM and 2.3 GHz Intel Core i5.",
"The performance is measured by peak signal-to-noise ratio (PSNR), and upscaling factor $s$ is set to 2 for all experiments.",
"RGB input images are first converted into $YC_BC_R$ color space with three channels.",
"$Y$ channel represents luma component which is processed in most SR algorithms.",
"Therefore, we upscale it independently by the proposed model.",
"To evaluate the result, we only compared the $Y$ channel by PSNR.",
"The remaining two channels $C_B$ and $C_R$ are enlarged by bicubic interpolation so that $YC_BC_R$ image can be transformed to the original RGB image for visualization.",
"Grayscale input images can be directly applied by the proposed method and evaluation.",
"Table: The PSNR (dB) results obtained through the experiments, HFSR is the proposed methodwith conventional refinement, while HFSR(multi-scale) is the proposed method with multi-scale Refinement.The bold means the highest score among the non-learning method while the brown color represents the second best score.The priority is to firstly tune the parameters in dictionary.",
"For AHF, $\\theta $ is sampled from 16 angles, which are distributed evenly on $[0, 2\\pi ]$ .",
"$b$ is sampled from $[-6, 6]$ with 12 isometric intervals.",
"$\\xi $ is sampled from $[0.1, 10^{-4}]$ .",
"For Sin function , 6 angles are ranged form $[0, \\pi ]$ for $\\theta $ , $b$ is sampled from $[0, 1, 2, 3, 4, 5, 6]$ and $a$ is sampled from $[2.5, 2.25, 2]$ .",
"For DCT function, $a$ is average sampled from $[0, 5]$ with 1 distance.",
"$b$ are sampled in the same way as $a$ .",
"Then the dictionary is composed by parameter combinations and excludes those whose norm of its corresponding constructed patch less than $1.0$ .",
"The total size of the dictionary in our experiment is 334.",
"The size of patches sampled from the LR image is $6 \\times 6$ and the overlap of the patch is set as 4.",
"Smaller size of the patch like $5\\times 5$ may work better when the image details are more complex but it is more time consuming.",
"In original refinement, all of the 6 iterations are operated on the patch with fixed upscaling factor 2.",
"In multi-scale refinement, patches are enlarged from size $6\\times 6$ to $12\\times 12$ where the upscaling factors are sampled from $[\\frac{7}{6},\\frac{8}{6},\\frac{9}{6},\\frac{10}{6},\\frac{11}{6},\\frac{12}{6}]$ , the number of iterations $iters_{\\bar{s}}$ in each scale with corresponding upscaling factor, in order, are $[1,1,1,1,0,2]$ .",
"Large sparse regularization parameters $\\lambda $ will degrade the quality for $\\alpha $ to approximate $p_L$ but lead better upscaling results for what it represents.",
"Small $\\lambda $ usually gets less reconstruct error for $p_L$ , but it is easy to lose the fidelity when recovering the HR image.",
"Hence, it is important to select an optimal value $\\lambda $ so as to balance the two effects above.",
"Ultimately, the sparse regularization $\\lambda _1$ for coarse sparse representation and $\\lambda _2$ for refinement are both tuned as $10^{-4}$ .",
"Figure: Results of images including baboon, comic and zebra with upscaling factor 2 (from top to bottom)and comparison of HR(Ground truth), NN, Bicubic, Glasner and proposed HFSR (from left to right)The experimental results from Table REF on the dataset Set14 show that HFSR performs much better than conventional interpolation-based methods, and multi-scale refinement is verified to improve PSNR score.",
"Furthermore, in Fig.",
"REF , we select three pictures for visualization in which HFSR outperforms the glanser [12] and find that all of them contain rich textures.",
"This is probably because our dictionary well contains the corresponding features and therefore can recover the patch robustly."
],
[
"Conclusion",
"This paper proposed a framework of using function-based dictionary for sparse representation in SR task.",
"In our HFSR model, AHF, DCT function and sine function are combined for patch approximation to form a hybrid function dictionary.",
"The dictionary is scalable without additional training, and this property is utilized to design the multi-scale refinement to improve the proposed algorithm.",
"The experiment is performed on the Set14 benchmark, and results show that the HFSR algorithm performs well on a certain type of images which contains complex details and contexts.",
"For future improvements, the dictionary comprising more functions can be explored.",
"Many interesting topics are remained, including using the learning-based method to fine tune parameters of the dictionary or apply HFSR to other domain such as image compression or image denoising.",
"To encourage future works of proposed algorithm and discover effects in other applications, we public all the source code and materials on website: https://github.com/Eulring/Hybrid-Function-Sparse-Representation."
]
] | 1906.04363 |
[
[
"Disorder-Induced Electronic Nematicity"
],
[
"Abstract We expose the theoretical mechanisms underlying disorder-induced nematicity in systems exhibiting strong fluctuations or ordering in the nematic channel.",
"Our analysis consists of a symmetry-based Ginzburg-Landau approach and associated microscopic calculations.",
"We show that a single featureless point-like impurity induces nematicity locally, already above the critical nematic transition temperature.",
"The persistence of fourfold rotational symmetry constrains the resulting disorder-induced nematicity to be inhomogeneous and spatially average to zero.",
"Going beyond the single impurity case, we discuss the effects of finite disorder concentrations on the appearance of nematicity.",
"We identify the conditions that allow disorder to enhance the nematic transition temperature, and we provide a concrete example.",
"The presented theoretical results can explain a large series of recent experimental discoveries of disorder-induced nematic order in iron-based superconductors."
],
[
"Ginzburg-Landau Theory: Phenomenological Analysis",
"Eq.",
"(1) of the main text is obtained by demanding that the free energy density is a real functional transforming according to the trivial irreducible representation (IR) of the ensuing point group.",
"Here, we assume a system with tetragonal and inversion symmetries present, which is described by the D$_{\\rm 4h}$ point group symmetry.",
"The free energy density transforms according to the A$_{\\rm 1g}$ IR of D$_{\\rm 4h}$ and is here also assumed invariant under time reversal.",
"The equation describing the nematic field is found via the Euler-Lagrange equation of motion (EOM): $\\frac{\\partial {\\cal F}}{\\partial N}=\\partial _x\\frac{\\partial {\\cal F}}{\\partial (\\partial _xN)}+\\partial _y\\frac{\\partial {\\cal F}}{\\partial (\\partial _yN)}$ and reads: $&&\\big [\\alpha (T-T_{\\rm nem})-c\\mathbf {\\nabla }^2\\big ]N(\\mathbf {r})+\\beta [N(\\mathbf {r})]^3\\quad \\numero \\\\&&\\qquad \\qquad =-g\\left(\\partial _x^2-\\partial _y^2\\right)V(\\mathbf {r})\\,.\\phantom{\\dag }\\quad \\qquad $ From the above, one notes that if the potential $V(\\mathbf {r})$ is homogeneous, i.e.",
"$V(\\mathbf {r})=V$ , the EOM includes only derivatives of $N$ and no other spatially-dependent functions or source terms.",
"Thus, for an infinite (bulk) system $N(\\mathbf {r})=N$ .",
"When $T>T_{\\rm nem}$ , the appearance of nematicity is disfavored and, thus, $N=0$ in the bulk.",
"In contrast, the presence of an inhomogeneous potential functions as a source of nematicity and allows for non-zero inhomogeneous solutions of $N(\\mathbf {r})$ ."
],
[
"Case Study: Single Impurity for $\\mathbf {T>T_{\\rm nem}}$",
"Above $T_{\\rm nem}$ , we drop the cubic term in the EOM in Eq.",
"(REF ) above, and obtain Eq.",
"(2) of the manuscript.",
"For a potential satisfying $V(\\mathbf {q})=V(|\\mathbf {q}|)$ , we set $q_x=q\\cos \\theta $ , $q_y=q\\sin \\theta $ , $x=r\\cos \\phi $ and $y=r\\sin \\phi $ , and find: $N(r,\\phi )=\\cos (2\\phi )\\int _{+\\infty }^0\\frac{q{\\rm d}q}{2\\pi }\\phantom{.",
"}\\frac{g}{c}\\frac{q^2V(q)}{q^2+\\xi _{\\rm nem}^{-2}}J_2(qr)\\,,$ with $J_2(z)$ the respective Bessel function of the first kind.",
"One notes the distinctive angular dependence of the spatial profile of the induced nematic order, which is fixed by the B$_{\\rm 1g}$ IR of $N$ , the $A_{\\rm 1g}$ IR of $V$ , and the fourfold-symmetric impurity profile.",
"We resolve the radial dependence in the case $V(\\mathbf {r})=V/r$ , and find: $N(r,\\phi )=\\frac{\\gamma }{c}\\left[I_{2}\\left(\\frac{r}{\\xi _{\\rm nem}}\\right)-L_{-2}\\left(\\frac{r}{\\xi _{\\rm nem}}\\right)\\right]\\cos (2\\phi ),$ where we introduced the modified Bessel and Struve functions, and defined $\\gamma =-\\pi gV/(2\\xi _{\\rm nem})$ .",
"Notably, the decaying function in the brackets yields $\\approx 1/2$ for $r=\\xi _{\\rm nem}$ ."
],
[
"Non-Induction of Net Nematicity by a C$_4$ -Symmetric Potential",
"In this section we explore whether there exists a term in the Ginzburg-Landau expansion which can induce a nonzero $N(\\mathbf {q}=\\mathbf {0})$ for an impurity-potential profile which preserves C$_4$ symmetry.",
"Consider the most general term: $\\int {\\rm d}\\mathbf {r}\\phantom{.",
"}[N(\\mathbf {r})]^{n}[V(\\mathbf {r})]^m\\big (\\partial _x^2-\\partial _y^2\\big )^{\\ell }V(\\mathbf {r})$ where, if $\\ell $ is odd, then $n=\\ell +2\\mathbb {N}$ .",
"We fix the spatial profile of $V$ to be C$_4$ -symmetric.",
"The above general term can be mapped to two distinct types of couplings: $\\int {\\rm d}\\mathbf {r}\\phantom{.",
"}[N(\\mathbf {r})]^{2n} \\phantom{\\dag }{\\rm and}\\phantom{\\dag }\\int {\\rm d}\\mathbf {r}\\phantom{.",
"}[N(\\mathbf {r})]^{2n+1}\\big (\\partial _x^2-\\partial _y^2\\big )V(\\mathbf {r})\\,.$ The respective equations of motion read: $N(\\mathbf {r})\\propto [N(\\mathbf {r})]^{2n-1}\\phantom{.",
"}{\\rm and}\\phantom{.",
"}N(\\mathbf {r})\\propto [N(\\mathbf {r})]^{2n}\\big (\\partial _x^2-\\partial _y^2\\big )V(\\mathbf {r})\\,.$ We Fourier transform the first equation and find: $&&N(\\mathbf {q}=\\mathbf {0})\\propto \\numero \\\\&&\\int {\\rm d}\\mathbf {p}_1\\ldots {\\rm d}\\mathbf {p}_{2n-1} N(\\mathbf {p}_1)\\ldots N(\\mathbf {p}_{2n-1})\\delta \\left(\\sum _s^{2n-1}\\mathbf {p}_s\\right)\\,.\\qquad $ Assuming that the components appearing on the rhs are given by: $\\bar{N}(\\mathbf {q})=\\frac{g}{c}\\frac{\\big (q_x^2-q_y^2\\big )V(\\mathbf {q})}{\\mathbf {q}^2+\\xi _{\\rm nem}^{-2}}\\equiv \\cos (2\\theta )\\phantom{.",
"}\\frac{g}{c}\\frac{q^2V(q)}{q^2+\\xi _{\\rm nem}^{-2}}\\,,$ where we set $q_x=q\\cos \\theta $ and $q_y=q\\sin \\theta $ , we find that the angular part of the integral is proportional to: $&&\\int _0^{2\\pi }{\\rm d}\\theta _1\\ldots \\int _0^{2\\pi }{\\rm d}\\theta _{2n-2}\\cos (2\\theta _1)\\ldots \\cos (2\\theta _{2n-2})\\cdot \\qquad \\numero \\\\&&\\left[\\sum _{s=1}^{2n-2}p_s^2\\cos \\big (2\\theta _s\\big )+\\sum _{s\\ne \\ell }^{2n-2}p_sp_\\ell \\cos \\big (\\theta _s+\\theta _{\\ell }\\big )\\right]=0\\,,\\qquad $ where we set $\\cos \\theta _s=p_{s,x}/p_s$ and $\\sin \\theta _s=p_{s,y}/p_s$ , with $p_s=|\\mathbf {p}_s|$ .",
"A similar treatment for the remaining equation also yields zero.",
"This result naturally confirms, that, a C$_4$ -symmetric spatial profile for the impurity potential cannot lead to net nematicity."
],
[
"Case Study: Single Impurity for $\\mathbf {T<T_{\\rm nem}}$",
"In order to explain the elongated clover-like spatial profile induced by the impurity in the bulk nematic phase ($T< T_{\\rm nem}$ ), we need to include higher order terms in the free energy described by Eq.",
"(REF ) of the SM.",
"To demonstrate how this elongation comes about, it is sufficient to solely retain the first term of Eq.",
"(5) presented in the main text.",
"The modified EOM has the form: $&&\\quad \\big [\\alpha (T-T_{\\rm nem})-c\\mathbf {\\nabla }^2\\big ]N(\\mathbf {r})+\\beta [N(\\mathbf {r})]^3\\quad \\numero \\\\&&\\phantom{\\dag }=-g\\big (\\partial _x^2-\\partial _y^2\\big )V(\\mathbf {r})+g^{\\prime }N(\\mathbf {r})V(\\mathbf {r})\\,.\\qquad \\qquad $ We separate the nematic field into two parts, i.e.",
"$N(\\mathbf {r})=N_{\\rm B}+\\delta N(\\mathbf {r})$ .",
"Here, $N_{\\rm B}$ denotes the value of the bulk nematic order parameter and is given by $\\beta N_{\\rm B}^2=\\alpha (T_{\\rm nem}-T)$ for $T<T_{\\rm nem}$ , while $\\delta N(\\mathbf {r})$ denotes the contribution stemming from the presence of the impurity.",
"For $|\\delta N(\\mathbf {r})|\\ll |N_{\\rm B}|$ we linearize the above EOM and obtain: $&&\\big [2\\alpha (T_{\\rm nem}-T)/c-\\mathbf {\\nabla }^2\\big ]\\delta N(\\mathbf {r})\\qquad \\numero \\\\&&\\phantom{\\dag }=-\\frac{g}{c}\\left(\\partial _x^2-\\partial _y^2\\right)V(\\mathbf {r})+\\frac{g^{\\prime }}{c}N_{\\rm B}V(\\mathbf {r})\\,.$ In the above, we retained the terms which lead to a $\\delta N(\\mathbf {r})$ which is linear in terms of the strength of the impurity potential.",
"Within this assumption, we dropped the term $\\delta N(\\mathbf {r})V(\\mathbf {r})$ which leads to higher-order contributions with respect to the strength of the impurity potential.",
"In the same line of arguments as the ones leading to Eq.",
"(REF ), we obtain a constant angular profile superimposed on the usual $\\cos (2\\phi )$ -form: $\\delta N(r,\\phi )&=&\\cos (2\\phi )\\int _{+\\infty }^0\\frac{q{\\rm d}q}{2\\pi }\\phantom{.",
"}\\frac{g}{c}\\frac{q^2V(q)}{q^2+\\xi _{\\rm nem}^{-2}}J_2(qr)\\quad \\numero \\\\&&\\phantom{\\dag }-N_{\\rm B}\\int ^{0}_{+\\infty }\\frac{q{\\rm d}q}{2\\pi }\\phantom{.",
"}\\frac{g^{\\prime }}{c}\\frac{V(q)}{q^2 + \\xi _{\\rm nem}^{-2}}J_0(qr)\\,,\\quad $ with the coherence length being given now by $\\xi _{\\rm nem}^{-2}=2\\alpha (T_{\\rm nem}-T)$ due to the contribution of the quartic term of the free energy.",
"In connection to Eq.",
"(REF ) of the SM, we find that for $V(\\mathbf {r})=V/r$ : $\\delta N(r,\\phi )&=&\\frac{\\gamma }{c}\\left[I_{2}\\left(\\frac{r}{\\xi _{\\rm nem}}\\right)-L_{-2}\\left(\\frac{r}{\\xi _{\\rm nem}}\\right)\\right]\\cos (2\\phi )\\numero \\\\&-&\\frac{\\gamma ^{\\prime }}{c}\\left[I_{0}\\left(\\frac{r}{\\xi _{\\rm nem}}\\right)-L_{0}\\left(\\frac{r}{\\xi _{\\rm nem}}\\right)\\right]N_{\\rm B}\\numero \\\\&\\equiv &f(r)\\cos (2\\phi )+h(r)N_{\\rm B}$ with $\\gamma ^{\\prime }=-\\pi g^{\\prime }V\\xi _{\\rm nem}/2$ .",
"From the above, one can read off the decaying functions $f(r)$ and $h(r)$ discussed in the main text.",
"This spatial profile does indeed lead to a profile on the same form as the anisotropic induced order in Fig.",
"2(b) of the main text.",
"Furthermore, note that it is the presence of the nonzero bulk nematic order parameter $N_{\\rm B}$ , that induces the anisotropy."
],
[
"Interaction in the Nematic Channel and Mean-Field Theory Decoupling",
"We assume the presence of the interaction $\\widehat{\\cal H}_{\\rm int}=-V_{\\rm nem}\\sum _{\\mathbf {R}}\\widehat{{\\cal O}}_{\\mathbf {R}}^2/2\\,,$ which contributes to the desired nematic channel.",
"In the above, we have introduced: $\\widehat{{\\cal O}}_{\\mathbf {R}}=\\sum _{\\mathbf {\\delta }}^{\\pm \\hat{\\mathbf {x}},\\pm \\hat{\\mathbf {y}}}f_{\\mathbf {\\delta }}\\left(c^{\\dag }_{\\mathbf {R}+\\mathbf {\\delta }}c_{\\mathbf {R}}+c^{\\dag }_{\\mathbf {R}}c_{\\mathbf {R}+\\mathbf {\\delta }}\\right)\\,,$ where we have defined the form factor $f_{\\pm \\hat{\\mathbf {x}}}=-f_{\\pm \\hat{\\mathbf {y}}}=1/4$ .",
"Note that the lattice constant has been set to unity.",
"We perform a mean-field decoupling of the interaction in the direct channel by introducing the nematic order parameter $N_{\\mathbf {R}}=-V_{\\rm nem}\\big <\\widehat{{\\cal O}}_{\\mathbf {R}}\\big >$ .",
"The latter steps led to Eq.",
"(9) of the main text.",
"In wavevector space we have $N_{\\mathbf {q}}=\\sum _{\\mathbf {R}}N_{\\mathbf {R}}e^{-i\\mathbf {q}\\cdot \\mathbf {R}}$ and the complete mean-field Hamiltonian reads: $\\widehat{{\\cal H}}=\\frac{1}{\\cal N}\\sum _{\\mathbf {q},\\mathbf {k}}c^{\\dag }_{\\mathbf {k}+\\mathbf {q}/2}\\big (\\varepsilon _{\\mathbf {k}}{\\cal N}\\delta _{\\mathbf {q},\\mathbf {0}}+V_{\\mathbf {q}}+N_{\\mathbf {q}}f_{\\mathbf {q},\\mathbf {k}}\\big )c_{\\mathbf {k}-\\mathbf {q}/2}$ with ${\\cal N}$ being the number of lattice sites, while the nematic form factor in wavevector space takes the form: $f_{\\mathbf {q},\\mathbf {k}}=\\frac{f_{\\mathbf {k}+\\mathbf {q}/2}+f_{\\mathbf {k}-\\mathbf {q}/2}}{2}\\phantom{\\dag }{\\rm with}\\phantom{\\dag }f_{\\mathbf {k}}=\\cos k_x-\\cos k_y\\,.\\phantom{.",
"}$ The mean-field Hamiltonian has to be supplemented with the self-constistency equation for the nematic order parameter, which reads $N_{\\mathbf {q}}&=&-V_{\\rm nem}\\sum _{\\mathbf {k}}f_{\\mathbf {q},\\mathbf {k}}\\big <c^{\\dag }_{\\mathbf {k}-\\mathbf {q}/2}c_{\\mathbf {k}+\\mathbf {q}/2}\\big >\\numero \\\\&\\equiv &-V_{\\rm nem}T\\sum _{k_n,\\mathbf {k}}f_{\\mathbf {q},\\mathbf {k}}G_{\\mathbf {k}+\\mathbf {q}/2,k_n;\\mathbf {k}-\\mathbf {q}/2,k_n}$ where we introduced the full single-particle fermionic Matsubara Green function: $G_{\\mathbf {k}+\\mathbf {q}/2,k_n;\\mathbf {k}-\\mathbf {q}/2,k_n}=-\\big <c_{\\mathbf {k}+\\mathbf {q}/2,k_n}c^{\\dag }_{\\mathbf {k}-\\mathbf {q}/2,k_n}\\big >\\,.$ In the above, $k_n=(2n+1)\\pi T$ ($k_{\\rm B}=1$ ) and the Matsubara Green function for the free electrons has the form $G^0_{\\mathbf {k},k_n}=1/(ik_n-\\varepsilon _{\\mathbf {k}})$ .",
"The above construction allows us to employ Dyson's equation in order to perform an expansion of the rhs of the self-consistency equation with respect to the nematic order parameter and/or the impurity potential."
],
[
"Ginzburg-Landau Theory: Microscopic Analysis",
"Given the above, here we show how the electro-nematic coefficient $g$ relates to the microscopic parameters for the disorder-free microscopic model under consideration.",
"We employ a perturbative expansion by employing the Dyson equation for the full Matsubara Green function which reads: $&&G_{\\mathbf {k}+\\mathbf {q}/2,k_n;\\mathbf {k}-\\mathbf {q}/2,k_n}=G^0_{\\mathbf {k},k_n}\\delta _{\\mathbf {q},\\mathbf {0}}\\numero \\\\&+&G^0_{\\mathbf {k}+\\mathbf {q}/2,k_n}\\sum _{\\mathbf {p}}U_{\\mathbf {p};\\mathbf {k}+\\mathbf {q}/2}G_{\\mathbf {k}+\\mathbf {q}/2-\\mathbf {p},k_n;\\mathbf {k}-\\mathbf {q}/2,k_n}\\,,\\qquad $ where we introduced $U_{\\mathbf {q};\\mathbf {k}}=\\big (V_{\\mathbf {q}}+N_{\\mathbf {q}}f_{\\mathbf {q},\\mathbf {k}}\\big )/{\\cal N}$ .",
"We obtain the lowest order contribution of $U$ by replacing the full Green function on the rhs by the bare one.",
"We find: $g_{\\mathbf {q}}=-\\frac{T}{\\cal N}\\sum _{k_n,\\mathbf {k}}f_{\\mathbf {q},\\mathbf {k}}G^0_{\\mathbf {k}+\\mathbf {q}/2,k_n}G^0_{\\mathbf {k}-\\mathbf {q}/2,k_n}\\,.$ To facilitate the calculations, we consider the continuum limit of our model and assume spinless single-band electrons with a parabolic dispersion $\\varepsilon (\\mathbf {k})=E_F\\big [\\left(k/k_F\\right)^2-1\\big ]$ with $\\mathbf {k}=(k_x,k_y)$ , $k=|\\mathbf {k}|$ and set $f(\\mathbf {k})=\\big (k_x^2-k_y^2\\big )/k_F^2$ .",
"The quantity of interest, after taking into account the symmetries of $\\varepsilon (\\mathbf {k}),f(\\mathbf {k})$ and restricting up to second order terms in $\\mathbf {q}$ , reads: $&&g(\\mathbf {q})\\approx -\\int \\frac{{\\rm d}\\mathbf {k}}{(2\\pi )^2}\\bigg \\lbrace n_F^{\\prime }[\\varepsilon (k)]\\numero \\\\&&\\quad +\\big [f(\\mathbf {k})\\big ]^2\\frac{1}{3}E_F^2n_F^{\\prime \\prime \\prime }[\\varepsilon (k)]\\bigg \\rbrace f(\\mathbf {q}/2)\\equiv g\\big (q_x^2-q_y^2\\big )\\,.\\qquad $"
],
[
"Self-Consistent Calculation of the Nematic Order Parameter",
"By means of the microscopic Hamiltonian in Eq.",
"(10) of the main text, we calculate the nematic order parameter self-consistently until we reach an accuracy of $10^{-6}$ , while keeping the electron density fixed.",
"The expectation values entering in the order parameter and the electron density are calculated by expressing the fermionic field operators in the diagonal basis of the Hamiltonian $c_{\\mathbf {R}}=\\sum _{m}\\gamma _m\\langle m|\\mathbf {R}\\rangle $ with the defining equation $\\widehat{\\cal H}\\gamma ^{\\dagger }_m|0\\rangle =E_m|m\\rangle $ .",
"This leads to the following simplified expressions for the order parameter, and electron density, respectively: $N_{\\mathbf {R}}&=-V_{\\rm nem}\\sum _{\\mathbf {\\delta },\\,m}f_{\\mathbf {\\delta }}\\langle \\mathbf {R}+\\mathbf {\\delta }|m\\rangle n_{F}(E_m)\\langle m|\\mathbf {R}\\rangle +{\\rm c.c.",
"}\\,,\\nonumber \\\\\\langle n\\rangle &=\\frac{1}{\\mathcal {N}}\\sum _{m}n_{F}(E_m)\\,.$"
],
[
"Disorder-Modified Stoner Criterion and the Resulting $\\mathbf {T_{\\rm nem}^{\\rm imp}}$",
"In the presence of dilute and uncorrelated identical impurities, the disorder may enhance the $T_{\\rm nem}$ .",
"This was shown in the main text by investigating the modified nematic Stoner criterion.",
"In Fig.",
"1 of the SM, we provide additional results for other electron-density values.",
"The electron density is calculated via: $\\langle n\\rangle =\\frac{1}{\\mathcal {N}}\\sum _{\\mathbf {k}}\\int _{-\\infty }^{\\infty }\\frac{{\\rm d}\\varepsilon }{2\\pi }\\frac{1}{\\tau _{\\mathbf {k}}}\\frac{n_{F}(\\varepsilon )}{(\\varepsilon -\\varepsilon _{\\mathbf {k}})^2+1/(2\\tau _{\\mathbf {k}})^2},$ which recovers its usual form $\\langle n\\rangle =\\sum _{\\mathbf {k}}n_{F}(\\varepsilon _{\\mathbf {k}})/\\mathcal {N}$ in the disorder-free case, i.e.",
"$\\tau _{\\mathbf {k}}\\rightarrow \\infty $ .",
"For these calculations finite size effects are diminishing for $\\mathcal {N} \\sim 40\\times 10^{3}$ .",
"In Fig.",
"1 we demonstrate two typical situations, in which, $T_{\\rm nem}$ becomes either enhanced or reduced.",
"This is reflected in the behavior of the quantity $\\delta \\chi _{\\rm nem}/\\chi ^0_{\\rm nem}\\equiv (\\chi _{\\rm nem}^{\\rm imp}-\\chi ^0_{\\rm nem})/\\chi ^0_{\\rm nem}$ which is depicted.",
"We first focus on $n_{\\rm imp}$ in the vicinity of $5\\%$ , i.e.",
"the optimal value discussed in the main text.",
"For the value $\\langle n\\rangle =0.51$ of the electron density, the Fermi energy is tuned very near the van Hove singularity (see Figs.",
"1(a,b)), which constitutes the sweet spot for the development of the nematic order parameter in the absence of disorder, since there, $\\chi ^0_{\\rm nem}$ obtains its maximum value.",
"From Fig.",
"1(c) we find that introducing disorder worsens the tendency of the system to develop a nematic order parameter as reflected in the negative values of $\\delta \\chi _{\\rm nem}/\\chi ^0_{\\rm nem}$ .",
"The addition of disorder broadens the spectral function, and the density of states (DOS) unavoidably becomes lowered, since contributions from low DOS $\\mathbf {k}$ points are taken into account.",
"In contrast, in the case $\\langle n\\rangle =0.53$ discussed in the main text, and also shown here, the broadening allows the DOS to increase by picking up contributions from the van Hove singularity, while at the same time avoiding significant contributions from other low DOS $\\mathbf {k}$ points.",
"Increasing the electron density to $\\langle n\\rangle =0.55$ shifts the Fermi level further away from the van Hove singularity and thus reduces its favorable impact on the DOS.",
"As a result, the nematic susceptibility drops and $\\delta \\chi _{\\rm nem}/\\chi ^0_{\\rm nem}$ is negative.",
"The balance between the contributions to the DOS originating from the van Hove singularity and the low DOS $\\mathbf {k}$ points is controlled by the concentration of impurities.",
"Varying $n_{\\rm imp}$ leads to a modification of the relative strength of the two competing contributions and allows the sign changes of $\\delta \\chi _{\\rm nem}/\\chi ^0_{\\rm nem}$ which are shown in Fig.",
"1(c) for $\\langle n\\rangle =0.55$ .",
"Figure: (a) Energy dispersion along the Γ-X{\\rm \\Gamma }-{\\rm X} line, (b) Fermi line in (k x ,k y )(k_x,k_y) space and (c) δχ nem /χ nem 0 \\delta \\chi _{\\rm nem}/\\chi ^0_{\\rm nem} as a function of n imp n_{\\rm imp}, all shown for different electron fillings 〈n〉=0.51,0.53,0.55\\langle n\\rangle =0.51,0.53,0.55.",
"Panel (c) reveals that disorder always has a negative impact on the nematic susceptibility when the Fermi level is tuned very near the van Hove singularity, as inferred for 〈n〉=0.51\\langle n\\rangle =0.51.",
"When the Fermi level is tuned suffieciently away from the van Hove singularity, the resulting nematic susceptibility can be either enhanced or reduced depending on the relative strength of the contributions to the density of states (DOS) stemming from the van Hove singularity and the low DOS 𝐤\\mathbf {k} points.",
"This ratio is controlled by the concentration of impurities n imp n_{\\rm imp}."
]
] | 1906.04422 |
[
[
"Residual Entropy and Spin Fractionalizations in the Mixed-Spin Kitaev\n Model"
],
[
"Abstract We investigate ground-state and finite temperature properties of the mixed-spin $(s, S)$ Kitaev model.",
"When one of spins is half-integer and the other is integer, we introduce two kinds of local symmetries, which results in a macroscopic degeneracy in each energy level.",
"Applying the exact diagonalization to several clusters with $(s, S)=(1/2, 1)$, we confirm the presence of this large degeneracy in the ground states, in contrast to the conventional Kitaev models.",
"By means of the thermal pure quantum state technique, we calculate the specific heat, entropy, and spin-spin correlations in the system.",
"We find that in the mixed-spin Kitaev model with $(s, S)=(1/2, 1)$, at least, the double peak structure appears in the specific heat and the plateau in the entropy at intermediate temperatures, indicating the existence of the spin fractionalization.",
"Deducing the entropy in the mixed-spin system with $s, S\\le 2$ systematically, we clarify that the smaller spin-$s$ is responsible for the thermodynamic properties at higher temperatures."
],
[
" Residual Entropy and Spin Fractionalizations in the Mixed-Spin Kitaev Model Akihisa Koga Department of Physics, Tokyo Institute of Technology, Meguro, Tokyo 152- 8551, Japan Joji Nasu Department of Physics, Yokohama National University, Hodogaya, Yokohama 240-8501, Japan We investigate ground-state and finite temperature properties of the mixed-spin $(s, S)$ Kitaev model.",
"When one of spins is half-integer and the other is integer, we introduce two kinds of local symmetries, which results in a macroscopic degeneracy in each energy level.",
"Applying the exact diagonalization to several clusters with $(s, S)=(1/2, 1)$ , we confirm the presence of this large degeneracy in the ground states, in contrast to the conventional Kitaev models.",
"By means of the thermal pure quantum state technique, we calculate the specific heat, entropy, and spin-spin correlations in the system.",
"We find that in the mixed-spin Kitaev model with $(s, S)=(1/2, 1)$ , at least, the double peak structure appears in the specific heat and the plateau in the entropy at intermediate temperatures, indicating the existence of the spin fractionalization.",
"Deducing the entropy in the mixed-spin system with $s, S\\le 2$ systematically, we clarify that the smaller spin-$s$ is responsible for the thermodynamic properties at higher temperatures.",
"Kitaev model [1] and its related models have attracted much interest in condensed matter physics since the possibility of the direction-dependent Ising interactions has been proposed in the realistic materials [2].",
"Among them, low temperature properties in the candidate materials such as $A_2{\\rm IrO_3}$ ($A={\\rm Na, K}$ ) [3], [4], [5], [6], [7], [8], [9] and $\\alpha $ -${\\rm RuCl_3}$ [10], [11], [12], [13], [14] have been examined extensively.",
"To clarify the experimental results, the roles of the Heisenberg interactions [15], [16], [17], off-diagonal interactions [18], [19], interlayer coupling [20], [21], [22], and the spin-orbit couplings [23] have been theoretically investigated for both ground state and finite temperature properties.",
"One of the important issues characteristic of the Kitaev models is the fractionalization of the spin degree of freedom.",
"In the Kitaev model with $S=1/2$ spins, the spins are exactly shown to be fractionalized into itinerant Majorana fermions and localized fluxes, which manifest themselves in the ground state and thermodynamic properties [24], [25].",
"It has been observed as the half-quantized thermal quantum Hall effects, which is a clear evidence of the Majorana quasiparticles fractionalized from quantum spins [14].",
"Recently, the Kitaev model with larger spins has theoretically been examined [26], [27], [28], [29], [30].",
"In the spin-$S$ Kitaev model, the specific heat exhibits double peak structure, and plateau appears in the temperature dependence of the entropy [28].",
"This suggests the existence of the fractionalization even in this generalized Kitaev model.",
"However, it is still hard to explain how the spin degree of freedom is divided in the generalized Kitaev models beyond the exactly solvable $S=1/2$ case [1], [24], [25].",
"The key to understand the “fractionalization” in the spin-$S$ Kitaev model should be the multiple entropy release phenomenon.",
"The half of spin entropy $\\sim \\frac{1}{2}\\ln (2S+1)$ in higher temperatures emerges with a broad peak in the specific heat.",
"Then, a question arises how the plateau structure appears in the entropy in the Kitaev model composed of multiple kinds of spins (the mixed-spin Kitaev model).",
"In other words, how is the many-body state realized in the system, with decreasing temperatures?",
"The extension to the mixed-spin models should be a potential to exhibit an intriguing nature of the ground states.",
"In fact, the mixed-spin quantum Heisenberg model has been examined [31], [32], [33], [34], [35], [36], [37], [38], and the topological nature of spins and lattice plays an important role in stabilizing the non-magnetic ground states.",
"Moreover, mixed-spin Kitaev model can be realized by replacing transition metal ions to other ions in the Kitaev candidate materials.",
"Therefore, it is desired to study this model to discuss the nature of the spin fractionalization in the Kitaev system.",
"Figure: (a) Mixed-spin Kitaev model on a honeycomb lattice.Solid (open) circles represent spin ss (SS).Red, blue, and green lines denote xx, yy, and zz bondsbetween nearest neighbor sites, respectively.",
"(b) Plaquette with sites marked 1-61-6 is shownfor the corresponding operatorW p W_p defined in Eq.",
"().In this manuscript, we investigate the mixed-spin Kitaev model, where two distinct spins $(s, S)\\;[s<S]$ are periodically arranged on the honeycomb lattice (see Fig.",
"REF ).",
"First, we show the existence of the $Z_2$ symmetry in each plaquette in the system.",
"In addition, by considering another local symmetry, we show that the macroscopic degeneracy exists in each energy level when one of the spins is half-integer and the other integer.",
"The exact diagonalization (ED) in the system with $(s,S)=(\\frac{1}{2},1)$ reveals that the ground state has a macroscopic degeneracy, which is consistent with the presence of the two kinds of local symmetries.",
"Using thermal pure quantum (TPQ) state methods [39], [40], we find that, at least, the double peak structure appears in the specific heat and the plateau appears at intermediate temperatures in the entropy, which are similar to those in the spin-$S$ Kitaev models [28].",
"From systematic calculations for the mixed-spin systems with $s, S\\le 2$ , we clarify that the smaller spin-$s$ is responsible for the high-temperature properties.",
"The deconfinement picture to explain the “spin fractionalization” in the Kitaev model is addressed.",
"We consider the Kitaev model on a honeycomb lattice, which is given by the following Hamiltonian as ${\\cal H} &=&-J\\sum _{\\langle i,j \\rangle _x}s_i^x S_j^x-J\\sum _{\\langle i,j \\rangle _y}s_i^y S_j^y-J\\sum _{\\langle i,j \\rangle _z}s_i^z S_j^z,$ where $s_i^\\alpha (S_i^\\alpha )$ is the $\\alpha (=x,y,z)$ component of a spin-$s(S)$ operator at the $i$ th site.",
"$J$ is the exchange constant between the nearest neighbor spin pairs $\\langle i,j \\rangle _\\gamma $ .",
"The model is schematically shown in Fig.",
"REF (a).",
"We consider here the following local Hermite operator defined on each plaquette $p$ as, $W_p &=& \\exp \\Big [i\\pi \\left(S_1^x+s_2^y+S_3^z+s_4^x+S_5^y+s_6^z\\right)-i\\pi \\eta \\Big ],$ where $\\eta =[3(s+S)]$ is a phase factor.",
"By using the following relation for the spin operators, $ e^{i\\pi S^\\alpha }S^\\beta e^{-i\\pi S^\\alpha }=(2\\delta _{\\alpha \\beta }-1)S^\\beta ,$ we find $[{\\cal H}, W_p]=0$ for each plaquette and $W_p^2=1$ .",
"Therefore, the mixed-spin Kitaev system has a $Z_2$ local symmetry.",
"It is known that this local $Z_2$ symmetry is important to understand ground state properties in the Kitaev model.",
"We wish to note that the local operator $W_p$ on a plaquette $p$ commutes with those on all other plaquettes in the spin-$S$ Kitaev models, while this commutation relation is not always satisfied in the present mixed-spin Kitaev model.",
"In fact, we obtain $[W_p, W_q]\\propto [e^{i\\pi (s_i^x+S_j^y)},e^{i\\pi (s_i^y+S_j^x)}]\\propto \\sin [\\pi (s_i^y-S_j^x)]$ when the plaquettes $p$ and $q$ share the same $z$ bond $\\langle ij\\rangle _z$ .",
"This means that the local operator does not commute with the adjacent ones in the mixed-spin Kitaev model with one of spins being half-integer and the other integer.",
"Instead, we introduce another local symmetry specific in this case.",
"When either $s$ or $S$ is half-integer and the other is integer, the Hilbert space is divided into subspaces specified by the set of the eigenvalues $w_p(=\\pm 1)$ of the $N_p(\\le N/6)$ local operators $W_p$ defined on the plaquettes $p\\in {\\cal P}$ , where ${\\cal P}$ is a set of the plaquettes whose corners are not shared with each other.",
"Now, we assume the presence of the local operator $R_p$ on a plaquette $p(\\in {\\cal P})$ so as to satisfy the conditions, $R_p^2=1$ and the following commutation relations $[{\\cal H}, R_p]=0$ , $[W_p, R_q]=0\\; (p\\ne q)$ , and $\\lbrace W_p, R_p\\rbrace =0$ .",
"In the case that half-integer and integer spins are mixed, such an operator can be introduced so that the spins located on its corners are inverted as ${\\bf S}_{2i-1}\\rightarrow -{\\bf S}_{2i-1},{\\bf s}_{2i}\\rightarrow -{\\bf s}_{2i}\\;(i=1,2,3)$ and the signs of the six exchange constants are changed on the bonds connecting with a corner site belonging to the plaquette, shown as the dashed lines in Fig.",
"REF (b).",
"When a wavefunction for the energy level $E$ is given by the set of $\\lbrace w_p\\rbrace $ as $|\\psi \\rangle =|\\psi ;\\lbrace w_1, w_2,\\cdots ,w_p,\\cdots \\rbrace \\rangle $ , we obtain ${\\cal H}|\\psi ^{\\prime }\\rangle =E|\\psi ^{\\prime }\\rangle $ with the wave function $|\\psi ^{\\prime }\\rangle =R_p|\\psi \\rangle =|\\psi ;\\lbrace w_1, w_2,\\cdots ,-w_p,\\cdots \\rbrace \\rangle $ .",
"Since the operators $R_p$ for arbitrary plaquettes in ${\\cal P}$ generates degenerate states, the presence of $R_p$ results in, at least, $2^{N/6}$ -fold degenerate ground states.",
"This qualitative difference in the spin magnitudes $s$ and $S$ can be confirmed in the small clusters.",
"By using the ED method, we obtain ground state properties in the twelve-site systems, as shown in Table REF .",
"We clearly find that, as for the ground-state degeneracy, the mixed-spin systems can be divided into three groups.",
"When both spins $s$ and $S$ are integer, the ground state is always singlet.",
"In the half-integer case, the four-fold degenerate ground state is realized in the $N=12a$ system, while the singlet ground state is realized in the $N=12b$ system.",
"This feature is essentially the same as ground state properties in the $S=1/2$ Kitaev model, where the ground-state degeneracy depends on the topology in the boundary condition.",
"By contrast, the eight-fold degenerate state is realized in the system with one of spins being half-integer and the other integer, which suggests the macroscopic degeneracy in the thermodynamic limit.",
"Table: Ground state energy E g E_g and its degeneracy N d N_din the mixed-spin (s,S)(s, S) Kitaev models with the twelve-site clusters.To confirm this, we focus on the mixed-spin system with $(s, S)=(1/2, 1)$ .",
"By using the ED method, we obtain the ground-state energies for several clusters up to 24 sites [see Fig.",
"REF (a)].",
"The obtained results are shown in Table REF .",
"It is clarified that a finite size effect slightly appears in the ground state energy, and its value is deduced as $E_g/JN=-0.335$ .",
"We also find that the ground state is $N_{\\cal S}(=N_d/2^{N_p})$ -fold degenerate in each subspace and its energy is identical in all subspaces ${\\cal S}[\\lbrace w_p\\rbrace ]$ except for the $N=18a$ system [41].",
"The large ground-state degeneracy $N_d\\ge 2^{N/6}$ is consistent with the above conclusion.",
"We also find that the first excitation energy $\\Delta $ is much smaller than the exchange constant $J$ , as shown in Table REF .",
"These imply the existence of multiple low-energy states in the system.",
"Table: Ground state profile for several clusters in the Kitaev modelwith (s,S)=(1/2,1)(s, S)=(1/2, 1).",
"N p N_p is the number of plaquettes,where the local operator W p W_p is diagonal in the basis set.N d N_d is the degeneracy in the ground state.Next, we consider thermodynamic properties in the Kitaev model.",
"It is known that there exist two energy scales in the $S=1/2$ Kitaev model [1], which clearly appear as double peak structure in the specific heat and a plateau in the entropy [24], [25].",
"Similar behavior has been reported in the spin-$S$ Kitaev model [28].",
"These suggest the existence of the fractionalization in the generalized spin-$S$ Kitaev model.",
"An important point is that the degrees of freedom for the high energy part depend on the magnitude of spins $\\sim (2S+1)^{N/2}$ .",
"On the other hand, in the mixed-spin case, it is unclear which spin is responsible for the high-temperature properties.",
"Here, we calculate thermodynamic quantities for twelve-site clusters, by diagonalizing the corresponding Hamiltonian.",
"Furthermore, we apply the TPQ state method [39], [40] to larger clusters.",
"In this calculation, the thermodynamic quantities are deduced by the statistical average of the results obtained from, at least, 25 independent TPQ states.",
"Here, we calculate specific heat $C(T)=dE(T)/dT$ , entropy $S(T)=S_\\infty -\\int _T^\\infty C(T^{\\prime })/T^{\\prime } dT^{\\prime }$ , and the nearest-neighbor spin-spin correlation $C_S(T)=\\langle s_i^\\alpha S_j^\\alpha \\rangle _\\alpha = -2E(T)/(3J)$ , where $S_\\infty = \\frac{1}{2} \\ln (2s+1)(2S+1)$ and $E(T)$ is the internal energy per site.",
"Figure: (a) Specific heat, (b) entropy, and (c) spin-spin correlationas a function of temperatures.Shaded areas stand for the standard deviation of the resultsobtained from the TPQ states.The results for the mixed-spin systems with $(s, S)=(1/2, 1)$ are shown in Fig.",
"REF .",
"We clearly find the multiple-peak structure in the specific heat.",
"Note that finite size effects appear only at low temperatures.",
"Therefore, our TPQ results for the 24 sites appropriately capture the high temperature properties ($T\\gtrsim 0.01J$ ) in the thermodynamic limit.",
"Then, we find the broad peak around $T_H\\sim 0.6J$ , which is clearly separated by the structure at low temperatures ($T<0.01J$ ).",
"Now, we focus on the corresponding entropy, which is shown in Fig.",
"REF (b).",
"This indicates that, decreasing temperature, the entropy monotonically decreases and the plateau structure is found around $T/J\\sim 0.1$ .",
"The released entropy is $\\sim \\frac{1}{2}\\ln 2$ , which is related to the smaller spin $(s=1/2)$ .",
"Therefore, multiple temperature scales do not appear at high temperatures although the system is composed of two kinds of spins $(s, S)$ .",
"However, it does not imply that only smaller spins are frozen and larger spins remain paramagnetic at the temperature since the spin-spin correlations develop around $T\\sim T_H$ and a quantum many-body spin state is formed, as shown in Fig.",
"REF (c).",
"We have also confirmed that local magnetic moments do not appear even in the wavefunction constructed by the superposition of the ground states with different configurations of $\\lbrace w_p\\rbrace $ .",
"By contrast, the value $\\frac{1}{2}\\ln 2$ reminds us of the high-temperature feature for itinerant Majorana fermions in spin-1/2 Kitaev model [24], [25].",
"Then, one expects that, in the mixed-spin $(s, S)$ Kitaev model, higher temperature properties are described by the smaller spin-$s$ Kitaev model, where degrees of freedom $\\sim (2s+1)^{1/2}$ are frozen at each site [28].",
"In the case, a peak structure appears in the specific heat and the plateau structure at $\\sim S_\\infty - \\ln (2s+1)/2$ in the entropy.",
"These interesting properties at higher temperatures will be examined systematically.",
"Further decrease of temperatures decreases the entropy and finally $S \\sim S_\\infty -\\ln 2$ at lower temperatures, as shown in Fig.",
"REF (b).",
"This may suggest that thermodynamic properties in this mixed-spin Kitaev model with $(s, S)=(1/2, 1)$ are governed by two kinds of fractional quasiparticles originating from the smaller $s=1/2$ spin by analogy with the spin fractionalization in the spin-1/2 Kitaev model.",
"In the case, the existence of the remaining entropy $S \\sim S_\\infty -\\ln 2$ should be consistent with macroscopic degeneracy in the ground state as discussed before.",
"However, our TPQ data have a large system size dependence at low temperatures, and conclusive results could not be obtained.",
"Therefore, a systematic analysis is desired to clarify the nature of low temperature properties.",
"To clarify the role of the smaller spins in the mixed-spin Kitaev models, we calculate the entropy in the systems with $s, S\\le 2$ and $N=12a$ by means of the TPQ state methods.",
"The results are shown in Fig.",
"REF .",
"Figure: (a) S-S ∞ S-S_\\infty in the generalized (s,S)(s, S) Kitaev modelat higher temperatures.Squares with lines represent data for the S=1/2S=1/2 Kitaev modelobtained from the Monte Carlo simulations , .Circles, triangles, and diamonds with lines representthe TPQ data for S=1,3/2S=1, 3/2, and 2 cases .",
"(b) (S-S ∞ )/ln(2s+1)(S-S_\\infty )/\\ln (2s+1) and C s /(sS)C_s/(sS) as a function of T/JST/JS.The plateau structure is clearly observed in the curve of the entropy in the mixed-spin Kitaev models.",
"In addition, we find that the plateau is located around $S=S_\\infty -\\frac{1}{2}\\ln (2s+1)$ , as expected above.",
"Therefore, we can say that, decreasing temperatures, the half of the degree of freedom in the smaller spin-$s$ are released.",
"This may be explained by the deconfined-spin picture in the Kitaev model.",
"In the picture, each spin $S$ is divided into two kinds of quasiparticles with distinct energy scales: $2S$ $L$ -quasiparticles and $2S$ $H$ -quasiparticles, which are dominant at lower and higher temperatures, respectively.",
"In the exactly solvable $S=1/2$ Kitaev model, $H$ - ($L$ -)quasiparticles are identical to itinerant Majorana fermions (localized fluxes).",
"In addition, this should explain the double peak structure in the specific heat of the spin-$S$ Kitaev model, each of which corresponds to the half entropy release [28].",
"In our mixed-spin $(s,S)$ system, the entropy release at higher temperatures can be interpreted as follows: $2s$ fractional $H$ -quasiparticles are present with the energy scale of $\\sim J$ .",
"On the other hand, remaining $H$ -quasiparticles originating from larger spin-$S$ posses the energy that is much smaller than $J$ due to the absense of the two-dimensional network.",
"Therefore, only $2s$ $H$ -quasiparticles form the many-body state at high temperatures, resulting in the plateau structure in the entropy.",
"Interestingly, the temperature $T^*$ characteristic of the plateau in the entropy, which may be defined such that $S(T^*)=S_\\infty -\\frac{1}{2}\\ln (2s+1)$ , depends on the magnitude of the larger spin.",
"In fact, we find that $T^*$ should be scaled by the larger spin $T^*\\sim JS$ , which is shown in Fig.",
"REF (b).",
"This is in contrast to the conventional temperature scale $T^{**}\\sim J\\sqrt{s(s+1)S(S+1)}$ , which is derived from the high-temperature expansion.",
"This discrepancy is common to the spin-$S$ Kitaev model [28], implying that quantum fluctuations are essential even in this temperature range in the mixed-spin Kitaev models.",
"As for the spin-spin correlation, decreasing temperatures, it develops around $T/JS\\sim 1$ and is almost saturated around $T^*$ , as shown in Fig.",
"REF (b).",
"This means that the many-body spin state is indeed realized at the temperature.",
"We also find that at low temperatures, the normalized spin-spin correlation $C_s/(sS)\\sim 0.4$ is less than unity when $s$ and $S$ are large.",
"This suggests that the quantum spin liquid state is, in general, realized in the generalized mixed-spin Kitaev model, which is consistent with the presence of magnetic fluctuations even in the classical limit [27].",
"In summary, we have studied the mixed-spin Kitaev model.",
"First, we have clarified the existence of the local $Z_2$ symmetry at each plaquette.",
"We could introduce an operator $R_p$ on the plaquette $p$ so as to (anti)commute with the Hamiltonian ($W_p$ ), which leads to the macroscopic degeneracy for each energy level in the mixed-spin system with one of spins being half-integer and the other integer.",
"Using the TPQ state methods for several clusters, we have found the double peak structure in the specific heat and plateau in the entropy, which suggests the existence of the fractionalization in the mixed-spin system.",
"Deducing the entropy in the mixed-spin system with $s, S\\le 2$ systematically, we have clarified that the smaller spin plays a crucial role in the thermodynamic properties at higher temperatures.",
"We expect that the present mixed-spin Kitaev systems are realizable in the real materials by substituting the magnetic ions in the Kitaev candidate materials to other magnetic ions with larger spins, and therefore, the present work should stimulate material researches for mixed-spin Kitaev systems.",
"Parts of the numerical calculations were performed in the supercomputing systems in ISSP, the University of Tokyo.",
"This work was supported by Grant-in-Aid for Scientific Research from JSPS, KAKENHI Grant Nos.",
"JP18K04678, JP17K05536 (A.K.",
"), JP16K17747, JP16H02206, JP18H04223 (J.N.",
")."
]
] | 1906.04335 |
[
[
"Localized Fourier Analysis for Graph Signal Processing"
],
[
"Abstract We propose a new point of view in the study of Fourier analysis on graphs, taking advantage of localization in the Fourier domain.",
"For a signal $f$ on vertices of a weighted graph $\\mathcal{G}$ with Laplacian matrix $\\mathcal{L}$, standard Fourier analysis of $f$ relies on the study of functions $g(\\mathcal{L})f$ for some filters $g$ on $I_\\mathcal{L}$, the smallest interval containing the Laplacian spectrum ${\\mathrm sp}(\\mathcal{L}) \\subset I_\\mathcal{L}$.",
"We show that for carefully chosen partitions $I_\\mathcal{L} = \\sqcup_{1\\leq k\\leq K} I_k$ ($I_k \\subset I_\\mathcal{L}$), there are many advantages in understanding the collection $(g(\\mathcal{L}_{I_k})f)_{1\\leq k\\leq K}$ instead of $g(\\mathcal{L})f$ directly, where $\\mathcal{L}_I$ is the projected matrix $P_I(\\mathcal{L})\\mathcal{L}$.",
"First, the partition provides a convenient modelling for the study of theoretical properties of Fourier analysis and allows for new results in graph signal analysis (\\emph{e.g.}",
"noise level estimation, Fourier support approximation).",
"We extend the study of spectral graph wavelets to wavelets localized in the Fourier domain, called LocLets, and we show that well-known frames can be written in terms of LocLets.",
"From a practical perspective, we highlight the interest of the proposed localized Fourier analysis through many experiments that show significant improvements in two different tasks on large graphs, noise level estimation and signal denoising.",
"Moreover, efficient strategies permit to compute sequence $(g(\\mathcal{L}_{I_k})f)_{1\\leq k\\leq K}$ with the same time complexity as for the computation of $g(\\mathcal{L})f$."
],
[
"Introduction",
"Graphs provide a generic representation for modelling and processing data that reside on complex domains such as transportation or social networks.",
"Numerous works combining both concepts from algebraic and spectral graphs with those from harmonic analysis (see for example [5], [6], [2] and references therein) have allowed to generalize fundamental notions from signal processing to the context of graphs thus giving rise to Graph Signal Processing (GSP).",
"For an introduction to this emerging field and a review of recent developments and results see [26] and [21].",
"In general, two types of problems can be distinguished according to whether the underlying graph is known or unknown.",
"The first case corresponds to the setup of a sampled signal at certain irregularly spaced points (intersections of a transportation network, nodes in a computer network, ...).",
"In the second case, a graph is constructed from the data itself, it is generally interpreted as a noisy realization of one or several distributions supported by a submanifold of the Euclidean space.",
"In this latter context, the theoretical submanifold is somehow approximated using standard methods such as $k$ -NN, $\\varepsilon $ -graph and their Gaussian weighted versions.",
"In any of these cases, the framework is actually similar: it consists of a graph (given by the application or by the data) and signals are real-valued functions defined on the vertices of the graph.",
"Notions of graph Fourier analysis for signals on graphs were introduced and studied over the past several years [26], [23], [27], [24].",
"The graph Fourier basis is given by the eigenbasis $(\\chi _\\ell )_\\ell $ of the Laplacian matrix $\\mathcal {L}$ .",
"The Graph Fourier Transform (GFT) consists in representing a signal $f$ in the Fourier basis $(\\langle f, \\chi _\\ell \\rangle )_\\ell $ , and by analogy with the standard case the eigenvalues of $\\mathcal {L}$ play the role of frequencies.",
"From this definition, it follows that many filtering techniques are written in terms of vectors $g(\\mathcal {L})f$ , for some filter functions $g$ which act on the spectrum of $\\mathcal {L}$ (scaling, selecting, ...).",
"Fourier analysis on graphs has been successfully applied to many different fields such as stationary signals on graphs [22], graph signal energy study [14], convolutional neural networks on graphs [10].",
"Graph wavelets are an important application of graph Fourier analysis, and several definitions of graph wavelets were proposed [7], [6], [13], [18], [31], [15].",
"When performing Fourier analysis of a signal, there is no guarantee that localization of a signal in the frequency domain (a.k.a Fourier domain) implies localization in the graph domain.",
"This phenomenon is illustrated by the fact that the eigenvectors corresponding to the upper part of Laplacian spectrum tend to be more oscillating than those from the bottom of the spectrum (see for example [32] for an illustration).",
"To overcome this problem, [16] developed a fairly general construction of a frame enjoying the usual properties of standard wavelets: each vector of the frame is defined as a function $g(s\\mathcal {L})\\delta _m$ (where $\\delta _m$ is a signal with zero values at every vertex except $m$ ) and is localized both in the graph domain and the spectral domain at fine scale $s$ .",
"The transform associated with this frame is named Spectral Graph Wavelet Transform (SGWT), and it was used in numerous subsequent works [30], [1], [15].",
"Signals which are sparse in the Fourier domain form an important class of graph signals.",
"Indeed, there is a tight relationship between sparsity in the Fourier domain and the notion of regularity of a signal $f$ on the vertices of a graph $\\mathcal {G}$ which comes from the Laplacian matrix $\\mathcal {L}$ of $\\mathcal {G}$ .",
"Intuitively, a smooth signal will not vary much between two vertices that are close in the graph.",
"This regularity property can be read in the Fourier domain: a very smooth signal will be correctly represented in the Fourier domain with a small number of eigenvectors associated with the lower spectral values; on the contrary, non-smooth signals (i.e.",
"highly oscillating) are represented with eigenvectors corresponding to the upper part of the spectrum.",
"Both the types of signal are said frequency sparse.",
"In this paper, we propose to exploit localization in the Fourier domain to improve graph Fourier analysis.",
"More precisely, we consider vectors of the form $g(\\mathcal {L}_{I_k})f$ instead of vectors $g(\\mathcal {L})f$ in graph Fourier analysis, where $\\mathcal {L}_{I_k}$ is defined as the matrix $\\mathcal {L}P_{I_k}(\\mathcal {L})$ and $P_{I_k}(\\mathcal {L})$ denotes the projection onto the eigenspaces whose eigenvalue is contained in subset $I_k$ .",
"Localized Fourier analysis is motivated by problems and properties defined on strict subsets of the spectrum ${\\rm sp}(\\mathcal {L})$ (e.g.",
"any problem defined in terms of frequency sparse graph signals).",
"As a central application fo Fourier localization, we introduce the Fourier localized counterpart of SGWT, that we call LocLets for Localized graph wavelets.",
"We prove that various frame constructions can be written in terms of LocLets, hence benefiting from all the advantages of localization discussed in this paper.",
"Defining $I_\\mathcal {L}$ as the smallest interval containing the entire spectrum ${\\rm sp}(\\mathcal {L})$ , the local Fourier analysis consists in choosing a suitable partition $I_\\mathcal {L}= \\sqcup _k I_k$ into subintervals on which standard Fourier analysis is performed.",
"Such an analysis on disjoint intervals naturally benefits from several interesting properties.",
"In particular, when $f$ is modeled by a Gaussian random vector with independent entries, the disjointness of subintervals preserves these properties in the sense that random variables $(g(\\mathcal {L}_{I_k})f)_k$ are still Gaussian and independent.",
"This simple observation has important consequences to study the graph problem at stake.",
"In this work, it allows us to propose some noise level estimator from the random variables sequence $(g(\\mathcal {L}_{I_k})f)_k$ , and to provide a theoretical analysis of the denoising problem.",
"Disjointness of subsets $(I_k)_k$ also provides simple strategies to parallelize Fourier analysis computations.",
"We also consider the general problem given by a noisy signal on a graph $\\widetilde{f} = f + \\xi $ , where $\\xi $ is some random Gaussian vector with noise level $\\sigma $ .",
"We provide results for two important tasks: the estimation of $\\sigma $ when the latter is unknown, and the denoising of noisy signal $\\widetilde{f}$ in order to recover signal $f$ .",
"We show that for frequency sparse signals, localization allows to adapt to the unknown Fourier support of signal $f$ .",
"Theoretical guarantees and practical experiments show that localized Fourier analysis can improve state-of-the-art denoising techniques, not only in precision of the estimator $\\widehat{f}$ of $f$ , but also in time computations.",
"We provide an efficient method to choose a partition $I_\\mathcal {L}= \\sqcup _k I_k$ for the Fourier localized vectors $g(\\mathcal {L}_{I_k})f$ to be sufficiently informative.",
"Using well-known techniques for efficient graph Fourier analysis (a.k.a Chebyshev filters approximations), we propose scalable methods to perform localized Fourier analysis with no computational overhead over standard fast Fourier graph analysis.",
"The paper is structured as follows.",
"Section presents the relevant notions and techniques necessary to perform localized Fourier analysis.",
"In Section REF , we introduce LocLets, the Fourier localized extension of SGWT.",
"Section is devoted to the study of the denoising problem for signals on graphs.",
"The section provides results about noise level estimation, and signal denoising.",
"Additional properties of LocLets, such as computational aspects and relationships with known wavelet transforms, are further developed in Section .",
"In Section , we analyze the experiments made to support the interesting properties of localized Fourier analysis highlighted in this paper.",
"Finally, the proofs are gathered in Section ."
],
[
"Localized Fourier analysis for graph signals",
"In this section, we introduce the central notion studied in this paper: localization of graph Fourier analysis.",
"First, we recall the relevant notions of graph Fourier analysis.",
"Then we introduce LocLets, an important application of Fourier localization to SGWT."
],
[
"Functional calculus and Fourier analysis for graph signals",
"Let $\\mathcal {G}=(\\mathcal {V},\\mathcal {E})$ be an undirected weighted graph with $\\mathcal {V}$ the set of vertices, $\\mathcal {E}$ the set of edges, $n=|\\mathcal {V}|$ the number of nodes (the size of $\\mathcal {G}$ ), and $(W_{ij})_{i,j\\le n}$ the weights on edges.",
"Let us introduce the diagonal degree matrix whose diagonal coefficients are given by $D_{ii}=\\sum _{1 \\le j \\le n} W_{ij}$ for $1 \\le i \\le n$ .",
"The resulting non-normalized Laplacian matrix $\\mathcal {L}$ of graph $\\mathcal {G}$ is defined as $\\mathcal {L}= D-W$ .",
"The $n$ non-negative eigenvalues of $\\mathcal {L}$ , counted without multiplicity, are denoted by $\\lambda _1, \\ldots , \\lambda _n$ in the decreasing order.",
"In the sequel, $\\mathrm {sp}(\\mathcal {L})$ stands for the spectrum of $\\mathcal {L}$ .",
"The corresponding eigenvectors are denoted $\\chi _1, \\ldots , \\chi _n$ .",
"Given a graph $\\mathcal {G}$ , the GFT of a real-valued function $f$ defined on the vertices of $\\mathcal {G}$ is nothing but the representation of $f$ in the orthonormal basis of eigenvectors of $\\mathcal {L}$ .",
"Namely, for a signal $f:\\mathcal {G}\\rightarrow \\mathbb {R}$ , the $\\ell $ -th Fourier coefficient of $f$ , denoted $\\widehat{f}(\\ell )$ , is given by $\\widehat{f} (\\ell ) = \\langle f, \\chi _{\\ell } \\rangle $ .",
"The Fourier support ${\\rm supp}(\\widehat{f})$ of signal $f$ is the set of indices $\\ell $ such that $\\widehat{f}(\\ell ) \\ne 0$ .",
"We will see in Section REF that graph wavelets can be defined in a similar manner.",
"Functional calculus is a powerful technique to study matrices which is the heart of GSP.",
"For a function $g$ defined on some domain $D_g$ , $\\mathrm {sp}(\\mathcal {L})\\subset D_g$ , functional calculus reads as $g(\\mathcal {L}) = \\sum _{1 \\le \\ell \\le n} g(\\lambda _{\\ell }) \\langle \\chi _{\\ell }, \\cdot \\rangle \\chi _{\\ell }.$ Interpreting the eigenvalues $\\lambda _\\ell $ , $\\ell =1, \\ldots , n$ , as the fundamental frequencies associated with a graph, the linear map $g(\\mathcal {L})$ is generally seen as a filter operator in terms of signal analysis.",
"Also, spectral projections of matrix $\\mathcal {L}$ can be explicited with the help of functional calculus setting $g=\\mathrm {1}_I$ .",
"More precisely, for any subset $I\\subset I_\\mathcal {L}$ , consider the map $P_{I}(\\mathcal {L})$ given by: $P_{I}(\\mathcal {L}) = \\sum _{1 \\le \\ell \\le n} \\mathrm {1}_I(\\lambda _\\ell ) \\langle \\chi _{\\ell }, \\cdot \\rangle \\chi _{l} = \\sum _{\\ell :~\\lambda _{\\ell } \\in I} \\langle \\chi _{\\ell }, \\cdot \\rangle \\chi _{\\ell }.$ Then, $P_I(\\mathcal {L})$ is nothing but the spectral projection on the linear subspace spanned by the eigevectors associated with the eigenvalues belonging to $I$ .",
"In the sequel, $n_I = | I \\cap \\mathrm {sp}(\\mathcal {L})|$ will stand for the number of eigenvalues contained in subset $I \\cap \\mathrm {sp}(\\mathcal {L})$ .",
"Spectral projections are a practical tool to focus on some part of the spectrum $\\mathrm {sp}(\\mathcal {L})$ .",
"More precisely, let $I_\\mathcal {L}= \\sqcup _{1\\le k \\le K} I_k$ be a partition of interval $I_\\mathcal {L}$ into disjoint subsets $(I_k)_k$ .",
"Since intervals $I_k$ are disjoints, functional analysis of $\\mathcal {L}$ reduces to that of its projections $\\mathcal {L}_{I_k} = \\mathcal {L}P_{I_k}(\\mathcal {L})$ in the sense of the identity: $g(\\mathcal {L})= \\sum _{1 \\le k \\le K} g(\\mathcal {L}_{I_k}).$ In this paper, one will study the extent to which Fourier analysis on large graphs is improved when considering local Fourier analysis on each subset $I_k$ instead of global Fourier analysis on $I_\\mathcal {L}$ ."
],
[
"LocLets: a localized version of SGWT",
"This section introduces an important application of the localized graph Fourier analysis, namely the notion of localized SGWT."
],
[
"Construction of a SGWT",
"Let $f:\\mathcal {G}\\rightarrow \\mathbb {R}$ be a signal on the graph $\\mathcal {G}$ .",
"Let $\\varphi , \\psi :\\mathbb {R}\\rightarrow \\mathbb {R}$ be respectively the scaling and kernel functions (a.k.a.",
"father and mother wavelet functions), and let $s_j > 0$ , $1\\le j\\le J$ , be some scale values.",
"The discrete SGWT is defined in [16] as follows: $\\mathcal {W} f = ( \\varphi (\\mathcal {L})f^{T} , \\psi (s_{1}\\mathcal {L})f^{T}, \\ldots , \\psi (s_J\\mathcal {L})f^T)^{T}.$ The adjoint matrix $\\mathcal {W}^{*}$ of $\\mathcal {W}$ is: $\\mathcal {W}^{*}(\\eta _{0}^{T}, \\eta _{1}^{T}, \\ldots , \\eta _J^T)^{T} = \\varphi (\\mathcal {L})\\eta _{0} + \\sum _{j=1}^J \\psi (s_{j}\\mathcal {L})\\eta _{j}.$ We also recall from [16] that a discrete transform reconstruction formula using SGWT coefficients $(c_{j,m})_{\\begin{array}{c}0\\le j\\le J\\\\ 1\\le m\\le n\\end{array}}$ is obtained by the formula $(\\mathcal {W}^{*} \\mathcal {W})^{-1} \\mathcal {W}^{*}(c_{j,m})_{j,m},$ where $(\\mathcal {W}^{*} \\mathcal {W})^{-1}$ stands for a pseudo-inverse of matrix $\\mathcal {W}^{*} \\mathcal {W}$ ."
],
[
"Definition of LocLets",
"Spectral graph wavelet functions are given by $(\\varphi (\\mathcal {L})\\delta _m , \\psi (s_j \\mathcal {L})\\delta _m )_{1\\le j\\le J, 1\\le m\\le n}$ .",
"We define a LocLet function to be the projection of a graph wavelet function onto a subset of eigenspaces of $\\mathcal {L}$ .",
"Definition 1 Let $(\\varphi (\\mathcal {L})\\delta _m , \\psi (s_j \\mathcal {L})\\delta _m )_{1\\le j\\le J, 1\\le m\\le n}$ be the family functions induced by a SGWT.",
"Then, for any subset $I \\subset I_\\mathcal {L}$ , $1\\le j\\le J$ and $1\\le m\\le n$ , set: $\\begin{split}& \\varphi _{m,I} = \\varphi (\\mathcal {L}_{I})\\delta _{m} = \\sum _{\\ell : \\lambda _{\\ell }\\in I} \\varphi (\\lambda _{\\ell }) \\widehat{ \\delta }_m (\\ell ) \\chi _{\\ell }\\\\&\\psi _{j,m,I} = \\psi (s_{j} \\mathcal {L}_{I})\\delta _{m} = \\sum _{\\ell : \\lambda _{\\ell }\\in I} \\psi (s_{j}\\lambda _{\\ell }) \\widehat{ \\delta }_m (\\ell ) \\chi _{\\ell }.\\end{split}$ The functions $(\\varphi _{m,I}, \\psi _{j,m,I})_{1\\le j\\le J, 1\\le m\\le n}$ are called Localized waveLets functions (LocLets).",
"The functions $\\varphi _{m,I}, \\psi _{j,m,I}$ are said to be localized at $I$ .",
"Let $I_\\mathcal {L}= \\sqcup _{1\\le k\\le K} I_k$ be some partition.",
"Then, the localized SGWT transfom of $f$ with respect to partition $(I_k)_{1\\le k\\le K}$ , denoted by $\\mathcal {W}^{(I_k)_k}f$ , is defined as the family $\\mathcal {W}^{(I_k)_k}f = (\\mathcal {W}^{I_k}f)_k$ where $\\mathcal {W}^{I_k}f = ( \\varphi (\\mathcal {L}_{I_k})f^{T} , \\psi (s_{1}\\mathcal {L}_{I_k})f^{T}, ... )^{T}, \\quad 1 \\le k \\le K.$ Similarly to Equation (REF ), the adjoint transform is given by $\\mathcal {W}^{I_k\\ast }(\\eta _{0}^{T}, \\eta _{1}^{T}, \\ldots , \\eta _J^T)^{T} = \\varphi (\\mathcal {L}_{I_k})\\eta _{0} + \\sum _{j=1}^J \\psi (s_{j}\\mathcal {L}_{I_k})\\eta _{j}, \\quad 1 \\le k \\le K.$ As already observed, localized SGWT of a signal $f$ contains more precise information about signal $f$ than its standard SGWT.",
"The latter can easily be obtained from the former since subsets $(I_k)_k$ are pairwise disjoint and formula $g(s\\mathcal {L}) = \\sum _{1\\le k\\le K}g(s\\mathcal {L}_{I_k})$ holds for all filter $g$ , and in particular for $g=\\varphi $ or $g=\\psi $ .",
"When the partition $I_\\mathcal {L}= \\sqcup _{1\\le k\\le K} I_k$ is carefully chosen, we show that the SGWT localization provides interesting features such as independence of random variables in denoising modelling, or considerable improvements in denoising tasks.",
"Remark 2 A different localization property is studied in [16].",
"The latter refers to the vanishing of coefficients $\\langle \\psi _{j,m}, \\delta _{m^{\\prime }}\\rangle $ when the geodesic distance in the graph between vertices $m$ and $m^{\\prime }$ is sufficiently large (see [16]).",
"This notion is a property observable in the graph domain (through impulse functions $\\delta _m$ and $\\delta _{m^{\\prime }}$ ), and is different from the notion of Fourier localization discussed in the current paper.",
"We refer to the localization property in [16] as graph domain localization."
],
[
"Local Fourier analysis and graph functions denoising",
"The denoising problem is stated as follows: given an observed noisy signal $\\widetilde{f}$ of the form $\\widetilde{f}=f+\\xi $ where $\\xi $ is a $n$ -dimensional Gaussian vector distributed as $\\mathcal {N}(0,\\sigma ^2 \\mathrm {Id})$ , provide an estimator of the a priori unknown signal $f$ .",
"This section shows how localized Fourier analysis helps in estimating the noise level $\\sigma $ when it is unknown, and in recovering the original signal $f$ when the latter is sparse in the Fourier domain.",
"In what follows, we will focus on random variables of the form $\\Vert P_{I_k}\\widetilde{f}_{I_k}\\Vert _2$ where $\\widetilde{f}$ is the noisy signal and $I_k$ is a subset in the partition $I_\\mathcal {L}= \\sqcup _k I_k$ .",
"To keep the notations light, $n_k$ , $f_k$ , $\\xi _k$ and $\\widetilde{f}_k$ will stand for $n_{I_k}$ , $P_{I_k}f$ , $P_{I_k}\\xi $ and $P_{I_k}\\widetilde{f}$ respectively.",
"In addition, the cumulative distribution function of a random variable $X$ will be denoted by $\\Phi _X$ ."
],
[
"Noise level estimation for frequency sparse signals",
"Since in real application the noise level $\\sigma $ remains unknown in general, new estimators $\\widehat{\\sigma }$ based on localization properties in the spectrum are introduced in the sequel."
],
[
"Noise level estimation from projections along ${\\rm sp}(\\mathcal {L})$",
"For any filter $g$ defined on $I_\\mathcal {L}$ and any subset $I\\subset I_\\mathcal {L}$ , simple computation gives rises to $\\mathbb {E}( \\widetilde{f}^T g(\\mathcal {L}_I) \\widetilde{f} )=f^Tg(\\mathcal {L}_I)f+\\sigma ^2 {\\rm Tr}(g(\\mathcal {L}_I)).$ Since both $\\widetilde{f}^T g(\\mathcal {L}_I) \\widetilde{f}$ and ${\\rm Tr}(g(\\mathcal {L}_I))$ are known, Equation (REF ) suggests building estimators from the expression $\\frac{\\widetilde{f}^T g(\\mathcal {L}_I) \\widetilde{f}}{ {\\rm Tr}(g(\\mathcal {L}_I)) }$ .",
"In [9], the noise level is estimated by $\\frac{\\widetilde{f}^T \\mathcal {L}\\widetilde{f}}{ {\\rm Tr}(\\mathcal {L}) }$ which can be seen as the graph analog of the Von Neumann estimator from [33].",
"The main drawback of this estimator is its bias.",
"Theoretically, without any assumption on the signal $f$ , the bias term $\\frac{f^T g(\\mathcal {L}_I) f }{{\\rm Tr}(g(\\mathcal {L}_I))}$ is minimized when $g=\\mathrm {1}_{\\lbrace \\lambda _{\\ell ^\\ast } \\rbrace }$ where $\\ell ^\\ast = \\mathrm {argmin} \\lbrace | \\widehat{f}(\\ell ) | : \\lambda _\\ell \\in \\mathrm {sp}(\\mathcal {L})\\rbrace $ .",
"The computation of such filters would require the complete reduction of $\\mathcal {L}$ which does not scale well with the size of the graph.",
"Instead, these ideal filters will be approximated by filters of the form $g=\\mathrm {1}_{I_k}$ , for $I_k$ a subset in the partition $I_\\mathcal {L}= \\sqcup _k I_k$ .",
"It is worth noting that with $k^\\ast = {\\rm argmin}_{k} \\Vert f_{k}\\Vert _2$ , the function $g^\\ast =\\mathrm {1}_{I_{k^\\ast }}$ achieves the minimal bias of the estimator among all filters of the form $g = \\sum _k \\alpha _k \\mathrm {1}_{I_k}$ .",
"Discarding some intervals $I_k$ with $n_k=0$ , it can be assumed without loss of generality that $n_k\\ne 0$ for all $1\\le k\\le K$ .",
"Also observe that the random variable $\\Vert \\widetilde{f}_{k}\\Vert _2^2$ can be decomposed as follows $\\Vert \\widetilde{f}_{k}\\Vert _2^2=\\Vert f_{k} \\Vert _2^2 + \\Vert \\xi _{k} \\Vert _2^2+2 \\langle f_{k} , \\xi _{k} \\rangle $ where $\\frac{\\Vert \\xi _{k} \\Vert _2^2}{\\sigma ^2}$ and $\\frac{\\langle f_{k} , \\xi _{k} \\rangle }{\\sigma }$ are random variables distributed as $\\chi ^2( n_k )$ and $\\mathcal {N}( 0 , \\Vert f_{k} \\Vert _2^2 )$ respectively.",
"Proposition 3 Let $(c_k)_{1 \\le k \\le K}$ be the sequence of non-negative random variables defined, for all $k=1, \\ldots , K$ , by $c_k=\\Vert \\widetilde{f}_{k}\\Vert _2^2/n_k$ .",
"Then, the random variables $c_1, \\ldots , c_K$ are independent; for all $k,k^\\prime $ such that $f_k=f_{k^\\prime }$ , $c_k$ and $c_{k^\\prime }$ are identically distributed if and only if $n_k = n_{k^{\\prime }}$ ; for $k$ such that $f_k=0$ , $c_k$ is distributed as $\\frac{\\sigma ^2}{n_k} \\Gamma _{n_k}$ where $\\Gamma _{n_k} \\sim \\chi ^2(n_k)$ ."
],
[
"The case of frequency-sparse signals",
"When the signal $f$ is sparse in the Fourier domain, the condition $f_k=0$ is met for most of the intervals $I_k\\subset I_\\mathcal {L}$ .",
"Let us define $I_f = \\sqcup _{k: I_k \\cap {\\rm supp}\\widehat{f} \\ne \\emptyset } I_k$ to be the union of subsets $I_k$ intersecting the Fourier support ${\\rm supp} (\\widehat{f})$ of $f$ .",
"Also, denote by $\\overline{I_f} = I_L \\backslash I_f$ its complement set.",
"In order to take advantage of Fourier sparsity, let us introduce the quantities $\\widehat{\\sigma }_{\\rm mean}$ and $\\widehat{\\sigma }_{\\rm med}$ as follows: $\\widehat{\\sigma }_{\\rm mean}(c)^2 = \\frac{1}{|\\lbrace k: I_k \\subset \\overline{I_f} \\rbrace |}\\sum _{k: I_k \\subset \\overline{I_f}} c_k \\quad \\textrm {and} \\quad \\widehat{\\sigma }_{\\rm med}(c)^2 = \\mathrm {median}_{k: I_k \\subset \\overline{I_f}}(c_k).$ The following concentration inequalities show that $\\widehat{\\sigma }_{\\rm mean}$ and $\\widehat{\\sigma }_{\\rm med}$ are natural estimators of the noise level $\\sigma $ .",
"Proposition 4 Let $K_f = |\\lbrace k: I_k \\subset \\overline{I_f} \\rbrace |$ , $n_0=\\min \\lbrace n_k: k, I_k \\subset \\overline{I_f} \\rbrace $ , $n_\\infty =\\max \\lbrace n_k:k, I_k \\subset \\overline{I_f} \\rbrace $ , $V_f = 2\\sigma ^4 \\sum _{k:I_k\\subset \\overline{I_f}} {1}/{n_k}$ and $B_f = {2\\sigma ^2}/n_0$ .",
"Then the following concentration inequalities hold: for all $t\\ge 0$ , $\\mathbb {P}\\left(\\widehat{\\sigma }_{\\rm mean}(c)^2 - \\sigma ^2 \\ge t\\right)\\le \\exp \\left(- \\frac{K_f^2 t^2}{V_f ( 1 + B_f + \\sqrt{ 1 + \\frac{2 B_f K_f t}{V_f} } )}\\right),$ and for all $0\\le t\\le \\sigma ^2$ , $\\mathbb {P}\\left(\\widehat{\\sigma }_{\\rm mean}(c)^2 - \\sigma ^2 \\le - t\\right)\\le \\exp \\left(- \\frac{K_f^2 t^2}{2 V_f}\\right);$ for all $t\\ge 0$ , with $\\beta =n_0/n_\\infty $ , $\\mathbb {P} \\left( \\widehat{\\sigma }^2_{\\rm med}\\ge \\beta ^{-1} \\sigma ^2 + 2\\sigma ^2 \\beta ^{-1} t \\right) \\le \\exp \\left( \\frac{K_f}{2} \\ln \\Big [ 4p^+(t)(1-p^+(t)) \\Big ] \\right),$ and for all $0\\le t\\le 1$ such that $p^-(t)\\le 1/2$ , $\\mathbb {P} \\left( \\widehat{\\sigma }^2_{\\rm med}\\le \\beta \\sigma ^2-\\sigma ^2 \\beta t \\right) \\le \\exp \\left( \\frac{K_f}{2} \\ln \\Big [ 4p^-(t)(1-p^-(t)) \\Big ] \\right),$ where $p^+(t)=\\mathbb {P}(\\Gamma _{n_\\infty } \\ge n_\\infty + 2n_\\infty t) \\quad \\textrm {and} \\quad p^-(t)=\\mathbb {P}(\\Gamma _{n_0} \\le n_0-n_0 t).$ Obviously, the Fourier support ${\\rm supp}(\\widehat{f})$ and the subset $\\overline{I_f}$ remain generally unknown in applications and have to be approximated.",
"Let us recall that the main issue for estimating $\\sigma $ comes from the bias term $\\frac{\\Vert f_{k}\\Vert _2^2}{n_{k}}$ in Equation (REF ), and in particular when the value $\\sigma ^2$ is negligible compared to $\\frac{ \\Vert f_{k}\\Vert _2^2}{n_{k}}$ .",
"Therefore, a suitable candidate to approximate $\\overline{I_f}$ will be some subset $\\overline{J_f} \\subset I_\\mathcal {L}$ for which the impact of larger values $\\frac{ \\Vert f_{k}\\Vert _2^2}{n_{k}}$ is minimized.",
"This is made clear by Proposition REF below.",
"The latter involves the following concentration bounds for Gaussian random variables: for all $0 < \\alpha < 1$ $ \\mathbb {P}( |\\langle f_{k}, \\xi _{k} \\rangle | \\ge t_{\\alpha ,\\sigma }\\Vert f_{k} \\Vert _2 ) \\le \\alpha \\quad \\textrm {where} \\quad t_{\\alpha , \\sigma } = \\sigma \\times \\sqrt{- 2\\ln \\left(\\frac{\\alpha }{4} \\right)}.$ Proposition 5 Let $0 < \\alpha < 1$ .",
"Let $t_{\\alpha , \\sigma }$ be defined by Equation (REF ).",
"Assume that $f_{\\ell } = 0$ and that the following inequality holds: $\\frac{\\Vert f_k\\Vert _2^2 + 2 t_{\\alpha , \\sigma } \\Vert f_k\\Vert _2}{\\sigma ^2}\\ge \\Phi _{ \\frac{n_k}{n_\\ell } \\Gamma _{n_\\ell } - \\Gamma _{n_k}}^{-1} \\left( 1 - \\frac{3\\alpha }{2} \\right).$ Then, the quantities $b_k = \\frac{\\Vert \\xi _{k}\\Vert _2^2 + \\Vert f_k\\Vert _2^2 + 2 \\langle \\xi _k , f_k \\rangle }{n_k} \\quad \\textrm {and} \\quad b_\\ell = \\frac{\\Vert \\xi _\\ell \\Vert _2^2}{n_\\ell }$ satisfy $\\mathbb {P}( b_k \\ge b_\\ell ) \\ge 1 - \\alpha $ .",
"By invariance under permutations, one may assume without loss of generality that the values $c_k$ are ordered in the decreasing order.",
"Proposition REF quantifies the fact that the highest values of $c_k$ correspond most likely to the indices $k$ for which $f_k \\ne 0$ .",
"Consequently, setting $\\overline{J_f}(r) = \\sqcup _{k \\in \\lbrace r, r+1, ... K-r \\rbrace } I_k$ for all $1\\le r\\le \\frac{K}{2}$ , the estimators introduced in Equation (REF ) may be rewritten replacing the unknown subset $\\overline{I_f}$ by its known approximation $\\overline{J_f}(r)$ .",
"So we define the estimators $\\widehat{\\sigma }_{\\rm mean}^{r}(c)^2 = \\frac{1}{| \\lbrace k: I_k\\subset \\overline{J_f}(r) \\rbrace |}\\sum _{k: I_k\\subset \\overline{J_f}(r)} c_k \\quad \\textrm {and} \\quad \\widehat{\\sigma }_{\\rm med}^{r}(c)^2 = \\mathrm {med}_{k: I_k \\subset \\overline{J_f}(r)}(c_k).$ It is worth noting that from the symmetry of the subset $\\overline{J}_f(r)$ , it follows that the value $\\widehat{\\sigma }_{\\rm med}^r$ actually does not depend on parameter $r$ , and one will write $\\widehat{\\sigma }_{\\rm med}$ in place of $\\widehat{\\sigma }_{\\rm med}^r$ ."
],
[
"Denoising Frequency Sparse Signals",
"Let us begin with a result that illustrates that localized Fourier analysis in $I_\\mathcal {L}$ provides strong benefits in noise reduction tasks when the underlying is frequency sparse.",
"Proposition 6 Assume $f=f_{I}$ for some subset $I\\subset I_{\\mathcal {L}}$ .",
"Then $\\mathbb {E}\\left[ \\big \\Vert f -\\widetilde{f}_{I} \\big \\Vert _2^{2} \\right] = \\mathbb {E} \\left[ \\big \\Vert f -\\widetilde{f} \\big \\Vert _2^{2} \\right]- \\sigma ^{2} \\big \\vert \\overline{I} \\cap {\\rm sp} (\\mathcal {L}) \\big \\vert .$ In particular, denoising of $\\widetilde{f}$ boils down to denoising of $\\widetilde{f}_{I}=f_{I}+\\xi _{I}$ .",
"While Proposition REF asserts a trivial denoising solution in the Fourier domain, i.e.",
"simply destroying the projection $\\widetilde{f}_{\\overline{I}}=\\xi _{\\overline{I}}$ , this approach is no longer that immediate when considering the graph domain observations since the Fourier support of $f$ is unknown in practice and needs to be estimated.",
"Based on the $\\chi ^2$ -statistics, Algorithm REF is designed for this purpose.",
"To the best of our knowledge, previous works that proposed method for Fourier support recovery for graph noisy signals [25] involve the complete eigendecomposition of matrix $\\mathcal {L}$ .",
"The methodology suggested below makes use of projectors on eigenspaces which can be approximating with Chebyshev polynomials as detailed in the next Section .",
"noisy signal $\\widetilde{f}$ , a subdivision $I_1, I_2, \\ldots , I_{K}$ , estimated $n_k = |I_k \\cap \\mathrm {sp}{L}|$ , $k=1, \\ldots , K$ , threshold $\\alpha \\in (0,1)$ $\\widetilde{f}_{I}=P_{I}(\\mathcal {L})\\widetilde{f}$ , where $I$ is an approximation of the Fourier support of $\\widetilde{f}$ for $k=1, \\ldots , K$ Compute $\\Vert \\widetilde{f}_{k}\\Vert ^2_{2}= \\Vert P_{I_{k}}(\\mathcal {L}) \\widetilde{f} \\Vert ^2_{2}$ ; Compute $p_k=\\mathbb {P}(\\sigma ^2 \\Gamma _{n_k} > \\Vert \\widetilde{f}_{k}\\Vert ^2_2) \\quad \\textrm {and} \\quad \\Gamma _{n_k} \\sim \\chi ^2(n_k);$ Compute $\\widetilde{f}_I = \\displaystyle \\sum _{ k:~p_k \\le \\alpha } P_{I_k}\\widetilde{f}$ .",
"Support approximation in the Fourier domain for noisy signal Heuristically, if $I$ contains the support of the Fourier transform of $f$ , on the complementary subset $\\overline{I}$ we only observe pure white Gaussian noise so that $\\Vert P_{\\overline{I}}\\widetilde{f} \\Vert _2^2=\\Vert \\widetilde{f}_{\\overline{I}} \\Vert _2^2$ is distributed as $\\sigma ^2 \\chi ^2(n_I)$ with $n_I=|\\overline{I} \\cap \\mathrm {sp}(L)|$ .",
"On the other hand, on $I$ the square of the Euclidean norm of a non-centered Gaussian vector is observed.",
"Consequently, the quantity $\\mathbb {P} \\left(\\chi ^2(n_I) > \\sigma ^{-2} \\Vert P_{I} \\widetilde{f} \\Vert ^2_2 \\right)$ is typically very close to zero whereas $\\mathbb {P} \\left(\\chi ^2(n-n_I) > \\sigma ^{-2} \\Vert P_{\\overline{I}} \\widetilde{f} \\Vert ^2_2 \\right)$ remains away from 0.",
"To put it in a nutshell, sliding a window along the spectrum of $\\mathcal {L}$ , Algorithm REF performs a series of $\\chi ^2$ -test.",
"With the objective to provide theoretical guarantees that $\\chi ^2$ -tests approach ${\\rm supp}(\\widehat{f})$ correctly, it is important to turn the condition on the $p_k$ -value into a condition involving only the values $\\Vert f_k\\Vert _2$ and $\\sigma $ .",
"The next lemma shows that for sufficiently large values of the ratio $\\frac{\\Vert f_k\\Vert _2}{\\sigma }$ , the inequality $p_k\\le \\alpha $ holds so that the corresponding components ${\\rm supp}( \\widehat{f}_k )$ of the Fourier domain are legitimately included in the support estimate $I$ .",
"Lemma 7 Let $0<\\alpha <1$ and let $\\Gamma _{n_k} , \\Gamma _{n_k}^\\prime $ be two i.i.d $\\chi ^2(n_k)$ random variables.",
"Assume that: $\\frac{\\Vert f_k \\Vert _2}{\\sigma } \\left( \\frac{\\Vert f_k\\Vert _2}{\\sigma } - 2\\frac{t_{\\alpha / 2,\\sigma }}{\\sigma } \\right) \\ge \\Phi _{\\Gamma _{n_k} - \\Gamma _{n_k}^\\prime }^{-1} \\left( 1- \\frac{\\alpha }{2} \\right),$ where $t_{\\alpha ,\\sigma }$ is defined by Equation (REF ).",
"Then $p_k \\le \\alpha $ .",
"In contrast to Lemma REF , the lemma below states that condition $p_k > \\alpha $ holds for sufficiently small values of ratio $\\sigma ^{-1}\\Vert f_k\\Vert _2$ .",
"Lemma 8 Let $0<\\alpha <1$ and let $\\Gamma _{n_k}$ be a $\\chi ^2(n_k)$ random variable.",
"For $0<\\beta <1$ , set $t_{\\beta , k} = \\sigma ^2 \\Phi _{\\Gamma _{n_k}}^{-1}(1-\\beta )$ .",
"Assume that $\\left( \\frac{\\Vert f_k\\Vert _2+\\sqrt{t_{\\beta , k}}}{\\sigma } \\right)^2 < \\Phi _{\\Gamma _{n_k}}^{-1} \\left( 1 - \\frac{\\alpha }{1-\\beta } \\right).$ Then $p_k > \\alpha $ .",
"Compared to Proposition REF , the result below quantifies the error resulting by approximating the support running Algorithm REF .",
"Note that the requirement to have a constant sequence $(n_k)_k$ is used for statement clarity but similar assertions hold for the case $n_k \\ne n_{k^{\\prime }}$ .",
"Proposition 9 Set $f_I = \\sum _{k: p_k \\le \\alpha } P_{I_k}f$ .",
"Assume that $n_k = n_1$ for all $1\\le k\\le K$ .",
"Then, the Fourier support approximation $\\ell _2$ -error satisfies $\\Vert f-f_I\\Vert _2^2 \\le |\\lbrace k, I_k\\subset I_f, p_k > \\alpha \\rbrace | \\left( t_{\\alpha / 2, \\sigma } + \\sqrt{ t_{\\alpha / 2, \\sigma }^2 + \\left( \\sigma \\Phi _{\\Gamma _{n_1} - \\Gamma _{n_1}^{\\prime }}^{-1} \\left( 1- \\frac{\\alpha }{2} \\right) \\right)^2} \\right)^2.$ the Noise $\\ell _2$ -error on Fourier support: $\\mathbb {E} \\Vert f_I - \\widetilde{f}_I \\Vert _2^2 = |\\lbrace k, p_k \\le \\alpha \\rbrace | n_1 \\sigma ^2 .$ Lemma REF asserts that the set $\\lbrace k, p_k > \\alpha \\rbrace $ is small when most of the values $\\Vert f_k\\Vert _2$ are large enough compared to noise level $\\sigma $ for $I_k \\cap {\\rm supp}\\widehat{f} \\ne \\emptyset $ .",
"In such a case, Fourier support approximation $\\ell _2$ -error is small.",
"Regarding the noise $\\ell _2$ -error, the inclusion $ \\lbrace k, p_k \\le \\alpha \\rbrace \\subset \\lbrace k, I_k\\cap {\\rm supp}\\widehat{f} \\ne \\emptyset \\rbrace $ holds by Lemma REF .",
"Moreover, Lemma REF asserts that the set $\\lbrace k, p_k \\le \\alpha \\rbrace $ contains the entire set $\\lbrace k, I_k \\cap {\\rm supp}\\widehat{f} \\ne \\emptyset \\rbrace $ for sufficiently large values of $\\sigma ^{-1}\\Vert f_k\\Vert _2$ when $I_k \\cap {\\rm supp}\\widehat{f} \\ne \\emptyset $ .",
"For such favorable situations, the noise $\\ell _2$ -error is exactly $ n_1 \\sigma |\\lbrace k, I_k \\subset I_f\\rbrace |$ , the amount of noise on the extended support $I_f$ .",
"$\\widetilde{f}$ , $\\alpha $ , $(I_k)_{k=1, \\ldots , K}$ , estimated $n_k = |I_k \\cap \\mathrm {sp}(\\mathcal {L})|$ , thresholds $t_1$ , $t_2$ estimator $\\widehat{f}$ of signal $f$ Apply Algorithm REF with $\\widetilde{f}$ , $\\alpha $ , $(I_k)_{k=1, \\ldots K}$ , estimated $n_k$ ; it outputs $\\widetilde{f_{I}}$ and $\\widetilde{f_{\\overline{I}}}$ ; Apply soft-thresholding with threshold $t_1$ to $\\mathcal {W}^{I}\\widetilde{f}$ and $t_2$ to $\\mathcal {W}^{\\overline{I}} \\widetilde{f}$ ; Apply the inverse LocLet transform to the soft-thresholded coefficients to obtain $\\widehat{f_I},\\widehat{f}_{\\widetilde{I}}$ ; Compute the estimator $\\widehat{f} = \\widehat{f}_{I} + \\widehat{f}_{\\overline{I}}$ ; LocLets thresholding estimation procedure The second step gives an estimate of the original signal using a thresholding procedure on each element $\\widetilde{f}_I$ and $\\widetilde{f}_{\\overline{I}}$ .",
"On the one hand, the methodology developed in [15] is prohibitive in terms of time and space complexity as soon as the underlying graphs become moderately large.",
"On the other hand, the fast SGWT remains an approximating procedure.",
"If a signal happens to be very frequency-sparse, then an even more optimal strategy is possible: first, the support $I$ in the frequency domain is approximated with the help of Algorithm REF ; then, the procedure of [15] is applied to $P_If$ (the low-rank part) and LocLets on $P_{\\overline{I}}(\\mathcal {L})f$ .",
"This idea is made precisely in Algorithm REF .",
"$\\widetilde{f}$ , $\\alpha $ , $(I_k)_{k=1, \\ldots , K}$ , estimated $n_k = |I_k \\cap \\mathrm {sp}(\\mathcal {L})|$ , thresholds $t_1$ , $t_2$ estimator $\\widehat{f}$ of signal $f$ Apply Algorithm REF with $\\widetilde{f}$ , $\\alpha $ , $(I_k)_{k=1, \\ldots K}$ , estimated $n_k$ ; it outputs $\\widetilde{f_{I}}$ and $\\widetilde{f_{\\overline{I}}}$ ; Compute Parseval Frame for $\\mathcal {L}_I$ ; Apply Parseval Frame thresholding with threshold $t_1$ to $\\widetilde{f}_I$ ; it outputs $\\widehat{f_I}$ ; Apply soft-thresholding with threshold $t_2$ to $\\mathcal {W}^{\\overline{I}} \\widetilde{f}$ ; Apply the inverse LocLet transform to the soft-thresholded coefficients to obtain $\\widehat{f}_{\\widetilde{I}}$ ; Compute the estimator $\\widehat{f} = \\widehat{f}_{I} + \\widehat{f}_{\\overline{I}}$ ; LocLets support approximation, and low-rank Parseval Frame thresholding procedure Estimator $\\widehat{f}$ produced in Algorithm REF satisfies a tighter oracle bound inequality than the one given in [15].",
"This theoretical guarantee is widely supported by our experiments described in Section .",
"Following notations from [15], we denote by $OB(f_I)$ the oracle bound obtained from an oracle estimator of $f_I$ from a noisy $\\widetilde{f}_I$ exploiting some knowledge about the unknown signal $f_I$ .",
"We refer to [15] for precise details.",
"Theorem 10 Let $I, \\widehat{f}$ be respectively the support approximation and the estimator of $f$ obtained from Algorithms REF and REF with threshold value $t_2 = 0$ .",
"Then we have $\\mathbb {E} \\Vert f-\\widehat{f} \\Vert _2^2 \\le \\mathbb {E} \\Vert f-f_I \\Vert _2^2+(2 \\log (n_I) + 1)(\\sigma ^2 + OB(f_I)).$ The right-hand side in the inequality of Theorem REF has a more explicit expression in terms of $\\alpha , \\sigma $ using Proposition REF .",
"Up to the error made by approximating the support with Algorithm REF , the $\\ell _2$ -risk is essentially bounded by the $\\ell _2$ -risk of the Parseval frame procedure from [15] on the low-rank projection $f_I$ of $f$ , that is $\\mathbb {E} \\Vert f-\\widehat{f}\\Vert _2^2 \\lesssim (2 \\log (n_I) + 1) (\\sigma ^2 + OB(f_I)).$ To conclude, Theorem REF provides a theoretical guarantee that the support approximation improves the denoising performances obtained from [15]."
],
[
"Properties of LocLets",
"In this section, we highlight important properties for the application of Fourier localization in practice.",
"First we discuss computational analysis, and methods to apply our techniques to large graphs.",
"Then we study the relationships of LocLets with well-known graph wavelet constructions."
],
[
"Fast LocLet Transform and Computational Analysis",
"In the case of large graphs, GSP requires a special care for being efficient since functional calculus relies a priori on the complete reduction of the Laplacian.",
"Actually, several efficient methods were designed to retrieve only partial information from the eigendecomposition as matrix reduction techniques (see for instance [20], [30]) or polynomial approximations [16], [28], [11].",
"In this paper, the widely adopted latter approach with Chebyshev polynomials approximation is preferred and briefly recalled below (we refer the reader to [28] for a brief but more detailed description of Chebyshev approximation)."
],
[
"Chebyshev approximations",
"Roughly speaking, the idea is to approximate the function $g$ with its Chebyshev expansion $g_N$ at order $N$ .",
"More precisely, the Chebyshev polynomials of the first kind $(T_i)_{i \\ge 0}$ are defined from the second order recursion $T_0(x) = 1, \\quad T_1(x) = x, \\quad |x|\\le 1, \\quad \\textrm {and} \\quad T_i(x) = xT_{i-1}(x) - T_{i-2}(x),$ for $i \\ge 2$ .",
"Then, the matrix $\\mathcal {L}$ is normalized as $\\widetilde{\\mathcal {L}} = \\frac{2}{\\lambda _1}\\mathcal {L}- I_n$ so that $\\mathrm {sp}(\\widetilde{\\mathcal {L}}) \\subset [-1, 1]$ .",
"This gives rise to some function $\\widetilde{g} : [-1, 1]\\rightarrow \\mathbb {R}$ with the property $g(\\mathcal {L}) = \\widetilde{g}(\\widetilde{\\mathcal {L}})$ .",
"In fact, $\\widetilde{g}(x) = g(\\frac{\\lambda _1}{2}(x+1))$ for all $x \\in [-1, 1]$ .",
"Then $g(\\mathcal {L})$ has the following truncated Chebyshev expansion $g(\\mathcal {L}) \\approx g_N(\\mathcal {L})$ : $g_N(\\mathcal {L}) = \\sum _{0\\le i\\le N} a_i(\\widetilde{g}) T_i(\\widetilde{\\mathcal {L}}),$ where $N$ is the maximal degree of polynomials $T_i$ used in the expansion, and $a_i(\\widetilde{g})$ is the $i$ -th coefficient in the $N$ -th order Chebyshev expansion of function $\\widetilde{g}$ .",
"Following [16], for any filter $g$ on $\\mathrm {sp}(\\mathcal {L})$ and any signal $f$ on graph $\\mathcal {G}$ , the approximation $g_N(\\mathcal {L})$ provides a vector value close to $g(\\mathcal {L})f$ with time complexity $O(|\\mathcal {E}| N)$ .",
"The object presented in the sequel involves in particular the spectral projection $P_I(\\mathcal {L})f$ of a signal $f$ for any subset $I\\subset I_\\mathcal {L}$ which can be derived from the Chebyshev expansion of the indicator function $g = \\mathrm {1}_I$ .",
"This observation actually appears in several recent works [11], [12].",
"More importantly for our study, this efficient estimation is part of the Hutchinson stochastic trace estimator technique [17], providing us with an effective method to estimate $n_I = \\rm {Tr}(\\mathcal {L}_I)$ .",
"Finally, the present paper focuses on the computation of a sequence $g(\\mathcal {L}_{I_k})_{1\\le k\\le K}$ (or its vector counterpart $g(\\mathcal {L}_{I_k})f$ ) instead of a single $g(\\mathcal {L})$ (resp.",
"$g(\\mathcal {L})f$ ).",
"While a naive estimation would suggest that the computational complexity is then multiplied by a factor $K$ compared to the complexity of the computation of $g(\\mathcal {L})$ , we argue in the following that there is in fact no significant computational overhead."
],
[
"Sharing Chebyshev polynomial matrices among filters",
"Let us assume that it is needed to compute the estimated values of $g_k(\\mathcal {L})f$ for a given signal $f$ for several filters $g_k$ , $k=0, \\ldots , K$ .",
"Then the following two-step strategy can be adopted: (1) pre-compute Chebyshev expansions $\\widetilde{g}_k(x) \\approx \\widetilde{g}_{k, N}(x) = \\sum _{0\\le i\\le N} a_i(\\widetilde{g_k}) T_i(x)$ for all $k=0, \\ldots , K$ ; independently, compute Chebyshev approximation vectors $T_i(\\widetilde{\\mathcal {L}})f$ for all $0\\le i\\le N$ ; (2) combine the previous results to compute the Chebyshev approximation $g_{k,N}(\\mathcal {L})f$ of $g_k(\\mathcal {L})f$ : $g_{k,N}(\\mathcal {L})f = \\sum _{0 \\le i \\le N} a_i(\\widetilde{g}_k) T_i(\\widetilde{\\mathcal {L}})f.$ The complexity of the first step is dominated by the $N$ matrix-vector multiplications required to obtain $T_i(\\widetilde{\\mathcal {L}})f$ .",
"So the first step has complexity $O(|\\mathcal {E}| N)$ .",
"The second step adds $N$ weighted matrices $a_i(\\widetilde{g}_k) T_i(\\widetilde{\\mathcal {L}})$ together, which is an operation of complexity $O(N n^2)$ at most.",
"As an important matter of fact, the overall complexity for this procedure is bounded by $O(|\\mathcal {E}| N + N n^2)$ , which is independent of the number of filters $g_k$ , and the same as for the computation of $g(\\mathcal {L})$ .",
"Sharing matrices among filters has several examples of applications in the current paper: Computation of $g( \\mathcal {L}_{I_k} )f$ for all $1\\le k \\le K$ : the equation $g(\\mathcal {L}_{I_k})f = g(\\mathrm {1}_{I_k}(\\mathcal {L}) \\mathcal {L})f$ holds so that we can consider filters $g_k(x) = g(\\mathrm {1}_{I_k}(x) x)$ .",
"Computation of $g(s \\mathcal {L})f$ for several scale values $s$ : consider filters of the form $g_s(x) = g(sx)$ .",
"Computation of $n_{I_k}$ for all $1\\le k \\le K$ : Hutchinson's stochastic estimation computes averages of $f_i^T P_{I_k}(\\mathcal {L})f_i$ for some random vectors $f_i$ ($i\\le n_H$ ) whose computational complexity is dominated by the approximation of vectors $P_{I_k}f_i$ .",
"Considering filters $g_k(x) = \\mathrm {1}_{I_k}(x)$ , and sharing random vectors $(f_i)_i$ among all approximations of $n_k$ , we end up with a complexity of $O(n_H N |\\mathcal {E}|)$ , independent of value $K$ .",
"In particular, Algorithm REF has complexity $O(n_H N |\\mathcal {E}| + N n^2)$ .",
"Indeed, its efficiency is calibrated on the computations of sequences $(\\Vert \\widetilde{f}_k\\Vert _2)_{1\\le k\\le K}$ and $(n_k)_{1\\le k\\le K}$ whose computational analysis was discussed previously.",
"It is worth observing that values $n_k$ do not depend on signal $f$ and should be estimated only once in the case where several signals $\\widetilde{f}_1 , \\widetilde{f}_2, \\ldots $ are to be denoised."
],
[
"Optimizing storage of LocLets coefficients",
"The storage of wavelet coefficients $(\\mathcal {W}^{I_k}f)_{1\\le k\\le K}$ requires a priori $K$ times the storage cost associated with the original transform $\\mathcal {W} f$ .",
"When matrix reduction techniques are used to compute wavelets transform [30], one may reduce the storage consumption of the localized SGWT by suitably choosing the impulse functions $(\\delta _{m})_m$ .",
"For instance, assume that for each subset $I_k$ a Lanczos basis $(v_{m}^{k})_m$ of the subspace spanned by $\\lbrace \\chi _\\ell , \\ell \\in I_k \\rbrace $ is given.",
"Then the size of sequences $(v_{m}^{k})_m$ and $(v_{m}^{k})_{m, k}$ are respectively of order $O(|I_k \\cap {\\rm sp}(\\mathcal {L})|)= O(n_k)$ and $O(n)$ .",
"Thus, with impulse functions $(v_{m}^{k})_n$ in place of $\\delta _m$ for transform $\\mathcal {W}^{I_k}$ , the storage requirements of localized transform $(\\mathcal {W}^{I_k}f)_{1\\le k\\le K}$ and the original one $\\mathcal {W} f$ are of the same order $O(J n)$ ."
],
[
"Connections with Well-know Frames",
"A family $\\mathfrak {F}=\\lbrace r_i \\rbrace _{i \\in I}$ of vectors of $\\mathbb {R}^\\mathcal {V}$ is a frame if there exist $A,B > 0$ satisfying for all $f \\in \\mathbb {R}^\\mathcal {V}$ $ A\\Vert f \\Vert ^2_2 \\le \\sum _{i \\in I} |\\langle f,r_i \\rangle |^2 \\le B \\Vert f\\Vert ^2_2.$ A frame is said to be tight if $A=B$ .",
"This section gives two examples of frames introduced in the literature which can be realized as a LocLets representation and thus benefit from the advantages given by localization in the spectrum."
],
[
"Parseval frames",
"Parseval frames are powerful representations to design wavelets with nice reconstruction properties [18], [15].",
"In this section, we investigate the extent to which Parseval frames can be obtained from some LocLet representation.",
"We show that for a particular choice of partition $I_\\mathcal {L}= \\sqcup _k I_k$ , there exist frames which are Parseval frames and composed only of LocLets functions.",
"A finite collection $(\\psi _j)_{j=0, \\ldots ,J}$ is a finite partition of unity on the compact $[0,\\lambda _1]$ if $\\psi _j : [0,\\lambda _1] \\rightarrow [0,1] \\quad \\textrm {for all} \\quad j \\le J \\quad \\textrm {and} \\quad \\forall \\lambda \\in [0,\\lambda _1], \\quad \\sum _{j=0}^J \\psi _j(\\lambda )=1.$ Given a finite partition of unity $(\\psi _j)_{j=0, \\ldots , J}$ , the Parseval identity implies that the following set of vectors is a tight frame: $\\mathfrak {F} = \\left\\lbrace \\sqrt{\\psi _j}(\\mathcal {L})\\delta _i, \\quad j=0, \\ldots , J, \\quad i \\in V \\right\\rbrace .$ Some constructions of partition of unity involve functions $(\\psi _j)_j$ that have almost pairwise disjoint supports i.e.",
"${\\rm supp}(\\psi _{j})\\cap {\\rm supp}(\\psi _{j^\\prime }) = \\emptyset $ as soon as $|j - j^\\prime |>1$ .",
"For such partition of unity, set $I_0 = {\\rm supp}(\\psi _{0})$ , $I_J = I_0 = {\\rm supp}(\\psi _{J})$ and $I_j = {\\rm supp}(\\psi _{j}) \\cap {\\rm supp}(\\psi _{j+1})$ for all $1\\le j\\le J-1$ .",
"Then, the sequence $(I_j)_{0\\le j\\le J}$ defines a finite partition of $[0, \\lambda _1]$ , $[0, \\lambda _1] = \\sqcup _{0\\le j\\le J} I_j$ , such that: $\\psi _0 \\mathrm {1}_{I_0} = \\mathrm {1}_{I_0},\\quad (\\psi _j + \\psi _{j+1}) \\mathrm {1}_{I_j} = \\mathrm {1}_{I_j}, \\quad 0< j < J, \\quad \\textrm {and} \\quad \\psi _J \\mathrm {1}_{I_J} = \\mathrm {1}_{I_J}.$ An alternative tight frame can be constructed using a LocLet representation as shown in the following proposition.",
"Proposition 11 Assume Equations (REF ) and (REF ) hold and set, for all $1 \\le k \\le J$ , $\\varphi _{n,k}=\\sqrt{\\psi _{0}}(\\mathcal {L}_{I_k})\\delta _n$ , $\\psi _{1,n,k}=\\sqrt{\\psi _{k}}(\\mathcal {L}_{I_k})\\delta _n$ and $\\psi _{2,n,k}=\\sqrt{\\psi _{k}}(\\mathcal {L}_{I_{k+1}})\\delta _n$ for all $1\\le k\\le J$ .",
"Then $(\\varphi _{n,k}, \\psi _{j,n,k})_{1\\le j\\le 2,~1\\le m\\le n,~1\\le k\\le J}$ is a tight frame.",
"The resulting tight frame of Proposition REF is actually frame of LocLets if additionally the functions $\\psi _j$ is of the form $\\psi _j = \\psi _1(s_j .",
")$ for some scale parameter $s_j$ , $1 \\le j \\le J$ .",
"This is typically the case for the frames introduced in [18], [15].",
"In these papers, the partition of unity is defined as follows: let $\\omega : \\mathbb {R}^+ \\rightarrow [0,1]$ be some function with support in $[0,1]$ , satisfying $\\omega \\equiv 1$ on $[0,b^{-1}]$ and set $\\psi _0(\\cdot )=\\omega (\\cdot )$ and for $j=1, \\ldots , J$ $\\psi _j(\\cdot )=\\omega (b^{-j} \\cdot )-\\omega (b^{-j+1} \\cdot ) \\quad \\textrm {with} \\quad J= \\left\\lfloor \\frac{\\log \\lambda _1}{\\log b} \\right\\rfloor + 2.$ In particular, the functions $\\psi _k$ have supports in intervals $J_{k}=[b^{k-2}, b^{k}]$ .",
"Thus, one may define disjoint intervals $(I_{k})_{k}$ as follows: $I_{k} = [b^{k-1}, b^{k}]$ .",
"We have $J_{k} = I_{k} \\cup I_{k+1} $ , so that Equations (REF ) hold whereas the scaling property $\\psi _j = \\psi _1(b^{-1} .",
")$ is straightforward.",
"By Proposition REF , the set of vectors $\\left\\lbrace \\sqrt{\\psi _{0}}(\\mathcal {L}_{I_k})\\delta _n, \\sqrt{\\psi _{1}}(s_k\\mathcal {L}_{I_k})\\delta _n, \\sqrt{\\psi _{1}}(s_k\\mathcal {L}_{I_{k+1}})\\delta _n, \\quad n,k \\right\\rbrace $ is a tight frame of LocLets.",
"Observe that the transform $(\\mathcal {W}^{I_k})_{I_k}$ each component $\\mathcal {W}^{I_k}$ of the LocLet transform $(\\mathcal {W}^{I_k})_{I_k}$ only admit two scale parameters $s_k, s_{k-1}$ ."
],
[
"Spectrum-adapted tight frames",
"Let us consider another family of tight frames tailored to the distribution of the Laplacian $\\mathcal {L}$ eigenvalues proposed in [1].",
"As shown below, these frames can be written in terms of a warped version of LocLets, and up to some approximation, in terms of (non-warped) LocLets.",
"First, let us briefly recall the construction from [1].",
"The notion of warped SGWT is introduced in [1] to adapt the kernel to the spectral distribution.",
"Given a warping function $\\omega : I_\\mathcal {L}\\rightarrow \\mathbb {R}$ , the warped SGWT is defined as: $\\mathcal {W}^{\\omega } f = ( \\varphi (\\omega (\\mathcal {L}) )f^{T} , \\psi (s_{1}\\omega (\\mathcal {L}) )f^{T}, \\ldots , \\psi (s_J \\omega (\\mathcal {L}))f^T)^{T}.$ As for our spectral localization, the objective of warping is to take benefits from the distribution of ${\\rm sp}(\\mathcal {L})$ along interval $I_\\mathcal {L}$ .",
"While the two techniques show similarities (e.g.",
"estimation of ${\\rm sp}(\\mathcal {L})$ distribution), they are meant to answer different problems: warped SGWT is a technique to adapt the whole spectrum to some task (e.g.",
"producing a tight frame), whereas localized SGWT is designed to answer problems related to localized subsets in the spectrum (e.g.",
"denoising a frequency sparse signal).",
"Here we show that the advantages of both LocLets and warped SGWT are obtained when the two methods are combined in a warped LocLet representation.",
"Let $\\omega $ be some warping function on $I_\\mathcal {L}$ chosen in the form $\\omega (\\cdot ) = \\log (C\\omega _0(\\cdot ))$ where $\\omega _0$ stands for the cumulative spectral distribution of $\\mathcal {L}$ and $C$ is some normalization constant as shown in [1].",
"Then, let $\\gamma >0$ be an upper bound on $\\rm {sp }(\\mathcal {L})$ and let $R,J$ be two integers such that $2 \\le R \\le J$ .",
"Setting $\\omega _{\\gamma , J, R} = \\frac{\\gamma }{J+1+R}$ , Corollary 2 in [1] asserts that the family $(g_{m,j})_{m,j}$ of functions defined below is a tight frame $g_{m,j} = \\sum _{\\ell } \\widehat{g}_j(\\lambda _\\ell ) \\widehat{\\delta }_m(\\ell ) \\chi _\\ell ,$ where functions $\\widehat{g}_j$ arise from some kernel $\\widehat{g}$ as $\\widehat{g}_j(\\lambda )= \\widehat{g}(\\omega (\\lambda ) - j\\omega _{\\gamma ,J,R} )= \\widehat{g} \\left( \\log \\frac{C\\omega (\\lambda )}{ e^{j\\omega _{\\gamma ,J,R}} } \\right).$ Typically in [1], the kernel $\\widehat{g}$ takes the form $\\widehat{g}(\\lambda )=\\left[ \\sum _{0\\le j\\le J} a_j \\cos \\left( 2\\pi j \\cos \\left( \\frac{\\lambda }{R \\omega _{\\gamma ,J,R}} + \\frac{1}{2} \\right) \\right) \\right]\\mathrm {1}_{[ -R\\omega _{\\gamma ,J,R}, 0]} (\\lambda ).$ for some sequence $(a_j)_j$ satisfying $\\sum _j (-1)^j a_j = 0$ .",
"The following proposition states that Equation (REF ) admits an alternative form involving only (warped) LocLets functions.",
"Proposition 12 Setting $\\psi (\\lambda )=\\widehat{g}(\\log (C \\lambda ) )$ for $\\lambda > 0$ , consider the family of warped LocLets defined for all $0 \\le k \\le R-1$ , $1\\le m\\le n$ and $1\\le j\\le J$ by $\\psi _{j,m,I_k} = \\sum _{\\ell \\in I_k} \\psi (s_j \\omega _0(\\lambda _\\ell )) \\widehat{\\delta }_m(\\ell ) \\chi _\\ell \\quad \\textrm {with} \\quad I_k = \\left[ \\frac{e^{(k-R)\\omega _{\\gamma ,J,R}}}{C},\\frac{e^{(k-R+1)\\omega _{\\gamma ,J,R}}}{C} \\right].$ Then, the following identity holds for all $j=1, \\ldots , J$ and all $m=1, \\ldots , n$ $g_{m,j} = \\sum _{1\\le k\\le R-1} \\psi _{j,m,I_k}.$"
],
[
"Experiments on some large matrices suite",
"This section details experiments made on large graphs to validate the Fourier localization techniques introduced in that paper.",
"After describing the experimental settings, we describe the outcomes of several experiments showing strong advantages in the use of Fourier localization in practice."
],
[
"Choice of spectral partition $I_\\mathcal {L}= \\sqcup _k I_k$",
"In order to keep the problem combinatorially tractable, it is necessary to reduce the choice of possible partitions of $I_\\mathcal {L}$ into subintervals $I_k$ .",
"That is why, the partitions considered in the sequel are regular in the sense that all intervals have the same length $\\lambda _1/K$ for some integer $K \\ge 1$ .",
"Thereafter, the parameter $K$ is chosen so that the eigenvalues are distributed as evenly as possible in each interval $I_k$ .",
"Without prior information, it is indeed natural not to favor one part of the spectrum over another.",
"Most importantly, in the view of the concentration property of the median around the noise level $\\sigma ^2$ of Proposition REF , it is essential to keep the parameter $\\beta $ as close to one as possible.",
"In order to implement the ideas above, it is necessary to estimate the spectral measure of $\\mathcal {L}$ which can be described by the so-called spectral density function: $\\varphi _\\mathcal {L}(\\lambda )=\\frac{1}{n} \\sum _{\\ell =1}^n \\delta (\\lambda -\\lambda _\\ell ) \\quad \\textrm {for all} \\quad \\lambda \\in I_\\mathcal {L}.$ There are several techniques for such an approximation among which the Kernel Polynomial Method (see, e.g.",
"[29], [34]).",
"The latter approximates the spectral density $\\varphi _\\mathcal {L}$ with the help of matrix Chebyshev expansion $(\\varphi _\\mathcal {L}^N)_N$ (see [19] for a detailed presentation).",
"Now, let $(I_k)_{1 \\le k \\le K}$ be some regular partition of $I_\\mathcal {L}$ and $(n_k)_{1 \\le k \\le K}$ be the corresponding numbers of eigenvalues in each $I_k$ .",
"Choosing the parameter $K \\ge 1$ so that the entropy defined by $E(K) = - \\sum _{1\\le k\\le K} \\frac{n_k}{n} \\log \\left(\\frac{n_k}{n}\\right)$ is maximal ensures that the eigenvalues are as equally distributed in each interval as possible.",
"In application, the Kernel Polynomial Method provides an approximation $n_k^N$ of $n_k$ and the corresponding empirical entropy $E_N(K)$ is used as a proxy for the theoretical one.",
"Empirically, the entropy increases logarithmically and then stabilizes from a certain elbow value $K_{\\rm elbow}$ as illustrated in Figure REF .",
"This elbow value is displayed in dashed lines in Figure REF .",
"In the experiments, we choose this value $K_{\\rm elbow}$ motivated by two reasons.",
"First, as the intervals become shorter it is more difficult to obtain a uniform repartition of the eigenvalues into those intervals.",
"The second reason is related to the quality of the estimate $n_k^N$ of $n_k$ as the sample size decreases.",
"To illustrate this fact, we consider the Mean Relative Error (MRE) defined by $\\mathrm {MRE}_N(K)=\\frac{\\sum _{1\\le k\\le K} | n_k - n_{k}^N |}{n}.$ As highlighted by Figure REF , the empirical entropy actually stabilizes when the Chebyshev approximation, in terms of MRE, is no longer sharp enough.",
"Figure: minnesota graph"
],
[
"The experimental settings",
"Following [12], we propose to validate our techniques on an extended suite of large matrices extracted from the Sparse Matrix Collection in [8].",
"Most of these matrices have an interpretation as the Laplacian matrix of a large graph.",
"We define matrix $\\mathcal {L}$ from the following matrices of the suite:si2 ($n=769$ ), minnesota ($n=2642$ ), cage9 ($n=3534$ ), saylr4 ($n=3564$ ) and net25 ($n=9520$ ).",
"We extend this graph collection with the well-studied swissroll graph Laplacian matrix ($n=1000$ ).",
"We sample randomly signals whose support are sparse in the Fourier domain.",
"We will use the notation $f_{i-j}$ for normalized signals supported on a sub-interval of $I_\\mathcal {L}$ containing exactly the eigenvalues $\\lambda _{i}, \\lambda _{i+1},\\ldots ,\\lambda _{j}$ .",
"As an example, $f_{n-n}$ is a constant signal while $f_{1-2}$ is a highly non-smooth signal supported on the eigenspaces of large eigenvalues $\\lambda _1, \\lambda _2$ .",
"For experiments, the signals were calculated from the knowledge of $\\mathrm {sp}(\\mathcal {L})$ , and relevant projections of random functions on the graph.",
"We have compared the performances of Algorithms REF and REF against the thresholding procedure described in [15].",
"As the denoising method in [15] requires the computation of the whole spectral decomposition of the Laplacian, it does not scale to large graphs.",
"We stress here that we provide a fair comparison with [15], only in terms of denoising performance, and with no computational considerations.",
"Moreover, we choose for LocLets to use the most naive thresholding procedure by considering a global and scale independent threshold level.",
"For all the experiments below, the SGWT and LocLets are built upon the scale and kernel functions giving rise to the Parseval frame of [15], whose construction is recalled in Section REF .",
"More precisely, set respectively $\\varphi =\\sqrt{\\zeta _0}$ and $\\psi =\\sqrt{\\zeta _1}$ for the scale and kernel functions with $\\zeta _0(x)=\\omega (x)$ , and $\\zeta _1(x)=\\omega (b^{-1}x)-\\omega (x)$ , where we choose $b=2$ and $\\omega $ is piecewise linear, vanishes on $[1,\\infty )$ and is constant equal to one on $(-\\infty ,b]$ .",
"The scales are of the form $s_j=b^{-j+1}$ for $j=1, \\ldots , J$ where $J$ is chosen similarly to [15].",
"In what follows, `PF' stands for Parseval Frame and refers to the estimator of [15]; the estimators implemented by Algorithm REF and Algorithm REF are referred to as `LLet' and `LLet+PF' respectively.",
"The notation `$\\mathrm {SNR}_{\\mathrm {in}}$ ' refers to the trivial model releasing the noisy signal $\\widetilde{f}$ , corresponding to the classical input noise level measurement, and serves as a worst-case baseline for other models.",
"Below, the latter methodology is shown to outperform all the others for very frequency-sparse signals.",
"It is also worth recalling that `LLet+PF' benefits from the dimension reduction property of LocLets.",
"More precisely, whereas the whole eigendecomposition of $\\mathcal {L}$ is required to apply `PF', for Parseval frame denoising in the context of `LLet+PF', only a low-rank spectral decomposition is needed, namely the decomposition of $\\mathcal {L}_I$ for $I$ the estimate of ${\\rm supp}\\widehat{f}$ .",
"For all our experiments, we set $\\alpha =0.001$ for Algorithm REF .",
"For the denoising experiment, we compute the best SNR result $r_{D}$ over a large grid of values $(t_1, t_2)$ , and for each denoising method $D$ with $D \\in \\lbrace `\\mathrm {SNR}_{\\mathrm {in}}\\textrm {^{\\prime },`PF^{\\prime },`LLet^{\\prime },`LLet+PF^{\\prime }} \\rbrace $ .",
"Then, we calculate two metrics: the maximum $\\mathrm {M}_{D}$ and average value $\\mu _{D}$ of the values $r_{D}$ over 10 random perturbations of the signal $f$ .",
"We recall that a good quality in denoising is reflected by a large value of the SNR metric."
],
[
"Noise level estimation",
"We have evaluated the performances of estimators $\\widehat{\\sigma }_{\\rm mean}^{r}$ and $\\widehat{\\sigma }_{\\rm med}$ in the estimation of the unknown noise level $\\sigma $ from 10 realizations of the noisy signal $\\widetilde{f} = f + \\xi $ for a given noise level $\\sigma $ .",
"Figure REF (resp.",
"Figure REF ) shows the best performances of each estimator on the minnesota (resp.",
"net25) graph for the non-regular but frequency sparse signal $f_{1392-1343}$ (resp.",
"$f=f_{4971-5020}$ ), when parameter $K$ ranges in $\\lbrace 5, 10, 20, 30, 40, 50 \\rbrace $ and for level of noise $\\sigma =0.01$ (resp.",
"$\\sigma =0.001$ ).",
"Figure: Performances of estimators σ ^ mean \\widehat{\\sigma }_{\\rm mean} (left) and σ ^ med \\widehat{\\sigma }_{\\rm med} (right) for minnesota graph, signal f 1392-1343 f_{1392-1343} and σ=0.01\\sigma =0.01.Figure REF illustrates that both estimators $\\widehat{\\sigma }_{\\rm mean}^{r}$ and $\\widehat{\\sigma }_{\\rm med}$ can provide good estimates of $\\sigma $ .",
"Best performances are obtained for values of parameter $K$ below the elbow value $K_{elbow}(minnesota)=22$ introduced in Section REF .",
"We observe that performances drop considerably if almost no localization is used (for instance, for parameter values $K=1$ or $K=2$ , $\\widehat{\\sigma }\\sim 0.021$ in the experiment of Figure REF , far from the performances for $K\\ge 5$ for estimating $\\sigma =0.01$ ).",
"Figure: Performances of estimators σ ^ mean r \\widehat{\\sigma }_{\\rm mean}^r (left) and σ ^ med r \\widehat{\\sigma }_{\\rm med}^r (right) for net25 graph, signal f 4971-5020 f_{4971-5020} and σ=0.001\\sigma = 0.001.Figure REF shows that localization is necessary, namely $K\\ge 10$ or even $K\\ge 20$ , in order to reach the best performances for the large net25 graph.",
"Contrary to experiments for the minnesota graph, estimators $\\widehat{\\sigma }_{\\rm mean}^{r}$ and $\\widehat{\\sigma }_{\\rm med}$ underestimate the value of $\\sigma $ .",
"Also, best values of $K$ range between 10 and 30 for net25 graph, compared to best values $K=5$ and $K=10$ for minnesota graph (see Figure REF ).",
"This illustrates the idea that noise level estimation strongly depends on the underlying graph structure.",
"As a consequence, a parameter $K$ selection has to take graph and signal information into account to be relevant.",
"Interestingly, the elbow values $K_{elbow}(minnesota)=22$ and $K_{elbow}(net25)=22$ provide performances which are not optimal, but close to the best possible ones.",
"In Figure REF , performances for various values of parameter $r$ are displayed for a fixed parameter $K=K_{elbow}(minnesota)$ .",
"While it is true that $\\widehat{\\sigma }_{\\rm mean}^{r}$ can perform better than $\\widehat{\\sigma }_{\\rm med}^{r}$ , it happens only for very specific values of $r$ , which a priori depend on the signal regularity.",
"Without any further parameter selection, these observations suggest using the most robust estimator $\\widehat{\\sigma }_{\\rm med}$ in practice.",
"Figure: Dependence on parameter rr for minnesota graph, signal f 1392-1343 f_{1392-1343}, K=22K=22 and σ=0.1\\sigma =0.1 (left), σ=0.01\\sigma =0.01 (right)."
],
[
"Sparse signal denoising",
"As a first denoising experiment, we have compared the performances of `LLet', `PF' and `LLet+PF' for a fixed value $K=K_{elbow}$ given by the rule of thumb described in Section REF .",
"For each matrix in the Extended Matrices Suite, we have experimented the denoising task on two frequency-sparse signals, one regular and the other non-regular.",
"Several values of noise level $\\sigma $ were used, corresponding to values of $\\mathrm {SNR}_{\\mathrm {in}}$ ranging in $[4, 18]$ .",
"Results from experiments are displayed in Tables REF and REF .",
"The first obvious observation is that `LLet+PF' performs better than its competitors in almost all situations.",
"The gain is sometimes considerable since we observed a gap of 5dB in $\\mu _{D}$ -metric between `LLet+PF' and its closest concurrent `PF' in some cases, and up to 7dB in $\\mathrm {M}_{D}$ -metric.",
"These experiments confirm the theoretical guarantees obtained in Theorem REF .",
"The benefits of localization are reduced for graph net25: Table REF shows that the more conservative choice $K=5$ is better than $K=25$ .",
"It appears that for net25, the spectrum $\\mathrm {sp}(\\mathcal {L})$ is localized at a small number of distinct eigenvalues, hence diminishing the advantages of localizing with our methods.",
"Table: SNR performance for Swissroll (n=1000n=1000, K=22K=22).Another interesting observation is that `LLet' may outperform `PF' in some specific signal and noise level configurations, as shown in Table REF .",
"This is a very favorable result for localized Fourier analysis, since `LLet' appears to be a technique which is more accurate and more efficient as well compared to `PF' in some situations.",
"However in many cases, `LLet' performances drop down compared to the more stable thresholding techniques `PF' and `LLet+PF', which use thresholds adapted to the wavelet basis.",
"Table: SNR performances for denoising task.We also provide experimental results to understand the extent to which our results depend on the partition size parameter $K$ .",
"A few remarks are suggested by Figure REF : The best performances are not obtained for the elbow value $K_{elbow}$ , suggesting searching for a more task-adapted size of partition $K$ .",
"Good performances persist for values of $K$ much larger than $K_{elbow}$ , and in particular for regular signals.",
"For large values of $K$ , there is a severe drop in performances.",
"As explained before, the error generated by Chebyshev's approximation grows with the number of intervals in the partition, which makes the approximation of the support more difficult.",
"Figure: SNR performance depending on parameter KK for minnesota graph, σ=0.01\\sigma =0.01, a regular signal f 2593-2642 f_{2593-2642} (left) and a non-regular signal f 1343-1392 f_{1343-1392} (right), over 10 realizations of noise."
],
[
"Conclusion and future works",
"We have introduced a novel technique to perform efficiently graph Fourier analysis.",
"This technique uses functional calculus to perform Fourier analysis on different subsets of the graph Laplacian spectrum.",
"In this paper, we have demonstrated that localization in the spectrum provides interesting improvements in theoretical results for some graph signal analysis tasks.",
"New estimators of the noise level were introduced taking advantage of the convenient modelling of the denoising problem given by localization, and for which concentration results were proved.",
"Localization allows also to study theoretically the denoising procedure with wavelets, and fits with the design of many well-known techniques (e.g.",
"tight frames for graph analysis).",
"Through many experiments, we have validated that localization techniques introduced in this paper improve on state-of-the-art methods for several standard tasks.",
"Although we provide a rule of thumb to choose a partition $I_\\mathcal {L}= \\sqcup _{1\\le k\\le K} I_k$ for which denoising results show good performances, experiments suggest that our elbow rule is not optimal in most cases.",
"There is certainly an interesting topic in searching for a suitable partition $I_\\mathcal {L}= \\sqcup _{1\\le k\\le K} I_k$ that would be more adapted to a specific task (e.g.",
"denoising).",
"To extend the current work, it would also be interesting to consider other common tasks in GSP, such as de-convolution or in-painting."
],
[
"Proofs",
"[Proof of Proposition REF ] We have $\\tilde{f}_k = f_k + \\xi _k$ , where $\\xi _k=\\sum _{\\ell :\\lambda _\\ell \\in I_k} \\widehat{\\xi }(\\ell ) \\chi _\\ell $ .",
"Random variables $(\\widehat{\\xi }(\\ell ))_\\ell $ are all distributed as $\\mathcal {N}(0,\\sigma ^2)$ and independent by orthogonality of the eigenbasis $(\\chi _\\ell )_\\ell $ .",
"In particular for $k\\ne k^\\prime $ , vectors $\\xi _k$ and $\\xi _{k^\\prime }$ are independent as expressions involving variables $\\widehat{\\xi }(\\ell )$ over disjoint subsets $I_k$ and $I_{k^\\prime }$ .",
"Thus the random variables $(c_k)_{1\\le k\\le K}$ are also independent.",
"When $n_k = n_{k^\\prime }$ , $\\xi _k$ and $\\xi _{k^\\prime }$ are identically distributed and the result follows from Equality (REF ).",
"When $n_k \\ne n_{k^\\prime }$ , we have $\\mathbb {E}(c_k) \\ne \\mathbb {E}(c_{k^\\prime })$ as the following equality holds for all $1\\le k\\le K$ : $\\mathbb {E}(c_{k}) = \\frac{\\Vert f_k \\Vert _2^2}{n_k} + \\sigma ^2.$ Since $(\\widehat{\\xi }(\\ell ))_\\ell $ are independent normal variables $\\mathcal {N}(0,\\sigma ^2)$ , the statement is clear from the expression $c_k = \\frac{1}{n_k} \\sum _{\\ell :\\lambda _\\ell \\in I_k} | \\widehat{\\xi }(\\ell ) |^2$ .",
"The following lemma is useful for the proof of Proposition REF .",
"Lemma 13 Let $Z \\sim \\mathcal {B}(n,p)$ for some parameters $n \\ge 1$ and $p\\le 1/2$ .",
"Then $\\mathbf {P}(Z \\ge \\lceil n/2 \\rceil ) \\le \\exp \\left( \\frac{n}{2} \\ln ( 4p(1-p)) \\right).$ A simple consequence of [4] implies that for all $n \\ge 1$ and all $a \\ge p$ $\\mathbf {P}(Z \\ge na) \\le \\left[ \\frac{(1-p)^{1-a}}{1-a} \\left( \\frac{1-a}{a} p \\right)^a \\right]^n.$ Now the result follows since $na=\\lceil n/2 \\rceil $ implies $\\frac{1}{2} \\le a \\le \\frac{1}{2}+\\frac{1}{n}$ so that $\\frac{(1-p)^{1-a}}{1-a} \\left( \\frac{1-a}{a} p \\right)^a \\le \\frac{(1-a)^{a-1}}{a^a}\\sqrt{p(1-p)} \\le \\sqrt{4p(1-p)}.$ [Proof of Proposition REF ] For $k$ such that $I_k \\subset \\overline{I_f}$ , we have $c_k = \\frac{\\sigma ^2}{n_k} \\Gamma _{n_k}$ , which follows the $\\Gamma (\\frac{n_k}{2}, \\frac{2\\sigma ^2}{n_k})$ distribution.",
"Then concentration inequalities for $\\widehat{\\sigma }_{\\rm mean}(c)^2$ are a direct consequence of Theorem 2.57 in [3], applied with $a_k = \\frac{n_k}{2}$ and $b_k = \\frac{2\\sigma ^2}{n_k}$ .",
"For all $k=1, \\ldots , K_f$ , we define $\\gamma ^-_k = \\Phi _{\\Gamma _{n_0}}^{-1} \\circ \\Phi _{\\Gamma _{n_k}} \\left(\\frac{n_k}{\\sigma ^2} c_k \\right) \\quad \\textrm {and} \\quad \\gamma ^+_k = \\Phi _{\\Gamma _{n_\\infty }}^{-1} \\circ \\Phi _{\\Gamma _{n_k}} \\left(\\frac{n_k}{\\sigma ^2} c_k \\right).$ As a matter of fact, $(\\gamma ^-)_{k=1, \\ldots , K_f}$ and $(\\gamma _k^+)_{k=1, \\ldots , K_f}$ are two sequences of i.i.d.",
"random variables with $\\gamma _1^- \\sim \\chi ^2(n_0)$ and $\\gamma ^+_1 \\sim \\chi ^2(n_\\infty )$ such that $\\forall k=1, \\ldots , K_f, \\quad \\gamma _k^- \\le \\frac{n_k}{\\sigma ^2} c_k \\le \\gamma ^+_k \\quad \\textrm {almost surely}.$ Then, for all $t>0$ , $\\mathbb {P} \\left( \\widehat{\\sigma }^2_{\\rm med} \\ge \\beta ^{-1} \\sigma ^2 + 2\\sigma ^2 \\beta ^{-1} t \\right) & = \\mathbb {P} \\left( \\sum _{k=1}^{K_f} \\mathrm {1}_{\\left\\lbrace c_k \\ge \\beta ^{-1} \\sigma ^2+2\\sigma ^2 \\beta ^{-1} t \\right\\rbrace } \\ge \\left\\lceil \\frac{K_f}{2} \\right\\rceil \\right) \\nonumber \\\\& \\le \\mathbb {P} \\left( \\sum _{k=1}^{K_f} \\mathrm {1}_{ \\left\\lbrace \\gamma _k^+ \\ge n_\\infty + 2 n_\\infty t \\right\\rbrace } \\ge \\left\\lceil \\frac{K_f}{2} \\right\\rceil \\right).$ Similarly, for all $t \\in (0,1)$ , $\\mathbb {P} \\left( \\widehat{\\sigma }^2_{\\rm med} \\le \\beta \\sigma ^2-\\sigma ^2 \\beta t \\right)& \\le \\mathbb {P} \\left( \\sum _{k=1}^{K_f} \\mathrm {1}_{ \\left\\lbrace \\gamma _k^- \\le n_0 - n_0 t \\right\\rbrace } \\ge \\left\\lceil \\frac{K_f}{2} \\right\\rceil \\right).$ To conclude, apply Lemma REF to Inequalities (REF ) and (REF ) to obtain our result.",
"[Proof of Proposition REF ] The concentration bound of Equation (REF ) implies that $\\mathbb {P}(b_k \\ge b_\\ell )& = \\mathbb {P}(n_k b_k \\ge n_k b_\\ell ) = \\mathbb {P} \\left( \\sigma ^{-2} \\left( \\frac{n_k}{n_\\ell } \\Vert \\xi _\\ell \\Vert _2^2 - \\Vert \\xi _k \\Vert _2^2 \\right) \\le \\sigma ^{-2} \\left(\\Vert f_k\\Vert _2^2 + 2 \\langle f_k, \\xi _k \\rangle \\right) \\right) \\\\& \\ge \\frac{\\alpha }{2}+\\mathbb {P} \\left( \\sigma ^{-2} \\left( \\frac{n_k}{n_\\ell } \\Vert \\xi _\\ell \\Vert _2^2 - \\Vert \\xi _k \\Vert _2^2 \\right) \\le \\sigma ^{-2} \\left(\\Vert f_k\\Vert _2^2 + 2 t_{\\alpha , \\sigma } \\Vert f_k \\Vert _2 \\right) \\right).$ Since subsets $I_k$ and $I_\\ell $ are disjoint, random variables $\\Vert \\xi _k \\Vert _2^2$ and $\\Vert \\xi _\\ell \\Vert _2^2$ are independent.",
"Thus, $\\frac{n_k}{n_\\ell } \\Vert \\xi _\\ell \\Vert _2^2 - \\Vert \\xi _k \\Vert _2^2$ is distributed as $\\frac{n_k}{n_\\ell } \\Gamma _{n_\\ell } - \\Gamma _{n_k}$ where $\\Gamma _{n_k}$ and $\\Gamma _{n_\\ell }$ are independent random variables with $\\Gamma _{n_k} \\sim \\chi ^2(n_k)$ and $\\Gamma _{n_\\ell } \\sim \\chi ^2(n_\\ell )$ .",
"Therefore, the statement of Proposition REF follows.",
"[Proof of Proposition REF ] First, the following equalities hold: $f-\\widetilde{f}= f-\\widetilde{f}_I + \\widetilde{f}_I - \\widetilde{f}= (f-\\widetilde{f})_I + \\widetilde{f}_{\\overline{I}}.$ As $(f-\\widetilde{f})_I$ and $\\widetilde{f}_{\\overline{I}}$ are orthogonal vectors, it follows that $\\left\\Vert f-\\widetilde{f} \\right\\Vert _2^{2} = \\left\\Vert f-\\widetilde{f}_I \\right\\Vert _2^{2} + \\left\\Vert \\widetilde{f}_{\\overline{I}} \\right\\Vert _2^{2}.$ It remains to notice that $\\mathbb {E}( \\vert \\vert \\widetilde{f}_{\\overline{I}} \\vert \\vert ^{2} ) = \\sigma ^{2} \\vert \\overline{I} \\cap \\mathrm {sp}(\\mathcal {L}) \\vert $ .",
"[Proof of Lemma REF ] By Equation (REF ) and the concentration bound of Equation (REF ), it follows that $\\begin{split}p_k & = \\mathbb {P}( \\sigma ^2 \\Gamma _{n_k} > \\Vert \\xi _{k}\\Vert _2^2 + \\Vert f_k\\Vert _2^2 + 2 \\langle f_k, \\xi _k\\rangle )\\\\&\\le \\frac{\\alpha }{2} + \\mathbb {P} \\left( \\sigma ^2 \\Gamma _{n_k} > \\Vert \\xi _{k}\\Vert _2^2 + \\Vert f_k\\Vert _2^2 - 2 \\Vert f_k\\Vert _2 t_{\\alpha / 2,\\sigma } \\right) \\\\& = \\frac{\\alpha }{2} + \\mathbb {P} \\left( \\sigma ^2 (\\Gamma _{n_k} - \\Gamma _{n_k}^\\prime ) > \\Vert f_k\\Vert _2^2 - 2 \\Vert f_k\\Vert _2 t_{\\alpha / 2,\\sigma } \\right) \\\\& = \\frac{\\alpha }{2} + 1 - \\Phi _{\\Gamma _{n_k} - \\Gamma _{n_k}^\\prime }( \\theta (f_k,\\alpha ,\\sigma ) ),\\end{split}$ where $\\theta (f_k,\\alpha ,\\sigma ) = \\sigma ^{-2}( \\Vert f_k\\Vert _2 - 2 t_{\\alpha / 2,\\sigma }) \\Vert f_k\\Vert _2 $ .",
"Consequently, $1 - \\Phi _{\\Gamma _{n_k} - \\Gamma _{n_k}^\\prime }( \\theta (f_k,\\alpha ,\\sigma ) ) \\le \\alpha /2$ and $p_k \\le \\alpha $ .",
"[Proof of Lemma REF ] Using an estimate on the $\\chi ^2(n_k)$ tail distribution and independence of $\\Gamma _{n_k}$ and $\\Gamma _{n_k}^\\prime = \\sigma ^{-2} \\Vert \\xi _k \\Vert _2^2$ , it follows $\\begin{split}p_k & \\ge \\mathbb {P} \\left( \\sigma ^2 \\Gamma _{n_k} > \\Vert \\xi _k\\Vert _2^2 + \\Vert f_k\\Vert _2^2 + 2\\langle \\xi _k, f_k \\rangle , \\Vert \\xi _k \\Vert _2^2 \\le t_{\\beta , k} \\right) \\\\& \\ge \\mathbb {P} \\left( \\sigma ^2 \\Gamma _{n_k} > \\Vert \\xi _k\\Vert _2^2 + \\Vert f_k\\Vert _2^2 + 2 \\Vert \\xi _k\\Vert _2 \\Vert f_k\\Vert _2, \\Vert \\xi _k \\Vert _2^2 \\le t_{\\beta , k} \\right) \\\\& = \\mathbb {P} \\left( \\sigma ^2 \\Gamma _{n_k} > ( \\Vert f_k\\Vert _2 + \\sqrt{t_{\\beta , k}} )^2,\\Vert \\xi _k \\Vert _2^2 \\le t_{\\beta , k} \\right) \\\\& = \\mathbb {P} \\left( \\sigma ^2 \\Gamma _{n_k} > ( \\Vert f_k\\Vert _2 + \\sqrt{t_{\\beta , k}})^2, \\sigma ^2 \\Gamma _{n_k}^\\prime \\le t_{\\beta , k} \\right) \\\\& \\ge \\mathbb {P} \\left( \\sigma ^2 \\Gamma _{n_k} > ( \\Vert f_k\\Vert _2 + \\sqrt{ t_{\\beta , k} } )^2 \\right) (1 - \\beta ) \\\\&\\ge \\frac{\\alpha }{1 - \\beta } \\times (1 - \\beta )=\\alpha .\\end{split}$ [Proof of Proposition REF ] To prove Inequality (REF ), first observe that $f = \\sum _{k: I_k\\subset I_f} f_k$ so that $f-f_I = \\sum _{k: I_k\\subset I_f} f_k - \\sum _{k: p_k \\le \\alpha } f_k.$ The summands which are not present in both terms are exactly those satisfying either $I_k\\subset I_f$ and $p_k>\\alpha $ or $I_k \\cap I_f = \\emptyset $ and $p_k\\le \\alpha $ .",
"Noting that $f_k = 0$ when $I_k \\cap I_f = \\emptyset $ , it comes $\\Vert f-f_I\\Vert _2^2 = \\sum _{k: I_k\\subset I_f, p_k > \\alpha } \\Vert f_k\\Vert _2^2.$ Applying Lemma REF for all indices $1\\le k \\le K$ satisfying $p_k > \\alpha $ , one deduce $\\Vert f_k \\Vert _2 < t_{\\alpha / 2, \\sigma } + \\sqrt{ t_{\\alpha / 2, \\sigma }^2 + \\left( \\sigma \\Phi _{\\Gamma _{n_k} - \\Gamma _{n_k}^{\\prime }}^{-1}( 1- \\frac{\\alpha }{2} ) \\right)^2 }.$ from which, since $n_k=n_1$ for all $k$ , Inequality (REF ) follows.",
"Since $\\sigma ^{-2}||\\xi _k||_2^2$ is distributed as a $\\chi ^2(n_1)$ random variable, the second Inequality (REF ) follows $\\mathbb {E} \\Vert f_I - \\widetilde{f}_I \\Vert _2^2 = \\sum _{k: p_k\\le \\alpha } \\mathbb {E} \\Vert \\xi _k\\Vert _2^2= |\\lbrace k, p_k \\le \\alpha \\rbrace | n_1 \\sigma ^2.$ [Proof of Theorem REF ] Since threshold value is $t_2 = 0$ on $\\overline{I}$ , $\\widehat{f} = \\widehat{f}_I$ .",
"Then, clearly $f_{\\overline{I}} (f_I - \\widehat{f}_I) = 0$ almost surely so that $\\mathbb {E} \\Vert f-\\widehat{f}\\Vert _2^2= \\mathbb {E} \\Vert f-\\widehat{f}_I\\Vert _2^2= \\mathbb {E} \\Vert f - f_I + f_I - \\widehat{f}_I \\Vert _2^2= \\mathbb {E} \\Vert f - f_I \\Vert _2^2 + \\mathbb {\\Vert } f_I - \\widehat{f}_I \\Vert _2^2.$ Applying Theorem 3 from [15] to $\\mathbb {E} \\Vert f_I-\\widehat{f_I} \\Vert _2^2$ yields our statement.",
"[Proof of Proposition REF ] Recalling that, for any function $g$ defined on $\\mathrm {sp}(\\mathcal {L})$ and any subset $I\\subset I_{\\mathcal {L}}$ , $\\sum _n | \\langle \\sqrt{g}(\\mathcal {L}_{I})\\delta _n , f \\rangle |^2 = \\Vert \\sqrt{g}(\\mathcal {L}_{I})f \\Vert _2^2 = \\langle g(\\mathcal {L}_{I})f, f \\rangle $ it follows by Equations (REF ) and (REF ).",
"$\\sum _{n,k} |\\langle \\varphi _{n,k}, f\\rangle |^2 + |\\langle \\psi _{1,n,k}, f\\rangle |^2 + |\\langle \\psi _{2,n,k}, f\\rangle |^2 \\\\= \\sum _{k} \\langle \\psi _0(\\mathcal {L}_{I_k}) f, f\\rangle + \\langle \\psi _k(\\mathcal {L}_{I_k}) f, f\\rangle + \\langle \\psi _k(\\mathcal {L}_{I_{k+1}}) f, f\\rangle \\\\= \\langle \\psi _0(\\mathcal {L}) f, f\\rangle + \\sum _k \\langle \\psi _k(\\mathcal {L}) f, f\\rangle = \\Vert f \\Vert _2^2$ [Proof of Proposition REF ] Remarking that $\\widehat{g}_j(\\lambda )=\\psi (s_j \\omega _0(\\lambda ) )$ with $s_j = e^{-j \\omega _{\\gamma ,J,R}}$ , Equation (REF ) implies that $g_{m,j} = \\sum _l \\psi (s_j \\omega _0(\\lambda _l)) \\widehat{\\delta }_m(l) \\chi _l.$ Setting $J_j = [C^{-1} e^{(j-R)\\omega _{\\gamma ,J,R}} , C^{-1} e^{j\\omega _{\\gamma ,J,R}} ]$ and recalling that ${\\rm supp}(\\widehat{g}) = [ -R\\omega _{\\gamma ,J,R}, 0]$ , it follows that $\\lambda \\in \\mathrm {supp}(\\widehat{g}_j)$ if and only if $\\omega _0(\\lambda ) \\in J_j$ if and only if $s_j\\omega _0(\\lambda ) \\in J_0$ .",
"Moreover, $J_0=\\sqcup _k I_k$ and $J_j=s_j^{-1} J_0$ yield $J_j=\\sqcup _k s_j^{-1} I_k$ with $s_j^{-1} I_k \\cap s^{-1}_{j^\\prime } I_{k^\\prime }$ excepted when $j=j^\\prime $ ad $k=k^\\prime $ .",
"Consequently, Equation (REF ) can be reformulated as $g_{m,j} = \\sum _{1\\le k\\le R-1} \\sum _{\\ell \\in I_k} \\psi (s_j \\omega _0(\\lambda _\\ell )) \\widehat{\\delta }_m(\\ell ) \\chi _\\ell .$"
]
] | 1906.04529 |
[
[
"Reddening map and recent star formation in the Magellanic Clouds based\n on OGLE IV Cepheids"
],
[
"Abstract In the present study, we examine reddening distribution across the LMC and SMC through largest data on Classical Cepheids provided by the OGLE Phase IV survey.",
"The V and I band photometric data of 2476 fundamental mode (FU) and 1775 first overtone mode (FO) Cepheids in the LMC and 2753 FU and 1793 FO Cepheids in the SMC are analyzed for their Period-Luminosity (P-L) relations.",
"We convert period of FO Cepheids to corresponding period of FU Cepheids before combining the two modes of Cepheids.",
"The reddening analysis is performed on 133 segments covering a total area of about 154.6 deg^2 in the LMC and 136 segments covering a total area of about 31.3 deg^2 in the SMC.",
"By comparing with well calibrated P-L relations of these two galaxies, we determine reddening E(V-I) in each segment.",
"Using reddening values in different segments across the LMC and SMC, reddening maps are constructed.",
"We find clumpy structures in the reddening distributions of the LMC and SMC.",
"From the reddening map of the LMC, highest reddening of E(V-I) = 0.466 mag is traced in the region centered at RA ~ 85.13 deg, DEC ~ -69.34 deg which is in close vicinity of the star forming HII region 30 Doradus.",
"In the SMC, maximum reddening of E(V-I) = 0.189 mag is detected in the region centered at RA ~ 12.10 deg, DEC ~ -73.07 deg.",
"The mean reddening values in the LMC are estimated as E(V-I) = 0.113+/-0.060 mag and E(B-V) = 0.091+/-0.050 mag; and that in the SMC are E(V-I) = 0.049+/-0.070 mag and E(B-V) = 0.038+/-0.053 mag.",
"The period-age relations are used to derive the age of the Cepheid populations in the LMC and SMC.",
"We investigate age and spatio-temporal distributions of Cepheids to understand the recent star formation history in the Magellanic Clouds (MCs) and found an evidence of a common enhanced Cepheid population in the MCs at around 200 Myr ago which appears to have occurred due to close encounter between the two clouds."
],
[
"Introduction",
"The LMC and SMC (together known as Magellanic Clouds, MCs) are among one of the most studied galaxies in the Universe due to their close proximity to the Galaxy as they are located at the distance of $\\approx $ 50 kpc and 60 kpc, respectively [108], [36], [49], [18].",
"They offer an excellent opportunity to strive many astronomical issues such as star formation, structure formation, distribution of interstellar medium through wide range of tracers and they have significantly advanced our understanding on galaxy evolution and dynamical interaction [33], [86], [75], [70], [64].",
"The MCs provide the ideal laboratory to probe the spatially resolved star formation history (SFH) and dust distributions since these galaxies are not just close enough to be resolved into stars but also moderately affected by the interstellar extinction and foreground Milky Way stars [79].",
"In recent times, the distribution of different stellar populations in the MCs has been studied by various authors [2], [72], [73], [65], [11], [10], [34], [31], [17], [95], [42], [41], [60], [45], [79], [14], among others.",
"For example, [11], [10] suggested the existence of homogeneous, old and metal-poor stellar halo in the LMC based on the velocity dispersion of RR Lyrae stars.",
"On the basis of spatial and age distributions of Cepheids reported in the Optical Gravitational Lensing Experiment (OGLE) Phase-III survey, [42] and [45] affirmed that a major star formation triggered in these two dwarf galaxies at $\\approx 200\\pm 50$ Myr ago.",
"[14] revealed a significant warp in the south west region of LMC outer disk towards SMC and an off-centered tilted LMC bar which they believe is consistent with a direct collision between the two clouds in agreement with the earlier studies [6], [12], [112], [111].",
"On the basis of VMC data in the near-infrared filters, [79] provided maps for the mass distribution of stars of different ages, and a total mass of stars ever formed in the SMC.",
"These kind of studies are possible due to availability of enormous amount of data generated by the large sky surveys in these directions like Massive Compact Halo Objects Survey [3], Optical Gravitational Lensing Experiment [101], Magellanic Clouds Photometric Survey [109], Two Micron All Sky Survey [87], VISTA survey of Magellanic Clouds system [17], Survey of the MAgellanic Stellar History [64], etc.",
"It has long been observed that reddening (a measurement of the selective total dust extinction) information is one of the important parameter to estimate the structure of the disk of the MCs as well as deriving SFH in these two dwarf galaxies.",
"For example, [95] suggested that the extra-planer features which are found both in front of and behind the disk could be in the plane of the disk itself if there would be an under-estimate or over-estimate of the extinction values in the direction of the LMC.",
"[79] derived the extinction $A_V$ varying between $\\sim $ 0.1 mag to 0.9 mag across the SMC and found that the high-extinction values follow the distribution of the youngest stellar populations.",
"By simultaneously solving SFH, mean distance, reddening, etc, they were able to get more reliable picture of how the mean distances, extinction values, SF rate, and metallicities vary across the SMC.",
"[14] presented $E(g-i)$ reddening map across the LMC disk through the colour of red clump (RC) stars and suggested that the majority of the reddening toward the non-central regions results from the Milky Way foreground, which acts as a dust screen on stars behind it.",
"They were able to explore the detailed three-dimensional structure of the LMC and detect a new stellar warp by using their accurate and precise two-dimensional reddening map in the LMC disk.",
"In our analysis, we use Classical Cepheids or Population I Cepheids (here onward we simply use the term Cepheids for these objects) to map the reddening and probe the recent SFH within the MCs.",
"Cepheids are relatively young massive stars in the core He-burning phase and occupy the space in a well defined instability strip in the H-R diagram.",
"These are regarded as an excellent tracer for the understanding of recent star formation in the host galaxy and one of the most important establishment of the cosmic distance ladder.",
"They play a pivotal role in structure studies of nearby external galaxies like MCs, M31, and M33, among others as they are found in plenty in these galaxies.",
"These pulsating variables, because of their large intrinsic brightness and periodic luminosity variation over time, can be easily detected in the outskirts of the Galactic disk and in nearby galaxies.",
"Cepheids pulsate in two modes, one called FU mode in which all parts of the system move sinusoidally with the same frequency and a fixed phase relation, and other are called FO Cepheids which pulsate with short period and small amplitude.",
"The distinction between FU and FO Cepheids can easily be made using their positions in the $P$ -$L$ diagram.",
"Cepheids pulsating in the FO mode are about 1 mag brighter and relatively lower amplitude than the FU mode for the same pulsation period [102], [9].",
"One can also distinguish the two modes through the Fourier parameters $R_{21}$ and $\\phi _{21}$ determined from the shape of their light curves.",
"During the past few decades, many studies have been directed towards the understanding structure and reddening distribution in the MCs employing the Cepheids [102], [65], [96], [42], [45], [40], [84].",
"In the present study, we primarily aim to understand the reddening variation across the MCs through multi-band $P$ -$L$ relations of the Cepheids by taking advantage of largest and most homogeneous sample of FU and FO Cepheids spread all across the MCs in the recently released catalogue of OGLE-IV survey.",
"Here, we construct large number of narrow sub-regions in these two galaxies and select significant number of Cepheids in each sub-region to draw respective P-L diagrams.",
"Furthermore, we convert period of FO Cepheids to the corresponding period of FU Cepheids in order to make a single $P$ -$L$ diagram with increased sample size of Cepheids.",
"Then using the multi-band $P$ -$L$ diagrams in each sub-region, we estimate the corresponding reddening values.",
"In this way, we obtain the reddening in all such sub-regions that are used to construct the reddening map which is of vital importance to study the dust distribution within different regions of these two galaxies.",
"We also estimate the age of Cepheids in order to probe the recent SFH in the two clouds.",
"This paper is structured as follows.",
"In Section , we describe the detail of the data used in the present study.",
"The spatial distribution of Cepheids in the MCs is described in Section .",
"The reddening determinations and construction of reddening maps are presented in Section .",
"Our analysis on the Cepheids age determination and recent SFH in the MCs is given in Section .",
"The discussion and conclusion of this study are summarized in Section ."
],
[
"Data",
"There has been plenty of surveys for Cepheids in the MCs, however, OGLEhttp://ogle.astrouw.edu.pl/ has revolutionized the field by producing thousands of Cepheid light curves in the Galaxy, the LMC and the SMC with high signal-to-noise ratio and determining very accurate parameters like periods, magnitudes and amplitudes.",
"OGLE has begun its survey in 1992 in its first phase, followed by three more phases appending additional sky coverage.",
"The fourth phase of OGLE survey has been carried out using a 32-chip mosaic CCD camera on a 1.3-m Warsaw University Telescope at Las Campanas Observatory, Chile between 2010 March and 2015 July which has increased observing capabilities by almost an order of magnitude compared to OGLE-III phase and covered over 3000 square degrees in the sky [90].",
"The complete detail of the reduction procedures, photometric calibrations and astrometric transformations is available in [103].",
"Recently OGLE-IV survey has released high quality photometric data in the $V$ and $I$ band from their observations of Magellanic System [89], [91].",
"This work utilizes the publicly available photometric catalog of variable stars consisting of 4704 Cepheids in the LMC and 4945 Cepheids in the SMC.",
"The LMC Cepheids sample consists of 2476 FU, 1775 FO, 26 second-overtone (SO), 95 double-mode FU/FO, 322 double-mode FO/SO, 1 double mode FO/Third overtone (TO),1 double mode SO/TO, and 8 triple-mode Cepheids [89].",
"Similarly, the SMC sample consists of 2753 $FU$ , 1793 $FO$ , 91 $SO$ , 68 $FU/FO$ , 239 $FO/SO$ , and one triple-mode Cepheids [91].",
"The sample is reported to be over 99% complete making it the most complete and least contaminated sample of Cepheids in the Magellanic System [41].",
"For the present study, we have only used the archival $V$ and $I$ band data containing the mean magnitude and period of FU and FO Cepheids for both the dwarf galaxies.",
"In total, we used simultaneous $V$ and $I$ band data of 4251 Cepheids in the LMC and 4546 Cepheids in the SMC.",
"The photometric uncertainty on OGLE-IV mean magnitudes is found to be $\\sigma _{I,V} = 0.007$ mag for brighter and longer period Cepheids ($0.7 \\le log~P \\le 1.5$ ) and $\\sigma _{I,V} = 0.02$ mag for fainter and shorter period Cepheids ($log~P < 0.7$ ) [40]."
],
[
"Spatial Distribution",
"During the last several years, a large number of studies based on different stellar populations have been carried out to understand the structure of the MCs [31], [106], [95], [42], [45], [59].",
"In the present study, we have a total of 4251 Cepheids in the LMC and 4546 Cepheids in the SMC, largest sample among any previous studies so far which provides a chance to examine their spatial distributions within the clouds in order to understand the geometry of the disk of these galaxies.",
"To carry out this analysis, we first converted the ($RA$ , $DEC$ ) coordinates to ($X$ , $Y$ ) coordinates using the relations given by [104].",
"Then we divided the LMC in $45\\times 45$ segments with a dimension of $0.4\\times 0.4$ kpc square for each segment having a mean spatial resolution of $\\approx $ 1.2 deg$^2$ .",
"For the SMC, we divided in $90\\times 90$ segments with a smaller dimension of $0.2\\times 0.2$ kpc square having a mean spatial resolution of $\\approx $ 0.22 deg$^2$ because SMC contains higher density of Cepheids in its relatively small area.",
"We kept the same spatial cell size for all subsequent analysis in our study.",
"The cell size was chosen in such a way that we can get enough Cepheids in each cell for a statistically meaningful determination of mean reddening and recent SFH measurements.",
"Our selection has resulted 389 segments covering a total area of about 456.7 deg$^2$ in the LMC and 562 segments covering a total area of about 131.3 deg$^2$ in the SMC.",
"We counted the total number of Cepheids in each segment in the LMC and SMC, and constructed the spatial map of the Cepheids distribution.",
"Figure: Two-dimensional spatial map in the LMC as a function of number of Cepheids measured in different segments.",
"North is up and east is to the left.",
"The location of the optical center of the LMC is shown by plus sign.Figure: Same as Figure but for the SMC.In Figure REF , we show the two-dimensional spatial distribution of Cepheids in each of 389 segments in the LMC.",
"The colour code in the map represents the number of Cepheids in each segment.",
"The observed spatial maps in the LMC shows some interesting structures.",
"The center of the Cepheids density region in the LMC is found to be at $X=-1.2$ , $Y=-0.8$ which corresponds to the coordinates $\\alpha \\sim 84^{o}.00$ and $\\delta \\sim -70^{o}.29$ .",
"It is apparent that the stellar overdensities do not match with the optical center of the LMC but shifted far away towards south-east direction.",
"The spatial distribution of Cepheids along the disk is very distinctive and a bar structure is conspicuous where most of the Cepheids are located.",
"It is also seen that the density structure of the bar region is not smooth.",
"Many Cepheids are distributed in the clumpy structure along the bar which is elongated in the east-west direction where eastern side of the bar shows higher density of Cepheids as compared to its western side.",
"On the basis of RC stars in the LMC, the presence of warp in the bar was also noticed by [92] indicating a dynamically disturbed structure for the LMC.",
"The spatial density of Cepheids in the LMC is found to be very poor in its northern arm.",
"In Figure REF , we present similar two-dimensional spatial map for the SMC constructed from 562 segments of our chosen size.",
"As one can see in the map, disk of the SMC seems to be quite irregular and asymmetric.",
"It is evident that the south-western region of the SMC is the most populated and centroid of the Cepheids density distribution is located at $\\alpha \\sim 12^{o}.73$ , $\\delta \\sim -73^{o}.07$ .",
"While dense region is much farther away in the LMC from its optical center, the centroid of the most populated region having Cepheids lies very close to the optical center in the SMC."
],
[
"Methodology of reddening determination",
"The dust and gas is heterogeneously distributed within the MCs and a differential extinction is found to exist in these two galaxies hence a constant extinction cannot be applied in deriving $P$ -$L$ relations across the LMC and SMC.",
"However, one can study the extinction variation in these galaxies by determining reddening in small sub-regions.",
"Since thousands of Cepheids are available in the MCs, we can divide both the galaxies into small segments containing tens of Cepheids in each zone.",
"For each segment, $P$ -$L$ diagram is plotted in the form of $m_\\lambda = a_\\lambda ~logP + ZP_\\lambda $ where $m_\\lambda $ , $a_\\lambda $ , $P$ , and $ZP_\\lambda $ denote apparent magnitude, $P$ -$L$ slope, period of Cepheid, and zero point, respectively for a given bandpass.",
"Once slope $a_\\lambda $ is known through calibrated $P$ -$L$ relations, one can obtain the $ZP_\\lambda $ .",
"As OGLE data is given in $V$ and $I$ band and there are many calibrated $P$ -$L$ relations for Cepheids in these bands derived in the past studies for the MCs [102], [51], [30], [81], [82], [63], [62], we used the most recent ones given by [63] for the LMC FU Cepheids as following: $\\ M_{V} = -2.769(\\pm 0.023)~logP + 17.115(\\pm 0.015)$ $\\ M_{I} = -2.961(\\pm 0.015)~logP + 16.629(\\pm 0.010)$ For the SMC FU Cepheids, we used the $P$ -$L$ relation given by [62] as following: $\\ M_{V} = -2.660(\\pm 0.040)~logP + 17.606(\\pm 0.028)$ $\\ M_{I} = -2.918(\\pm 0.031)~logP + 17.127(\\pm 0.022)$ where $M_{V}$ and $M_{I}$ denote absolute magnitudes in $V$ and $I$ band, respectively.",
"We note here that no significant metallicity gradient has been seen across the LMC and SMC [32], [16], [69], [29], [20], [45] hence any variation in the MCs $P$ -$L$ relations due to metallicity variation is not considered in the present study.",
"Even if we accept any variation in the metallicity at all as reported in some previous studies [99], [13], [48], [23] the difference in reddening is found to be $\\sim $ 0.001 mag which is insignificant in comparison of the scatter in the $P$ -$L$ diagrams itself.",
"Moreover, there are many studies reported the non-linearity in the $P$ -$L$ relations in these galaxies [27], [102], [85], [47], [82], [88], [98], [96], [7].",
"A wide range of break points has been reported at different places in the $P$ -$L$ diagrams (1.0, 2.5, 2.9, 3.55, and 10 day) which also varies in different bands and different overtones.",
"However, we did not consider any breaks in the $P$ -$L$ diagrams because uncertainties from photometry and low number statistics in the present data were larger than the uncertainties from breaks.",
"With these assumptions, we determine the reddening in large number of segments across these two galaxies.",
"Figure: The PP-LL diagrams in the LMC for the FU and FO Cepheids in the left panels.",
"The right panels show the combined FU and FO Cepheids after converting period of FO Cepheids to corresponding periods of FU Cepheids.",
"Upper panels are for VV filter and lower panels are for II filter.",
"Points shown in blue color are for FU Cepheids and red color are for FO Cepheids.It has been however observed through the $P$ -$L$ diagram of FU and FO Cepheids that for the same pulsation period FU Cepheids have greater magnitude as compared to FO Cepheids and they follow two different $P$ -$L$ relations.",
"Hence, to combine the FU and FO Cepheids together in a single $P$ -$L$ diagram, one requires to first convert period of FO Cepheids into the corresponding period of FU Cepheids which requires a well defined relation connecting the two periods.",
"As vast majority of Cepheids pulsates only in a single mode, there are only a limited number of Cepheids which pulsate in two modes simultaneously [53].",
"Using the high resolution spectroscopic observations of 17 such double mode Galactic Cepheids, [97] derived a relation for transformation of FO and FU periods as following: $\\frac{P_{1}}{P_{0}} = -0.0143~logP_{0}-0.0265~\\left[\\frac{Fe}{H}\\right]+0.7101$ $~~~~~~~~~\\pm 0.0025~~~~~~~~~\\pm 0.0044~~~~~~~~~~~\\pm 0.0014$ where $P_{1}$ and $P_{0}$ denote period of FO Cepheid and corresponding period of FU Cepheid, respectively.",
"$\\left[\\frac{Fe}{H}\\right]$ denotes the metallicity.",
"We converted periods of FO Cepheids to those of the corresponding periods of FU Cepheids to draw $P$ -$L$ diagrams of combined sample of FU and FO Cepheids.",
"Here, we considered a mean present-day metallicity of $-0.34\\pm 0.03$ dex for the LMC based on Cepheids [49] and $-0.70\\pm 0.07$ dex for the SMC based on supergiants [37], [105].",
"Figure: Same as Figure but for the SMC.In Figures REF and REF , we illustrate $V$ and $I$ band $P$ -$L$ diagrams in the LMC and SMC, respectively.",
"In each diagram, we show FU and FO Cepheids independently in the left panel and combined FU and FO Cepheids in the right panel after the period conversion.",
"It is conspicuous that FU and FO Cepheids follow two different $P$ -$L$ relations and after converting FO periods to corresponding FU periods using eq.",
"REF , they fall on a single $P$ -$L$ relation in their respective filters in both the LMC and SMC, as can be seen in the right plots of Figures REF and REF .",
"This approach allows us to combine two different modes of Cepheids in a single P-L relation which in turn gives us a larger sample of Cepheids for the study.",
"We then draw multi-band period versus magnitude diagrams in each segment and fit the calibrated $P$ -$L$ relation to determine the corresponding intercept.",
"The difference in intercepts between calibrated and observed $P$ -$L$ relations yields the $ZP$ value for a given passband.",
"$ZP = ZP_{\\sl observed} - ZP_{\\sl calibrated}$ Here, $ZP_{calibrated}$ in the $V$ and $I$ bands are respectively taken as 17.115 mag and 16.629 mag for the LMC and 17.606 and 17.127 for the SMC as given in the eqs.",
"2 to 5.",
"The $ZP_{observed}$ varies for each segment mainly according to their relative dust extinction, whereas $P$ -$L$ slope is almost fixed in a galaxy.",
"Once we know the $ZP$ values in two different passbands, we derive the reddening in the selected region by subtracting them as follows: $E(V-I) = ZP_V - ZP_I$ where $ZP_V$ and $ZP_I$ are the zero points in $V$ and $I$ band, respectively.",
"Assuming that the MCs Cepheids follow the same reddening law as that of Galactic Cepheids, we can determine reddening $E(B-V)$ using the following reddening ratio [107]: $E(B-V) = E(V-I)/1.32$ In this way, one can estimate reddening $E(V-I)$ and $E(B-V)$ in each segment of the LMC and SMC.",
"The uncertainties in $E(V-I)$ and $E(B-V))$ values are calculated by combining the error in the $ZP$ values and those in the P-L slopes.",
"It is important to note here that one does not require any prior information on the distance of the host galaxy of Cepheids to map the reddening when their multi-band photometry is available.",
"Since we have $V$ and $I$ band photometry for thousands of Cepheids in the MCs, it ideally suited us to study the reddening in detail in these two nearby clouds which is crucial to understand the dust distribution and probe the recent SFH in the MCs."
],
[
"Reddening in the MCs",
"Although one can measure reddening corresponds to each Cepheid but the mean magnitude of individual Cepheid determined through the photometric light curves may contain some uncertainty which propagates in the reddening estimation.",
"Therefore, to carry out analysis, we made small segments in the LMC and SMC as defined in Section .",
"To draw $P$ -$L$ relations, we need to select those segments which have a larger number of Cepheids lying in the same direction in order to determine more precise value of mean reddening with lesser uncertainty in any given direction.",
"Therefore only those segments were considered which contain a minimum of 10 Cepheids (3 $\\times $ Poissonian error) in at least one passband."
],
[
"The LMC",
"We selected a total 133 segments with an average angular resolution of $\\approx $ 1.2 deg$^2$ covering a total area of about 154.6 deg$^2$ in the LMC.",
"The maximum number of Cepheids among selected segments is found to be 144 in the LMC.",
"Keeping the slopes fixed in the $V$ and $I$ band $P$ -$L$ diagrams as given in eqs.",
"REF and REF , we estimated $ZP$ values for each segment using a least-square fit to the observed period versus magnitude diagrams.",
"It should be noted here that uncertainties in the measurements of apparent $V$ and $I$ magnitudes are not provided in the OGLE-IV catalogue hence individual photometric errors are not considered in the fit.",
"Figure: The PP-LL relations for a sub-region in the LMC in VV band (upper panel) and II band (lower panel).",
"Here, the continuous lines represent best linear fit with fixed slope of -2.769 for VV band and -2.961 for II band, respectively.",
"The dashed lines represent the 3σ\\sigma cut lines and Cepheids outside 3σ\\sigma lines are shown by open circles which are not considered in the best fit.Table: Table provides reddening in 133 different segments of the LMC.",
"Here, Ist column represents ID of the segment, 2nd and 3rd columns give central (X,Y) coordinate, 4th and 5th columns give corresponding (RA,DEC) coordinates.",
"The last 3 columns give reddening E(V-I)E(V-I), E(B-V)E(B-V) and error in the E(V-I)E(V-I).In Figure REF , we show one such randomly selected $P$ -$L$ diagram, each in $V$ and $I$ band.",
"We here apply $3\\sigma $ cut to reduce the contamination in iterative manner unless all the Cepheids fall within $3\\sigma $ lines of the given $P$ -$L$ slope where $\\sigma $ is the rms derived from fitting $P$ -$L$ relation to the observed data points.",
"In Figure REF , the best fit slope is drawn by continuous line while final 3-$\\sigma $ deviation is shown by dashed lines on both sides of the best fit.",
"It should be noted here that some intrinsic dispersion in the $P$ -$L$ diagram is obvious due to the finite width of the Cepheid instability strip as well as contamination due to blending and crowding effects [80], [44], [43].",
"On an average the dispersion in the LMC Cepheid $P$ -$L$ diagrams is found to be $\\lesssim 0.06$ mag in both $V$ and $I$ band.",
"We note here that an uncertainty in the LMC metallicity of 0.03 dex [49] used in eq.",
"REF gives only an added 0.001 mag error in the $ZP$ values determined through combined (FU+FO) $P$ -$L$ relations which is much smaller than the typical error on the $ZP$ estimations.",
"Using the zero point intercepts in $V$ and $I$ band, we determined the reddening $E(V-I)$ and $E(B-V)$ values as described in eqs.",
"REF and REF for all the 133 segments in the LMC.",
"Table REF lists the central coordinates of all the selected segments in the LMC, corresponding to which reddening values $E(V-I)$ and $E(B-V)$ are given in columns 6 and 7.",
"The complete table is available online or from the lead author.",
"The value of reddening $E(V-I)$ in different segments of the LMC varies from 0.041 mag to 0.466 mag with a mean value of $0.134\\pm 0.006$ mag.",
"The histogram shown in Figure REF illustrates the distribution of reddening $E(V-I)$ in the LMC.",
"Table: A summary of recent reddening measurements using different stellar populations in the LMC.The probability distribution function of interstellar column density measurements is known to be close to log-normal in the solar neighborhood [54].",
"Hence we draw a log-normal profile in the reddening distribution of the LMC.",
"The best profile illustrated by dashed line is shown in Figure REF that shows it matches the distribution with exquisite accuracy.",
"The uncertainty in the reddening $E(V-I)$ was determined from the $\\sigma $ -value in the best fit profile and error in the $ZP$ values in $V$ and $I$ band.",
"First, we calculated error in each value of $ZP$ using the uncertainties in the slope and zero points of the $P$ -$L$ relation in both the filters.",
"Then, combined error in the estimation of $E(V-I)$ has been determined through the error propagation $\\sigma (E(V-I)) = \\sqrt{(\\sigma (ZP_V)^2 + \\sigma (ZP_I)^2 + \\sigma (fit)^2}$ We determined error in each value of $E(V-I)$ that is given in the last column of Table 1.",
"A mean error in the $E(V-I)$ values was taken corresponding to the peak in the distribution of $\\sigma (E(V-I))$ values obtained in all the 136 segments.",
"The mean value of reddening using the log-normal profile fit is found to be $E(V-I)=0.113\\pm 0.060$ mag.",
"The corresponding mean value of $E(B-V)$ is estimated as $0.091\\pm 0.050$ mag.",
"Figure: Histogram of reddening values in 133 segments of the LMC shown by continuous line.",
"The dashed line represents the best fit with a log-normal distribution.The reddening in the LMC has been estimated by several authors in the past and we summarize some of the recent reddening measures in $E(B-V)$ or $E(V-I)$ using different stellar populations in Table REF .",
"Among some of the most recent estimates of reddening in the LMC, the RC stars used by [35] indicated a mean reddening of the LMC as $E(V-I)=0.09\\pm 0.07$ mag while with RR Lyrae stars a median value of $E(V-I)=0.11\\pm 0.06$ mag was obtained.",
"[40] provided reddening estimates derived through Cepheids at 7 different positions across the LMC and found that $E(B-V)$ varies in the range of $0.08\\pm 0.03$ mag to $0.12\\pm 0.02$ mag.",
"Furthermore, a broader range of $E(B-V)$ from $0.10\\pm 0.02$ mag to $0.16\\pm 0.02$ mag was given by [71] using detached eclipsing binaries found in the OGLE data.",
"[14] constructed 2D reddening map of LMC disk with help of red clump (RC) stars observed in the Survey of the MAgellanic Stellar History (SMASH) and reported an average $E(g-i)=0.15\\pm 0.05$ mag.",
"Using the conversion equations from [1], this reddening corresponds to $E(B-V)=0.093\\pm 0.031$ mag.",
"From the Table REF , it is found that the mean LMC reddening varies from $E(B-V) \\approx 0.031$ mag to $E(B-V) \\approx 0.353$ mag.",
"The average LMC reddening $E(V-I)=0.113\\pm 0.060$ mag and $E(B-V)=0.091\\pm 0.050$ mag estimated in the present study is in good agreement with the recently reported average reddening measurements using stellar populations of different ages in the LMC."
],
[
"The SMC",
"Employing the same approach, we selected 136 segments in the SMC with an average angular resolution of $\\approx $ 0.22 deg$^2$ covering a total area of about 31.3 deg$^2$ .",
"The maximum number of Cepheids among selected segments is found to be 80 in the SMC.",
"We plot period versus magnitude diagrams for each segment and drawn a least-square linear fit keeping the slopes fixed in the $V$ and $I$ band $P$ -$L$ diagrams as given in eqs.",
"REF and REF .",
"We also applied here $3\\sigma $ iterative rejection criteria to remove the outliers.",
"In Figure REF , we show one such randomly selected $P$ -$L$ diagram along with the best fit line, both in $V$ and $I$ band.",
"We also illustrate 3-$\\sigma $ cut lines on both sides of the best fit to represent the acceptable deviations in the $P$ -$L$ relation.",
"Employing the same approach as that of the LMC, we determined the reddening values of $E(V-I)$ and $E(B-V)$ in each of 136 segments in the SMC.",
"Here, we found 9 segments (6.6%) have unphysical negative values of reddening and we assigned them zero reddening within the given uncertainties.",
"It should also be noted here that the uncertainty in the SMC metallicity of 0.07 dex [37], [105] used in eq.",
"REF results only an additional 0.001 mag error in the $ZP$ values which is negligible in comparison of the combined uncertainty in the reddening estimates.",
"Table REF provides reddening values $E(V-I)$ and corresponding $E(B-V)$ in the columns 6 and 7 for each segment in the SMC.",
"The complete table is available online or from the lead author.",
"The value of reddening $E(V-I)$ in different segments within the SMC varies from 0.0 to 0.189 mag with a mean value of $0.064\\pm 0.008$ mag.",
"The histogram shown in Figure REF illustrates the distribution of reddening $E(V-I)$ across the SMC.",
"After fitting log-normal profile in the histogram, mean value of reddening is estimated to be $E(V-I) = 0.049\\pm 0.070$ mag.",
"The equivalent reddening $E(B-V)$ in the SMC is estimated to be $0.038\\pm 0.053$ mag.",
"Figure: Same as Figure but for the SMC with fixed slope of -2.66 for VV band and -2.918 for II band, respectively .Figure: Same as Figure but for the 136 segments of the SMC.Table: Same as Table but for the 136 degments of the SMC.Table: A summary of recent reddening measurements using different stellar populations in the SMC.The reddening in the SMC has been studied by several authors in the past through reddening measures in $E(B-V)$ or $E(V-I)$ using different tracers and we provide mean reddening values obtained in some of the recent studies in Table REF .",
"Among the most recent analysis, [35] obtained the mean value of $E(V-I)=0.07\\pm 0.06$ mag through 1529 RRab stars of OGLE-III survey and $E(V-I)=0.04\\pm 0.06$ mag using RC stars.",
"A mean reddening of $E(B-V)=0.071\\pm 0.004$ mag was reported by [84] using the sample of 92 SMC Cepheids with period greater than 6 days.",
"[59] estimated a reddening of $E(V-I) = 0.06\\pm 0.06$ mag using the VISTA and OGLE-IV survey data of 2997 fundamental mode RR Lyrae stars.",
"Using infrared colour-magnitude diagrams on the deep VISTA survey data, [79] inferred a large extinction in the central region of the SMC, however, found a relatively lower extinction in the external regions of the SMC.",
"If we compare our present estimates of $E(V-I)=0.049\\pm 0.070$ mag and $E(B-V)=0.038\\pm 0.053$ mag with the recent reddening values given in Table REF , we find that our analysis is in broad agreement with the previous studies except that of [96] which yield higher reddening value of $E(B-V)=0.096\\pm 0.080$ mag from their study of SMC Cepheids.",
"It is also observed from the Tables REF and REF that the mean reddening values obtained in the MCs through different stellar populations do not show any significant variation among different studies."
],
[
"Reddening Maps in the LMC and SMC",
"The reddening maps are often generated to remove the effects of reddening from the optical and UV images of different regions of the galaxies.",
"They also have wide implications in understanding the star formation rates and recent SFH in the galaxies.",
"In the following subsections, we individually focus our attention to the reddening distributions across the LMC and SMC."
],
[
"The LMC",
"An examination of individual estimates of reddening values determined through $P$ -$L$ relations, we found that the reddening varies from one region to another region of the cloud.",
"While in some regions of the LMC, reddening is negligibly small but in some regions it is found to be significantly high.",
"The reddening values estimated for the 133 selected segments were used to construct a reddening map on a rectangular grid using the central $(X,Y)$ of all segments.",
"Figure REF presents the reddening map derived through the $P$ -$L$ diagrams of the Cepheids, showing the mean reddening along the line of sight toward each segment.",
"Here, a smoothed reddening distribution was adopted to construct the map.",
"The optical center of LMC ($\\alpha = 05^h 19^m 38^s \\equiv 79^{o}.91$ and $\\delta = -69^{o} 27^{^{\\prime }} 5^{^{\\prime \\prime }}.2 \\equiv -69^{o}.45$ ; [19]) is shown in the figure.",
"The quality of reddening map is basically regulated by the size of the segments as well as number of Cepheids available in these regions.",
"The sparse spatial distributions of the LMC Cepheids in the outer regions, as shown in Figure REF , lend a poor resolution to the reddening map in the outer regions of the LMC as seen in Figure REF .",
"Figure: The reddening maps emanated through the PP-LL diagrams of the Cepheids in the LMC.",
"North is up and east is to the left.",
"The optical center of the LMC is shown by plus sign.",
"The colour bar represents the interpolated reddening E(V-I)E(V-I).The reddening map of the LMC exhibits a non-uniform distribution of dust across the LMC.",
"With respect to the optical center in the LMC, a lopsidedness in the reddening distribution towards the eastern regions is noticed.",
"While the central region of the LMC contains low values of reddening having $E(V-I)=0.113\\pm 0.045$ mag, the most promising region in the LMC reddening map is seen towards the north-east region of the LMC outer disk centred at $X=-1.6$ kpc, $Y=0.0$ kpc from the center, which corresponds to $\\alpha \\sim 85^{o}.13,~\\delta \\sim -69^{o}.34$ having $E(V-I) = 0.466$ mag.",
"Interestingly, the adjacent region centred around $\\alpha \\sim 84^o$ , $\\delta \\sim -70^o$ contains the highest concentration of Cepheids in our sample.",
"This region is most likely associated with the star forming HII region 30 Doradus (Tarantula Nebula or NGC 2070) centred at $\\alpha = 84^o.5, \\delta = -69^o.1$ [38].",
"This is the most active star forming region in the bar of the LMC [50], [100].",
"The same region was identified as having highest reddening by [65] and [40] as well as having highest concentration of young cluster populations by [31].",
"This region is lying in the direction of compact massive cluster RMC 136a that contains a large concentration of young star clusters [31], [28].",
"[66] found that the 30 Dor is located in the region where the LMC bar joins the H I arms and such locations are prone to enhance star formation activity due to high concentrations of gas and to the shocks induced by the internal dynamical processes [8].",
"In the bar region across the LMC disk, reddening is relatively low and a mean value of $E(V-I) = 0.153$ mag was estimated.",
"On the other side of the LMC bar, few extincted regions are found having moderate reddening.",
"The reddening map, in general, is in good agreement with the maps of HI column density [55] and MACHO Cepheids [65].",
"The geometry of the LMC disk was studied by [65] using the MACHO and 2MASS Cepheids data and estimated the mean reddening $E(B-V)$ by three different methods.",
"All the three estimated reddening values were consistent with each other and the variance-weighted average reddening was found to be $E(B-V)=0.14\\pm 0.02$ mag.",
"It is found that the reddening maps of the LMC have been constructed in numerous studies in the past using different kind of stellar populations.",
"In order to compare our reddening map with some of the recent studies in the LMC, we took advantage of four such reddening maps in the LMC constructed in the optical bands for which archival data is available [93], [35], [60], [40].",
"We performed a cell-by-cell comparison of our reddening map with these maps after cross-examinations of central ($\\alpha $ , $\\delta $ ) values of each segment with the reddening values given in same region in these maps.",
"The histograms of the comparisons between our reddening map and these maps are illustrated in Figure REF .",
"We discuss these comparisons in some detail as follows.",
"The reddening $E(V-I)$ values in the LMC is estimated by [93] for 1123 locations and found $E(V-I)$ in the LMC varies between 0.1 to 0.3 mag.",
"Her study shows the total average LMC bar reddening is $E(V-I)=0.08\\pm 0.04$ mag and eastern regions have higher reddening as compared to western regions.",
"We carried out cell-by-cell comparison of $E(V-I)$ obtained in our study with that of the [93] as shown in Figure REF (a) with the dotted line which shows that reddening values estimated by [93] is marginally underestimated in comparison of our studies.",
"[35] estimated reddening $E(V-I)$ using RR Lyrae stars and RC stars form OGLE-III data.",
"They reported low reddening in central bar regions of the LMC and higher reddening towards 30 Doradus region s has also been noticed in the present study.",
"In Figure REF (a), we show comparison of our estimated reddening values with those of [35] with continuous line and both the studies are found in close agreement.",
"[60] studied 1072 star clusters in the LMC using OGLE-III data and a semi-automated quantitative method was used to estimate the age and reddening of these clusters.",
"Their $E(V-I)$ values range from 0.05 to 0.50 mag with maximum reddening lying between 0.1 to 0.3 mag.",
"The distribution of cell-by-cell comparison shown in Figure REF (a) with dashed line peaks at $\\sim $ -0.28 mag, which means that the reddening given by [60] is much higher in comparison of any other studies carried out in the LMC in recent times.",
"[40] investigated LMC disk using Cepheids in OGLE-IV data.",
"In the Figure REF (b), we illustrate comparison between $E(B-V)$ values derived in present study and [40].",
"The histogram shows that two studies are in excellent agreement within the quoted uncertainties.",
"This result also endorse our approach of combining two modes of Cepheids in a single $P$ -$L$ relation in order to estimate reddening measurements.",
"Figure: (a) Upper panel shows the distributions of the ΔE(V-I)\\Delta E(V-I) i.e.",
"past studies results subtracted from our results in the LMC as mentioned on the top-left corner, (b) Lower panel shows the difference ΔE(B-V)\\Delta E(B-V) between our reddening estimates and those of .In general, we find our reddening map is in excellent agreement with the recent reddening studies in the optical region except that of the [60].",
"The histograms shown in Figure REF also provide opportunity to examine the difference in the reddening estimates traced through different stellar populations.",
"It is seen from Figure REF that while [93] and [35] yield smaller reddening values from the older stellar populations RC and RR Lyrae stars, the map constructed by [60] gives higher reddening values using the young star clusters in comparison of Cepheid variables which are relatively intermediate age stellar populations.",
"It is also found that [40], who also used Cepheids in their study like in the present analysis, has similar reddening estimates as ours.",
"In general, younger populations like open clusters are likely to be more embedded in the dusty clouds hence provide larger reddening than the older populations like RC stars and RR Lyraes."
],
[
"The SMC",
"The reddening values determined for 136 selected segments within the SMC was used to construct a reddening map which is illustrated in Figure REF .",
"The optical center of SMC ($\\alpha = 00^h 52^m 12^s.5 \\equiv 13^{o}.05$ and $\\delta = -72^{o} 49^{^{\\prime }} 43^{^{\\prime \\prime }} \\equiv -72^{o}.82$ ; [19]) is shown in the same figure.",
"The reddening map exhibits a non-uniform and highly clumpy structure across the SMC.",
"The peripheral regions of the SMC show smaller reddening in comparison of the central regions.",
"The reddening is larger in the south west parts of the SMC.",
"The largest reddening in the SMC is found to be $E(V-I) = 0.189$ in the region centred at $X$ =0.3, $Y$ =-0.3 which corresponds to $\\alpha \\sim 12^{o}.10,~\\delta \\sim -73^{o}.07$ .",
"On an average, the reddening in the SMC is found to be $E(V-I) = 0.049\\pm 0.070$ mag and corresponding $E(B-V) = 0.038\\pm 0.053$ mag which are substantially smaller in comparison of the mean reddening values in the LMC.",
"If we compare the reddening maps in the LMC and SMC shown in Figures REF and REF with that of the spatial distributions of Cepheids given in Figures REF and REF , it is interesting to note that the reddening structures are found to be slightly correlated with those of the spatial distributions of Cepheids in both the galaxies, particularly in the SMC where higher reddening has been observed in the close vicinity of the densely populated regions of Cepheids.",
"Figure: Same as Figure but for the SMC.Figure: (a) Upper panel shows the distributions of ΔE(V-I)\\Delta E(V-I) i.e.",
"past reddening estimates subtracted from our estimate in the SMC as mentioned on the top-left corner, (b) Lower panel shows the difference of E(B-V)E(B-V) from subtracted from our reddening estimate.Figure: Lower panel shows the histogram of the difference ΔE(V-I)\\Delta ~E(V-I) between the present study and those of drawn by continuous line and drawn by dashed line.",
"Upper panel shows the heat map representing the difference in reddening divided by reddening uncertainty in each segment between the present reddening estimates and that of the in the left panel and same comparison between our values and in the right panel.",
"The difference is obtained by subtracting other results out of our estimates.The reddening maps of the SMC have been presented by many authors in the past using different kind of stellar populations [83], [110], [22], [35], [96], [79], [61].",
"In the following part, we discuss a comparative study of our reddening map with some of the recent studies carried out in the SMC.",
"[35] studied the SMC reddening $E(V-I)$ using RR Layrae and RC stars from the OGLE-III data.",
"The continuous line in Figure REF shows difference between reddening $E(V-I)$ estimated by us and [35] using RC stars.",
"The distribution peaks at 0.02 mag suggesting that [35] reddening values determined are slightly smaller in comparison of our study.",
"[61] studied 179 open clusters within the SMC using the same techniques as in [61] carried out for the LMC open clusters.",
"We show comparison of present reddening estimates with the [61] in Figure REF as a dashed line.",
"The distribution peaks at -0.08 mag, which means that [61] reddening values are larger in comparison of our reddening estimates.",
"A similar study carried out earlier in the LMC has also shown that their reddening estimates are higher in comparison of the present estimates.",
"[79] constructed the reddening map using 14 deep tile images taken in the $YJK_s$ filters under VMC survey and found a range of extinction ($A_V$ ) between $\\sim $ 0.1 mag (external regions) and 0.9 mag (inner regions).",
"On a comparative study between ours and [79] reddening maps in the 32 common regions, we found that our $E(B-V)$ values are smaller in comparison of their study.",
"In the lower panel of Figure REF , we present histogram of the differences between two reddening estimates which shows a range of variations peaking around -0.08 mag.",
"We however note that their study is based on near-IR data whereas our reddening estimates are based on the optical data.",
"It is evident from the histograms in Figure REF that [35] which used older stellar population found lower reddening values while [61] which used younger stellar population retrieved higher reddening values in comparison of our reddening estimates in the SMC, a result similar to our previous comparison for the LMC.",
"It is therefore quite clear from these comparative studies that the low reddening values follow the distribution of the older stellar populations and younger populations cater higher reddening in the MCs.",
"Figure: Same as Figure but for the SMC.",
"Here, comparisons are made between our maps and those obtained through and PP-LL relations."
],
[
"Effect on the Reddening Maps due to different $P$ -{{formula:d6fc775f-fe70-4aa2-8069-c28a5d227718}} relations",
"In the present study, we used most recent $P$ -$L$ relations given by [63] and [62] to determine reddening estimates in the regions of the LMC and SMC, respectively.",
"There are many other $P$ -$L$ relations reported in the previous studies such as [51], [102], [30], [81], [82], [63], [62].",
"To investigate whether reddening distribution across the LMC and SMC determined in the present study changes due to different $P$ -$L$ relations, we cross-examined present reddening maps with those using the [102] $P$ -$L$ relations as well independent set of P-L relations given by [81] for LMC and [82] for the SMC.",
"We used the same methodology as discussed previously in Section REF to find reddening values in the MCs and constructed the reddening maps for both the LMC and SMC.",
"In the lower panel of Figure REF , we illustrate the histogram of the difference between present estimates with those determined through previous $P$ -$L$ relations given by [102] and [81] by continuous and dashed lines, respectively.",
"While we yield slightly lower values of $E(V-I)$ using [102], we acquired slightly larger $E(V-I)$ values in comparison of [81], [82].",
"It is seen that the reddening estimates in the LMC obtained through [63] in the present study are in between to those obtained through [102] and [81] $P$ -$L$ relations but this variation is less than 0.02 mag which is quite small in comparison of the uncertainties involved in the reddening estimations itself.",
"For a quantitative verification of our reddening values, we also present significance of the difference among different reddening maps in the upper panel of Figures REF which illustrates the difference in reddening divided by the reddening uncertainty in each segment.",
"Here, we determine comparison of our values with those obtained through the [102] and [81] $P$ -$L$ relations in the left and right side, respectively.",
"These maps suggest an excellent quantitative agreement between the reddening maps obtained through different $P$ -$L$ relations in the LMC except few isolated segments as seen by blue squares in Figure REF .",
"A similar comparison between the reddening values obtained in the present study using [62] $P$ -$L$ relations and those obtained through [102] and [82] $P$ -$L$ relations are illustrated in Figure REF for the SMC.",
"It is conspicuous from the differential plots that the present estimates in the SMC are slightly smaller than those obtained through both [102] and [82] $P$ -$L$ relations however differences lie between 0.02 mag to 0.03 mag which can be considered as non-significant considering the uncertainties involved in the estimation of these values.",
"In general, we conclude that the change in Cepheids P-L relations does not make any noticeable difference in the reddening maps of these two clouds."
],
[
"Recent star formation history in the MCs",
"Recent studies shows that major star formation events took place in the MCs at several epochs ranging from few Gyrs to few Myrs ago [33], [74], [79] though with varying star formation rates from field to field [15], [78], [77].",
"Therefore a comprehensive study of Cepheids provides a unique opportunity to probe the recent SFH in the MCs as these are relatively young population and most of them have ages less than few hundred Myr.",
"Therefore the age and spatial-temporal distributions of Cepheids along with MCs structural parameters may provide important information about the formation history of the Magellanic System."
],
[
"Age Distribution",
"The $P$ -$L$ and mass-luminosity relations of Cepheids imply that longer period Cepheids have higher luminosities and are more massive which means relatively shorter life span for these pulsating stars.",
"Therefore, the period and age of Cepheids have an obvious connection.",
"As Cepheids typically have ages in the range of roughly 30 to 600 Myr, a study of age distribution of Cepheids can be used to reconstruct the recent SFH within the MCs in last few tens to few hundreds Myr.",
"Since pulsation period of Cepheids is the only quantity which can be precisely determined from the observations of the pulsating stars, their ages can be determined with a good accuracy using the PA relations.",
"To determine ages of Cepheids from their periods,[57], [26], [25] proposed many semi-empirical relations and [9] provided theoretical period-age (PA) and and period-age-colour (PAC) relations.",
"Recently [42] used mean periods of 74 LMC Cepheids found in 25 different open clusters and corresponding cluster ages taken from [68] to draw an improved PA relation in the LMC.",
"The empirical PA relation derived by [42] for the Cepheids in the LMC is given as $\\log ({\\rm t}) = 8.60(\\pm 0.07) - 0.77(\\pm 0.08)~\\log ({\\rm P})$ where age denoted by $t$ is in years and P is the period given in days.",
"The reliability of the above PA relation has been examined in [42] by comparing this relation with the [9] theoretical PA relation determined on the basis of evolutionary and pulsation models covering a broad range of stellar masses and chemical compositions.",
"We found a reasonable agreement between the two relations given for the LMC.",
"Cepheids PA relation however varies for the galaxies having difference metallicities and as we have not derived any such PA relation for the SMC, we considered here theoretical PA relation given by the [9] for the SMC.",
"Since we have already converted period of FO Cepheids to corresponding FU Cepheids, we only used PA relation of FU Cepheids for the known metallicity of SMC ($z$ =0.004): $\\log ({\\rm t}) = 8.49(\\pm 0.09) - 0.79(\\pm 0.01)~\\log ({\\rm P})$ We used above PA relations to determine the age of each Cepheid in the LMC and SMC.",
"As error in the period of Cepheids is reported to be less than 0.001%, we have not taken them into account in the subsequent age conversion but the typical average error in age is estimated to be $\\sim $ 40 Myr due to uncertainty in the PA relations given by [9] and [42].",
"Although resulting ages are in the range of $\\log ({\\rm t/yr})$ = 6.96 to 8.96 with a mean $\\log ({\\rm t/yr})$ of 8.21 for the LMC but three-fourth Cepheids population in the LMC are distributed between $\\log ({\\rm t/yr})$ of 8.0 to 8.4.",
"Similarly, ages for the SMC Cepheids ranges from $\\log ({\\rm t/yr})$ = 6.66 to 8.86 with a mean age of 8.27, about three-fourth of them are confined between $\\log ({\\rm t/yr})$ of 8.1 to 8.5.",
"One can see that the SMC Cepheids are on average slightly older in comparison of the LMC Cepheids.",
"We determined the distribution of Cepheids in a bin width of 0.05 dex (on a logarithmic scale) in the LMC and SMC which are respectively shown in Figures REF and REF .",
"We see a pronounced peak in the age distribution of Cepheids in the LMC that can be represented by a Gaussian-like profile.",
"In Figure REF , we show the best fit Gaussian distribution in the histogram that gives a peak at $\\log ({\\rm t/yr}) = 8.21\\pm 0.11$ .",
"However, a slightly broad profile is evident in the age distribution of Cepheids in the SMC which may be represented by a bi-modal Gaussian profile as has been drawn by [41].",
"Though we do not observe a clear bimodality in the age distribution of SMC Cepheids but a double-Gaussian profile still gives a better fit than a single-Gaussian therefore we prefer the earlier one.",
"A best fit double-Gaussian profile in the age histogram, which is shown by a dashed line in Figure REF , gives a prominent primary peak at $\\log ({\\rm t/yr}) = 8.36\\pm 0.08$ and a small secondary peak at $\\log ({\\rm t/yr}) = 8.17\\pm 0.08$ .",
"The peak for the younger Cepheids is not strong enough to draw any firm conclusion although it is coinciding with the primary peak in the LMC.",
"We note here that the final error in age represents the combined uncertainty in the PA relation as well as error corresponds to the mean age estimation in the Gaussian fit.",
"The maxima in age distributions of Cepheids indicate a rapid enhancement of Cepheid formation at around $162_{-36}^{+46}$ Myr for the LMC while it is $229_{-39}^{+46}$ Myr for the SMC.",
"The rapid enhancement of Cepheids around 200 Myr ago (within 1-$\\sigma $ uncertainty) in the LMC and SMC thus pointed to a major star formation episode at that time, most likely due to a close encounter between the two clouds or their interactions with the Galaxy stem from their multiple pericentric passages as they orbit the Milky Way.",
"Figure: The age distribution of Cepheids in the LMC.",
"The best fit Gaussian profile is shown by a dashed red line.Figure: Same as Figure but for the SMC.",
"The best fit of double Gaussian is shown as a dashed red line and individual Gaussian are shown by dotted green line.The oscillation between rise and fall in the star formation rate depends upon whether these two clouds are approaching or receding [31], [45].",
"This repeated interaction between LMC and gas-rich SMC lead to episodic star formations in these two dwarf galaxies which are locked in tidal interaction.",
"It is believed that the Magellanic Bridge and the Magellanic Stream might have formed due to such interactions in the past between the two segments of the MCs [5], [59].",
"As a consequence of these frequent encounters, the tidal stripping of stars and gas/material from the gaseous disk takes place in the Magellanic System.",
"In fact gas in the Magellanic Bridge is thought to have been largely stripped from the SMC as a consequence of its close interactions with the LMC at about 200 Myr ago [23], [56].",
"Considering several previous studies on the interaction of the clouds [6], [112], [111] there are ample evidences of a direct collision between two clouds with an impact parameter of few kpc [67], [112].",
"These studies along with many previous studies confirm epochs of recent star formation in the MCs albeit at slightly different ages.",
"Using open star clusters in the SMC, [73] and [31] found a peak at 160 Myr through their age distribution.",
"A similar conclusion was also drawn by [96] using OGLE-III data based on [9] PAC relation and [41] using OGLE-IV data based on [9] PA relation.",
"In a recent investigation by [76] using the data from VISTA near-infrared YJKs survey of the Magellanic System (VMC), they also predicted a close encounter or a direct collision between the two cloud components some 200 Myr ago and confirm the presence of a Counter-Bridge.",
"On the theoretical side, there have been several studies to infer cloud-cloud interaction in the MCs.",
"In some of the recent models proposed by [4], [46], [21], [6] and [112], they suggested the last cloud-cloud collision within the Magellanic System had occurred about 100-300 Myr ago causing an enhanced star formation activities in these two dwarf galaxies that might have resulted the formation of Magellanic Stream.",
"In fact mutual interactions between the clouds and subsequent tidal stripping of material from the SMC are believed to be the most likely reason for the formation of Magellanic Bridge [6], [21], [59].",
"This would also mean that Magellanic stream and Magellanic bridge stellar populations should contain stars from both the components of the MCs.",
"On the other hand, it was also suggested by [31] (and references therein) that not only the frequent tidal interaction between the two clouds but stellar winds and supernova explosions may also induce episodic star formation in these two dwarf galaxies."
],
[
"Spatio-temporal Distribution",
"The spatial distribution of Cepheids as a function of age is shown in Figures REF and REF for the LMC and SMC, respectively.",
"The figures show that the Cepheids are distributed all over the LMC while preferential distributions of Cepheids is seen in the SMC.",
"From the examination of age map of the LMC, complex and patchy nature in age distribution is quite evident where both young and old Cepheid populations are distributed in small structures across the cloud.",
"We also see a gradual change in the ages of Cepheids from central to peripheral regions where inner regions have lower ages and outer regions have pockets of higher ages as has been noticed in the last star formation event (LSFE) map presented by the [39] through the determination of the Main-Sequence (MS) turn-off point in the colour-magnitude diagram.",
"A similar structure has also been observed by [58] who suggested a stellar population gradient in the LMC disk where younger stellar populations are more centrally concentrated.",
"They also proposed an outside-in quenching of the star formation in the outer LMC disk ($R_{GC}$ = 3.5-6.2 kpc) which might be associated with the variation of the size of HI disk as a result from gas depletion due to star formation or ram-pressure stripping, or from the compression of the gas disc as ram pressure from the Milky Way halo acted on the LMC interstellar medium.",
"Figure: Spatial distributions of the Cepheids in the 389 segments of the LMC as a function of their mean log(t/ yr )\\log ({\\rm t/yr}).",
"North is up and east is to the left.",
"The locations of the optical center of LMC is shown by plus sign.In the SMC, age map of Cepheids shows a systematic distribution where younger Cepheids lie towards inner region and older Cepheids are mainly confined towards peripheral regions.",
"This suggest an inwards quenching of star formation in the SMC.",
"[76] also found that young and old Cepheids have different geometric distributions in the SMC.",
"They observed that closer Cepheids are preferentially distributed in the eastern regions of the SMC which are off-centred in the direction of LMC owing to the tidal interaction between the two clouds [59].",
"Figure: Same as Figure but for the 562 segments in the SMC.If we compare frequency distribution map of the Cepheids as shown in Figures REF and REF with the similar maps made for the star clusters identified in [31] (see Figure 9 of [42] for the LMC, and Figure 5 of [45] for the SMC), we notice that the clumps of Cepheids do not coincide with the clumps of the star clusters in both the component of the MCs and a mutual avoidance of clumps of the Cepheids and star clusters is present within the MCs."
],
[
"Discussion and Conclusions",
"The nearby LMC and SMC are the two galaxies for which the highest number of Cepheids are detected, that too with an excellent data quality.",
"The main motive of this paper was thus to exploit the available $V$ and $I$ band data of close to nine thousand Cepheids in the MCs.",
"The independent reddening values determined through the multi-wavelength $P$ -$L$ relations of Cepheids are important to construct reddening map which is necessary to understand the dust distributions within different regions of the host galaxy.",
"In the present study $V$ and $I$ band photometric data of Cepheids provided in the OGLE-IV photometric survey was analysed to understand the reddening distributions across the LMC and SMC and subsequently to draw reddening maps in these two nearby galaxies.",
"We used 2476 FU and 1775 FO Cepheids in the LMC and 2753 FU and 1793 FO Cepheids in the SMC in the present analysis.",
"In the present study there is one major addition that instead of studying individual $P$ -$L$ relations for FU and FO Cepheids, we combined these two modes of pulsating stars, after converting periods of FO Cepheids to the corresponding periods of FU Cepheids.",
"This has increased our sample size of Cepheids in order to draw $P$ -$L$ diagrams that has allowed us to make small size segments within the galaxies hence better resolution of the reddening maps.",
"To reduce statistical error in the reddening estimation, we selected only those segments which contain a minimum of 10 Cepheids.",
"We drew a best fit $P$ -$L$ relation in both $V$ and $I$ band data in all the segments of the LMC as well as those of the SMC.",
"Using the well calibrated $P$ -$L$ relations for the LMC and SMC, we estimated the reddening $E(V-I)$ in each segment of both the galaxies.",
"We found that the reddening $E(V-I)$ varies from 0.041 mag to 0.466 mag in the LMC and 0.00 to 0.189 mag in the SMC.",
"The mean value of reddening obtained through best fit log-normal profile was found to be $E(V-I)=0.113\\pm 0.060$ mag and $E(B-V)=0.091\\pm 0.050$ mag for the LMC.",
"The mean value of reddening obtained through similar approach in the SMC was estimated to be $E(V-I)=0.049\\pm 0.070$ mag and $E(B-V)=0.038\\pm 0.053$ mag.",
"Using the reddening distributions of 133 segments in the LMC and 136 segments in the SMC, we constructed reddening maps with a cell size of 0.4$\\times $ 0.4 square kpc in the LMC and 0.2$\\times $ 0.2 square kpc in the SMC.",
"This provides an average angular resolution of about 1.2 deg$^2$ covering an area over 150 deg$^2$ in the LMC and angular resolution of about 0.22 deg$^2$ covering an area of over 30 deg$^2$ in the SMC.",
"The reddening map of the LMC shows a heterogeneous distribution having small reddening in the bar region and higher reddening towards north-east region.",
"We did not find any significant reddening in the central region of the LMC.",
"The highest reddening in the LMC having $E(V-I)=0.466$ mag was traced in the north east region located at $\\alpha \\sim 85^{o}.13,~\\delta \\sim -69^{o}.34$ which seems to be associated with the most active star forming HII region 30 Doradus situated at $\\alpha \\sim 84^o$ , $\\delta \\sim -70^o$ that also contains the highest concentration of Cepheids in our sample.",
"In case of the SMC, reddening map exhibits a non-uniform and highly clumpy structure across the cloud.",
"In general, smaller reddening was found around central regions of the SMC but larger reddening was seen in the south-west region away from the optical center.",
"The largest reddening in the SMC is found to be $E(V-I) = 0.189$ in the region centred at $\\alpha \\sim 12^{o}.10,~\\delta \\sim -73^{o}.07$ .",
"The peripheral regions of the SMC have shown very small reddening.",
"If we compare our reddening maps with those of the spatial maps in the MCs, we found a broad correlation between the denser regions to the reddened structures which is found to be more closely related in the SMC where higher reddening has been found in the close vicinity of the densely populated regions of Cepheids.",
"The comparison of our reddening maps with the some recent optical reddening maps has also been carried out, most of them matches well with our results although few of them are found to be overestimated or underestimated in comparison of the present study.",
"There are various stellar populations which are used to define the reddening map but shows some discrepancies among different populations because different stellar populations experience different dust.",
"It is well expected that the reddening estimates through early type stars or star forming regions yield higher reddening in comparison of intermediate or old age stellar populations.",
"Different stellar populations have different spatial distributions and most of them are non-axisymmetric.",
"Furthermore, no significant variation was noticed in the reddening maps when we used different sets of $P$ -$L$ relations which demonstrate the stability of present reddening maps.",
"The unprecedented large data set on Cepheids in the OGLE-IV survey led to refine our knowledge about the spatial and age distributions of Cepheids within the MCs.",
"We found that Cepheids in the LMC and SMC are concentrated in the south-east and south-west regions, respectively.",
"While dense population of Cepheids lies in close vicinity of the optical center of the SMC, it is substantially shifted from the optical center of the LMC.",
"The western region of the LMC bar is densely populated for Cepheids in comparison of the eastern region and northern arm of the LMC reveals a very poor spatial density.",
"To explore the recent star formation activity across the MCs, we also estimated the ages of Cepheids taking advantage of known PA relations in the literature.",
"The age distribution in the LMC shows a Gaussian profile having peak at $\\log ({\\rm t/yr}) = 8.21\\pm 0.11$ , the age distribution in the SMC displays a prominent peak at $\\log ({\\rm t/yr}) = 8.36\\pm 0.08$ although it shows a weak bimodal distribution.",
"The age maxima in the LMC is found to be very close to that of the SMC which suggests a common enhancement of star formation had happened in these two galaxies sometime around 200 Myr ago.",
"The most likely scenario for this simultaneous star formation burst is thought to be resulted due to cloud-cloud encounter between the LMC and SMC.",
"A similar distribution has also been noticed in earlier studies using different stellar populations and our result also supports the current theoretical scenario predicting a close encounter between the Clouds.",
"Although cloud-cloud collisions are well expected between these two nearby galaxies due to their tidal interactions but any external phenomenons like stellar winds and supernova explosions cannot be ruled out as a cause of enhanced star formations in these systems.",
"We noticed a slightly preferential distribution in the SMC where relatively older Cepheids were observed towards the peripheral regions.",
"It was interesting to note that eastern part of the SMC possessed most of younger Cepheids which indicates that the eastern region of the galaxy may be relatively younger.",
"As MCs have shown evidence of undergoing numerous star formation episodes in the past ranging from few Myrs to few Gyrs, it is absolutely necessary to study these two nearby clouds with different stellar populations as life span of these tracers also varies from few Myrs to few Gyrs.",
"Moreover the properties of Clouds vary both in spatial distributions as well as a function of stellar population.",
"Once derived, they can provide important clues to understand the outlying mechanisms of galaxy interactions that in turn drive the star formations over varying time scales.",
"Therefore, in order to retrieve a complete picture of the Magellanic System that comprises LMC, SMC, Magellanic Bridge and Magellanic Stream, a combination of various data samples and multi-band catalogues of different stellar populations, particularly extracted through large sky surveys, is of vital importance."
],
[
"Acknowledgments",
"We are grateful to the referee for providing helpful comments that significantly improved this paper.",
"We are thankful to P. K. Nayak for providing their reddening estimates in the SMC before publication, Jeewan C. Pandey for his useful suggestions and Smitha Subramanian for pointing out few mistakes in the early stage of the draft.",
"We also acknowledge Aurobinda Ghosh who repeated part of the present work during his summer project sponsored by the Indian Academy of Sciences (IASc), Bangalore through the grant no.",
"IAS-SRFP 2018.",
"This publication makes use of data products from the OGLE archive."
]
] | 1906.04481 |
[
[
"Effective Dynamics of 2D Bloch Electrons in External Fields Derived From\n Symmetry"
],
[
"Abstract We develop a comprehensive theory for the effective dynamics of Bloch electrons based on symmetry.",
"We begin with a scheme to systematically derive the irreducible representations (IRs) characterizing the Bloch functions.",
"Starting from a tight-binding (TB) approach, we decompose the TB basis functions into localized symmetry-adapted atomic orbitals and crystal-periodic symmetry-adapted plane waves.",
"Each of these subproblems is independent of the details of a particular crystal structure and it is largely independent of the other subproblem, hence permitting for each subproblem an independent universal solution.",
"Taking monolayer MoS$_2$ and few-layer graphene as examples, we tabulate the symmetrized $p$ and $d$ orbitals as well as the symmetrized plane wave spinors for these systems.",
"The symmetry-adapted basis functions block-diagonalize the TB Hamiltonian such that each block yields eigenstates transforming according to one of the IRs of the group of the wave vector $G_k$.",
"For many crystal structures, it is possible to define multiple distinct coordinate systems such that for wave vectors $k$ at the border of the Brillouin zone the IRs characterizing the Bloch states depend on the coordinate system, i.e., these IRs of $G_k$ are not uniquely determined by the symmetry of a crystal structure.",
"The different coordinate systems are related by a coordinate shift that results in a rearrangement of the IRs of $G_k$ characterizing the Bloch states.",
"We illustrate this rearrangement with three coordinate systems for MoS$_2$ and tri-layer graphene.",
"Using monolayer MoS$_2$ as an example, we combine the symmetry analysis of its bulk Bloch states with the theory of invariants to construct a generic multiband Hamiltonian for electrons near the $K$ point of the Brillouin zone.",
"The Hamiltonian includes the effect of spin-orbit coupling, strain and external electric and magnetic fields."
],
[
"Introduction",
"Near a band extremum, the electron dynamics in a crystalline solid resembles the dynamics of free electrons in the absence of the periodic crystal potential.",
"In the multiband envelope-function approximation (EFA) the electrons are characterized by an $N \\times N$ Hamiltonian $\\mathcal {H}$ for $N$ -component spinors conceptually similar to relativistic electrons described by the Dirac equation [1], [2], [3], [4], [5].",
"The simplest approach within the EFA is the effective-mass approximation (EMA) that represents the electron dynamics by a Schrödinger equation with effective mass $m^\\ast $ reflecting the curvature of the band dispersion $E({\\mathbf {\\mathrm {k}}})$ .",
"External electric and magnetic fields ${\\mathbf {\\mathcal {E}}}$ and ${\\mathbf {\\mathcal {B}}}$ break the lattice periodicity of the crystal structure.",
"It is an important advantage of EFA and EMA that they allow one to incorporate the field ${\\mathbf {\\mathcal {E}}}$ by adding the corresponding scalar potential $\\Phi $ to the diagonal of the Hamiltonian, and the operator of crystal momentum $\\hbar {\\mathbf {\\mathrm {k}}} = -i\\hbar \\nabla $ is replaced by $-i\\hbar \\nabla + e{\\mathbf {\\mathrm {A}}}$ , where ${\\mathbf {\\mathrm {A}}}$ is the vector potential for the magnetic field ${\\mathbf {\\mathcal {B}}}$ .",
"Other perturbations such as spin-orbit coupling, strain and electron-phonon coupling can likewise be included in the Hamiltonian [3].",
"This is a major reason why EFA and EMA are very popular for theoretical studies of both bulk semiconductors (e.g., Refs.",
"[6], [7], [3], [8], [9], [10], [11]) and semiconductor quantum structures (e.g., Refs.",
"[4], [12], [5], [13], [14], [15], [16]).",
"The form of the Hamiltonian $\\mathcal {H}$ depends on the symmetry of the crystal structure and more specifically on the symmetry of the bulk electronic states that are included in $\\mathcal {H}$ [17], [18], [3].",
"The relevant symmetry group for states with wave vector ${\\mathbf {\\mathrm {k}}}$ is the point group $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ which includes those symmetry elements of the crystallographic point group $\\mathcal {G}_0$ (crystal class) which either leave ${\\mathbf {\\mathrm {k}}}$ unchanged or map ${\\mathbf {\\mathrm {k}}}$ onto an equivalent wave vector.",
"The symmetry of individual states at ${\\mathbf {\\mathrm {k}}}$ is characterized by the respective irreducible representations (IRs) of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ according to which these states transform.",
"The general form of the Hamiltonian $\\mathcal {H}$ including its dependence on, e.g., spin-orbit coupling, strain and external fields can then be derived from its invariance under $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ [17], [3].",
"Here we develop a general theory to determine the IRs of Bloch functions for a given wave vector ${\\mathbf {\\mathrm {k}}}$ , focusing for clarity on symmorphic space groups.",
"Using a tight-binding (TB) approach along with the fact that the atomic orbitals are localized in the vicinity of the atomic sites we demonstrate that the TB basis functions can be factorized into localized symmetry-adapted atomic orbitals and crystal-periodic symmetry-adapted plane waves.",
"Each of these two subproblems permits a universal classification, independent of the details of a particular crystal structure and also largely independent of the other subproblem.",
"The symmetrized atomic orbitals depend only on the angular momentum of the atomic orbitals and the point group $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ of the wave vector ${\\mathbf {\\mathrm {k}}}$ ; but these orbitals are independent of the specific type of atom and the details of the crystal structure.",
"The symmetrized plane waves form discrete Bloch functions that depend on the wave vector ${\\mathbf {\\mathrm {k}}}$ and the Wyckoff positions of the atoms in a crystal structure; but they are independent of the type of atoms occupying these positions.",
"The symmetry-adapted basis functions block-diagonalize the TB Hamiltonian such that each block yields eigenstates transforming according to one of the IRs of the group of the wave vector $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ .",
"Given the symmetry group $G$ of a quantum system, the IRs of $G$ are generally assumed to provide a distinct label for the eigenstates of the Hamiltonian, as noted by Wigner: “The representation of the group of the Schrödinger equation which belongs to a particular eigenvalue is uniquely determined up to a similarity transformation.” (Ref.",
"[19], p. 110, highlighting adopted from Ref.",
"[19]).",
"This uniqueness of the IRs is immediately relevant for many physical properties of a physical system that depend on the symmetry of its electronic states.",
"For example, the Wigner-Eckart theorem allows one to express the selection rules for optical transitions in terms of the IRs of the initial and final states between which a transition occurs [19].",
"Similarly, the EFA Hamiltonians $\\mathcal {H}$ depend on the IRs of the bands described by $\\mathcal {H}$ , as noted above.",
"We demonstrate that the IRs characterizing the Bloch eigenstates in certain crystals including transition metal dichalcogenides (TMDCs) are not unique, but they depend on the coordinate system used to describe the space group symmetries of these materials [20], [21], [22], [23].",
"We show that distinct valid coordinate systems are related by a coordinate shift that defines a rearrangement representation.",
"The IRs of the electronic states in the different coordinate systems are then related via a rearrangement lemma that facilitates the symmetry analysis of Bloch states.",
"Also, we show how important physics including optical selection rules and EFA Hamiltonians $\\mathcal {H}$ , despite the rearrangement of band IRs, does not depend on the coordinate system being used.",
"Our general theory applies to any crystalline material.",
"For a detailed example, we focus on a monolayer of the TMDC MoS$_2$ .",
"TMDCs are of the general form $TX_2$ , where $T$ is a transition-metal such as Mo or W and $X$ is a chalcogen which can be S, Se, or Te.",
"Three-dimensional (3D) bulk $TX_2$ consists of covalently bonded 2D monolayers coupled vertically by weak van der Waals forces [24], making it possible to obtain monolayers via, e.g., mechanical exfoliation [25].",
"Electronic band structure calculations have shown that bulk 2H-MoS$_2$ is a semiconductor [24].",
"More recently, optical spectroscopy [25] and theoretical studies [26], [27], [28] found that decreasing the number of layers changes the fundamental gap from indirect to direct in the limit of a single monolayer.",
"The spin-dependent dispersion of monolayer TMDCs has been studied using TB [29], [30] and ${\\mathbf {\\mathrm {k}}} \\cdot {\\mathbf {\\mathrm {p}}}$ methods [31], [16].",
"See Refs.",
"[32], [33], [34], [35] for general reviews of 2D TMDCs.",
"In this paper, we combine our symmetry analysis for the bulk Bloch states in monolayer MoS$_2$ with the theory of invariants [3] to derive a generic multiband EFA Hamiltonian for electrons near the ${\\mathbf {\\mathrm {K}}}$ point of the Brillouin zone (BZ).",
"The Hamiltonian includes the effect of strain, external electric and magnetic fields, spin and valley degrees of freedom.",
"For comparison, we also perform a symmetry analysis for few-layer graphene [14] which confirms earlier work [36], [37].",
"We note that our work expands on the theory of IRs for point and space groups [38], [3], [39].",
"It is conceptually rather different from recent work on band representations [40], [41], [42].",
"In Sec.",
", we develop the general theory of the symmetry of TB Bloch functions.",
"The decomposition of TB wave functions is discussed in Sec.",
"REF followed by detailed discussions of the symmetrized atomic orbitals (Sec.",
"REF ) and symmetrized plane waves (Sec.",
"REF ).",
"The rearrangement of the IRs of Bloch states under a change of the coordinate system is discussed in Sec.",
"REF .",
"We use the general formalism of Sec.",
"to derive the symmetry of bulk Bloch states in monolayer TMDCs (Sec. )",
"such as MoS$_2$ and to few-layer graphene (Sec. ).",
"We show in Sec.",
"how optical selection rules are not affected by the rearrangement of IRs under a change of coordinate system.",
"In Sec.",
"we derive the generic invariant expansion of the EFA Hamiltonian $\\mathcal {H}$ for MoS$_2$ .",
"Section contains our conclusions."
],
[
"Symmetry of Bloch Functions",
"Very generally, the eigenstates of a Hamiltonian transform according to an IR of the symmetry group of the Hamiltonian.",
"In band theory it is thus an important goal to determine the IRs of the energy bands $E_n ({\\mathbf {\\mathrm {k}}})$ and corresponding Bloch functions $\\Psi _{n {\\mathbf {\\mathrm {k}}}} ({\\mathbf {\\mathrm {r}}})$ , where the symmetry group $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ at a given wave vector ${\\mathbf {\\mathrm {k}}}$ is called the point group of the wave vector ${\\mathbf {\\mathrm {k}}}$ .",
"In this section we discuss a general method for determining the transformation properties of Bloch functions with a certain wave vector ${\\mathbf {\\mathrm {k}}}$ , which allows us to determine the corresponding IRs of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ .",
"We also discuss a rearrangement lemma for the IRs characterizing the Bloch functions in a crystal.",
"Applications to specific materials such as monolayer MoS$_2$ will be discussed in subsequent sections."
],
[
"The group of the wave vector",
"In the following, we will repeatedly need to evaluate the action of a point symmetry operation $g$ on a plane wave $\\exp ( i {\\mathbf {\\mathrm {k}}} \\cdot {\\mathbf {\\mathrm {r}}})$ .",
"Here, $g$ can be represented via an orthogonal $3 \\times 3$ matrix ${\\mathbf {\\mathrm {g}}}$ .",
"(In the context of quasi-2D materials discussed below ${\\mathbf {\\mathrm {g}}}$ becomes a $2 \\times 2$ matrix.)",
"Thus we have $g \\, \\exp ( i {\\mathbf {\\mathrm {k}}} \\cdot {\\mathbf {\\mathrm {r}}})= \\exp ( i {\\mathbf {\\mathrm {k}}} \\cdot {\\mathbf {\\mathrm {g}}} \\cdot {\\mathbf {\\mathrm {r}}})\\equiv \\exp ( i {\\mathbf {\\mathrm {k}}} \\cdot \\mathbf {\\mathrm {r}}^{\\prime })$ with $\\mathbf {\\mathrm {r}}^{\\prime } = {\\mathbf {\\mathrm {g}}} \\cdot {\\mathbf {\\mathrm {r}}} .$ Note that when transforming the position vector ${\\mathbf {\\mathrm {r}}}$ , the wave vector ${\\mathbf {\\mathrm {k}}}$ is a fixed parameter characterizing the plane wave $\\exp ( i {\\mathbf {\\mathrm {k}}} \\cdot {\\mathbf {\\mathrm {r}}})$ that does not change under $g$ .",
"Nonetheless, since $g$ is an orthogonal transformation, we can also write Eq.",
"(REF ) as $g \\, \\exp ( i {\\mathbf {\\mathrm {k}}} \\cdot {\\mathbf {\\mathrm {r}}})& = \\exp [ i ({\\mathbf {\\mathrm {g}}}^{-1} \\cdot {\\mathbf {\\mathrm {k}}}) \\cdot {\\mathbf {\\mathrm {r}}} ] \\\\& = \\exp ( i \\mathbf {\\mathrm {k}}^{\\prime } \\cdot {\\mathbf {\\mathrm {r}}})$ with $\\mathbf {\\mathrm {k}}^{\\prime } = {\\mathbf {\\mathrm {g}}}^{-1} \\cdot {\\mathbf {\\mathrm {k}}} .$ Thus we can evaluate $g$ either by transforming the position vector ${\\mathbf {\\mathrm {r}}}$ or by inversely transforming the wave vector ${\\mathbf {\\mathrm {k}}}$ .",
"In the group theory of crystallographic space groups, the point-group symmetries $g$ of Bloch functions $\\Psi _{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {r}}})$ with wave vector ${\\mathbf {\\mathrm {k}}}$ form the point group $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ of the wave vector ${\\mathbf {\\mathrm {k}}}$ [38], [3], [39].",
"Given the point group $\\mathcal {G}_0$ of a crystal structure, the group $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ is defined by the condition that it contains the symmetry elements of $\\mathcal {G}_0$ that map ${\\mathbf {\\mathrm {k}}}$ onto a vector $\\mathbf {\\mathrm {k}}^{\\prime }$ such that $\\mathbf {\\mathrm {k}}^{\\prime } = {\\mathbf {\\mathrm {g}}}^{-1} \\cdot {\\mathbf {\\mathrm {k}}} = {\\mathbf {\\mathrm {k}}} + {\\mathbf {\\mathrm {b}}}_g ,$ where ${\\mathbf {\\mathrm {b}}}_g$ is a reciprocal lattice vector with the possibility ${\\mathbf {\\mathrm {b}}}_g = 0$ .",
"Indeed, since $g$ represents point group operations, we can have ${\\mathbf {\\mathrm {b}}}_g \\ne 0$ only if ${\\mathbf {\\mathrm {k}}}$ is from the border of the BZ.",
"For positions ${\\mathbf {\\mathrm {r}}} = {\\mathbf {\\mathrm {a}}}$ that are lattice vectors we have $g \\, \\exp ( i {\\mathbf {\\mathrm {k}}} \\cdot {\\mathbf {\\mathrm {a}}})= \\exp ( i \\mathbf {\\mathrm {k}}^{\\prime } \\cdot {\\mathbf {\\mathrm {a}}})= \\exp ( i {\\mathbf {\\mathrm {k}}} \\cdot {\\mathbf {\\mathrm {a}}})$ by definition of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ ."
],
[
"Tight-binding Hamiltonian",
"We denote the TB basis functions (that are Bloch functions) as $\\Phi _{\\nu {\\mathbf {\\mathrm {k}}}}^{W \\mu } ({\\mathbf {\\mathrm {r}}}) = \\frac{e^{i {\\mathbf {\\mathrm {k}}} \\cdot {\\mathbf {\\mathrm {r}}}}}{\\sqrt{N}}\\sum _j e^{-i {\\mathbf {\\mathrm {k}}} \\cdot ({\\mathbf {\\mathrm {r}}}-{\\mathbf {\\mathrm {R}}}_j^{W \\mu })} \\, \\phi _\\nu ^W ({\\mathbf {\\mathrm {r}}} -{\\mathbf {\\mathrm {R}}}_j^{W \\mu }) ,$ where $\\phi _\\nu ^W ({\\mathbf {\\mathrm {r}}}-{\\mathbf {\\mathrm {R}}}_j^{W \\mu })$ are the atomic orbitals of type $\\nu $ centered about the positions ${\\mathbf {\\mathrm {R}}}_j^{W \\mu }$ of the atoms.",
"The label $W$ denotes the Wyckoff letter of the atomic positions of the crystal structure [43].",
"The label $\\mu $ has values $\\mu = 1, \\dots , m$ , where $m$ is the multiplicity of $W$ .",
"The index $j$ labels the unit cells of the crystal structure; it runs through the positions in a Bravais lattice.",
"The matrix elements of the TB Hamiltonian can then be written as $H ({\\mathbf {\\mathrm {k}}})_{\\nu \\nu ^{\\prime }}^{W \\mu W^{\\prime } \\mu ^{\\prime }} & = \\int \\Phi _{\\nu {\\mathbf {\\mathrm {k}}}}^{W \\mu \\ast } ({\\mathbf {\\mathrm {r}}}) \\, H \\, \\Phi _{\\nu ^{\\prime } {\\mathbf {\\mathrm {k}}}}^{W^{\\prime } \\mu ^{\\prime }} ({\\mathbf {\\mathrm {r}}}) \\, d^3r\\\\& = \\epsilon _\\nu ^W \\delta _{\\nu \\nu ^{\\prime }} \\, \\delta _{W W^{\\prime }} \\, \\delta _{\\mu \\mu ^{\\prime }} + \\mathop {\\mmlmultiscripts{\\sum {\\mmlnone }{\\prime }}}\\limits _{jj^{\\prime }} t_{\\nu \\nu ^{\\prime } jj^{\\prime }}^{W W^{\\prime } \\mu \\mu ^{\\prime }} ,$ where $\\epsilon _\\nu ^W & \\equiv \\int \\phi _\\nu ^{W \\ast } ({\\mathbf {\\mathrm {r}}} -{\\mathbf {\\mathrm {R}}}_j^{W \\mu }) \\, H \\, \\phi _\\nu ^W ({\\mathbf {\\mathrm {r}}} -{\\mathbf {\\mathrm {R}}}_j^{W \\mu }) \\, d^3r \\\\& = \\int \\phi _\\nu ^{W \\ast } ({\\mathbf {\\mathrm {r}}}) \\, H \\, \\phi _\\nu ^W ({\\mathbf {\\mathrm {r}}}) \\, d^3r$ denotes the on-site energies for the atomic orbitals (that do not depend on the indices $\\mu $ and $j$ ) and $t_{\\nu \\nu ^{\\prime } jj^{\\prime }}^{W W^{\\prime } \\mu \\mu ^{\\prime }}& \\equiv \\; e^{-i {\\mathbf {\\mathrm {k}}} \\cdot ({\\mathbf {\\mathrm {R}}}_j^{W \\mu } - {\\mathbf {\\mathrm {R}}}_{j^{\\prime }}^{W^{\\prime } \\mu ^{\\prime }})} \\,\\nonumber \\\\ & \\times \\int \\phi _\\nu ^{W \\ast } ({\\mathbf {\\mathrm {r}}} -{\\mathbf {\\mathrm {R}}}_j^{W \\mu }) \\, H \\, \\phi _{\\nu ^{\\prime }}^{W^{\\prime }} ({\\mathbf {\\mathrm {r}}} -{\\mathbf {\\mathrm {R}}}_{j^{\\prime }}^{W^{\\prime } \\mu ^{\\prime }}) \\, d^3r$ are the hopping integrals.",
"The prime on the summation sign in Eq.",
"(REF ) indicates that the sum excludes the on-site term $\\epsilon _\\nu ^W$ .",
"The TB approximation implies that this sum is restricted to $n$ th-nearest neighbors with a small value of $n$ .",
"The hopping integrals can be written in terms of the Slater-Koster parameters [44] for hopping integrals between atomic orbitals at positions ${\\mathbf {\\mathrm {R}}}_j^{W \\mu }$ and ${\\mathbf {\\mathrm {R}}}_{j^{\\prime }}^{W^{\\prime } \\mu ^{\\prime }}$ ."
],
[
"Decomposition of TB wave functions",
"Generally, the basis functions (REF ) for a given wave vector ${\\mathbf {\\mathrm {k}}}$ transform according to a representation $\\Gamma ^\\Phi _{{\\mathbf {\\mathrm {k}}} W}$ of the group of the wave vector $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ that need not be irreducible.",
"(Here the generic superscript $\\Phi $ accounts for the fact that multiple atomic orbitals with different indices $\\nu $ may transform jointly according to the same representation.",
"By definition of Wyckoff letters $W$ , orbitals at different positions $\\mu $ of a given Wyckoff letter $W$ transform jointly according to one representation.)",
"As a consequence of the Wigner-Eckart theorem [19], [45] and the fact that $H$ transforms according to the identity representation $\\Gamma _1$ of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ , the hopping integrals (REF ) vanish when the product of the representations $\\Gamma ^\\Phi _{{\\mathbf {\\mathrm {k}}} W}$ and $\\Gamma ^{\\Phi ^{\\prime }}_{{\\mathbf {\\mathrm {k}}} W^{\\prime }}$ does not contain the identity representation.",
"In the following, we present a general scheme for transforming the set of basis functions (REF ) into a symmetry-adapted set of basis functions, where each function transforms irreducibly under $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ , so that two such basis functions only couple when they both transform according to the same IR of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ .",
"This scheme is based on a decomposition of the basis functions into symmetry-adapted plane waves and symmetry-adapted atomic orbitals.",
"We denote $\\tilde{\\phi }_{\\nu {\\mathbf {\\mathrm {k}}}}^W ({\\mathbf {\\mathrm {r}}} -{\\mathbf {\\mathrm {R}}}_j^{W \\mu })\\equiv e^{-i {\\mathbf {\\mathrm {k}}} \\cdot ({\\mathbf {\\mathrm {r}}} -{\\mathbf {\\mathrm {R}}}_j^{W \\mu })}\\, \\phi _\\nu ^W ({\\mathbf {\\mathrm {r}}} -{\\mathbf {\\mathrm {R}}}_j^{W \\mu }),$ so that Eq.",
"(REF ) becomes $\\Phi _{\\nu {\\mathbf {\\mathrm {k}}}}^{W \\mu } ({\\mathbf {\\mathrm {r}}}) = \\frac{e^{i {\\mathbf {\\mathrm {k}}} \\cdot {\\mathbf {\\mathrm {r}}}}}{\\sqrt{N}} \\sum _j \\tilde{\\phi }_{\\nu {\\mathbf {\\mathrm {k}}}}^W ({\\mathbf {\\mathrm {r}}} - {\\mathbf {\\mathrm {R}}}_j^{W \\mu }).$ Assuming for conceptual simplicity that the atomic orbitals $\\phi _\\nu ^W$ are localized over a region much smaller than the nearest-neighbor distance This assumption facilitates a discussion of the symmetry of the basis functions (REF ).",
"Functions (REF ) with a finite overlap between nearest neighbors that are needed in a TB calculation must have the same symmetry as the simplified functions discussed here.",
"The symmetry of these functions cannot change discontinuously when the overlap between the atomic orbitals is switched off., the functions $\\tilde{\\phi }_{\\nu {\\mathbf {\\mathrm {k}}}}^W ({\\mathbf {\\mathrm {r}}} - {\\mathbf {\\mathrm {R}}}_j^{W \\mu })$ are only nonzero for ${\\mathbf {\\mathrm {r}}}$ close to ${\\mathbf {\\mathrm {R}}}_j^{W \\mu }$ .",
"Hence, in the vicinity of any atomic position ${\\mathbf {\\mathrm {R}}}_{j}^{W \\mu }$ , i.e., for ${\\mathbf {\\mathrm {r}}} \\equiv {\\mathbf {\\mathrm {R}}}_{j}^{W \\mu } + \\delta {\\mathbf {\\mathrm {r}}}$ with small $\\delta {\\mathbf {\\mathrm {r}}}$ , we have $\\tilde{\\phi }_{\\nu {\\mathbf {\\mathrm {k}}}}^W ({\\mathbf {\\mathrm {r}}} - {\\mathbf {\\mathrm {R}}}_j^{W \\mu })\\approx \\phi _\\nu ^W ({\\mathbf {\\mathrm {r}}} - {\\mathbf {\\mathrm {R}}}_j^{W \\mu }).$ Therefore, the TB basis function $\\Phi _{\\nu {\\mathbf {\\mathrm {k}}}}^{W \\mu } ({\\mathbf {\\mathrm {r}}})$ can be approximated as $\\Phi _{\\nu {\\mathbf {\\mathrm {k}}}}^{W \\mu } ({\\mathbf {\\mathrm {r}}})& \\approx \\frac{e^{i {\\mathbf {\\mathrm {k}}} \\cdot {\\mathbf {\\mathrm {r}}}}}{\\sqrt{N}}\\sum _j \\phi _\\nu ^W ({\\mathbf {\\mathrm {r}}} - {\\mathbf {\\mathrm {R}}}_j^{W \\mu }) \\\\& = e^{i {\\mathbf {\\mathrm {k}}} \\cdot {\\mathbf {\\mathrm {r}}}} \\, \\mathcal {A}_\\nu ^{W \\mu } ({\\mathbf {\\mathrm {r}}})$ with atomic functions $\\mathcal {A}_\\nu ^{W \\mu } ({\\mathbf {\\mathrm {r}}})= \\frac{1}{\\sqrt{N}} \\sum _j \\phi _\\nu ^W ({\\mathbf {\\mathrm {r}}} - {\\mathbf {\\mathrm {R}}}_j^{W \\mu })$ independent of the wave vector ${\\mathbf {\\mathrm {k}}}$ .",
"For strongly localized atomic orbitals and positions ${\\mathbf {\\mathrm {r}}} = {\\mathbf {\\mathrm {R}}}_j^{W \\mu } + \\delta {\\mathbf {\\mathrm {r}}}$ we have $\\mathcal {A}_\\nu ^{W \\mu } ({\\mathbf {\\mathrm {r}}} = {\\mathbf {\\mathrm {R}}}_j^{W \\mu } + \\delta {\\mathbf {\\mathrm {r}}})= \\frac{1}{\\sqrt{N}} \\sum _j \\phi _\\nu ^W (\\delta {\\mathbf {\\mathrm {r}}})\\propto \\phi _\\nu ^W (\\delta {\\mathbf {\\mathrm {r}}}),$ that is, near an atomic site ${\\mathbf {\\mathrm {R}}}_j^{W \\mu }$ , the atomic functions $\\mathcal {A}_\\nu ^{W \\mu } ({\\mathbf {\\mathrm {r}}})$ are simply proportional to $\\phi _\\nu ^W (\\delta {\\mathbf {\\mathrm {r}}})$ , independent of the index $\\mu $ .",
"Therefore, the atomic functions $\\mathcal {A}_\\nu ^{W \\mu } ({\\mathbf {\\mathrm {r}}} = {\\mathbf {\\mathrm {R}}}_j^{W \\mu } + \\delta {\\mathbf {\\mathrm {r}}})$ have the same symmetry properties as the atomic orbitals $\\phi _\\nu ^W (\\delta {\\mathbf {\\mathrm {r}}})$ .",
"The plane wave $e^{i {\\mathbf {\\mathrm {k}}} \\cdot {\\mathbf {\\mathrm {r}}}}$ for positions ${\\mathbf {\\mathrm {r}}} = {\\mathbf {\\mathrm {R}}}_j^{W \\mu } + \\delta {\\mathbf {\\mathrm {r}}}$ near an atomic site ${\\mathbf {\\mathrm {R}}}_j^{W \\mu }$ is approximately given by $\\exp [i {\\mathbf {\\mathrm {k}}} \\cdot ({\\mathbf {\\mathrm {R}}}_j^{W \\mu } + \\delta {\\mathbf {\\mathrm {r}}})]\\approx \\exp (i {\\mathbf {\\mathrm {k}}} \\cdot {\\mathbf {\\mathrm {R}}}_j^{W \\mu })\\equiv q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{W \\mu }) ,$ where $q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{W \\mu })$ denotes the plane wave with wave vector ${\\mathbf {\\mathrm {k}}}$ associated with the Wyckoff position ${\\mathbf {\\mathrm {R}}}_j^{W \\mu }$ for fixed $W$ and $\\mu $ , but $j$ runs over all positions in a Bravais lattice.",
"These discrete quantities $q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{W \\mu })$ will be discussed in more detail in Sec.",
"REF .",
"The TB basis function thus can be factorized (ignoring normalization) $\\Phi _{\\nu {\\mathbf {\\mathrm {k}}}}^{W \\mu } ({\\mathbf {\\mathrm {r}}}= {\\mathbf {\\mathrm {R}}}_j^{W \\mu } + \\delta {\\mathbf {\\mathrm {r}}} )\\approx q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{W \\mu }) \\, \\phi _\\nu ^W (\\delta {\\mathbf {\\mathrm {r}}}).$ This expression will be analyzed further in the following sections.",
"As a side remark, we note that the eigenfunctions of the TB Hamiltonian (REF ) for the band $n$ and wave vector ${\\mathbf {\\mathrm {k}}}$ expressed in terms of the basis functions (REF ) take the form $\\Psi _{n {\\mathbf {\\mathrm {k}}}} ({\\mathbf {\\mathrm {r}}})= \\sum _{W \\mu \\nu } \\psi _{\\nu n{\\mathbf {\\mathrm {k}}}}^{W \\mu } \\,\\Phi _{\\nu {\\mathbf {\\mathrm {k}}}}^{W \\mu } ({\\mathbf {\\mathrm {r}}})$ with expansion coefficients $\\psi _{\\nu n {\\mathbf {\\mathrm {k}}}}^{W \\mu }$ .",
"These eigenfunctions permit a factorization similar to Eq.",
"(REF ) $\\Psi _{n {\\mathbf {\\mathrm {k}}}} ({\\mathbf {\\mathrm {r}}}= {\\mathbf {\\mathrm {R}}}_j^{W \\mu } + \\delta {\\mathbf {\\mathrm {r}}})\\approx \\sum _{W \\mu } q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{W \\mu })\\sum _\\nu \\psi _{\\nu n {\\mathbf {\\mathrm {k}}}}^{W \\mu } \\; \\phi _\\nu ^W (\\delta {\\mathbf {\\mathrm {r}}}) .$ However, the discussion of the TB wave functions $\\Psi _{n {\\mathbf {\\mathrm {k}}}} ({\\mathbf {\\mathrm {r}}})$ is greatly simplified if instead of the basis functions (REF ) we use symmetry-adapted basis functions to be discussed in the following."
],
[
"Symmetry-adapted basis functions",
"The main advantage of the approximate expression (REF ) lies in the fact that the function (REF ) has the same symmetry properties as the TB basis function (REF ).",
"Yet the factorization in Eq.",
"(REF ) allows one to discuss the symmetry of the plane waves $q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{W \\mu })$ (characterized by a representation $\\Gamma ^q_{{\\mathbf {\\mathrm {k}}} W}$ of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ , see Sec.",
"REF ) separate from the symmetry of the atomic orbitals $\\phi _\\nu ^W (\\delta {\\mathbf {\\mathrm {r}}})$ (characterized by a representation $\\Gamma ^\\phi _{{\\mathbf {\\mathrm {k}}} W}$ of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ , see Sec.",
"REF ).",
"Often, these representations are irreducible, though generally they can be reducible.",
"The TB basis functions (REF ) at one Wyckoff letter $W$ then transform according to the product representation $\\Gamma ^\\Phi _{{\\mathbf {\\mathrm {k}}} W} \\equiv \\Gamma ^q_{{\\mathbf {\\mathrm {k}}} W} \\times \\Gamma ^\\phi _{{\\mathbf {\\mathrm {k}}} W}.$ The representation $\\Gamma ^\\Phi _{{\\mathbf {\\mathrm {k}}} W}$ may be reducible (even if $\\Gamma ^q_{{\\mathbf {\\mathrm {k}}} W}$ and $\\Gamma ^\\phi _{{\\mathbf {\\mathrm {k}}} W}$ are irreducible), giving the decomposition $\\Gamma ^\\Phi _{{\\mathbf {\\mathrm {k}}} W} = \\sum _I a_I \\Gamma _I ,$ where $a_I \\ge 0$ are the multiplicities with which the IRs $\\Gamma _I$ of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ appear in $\\Gamma ^\\Phi _{{\\mathbf {\\mathrm {k}}} W}$ .",
"Generally, the IRs $\\Gamma _I$ contained in $\\Gamma ^\\Phi _{{\\mathbf {\\mathrm {k}}} W}$ give symmetry-adapted basis functions $\\Phi _{\\mathbf {\\mathrm {k}}}^{W I \\alpha \\beta } ({\\mathbf {\\mathrm {r}}})= \\sum _{\\mu \\nu } \\mathcal {C}_{I \\alpha \\beta }^{W \\mu \\nu {\\mathbf {\\mathrm {k}}}} \\;\\Phi _{\\nu {\\mathbf {\\mathrm {k}}}}^{W \\mu } ({\\mathbf {\\mathrm {r}}}) ,$ where $\\alpha = 1, 2, \\ldots , a_I$ and $\\beta $ labels the different functions transforming jointly according to $\\Gamma _I$ .",
"The coefficients $\\mathcal {C}_{I \\alpha \\beta }^{W \\mu \\nu {\\mathbf {\\mathrm {k}}}}$ represent the weights with which the functions $\\Phi _{\\nu {\\mathbf {\\mathrm {k}}}}^{W \\mu } ({\\mathbf {\\mathrm {r}}})$ contribute to the symmetry-adapted basis functions $\\Phi _{\\mathbf {\\mathrm {k}}}^{W I \\alpha \\beta } ({\\mathbf {\\mathrm {r}}})$ .",
"Given symmetry-adapted plane waves transforming according to the IR $\\Gamma _J$ and atomic orbitals transforming according to the IR $\\Gamma _{J^{\\prime }}$ (see discussion below), the expansion coefficients $\\mathcal {C}_{I \\alpha \\beta }^{W \\mu \\nu {\\mathbf {\\mathrm {k}}}}$ are given by the Clebsch-Gordan coefficients for coupling $\\Gamma _J$ and $\\Gamma _{J^{\\prime }}$ to obtain $\\Gamma _I$ .",
"For the symmetry-adapted basis functions (REF ), the matrix elements (REF ) of the TB Hamiltonian become block-diagonal with respect to different IRs $\\Gamma _I$ $H ({\\mathbf {\\mathrm {k}}})_{\\alpha \\alpha ^{\\prime } \\beta \\beta ^{\\prime }}^{W W^{\\prime } I I^{\\prime }}= \\delta _{I I^{\\prime }} \\Bigl [\\epsilon _{I \\alpha }^W \\, \\delta _{W W^{\\prime }} \\,\\delta _{\\alpha \\alpha ^{\\prime }} \\, \\delta _{\\beta \\beta ^{\\prime }}+ \\mathop {\\mmlmultiscripts{\\sum {\\mmlnone }{\\prime }}}\\limits _{jj^{\\prime }} t_{\\beta \\beta ^{\\prime } \\, j j^{\\prime }}^{W W^{\\prime } I \\, \\alpha \\alpha ^{\\prime }}\\Bigr ] .$ Here $\\epsilon _{I \\alpha }^W \\equiv \\int \\Phi _{\\mathbf {\\mathrm {k}}}^{W I \\alpha \\beta \\ast } ({\\mathbf {\\mathrm {r}}}) \\, H \\, \\Phi _{\\mathbf {\\mathrm {k}}}^{W I \\alpha \\beta } ({\\mathbf {\\mathrm {r}}}) \\, d^3r$ denotes the on-site energies.",
"These can always be made diagonal in the index $\\alpha $ by a suitable definition of the $a_I$ sets of basis functions (REF ) transforming according to $\\Gamma _I$ .",
"The hopping matrix elements become $t_{\\beta \\beta ^{\\prime } \\, j j^{\\prime }}^{W W^{\\prime } I \\, \\alpha \\alpha ^{\\prime }}\\equiv \\int \\Phi _{\\mathbf {\\mathrm {k}}}^{W I \\alpha \\beta \\ast } ({\\mathbf {\\mathrm {r}}}) \\, H \\, \\Phi _{\\mathbf {\\mathrm {k}}}^{W^{\\prime } I \\alpha ^{\\prime } \\beta ^{\\prime }} ({\\mathbf {\\mathrm {r}}}^{\\prime }) \\, d^3r$ with ${\\mathbf {\\mathrm {r}}} = {\\mathbf {\\mathrm {R}}}_j^{W \\mu } + \\delta {\\mathbf {\\mathrm {r}}}$ and ${\\mathbf {\\mathrm {r}}}^{\\prime } = {\\mathbf {\\mathrm {R}}}_{j^{\\prime }}^{W^{\\prime } \\mu ^{\\prime }} + \\delta {\\mathbf {\\mathrm {r}}}$ .",
"We remark that in actual TB models it may happen that a block (REF ) can be further decomposed into subblocks if symmetry-allowed couplings between distant neighbors are ignored within the TB approximation.",
"Each block (REF ) of the TB Hamiltonian yields eigenfunctions $\\Psi _{n {\\mathbf {\\mathrm {k}}}}^{I \\beta } ({\\mathbf {\\mathrm {r}}})= \\sum _{W, \\alpha } C_{n {\\mathbf {\\mathrm {k}}}}^{W I \\alpha }\\Phi _{\\mathbf {\\mathrm {k}}}^{W I \\alpha \\beta } ({\\mathbf {\\mathrm {r}}})$ with expansion coefficients $C_{n {\\mathbf {\\mathrm {k}}}}^{W I \\alpha \\beta }$ that transform irreducibly according to the $\\beta $ th component of the IR $\\Gamma _I$ of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ .",
"To proceed, we discuss first the symmetry of the atomic orbitals $\\phi _\\nu ^W (\\delta {\\mathbf {\\mathrm {r}}})$ , followed by a discussion of the symmetry of the plane waves $q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{W \\mu })$ ."
],
[
"Transformation of atomic orbitals $\\phi _\\nu ^W (\\delta \\mathbf {r})$",
"We can study the symmetry of the atomic function $\\mathcal {A}_\\nu ^{W \\mu } ({\\mathbf {\\mathrm {r}}})$ in the vicinity of the atomic positions ${\\mathbf {\\mathrm {R}}}_j^{W \\mu }$ by only looking at the orbitals $\\phi _\\nu ^W (\\delta {\\mathbf {\\mathrm {r}}})$ .",
"Obviously, this problem is independent of the index $\\mu $ , the band index $n$ , and the wave vector ${\\mathbf {\\mathrm {k}}}$ though, of course, we are generally interested in the transformational behavior of these orbitals with respect to the group $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ of the wave vector ${\\mathbf {\\mathrm {k}}}$ .",
"For symmetry operations $g \\in \\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ we have $g \\, \\phi _\\nu ^W (\\delta {\\mathbf {\\mathrm {r}}})= \\phi _\\nu ^W (g \\, \\delta {\\mathbf {\\mathrm {r}}})= \\sum _{\\nu ^{\\prime }} \\mathcal {D}^\\phi _{{\\mathbf {\\mathrm {k}}} W} (g)_{\\nu \\nu ^{\\prime }}\\phi _{\\nu ^{\\prime }}^W (\\delta {\\mathbf {\\mathrm {r}}}) ,$ where $\\Gamma ^\\phi _{{\\mathbf {\\mathrm {k}}} W} = \\lbrace \\mathcal {D}^\\phi _{{\\mathbf {\\mathrm {k}}} W} (g): g \\in \\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}\\rbrace $ is the representation describing the transformation of the atomic orbitals $\\phi _\\nu ^W$ .",
"As usual, we assume that the atomic orbitals are characterized by some orbital angular momentum $l=0, 1, 2, \\ldots $ , so that the matrices $\\mathcal {D}^\\phi _{{\\mathbf {\\mathrm {k}}} W} (g)$ acquire a block structure corresponding to different values of $l$ , indicating that there exists no mixing between atomic orbitals with different angular momenta under symmetry transformations $g \\in \\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ .",
"In general, the representation $\\Gamma ^\\phi _{{\\mathbf {\\mathrm {k}}} W}$ is reducible The $2l+1$ atomic orbitals for a given magnitude $l$ of orbital angular momentum transform according to an IR of the full rotation group.",
"The compatibility tables for decomposing the IRs of the full rotation group into IRs of the crystallographic point groups are listed in Ref. [56].",
"Here we require a more complete solution of this problem where we also construct the linear combinations of angular-momentum eigenstates that transform irreducibly according to the different IRs of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ .",
"This problem does not depend on, e.g., the principal quantum numbers of these orbitals.",
"We note that for sufficiently large $l$ (for any $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ if $l \\ge 6$ ) these $2l+1$ atomic orbitals contribute to all (even or odd if $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ includes inversion) IRs of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ , which limits possibilities to decompose a TB Hamiltonian into blocks describing parts of the full band structure..",
"The projection operators (REF ) yield symmetry-adapted atomic orbitals $\\phi ^W_{I \\beta } (\\delta {\\mathbf {\\mathrm {r}}})$ transforming like component $\\beta $ of the IR $\\Gamma _I$ of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ , $\\phi ^W_{I \\beta } (\\delta {\\mathbf {\\mathrm {r}}})& \\equiv \\Pi _{I \\beta } \\, \\phi _\\nu ^W (\\delta {\\mathbf {\\mathrm {r}}}) \\\\& = \\frac{n_I}{h}\\sum _g \\mathcal {D}_I (g)^\\ast _{\\beta \\beta }\\: g \\, \\phi _\\nu ^W (\\delta {\\mathbf {\\mathrm {r}}}) \\\\& = \\sum _{\\nu ^{\\prime }} c^{W \\nu ^{\\prime }}_{I \\beta } \\phi _{\\nu ^{\\prime }}^W (\\delta {\\mathbf {\\mathrm {r}}})$ with expansion coefficients $c^{W \\nu ^{\\prime }}_{I \\beta }= \\frac{n_I}{h} \\sum _g \\mathcal {D}_I (g)^\\ast _{\\beta \\beta }\\, \\mathcal {D}^\\phi _{{\\mathbf {\\mathrm {k}}} W} (g)_{\\nu \\nu ^{\\prime }} .$ We note that this analysis applies to the spinless case when the angular part of the atomic orbitals is given by the usual spherical harmonics $Y_l^m$ .",
"It can likewise be used in the spin-dependent case when the angular part of the atomic orbitals is given by spin-angular functions and the projection operators $\\Pi _{I \\beta }$ project on the double-group representations of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ ."
],
[
"Transformation of plane waves $q_\\mathbf {k} (\\mathbf {R}_j^{W \\mu })$",
"While the IRs of TB eigenfunctions (REF ) depend on the band index $n$ , the symmetry of the plane waves $q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{W \\mu })$ can be discussed independent of the index $n$ .",
"Very generally, for a given Wyckoff letter $W$ , the positions ${\\mathbf {\\mathrm {R}}}_j^{W \\mu }$ transform among themselves under the operations of the space group.",
"Hence, using the matrix ${\\mathbf {\\mathrm {g}}}$ defined in Eq.",
"(REF ), we have $g \\, {\\mathbf {\\mathrm {R}}}_j^{W \\mu }= {\\mathbf {\\mathrm {g}}} \\cdot {\\mathbf {\\mathrm {R}}}_j^{W \\mu }\\equiv {\\mathbf {\\mathrm {R}}}_{j^{\\prime }}^{W \\mu ^{\\prime }} ,$ i.e., a symmetry transformation $g \\in \\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ generally changes both $\\mu $ and $j$ .",
"We rewrite this as ${\\mathbf {\\mathrm {g}}} \\cdot {\\mathbf {\\mathrm {R}}}_j^{W \\mu }= {\\mathbf {\\mathrm {R}}}_j^{W \\mu ^{\\prime }}+ \\left[ {\\mathbf {\\mathrm {R}}}_{j^{\\prime }}^{W \\mu ^{\\prime }} - {\\mathbf {\\mathrm {R}}}_j^{W \\mu ^{\\prime }} \\right] ,$ where the term in square brackets is a lattice vector.",
"Applying the operation $g$ to the plane wave $q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{W \\mu })$ yields $g \\, q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{W \\mu })& = \\exp ( i {\\mathbf {\\mathrm {k}}} \\cdot {\\mathbf {\\mathrm {R}}}_{j^{\\prime }}^{W \\mu ^{\\prime }})\\\\& = q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{W \\mu ^{\\prime }})\\exp [ i {\\mathbf {\\mathrm {k}}} \\cdot ({\\mathbf {\\mathrm {R}}}_{j^{\\prime }}^{W \\mu ^{\\prime }} - {\\mathbf {\\mathrm {R}}}_j^{W \\mu ^{\\prime }})] .$ Hence, the symmetry operation $g$ generally maps the plane wave $q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{W \\mu })$ onto $q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{W \\mu ^{\\prime }})$ multiplied by a phase factor.",
"To analyze the mappings (REF ) further, we interpret the discrete plane waves $\\lbrace q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{W \\mu }) : \\mu = 1, \\ldots , m \\rbrace $ as basis vectors in an $m$ -dimensional vector space.",
"Relative to this basis, we can express arbitrary points as $m$ -component spinors.",
"It facilitates this analysis to introduce $m$ -component base spinors $\\lbrace \\mathcal {q}_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{W \\mu }) : \\mu = 1, \\ldots , m \\rbrace $ with components $\\nu $ equal to $\\delta _{\\mu \\nu }$ .",
"Using these base spinors, a plane wave at positions ${\\mathbf {\\mathcal {R}}}_j^W \\equiv \\lbrace {\\mathbf {\\mathrm {R}}}_j^{W \\mu }: \\mu = 1, \\ldots , m\\rbrace $ becomes ${Q}_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathcal {R}}}_j^W) = \\sum _\\mu \\mathcal {q}_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{W \\mu })= \\left( \\begin{array}{s{0.15em}cs{0.15em}} 1 \\\\[-1ex] \\vdots \\\\ 1\\end{array} \\right) .$ Applying a symmetry transformation $g$ to ${Q}_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathcal {R}}}_j^W)$ gives $g \\, {Q}_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathcal {R}}}_j^W)= \\mathcal {D}^q_{{\\mathbf {\\mathrm {k}}} W} (g) \\, {Q}_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathcal {R}}}_j^W) ,$ where $\\mathcal {D}^q_{{\\mathbf {\\mathrm {k}}} W} (g)$ is an $m \\times m$ matrix corresponding to the group element $g$ .",
"Each row and each column of $\\mathcal {D}^q_{{\\mathbf {\\mathrm {k}}} W} (g)$ has only one nonzero matrix element which, according to Eq.",
"(REF ), is given by $\\mathcal {D}^q_{{\\mathbf {\\mathrm {k}}} W} (g)_{\\mu ^{\\prime } \\mu }= \\exp [ i {\\mathbf {\\mathrm {k}}} \\cdot ({\\mathbf {\\mathrm {R}}}_{j^{\\prime }}^{W \\mu ^{\\prime }} - {\\mathbf {\\mathrm {R}}}_j^{W \\mu ^{\\prime }})] ,$ where the $g$ dependence on the right-hand side is given by Eq.",
"(REF ).",
"We show in the following that $\\Gamma ^q_{{\\mathbf {\\mathrm {k}}} W} \\equiv \\lbrace \\mathcal {D}^q_{{\\mathbf {\\mathrm {k}}} W} (g) : g \\in \\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}\\rbrace $ defines an $m$ -dimensional representation of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ , i.e., for any two group elements $g_1, g_2 \\in \\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ we have $\\mathcal {D}^q_{{\\mathbf {\\mathrm {k}}} W} (g_2) \\mathcal {D}^q_{{\\mathbf {\\mathrm {k}}} W} (g_1)= \\mathcal {D}^q_{{\\mathbf {\\mathrm {k}}} W} (g_2 g_1).$ Given a position ${\\mathbf {\\mathrm {R}}}_{j_0}^{W \\mu _0}$ and using Eq.",
"(REF ), we have $g_1 {\\mathbf {\\mathrm {R}}}_{j_0}^{W \\mu _0}& = {\\mathbf {\\mathrm {g}}}_1 \\cdot {\\mathbf {\\mathrm {R}}}_{j_0}^{W \\mu _0} \\nonumber \\\\& \\equiv {\\mathbf {\\mathrm {R}}}_{j_1}^{W \\mu _1}= {\\mathbf {\\mathrm {R}}}_{j_0}^{W \\mu _1}+ \\left[ {\\mathbf {\\mathrm {R}}}_{j_1}^{W \\mu _1} - {\\mathbf {\\mathrm {R}}}_{j_0}^{W \\mu _1} \\right] , \\\\g_2 {\\mathbf {\\mathrm {R}}}_{j_0}^{W \\mu _1}& = {\\mathbf {\\mathrm {g}}}_2 \\cdot {\\mathbf {\\mathrm {R}}}_{j_0}^{W \\mu _1} \\nonumber \\\\& \\equiv {\\mathbf {\\mathrm {R}}}_{j_2}^{W \\mu _2}= {\\mathbf {\\mathrm {R}}}_{j_0}^{W \\mu _2}+ \\left[ {\\mathbf {\\mathrm {R}}}_{j_2}^{W \\mu _2} - {\\mathbf {\\mathrm {R}}}_{j_0}^{W \\mu _2} \\right] ,$ so that the phase (REF ) for the group elements $g_1$ and $g_2$ becomes $\\mathcal {D}^q_{{\\mathbf {\\mathrm {k}}} W} (g_1)_{\\mu _1 \\mu _0}& = \\exp [ i {\\mathbf {\\mathrm {k}}} \\cdot ({\\mathbf {\\mathrm {R}}}_{j_1}^{W \\mu _1} - {\\mathbf {\\mathrm {R}}}_{j_0}^{W \\mu _1})] , \\\\\\mathcal {D}^q_{{\\mathbf {\\mathrm {k}}} W} (g_2)_{\\mu _2 \\mu _1}& = \\exp [ i {\\mathbf {\\mathrm {k}}} \\cdot ({\\mathbf {\\mathrm {R}}}_{j_2}^{W \\mu _2} - {\\mathbf {\\mathrm {R}}}_{j_0}^{W \\mu _2})].$ The product $g_2 \\, g_1$ , i.e., the transformation $g_1$ followed by $g_2$ , is also an element of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ , giving $g_2 \\, g_1 \\, q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_{j_0}^{W \\mu _0})& = \\exp (i {\\mathbf {\\mathrm {k}}} \\cdot g_2 {\\mathbf {\\mathrm {R}}}_{j_1}^{W \\mu _1})\\\\& = \\exp (i {\\mathbf {\\mathrm {k}}} \\cdot g_2 {\\mathbf {\\mathrm {R}}}_{j_0}^{W \\mu _1})\\exp [i {\\mathbf {\\mathrm {k}}} \\cdot g_2 ({\\mathbf {\\mathrm {R}}}_{j_1}^{W \\mu _1}- {\\mathbf {\\mathrm {R}}}_{j_0}^{W \\mu _1})]\\\\& = \\exp (i {\\mathbf {\\mathrm {k}}} \\cdot {\\mathbf {\\mathrm {R}}}_{j_0}^{W \\mu _2})\\exp [i {\\mathbf {\\mathrm {k}}} \\cdot ( {\\mathbf {\\mathrm {R}}}_{j_2}^{W \\mu _2}- {\\mathbf {\\mathrm {R}}}_{j_0}^{W \\mu _2})]\\, \\exp [i {\\mathbf {\\mathrm {k}}} \\cdot g_2 ({\\mathbf {\\mathrm {R}}}_{j_1}^{W \\mu _1}- {\\mathbf {\\mathrm {R}}}_{j_0}^{W \\mu _1})] .$ As ${\\mathbf {\\mathrm {R}}}_{j_1}^{W \\mu _1} - {\\mathbf {\\mathrm {R}}}_{j_0}^{W \\mu _1}$ is a lattice vector, it follows from Eq.",
"(REF ) $g_2 \\, g_1 \\, q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_{j_0}^{W \\mu _0})= q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_{j_0}^{W \\mu _2})\\, \\mathcal {D}^q_{{\\mathbf {\\mathrm {k}}} W} (g_2 \\, g_1)_{\\mu _2 \\mu _0}$ with $\\mathcal {D}^q_{{\\mathbf {\\mathrm {k}}} W} (g_2 \\, g_1)_{\\mu _2 \\mu _0} = \\exp [i {\\mathbf {\\mathrm {k}}} \\cdot ( {\\mathbf {\\mathrm {R}}}_{j_2}^{W \\mu _2} - {\\mathbf {\\mathrm {R}}}_{j_0}^{W \\mu _2} + {\\mathbf {\\mathrm {R}}}_{j_1}^{W \\mu _1} - {\\mathbf {\\mathrm {R}}}_{j_0}^{W \\mu _1})] .$ This confirms Eq.",
"(REF ).",
"We remark that for ${\\mathbf {\\mathrm {k}}} = 0$ the representation $\\Gamma ^q_{{\\mathbf {\\mathrm {k}}} W}$ is known as permutation representation [48] or equivalence representation [39], where it characterizes the permutations of $m$ objects under the symmetry operations $g \\in \\mathcal {G}_0$ .",
"Using the fact that $g$ is an orthogonal transformation, we can also write Eq.",
"(REF ) as $\\mathcal {D}^q_{{\\mathbf {\\mathrm {k}}} W} (g)_{\\mu ^{\\prime } \\mu }& = \\exp [ i {\\mathbf {\\mathrm {k}}} \\cdot ({\\mathbf {\\mathrm {R}}}_{j^{\\prime }}^{W \\mu ^{\\prime }} - {\\mathbf {\\mathrm {R}}}_j^{W \\mu ^{\\prime }})]\\\\& = \\exp [ i {\\mathbf {\\mathrm {k}}} \\cdot ({\\mathbf {\\mathrm {g}}} \\cdot {\\mathbf {\\mathrm {R}}}_j^{W \\mu } - {\\mathbf {\\mathrm {R}}}_j^{W \\mu ^{\\prime }})]\\\\& = \\exp \\lbrace i [({\\mathbf {\\mathrm {g}}}^{-1} \\cdot {\\mathbf {\\mathrm {k}}}) \\cdot {\\mathbf {\\mathrm {R}}}_j^{W \\mu } - {\\mathbf {\\mathrm {k}}} \\cdot {\\mathbf {\\mathrm {R}}}_j^{W \\mu ^{\\prime }}] \\rbrace .$ For ${\\mathbf {\\mathrm {k}}}$ inside the Brillouin zone, where ${\\mathbf {\\mathrm {g}}}^{-1} \\cdot {\\mathbf {\\mathrm {k}}} = {\\mathbf {\\mathrm {k}}}$ by definition of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ , the nonzero matrix elements of $\\mathcal {D}^q_{{\\mathbf {\\mathrm {k}}} W} (g)$ therefore become $\\mathcal {D}^q_{{\\mathbf {\\mathrm {k}}} W} (g)_{\\mu ^{\\prime } \\mu }\\equiv \\exp [ i {\\mathbf {\\mathrm {k}}} \\cdot ( {\\mathbf {\\mathrm {R}}}_j^{W \\mu } - {\\mathbf {\\mathrm {R}}}_j^{W \\mu ^{\\prime }})].$ We see that the representation is generally nontrivial for $m>1$ .",
"However, for $m=1$ , we have ${\\mathbf {\\mathrm {R}}}_j^{W \\mu } = {\\mathbf {\\mathrm {R}}}_j^{W \\mu ^{\\prime }}$ , so that $q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{W \\mu })$ transforms according to the identity representation $\\Gamma _1$ ."
],
[
"Wyckoff positions with multiplicity $m = 1$",
"We consider first the special case of Wyckoff positions with multiplicity $m=1$ .",
"Here we drop the index $\\mu $ , denoting atomic positions as ${\\mathbf {\\mathrm {R}}}_j^W$ .",
"This case is equivalent to atomic positions ${\\mathbf {\\mathrm {R}}}_j^W$ forming a Bravais lattice.",
"Note also that multiplicities $m=1$ occur only for symmorphic space groups [43].",
"Here, plane waves ${Q}_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^W) = \\mathcal {q}_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^W)$ transform according to the one-dimensional IR $\\mathcal {D}^q_{{\\mathbf {\\mathrm {k}}} W} (g)= \\exp [ i \\, {\\mathbf {\\mathrm {k}}} \\cdot ({\\mathbf {\\mathrm {R}}}_{j^{\\prime }}^W - {\\mathbf {\\mathrm {R}}}_j^W)] .$ We saw in Eq.",
"(REF ) that for wave vectors ${\\mathbf {\\mathrm {k}}}$ inside the BZ this becomes the identity representation $\\Gamma _1$ .",
"It is illuminating to rederive this result by writing Eq.",
"(REF ) as $\\mathcal {D}^q_{{\\mathbf {\\mathrm {k}}} W} (g)& = \\exp [ i \\, ({\\mathbf {\\mathrm {k}}} \\cdot {\\mathbf {\\mathrm {g}}} - {\\mathbf {\\mathrm {k}}}) \\cdot {\\mathbf {\\mathrm {R}}}_j^W] \\\\& = \\exp [ i \\, ({\\mathbf {\\mathrm {g}}}^{-1} \\cdot {\\mathbf {\\mathrm {k}}} - {\\mathbf {\\mathrm {k}}}) \\cdot {\\mathbf {\\mathrm {R}}}_j^W] .$ It follows from Eq.",
"(REF ) that $\\mathbf {\\mathrm {k}}^{\\prime } = {\\mathbf {\\mathrm {g}}}^{-1} \\cdot {\\mathbf {\\mathrm {k}}} = {\\mathbf {\\mathrm {k}}} + {\\mathbf {\\mathrm {b}}}_g $ with a reciprocal lattice vector ${\\mathbf {\\mathrm {b}}}_g$ .",
"Thus Eq.",
"(REF ) describing the effect of $g$ in real space is equivalent to $\\mathcal {D}^q_{{\\mathbf {\\mathrm {k}}} W} (g)= \\exp [ i \\, {\\mathbf {\\mathrm {b}}}_g \\cdot {\\mathbf {\\mathrm {R}}}_j^W]$ describing the effect of $g$ in reciprocal space.",
"Hence, for $m=1$ plane waves ${Q}_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^W)$ transform under $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ in a nontrivial way only if the vector ${\\mathbf {\\mathrm {k}}}$ is from the border of the BZ when ${\\mathbf {\\mathrm {k}}}^{\\prime } = {\\mathbf {\\mathrm {g}}}^{-1} \\cdot {\\mathbf {\\mathrm {k}}}$ and ${\\mathbf {\\mathrm {k}}}$ can differ by a reciprocal lattice vector ${\\mathbf {\\mathrm {b}}}_g \\ne 0$ .",
"Otherwise, ${\\mathbf {\\mathrm {k}}}^{\\prime } = {\\mathbf {\\mathrm {k}}}$ implies that ${Q}_{\\mathbf {\\mathrm {k}}}^W ({\\mathbf {\\mathrm {R}}}_j^W)$ transforms according to $\\Gamma _1$ of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ .",
"If a space group has Wyckoff positions $W \\ne W^{\\prime }$ each with multiplicity $m=1$ , we can compare the IRs of the plane waves at the positions ${\\mathbf {\\mathrm {R}}}_j^W$ and ${\\mathbf {\\mathrm {R}}}_j^{W^{\\prime }}$ .",
"We have $\\mathcal {D}^q_{{\\mathbf {\\mathrm {k}}} W} (g) & = \\exp ( i {\\mathbf {\\mathrm {b}}}_g \\cdot {\\mathbf {\\mathrm {R}}}_j^W ) ,\\\\\\mathcal {D}^q_{{\\mathbf {\\mathrm {k}}} W^{\\prime }} (g) & = \\exp ( i {\\mathbf {\\mathrm {b}}}_g \\cdot {\\mathbf {\\mathrm {R}}}_j^{W^{\\prime }} ).$ Hence $\\mathcal {D}^q_{{\\mathbf {\\mathrm {k}}} W} (g) = \\mathcal {D}^q_{{\\mathbf {\\mathrm {k}}} W^{\\prime }} (g)\\exp [ i {\\mathbf {\\mathrm {b}}}_g \\cdot ({\\mathbf {\\mathrm {R}}}_j^W - {\\mathbf {\\mathrm {R}}}_j^{W^{\\prime }}) ] .$ For $W \\ne W^{\\prime }$ , the vector ${\\mathbf {\\mathrm {R}}}_j^W - {\\mathbf {\\mathrm {R}}}_j^{W^{\\prime }}$ is not equal to a lattice vector, so that for nonzero ${\\mathbf {\\mathrm {b}}}_g$ (i.e., for ${\\mathbf {\\mathrm {k}}}$ on the boundary of the Brillouin zone) we generally have $\\Gamma ^q_{{\\mathbf {\\mathrm {k}}} W} \\ne \\Gamma ^q_{{\\mathbf {\\mathrm {k}}} W^{\\prime }} .$ This implies that a nontrivial change of the coordinate system which requires a relabeling of the Wyckoff letters associated with atomic positions changes the IRs of the plane waves at these positions.",
"This relabeling of IR assignments will be discussed in Sec.",
"REF .",
"As a simple example for Eq.",
"(REF ), consider the case where the positions ${\\mathbf {\\mathrm {R}}}_j^W$ are equal to lattice vectors, i.e., one of the positions ${\\mathbf {\\mathrm {R}}}_j^W$ is at the origin of the coordinate system.",
"Hence, Eq.",
"(REF ) gives $\\Gamma ^q_{{\\mathbf {\\mathrm {k}}} W} = \\lbrace \\mathcal {D}^q_{{\\mathbf {\\mathrm {k}}} W} (g) =1 : g \\in \\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}\\rbrace $ , i.e., the plane wave ${Q}_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^W)$ transforms according to the identity representation.",
"On the other hand, the IR $\\Gamma ^q_{{\\mathbf {\\mathrm {k}}} W^{\\prime }}$ at a different Wyckoff position $W^{\\prime }$ can never be the identity representation.",
"Examples of Wyckoff positions with multiplicity $m=1$ are the positions occupied by the Mo atoms and the center of the hexagon in monolayer MoS$_2$ .",
"The plane wave at these positions transforms according to different IRs.",
"In a certain coordinate system where one of these two inequivalent positions are located at the origin, the plane waves at that Wyckoff position transform as the identity representation, while the plane waves at the other Wyckoff position transform according to a different IR.",
"Another example is given by the inequivalent IRs of the plane waves at the positions of C atoms in the central layer of graphene with odd number of layers such as the trilayer graphene discussed in Sec.",
"REF ."
],
[
"Wyckoff positions with multiplicity $m > 1$",
"For Wyckoff positions with multiplicity $m > 1$ the simple analysis based on Eq.",
"(REF ) is not valid as it does not keep track of how positions ${\\mathbf {\\mathrm {R}}}_j^{W \\mu }$ are mapped onto each other by symmetry operations $g \\in \\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ .",
"Instead we need to use the plane-wave spinors (REF ).",
"For multiplicity $m > 1$ , the representation $\\Gamma ^q_{{\\mathbf {\\mathrm {k}}} W}$ characterizing the plane-wave spinors ${Q}_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathcal {R}}}_j^W)$ is generally reducible.",
"Using the projection operator (REF ), we can construct linear combinations $\\mathcal {Q}_{\\mathbf {\\mathrm {k}}}^{I \\beta } ({\\mathbf {\\mathcal {R}}}_j^W)$ of the base spinors $\\mathcal {q}_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{W \\mu })$ transforming like component $\\beta $ of the IR $\\Gamma _I$ of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ contained in $\\Gamma ^q_{{\\mathbf {\\mathrm {k}}} W}$ $\\mathcal {Q}_{\\mathbf {\\mathrm {k}}}^{I\\beta } ({\\mathbf {\\mathcal {R}}}_j^W)& \\equiv \\Pi _{\\mathbf {\\mathrm {k}}}^{I \\beta } \\, {Q}_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathcal {R}}}_j^W) \\\\[1ex]& = \\frac{n_I}{h}\\sum _g \\mathcal {D}_I (g)^\\ast _{\\beta \\beta } \\: g \\, {Q}_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathcal {R}}}_j^W)\\\\& = \\frac{n_I}{h} \\sum _g \\mathcal {D}_I (g)^\\ast _{\\beta \\beta } \\, \\mathcal {D}^q_{{\\mathbf {\\mathrm {k}}} W} (g) \\, {Q}_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathcal {R}}}_j^W) ,$ where we used Eq.",
"(REF ).",
"This yields $\\mathcal {Q}_{\\mathbf {\\mathrm {k}}}^{I\\beta } ({\\mathbf {\\mathcal {R}}}_j^W)= \\sum _m u_{{\\mathbf {\\mathrm {k}}} W \\mu }^{I\\beta } \\, \\mathcal {q}_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{W \\mu })= \\left( \\begin{array}{s{0.15em}cs{0.15em}} u_{{\\mathbf {\\mathrm {k}}} W 1}^{I\\beta } \\\\[-0.5ex] \\vdots \\\\ u_{{\\mathbf {\\mathrm {k}}} W m}^{I\\beta } \\end{array} \\right)$ with expansion coefficients $u_{{\\mathbf {\\mathrm {k}}} W \\mu }^{I\\beta }= \\frac{n_I}{h} \\sum _g \\mathcal {D}_I (g)^\\ast _{\\beta \\beta } \\,\\sum _{\\tilde{\\mu }} \\mathcal {D}^q_{{\\mathbf {\\mathrm {k}}} W} (g)_{\\mu \\tilde{\\mu }}$ that completely characterize each symmetrized spinor $\\mathcal {Q}_{\\mathbf {\\mathrm {k}}}^{I \\beta } ({\\mathbf {\\mathcal {R}}}_j^W)$ .",
"Upon translation by a lattice vector ${\\mathbf {\\mathrm {a}}}$ , the plane waves $q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{W \\mu })$ acquire a phase $\\exp (i {\\mathbf {\\mathrm {k}}} \\cdot {\\mathbf {\\mathrm {a}}})$ $q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{W \\mu } + {\\mathbf {\\mathrm {a}}})= \\exp (i {\\mathbf {\\mathrm {k}}} \\cdot {\\mathbf {\\mathrm {a}}}) \\, q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{W \\mu }) .$ This implies that $\\mathcal {q}_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{W \\mu })$ represents (for each $\\mu $ ) a discrete Bloch function for wave vector ${\\mathbf {\\mathrm {k}}}$ .",
"Similarly, linear combinations of these base spinors including the spinors ${Q}_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathcal {R}}}_j^W)$ and $\\mathcal {Q}_{\\mathbf {\\mathrm {k}}}^{I\\beta } ({\\mathbf {\\mathcal {R}}}_j^W)$ are thus discrete Bloch functions for wave vector ${\\mathbf {\\mathrm {k}}}$ .",
"The expansion coefficients $u_{{\\mathbf {\\mathrm {k}}} W \\mu }^{I\\beta }$ take the role of lattice-periodic functions for these discrete Bloch functions.",
"The projection (REF ) is valid for all wave vectors ${\\mathbf {\\mathrm {k}}}$ in the Brillouin zone (though trivial for $m=1$ when ${\\mathbf {\\mathrm {k}}}$ is inside the Brillouin zone, as noted above).",
"In general, the projectors $\\Pi _{\\mathbf {\\mathrm {k}}}^{I\\beta }$ decompose a plane-wave spinor ${Q}_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathcal {R}}}_j^W)$ into multiple Bloch functions $\\mathcal {Q}_{\\mathbf {\\mathrm {k}}}^{I\\beta } ({\\mathbf {\\mathcal {R}}}_j^W)$ corresponding to different IRs $\\Gamma _I$ of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ .",
"Yet we often have sets of special positions ${\\mathbf {\\mathcal {R}}}_{W j}^{I {\\mathbf {\\mathrm {k}}}} = \\lbrace {\\mathbf {\\mathrm {R}}}_{W \\mu j}^{I {\\mathbf {\\mathrm {k}}}} : \\mu = 1, \\ldots , m\\rbrace $ within the unit cell where only the Bloch functions $\\mathcal {Q}_{\\mathbf {\\mathrm {k}}}^{I\\beta } ({\\mathbf {\\mathcal {R}}}_j^W)$ for one IR $\\Gamma _I$ are nonzero, but all other projections vanish.",
"This greatly simplifies further discussion of TB Bloch functions at positions ${\\mathbf {\\mathcal {R}}}_{W j}^{I {\\mathbf {\\mathrm {k}}}}$ .",
"The positions ${\\mathbf {\\mathcal {R}}}_{W j}^{I {\\mathbf {\\mathrm {k}}}}$ are characterized by two different groups, the group describing the site symmetry [43] denoted as $\\mathcal {G}_{W}$ and the group of the wave vector $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ .",
"Often the positions ${\\mathbf {\\mathcal {R}}}_{W j}^{I {\\mathbf {\\mathrm {k}}}}$ with nontrivial $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ coincide with Wyckoff positions with a nontrivial $\\mathcal {G}_{W}$ .",
"For Wyckoff positions with multiplicity $m=1$ , we always have $\\mathcal {G}_{W}= \\mathcal {G}_0$ , so that $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}\\subseteq \\mathcal {G}_{W}$ .",
"However, for $m>1$ we will find below that in general there is no simple relation between the group $\\mathcal {G}_{W}$ characterizing such special positions ${\\mathbf {\\mathcal {R}}}_{W j}^{I {\\mathbf {\\mathrm {k}}}}$ and the group of the wave vector $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ for which this happens In SLG, the carbon atoms with multiplicity $m=2$ are characterized by the site symmetry group $\\mathcal {G}_{W}= D_{3h}$ , see Table REF .",
"Here, likewise, the group of the wave vector at the ${\\mathbf {\\mathrm {K}}}$ point is $\\mathcal {G}_{{\\mathbf {\\mathrm {K}}}} = D_{3h}$ and symmetrized plane waves at ${\\mathbf {\\mathrm {K}}}$ transform according to the two-dimensional IR $\\Gamma _6$ of $D_{3h}$ , see Table REF .",
"On the other hand, in BLG, the carbon atoms at positions ${\\mathbf {\\mathrm {R}}}_j^{d1},{\\mathbf {\\mathrm {R}}}_j^{d2}$ have site symmetry $\\mathcal {G}_{W}= C_{3v}$ , whereas the symmetrized plane waves at ${\\mathbf {\\mathrm {K}}}$ for these positions transform according to the two-dimensional IR $\\Gamma _3$ of $D_3$ .",
"Neither of the groups $C_{3v}$ and $D_3$ can be viewed as a subgroup of the other one..",
"The symmetrized plane waves $\\mathcal {Q}_{\\mathbf {\\mathrm {k}}}^{I\\beta } ({\\mathbf {\\mathcal {R}}}_j^W)$ including the positions ${\\mathbf {\\mathcal {R}}}_{W j}^{I {\\mathbf {\\mathrm {k}}}}$ are universal features of each space group, independent of the “atomistic realization” of a space group in different crystal structures (e.g., the number and positions of atoms in a unit cell).",
"They apply both to spinless models and models that include the spin degree of freedom.",
"We note that the symmetrized plane waves $\\mathcal {Q}_{\\mathbf {\\mathrm {k}}}^{I\\beta } ({\\mathbf {\\mathcal {R}}}_j^W)$ introduced here in the context of the TB approximation for Wyckoff positions ${\\mathbf {\\mathcal {R}}}_j^W$ are conceptually different from the symmetrized plane waves discussed previously in the context of the nearly-free electron approximation, see, e.g., Refs.",
"[38], [50], [39]."
],
[
"Rearrangement of IRs of Bloch states under a change of the coordinate system",
"The Bloch states in certain crystal structures are characterized by IRs of the group $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ that depend in a nontrivial way on the location of the origin or the orientation of the coordinate system relative to the position of the atoms [20], [21], [22], [23].",
"Cornwell [22] has given a general discussion of the origin dependence of the symmetry labeling of electron states in such systems.",
"Here we review and extend these findings, focusing on symmorphic space groups and adopting a notation matching other parts of this study.",
"We show that for different choices of the origin we get a rearrangement of the IRs of Bloch states.",
"We exploit this rearrangement lemma when discussing band symmetries for specific materials further below.",
"We consider a crystal structure with space group $\\mathcal {G}$ .",
"For the coordinate system ${\\mathbf {\\mathrm {r}}} = (x,y,z)$ , the lattice-periodic (single-electron) Hamiltonian is $H ({\\mathbf {\\mathrm {r}}} )$ .",
"The eigenfunctions of $H ({\\mathbf {\\mathrm {r}}} )$ are Bloch functions $\\Psi _{n {\\mathbf {\\mathrm {k}}}}^{I \\beta } ({\\mathbf {\\mathrm {r}}})$ , obeying the eigenvalue equation $H ({\\mathbf {\\mathrm {r}}} )\\, \\Psi _{n {\\mathbf {\\mathrm {k}}}}^{I \\beta } ({\\mathbf {\\mathrm {r}}})= E_n ({\\mathbf {\\mathrm {k}}}) \\, \\Psi _{n {\\mathbf {\\mathrm {k}}}}^{I \\beta } ({\\mathbf {\\mathrm {r}}})$ with energy $E_n ({\\mathbf {\\mathrm {k}}})$ .",
"For a given wave vector ${\\mathbf {\\mathrm {k}}}$ , the index $I$ denotes the IR $\\Gamma _I$ of the point group $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ of the wave vector, to be discussed in more detail below.",
"The eigenfunction $\\Psi _{n {\\mathbf {\\mathrm {k}}}}^{I \\beta } ({\\mathbf {\\mathrm {r}}})$ transforms according to the $\\beta $ th component of the IR $\\Gamma _I$ .",
"For brevity, we drop in this section the band index $n$ .",
"We denote coordinate transformations using the Seitz notation as $\\lbrace g \\rbrace $ , where $g$ is a (proper or improper) rotation that is followed by a translation ${\\mathbf {\\mathrm {\\tau }}}$ .",
"We seek to identify a pure translation $T\\equiv \\lbrace \\rbrace $ of the coordinate system ${\\mathbf {\\mathrm {r}}}$ , where ${\\mathbf {\\mathrm {\\tau }}}$ equals a fraction of a lattice vector such that for the shifted, primed coordinated system ${\\mathbf {\\mathrm {r}}}^{\\prime } = (x^{\\prime },y^{\\prime },z^{\\prime })$ the crystal structure has the same space group symmetry $\\mathcal {G}$ as for the unprimed coordinate system ${\\mathbf {\\mathrm {r}}} = (x,y,z)$ .",
"The translation $T$ transforms the Bloch function $\\Psi _{{\\mathbf {\\mathrm {k}}}}^{I \\beta } ({\\mathbf {\\mathrm {r}}})$ into $\\Psi _{{\\mathbf {\\mathrm {k}}}}^{I^{\\prime } \\beta } ({\\mathbf {\\mathrm {r}}}^{\\prime })\\equiv T\\, \\Psi _{{\\mathbf {\\mathrm {k}}}}^{I \\beta } ({\\mathbf {\\mathrm {r}}}).$ The index $I^{\\prime } \\ne I$ will be justified below.",
"As $T= \\lbrace \\rbrace $ commutes with primitive translations $\\lbrace {a} \\rbrace $ we get $\\lbrace {a} \\rbrace \\, \\Psi _{{\\mathbf {\\mathrm {k}}}}^{I^{\\prime } \\beta } ({\\mathbf {\\mathrm {r}}}^{\\prime })= \\exp (-i {\\mathbf {\\mathrm {k}}} \\cdot {\\mathbf {\\mathrm {a}}}) \\, \\Psi _{{\\mathbf {\\mathrm {k}}}}^{I^{\\prime } \\beta } ({\\mathbf {\\mathrm {r}}}^{\\prime }) ,$ so that the transformed Bloch function $\\Psi _{{\\mathbf {\\mathrm {k}}}}^{I^{\\prime } \\beta } ({\\mathbf {\\mathrm {r}}}^{\\prime })$ has, indeed, the same wave vector ${\\mathbf {\\mathrm {k}}}$ as $\\Psi _{{\\mathbf {\\mathrm {k}}}}^{I \\beta } ({\\mathbf {\\mathrm {r}}})$ .",
"Furthermore, the Hamiltonian in the primed coordinate system becomes $H ({\\mathbf {\\mathrm {r}}} ^{\\prime }) = T\\, H ({\\mathbf {\\mathrm {r}}} )\\, T^{-1} .$ Thus $H ({\\mathbf {\\mathrm {r}}} ^{\\prime }) \\, \\Psi _{{\\mathbf {\\mathrm {k}}}}^{I^{\\prime } \\beta } ({\\mathbf {\\mathrm {r}}}^{\\prime })& = T\\, H ({\\mathbf {\\mathrm {r}}} )\\, \\Psi _{{\\mathbf {\\mathrm {k}}}}^{I \\beta } ({\\mathbf {\\mathrm {r}}}) \\\\& = E({\\mathbf {\\mathrm {k}}}) \\, \\Psi _{{\\mathbf {\\mathrm {k}}}}^{I^{\\prime } \\beta } ({\\mathbf {\\mathrm {r}}}^{\\prime }) ,$ so that the transformed function $\\Psi _{{\\mathbf {\\mathrm {k}}}}^{I^{\\prime } \\beta } ({\\mathbf {\\mathrm {r}}}^{\\prime })$ is an eigenfunction of $H ({\\mathbf {\\mathrm {r}}} ^{\\prime })$ with the same eigenvalue $E({\\mathbf {\\mathrm {k}}})$ as $\\Psi _{{\\mathbf {\\mathrm {k}}}}^{I \\beta } ({\\mathbf {\\mathrm {r}}})$ .",
"Given the space group $\\mathcal {G}$ of the crystal, the invariance of $H ({\\mathbf {\\mathrm {r}}} )$ under the symmetry operations $\\mathcal {T}\\equiv \\lbrace g {a} \\rbrace \\in \\mathcal {G}$ reads $\\mathcal {T}\\, H ({\\mathbf {\\mathrm {r}}} )\\, \\mathcal {T}^{-1} = H ({\\mathbf {\\mathrm {r}}} )\\quad \\forall \\mathcal {T}\\in \\mathcal {G}.$ For the Hamiltonian $H ({\\mathbf {\\mathrm {r}}} ^{\\prime })$ in the primed coordinate system we get $\\mathcal {T}\\, H ({\\mathbf {\\mathrm {r}}} ^{\\prime }) \\, \\mathcal {T}^{-1} =T\\left( T^{-1} \\, \\mathcal {T}\\, T\\right) H ({\\mathbf {\\mathrm {r}}} )\\left( T^{-1} \\, \\mathcal {T}\\, T\\right)^{-1} T^{-1} .$ Hence the primed Hamiltonian $H ({\\mathbf {\\mathrm {r}}} ^{\\prime })$ obeys an invariance condition analogous to Eq.",
"(REF ) if $( T^{-1} \\, \\mathcal {T}\\, T)$ commutes with $H ({\\mathbf {\\mathrm {r}}} )$ $[ T^{-1} \\, \\mathcal {T}\\, T, H ({\\mathbf {\\mathrm {r}}} )] = 0 .$ This requires in turn, given Eq.",
"(REF ), that for all $\\mathcal {T}\\in \\mathcal {G}$ $T^{-1} \\, \\mathcal {T}\\, T& = \\lbrace {\\mathbf {\\mathrm {\\tau }}} \\rbrace \\, \\lbrace g {a} \\rbrace \\, \\lbrace \\rbrace \\\\& = \\lbrace g g̰\\, {\\mathbf {\\mathrm {\\tau }}}- {\\mathbf {\\mathrm {\\tau }}}+ {\\mathbf {\\mathrm {a}}} \\rbrace $ is an element of the space group $\\mathcal {G}$ , so that $g\\, {\\mathbf {\\mathrm {\\tau }}}- {\\mathbf {\\mathrm {\\tau }}}$ must be equal to a lattice vector ${\\mathbf {\\mathrm {a}}}^{\\prime }$ of the crystal $g\\, {\\mathbf {\\mathrm {\\tau }}}- {\\mathbf {\\mathrm {\\tau }}}= {\\mathbf {\\mathrm {a}}}^{\\prime }\\quad \\forall \\, g \\equiv \\lbrace g \\rbrace \\in \\mathcal {G}_0,$ where $\\mathcal {G}_0$ is the point group corresponding to the space group $\\mathcal {G}$ .",
"A nontrivial solution to this problem is a vector ${\\mathbf {\\mathrm {\\tau }}}$ that is not equal to a lattice vector ${\\mathbf {\\mathrm {a}}}$ .",
"Equation (REF ) defines the allowed shifts ${\\mathbf {\\mathrm {\\tau }}}$ (up to a lattice vector) that provide alternative descriptions of a crystal structure with space group $\\mathcal {G}$ .",
"We obtain nontrivial solutions to Eq.",
"(REF ), for example, if a crystal consists of atoms at Wyckoff positions $W = A, B, \\ldots $ each with multiplicity $m=1$ In Cornwell's notation [22], different Wyckoff positions each with multiplicity $m=1$ are positions each forming a Bravais lattice.. We denote these positions in the unit cell as ${\\mathbf {\\mathrm {t}}}_\\mathrm {A}$ , ${\\mathbf {\\mathrm {t}}}_\\mathrm {B}$ , ${\\mathbf {\\mathrm {t}}}_\\mathrm {C}$ , $\\dots $ , respectively.",
"By definition of the space group $\\mathcal {G}$ , these positions ${\\mathbf {\\mathrm {t}}}_W$ obey the condition $g \\, {\\mathbf {\\mathrm {t}}}_W - {\\mathbf {\\mathrm {t}}}_W = {\\mathbf {\\mathrm {a}}}^{\\prime }\\qquad \\forall \\, g \\in \\mathcal {G}_0,\\quad W = \\mathrm {A}, \\mathrm {B}, \\mathrm {C}, \\ldots $ with lattice vectors ${\\mathbf {\\mathrm {a}}}^{\\prime }$ .",
"Hence, any linear combination of these position vectors $\\lbrace {\\mathbf {\\mathrm {t}}}_W\\rbrace $ with integer prefactors (e.g., ${\\mathbf {\\mathrm {\\tau }}}= {\\mathbf {\\mathrm {t}}}_W - {\\mathbf {\\mathrm {t}}}_{W^{\\prime }}$ with $W \\ne W^{\\prime }$ ) yields a translation ${\\mathbf {\\mathrm {\\tau }}}$ consistent with Eq.",
"(REF ).",
"In the unprimed coordinate system the eigenfunctions $\\Psi _{{\\mathbf {\\mathrm {k}}}}^{I \\beta } ({\\mathbf {\\mathrm {r}}})$ transform according to the $\\beta $ th component of an IR $\\Gamma _I$ of the point group $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ of the wave vector ${\\mathbf {\\mathrm {k}}}$ $\\lbrace g \\rbrace \\, \\Psi _{{\\mathbf {\\mathrm {k}}}}^{I \\beta } ({\\mathbf {\\mathrm {r}}})= \\sum _{\\beta ^{\\prime }} \\mathcal {D}_I (g)_{\\beta ^{\\prime } \\beta } \\, \\Psi _{{\\mathbf {\\mathrm {k}}}}^{I \\beta ^{\\prime }} ({\\mathbf {\\mathrm {r}}})$ with representation matrices $\\mathcal {D}_I (g)_{\\beta ^{\\prime } \\beta }$ .",
"We can evaluate the action of a symmetry operation $g \\in \\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ on primed Bloch functions $\\Psi _{{\\mathbf {\\mathrm {k}}}}^{I^{\\prime } \\beta } ({\\mathbf {\\mathrm {r}}}^{\\prime })$ as follows $\\makebox{[}2em][l]{\\lbrace g \\rbrace \\, \\Psi _{{\\mathbf {\\mathrm {k}}}}^{I^{\\prime } \\beta } ({\\mathbf {\\mathrm {r}}}^{\\prime })}\\nonumber \\\\& = \\lbrace g \\rbrace \\, \\lbrace \\rbrace \\, \\Psi _{{\\mathbf {\\mathrm {k}}}}^{I \\beta } ({\\mathbf {\\mathrm {r}}}) \\\\& = \\lbrace \\rbrace \\, \\lbrace g̰ \\, {\\mathbf {\\mathrm {\\tau }}}- {\\mathbf {\\mathrm {\\tau }}} \\rbrace \\, \\lbrace g \\rbrace \\, \\Psi _{{\\mathbf {\\mathrm {k}}}}^{I \\beta } ({\\mathbf {\\mathrm {r}}}) \\\\& = \\sum _{\\beta ^{\\prime }} \\mathcal {D}_I (g)_{\\beta ^{\\prime } \\beta }\\, \\lbrace \\rbrace \\, \\lbrace g̰ \\, {\\mathbf {\\mathrm {\\tau }}}- {\\mathbf {\\mathrm {\\tau }}} \\rbrace \\, \\Psi _{{\\mathbf {\\mathrm {k}}}}^{I \\beta ^{\\prime }} ({\\mathbf {\\mathrm {r}}}) \\\\& = \\sum _{\\beta ^{\\prime }} \\mathcal {D}_I (g)_{\\beta ^{\\prime } \\beta }\\, \\exp [-i{\\mathbf {\\mathrm {k}}} \\cdot (g \\, {\\mathbf {\\mathrm {\\tau }}}- {\\mathbf {\\mathrm {\\tau }}})]\\, \\lbrace \\rbrace \\, \\Psi _{{\\mathbf {\\mathrm {k}}}}^{I \\beta ^{\\prime }} ({\\mathbf {\\mathrm {r}}}) \\\\& = \\sum _{\\beta ^{\\prime }} \\mathcal {D}_{I^{\\prime }} (g)_{\\beta ^{\\prime } \\beta }\\, \\Psi _{{\\mathbf {\\mathrm {k}}}}^{I^{\\prime } \\beta ^{\\prime }} ({\\mathbf {\\mathrm {r}}}) ,$ where the primed representation matrices $\\mathcal {D}_{I^{\\prime }} (g)_{\\beta ^{\\prime } \\beta }$ become $\\mathcal {D}_{I^{\\prime }} (g)_{\\beta ^{\\prime } \\beta } = \\mathcal {D}_I (g)_{\\beta ^{\\prime } \\beta }\\, \\exp [-i{\\mathbf {\\mathrm {k}}} \\cdot (g \\, {\\mathbf {\\mathrm {\\tau }}}- {\\mathbf {\\mathrm {\\tau }}})] .$ We denote the phase factors in Eq.",
"(REF ) by $\\mathcal {D}_{\\mathbf {\\mathrm {\\tau }}}^{\\mathbf {\\mathrm {k}}} (g)= \\exp [-i{\\mathbf {\\mathrm {k}}} \\cdot (g \\, {\\mathbf {\\mathrm {\\tau }}}- {\\mathbf {\\mathrm {\\tau }}})].$ Using Eq.",
"(REF ), this becomes $\\mathcal {D}_{\\mathbf {\\mathrm {\\tau }}}^{\\mathbf {\\mathrm {k}}} (g) = \\exp (-i {\\mathbf {\\mathrm {b}}}_g \\cdot {\\mathbf {\\mathrm {\\tau }}})$ with a reciprocal lattice vector ${\\mathbf {\\mathrm {b}}}_g = g^{-1} {\\mathbf {\\mathrm {k}}} - {\\mathbf {\\mathrm {k}}}.$ The phase $\\mathcal {D}_{\\mathbf {\\mathrm {\\tau }}}^{\\mathbf {\\mathrm {k}}} (g)$ is therefore nontrivial when ${\\mathbf {\\mathrm {b}}}_g \\ne 0$ , which can only happen at the border of the Brillouin zone.",
"We show in the next paragraph that $\\Gamma _{\\mathbf {\\mathrm {\\tau }}}\\equiv \\lbrace \\mathcal {D}_{\\mathbf {\\mathrm {\\tau }}}^{\\mathbf {\\mathrm {k}}} (g) : g \\in \\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}\\rbrace $ defines a one-dimensional IR of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ (for every wave vector ${\\mathbf {\\mathrm {k}}}$ in the Brillouin zone).",
"Therefore, the IR $\\Gamma _{I^{\\prime }}$ of a Bloch function in the primed coordinate system is given by $\\Gamma _{\\mathbf {\\mathrm {\\tau }}}$ times the IR $\\Gamma _I$ of the Bloch function in the unprimed coordinate system, so that Eq.",
"(REF ) becomes $\\Gamma _{I^{\\prime }} = \\Gamma _{\\mathbf {\\mathrm {\\tau }}}\\times \\Gamma _I.$ The rearrangement lemma for IRs discussed in Appendix applied to Eq.",
"(REF ) shows that, unless we have the trivial case that $\\Gamma _{\\mathbf {\\mathrm {\\tau }}}$ is the identity representation, each IR $\\Gamma _I$ of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ in the unprimed coordinate system is mapped on an IR $\\Gamma _{I^{\\prime }} \\ne \\Gamma _I$ in the primed coordinate system.",
"Hence we call Eq.",
"(REF ) the rearrangement lemma for the IRs of Bloch states and $\\Gamma _{\\mathbf {\\mathrm {\\tau }}}$ the rearrangement representation (RAR) for the coordinate shift ${\\mathbf {\\mathrm {\\tau }}}$ .",
"Examples for this rearrangement of IRs of Bloch states will be given below when we study the symmetries of the Bloch functions in monolayer MoS$_2$ (Sec.",
"REF ) and trilayer graphene (Sec.",
"REF ).",
"It follows from Eq.",
"(REF ) that only at the border of the Brillouin zone the IR labeling of Bloch states can depend on the origin of the coordinate system [22].",
"Also, Eq.",
"(REF ) implies $\\Gamma _{-{\\mathbf {\\mathrm {\\tau }}}} = \\Gamma _{\\mathbf {\\mathrm {\\tau }}}^\\ast $ .",
"Generally, the shift ${\\mathbf {\\mathrm {\\tau }}}$ is defined up to a lattice vector ${\\mathbf {\\mathrm {a}}}$ .",
"It follows immediately from Eq.",
"(REF ) that ${\\mathbf {\\mathrm {\\tau }}}$ and $\\tilde{{\\mathbf {\\mathrm {\\tau }}}} \\equiv {\\mathbf {\\mathrm {\\tau }}}+ {\\mathbf {\\mathrm {a}}}$ define the same RAR $\\Gamma _{\\mathbf {\\mathrm {\\tau }}}$ .",
"To show that $\\Gamma _{\\mathbf {\\mathrm {\\tau }}}= \\lbrace \\mathcal {D}_{\\mathbf {\\mathrm {\\tau }}}^{\\mathbf {\\mathrm {k}}} (g) : g \\in \\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}\\rbrace $ defines a one-dimensional IR of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ we consider two group elements $g_i \\in \\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ ($i = 1,2$ ).",
"According to Eq.",
"(REF ), the transformations $g_i \\, {\\mathbf {\\mathrm {\\tau }}}$ differ from ${\\mathbf {\\mathrm {\\tau }}}$ by lattice vectors ${\\mathbf {\\mathrm {a}}}_i$ $g_i \\, {\\mathbf {\\mathrm {\\tau }}}= {\\mathbf {\\mathrm {\\tau }}}+ {\\mathbf {\\mathrm {a}}}_i ,$ so that $\\mathcal {D}_{\\mathbf {\\mathrm {\\tau }}}^{\\mathbf {\\mathrm {k}}} ( g_i )= \\exp [-i{\\mathbf {\\mathrm {k}}} \\cdot (g_i \\, {\\mathbf {\\mathrm {\\tau }}}- {\\mathbf {\\mathrm {\\tau }}})]= \\exp ( - i {\\mathbf {\\mathrm {k}}} \\cdot {\\mathbf {\\mathrm {a}}}_i )$ and $\\mathcal {D}_{\\mathbf {\\mathrm {\\tau }}}^{\\mathbf {\\mathrm {k}}} (g_1) \\mathcal {D}_{\\mathbf {\\mathrm {\\tau }}}^{\\mathbf {\\mathrm {k}}} (g_2) = \\exp [ - i {\\mathbf {\\mathrm {k}}} \\cdot ({\\mathbf {\\mathrm {a}}}_1 + {\\mathbf {\\mathrm {a}}}_2)].$ We can also write $g_1 g_2 \\, {\\mathbf {\\mathrm {\\tau }}}$ as $g_1 g_2 \\, {\\mathbf {\\mathrm {\\tau }}}& = g_1 \\, {\\mathbf {\\mathrm {\\tau }}}+ g_1 \\, {\\mathbf {\\mathrm {a}}}_2\\\\& = {\\mathbf {\\mathrm {\\tau }}}+ {\\mathbf {\\mathrm {a}}}_1 + {\\mathbf {\\mathrm {a}}}_2 + g_1 \\, {\\mathbf {\\mathrm {a}}}_2 - {\\mathbf {\\mathrm {a}}}_2 .$ Hence $\\mathcal {D}_{\\mathbf {\\mathrm {\\tau }}}^{\\mathbf {\\mathrm {k}}} ( g_1 g_2 )& = \\exp [-i{\\mathbf {\\mathrm {k}}} \\cdot (g_1 g_2 \\, {\\mathbf {\\mathrm {\\tau }}}- {\\mathbf {\\mathrm {\\tau }}})]\\\\& = \\exp [ - i {\\mathbf {\\mathrm {k}}} \\cdot ({\\mathbf {\\mathrm {a}}}_1 + {\\mathbf {\\mathrm {a}}}_2 + g_1 \\, {\\mathbf {\\mathrm {a}}}_2 - {\\mathbf {\\mathrm {a}}}_2)]\\\\& = \\mathcal {D}_{\\mathbf {\\mathrm {\\tau }}}^{\\mathbf {\\mathrm {k}}} (g_1) \\mathcal {D}_{\\mathbf {\\mathrm {\\tau }}}^{\\mathbf {\\mathrm {k}}} (g_2) \\exp [ - i {\\mathbf {\\mathrm {k}}} \\cdot (g_1 \\, {\\mathbf {\\mathrm {a}}}_2 - {\\mathbf {\\mathrm {a}}}_2)].$ We get similar to Eq.",
"(REF ) and using Eq.",
"(REF ) $\\exp [ - i {\\mathbf {\\mathrm {k}}} \\cdot (g_1 {\\mathbf {\\mathrm {a}}}_2 - {\\mathbf {\\mathrm {a}}}_2)]& = \\exp [ - i (g_1^{-1} {\\mathbf {\\mathrm {k}}} - {\\mathbf {\\mathrm {k}}}) \\cdot {\\mathbf {\\mathrm {a}}}_2)] \\\\& = \\exp [ - i {\\mathbf {\\mathrm {b}}}_{g_1} \\cdot {\\mathbf {\\mathrm {a}}}_2] = 1 .$ This gives us finally $\\mathcal {D}_{\\mathbf {\\mathrm {\\tau }}}^{\\mathbf {\\mathrm {k}}} ( g_1 g_2 )= \\mathcal {D}_{\\mathbf {\\mathrm {\\tau }}}^{\\mathbf {\\mathrm {k}}} (g_1) \\mathcal {D}_{\\mathbf {\\mathrm {\\tau }}}^{\\mathbf {\\mathrm {k}}} (g_2) ,$ so that, indeed, $\\Gamma _{\\mathbf {\\mathrm {\\tau }}}= \\lbrace \\mathcal {D}_{\\mathbf {\\mathrm {\\tau }}}^{\\mathbf {\\mathrm {k}}} (g) : g \\in \\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}\\rbrace $ is a one-dimensional IR of the group $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ .",
"The above argument [22] is based on the full Bloch functions $\\Psi _{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {r}}})$ .",
"We showed in Eq.",
"(REF ) that in a TB description, these Bloch functions can be factorized as $\\Phi _{\\nu {\\mathbf {\\mathrm {k}}}}^{W \\mu } ({\\mathbf {\\mathrm {r}}}) = q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{W \\mu }) \\, \\phi _\\nu ^W (\\delta {\\mathbf {\\mathrm {r}}})$ , so that the symmetry of the plane waves $q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{W \\mu })$ can be discussed separately from the symmetry of the atomic orbitals $\\phi _\\nu ^W (\\delta {\\mathbf {\\mathrm {r}}})$ .",
"The orbitals $\\phi _\\nu ^W (\\delta {\\mathbf {\\mathrm {r}}})$ only depend on $\\delta {\\mathbf {\\mathrm {r}}}$ but not on the actual positions ${\\mathbf {\\mathrm {R}}}_j^{W \\mu }$ .",
"Hence it follows immediately that the symmetry of the orbitals $\\phi _\\nu ^W (\\delta {\\mathbf {\\mathrm {r}}})$ is independent of the coordinate system used, see also Eq.",
"(REF ).",
"Only the representation $\\Gamma ^q_{{\\mathbf {\\mathrm {k}}} W}$ of the plane waves $q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{W \\mu })$ depends, in general, on the coordinate system.",
"A translation of the coordinate system by ${\\mathbf {\\mathrm {\\tau }}}$ maps the positions ${\\mathbf {\\mathrm {R}}}_j^{W \\mu }$ onto the atomic position ${\\mathbf {\\mathrm {R}}}_j^{W^{\\prime } \\mu } = {\\mathbf {\\mathrm {R}}}_j^{W \\mu } - {\\mathbf {\\mathrm {\\tau }}}$ .",
"The new coordinate system is valid if and only if $g \\, {\\mathbf {\\mathrm {\\tau }}}- {\\mathbf {\\mathrm {\\tau }}}$ are lattice vectors for all $g \\in \\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ [see Eq.",
"(REF )], so that the plane waves transform as $\\Gamma ^q_{{\\mathbf {\\mathrm {k}}} W^{\\prime }}$ .",
"We have ${\\mathbf {\\mathrm {R}}}_{j^{\\prime }}^{W^{\\prime } \\mu ^{\\prime }}& \\equiv g \\, {\\mathbf {\\mathrm {R}}}_j^{W^{\\prime } \\mu } = g \\, ( {\\mathbf {\\mathrm {R}}}_j^{W \\mu } - {\\mathbf {\\mathrm {\\tau }}})\\\\& = g \\, {\\mathbf {\\mathrm {R}}}_j^{W \\mu } - g \\, {\\mathbf {\\mathrm {\\tau }}}\\\\& = {\\mathbf {\\mathrm {R}}}_{j^{\\prime }}^{W \\mu ^{\\prime }} - g \\, {\\mathbf {\\mathrm {\\tau }}},$ so that for the transformed Wyckoff letter $W^{\\prime }$ , Eq.",
"(REF ) becomes $\\mathcal {D}_{\\mathbf {\\mathrm {k}}}^{W^{\\prime }} (g)_{\\mu ^{\\prime } \\mu }& = \\exp [ i {\\mathbf {\\mathrm {k}}} \\cdot ({\\mathbf {\\mathrm {R}}}_{j^{\\prime }}^{W^{\\prime } \\mu ^{\\prime }} - {\\mathbf {\\mathrm {R}}}_j^{W^{\\prime } \\mu ^{\\prime }})]\\\\& = \\exp [ i {\\mathbf {\\mathrm {k}}} \\cdot ( {\\mathbf {\\mathrm {R}}}_{j^{\\prime }}^{W \\mu ^{\\prime }} - g \\, {\\mathbf {\\mathrm {\\tau }}}- {\\mathbf {\\mathrm {R}}}_j^{W \\mu } + {\\mathbf {\\mathrm {\\tau }}})]\\\\& = \\exp [ i {\\mathbf {\\mathrm {k}}} \\cdot ( {\\mathbf {\\mathrm {R}}}_{j^{\\prime }}^{W \\mu ^{\\prime }} - {\\mathbf {\\mathrm {R}}}_j^{W \\mu } )]\\exp [ - i {\\mathbf {\\mathrm {k}}} \\cdot ( g \\, {\\mathbf {\\mathrm {\\tau }}}- {\\mathbf {\\mathrm {\\tau }}})]\\\\& = \\mathcal {D}^q_{{\\mathbf {\\mathrm {k}}} W} (g)_{\\mu ^{\\prime } \\mu } \\,\\mathcal {D}_{\\mathbf {\\mathrm {\\tau }}}^{\\mathbf {\\mathrm {k}}} (g)$ with $\\mathcal {D}_{\\mathbf {\\mathrm {\\tau }}}^{\\mathbf {\\mathrm {k}}} (g)$ given by Eq.",
"(REF ).",
"This gives us the rearrangement lemma for the representations of plane waves $\\Gamma ^q_{{\\mathbf {\\mathrm {k}}} W^{\\prime }} = \\Gamma _{\\mathbf {\\mathrm {\\tau }}}\\times \\Gamma ^q_{{\\mathbf {\\mathrm {k}}} W}.$ Hence we confirm that the nontrivial case $\\Gamma ^q_{{\\mathbf {\\mathrm {k}}} W^{\\prime }} \\ne \\Gamma ^q_{{\\mathbf {\\mathrm {k}}} W}$ requires that $\\Gamma _{\\mathbf {\\mathrm {\\tau }}}= \\lbrace \\mathcal {D}_{\\mathbf {\\mathrm {\\tau }}}^{\\mathbf {\\mathrm {k}}} (g) : g \\in \\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}\\rbrace $ is not the identity representation.",
"As mentioned above, the symmetry of plane waves $q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{W \\mu })$ is a universal problem for each space group $\\mathcal {G}$ , independent of the detailed realization of a crystal structure.",
"This holds, in particular, if the atoms are located at positions ${\\mathbf {\\mathrm {R}}}_{W \\mu j}^{I {\\mathbf {\\mathrm {k}}}}$ where the $\\mathcal {q}_{\\mathbf {\\mathrm {k}}}^{I\\beta } ({\\mathbf {\\mathrm {R}}}_{W \\mu j}^{I {\\mathbf {\\mathrm {k}}}})$ transforms according to only one IR $\\Gamma _I$ .",
"Hence it is possible to discuss the rearrangement lemma for the IRs of Bloch states independent of a particular crystal structure, but it depends only on the space group $\\mathcal {G}$ .",
"Among all 230 space groups, 159 contain Wyckoff sites with origin-dependent site symmetries [52].",
"Though a necessary criterion, it is however not a sufficient criterion for a rearrangement of the IRs of Bloch states under a change of the coordinate system [22]."
],
[
"Effect of time reversal",
"In the absence of an external magnetic field, the eigenfunctions of the TB Hamiltonian obey time-reversal symmetry $\\Theta $ .",
"This implies that if an eigenfunction $\\Psi _{{\\mathbf {\\mathrm {k}}}}^{I \\beta } ({\\mathbf {\\mathrm {r}}})$ with energy $E({\\mathbf {\\mathrm {k}}})$ transforms according to the $\\beta $ th component of the IR $\\Gamma _I$ of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ , the time-reversed wave function $\\Theta \\, \\Psi _{{\\mathbf {\\mathrm {k}}}}^{I \\beta } ({\\mathbf {\\mathrm {r}}}) = \\Psi _{{\\mathbf {\\mathrm {k}}}}^{I \\beta } ({\\mathbf {\\mathrm {r}}})^\\ast = \\Psi _{-{\\mathbf {\\mathrm {k}}}}^{\\tilde{I}, \\beta ^{\\prime }} ({\\mathbf {\\mathrm {r}}})$ [which is likewise an eigenfunction of the Hamiltonian with energy $E(-{\\mathbf {\\mathrm {k}}}) = E({\\mathbf {\\mathrm {k}}})$ ] transforms according to the $\\beta ^{\\prime }$ th component of the complex conjugate IR $\\Gamma _I^\\ast $ of $\\mathcal {G}_{-{\\mathbf {\\mathrm {k}}}} = \\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ .",
"Therefore, if the eigenfunctions of the TB Hamiltonian for some energy $E ({\\mathbf {\\mathrm {k}}})$ contain atomic orbitals transforming according to an IR $\\Gamma _J^\\phi $ of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ and symmetry-adapted plane waves transforming according to $\\Gamma _{J^{\\prime }}^q$ , the eigenfunctions for wave vector $-{\\mathbf {\\mathrm {k}}}$ with energy $E (-{\\mathbf {\\mathrm {k}}}) = E ({\\mathbf {\\mathrm {k}}})$ contain atomic orbitals transforming according to the complex conjugate IR $\\Gamma _J^{\\phi \\ast }$ and plane waves transforming according to $\\Gamma _{J^{\\prime }}^{q \\ast }$ .",
"Here, the symmetry-adapted atomic orbitals at $-{\\mathbf {\\mathrm {k}}}$ are the complex conjugates of the corresponding atomic orbitals at ${\\mathbf {\\mathrm {k}}}$ .",
"We obtain the symmetry-adapted plane waves at $-{\\mathbf {\\mathrm {k}}}$ from the corresponding plane waves at ${\\mathbf {\\mathrm {k}}}$ by replacing ${\\mathbf {\\mathrm {k}}} \\rightarrow -{\\mathbf {\\mathrm {k}}}$ .",
"Degeneracies of Bloch states due to time-reversal symmetry are discussed in Refs.",
"[53], [38], [3].",
"In general, three cases must be distinguished The classification of representations under time reversal adopted here from Ref.",
"[3] is the most convenient for physical applications, but it differs from that customary in courses on group theory, where real representations are assigned to case (a) and complex inequivalent and equivalent representations to cases (b) and (c).",
"The two classifications agree for single-group representations, but cases (a) and (c) must be interchanged for double-group representations..",
"In case (a), eigenfunctions $\\Psi $ and $\\Theta \\, \\Psi $ of the crystal Hamiltonian $H ({\\mathbf {\\mathrm {r}}} )$ are linearly dependent.",
"In case (b), eigenfunctions $\\Psi $ and $\\Theta \\, \\Psi $ of $H ({\\mathbf {\\mathrm {r}}} )$ are linearly independent and transform according to inequivalent representations $\\Gamma _I$ and $\\Gamma _I^\\ast $ , i.e., $\\chi _I (g) \\ne \\chi _I^\\ast (g)$ for some $g \\in \\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ .",
"Finally, in case (c), eigenfunctions $\\Psi $ and $\\Theta \\, \\Psi $ of $H ({\\mathbf {\\mathrm {r}}} )$ are linearly independent and transform according to equivalent representations $\\Gamma _I$ and $\\Gamma _I^\\ast $ , i.e., $\\chi _I (g) = \\chi _I^\\ast (g)$ for all $g \\in \\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ .",
"In cases (b) and (c) invariance under time reversal causes additional degeneracy.",
"We have for symmorphic space groups [53], [38], [3] $\\frac{f}{h} \\sum _{g \\in \\mathcal {G}_0} \\chi _I (g^2)\\; \\delta _{g {\\mathbf {\\mathrm {k}}} + {\\mathbf {\\mathrm {k}}}, {\\mathbf {\\mathrm {b}}}}= \\left\\lbrace \\begin{array}{rs{1em}L}\\Theta ^2 & case (a) \\\\[0.5ex]0 \\hspace*{5.50003pt} & case (b) \\\\[0.5ex]- \\Theta ^2 & case (c)\\end{array} \\right.",
",$ where $f$ is the number of points of the star of ${\\mathbf {\\mathrm {k}}}$ , $h$ is the order of the crystallographic point group $\\mathcal {G}_0$ of the crystal, $\\chi _I (g)$ are the characters of the IR $\\Gamma _I$ of the point group $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ of the wave vector ${\\mathbf {\\mathrm {k}}}$ , and $g {\\mathbf {\\mathrm {k}}} + {\\mathbf {\\mathrm {k}}}$ may be zero or a reciprocal lattice vector ${\\mathbf {\\mathrm {b}}}$ .",
"We have $\\Theta ^2 = +1$ for single-group representations and $\\Theta ^2 = -1$ for double-group representations.",
"The criterion (REF ) applies to the IRs $\\Gamma _I$ of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ independent of the origin of the coordinate system.",
"If a crystal structure permits a change of the coordinate system characterized by a vector ${\\mathbf {\\mathrm {\\tau }}}$ with RAR $\\Gamma _{\\mathbf {\\mathrm {\\tau }}}$ , a Bloch function transforming according to the IR $\\Gamma _I$ of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ in the old coordinate system transforms according to $\\Gamma _{I^{\\prime }} = \\Gamma _{\\mathbf {\\mathrm {\\tau }}}\\times \\Gamma _I$ in the new coordinate, see Eq.",
"(REF ).",
"Therefore, the IRs $\\Gamma _I$ and $\\Gamma _{I^{\\prime }}$ of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ must fall into the same category according to Eq.",
"(REF ).",
"In a more detailed analysis [55], [38], [3], for each of the cases (a), (b), and (c) three possibilities must be distinguished: (1) the points ${\\mathbf {\\mathrm {k}}}$ and $-{\\mathbf {\\mathrm {k}}}$ are equivalent, i.e., ${\\mathbf {\\mathrm {k}}} = - {\\mathbf {\\mathrm {k}}} + {\\mathbf {\\mathrm {b}}}$ ; (2) ${\\mathbf {\\mathrm {k}}}$ is not equivalent to $-{\\mathbf {\\mathrm {k}}}$ , but the space group contains an element $R$ which maps ${\\mathbf {\\mathrm {k}}}$ onto $-{\\mathbf {\\mathrm {k}}}$ $R {\\mathbf {\\mathrm {k}}} = - {\\mathbf {\\mathrm {k}}};$ (3) the points ${\\mathbf {\\mathrm {k}}}$ and $-{\\mathbf {\\mathrm {k}}}$ are in different stars.",
"For the systems discussed below, case (1) applies to the ${\\mathbf {\\mathrm {\\Gamma }}}$ and ${\\mathbf {\\mathrm {M}}}$ points of the BZ, whereas case (2) applies to the ${\\mathbf {\\mathrm {K}}}$ points."
],
[
"Band Symmetries in MoS$_2$",
"Having derived a systematic theory for the symmetry of TB Bloch functions, we now apply this theory to several quasi-2D materials.",
"Our main focus is on monolayer MoS$_2$ .",
"For comparison, we also discuss single-layer (SLG), bilayer (BLG), and trilayer (TLG) graphene in the next section."
],
[
"Crystal structure of MoS$_2$",
"The crystal structure of single-layer MoS$_2$ is shown in Fig.",
"REF .",
"It is characterized by the point group $D_{3h}$ and space group $P\\bar{6} m 2$ (# 187).",
"Three Wyckoff positions have multiplicity $m=1$ , the positions of the Mo atom, the midpoint between a pair of top and bottom S atoms, and the center of the hexagon.",
"Hence, as discussed in Sec.",
"REF , three choices $\\alpha = a,b,c$ emerge for the origin of the coordinate system: ($a$ ) origin at the center of the hexagon [Fig.",
"REF (a)], ($b$ ) origin at a Mo atom [Fig.",
"REF (b)], and ($c$ ) origin at the midpoint between a top and bottom S atom [Fig.",
"REF (c)].",
"In either case, the Mo atoms have site symmetry $D_{3h}$ and Wyckoff multiplicity $m=1$ .",
"Yet the Wyckoff letters for these positions listed in Table REF depend on the coordinate system [52].",
"For coordinate system ($a$ ), the Mo and S atoms have Wyckoff letters $e$ and $h$ , respectively, whereas these letters become $a$ and $i$ in coordinate system ($b$ ), and $c$ and $g$ in coordinate system ($c$ ).",
"The S atoms have site symmetry $C_{3v}$ and multiplicity $m=2$ .",
"The positions of Mo and S atoms in unit cell $j$ are denoted by ${\\mathbf {\\mathrm {R}}}_j^{\\textrm {Mo} (\\alpha )}$ and ${\\mathbf {\\mathrm {R}}}_j^{\\textrm {S} \\mu (\\alpha )}$ , respectively.",
"For the S atom, the top (bottom) atoms are labeled $\\mu =1$ ($\\mu = 2$ ).",
"There are yet other coordinate systems that can be used for MoS$_2$ .",
"For example, Ref.",
"[31] used a coordinate system that differs from coordinate system ($a$ ) by a reflection about the $xz$ plane.",
"Here, we do not consider these additional coordinate systems [23].",
"Figure: Crystal structure of single-layer MoS 2 _2.",
"Three coordinate systems α=a,b,c\\alpha =a, b, c are considered with (a) the origin located at the center of a hexagon, (b) origin at an Mo atom, and (c) origin at the midpoint between top and bottom S atoms.",
"The atomic positions of Mo and S in unit cell jj are denoted by R j Mo (α) {\\mathbf {\\mathrm {R}}}_j^{\\textrm {Mo}(\\alpha )} and R j Sμ(α) {\\mathbf {\\mathrm {R}}}_j^{\\textrm {S}\\mu (\\alpha )}, respectively.",
"For the S atom, the top (bottom) atoms are labeled μ=1\\mu =1 (μ=2\\mu = 2).",
"The primitive lattice vectors are denoted a 1 {\\mathbf {\\mathrm {a}}}_1 and a 2 {\\mathbf {\\mathrm {a}}}_2.",
"The shaded region shows a unit cell (j=1j=1).",
"The vectors t Mo α {\\mathbf {\\mathrm {t}}}_\\mathrm {Mo}^\\alpha and t S α {\\mathbf {\\mathrm {t}}}_\\mathrm {S}^\\alpha give the positions of the Mo and S atoms within a unit cell.",
"The dashed axes (1)(1), (2)(2), and (3)(3) are the twofold rotation axes of the point group D 3h D_{3h}.",
"(d) Three-dimensional illustration of single-layer MoS 2 _2.",
"(e) The first Brillouin zone.Table: Site symmetries in monolayer MoS 2 _2 and SLG, BLG, and TLG.",
"We include here the Wyckoff positions occupied by atoms as well as the unoccupied center of the hexagon denoted by R j center {\\mathbf {\\mathrm {R}}}_j^{\\mathrm {center}}.",
"The coordinate systems α=a,b,c\\alpha = a, b, c for MoS 2 _2 and TLG are depicted in Figs.",
"and , respectively.The Brillouin zone for single-layer MoS$_2$ is shown in Fig.",
"REF (e).",
"In the following, we will focus on the ${\\mathbf {\\mathrm {\\Gamma }}}$ point ${\\mathbf {\\mathrm {k}}} = 0$ , the ${\\mathbf {\\mathrm {K}}}$ , and the ${\\mathbf {\\mathrm {M}}}$ points.",
"The star of the ${\\mathbf {\\mathrm {K}}}$ point includes two inequivalent wave vectors denoted ${\\mathbf {\\mathrm {K}}}$ and ${\\mathbf {\\mathrm {K}}}^{\\prime }$ .",
"The star of the ${\\mathbf {\\mathrm {M}}}$ point includes three inequivalent wave vectors denoted ${\\mathbf {\\mathrm {M}}}_1$ , ${\\mathbf {\\mathrm {M}}}_2$ , and ${\\mathbf {\\mathrm {M}}}_3$ .",
"The point group of single-layer MoS$_2$ is $D_{3h}$ .",
"This is also the point group of the wave vector at the ${\\mathbf {\\mathrm {\\Gamma }}}$ point.",
"It contains a $120^\\circ $ counterclockwise rotation $C_3$ about the $z$ axis.",
"The reflection plane of $\\sigma _h$ is the $xy$ plane and $S_3 = \\sigma _h C_3$ .",
"The rotation axes of the three twofold rotations $C_2^{\\prime (i)}$ are the axes $i=1,2,3$ shown in Fig.",
"REF .",
"These axes are also shown as dashed lines in Fig.",
"REF (a).",
"The reflection plane of $\\sigma _v^{(i)}$ is the plane passing through the axis $i$ and the $z$ axis.",
"The characters of $D_{3h}$ are listed in Table REF (Appendix ).",
"We label the IRs of the crystallographic point groups following Koster et al.",
"[56].",
"Figure: Groups of the wave vector in monolayer MoS 2 _2 and TLG.",
"(a) The point Γ{\\mathbf {\\mathrm {\\Gamma }}} has the point group D 3h D_{3h} with the zz axis (out of plane) as the threefold rotation axis.",
"The dashed lines (ii) are the axes for twofold rotations C 2 '(i) C_2^{\\prime (i)} with i=1,2,3i=1,2,3.",
"The reflection σ v (i) \\sigma _v^{(i)} is about a plane that includes the dashed axis ii and the zz axis.",
"(b) The points K{\\mathbf {\\mathrm {K}}} and K ' {\\mathbf {\\mathrm {K}}}^{\\prime } have the point group C 3h C_{3h} with threefold rotations about the zz axis.",
"The reflection plane of σ h \\sigma _h is the xyxy plane.",
"The dotted lines indicate the twofold rotation axes that appear in the point group D 3h D_{3h} but are not symmetry elements of C 3h C_{3h}.",
"[(c)–(e)] The points M 1 {\\mathbf {\\mathrm {M}}}_1, M 2 {\\mathbf {\\mathrm {M}}}_2, and M 1 {\\mathbf {\\mathrm {M}}}_1 have the point group C 2v C_{2v}.",
"The dashed line is the axis of the twofold rotation C 2 C_2.",
"The reflection plane of σ v \\sigma _v is the xyxy plane.",
"The reflection plane of σ v ' \\sigma _v^{\\prime } contains the dashed line and the zz axis.At the ${\\mathbf {\\mathrm {K}}}$ point [Fig.",
"REF (b)], the point group of the wave vector becomes $\\mathcal {G}_{{\\mathbf {\\mathrm {K}}}} = C_{3h}$ whose characters are listed in Table REF (Appendix ).",
"Finally, at the inequivalent points ${\\mathbf {\\mathrm {M}}}_i$ ($i=1,2,3$ ), the group of the wave vector is $\\mathcal {G}_{{\\mathbf {\\mathrm {M}}}} = C_{2v}$ , the character table of which is reproduced in Table REF (Appendix ).",
"This group contains the twofold rotation $C_2$ , the axis of which is indicated as dashed line in Figs.",
"REF (c)–REF (e), the reflection $\\sigma _v$ about the $xy$ plane, and the reflection $\\sigma _v^{\\prime }$ for which the reflection plane includes the dashed line and the $z$ axis.",
"The primitive lattice vectors ${\\mathbf {\\mathrm {a}}}_1$ and ${\\mathbf {\\mathrm {a}}}_2$ are ${\\mathbf {\\mathrm {a}}}_1 =\\frac{a}{2}\\left( \\begin{array}{s{0.15em}cs{0.15em}}1\\\\\\sqrt{3}\\end{array} \\right),\\quad {\\mathbf {\\mathrm {a}}}_2 = \\frac{a}{2} \\left( \\begin{array}{s{0.15em}cs{0.15em}}1\\\\-\\sqrt{3}\\end{array} \\right),$ where $a$ is the lattice constant.",
"Ignoring for brevity the $z$ component, the positions ${\\mathbf {\\mathrm {t}}}_\\mathrm {Mo}^\\alpha $ of Mo and ${\\mathbf {\\mathrm {t}}}_\\mathrm {S}^\\alpha $ of S in the unit cell are ${\\mathbf {\\mathrm {t}}}_\\mathrm {Mo}^a & = \\frac{a}{2} \\left( \\begin{array}{s{0.15em}cs{0.15em}}1\\\\\\frac{1}{\\sqrt{3}}\\end{array} \\right) , & {\\mathbf {\\mathrm {t}}}_\\mathrm {S}^a & = \\frac{a}{2} \\left( \\begin{array}{s{0.15em}cs{0.15em}}1 \\\\ -\\frac{1}{\\sqrt{3}}\\end{array} \\right) , \\\\{\\mathbf {\\mathrm {t}}}_\\mathrm {Mo}^b & = \\frac{a}{2}\\left( \\begin{array}{s{0.15em}cs{0.15em}} 0 \\\\ 0 \\end{array} \\right) , &{\\mathbf {\\mathrm {t}}}_\\mathrm {S}^b & = \\frac{a}{2}\\left( \\begin{array}{s{0.15em}cs{0.15em}}0 \\\\ - \\frac{2}{\\sqrt{3}}\\end{array} \\right) ,\\\\{\\mathbf {\\mathrm {t}}}_\\mathrm {Mo}^c & = \\frac{a}{2} \\left( \\begin{array}{s{0.15em}cs{0.15em}} 0 \\\\ \\frac{2}{\\sqrt{3}}\\end{array} \\right) , & {\\mathbf {\\mathrm {t}}}_\\mathrm {S}^c & = \\frac{a}{2} \\left( \\begin{array}{s{0.15em}cs{0.15em}} 0 \\\\ 0 \\end{array} \\right) ,$ where the superscript $\\alpha = a, b, c$ denotes the coordinate system.",
"The primitive vectors ${\\mathbf {\\mathrm {b}}}_1$ and ${\\mathbf {\\mathrm {b}}}_2$ of the reciprocal lattice are ${\\mathbf {\\mathrm {b}}}_1 =\\frac{2 \\pi }{a}\\left( \\begin{array}{s{0.15em}cs{0.15em}}1\\\\1/\\sqrt{3}\\end{array} \\right),\\quad {\\mathbf {\\mathrm {b}}}_2 =\\frac{2 \\pi }{a}\\left( \\begin{array}{s{0.15em}cs{0.15em}}1\\\\-1/\\sqrt{3}\\end{array} \\right).$ The two inequivalent corner points of the Brillouin zone are ${\\mathbf {\\mathrm {K}}} =\\frac{2\\pi }{a}\\left( \\begin{array}{s{0.15em}cs{0.15em}}2/3\\\\0\\end{array} \\right),\\quad {\\mathbf {\\mathrm {K}}}^{\\prime } =\\frac{2\\pi }{a}\\left( \\begin{array}{s{0.15em}cs{0.15em}}-2/3\\\\0\\end{array} \\right),$ and the ${\\mathbf {\\mathrm {M}}}$ points are $\\nonumber {\\mathbf {\\mathrm {M}}}_1 & = \\frac{\\pi }{a}\\left( \\begin{array}{s{0.15em}cs{0.15em}}1\\\\1/\\sqrt{3}\\end{array} \\right),\\quad {\\mathbf {\\mathrm {M}}}_2 =\\frac{2\\pi }{a}\\left( \\begin{array}{s{0.15em}cs{0.15em}}0\\\\1/\\sqrt{3}\\end{array} \\right),\\\\{\\mathbf {\\mathrm {M}}}_3 & = \\frac{\\pi }{a}\\left( \\begin{array}{s{0.15em}cs{0.15em}}-1\\\\1/\\sqrt{3}\\end{array} \\right),$ see Fig.",
"REF (e)."
],
[
"Rearrangement of IRs of Bloch states in MoS$_2$",
"The crystal structure of monolayer MoS$_2$ can be described by three different coordinate systems $\\alpha = a, b, c$ shown in Fig.",
"REF .",
"This provides an example for the rearrangement of the IRs of Bloch states discussed in general terms in Sec.",
"REF .",
"The coordinate systems $\\alpha $ and $\\beta $ are related via a translation ${\\mathbf {\\mathrm {\\tau }}}_{\\alpha \\beta }$ .",
"For the shifts $a \\rightarrow b$ , $b \\rightarrow c$ , and $c \\rightarrow a$ , the translation vectors (apart from a lattice vector) are given by ${\\mathbf {\\mathrm {\\tau }}}_{a b} = {\\mathbf {\\mathrm {\\tau }}}_{b c} = {\\mathbf {\\mathrm {\\tau }}}_{c a}= \\frac{a}{2}\\left( \\begin{array}{s{0.15em}cs{0.15em}}1 \\\\ 1/\\sqrt{3}\\end{array} \\right)$ and ${\\mathbf {\\mathrm {\\tau }}}_{\\beta \\alpha } = - {\\mathbf {\\mathrm {\\tau }}}_{\\alpha \\beta }$ .",
"For wave vectors ${\\mathbf {\\mathrm {k}}}$ inside the BZ such as the ${\\mathbf {\\mathrm {\\Gamma }}}$ point as well as for the ${\\mathbf {\\mathrm {M}}}$ points, the RARs $\\Gamma _{\\alpha \\beta }^{\\mathbf {\\mathrm {\\Gamma }}}$ and $\\Gamma _{\\alpha \\beta }^{\\mathbf {\\mathrm {M}}}$ are given by Eq.",
"(REF ) with ${\\mathbf {\\mathrm {b}}}_g =0$ for all $g \\in \\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ .",
"These RARs are, therefore, given by the identity representation $\\Gamma _1$ , i.e., at both the ${\\mathbf {\\mathrm {\\Gamma }}}$ and ${\\mathbf {\\mathrm {M}}}$ points the labeling of Bloch states is independent of the coordinate system.",
"However, a shift of the coordinate system rearranges the IRs at the ${\\mathbf {\\mathrm {K}}}$ point.",
"Using Eq.",
"(REF ) at the ${\\mathbf {\\mathrm {K}}}$ point, where the group of the wave vector is $C_{3h}$ , we get ${\\mathbf {\\mathrm {b}}}_E & = {\\mathbf {\\mathrm {b}}}_{\\sigma _h} = 0 , \\\\{\\mathbf {\\mathrm {b}}}_{C_3} & = {\\mathbf {\\mathrm {b}}}_{S_3} = - {\\mathbf {\\mathrm {b}}}_1= - \\frac{2 \\pi }{a} \\left( \\begin{array}{s{0.15em}cs{0.15em}} 1 \\\\ 1/\\sqrt{3}\\end{array} \\right) , \\\\{\\mathbf {\\mathrm {b}}}_{C_3^{-1}}& = {\\mathbf {\\mathrm {b}}}_{S_3^{-1}} = - {\\mathbf {\\mathrm {b}}}_2= - \\frac{2 \\pi }{a} \\left( \\begin{array}{s{0.15em}cs{0.15em}} 1 \\\\ - 1/\\sqrt{3} \\end{array} \\right).$ For the shifts $a \\rightarrow b$ , $b \\rightarrow c$ , and $c \\rightarrow a$ , we thus have using Eq.",
"(REF ) $\\mathcal {D}_{\\alpha \\beta }^{\\mathbf {\\mathrm {K}}} (E)& = \\mathcal {D}_{\\alpha \\beta }^{\\mathbf {\\mathrm {K}}} (\\sigma _h)= \\exp (-i {\\mathbf {\\mathrm {b}}}_E \\cdot {\\mathbf {\\mathrm {\\tau }}}_{\\alpha \\beta } ) = 1 , \\\\\\mathcal {D}_{\\alpha \\beta }^{\\mathbf {\\mathrm {K}}} (C_3)& = \\mathcal {D}_{\\alpha \\beta }^{\\mathbf {\\mathrm {K}}} (S_3)= \\exp (-i {\\mathbf {\\mathrm {b}}}_{C_3} \\cdot {\\mathbf {\\mathrm {\\tau }}}_{\\alpha \\beta } ) = \\omega ^{-4} , \\\\\\mathcal {D}_{\\alpha \\beta }^{\\mathbf {\\mathrm {K}}} (C_3^{-1})& = \\mathcal {D}_{\\alpha \\beta }^{\\mathbf {\\mathrm {K}}} (S_3^{-1})= \\exp (-i {\\mathbf {\\mathrm {b}}}_{C_3^{-1}} \\cdot {\\mathbf {\\mathrm {\\tau }}}_{\\alpha \\beta } ) = \\omega ^4$ with $\\omega \\equiv \\exp (i\\pi /6)$ .",
"This implies $\\Gamma _{ab}^{\\mathbf {\\mathrm {K}}} = \\Gamma _{bc}^{\\mathbf {\\mathrm {K}}} = \\Gamma _{ca}^{\\mathbf {\\mathrm {K}}} = \\Gamma _3$ .",
"Since ${\\mathbf {\\mathrm {\\tau }}}_{\\beta \\alpha } = - {\\mathbf {\\mathrm {\\tau }}}_{\\alpha \\beta }$ , we have $\\Gamma _{ba}^{\\mathbf {\\mathrm {K}}} = \\Gamma _{cb}^{\\mathbf {\\mathrm {K}}} = \\Gamma _{ac}^{\\mathbf {\\mathrm {K}}} = \\Gamma _3^\\ast = \\Gamma _2$ .",
"Hence, at the ${\\mathbf {\\mathrm {K}}}$ point, for a Bloch state transforming in one coordinate system according to a certain IR, we can multiply this IR with either $\\Gamma _2$ or $\\Gamma _3$ to obtain the IR of the same Bloch state in a different coordinate system.",
"The multiplication table for the IRs of $C_{3h}$ is reproduced in Table REF (Appendix ).",
"Table REF shows how the IRs of $C_{3h}$ are rearranged when we go from one of the three coordinate systems $\\alpha =a, b, c$ to a different coordinate system $\\beta $ .",
"At the ${\\mathbf {\\mathrm {K}}}^{\\prime }$ point, Eq.",
"(REF ) gives $\\Gamma _{\\alpha \\beta }^{{\\mathbf {\\mathrm {K}}}^{\\prime }}= \\Gamma _{\\alpha \\beta }^{-{\\mathbf {\\mathrm {K}}}}= \\Gamma _{\\alpha \\beta }^{{\\mathbf {\\mathrm {K}}} \\ast }.$ Table: Rearrangement of the IRs of C 3h C_{3h} at the K{\\mathbf {\\mathrm {K}}} point of monolayer MoS 2 _2 when we go from one of the three coordinate systems α=a,b,c\\alpha =a, b, c to a different coordinate system β\\beta (see Fig.",
")."
],
[
"Atomic orbitals $\\phi _\\nu ^W(\\delta \\mathbf {r})$ at {{formula:90790d36-30ad-44ab-a654-1aef7f1d3ad3}} , {{formula:a4d8f2e9-df41-405a-bd0d-3d86d7793cca}} , and {{formula:21e6af6c-23a8-4526-b5e6-53410d4a26fe}}",
"The conduction and valence bands in MoS$_2$ are dominated by Mo $d$ and S $p$ orbitals [24].",
"At the points ${\\mathbf {\\mathrm {\\Gamma }}}$ , ${\\mathbf {\\mathrm {K}}}$ , and ${\\mathbf {\\mathrm {M}}}$ , the groups of the wave vectors $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ are $D_{3h}$ , $C_{3h}$ , and $C_{2v}$ , respectively, with character tables reproduced in Tables REF , REF , and REF .",
"We use Eq.",
"(REF ) to project these functions onto functions transforming according to the IRs of the various groups $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ .",
"The relevant symmetry operations are defined in Fig.",
"REF and Table REF considering both polar vectors ${\\mathbf {\\mathrm {P}}}$ such as position ${\\mathbf {\\mathrm {r}}}$ and axial vectors ${\\mathbf {\\mathrm {A}}}$ .",
"The two inequivalent points ${\\mathbf {\\mathrm {K}}}$ and ${\\mathbf {\\mathrm {K}}}^{\\prime }$ are related by a vertical reflection that transforms the component $x$ into $-x$ while keeping the $y$ and $z$ components fixed.",
"Table REF summarizes the IRs of the $p$ and $d$ orbitals.",
"We note that these results are fully consistent with the full rotation group compatibility tables in Ref. [56].",
"Table: Mapping of polar vectors P{\\mathbf {\\mathrm {P}}} and axial vectors A{\\mathbf {\\mathrm {A}}} under g=C 3 g = C_3, C 3 -1 C_3^{-1}, σ v (i) \\sigma _v^{(i)}, and σ h \\sigma _h, where the images are expressed in terms of the original components of these vectors.",
"C 3 C_3 is a 120 ∘ 120^\\circ counterclockwise rotation about the zz axis while σ v (i) \\sigma _v^{(i)} is a reflection about the plane containing the zz axis and the axis ii defined in Fig. .",
"We have ω≡exp(iπ/6)\\omega \\equiv \\exp (i\\pi /6).",
"Note S 3 ±1 =σ h ×C 3 ±1 S_3^{\\pm 1} = \\sigma _h \\times C_3^{\\pm 1} and C 2 '(i) =σ h ×σ v (i) C_2^{\\prime (i)} = \\sigma _h \\times \\sigma _v^{(i)}.Table: Symmetry-adapted pp and dd atomic orbitals for the point groups D 3h D_{3h}, C 3h C_{3h} and C 2v C_{2v} with coordinate systems defined in Fig.",
"(MoS 2 _2 and TLG).",
"For C 2v C_{2v}, the symmetrized atomic orbital φ ν [i] \\phi _\\nu ^{[i]} corresponds to the coordinate systems used for the point M i {\\mathbf {\\mathrm {M}}}_i in Figs.",
"(c)–(e).",
"The orbital [i(j)][i(j)] takes the upper (lower) sign.",
"The IRs of the atomic orbitals listed here are consistent with the compatibility relations in Table .Table: Compatibility relations for the IRs of D 3h D_{3h} and the IRs of its subgroups C 3h C_{3h} and C 2v C_{2v} using the coordinate system in Fig. .",
"The three different orientations of coordinate systems for C 2v C_{2v} in Figs.",
"(c)–(E) follow the same compatibility relations.Table: Symmetrized plane waves 𝒬 k Iβ (ℛ j W )\\mathcal {Q}_{\\mathbf {\\mathrm {k}}}^{I \\beta } ({\\mathbf {\\mathcal {R}}}_j^W) for monolayer MoS 2 _2 and SLG, BLG, and TLG at the Γ{\\mathbf {\\mathrm {\\Gamma }}}, K{\\mathbf {\\mathrm {K}}}, and M{\\mathbf {\\mathrm {M}}} points for the positions occupied by atoms.",
"The symmetrized plane waves 𝒬 K ' Iβ (ℛ j W )\\mathcal {Q}_{{\\mathbf {\\mathrm {K}}}^{\\prime }}^{I \\beta } ({\\mathbf {\\mathcal {R}}}_j^W) at K ' =-K{\\mathbf {\\mathrm {K}}}^{\\prime } = - {\\mathbf {\\mathrm {K}}} are obtained from the expressions given for K{\\mathbf {\\mathrm {K}}} by replacing K{\\mathbf {\\mathrm {K}}} by K ' {\\mathbf {\\mathrm {K}}}^{\\prime }.",
"The corresponding IRs are the complex conjugates of the IRs at K{\\mathbf {\\mathrm {K}}}.",
"The group 𝒢 k \\mathcal {G}_{{\\mathbf {\\mathrm {k}}}} of the wave vector and the plane wave IRs are shown.",
"At the K{\\mathbf {\\mathrm {K}}} point, we distinguish between the three coordinate systems α=a,b,c\\alpha = a, b, c for MoS 2 _2 and TLG with atomic positions R j W(α)μ {\\mathbf {\\mathrm {R}}}_j^{W (\\alpha ) \\mu } depicted in Figs.",
"and , respectively.",
"The IRs for the coordinate systems α\\alpha are then denoted by Γ i/j/k \\Gamma _{i/j/k}.",
"The definition of the symmetrized plane waves at the points M i {\\mathbf {\\mathrm {M}}}_i of SLG and BLG contains prefactors γ i \\gamma _i, where γ 1 =γ 3 =-1\\gamma _1 = \\gamma _3 = -1, and γ 2 =+1\\gamma _2 = +1."
],
[
"Transformation of plane waves $q_\\mathbf {k} (\\mathbf {R}_j^{\\mathrm {Mo} (\\alpha )})$",
"We now determine the IRs of the plane waves $q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{\\textrm {Mo} (\\alpha )})$ for the coordinate systems $\\alpha = a, b, c$ at the ${\\mathbf {\\mathrm {\\Gamma }}}$ , ${\\mathbf {\\mathrm {K}}}$ , and ${\\mathbf {\\mathrm {M}}}$ points of the Brillouin zone.",
"Since the Wyckoff letter corresponding to the positions of the Mo atoms has multiplicity $m=1$ , we can use either Eq.",
"(REF ) or Eq.",
"(REF ) to determine the phase $\\mathcal {D}_{\\mathbf {\\mathrm {k}}}^{\\mathrm {Mo} (\\alpha )} (g)$ acquired by the plane waves $q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{\\textrm {Mo} (\\alpha )})$ under a transformation $g$ .",
"We can then derive the IRs of the plane waves using the projection operators (REF ).",
"The results are summarized in Table REF ."
],
[
"Transformation of plane waves $q_\\mathbf {k} (\\mathbf {R}_j^{\\mathrm {S} \\mu (\\alpha )})$",
"The S atoms are located at Wyckoff positions with multiplicity $m=2$ , so that we represent the plane wave at the positions ${\\mathbf {\\mathrm {R}}}_j^{\\mathrm {S} \\mu (\\alpha )}$ as a two-component spinor ${Q}_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathcal {R}}}_j^{S (\\alpha )})= \\mathcal {q}_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_{j}^{S1(\\alpha )})+ \\mathcal {q}_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_{j}^{S2(\\alpha )})\\equiv \\left( \\begin{array}{s{0.15em}cs{0.15em}}1 \\\\ 1\\end{array} \\right).$ We can then use Eq.",
"(REF ) to determine the phases acquired under symmetry transformations.",
"To obtain the plane wave IRs, it is again advantageous to consider the simplest coordinate system.",
"For the S atoms, this is coordinate system $(c)$ where the origin of the coordinate system is at the midpoint between the S atoms in the top and bottom layer of a unit cell.",
"In this case, the transformation $g$ maps ${\\mathbf {\\mathrm {R}}}_1^{\\mathrm {S} \\mu (c)}$ either onto itself or onto ${\\mathbf {\\mathrm {R}}}_1^{\\mathrm {S} \\mu ^{\\prime } (c)}$ with $\\mu \\ne \\mu ^{\\prime }$ , so that Eq.",
"(REF ) becomes $\\mathcal {D}_{\\mathbf {\\mathrm {k}}}^{\\mathrm {S} (c)} (g)_{\\mu ^{\\prime } \\mu }= \\exp [ i {\\mathbf {\\mathrm {k}}} \\cdot ({\\mathbf {\\mathrm {R}}}_1^{\\mathrm {S} \\mu ^{\\prime } (c) } - {\\mathbf {\\mathrm {R}}}_1^{\\mathrm {S} \\mu ^{\\prime } (c)})]= 1$ for all $g \\in \\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ .",
"We can then determine the IRs of the plane waves for the coordinate systems $(a)$ and $(b)$ by using the respective RAR derived in Sec.",
"REF .",
"The results are summarized in Table REF ."
],
[
"IRs of Bloch states in MoS$_2$",
"The full symmetry-adapted Bloch functions are written as products of symmetrized plane waves and symmetrized atomic orbitals.",
"The five symmetry-adapted $d$ orbitals of the Mo atom times the plane wave $q_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{\\mathrm {Mo} (\\alpha )})$ and the three symmetry-adapted $p$ orbitals of the S atoms times the plane waves $\\mathcal {q}_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{\\mathrm {S}1 (\\alpha )}) \\pm \\mathcal {q}_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{\\mathrm {S}2 (\\alpha )})$ therefore comprises eleven symmetry-adapted basis functions for MoS$_2$ [30].",
"The corresponding IRs are listed in Table REF for the ${\\mathbf {\\mathrm {\\Gamma }}}$ , ${\\mathbf {\\mathrm {K}}}$ , and ${\\mathbf {\\mathrm {M}}}$ points.",
"We list in Tables REF , REF , and REF the sets of Bloch states transforming according to an IR of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ for the wave vectors ${\\mathbf {\\mathrm {k}}} = {\\mathbf {\\mathrm {\\Gamma }}} , {\\mathbf {\\mathrm {K}}}, {\\mathbf {\\mathrm {M}}}$ .",
"Using these symmetrized Bloch functions, the TB Hamiltonian for a wave vector ${\\mathbf {\\mathrm {k}}}$ can be written in a block-diagonal form, where each block refers to the basis functions transforming according to an IR $\\Gamma _I$ of $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ The symmetry-adapted bases derived here for MoS$_2$ define unitary transformations for block-diagonalizing a MoS$_2$ TB Hamiltonian that agree with the unitary transformations discussed in Ref. [30]..",
"To classify the additional degeneracy of the Bloch states due to time-reversal symmetry, we evaluate Eq.",
"(REF ).",
"All IRs of the space group for the stars $\\lbrace {\\mathbf {\\mathrm {\\Gamma }}}\\rbrace $ , $\\lbrace {\\mathbf {\\mathrm {K}}}\\rbrace $ , and $\\lbrace {\\mathbf {\\mathrm {M}}}\\rbrace $ belong to case $(a)$ .",
"Table: IRs of the plane waves (Γ k q \\Gamma ^q_{\\mathbf {\\mathrm {k}}}), the atomic orbitals φ ν \\phi _\\nu (Γ k φ \\Gamma ^\\phi _{\\mathbf {\\mathrm {k}}}), and the full Bloch functions (Γ k Φ =Γ k q ×Γ k φ \\Gamma ^\\Phi _{\\mathbf {\\mathrm {k}}} = \\Gamma ^q_{\\mathbf {\\mathrm {k}}} \\times \\Gamma ^\\phi _{\\mathbf {\\mathrm {k}}}) for the Mo and S atoms at the points k=Γ{\\mathbf {\\mathrm {k}}} = {\\mathbf {\\mathrm {\\Gamma }}}, K{\\mathbf {\\mathrm {K}}}, and M i {\\mathbf {\\mathrm {M}}}_i.",
"At K{\\mathbf {\\mathrm {K}}}, we distinguish between the three coordinate systems α=a,b,c\\alpha = a, b ,c .",
"For the plane waves at K ' {\\mathbf {\\mathrm {K}}}^{\\prime }, the IR Γ K ' (i/j/k) q \\Gamma ^q_{{\\mathbf {\\mathrm {K}}}^{\\prime } (i/j/k)} is the complex conjugate of the IR Γ K(i/j/k) q \\Gamma ^q_{{\\mathbf {\\mathrm {K}}} (i/j/k)}.",
"At the points M i {\\mathbf {\\mathrm {M}}}_i (i=1,2,3i=1,2,3), the atomic orbitals are denoted by φ ν [i] \\phi _\\nu ^{[i]}.Table: Symmetry-adapted TB Bloch functions in MoS 2 _2 at k=Γ{\\mathbf {\\mathrm {k}}} = {\\mathbf {\\mathrm {\\Gamma }}} with group of the wave vector 𝒢 Γ =D 3h \\mathcal {G}_{{\\mathbf {\\mathrm {\\Gamma }}}} = D_{3h}.",
"The Bloch functions are written as a product of the plane wave 𝓆 Γ (R j Mo )\\mathcal {q}_{\\mathbf {\\mathrm {\\Gamma }}} ({\\mathbf {\\mathrm {R}}}_j^{\\textrm {Mo}}) and dd orbitals for Mo atoms and the plane waves 𝓆 Γ ± (R j S )=𝓆 Γ (R j S1 )±𝓆 Γ (R j S2 )\\mathcal {q}_{\\mathbf {\\mathrm {\\Gamma }}}^\\pm ({\\mathbf {\\mathrm {R}}}_j^\\textrm {S}) = \\mathcal {q}_{\\mathbf {\\mathrm {\\Gamma }}} ({\\mathbf {\\mathrm {R}}}_j^{\\textrm {S}1}) \\pm \\mathcal {q}_{\\mathbf {\\mathrm {\\Gamma }}} ({\\mathbf {\\mathrm {R}}}_j^{\\textrm {S}2}) and pp orbitals for S atoms.",
"Also, 𝓆 Γ (R j W ){d μ ,d ν }\\mathcal {q}_{\\mathbf {\\mathrm {\\Gamma }}} ({\\mathbf {\\mathrm {R}}}_j^W) \\, \\lbrace d_\\mu , d_\\nu \\rbrace is a short-hand notation for the pair of Bloch functions {𝓆 Γ (R j W )d μ ,𝓆 Γ (R j W )d ν }\\lbrace \\mathcal {q}_{\\mathbf {\\mathrm {\\Gamma }}} ({\\mathbf {\\mathrm {R}}}_j^W) \\, d_\\mu , \\mathcal {q}_{\\mathbf {\\mathrm {\\Gamma }}} ({\\mathbf {\\mathrm {R}}}_j^W) \\, d_\\nu \\rbrace .",
"The last column indicates the degeneracy of Bloch states due to time-reversal symmetry discussed in Sec.",
".Table: Symmetry-adapted TB Bloch functions in MoS 2 _2 at k=K,K ' {\\mathbf {\\mathrm {k}}} = {\\mathbf {\\mathrm {K}}}, {\\mathbf {\\mathrm {K}}}^{\\prime } with group of the wave vector 𝒢 K =𝒢 K ' =C 3h \\mathcal {G}_{{\\mathbf {\\mathrm {K}}}} = \\mathcal {G}_{{\\mathbf {\\mathrm {K}}}^{\\prime }} = C_{3h}.",
"The Bloch functions are written as a product of the plane wave 𝓆 k (R j Mo )\\mathcal {q}_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{\\textrm {Mo}}) and dd orbitals for Mo atoms and the plane waves 𝓆 k ± (R j S )=𝓆 k (R j S1 )±𝓆 k (R j S2 )\\mathcal {q}_{\\mathbf {\\mathrm {k}}}^\\pm ({\\mathbf {\\mathrm {R}}}_j^\\textrm {S}) = \\mathcal {q}_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{\\textrm {S}1}) \\pm \\mathcal {q}_{\\mathbf {\\mathrm {k}}} ({\\mathbf {\\mathrm {R}}}_j^{\\textrm {S}2}) and pp orbitals for S atoms.",
"The IRs Γ i/j/k \\Gamma _{i/j/k} correspond to the coordinate system α=a/b/c\\alpha = a/b/c in Fig.",
".Table: Symmetry-adapted TB Bloch functions in MoS 2 _2 at k=M{\\mathbf {\\mathrm {k}}} = {\\mathbf {\\mathrm {M}}} with group of the wave vector 𝒢 M =C 2v \\mathcal {G}_{{\\mathbf {\\mathrm {M}}}} = C_{2v}.",
"The Bloch functions are written as a product of the plane wave 𝓆 M (R j Mo )\\mathcal {q}_{\\mathbf {\\mathrm {M}}} ({\\mathbf {\\mathrm {R}}}_j^{\\textrm {Mo}}) and dd orbitals for Mo atoms and the plane waves 𝓆 M ± (R j S )=𝓆 M (R j S1 )±𝓆 M (R j S2 )\\mathcal {q}_{\\mathbf {\\mathrm {M}}}^\\pm ({\\mathbf {\\mathrm {R}}}_j^\\textrm {S}) = \\mathcal {q}_{\\mathbf {\\mathrm {M}}} ({\\mathbf {\\mathrm {R}}}_j^{\\textrm {S}1}) \\pm \\mathcal {q}_{\\mathbf {\\mathrm {M}}} ({\\mathbf {\\mathrm {R}}}_j^{\\textrm {S}2}) and pp orbitals for S atoms."
],
[
"Band Symmetries in Few-Layer Graphene",
"We can also apply the general formalism in Sec.",
"to identify the band symmetries in other quasi-2D materials such as SLG, BLG, and TLG."
],
[
"Crystal structure of few-layer graphene",
"Like the crystal structure of monolayer MoS$_2$ , the crystal structures of SLG, BLG, and TLG belong to the hexagonal crystal system.",
"Therefore, we use the same expressions for the primitive lattice vectors [Eq.",
"(REF )] and reciprocal lattice vectors [Eq.",
"(REF )]; and we have the same high-symmetry points in the BZ denoted ${\\mathbf {\\mathrm {\\Gamma }}}$ , ${\\mathbf {\\mathrm {K}}}$ [Eq.",
"(REF )], and ${\\mathbf {\\mathrm {M}}}$ [Eq.",
"(REF )].",
"The space groups for few-layer graphene are listed in Table REF .",
"This table also contains the site symmetries of the high-symmetry points for these crystal structures.",
"Figure REF shows the crystal structure of SLG.",
"It is characterized by the point group $D_{6h}$ (space group $P6/mmm$ , # 191).",
"The carbon atoms form two distinct Bravais lattices denoted as sublattices $A$ and $B$ .",
"The atomic positions denoted by $\\lbrace {\\mathbf {\\mathrm {R}}}_j^{c1},{\\mathbf {\\mathrm {R}}}_j^{c2}\\rbrace $ have site symmetries characterized by the point group $D_{3h}$ and Wyckoff letter $c$ .",
"The center of the hexagon, characterized by site symmetry $D_{6h}$ is the only Wyckoff position with multiplicity $m=1$ .",
"This point is the origin of the coordinate system for this crystal structure.",
"The positions of the C atoms in the unit cell are given by ${\\mathbf {\\mathrm {t}}}_{c_1} = \\frac{a}{2}\\left( \\begin{array}{s{0.15em}cs{0.15em}}1 \\\\ -\\frac{1}{\\sqrt{3}}\\end{array} \\right), \\quad {\\mathbf {\\mathrm {t}}}_{c_2} = \\frac{a}{2}\\left( \\begin{array}{s{0.15em}cs{0.15em}}1 \\\\ \\frac{1}{\\sqrt{3}}\\end{array} \\right).$ Figure: (a) Crystal structure of single-layer graphene characterized by the point group D 6h D_{6h}.",
"The shaded region shows a unit cell (j=1j=1).",
"The C atoms are located at Wyckoff position cc with multiplicity m=2m=2, hence the label c1c1 and c2c2 on one unit cell.",
"The primitive lattice vectors are denoted a 1 {\\mathbf {\\mathrm {a}}}_1 and a 2 {\\mathbf {\\mathrm {a}}}_2.",
"The positions of C atoms in unit cell jj are denoted by R j c1 {\\mathbf {\\mathrm {R}}}_j^{c1} and R j c2 {\\mathbf {\\mathrm {R}}}_j^{c2}.",
"The vectors t c1 {\\mathbf {\\mathrm {t}}}_{c1} and t c2 {\\mathbf {\\mathrm {t}}}_{c2} give the positions of C atoms in the unit cell.",
"(b) The first Brillouin zone with primitive reciprocal lattice vectors b 1 {\\mathbf {\\mathrm {b}}}_1 and b 2 {\\mathbf {\\mathrm {b}}}_2.The point group $D_{6h}$ of the crystal, which also characterizes the ${\\mathbf {\\mathrm {\\Gamma }}}$ point of the BZ, contains twofold, threefold and sixfold rotations $C_2$ , $C_3$ and $C_6$ where the $z$ axis is the rotation axis.",
"The rotation axes of the three twofold rotation $C_2^{\\prime (i)}$ and $C_2^{\\prime \\prime (jk)}$ are the corresponding dashed lines shown in Fig.",
"REF (a).",
"The reflection planes for $\\sigma _v^{(i)}$ and $\\sigma _d^{(ij)}$ are perpendicular to the $xy$ plane passing through the corresponding dashed lines.",
"The reflection $\\sigma _h$ is along the $xy$ plane and $S_n = C_n \\sigma _h$ .",
"At the ${\\mathbf {\\mathrm {K}}}$ point, the group of the wave vector is $\\mathcal {G}_{{\\mathbf {\\mathrm {K}}}} = D_{3h}$ containing three twofold rotations $C_2^{\\prime (ij)}$ about the corresponding dashed axes in Fig.",
"REF (b).",
"The reflection plane of $\\sigma _v^{(ij)}$ is perpendicular to the $xy$ plane along the rotation axis of $C_2^{\\prime (ij)}$ .",
"The threefold rotation axis is the $z$ axis, and $\\sigma _h$ is a reflection about the $xy$ plane.",
"The group of the wave vector at the ${\\mathbf {\\mathrm {M}}}_i$ points is $\\mathcal {G}_{{\\mathbf {\\mathrm {M}}}} = D_{2h}$ containing the symmetry operations $C_2$ , $C_2^{\\prime }$ , $C_2^{\\prime \\prime }$ , $\\sigma _v$ , $\\sigma _v^{\\prime }$ , and $\\sigma _v^{\\prime \\prime }$ with rotation axes and reflection planes shown in Figs.",
"REF (c)–REF (e).",
"The character table for $D_{2h}$ is reproduced in Table REF (Appendix ).",
"Figure: Groups of the wave vector in SLG.",
"(a) The point Γ{\\mathbf {\\mathrm {\\Gamma }}} has the point group D 6h D_{6h} with the zz axis (out of plane) as the axis for the nn-fold rotations C n C_n (n=2,3,6n = 2, 3, 6) and the xyxy plane as the reflection plane for σ h \\sigma _h.",
"The dashed lines (ii and ijij) are the axes for twofold rotations C 2 '(i) C_2^{\\prime (i)} and C 2 ''(ij) C_2^{\\prime \\prime (ij)} with i,j=1,2,3i,j = 1, 2, 3.",
"The reflection σ v (i) \\sigma _v^{(i)} [σ d (ij) \\sigma _d^{(ij)}] is about a plane that includes the corresponding dashed axis and the zz axis.",
"(b) The points K{\\mathbf {\\mathrm {K}}} and K ' {\\mathbf {\\mathrm {K}}}^{\\prime } have the point group D 3h D_{3h} with threefold rotations about the zz axis.",
"The dashed lines are the axes for the twofold rotations C 2 '(ij) C_2^{\\prime (ij)}.",
"The reflection plane of σ v (ij) \\sigma _v^{(ij)} contains the corresponding dashed lines and the zz axis.",
"The reflection plane of σ h \\sigma _h is the xyxy plane.",
"The dotted lines indicate the twofold rotation axes that appear in the point group D 6h D_{6h} but are not symmetry elements of D 3h D_{3h}.",
"[(c)–(e)] The points M 1 {\\mathbf {\\mathrm {M}}}_1, M 2 {\\mathbf {\\mathrm {M}}}_2, and M 1 {\\mathbf {\\mathrm {M}}}_1 have the point group D 2h D_{2h}.",
"The rotation axis of C 2 C_2 is the zz axis and the reflection plane of σ v \\sigma _v is the xyxy plane.",
"The dashed lines are the axes of the twofold rotations C 2 ' C_2^{\\prime } and C 2 '' C_2^{\\prime \\prime }.",
"The reflection planes of σ v ' \\sigma _v^{\\prime } and σ v '' \\sigma _v^{\\prime \\prime } contain the corresponding dashed line and the zz axis."
],
[
"Bilayer graphene",
"The point group $D_{3d}$ (space group $P\\bar{3}m1$ , # 164) characterizes BLG as shown in Fig.",
"REF .",
"The only Wyckoff position with multiplicity $m=1$ is the midpoint of two C atoms on top of each other.",
"We use this point as the origin of the coordinate system.",
"The atomic positions in BLG are the Wyckoff positions $c$ and $d$ , each with multiplicity $m=2$ and site symmetry $C_{3v}$ .",
"The two atoms in one Wyckoff letter are labeled $\\mu =1$ ($\\mu = 2$ ) for the atom in the top (bottom) layer.",
"The layers are arranged in an $AB$ stacking, so that the atomic position ${\\mathbf {\\mathrm {R}}}_j^{c1}$ is located on top of ${\\mathbf {\\mathrm {R}}}_j^{c2}$ .",
"Ignoring the $z$ component, the positions of the C atoms in the unit cell are ${\\mathbf {\\mathrm {t}}}_{c \\mu } = \\frac{a}{2}\\left( \\begin{array}{s{0.15em}cs{0.15em}} 0 \\\\ 0 \\end{array} \\right), \\quad {\\mathbf {\\mathrm {t}}}_{d 1} = \\frac{a}{2}\\left( \\begin{array}{s{0.15em}cs{0.15em}}0 \\\\ \\frac{2}{\\sqrt{3}}\\end{array} \\right), \\quad {\\mathbf {\\mathrm {t}}}_{d 2} = \\frac{a}{2}\\left( \\begin{array}{s{0.15em}cs{0.15em}}1 \\\\ \\frac{1}{\\sqrt{3}}\\end{array} \\right) .$ The threefold proper and sixfold improper rotation axis is the $z$ axis.",
"The axis of the twofold rotation $C_2^{\\prime (ij)}$ is the corresponding dashed line in Fig.",
"REF (a).",
"The three reflection planes corresponding to $\\sigma _d^{(i)}$ are perpendicular to the $xy$ plane passing through the corresponding dashed line.",
"The group of the wave vector is $\\mathcal {G}_{{\\mathbf {\\mathrm {K}}}} = D_3$ at the ${\\mathbf {\\mathrm {K}}}$ point with threefold rotations about the $z$ axis and three twofold rotations $C_2^{\\prime (jk)}$ about the corresponding dashed line in Fig.",
"REF (b).",
"At the ${\\mathbf {\\mathrm {M}}}_i$ points, the group of the wave vector is $\\mathcal {G}_{{\\mathbf {\\mathrm {M}}}} = C_{2h}$ , where $\\sigma _h$ is perpendicular to the $xy$ plane passing through the corresponding dashed line in Figs.",
"REF (c)–REF (e), and the twofold rotation $C_2$ is about the corresponding dashed line.",
"The character table for $C_{2h}$ is reproduced in Table REF (Appendix ).",
"Figure: (a) Crystal structure of bilayer graphene characterized by the point group D 3d D_{3d}.",
"The shaded region shows a unit cell (j=1j=1).",
"The C atoms are located on Wyckoff positions cc and dd each with multiplicity m=2m=2.",
"The atomic position in unit cell jj are denoted by R j cμ {\\mathbf {\\mathrm {R}}}_j^{c \\mu } and R j dμ {\\mathbf {\\mathrm {R}}}_j^{d \\mu } with μ=1\\mu =1 (μ=2\\mu = 2) corresponding to the atom at the top (bottom) layer.",
"The vectors t cμ {\\mathbf {\\mathrm {t}}}_{c \\mu } and t dμ {\\mathbf {\\mathrm {t}}}_{d \\mu } give the positions of the C atoms within a unit cell.",
"The primitive lattice vectors are denoted by a 1 {\\mathbf {\\mathrm {a}}}_1 and a 2 {\\mathbf {\\mathrm {a}}}_2.",
"(b) The first Brillouin zone with primitive reciprocal lattice vectors b 1 {\\mathbf {\\mathrm {b}}}_1 and b 2 {\\mathbf {\\mathrm {b}}}_2.Figure: Groups of the wave vector in BLG.",
"(a) The point Γ{\\mathbf {\\mathrm {\\Gamma }}} has the point group D 3d D_{3d} with the zz axis (out of plane) as the axis for the threefold proper rotation C 3 C_3 and the sixfold improper rotation S 6 S_6.",
"The green dashed lines ijij are the axes for twofold rotations C 2 ''(ij) C_2^{\\prime \\prime (ij)} with i,j=1,2,3i,j = 1, 2, 3.",
"The reflection σ d (i) \\sigma _d^{(i)} is about a plane that includes the corresponding blue dashed axis and the zz axis.",
"(b) The points K{\\mathbf {\\mathrm {K}}} and K ' {\\mathbf {\\mathrm {K}}}^{\\prime } have the point group D 3 D_3 with threefold rotations about the zz axis.",
"The dashed lines are the axes for the twofold rotations C 2 '(ij) C_2^{\\prime (ij)}.",
"The dotted lines indicate the reflection planes that appear in the point group D 3d D_{3d} but are not symmetry elements of D 3 D_3.",
"[(c)–(e)] The points M 1 {\\mathbf {\\mathrm {M}}}_1, M 2 {\\mathbf {\\mathrm {M}}}_2, and M 1 {\\mathbf {\\mathrm {M}}}_1 have the point group C 2h C_{2h}.The green dashed line is the axis of the twofold rotation C 2 C_2.",
"The reflection plane of σ h \\sigma _h contains the blue dashed line and the zz axis."
],
[
"Trilayer graphene",
"Last, Fig.",
"REF shows the crystal structure of TLG.",
"This system has the same space group $P \\bar{6} m 2$ , # 187 as monolayer MoS$_2$ (point group $D_{3h}$ ).",
"We designate the Wyckoff positions of the carbon atoms in the middle layer as $A$ and $B$ with site symmetry group $D_{3h}$ , and the remaining positions as $A^{\\prime }\\mu $ and $B^{\\prime }\\mu $ with $\\mu =1$ ($\\mu =2$ ) for the top (bottom) layer with site symmetry group $C_{3v}$ .",
"The points $A$ and $B$ as well as the center of the hexagon in the middle layer are Wyckoff positions with multiplicity $m=1$ , which we use as the origin of the three coordinate systems defined in Fig.",
"REF .",
"We therefore associate with each Wyckoff letter $W$ an index $(\\alpha )$ corresponding to the coordinate systems $\\alpha = a, b, c$ .",
"Ignoring the $z$ component, the positions of the C atoms in the unit cell for the three coordinate systems are ${\\mathbf {\\mathrm {t}}}_A^a & = \\frac{a}{2} \\left( \\begin{array}{s{0.15em}cs{0.15em}} 1 \\\\ -\\frac{1}{\\sqrt{3}} \\end{array} \\right) , &{\\mathbf {\\mathrm {t}}}_{A^{\\prime } \\mu }^a & = \\frac{a}{2} \\left( \\begin{array}{s{0.15em}cs{0.15em}} 0 \\\\ 0 \\end{array} \\right), \\\\{\\mathbf {\\mathrm {t}}}_B^a & = \\frac{a}{2} \\left( \\begin{array}{s{0.15em}cs{0.15em}} 1 \\\\ \\frac{1}{\\sqrt{3}} \\end{array} \\right), &{\\mathbf {\\mathrm {t}}}_{B^{\\prime } \\mu }^a & = \\frac{a}{2}\\left( \\begin{array}{s{0.15em}cs{0.15em}} 1 \\\\ -\\frac{1}{\\sqrt{3}} \\end{array} \\right) ,\\\\{\\mathbf {\\mathrm {t}}}_A^b & = \\frac{a}{2} \\left( \\begin{array}{s{0.15em}cs{0.15em}} 0 \\\\ -\\frac{2}{\\sqrt{3}} \\end{array} \\right), &{\\mathbf {\\mathrm {t}}}_{A^{\\prime } \\mu }^b & = \\frac{a}{2} \\left( \\begin{array}{s{0.15em}cs{0.15em}} -1 \\\\ -\\frac{1}{\\sqrt{3}} \\end{array} \\right), \\\\{\\mathbf {\\mathrm {t}}}_B^b & = \\frac{a}{2} \\left( \\begin{array}{s{0.15em}cs{0.15em}} 0 \\\\ 0 \\end{array} \\right), &{\\mathbf {\\mathrm {t}}}_{B^{\\prime } \\mu }^b & = \\frac{a}{2} \\left( \\begin{array}{s{0.15em}cs{0.15em}} 0 \\\\ -\\frac{2}{\\sqrt{3}} \\end{array} \\right), \\\\{\\mathbf {\\mathrm {t}}}_A^c & = \\frac{a}{2} \\left( \\begin{array}{s{0.15em}cs{0.15em}} 0 \\\\ 0 \\end{array} \\right), &{\\mathbf {\\mathrm {t}}}_{A^{\\prime } \\mu }^c & = \\frac{a}{2} \\left( \\begin{array}{s{0.15em}cs{0.15em}} -1 \\\\ \\frac{1}{\\sqrt{3}} \\end{array} \\right), \\\\{\\mathbf {\\mathrm {t}}}_B^c & = \\frac{a}{2} \\left( \\begin{array}{s{0.15em}cs{0.15em}} 0 \\\\ \\frac{2}{\\sqrt{3}} \\end{array} \\right), &{\\mathbf {\\mathrm {t}}}_{B^{\\prime } \\mu }^c & = \\frac{a}{2} \\left( \\begin{array}{s{0.15em}cs{0.15em}} 0 \\\\ 0 \\end{array} \\right),$ where the superscripts $\\alpha = a, b, c$ denote the coordinate system.",
"The positions of the six atoms in unit cell $j$ are denoted by ${\\mathbf {\\mathrm {R}}}_j^A$ , ${\\mathbf {\\mathrm {R}}}_j^B$ , ${\\mathbf {\\mathrm {R}}}_j^{A^{\\prime }\\mu }$ and ${\\mathbf {\\mathrm {R}}}_j^{B^{\\prime }\\mu }$ .",
"In standard notation [58], the positions $A$ , $B$ , $A^{\\prime }\\mu $ and $B^{\\prime }\\mu $ are characterized by the Wyckoff letters $c$ , $e$ , $g$ , and $h$ respectively in coordinate system $(a)$ , by $e$ , $a$ , $h$ , and $i$ in coordinate system $(b)$ , and by $a$ , $c$ , $i$ , and $g$ in coordinate system $(c)$ , see Table REF .",
"Figure: Crystal structure of trilayer graphene.",
"Three coordinate systems α=a,b,c\\alpha = a, b, c are considered with (a) the origin located at the atom AA, (b) origin at atom BB, and (c) origin at the midpoint between B ' 1B^{\\prime }1 and B ' 2B^{\\prime }2.",
"The dashed axes (1)(1), (2)(2), and (3)(3) are the twofold rotation axes of the point group D 3h D_{3h}.",
"The shaded region shows a unit cell (j=1j=1).",
"The vectors t A α {\\mathbf {\\mathrm {t}}}_{A}^\\alpha , t B α {\\mathbf {\\mathrm {t}}}_{B}^\\alpha , t A ' α {\\mathbf {\\mathrm {t}}}_{A^{\\prime }}^\\alpha , and t B ' α {\\mathbf {\\mathrm {t}}}_{B^{\\prime }}^\\alpha give the positions of the C atoms labeled AA, BB, A ' μA^{\\prime }\\mu , and B ' μB^{\\prime }\\mu within a unit cell, respectively.",
"For the A ' A^{\\prime } and B ' B^{\\prime } atom, the top (bottom) atoms are labeled μ=1\\mu =1 (μ=2\\mu = 2).",
"The positions of these atoms in unit cell jj are denoted by R j A(α) {\\mathbf {\\mathrm {R}}}_j^{A(\\alpha )}, R j B(α) {\\mathbf {\\mathrm {R}}}_j^{B (\\alpha )}, R j A ' μ(α) {\\mathbf {\\mathrm {R}}}_j^{A^{\\prime } \\mu (\\alpha )}, and R j B ' μ(α) {\\mathbf {\\mathrm {R}}}_j^{B^{\\prime } \\mu (\\alpha )}, respectively.",
"(d) The first Brillouin zone.The point group $D_{3h}$ of the crystal structure of TLG is the same as for MoS$_2$ with coordinate system shown in Fig.",
"REF (a).",
"The group of the wave vector at ${\\mathbf {\\mathrm {K}}}$ is $\\mathcal {G}_{{\\mathbf {\\mathrm {K}}}} = C_{3h}$ with coordinate system shown in Fig.",
"REF (b).",
"At the three points ${\\mathbf {\\mathrm {M}}}_i$ the group of the wave vector is $\\mathcal {G}_{{\\mathbf {\\mathrm {M}}}} = C_{2v}$ with coordinate system shown in Figs.",
"REF (c)–REF (e), which contains the twofold rotation $C_2$ about the dashed line, reflection $\\sigma _v^{\\prime }$ about the dashed line and the $z$ axis, and reflection $\\sigma _v$ about the $xy$ plane."
],
[
"IRs of Bloch states in graphene",
"In graphene, the bands near the Fermi level are dominated by the $p$ orbitals of the C atoms.",
"For SLG, using the coordinate systems in Fig.",
"REF , the IRs of the symmetry-adapted $p$ orbitals are listed in Table REF for the points ${\\mathbf {\\mathrm {\\Gamma }}}$ , ${\\mathbf {\\mathrm {K}}}$ , and ${\\mathbf {\\mathrm {M}}}$ with groups of the wave vector $D_{6h}$ , $D_{3h}$ , and $D_{2h}$ , respectively.",
"For BLG, Fig.",
"REF shows the coordinate systems used for the points ${\\mathbf {\\mathrm {\\Gamma }}}$ , ${\\mathbf {\\mathrm {K}}}$ , and ${\\mathbf {\\mathrm {M}}}$ with group of the wave vector $D_{3d}$ , $D_3$ , and $C_{2h}$ , respectively.",
"The IRs of the $p$ orbitals at these points are listed in Table REF .",
"The coordinate systems used for TLG are the same as the ones for MoS$_2$ (Fig.",
"REF ), so that the IRs of the $p$ orbitals can be taken from Table REF .",
"The symmetry-adapted plane waves with the corresponding IRs are summarized in Table REF .",
"Table: Symmetry-adapted pp orbitals for the point groups D 6h D_{6h}, D 3h D_{3h}, and D 2h D_{2h} for the coordinate systems shown in Fig.",
"(SLG) and D 3d D_{3d}, D 3 D_3, and C 2h C_{2h} for the coordinate system shown in Fig.",
"(BLG).",
"For D 2h D_{2h} and C 2h C_{2h}, the symmetrized atomic orbital φ ν [i] \\phi _\\nu ^{[i]} corresponds to the coordinate systems used for the point M i {\\mathbf {\\mathrm {M}}}_i in Figs.",
"(c)–(e) and (c)–(e), respectively.",
"The orbital [i(j)][i(j)] takes the upper (lower) sign.The IRs of the full Bloch functions for SLG, BLG, and TLG are listed in Table REF .",
"The sets of Bloch states transforming as an IR in $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ for the wave vectors ${\\mathbf {\\mathrm {k}}} = {\\mathbf {\\mathrm {\\Gamma }}} , {\\mathbf {\\mathrm {K}}}, {\\mathbf {\\mathrm {M}}}$ are listed in Tables REF , REF , and REF for SLG, Tables REF , REF , and REF for BLG, and Tables REF , REF , and REF for TLG.",
"Table: IRs of the plane waves (Γ k q \\Gamma ^q_{\\mathbf {\\mathrm {k}}}), the atomic orbitals φ ν \\phi _\\nu (Γ k φ \\Gamma ^\\phi _{\\mathbf {\\mathrm {k}}}), and the full Bloch functions (Γ k Φ =Γ k q ×Γ k φ \\Gamma ^\\Phi _{\\mathbf {\\mathrm {k}}} = \\Gamma ^q_{\\mathbf {\\mathrm {k}}} \\times \\Gamma ^\\phi _{\\mathbf {\\mathrm {k}}}) for SLG, BLG, and TLG at the points k=Γ{\\mathbf {\\mathrm {k}}} = {\\mathbf {\\mathrm {\\Gamma }}}, K{\\mathbf {\\mathrm {K}}}, and M i {\\mathbf {\\mathrm {M}}}_i.",
"The positions of C atoms in SLG are characterized by the Wyckoff letter cc with multiplicity m=2m=2.",
"For BLG, the atomic positions have Wyckoff letters cc and dd each with multiplicity m=2m=2.",
"For TLG, the Wyckoff letters of the atomic positions are denoted AA and BB with multiplicity m=1m=1, and A ' A^{\\prime } and B ' B^{\\prime } with multiplicity m=2m=2.",
"For TLG at K{\\mathbf {\\mathrm {K}}}, we need to distinguish between the three coordinate systems α=a,b,c\\alpha = a, b ,c .",
"The IRs of 𝒢 K \\mathcal {G}_{{\\mathbf {\\mathrm {K}}}} and 𝒢 K ' \\mathcal {G}_{{\\mathbf {\\mathrm {K}}}^{\\prime }} in SLG and BLG are real, so that Γ K q =Γ K ' q \\Gamma ^q_{\\mathbf {\\mathrm {K}}} = \\Gamma ^q_{{\\mathbf {\\mathrm {K}}}^{\\prime }} and Γ K φ =Γ K ' φ \\Gamma ^\\phi _{\\mathbf {\\mathrm {K}}} = \\Gamma ^\\phi _{{\\mathbf {\\mathrm {K}}}^{\\prime }}, whereas in TLG the IR Γ K ' (i/j/k) q \\Gamma ^q_{{\\mathbf {\\mathrm {K}}}^{\\prime } (i/j/k)} is the complex conjugate of Γ K(i/j/k) q \\Gamma ^q_{{\\mathbf {\\mathrm {K}}} (i/j/k)}.",
"At the points M i {\\mathbf {\\mathrm {M}}}_i (i=1,2,3i=1,2,3), the atomic orbitals are denoted by φ ν [i] \\phi _\\nu ^{[i]}.Table: Symmetry-adapted TB Bloch functions in SLG at k=Γ{\\mathbf {\\mathrm {k}}} = {\\mathbf {\\mathrm {\\Gamma }}} with group of the wave vector 𝒢 Γ =D 6h \\mathcal {G}_{{\\mathbf {\\mathrm {\\Gamma }}}} = D_{6h}.",
"The C atoms are located at Wyckoff position cc of multiplicity 2.",
"The Bloch functions are written as a product of the plane waves 𝓆 Γ ± (R j c )=𝓆 Γ (R j c1 )±𝓆 Γ (R j c2 )\\mathcal {q}_{\\mathbf {\\mathrm {\\Gamma }}}^\\pm ({\\mathbf {\\mathrm {R}}}_j^c) = \\mathcal {q}_{\\mathbf {\\mathrm {\\Gamma }}} ({\\mathbf {\\mathrm {R}}}_j^{c1}) \\pm \\mathcal {q}_{\\mathbf {\\mathrm {\\Gamma }}} ({\\mathbf {\\mathrm {R}}}_j^{c2}) and the pp orbitals of the C atoms.",
"Also, 𝓆 Γ (R j W ){p x ,p y }\\mathcal {q}_{\\mathbf {\\mathrm {\\Gamma }}} ({\\mathbf {\\mathrm {R}}}_j^W) \\, \\lbrace p_x , p_y \\rbrace is a short-hand notation for the pair of Bloch functions {𝓆 Γ (R j W )p x ,𝓆 Γ (R j W )p y }\\lbrace \\mathcal {q}_{\\mathbf {\\mathrm {\\Gamma }}} ({\\mathbf {\\mathrm {R}}}_j^W) \\, p_x , \\mathcal {q}_{\\mathbf {\\mathrm {\\Gamma }}} ({\\mathbf {\\mathrm {R}}}_j^W) \\, p_y \\rbrace .",
"The last column indicates the degeneracy of Bloch states due to time-reversal symmetry discussed in Sec.",
".Table: Symmetry-adapted TB Bloch functions in SLG at k=K{\\mathbf {\\mathrm {k}}} = {\\mathbf {\\mathrm {K}}} with group of the wave vector 𝒢 K =D 3h \\mathcal {G}_{{\\mathbf {\\mathrm {K}}}} = D_{3h}.",
"The same form of the symmetry-adapted TB Bloch functions and corresponding IRs work for k=K ' {\\mathbf {\\mathrm {k}}} = {\\mathbf {\\mathrm {K}}}^{\\prime }, that is we replace K{\\mathbf {\\mathrm {K}}} with K ' {\\mathbf {\\mathrm {K}}}^{\\prime }.Table: Symmetry-adapted TB Bloch functions in SLG at k=M{\\mathbf {\\mathrm {k}}} = {\\mathbf {\\mathrm {M}}} with group of the wave vector 𝒢 M =D 2h \\mathcal {G}_{{\\mathbf {\\mathrm {M}}}} = D_{2h}.Table: Symmetry-adapted TB Bloch functions in BLG at k=Γ{\\mathbf {\\mathrm {k}}} = {\\mathbf {\\mathrm {\\Gamma }}} with group of the wave vector 𝒢 Γ =D 3d \\mathcal {G}_{{\\mathbf {\\mathrm {\\Gamma }}}} = D_{3d}.",
"The C atoms are located at Wyckoff positions W=c,dW = c, d each of multiplicity 2. 𝓆 Γ ± (R j W ){p x ,p y }\\mathcal {q}_{\\mathbf {\\mathrm {\\Gamma }}}^\\pm ({\\mathbf {\\mathrm {R}}}_j^W) \\, \\lbrace p_x , p_y \\rbrace is a short-hand notation for the pair of Bloch functions {𝓆 Γ ± (R j W )p x ,𝓆 Γ ± (R j W )p y }\\lbrace \\mathcal {q}_{\\mathbf {\\mathrm {\\Gamma }}}^\\pm ({\\mathbf {\\mathrm {R}}}_j^W) \\, p_x , \\mathcal {q}_{\\mathbf {\\mathrm {\\Gamma }}}^\\pm ({\\mathbf {\\mathrm {R}}}_j^W) \\, p_y \\rbrace .Table: Symmetry-adapted TB Bloch functions in BLG at k=K{\\mathbf {\\mathrm {k}}} = {\\mathbf {\\mathrm {K}}} with group of the wave vector 𝒢 K =D 3 \\mathcal {G}_{{\\mathbf {\\mathrm {K}}}} = D_3.",
"𝓆 K ± (R j W ){p x ,p y }\\mathcal {q}_{\\mathbf {\\mathrm {K}}}^\\pm ({\\mathbf {\\mathrm {R}}}_j^W) \\, \\lbrace p_x , p_y \\rbrace is a short-hand notation for the pair of Bloch functions {𝓆 K ± (R j W )p x ,𝓆 K ± (R j W )p y }\\lbrace \\mathcal {q}_{\\mathbf {\\mathrm {K}}}^\\pm ({\\mathbf {\\mathrm {R}}}_j^W) \\, p_x , \\mathcal {q}_{\\mathbf {\\mathrm {K}}}^\\pm ({\\mathbf {\\mathrm {R}}}_j^W) \\, p_y \\rbrace .",
"The same form of the symmetry-adapted TB Bloch functions and corresponding IRs work for k=K ' {\\mathbf {\\mathrm {k}}} = {\\mathbf {\\mathrm {K}}}^{\\prime }, that is we replace K{\\mathbf {\\mathrm {K}}} with K ' {\\mathbf {\\mathrm {K}}}^{\\prime }.Table: Symmetry-adapted TB Bloch functions in BLG at k=M{\\mathbf {\\mathrm {k}}} = {\\mathbf {\\mathrm {M}}} with group of the wave vector 𝒢 M =C 2h \\mathcal {G}_{{\\mathbf {\\mathrm {M}}}} = C_{2h}.Table: Symmetry-adapted TB Bloch functions in TLG at k=Γ{\\mathbf {\\mathrm {k}}} = {\\mathbf {\\mathrm {\\Gamma }}} with group of the wave vector 𝒢 Γ =D 3h \\mathcal {G}_{{\\mathbf {\\mathrm {\\Gamma }}}} = D_{3h}.",
"The C atoms are located at Wyckoff positions W=A,BW = A, B (W=A ' ,B ' )(W = A^{\\prime }, B^{\\prime }) of multiplicity 1 (2).",
"𝓆 Γ (R j W ){p x ,p y }\\mathcal {q}_{\\mathbf {\\mathrm {\\Gamma }}} ({\\mathbf {\\mathrm {R}}}_j^W) \\, \\lbrace p_x , p_y \\rbrace is a short-hand notation for the pair of Bloch functions {𝓆 Γ (R j W )p x ,𝓆 Γ (R j W )p y }\\lbrace \\mathcal {q}_{\\mathbf {\\mathrm {\\Gamma }}} ({\\mathbf {\\mathrm {R}}}_j^W) \\, p_x , \\mathcal {q}_{\\mathbf {\\mathrm {\\Gamma }}} ({\\mathbf {\\mathrm {R}}}_j^W) \\, p_y \\rbrace .Table: Symmetry-adapted TB Bloch functions in TLG at k=K,K ' {\\mathbf {\\mathrm {k}}} = {\\mathbf {\\mathrm {K}}}, {\\mathbf {\\mathrm {K}}}^{\\prime } with group of the wave vector 𝒢 K =𝒢 K ' =C 3h \\mathcal {G}_{{\\mathbf {\\mathrm {K}}}} = \\mathcal {G}_{{\\mathbf {\\mathrm {K}}}^{\\prime }} = C_{3h}.",
"The IRs Γ i/j/k \\Gamma _{i/j/k} correspond to the coordinate system α=a/b/c\\alpha = a/b/c in Fig. .",
"The IRs at K ' {\\mathbf {\\mathrm {K}}}^{\\prime } are the complex conjugates of the IRs at K{\\mathbf {\\mathrm {K}}}.Table: Symmetry-adapted TB Bloch functions in TLG at k=M{\\mathbf {\\mathrm {k}}} = {\\mathbf {\\mathrm {M}}} with group of the wave vector 𝒢 M =C 2v \\mathcal {G}_{{\\mathbf {\\mathrm {M}}}} = C_{2v}."
],
[
"Selection Rules: Effect of Band IR Rearrangement",
"We show in the following that the selection rules for the observable matrix elements of a Hermitian operator $\\mathcal {O}$ taken between Bloch states are not affected by the rearrangement of band IRs discussed in Sec.",
"REF provided the perturbation of a crystal represented by the operator $\\mathcal {O}$ preserves translational invariance.",
"This condition for the operator $\\mathcal {O}$ is certainly obeyed by the dipole operator $\\mathcal {O} = {\\mathbf {\\mathrm {r}}}$ representing optical transitions, but it does not apply to localized perturbations such as point defects for which anyway the specific location of the defect plays a crucial role [3].",
"We consider a translation ${\\mathbf {\\mathrm {\\tau }}}$ of the coordinate system as introduced in Sec.",
"REF , so that $T= \\lbrace \\rbrace $ is a unitary operator with $T^\\dagger = T^{-1}$ .",
"This transforms both the states and operators.",
"A state $| {\\psi _{J \\beta }} \\rangle $ in the old coordinate system transforming according to the IR $\\Gamma _J$ becomes $| {\\psi _{J^{\\prime } \\beta ^{\\prime }}} \\rangle = T| {\\psi _{J \\beta }} \\rangle $ transforming according to $\\Gamma _{J^{\\prime }} = \\Gamma _{\\mathbf {\\mathrm {\\tau }}}\\times \\Gamma _J$ , with $\\Gamma _{\\mathbf {\\mathrm {\\tau }}}= \\lbrace \\mathcal {D}_{\\mathbf {\\mathrm {\\tau }}}^{\\mathbf {\\mathrm {k}}} (g) = \\exp (-i {\\mathbf {\\mathrm {b}}}_g \\cdot {\\mathbf {\\mathrm {\\tau }}}) : g \\in \\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}\\rbrace $ [Eq.",
"(REF )].",
"Similarly, $\\langle {\\psi _{I \\alpha }} |$ transforming according to the IR $\\Gamma _I^\\ast $ becomes $\\langle {\\psi _{I^{\\prime } \\alpha ^{\\prime }}} | = \\langle {\\psi _{I \\alpha }} | T^\\dagger $ transforming according to $\\Gamma _{I^{\\prime }}^\\ast = \\Gamma _{\\mathbf {\\mathrm {\\tau }}}^\\ast \\times \\Gamma _I^\\ast $ .",
"Invariance of the operator $\\mathcal {O}$ under translations $T$ implies $[\\mathcal {O}, T] = 0$ , or $\\mathcal {O}^{\\prime } \\equiv T\\, \\mathcal {O} \\, T^{-1} = \\mathcal {O} ,$ so that both $\\mathcal {O}$ and $\\mathcal {O}^{\\prime }$ transform according to the same representation (which need not be irreducible) $\\Gamma _\\mathcal {O} = \\Gamma _{\\mathcal {O}^{\\prime }} .$ Hence, we get (note $\\Gamma _{\\mathbf {\\mathrm {\\tau }}}^\\ast \\times \\Gamma _{\\mathbf {\\mathrm {\\tau }}}= \\Gamma _1$ ) $\\Gamma _{I^{\\prime }}^\\ast \\times \\Gamma _{\\mathcal {O}^{\\prime }} \\times \\Gamma _{J^{\\prime }}= \\Gamma _I^\\ast \\times \\Gamma _{\\mathcal {O}} \\times \\Gamma _J ,$ i.e., in the unprimed and primed coordinate system we get the same selection rules.",
"In the primed coordinate system, the matrix elements become $\\mathcal {O}_{\\alpha ^{\\prime } \\beta ^{\\prime }}^{\\prime }& = \\langle {\\psi _{I^{\\prime } \\alpha ^{\\prime }} | \\mathcal {O}^{\\prime } | \\psi _{J^{\\prime } \\beta ^{\\prime }}} \\rangle \\\\& = \\langle {\\psi _{I \\alpha } | T^{-1} T\\, \\mathcal {O}\\, T^{-1} T\\,| \\psi _{J \\beta }} \\rangle \\\\& = \\langle {\\psi _{I \\alpha } | \\mathcal {O} | \\psi _{J \\beta }} \\rangle \\\\& = \\mathcal {O}_{\\alpha \\beta } .$ Note that for multidimensional IRs $\\Gamma _J$ and $\\Gamma _{J^{\\prime }} = \\Gamma _{\\mathbf {\\mathrm {\\tau }}}\\times \\Gamma _J$ , we can always choose the representation matrices such that $T| {\\psi _{J \\beta }} \\rangle = | {\\psi _{J^{\\prime } \\beta }} \\rangle $ , and similarly $\\langle {\\psi _{I \\alpha }} | T^\\dagger = \\langle {\\psi _{I^{\\prime } \\alpha }} |$ , so that Eq.",
"() becomes $\\mathcal {O}_{\\alpha \\beta }^{\\prime } = \\mathcal {O}_{\\alpha \\beta }$ ."
],
[
"General theory",
"The Bloch eigenstates $\\Psi _{n {\\mathbf {\\mathrm {k}}}}^{I \\beta } ({\\mathbf {\\mathrm {r}}})$ at a wave vector ${\\mathbf {\\mathrm {k}}}$ transform according to an IR $\\Gamma _I$ of the group of the wave vector $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ .",
"The knowledge of these IRs suffices to construct the general form of the effective Hamiltonian characterizing the Bloch eigenstates near the expansion point ${\\mathbf {\\mathrm {k}}}$ consistent with the symmetry operations in $\\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ .",
"This method is known as the theory of invariants [3].",
"The Hamiltonian can be expressed in terms of a general tensor operator denoted as ${\\mathbf {\\mathcal {K}}}$ that may depend on, e.g., the kinetic wave vector ${\\mathbf {\\mathrm {\\kappa }}}$ measured from ${\\mathbf {\\mathrm {k}}}$ , external electric and magnetic fields ${\\mathbf {\\mathcal {E}}}$ and ${\\mathbf {\\mathcal {B}}}$ , strain ${\\mathbf {\\mathrm {\\epsilon }}}$ and spin ${\\mathbf {\\mathrm {S}}}$ .",
"For all group elements $g \\in \\mathcal {G}_{{\\mathbf {\\mathrm {k}}}}$ , the Hamiltonian $\\mathcal {H} ({\\mathbf {\\mathcal {K}}})$ obeys the invariance condition [3] $\\mathcal {D} (g) \\mathcal {H} (g^{-1} {\\mathbf {\\mathcal {K}}}) \\mathcal {D}^{-1} (g) = \\mathcal {H} ({\\mathbf {\\mathcal {K}}}).$ According to the theory of invariants, each block $\\mathcal {H}_{I I^{\\prime }}$ of the matrix $\\mathcal {H}$ corresponding to a pair of bands transforming according to the IRs $\\Gamma _I$ and $\\Gamma _{I^{\\prime }}$ has the form $\\mathcal {H}_{I I^{\\prime }} ({\\mathbf {\\mathcal {K}}})= \\sum _{i,J}a_{iJ}^{I I^{\\prime }} \\sum _{l=1}^{L_J} X_l^J \\mathcal {K}^{i,J*}_l ,$ where $J$ labels the $L_J$ -dimensional IRs $\\Gamma _J$ contained in the product representation $\\Gamma _I^\\ast \\times \\Gamma _{I^{\\prime }}$ , the basis matrices $X_l^J$ and the irreducible tensor operators $\\mathcal {K}_J^i$ constructed from the perturbations ${\\mathbf {\\mathcal {K}}}$ transform according to the IR $\\Gamma _J$ , and $a_{iJ}^{I I^{\\prime }}$ are constant prefactors.",
"The index $i$ labels the irreducible tensor operators transforming as $\\Gamma _J$ .",
"In general, we have multiple blocks $\\mathcal {H}_{I I^{\\prime }} ({\\mathbf {\\mathcal {K}}})$ corresponding to different bands $n$ and $n^{\\prime }$ transforming according to $\\Gamma _I$ and $\\Gamma _{I^{\\prime }}$ .",
"To simplify the notation, we drop these additional indices."
],
[
"Effect of band IR rearrangements",
"A translation of the coordinate system by ${\\mathbf {\\mathrm {\\tau }}}$ changes the IR of an eigenfunction $\\Psi _{n{\\mathbf {\\mathrm {k}}}}^I ({\\mathbf {\\mathrm {r}}})$ from $\\Gamma _I$ to $\\Gamma _J = \\Gamma _{\\mathbf {\\mathrm {\\tau }}}\\times \\Gamma _I$ , see Eq.",
"(REF ).",
"We have $\\Gamma _J^\\ast \\times \\Gamma _{J^{\\prime }}& = (\\Gamma _{\\mathbf {\\mathrm {\\tau }}}\\times \\Gamma _I)^\\ast \\times (\\Gamma _{\\mathbf {\\mathrm {\\tau }}}\\times \\Gamma _{I^{\\prime }})\\\\& = \\Gamma _I^\\ast \\times \\Gamma _{I^{\\prime }} ,$ where we used $\\Gamma _{\\mathbf {\\mathrm {\\tau }}}^\\ast \\times \\Gamma _{\\mathbf {\\mathrm {\\tau }}}= \\Gamma _1$ for one-dimensional IRs $\\Gamma _{\\mathbf {\\mathrm {\\tau }}}$ .",
"Therefore, the IRs contained in $\\Gamma _J^\\ast \\times \\Gamma _{J^{\\prime }}$ are equal to the IRs contained in $\\Gamma _I^\\ast \\times \\Gamma _{I^{\\prime }}$ .",
"This implies that translations ${\\mathbf {\\mathrm {\\tau }}}$ of the coordinate system resulting in a rearrangement of the IRs assigned to the eigenfunctions do not affect the invariant expansion of the Hamiltonian $\\mathcal {H} ({\\mathbf {\\mathcal {K}}})$ ."
],
[
"Invariant Hamiltonian for MoS$_2$",
"As an application of the theory of invariants, we consider monolayer MoS$_2$ .",
"In this material, the lowest conduction and highest valence band are at the points ${\\mathbf {\\mathrm {K}}}$ and ${\\mathbf {\\mathrm {K}}}^{\\prime }$ [29], hence we focus on these points (where $\\mathcal {G}_{{\\mathbf {\\mathrm {K}}}} = C_{3h}$ ).",
"Table REF lists the mapping of axial (${\\mathbf {\\mathrm {A}}}$ ) and polar vectors (${\\mathbf {\\mathrm {P}}}$ ) under the relevant symmetry operations.",
"This allows one to confirm the examples for basis functions listed for $C_{3h}$ in Table REF .",
"Crystal momentum ${\\mathbf {\\mathrm {\\kappa }}}$ and an electric field ${\\mathbf {\\mathcal {E}}}$ transform like polar vectors, whereas spin ${\\mathbf {\\mathrm {S}}}$ and a magnetic field ${\\mathbf {\\mathcal {B}}}$ transform like axial vectors.",
"Hence we immediately obtain from Table REF the lowest-order tensor operators listed in the second column of Table REF .",
"In general, we obtain higher-order tensor operators using the Clebsch-Gordan coefficients that are tabulated in, e.g., Ref. [56].",
"However, this procedure is greatly simplified if all IRs of the relevant group are one-dimensional, which holds for $C_{3h}$ .",
"In such a case, the higher-order tensor operators can be constructed using the multiplication table for the IRs that is reproduced for $C_{3h}$ in Table REF .",
"Irreducible tensor operators are generally not unique.",
"Given two irreducible tensor operators $\\mathcal {K}$ and $\\mathcal {K}^{\\prime }$ transforming according to the same IR $\\Gamma _I$ , any linear combinations of these tensors transforms likewise irreducibly according to $\\Gamma _I$ [5].",
"We exploit this freedom to choose linear combinations of irreducible tensors such as $\\kappa _-^3$ and $\\kappa _+^3$ that have also a well-defined behavior under time reversal symmetry (see Sec.",
"REF ).",
"We can also consider the effects of strain.",
"When stress deforms a crystalline solid, the symmetry of the system is altered which changes the energy spectrum of the material.",
"Suppose under a deformation a point ${\\mathbf {\\mathrm {r}}}$ in a solid undergoes a displacement ${\\mathbf {\\mathrm {u}}} ({\\mathbf {\\mathrm {r}}})$ .",
"For small homogeneous strain, the symmetric strain tensor is defined as ($i,j = x,y,z$ ) [59] $\\epsilon _{ij}=\\frac{1}{2}\\left( \\frac{\\partial u_i}{\\partial r_j}+\\frac{\\partial u_j}{\\partial r_i}+\\frac{\\partial u_k}{\\partial r_i}\\frac{\\partial u_k}{\\partial r_j}\\right).$ However, considering MoS$_2$ as a quasi-2D material, strain due to a perpendicular stress component is not relevant.",
"The components $\\epsilon _{ij}$ transform like the symmetrized products $\\lbrace \\kappa _i, \\kappa _j \\rbrace $ [3] so that we get the lowest-order operators listed in the second column of Table REF while mixed higher-order tensor operators are listed in the third column of Table REF .",
"Since the IRs of $C_{3h}$ are all one-dimensional, the Hamiltonian blocks (REF ) at the points ${\\mathbf {\\mathrm {K}}}$ and ${\\mathbf {\\mathrm {K}}}^{\\prime }$ of MoS$_2$ are one-dimensional.",
"The $1\\times 1$ basis matrices $X_1^J$ can be absorbed into the prefactors $a_{iJ}^{I I^{\\prime }}$ .",
"Hence, Eq.",
"(REF ) can be simplified to $\\mathcal {H}_{I I^{\\prime }} ({\\mathbf {\\mathcal {K}}}) = \\sum _ia_i^{I I^{\\prime }} \\mathcal {K}^{i,J*}_l .$ The IRs corresponding to the eleven bands at ${\\mathbf {\\mathrm {K}}}$ and ${\\mathbf {\\mathrm {K}}}^{\\prime }$ that are dominated by the Mo $d$ and S $p$ orbitals are listed in Table REF .",
"Here we focus on constructing a generic $6 \\times 6$ Hamiltonian consisting of a sequence of bands transforming as $\\Gamma _{1/3/2}, \\Gamma _{2/1/3}, \\Gamma _{3/2/1}, \\Gamma _{4/6/5}, \\Gamma _{5/4/6}$ , and $\\Gamma _{6/5/4}$ , respectively (realized, e.g., by the bands $v_1, c_1, c_3, c_2, v_3$ , and $v_2$ ; additional bands will replicate the behavior obtained for these bands).",
"For these bands, the Hamiltonian matrix elements $\\mathcal {H}^{{\\mathbf {\\mathrm {K}}}}({\\mathbf {\\mathcal {K}}})_{I I^{\\prime }}$ contain tensors $\\mathcal {K}$ transforming as $(\\Gamma _I^\\ast \\times \\Gamma _{I^{\\prime }}) = \\left( \\begin{array}{s{0.15em}*{11}{c}s{0.15em}}\\Gamma _1 & \\Gamma _2 & \\Gamma _3 & \\Gamma _4 & \\Gamma _5 & \\Gamma _6 \\\\\\Gamma _3 & \\Gamma _1 & \\Gamma _2 & \\Gamma _6 & \\Gamma _4 & \\Gamma _5 \\\\\\Gamma _2 & \\Gamma _3 & \\Gamma _1 & \\Gamma _5 & \\Gamma _6 & \\Gamma _4 \\\\\\Gamma _4 & \\Gamma _5 & \\Gamma _6 & \\Gamma _1 & \\Gamma _2 & \\Gamma _3 \\\\\\Gamma _6 & \\Gamma _4 & \\Gamma _5 & \\Gamma _3 & \\Gamma _1 & \\Gamma _2 \\\\\\Gamma _5 & \\Gamma _6 & \\Gamma _4 & \\Gamma _2 & \\Gamma _3 & \\Gamma _1 \\end{array} \\right),$ see Table REF (Appendix ).",
"Table: Irreducible tensor operators for the point group C 3h C_{3h}.",
"For off-diagonal terms in the Hamiltonian ℋ(𝒦)\\mathcal {H} ({\\mathbf {\\mathcal {K}}}), using the phase conventions in Table , the bold-face tensor operators give rise to invariants with purely real prefactors, while the remaining tensor operators give rise to invariants with purely imaginary prefactors.",
"At the same time, these phase conventions and time-reversal symmetry imply that invariants appearing on the diagonal of ℋ(𝒦)\\mathcal {H} ({\\mathbf {\\mathcal {K}}}) (which transform as Γ 1 \\Gamma _1) can only be formed from tensor operators listed in bold, the remaining tensor operators correspond to invariants that are forbidden by time reversal symmetry.",
"Tensors transforming according to Γ 3 \\Gamma _3 and Γ 6 \\Gamma _6 are the Hermitean adjoint of the tensors transforming according to Γ 2 \\Gamma _2 and Γ 5 \\Gamma _5, respectively.",
"Notation: V ± =V x ±iV y V_\\pm = V_x \\pm iV_y with V=κ,ℬ,ℰ{\\mathbf {\\mathrm {V}}} = {\\mathbf {\\mathrm {\\kappa }}}, {\\mathbf {\\mathcal {B}}}, {\\mathbf {\\mathcal {E}}}, and ϵ ± =ϵ xx -ϵ yy ±2iϵ xy \\epsilon _\\pm = \\epsilon _{xx}-\\epsilon _{yy}\\pm 2i\\epsilon _{xy}, ϵ ∥ =ϵ xx +ϵ yy \\epsilon _\\Vert =\\epsilon _{xx}+\\epsilon _{yy}.Table: IRs Γ i/j/k \\Gamma _{i/j/k} for the eleven TB bands in MoS 2 _2 obtained from Mo dd and S pp orbitals at the K{\\mathbf {\\mathrm {K}}} point for coordinate systems α=a/b/c\\alpha = a/b/c in Fig. .",
"We denote the bands by c 4 ,...,c 1 ,v 1 ,...,v 7 c_4, \\ldots , c_1 , v_1, \\ldots , v_7 arranged in order of decreasing energy.",
"The seven bands c 3 ,...,v 4 c_3, \\ldots , v_4 are used in Ref.",
", while the three bands c 3 c_3, c 1 c_1, and v 1 v_1 are used in Ref. .",
"The main (second) atomic orbital for each band is given in the third (fourth) column (compare Table )."
],
[
"Time reversal",
"At the points ${\\mathbf {\\mathrm {K}}}$ and ${\\mathbf {\\mathrm {K}}}^{\\prime } = -{\\mathbf {\\mathrm {K}}}$ with $\\mathcal {G}_{{\\mathbf {\\mathrm {K}}}} = C_{3h}$ , the Bloch functions $\\Psi _{{\\mathbf {\\mathrm {K}}}}^I$ and the corresponding time reversed functions $\\Theta \\, \\Psi _{{\\mathbf {\\mathrm {K}}}}^I$ are linearly dependent on each other [case $(a)$ according to Eq.",
"(REF ), see also Table REF ].",
"Also, the eigenfunctions at ${\\mathbf {\\mathrm {K}}}$ can be mapped onto the eigenfunctions at $- {\\mathbf {\\mathrm {K}}}$ by a vertical reflection $R = \\sigma _v^{(2)}$ or a $180^{\\circ } $ rotation $C_2^{\\prime (2)}$ , which is case (2) as defined in Eq.",
"(REF ).",
"[These two operations $R$ are elements of the group $D_{3h}$ , the point group of MoS$_2$ , see Fig.",
"REF (a).]",
"Hence, for a Bloch state $\\Psi _{{\\mathbf {\\mathrm {K}}}}^I$ of band $I$ , the time reversed state $\\Theta \\, \\Psi _{{\\mathbf {\\mathrm {K}}}}^I$ and the spatially transformed state $R \\, \\Psi _{{\\mathbf {\\mathrm {K}}}}^I$ , with $R = \\sigma _v^{(2)}, C_2^{\\prime (2)}$ , obey the linear relation $\\Theta \\, \\Psi _{{\\mathbf {\\mathrm {K}}}}^I= (\\Psi _{{\\mathbf {\\mathrm {K}}}}^I) ^\\ast = t_R^I R \\, \\Psi _{{\\mathbf {\\mathrm {K}}}}^I ,$ where $t_R^I$ is a phase factor (a unitary matrix if the dimensions of the IRs $\\Gamma _I$ and $\\Gamma _I^\\ast $ was larger than one) that depends on the choice for the operation $R$ .",
"For the phase conventions used in Table REF , we have $t_{\\sigma _v^{(2)}}^I & = \\left\\lbrace \\begin{array}{rs{1em}l}1, & I = 1, 2, 3 \\\\-1, & I = 4, 5, 6 ,\\end{array}\\right.",
"\\\\t_{C_2^{\\prime (2)}}^I & = 1, \\quad I = 1, \\ldots , 6 .$ The matrix $\\mathcal {H} ({\\mathbf {\\mathcal {K}}})$ must then satisfy the additional condition $(\\mathcal {t}_R)^{-1} \\, \\mathcal {H} (R^{-1} \\, {\\mathbf {\\mathcal {K}}}) \\, \\mathcal {t}_R= \\mathcal {H}^\\ast (\\zeta \\, {\\mathbf {\\mathcal {K}}})= \\mathcal {H}^t (\\zeta \\, {\\mathbf {\\mathcal {K}}}) .$ where $\\mathcal {t}_R$ is a diagonal matrix with elements $(\\mathcal {t}_R)_{II} = t_R^I$ , $\\zeta =+1$ ($-1$ ) for quantities that are even (odd) under time reversal such as ${\\mathbf {\\mathcal {E}}}$ and ${\\mathbf {\\mathrm {\\epsilon }}}$ (${\\mathbf {\\mathrm {\\kappa }}}$ , ${\\mathbf {\\mathcal {B}}}$ , and ${\\mathbf {\\mathrm {S}}}$ ), $\\ast $ denotes complex conjugation and $t$ transposition.",
"The components of polar vectors ${\\mathbf {\\mathrm {P}}}$ and axial vectors ${\\mathbf {\\mathrm {A}}}$ transform under $\\sigma _v^{(2)}$ and $C_2^{\\prime (2)}$ as follows (see Table REF ) $\\sigma _v^{(2)} \\, P_\\pm & = -P_\\mp , &C_2^{\\prime (2)} \\, P_\\pm & = -P_\\mp , \\\\\\sigma _v^{(2)} \\, P_z & = P_z, &C_2^{\\prime (2)} \\, P_z & = -P_z, \\\\\\sigma _v^{(2)} \\, A_\\pm & = A_\\mp , &C_2^{\\prime (2)} \\, A_\\pm & = -A_\\mp , \\\\\\sigma _v^{(2)} \\, A_z & = -A_z, &C_2^{\\prime (2)} \\, A_z & = -A_z .$ Table: Linear relation between the Bloch functions at K ' =-K{\\mathbf {\\mathrm {K}}}^{\\prime } = - {\\mathbf {\\mathrm {K}}} obtained via time reversal Θ\\Theta and via a spatial transformation RR applied to a Bloch function Ψ K I \\Psi _{{\\mathbf {\\mathrm {K}}}}^I at wave vector k=K{\\mathbf {\\mathrm {k}}} = {\\mathbf {\\mathrm {K}}}.",
"The K{\\mathbf {\\mathrm {K}}} point in the Brillouin zone is mapped onto -K- {\\mathbf {\\mathrm {K}}} by a vertical reflection R=σ v (2) R = \\sigma _v^{(2)} and by a rotation R=C 2 '(2) R = C_2^{\\prime (2)}, see Fig.",
"(a).It follows from Eq.",
"(REF ) whether off-diagonal terms in $\\mathcal {H} ({\\mathbf {\\mathcal {K}}})$ have real or imaginary prefactors.",
"Tensor operators that give rise to invariants with real prefactors are marked in bold in Table REF .",
"Furthermore, condition (REF ) provides a general criterion which terms are allowed by time-reversal symmetry on the diagonal of the Hamiltonian [when the tensor operators must transform according to the identity IR $\\Gamma _1$ , see Eq.",
"(REF )].",
"Here, our phase conventions imply that invariants (with real prefactors) can only be formed from tensor operators marked in bold in Table REF .",
"Thus, for example, on the diagonal of the Hamiltonian the third-order trigonal term $\\kappa _+^3 + \\kappa _-^3 = 2 \\kappa _x(\\kappa _x^2 - 3\\kappa _y^2)$ , as well as the field-dependent terms $i\\kappa _- \\mathcal {E}_+ - i\\kappa _+ \\mathcal {E}_- = 2 (\\kappa _y \\mathcal {E}_x - \\kappa _x \\mathcal {E}_y)$ and $\\mathcal {B}_+S_- + \\mathcal {B}_-S_+ = 2 (\\mathcal {B}_x S_x + \\mathcal {B}_y S_y)$ are allowed by symmetry and thus present in the Hamiltonian.",
"However, the terms $i\\kappa _-^3 -i\\kappa _+^3 = 2 \\kappa _y (3\\kappa _x^2 - \\kappa _y^2)$ , $\\kappa _+ \\mathcal {E}_- + \\kappa _- \\mathcal {E}_+ = 2 (\\kappa _x \\mathcal {E}_x + \\kappa _y \\mathcal {E}_y)$ and $i\\mathcal {B}_-S_+ -i\\mathcal {B}_+S_- = 2 (\\mathcal {B}_y S_x - \\mathcal {B}_x S_y)$ are allowed by spatial symmetry; but these terms are forbidden by time-reversal symmetry and hence do not appear in the Hamiltonian.",
"(They are allowed, though, as off-diagonal terms coupling different bands transforming according to the same IR, when these terms will have imaginary prefactors.)",
"Using the transformation $R$ , one can derive the effective Hamiltonian for the valley ${\\mathbf {\\mathrm {K}}}^{\\prime }$ using the transformation $\\mathcal {H}^{{\\mathbf {\\mathrm {K}}}^{\\prime }} ({\\mathbf {\\mathcal {K}}}) = \\mathcal {H}^{\\mathbf {\\mathrm {K}}} (R^{-1} \\, {\\mathbf {\\mathcal {K}}}).$ Alternatively, time reversal can be used, see Eq.",
"(REF ).",
"Note that the definition of $\\mathcal {H}^{{\\mathbf {\\mathrm {K}}}^{\\prime }} ({\\mathbf {\\mathcal {K}}})$ relative to $\\mathcal {H}^{{\\mathbf {\\mathrm {K}}}} ({\\mathbf {\\mathcal {K}}})$ depends on the phase conventions used for the basis functions $\\Psi _{{\\mathbf {\\mathrm {K}}}^{\\prime }}^I$ relative to the phase conventions used for $\\Psi _{{\\mathbf {\\mathrm {K}}}}^I$ ."
],
[
"Analysis of the invariant expansion",
"The invariant Hamiltonian at the point ${\\mathbf {\\mathrm {K}}}$ of the BZ becomes in lowest order of the wave vector ${\\mathbf {\\mathrm {\\kappa }}}$ and spin-orbit coupling (spin ${\\mathbf {\\mathrm {S}}}$ ) $\\mathcal {H} ({\\mathbf {\\mathrm {\\kappa }}}, {\\mathbf {\\mathrm {S}}})= \\mathcal {H}_0 + \\mathcal {H}_\\kappa + \\mathcal {H}_\\mathrm {so}$ with $\\mathcal {H}_0 & =\\left( \\begin{array}{s{0.15em}c*{5}{s{0.2em}c}s{0.15em}}E_1 & 0 & 0 & 0 & 0 & 0\\\\0 & E_2 & 0 & 0 & 0 & 0 \\\\0 & 0 & E_3 & 0 & 0 & 0 \\\\0 & 0 & 0 & E_4 & 0 & 0 \\\\0 & 0 & 0 & 0 & E_5 & 0 \\\\0 & 0 & 0 & 0 & 0 & E_6\\end{array} \\right) , \\\\\\mathcal {H}_{{\\mathbf {\\mathrm {\\kappa }}}} & =\\left( \\begin{array}{s{0.15em}c*{5}{s{0.2em}c}s{0.15em}}0 & \\gamma _{12} \\kappa _+ & \\gamma _{13} \\kappa _- & 0 & 0 & 0\\\\\\gamma _{12} \\kappa _- & 0 & \\gamma _{23} \\kappa _+ & 0 & 0 & 0 \\\\\\gamma _{13} \\kappa _+ & \\gamma _{23} \\kappa _- & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & \\gamma _{45} \\kappa _+ & \\gamma _{46} \\kappa _- \\\\0 & 0 & 0 & \\gamma _{45} \\kappa _- & 0 & \\gamma _{56} \\kappa _+ \\\\0 & 0 & 0 & \\gamma _{46} \\kappa _+ & \\gamma _{56} \\kappa _- & 0\\end{array} \\right) , \\\\\\mathcal {H}_\\mathrm {so}& = \\left( \\begin{array}{s{0.15em}c*{5}{s{0.2em}c}s{0.15em}}\\lambda _{11} S_z & 0 & 0 & 0 & \\lambda _{15} S_+ & \\lambda _{16} S_-\\\\0 & \\lambda _{22} S_z & 0 & \\lambda _{24} S_- & 0 & \\lambda _{26} S_+ \\\\0 & 0 & \\lambda _{33} S_z & \\lambda _{34} S_+ & \\lambda _{35} S_- & 0 \\\\0 & \\lambda _{24} S_+ & \\lambda _{34} S_- & \\lambda _{44} S_z & 0 & 0 \\\\\\lambda _{15} S_- & 0 & \\lambda _{35} S_+ & 0 & \\lambda _{55} S_z & 0 \\\\\\lambda _{16} S_+ & \\lambda _{26} S_- & 0 & 0 & 0 & \\lambda _{66} S_z\\end{array} \\right) ,$ where $E_I$ , $\\gamma _{ij}$ , and $\\lambda _{ij}$ are material-dependent real parameters.",
"The lowest-order strain-dependent terms become $\\mathcal {H}_{{\\mathbf {\\mathrm {\\epsilon }}}} =\\left( \\begin{array}{s{0.15em}c*{5}{s{0.2em}c}s{0.15em}}\\xi _{11} \\epsilon _\\Vert & \\xi _{12} \\epsilon _- & \\xi _{13} \\epsilon _+ & 0 & 0 & 0\\\\\\xi _{12} \\epsilon _+ & \\xi _{22} \\epsilon _\\Vert & \\xi _{23} \\epsilon _- & 0 & 0 & 0 \\\\\\xi _{13} \\epsilon _- & \\xi _{23} \\epsilon _+ & \\xi _{33} \\epsilon _\\Vert & 0 & 0 & 0 \\\\0 & 0 & 0 & \\xi _{44} \\epsilon _\\Vert & \\xi _{45} \\epsilon _- & \\xi _{46} \\epsilon _+ \\\\0 & 0 & 0 & \\xi _{45} \\epsilon _+ & \\xi _{55} \\epsilon _\\Vert & \\xi _{56} \\epsilon _- \\\\0 & 0 & 0 & \\xi _{46} \\epsilon _- & \\xi _{56} \\epsilon _+ & \\xi _{66} \\epsilon _\\Vert \\end{array} \\right)$ with real parameters $\\xi _{ij}$ .",
"The effect of electric and magnetic fields ${\\mathbf {\\mathcal {E}}}$ and ${\\mathbf {\\mathcal {B}}}$ can also be included in the Hamiltonian (REF ).",
"For the electric field, we add on the diagonal a scalar potential $e{\\mathbf {\\mathcal {E}}}\\cdot {\\mathbf {\\mathrm {r}}}$ and we replace crystal momentum by kinetic momentum $\\hbar {\\mathbf {\\mathrm {\\kappa }}}=-i\\hbar {\\mathbf {\\mathrm {\\nabla }}} +e{\\mathbf {\\mathrm {A}}}$ , where ${\\mathbf {\\mathrm {A}}}$ is the vector potential due to the magnetic field.",
"In addition, we may have terms that depend explicitly on the fields ${\\mathbf {\\mathcal {E}}}$ and ${\\mathbf {\\mathcal {B}}}$ .",
"The in-plane components $\\mathcal {E}_x$ and $\\mathcal {E}_y$ transform spatially like the wave vector components $\\kappa _x$ and $\\kappa _y$ .",
"However, ${\\mathbf {\\mathcal {E}}}$ is even under time reversal symmetry, whereas ${\\mathbf {\\mathrm {\\kappa }}}$ is odd.",
"In lowest order, the ${\\mathbf {\\mathcal {E}}}$ -dependent terms become $\\mathcal {H}_{{\\mathbf {\\mathrm {\\mathcal {E}}}}} =\\left( \\begin{array}{s{0.15em}c*{5}{s{0.2em}c}s{0.15em}}0 & \\eta _{12} \\mathcal {E}_+ & \\eta _{13} \\mathcal {E}_- & \\eta _{14} \\mathcal {E}_z & 0 & 0\\\\\\eta ^\\ast _{12} \\mathcal {E}_- & 0 & \\eta _{23} \\mathcal {E}_+ & 0 & \\eta _{25} \\mathcal {E}_z & 0 \\\\\\eta ^\\ast _{13} \\mathcal {E}_+ & \\eta ^\\ast _{23} \\mathcal {E}_- & 0 & 0 & 0 & \\eta _{36} \\mathcal {E}_z \\\\\\eta ^\\ast _{14} \\mathcal {E}_z & 0 & 0 & 0 & \\eta _{45} \\mathcal {E}_+ & \\eta _{46} \\mathcal {E}_- \\\\0 & \\eta ^\\ast _{25} \\mathcal {E}_z & 0 & \\eta ^\\ast _{45} \\mathcal {E}_- & 0 & \\eta _{56} \\mathcal {E}_+ \\\\0 & 0 & \\eta ^\\ast _{36} \\mathcal {E}_z & \\eta ^\\ast _{46} \\mathcal {E}_+ & \\eta ^\\ast _{56} \\mathcal {E}_- & 0\\end{array} \\right)$ with imaginary prefactors $\\eta _{ij}$ .",
"Since ${\\mathbf {\\mathcal {B}}}$ transforms in the same way as the spin ${\\mathbf {\\mathrm {S}}}$ both under spatial transformations and time reversal, the ${\\mathbf {\\mathcal {B}}}$ -dependent terms are similar to Eq.",
"() $\\mathcal {H}_{{\\mathbf {\\mathcal {B}}}}= \\left( \\begin{array}{s{0.15em}c*{5}{s{0.2em}c}s{0.15em}}\\beta _{11} \\mathcal {B}_z & 0 & 0 & 0 & \\beta _{15} \\mathcal {B}_+ & \\beta _{16} \\mathcal {B}_-\\\\0 & \\beta _{22} \\mathcal {B}_z & 0 & \\beta _{24} \\mathcal {B}_- & 0 & \\beta _{26} \\mathcal {B}_+ \\\\0 & 0 & \\beta _{33} \\mathcal {B}_z & \\beta _{34} \\mathcal {B}_+ & \\beta _{35} \\mathcal {B}_- & 0 \\\\0 & \\beta _{24} \\mathcal {B}_+ & \\beta _{34} \\mathcal {B}_- & \\beta _{44} \\mathcal {B}_z & 0 & 0 \\\\\\beta _{15} \\mathcal {B}_- & 0 & \\beta _{35} \\mathcal {B}_+ & 0 & \\beta _{55} \\mathcal {B}_z & 0 \\\\\\beta _{16} \\mathcal {B}_+ & \\beta _{26} \\mathcal {B}_- & 0 & 0 & 0 & \\beta _{66} \\mathcal {B}_z\\end{array} \\right)$ with real parameters $\\beta _{ij}$ .",
"We can project the multiband Hamiltonian (REF ) on a subspace containing the bands of interest using quasidegenerate perturbation theory or Löwdin partitioning [5].",
"This gives rise to higher-order terms in the effective Hamiltonian that may include also mixed terms proportional to products of ${\\mathbf {\\mathrm {\\kappa }}}$ , ${\\mathbf {\\mathrm {S}}}$ , ${\\mathbf {\\mathrm {\\epsilon }}}$ , ${\\mathbf {\\mathcal {E}}}$ , and ${\\mathbf {\\mathcal {B}}}$ .",
"In particular, in the presence of a magnetic field ${\\mathbf {\\mathcal {B}}}$ , the components of crystal momentum $\\hbar {\\mathbf {\\mathrm {\\kappa }}} = - i\\hbar {\\mathbf {\\mathrm {\\nabla }}} + e{\\mathbf {\\mathrm {A}}}$ do not commute, $\\left[ \\kappa _x, \\kappa _y \\right]=(e/i\\hbar )\\mathcal {B}_z$ , so that antisymmetrized products of $\\kappa _x$ and $\\kappa _y$ appearing in higher-order perturbation theory give rise to terms proportional to $\\mathcal {B}_z$ .",
"Similarly, terms proportional to in-plane electric fields $\\mathcal {E}_x, \\mathcal {E}_y$ appear because of the presence of the scalar potential $e{\\mathbf {\\mathcal {E}}} \\cdot {\\mathbf {\\mathrm {r}}}$ and the relation $\\left[ r_i, \\kappa _j \\right]=i\\delta _{ij}$ ."
],
[
"Conclusions",
"Starting from a TB approach, we have developed a comprehensive theory to derive the IRs characterizing the Bloch eigenstates in a crystal by decomposing the TB basis functions into localized symmetry-adapted atomic orbitals and crystal-periodic symmetry-adapted plane waves.",
"Both the symmetry-adapted atomic orbitals and the symmetry-adapted plane waves can easily be tabulated, thus accelerating the design and exploration of new materials.",
"The symmetry-adapted basis functions block-diagonalize the TB Hamiltonian, which naturally facilitates further analysis of the band structure.",
"While our analysis focused for clarity on symmorphic space groups, our theory can readily be generalized to nonsymmorphic groups.",
"The present work was motivated by the goal to develop a systematic theory of effective multiband Hamiltonians for the dynamics of Bloch electrons in external fields that break the symmetry of the crystal structure.",
"Yet our general symmetry analysis of Bloch states will likely be useful for other applications, too.",
"We appreciate stimulating discussions with G. Burkard, A. Kormányos, and U. Zülicke This work was supported by the NSF under Grant No.",
"DMR-1310199.",
"Work at Argonne was supported by DOE BES under Contract No.",
"DE-AC02-06CH11357."
],
[
"Projection operators",
"A general method to identify the IRs of the eigenstates of a Hamiltonian uses projection operators [19], [38], [3], [39].",
"Given a symmetry group $G$ with IRs $\\Gamma _I$ , we can project a general function $f({\\mathbf {\\mathrm {r}}})$ onto its components $f_{I \\beta } ({\\mathbf {\\mathrm {r}}})$ transforming according to the $\\beta $ th component of the IR $\\Gamma _I$ of $G$ .",
"Here the projection operators $\\Pi _{I \\beta }$ are $\\Pi _{I \\beta } \\equiv \\frac{n_I}{h} \\sum _g \\: \\mathcal {D}_I(g)^\\ast _{\\beta \\beta } \\, P(g) ,$ where $h$ is the order of the group $G$ , $n_I$ is the dimensionality of the IR $\\Gamma _I$ , $\\mathcal {D}_I(g)$ are the representation matrices for $\\Gamma _I$ , and $P(g)$ are the symmetry operators corresponding to $g \\in G$ .",
"Often, we denote $P(g)$ simply as $g$ .",
"If we are not interested in a particular component $\\beta $ of $\\Gamma _I$ , we can use the “coarse-grained” projection operators $\\Pi _I \\equiv \\sum _\\beta \\: \\Pi _{I \\beta }= \\frac{n_I}{h} \\sum _g \\: \\chi _I(g)^\\ast \\, P(g),$ where $\\chi _I (g) \\equiv \\operatorname{tr}\\mathcal {D}_I(g)$ are the characters for $\\Gamma _I$ .",
"For one-dimensional IRs, the operators $\\Pi _I$ become equivalent to $\\Pi _{I \\beta }$ .",
"The projection operators obey the completeness relation $\\sum _{I, \\beta } \\: \\Pi _{I \\beta } = \\sum _I \\: \\Pi _I = .$"
],
[
"Rearrangement lemma for IRs",
"Generally, if $\\Gamma _0$ is a one-dimensional IR of a group $G$ , then for any IR $\\Gamma _I$ of $G$ , the product representation $\\Gamma _{I^{\\prime }} = \\Gamma _0 \\times \\Gamma _I$ is irreducible.",
"If two IRs $\\Gamma _I$ and $\\Gamma _J$ of $G$ are (in)equivalent, then the IRs $\\Gamma _{I^{\\prime }} = \\Gamma _0 \\times \\Gamma _I$ and $\\Gamma _{J^{\\prime }} = \\Gamma _0 \\times \\Gamma _J$ are also (in)equivalent.",
"Hence, a multiplication of the IRs of $G$ by $\\Gamma _0$ simply rearranges the sequence of IRs of $G$ .",
"These statements can be proven as follows: For the element $g \\in G$ , we denote the characters for the IRs $\\Gamma _0$ and $\\Gamma _I$ by $\\chi _0(g)$ and $\\chi _I(g)$ , respectively.",
"Hence the character of the representation $\\Gamma _{I^{\\prime }}$ becomes $\\chi _{I^{\\prime }} (g) = \\chi _0(g) \\, \\chi _I(g)$ .",
"Since $\\Gamma _0$ is one-dimensional, its characters are also its unitary representation matrices obeying $|\\chi _0(g)|^2 = 1$ .",
"Using the orthogonality relations for characters we get $\\sum _g |\\chi _{I^{\\prime }} (g) |^2& = \\sum _g |\\chi _0(g) \\, \\chi _I(g)|^2\\\\& = \\sum _g |\\chi _0(g)|^2 \\, |\\chi _I(g)|^2\\\\& = \\sum _g |\\chi _I(g)|^2 = h ,$ where $h$ is the order of the group.",
"Hence $\\Gamma _{I^{\\prime }} = \\Gamma _0 \\times \\Gamma _I$ is irreducible if $\\Gamma _I$ is irreducible.",
"Now consider two IRs $\\Gamma _{I^{\\prime }} = \\Gamma _0 \\times \\Gamma _I$ and $\\Gamma _{J^{\\prime }} = \\Gamma _0 \\times \\Gamma _J$ .",
"Using again the orthogonality relations for characters we get $\\sum _g \\chi _{I^{\\prime }}^\\ast (g) \\chi _{J^{\\prime }} (g)& = \\sum _g \\chi _I^\\ast (g) \\, \\chi _0^\\ast (g) \\, \\chi _0(g) \\, \\chi _J(g)\\\\& = \\sum _g \\chi _I^\\ast (g) \\, \\chi _J(g) = h\\, \\delta _{IJ},$ so that $\\Gamma _{I^{\\prime }}$ and $\\Gamma _{J^{\\prime }}$ are indeed (in)equivalent if $\\Gamma _I$ and $\\Gamma _J$ are (in)equivalent."
],
[
"Group tables",
"Character tables and basis functions for the groups $D_{3h}$ , $D_{2h}$ , $C_{3h}$ , $C_{2v}$ , and $C_{2h}$ are reproduced in Tables REF – REF following the designations by Koster et al.",
"[39].",
"The second column in each table gives the designations for the IRs used by Dresselhaus et al.",
"[39].",
"Examples of basis functions that transform irreducibly according to the different IRs are listed in the last column.",
"These functions are expressed in terms of the cartesian components of polar (P) and axial (A) vectors using the coordinate systems defined for the respective groups in Figs.",
"REF , REF , and REF .",
"Finally, we reproduce in Table REF the multiplication table for the IRs of the group $C_{3h}$ [56].",
"Table: Character table and basis functions for the group D 3h D_{3h} .",
"The coordinate system for the basis functions is defined in Fig.",
"(a).Table: Character table and basis functions for the group D 2h D_{2h} .",
"The coordinate system for the basis functions is defined in Fig.",
"(c-e).Table: Character table and basis functions for the group C 3h C_{3h} with ω≡exp(iπ/6)\\omega \\equiv \\exp (i\\pi /6) .",
"The coordinate system for the basis functions is defined in Fig.",
"(b).Table: Character table and basis functions for the group C 2v C_{2v} .",
"The coordinate system for the basis functions is defined in Fig.",
"(c-e).Table: Character table and basis functions for the group C 2h C_{2h} .",
"The coordinate system for the basis functions is defined in Fig.",
"(c-e).Table: Multiplication table for the IRs of the group C 3h C_{3h} ."
]
] | 1906.04391 |
[
[
"The shape of shortest paths in random spatial networks"
],
[
"Abstract In the classic model of first passage percolation, for pairs of vertices separated by a Euclidean distance $L$, geodesics exhibit deviations from their mean length $L$ that are of order $L^\\chi$, while the transversal fluctuations, known as wandering, grow as $L^\\xi$.",
"We find that when weighting edges directly with their Euclidean span in various spatial network models, we have two distinct classes defined by different exponents $\\xi=3/5$ and $\\chi = 1/5$, or $\\xi=7/10$ and $\\chi = 2/5$, depending only on coarse details of the specific connectivity laws used.",
"Also, the travel time fluctuations are Gaussian, rather than Tracy-Widom, which is rarely seen in first passage models.",
"The first class contains proximity graphs such as the hard and soft random geometric graph, and the $k$-nearest neighbour random geometric graphs, where via Monte Carlo simulations we find $\\xi=0.60\\pm 0.01$ and $\\chi = 0.20\\pm 0.01$, showing a theoretical minimal wandering.",
"The second class contains graphs based on excluded regions such as $\\beta$-skeletons and the Delaunay triangulation and are characterised by the values $\\xi=0.70\\pm 0.01$ and $\\chi = 0.40\\pm 0.01$, with a nearly theoretically maximal wandering exponent.",
"We also show numerically that the KPZ relation $\\chi = 2\\xi -1$ is satisfied for all these models.",
"These results shed some light on the Euclidean first passage process, but also raise some theoretical questions about the scaling laws and the derivation of the exponent values, and also whether a model can be constructed with maximal wandering, or non-Gaussian travel fluctuations, while embedded in space."
],
[
"Introduction",
"Many complex systems assume the form of a spatial network [1], [2].",
"Transport networks, neural networks, communication and wireless sensor networks, power and energy networks, and ecological interaction networks are all important examples where the characteristics of a spatial network structure are key to understanding the corresponding emergent dynamics.",
"Shortest paths form an important aspect of their study.",
"Consider for example the appearance of bottlenecks impeding traffic flow in a city [3], [4], the emergence of spatial small worlds [5], [6], bounds on the diameter of spatial preferential attachment graphs [7], [8], [9], the random connection model [10], [11], [12], [13], or in spatial networks generally [14], [15], as well as geometric effects on betweenness centrality measures in complex networks [11], [16], and navigability [17].",
"First passage percolation (FPP) [18] attempts to capture these features with a probabilistic model.",
"As with percolation [19], the effect of spatial disorder is crucial.",
"Compare this to the elementary random graph [20].",
"In FPP one usually considers a deterministic lattice such as $\\mathbb {Z}^{d}$ with independent, identically distributed weights, known as local passage times, on the edges.",
"With a fluid flowing outward from a point, the question is, what is the minimum passage time over all possible routes between this and another distant point, where routing is quicker along lower weighted edges?",
"More than 50 years of intensive study of FPP has been carried out [21].",
"This has lead to many results such as the subadditive ergodic theorem, a key tool in probability theory, but also a number of insights in crystal and interface growth [22], the statistical physics of traffic jams [19], and key ideas of universality and scale invariance in the shape of shortest paths [23].",
"As an important intersection between probability and geometry, it is also part of the mathematical aspects of a theory of gravity beyond general relativity, and perhaps in the foundations of quantum mechanics, since it displays fundamental links to complexity, emergent phenomena, and randomness in physics [24], [25].",
"A particular case of FPP is the topic of this article, known as Euclidean first passage percolation (EFPP).",
"This is a probabilistic model of fluid flow between points of a $d$ -dimensional Euclidean space, such as the surface of a hypersphere.",
"One studies optimal routes from a source node to each possible destination node in a spatial network built either randomly or deterministically on the points.",
"Introduced by Howard and Newman much later in 1997 [26] and originally a weighted complete graph, we adopt a different perspective by considering edge weights given deterministically by the Euclidean distances between the spatial points themselves.",
"This is in sharp contrast with the usual FPP problem, where weights are i.i.d.",
"random variables.",
"Howard's model is defined on the complete graph constructed on a point process.",
"Long paths are discouraged by taking powers of interpoint distances as edge weights.",
"The variant of EFPP we study is instead defined on a Poisson point process in an unbounded region (by definition, the number of points in a bounded region is a Poisson random variable, see for example [27]), but with links added between pairs of points according to given rules [28], [29], rather than the totality of the weighted complete graph.",
"More precisely, the model we study in this paper is defined as follows.",
"We take a random spatial network such as the random geometric graph constructed over a simple Poisson point process on a flat torus, and weight the edges with their Euclidean length (see Fig.",
"REF ).",
"Figure: Illustration of the problem on a small network.",
"Thenetwork is constructed over a set of points denoted by circleshere and the edges are denoted by lines.",
"For a pair ofnodes (x,y)(x,y) we look for the shortest path (shown here by adotted line) where the length ofthe path is given by the sum of all edges length:d(x,y)=|x-a|+|a-b|+|b-c|+|c-d|+|d-y|d(x,y)=|x-a|+|a-b|+|b-c|+|c-d|+|d-y|.We then study the random length and transversal deviation of the shortest paths between two nodes in the network, denoted $x$ and $y$ , conditioned to lie at mutual Euclidean separation $|x-y|$ , as a function of the point process density and other parameters of the model used (here and in the following $|x|$ denotes the usual norm in euclidean space).",
"The study of the scaling with $|x-y|$ of the length and the deviation allow to define the fluctuation and wandering exponents (see precise definitions below).",
"We will consider a variety of networks such as the random geometric graph with unit disk and Rayleigh fading connection functions, the $k$ -nearest neighbour graph, the Delaunay triangulation, the relative neighbourhood graph, the Gabriel graph, and the complete graph with (in this case only) the edge weights raised to the power $\\alpha >1$ .",
"We describe these models in more detail in Section .",
"To expand on two examples, the random geometric graph (RGG) is a spatial network in which links are made between all pair of points with mutual separation up to a threshold.",
"This has applications in e.g.",
"wireless network theory, complex engineering systems such as smart grid, granular materials, neuroscience, spatial statistics, topological data analyis, and complex networks [30], [31], [32], [33], [10], [34], [35], [36], [37], [38].",
"This paper is structured as follows.",
"We first recap known results obtained for both the FPP and Euclidean FPP in Section .",
"We also discuss previous literature for the FPP in non-typical settings such as random graphs and tessellations.",
"The reader eager to view the results can skip this section at first reading, apart from the definitions of A, however the remaining background is very helpful for appreciating the later discussion.",
"In the Section we introduce the various spatial networks studied here, and in Section we present the numerical method and our new results on the EFPP model on random graphs.",
"In particular, due to arguments based on scale invariance, we expect the appearance of power laws and universal exponents [23].",
"We reveal the scaling exponents of the geodesics for the complete graph and for the network models studied here, and also show numerical results about the travel time and transversal deviation distribution.",
"In particular, we find distinct exponents from the KPZ class (see for example [39] and references therein) which has wandering and fluctuation exponents $\\xi =2/3$ and $\\chi =1/3$ , respectively.",
"Importantly, we conjecture and numerically corroborate a Gaussian central limit theorem for the travel time fluctuations, on the scale $t^{1/5}$ for the RGG and the other proximity graphs, and $t^{2/5}$ for the Delaunay triangulation and other excluded region graphs, which is also distinct from KPZ where the Tracy-Widom distribution, and the scale $t^{1/3}$ , is the famous outcome.",
"Finally, in Section we present some analytic ideas which help explain the distinction between universality classes.",
"We then conclude and discuss some open questions in Section ."
],
[
"Background: FPP and EFPP",
"In EFPP, we first construct a Poisson point process in $\\mathbb {R}^d$ which forms the basis of an undirected graph.",
"A fluid or current then flows outward from a single source at a constant speed with a travel time along an edge given by a power $\\alpha \\ge 1$ of the Euclidean length of the edge along which it travels [26].",
"See Fig.",
"REF , where the model is shown on six different random spatial network models.",
"Euclidean FPP on a large family of connected random geometric graphs has been studied in detail by Hirsch, Neuhäuser, Gloaguen and Schmidt in [40], [32], [41] and the closely related works [40], [42], [43], [32], [44], [41], [45], and references therein.",
"Developing FPP in this setting, Santalla et al [46] recently studied the model on spatial networks, as we do here.",
"Instead of EFPP, they weight the edges of the Delaunay triangulation, and also the square lattice, with i.i.d.",
"variates, for example $\\text{U}[a,b]$ for $a,b>0$ , and proceed to numerically verify the existence of the KPZ class for the geodesics, see e.g.",
"[47], and the earlier work of Pimentel [48] giving the asymptotic first passage times for the Delaunay triangulation with i.i.d weights.",
"Moreover, FPP on small-world networks and Erdős-Renyi random graphs are studied by Bhamidi, van der Hofstad and Hooghiemstra in [49], who discuss applications in diverse fields such as magnetism [50], wireless ad hoc networks [51], [12], [10], competition in ecological systems [52], and molecular biology [53].",
"See also their work specifically on random graphs [54].",
"Optimal paths in disordered complex networks, where disorder is weighting the edges with i.i.d.",
"random variables, is studied by Braunstein et al.",
"in [55], and later by Chen et al.",
"in [56].",
"We also point to the recent analytic results of Bakhtin and Wu, who have provided a good lower bound rate of growth of geodesic wandering, which in fact we find to be met with equality in the random geometric graph [57].",
"To highlight the difference between these results and our own, we have edge weights which are not independent random variables, but interpoint distances.",
"As far as we are aware, this has not been addressed directly in the literature."
],
[
"First passage percolation",
"Given i.i.d weights, paths are sums of i.i.d.",
"random variables.",
"The lengths of paths between pairs of points can be considered to be a random perturbation of the plane metric.",
"In fact, these lengths, and the corresponding transversal deviations of the geodesics, have been the focus of in-depth research over the last 50 years [21].",
"They exist as minima over collections of correlated random variables.",
"The travel times are conjectured (in the i.i.d.)",
"case to converge to the Tracy-Widom distribution (TW), found throughout various models of statistical physics, see e.g.",
"[46].",
"This links the model to random matrix theory, where $\\beta $ -TW appears as the limiting distribution of the largest eigenvalue of a random matrix in the $\\beta $ -Hermite ensemble, where the parameter $\\beta $ is 1,2 or 4 [58].",
"The original FPP model is defined as follows.",
"We consider vertices in the $d$ -dimensional lattice $\\mathbb {L}^{d} = (\\mathbb {Z}^{d},E^{d})$ where $E^{d}$ is the set of edges.",
"We then construct the function $\\tau : E^{d} \\rightarrow (0,\\infty )$ which gives a weight for each edge and are usually assumed to be identically independently distributed random variables.",
"The passage time from vertices $x$ to $y$ is the random variable given as the minimum of the sum of the $\\tau $ 's over all possible paths $P$ on the lattice connecting these points, T(x,y)=PP(e) This minimum path is a geodesic, and it is almost surely unique when the edge weights are continuous.",
"The average travel time is proportional to the distance E(T(x,y))|x-y| where here and in the following we denote the average of a quantity by $\\mathbb {E}\\left(\\cdot \\right)$ , and where $a \\sim b$ means $a$ converges to $Cb$ with $C$ a constant independent of $x,y$ , as $|x-y| \\rightarrow \\infty $ .",
"More generally, if the ratio of the geodesic length and the Euclidean distance is less than a finite number $t$ (the maximum value of this ratio is called the stretch), the network is a $t$-spanner [59].",
"Many important networks are $t$ -spanners, including the Delaunay triangulation of a Poisson point process, which has $\\pi /2< t < 1.998$ [60], [61].",
"The variance of the passage time over some distance $|x-y|$ is also important, and scales as Var(T(x,y)) |x-y|2, The maximum deviation $D(x,y)$ of the geodesic from the straight line from $x$ to $y$ is characterised by the wandering exponent $\\xi $ , i.e.",
"E(D(x,y))|x-y| for large $|x-y|$ .",
"Knowing $\\xi $ informs us about the geometry of geodesics between two distant points.",
"See Fig.",
"REF for an illustration of wandering on different networks.",
"The other exponent, $\\chi $ , informs us about the variance of their random length.",
"Another way to see this exponent is to consider a ball of radius $R$ around any point.",
"For large $R$ , the ball has an average radius proportional to $R$ and the fluctuations around this average grow as $R^\\chi $ [46].",
"With $\\chi <1$ the fluctuations die away $R \\rightarrow \\infty $ , leading to the shape theorem, see e.g.",
"[21].",
"According to Benjamini, Kalai and Schramm, $\\text{Var}\\left(T(x,y)\\right)$ grows sub-linearly with $|x-y|$ [62], a major theoretical step in characterising their scaling behaviour.",
"With $C$ some constant which depends only on the distribution of edge weights and the dimension $d$ , they prove that $\\text{Var}\\left(T\\left(x,y\\right)\\right) \\le C |x-y|/\\log |x-y|.$ The numerical evidence, in fact, shows this follows the non-typical scaling law $|x-y|^{2/3}$ .",
"Transversal fluctuations also scale as $|x-y|^{2/3}$ [21].",
"In this case, the fluctuations of $T$ are asymptotic to the TW distribution.",
"According to recent results of Santalla et al.",
"[63], curved spaces lead to similar fluctuations of a subtly different kind: if we embed the graph on the surface of a cylinder, the distribution changes from the largest eigenvalue of the GUE, to GOE, ensembles of random matrix theory.",
"When we see a sum of random variables, it is natural to conjecture a central limit theorem, where the fluctuations of the sum, after rescaling, converge to the standard normal distribution in some limit, in this case as $|x-y|\\rightarrow \\infty $ .",
"Durrett writes in a review that “...novice readers would expect a central limit theorem being proved,...however physicists tell us that in two dimensions, the standard deviation is of order $|x-y|^{1/3}$ ”, see [62].",
"This suggests that one does not have a Gaussian central limit theorem for the travel time fluctuations in the usual way.",
"This has been rigourously proven [64], [65], [66]."
],
[
"Scaling exponents",
"A well-known result in the 2d lattice case [67] is that $\\chi =1/3$ , $\\xi =2/3$ .",
"Also, another belief is that $\\chi =0$ for dimensions $d$ large enough.",
"Many physicists, see for example [68], [67], [69], [70], [71], [72], [73], also conjecture that independently from the dimension, one should have the so-called KPZ relation between these exponents $\\chi = 2\\xi - 1$ This is connected with the KPZ universality class of random geometry, apparent in many physical situations, including traffic and data flows, and their respective models, including the corner growth model, ASEP, TASEP, etc [74], [19], [75].",
"In particular, FPP is in direct correspondence with important problems in statistical physics [39] such as the directed polymer in random media (DPRM) and the KPZ equation, in which case the dynamical exponent $z$ corresponds to the wandering exponent $\\xi $ defined for the FPP [76], [46]."
],
[
"Bounds on the exponents",
"The situation regarding exact results is more complex [47], [21].",
"The majority of results are based on the model on $\\mathbb {Z}^d$ .",
"Kesten [77] proved that $\\chi \\le 1/2$ in any dimension, and for the wandering exponent $\\xi $ , Licea et al.",
"[78] gave some hints that possibly $\\xi \\ge 1/2$ in any dimension and $\\xi \\ge 3/5$ for $d=2$ .",
"Concerning the KPZ relation, Wehr and Aizenman [79] and Licea et al [78] proved the inequality $\\chi \\ge (1-d\\xi )/2$ in $d$ dimensions.",
"Newman and Piza [80] gave some hints that possibly $\\chi \\ge 2\\xi -1$ .",
"Finally, Chatterjee [47] proved Eq.",
"REF for $\\mathbb {Z}^d$ with independent and identically distributed random edge weights, with some restrictions on distributional properties of the weights.",
"These were lifted by independent work of Auffinger and Damron [21]."
],
[
"Euclidean first passage percolation",
"Euclidean first passage percolation [26] adopts a very different perspective from FPP by considering a fluid flowing along each of the links of the complete graph on the points at some weighted speed given by a function, usually a power, of the Euclidean length of the edge.",
"We ask, between two points of the process at large Euclidean distance $|{x-y}|$ , what is the minimum passage time over all possible routes.",
"More precisely, the original model of Howard and Newman goes as follows.",
"Given a domain $\\mathcal {V}$ such as the Euclidean plane, and $\\mathrm {d}x$ Lesbegue measure on $\\mathcal {V}$ , consider a Poisson point process $\\mathcal {X} \\subset \\mathcal {V}$ of intensity $\\rho \\mathrm {d}x$ , and the function $\\phi : \\mathbb {R}^{+} \\rightarrow \\mathbb {R}^{+}$ satisfying $\\phi (0)=0$ , $\\phi (1)=1$ , and strict convexity.",
"We denote by $K_{\\mathcal {X}}$ the complete graph on $\\mathcal {X}$ .",
"We assign to edges $e = \\lbrace q,q^{\\prime }\\rbrace $ connecting points $q$ and $q^{\\prime }$ the weights $\\tau (e) = \\phi (|{q-q^{\\prime }}|)$ , and a natural choice is $\\phi (x) = x^{\\alpha }, \\quad \\alpha > 1$ The reason for $\\alpha >1$ is that the shortest path is otherwise the direct link, so this introduces non-trivial geodesics.",
"The first work on a Euclidean model of FPP concerned the Poisson-Voronoi tessellation of the $d$ -dimensional Euclidean space by Vahidi-Asl and Wierman in 1992 [81].",
"This sort of generalisation is a long term goal of FPP [21].",
"Much like the lattice model with i.i.d.",
"weights, the model is rotationally invariant.",
"The corresponding shape theorem, discussed in [21], leads to a ball.",
"The existence of bigeodesics (two paths, concatenated, which extend infinitely in two distinct directions from the origin, with the geodesic between the endpoints remaining unchanged), the linear rate of the local growth dynamics (the wetted region grows linearly with time), and the transversal fluctuations of the random path or surface are all summarised in [82]."
],
[
"Bounds on the exponents",
"Licea et al [78] showed that for the standard first-passage percolation on $\\mathbb {Z}^d$ with $d\\ge 2$ , the wandering exponent satisfies $ \\xi (d)\\ge 1/2$ and specifically (2)3/5 In Euclidean FPP, however, these bounds do not hold, and we have [83], [84] 1d+13/4 and, for the wandering exponent, 1-(d-1)2.",
"Combining these different results then yields, for $d=2$ 1/8 1/33/4 Since the KPZ relation of Eq.",
"REF apparently holds in our setting, the lower bound for $\\chi $ implies then a better bound for $\\xi $ , namely 33+d which in the two dimensional case leads to $\\xi \\ge 3/5$ , the same result as in the standard FPP.",
"Also, the `rotational invariance' of the Poisson point process implies the KPZ relation Eq.",
"REF is satisfied in each spatial network we study.",
"We numerically corroborate this in Section .",
"See for example [21] for a discussion of the generality of the relation, and the notion of rotational invariance."
],
[
"EFPP on a spatial network",
"This is the model that we are considering here.",
"Instead of taking as in the usual EFPP into account all possible edges with an exponent $\\alpha >1$ in Eq.",
"REF , we allow only some edges between the points and take the weight proportional to their length (ie.",
"$\\alpha =1$ here).",
"This leads to a different model, but apparently universal properties of the geodesics.",
"We therefore move beyond the weighted complete graph of Howard and Newman, and consider a large class of spatial networks, including the random geometric graph (RGG), the $k$ -nearest neighbour graph (NNG), the $\\beta $ -skeleton (BS), and the Delaunay triangulation (DT).",
"We introduce them in Section .",
"We consider in this study spatial networks constructed over a set of random points.",
"We focus on the most straightforward case, and consider a stationary Poisson point process in the $d$ -dimensional Euclidean space, taking $d=2$ .",
"This constitutes a Poisson random number of points, with expectation given by $\\rho $ per unit area, distributed uniformly at random.",
"We do not discuss here typical generalisations, such as to the Gibbs process, or Papangelou intensities [30].",
"First, we will consider the complete graph as in the usual EFPP, with edges weighted according to the details of Sec.",
"REF .",
"We will then consider the four distinct excluded region graphs defined below.",
"Note that some of these networks actually obey inclusion relations, see for example [15].",
"We have RNG GG DT where RNG stands for the relative neighborhood graph, GG for the Gabriel graph, and DT for the Delaunay triangulation.",
"This nested relation trivially implies the following inequality RNGGGDT as adding links can only decrease the wandering exponent.",
"We are not aware of a similar relation for $\\chi $ .",
"We will also consider three distinct proximity graphs such as the hard and soft RGG, and the $k$ -nearest neighbour graph."
],
[
"Proximity graphs",
"The main idea for constructing these graphs is that two nodes have to be sufficiently near in order to be connected.",
"The usual random geometric graph is defined in [29] and was introduced by Gilbert [85] who assumes that points are randomly located in the plane and have each a communication range $r$ .",
"Two nodes are connected by an edge if they are separated by a distance less than $r$ .",
"We also have the following variant: the soft random geometric graph [86], [87], [10].",
"This is a graph formed on $\\mathcal {X} \\subset \\mathbb {R}^{d}$ by adding an edge between distinct pairs of $\\mathcal {X}$ with probability $H(|x-y|)$ where $H: \\mathbb {R}^{+} \\rightarrow \\left[0,1\\right]$ is called the connection function, and $|x-y|$ is Euclidean distance.",
"We focus on the case of Rayleigh fading where, with $\\gamma > 0$ a parameter and $\\eta > 0$ the path loss exponent, the connection function, with $|x-y|>0$ , is given by $H(|{x-y}|) = \\exp \\left(-\\gamma |{x-y}|^{\\eta }\\right)$ and is otherwise zero.",
"This choice is discussed in [33].",
"This graph is connected with high probability when the mean degree is proportional to the logarithm of the number of nodes in the graph.",
"For the hard RGG, this is given by $\\rho \\pi r^2$ , and otherwise the integral of the connectivity function over the region visible to a node in the domain, scaled by $\\rho $ [87].",
"As such, the graph must have a very large typical degree to connect."
],
[
"$k$ -Nearest Neighbour Graph",
"For this graph, we connect points to their $k \\in \\mathbb {N}$ nearest neighbours.",
"When $k=1$ , we obtain the nearest neighbour graph (1-NNG), see e.g.",
"[88].",
"The model is notably different from the RGG because local fluctuations in the density of nodes do not lead to local fluctuations in the degrees.",
"The typical degree is much lower than the RGG when connected [88], though still remains disconnected on a random, countably infinite subset of the $d$ -dimensional Euclidean space, since isolated subgraph exist.",
"For large enough $k$ , the graph contains the RGG as a subgraph.",
"See Section REF for further discussion.",
"The main idea here for constructing these graphs is that two nodes will be connected if some region between them is empty of points.",
"See Fig.",
"REF for a depiction of the geometry of the lens regions for $\\beta -$ skeletons."
],
[
"Delaunay triangulation",
"The Delaunay triangulation of a set of points is the dual graph of their Voronoi tessellation.",
"One builds the graph by trying to match disks to pairs of points, sitting just on the perimeter, without capturing other points of the process within their bulk.",
"If and only if this can be done, those points are joined by an edge.",
"The triangular distance Delaunay graph can be similarly constructed with a triangle, rather than a disk, but we expect universal exponents.",
"For each simplex within the convex hull of the triangulation, the minimum angle is maximised, leading in general to more realistic graphs.",
"It is also a t-spanner [59], such that with $d=2$ we have the geodesic between two points of the plane along edges of the triangulation to be no more than $t<1.998$ times the Euclidean separation [61].",
"The DT is necessarily connected."
],
[
"$\\beta $ -skeleton",
"The lune-based $\\beta $ -skeleton is a way of naturally capturing the shape of points [89].",
"see Fig.",
"REF .",
"A lune is the intersection of two disks of equal radius, and has a midline joining the centres of the disks and two corners on its perpendicular bisector.",
"For $\\beta \\le 1$ , we define the excluded region of each pair of points $(x,y)$ to be the lune of radius $|x-y| / 2 \\beta $ with corners at $x$ and $y$ .",
"For $\\beta \\ge 1$ we use instead the lune of radius $\\beta |x-y|/2$ , with $x$ and $y$ on the midline.",
"For each value of beta we construct an edge between each pair of points if and only if its excluded region is empty.",
"For $\\beta =1$ , the excluded region is a disk and the beta-skeleton is called the Gabriel graph (GG), whilst for $\\beta =2$ we have the relative neighbourhood graph (RNG).",
"For $\\beta \\le 2$ , the graph is necessarily connected.",
"Otherwise, it is typically disconnected."
],
[
"Numerical setup",
"Given the models in the previous section, we numerically evaluate the scaling exponents $\\chi $ and $\\xi $ , as well as the distribution of the travel time fluctuations.",
"We now describe the numerical setup.",
"With density of points $\\rho >0$ , and a small tolerance $\\epsilon $ , we consider the rectangle domain $\\mathcal {V} = [-w/2-\\epsilon /2,w/2+\\epsilon /2] \\times [-h/2,h/2],$ and place a $n \\sim \\text{Pois}\\left(h\\left(w+\\epsilon \\right) \\rho \\right)$ points uniformly at random in $\\mathcal {V}$ .",
"Then, on these random points, we build a spatial network by connecting pairs of points according to the rules of either the NNG, RGG, $\\beta $ -skeleton for $\\beta =1,2$ , the DT, or the weighted complete graph of EFPP.",
"Two extra points are fixed near the boundary arcs at $(-w/2,0)$ and $(w/2,0)$ , and the Euclidean geodesic is then identified using a variant of Djikstra's algorithm, implemented in Mathematica 11.",
"The tolerance $\\epsilon $ is important for the Soft RGG, since this graph can display geodesics which reach backwards from their starting point, or beyond their destination, before hopping back.",
"We set $\\epsilon = w/10$ .",
"This process is repeated for $N=2000$ graphs, each time taking only a single sample of the geodesic length over the span $w$ between the fixed points on the boundary.",
"This act of taking only a single path is done to avoid any small correlations between their statistics, since the exponents are vulnerable to tiny errors given we need multiple significant figures of precision to draw fair conclusions.",
"It also allows us to use smaller domains.",
"The relatively small value for $N$ is sufficient to determine the exponents at the appropriate computational speed for the larger graphs.",
"The approach in [46] involves rotating the point process before each sample is taken, which is valid alternative method, but we, instead, aim for maximium accuracy given the exponents are not previously conjectured, and therefore need to be determined with exceptional sensitivity, rather than at speed.",
"Note that the fits that we are doing here are over the same typical range as in this work [46].",
"We then increase $w$ , in steps of 3 units of distance, and repeat, until we have statistics of all $w$ , up to the limit of computational feasibility.",
"This varies slightly between models.",
"The RGGs are more difficult to simulate due to their known connectivity constraint where vertex degrees must approach infinity, see e.g.",
"[29].",
"Thus we cannot simulate connected graphs to the same limits of Euclidean span as with the other models.",
"We are then able to relate the mean and standard deviation of the passage time, as well as the mean wandering, to $w$ , at various $\\rho $ , and for each model.",
"For example, the left hand plots in Fig.",
"REF show that the typical passage time $\\mathbb {E}T(x,y) \\sim w$ , i.e.",
"grows linearly with $w$ , for all networks [14], [15], [10].",
"The standard error is shown, but is here not clearly distinguishable from the symbols.",
"We ensure $h$ is large enough to stop the geodesics hitting the boundary, so we use a domain of height equal to the mean deviation $\\mathbb {E}D(w)$ , plus six standard deviations.",
"The key computational difficulty here is the memory requirement for large graphs, of which all $N$ are stored simultaneously, and mapped in parallel on a Linux cluster over a function which measures the path statistic.",
"This parallel processing is used to speed up the computation of the geodesics lengths and wandering."
],
[
"Scaling exponents",
"The results are shown in Figs.",
"REF .",
"These plots, shown in loglog, reveal a power law behaviour of $T$ and $D$ , and the linear growth of typical travel time with Euclidean span.",
"We then compute the exponents to two significant figures using a nonlinear model fit, based on the model $a |x-y|^{b}$ , and then determining the parameters $a,b$ using the quasi-Newton method in Mathematica 11.",
"Figure: The three statistics we observe, expected travel time(a) and (d), expected wandering (b) and (e), and standard deviation of the travel time (c) and (f).",
"The power law exponents are indicated in thelegend.",
"Error bars of one standard deviation are shown for eachpoint.",
"The top plots show the results from the models in firstuniversality class, while the lower plots show the secondclass.",
"The RGG and NNG are distinguished with different colours(green and blue), as are EFPP on the complete graph, the DT, and the two beta skeletons(Gabriel graph, and relative neighbourhood graph).",
"The point process density ρ\\rho points per unit area is given for each model.Our numerical results suggest that we can distinguish two classes of spatial network models by the scaling exponents of their Euclidean geodesics.",
"The proximity graphs (hard and soft RGG, and $k$ -NNG) are in one class, with exponents RGG,NNG=0.20 0.01 RGG,NNG=0.60 0.01 whereas the excluded region graphs (the $\\beta $ -skeletons and Delaunay triangulation), and Howard's EFPP model with $\\alpha >1$ , are in another class with DT,-skel,EFPP=0.40 0.01 DT,-skel,EFPP=0.70 0.01 Clearly, the KPZ relation of Eq.",
"REF is satisfied up to the numerical accuracy which we are able to achieve.",
"We corroborate that this is independent of the density of points and connection range scaling, given the graphs are connected.",
"The exponents hold asymptotically i.e.",
"large inter-point distances.",
"Thus we conjecture Var(T(x,y)) |x-y|4/5, E(D(x,y)) |x-y|7/10 for the proximity graphs (the DT and the $\\beta $ -skeletons for all $\\beta $ ), and, for the RGGs and the $k$ -NNG, Var(T(x,y)) |x-y|2/5, E(D(x,y)) |x-y|3/5, We summarize these new results in Table REF .",
"Table: Exponents ξ\\xi and χ\\chi , and passage time distribution forthe various networks considered.It is surprising that these exponents are apparently rational numbers.",
"In Bernoulli continuum percolation, for example, the threshold connection range for percolation is not known, but not thought to be rational, as it is with bond percolation on the integer lattice [29].",
"Exact exponents are not necessarily expected in the continuum setting of this problem, which suggests there is more to be said about the classification of first passage process via this method."
],
[
"Travel time fluctuations",
"We see numerically that the travel time distribution is a normal for most cases (see Fig.",
"REF ).",
"We summarise these results in the Table REF and in Fig.",
"REF we show the skewness and kurtosis for the travel time fluctuations, computed for the different networks.",
"For a Gaussian distribution, the skewness is 0 and the kurtosis equal to 3, while the Tracy-Widom distribution displays other values.",
"We provide some detail of the distribution of $T$ for each model from the proximity class in Fig.",
"REF .",
"This is compared against four test distributions, the Gaussian orthogonal, unitary and symplectic Tracy Widom distributions, and also the standard normal distribution.",
"Figure: Travel time distributions for the DT (a)-(c), RNG (d)-(f), and Gabriel (g)-(i)graphs, compared with the GUE and GSE Tracy-Widom ensembles,and the Gaussian distribution.",
"The point process density ρ\\rho points per unit area is given for each model.",
"The slight skew of the TWdistribution is not present in the data.This makes the case of EFPP on spatial networks one of only a few special cases where Gaussian fluctuations in fact occur.",
"Auffinger and Damron go into detail concerning each of the remaining cases in [21].",
"One example, reviewed extensively by Chaterjee and Dey [47], is when geodesics are constrained to lie within thin cylinders i.e.",
"ignore paths which traverse too far, and thus select the minima from a subdomain.",
"This result could shed some light on their questions, though in what way it is not clear.",
"We also highlight that Tracy-Widom is thought to occur in problems where matrices represent collections of totally uncorrelated random variables [90].",
"In the case of EFPP, we have the interpoint distances of a point process, which lead to spatially correlated interpoint distances, so the adjacency matrix does not contain i.i.d.",
"values.",
"This potentially leads to the loss of Tracy-Widom.",
"However, we also see some cases of $N \\times N$ large complex correlated Wishart matrices leading to TW for at least one of their eigenvalues, and with convergence at the scale $N^{2/3}$ [91]."
],
[
"Transversal fluctuations",
"The transversal deviation distribution results appear beside our evaluation of the scaling exponents, in Fig.",
"REF .",
"All the models produce geodesics with the same transversal fluctuation distribution, despite distinct values of $\\xi $ .",
"The fluctuations are also distinct from the Brownian bridge (a geometric brownian motion constrained to start and finish at two fixed position vectors in the plane), running between the midpoints of the boundary arcs [19].",
"It is a key open question to provide some information about this distribution, as it is rarely studied in any FPP model, as far as we are aware of the literature.",
"A key work is Kurt Johannson's, where the wandering exponent is derived analytically in a variant of oriented first passage percolation.",
"One might ask if a similar variant of EFPP might be possible [64].",
"Figure: Skewness (a) and kurtosis (b), for the travel timefluctuations, computed for each network model.",
"For aGaussian distribution, the skewness is 0 and the kurtosisequal to 3, values that we indicate by dashed black lines.",
"The point process density ρ\\rho points per unit area is given for each model.",
"TheTracy-Widom distribution has only marginally different moments to the normal, also shown by dashed black lines, with labels added to distinguish each specific distribution (GOE, GUE or GSE), as well as the Gaussian."
],
[
"Discussion",
"The main results of our investigation are the new rational exponents $\\chi $ and $\\xi $ for the various spatial models, and the discovery of the unusual Gaussian fluctuations of the travel time.",
"We found that for the different spatial networks the KPZ relation holds and known bounds are satisfied.",
"Also, due to known relations and the the KPZ law we have 33+d34 It is surprising to find a large class of networks, in particular the Delaunay triangulation, that displays an exponent $\\xi = 7/10$ and points to the question of the existence of another class of graphs which display the theoretically maximal $\\xi =3/4$ .",
"Both immediately present a number of open questions and topics of further research which may shed light on the first passage process on spatial networks.",
"We list below a number of questions that we think are important."
],
[
"Gaussian travel time fluctuations",
"We are not able to conclude that all the models in the proximity graph class $\\chi =3/5$ , $\\xi =1/5$ have Gaussian fluctuations in the travel time.",
"This is for a technical reason.",
"All the models we study are either connected with probability one, such as the DT or $\\beta $ -skeleton with $\\beta \\le 2$ , or have a connection probability which goes to one in some limit.",
"We require connected graphs, or paths do not span the boundary arcs, and the exponents are not well defined.",
"Thus, the difficult models to simulate are the HRGG, SRGG and $k$ -NNG, since these are in fact disconnected with probability one without infinite expected degrees i.e.",
"the dense limit of Penrose, see [29], or with the fixed degree of the $k$ -NNG $k = \\Theta (\\log n)$ and $n \\rightarrow \\infty $ in a domain with fixed density and infinite volume.",
"Otherwise we have isolated vertices, or isolated subgraphs, respectively.",
"However, the $k$ -NNG has typically shorter connection range i.e.",
"in terms of the longest edge, and shortest non-edge, where the `length of a non-edge' is the corresponding interpoint distance between the disconnected vertices [88].",
"So the computations used to produce these graphs and then evaluate their statistical properties are significantly less demanding.",
"Thus, the HRGG is computationally intractable in the necessary dense limit, so we are unable to verify the fluctuations of either $T$ or $D$ .",
"However, we can see a skewness and kurtosis for $T(|x-y|)$ which are monotonically decreasing with $|x-y|$ , towards the hypothesised limiting Gaussian statistics, at least for the limited Euclidean span we can achieve.",
"Given the $k$ -NNG is in the same class, we are left to conjecture Gaussian fluctuations hold throughout all the spatial models described in Section .",
"It remains an open question to identify any exceptional models where this does not hold."
],
[
"Percolation and connectivity",
"If we choose two points at a fixed Euclidean distance, then simulate a Poisson point process in the rest of the $d$ -dimensional plane, construct the relevant graph, and consider the probability that both points are in the giant component, this is effectively a positive constant for reasonable distances, assuming that we are above the percolation transition.",
"At small distances, the two events are positively correlated.",
"Thus, one can condition on this event, and therefore when simulating, discount results where the Euclidean geodesic does not exist.",
"This defines FPP on the giant component of a random graph.",
"It's not clear from our experiments whether the rare isolated nodes, or occasionally larger isolated clusters, either in the RGGs or $k$ -NNG, affect the exponents.",
"One similar model system would be the Lorentz gas: put disks of constant radius in the plane, perhaps at very low density, and seek the shortest path between two points that does not intersect the disks.",
"The exponents $\\chi $ and $\\xi $ for this setting are not known [92], [19].",
"An alternative to giant component FPP would be to condition on the two points being connected to each other.",
"This would be almost identical for the almost connected regime, but weird below the percolation transition.",
"In that case the event we condition on would have a probability decaying exponentially with distance, and the point process would end up being extremely special for the path to even exist.",
"For example, an extremely low density RGG would be almost empty apart from a path of points connecting the end points, with a minimum number of hops."
],
[
"Betweenness centrality",
"The extent to which nodes take part in shortest paths throughout a network is known as betweenness centrality [1], [4].",
"We ask to what extent knowledge of wandering can lead to understanding the centrality of nodes.",
"The variant node shortest path betweenness centrality is highest for nodes in bottlenecks.",
"Can this centrality index be analytically understood in terms of the power law scaling of $D$ ?",
"Is the exponent directly relevant to its large scale behaviour?",
"In order to illustrate more precisely this question, let $G$ be the graph formed on a point process $\\mathcal {X}$ by joining pairs of points with probability $H(|x-y|)$ .",
"Consider $\\sigma _{xy}$ to be the number of shortest paths in $G$ which join vertices $x$ and $y$ in $G$ , and $\\sigma _{xy}(z)$ to be the number of shortest paths which join $x$ to $y$ in $G$ , but also run through $z$ , then with $\\ne $ indicating a sum over unordered pairs of vertices not including $z$ , define the betweenness centrality $g(z)$ of some vertex $z$ in $G$ to be $g(z)=\\sum _{i \\ne j \\ne k} \\frac{\\sigma _{ij}(z)}{\\sigma _{ij}}$ In the continuous limit for dense networks we can discuss the betweenness centrality and we recall some of the results of [11].",
"More precisely, we define $\\chi _{xy}(z)$ as the indicator which gives unity whenever $z$ intersects the shortest path connecting the $d$ -dimensional positions $x,y \\in \\mathcal {V}$ .",
"Then the normalised betweenness $g(z)$ is given by $g(z) = \\frac{1}{\\int _{\\mathcal {V}^{2}} \\chi _{xy}(\\mathbf {0})\\mathrm {d}x\\mathrm {d}y}\\int _{\\mathcal {V}^{2}} \\chi _{xy}(z) \\mathrm {d}x \\mathrm {d}y$ Based on the assumption that there exists a single topological geodesic as $\\rho \\rightarrow \\infty $ , and that this limit also results in an infinitesimal wandering of the path from a straight line segment, an infinite number of points of the process lying on this line segment intersect the topological geodesic as $\\rho \\rightarrow \\infty $ , assuming the graph remains connected, and so $\\chi _{xy}(z)$ can then written as a delta function of the transverse distance from $z$ to the straight line from $x$ to $y$ .",
"The betweenness can then be computed and we obtain [11] (normalised by its maximum value at $g(0)$ ) $g(\\epsilon ) =\\frac{2}{\\pi }\\left(1-\\epsilon ^{2}\\right)E\\left(\\epsilon \\right)$ where $E\\left(k\\right)=\\int _{0}^{\\pi /2}d\\theta \\left(1-k^{2}\\sin ^{2}\\left(\\theta \\right)\\right)^{1/2}$ is the complete elliptic integral of the second kind.",
"We have also rescaled such that $\\epsilon $ is in units of $R$ .",
"Take $D(x,y)$ to be the maximum deviation from the horizontal of the Euclidean geodesic.",
"Numerical simulations suggest that $\\mathbb {E}D(x,y) = C|{x-y}|^{\\xi }$ where the expectation is taken over all point sets $\\mathcal {X}$ .",
"The `thin cylinders' are given by a Heaviside Theta function, so assume that the characteristic function is no longer a delta spike, but a wider cylinder $\\chi _{xy}(z) = \\theta \\left(D(x,y)-z_{\\perp }\\right)$ where $z_{\\perp }$ is the magnitude of the perpendicular deviation of the position $z$ from $\\text{hull}(x,y)$ .",
"We then have that $g(z) = \\frac{1}{\\int _{\\mathcal {V}^{2}}\\theta \\left(D-\\mathbf {0}_{\\perp }\\right)\\mathrm {d}x \\mathrm {d}y}\\int _{\\mathcal {V}^{2}} \\theta \\left(D-z_{\\perp }\\right) \\mathrm {d}V$ (where $\\mathbf {0}$ is the transverse vector computed for the origin).",
"This quantity is certainly difficult to estimate, but provides a starting point for computing finite density corrections to the result of [11].",
"The boundary of the domain is crucial in varying the centrality, which is something we ignore here.",
"Without an enclosing boundary, such as with networks embedded into spheres or tori, the typical centrality at a position in the domain is uniform, since no point is clearly distinguishible from any other.",
"This is discussed in detail on [11].",
"In fact, a significant amount of recent work on random geometric networks has highlighted the importance of the enclosing boundary [33], [86]."
],
[
"Conclusions",
"We have shown numerically that there are two distinct universality classes in Euclidean first passage percolation on a large class of spatial networks.",
"These two classes correspond to the following two broad classes of networks: firstly, based on joining vertices according to critical proximity, such as in the RGG and the NNG, and secondly, based on graphs which connect vertices based on excluded regions, as in the lune-based $\\beta $ -skeletons, or the DT.",
"Heuristically, the most efficient way to connect two points is via the nearest neighbour, which suggests that $\\xi $ for proximity graphs should on the whole be smaller than for exclusion-based graphs, which is in agreement with our numerical observations.",
"The passage times show Gaussian fluctuations in all models, which we are able to numerically verify.",
"This is a clear distinction between EFPP and FPP.",
"After similar results of Chaterjee and Dey [47], it remains an open question why this happens, and also of course how to rigorously prove it.",
"We also briefly discussed notions of the universality of betweenness centrality in spatial networks, which is influenced by the wandering of shortest paths.",
"A number of open questions remain about the range of possible universal exponents which could exist on spatial networks, whose characterisation would shed light on the interplay between the statistical physics of random networks, and their spatial counterparts, in way which could reveal deep insights about universality and geometry more generally.",
"Acknowledgements The authors wish to thank Márton Balázs and Bálint Tóth for a number of very helpful discussions, as well as Ginestra Bianconi at QMUL, Jürgen Jost at MPI Leipzig, and the School of Mathematics at the University of Bristol, who provided generous hosting for APKG while carrying out various parts of this research.",
"This work was supported by the EPSRC project “Spatially Embedded Networks” (grant EP/N002458/1).",
"APKG was partly supported by the EPSRC project “Random Walks on Random Geometric Networks” (grant EP/N508767/1).",
"All underlying data are reproduced in full within the paper."
]
] | 1906.04314 |
[
[
"Morphisms, direct sums and tensor products of K\\\"ahler-Poisson algebras"
],
[
"Abstract In this paper we introduce the concept of morphisms of K\\\"ahler-Poisson algebras and study their algebraic properties.",
"In particular, we find conditions, in terms of the metric, for two algebras to be isomorphic, and we introduce direct sums and tensor products of K\\\"ahler-Poisson algebras.",
"We provide detailed examples to illustrate the novel concepts."
],
[
"Introduction",
"The study of geometry via Poisson algebras goes back to two centuries ago through the works of Lagrange, Poisson and Lie.",
"Poisson [16] invented his brackets as a tool for classical dynamics, Jacobi [10] realized the importance of these brackets and studied their algebraic properties, and Lie [15] began the study of their geometry.",
"The study of geometry via Poisson algebras has experienced an amazing development since the $1980^,$ s, starting with the foundational work of Weinstein [18] on Poisson manifolds.",
"Since then many authors have studied the geometric and algebraic properties of symplectic and Poisson manifolds (see e.g.",
"[5], [8], [9], [11], [14]).",
"Later, Kontsevich [11] has shown that the classification of formal deformations of the algebra of functions for any manifold $\\Sigma $ is equivalent to the classification of formal families of Poisson structures on $\\Sigma $ .",
"However, metric aspects of Poisson manifolds have not been investigated to the same extent.",
"In [3] we began to study these metric aspects by defining Kähler-Poisson algebras as algebraic analogues of the algebra of functions on Kähler manifolds (the concept of Kähler manifolds was introduced by E. Kähler [12]).",
"This study of metric aspects was motivated by the results in [1], [2], where many aspects of the differential geometry of embedded Riemannian manifolds $\\Sigma $ can be formulated in terms of the Poisson structure of the algebra of functions of $\\Sigma $ .",
"In [3] we showed that “the Kähler–Poisson condition”, being the crucial identity in the definition of Kähler-Poisson algebras, allowed for an identification of geometric objects in the Poisson algebra which share crucial properties with their classical counterparts.",
"For instance, we proved the existence of a unique Levi-Civita connection on the module generated by the inner derivations of the Kähler-Poisson algebra, and show that the curvature operator has all the classical symmetries.",
"It is generally interesting to ask how many different (up to isomorphism) Kähler-Poisson structures do there exist on a given Poisson algebra?",
"The aim of this paper is to explore further algebraic properties of Kähler-Poisson algebras.",
"In particular, we find appropriate definitions of morphisms of Kähler-Poisson algebras.",
"We illustrate with examples when two Kähler-Poisson algebras are isomorphic, for instance, we begin by taking algebras $\\mathcal {A}$ and $\\mathcal {A}^{\\prime }$ , where $\\mathcal {A}$ is finitely generated algebra, $\\mathcal {A}^{\\prime }=\\mathcal {A}$ and we consider different set of generators for the Kähler-Poisson algebra structures of $\\mathcal {A}$ and $\\mathcal {A}^{\\prime }$ .",
"We then use the concept of morphism to define subalgebras of Kähler-Poisson algebras.",
"Again, we present examples to understand subalgebras of Kähler-Poisson algebras.",
"Finally, we introduce direct sums and tensor products of Kähler-Poisson algebras together with their basic properties."
],
[
"Kähler-Poisson algebras",
"We begin this section by recalling the main object of our investigation.",
"In [3] we introduce Kähler-Poisson algebras over the field $\\mathbb {K}$ (either $\\mathbb {R}$ or $\\mathbb {C}$ ).",
"Let us consider a Poisson algebra $(\\mathcal {A},\\lbrace .,.\\rbrace )$ , over a field $\\mathbb {K}$ and let $\\lbrace x^1,...,x^m\\rbrace $ be a set of distinguished elements of $\\mathcal {A}$ .",
"These elements play the role of functions providing an embedding into $\\mathbb {R}^m$ for Kähler manifolds.",
"Kähler manifolds and their properties have been extensively studied (see e.g.",
"[5], [14], [18], [19]).",
"Let us recall the definition of Kähler-Poisson algebras together with a few basic results.",
"Definition 2.1 Let $(\\mathcal {A},\\lbrace \\cdot ,\\cdot \\rbrace )$ be a Poisson algebra over $\\mathbb {K}$ and let ${x^1,...,x^m} \\in \\mathcal {A}$ .",
"Given a symmetric $m \\times m$ matrix $g=(g_{ij})$ with entries $g_{ij}\\in \\mathcal {A}$ , for $i,j=1,...,m$ , we say that the triple $\\mathcal {K}=(\\mathcal {A},g,\\lbrace x^1,...,x^m\\rbrace )$ is a Kähler–Poisson algebra if there exists $\\eta \\in \\mathcal {A}$ such that $ \\sum \\limits _{i,j,k,l}^m\\eta \\lbrace a,x^i\\rbrace g_{ij}\\lbrace x^j,x^k\\rbrace g_{kl}\\lbrace x^l,b\\rbrace =-\\lbrace a,b\\rbrace $ for all $a,b$ $\\in $ $\\mathcal {A}$ .",
"We call equation (REF ) “the Kähler–Poisson condition”.",
"Given a Kähler-Poisson algebra $\\mathcal {K}=(\\mathcal {A},g,\\lbrace x^1,...,x^m\\rbrace )$ , let $\\mathfrak {g}$ denote the $\\mathcal {A}$ -module generated by all $\\emph {inner derivations}$ , i.e.",
"$\\mathfrak {g}=\\lbrace a_1\\lbrace c^1,.\\rbrace +...+a_N\\lbrace c^N,.\\rbrace :$ $a_i,c^i \\in \\mathcal {A}$ and $N\\in \\mathbb {N}\\rbrace $ .",
"It is a standard fact that $\\mathfrak {g}$ is a Lie algebra over $\\mathbb {K}$ with respect to the bracket $[\\alpha ,\\beta ](a)=\\alpha (\\beta (a))-\\beta (\\alpha (a))$ , where $\\alpha ,\\beta \\in \\mathfrak {g}$ and $a \\in \\mathcal {A}$ (see e.g.",
"[7], [13]).",
"It was shown in [3] that $\\mathfrak {g}$ is a projective module and that every Kähler–Poisson algebra is a Lie-Rinehart algebra.",
"More details for Lie-Rinehart algebra can be found in [9], [17].",
"Moreover, it follows that the matrix $g$ defines a metric on $\\mathfrak {g}$ by $g(\\alpha ,\\beta )=\\alpha (x^i)g_{ij}\\beta (x^j)$ , for $\\alpha ,\\beta \\in \\mathfrak {g}$ .",
"Let us now introduce some notation for Kähler–Poisson algebras.",
"We set $\\mathbb {\\mathcal {P}}^{ij}$ =$\\lbrace x^i,x^j\\rbrace $ and $\\mathbb {\\mathcal {P}}^i(a)$ =$\\lbrace x^i,a\\rbrace $ for $a\\in \\mathcal {A}$ , as well as $\\mathcal {D}^{ij}=\\eta \\lbrace x^i,x^l\\rbrace g_{lk}\\lbrace x^j,x^k\\rbrace $ $\\mathcal {D}^{i}(a) =\\eta \\lbrace x^k,a\\rbrace g_{kl}\\lbrace x^l,x^i\\rbrace $ .",
"Note that $\\mathcal {D}^{ij}=\\mathcal {D}^{ji}$ .",
"The metric $g$ will be used to lower indices in analogy with differential geometry.",
"E.g.",
"$\\mathcal {P}^i_{\\;\\;j}=\\mathcal {P}^{ik}g_{kj} \\quad \\mathcal {D}^i_{\\;\\;j}=\\mathcal {D}^{ik}g_{kj} \\quad \\mathcal {D}_i=g_{ij}\\mathcal {D}^j$ .",
"In this notation, (REF ) can be stated as $\\mathcal {D}_{i}(a)\\mathcal {P}^{i}(b)=\\lbrace a,b\\rbrace .$ Furthermore, one immediately derives the following identities $\\mathcal {D}^{ij}\\mathcal {P}_j(a)=\\mathcal {P}^i(a),\\\\\\\\\\\\\\ \\mathcal {P}^{ij}\\mathcal {D}_j(a)=\\mathcal {P}^i(a)\\\\\\\\\\ \\text{and} \\\\\\\\\\ \\mathcal {D}^{i}_j\\mathcal {D}^{jk}=\\mathcal {D}^{jk}.$ If we denote by $\\mathcal {P}$ the matrix with entries $\\mathcal {P}^{ij}$ , the Kähler-Poisson condition (REF ) in Definition REF , can be written in matrix notation as $\\eta \\mathcal {P}g\\mathcal {P}g\\mathcal {P}=-\\mathcal {P}$ , for an algebra $\\mathcal {A}$ generated by $\\lbrace x^1,...,x^m\\rbrace $ .",
"Let us also recall Example 6.1 in [3] (see also [4]).",
"This example shows that any Poisson algebra generated by two elements can be endowed with a Kähler-Poisson algebra structure.",
"Example 2.2 [3] Let $\\mathcal {A}$ be a Poisson algebra generated by two elements $x$ and $y\\in \\mathcal {A}$ and let $\\mathcal {P}=\\begin{pmatrix}0 & \\lbrace x,y\\rbrace \\\\-\\lbrace x,y\\rbrace & 0 \\\\\\end{pmatrix}$ .",
"It is easy to check that for an arbitrary symmetric matrix with ($\\det (g) \\ne 0$ ) $g=\\begin{pmatrix}a & c \\\\c & b \\\\\\end{pmatrix}$ one obtains $\\mathcal {P}g\\mathcal {P}g &=\\begin{pmatrix}c\\lbrace x,y\\rbrace & b\\lbrace x,y\\rbrace \\\\-a\\lbrace x,y\\rbrace & -c\\lbrace x,y\\rbrace \\\\\\end{pmatrix}\\begin{pmatrix}c\\lbrace x,y\\rbrace & b\\lbrace x,y\\rbrace \\\\-a\\lbrace x,y\\rbrace & -c\\lbrace x,y\\rbrace \\\\\\end{pmatrix}\\\\&=\\begin{pmatrix}(c\\lbrace x,y\\rbrace )^2-ab\\lbrace x,y\\rbrace ^2& 0 \\\\0 & (c\\lbrace x,y\\rbrace )^2-ab\\lbrace x,y\\rbrace ^2\\\\\\end{pmatrix}.$ Therefore $\\mathcal {P}g\\mathcal {P}g\\mathcal {P} &=\\begin{pmatrix}0& \\lbrace x,y\\rbrace ( (c\\lbrace x,y\\rbrace )^2-ab\\lbrace x,y\\rbrace ^2) \\\\-\\lbrace x,y\\rbrace ( (c\\lbrace x,y\\rbrace )^2-ab\\lbrace x,y\\rbrace ^2) & 0\\\\\\end{pmatrix}\\\\&= -\\lbrace x,y\\rbrace ^2(ab-c^2)\\mathcal {P}\\\\&= -\\lbrace x,y\\rbrace ^2 \\det (g)\\mathcal {P},$ giving $\\eta =(\\lbrace x,y\\rbrace ^2\\det (g))^{-1}$ , provided that the inverse exists.",
"Thus, as long as $\\lbrace x,y\\rbrace ^2 \\det (g)$ is not a zero-divisor, one may localize $\\mathcal {A}$ to obtain a Kähler-Poisson algebra $\\mathcal {K}=(\\mathcal {A}[(\\lbrace x,y\\rbrace ^2\\det (g))^{-1}],g,\\lbrace x,y\\rbrace )$ ."
],
[
"Homomorphisms of Kähler-Poisson algebras",
"In this section we are interested in studying maps between Kähler-Poisson algebras.",
"Given a Poisson algebra, we will investigate isomorphism classes of Kähler-Poisson algebra structures on the Poisson algebra.",
"As the definition of a Kähler-Poisson algebra involves the choice of a set of distinguished elements, we will require a morphism to respect the subalgebra generated by these elements.",
"To this end, we start by making the following definition: Definition 3.1 Given a Kähler-Poisson algebra $\\mathcal {K}=(\\mathcal {A},g,\\lbrace x^1,...,x^m\\rbrace )$ on a Poisson algebra $(\\mathcal {A},\\lbrace \\cdot ,\\cdot \\rbrace )$ , let $\\mathcal {A}_\\text{fin} \\subseteq \\mathcal {A}$ denote the Poisson subalgebra generated by $\\lbrace x^1,...,x^m\\rbrace $ .",
"We now introduce morphisms of Kähler-Poisson algebras in the following way: Definition 3.2 Let $\\mathcal {K}=(\\mathcal {A},g,\\lbrace x^1,...,x^m\\rbrace )$ and $\\mathcal {K}^{\\prime }=(\\mathcal {A}^{\\prime },g^{\\prime },\\lbrace y^1,...,{y^m}^{\\prime }\\rbrace )$ be Kähler-Poisson algebras together with their modules of derivations $\\mathfrak {g}$ and $\\mathfrak {g}^{\\prime }$ , respectively.",
"A (homo)morphism of Kähler-Poisson algebras is a pair of maps $(\\phi ,\\psi )$ , with $\\phi :\\mathcal {A}\\rightarrow \\mathcal {A}^{\\prime }$ a Poisson algebra homomorphism and $\\psi :\\mathfrak {g}\\rightarrow \\mathfrak {g}^{\\prime }$ a Lie algebra homomorphism, such that $\\psi (a\\alpha )=\\phi (a)\\psi (\\alpha )$ , $\\phi (\\alpha (a))=\\psi (\\alpha )(\\phi (a))$ , $\\phi (g(\\alpha ,\\beta ))=g^{\\prime }(\\psi (\\alpha ),\\psi (\\beta ))$ , $\\phi (\\mathcal {A}_\\text{fin})\\subseteq \\mathcal {A}^{\\prime }_{\\text{fin}} $ , for all $a \\in \\mathcal {A}$ and $\\alpha ,\\beta \\in \\mathfrak {g}$ .",
"Remark 3.3 Note that a morphism of Kähler-Poisson algebras is also a morphism of the underlying Lie-Rinehart algebras.",
"Furthermore, in next Proposition we show that the composition of two Kähler-Poisson algebras morphisms is a morphism of Kähler-Poisson algebras as we expect and require.",
"Proposition 3.4 Let $\\mathcal {K}=(\\mathcal {A},g,\\lbrace x^1,...,x^m\\rbrace )$ and $\\mathcal {K}^{\\prime }=(\\mathcal {A}^{\\prime },g^{\\prime },\\lbrace y^1,...,{y^m}^{\\prime }\\rbrace )$ be Kähler-Poisson algebras together with their corresponding modules of derivations $\\mathfrak {g}$ , $\\mathfrak {g}^{\\prime }$ and let $\\mathcal {K^{\\prime \\prime }}=(\\mathcal {A^{\\prime \\prime }},g^{\\prime \\prime },\\lbrace z^1,...,z^m{^{\\prime \\prime }}\\rbrace )$ be a Kähler-Poisson algebra with its module of derivations $\\mathfrak {g}^{\\prime \\prime }$ .",
"If $(\\phi ,\\psi ):\\mathcal {K}\\rightarrow \\mathcal {K}^{\\prime }$ and $(\\phi ^{\\prime },\\psi ^{\\prime }):\\mathcal {K}^{\\prime }\\rightarrow \\mathcal {K}^{\\prime \\prime }$ are homomorphisms of Kähler-Poisson algebras then $(\\phi ^{\\prime }\\circ \\phi ,\\psi ^{\\prime }\\circ \\psi ):\\mathcal {K}\\rightarrow \\mathcal {K}^{\\prime \\prime }$ is a homomorphism of Kähler-Poisson algebras.",
"Let $\\hat{\\phi }=\\phi ^{\\prime }\\circ \\phi $ and $\\hat{\\psi }=\\psi ^{\\prime }\\circ \\psi $ , where $\\phi :\\mathcal {A}\\rightarrow \\mathcal {A}^{\\prime }$ , $\\phi ^{\\prime }:\\mathcal {A}^{\\prime } \\rightarrow \\mathcal {A}^{\\prime \\prime }$ , $\\psi :\\mathfrak {g}\\rightarrow \\mathfrak {g}^{\\prime }$ and $\\psi ^{\\prime }:\\mathfrak {g}^{\\prime }\\rightarrow \\mathfrak {g}^{\\prime \\prime }$ , giving $\\hat{\\phi }:\\mathcal {A} \\rightarrow \\mathcal {A}^{\\prime \\prime }$ and $\\hat{\\psi }:\\mathfrak {g}\\rightarrow \\mathfrak {g}^{\\prime \\prime }$ .",
"To prove that $(\\hat{\\phi },\\hat{\\psi })$ is a homomorphism of Kähler-Poisson algebras we show that $(\\hat{\\phi },\\hat{\\psi })$ satisfies properties (1-4) in Definition REF .",
"(1) $\\hat{\\psi }(a\\alpha )=\\hat{\\phi }(a)\\hat{\\psi }(\\alpha )$ , for all $\\alpha \\in \\mathfrak {g}$ and $a \\in \\mathcal {A}$ .",
"$\\psi ^{\\prime }(\\psi (a\\alpha ))-\\phi ^{\\prime }(\\phi (a))\\psi ^{\\prime }(\\psi (\\alpha ))&=\\psi ^{\\prime }(\\phi (a)\\psi (\\alpha ))-\\phi ^{\\prime }(\\phi (a))\\psi ^{\\prime }(\\psi (\\alpha ))\\\\&=\\phi ^{\\prime }(\\phi (a))\\psi ^{\\prime }(\\psi (\\alpha ))-\\phi ^{\\prime }(\\phi (a))\\psi ^{\\prime }(\\psi (\\alpha ))=0.$ (2) $\\hat{\\phi }(\\alpha (a))=\\hat{\\psi }(\\alpha )(\\hat{\\phi }(a))$ , for all $\\alpha \\in \\mathfrak {g}$ and $a \\in \\mathcal {A}$ .",
"$\\hat{\\phi }(\\alpha (a))-\\hat{\\psi }(\\alpha )(\\hat{\\phi }(a)) &=\\phi ^{\\prime }(\\phi (\\alpha (a)))-\\psi ^{\\prime }(\\psi (\\alpha ))(\\phi ^{\\prime }(\\phi (a)))\\\\&=\\phi ^{\\prime }(\\psi (\\alpha )(\\phi (a)))-\\psi ^{\\prime }(\\psi (\\alpha ))(\\phi ^{\\prime }(\\phi (a)))\\\\&=\\psi ^{\\prime }(\\psi (\\alpha )(\\phi ^{\\prime }(\\phi (a)))-\\psi ^{\\prime }(\\psi (\\alpha ))(\\phi ^{\\prime }(\\phi (a)))=0.$ (3) $\\hat{\\phi }(g(\\alpha ,\\beta ))=g^{\\prime \\prime }(\\hat{\\psi }(\\alpha ),\\hat{\\psi }(\\beta ))$ , for $\\alpha ,\\beta \\in \\mathcal {A}$ .",
"$\\hat{\\phi }(g(\\alpha ,\\beta ))-g^{\\prime \\prime }(\\hat{\\psi }(\\alpha ),\\hat{\\psi }(\\beta )) &=\\phi ^{\\prime }(\\phi (g(\\alpha ,\\beta )))-g^{\\prime \\prime }(\\psi ^{\\prime }(\\psi (\\alpha )),\\psi ^{\\prime }(\\psi (\\beta )))\\\\&=\\phi ^{\\prime }(g^{\\prime }(\\psi (\\alpha ),\\psi (\\beta )))-g^{\\prime \\prime }(\\psi ^{\\prime }(\\psi (\\alpha )),\\psi ^{\\prime }(\\psi (\\beta )))\\\\&=g^{\\prime \\prime }(\\psi ^{\\prime }(\\psi (\\alpha )),\\psi ^{\\prime }(\\psi (\\beta )))-g^{\\prime \\prime }(\\psi ^{\\prime }(\\psi (\\alpha )),\\psi ^{\\prime }(\\psi (\\beta )))=0.$ (4) $\\phi ^{\\prime }(\\phi (\\mathcal {A}_\\text{fin}))\\subseteq \\mathcal {A}^{\\prime \\prime }_{\\text{fin}} $ , since, $\\phi (\\mathcal {A}_\\text{fin})\\subseteq \\mathcal {A}^{\\prime }_{\\text{fin}} $ and $\\phi ^{\\prime }(\\mathcal {A}^{\\prime }_\\text{fin})\\subseteq \\mathcal {A}^{\\prime \\prime }_{\\text{fin}} $ .",
"Therefore, $(\\hat{\\phi },\\hat{\\psi })$ is a homomorphism of Kähler-Poisson algebras.",
"Remark 3.5 Note that, the composition of Kähler-Poisson algebra morphism is associative and the identity $(\\operatorname{id}_\\mathcal {A},\\operatorname{id}_\\mathfrak {g})$ of $\\mathcal {K}=(\\mathcal {A},g,\\lbrace x^1,...,x^m\\rbrace )$ with $\\operatorname{id}_\\mathcal {A}:\\mathcal {A}\\rightarrow \\mathcal {A}$ and $\\operatorname{id}_\\mathfrak {g}:\\mathfrak {g}\\rightarrow \\mathfrak {g}$ is a morphism of Kähler-Poisson algebras.",
"Remark 3.6 An isomorphism of Kähler-Poisson algebras is a morphism $(\\phi ,\\psi )$ of Kähler-Poisson algebras such that $\\phi $ is a Poisson algebra isomorphism and $\\phi (\\mathcal {A}_\\text{fin})= \\mathcal {A}^{\\prime }_{\\text{fin}}$ .",
"Observe that, $\\psi $ can always be constructed from $\\phi $ .",
"When two Kähler-Poisson algebras $\\mathcal {K}$ and $\\mathcal {K}^{\\prime }$ are isomorphic, we write $\\mathcal {K}\\cong \\mathcal {K}^{\\prime }$ .",
"Before continuing, we need to introduce some notation.",
"Let $(\\mathcal {A},\\lbrace .,.\\rbrace )$ and $(\\mathcal {A}^{\\prime },\\lbrace .,.\\rbrace ^{\\prime })$ be Poisson algebras and let $x^i\\in \\mathcal {A}$ for $i=1,...,m.$ If $p \\in \\mathcal {A}$ is a polynomial in $\\lbrace x^1,...,x^m\\rbrace $ then, using Leibniz rule, one computes $\\lbrace p,a\\rbrace =\\frac{\\partial {p}}{\\partial {x^i}}\\lbrace x^i,a\\rbrace $ where $\\frac{\\partial {p}}{\\partial {x^i}}$ denotes the formal derivative of the polynomial $p$ with respect to the variable $x^i$ .",
"Note that, in general, $\\frac{\\partial {p}}{\\partial {x^i}}$ is itself not well-defined in the algebra, since there might exist several different (but equivalent) representations of $p$ as a polynomial in $x^1,...,x^m$ , and the formal derivative then yields several, possibly non-equivalent, elements of the algebra.",
"However, the combination in (REF ) is always well-defined, and gives the same result for all representations of $p$ .",
"Given a matrix $M=(m_{ij})$ over $\\mathcal {A}$ , we set $\\phi (M)=(\\phi (m_{ij}))$ .",
"Given a morphism $(\\phi ,\\psi ):(\\mathcal {A},g,\\lbrace x^1,...,x^m\\rbrace )\\rightarrow (\\mathcal {A}^{\\prime },g^{\\prime },\\lbrace y^1,...,{y^m}^{\\prime }\\rbrace )$ , it will be convenient to introduce the notation ${A^i}_\\alpha =\\frac{\\partial \\phi (x^i)}{\\partial y^\\alpha }$ (keeping in mind that this is not well-defined by itself); recall that if $(\\phi ,\\psi )$ is a morphism of Kähler-Poisson algebras, then $\\phi (\\mathcal {A}_\\text{fin})\\subseteq \\mathcal {A}^{\\prime }_{\\text{fin}}$ , ensuring that $\\phi (x^i)$ is indeed a polynomial in $y^1,...,{y^m}^{\\prime }$ .",
"This notation allows us to write $\\phi (\\lbrace x^i,x^j\\rbrace )=\\lbrace \\phi (x^i),\\phi (x^j)\\rbrace ^{\\prime }={A^i}_\\alpha \\lbrace y^\\alpha ,y^\\beta \\rbrace ^\\prime {A^j}_\\beta $ in matrix notation: $\\phi (\\mathcal {P})=A\\mathcal {P}^\\prime A^T$ , where $\\mathcal {P}=(\\lbrace x^i,x^j\\rbrace )$ and $\\mathcal {P}^\\prime =(\\lbrace y^\\alpha ,y^\\beta \\rbrace ^\\prime )$ .",
"Given two Kähler-Poisson algebras $\\mathcal {K}=(\\mathcal {A},g,\\lbrace x^1,...,x^m\\rbrace )$ and $\\mathcal {K}^{\\prime }=(\\mathcal {A}^{\\prime },g^{\\prime },\\lbrace y^1,...,{y^m}^{\\prime }\\rbrace )$ , we would like to understand when they are isomorphic.",
"In the following, we shall consider a number of examples in order to explore when Kähler-Poisson algebras are isomorphic.",
"From now on, we consider only finitely generated Poisson algebras that have been properly localized (cf.",
"Example REF ) to allow the construction of a Kähler-Poisson algebra.",
"We begin by taking finitely generated algebras $\\mathcal {A}$ and $\\mathcal {A}^{\\prime }$ , where $\\mathcal {A}=\\mathcal {A}^{\\prime }$ and we consider different set of generators for the Kähler-Poisson algebra structures of $\\mathcal {A}$ and $\\mathcal {A}^{\\prime }$ .",
"Let us start with the following example.",
"Example 3.7 Let $(\\mathcal {A},\\lbrace .,.\\rbrace )$ be a Poisson algebra generated by two elements $x$ and $y$ .",
"From Example REF we know that $\\mathcal {K}=(\\mathcal {A}[(\\lbrace x,y\\rbrace ^2\\det (g))^{-1}],g,\\lbrace x,y\\rbrace )$ is a Kähler-Poisson algebra for an arbitrary symmetric matrix $g= \\begin{pmatrix}g_{11} & g_{12} \\\\g_{12} & g_{22}\\\\\\end{pmatrix}$ $, $ (g)0$ with $ =({x,y}2(g))-1$.$ Consider $\\mathcal {K}^{\\prime }=(\\mathcal {A}[(\\lbrace x,y\\rbrace ^2\\det (g))^{-1}],h,\\lbrace x+y,x-y\\rbrace )$ , with a symmetric matrix $h= \\begin{pmatrix}h_{11} & h_{12} \\\\h_{12} & h_{22} \\\\\\end{pmatrix} $ , $\\det (h)\\ne 0$ .",
"We have $ \\mathcal {P}^{\\prime }= \\begin{pmatrix}\\lbrace x+y,x+y\\rbrace & \\lbrace x+y,x-y\\rbrace \\\\\\lbrace x+y,x-y\\rbrace &\\lbrace x-y,x-y\\rbrace \\\\\\end{pmatrix}=\\begin{pmatrix}0 & -2\\lbrace x,y\\rbrace \\\\2\\lbrace x,y\\rbrace &0\\\\\\end{pmatrix}.",
"$ It is easy to check that for the symmetric matrix $h$ we obtain $\\mathcal {P}^{\\prime }h\\mathcal {P}^{\\prime }h\\mathcal {P}^{\\prime }=-4\\lbrace x,y\\rbrace ^2 \\det (h)\\mathcal {P}^{\\prime },$ giving $\\eta ^\\prime =(4\\lbrace x,y\\rbrace ^2 \\det (h))^{-1}$ .",
"For each metric $g$ in the definition of $\\mathcal {K}$ , we find a suitable matrix $h$ such that $\\mathcal {K}\\cong \\mathcal {K}^{\\prime }$ .",
"From $\\mathfrak {g}=\\lbrace a_1\\lbrace x,\\cdot \\rbrace +a_2\\lbrace y,\\cdot \\rbrace :a_1,a_2\\in \\mathcal {A}\\rbrace $ , and $\\mathfrak {g}^{\\prime }&=\\lbrace a_1\\lbrace x+y,\\cdot \\rbrace +a_2\\lbrace x-y,\\cdot \\rbrace :a_1,a_2\\in \\mathcal {A}\\rbrace \\\\&=\\lbrace a_1\\lbrace x,\\cdot \\rbrace +a_1\\lbrace y,\\cdot \\rbrace +a_2\\lbrace x,\\cdot \\rbrace -a_2\\lbrace y,\\cdot \\rbrace : a_1,a_2 \\in \\mathcal {A}\\rbrace \\\\&=\\lbrace (a_1+a_2)\\lbrace x,\\cdot \\rbrace +(a_1-a_2)\\lbrace y,\\cdot \\rbrace : a_1,a_2\\in \\mathcal {A}\\rbrace ,$ we see that $\\mathfrak {g}=\\mathfrak {g}^{\\prime }$ .",
"Since we require that $\\mathcal {K}\\cong \\mathcal {K}^{\\prime }$ , we define maps $\\phi :\\mathcal {A}\\rightarrow \\mathcal {A}$ and $\\psi :\\mathfrak {g}\\rightarrow \\mathfrak {g}$ satisfying properties (1-4).",
"We choose $\\phi =id$ , $\\psi =id$ and find a suitable choice of matrix $h$ yielding that $(\\phi ,\\psi )$ is an isomorphism of Kähler Poisson algebras.",
"Now, we check that properties (1-4) in Definition REF are satisfied.",
"$(1)$ $\\phi (a)\\psi (\\alpha )=a\\alpha =\\psi (a\\alpha )$ $(2)$ $\\psi (\\alpha )(\\phi (a))=\\alpha (a)=\\phi (\\alpha (a))$ $(3)$ To show that $g(\\alpha ,\\beta )=h(\\alpha ,\\beta )$ , we start from the left hand side $\\phi (g(\\alpha ,\\beta ))&=\\alpha (x^i)g_{ij}\\beta (x^j)\\\\&= \\alpha (x)g_{11}\\beta (x)+\\alpha (x)g_{12}\\beta (y)+\\alpha (y)g_{21}\\beta (x)+\\alpha (y)g_{22}\\beta (y).$ From the right hand side we get $h(\\alpha ,\\beta ) &= \\alpha (x^i)h_{ij}\\beta (x^j)\\\\&=\\alpha (x+y)h_{11}\\beta (x+y)+\\alpha (x+y)h_{12}\\beta (x-y)+\\alpha (x-y)h_{21}\\beta (x+y)\\\\& \\quad +\\alpha (x-y)h_{22}\\beta (x-y)\\\\&=\\alpha (x)h_{11}\\beta (x)+\\alpha (x)h_{11}\\beta (y)+\\alpha (y)h_{11}\\beta (x)+\\alpha (y)h_{11}\\beta (y)+\\alpha (x)h_{12}\\beta (x)+\\\\& \\quad -\\alpha (x)h_{12}\\beta (y)+\\alpha (y)h_{12}\\beta (x)-\\alpha (y)h_{12}\\beta (y)+\\alpha (x)h_{21}\\beta (x)+\\alpha (x)h_{21}\\beta (y)\\\\&\\quad -\\alpha (y)h_{21}\\beta (x)-\\alpha (y)h_{21}\\beta (y)+\\alpha (x)h_{22}\\beta (x)-\\alpha (x)h_{22}\\beta (y)-\\alpha (y)h_{22}\\beta (x)\\\\&\\quad +\\alpha (y)h_{22}\\beta (y)\\\\&=\\alpha (x)h_{11}\\beta (x)+\\alpha (x)h_{11}\\beta (y)+\\alpha (y)h_{11}\\beta (x)+\\alpha (y)h_{11}\\beta (y)+2\\alpha (x)h_{12}\\beta (x)\\\\& \\quad -2\\alpha (y)h_{12}\\beta (y)+\\alpha (x)h_{22}\\beta (x)-\\alpha (x)h_{22}\\beta (y)-\\alpha (y)h_{22}\\beta (x)+\\alpha (y)h_{22}\\beta (y).$ We conclude that $&\\begin{aligned}\\alpha (x)\\beta (x): h_{11}+2h_{12}+h_{22}=g_{11}\\end{aligned}\\\\&\\begin{aligned}\\alpha (x)\\beta (y): h_{11}-h_{22}=g_{12}\\end{aligned}\\\\&\\begin{aligned}\\alpha (y)\\beta (x): h_{11}-h_{22}=g_{21}\\end{aligned}\\\\&\\begin{aligned}\\alpha (y)\\beta (y): h_{11}-2h_{12}+h_{22}=g_{22}.\\end{aligned}$ From () we obtain, $ h_{11}=2h_{12}-h_{22}+g_{22} $ and setting this in (REF ) we get $ 2h_{12}-h_{22}+g_{22}+2h_{12}+h_{22}=g_{11}$ $4h_{12}=g_{11}-g_{22}$ $h_{12}=\\frac{g_{11}-g_{22}}{4}$ .",
"From () we get $h_{22}=h_{11}-g_{12}$ , which in (REF ) gives $h_{11}+2(\\frac{g_{11}-g_{22}}{4})+h_{11}-g_{12} =g_{11}$ $2h_{11}+(\\frac{g_{11}-g_{22}}{2})-g_{12} =g_{11}$ $4h_{11}+g_{11}-g_{22} =2(g_{11}+g_{12})$ $4h_{11}=g_{11}+2g_{12}+g_{22}$ $h_{11}=\\frac{g_{11}+2g_{12}+g_{22}}{4}$ , which implies that $h_{22}=h_{11}-g_{12}=\\frac{g_{11}+2g_{12}+g_{22}}{4}-g_{12}=\\frac{g_{11}-2g_{12}+g_{22}}{4}.$ Now, the symmetric matrix $h$ becomes $h= \\begin{pmatrix}\\frac{g_{11}+2g_{12}+g_{22}}{4} &\\frac{g_{11}-g_{22}}{4} \\\\\\frac{g_{11}-g_{22}}{4} & \\frac{g_{11}-2g_{12}+g_{22}}{4} \\\\\\end{pmatrix} $ , giving $\\det (h)&=\\frac{1}{16}\\Big [(g_{11}+2g_{12}+g_{22})(g_{11}-2g_{12}+g_{22})-(g_{11}-g_{22})^2\\Big ]\\\\&=\\frac{1}{16}\\Big [(g_{11})^2-2g_{11}g_{12}+g_{11}g_{22}+2g_{11}g_{12}-4(g_{12})^2+2g_{12}g_{22}+g_{11}g_{22}\\\\&\\quad -2g_{12}g_{22}+(g_{22})^2-(g_{11})^2-(g_{22})^2+2g_{11}g_{22}\\Big ]\\\\&=\\frac{1}{16}(4g_{11}g_{22}-4(g_{12})^2)=\\frac{1}{4}(g_{11}g_{22}-(g_{12})^2)=\\frac{1}{4}\\det (g).$ Inserting $\\det (h)$ in equation REF we get $\\mathcal {P}^{\\prime }h\\mathcal {P}^{\\prime }h\\mathcal {P}^{\\prime }=-4\\lbrace x,y\\rbrace ^2 \\frac{1}{4}\\det (g)\\mathcal {P}^{\\prime }=-\\lbrace x,y\\rbrace ^2\\det (g)\\mathcal {P}^{\\prime },$ giving $\\eta ^{\\prime }=(\\lbrace x,y\\rbrace ^2\\det (g))^{-1}$ .",
"Therefore $\\eta ^\\prime =\\eta $ .",
"We conclude that $g(\\alpha ,\\beta )=h(\\alpha ,\\beta )$ .",
"$(4)$ It is easy to see that $\\phi (\\mathcal {A}_\\text{fin})\\subseteq \\mathcal {A}^{\\prime }_{\\text{fin}} $ .",
"Hence, $\\mathcal {K}\\cong \\mathcal {K}^{\\prime }$ if $h=\\begin{pmatrix}\\frac{g_{11}+2g_{12}+g_{22}}{4} &\\frac{g_{11}-g_{22}}{4} \\\\\\frac{g_{11}-g_{22}}{4} & \\frac{g_{11}-2g_{12}+g_{22}}{4} \\\\\\end{pmatrix}.$ Note that the above example extends to the case where we choose more (dependent) generators of the algebra, giving many possible presentations of the same Kähler-Poisson algebra.",
"Next, let us explore the case when we choose a different number of generators for the same algebra.",
"Example 3.8 Let $(\\mathcal {A},\\lbrace \\cdot ,\\cdot \\rbrace )$ be a Poisson algebra generated by two elements $x$ and $y$ .",
"Consider the Kähler-Poisson algebra $\\mathcal {K}=(\\mathcal {A}[(\\lbrace x,y\\rbrace ^2\\det (g))^{-1}],g,\\lbrace x,y\\rbrace )$ , with an arbitrary symmetric matrix $g= \\begin{pmatrix}g_{11} & g_{12} \\\\g_{12} & g_{22}\\\\\\end{pmatrix}$$, $ (g)0$ and$ =({x,y}2(g))-1$.$ Let $\\mathcal {K}^{\\prime }=(\\mathcal {A}[(\\lbrace x,y\\rbrace ^2\\det (g))^{-1}],h,\\lbrace x,y,x\\rbrace )$ with $\\mathcal {P}^{\\prime }= \\begin{pmatrix}0 & \\lambda & 0 \\\\-\\lambda &0 & -\\lambda \\\\0 & \\lambda & 0 \\\\\\end{pmatrix} $ , where $\\lambda =\\lbrace x,y\\rbrace $ .",
"It is easy to check that for the symmetric matrix h $h= \\begin{pmatrix}\\frac{1}{4}g_{11} & \\frac{1}{2}g_{12} & \\frac{1}{4}g_{11} \\\\\\frac{1}{2}g_{12} & g_{22} & \\frac{1}{2}g_{12} \\\\\\frac{1}{4}g_{11} & \\frac{1}{2}g_{12} & \\frac{1}{4}g_{11} \\\\\\end{pmatrix} $ , one obtains $\\mathcal {P}^{\\prime }h\\mathcal {P}^{\\prime }h\\mathcal {P}^{\\prime }=-\\lbrace x,y\\rbrace ^2\\det (g)\\mathcal {P}^{\\prime }$ , giving $\\eta ^{\\prime }=(\\lbrace x,y\\rbrace ^2\\det (g))^{-1}$ .",
"We conclude that $\\eta ^{\\prime }=\\eta $ .",
"Now, we show that $\\mathcal {K}\\cong \\mathcal {K}^{\\prime }$ .",
"From $\\mathfrak {g}=\\lbrace a_1\\lbrace x,\\cdot \\rbrace +a_2\\lbrace y,\\cdot \\rbrace :a_1,a_2\\in \\mathcal {A}\\rbrace $ and $\\mathfrak {g}^{\\prime }&=\\lbrace a_1\\lbrace x,\\cdot \\rbrace +a_2\\lbrace y,\\cdot \\rbrace +a_3\\lbrace x,\\cdot \\rbrace :a_1,a_2,a_3\\in \\mathcal {A}\\rbrace \\\\&=\\lbrace (a_1+a_3)\\lbrace x,\\cdot \\rbrace +a_2\\lbrace y,\\cdot \\rbrace : a_1,a_2,a_3 \\in \\mathcal {A}\\rbrace ,$ we see that $\\mathfrak {g}=\\mathfrak {g}^{\\prime }$ .",
"We define maps $\\phi :\\mathcal {A}\\rightarrow \\mathcal {A}$ and $\\psi :\\mathfrak {g}\\rightarrow \\mathfrak {g}$ by choosing $\\phi =id$ and $\\psi =id$ , and we find a suitable choice of matrix $h$ such that $(\\phi ,\\psi )$ is an isomorphism of Kähler Poisson algebras.",
"We again check properties (1-4) in Definition REF are satisfied.",
"$(1)$ $\\phi (a)\\psi (\\alpha )=a\\alpha =\\psi (a\\alpha )$ .",
"$(2) $ $\\psi (\\alpha )(\\phi (a))=\\alpha (a)=\\phi (\\alpha (a))$ .",
"$(3)$ To show that $\\phi (g(\\alpha ,\\beta ))=h(\\psi (\\alpha ),\\psi (\\beta ))$ , we start from the right hand side $h(\\psi (\\alpha ),\\psi (\\beta )) & \\nonumber =h(\\alpha ,\\beta )\\\\& =\\nonumber \\alpha (x^i)h_{ij}\\beta (x^j)\\\\&=\\nonumber \\alpha (x)h_{11}\\beta (x)+\\alpha (x)h_{12}\\beta (y)+\\alpha (y)h_{13}\\beta (x)+\\alpha (y)h_{21}\\beta (x)+\\alpha (y)h_{22}\\beta (y)\\\\& \\nonumber \\quad +\\alpha (y)h_{23}\\beta (x)+\\alpha (x)h_{31}\\beta (x)+\\alpha (x)h_{32}\\beta (y)+\\alpha (x)h_{33}\\beta (x)\\\\&=\\nonumber \\frac{1}{4}\\alpha (x)g_{11}\\beta (x)+\\frac{1}{2}\\alpha (x)g_{12}\\beta (y)+\\frac{1}{4}\\alpha (x)g_{11}\\beta (x)+\\frac{1}{2}\\alpha (y)g_{12}\\beta (x)+\\alpha (y)g_{22}\\beta (y)\\\\& \\nonumber \\quad +\\frac{1}{2}\\alpha (y)g_{12}\\beta (x)+\\frac{1}{4}\\alpha (x)g_{11}\\beta (x)+\\frac{1}{2}\\alpha (x)g_{12}\\beta (y)+\\frac{1}{4}\\alpha (x)g_{11}\\beta (x)\\\\&=\\alpha (x)g_{11}\\beta (x)+\\alpha (x)g_{12}\\beta (y)+\\alpha (y)g_{12}\\beta (x)+\\alpha (y)g_{22}\\beta (y).$ From the left hand side we get $\\phi (g(\\alpha ,\\beta ))&=\\alpha (x^i)g_{ij}\\beta (x^j)\\\\&= \\alpha (x)g_{11}\\beta (x)+\\alpha (x)g_{12}\\beta (y)+\\alpha (y)g_{21}\\beta (x)+\\alpha (y)g_{22}\\beta (y),$ and we conclude that $\\phi (g(\\alpha ,\\beta ))=h(\\psi (\\alpha ),\\psi (\\beta ))$ .",
"$(4)$ It is easy to see that $\\phi (\\mathcal {A}_\\text{fin})\\subseteq \\mathcal {A}^{\\prime }_{\\text{fin}} $ .",
"Therefore, $\\mathcal {K}\\cong \\mathcal {K}^{\\prime }$ for the matrix $h$ with entries as in (REF ).",
"The above examples show that two Kähler-Poisson algebras can be isomorphic when considering different set of generators of the same finitely generated Poisson algebra."
],
[
"Properties of isomorphisms",
"We are interested in studying properties of more general isomorphisms for Kähler-Poisson algebras.",
"Let Let $\\mathcal {K}=(\\mathcal {A},g,\\lbrace x^1,...,x^m\\rbrace )$ and $\\mathcal {K}^{\\prime }=(\\mathcal {A}^{\\prime },g^{\\prime },\\lbrace y^1,...,{y^m}^{\\prime }\\rbrace )$ be Kähler-Poisson algebras, and assume that there exists a Poisson algebra isomorphism $\\phi :(\\mathcal {A},\\lbrace .,.\\rbrace )\\rightarrow (\\mathcal {A}^{\\prime },\\lbrace .,.\\rbrace ^{\\prime })$ .",
"When does there exist a map $\\psi :\\mathfrak {g}\\rightarrow \\mathfrak {g}^{\\prime }$ such that $(\\phi ,\\psi )$ is an isomorphism of Kähler-Poisson algebras?.",
"The following result provides an answer to this question.",
"Proposition 3.9 Let $\\mathcal {K}=(\\mathcal {A},g,\\lbrace x^1,...,x^m\\rbrace )$ and $\\mathcal {K}^{\\prime }=(\\mathcal {A}^{\\prime },g^{\\prime },\\lbrace y^1,...,{y^m}^{\\prime }\\rbrace )$ be Kähler-Poisson algebras.",
"Then $\\mathcal {K}$ and $\\mathcal {K}^{\\prime }$ are isomorphic if and only if there exists a Poisson algebra isomorphism $\\phi :\\mathcal {A}\\rightarrow \\mathcal {A}^{\\prime }$ such that $\\phi (\\mathcal {A}_\\text{fin})= \\mathcal {A}^{\\prime }_\\text{fin} $ , and $\\mathcal {P}^{\\prime }g^{\\prime }\\mathcal {P}^{\\prime }&=\\mathcal {P}^{\\prime }A^T\\phi (g)A\\mathcal {P}^{\\prime },$ where ${A^i}_\\alpha =\\frac{\\partial \\phi (x^i)}{\\partial y^\\alpha }$ and $(\\mathcal {P}^{\\prime })^{\\alpha \\beta }=\\lbrace y^\\alpha ,y^\\beta \\rbrace ^{\\prime }$ .",
"Assume that $\\mathcal {K}\\cong \\mathcal {K}^{\\prime }$ .",
"Then we need to show that $\\mathcal {P}^{\\prime }g^{\\prime }\\mathcal {P}^{\\prime }&=\\mathcal {P}^{\\prime }A^T\\phi (g)A\\mathcal {P}^{\\prime }.$ Let us start by computing $\\phi (\\mathcal {P}^{ij})$ $\\phi (\\mathcal {P}^{ij})&=\\phi (\\lbrace x^i,x^j\\rbrace )=\\lbrace \\phi (x^i),\\phi (x^j)\\rbrace ^{\\prime }=\\frac{\\partial \\phi (x^i)}{\\partial y^\\alpha }\\lbrace y^\\alpha ,\\phi (x^i)\\rbrace ^{\\prime }\\\\&=\\frac{\\partial \\phi (x^i)}{\\partial y^\\alpha }\\lbrace y^\\alpha ,y^\\beta \\rbrace ^{\\prime }\\frac{\\partial \\phi (x^j)}{\\partial y^\\beta }={A^i}_\\alpha (\\mathcal {P}^{\\prime })^{\\alpha \\beta } {A^j}_\\beta .$ Now, in order to prove (REF ), we let $\\gamma _i=\\lbrace x^i,.\\rbrace $ and $\\gamma _j=\\lbrace x^j,.\\rbrace $ then $\\phi (g(\\gamma _i,\\gamma _j)) &=\\phi (\\lbrace x^i,x^k\\rbrace g_{kl}\\lbrace x^j,x^l\\rbrace ) =\\phi (\\mathcal {P}^{ik}g_{kl}\\mathcal {P}^{jl}).$ From $(2)$ in Definition REF it follows that $\\psi (\\gamma _i)(b)=\\phi (\\lbrace x^i,\\phi ^{-1}(b)\\rbrace )=\\lbrace \\phi (x^i),b\\rbrace ^{\\prime }$ , which implies that, $\\psi (\\gamma _i)=\\lbrace \\phi (x^i),\\cdot \\rbrace ^{\\prime }$ and in the same way we get $\\psi (\\gamma _j)=\\lbrace \\phi (x^j),\\cdot \\rbrace ^{\\prime }$ .",
"Furthermore, $\\phi (\\mathcal {P}^{ik}g_{kl}\\mathcal {P}^{jl})&=\\lbrace \\phi (x^i),y^\\alpha \\rbrace ^{\\prime }{A^k}_\\alpha \\phi (g_{kl})\\lbrace \\phi (x^j),y^\\beta \\rbrace ^{\\prime } {A^l}_\\beta ,$ and $g^{\\prime }(\\psi (\\gamma _i),\\psi (\\gamma _j))&=\\lbrace \\phi (x^i),y^\\alpha \\rbrace ^{\\prime }g^{\\prime }_{\\alpha \\beta }\\lbrace \\phi (x^j),y^\\beta \\rbrace ^{\\prime }.$ From $(3)$ in Definition REF one obtains $\\lbrace \\phi (x^i),y^\\alpha \\rbrace ^{\\prime }({A^k}_\\alpha \\phi (g_{kl}){A^l}_\\beta -g^{\\prime }_{\\alpha \\beta })\\lbrace \\phi (x^j),y^\\beta \\rbrace ^{\\prime }=0.$ Note that if $\\lbrace \\phi (x^i),y^\\beta \\rbrace C_\\beta =0$ , for $C_\\beta \\in \\mathcal {A}^{\\prime }$ , then $\\lbrace y^\\alpha ,y^\\beta \\rbrace ^{\\prime }C_\\beta &= \\phi (\\lbrace \\phi ^{-1}(y^\\alpha ),\\phi ^{-1}(y^\\alpha )\\rbrace )C_\\beta =\\phi \\Big ( \\frac{\\partial \\phi ^{-1}(y^\\alpha )}{\\partial x^i}\\lbrace x^i,\\phi ^{-1}(y^\\beta )\\rbrace \\Big )C_\\beta \\\\&=\\phi \\Big (\\frac{\\partial \\phi ^{-1}(y^\\alpha )}{\\partial x^i}\\Big )\\phi \\Big (\\lbrace x^i,\\phi ^{-1}(y^\\beta )\\rbrace \\Big )C_\\beta =\\phi \\Big (\\frac{\\partial \\phi ^{-1}(y^\\alpha )}{\\partial x^i}\\Big )\\lbrace \\phi (x^i),y^\\beta \\rbrace ^{\\prime }C_\\beta =0.$ Therefore, equation (REF ) yields $\\lbrace y^\\gamma ,y^\\alpha \\rbrace ^\\prime ({A^k}_\\alpha \\phi (g_{kl}){A^l}_\\beta -g^{\\prime }_{\\alpha \\beta })\\lbrace \\phi (x^j),y^\\beta \\rbrace ^\\prime =0$ , and furthermore $\\lbrace y^\\gamma ,y^\\alpha \\rbrace ^{\\prime }({A^k}_\\alpha \\phi (g_{kl}){A^l}_\\beta -g^{\\prime }_{\\alpha \\beta })\\lbrace y^\\delta ,y^\\beta \\rbrace ^\\prime =0$ .",
"In matrix notation this becomes $\\mathcal {P}^{\\prime }A\\phi (g)A^T\\mathcal {P}^{\\prime }=\\mathcal {P}^{\\prime }g^{\\prime }\\mathcal {P}^{\\prime }.$ To prove the converse, we assume that $\\phi :\\mathcal {A}\\rightarrow \\mathcal {A}^{\\prime }$ ($\\phi (a)=a^{\\prime }$ ) is an isomorphism such that $\\phi (\\mathcal {A}_\\text{fin})=\\mathcal {A}^\\prime _{\\text{fin}}$ and (REF ) holds.",
"First, we need to define the map $\\psi $ .",
"Since $\\phi $ is an isomorphism of Poisson algebras, we may define $\\psi $ as: $\\psi (\\gamma _i)(a^{\\prime }):=\\phi (\\alpha (\\phi ^{-1}(a^{\\prime })))$ , which clearly fulfills: $\\phi (\\gamma _i(a))=\\psi (\\gamma _i)(\\phi (a))$ .",
"With a slight abuse of notation, let us show that $\\psi (\\alpha ) \\in \\mathfrak {g}^{\\prime }$ for $\\alpha \\in \\mathfrak {g} $ .",
"Writing $\\alpha =a_i\\lbrace b^i,.\\rbrace $ and $\\beta =b_i\\lbrace b^i,.\\rbrace $ as inner derivations in $\\mathfrak {g}$ one obtains $\\psi (\\alpha )(a^{\\prime })=\\phi (\\alpha (\\phi ^{-1}(a^{\\prime }))) =\\phi (a_i\\lbrace b^i,\\phi ^{-1}(a^{\\prime })\\rbrace ) =\\phi (a_i)\\lbrace \\phi (b^i),a^{\\prime }\\rbrace ^{\\prime },$ which implies that $\\psi (\\alpha )=\\phi (a_i)\\lbrace \\phi (b^i),.\\rbrace ^{\\prime }\\in \\mathfrak {g}^{\\prime }$ .",
"Similarly, one obtains $\\psi (\\beta )=\\phi (b_i)\\lbrace \\phi (b^i),.\\rbrace ^{\\prime } \\in \\mathfrak {g}^{\\prime }$ .",
"Secondly, we show that $\\psi $ is a Lie algebra isomorphism: $\\text{I)} \\qquad [\\psi (\\alpha ),\\psi (\\beta )](a^{\\prime })&=\\psi (\\alpha )\\big (\\psi (\\beta )(a^{\\prime })\\big )-\\psi (\\beta )\\big (\\psi (\\alpha )(a^{\\prime })\\big ) \\\\&=\\psi (\\alpha )\\big (\\phi (\\beta (\\phi ^{-1}(a^{\\prime })))\\big )-\\psi (\\beta )\\big (\\phi (\\alpha (\\phi ^{-1}(a^{\\prime })))\\big )\\\\&=\\phi \\big (\\alpha (\\beta (\\phi ^{-1}(a^{\\prime })))\\big )-\\phi \\big (\\beta (\\alpha (\\phi ^{-1}(a^{\\prime })))\\big ) \\\\&=\\phi \\big (\\alpha (\\beta (\\phi ^{-1}(a^{\\prime })))-\\beta (\\alpha (\\phi ^{-1}(a^{\\prime })))\\big )\\\\&=\\phi \\big ([\\alpha ,\\beta ](\\phi ^{-1}(a^{\\prime })\\big ) \\\\&=\\psi ([\\alpha ,\\beta ])(a^{\\prime }).$ $\\text{II)} \\quad \\psi (\\alpha )(a^{\\prime })+\\psi (\\beta )(a^{\\prime })&=\\phi \\big (\\alpha (\\phi ^{-1}(a^{\\prime }))\\big ) +\\phi \\big (\\beta (\\phi ^{-1}(a^{\\prime }))\\big )= \\phi \\big (\\alpha (\\phi ^{-1}(a^{\\prime }) +\\beta (\\phi ^{-1}(a^{\\prime }))\\big )\\\\&= \\phi \\big ((\\alpha +\\beta )(\\phi ^{-1}(a^{\\prime }))\\big )=\\psi (\\alpha +\\beta )(a^{\\prime }).$ Therefore $\\psi $ is a Lie algebra homomorphism.",
"Now, we show that the map $\\psi ^{-1}$ defined by $\\psi ^{-1}(\\alpha ^{\\prime })(a):=\\phi ^{-1}(\\alpha ^{\\prime }(\\phi (a)))$ , for all $a \\in \\mathcal {A}$ and $\\alpha ^{\\prime } \\in \\mathfrak {g}^{\\prime }$ is indeed the inverse of $\\psi $ : $\\psi ^{-1}\\big (\\psi (\\alpha )\\big )(a)=\\phi ^{-1}\\Big (\\psi (\\alpha )\\big (\\phi (a)\\big )\\Big ) = \\phi ^{-1}\\Big (\\phi \\big (\\alpha (\\phi ^{-1}\\phi (a))\\big )\\Big ) = \\alpha (a)$ and $\\psi \\big (\\psi ^{-1}(\\alpha ^{\\prime })\\big )(a^{\\prime })=\\phi \\Big (\\psi ^{-1}(\\alpha ^{\\prime })\\big (\\phi ^{-1}(a^{\\prime })\\big )\\Big ) = \\phi \\Big (\\phi ^{-1}\\big (\\alpha ^{\\prime }(\\phi \\phi ^{-1}(a^{\\prime }))\\big )\\Big ) = \\alpha ^{\\prime }(a^{\\prime }).$ Finally, we show that $(\\phi ,\\psi )$ is a morphism of Kähler-Poisson algebras.",
"$(1)$ $\\phi (a)\\psi (\\alpha )(a^{\\prime })=\\phi (a)\\phi (\\alpha (\\phi ^{-1}(a^{\\prime }))) =\\phi (a\\alpha (\\phi ^{-1}(a^{\\prime })))=\\psi (a\\alpha )(a^{\\prime }).$ $(2)$ $\\psi (\\alpha )(\\phi (a))=\\phi (\\alpha (\\phi ^{-1}(\\phi (a)))) =\\phi (\\alpha (a)).$ $(3)$ For $\\alpha =\\alpha _i\\lbrace x^i,.\\rbrace $ and $\\beta =\\beta _i\\lbrace x^i,.\\rbrace $ one gets $\\phi (g(\\alpha ,\\beta ))&=\\phi (\\alpha _i\\lbrace x^i,x^k\\rbrace g_{kl}\\beta _j\\lbrace x^j,x^l\\rbrace ) =\\phi (\\alpha _i)\\phi (\\lbrace x^i,x^k\\rbrace )\\phi (g_{kl})\\phi (\\beta _j)\\phi (\\lbrace x^j,x^l\\rbrace )\\\\&=\\phi (\\alpha _i)\\phi (\\mathcal {P}^{ik})\\phi (g_{kl})\\phi (\\beta _j)\\phi (\\mathcal {P}^{jl}) =\\phi (\\alpha _i)(A\\mathcal {P}^{\\prime }A^T)^{ik}\\phi (g_{kl})\\phi (\\beta _j)(A\\mathcal {P}^{\\prime }A^T)^{jl}\\\\&=\\phi (\\alpha _i)\\phi (\\beta _j)\\big (A\\mathcal {P}^{\\prime }A^T\\phi (g)A(-\\mathcal {P}^{\\prime })A^T\\big )^{ij} =-\\phi (\\alpha _i)\\phi (\\beta _j)\\big (A\\mathcal {P}^{\\prime }A^T\\phi (g)A\\mathcal {P}^{\\prime }A^T\\big )^{ij}.$ By using $\\mathcal {P}^{\\prime }g^{\\prime }\\mathcal {P}^{\\prime }=\\mathcal {P}^{\\prime }A^T\\phi (g)A\\mathcal {P}^{\\prime }$ , one obtains $\\phi (g(\\alpha ,\\beta ))&=-\\phi (\\alpha _i)\\phi (\\beta _j)(A\\mathcal {P}^{\\prime }g^{\\prime }\\mathcal {P}^{\\prime }A^T)^{ij}=-\\phi (\\alpha _i)\\phi (\\beta _j){A^i}_\\alpha \\lbrace y^\\alpha ,y^\\beta \\rbrace g^{\\prime }_{\\beta \\gamma } \\lbrace y^\\gamma ,y^\\delta \\rbrace {A^j}_\\delta \\\\&=-\\phi (\\alpha _i)\\phi (\\beta _j) \\lbrace \\phi (x^i),y^\\beta \\rbrace g^{\\prime }_{\\beta \\gamma } \\lbrace y^\\gamma ,\\phi (x^j)\\rbrace =\\psi (\\alpha )(y^\\beta )g^{\\prime }_{\\beta \\gamma }\\psi (\\beta )(y^\\gamma )\\\\&=g^{\\prime }(\\psi (\\alpha ),\\psi (\\beta )).$ Note that $(4)$ is true by assumption.",
"As an illustration, Proposition REF gives us another method to do the calculations of Example REF .",
"We have seen in Example REF that two Kähler-Poisson algebras $\\mathcal {K}=(\\mathcal {A},g,\\lbrace x^1,...,x^m\\rbrace )$ and $\\mathcal {K}^{\\prime }=(\\mathcal {A},g^{\\prime },\\lbrace y^1,...,{y^m}^{\\prime }\\rbrace )$ , where $\\mathcal {A}$ is a finitely generated algebra, can be isomorphic when considering different sets of generators for the Kähler-Poisson algebra structures of $\\mathcal {A}$ and $\\mathcal {A}^{\\prime }$ .",
"Example 3.10 (Continuation of REF ) Proposition REF tells us that if we set $h=A^T\\phi (g)A$ , then $\\mathcal {P^\\prime }h\\mathcal {P^\\prime }=\\mathcal {P^\\prime }A^T\\phi (g)A\\mathcal {P^\\prime }$ implying that $\\mathcal {K}\\cong \\mathcal {K}^{\\prime }$ .",
"Let $y^1=x+y$ and $y^2=x-y$ .",
"Hence $x=\\frac{1}{2}(y^1+y^2)$ and $y=\\frac{1}{2}(y^1-y^2)$ .",
"We compute the matrix ${A^i}_\\alpha =\\frac{\\partial x^i}{\\partial y^\\alpha }$ , with $\\phi =id$ : ${A^1}_1=\\frac{\\partial x^1}{\\partial y^1}=\\frac{\\partial (\\frac{1}{2}(y^1+y^2))}{\\partial y^1}=\\frac{1}{2}$ , ${A^1}_2=\\frac{\\partial x^1}{\\partial y^2}=\\frac{\\partial (\\frac{1}{2}(y^1+y^2))}{\\partial y^2}=\\frac{1}{2}$ , ${A^2}_1=\\frac{\\partial x^2}{\\partial y^1}=\\frac{\\partial y}{\\partial y^1}=\\frac{\\partial (\\frac{1}{2}(y^1-y^2))}{\\partial y^1}=\\frac{1}{2}$ , ${A^2}_1=\\frac{\\partial x^2}{\\partial y^2}=\\frac{\\partial y}{\\partial y^2}=\\frac{\\partial (\\frac{1}{2}(y^1-y^2))}{\\partial y^2}=-\\frac{1}{2}$ .",
"Therefore the matrix $A$ becomes $A=\\begin{pmatrix}\\frac{1}{2}& \\frac{1}{2}\\\\\\frac{1}{2}& -\\frac{1}{2}\\\\\\end{pmatrix}$ and $h&=\\begin{pmatrix}\\frac{1}{2}& \\frac{1}{2}\\\\\\frac{1}{2}& -\\frac{1}{2}\\\\\\end{pmatrix}\\begin{pmatrix}g_{11}& g_{22}\\\\g_{21}& g_{22}\\\\\\end{pmatrix}\\begin{pmatrix}\\frac{1}{2}& \\frac{1}{2}\\\\\\frac{1}{2}& -\\frac{1}{2}\\\\\\end{pmatrix}=\\begin{pmatrix}\\frac{1}{2}(g_{11}+g_{21})& \\frac{1}{2}(g_{12}+g_{22})\\\\\\frac{1}{2}(g_{11}-g_{21})& \\frac{1}{2}(g_{12}-g_{22})\\\\\\end{pmatrix}\\begin{pmatrix}\\frac{1}{2}& \\frac{1}{2}\\\\\\frac{1}{2}& -\\frac{1}{2}\\\\\\end{pmatrix}\\\\&=\\begin{pmatrix}\\frac{1}{4}(g_{11}+g_{21})+\\frac{1}{4}(g_{12}+g_{22})& \\frac{1}{4}(g_{11}+g_{21})-\\frac{1}{4}(g_{12}+g_{22})\\\\\\frac{1}{4}(g_{11}-g_{21})+\\frac{1}{4}(g_{12}-g_{22})& \\frac{1}{4}(g_{11}-g_{21})-\\frac{1}{4}(g_{12}-g_{22})\\\\\\end{pmatrix}\\\\&=\\begin{pmatrix}\\frac{1}{4}(g_{11}+2g_{12}+g_{22})& \\frac{1}{4}(g_{11}-g_{22})\\\\\\frac{1}{4}(g_{11}-g_{22})& \\frac{1}{4}(g_{11}-2g_{12}+g_{22})\\\\\\end{pmatrix},$ which agrees with the result in Example REF .",
"Also $\\phi (\\mathcal {A}_\\text{fin})=\\phi (\\mathcal {A^\\prime _{\\text{fin}}})$ , since $\\phi =id_\\mathcal {A}$ .",
"Therefore we can now use Proposition REF to conclude that $\\mathcal {K}\\cong \\mathcal {K}^{\\prime }$ .",
"In the above examples of isomorphism $\\mathcal {K}\\cong \\mathcal {K}^{\\prime }$ , we noted that $\\eta =\\eta ^\\prime $ .",
"The next proposition shows that this is indeed true in the generic case.",
"Proposition 3.11 Let $\\mathcal {K}=(\\mathcal {A},g,\\lbrace x^1,...,x^m\\rbrace )$ and $\\mathcal {K}^{\\prime }=(\\mathcal {A}^{\\prime },g^{\\prime },\\lbrace y^1,...,{y^m}^{\\prime }\\rbrace )$ be Kähler-Poisson algebras and let $(\\phi ,\\psi ):\\mathcal {K}\\rightarrow \\mathcal {K}^{\\prime }$ be an isomorphism of Kähler-Poisson algebras.",
"If $\\eta \\mathcal {P}g\\mathcal {P}g\\mathcal {P}=-\\mathcal {P} \\quad \\text{and}\\quad \\eta ^{\\prime }\\mathcal {P}^{\\prime }g^{\\prime }\\mathcal {P}^{\\prime }g^{\\prime }\\mathcal {P}^{\\prime }=-\\mathcal {P}^{\\prime }$ then $(\\phi (\\eta )-\\eta ^{\\prime })\\mathcal {P}^\\prime =0$ .",
"From Proposition REF we have $\\mathcal {P}^{\\prime }g^{\\prime }\\mathcal {P}^{\\prime }=\\mathcal {P}^{\\prime }A^T\\phi (g)A\\mathcal {P}^{\\prime }.$ Starting from $\\eta \\mathcal {P}g\\mathcal {P}g\\mathcal {P}=-\\mathcal {P}$ and using that $\\phi (\\mathcal {P})=A\\mathcal {P}^{\\prime }A^T$ and $\\mathcal {P}^{\\prime }g^{\\prime }\\mathcal {P}^{\\prime }=\\mathcal {P}^{\\prime }A^T\\phi (g)A\\mathcal {P}^{\\prime }$ , one has $-\\phi (\\mathcal {P})=\\phi (\\eta )\\phi (\\mathcal {P}g\\mathcal {P}g\\mathcal {P}).$ Multiplying both sides by $\\eta ^{\\prime }$ , where $\\eta ^{\\prime }\\mathcal {P}^{\\prime }g^{\\prime }\\mathcal {P}^{\\prime }g^{\\prime }\\mathcal {P}^{\\prime }=-\\mathcal {P}^{\\prime }$ , we obtain $-\\eta ^{\\prime }\\phi (\\mathcal {P})&=\\phi (\\eta )\\phi (\\mathcal {P}g\\mathcal {P}g\\mathcal {P})\\eta ^{\\prime }=\\phi (\\eta )A\\mathcal {P}^{\\prime }A^T\\phi (g)A\\mathcal {P}^{\\prime }A^T\\phi (g)A\\mathcal {P}^{\\prime }A^T\\eta ^{\\prime }\\\\&=\\phi (\\eta )A\\mathcal {P}^{\\prime }g^{\\prime }\\mathcal {P}^{\\prime }A^T\\phi (g)A\\mathcal {P}^{\\prime }A^T\\eta ^{\\prime }=\\phi (\\eta )A\\eta ^{\\prime }\\mathcal {P}^{\\prime }g^{\\prime }\\mathcal {P}^{\\prime }g^{\\prime }\\mathcal {P}^{\\prime }A^T=-\\phi (\\eta )A\\mathcal {P}^{\\prime }A^T.$ Hence, one obtains $\\eta ^{\\prime }A\\mathcal {P}^{\\prime }A^T=\\phi (\\eta )A\\mathcal {P}^{\\prime }A^T,$ which implies that$(\\eta ^{\\prime }-\\phi (\\eta ))\\mathcal {P}^{\\prime }=0,$ using the same argument as in the proof of Proposition REF Thus, if at least one of $(\\mathcal {P^{\\prime }})^{\\alpha \\beta }=\\lbrace y^\\alpha ,y^\\beta \\rbrace $ is not a zero divisor, then Proposition REF implies that $\\phi (\\eta )=\\eta ^\\prime $ ."
],
[
"Subalgebras of Kähler-Poisson algebras",
"As shown in Section 3, a morphism of Kähler-Poisson algebras is a pair of maps $(\\phi ,\\psi )$ , with $\\phi :\\mathcal {A}\\rightarrow \\mathcal {A}^{\\prime }$ a Poisson algebra homomorphism and $\\psi :\\mathfrak {g}\\rightarrow \\mathfrak {g}^{\\prime }$ a Lie algebra homomorphism, satisfying certain conditions.",
"In this section, we present some algebraic properties of Kähler-Poisson algebras.",
"In particular, we introduce subalgebras of Kähler-Poisson algebras, which we define by using the concept of morphisms.",
"Definition 4.1 Let $\\mathcal {K}=(\\mathcal {A},g,\\lbrace x^1,...,x^m\\rbrace )$ and $\\mathcal {K}^{\\prime }=(\\mathcal {A}^{\\prime },g^{\\prime },\\lbrace y^1,...,{y^m}^{\\prime }\\rbrace )$ be Kähler-Poisson algebras together with their corresponding modules of derivations $\\mathfrak {g}$ and $\\mathfrak {g}^{\\prime }$ , respectively.",
"$\\mathcal {K}$ is a Kähler-Poisson subalgebra of $\\mathcal {K}^{\\prime }$ if: $\\mathcal {A}$ is a Poisson subalgebra of $\\mathcal {A}^{\\prime }$ , $(\\operatorname{id}|_\\mathcal {A},\\operatorname{id}|_\\mathfrak {g})$ is a homomorphism of Kähler-Poisson algebras,where $\\operatorname{id}|_\\mathcal {A}$ and $\\operatorname{id}|_\\mathfrak {g}$ denotes the identity maps restricted to $\\mathcal {A}$ and $\\mathfrak {g}$ , respectively.",
"Given two Kähler-Poisson algebras $\\mathcal {K}=(\\mathcal {A},g,\\lbrace x^1,...,x^m\\rbrace )$ and $\\mathcal {K}^{\\prime }=(\\mathcal {A}^{\\prime },g^{\\prime },\\lbrace y^1,...,{y^m}^{\\prime }\\rbrace )$ , we illustrate with examples when $\\mathcal {K}$ is a Kähler-Poisson subalgebra of $\\mathcal {K}^{\\prime }$ , where $\\mathcal {A}$ is a proper Poisson subalgebra of a finitely generated algebra $\\mathcal {A}^{\\prime }$ .",
"We consider different set of generators for the Kähler-Poisson algebra structures of $\\mathcal {A}$ and $\\mathcal {A}^{\\prime }$ .",
"Note that the property (B) in Definition REF determines the metric $g$ on $\\mathcal {K}$ .",
"Example 4.2 Let $\\mathcal {K}=(\\mathcal {A},g,\\lbrace x,y\\rbrace )$ and $\\mathcal {K}^{\\prime }=(\\mathcal {A}^{\\prime },g^{\\prime },\\lbrace x,y,z\\rbrace )$ be Kähler-Poisson algebras, where $\\mathcal {A}$ is a subalgebra of $\\mathcal {A}^{\\prime }$ generated by $\\lbrace x,y\\rbrace $ and $\\mathcal {A}^{\\prime }$ generated by $\\lbrace x,y,z\\rbrace $ .",
"Let $\\lbrace x,y\\rbrace =p(x,y)$ and $\\lbrace x,z\\rbrace =\\lbrace y,z\\rbrace =0$ .",
"Since $\\mathcal {A}$ is a Poisson subalgebra of $\\mathcal {A}^{\\prime }$ , to show that $\\mathcal {K}$ is a Kähler-Poisson subalgebra of $\\mathcal {K}^{\\prime }$ , we only need to show that $(\\operatorname{id}|_\\mathcal {A},\\operatorname{id}|_\\mathfrak {g})$ satisfies the properties in Definition REF .",
"This fact determines the metric $g$ on $\\mathcal {K}$ .",
"$(1)$ $\\psi (a\\alpha )=a\\alpha =\\phi (a)\\psi (\\alpha )$ .",
"$(2)$ $\\phi (\\alpha (a))=\\alpha (a)=\\psi (\\alpha )(\\phi (a))$ .",
"$(3)$ We see that $\\phi (g(\\alpha ,\\beta ))=g^{\\prime }(\\psi (\\alpha ),\\psi (\\beta ))$ .",
"$\\phi (g(\\alpha ,\\beta ))&=\\sum _{i,j=1}^{2}\\alpha (x^i)g_{ij}\\beta (x^j)\\\\&= \\alpha (x^1)g_{11}\\beta (x^1)+\\alpha (x^1)g_{12}\\beta (x^2)+\\alpha (x^2)g_{21}\\beta (x^1)+\\alpha (x^2)g_{22}\\beta (x^2) \\\\&= \\alpha (x)g_{11}\\beta (x)+\\alpha (x)g_{12}\\beta (y)+\\alpha (y)g_{21}\\beta (x)+\\alpha (y)g_{22}\\beta (y),$ and $g^{\\prime }(\\psi (\\alpha ),\\psi (\\beta ))&=g^{\\prime }(\\alpha ,\\beta )=\\sum _{i,j=1}^{3}\\alpha (x^i)g_{ij}^{\\prime }\\beta (x^j)\\\\&=\\alpha (x^1)g_{11}^{\\prime }\\beta (x^1)+\\alpha (x^1)g_{12}^{\\prime }\\beta (x^2)+\\alpha (x^1)g_{13}^{\\prime }\\beta (x^3)+\\alpha (x^2)g_{21}^{\\prime }\\beta (x^1)\\\\&\\quad +\\alpha (x^2)g_{22}^{\\prime }\\beta (x^2)+\\alpha (x^2)g_{23}^{\\prime }\\beta (x^3)+\\alpha (x^3)g_{31}^{\\prime }\\beta (x^1)+\\alpha (x^3)g_{32}^{\\prime }\\beta (x^2)\\\\&\\quad +\\alpha (x^3)g_{33}^{\\prime }\\beta (x^3)\\\\&= \\alpha (x)g_{11}^{\\prime }\\beta (x)+\\alpha (x)g_{12}^{\\prime }\\beta (y)+\\alpha (x)g_{13}^{\\prime }\\beta (z)+\\alpha (y)g_{21}^{\\prime }\\beta (x)+\\alpha (y)g_{22}^{\\prime }\\beta (y)\\\\& \\quad +\\alpha (y)g_{23}^{\\prime }\\beta (z)\\alpha (z)g_{31}^{\\prime }\\beta (x)+\\alpha (z)g_{32}^{\\prime }\\beta (y)+\\alpha (z)g_{33}^{\\prime }\\beta (z)\\\\&= \\alpha (x)g_{11}^{\\prime }\\beta (x)+\\alpha (x)g_{12}^{\\prime }\\beta (y)+\\alpha (y)g_{21}^{\\prime }\\beta (x)+\\alpha (y)g_{22}^{\\prime }\\beta (y),$ since, $\\alpha =a_i\\lbrace x^i,.\\rbrace =a_1\\lbrace x,.\\rbrace +a_2\\lbrace y,.\\rbrace \\in \\mathfrak {g}$ , we get $\\alpha (z)=a_1\\lbrace x,z\\rbrace +a_2\\lbrace y,z\\rbrace =0$ .",
"Similarly, $\\beta (z)=0$ .",
"By comparing both sides $\\phi (g(\\alpha ,\\beta ))$ and $g^{\\prime }(\\psi (\\alpha ),\\psi (\\beta ))$ , we conclude that $g_{11}=g_{11}^{\\prime }$ ,$g_{12}=g_{12}^{\\prime }$ ,$g_{21}=g_{21}^{\\prime }$ and $g_{22}=g_{22}^{\\prime }$ .",
"Therefore, $\\phi (g(\\alpha ,\\beta ))=g^{\\prime }(\\psi (\\alpha ),\\psi (\\beta ))$ .",
"$(4)$ By construction we have that $\\phi (\\mathcal {A}_\\text{fin})\\subseteq \\mathcal {A}^{\\prime }_{\\text{fin}} $ .",
"Therefore, $\\mathcal {K}$ is a Kähler-Poisson subalgebra of $\\mathcal {K}^{\\prime }$ and $g$ obtained from $g^{\\prime }$ as above.",
"Let us take another example with different number of generators.",
"Example 4.3 Let $\\mathcal {K}=(\\mathcal {A},g,\\lbrace x,y\\rbrace )$ and $\\mathcal {K}^{\\prime }=(\\mathcal {A}^{\\prime },g^{\\prime },\\lbrace x,y,z,w\\rbrace )$ be Kähler-Poisson algebras, where $\\mathcal {A}$ is a subalgebra of $\\mathcal {A}^{\\prime }$ generated by $\\lbrace x,y\\rbrace $ and $\\mathcal {A}^{\\prime }$ generated by $\\lbrace x,y,z,w\\rbrace $ .",
"Let $\\lbrace x,y\\rbrace =p(x,y)$ , $\\lbrace z,w\\rbrace =q(z,w)$ and $\\lbrace x,z\\rbrace =\\lbrace x,w\\rbrace =\\lbrace y,z\\rbrace =\\lbrace y,w\\rbrace =0$ .",
"To show that $\\mathcal {K}$ is a Kähler-Poisson subalgebra of $\\mathcal {K}^{\\prime }$ we only need to show that $(\\operatorname{id}|_\\mathcal {A},\\operatorname{id}|_\\mathfrak {g})$ satisfies the properties in Definition REF .",
"This fact determines the metric $g$ on $\\mathcal {K}$ .",
"$(1)$ $\\psi (a\\alpha )=a\\alpha =\\phi (a)\\psi (\\alpha )$ .",
"$(2)$ $\\phi (\\alpha (a))=\\alpha (a)=\\psi (\\alpha )(\\phi (a))$ .",
"$(3)$ We see that $\\phi (g(\\alpha ,\\beta ))=g^{\\prime }(\\psi (\\alpha ),\\psi (\\beta ))$ , the left hand side is $\\phi (g(\\alpha ,\\beta ))&=\\sum _{i,j=1}^{2}\\alpha (x^i)g_{ij}\\beta (x^j)\\\\&= \\alpha (x^1)g_{11}\\beta (x^1)+\\alpha (x^1)g_{12}\\beta (x^2)+\\alpha (x^2)g_{21}\\beta (x^1)+\\alpha (x^2)g_{22}\\beta (x^2) \\\\&= \\alpha (x)g_{11}\\beta (x)+\\alpha (x)g_{12}\\beta (y)+\\alpha (y)g_{21}\\beta (x)+\\alpha (y)g_{22}\\beta (y),$ and the right hand side is $g^{\\prime }(\\psi (\\alpha ),\\psi (\\beta ))&=g^{\\prime }(\\alpha ,\\beta )=\\sum _{i,j=1}^{4}\\alpha (x^i)g_{ij}^{\\prime }\\beta (x^j)\\\\&=\\alpha (x^1)g_{11}^{\\prime }\\beta (x^1)+\\alpha (x^1)g_{12}^{\\prime }\\beta (x^2)+\\alpha (x^1)g_{13}^{\\prime }\\beta (x^3)+\\alpha (x^1)g_{14}^{\\prime }\\beta (x^4) \\\\& \\quad +\\alpha (x^2)g_{21}^{\\prime }\\beta (x^1)+\\alpha (x^2)g_{22}^{\\prime }\\beta (x^2)+\\alpha (x^2)g_{23}^{\\prime }\\beta (x^3)+\\alpha (x^2)g_{24}^{\\prime }\\beta (x^4) \\\\& \\quad +\\alpha (x^3)g_{31}^{\\prime }\\beta (x^1)+\\alpha (x^3)g_{32}^{\\prime }\\beta (x^2)+\\alpha (x^3)g_{33}^{\\prime }\\beta (x^3)+\\alpha (x^3)g_{34}^{\\prime }\\beta (x^4)\\\\& \\quad +\\alpha (x^4)g_{41}^{\\prime }\\beta (x^1)+\\alpha (x^4)g_{42}^{\\prime }\\beta (x^2)+\\alpha (x^4)g_{43}^{\\prime }\\beta (x^3)+\\alpha (x^4)g_{44}^{\\prime }\\beta (x^4)\\\\&= \\alpha (x)g_{11}^{\\prime }\\beta (x)+\\alpha (x)g_{12}^{\\prime }\\beta (y)+\\alpha (x)g_{13}^{\\prime }\\beta (z)+\\alpha (x)g_{14}^{\\prime }\\beta (w) \\\\& \\quad +\\alpha (y)g_{21}^{\\prime }\\beta (x)+\\alpha (y)g_{22}^{\\prime }\\beta (y)+\\alpha (y)g_{23}^{\\prime }\\beta (z)+\\alpha (y)g_{24}^{\\prime }\\beta (w) \\\\& \\quad +\\alpha (z)g_{31}^{\\prime }\\beta (x)+\\alpha (z)g_{32}^{\\prime }\\beta (y)+\\alpha (z)g_{33}^{\\prime }\\beta (z)+\\alpha (z)g_{34}^{\\prime }\\beta (w)\\\\& \\quad +\\alpha (w)g_{41}^{\\prime }\\beta (x)+\\alpha (w)g_{42}^{\\prime }\\beta (y)+\\alpha (w)g_{43}^{\\prime }\\beta (z)+\\alpha (w)g_{44}^{\\prime }\\beta (w) \\\\&= \\alpha (x)g_{11}^{\\prime }\\beta (x)+\\alpha (x)g_{12}^{\\prime }\\beta (y)+\\alpha (y)g_{21}^{\\prime }\\beta (x)+\\alpha (y)g_{22}^{\\prime }\\beta (y),$ since, $\\alpha =a_i\\lbrace x^i,.\\rbrace =a_1\\lbrace x,.\\rbrace +a_2\\lbrace y,.\\rbrace \\in \\mathfrak {g}$ , we get $\\alpha (z)=a_1\\lbrace x,z\\rbrace +a_2\\lbrace y,z\\rbrace =0$ .",
"Similarly, $\\alpha (w)=\\beta (z)=\\beta (w)=0$ .",
"By comparing both sides $\\phi (g(\\alpha ,\\beta ))$ and $g^{\\prime }(\\psi (\\alpha ),\\psi (\\beta ))$ , we conclude that $g_{11}=g_{11}^{\\prime }$ ,$g_{12}=g_{12}^{\\prime }$ ,$g_{21}=g_{21}^{\\prime }$ and $g_{22}=g_{22}^{\\prime }$ .",
"Therefore, $\\phi (g(\\alpha ,\\beta ))=g^{\\prime }(\\psi (\\alpha ),\\psi (\\beta ))$ .",
"$(4)$ By construction we have that $\\phi (\\mathcal {A}_\\text{fin})\\subseteq \\mathcal {A}^{\\prime }_{\\text{fin}} $ .",
"Therefore, $\\mathcal {K}$ is a Kähler-Poisson subalgebra of $\\mathcal {K}^{\\prime }$ and $g$ obtained from $g^{\\prime }$ as above.",
"Above we have given examples of when $\\mathcal {K}=(\\mathcal {A},g,\\lbrace x^1,...,x^m\\rbrace )$ is a Kähler-Poisson subalgebra of $\\mathcal {K}^{\\prime }=(\\mathcal {A}^{\\prime },g^{\\prime },\\lbrace y^1,...,{y^m}^{\\prime }\\rbrace )$ , where $\\mathcal {A}$ is a Poisson subalgebra of $\\mathcal {A}^{\\prime }$ .",
"Next proposition shows that, in general a morphism $(\\phi ,\\psi ):(\\mathcal {K},\\mathfrak {g})\\rightarrow (\\mathcal {K}^{\\prime },\\mathfrak {g}^{\\prime })$ induces a Kähler-Poisson subalgebra of $\\mathcal {K}^{\\prime }$ , denoted by $\\operatorname{Im}(\\phi ,\\psi )$ .",
"Proposition 4.4 Let $\\mathcal {K}=(\\mathcal {A},g,\\lbrace x^1,...,x^m\\rbrace )$ and $\\mathcal {K}^{\\prime }=(\\mathcal {A}^{\\prime },g^{\\prime },\\lbrace y^1,...,{y^m}^{\\prime }\\rbrace )$ be Kähler-Poisson algebras, and let $(\\phi ,\\psi ):\\mathcal {K}\\rightarrow \\mathcal {K}^{\\prime }$ be a homomorphism of Kähler-Poisson algebras.",
"If $y^J\\in \\phi (\\mathcal {A})$ for $J=1,...,m^{\\prime }$ then $\\operatorname{Im}(\\phi ,\\psi )=(\\phi (\\mathcal {A}),\\tilde{g},\\lbrace \\phi (x^1),\\phi (x^2),...,\\phi (x^m)\\rbrace )$ is a Kähler-Poisson subalgebra of $\\mathcal {K}^{\\prime }$ , where $\\tilde{g}_{kl}=\\phi (\\eta \\mathcal {P}_{km})\\lbrace \\phi (x^m),y^J\\rbrace ^{\\prime }g^{\\prime }_{JM}\\phi (\\eta \\mathcal {P}_{ln})\\lbrace \\phi (x^n),y^M\\rbrace ^{\\prime }.$ First, we show that $\\tilde{g}$ is symmetric.",
"$\\tilde{g}_{kl}&=\\phi (\\eta \\mathcal {P}_{km})\\lbrace \\phi (x^m),y^J\\rbrace ^{\\prime }g^{\\prime }_{JM}\\phi (\\eta \\mathcal {P}_{ln})\\lbrace \\phi (x^n),y^M\\rbrace ^{\\prime }\\\\&=\\phi (\\eta \\mathcal {P}_{ln})\\lbrace \\phi (x^n),y^M\\rbrace ^{\\prime }g^{\\prime }_{JM}\\phi (\\eta \\mathcal {P}_{km})\\lbrace \\phi (x^m),y^M\\rbrace ^{\\prime }\\\\&=\\phi (\\eta \\mathcal {P}_{ln})\\lbrace \\phi (x^n),y^M\\rbrace ^{\\prime }g^{\\prime }_{MJ}\\phi (\\eta \\mathcal {P}_{km})\\lbrace \\phi (x^m),y^M\\rbrace ^{\\prime },$ since, $g^{\\prime }_{JM}=g^{\\prime }_{MJ}$ we obtain $\\tilde{g}_{kl}=\\phi (\\eta \\mathcal {P}_{lm})\\lbrace \\phi (x^m),y^J\\rbrace ^{\\prime }g^{\\prime }_{JM}\\phi (\\eta \\mathcal {P}_{kn})\\lbrace \\phi (x^n),y^M\\rbrace ^{\\prime }=\\tilde{g}_{lk}$ .",
"As $\\phi (\\mathcal {A})$ is a subalgebra of $\\mathcal {A}^{\\prime }$ , to show that $\\operatorname{Im}(\\phi ,\\psi )$ is a subalgebra of $\\mathcal {K}^{\\prime }$ , we need to show that $(\\operatorname{id}|_{\\phi (\\mathcal {A})},\\operatorname{id}|_{\\tilde{\\mathfrak {g}}})$ is a morphism of Kähler-Poisson algebras, where $\\tilde{\\mathfrak {g}}$ is module of derivations of $\\operatorname{Im}(\\phi ,\\psi )$ .",
"Now, let $\\alpha =\\alpha _i\\lbrace \\phi (x^i),.\\rbrace ^{\\prime }$ and $\\beta =\\beta _j\\lbrace \\phi (x^j),.\\rbrace ^{\\prime }$ be arbitrary elements of $\\mathfrak {\\tilde{g}}$ as usual, we have $\\psi (a\\alpha )=a\\alpha =\\phi (a)\\psi (\\alpha )$ , for all $\\alpha \\in \\tilde{\\mathfrak {g}}$ and $a \\in \\phi (\\mathcal {A})$ .",
"$\\phi (\\alpha (a))=\\alpha (a)=\\psi (\\alpha )(\\phi (a))$ , for all $\\alpha \\in \\tilde{\\mathfrak {g}}$ and $a \\in \\phi (\\mathcal {A})$ .",
"$\\phi (\\tilde{g}(\\alpha ,\\beta ))=g^{\\prime }(\\psi (\\alpha ),\\psi (\\beta ))=g^{\\prime }(\\alpha ,\\beta )$ , for all $\\alpha ,\\beta \\in \\tilde{\\mathfrak {g}}$ and $a \\in \\phi (\\mathcal {A})$ .",
"$ \\begin{split}\\phi (\\tilde{g}(\\alpha ,\\beta ))-g^{\\prime }(\\alpha ,\\beta ) =& \\alpha _i\\lbrace \\phi (x^i),\\phi (x^k)\\rbrace ^{\\prime }\\tilde{g}_{kl} \\beta _j\\lbrace \\phi (x^j),\\phi (x^l)\\rbrace ^{\\prime }\\\\& -\\alpha _i\\lbrace \\phi (x^i),y^J\\rbrace ^{\\prime }g^{\\prime }_{JM} \\beta _j\\lbrace \\phi (x^j),y^M\\rbrace ^{\\prime }.\\end{split}$ Since $y^J\\in \\phi (\\mathcal {A})$ and for each $J$ , we can find $\\hat{y}^J\\in \\mathcal {A}$ such that $\\phi (\\hat{y}^J)=y^J$ .",
"We can write $\\tilde{g}_{kl}=\\phi (\\eta \\mathcal {P}_{km})\\lbrace \\phi (x^m),\\phi (\\hat{y}^J)\\rbrace ^{\\prime }g^{\\prime }_{JM}\\phi (\\eta \\mathcal {P}_{ln})\\lbrace \\phi (x^n),\\phi (\\hat{y}^M)\\rbrace ^{\\prime }$ .",
"We compute $\\phi (\\tilde{g}(\\alpha ,\\beta ))&=\\phi \\Big (\\eta \\alpha _i\\lbrace x^i,x^k\\rbrace g_{km}\\lbrace x^m,x^p\\rbrace g_{pq}\\lbrace x^q,\\hat{y}^J\\rbrace \\Big )g^{\\prime }_{JM}\\phi \\Big (\\eta \\beta _j\\lbrace x^j,x^l\\rbrace g_{lm}\\lbrace x^m,x^r\\rbrace g_{rs}\\lbrace x^s,\\hat{y}^M\\rbrace \\Big )\\\\&=\\alpha _i\\lbrace \\phi (x^i),y^J\\rbrace ^{\\prime }g^{\\prime }_{JM}\\beta _j\\lbrace \\phi (x^j),y^M\\rbrace ^{\\prime },$ by (REF ) we get $ \\phi (\\tilde{g}(\\alpha ,\\beta ))=g^{\\prime }(\\alpha ,\\beta )$ .",
"$\\phi (\\mathcal {A})_{\\text{fin}}=\\phi (\\mathcal {A}_{\\text{fin}})\\subseteq \\mathcal {A}^{\\prime }_{\\text{fin}} $ , since, $\\phi (\\mathcal {A})_{\\text{fin}}$ is generated by $\\lbrace \\phi (x^1),...,\\phi (x^m)\\rbrace $ Therefore, $(\\operatorname{id}_{\\phi (\\mathcal {A})},\\operatorname{id}_{\\tilde{\\mathfrak {g}}})$ is a morphism of Kähler-Poisson algebras, and $\\operatorname{Im}(\\phi ,\\psi )$ is a subalgebra of $\\mathcal {K}^{\\prime }$ ."
],
[
"Direct sums and tensor products of Kähler-Poisson algebras",
"Let $\\mathcal {K}=(\\mathcal {A},g,\\lbrace x^1,...,x^m\\rbrace )$ and $\\mathcal {K}^{\\prime }=(\\mathcal {A}^{\\prime },g^{\\prime },\\lbrace y^1,...,{y^m}^{\\prime }\\rbrace )$ be Kähler-Poisson algebras.",
"We have seen in Section 4, when $\\mathcal {K}$ is Kähler-Poisson subalgebra of $\\mathcal {K}^{\\prime }$ .",
"In this section, we are interested in defining operations of Kähler-Poisson algebras.",
"In particular, we introduce direct sums and tensor products of Kähler-Poisson algebras and study properties of these operations.",
"We prove that, direct sums and tensor products of two Kähler-Poisson algebras are Kähler-Poisson algebras and we show that $\\mathcal {K}$ and $\\mathcal {K}^{\\prime }$ are subalgebras of the direct sum $\\mathcal {K}\\oplus \\mathcal {K}^{\\prime }$ .",
"Let us first recall some basic facts of direct sums and tensor products of Poisson algebras.",
"Firstly, we recall that the tensor product $\\mathcal {A}\\otimes \\mathcal {A}^{\\prime }$ of two Poisson algebras $\\mathcal {A}$ and $\\mathcal {A}^{\\prime }$ is a Poisson algebra with Poisson products $\\lbrace a_1\\otimes a_2,b_1\\otimes b_2\\rbrace =\\lbrace a_1,b_1\\rbrace \\otimes a_2 b_2+a_1b_1\\otimes \\lbrace a_2,b_2\\rbrace ,$ for $a_1,b_1\\in \\mathcal {A}$ and $a_2,b_2\\in \\mathcal {A}^{\\prime }$ (see [6]).",
"Secondly, the direct sum of two Poisson algebras $\\mathcal {A}\\oplus \\mathcal {A}^{\\prime }$ is a Poisson algebra with Poisson product.",
"$\\lbrace (a_1,a_2),(b_1,b_2)\\rbrace =(\\lbrace a_1,b_1\\rbrace ,\\lbrace a_2,b_2\\rbrace )$ for $a_1,b_1\\in \\mathcal {A}$ and $a_2,b_2\\in \\mathcal {A}^{\\prime }$ .",
"In next proposition, we see that direct sums of Kähler-Poisson algebras are Kähler-Poisson algebra.",
"Proposition 5.1 Let $\\mathcal {K}=(\\mathcal {A},g,\\lbrace x^1,...,x^m\\rbrace )$ and $\\mathcal {K}^{\\prime }=(\\mathcal {A}^{\\prime },g^{\\prime },\\lbrace y^1,...,{y^m}^{\\prime }\\rbrace )$ be Kähler-Poisson algebras, and set $\\mathcal {K}\\oplus \\mathcal {K}^{\\prime }=(\\mathcal {A}\\oplus \\mathcal {A}^{\\prime },\\hat{g},\\lbrace z^1,...,z^{m+m^{\\prime }}\\rbrace )$ where $z^I ={\\left\\lbrace \\begin{array}{ll}(x^I,0) & \\quad \\text{if } I\\in \\lbrace 1,...,m\\rbrace \\\\(0,y^{I-m}) & \\quad \\text{if } I\\in \\lbrace m+1,...,m+m^{\\prime }\\rbrace ,\\end{array}\\right.",
"}$ and $\\hat{g}_{IJ} ={\\left\\lbrace \\begin{array}{ll}(g_{IJ},0) & \\quad \\text{if } I,J\\in \\lbrace 1,...,m\\rbrace \\\\(0,g^{\\prime }_{I-m,J-m}) & \\quad \\text{if } I,J\\in \\lbrace m+1,...,m+m^{\\prime }\\rbrace \\\\(0,0) & \\quad \\text{otherwise.}\\end{array}\\right.",
"}$ Then $\\mathcal {K}\\oplus \\mathcal {K}^{\\prime }$ is a Kähler-Poisson algebra.",
"Since $\\mathcal {K}=(\\mathcal {A},g,\\lbrace x^1,...,x^m\\rbrace )$ and $\\mathcal {K}^{\\prime }=(\\mathcal {A}^{\\prime },g^{\\prime },\\lbrace y^1,...,{y^m}^{\\prime }\\rbrace )$ are Kähler-Poisson algebras then there exists $\\eta \\in \\mathcal {A}$ and $\\eta ^{\\prime }\\in \\mathcal {A}^{\\prime }$ such that $&\\sum \\limits _{i,j,k,l=1}^m\\eta \\lbrace a_1,x^i\\rbrace g_{ij}\\lbrace x^j,x^k\\rbrace g_{kl}\\lbrace x^l,b_1\\rbrace =-\\lbrace a_1,b_1\\rbrace \\\\&\\sum \\limits _{\\alpha ,\\beta ,\\gamma ,\\delta =1}^{m^\\prime }\\eta ^{\\prime }\\lbrace a_2,y^\\alpha \\rbrace ^{\\prime }g^{\\prime }_{\\alpha \\beta }\\lbrace y^\\beta ,y^\\gamma \\rbrace ^{\\prime }g^{\\prime }_{\\gamma \\delta }\\lbrace y^\\delta ,b_2\\rbrace ^{\\prime }=-\\lbrace a_2,b_2\\rbrace ^{\\prime }.$ where $a_1,b_1\\in \\mathcal {A}$ and $a_2,b_2\\in \\mathcal {A}^{\\prime }$ .",
"We would like to show that $\\mathcal {K}\\oplus \\mathcal {K}^{\\prime }$ satisfies the Kähler-Poisson condition (REF ); that is $\\sum \\limits _{I,J,K,L}^{m+m^{\\prime }} (\\eta ,\\eta ^{\\prime })\\lbrace (a_1,a_2),z^I\\rbrace \\hat{g}_{IJ}\\lbrace z^J,z^K\\rbrace \\hat{g}_{KL}\\lbrace z^L,(b_1,b_2)\\rbrace =-\\lbrace (a_1,a_2),(b_1,b_2)\\rbrace $ for $a_1,b_1\\in \\mathcal {A}$ and $a_2,b_2\\in \\mathcal {A}^{\\prime }$ .",
"Starting from the left hand side $\\sum _{I,J,K,L}^{m+m^{\\prime }} (\\eta ,&\\eta ^{\\prime })\\lbrace (a_1,a_2),z^I\\rbrace \\hat{g}_{IJ}\\lbrace z^J,z^K\\rbrace \\hat{g}_{KL}\\lbrace z^L,(b_1,b_2)\\rbrace \\\\&=\\sum \\limits _{J,K,L=1}^{m+m^{\\prime }}\\sum \\limits _{i=1}^m(\\eta ,\\eta ^{\\prime })\\lbrace (a_1,a_2),z^i\\rbrace \\hat{g}_{iJ}\\lbrace z^J,z^K\\rbrace \\hat{g}_{KL}\\lbrace z^L,(b_1,b_2)\\rbrace \\\\&+\\sum \\limits _{J,K,L=1}^{m+m^{\\prime }}\\sum \\limits _{\\alpha =1}^{m^\\prime }(\\eta ,\\eta ^{\\prime })\\lbrace (a_1,a_2),z^{\\alpha +m}\\rbrace \\hat{g}_{\\alpha +m,J}\\lbrace z^J,z^K\\rbrace \\hat{g}_{KL}\\lbrace z^L,(b_1,b_2)\\rbrace \\\\&=\\sum \\limits _{K,L=1}^{m+m^{\\prime }}\\sum \\limits _{i,j=1}^m(\\eta ,\\eta ^{\\prime })\\lbrace (a_1,a_2),z^i\\rbrace \\hat{g}_{ij}\\lbrace z^j,z^K\\rbrace \\hat{g}_{KL}\\lbrace z^L,(b_1,b_2)\\rbrace \\\\&+\\sum \\limits _{K,L=1}^{m+m^{\\prime }}\\sum \\limits _{\\alpha ,\\beta =1}^{m^\\prime }(\\eta ,\\eta ^{\\prime })\\lbrace (a_1,a_2),z^{\\alpha +m}\\rbrace \\hat{g}_{\\alpha +m,\\beta +m}\\lbrace z^{\\beta +m},z^K\\rbrace \\hat{g}_{KL}\\lbrace z^L,(b_1,b_2)\\rbrace ,$ since $\\hat{g}_{iK}=0$ when $K > m$ , and $\\hat{g}_{m+\\alpha ,K}=0$ when $K\\le m$ .",
"Moreover, since $\\lbrace z^i,z^K\\rbrace =0$ and $\\lbrace z^{\\alpha +m},z^L\\rbrace =0$ if $1\\le i\\le m$ , $1\\le \\alpha \\le m^{\\prime }$ , $K> m$ and $L\\le m$ , then $ \\sum \\limits _{I,J,K,L}^{m+m^{\\prime }} (\\eta ,&\\eta ^{\\prime })\\lbrace (a_1,a_2),z^I\\rbrace \\hat{g}_{IJ}\\lbrace z^J,z^K\\rbrace \\hat{g}_{KL}\\lbrace z^L,(b_1,b_2)\\rbrace \\\\&=\\sum \\limits _{i,j,k,l=1}^m(\\eta ,\\eta ^{\\prime })\\lbrace (a_1,a_2),z^i\\rbrace \\hat{g}_{ij}\\lbrace z^j,z^k\\rbrace \\hat{g}_{kl}\\lbrace z^l,(b_1,b_2)\\rbrace \\\\&+\\sum \\limits _{\\alpha ,\\beta ,\\gamma ,\\delta =1}^{m^\\prime }(\\eta ,\\eta ^{\\prime })\\lbrace (a_1,a_2),z^{\\alpha +m}\\rbrace \\hat{g}_{\\alpha +m,\\beta +m}\\lbrace z^{\\beta +m},z^{\\gamma +m}\\rbrace \\hat{g}_{\\gamma +m,\\delta +m}\\lbrace z^{\\delta +m},(b_1,b_2)\\rbrace .$ and $ \\sum \\limits _{I,J,K,L}^{m+m^{\\prime }}(\\eta ,&\\eta ^{\\prime })\\lbrace (a_1,a_2),z^I\\rbrace \\hat{g}_{IJ}\\lbrace z^J,z^K\\rbrace \\hat{g}_{KL}\\lbrace z^L,(b_1,b_2)\\rbrace \\\\&=\\sum \\limits _{i,j,k,l=1}^m(\\eta ,\\eta ^{\\prime })\\lbrace (a_1,a_2),(x^i,0)\\rbrace (g_{ij},0)\\lbrace (x^j,0),(x^k,0)\\rbrace (g_{kl},0)\\lbrace (x^l,0),(b_1,b_2)\\rbrace \\\\&+\\sum \\limits _{\\alpha ,\\beta ,\\gamma ,\\delta =1}^{m^\\prime }(\\eta ,\\eta ^{\\prime })\\lbrace (a_1,a_2),(0,y^\\alpha )\\rbrace (0,g^{\\prime }_{\\alpha \\beta }\\lbrace (0,y^\\beta ),(0,y^\\gamma )\\rbrace (0,g^{\\prime }_{\\gamma \\delta })\\lbrace (0,y^\\delta ),(b_1,b_2)\\rbrace \\\\&=\\sum \\limits _{i,j,k,l=1}^m(\\eta ,\\eta ^{\\prime })(\\lbrace a_1,x^i\\rbrace ,0)(g_{ij},0)(\\lbrace x^j,x^k\\rbrace ,0)(g_{kl},0)(\\lbrace x^l,b_1\\rbrace ,0)\\\\&+\\sum \\limits _{\\alpha ,\\beta ,\\gamma ,\\delta =1}^{m^\\prime }(\\eta ,\\eta ^{\\prime })(0,\\lbrace a_2,y^\\alpha \\rbrace ^{\\prime })(0,g^{\\prime }_{\\alpha \\beta })(0,\\lbrace y^\\beta ,y^\\gamma \\rbrace ^{\\prime })(0,g^{\\prime }_{\\gamma \\delta })(0,\\lbrace y^\\delta ,b_2\\rbrace ^{\\prime })\\\\&=\\sum \\limits _{i,j,k,l=1}^m(\\eta \\lbrace a_1,x^i\\rbrace g_{ij}\\lbrace x^j,x^k\\rbrace g_{kl}\\lbrace x^l,b_1\\rbrace ,0)+\\sum \\limits _{\\alpha ,\\beta ,\\gamma ,\\delta =1}^{m^\\prime }(0,\\eta ^{\\prime }\\lbrace a_2,y^\\alpha \\rbrace ^{\\prime }g^{\\prime }_{\\alpha \\beta }\\lbrace y^\\beta ,y^\\gamma \\rbrace ^{\\prime }g^{\\prime }_{\\gamma \\delta }\\lbrace y^\\delta ,b_2\\rbrace ^{\\prime })\\\\&=(-\\lbrace a_1,b_1\\rbrace ,0)+(0,-\\lbrace a_2,b_2\\rbrace ^{\\prime })=-(\\lbrace a_1,b_1\\rbrace ,\\lbrace a_2,b_2\\rbrace ^{\\prime })=-\\lbrace (a_1,a_2),(b_1,b_2)\\rbrace ,$ since $\\mathcal {K}$ and $\\mathcal {K}^{\\prime }$ are Kähler-Poisson algebras.",
"Therefore, $\\mathcal {K}\\oplus \\mathcal {K}^{\\prime }$ is a Kähler-Poisson algebra.",
"Next, given two Kähler-Poisson algebras $ \\mathcal {K}$ and $\\mathcal {K}^{\\prime }$ , we show that they are Kähler-Poisson subalgebras of the direct sum $\\mathcal {K}\\oplus \\mathcal {K}^{\\prime }$ .",
"Remark 5.2 The module of inner derivations of $\\mathcal {A}\\oplus \\mathcal {A}^{\\prime }$ can be written as: $\\mathfrak {g}\\oplus \\mathfrak {g}^{\\prime }=\\lbrace (\\alpha ,\\beta ):\\alpha \\in \\mathfrak {g}\\text{ and }\\beta \\in \\mathfrak {g}^{\\prime }\\rbrace $ since given $\\tilde{\\alpha }\\in \\mathfrak {g}\\oplus \\mathfrak {g}^{\\prime }$ $\\tilde{\\alpha }(c,d)=\\sum _{I=1}^{m+m^{\\prime }}(a_I,b_I)\\lbrace z^I,(c,d)\\rbrace &=\\sum _{I=1}^{m}(a_I\\lbrace x^I,c\\rbrace ,0)+\\sum _{I=m+1}^{m+m^{\\prime }}(0,b_I\\lbrace y^{I-m},d\\rbrace )\\\\&=(\\alpha (c),\\beta (d))=(\\alpha ,\\beta )(c,d),$ with $\\alpha = \\sum _{I=1}^ma_I\\lbrace x^I,\\cdot \\rbrace \\quad \\text{and}\\quad \\beta = \\sum _{I=1}^{m^{\\prime }}b_{I+m}\\lbrace y^I,\\cdot \\rbrace .$ Proposition 5.3 Let $\\mathcal {K}=(\\mathcal {A},g,\\lbrace x^1,...,x^m\\rbrace )$ and $\\mathcal {K}^{\\prime }=(\\mathcal {A}^{\\prime },g^{\\prime },\\lbrace y^1,...,{y^m}^{\\prime }\\rbrace )$ be Kähler-Poisson algebras, then $\\mathcal {K}$ and $\\mathcal {K}^{\\prime }$ are Kähler-Poisson subalgebras of $\\mathcal {K}\\oplus \\mathcal {K}^{\\prime }$ .",
"To prove this proposition for $\\mathcal {K}$ and $\\mathcal {K}^{\\prime }$ , we first show the following results: Lemma 5.4 Let $\\mathcal {K}=(\\mathcal {A},g,\\lbrace x^1,...,x^m\\rbrace )$ and $\\mathcal {K}^{\\prime }=(\\mathcal {A}^{\\prime },g^{\\prime },\\lbrace y^1,...,{y^m}^{\\prime }\\rbrace )$ be Kähler-Poisson algebras, and let $\\tilde{\\mathcal {A}}=\\lbrace (a,0):a\\in \\mathcal {A}\\rbrace $ .",
"Then $\\tilde{\\mathcal {K}}=(\\tilde{\\mathcal {A}},\\tilde{g},\\lbrace (x^1,0),...,(x^m,0)\\rbrace )$ is a Kähler-Poisson subalgebra of $\\mathcal {K}\\oplus \\mathcal {K}^{\\prime }$ , where $\\tilde{g}_{ij}=\\big ((g_{ij},0)\\big )$ .",
"Firstly, $\\tilde{\\mathcal {A}}$ is a Poisson subalgebra of $\\mathcal {A}\\oplus \\mathcal {A}^{\\prime }$ .",
"Denote the derivation module $\\tilde{\\mathfrak {g}}=\\lbrace (\\alpha ,0)\\in \\mathfrak {g}\\oplus \\mathfrak {g}^{\\prime }:\\alpha \\in \\mathfrak {g}\\rbrace $ as a submodule of $\\mathfrak {g}\\oplus \\mathfrak {g}^{\\prime }$ .",
"Secondly, we show that $(\\text{id}|_{\\tilde{A}},\\text{id}|_{\\tilde{\\mathfrak {g}}})$ is a morphism of Kähler-Poisson algebras.",
"Let $\\phi =\\text{id}|_{\\tilde{A}}$ and $\\psi =\\text{id}|_{\\tilde{\\mathfrak {g}}}$ , then by Definition REF we get $\\psi ((a,0)(\\alpha ,0))=\\psi (a\\alpha ,0)=(a\\alpha ,0)$ and $\\phi ((a,0))\\psi ((\\alpha ,0))=(a,0)(\\alpha ,0)=(a\\alpha ,0)$ .",
"$\\phi (\\alpha (a,0))=\\phi (a\\alpha ,0)=(a\\alpha ,0)$ and $\\psi (\\alpha ,0)(\\phi (a,0))=(\\alpha ,0)(a,0)=(a\\alpha ,0)$ .",
"We see that $\\phi (\\tilde{g}((\\alpha ,0),(\\beta ,0)))=\\hat{g}(\\psi (\\alpha ,0),\\psi (\\beta ,0))$ .",
"$\\phi (\\tilde{g}((\\alpha ,0),(\\beta ,0)))&=\\sum _{i,j=1}^{m}((\\alpha ,0)(x^i,0)(g_{ij},0)(\\beta ,0)(x^j,0))\\\\&=\\sum _{i,j=1}^{m}(\\alpha (x^i)g_{ij}\\beta (x^j),0),$ and $\\hat{g}(\\psi (\\alpha ,0),\\psi (\\beta ,0))=\\hat{g}((\\alpha ,0),(\\beta ,0))=(\\alpha ,0)(\\tilde{x}^I,0)\\hat{g}_{IJ}(\\beta ,0)(\\tilde{x}^J,0).$ Since $\\hat{g}_{IJ}$ is a diagonal block matrix, we get $\\hat{g}(\\psi (\\alpha ,0),\\psi (\\beta ,0))&=\\sum _{I,J=1}^{m}(\\alpha ,0)(x^I,0)({g}_{IJ},0)(\\beta ,0)(x^J,0)\\\\&+\\sum _{I,J=m+1}^{m+m^{\\prime }}(\\alpha ,0)(0,y^I)(0,{g^{\\prime }}_{I-m,J-m})(\\beta ,0)(0,y^J)$ since $(\\alpha ,0)(0,y^J)=0$ , we get $\\hat{g}(\\psi (\\alpha ,0),\\psi (\\beta ,0))=\\sum _{i,j=1}^{m}(\\alpha (x^i),0)(g_{ij},0)(\\beta (x^j),0)=\\sum _{i,j=1}^{m}(\\alpha (x^i)g_{ij}\\beta (x^j),0).$ By construction, we have $\\phi (\\tilde{\\mathcal {A}}_\\text{fin})\\subseteq (\\mathcal {A}\\oplus \\mathcal {A}^{\\prime })_{\\text{fin}} $ .",
"Therefore, $\\tilde{\\mathcal {K}}$ is a Kähler-Poisson subalgebra of $\\mathcal {K}\\oplus \\mathcal {K}^{\\prime }$ .",
"Lemma 5.5 Let $\\mathcal {K}=(\\mathcal {A},g,\\lbrace x^1,...,x^m\\rbrace )$ and $\\tilde{\\mathcal {K}}=(\\tilde{\\mathcal {A}},\\tilde{g},\\lbrace (x^1,0),...,(x^m,0)\\rbrace )$ be Kähler-Poisson algebras as in Lemma REF , then $\\mathcal {K}$ is isomorphic to $\\tilde{\\mathcal {K}}$ .",
"We see that there is a morphism $\\phi :\\mathcal {A}\\rightarrow \\tilde{\\mathcal {A}}$ and $\\psi :\\mathfrak {g}\\rightarrow \\tilde{\\mathfrak {g}}$ defined by $\\phi (c)=(c,0)$ ($\\phi $ is a Poisson algebra isomorphism, see Remark REF ), for all $c\\in \\mathcal {A}$ and let $\\alpha =a_i\\lbrace x^i,.\\rbrace $ we define $\\psi (\\alpha )=(a_i,0)\\lbrace (x^i,0),.\\rbrace $ and $\\psi (\\beta )=(b_i,0)\\lbrace (x^i,0),.\\rbrace $ for all $\\alpha ,\\beta \\in \\mathfrak {g}$ , we get $\\phi (c)\\psi (\\alpha )=(c,0)(a_i,0)\\lbrace (x^i,0),.\\rbrace =(ca_i,0)\\lbrace (x^i,0),.\\rbrace =\\psi (c\\alpha )$ .",
"$\\psi (\\alpha )(\\phi (a))=(a_i,0)\\lbrace (x^i,0),(c,0)\\rbrace =(a_i,0)(\\lbrace x^i,c\\rbrace ,0)=(a_i\\lbrace x^i,c\\rbrace ,0)=\\phi (a_i\\lbrace x^i,c\\rbrace )=\\phi (\\alpha (c))$ .",
"We see that $\\phi (g(\\alpha ,\\beta ))=\\tilde{g}(\\psi (\\alpha ),\\psi (\\beta ))$ since $\\phi (g(\\alpha ,\\beta ))=\\sum _{i,j=1}^{m}(\\alpha (x^i)g_{ij}\\beta (x^j),0),$ and $\\tilde{g}(\\psi (\\alpha ),\\psi (\\beta ))&=\\tilde{g}((a_l,0)\\lbrace (x^l,0),.\\rbrace ,(b_k,0)\\lbrace (x^k,0),.\\rbrace )\\\\&=(a_l,0)\\lbrace (x^l,0),(x^i,0)\\rbrace \\tilde{g}_{ij}(b_k,0)\\lbrace (x^k,0),(x^j,0)\\rbrace \\\\&=(a_l,0)(\\lbrace x^l,x^i\\rbrace ,0)(g_{ij},0)(b_k,0)(\\lbrace x^k,x^j\\rbrace ,0)\\\\&=(a_l\\lbrace x^l,x^i\\rbrace ,0)(g_{ij},0)(b_k\\lbrace x^k,x^j\\rbrace ,0)\\\\&=(\\alpha (x^i),0)(g_{ij},0)(\\beta (x^j),0)=(\\alpha (x^i)g_{ij}\\beta (x^j),0).$ By construction, we have $\\phi (\\mathcal {A}_\\text{fin})=\\tilde{\\mathcal {A}}_{\\text{fin}} $ .",
"Therefore, $\\mathcal {K}$ isomorphic to $\\tilde{\\mathcal {K}}$ .",
"To prove Proposition REF we need the above results.",
"Lemma REF says that $\\tilde{\\mathcal {K}}=(\\tilde{\\mathcal {A}},\\tilde{g},\\lbrace (x^1,0),...,(x^m,0)\\rbrace )$ is a Kähler-Poisson subalgebra of $\\mathcal {K}\\oplus \\mathcal {K}^{\\prime }$ and Lemma REF says that $\\mathcal {K}$ is isomorphic to $\\tilde{\\mathcal {K}}$ .",
"Therefore, we can consider $\\mathcal {K}$ to be a Kähler-Poisson subalgebra of $\\mathcal {K}\\oplus \\mathcal {K}^{\\prime }$ .",
"Observe that, in the same way, $\\mathcal {K}^{\\prime }$ is a Kähler-Poisson subalgebra of $\\mathcal {K}\\oplus \\mathcal {K}^{\\prime }$ .",
"The next proposition shows that the tensor product of Kähler-Poisson algebras is Kähler-Poisson algebra under certain conditions.",
"Proposition 5.6 Let $\\mathcal {K}=(\\mathcal {A},g,\\lbrace x^1,...,x^m\\rbrace )$ and $\\mathcal {K}^{\\prime }=(\\mathcal {A}^{\\prime },g^{\\prime },\\lbrace y^1,...,{y^m}^{\\prime }\\rbrace )$ with $\\sum \\limits _{i,j,k,l=1}^m\\eta \\lbrace a_1,x^i\\rbrace g_{ij}\\lbrace x^j,x^k\\rbrace g_{kl}\\lbrace x^l,b_1\\rbrace =-\\lbrace a_1,b_1\\rbrace $ and $\\sum \\limits _{\\alpha ,\\beta ,\\gamma ,\\delta =1}^{m^\\prime }\\eta ^{\\prime }\\lbrace a_2,y^\\alpha \\rbrace ^{\\prime }g^{\\prime }_{\\alpha \\beta }\\lbrace y^\\beta ,y^\\gamma \\rbrace ^{\\prime }g^{\\prime }_{\\gamma \\delta }\\lbrace y^\\delta ,b_2\\rbrace ^{\\prime }=-\\lbrace a_2,b_2\\rbrace ^{\\prime }$ be Kähler-Poisson algebras, and assume that there exist $\\rho \\in \\mathcal {A}$ and $\\rho ^{\\prime } \\in \\mathcal {A}^{\\prime }$ such that $\\rho ^2=\\eta $ and $\\rho ^{\\prime 2}=\\eta ^{\\prime }$ .",
"Set $\\mathcal {K}\\otimes \\mathcal {K}^{\\prime }=(\\mathcal {A}\\otimes \\mathcal {A}^{\\prime },\\tilde{g},\\lbrace z^1,...,z^{m+m^{\\prime }}\\rbrace )$ where $z^I ={\\left\\lbrace \\begin{array}{ll}x^I\\otimes 1 & \\quad \\text{if } I\\in \\lbrace 1,...,m\\rbrace \\\\1\\otimes y^{I-m} & \\quad \\text{if } I\\in \\lbrace m+1,...,m+m^{\\prime }\\rbrace ,\\end{array}\\right.",
"}$ and $\\tilde{g}_{IJ} ={\\left\\lbrace \\begin{array}{ll}\\rho g_{IJ}\\otimes 1 & \\quad \\text{if } I,J\\in \\lbrace 1,...,m\\rbrace \\\\1\\otimes \\rho ^{\\prime }g^{\\prime }_{I-m,J-m} & \\quad \\text{if } I,J\\in \\lbrace m+1,...,m+m^{\\prime }\\rbrace \\\\0 & \\quad \\text{otherwise}.\\end{array}\\right.",
"}$ Then $\\mathcal {K}\\otimes \\mathcal {K}^{\\prime }$ is a Kähler-Poisson algebra.",
"Let us show that $\\mathcal {K}\\otimes \\mathcal {K}^{\\prime }$ satisfies the Kähler-Poisson condition: $\\sum \\limits _{I,J,K,L}^{m+m^{\\prime }} \\lbrace a_1\\otimes a_2,z^I\\rbrace \\tilde{g}_{IJ}\\lbrace z^J,z^K\\rbrace \\tilde{g}_{KL}\\lbrace z^L,b_1\\otimes b_2\\rbrace =-\\lbrace a_1\\otimes a_2,b_1\\otimes b_2\\rbrace $ for $a_1,b_1\\in \\mathcal {A}$ and $a_2,b_2\\in \\mathcal {A}^{\\prime }$ .",
"Starting from the left hand side $&\\sum _{I,J,K,L}^{m+m^{\\prime }}\\lbrace a_1\\otimes a_2,z^I\\rbrace \\tilde{g}_{IJ}\\lbrace z^J,z^K\\rbrace \\tilde{g}_{KL}\\lbrace z^L,b_1\\otimes b_2\\rbrace \\\\&=\\sum \\limits _{J,K,L=1}^{m+m^{\\prime }}\\sum \\limits _{i=1}^m\\lbrace a_1\\otimes a_2,z^i\\rbrace \\tilde{g}_{iJ}\\lbrace z^J,z^K\\rbrace \\tilde{g}_{KL}\\lbrace z^L,b_1\\otimes b_2\\rbrace \\\\&+\\sum _{J,K,L=1}^{m+m^{\\prime }}\\sum _{\\alpha =1}^{m^{\\prime }}\\lbrace a_1\\otimes a_2,z^{\\alpha +m}\\rbrace \\tilde{g}_{\\alpha +m,J}\\lbrace z^J,z^K\\rbrace \\tilde{g}_{KL}\\lbrace z^L,b_1\\otimes b_2\\rbrace \\\\&=\\sum _{K,L=1}^{m+m^{\\prime }}\\sum _{i,j=1}^m\\lbrace a_1\\otimes a_2,z^i\\rbrace \\tilde{g}_{ij}\\lbrace z^j,z^K\\rbrace \\tilde{g}_{KL}\\lbrace z^L,b_1\\otimes b_2\\rbrace \\\\&+\\sum \\limits _{K,L=1}^{m+m^{\\prime }}\\sum \\limits _{\\alpha ,\\beta =1}^{m^\\prime }\\lbrace a_1\\otimes a_2,z^{\\alpha +m}\\rbrace \\tilde{g}_{\\alpha +m,\\beta +m}\\lbrace z^{\\beta +m},z^K\\rbrace \\tilde{g}_{KL}\\lbrace z^L,b_1\\otimes b_2\\rbrace $ since $\\tilde{g}_{iK}=0$ when $K > m$ , and $\\tilde{g}_{m+\\alpha ,K}=0$ when $K\\le m$ .",
"Moreover, since $\\lbrace z^i,z^K\\rbrace =\\lbrace x^i\\otimes 1,1\\otimes y^{K-m}\\rbrace =0$ and $\\lbrace z^{\\alpha +m},z^L\\rbrace =\\lbrace 1\\otimes y^{\\alpha },x^L\\otimes 1\\rbrace =0$ if $1\\le i\\le m$ , $1\\le \\alpha \\le m^{\\prime }$ , $K> m$ and $L\\le m$ , then $\\sum \\limits _{I,J,K,L}^{m+m^{\\prime }}&\\lbrace a_1\\otimes a_2,z^I\\rbrace \\tilde{g}_{IJ}\\lbrace z^J,z^K\\rbrace \\tilde{g}_{KL}\\lbrace z^L,b_1\\otimes b_2\\rbrace \\\\&=\\sum \\limits _{i,j,k,l=1}^m\\lbrace a_1\\otimes a_2,z^i\\rbrace \\tilde{g}_{ij}\\lbrace z^j,z^k\\rbrace \\tilde{g}_{kl}\\lbrace z^l,b_1\\otimes b_2\\rbrace \\\\&+\\sum \\limits _{\\alpha ,\\beta ,\\gamma ,\\delta =1}^{m^\\prime }\\lbrace a_1\\otimes a_2,z^{\\alpha +m}\\rbrace \\tilde{g}_{\\alpha +m,\\beta +m}\\lbrace z^{\\beta +m},z^{\\gamma +m}\\rbrace \\tilde{g}_{\\gamma +m,\\delta +m}\\lbrace z^{\\delta +m},b_1\\otimes b_2\\rbrace ,$ and $ \\sum \\limits _{I,J,K,L}^{m+m^{\\prime }} &\\lbrace a_1\\otimes a_2,z^I\\rbrace \\tilde{g}_{IJ}\\lbrace z^J,z^K\\rbrace \\tilde{g}_{KL}\\lbrace z^L,b_1\\otimes b_2\\rbrace \\\\&=\\sum \\limits _{i,j,k,l=1}^m\\lbrace a_1\\otimes a_2,x^i\\otimes 1\\rbrace \\rho g_{ij}\\otimes 1\\lbrace x^j\\otimes 1,x^k\\otimes 1\\rbrace \\rho g_{kl}\\otimes 1\\lbrace x^l\\otimes 1,b_1\\otimes b_2\\rbrace \\\\&+\\sum \\limits _{\\alpha ,\\beta ,\\gamma ,\\delta =1}^{m^\\prime }\\lbrace a_1\\otimes a_2,1\\otimes y^\\alpha \\rbrace 1\\otimes \\rho ^{\\prime }g^{\\prime }_{\\alpha \\beta }\\lbrace 1\\otimes y^\\beta ,1\\otimes y^\\gamma \\rbrace 1\\otimes \\rho ^{\\prime }g^{\\prime }_{\\gamma \\delta }\\lbrace 1\\otimes y^\\delta ,b_1\\otimes b_2\\rbrace \\\\&=\\sum \\limits _{i,j,k,l=1}^m(\\lbrace a_1,x^i\\rbrace \\otimes a_2)(\\rho g_{ij}\\otimes 1)(\\lbrace x^j,x^k\\rbrace \\otimes 1)(\\rho g_{kl}\\otimes 1)(\\lbrace x^l,b_1\\rbrace \\otimes b_2)\\\\&+\\sum \\limits _{\\alpha ,\\beta ,\\gamma ,\\delta =1}^{m^\\prime }(a_1\\otimes \\lbrace a_2,y^\\alpha \\rbrace ^{\\prime })(1\\otimes \\rho ^{\\prime }g^{\\prime }_{\\alpha \\beta })(1\\otimes \\lbrace y^\\beta ,y^\\gamma \\rbrace ^{\\prime })(1\\otimes \\rho ^{\\prime }g^{\\prime }_{\\gamma \\delta })(b_1\\otimes \\lbrace y^\\delta , b_2\\rbrace ^{\\prime })\\\\&=\\sum \\limits _{i,j,k,l=1}^m\\rho ^2\\lbrace a_1,x^i\\rbrace g_{ij}\\lbrace x^j,x^k\\rbrace g_{kl}\\lbrace x^l,b_1\\rbrace \\otimes a_2 b_2\\\\&+\\sum \\limits _{\\alpha ,\\beta ,\\gamma ,\\delta =1}^{m^\\prime } a_1b_1 \\otimes \\rho ^{\\prime 2}\\lbrace a_2,y^\\alpha \\rbrace ^{\\prime }g^{\\prime }_{\\alpha \\beta }\\lbrace y^\\beta ,y^\\gamma \\rbrace ^{\\prime }g^{\\prime }_{\\gamma \\delta }\\lbrace y^\\delta , b_2\\rbrace ^{\\prime }\\\\&=\\sum \\limits _{i,j,k,l=1}^m(\\eta \\lbrace a_1,x^i\\rbrace g_{ij}\\lbrace x^j,x^k\\rbrace g_{kl}\\lbrace x^l,b_1\\rbrace \\otimes a_2 b_2)\\\\&+\\sum \\limits _{\\alpha ,\\beta ,\\gamma ,\\delta =1}^{m^\\prime }( a_1b_1 \\otimes \\eta ^{\\prime }\\lbrace a_2,y^\\alpha \\rbrace ^{\\prime }g^{\\prime }_{\\alpha \\beta }\\lbrace y^\\beta ,y^\\gamma \\rbrace ^{\\prime }g^{\\prime }_{\\gamma \\delta }\\lbrace y^\\delta , b_2\\rbrace ^{\\prime }),$ since $\\rho ^2=\\eta $ and $\\rho ^{\\prime 2}=\\eta ^{\\prime }$ .",
"Finally, since $\\mathcal {K}$ and $\\mathcal {K}^{\\prime }$ are Kähler-Poisson algebras we get $\\sum \\limits _{I,J,K,L}^{m+m^{\\prime }}\\lbrace a_1\\otimes a_2,z^I\\rbrace \\tilde{g}_{IJ}\\lbrace z^J,z^K\\rbrace \\tilde{g}_{KL}\\lbrace z^L,b_1\\otimes b_2\\rbrace &=-\\lbrace a_1,b_1\\rbrace \\otimes a_2b_2-a_1b_1\\otimes \\lbrace a_2,b_2\\rbrace ^{\\prime }\\\\&=-\\lbrace a_1\\otimes a_2,b_1\\otimes b_2\\rbrace .$ Therefore, $\\mathcal {K}\\otimes \\mathcal {K}^{\\prime }$ is a Kähler-Poisson algebra."
],
[
"Summary",
"In this paper, we have introduced the concept of morphism of Kähler-Poisson algebras.",
"We have recalled a few results from [3], in order to motivate and understand the concept of morphism of Kähler-Poisson algebras.",
"We have studied properties of isomorphisms for Kähler-Poisson algebras and we illustrates with examples when two Kähler-Poisson algebras are isomorphic.",
"We have used the concept of morphism to define subalgebras of Kähler-Poisson algebras and we have presented examples when $\\mathcal {K}$ is a Kähler-Poisson subalgebra of $\\mathcal {K}^{\\prime }$ , where $\\mathcal {A}$ is a proper Poisson subalgebra of a finitely generated algebra $\\mathcal {A}^{\\prime }$ .",
"Finally, in Section 5, we have introduced direct sums and tensor products of Kähler-Poisson algebras and properties of these operations.",
"There are many open questions that one would like to investigate in future work.",
"For instance, is there a natural way to study the moduli spaces of Poisson algebras; i.e.",
"how many (non-isomorphic) Kähler-Poisson structures does there exist on a given Poisson algebra?"
],
[
"Acknowledgements",
"I would like to thank my supervisor J. Arnlind for fruitful discussions and helpful comments.",
"I would also like to thank my co-supervisor M. Izquierdo for ideas and discussions.",
"The results in Section 3 are a part of my licentiate thesis [4]."
]
] | 1906.04519 |
[
[
"Transfer Learning for Ultrasound Tongue Contour Extraction with\n Different Domains"
],
[
"Abstract Medical ultrasound technology is widely used in routine clinical applications such as disease diagnosis and treatment as well as other applications like real-time monitoring of human tongue shapes and motions as visual feedback in second language training.",
"Due to the low-contrast characteristic and noisy nature of ultrasound images, it might require expertise for non-expert users to recognize tongue gestures.",
"Manual tongue segmentation is a cumbersome, subjective, and error-prone task.",
"Furthermore, it is not a feasible solution for real-time applications.",
"In the last few years, deep learning methods have been used for delineating and tracking tongue dorsum.",
"Deep convolutional neural networks (DCNNs), which have shown to be successful in medical image analysis tasks, are typically weak for the same task on different domains.",
"In many cases, DCNNs trained on data acquired with one ultrasound device, do not perform well on data of varying ultrasound device or acquisition protocol.",
"Domain adaptation is an alternative solution for this difficulty by transferring the weights from the model trained on a large annotated legacy dataset to a new model for adapting on another different dataset using fine-tuning.",
"In this study, after conducting extensive experiments, we addressed the problem of domain adaptation on small ultrasound datasets for tongue contour extraction.",
"We trained a U-net network comprises of an encoder-decoder path from scratch, and then with several surrogate scenarios, some parts of the trained network were fine-tuned on another dataset as the domain-adapted networks.",
"We repeat scenarios from target to source domains to find a balance point for knowledge transfer from source to target and vice versa.",
"The performance of new fine-tuned networks was evaluated on the same task with images from different domains."
],
[
"Introduction",
"Ultrasound imaging is safe, relatively affordable, and capable of real-time performance.",
"This technology has been utilized for many real-time medical applications.",
"Recently, ultrasound is used for visualizing and characterizing human tongue shape and motion in a real-time speech to study healthy or impaired speech production in applications such as visual second language training [4] or silent speech interfaces [2].",
"However, it requires expertise for non-expert users to recognize tongue shape and motion in noisy and low-contrast ultrasound data.",
"To address this problem and to have a quantitative analysis, tongue surface (dorsum) can be extracted, tracked, and visualized superimposed on the whole tongue region.",
"Delineating the tongue surface from each frame is a cumbersome, subjective, and error-prone task.",
"Moreover, the rapidity and complexity of tongue gestures have made it a challenging task, and manual segmentation is not a feasible solution for real-time applications.",
"Over the years, several image processing techniques, such as active contour models [7], have shown their capability for automatic tongue contour extraction.",
"In many of those traditional methods, manual labelling, initialization, monitoring, and manipulation are frequently needed.",
"Furthermore, those methods are computationally expensive whereas the image gradient should be calculated for each frame [5].",
"In recent years, convolutional neural networks (CNN) have been the method of choice for medical image analysis with outstanding results [8].",
"In a few studies, automatic tongue contour extraction [14] and tracking [10] using CNN have been already investigated.",
"In spite of their excellent results on a specific dataset from one ultrasound device, the generalization of those methods on test data with different distributions from different ultrasound device is often not investigated and evaluated.",
"Although ultrasound tongue datasets have different distributions, there is always a correlation between movements of the tongue and its possible positions in the mouth.",
"Therefore, domain adaptation might provide a universal solution for automatic, real-time tongue contour extraction, applicable to the majority of ultrasound datasets.",
"In transfer learning [12], a marginal probability distribution $P(X)$ , where $X = \\left\\lbrace x_{1}, ..., x_{n} \\right\\rbrace $ defined on a feature space of $R$ can be used for expressing a domain $D$ .",
"On a specific domain $D = \\left\\lbrace R, P(X) \\right\\rbrace $ , comprises of pair of a label space $Y$ and an objective function $f(\\cdot )$ in a form of $T = \\left\\lbrace Y, f(\\cdot ) \\right\\rbrace $ , the objective function $f(\\cdot )$ can be optimized and learned the training data, which consists of pairs $\\left\\lbrace x_{i}, y_{i} \\right\\rbrace $ , where $x_{i} \\in X$ and $y_{i} \\in Y$ in a supervised fashion using one CNN model.",
"After termination of the optimization process, the trained CNN model denoted by $\\tilde{f}(\\cdot )$ can predict the label of a new instance $x$ .",
"Transfer learning is defined as the procedure of enhancing the target prediction function $f_{T}(\\cdot )$ in $D_{T}$ using the information in $D_{S}$ and $T_{S}$ , whereas a source domain $D_{s}$ with a learning task $T_{S}$ and a target domain $D_{T}$ with learning task $T_{T}$ are given and $D_{S} \\ne D_{T}$ , or $T_{S} \\ne T_{T}$ [3].",
"Therefore, the prediction function $\\tilde{f}_{ST}(\\cdot )$ first is trained on the source domain $D_{S}$ and then fine-tuned for the target domain $D_{T}$ .",
"Conversely, $\\tilde{f}_{TS}(\\cdot )$ is initially trained for the target task, and then it is domain-adapted on the source dataset.",
"Fully convolutional networks (FCNs) consist of consecutive convolutional and pooling layers (encoder), and one up-sampling layer (decoder) was successfully exploited for the semantic segmentation problem in a study by [9].",
"Due to the loss of information in polling layers, only one layer of up-sampling cannot retrieve the input-sized resolution in the output prediction map.",
"Concatenation of feature maps from deconvolutional layers (DeconvNet) [11] and encoder layers in U-net [13] improved significantly the segmentation accuracy.",
"The encoder of U-net learns simple visual image features especially in the first few layers, while the decoder aims to reconstruct the input-sized output prediction map from the complicated, abstract, and task-dependent features of the last layer of the encoder.",
"Although encoder-decoder models like U-net have been used for tongue contour extraction, still it is not obvious how much knowledge is preserved during the transfer learning process for domain adaptation.",
"In this study, the performance of the U-net in different scenarios was analyzed to answer some fundamental questions in domain adaptation.",
"We investigated how many layers from the decoder part should be fine-tuned to achieve the best segmentation accuracy in both the source and target domain at the same time (we called that a balanced point).",
"Furthermore, the efficacy of dataset size in target domain along with the skip operation and concatenation on the performance of the U-net were explored on the problem of ultrasound tongue contour extraction.",
"Ultrasound video frames were randomly selected from recorded videos of a linear transducer connected to a Sonix Tablet ultrasound device at the University of Ottawa as well as videos from Seeing Speech project [6].",
"Using informed undersampling method [1], we generated two 2050 image datasets with different distributions, dataset I (uOttawa) and dataset II (SeeingSpeech).",
"In this method, an average intensity image is calculated over the entire dataset, then one score is assigned for each frame depends on its intensity distance to the average image.",
"After sorting the data by ranking order, we selected 2000 frames with the highest rank (high variance) and 50 images with the lowest grade (low variation).",
"Ground truth labels corresponding to each data was annotated semi-automatically by two experts using our custom annotation software.",
"Off-line augmentation comprises of natural transformations in ultrasound data (e.g., horizontal flipping, restricted rotation and zooming) was employed to create larger datasets (50K for each).",
"We split each dataset into training, validation, and test sets using $\\%90/5/5$ ratios."
],
[
"Network Architecture and Training",
"Fully convolutional networks (FCNs) can be considered as dense classification networks (e.g., VGG-nets) with consecutive convolutional and pooling layers such that a fully convolutional layer substitutes the fully connected layer (e.g., softmax in the last layer).",
"Similarly, DeconvNet is an FCN network with several deconvolutional layers in the up-sampling path.",
"In U-net [13] which is a DeconvNet architecture, feature maps (coarse contextual information) skips from each down-sampling layer to concatenate with deconvolutional layers for increasing the accuracy of output segmentation.",
"Structural details of U-net have been presented in Fig.",
"REF .",
"Figure: An overview of network structures.",
"Numbers in circles show several scenarios for finding the best model for domain adaptation.The DeconvNet comprises of 9 double convolutional layers of $3\\times 3$ filters with Rectified Linear Unit (ReLU) activation function as non-linearity.",
"Activations of all layers were normalized using batch normalization layers to speed up the convergence.",
"In the downsampling path, there are four max-pooling layers for the sake of translation invariance and saving memory by decreasing learnable parameters.",
"In contrast, in the up-sampling path, there are four deconvolutional layers which retrieve the original receptive filed and spatial resolution.",
"Finally, the high-level reasoning is done by a fully convolutional layer at the end layer.",
"Network models were deployed using the publicly available TensorFlow framework on Keras API as the backend library.",
"For initialization of network parameters, randomly distributed values have been selected.",
"Adam optimization was chosen with a fixed momentum value of 0.9 for finding the optimum solution on a binary cross-entropy loss function.",
"Each network model was trained using the mini-batch method employing one NVIDIA 1080 GPU unit which was installed on a Windows PC with Core i7, 4.2 GHz speed, and 32 GB of RAM.",
"Our results from hyper-parameter tuning revealed that, besides network architecture size, learning rate has the most significant effect on the performance of each architecture in terms of accuracy.",
"Testing fixed and scheduled decaying learning rate showed that the variable learning rate might provide better results, but it requires different initialization of decay factor and decay steps.",
"Therefore, for the sake of a fair comparison, we only reported results using fixed learning rates.",
"To alleviate the over-fitting problem, we regularized our networks by drop-out rate of 0.5.",
"Networks were trained for a maximum of five epochs, each of which for 5000 iterations and mini-batch size of 10."
],
[
"Domain Adaptation Scenarios",
"Models for $\\tilde{f}_{ST}(.",
")$ were built from several scenarios with transferring the learned weights from $\\tilde{f}_{S}$ when we froze the encoder and some parts of the decoder sections.",
"Specifically, in scenario I, we transferred weights of the whole encoder as well as portions of the DeconvNet decoder as $\\tilde{f}_{S}$ which was learned on the dataset I ($D_{S}$ ), then we froze those sections up to the $i$ th deconvolutional layer and fine-tuned the remaining ($4-i$ ) deconvolutional layers using the Dataset II ($D_{T}$ ) (see the circled numbers in Fig REF ).",
"In scenario II, we investigated the opposite transferring case by switching source and target datasets to build model $\\tilde{f}_{TS}(.",
")$ to see the effect of negative transferring.",
"In similar scenarios, we repeated the same experiments by considering the impact of skip operator and concatenation in U-net to investigate the effect of transferring knowledge by injecting feature maps to the decoder from the encoder.",
"To evaluate models, we investigated and compared different scenarios of tongue contour extraction as described in the previous section.",
"In each situation, we first trained the whole DeconvNet and U-net on the source domains (named base models) and directly apply them on two source and target domains to see the weakness of each model in terms of generalization from one domain to another.",
"From table REF , as it was anticipated, in both scenarios, base networks predicted better instances for their source domains than their target domains.",
"The result of each scenario related to DeconvNet and U-net have been presented in table REF .",
"Results of the table reveal that on average fine-tuning the whole decoder section is the best for achieving the best accuracy in target domain while the negative transferring can be seen clearly in these cases.",
"For instance, in the scenario I, in case of U-net base model, it achieved a Dice coefficient of 0.6884 for the source domain and 0.4664 for the target domain.",
"At the same time, when the whole decoder fine-tuned a better Dice coefficient of 0.6306 was achieved in the target domain and 0.5818 in the source domain.",
"As it can be seen, by freezing more layer in the decoder section (conv7, conv8, and conv9) the difference between the Dice coefficient values in source and target domains significantly increases.",
"For the case of DeconvNet, this is not true and the difference decrease in higher layers.",
"Table REF also indicate considerable result improvement in the scenario I for the U-net compare to the DeconvNet due to the concatenation and skip operation.",
"Table: Quantitative results of each scenario.",
"Negative knowledge transferring can be seen in the two first columns for both models.To identify the sufficient size of the target dataset for transfer learning, in separate experiments, we turned two transferred U-net models (encoder and conv 9) on three datasets with sizes of 100, 1000, and 10000.",
"We used the same network architecture and training procedure among the different experiments.",
"Figure REF shows the difference values between dice coefficients and cross-entropy losses in source and target domains for scenario I.",
"It can be seen that more data samples enhance the performance of each model in terms of accuracy.",
"Figure: Effect of increasing dataset sizes on the accuracy of two transferred models in scenario I.",
"Each column shows the difference of cross-entropy loss and dice-coefficient between source and target domains.Fig REF illustrates the qualitative results of the scenario I for U-net model applied on a test instance.",
"The U-net (base model) was trained on the set of images from the source domain ($\\tilde{f}_{S}$ ), achieved a Dice coefficient of 0.65 and Binary cross-entropy loss of 0.28 while for the same model, the value of Dice score and loss for the target domain was 0.50 and 0.49 without fine-tuning.",
"It means that although the result of the target domain is not significant, U-net base model can still predict instances in both source and target domains.",
"Nevertheless, in case of real-time testing when some frames contain rapid tongue movement along with noisy dorsum region with artifacts (see Fig.",
"REF .c), the model fails in prediction for the target domain.",
"Our experimental results revealed that fine-tuning of the whole decoder of the U-net alleviates this problem significantly.",
"For instance, dice score and loss values approximately become 0.58 and 0.34 for both source and target domains when the whole decoder fine-tuned on the target domain.",
"In general, we observed a balance point for the number of refined layers considering both source and target domains.",
"On the balance point, the model can achieve similar acceptable results in both source, and target domains whereas the segmentation accuracy is worse than the model's performance on only one domain (see Fig.",
"REF .d).",
"Figure: Prediction results of U-net in scenario I D S →D T D_{S} \\rightarrow D_{T}.",
"(a) sample data, (b) corresponding ground truth labels, (c) prediction result of f ˜ S \\tilde{f}_{S} (U-net base), (d) prediction result of f ˜ ST \\tilde{f}_{ST} when the whole decoder was fine-tuned on D T D_{T}."
],
[
"Discussion and Conclusions",
"In transfer learning literature, researchers usually focus on finding $\\tilde{f}_{S}$ which demonstrates a decent performance on a source domain $D_{S}$ .",
"Then they try domain adaptation from source to target $D_{T}$ task to find $\\tilde{f}_{ST}$ .",
"However, a reliable and universal method is the one which can provide acceptable results in the opposite path from target to source domain as well.",
"Our experimental results showed that there is a balance point for U-net model where it can provide reasonable predictions on both the source and the target domain ($\\tilde{f}_{ST} \\sim \\tilde{f}_{TS}$ ).",
"For instance, transferring the whole decoder of U-net on the target domain, it provided Binary cross-entropy loss values of 0.3034 and 0.3129 for source and target test data.",
"Furthermore, qualitative study shows that domain adaptation can improve segmentation result for frames with significant noise and artifacts.",
"Impact of using skip operator and concatenation and increasing dataset size in target domain indicate a slight improvement in final results.",
"In contrast with other research fields with large datasets, the size of a usual ultrasound tongue dataset is not more than $\\sim 200K$ frames, and it makes more sense to fine-tune one model on several datasets to find the knowledge balance point as a universal model for use in real-time applications on various ultrasound devices.",
"Using smaller learning rates in the target domains might increase the accuracy of the source and target domain segmentation further on the balance point."
]
] | 1906.04301 |
[
[
"CVPR19 Tracking and Detection Challenge: How crowded can it get?"
],
[
"Abstract Standardized benchmarks are crucial for the majority of computer vision applications.",
"Although leaderboards and ranking tables should not be over-claimed, benchmarks often provide the most objective measure of performance and are therefore important guides for research.",
"The benchmark for Multiple Object Tracking, MOTChallenge, was launched with the goal to establish a standardized evaluation of multiple object tracking methods.",
"The challenge focuses on multiple people tracking, since pedestrians are well studied in the tracking community, and precise tracking and detection has high practical relevance.",
"Since the first release, MOT15, MOT16 and MOT17 have tremendously contributed to the community by introducing a clean dataset and precise framework to benchmark multi-object trackers.",
"In this paper, we present our CVPR19 benchmark, consisting of 8 new sequences depicting very crowded challenging scenes.",
"The benchmark will be presented at the 4th BMTT MOT Challenge Workshop at the Computer Vision and Pattern Recognition Conference (CVPR) 2019, and will evaluate the state-of-the-art in multiple object tracking whend handling extremely crowded scenarios."
],
[
"Introduction",
"Since its first release in 2014, MOTChallenge has attracted more than $1,000$ active users who have successfully submitted their trackers and detectors to five different challenges, spanning 44 sequences with $2.7M$ bounding boxes over a total length of $36k$ seconds.",
"As evaluating and comparing multi-target tracking methods is not trivial (cf. e.g.",
"[10]), MOTChallenge provides carefully annotated datasets and clear metrics to evaluate the performance of tracking algorithms and pedestrian detectors.",
"Parallel to the MOTChallenge all-year challenges, we organize workshop challenges on multi-object tracking for which we often introduce new data.",
"Figure: An overview of the CVPR19 dataset.",
"The dataset consists of 8 different sequences from 3 different scenes.",
"The test dataset has two known and one unknown scene.",
"Top: training sequences; bottom: test sequences.In this paper, we introduce the CVPR19 benchmark, consisting of 8 novel sequences out of 3 very crowded scenes.",
"All sequences have been carefully selected and annotated according to the evaluation protocol of previous challenges , [8].",
"This benchmark addresses the challenge of very crowded scenes in which the density can reach values of 246 pedestrians per frame.",
"The sequences were filmed in both indoor and outdoor locations, and include day and night time shots.",
"Figure REF shows the split of the sequences of the three scenes into training and testing sets.",
"The testing data consists of sequences from known as well as from an unknown scenes in order to measure the genralization capabilities of both detectors and trackers.",
"We make available the images for all sequences, the ground truth annotations for the training set as well as a set of public detections (obtained from a Faster R-CNN trained on the training data) for the tracking challenge.",
"The CVPR19 challenges and all data, current ranking and submission guidelines can be found at: ht tp://www.motchallenge.net/ Note that the submission to the CVPR19 challenges will be temporary and will close shortly before the workshop at CVPR.",
"However, the data will be the foundation of a novel release of MOT19 later this year.",
"Figure: we provide for the challenges.",
"Left: Image of each frame of the sequences; middle: ground truth labels including all classes.",
"Only provided for training set; right: public detections from trained Faster R-CNN."
],
[
"Annotation rules",
"For the annotation of the dataset, we follow the protocol introduced in MOT16, ensuring that every moving person or vehicle within each sequence is annotated with a bounding box as accurately as possible.",
"In the following, we define a clear protocol that was obeyed throughout the entire dataset to guarantee consistency."
],
[
"Target class",
"In this benchmark, we are interested in tracking moving objects in videos.",
"In particular, we are interested in evaluating multiple people tracking algorithms, hence, people will be the center of attention of our annotations.",
"We divide the pertinent classes into three categories: (i) moving pedestrians; (ii) people that are not in an upright position, not moving, or artificial representations of humans; and (iii) vehicles and occluders.",
"In the first group, we annotate all moving pedestrians that appear in the field of view and can be determined as such by the viewer.",
"Furthermore, if a person briefly bends over or squats, e.g., to pick something up or to talk to a child, they shall remain in the standard pedestrian class.",
"The algorithms that submit to our benchmark are expected to track these targets.",
"In the second group, we include all people-like objects whose exact classification is ambiguous and can vary depending on the viewer, the application at hand, or other factors.",
"We annotate all static people, e.g., sitting, lying down, or do stand still at the same place over the whole sequence.",
"The idea is to use these annotations in the evaluation such that an algorithm is neither penalized nor rewarded for tracking, e.g., a sitting or not moving person.",
"In the third group, we annotate all moving vehicles such as cars, bicycles, motorbikes and non-motorized vehicles (e.g., strollers), as well as other potential occluders.",
"These annotations will not play any role in the evaluation, but are provided to the users both for training purposes and for computing the level of occlusion of pedestrians.",
"Static vehicles (parked cars, bicycles) are not annotated as long as they do not occlude any pedestrians."
],
[
"Datasets",
"The dataset for the new benchmark has been carefully selected to challenge trackers and detectors on extremely crowded scenes.",
"In contrast to previous challenges, some of the new sequences show a pedestrian density of 246 pedestrians per frame.",
"In Fig.",
"REF and Tab.",
"REF , we show an overview of the sequences included in the benchmark.",
"Table: Overview of the sequences currently included in the CVPR19 benchmark.Table: Overview of the types of annotations currently found in the CVPR19 benchmark."
],
[
"CVPR 19 sequences",
"We have compiled a total of 8 sequences, of which we use half for training and half for testing.",
"The annotations of the testing sequences will not be released in order to avoid (over)fitting of the methods to the specific sequences.",
"The sequences are filmed in three different scenes.",
"Several sequences are filmed per scene and distributed in the train and test sets.",
"One of the scenes though, is reserved for test time, in order to challenge the generalization capabilities of the methods.",
"The new data contains circa 3 times more bounding boxes for training and testing compared to MOT17.",
"All sequences are filmed in high resolution from an elevated viewpoint, and the mean crowd density reaches 246 pedestrians per frame which 10 times denser when compared to the first benchmark release.",
"Hence, we expect the new sequences to be more challenging for the tracking community and to push the models to their limits when it comes to handling extremely crowded scenes.",
"In Tab.",
"REF , we give an overview of the training and testing sequence characteristics for the challenge, including the number of bounding boxes annotated.",
"Aside from pedestrians, the annotations also include other classes like vehicles or bicycles, as detailed in Sec.",
".",
"In Tab.",
"REF , we detail the types of annotations that can be found in each sequence of CVPR19."
],
[
"Detections",
"We trained a Faster R-CNN [11] with ResNet101 [4] backbone on the CVPR19 training sequences, obtaining the detection results presented in Table REF .",
"This evaluation follows the standard protocol for the CVPR19 challenge and only accounts for pedestrians.",
"Static persons and other classes are not considered and filtered out from both, the detections, as well as the ground truth.",
"A detailed breakdown of detection bounding boxes on individual sequences is provided in Tab.",
"REF .",
"Table: Detection bounding box statistics.For the tracking challenge, we provide these public detections as a baseline to be used for training and testing of the trackers.",
"For the CVPR19 challenge, we will only accept results on public detections.",
"When later the benchmark will be open for continuous submissions, we will accept both public as well as private detections."
],
[
"Data format",
"All images were converted to JPEG and named sequentially to a 6-digit file name (e.g.",
"000001.jpg).",
"Detection and annotation files are simple comma-separated value (CSV) files.",
"Each line represents one object instance and contains 9 values as shown in Tab.",
"REF .",
"The first number indicates in which frame the object appears, while the second number identifies that object as belonging to a trajectory by assigning a unique ID (set to $-1$ in a detection file, as no ID is assigned yet).",
"Each object can be assigned to only one trajectory.",
"The next four numbers indicate the position of the bounding box of the pedestrian in 2D image coordinates.",
"The position is indicated by the top-left corner as well as width and height of the bounding box.",
"This is followed by a single number, which in case of detections denotes their confidence score.",
"The last two numbers for detection files are ignored (set to -1).",
"Table: Data format for the input and output files, both for detection (DET) and annotation/ground truth (GT) files.Table: Label classes present in the annotation files and ID appearing in the 7 th ^\\text{th} column of the files as described in Tab.",
".An example of such a 2D detection file is: 1, -1, 794.2, 47.5, 71.2, 174.8, 67.5, -1, -1 1, -1, 164.1, 19.6, 66.5, 163.2, 29.4, -1, -1 1, -1, 875.4, 39.9, 25.3, 145.0, 19.6, -1, -1 2, -1, 781.7, 25.1, 69.2, 170.2, 58.1, -1, -1 For the ground truth and results files, the 7$^\\text{th}$ value (confidence score) acts as a flag whether the entry is to be considered.",
"A value of 0 means that this particular instance is ignored in the evaluation, while a value of 1 is used to mark it as active.",
"The 8$^\\text{th}$ number indicates the type of object annotated, following the convention of Tab.",
"REF .",
"The last number shows the visibility ratio of each bounding box.",
"This can be due to occlusion by another static or moving object, or due to image border cropping.",
"An example of such an annotation 2D file is: 1, 1, 794.2, 47.5, 71.2, 174.8, 1, 1, 0.8 1, 2, 164.1, 19.6, 66.5, 163.2, 1, 1, 0.5 2, 4, 781.7, 25.1, 69.2, 170.2, 0, 12, 1.",
"In this case, there are 2 pedestrians in the first frame of the sequence, with identity tags 1, 2.",
"In the second frame, we can see a static person (class 7), which is to be considered by the evaluation script and will neither count as a false negative, nor as a true positive, independent of whether it is correctly recovered or not.",
"Note that all values including the bounding box are 1-based, i.e.",
"the top left corner corresponds to $(1,1)$ .",
"To obtain a valid result for the entire benchmark, a separate CSV file following the format described above must be created for each sequence and called “Sequence-Name.txt”.",
"All files must be compressed into a single ZIP file that can then be uploaded to be evaluated.",
"Table: Overview of performance of Faster R-CNN detector trained on the CVPR19 training dataset."
],
[
"Evaluation",
"Our framework is a platform for fair comparison of state-of-the-art tracking methods.",
"By providing authors with standardized ground truth data, evaluation metrics and scripts, as well as a set of precomputed detections, all methods are compared under the exact same conditions, thereby isolating the performance of the tracker from everything else.",
"In the following paragraphs, we detail the set of evaluation metrics that we provide in our benchmark."
],
[
"Evaluation metrics",
"In the past, a large number of metrics for quantitative evaluation of multiple target tracking have been proposed [13], [14], [2], [12], [16], [6].",
"Choosing “the right” one is largely application dependent and the quest for a unique, general evaluation metric is still ongoing.",
"On the one hand, it is desirable to summarize the performance into one single number to enable a direct comparison.",
"On the other hand, one might not want to lose information about the individual errors made by the algorithms and provide several performance estimates, which precludes a clear ranking.",
"Following a recent trend [9], [1], [15], we employ two sets of measures that have established themselves in the literature: The CLEAR metrics proposed by Stiefelhagen et al.",
"[14], and a set of track quality measures introduced by Wu and Nevatia [16].",
"The evaluation scripts used in our benchmark are publicly available.http://motchallenge.net/devkit There are two common prerequisites for quantifying the performance of a tracker.",
"One is to determine for each hypothesized output, whether it is a true positive (TP) that describes an actual (annotated) target, or whether the output is a false alarm (or false positive, FP).",
"This decision is typically made by thresholding based on a defined distance (or dissimilarity) measure $d$ (see Sec.",
"REF ).",
"A target that is missed by any hypothesis is a false negative (FN).",
"A good result is expected to have as few FPs and FNs as possible.",
"Next to the absolute numbers, we also show the false positive ratio measured by the number of false alarms per frame (FAF), sometimes also referred to as false positives per image (FPPI) in the object detection literature.",
"Figure: Four cases illustrating tracker-to-target assignments.",
"(a) An ID switchoccurs when the mapping switches from the previously assigned red trackto the blue one.",
"(b) A track fragmentation is counted in frame 3 becausethe target is tracked in frames 1-2, then interrupts, and thenreacquires its `tracked' status at a later point.",
"A new (blue) track hypothesis alsocauses an ID switch at this point.",
"(c) Although the tracking results isreasonably good, an optimal single-frame assignment in frame 1 ispropagated through the sequence, causing 5 missed targets (FN) and 4false positives (FP).",
"Note that no fragmentations are counted in frames3 and 6 because tracking of those targets is not resumed at a laterpoint.",
"(d) A degenerate case illustrating that target re-identificationis not handled correctly.",
"An interrupted ground truth trajectory willtypically cause a fragmentation.",
"Also note the less intuitive ID switch,which is counted because blue is the closest target in frame 5 that isnot in conflict with the mapping in frame 4.Obviously, it may happen that the same target is covered by multiple outputs.",
"The second prerequisite before computing the numbers is then to establish the correspondence between all annotated and hypothesized objects under the constraint that a true object should be recovered at most once, and that one hypothesis cannot account for more than one target.",
"For the following, we assume that each ground truth trajectory has one unique start and one unique end point, i.e.",
"that it is not fragmented.",
"Note that the current evaluation procedure does not explicitly handle target re-identification.",
"In other words, when a target leaves the field-of-view and then reappears, it is treated as an unseen target with a new ID.",
"As proposed in [14], the optimal matching is found using Munkre's (a.k.a.",
"Hungarian) algorithm.",
"However, dealing with video data, this matching is not performed independently for each frame, but rather considering a temporal correspondence.",
"More precisely, if a ground truth object $i$ is matched to hypothesis $j$ at time $t-1$ and the distance (or dissimilarity) between $i$ and $j$ in frame $t$ is below $t_d$ , then the correspondence between $i$ and $j$ is carried over to frame $t$ even if there exists another hypothesis that is closer to the actual target.",
"A mismatch error (or equivalently an identity switch, IDSW) is counted if a ground truth target $i$ is matched to track $j$ and the last known assignment was $k\\ne j$ .",
"Note that this definition of ID switches is more similar to [6] and stricter than the original one [14].",
"Also note that, while it is certainly desirable to keep the number of ID switches low, their absolute number alone is not always expressive to assess the overall performance, but should rather be considered in relation to the number of recovered targets.",
"The intuition is that a method that finds twice as many trajectories will almost certainly produce more identity switches.",
"For that reason, we also state the relative number of ID switches, which is computed as IDSW / Recall.",
"These relationships are illustrated in Fig.",
"REF .",
"For simplicity, we plot ground truth trajectories with dashed curves, and the tracker output with solid ones, where the color represents a unique target ID.",
"The grey areas indicate the matching threshold (see next section).",
"Each true target that has been successfully recovered in one particular frame is represented with a filled black dot with a stroke color corresponding to its matched hypothesis.",
"False positives and false negatives are plotted as empty circles.",
"See figure caption for more details.",
"After determining true matches and establishing correspondences it is now possible to compute the metrics.",
"We do so by concatenating all test sequences and evaluating on the entire benchmark.",
"This is in general more meaningful instead of averaging per-sequences figures due to the large variation in the number of targets."
],
[
"Distance measure",
"In the most general case, the relationship between ground truth objects and a tracker output is established using bounding boxes on the image plane.",
"Similar to object detection [3], the intersection over union (a.k.a.",
"the Jaccard index) is usually employed as the similarity criterion, while the threshold $t_d$ is set to $0.5$ or $50\\%$ ."
],
[
"Target-like annotations",
"People are a common object class present in many scenes, but should we track all people in our benchmark?",
"For example, should we track static people sitting on a bench?",
"Or people on bicycles?",
"How about people behind a glass?",
"We define the target class of CVPR19 as all upright walking people that are reachable along the viewing ray without a physical obstacle, i.e.",
"reflections, people behind a transparent wall or window are excluded.",
"We also exclude from our target class people on bicycles or other vehicles.",
"For all these cases where the class is very similar to our target class (see Figure REF ), we adopt a similar strategy as in [7].",
"That is, a method is neither penalized nor rewarded for tracking or not tracking those similar classes.",
"Since a detector is likely to fire in those cases, we do not want to penalize a tracker with a set of false positives for properly following that set of detections, i.e.",
"of a person on a bicycle.",
"Likewise, we do not want to penalize with false negatives a tracker that is based on motion cues and therefore does not track a sitting person.",
"Figure: The annotations include different classes.",
"The target class are pedestrians (left).",
"Besides pedestrians there exist special classes in the data such as static person and non-motorized vehicles (non mot vhcl).",
"However, these classes are filter out during evaluation and do not effect the test score.",
"Thirdly, we annotate occluders and crowds.In order to handle these special cases, we adapt the tracker-to-target assignment algorithm to perform the following steps: At each frame, all bounding boxes of the result file are matched to the ground truth via the Hungarian algorithm.",
"In contrast to MOT17 we account for the very crowded scenes and exclude result boxes that overlap $>75\\%$ with one of these classes (distractor, static person, reflection, person on vehicle) are removed from the solution.",
"During the final evaluation, only those boxes that are annotated as pedestrians are used."
],
[
"Multiple Object Tracking Accuracy",
"The MOTA [14] is perhaps the most widely used metric to evaluate a tracker's performance.",
"The main reason for this is its expressiveness as it combines three sources of errors defined above: $\\text{MOTA} =1 - \\frac{\\sum _t{(\\text{FN}_t + \\text{FP}_t + \\text{IDSW}_t})}{\\sum _t{\\text{GT}_t}},$ where $t$ is the frame index and GT is the number of ground truth objects.",
"We report the percentage MOTA $(-\\infty , 100]$ in our benchmark.",
"Note that MOTA can also be negative in cases where the number of errors made by the tracker exceeds the number of all objects in the scene.",
"Even though the MOTA score gives a good indication of the overall performance, it is highly debatable whether this number alone can serve as a single performance measure.",
"Robustness.",
"One incentive behind compiling this benchmark was to reduce dataset bias by keeping the data as diverse as possible.",
"The main motivation is to challenge state-of-the-art approaches and analyze their performance in unconstrained environments and on unseen data.",
"Our experience shows that most methods can be heavily overfitted on one particular dataset, and may not be general enough to handle an entirely different setting without a major change in parameters or even in the model.",
"To indicate the robustness of each tracker across all benchmark sequences, we show the standard deviation of their MOTA score."
],
[
"Multiple Object Tracking Precision",
"The Multiple Object Tracking Precision is the average dissimilarity between all true positives and their corresponding ground truth targets.",
"For bounding box overlap, this is computed as $\\text{MOTP} =\\frac{\\sum _{t,i}{d_{t,i}}}{\\sum _t{c_t}},$ where $c_t$ denotes the number of matches in frame $t$ and $d_{t,i}$ is the bounding box overlap of target $i$ with its assigned ground truth object.",
"MOTP thereby gives the average overlap between all correctly matched hypotheses and their respective objects and ranges between $t_d:= 50\\%$ and $100\\%$ .",
"It is important to point out that MOTP is a measure of localization precision, not to be confused with the positive predictive value or relevance in the context of precision / recall curves used, e.g., in object detection.",
"In practice, it mostly quantifies the localization accuracy of the detector, and therefore, it provides little information about the actual performance of the tracker."
],
[
"Track quality measures",
"Each ground truth trajectory can be classified as mostly tracked (MT), partially tracked (PT), and mostly lost (ML).",
"This is done based on how much of the trajectory is recovered by the tracking algorithm.",
"A target is mostly tracked if it is successfully tracked for at least $80\\%$ of its life span.",
"Note that it is irrelevant for this measure whether the ID remains the same throughout the track.",
"If a track is only recovered for less than $20\\%$ of its total length, it is said to be mostly lost (ML).",
"All other tracks are partially tracked.",
"A higher number of MT and few ML is desirable.",
"We report MT and ML as a ratio of mostly tracked and mostly lost targets to the total number of ground truth trajectories.",
"In certain situations one might be interested in obtaining long, persistent tracks without gaps of untracked periods.",
"To that end, the number of track fragmentations (FM) counts how many times a ground truth trajectory is interrupted (untracked).",
"In other words, a fragmentation is counted each time a trajectory changes its status from tracked to untracked and tracking of that same trajectory is resumed at a later point.",
"Similarly to the ID switch ratio (cf.",
"Sec.",
"REF ), we also provide the relative number of fragmentations as FM / Recall."
],
[
"Tracker ranking",
"As we have seen in this section, there are a number of reasonable performance measures to assess the quality of a tracking system, which makes it rather difficult to reduce the evaluation to one single number.",
"To nevertheless give an intuition on how each tracker performs compared to its competitors, we compute and show the average rank for each one by ranking all trackers according to each metric and then averaging across all performance measures."
],
[
"Conclusion and Future Work",
"We have presented a new challenging set of sequences within the MOTChallenge benchmark our CVPR19 Workshop.",
"Theses sequences contain a large number of targets to be tracked and the scenes are substantially more crowded when compared to previous MOTChallenge releases.",
"The scenes are carefully chosen and included indoor/ outdoor and day/ night scenarios.",
"The CVPR19 challenge is the foundation of a new bigger benchmark MOT19 that will presented later this year.",
"We believe that the CVPR19 release within the already established MOTChallenge benchmark provides a fairer comparison of state-of-the-art tracking methods, and challenges researchers to develop more generic methods that perform well in unconstrained environments and on very crowded scenes.",
"This challenge will only be active until the CVPR Workshop 2019, when the dataset will be used for a new ongoing MOT19 challenge."
]
] | 1906.04567 |
[
[
"Learning robust visual representations using data augmentation\n invariance"
],
[
"Abstract Deep convolutional neural networks trained for image object categorization have shown remarkable similarities with representations found across the primate ventral visual stream.",
"Yet, artificial and biological networks still exhibit important differences.",
"Here we investigate one such property: increasing invariance to identity-preserving image transformations found along the ventral stream.",
"Despite theoretical evidence that invariance should emerge naturally from the optimization process, we present empirical evidence that the activations of convolutional neural networks trained for object categorization are not robust to identity-preserving image transformations commonly used in data augmentation.",
"As a solution, we propose data augmentation invariance, an unsupervised learning objective which improves the robustness of the learned representations by promoting the similarity between the activations of augmented image samples.",
"Our results show that this approach is a simple, yet effective and efficient (10 % increase in training time) way of increasing the invariance of the models while obtaining similar categorization performance."
],
[
"Introduction",
"Deep artificial neural networks (DNNs) have borrowed much inspiration from neuroscience and are, at the same time, the current best model class for predicting neural responses across the visual system in the brain [9], [11].",
"Yet, despite consensus about the benefits of a closer integration of deep learning and neuroscience [2], [12], important differences remain.",
"Here, we investigate a representational property that is well established in the neuroscience literature on the primate visual system: the increasing robustness of neural responses to identity-preserving image transformations.",
"While early areas of the ventral stream are strongly affected by variation in e.g.",
"object size, position or illumination, later levels of processing are increasingly robust to such changes [8].",
"The cascaded achievement of invariance to such identity-preserving transformations has been proposed as a key mechanisms for obtaining robust object recognition [4], [17].",
"Learning such invariant representations has been a desired objective since the early days of artificial neural networks [14].",
"Accordingly, a myriad of techniques have been proposed to attempt to achieve tolerance to different types of transformations (see [3] for a review).",
"Interestingly, recent theoretical work has shown that invariance to “nuisance factors” should naturally emerge from the optimization process [1].",
"Nevertheless, DNNs are still not robust to identity-preserving transformations, including simple image translations [19], or more elaborate adversarial attacks [16], in which small changes, imperceptible to the human brain, can alter the classification output of the network.",
"In this regard, there is growing evidence that DNNs may exploit highly discriminative features that do not match human perception [7].",
"Extending this line of research, we use image perturbations using the data augmentation framework [5] to show that DNNs, despite being trained on augmented data, are not sufficiently robust to such transformations.",
"Inspired by the increasing invariance observed along the primate ventral visual stream, we subsequently propose a simple, yet effective and efficient mechanism to improve the robustness of the representations: we include an additional term in the objective function that encourages the similarity between augmented examples within each batch."
],
[
"Methods",
"This section presents the procedure to empirically measure the invariance of the representations of a convolutional neural network and our proposal to improve the invariance."
],
[
"Model, data and training parameters",
"As a test bed for our hypotheses and proposal we use the all convolutional network, All-CNN [15], a well-known architecture which achieves good performance in spite of being much shallower than other architectures, and thus faster to train and more convenient for the analysis.",
"It consists of 9 convolutional layers, with a total of 1.3 million parameters.",
"Our model is identical to All-CNN-C in the original paper, except that we remove the explicit regularizers—weight decay and dropout—following the conclusions from [5].",
"We also keep the original training hyperparameters: 350 epochs, initial learning rate of 0.01 and batch size of 128.",
"We train on the highly benchmarked data set for object recognition CIFAR-10 [10] and apply heavier data augmentation than in the original paper.",
"Specifically, we use the heavier training and evaluation scheme described by [5], which includes random affine transformations and contrast and brightness adjustment."
],
[
"Evaluation of invariance",
"To measure the invariance of the learned features under the influence of identity-preserving image transformations we compare the activations of a given image with the activations of a data augmented version of the same image.",
"Consider the activations of an input image $x$ at layer $l$ of a neural network, which can be described by a function $f^{(l)}(x) \\in \\mathbb {R}^{D^{(l)}}$ .",
"We can define the distance between the activations of two input images $x_{i}$ and $x_{j}$ by their mean square difference: $d^{(l)}(x_{i}, x_{j}) = \\frac{1}{D^{(l)}}\\sum _{k=1}^{D^{(l)}}(f_{k}^{(l)}(x_{i}) - f_{k}^{(l)}(x_{j}))^2$ Following this, we compute the mean squared difference between every $f^{(l)}(x_i)$ and a random transformation of $x_i$ , that is $d^{(l)}(x_{i}, G(x_{i}))$ .",
"In this case, we define $G(x)$ as the data augmentation scheme that can take any of the extreme values of each transformation in the heavier scheme, after halving the parameter ranges.",
"This is to ensure the same level of augmentation in all comparisons, while preventing too extreme transformations.",
"The assessment of the similarity between the activations of an image $x_i$ and of its augmented versions $G(x_{i})$ was normalised by the similarity with the other, different images, reminiscent of an image identification problem.",
"We define the invariance score $\\sigma _{i}^{(l)}$ of the transformation $G(x_{i})$ at layer $l$ of a model, with respect to a data set of size $N$ , as follows:: $\\sigma _{i}^{(l)} = 1 - \\frac{d^{(l)}(x_{i}, G(x_{i}))}{\\frac{1}{N}\\sum _{j=1}^{N}d^{(l)}(x_{i}, x_{j})}$ The invariance $\\sigma _{i}^{(l)}$ takes the maximum value of 1 if the activations of $x_{i}$ and its transformed version $G(x_{i})$ are identical.",
"To assess the overall invariance of a model post training, we calculate $\\sigma _{i}^{(l)}$ for the 10,000 test images of CIFAR-10, with respect to five different random transformations.",
"In Figure REF we show the distribution of $\\sigma _{i}^{(l)}$ at each layer of All-CNN-C.",
"Figure: Distributon of tnvariance score at each layer of the baseline model and the model trained data augmentation invariance."
],
[
"Data augmentation invariance",
"Most CNNs trained for object categorization are optimized through mini-batch gradient descent (SGD), that is the weights are updated iteratively by computing the loss of a batch $\\mathcal {B}$ of examples, instead of the whole data set at once.",
"The models are typically trained for a number of epochs, $E$ , which is a whole pass through the entire training data set of size $N$ .",
"That is, the weights are updated $K=\\frac{N}{|\\mathcal {B}|}$ times each epoch.",
"Data augmentation introduces variability into the process by performing a different, stochastic transformation of the data every time an example is fed into the network.",
"However, with standard data augmentation, the model has no information about the identity of the images, that is, that different augmented examples, seen at different epochs, separated by $\\frac{N}{|\\mathcal {B}|}$ iterations on average, correspond to the same seed data point.",
"We believe this information may be valuable and useful to learn better representations in a self-supervised manner.",
"For example, the high temporal correlation of the stimuli that reach the visual cortex may play a crucial role in the creation of robust connections [18].",
"In order to make use of this information in an unsupervised way, we propose to perform data augmentation within the batches by constructing the batches to include $M$ transformations of each example (see [6] for a similar idea).",
"Additionally, we propose to modify the loss function to include an additional term that accounts for the invariance of the feature maps across multiple image samples.",
"Considering the difference between the activations at layer $l$ of two images, $d^{(l)}(x_{i}, x_{j})$ , defined in Equation REF , we define the data augmentation invariance loss at layer $l$ for a given batch $\\mathcal {B}$ as follows: $\\mathcal {L}_{inv}^{(l)} = \\frac{\\sum _{k}\\frac{1}{|\\mathcal {S}_{k}|^2}\\sum _{x_i, x_j \\in \\mathcal {S}_{k}}d^{(l)}(x_{i}, x_{j})}{\\frac{1}{|\\mathcal {B}|^2}\\sum _{x_i, x_j \\in \\mathcal {B}}d^{(l)}(x_{i}, x_{j})}$ where $\\mathcal {S}_{k}$ is the set of samples in the batch $\\mathcal {B}$ that are augmented versions of the same seed sample $x_k$ .",
"This loss term intuitively represents the average difference of the activations between the sample pairs that correspond to the same source image, relative to the average difference of all pairs.",
"A convenient property of this definition is that $\\mathcal {L}_{inv}$ does not depend on the batch size nor the number of in-batch augmentations $M=|\\mathcal {S}_{k}|$ .",
"Furthermore, it can be efficiently implemented using matrix operations.",
"Since we want to achieve image invariance at $L$ layers of the network and jointly train for object recognition, we define the total loss as follows: $\\mathcal {L} = (1 - \\alpha )\\mathcal {L}_{obj} + \\sum _{l=1}^{L}\\alpha ^{(l)}\\mathcal {L}_{inv}^{(l)}$ where $\\sum _{l=1}^{L}\\alpha ^{(l)} = \\alpha $ and $\\mathcal {L}_{obj}$ is the loss associated with the object recognition objective, typically the cross-entropy between the object labels and the output of a softmax layer.",
"All the results we report in this paper have been obtained by setting $\\alpha =0.1$ and distributing the coefficients across the layers according to an exponential law, such that $\\alpha ^{(l=L)}= 10\\alpha ^{(l=1)}$ .",
"This aims at simulating a probable response along the ventral visual stream, where higher regions are more invariant than the early visual cortexIt is beyond the scope of this paper to analyze the sensitivity of the hyperparameters $\\alpha ^{(l)}$ , but we have not observed a significant impact in the classification performance by using other distributions.."
],
[
"Results",
"One of the contributions of this paper is to empirically test in how far convolutional neural networks produce invariant representations under the influence of identity-preserving transformations of the input images.",
"Figure REF shows the invariance scores, as defined in Equation REF , across network layers.",
"Despite the presence of data augmentation during training, which implies that the network may learn augmentation-invariant transformations, the representations of the baseline model (red boxes) do not increase in invariance beyond the pixel space.",
"As a solution, we have proposed a simple, unsupervised modification of the loss function to encourage the learning of data augmentation-invariant features.",
"As can be seen in Figure REF (blue boxes), our data augmentation mechanism pushed network representations to become increasingly more robust with network depth.",
"One exception is the top, 'readout' layer, likely because the features are dominated by the categorization objective.",
"In order to better understand the effect of the data augmentation invariance, we plotted the learning dynamics of the invariance loss at each layer.",
"In Figure REF , we can see that in the baseline model, the invariance loss keeps increasing over the course of training.",
"In contrast, when the loss is added to the optimization objective, the loss drops for all but the last layer.",
"Unexpectedly, the invariance loss increased during the first epochs and only then started to decrease.",
"While further investigations are required, these two phases may correspond to the compression and diffusion phases proposed by [13].",
"Figure: Dynamics of the data augmentation invariance loss ℒ inv (l) \\mathcal {L}_{inv}^{(l)} during training.",
"The axis of abscissas (epochs) is scaled quadratically to better appreciate the dynamics at the first epochs.",
"The same random initialization was used for both models.In terms of efficiency, adding terms to the objective function implies an overhead of the computations.",
"However, since the pairwise distances can be efficiently computed at each batch through matrix operations, the training time is only increased by about 10 %.",
"Finally, the improved invariance comes at no cost in the categorization performance, as the network trained with data augmentation invariance achieves similar classification performance to the baseline model—test accuracy baseline: 91.5 %; test accuracy data augmentation invariance: 92.2 %)."
],
[
"Conclusions",
"In this work we have empirically shown that the features learned by a prototypical convolutional neural networks are not invariant to identity-preserving image transformations despite being part of the training procedure.",
"This property is fundamentally different to the primate ventral visual stream, where neural populations have been found to be increasingly robust to changes in view or lighting conditions of the same object [4].",
"Taking inspiration from this property of the visual cortex, we have proposed an unsupervised objective to encourage learning more robust features, using data augmentation as the framework to perform identity-preserving transformations on the input data.",
"We created mini-batches with $M$ augmented versions of each image and modified the loss function to maximize the similarity between the activations of the same seed images.",
"Data augmentation invariance effectively produces more robust representations, unlike standard models optimized only for object categorization, at no cost in classification performance.",
"Future work will investigate whether this increased robustness also allows for better modelling of neural data."
],
[
"Acknowledgments",
"This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 641805, from the Cambridge Commonwealth, European and International Trust, and the DFG."
]
] | 1906.04547 |
[
[
"Quantum satellites and tests of relativity"
],
[
"Abstract Deployment of quantum technology in space provides opportunities for new types of precision tests of gravity.",
"On the other hand, the operational demands of such technology can make previously unimportant effects practically relevant.",
"We describe a novel optical interferometric red-shift measurement and a measurement scheme designed to witness possible spin-gravity coupling effects."
],
[
"Introduction",
"Amazing experimental progress in quantum sensing and quantum communications together with satellite deployment of quantum technologies have ushered in a new era of experimental physics in outer space.",
"The success of the first space based quantum key distribution experiments performed with the Micius satellite [1] is expected to be soon followed by European and North American missions.",
"At the same time current missions, such as LAGEOS-2, BEACON-C and LCT on Alphasat I-XL, are adapted for quantum optics experiments [2], [3].",
"While the primary goal of these space-based platforms is to provide links for global quantum key distribution, the missions also envisage substantial scientific programs.",
"These experiments have the exciting potential to open up new tests of fundamental physics by enabling new searches for signatures of quantum gravity and/or physics beyond the standard model [4].",
"On the other hand, the ambitious precision and stability goals [5] are likely to turn the questions of gravitational and inertial effects on spin into practical questions.",
"Here we describe how these technologies can be affected and used to test the Einstein Equivalence Principle (EEP).",
"The principle comprises three statements [6], [7], [8].",
"The first — Weak Equivalence Principle (WEP) — states that the trajectory of a freely falling test body is independent of its internal composition.",
"Closely related to the WEP is the Einstein elevator: if all bodies fall with the same acceleration in an external gravitational field, then to an observer in a small freely falling lab in the same gravitational field, they appear unaccelerated [7].",
"The remaining two statements deal with outcomes of non-gravitational experiments performed in freely falling laboratories where self-gravitational effects are negligible.",
"The second statement — Local Lorentz Invariance — asserts that such experiments are independent of the velocity of the laboratory where the experiment takes place.",
"The third statement — Local Position Invariance (LPI) — asserts that “the outcome of any local non-gravitational experiment is independent of where and when in the universe it is performed” [6].",
"In Sec.",
"II we outline a novel all-optical test of LPI.",
"In Sec.",
"III we discuss the inertial and aspects of the spin-gravity coupling and suggest the weak valued amplification scheme for detecting some of these effects."
],
[
"Optical test of position invariance",
"Tests of the “when” part of LPI bound the variability of the non-gravitational constants over cosmological time scales [9].",
"The “where” part was expressed in Einstein's analysis of what in modern terms is a comparison of two identical frequency standards in two different locations in a static gravitational field.",
"The so-called red-shift implied by the LPI affects the locally measured frequencies of a spectral line that is emitted at location 1 with $\\omega _{11}$ and then detected at location 2 with $\\omega _{12}$ .",
"The red-shift can be parameterized at the leading post-Newtonian order as $\\Delta \\omega / \\omega _{11} = (1 + \\alpha ) (U_2 - U_1) + \\mathcal {O}(c^{-3}),$ where $\\Delta \\omega := \\omega _{12} - \\omega _{11}$ , $U_i:=-\\phi _i/c^2$ has the opposite sign of the Newtonian gravitational potential $\\phi _i$ at the emission (1) and detection (2), while $\\alpha \\ne 0$ accounts for possible violations of LPI.",
"In principle, $\\alpha $ may depend on the nature of the clock that is used to measure the red-shift [6].",
"The standard model extension includes variously constrained parameters that predict LPI violation [10], [11].",
"Alternative theories of gravity not ruled out by current data also predict $\\alpha \\ne 0$ [6], [12].",
"A typical red-shift experiment involves a pair of clocks, naturally occurring [13] or specially-designed [14], [15], [16], whose readings are communicated by electromagnetic (EM) radiation.",
"The resulting estimates of $\\alpha $ are based on comparison of fermion-based standards.",
"Hence, different types of experiments, which employ a single EM-source and compare optical phase differences between beams of light traversing different paths in a gravitational field, provide a complementary test of LPI.",
"Such an all-optical experiment was proposed as a possible component of the QEYSSAT mission [4].",
"A photon time-bin superposition is sent from a ground station on Earth to a spacecraft, both equipped with an interferometer of imbalance $l$ , in order to temporally recombine the two time-bins and obtain an interference pattern depending on the gravitational phase-shift: $\\varphi _{\\rm gr} = \\frac{\\Delta \\omega }{\\omega } \\frac{2\\pi }{\\lambda } l \\approx (1+\\alpha ) \\frac{2\\pi }{\\lambda }\\frac{gh l }{c^2} \\ ,$ where $g$ is the Earth's gravity, $h$ the satellite altitude and $\\lambda = 2\\pi c/\\omega $ the sent wavelength.",
"For $\\alpha = 0$ , this phase-shift is of the order of few radians supposing $l = 6$ km, $\\lambda = 800$ nm and $h = 400$ km [4].",
"However, a careful analysis of this optical COW-like experiment [17] revealed that first-order Doppler effects are roughly $10^5$ times stronger than the desired signal $\\varphi _{\\rm gr}$ from which $\\alpha $ would be estimated.",
"This first-order Doppler effect was recently measured by exploiting large-distance precision interferometry along space channels [2], which constitute a resource for performing fundamental tests of quantum mechanics in space and space-based quantum cryptography.",
"We propose [18] a new gravitational red-shift experiment, which uses a single EM-source and a double large-distance interferometric measurement performed at two different gravitational potentials.",
"By comparing the phase-shifts obtained at a satellite and on Earth, it is possible to cancel the first-order Doppler effect.",
"Thus, this experimental proposal allows for a bound on $\\alpha $ quantifying the violation of LPI in the EM-sector with a precision on the order of $10^{-5}$ .",
"Figure: Top: A schematic diagram of the proposed experiment.Both the ground station (GS) and spacecraft (SC) are equipped with a MZI of equal delay line ll and an adaptive optics system for fibre injection.Bottom: The geometry of the GS and SC used in the experiment, where v → 1 \\vec{v}_1 is the velocity of the GS at the emission location and potential U 1 U_1;v → 2 \\vec{v}_2 is the velocity of the SC at the detection location on the satellite and potential U 2 U_2; v → 3 \\vec{v}_3 is the velocity of the ground station at the detectionof the beam retro-reflected by the SC, which occurs at a potential U 3 =U 1 U_3=U_1.This proposal [2] is comprised of an interferometric measurement obtained by sending a light pulse through a cascade of two fiber-based Mach-Zehnder interferometers (MZI) of equal temporal imbalance $\\tau _l$ .",
"After the first MZI, the pulse is split into two temporal modes, called short ($\\mathcal {S}$ ) and long ($\\mathcal {L}$ ) depending on the path taken in the first MZI.",
"The equal imbalance of the two MZIs guarantees that the two pulses are recombined at the output of the second MZI, where they are detected.",
"Such a satellite interferometry experiment setup is sketched in .",
"The combination of the possible paths the pulses may take leads to a characteristic detection pattern comprised of three possible arrival times for each pulse, as depicted in the insets of the upper picture in Fig.",
"REF .",
"The first (third) peak corresponds to the pulses that took the $\\mathcal {S}$ ($\\mathcal {L}$ ) path in both the MZIs, while the mid peak is due to the pulse that took the $\\mathcal {S}$ path in the first interferometer and the $\\mathcal {L}$ path in the second interferometer, or vice versa.",
"Hence, interference is expected only in the central peak due to the indistinguishability of these latter two possibilities.",
"A successful realization of the experiment depends on a number of important technical aspects that are described in detail in [18].",
"A bound on $\\alpha $ is retrieved from the difference of the two phase-shifts, $\\varphi _{\\rm SC}$ and $\\varphi _{\\rm GS}$ , that are obtained from interferometric measurements on the spacecraft and ground station, respectively.",
"As just described, the interfering beams take different paths in the passage through the two MZIs.",
"At the satellite, the beam that took the $\\mathcal {L}$ path on Earth and the $\\mathcal {S}$ path on the spacecraft interferes with the beam that passed took the $\\mathcal {S}$ path on Earth and then took the $\\mathcal {L}$ one on the spacecraft.",
"This interference is a result of the phase difference $\\varphi _{\\rm SC}$ .",
"Analogously, at the ground station (GS) the beams that were delayed on the Earth before and after their round trip to the spacecraft (SC) will also interfere because of the phase difference $\\varphi _{\\rm GS}$ .",
"The signal from which a bound on $\\alpha $ is obtained is a linear combination of the two measured phase-shifts $\\varphi _{\\rm SC}= (\\omega _{12}-\\omega _{11})\\tau _l, \\quad \\varphi _{\\rm GS}= (\\omega _{13}-\\omega _{11})\\tau _l,$ where $\\omega _{11}$ is the proper central frequency of the emitted signal at the GS and $\\omega _{13}$ is the frequency after the round trip, and the proper delay time $\\tau _l$ is the same in both frames.",
"The standard second-order expression for the frequencies detected at the satellite is $\\frac{\\omega _{12}}{\\omega _0} = {\\left( \\frac{1-U_1-\\tfrac{1}{2}\\beta _1^2}{1-U_2-\\tfrac{1}{2}\\beta _2^{2}} \\right) \\left( \\frac{1-{\\hat{n}}_{12}\\cdot {\\vec{\\beta }}_2 }{1-{\\hat{n}}_{12}\\cdot {\\vec{\\beta }}_1 } \\right)},$ and at the ground station after a go-return trip $\\frac{\\omega _{13}}{\\omega _0}= \\left(\\frac{1-{\\hat{n}}_{23} \\cdot {\\vec{\\beta }}_3 }{1-{\\hat{n}}_{23}\\cdot {\\vec{\\beta }}_2 }\\right)\\left(\\frac{1-{\\hat{n}}_{12}\\cdot {\\vec{\\beta }}_2 }{1-{\\hat{n}}_{12}\\cdot {\\vec{\\beta }}_1}\\right),$ where ${\\vec{\\beta }}_i=\\vec{v}/c$ .",
"The first-order Doppler terms are eliminated by manipulating the corresponding data sets from the GS and SC in a manner similar to time-delay interferometry techniques [19] and those used in the Gravity Probe A experiment [15].",
"The key feature allowing for this elimination is that the ratio of the first-order Doppler effect contributions to the two signals, $\\varphi _{\\rm SC}$ and $\\varphi _{\\rm GS}$ , is exactly two [18].",
"Hence the target signal is $S=\\varphi _{\\rm SC}-\\tfrac{1}{2}\\varphi _{\\rm GS},$ leading to $ \\frac{S}{\\omega _0 \\tau _l}& =(1+\\alpha ) (U_2-U_1)\\nonumber \\\\&+\\tfrac{1}{2}(\\vec{\\beta }_1-\\vec{\\beta }_2)^2 -(\\mathfrak {d}_1 -\\mathfrak {d}_2)^2 - {T} \\hat{n}_{12}\\cdot {\\vec{a}}_1/c, $ where $\\vec{\\beta _i}=\\vec{v}_i/c$ , $\\mathfrak {d}_i=\\hat{n}_{12}\\cdot \\vec{\\beta _i}$ , ${\\vec{a}}_1$ is the centripetal acceleration at the GS, and $T$ is the upward propagation time."
],
[
"Weak equivalence principle and orbiting clocks",
"Matter of the Standard Model is characterized by two parameters of the irreducible representations of the Poincaré group: mass and spin (or helicity).",
"General relativity is a universal interaction theory about masses [20], like the Newtonian gravity, with polarization effects implicitly omitted from the WEP.",
"Precision measurements up-to-date have not revealed spin-gravity coupling, but it is clearly conceivable [20].",
"Regardless of their origins, spin-gravity coupling terms provide effective corrections to the Hamiltonian in the limit of weak gravity and non-relativistic motion.",
"The leading terms of the Hamiltonian of a free spin-$\\frac{1}{2}$ particle that take into account the effects of rotation of the reference frame with angular velocity $\\vec{\\omega }$ and acceleration $\\vec{a}$ (or a uniform gravitational field) can be represented as $H=H_\\mathrm {cl}+H_\\mathrm {rel}+H_\\sigma +H_\\mathrm {ext}.$ The first three terms on the right hand side are obtained by performing the standard Foldy-Wouthuysen transformation and taking the non-relativistic limit [21].",
"The term $ H_\\mathrm {cl}$ represents the standard Hamiltonian of a free non-relativistic particle in a non-inertial frame, $H_\\mathrm {rel}$ describes the higher-order relativistic corrections that do not involve spin, and $H_\\sigma =-\\tfrac{1}{2}\\hbar {\\vec{\\omega }}\\cdot {\\vec{\\sigma }}+\\frac{\\hbar }{4mc^2}\\vec{\\sigma }\\cdot ({\\vec{a}}\\times \\vec{p}).$ Finally, the term $H_\\mathrm {ext}=\\frac{\\hbar k}{2c}{\\vec{a}}\\cdot {\\vec{\\sigma }},$ represents the spin-accelerating (or spin-gravity) coupling.",
"It is a limiting form of the simplest phenomenological addition to the Dirac equation that breaks the WEP [22].",
"For the value $k=1$ it results from a particular version of the Foldy-Wouthuysen transformation [23].",
"While commonly considered a mathematical artefact of this transformation, the term naturally arises in gravitationally inspired Standard Model extensions.",
"The Mashhoon term $-\\tfrac{1}{2}\\hbar {\\vec{\\omega }}\\cdot {\\vec{\\sigma }}$ was recently detected by using neutron polarimetry [24].",
"On the other hand, only model-dependent bounds on $k$ in $H_\\mathrm {ext}$ were obtained by a variety of techniques [25], including the optical magnetometery [26].",
"The spin dependent terms are small under normal conditions.",
"On the Earth's surface $\\hbar g/c=2.15\\times 10^{-23}$ eV, which is equivalent to an effective magnetic field of $3.7\\times 10^{-19}$ Tl, still several orders of magnitude below the peak sensitivity of optical magnetometery.",
"The spin-rotation term is significantly larger, since already on the ground $\\omega c/g=2.22\\times 10^3$ .",
"It will be about an order of magnitude stronger for low-orbit satellites that are planned to carry entangled optical clocks [5] aiming to establish the precision of $10^{-18} - 10^{-20}$ , making it a factor to consider in the clock design.",
"A potentially promising way of detecting these effects is via so-called weak amplification [27].",
"Weak value amplification involves two systems (typically referred to as “system” and “meter”) that can interact via an interaction Hamiltonian of the form $q\\delta (t-t_0)\\hat{A}\\otimes \\hat{p}$ .",
"The bipartite system-meter is prepared in an initial state $|{s_i}\\rangle \\otimes |{m_i}\\rangle $ , following which the two are allowed to interact for a small time that includes $t_0$ .",
"Following this, the system is measured and measurements corresponding to a post-selected system state $|{s_f}\\rangle $ are considered.",
"This pre- and post-selection induces a “kick” in the meter state, given by the evolution $e^{-iq \\mathcal {A}_w \\hat{p}}|{m_i}\\rangle $ , where $\\mathcal {A}_w\\equiv {\\langle {S_f}| A_s|{S_i}\\rangle }/{\\langle S_f\\vert S_i\\rangle }$ .",
"The key insight here is that since $\\langle S_f\\vert S_i\\rangle $ can be a small number, the measurement of $q$ is influenced by a large multiplicative factor $\\mathcal {A}_w$ .",
"A subsequent measurement of the meter reveals the desired parameter $q$ .",
"Trapped atoms are potentially promising system to implement this scheme [28].",
"The simplest model of such a set-up consists of two species of spins interacting with each other via a simple exchange force, subject to the additional terms implied above, $H= J(\\sigma _{1}^{S}\\otimes \\sigma _{1}^{M})+ \\frac{\\hbar g}{2c}\\Big (h_{x}(\\sigma _{1}^{S}+\\sigma _{1}^{M})+\\\\(h_{z}+1)(\\sigma _{3}^{S}+\\sigma _{3}^{M}) + h_{y}(\\sigma _{2}^{S}+\\sigma _{2}^{M})\\Big ),$ which we write as $H = (\\hbar \\lambda /t)H_{0} + H_{1}$ where $H_{1}$ is the term proportional to $g$ and $\\lambda = Jt/\\hbar $ for convenience, with $h_{i}=-c\\omega _{i}/g$ .",
"Analysis of the unitary evolution that is followed by post-selection indicates that for realistic parameter values the inertial and gravitational effects are within the sensitivity range of the optical magnetometery [28].",
"The work of DRT is supported by the grant FA2386-17-1-4015 of AOARD.",
"ARHS was supported by the Natural Sciences and Engineering Research Council of Canada and the Dartmouth College Society of Fellows.",
"We thank Costantino Agnesi, L.C.",
"Kwek and Alex Ling for useful discussions.",
"We acknowledge the International Laser Ranging Service (ILRS) for SLR data and software."
]
] | 1906.04415 |
[
[
"Nonuniform quantum-confined states and visualization of hidden defects\n in thin Pb(111) films"
],
[
"Abstract The spatial distribution of the differential conductance for ultrathin Pb films grown on Si(111)7x7 substrate is studied by means of low-temperature scanning tunneling microscopy and spectroscopy.",
"The formation of the quantum--confined states for conduction electrons and, correspondingly, the appearance of local maxima of the differential tunneling conductance are typical for Pb films; the energy of such states is determined mainly by the local thickness of Pb film.",
"We demonstrate that the magnitude of the tunneling conductivity within atomically flat terraces can be spatially nonuniform and the period of the small-scale modulation coincides with the period of Si(111)7x7 reconstruction.",
"For relatively thick Pb films we observe large-scale inhomogeneities of the tunneling conductance, which reveal itself as a gradual shift of the quantized levels at a value of the order of 50 meV at distances of the order of 100 nm.",
"We believe that such large-scale variations of the tunneling conductance and, respectively, local density of states in Pb films can be related to presence of internal defects of crystalline structure, for instance, local electrical potentials and stresses."
],
[
"Introduction",
"Reduction in size of logical elements, sensors and conducting wires connecting them leads to that their transport properties can be influenced by discreteness of electrical charge, spatial disorder as well as quantum–size effects [1].",
"Ultrathin Pb films and islands appear to be convenient objects for the investigation of quantum–size effect in metallic nanostructures ([2]–[14] and references therein).",
"Quantized electron states can be investigated by low–temperature scanning tunneling microscopy and spectroscopy (STM/STS) [2]–[9], transport measurements [10], [11] and photoemission studies [12]–[14].",
"An appearance of peaks of the differential conductance at certain bias voltages of the sample $U_n$ and maxima of conductance at certain values of the potential difference as well as an appearance of maxima of photoemission for certain energies of photon were demonstrated for thin Pb films.",
"In particular, the set of the energy values, corresponding of the local maxima of tunneling conductivity with respect to the Fermi level $E^{\\,}_n=eU^{\\,}_n+E^{\\,}_F$ , depend on the local thickness of Pb film and relate to the quantized energy levels in a one–dimensional potential well with boundaries at the interfaces 'metal—vacuum' and 'metal—substrate' (Fig.",
"REF ).",
"Usually the observed effects are interpreted in terms of the resonant tunneling through quasi–stationary energy levels at $E\\approx E^{\\,}_n$ .",
"The energy spectrum $E^{\\,}_n$ of a particle in the one–dimensional well with the constant potential is defined by the Bohr–Sommerfeld quantization rule [4]: $\\varphi ^{\\,}_1 + \\varphi ^{\\,}_2 + 2k^{\\,}_{\\perp ,n} d = 2\\pi n.$ Here $\\varphi ^{\\,}_1$ and $\\varphi ^{\\,}_2$ are the phase shifts for the electronic wave function reflected from the upper and lower interfaces, respectively; $k^{\\,}_{\\perp ,n}$ is the spectrum of allowed values of the wave vector transverse with respect to the interfaces; $d$ is film thickness, $n=0, 1,..$ is an integer–valued index.",
"A theory of the quantum–size effects in Pb(111) films within ab-initio models was presented in [15].",
"It should be mentioned that besides Pb films the quantum–size effects were also observed in Ag, Cu, In, Sn and Sb [12], [16], [17], [18], [19], [20].",
"Figure: (color online) Schematic view of a STM tip, a disordered wetting Pb layer, a single–crystalline Pb island with atomically flat terraces as well as a spatial structure of standing electronic waves inside the island for a certain energy E * E^* close to E F E^{\\,}_F, the parameter nn characterises the number of the half–waves.",
"Note that for the chosen value E * E^* the standing waves are absent for the terrace of the local thickness (N+1)d ML (N+1)\\,d^{\\,}_{ML}.We would like to emphasize that the investigation of peculiarities of the resonant electron tunneling in solid–state nanostructures seems to be important diagnostic instrument similar to other techniques based on interference of waves (e.g., optical, mechanical, electronic etc.).",
"The visualization of atomic structure of the lower interface under metallic layer was demonstrated in [4] and this effect is based on the dependence of $\\varphi ^{\\,}_2$ on the lateral coordinates.",
"The possibility to visualize defects such as monoatomic steps in the substrate and the inclusions of other materials under metallic layer, which were invisible in a topographic image, was shown in [9].",
"The estimates indicate that the thickness of a monolayer $d^{\\,}_{ML}$ for the Pb(111) surface is equal to 0.285 nm, the Fermi wavelength $\\lambda ^{\\,}_F$ is equal to 0.394 nm, therefore the ratio $\\lambda ^{\\,}_F/d^{\\,}_{ML}$ is close to 4/3 [5].",
"Consequently, the following conclusion should be valid for the electronic states in the form of standing waves with $E\\simeq E^{\\,}_F$ [5], [7], [9]: the energy of the state with the number of zeros $n$ for the film with the local thickness $Nd^{\\,}_{ML}$ should be close to the energy of the state with the number of zeros $n+3$ for the film with the local thickness $(N+2)\\,d^{\\,}_{ML}$ (Fig.",
"REF ).",
"Indeed during the measurements at the certain energy the relocation of the STM–tip from one area to another area whose local thicknesses differ in $d^{\\,}_{ML}$ should result in a drastic change of the differential conductance [3], [5], [9], what makes possible to reveal the areas with even or odd numbers of monolayers [9].",
"The diagrams $U^{\\,}_n-d$ allows one to recover the dependence $E(k^{\\,}_\\perp )$ and to estimate the thickness of the wetting layer, the effective mass and speed of electrons [3], [5], [6], [9].",
"Based on the analysis of the dependence of the width of the resonant tunneling peaks on temperature, the estimates of lifetime for different scattering mechanisms were derived [6].",
"This paper is devoted to the experimental investigation of spatial inhomogeneity of the differential tunneling conductance for thin Pb films by means of low–temperature STM/STS.",
"It allows us to find the correlation between local electronic properties and locations of structural defects.",
"It is worth to note that during STM–investigation of large areas a map of feedback–loop signal which is usually associated with a topographic image can be distorted because of different scanning speed along the $x-$ and $y-$ axes and variation in temperature of a piezo–scanner, what results in uncontrolled shift of the tip along the $z-$ axis.",
"In particular, the visible height of monoatomic step in the topographic image may differ from the ideal value and the atomically flat terraces may look like not–flat.",
"We propose a method for visualization of areas with apparent and hidden defects based on simultaneous measurements of topography and differential tunneling conductance directly during scanning process.",
"It makes possible to discriminate the images with artifacts caused by instrumental and processing imperfections and the images with real defects.",
"We believe that observed large–scale inhomogeneities of the differential tunneling conductance on the terraces of nominally constant height can be related with defects of crystal structure and thus point to, for instance, local electrical potentials and stresses.",
"Figure: (color online) (a) STM image of surface of the Pb island (175×\\times 175 nm 2 ^2, average sample potential U 0 =500U^{\\,}_0=500 mV, average tunneling current I 0 =400I^{\\,}_0=400 pA), the dashed line in the lower part of the image depicts the projection of the dislocation loop on the sample surface.",
"Hereafter the symbols ⨂\\bigotimes mark the reference points, which were used for the alignment of the image.",
"(b) The dependence G(U 0 )G(U^{\\,}_0) for points within areas I and II; the vertical dashed lines correspond to the values U 0 U^{\\,}_0 used for the maps in the panels (c) and (d).",
"(c,d) The maps of the differential conductance G(x,y,U 0 )G(x,y,U^{\\,}_0) for the same area shown in (a), acquired at U 0 =500U^{\\,}_0=500 mV (c) and U 0 =600U^{\\,}_0=600 mV (d); U 1 =40U^{\\,}_1=40 mV, f 0 =7285f^{\\,}_0=7285 Hz.",
"Lighter shades correspond to higher tunneling conductance, darker shades correspond to the lower conductance"
],
[
"Experimental procedure",
"Investigation of the electro–physical properties of Pb nanostructures is carried out on the UHV LT SPM Omicron Nanotechnology setup.",
"Thermal deposition of Pb (Alfa Aesar, purity 99.99%) is performed on the reconstructed surface Si(111)7$\\times $ 7 at room temperature and pressure $3\\cdot 10^{-10}$ mbar at a rate of the order of 0.5 nm/min, the deposition time is varied from 5 to 40 min.",
"The topography of the Pb islands is studied by STM at a temperature of 78 K in the regime of constant tunneling current $I$ at a given bias potential $U$ of the sample relative to the tip of a tunneling microscope.",
"Etched tungsten wires with apex cleaned by electron bombardment in ultra–high vacuum are used as tips.",
"All topographic images are processed by a subtraction of a plane defined by three reference points in order to reduce global tilt.",
"The electronic properties of the Pb islands are investigated by single point tunneling spectroscopy, consisting of measurements of series of the dependences $I(U)$ and $G(U)$ at a fixed position of the tip, where $G\\equiv dI/dU$ is the differential tunneling conductivity of the tip–sample contact.",
"In addition, maps of the local differential conductance are obtained by means of modulation scanning tunneling spectroscopy.",
"Using a Stanford Research SR830 lock–in amplifier we measure the amplitude of the ac-component of the tunneling current, which appear for the modulated bias potential $U=U^{\\,}_0+U^{\\,}_1\\,\\cos (2\\pi f^{\\,}_0t)$ , where $f^{\\,}_0=7285$ Hz.",
"Apparently that under the condition $U^{\\,}_1\\ll U^{\\,}_0$ the amplitude of the oscillations of the tunneling current at the modulation frequency $f^{\\,}_0$ is proportional to the differential conductance $G(U^{\\,}_0)$ .",
"Provided that $f^{\\,}_0$ significantly exceeds the threshold frequency of the feedback loop ($\\sim 200$ Hz), the modulated potential applied to the sample should not result in the appearance of any artifacts on the topographic images.",
"Such technique [9] allows us to synchronously obtain both topographic images in the regime of constant average current $I^{\\,}_0$ and the dependence of $G$ on the lateral coordinates $x$ and $y$ at the given value $U^{\\,}_0$ ."
],
[
"Results and discussion",
"The growth of Pb on the Si(111)7$\\times $ 7 surface at room temperature is known to occur through the Stranski–Krastanov mechanism: first a disordered wetting Pb layer with the thickness of the order of 1 nm is formed, then two-dimensional Pb islands with the upper facets corresponding to the (111) plane start to grow.",
"It was shown [2]–[9] that the local tunneling conductance $G$ depends on $U^{\\,}_0$ in non-monotonous way (Fig.",
"REF b).",
"Particularly, the values $U^{\\,}_n$ corresponding to the peaks on $G(U^{\\,}_0)$ , depend on the local thickness of Pb film and on the boundary conditions for an electron wave function.",
"The topographic image of the surface of the Pb island (panel a) and the differential conductance maps $G(x,y,U^{\\,}_0)$ at two different energies, acquired simultaneously with the topographic image at forward (panel c) and backward (panel d) scanning directions, are shown in Fig.",
"REF .",
"Since the interval $\\Delta E$ between the neighbour maxima at the $G(U^{\\,}_0)$ dependence is equal to 185 meV (Fig.",
"REF b) and the Fermi velocity is equal to $v^{\\,}_F\\approx 1.8\\cdot 10^8$ cm/s [2], [9], one could estimate the local thickness of this island $d\\simeq \\pi \\hbar v^{\\,}_F/\\Delta E \\simeq 19$ nm or approximately 70 monolayers.",
"The local thickness of the Pb island in the area I exceeds the local thicknesses in the areas II and III by one and two monolayers, respectively.",
"As a consequence, the local tunneling conductances in the areas I and III are practically equal at two different energies and both differ from the conductance in the area II (Fig.",
"REF c,d).",
"Note that a gradual variation in the height in the vicinity of the center of the screw dislocation does not lead in a gradual change of the differential conductance.",
"Indeed, the conductance changes drastically upon crossing a line invisible at the topographic image which corresponds to the hidden part of the dislocation loop inside the Pb film (dashed line in Fig.",
"REF a).",
"Since in the regions where the dislocation line is parallel to the surface such dislocation line has to be either an edge dislocation or a mismatch dislocation, the number of the monolayers changes by one upon relocation from the area III to the area IV.",
"However in the vicinity of the dashed line the change in numbers of the monolayers occurs at constant height of the film, therefore the electronic concentration $n$ should be changed abruptly.",
"Taking into account that $E^{\\,}_F=(\\hbar ^2/2m^*)\\,(3\\pi ^2n)^{2/3}$ in the model of free electron gas [21], the bottom of the conduction band in the area IV should be shifted down at the value of the order of $\\delta E^{\\,}_0 \\sim 2E^{\\,}_F/(3N)=90$ meV in order to ensure the constancy of the Fermi level, where $m^*$ is the effective mass which is close to the mass of free electron for Pb films in the (111) direction [9], $E^{\\,}_F\\simeq 9.47$ eV is the Fermi energy for bulk Pb [21], $N\\simeq 70$ is the number of the monolayers in the considered area.",
"Since $\\delta E^{\\,}_0$ is close to the half of $\\Delta E$ , the relocation across the invisible part of the dislocation loop should be accompanied by a sharp change in brightness for the maps of the tunneling conductance.",
"Figure: (color online)(a) STM image of the surface of Pb island (35×\\times 35 nm 2 ^2, U 0 =500U^{\\,}_0=500 mV, I 0 =200I^{\\,}_0=200 pA).",
"(b) Spatial dependence of the tunneling conductance G(x,y,U 0 )G(x,y,U^{\\,}_0) at U 0 =500U^{\\,}_0=500 mV, U 1 =40U^{\\,}_1=40 mV.",
"(c,d) Amplitude of the Fourier components for the topographic image (a) and for the map of the differential conductance (b), respectively; circles depict the Fourier maxima of the first orderFigure: (color online) (a) Map of the differential conductance GG for a Pb island (11.6×\\times 11.6 nm 2 ^2, U 0 =490U^{\\,}_0=490 mV, I 0 =200I^{\\,}_0=200 pA).",
"(b) Series of the single-point spectroscopical lines G(U 0 )G(U^{\\,}_0) acquired for several neighbour locations at x=7x=7 nm and different yy values; these lines are shifted vertically for clarity.",
"Thick blue line corresponds to the dependence 〈G(U 0 )〉\\langle G(U^{\\,}_0)\\rangle ; dashed line is the estimate of non-resonant background B(U 0 )B(U^{\\,}_0).",
"(c,d) The difference of the local conductance G(x,y,U 0 )G(x,y,U^{\\,}_0) and the non-resonant background B(U 0 )B(U^{\\,}_0) as a function of the bias voltage U 0 U^{\\,}_0 and the yy coordinate, for the locations marked in (a): x=3x=3 nm (c) and x=7x=7 nm (d).",
"Brightness is proportional to the difference G(U 0 )-B(U 0 )G(U^{\\,}_0)-B(U^{\\,}_0).",
"Dashed lines correspond to the U 0 U^{\\,}_0 value, which corresponds to the map (a).Figure: (color online)(a) STM image of the surface of Pb island (460×\\times 460 nm 2 ^2, U 0 =700U^{\\,}_0=700 mV, I 0 =400I_0=400 pA).",
"(b) Map of differential conductance G(x,y,U 0 )G(x,y,U^{\\,}_0) for the same area; U 0 =700U^{\\,}_0=700 mV, U 1 =40U^{\\,}_1=40 mV, arrow shows the position of the step of monoatomic height in the substrate.",
"(c,d) Profiles of the topographic image and the differential conductance along the dash line A–B, dashed lines correspond to the heights of the Pb terraces.",
"(e,f) Profiles of the topographic image and the differential conductance along the dash line C–D, dashed circles indicate the unavoidable artifacts of processing of the topographic imageFigure: (color online)(a) STM image of the surface of Pb island(230×\\times 210 nm 2 ^2, U 0 =900U^{\\,}_0=900 mV, I 0 =200I^{\\,}_0=200 pA).",
"The three points marked in the figure used for levelling.",
"(b) Map of the differential conductance G(x,y)G(x,y) for the same area acquired at U 0 =900U^{\\,}_0=900 mV, U 1 =40U^{\\,}_1=40 mV.",
"(c) Profiles of the differential conductance along the vertical lines I–II and III–IV.",
"(d,e) Difference of the local conductance G(x,y,U 0 )G(x,y,U^{\\,}_0) and the non-resonant background B(U 0 )B(U^{\\,}_0) as a function of the bias U 0 U_0 and the y-y-coordinate along the lines I–II (d) and III–IV (e).",
"Brightness is proportional to the value G(U 0 )-B(U 0 )G(U^{\\,}_0)-B(U^{\\,}_0).",
"Vertical dash lines correspond to the value U 0 U^{\\,}_0 used for the acquisition of the map in (a).",
"(f) Difference of the local conductance G(x,y,U 0 )G(x,y,U^{\\,}_0) and the non-resonant background B(U 0 )B(U^{\\,}_0) as a function of bias U 0 U_0 and the x-x-coordinate along the line V–VI.Note that the tunneling conductance even within single terrace is not strictly constant.",
"Figure REF shows the topography (a) and the map of conductance (b) for the Pb island and it has the thickness about six monolayers over the wetting layer according to our estimates.",
"This topographic image contains signatures of hexagon lattice what is confirmed by presence of well-defined peaks of the first and second orders on the map of amplitudes of the Fourier components (Fig.",
"REF c).",
"The spatial modulation with the same wave vectors takes place for the tunneling conductance (Fig.",
"REF b and d).",
"One can suggest that the observed periodicity is conditioned by an influence of the crystalline structure on the tunneling density of states in Pb films, since the period of modulation coincides with the period of the reconstruction Si(111)7$\\times $ 7.",
"Such effect can be related with the variation of the phase of electron wave reflected from the interface 'metal–substrate' at different points of the surface and can reveal itself as a periodic shift of the peaks of the tunneling conductance as well as the Moiré contrast [3], [4].",
"For a more careful investigation of the short–scale inhomogeneities of the differential conductance we consider the area 11.6$\\times $ 11.6 nm$^2$ of atomically-flat terrace of the Pb island with a thickness of 60–70 monolayers.",
"For this island we perform the series of measurement at the grid 32$\\times $ 32 (grid spectroscopy) with a step of 0.36 nm.",
"The spatial dependence of the conductance on the coordinates $x$ and $y$ at $U^{\\,}_0=490$ mV is shown in Fig.",
"REF a.",
"Several typical local dependences $G(U_0)$ are shown in Fig.",
"REF b.",
"Note that depending on the measurement location the local tunneling spectra contain either the set of well-defined peaks or these peaks are badly distinguishable.",
"For analysis of the dependence of the position and amplitude of the resonant peaks on energy and coordinates it is convenient to remove a non-resonant background.",
"With this purpose all 1024 spectral curves were averaged over the scanning area and then the mean conductance $\\langle G(U^{\\,}_0)\\rangle $ (the thick solid line in Fig.",
"REF b) was approximated by a third-order polynomial dependence in order to exclude any effects of quantum–confined states.",
"This approximation polynomial $B(U^{\\,}_0)$ (background) is shown in Fig.",
"REF b by dashed line.",
"The differences of the local conductance from the non-resonant background $B(U^{\\,}_0)$ as a function of bias $U^{\\,}_0$ and the coordinate $y$ for two values $x=3$ nm (c) and $x=7$ nm (d) are shown in Fig.",
"REF .",
"It is easy to see that the areas with pronounced peaks of the differential conductance alternate with the areas with no peaks, unclear peaks or peaks shifted to different energy.",
"Figure REF shows the topographic image and the map of the differential conductance for the Pb islands, where the terraces of monoatomic height have a form of concentric circles.",
"It should be noted that the conductance in the areas I and IV is close to the maximal value, while the conductance in the areas II and III is close to the minimal value.",
"It allows us to conclude that there is an invisible step of monoatomic height in the substrate what results in a drastic change in conductance within single Pb terrace (paths I–III and II–IV in Fig.",
"REF b).",
"Besides that we found out the terraces with a gradual change in conductance at a given energy (for instance, paths V–VI in Fig.",
"REF b and I–III and II–IV in Fig.",
"REF b).",
"The appearance of the regions with gradual change of differential conductance looks quite surprising since in elementary models a film thickness should be equal to an integer number of monolayers and, consequently, the tunneling conductance should varies discretely.",
"Note that the appearance of the gradual contrast at the maps $G(x,y,U^{\\,}_0)$ cannot be related to a modification of a tip apex during measurements, since the areas with sharp and edges are observed simultaneously.",
"Figure REF c, d show the cross-sections of the topographic image and the map of the differential conductance along the A–B line, which is close to the direction of fast scanning.",
"It is easy to see that the gradual change of conductivity marked by the circle in Fig.",
"REF d corresponds to the gradual variation of the height of the order $0.2\\,d^{\\,}_{ML}$ at the topographic image marked by the circle in Fig.",
"REF c. We think that the observed phenomenon is related to the presence of internal stress in Pb film which affect both the actual height of the terraces and the energy of the bottom of the conduction band.",
"To the contrary, the cross-section of the map of the differential conductance along the C–D line close to the direction of slow scanning is a function which has two limiting values (Fig.",
"REF f).",
"As a consequence, the local thickness of the Pb film along this line should be varied in a quantized way and the complicated shape of the profile along the same line (dashed circles in Fig.",
"REF e) is apparently an artifact caused by imperfection of both piezo–scanner and procedure of the compensation of the global tilt.",
"In order to study the peculiarities of the differential conductance for Pb films with gradual large–scale inhomogeneities as a function of the coordinates and the energy we investigate the area of the Pb island with three monoatomic steps; the height of this island is about 60 monolayers.",
"The topography of this islands is shown in Fig.",
"REF a.",
"A detailed analysis of the cross-sections along the lines I–II and III–IV profiles points to a monotonous variation of the height of the terraces at about $0.2\\,d^{\\,}_{ML}$ in the interval from $y=0$ to $y=100$ nm, what can be easily recognized in Fig.",
"REF a by a change of colors.",
"The map of the differential conductance (Fig.",
"REF b) evidences about the presence of sharp boundaries, for instance, upon relocating from the area I to the area III, whose heights differ by one monolayer.",
"However upon the tip relocation from the area I to the area II (or III–IV) a gradual change of the tunneling conductance takes place: the conductance at $U^{\\,}_0=900$ mV decreases along the line I–II and it increases along the line III–IV (Fig.",
"REF b,c).",
"In the same area the series of the current – voltage characteristics and the spectra of the differential tunneling conductance are obtained at the grid 32$\\times $ 32 and then a non-resonant background is subtracted using the procedure described above.",
"The results of these measurements indicate that there is the gradual shift of the levels of the quantum-confined states towards higher energy of the order of 50 mV happens along the $y-$ axis (Fig.",
"REF d,e).",
"In the other words, we observe the gradual transition from the local maximum on the $G(U^{\\,}_0)$ dependence at the energy 900 mV (vertical line in Fig.",
"REF d) to the local minimum provided that the tip is shifted along the line I–II, what corresponds to the decrease in the tunneling conductance (Fig.",
"REF b,c).",
"Similarly the shift along the line III–IV at the energy 900 mV causes the gradual increase of the conductance (Fig.",
"REF e).",
"It would be noted that upon relocation in the horizontal direction between the areas V and VI a position-independent differential conductivity is observed with abrupt change at the terrace edge (Fig.",
"REF f).",
"Consequently, the monotonous change of the height of the terrace is accompanied by the changes of electron properties of the sample and results in a systematic shift of the quantized quantum–confined levels in the interval from $y=0$ to $y=100$ nm.",
"Note that the observed shift of the energy levels is close to the estimate of the shift of the bottom of the conduction band $\\delta E^{\\,}_0$ caused by the change of the local electron density.",
"Turning back to the simplest model (REF ) of the localised electron states in a one-dimensional potential well, one can state that the gradual shift of the quantum–confined levels can be caused by, first, a monotonous change in the thickness of Pb layer $d(x,y)$ , second, a change in the energy of the bottom of the conduction band $E^{\\,}_0(x,y)$ and, third, a change in the boundary conditions at the interface 'metal—substrate'.",
"The last circumstance is probably responsible for the small–scale inhomogeneity of the electronic properties.",
"We suppose that mechanical stress of the crystalline structure, what arise during growth process of the Pb structures and can result in the change both the energy $E^{\\,}_0$ and the height of terraces, is the most probable origin of appearing of the areas with the gradual inhomogeneity of the tunneling conductance."
],
[
"Conclusion",
"We demonstrate that the change in the local thickness of the Pb film by one monolayer due to monoatomic steps at the lower or upper interfaces results in abrupt spatial variation (with typical length scale of the order of several nm) of the average differential conductance at the given energy.",
"The observed small–scale modulation of the tunneling conductance (with typical period $\\sim 3$ nm) is related with an influence of the periodic potential of the substrate, which is the reconstruction Si(111)7$\\times $ 7.",
"Besides that the large-scale variations of the differential tunneling conductance within single terrace of the Pb island are observed, manifesting as the gradual change of the quantum-confined energy levels at a value of the order of 50 nm at the lateral scales of the order of 100 nm.",
"A possible reason of the appearance of the large–scale variations is the spatially–inhomogeneous internal stress in thin Pb films, which can result in non-quantized changes in the thickness of the Pb layer different from the integer number of monolayers.",
"Systematic investigation of the dependence of differential conductivity on the coordinates and energy is a convenient method for studying of internal defects in Pb nanostructures."
],
[
"Acknowlednements",
"The authors thanks D. Yu.",
"Roditchev and A. N. Chaika for fruitful discussions.",
"The work was performed with the use of equipment at the Common Research Center 'Physics and Technology of Micro- and Nanostructures' at Institute for Physics of Microstructures RAS (IPM RAS).",
"The work was supported partly by the Presidium of RAS under the program 0035-2018-0019 (sample preparation), partly by the RFBR grant 19-02-00528 (STM-STS measurements), partly by the Governmental program for IPM RAS in 2019 (interpretation of results) and partly by the Governmental program for the Institute for Solid State Physics RAS in 2019 (interpretation of results).",
"D. K. Ferry and S. M. Goodnick, Transport in nanostructures.",
"Cambridge University Press, 2nd ed.",
"(2009), 670 p. I.",
"B. Altfeder, K. A. Matveev, and D. M. Chen, Phys.",
"Rev.",
"Lett.",
"78, 2815 (1997).",
"I.",
"B. Altfeder, D. M. Chen, K. A. Matveev, Phys.",
"Rev.",
"Lett.",
"80, 4895 (1998).",
"I.",
"B. Altfeder, V. Narayanamurti, D. M. Chen, Phys.",
"Rev.",
"Lett.",
"88, 206801 (2002).",
"W. B. Su, S. H. Chang, W. B. Jian, C. S. Chang, L. J. Chen, and T. T. Tsong, Phys.",
"Rev.",
"Lett.",
"86, 5116 (2001).",
"I-Po Hong, C. Brun, F. Patthey, I. Yu.",
"Sklyadneva, X. Zubizarreta, R. Heid, V. M. Silkin, P. M. Echenique, K. P. Bohnen, E. V. Chulkov, and W.-D. Schneider, Phys.",
"Rev.",
"B 80, 081409 (2009).",
"D. Eom, S. Qin, M. Y. Chou, and C. K. Shih, Phys.",
"Rev.",
"Lett.",
"96, 027005 (2006).",
"K. Wang, X. Zhang, M. M. T. Loy, T.-C. Chiang, and X. Xiao, Phys.",
"Rev.",
"Lett.",
"102, 076801 (2009).",
"S. S. Ustavshchikov, A. V. Putilov, A. Yu.",
"Aladyshkin, JETP Lett.",
"106, 491 (2017).",
"M. Jalochowski and E. Bauer, Phys.",
"Rev.",
"B 38, 5272 (1988).",
"N. Miyata, K. Horikoshi, T. Hirahara, S. Hasegawa, C. M. Wei, and I. Matsuda, Phys.",
"Rev.",
"B 78, 245405 (2008).",
"J. H. Dil, J. W. Kim, Th.",
"Kampen, K. Horn, and A. R. H. F. Ettema, Phys.",
"Rev.",
"B 73, 161308 (2006).",
"A. Mans, J. H. Dil, A. R. H. F. Ettema, and H. H. Weitering, Phys.",
"Rev.",
"B 66, 195410 (2002).",
"D. A. Ricci, Y. Liu, T. Miller, and T.-C. Chiang, Phys.",
"Rev.",
"B 79, 195433 (2009).",
"C. M. Wei and M. Y. Chou, Phys.",
"Rev.",
"B 66,233408 (2002).",
"I.",
"B. Altfeder, X. Liang, T. Yamada, D. M. Chen, and V. Narayanamurti, Phys.",
"Rev.",
"Lett.",
"92, 226404 (2004).",
"M. Milun, P. Pervan, and D. P. Woodruff, Rep. Prog.",
"Phys.",
"65, 99 (2002).",
"T.-C. Chiang, Surf.",
"Sci.",
"Rep. 39, 181 (2000).",
"Yu. F.",
"Komnik, E. I. Bukhshtab, JETP Lett.",
"6, 58 (1967).",
"Yu. F.",
"Komnik, E. I. Bukhshtab, JETP Lett.",
"8, 4 (1968).",
"N. W. Ashcroft, N. D. Mermin, Solid State Physics.",
"vol.",
"1, Holt, Rinehart and Winston, New York (1976)."
]
] | 1906.04487 |
[
[
"PAN: Projective Adversarial Network for Medical Image Segmentation"
],
[
"Abstract Adversarial learning has been proven to be effective for capturing long-range and high-level label consistencies in semantic segmentation.",
"Unique to medical imaging, capturing 3D semantics in an effective yet computationally efficient way remains an open problem.",
"In this study, we address this computational burden by proposing a novel projective adversarial network, called PAN, which incorporates high-level 3D information through 2D projections.",
"Furthermore, we introduce an attention module into our framework that helps for a selective integration of global information directly from our segmentor to our adversarial network.",
"For the clinical application we chose pancreas segmentation from CT scans.",
"Our proposed framework achieved state-of-the-art performance without adding to the complexity of the segmentor."
],
[
"Introduction",
"Segmentation has been a major area of interest within the fields of computer vision and medical imaging for years.",
"Owing to their success, deep learning based algorithms have become the standard choice for semantic segmentation in the literature.",
"Most state-of-the-art studies model segmentation as a pixel-level classification problem [2], [3], [4].",
"Pixel-level loss is a promising direction but, it fails to incorporate global semantics and relations.",
"To address this issue researchers have proposed a variety of strategies.",
"A great deal of previous research uses a post-processing step to capture pairwise or higher level relations.",
"Conditional Random Field (CRF) was used in [2] as an offline post-processing step to modify edges of objects and remove false positives in CNN output.",
"In other studies, to avoid offline post-processing and provide an end-to-end framework for segmentation, mean-field approximate inference for CRF with Gaussian pairwise potentials was modeled through Recurrent Neural Network (RNN) [17].",
"In parallel to post processing attempts, another branch of research tried to capture this global context through multi-scale or pyramid frameworks.",
"In [2], [3], [4], several spatial pyramid pooling at different scales with both conventional convolution layers and Atrous convolution layers were used to keep both contextual and pixel-level information.",
"Despite such efforts, combining local and global information in an optimal manner is not a solved problem, yet.",
"Following by the seminal work by Goodfellow et.al.",
"in [7] a great deal of research has been done on adversarial learning [15], [8], [14], [10].",
"Specific to segmentation, for the first time, Luc et.",
"al.",
"[8] proposed the use of a discriminator along with a segmentor in an adversarial min-max game to capture long-range label consistencies.",
"In another study SegAN was introduced, in which the segmentor plays the role of generator being in a min-max game with a discriminator with a multi-scale L1 loss [14].",
"A similar approach was taken for structure correction in chest X-rays segmentation in [5].",
"A conditional GAN approach was taken in [10] for brain tumor segmentation.",
"In this paper, we focused on the challenging problem of pancreas segmentation from CT images, although our framework is generic and can be applied to any 3D object segmentgation problem.",
"This particular application has unique challenges due to the complex shape and orientation of pancreas, having low contrast with neighbouring tissues and relatively small and varying size.",
"Pancreas segmentation were studied widely in the literature.",
"Yu et al.",
"introduced a recurrence saliency transformation network, which uses the information from previous iteration as a spatial weight for current iteration [16].",
"In another attempt, U-Net with an attention gate was proposed in [9].",
"Similarly, a two-cascaded-stage based method was used to localize and segment pancreas from CT scans in [13].",
"A prediction-segmentation mask was used in [18] for constraining the segmentation with a coarse-to-fine strategy.",
"Furthermore, a segmentation network with RNN was proposed in [1] to capture the spatial information among slices.",
"The unique challenges of pancreas segmentation (complex shape and small organ) shifted the literature towards methods with coarse-to-fine and multi-stage frameworks, promising but computationally expensive.",
"Summary of our contributions: The current literature on segmentation fails to capture 3D high-level shape and semantics with a low-computation and effective framework.",
"In this paper, for the fist time in the literature, we propose a projective adversarial network (PAN) for segmentation to fill this research gap.",
"Our method is able to capture 3D relations through 2D projections of objects, without relying on 3D images or adding to the complexity of the segmentor.",
"Furthermore, we introduce an attention module to selectively integrate high-level, whole-image features from the segmentor into our adversarial network.",
"With comprehensive evaluations, we showed that our proposed framework achieves the state-of-the-art performance on publicly available CT pancreas segmentation dataset [11] even when a simple encoder-decoder network was used as segmentor."
],
[
"Method",
"Our proposed method is built upon the adversarial networks.",
"The proposed framework's overview is illustrated in Figure REF .",
"We have three networks: a segmentor ($S$ in Figure REF ), which is our main network and was used during the test phase, and two adversarial networks ($D_{s}$ and $D_{p}$ in Figure REF ), each with a specific task.",
"The first adversarial network ($D_{s}$ ) captures high-level spatial label contiguity while the second adversarial network ($D_{p}$ ) enforces the 3D semantics through a 2D projection learning strategy.",
"The adversarial networks were used only during the training phase to boost the performance of the segmentor without adding to its complexity.",
"Figure: The proposed framework consists of a segmentor SS and two adversarial networks, D s D_{s} and D p D_{p}.",
"SS was trained with a hybrid loss from D s D_{s}, D p D_{p} and the ground-truth."
],
[
"Segmentor (S)",
"Our base network is a simple fully convolutoinal network with an encoder-decoder architecture.",
"The input to the segmentor is a 2D grey-scale image and the output is a pixel-level probability map.",
"The probability map shows probability of presence of the object at each pixel.",
"We use a hybrid loss function (explained in details in Section REF ) to update the parameters our segmentor ($S$ ).",
"This loss function is composed of three terms enforcing: (1) pixel-level high-resolution details, (2) spatial and high-range label continuity, (3) 3D shape and semantics, through our novel projective learning strategy.",
"As can be seen in Figure REF , the segmentor contains 10 conv layers in the encoder, 10 conv layers in the decoder and 4 conv layers as the bottleneck.",
"The last conv layer is a $1\\times 1$ conv layer with the channel output of 1, combining channel-wise information in the highest scale.",
"This layer is followed by a sigmoid function to create the notion of probability."
],
[
"Adversarial Networks",
"Our adversarial networks are designed with the goal of compensating for the missing global relations and correcting higher-order inconsistencies, produced by a single pixel-level loss.",
"Each of these networks produces an adversarial signal and apply it to the segmentor as a term in the overall loss function (Equation REF ).",
"The details of each network is described below: Spatial semantics network ($\\textbf {D}_{\\textbf {s}}$ ): This network is designed to capture spatial consistencies within each frame.",
"The input to this network is either the segmented object by the ground-truth or by the segmentor's prediction.",
"The Spatial semantics network ($D_{s}$ ) is trained to discriminate between these two inputs with a binary cross-entropy loss, formulated as in Equation REF .",
"The adversarial signal produced by the negative loss of $D_{s}$ to $S$ forces $S$ to produce predictions closer to ground-truth in terms of spatial semantics.",
"As illustrated in Figure REF top right, $D_{s}$ has a two-branch architecture with a late fusion.",
"The top branch processes the segmented objects by ground-truth or segmentor's prediction.",
"We propose an extra branch of processing, getting the bottleneck features corresponding to the original gray-scale input image, and passing them to an attention module for an information selection.",
"The processed features are then concatenated with the first branch and passed through the shared layers.",
"We believe that having the high-level features of whole image along with the segmentations improves the performance of $D_{s}$ .",
"Our attention module learns where to attend in the feature space to have a more discriminative information selection and processing.",
"The details of the attention module are described in the following.",
"Figure: Attention module assigns a weight to each feature allowing for a soft selection of information.Attention module ($\\textbf {A}$ ): We feed the high-level features form the segmentor's bottleneck to $D_{s}$ .",
"These features contain global information about the whole frame.",
"We use a soft-attention mechanism, in which our attention module assigns a weight to each feature based on its importance for discrimination.",
"The attention module gets the features with shape $w\\times h\\times c$ , as input, and outputs a weight set with a shape of $w\\times h\\times 1$ .",
"$A$ is composed of two $1\\times 1$ convolution layers followed by a softmax layer (Figure REF ).",
"The softmax layer introduces the notion of soft selection to this module.",
"The output of $A$ is then multiplied to the features before being passed to the rest of the network.",
"Projective network ($\\textbf {D}_{\\textbf {p}}$ ): Any 3D object can be projected into 2D planes from specific viewpoints, resulting in multiple 2D images.",
"The 2D projection contains 3D semantics information, to be retrieved.",
"In this section, we introduce our projective network ($D_{p}$ ).",
"The main task of $D_{p}$ is to capture 3D semantics without relying on 3D data and from the 2D projections.",
"Inducing 3D shapes form 2D images has previously been done for 3D shape generation [6].",
"Unlike existing notions, however, in this paper we propose 3D semantics induction from 2D projections, to benefit segmentation for the first time in the literature.",
"The projection module ($P$ ) projects a 3D volume (V) on a 2D plane as: $ P((i,j),V) = 1 - \\exp ^{-\\sum _{k}V(i,j,k)},$ where each pixel in the 2D projection $P((i,j),V)$ gets a value in the range of $[0,1]$ based on the number of voxel occupancy in the third dimension of corresponding $3D$ volume ($V$ ).",
"For the sake of simplicity, we refer to the projection of a 3D volume $V$ as $P(V)$ .",
"We pass each 3D image through our segmentor ($S$ ) slice by slice and stack the corresponding prediction maps.",
"Then, these maps are fed to the projection module ($P$ ) and are projected in the axial view.",
"The input to $D_{p}$ is either the projected ground-truth or projected prediction map produced by $S$ .",
"$D_{p}$ is trained to discriminate these inputs using the loss function defined in Equation .",
"The adversarial term produced by $D_{p}$ in Equation REF forces $S$ to create predictions which are closer to ground-truth in terms of 3D semantics.",
"Incorporating $D_{p}$ as an adversarial network to our segmentation framework helps $S$ to capture 3D information through a very simple 2D architecture and without adding to its complexity in the test time."
],
[
"Adversarial training",
"To train our framework, we use a hybrid loss function, which is a weighted sum of three terms.",
"For a dataset of $N$ training samples of images and ground truths $(I_{n},y_{n})$ , we define our hybrid loss function as: $ l_{hybrid} = \\sum _{n=1}^{N} l_{bce}(S(I_{n}),y_{n}) - \\lambda l_{D_{s}} - \\beta l_{D_{p}},$ where $l_{D_{s}}$ and $l_{D_{p}}$ are the losses corresponding to $D_{s}$ and $D_{p}$ and $S(I_{n})$ is the segmentor's prediction.",
"The first term in Equation REF is a weighted binary cross-entropy loss.",
"This loss is the state-of-the-art loss function for semantic segmentation and for a grey-scale image $I$ with size $H\\times W\\times 1$ is defined as: $ l_{bce}(\\hat{y},y) = - \\sum _{i=1}^{H\\times W} (wy_{i}\\log {\\hat{y}_{i}} + (1 - y_{i})\\log {(1 - \\hat{y}_{i})}),$ where $w$ is the weight for positive samples, $y$ is the ground-truth label and $\\hat{y}$ is the network's prediction.",
"Equation REF encourages $S$ to produce predictions similar to ground-truth and penalizes each pixel independently.",
"High-order relations and semantics cannot be captured by this term.",
"To account for this drawback, the second and third terms are added to train our auxiliary networks.",
"$l_{D_{s}}$ and $l_{D_{p}}$ are defined below, respectively: $ l_{D_{s}} = l_{bce}(D_{s}(I_{n},y_{n}),1) + l_{bce}(D_{s}(I_{n},S(I_{n})),0),\\\\ l_{D_{p}} = l_{bce}(D_{p}(P_{I_{n}},P{y_{n}}),1) + l_{bce}(D_{p}(P_{I_{n}},P_{S(I_{n})}),0).$ Here $P$ is the projection module, $l_{bce}$ is the binary cross-entropy loss with $w=1$ in Equation REF corresponding to a single number (0 or 1) as the output.",
"We evaluated the efficacy of our proposed system with the challenging problem of pancreas segmentation.",
"This particular problem was selected due to the complex and varying shape of pancreas and relatively more difficult nature of the segmentation problem compared to other abdominal organs.",
"In our experiments we show that our proposed framework outperforms other state-of-the-art methods and captures the complex 3D semantics with a simple encoder-decoder.",
"Furthermore, we have created an extensive comparison to some baselines, designed specifically to show the effects of each block of our framework.",
"Data and evaluation: We used the publicly available TCIA CT dataset from NIH [11].",
"This dataset contains a total of 82 CT scans.",
"The resolution of scans is $512\\times 512\\times Z$ , $Z \\in [181,466]$ is the number of slices in the axial plane.",
"The voxels spacing ranges from $0.5mm$ to $1.0mm$ .",
"We used a randomly selected set of 62 images for training and 20 for testing to perform a 4-fold cross-validation.",
"Dice Similarity Coefficient (DSC) is used as the metric of evaluation.",
"Comparison to baselines: Table: Comparison with baselines.To show the effect of each building block of our framework we designed an extensive set of experiments.",
"In our experiments we start from only training a single segmentor (S) and go to our final proposed framework.",
"Furthermore, we show comparison of encoder-decoder architecture with other state-of-the-art semantic segmentation architectures.",
"Table REF shows the results adding of each building block of our framework.",
"The eccoder-decoder architecture is the one showed in Figure REF as $S$ , while the Atrous pyramid architecture is similar to the recent work of [4].",
"This architecture is currently state-of-the-art for semantic segmentation.",
"In which an Atrous pyramid is used to capture global context.",
"We added an Atrous pyramid with 5 different scales: 4 Atrous convolutions at rates of $1,2,6,12$ , with the global image pooling.",
"We also replaced the decoder with 2 simple upsampling and conv layers similar to the main paper [4].",
"We refer the readers to the main paper for more details about this architecture due to space limitations [4].",
"We found out having an extensive processing in the decoder improves the results compared to the Atrous pyramid architecture (possibly a better choice for segmentation of objects at multiple scales).",
"This is because our object of interest is relatively small.",
"Moreover, we showed that adding a spatial adversarial notwork ($D_{s}$ ) can boost the performance of $S$ dramatically, in our task.",
"Introducing attention ($A$ ) helps for a better information selection (as described in section REF ) and boosts the performance further.",
"Finally, our best results is achieved by adding the projective adversarial network ($D_{p}$ ), which adds integration of 3D semantics into the framework.",
"This supports our hypothesis that our segmentor has enough capacity in terms of parameters to capture all this information and with proper and explicit supervision can achieve state-of-the-art results.",
"Comparison to the state-of-the-art: We provide the comparison of our method's performance with current state-of-the-art literature on the same TCIA CT dataset for pancreas segmentation.",
"As can be seen from experimental validation, our method outperforms the state-of-the-art with dice scores, provides better efficiency (less computational burden).",
"Of a note, the proposed algorithm's least achievement is consistently higher than the state of the art methods.",
"Table: Comparison with state-of-the-art on TCIA dataset."
],
[
"Conclusion",
"In this paper we proposed a novel adversarial framework for 3D object segmentation.",
"We introduced a novel projective adversarial network, inferring 3D shape and semantics form 2D projections.",
"The motivation behind our idea is that integration of 3D information through a fully 3D network, having all slices as input, is computationally infeasible.",
"Possible workarounds are: 1)down-sampling the data or 2)sacrificing number of parameters, which are sacrificing information or computational capacity, respectively.",
"We also introduced an attention module to selectively pass whole-frame high-level feature from the segmentor's bottleneck to the adversarial network, for a better information processing.",
"We showed that with proper and guided supervision through adversarial signals a simple encoder-decoder architecture, with enough parameters, achieves state-of-the-art performance on the challenging problem of pancreas segmentation.",
"We achieved a dice score of $\\textbf {85.53\\%}$ , which is state-of-the art performance on pancreas segmentation task, outperforming previous methods.",
"Furthermore, we argue that our framework is general and can be applied to any 3D object segmentation problem and is not specific to a single application."
]
] | 1906.04378 |
[
[
"Few-Shot Point Cloud Region Annotation with Human in the Loop"
],
[
"Abstract We propose a point cloud annotation framework that employs human-in-loop learning to enable the creation of large point cloud datasets with per-point annotations.",
"Sparse labels from a human annotator are iteratively propagated to generate a full segmentation of the network by fine-tuning a pre-trained model of an allied task via a few-shot learning paradigm.",
"We show that the proposed framework significantly reduces the amount of human interaction needed in annotating point clouds, without sacrificing on the quality of the annotations.",
"Our experiments also suggest the suitability of the framework in annotating large datasets by noting a reduction in human interaction as the number of full annotations completed by the system increases.",
"Finally, we demonstrate the flexibility of the framework to support multiple different annotations of the same point cloud enabling the creation of datasets with different granularities of annotation."
],
[
"Introduction",
"Two dimensional images have been the most popular digital representation of the world however, point cloud data is increasingly gaining center stage with applications in autonomous driving, robotics and augmented reality.",
"While synthetic point cloud datasets have been around for some time [3], prevalance of depth cameras such as [7] and [16] has led to creation of large 3D datasets [5] created from applying techniques from [11] on depth scans.",
"Finally, we have also seen a number of point cloud datasets created using LIDAR scans of outdoor environments such as [6], [1].",
"The intensity and geometric information in point clouds provide a more detailed digital description of the world than images but their value in algorithmic analysis is fully realised when the points have an associated semantic label.",
"However, annotating 3D point clouds is a time-consuming and labour intensive process owing to the size of the datasets and the limitations of the 2D interfaces in manipulating 3D data.",
"The problem of providing a label of each point in a point cloud has been tackled via a host of fully automatic approaches in the domain of point cloud segmentation [14] [8].",
"While these approaches are successful in delineating large structures such as buildings, roads and vehicles, they perform poorly on finer details in the 3D models.",
"Besides, most of these approaches use supervised learning methods which in-turn rely on labelled datasets making it a chicken and egg problem.",
"Thus, most existing datasets [1] [6] [13] have been annotated via dominantly manual systems to ensure accuracy and to avoid algorithmic biases in the produced datasets.",
"The large investment required in terms of human effort in generating the annotations severely limit both the significance and prevalence of point cloud datasets that are available for the community.",
"Annotating large scale datasets is a natural use case for fusing human and algorithmic intelligence.",
"Annotations inherently rely on a human definition and are also representative of semantic patterns that can be identified by an algorithm.",
"Thus, we observe an active field of research which seeks to fuse human and algorithmic actors in one overarching framework to aid in annotating datasets.",
"Most notable in the context of point cloud annotation is [15] which proposes an active learning framework for region annotations in datasets with repeating shapes.",
"However, this method is limited by allowing for annotating only in certain 2D views the point clouds.",
"Our proposed framework allows for annotation in full 3D thus allowing for finer annotation of the point cloud and the ability to work with less structured real-world point clouds as opposed to relatively noise-free synthetic point clouds.",
"Figure: This figure illustrates our pipeline.",
"The first step starts with a partial sparse annotation by a human, followed by a region growing step using 3D geometric cues.",
"We then iterate between few-shot learning using newly available annotations and sparse correction of predictions via human annotator to obtain final segmentation outputs.In this work, we propose a human-in-loop learning approach that fuses together manual annotation, algorithmic propagation and capitalises on existing 3D datasets for improving semantic understanding.",
"Our method starts with a partial sparse annotation by a human, followed by a region growing step using 3D geometric cues.",
"We then iterate over the following steps: a) Model fine tuning using newly available annotations b) Model prediction of annotated point cloud c) Sparse correction of predictions via human annotator.",
"Figure REF gives a snapshot of our annotation approach.",
"In the next couple of sections we go over our methodology in more details followed by a discussion of our results and future work."
],
[
"Methodology",
"This system is primarily focused on providing an annotation framework to create datasets of point clouds with ground truth semantic labels for each point.",
"For a given point cloud, our method starts with sparse manual annotation and then iterates between two main steps: few-shot learning and manual correction.",
"The manual annotations are provided by marking few representative points for each part to be labelled in the point cloud.",
"These labels are propagated across the point cloud using geometric cues which is used to train the network.",
"The final step involves correcting network mispredictions, which is used to further guide the training process.",
"For the initial point clouds to be annotated, these steps are iterated over multiple times but as more point clouds are annotated using this framework, the method converges to relying only on the initial set of manual annotations (or no annotations at all) to make more accurate annotation predictions."
],
[
"Manual Annotations and Region Growing",
"The decomposition of a point cloud into semantically meaningful parts/regions is an open-ended problem as the concept of an annotation is context dependent [2].",
"Owing to this ambiguity, the first step of our annotation pipeline is to allow the user to determine the number of possible classes that exist in the segmentation of point clouds in the dataset.",
"The framework of annotation, learning and correction also provides the flexibility to have different number of segmentations for the same point cloud allowing for creating datasets with varying granularities of segmentation as in [10].",
"The user initially provides labels to a point or a small group of points for each of the classes in the point cloud.",
"Thereafter, human provided annotations are automatically propagated to few unlabelled points by exploiting geometry of the point cloud.",
"We believe that relying on geometric attributes like surface normals, smoothness, curvature and color (if available) would simplify the goal of segmentation as decomposing the point cloud into locally smooth regions enclosed by sharp boundaries.",
"These segmentations also often end up matching with human perception and can be used as an initial training example for the learning pipeline.",
"For this reason, we use cues like surface normals to group spatially close points as belonging to the same region.",
"We also experimented with color based region growing, K-Nearest Neighbour (KNN) and Fixed Distance Neighbor (FDN) [9] based region growing methods which end up being faster than surface normal based region growing methods without compromising on accuracy of the overall system.",
"Region growing approaches reduce annotator overhead of selecting multiple points by giving a geometry aware selection mechanism."
],
[
"Few-shot Learning",
"The goal of few-shot learning optimization in this context is to rely on minimal human supervision to improve segmentation accuracy.",
"It is for this reason that we obtain the initial set of ground truth labels for training from manual annotations and use region growing methods for further supervision.",
"We use very conservative thresholds for the region growing methods to avoid noisy ground truth labels.",
"We also use a pre-trained network to bootstrap the training process and reduce the amount of human effort in correction phase.",
"The pre-trained network to be used in this system can be any segmentation network, pre-trained on an existing dataset in a similar domain.",
"For our experiments, we used PointNet [12] pre-trained on ShapeNet [4] to bootstrap the training.",
"We fine-tune the base network iteratively using limited supervision in the form of annotation and correction provided by human in the loop.",
"We also dynamically adapt the base network depending on the number of segmentation classes in the point cloud.",
"The initial seed acquired from manual annotation and region growing gives a partially labelled point cloud that is used to fine-tune the base network.",
"The model leverages the prior semantic understanding in the pre-trained network alongside the supervision of partially labelled points in the entire point cloud to provide meaningful segmentations in the first stage.",
"We rely on the human annotators to compensate for network mispredictions by assigning new label to points with incorrect segmentations.",
"Subsequently, we fine-tune further with all the labels (initial seed + corrections) that the human annotator has provided so far.",
"This process continues until all points are labelled correctly - as verified by the human annotator.",
"At this stage, we retrain the network with all the points in the point cloud - which allows us propagate these labels to newer point clouds of the same shape in the dataset.",
"Figure REF illustrates a sample of the results from this loop of user feedback and finetuning.",
"Figure: Figure to show effects of few-shot learning in 3 class segmentation of a chair.",
"From left to right i) Manual annotation with region growing ii) Predictions of the network after fine tuning.",
"Notice the spillage of labels at the boundary which is resolved after correction and final learning step iii) Partially corrected point cloud from the user iv) Final prediction after fine tuning with corrections"
],
[
"Smoothness Loss",
"We formulate segmentation as a per-point classification problem similar to the setup of PointNet [12] including global and local feature aggregation.",
"We also use transformation network to ensure that the network predictions are agnostic to rigid transformations of point cloud.",
"We further leverage smoothness of the shape to favor regions that are compact and continuous.",
"Overall, the network loss can be formulated as: $\\mathcal {L} = L_{segment} + \\alpha L_{transform} + \\beta L_{smooth}$ We use smoothness loss in addition to the segmentation cross entropy loss to encourage adjacent points to have similar labels.",
"The smoothness loss is formulated as follows: $L_{smooth} = \\sum _{i=1}^{N}\\sum _{j=i}^{N}D_{KL} \\left( p_i \\Vert p_j \\right) e^{-\\frac{\\sqrt{\\left\\Vert (pos_{p_i} - pos_{p_j})\\right\\Vert _2^2}}{\\sigma }}$ The smoothness term is computed as pairwise Kullback-Leibler (KL) divergence of predictions exponentially weighted on eucledean distance between any two points in the point cloud.",
"$\\sigma $ is set to the variance of pairwise distance between all points to capture point cloud density in the loss term.",
"The smoothness term in this context is expected to capture and minimize relative entropy between neighboring points in the point cloud.",
"This term ends up dominating total loss if nearby points have divergent logits.",
"Points which are far from each other end up contributing very little to the overall loss term, regardless of their logits owing to high pairwise distances between them.",
"Figure REF shows a qualitative example for the effect of the smoothness loss.",
"Figure: Illustration to show effect of the smoothness loss.",
"From left to right i) Manual annotation with region growing ii) Predictions of the network without smoothness loss iii) Network predictions with smoothness lossThe first stage of segmentation output requires less human cognitive effort for correction if the smoothening term is added to loss computation as it has been observed through our experiments.",
"The weights of smoothness loss term is subsequently dropped after getting further supervision from the user."
],
[
"Results",
"In this section, we discuss the experimental setup to validate the effectiveness of our framework by investigating its utility to create new datasets against completely manual or semi-automatic methods.",
"Additionally, we have also investigated the improvement in annotation efficiency as the total number of annotated point clouds increase.",
"Dataset.",
"To test the robustness and ease of adapting to our framework, we aim to use it to create a massive and diverse dataset of synthetic and reconstructed point clouds.",
"Towards this goal, we have created part segmentations of reconstructed point clouds taken from A Large Dataset of Object Scans [5].",
"Qualitative results for segmentations are shown in Figure REF .",
"The framework showed remarkable improvement in human annotation efficiency as measured in number of clicks required for manual annotation and correction which is discussed in subsequent parts of this section.",
"Figure: Qualitative results for part segmentation on reconstructed point clouds from Large Dataset of Object Scans .",
"The results are shown for segmentation of noisy shapes in potted plant and chair class into two and three classes respectively using our framework.Granularity.",
"The framework also provides the flexibility to annotate with different number of classes for the same shape.",
"The user selects sparse points for each of the classes in the first stage and this information is dynamically incorporated in the training process by re-initializing the last layer to accommodate different number of classes.",
"Qualitative segmentation outputs are illustrated in Figure REF .",
"Figure: Qualitative results for part segmentation on the same shape with different granularities."
],
[
"Annotation Efficiency Improvements",
"Existing semi-supervised methods [15] use amount of supervision and accuracy as evaluation metrics to measure performance.",
"We follow suit and compare the amount of supervision needed to completely annotate a point cloud in our framework as opposed to completely manual methods.",
"In Table REF we compare the number of clicks by the annotator required in our framework compared to a naive nearest neighbour painting based manual approach.",
"Table: Average number of clicks taken to annotate point clouds with varying granularities in terms of number of parts for the same shape.",
"We notice a significant reduction in number of clicks in comparison to manual methods and even our method without the smoothness constraint.With subsequent complete annotations of point clouds coming from the same dataset, we expect a reduction in the human supervision needed in order to have a scalable system.",
"As we incrementally train the network on a progressively complete annotation of the point clouds, the model adapts to the properties of the new domain represented by the dataset.",
"Thus, we are able to predict a more accurate segmentation of the point cloud in the initial iterations thereby cutting down on the total number of user correction steps needed.",
"This is validated via our experiments as illustrated in Figure REF .",
"Figure: Number of clicks taken to annotate subsequent point clouds using our framework.",
"We see a reduction in number of clicks needed as more point clouds of the new dataset are annotated.While previous works measure the amount of user supervision based on invested time, we focused on quantifying supervision via number of clicks.",
"Through our experiments we observed that time taken to annotate a point clouds reduces with the number of point clouds annotated even in completely manual methods.",
"This is because a large part of the time taken in annotating goes in manipulating the point clouds on a 2D tool.",
"As the annotators label more point clouds, they get more accustomed to the tool and the relevant manipulation interactions, reducing the overall time they need in annotating subsequent point clouds.",
"On the other hand, the number of clicks needed depend more on complexity of the point cloud and the number of classes to be annotated instead of the number of previous point clouds annotated in the system making it a suitable metric for evaluation."
],
[
"Conclusion",
"We provide a scalable interactive learning framework that can be used to annotate large point cloud datasets.",
"By fusing together three different cues (human annotations, learnt semantic similarity and geometric consistencies) we are able to obtain accurate annotations with fewer human interactions.",
"We note that while the number of clicks are a useful proxy for the quantum of human interaction needed, it is also important to study the amount of time needed for each click as it adds to overall human time investment needed in annotating a dataset.",
"Significant leaps in reducing the cognitive overload for a human annotator can be made by replacing 2D user interfaces with spatial user interfaces facilitated via virtual reality systems as they make point cloud manipulation and visualization more natural of the annotator."
]
] | 1906.04409 |
[
[
"End-to-End CAD Model Retrieval and 9DoF Alignment in 3D Scans"
],
[
"Abstract We present a novel, end-to-end approach to align CAD models to an 3D scan of a scene, enabling transformation of a noisy, incomplete 3D scan to a compact, CAD reconstruction with clean, complete object geometry.",
"Our main contribution lies in formulating a differentiable Procrustes alignment that is paired with a symmetry-aware dense object correspondence prediction.",
"To simultaneously align CAD models to all the objects of a scanned scene, our approach detects object locations, then predicts symmetry-aware dense object correspondences between scan and CAD geometry in a unified object space, as well as a nearest neighbor CAD model, both of which are then used to inform a differentiable Procrustes alignment.",
"Our approach operates in a fully-convolutional fashion, enabling alignment of CAD models to the objects of a scan in a single forward pass.",
"This enables our method to outperform state-of-the-art approaches by $19.04\\%$ for CAD model alignment to scans, with $\\approx 250\\times$ faster runtime than previous data-driven approaches."
],
[
"Introduction",
"In recent years, RGB-D scanning and reconstruction has seen significant advances, driven by the increasing availability of commodity range sensors such as the Microsoft Kinect, Intel RealSense, or Google Tango.",
"State-of-the-art 3D reconstruction approaches can now achieve impressive capture and reconstruction of real-world environments [18], [25], [26], [37], [37], [4], [7], spurring forth many potential applications of this digitization, such as content creation, or augmented or virtual reality.",
"Such advances in 3D scan reconstruction have nonetheless remained limited towards these use scenarios, due to geometric incompleteness, noise and oversmoothing, and lack of fine-scale sharp detail.",
"In particular, there is a notable contrast in such reconstructed scan geometry in comparison to the clean, sharp 3D models created by artists for visual and graphics applications.",
"With the increasing availability of synthetic CAD models [3], we have the opportunity to reconstruct a 3D scan through CAD model shape primitives; that is, finding and aligning similar CAD models from a database to each object in a scan.",
"Such a scan-to-CAD transformation enables construction of a clean, compact representation of a scene, more akin to artist-created 3D models to be consumed by mixed reality or design applications.",
"Here, a key challenge lies in finding and aligning similar CAD models to scanned objects, due to strong low-level differences between CAD model geometry (clean, complete) and scan geometry (noisy, incomplete).",
"Current approaches towards this problem thus often operate in a sparse correspondence-based fashion [21], [1] in order to establish reasonable robustness under such differences.",
"Unfortunately, such approaches, in order to find and align CAD models to an input scan, thus involve several independent steps of correspondence finding, correspondence matching, and finally an optimization over potential matching correspondences for each candidate CAD model.",
"With such decoupled steps, there is a lack of feedback through the pipeline; e.g., correspondences can be learned, but they are not informed by the final alignment task.",
"In contrast, we propose to predict symmetry-aware dense object correspondences between scan and CADs in a global fashion.",
"For an input scan, we leverage a fully-convolutional 3D neural network to first detect object locations, and then from each object location predict a uniform set of dense object correspondences and object symmetry are predicted, along with a nearest neighbor CAD model; from these, we introduce a differentiable Procrustes alignment, producing a final set of CAD models and 9DoF alignments to the scan in an end-to-end fashion.",
"Our approach outperforms state-of-the-art methods for CAD model alignment by $19.04\\%$ for real-world 3D scans.",
"Our approach is the first, to the best of our knowledge, to present an end-to-end scan-to-CAD alignment, constructing a CAD model reconstruction of a scene in a single forward pass.",
"In summary, we propose an end-to-end approach for scan-to-CAD alignment featuring: a novel differentiable Procrustes alignment loss, enabling end-to-end CAD model alignment to a 3D scan, symmetry-aware dense object correspondence prediction, enabling robust alignment even under various object symmetries, and CAD model alignment for a scan of a scene in a single forward pass, enabling very efficient runtime ($< 3$ s on real-world scan evaluation)"
],
[
"RGB-D Scanning and Reconstruction",
"3D scanning methods have a long research history across several communities, ranging from offline to real-time techniques.",
"In particular, RGB-D scanning has become increasingly popular, due to the increasing availability of commodity range sensors.",
"A very popular reconstruction technique is the volumetric fusion approach by Curless and Levoy [5], which has been materialized in many real-time reconstruction frameworks such as KinectFusion [18], [25], Voxel Hashing [26] or BundleFusion [7], as well as in the context of state-of-the-art offline reconstruction methods [4].",
"An alternative to these voxel-based scene representations is based on surfels [20], that has been used by ElasticFusion [37] to realize loop closure updates.",
"These works have led to RGB-D scanning methods that feature robust, global tracking and can capture very large 3D environments.",
"However, although these methods can achieve stunning results in RGB-D capture and tracking, the quality of reconstructed 3D geometry nonetheless remains far from from artist-created 3D content, as the reconstructed scans are partial, and contain noise or oversmoothing from sensor quality or small camera tracking errors."
],
[
"3D Features for Shape Alignment and Retrieval",
"An alternative to bottom-up 3D reconstruction from RGB-D scanning techniques is to find high-quality CAD models that can replace the noisy and incomplete geometry from a 3D scan.",
"Finding and aligning these CAD models inevitably requires 3D feature descriptors to find geometric matches between the scan and the CAD models.",
"Traditionally, these descriptors were hand-crafted, and often based on a computation of histograms (e.g., point normals), such as FPFH [29], SHOT [35], or point-pair features [11].",
"More recently, with advances in deep neural networks, these descriptors can be learned, for instance based on an implicit signed distance field representation [40], [9], [10].",
"A typical pipeline for CAD-to-scan alignments builds on these descriptors; i.e., the first step is to find 3D feature matches and then use a variant of RANSAC or PnP to compute 6DoF or 9Dof CAD model alignments.",
"This two-step strategy has been used by Slam++ [30], Li et al.",
"[21], Shao et al.",
"[31], but also by the data-driven work by Nan et al.",
"[24] and the recent Scan2CAD approach [1].",
"Other approaches rely only on single RGB or RGB-D frame input, but use a similar two-step alignment strategy [22], [19], [33], [17], [12], [41] images.",
"While these methods are related, their focus is difference as we address geometric alignment independent of RGB information.",
"A fundamental limitation of these two-step pipelines is the decoupled nature of feature matching and alignment computation.",
"This inherently limits the ability of data-driven descriptors, as they remain unaware of the used optimization algorithm.",
"In our work, we propose an end-to-end alignment algorithm where correspondences are trained through gradients from an differentiable Procrustes optimizer."
],
[
"Shape Retrieval Challenges and RGB-D Datasets",
"In the context of 2D object alignment methods several datasets provide alignment annotations between RGB images and CAD models, including the PASCAL 3D+ [39], ObjectNet3D [38], the IKEA objects [22], and Pix3D [33]; however, no geometric information is given in the query images.",
"A very popular series of challenges in the context of shape retrieval is the SHREC, which is organized as part of Eurographics 3DOR [16], [28]; the tasks include matching object instances from ScanNet [6] and SceneNN [15] to ShapeNet objects [3].",
"Scan2CAD [1] is a very recent effort that provides accurate CAD alignment annotations on top of ScanNet [6] using ShapeNet models [3], based on roughly 100k manually annotated correspondences.",
"In addition to evaluating our method on the Scan2CAD test dataset, we also create an alignment benchmark on the synthetic SUNCG [32] dataset."
],
[
"Overview",
"The goal of our method is to align a set of CAD models to the objects of an input 3D scan.",
"That is, for an input 3D scan $\\mathbb {S}$ of a scene, and a set of 3D CAD models $\\mathbb {M}=\\lbrace m_i\\rbrace $ , we aim to find 9DoF transformations $T_i$ (3 degrees each for translation, rotation, and scale) for each CAD model $m_i$ such that it aligns with a semantically matching object $\\mathbb {O} = \\lbrace o_j\\rbrace $ in $\\mathbb {S}$ .",
"This results in a complete, clean, CAD representation of the objects of a scene, as shown in Figure REF .",
"To this end, we propose an end-to-end 3D CNN-based approach to simultaneously retrieve and align CAD models to the objects of a scan in a single pass, for scans of varying sizes.",
"This end-to-end formulation enables the final alignment process to inform learning of scan-CAD correspondences.",
"To enable effective learning of scan-CAD object correspondences, we propose to use symmetry-aware object correspondences (SOCs), which establish dense correspondences between scan objects and CAD models, and are trained by our differentiable Procrustes alignment loss.",
"Then for an input scan $\\mathbb {S}$ represented by volumetric grid encoding a truncated signed distance field, our model first detects object center locations as heatmap predictions over the volumetric grid and corresponding bounding box sizes for each object location.",
"The bounding box represents the extent of the underlying object.",
"From these detected object locations, we use the estimated bounding box size to crop out the neighborhood region around the object center from the learned feature space in order to predict our SOC correspondences to CAD models.",
"From this neighborhood of feature information, we then predict SOCs.",
"These densely establish correspondences for each voxel in the object neighborhood to CAD model space.",
"In order to be invariant to potential reflection and rotational symmetries, which could induce ambiguity in the correspondences, we simultaneously estimate the symmetry type of the object.",
"We additionally predict a binary mask to segment the object instance from background clutter in the neighborhood, thus informing the set of correspondences to be used for the final alignment.",
"To find a CAD model corresponding to the scan object, we jointly learn an object descriptor which is used to retrieve a semantically similar CAD model from a database.",
"Finally, we introduce a differentiable Procrustes alignment, enabling an fully end-to-end formulation, where learned scan object-CAD model SOC correspondences can be informed by the final alignment process, achieving efficient and accurate 9DoF CAD model alignment for 3D scans."
],
[
"Network Architecture",
"Our network architecture is shown in Figure REF .",
"It is designed to operate on 3D scans of varying sizes, in a fully-convolutional manner.",
"An input scan is given by a volumetric grid encoding a truncated signed distance field, representing the scan geometry.",
"To detect objects in a scan and align CAD models to them, we structure the network around a backbone, from which features can then extracted to predict individual SOCs, informing the final alignment process.",
"The backbone of the network is structured in an encoder-decoder fashion, and composed of a series of ResNet blocks [13].",
"The bottleneck volume is spatially reduced by a factor of 16 from the input volume, and is decoded to the original resolution through transpose convolutions.",
"The decoder is structured symmetrically to the encoder, but with half the feature channels, which we empirically found to produce faster convergence and more accurate performance.",
"The output of the decoder is used to predict an objectness heatmap, identifying potential object locations, which is employed to inform bounding box regression for object detection.",
"The predicted object bounding boxes are used to crop and extract features from the output of the second decoder layer, which then inform the SOCs predictions.",
"The features used to inform the SOC correspondence are extracted from the second block of the decoder, whose feature map spatial dimensions are $1/4$ of the original input dimension."
],
[
"Object Detection",
"We first detect objects, predicting bounding boxes for the objects in a scan, which then inform the SOC predictions.",
"The output of the backbone decoder predicts heatmaps representing objectness probability over the full volumetric grid (whether the voxel is a center of an object).",
"We then regress object bounding boxes corresponding to these potential object center locations.",
"For object bounding boxes predictions, we regress a 3-channel feature map, with each 3-dimensional vector corresponding to the bounding box extent size, and regressed using an $\\ell _2$ loss.",
"Objectness is predicted as a heatmap, encoding voxel-wise probabilities as to whether each voxel is a center of an object.",
"To predict a location heatmap $H_1$ , we additionally employ two proxy losses, using a second heatmap prediction $H_2$ as well as a predicted offset field $O$ .",
"$H_1$ and $H_2$ are two 1-channel heatmaps designed to encourage high recall and precision, respectively, and $O$ is a 3-channel grid representing an offset field to the nearest object center.",
"The objectness heatmap loss is: $\\mathcal {L}_{OD} = 2.0\\cdot \\mathcal {L}_{\\textrm {recall}} + 10.0\\cdot \\mathcal {L}_{\\textrm {precision}} + 10.0\\cdot \\mathcal {L}_{\\textrm {offset}}$ The weights for each component in the loss are designed to bring the losses numerically to approximately the same order of magnitude.",
"$\\mathcal {L}_{\\textrm {recall}}$ aims to achieve high recall.",
"It operates on the prediction $H_1$ , on which we apply a sigmoid activation and calculate the loss via binary-cross entropy (BCE).",
"This loss on its own tends to establish a high recall, but also blurry predictions.",
"$\\mathcal {L}_{\\textrm {recall}} &= \\sum _{x \\in \\Omega } \\text{BCE}(\\sigma (H_1(x)), H_\\text{GT}(x)) \\\\H_1 &: \\Omega \\rightarrow [0,1], \\quad \\sigma : \\textrm {sigmoid} $ $\\mathcal {L}_{\\textrm {precision}}$ aims to achieve high precision.",
"It operates on the prediction $H_2$ , on which we apply a softmax activation and calculate the loss via negative log-likelihood (NLL).",
"Due to the softmax, this loss encourages highly localized predictions in the output volume, which helps to attain high precision.",
"$\\mathcal {L}_{\\textrm {recall}} &= \\sum _{x \\in \\Omega } \\text{NLL}(\\sigma (H_2(x)), H_\\text{GT}(x)) \\\\H_2 &: \\Omega \\rightarrow [0,1], \\quad \\sigma : \\textrm {softmax}$ $\\mathcal {L}_{\\textrm {offset}}$ is a regression loss on the predicted a 3D offset field $O$ , following [27].",
"Each voxel of $O$ represents a 3-dimensional vector that points to the nearest object center.",
"This regression loss is used as a proxy loss to support the other two classification losses.",
"$\\mathcal {L}_{\\textrm {offset}} &= \\sum _{x \\in \\Omega } \\Vert O(x) - O_\\text{GT}(x) \\Vert _2^2 \\\\O &: \\Omega \\rightarrow \\mathbb {R}^3 $"
],
[
"Predicting SOCs",
"SOCs are dense, voxel-wise correspondences to CAD models.",
"Hence, they are defined as $ \\text{{SOC}} : \\Omega \\rightarrow \\Psi $ where $\\Psi $ depicts a closed space often as $\\Psi \\in [0,1]^3$ or in our case with ShapeNet $\\Psi \\in [-0.5,0.5]^3$ ; generally $\\Psi $ depends on how the CAD models are normalized.",
"In summary $\\text{{SOC}}(x_{\\text{scan}})$ indicates the (normalized) coordinate in a CAD model of the correspondence to the given scan voxel $x_{\\text{scan}}$ .",
"SOCs are predicted using features cropped from the network backbone.",
"For each detected object, we crop a region with the extend of the predicted bounding box volume $\\mathcal {F}$ from the feature map of the second upsampling layer to inform our dense, symmetry-aware object correspondences.",
"This feature volume $\\mathcal {F}$ is first fitted through tri-linear interpolation into a uniform voxel grid of size $48^3$ before streaming into different prediction heads.",
"SOCs incorporate several output predictions: a volume of dense correspondences from scan space to CAD object space, an instance segmentation mask, and a symmetry classification.",
"The dense correspondences, which map to CAD object space, implicitly contain CAD model alignment information.",
"These correspondences are regressed as CAD object space coordinates, similar to [36], with the CAD object space defined as a uniform grid centered around the object, with coordinates normalized to $[-0.5,0.5]$ .",
"These coordinates are regressed using an $\\ell _2$ loss.",
"In order to avoid ambiguities in correspondence that could be induced by object symmetries, we predict the symmetry class of the object for common symmetry classes for furniture objects: two-fold rotational symmetry, four-fold rotational symmetry, infinite rotational symmetry, and no symmetry.",
"Finally, to constrain the correspondences used for alignment to the scan object, we additionally predict a binary mask indicating the instance segmentation of the object, which is trained using a binary cross entropy loss."
],
[
"Retrieval",
"To retrieve a similar CAD model to the detected object, we use the cropped feature neighborhood $\\mathcal {F}$ to train an object descriptor for the scan region, using a series of 3D convolutions to reduce the feature dimensionality to $8\\times 4^3$ .",
"This resulting 512-dimensional object descriptor is then constrained to match the latent vector of an autoencoder trained on the CAD model dataset, with latent spaces constrained by an $\\ell _2$ loss.",
"This enables retrieval of a semantically similar CAD model at test time through a nearest neighbor search using the object descriptor."
],
[
"9DoF Alignment",
"Our differentiable 9DoF alignment enables training for CAD model alignment in an end-to-end fashion, thereby informing learned correspondences of the final alignment objective.",
"To this end, we leverage a differentiable Procrustes loss on the masked correspondences given by the SOC predictions to find the rotation alignment.",
"That is, we aim to find a rotation matrix $R$ which brings together the CAD and scan correspondence points $P_c, P_s$ : $R = \\textrm {argmin}_\\Omega || \\Omega P_c - P_s||_F,\\quad \\quad \\Omega \\in SO_3$ This is solved through a differentiable SVD of $P_sP_c^T = U\\Sigma V^T$ , with $R=U \\left[ {\\begin{matrix} 1 & & \\\\ & 1 & \\\\ & & d\\end{matrix}} \\right]V^T$ , $d = \\text{det}(VU^T)$ .",
"For scale and translation, we directly regress the scale using a series of 2 3D downsampling convolutions on $\\mathcal {F}$ , and the translation using the detected object center locations.",
"Note that an object center is the geometric center of the bounding box."
],
[
"Data",
"Input scan data is represented by its truncated signed distance field (TSDF) encoded in a volumetric grid and generated through volumetric fusion [5] (we use voxel size = 3cm, truncation = 15cm).",
"The CAD models used to train the autoencoder to produce a latent space for scan object descriptor training are represented as unsigned distance fields (DF), using the level-set generation toolkit by Batty [2].",
"To train our model for CAD model alignment for real scan data, we use the Scan2CAD dataset introduced by [1].",
"These Scan2CAD annotations provide 1506 scenes for training.",
"Using upright rotation augmentation, we augment the number of training samples by 4 ($90^\\circ $ increments with $20^\\circ $ random jitter).",
"We train our network using full scenes as input, with batch size of 1.",
"For SOC prediction at train time the batch size is equal to the number of groundtruth objects in the given scene as crops are only performed around groundtruth object centers.",
"Only large scenes during training are randomly cropped to $400\\times 400\\times 64$ to meet memory requirements.",
"We found that training using 1 scene per batch generally yields stable convergence behavior.",
"For CAD model alignment to synthetic scan data, we use the SUNCG dataset [32], where we virtually scan the scenes following [8], [14] to produce input partial TSDF scans.",
"The training process for synthetic SUNCG scan data is identical to training with real data.",
"See supplemental material for further details."
],
[
"Optimization",
"We use an SGD optimizer with a batch size of 1 scene and an initial learning rate of 0.002, which is decayed by 0.5 every $20K$ iterations.",
"We train for $50K$ iterations until convergence, which typically totals to 48 hours.",
"For object retrieval, we pre-train an autoencoder on all ShapeNetCore CAD models, trained to reconstruct their distance fields at $32^3$ .",
"This CAD autoencoder is trained with a batch size of 16 for $30K$ iterations.",
"We then train the full model with pre-calculated object descriptors for all ShapeNet models for CAD model alignment, with the CAD autoencoder latent space constraining the object descriptor training for retrieval."
],
[
"Results",
"We evaluate our proposed end-to-end approach for CAD model alignment in comparison to state of the art as well as with an ablation study analyzing our differentiable Procrustes alignment loss and various design choices.",
"We evaluate on real-world scans using the Scan2CAD dataset [1].",
"We use the evaluation metric proposed by Scan2CAD [1]; that is, the ground truth CAD model pool is available as input, and a CAD model alignment is considered to be successful if the category of the CAD model matches that of the scan object and the alignment falls within 20cm, $20^\\circ $ , and $20\\%$ for translation, rotation, and scale, respectively.",
"For further evaluation on synthetic scans, we refer to the supplemental material.",
"In addition to evaluating CAD model alignment using the Scan2CAD [1] evaluation metrics, we also evaluate our approach on an unconstrained scenario with 3000 random CAD models as a candidate pool, shown in Figure REF .",
"In this scenario, we maintain robust CAD model alignment accuracy with a much larger set of possible CAD models."
],
[
"Comparison to state of the art.",
"Table REF evaluates our approach against several state-of-the-art methods for CAD model alignment, which establish correspondences and alignment independently of each other.",
"In particular, we compare to several approaches leveraging handcrafted feature descriptors: FPFH [29], SHOT [35], Li et al.",
"[21], as well as learned feature descriptors: 3DMatch [40], Scan2CAD [1].",
"We follow these descriptors with RANSAC to obtain final alignment estimation, except for Scan2CAD, where we use the proposed alignment optimization.",
"Our end-to-end formulation, where correspondence learning can be informed by the alignment, outperforms these decoupled approaches by over $19.04\\%$ .",
"Figure REF shows qualitative visualizations of our approach in comparison to these methods."
],
[
"How much does the differentiable Procrustes alignment loss help?",
"We additionally analyze the effect of our differentiable Procrustes loss.",
"In Table REF , we compare several different alignment losses.",
"As a baseline, we train our model to directly regress the 9DoF alignment parameters with an $\\ell _2$ .",
"We then evaluate our approach with (final) and without (no Procrustes) our differentiable Procrustes loss.",
"For CAD model alignment to 3D scans, our differentiable Procrustes alignment notably improves performance, by over $14.98\\%$ ."
],
[
"How much does SOC prediction help?",
"We evaluate our SOC prediction on CAD model alignment in Table REF .",
"We train our model with (final) and without (no SOCs) SOC prediction as well as with coordinate correspondence prediction but without symmetry (no symmetry).",
"We observe that our SOC prediction significantly improves performance, by over $20.75\\%$ .",
"Establishing SOCs is fundamental to our approach, as dense correspondences can produce more reliable alignment, and unresolved symmetries can lead to ambiguities and inconsistencies in finding object correspondences.",
"In particular, we also evaluate the effect of symmetry classification in our SOCs; explicitly predicting symmetry yields a performance improvement of $10.21\\%$ ."
],
[
"What is the effect of using an anchor mechanism for object detection?",
"In Table REF , we also compare our CAD model alignment approach with (final) and without (no anchor) using anchors for object detection, where without anchors we predict only object center locations as a probability heatmap over the volumetric grid of the scan, but do not regress bounding boxes, and thus only crop a fixed neighborhood for the following SOCs and alignment.",
"We observe that by employing bounding box regression, we can improve CAD model alignment performance, as this facilitates scale estimation and allows correspondence features to encompass the full object region.",
"Figure: Our end-to-end CAD model alignment approach applied to an unconstrained set of candidate CAD models; here, we use a set of 3000 randomly selected CAD models from ShapeNetCore .Our approach maintains robust CAD model alignment performance in such a scenario which is often reflected in real-world applications."
],
[
"Limitations",
"Although our approach shows significant improvements compared to state of the art, we believe there directions for improvement.",
"Currently, we focus on the objects in a scan, but do not consider structural components such as walls and floors.",
"We believe, however, that our method could be expanded to detect and match plane segments in the spirit of structural layout detection such as PlaneRCNN [23].",
"In addition, we currently only consider the geometry of the scan or CAD; however, it is an interesting direction to consider finding matching textures in order to better visually match the appearance of a scan.",
"Finally, we hope to incorporate our alignment algorithm in an online system that can work at interactive rates and give immediate feedback to the scanning operator."
],
[
"Conclusion",
"We have presented an end-to-end approach that automatically aligns CAD models with commodity 3D scans, which that is facilitated with symmetry-aware correspondences and a differentiable Procrustes algorithm.",
"We show that by jointly training the correspondence prediction with direct, end-to-end alignment, our method is able to outperform existing state of the art by over $19.04\\%$ in alignment accuracy.",
"In addition, our approach is roughly $250\\times $ faster than previous data-driven approaches and thus could be easily incorporated into an online scanning system.",
"Overall, we believe that this is an important step towards obtaining clean and compact representations from 3D scans, and we hope it will open up future research in this direction."
],
[
"Acknowledgements",
"We would like to thank Justus Thies and Jürgen Sturm for valuable feedback.",
"This work is supported by Occipital, the ERC Starting Grant Scan2CAD (804724), a Google Faculty Award, an Nvidia Professorship Award, and the ZD.B.",
"We would also like to thank the support of the TUM-IAS, funded by the German Excellence Initiative and the European Union Seventh Framework Programme under grant agreement n° 291763, for the TUM-IAS Rudolf Mößbauer Fellowship"
],
[
"Online Benchmark",
"In this appendix, we provide additional results, including measurements on the hidden test set of the Scan2CAD benchmark [1].",
"Specifically, we provide a quantitative comparison in Tab.",
"REF , which was submitted to official benchmark website on March 29th, 2019.",
"In addition, we show qualitative results of our approach in Fig.",
"REF ."
],
[
"SUNCG",
"We conduct experiments on the SUNCG dataset [32] to verify the effectiveness of our method.",
"For training and evaluation, we create virtual scans of the synthetic scenes, where we simulate a large-scale indoor 3D reconstruction by using rendered depth frames similar to [14], [8] with the distinction that we add noise to the synthetic depth frames in the fusion process.",
"The voxel resolution for the generated SDF grids is at $4.68cm$ .",
"The ground truth models are provided by the SUNCG scenes, where we discard any objects that have not been seen during the virtual scanning (no occupancy in the scanned SDF).",
"We show a quantitative evaluation in Tab.",
"REF , where we outperform the current state-of-the-art method Scan2CAD [1] by a significant margin.",
"We show that our method can align CAD models robustly through all classes.",
"Additionally, we see that our Procrustes loss notably improves overall alignment accuracy.",
"In particular, for less frequent CAD models (e.g., those summarized in other), we observe a considerable improvement in alignment accuracy.",
"Fig.",
"REF shows qualitative results on scanned SUNCG scenes.",
"Our end-to-end approach is able to handle large indoor scenes with complex furniture arrangements better than baseline methods.",
"Table: CAD alignment accuracy comparison (%\\%) on SUNCG .We compare to state-of-the-art handcrafted feature descriptors FPFH , SHOT as well as a learning based method Scan2CAD for CAD model alignment.",
"Note that the Procrustes loss considerably improves overall alignment accuracy.Figure: NO_CAPTION"
]
] | 1906.04201 |
[
[
"Magnetic Skyrmion State in Janus Monolayers of Chromium Trihalides\n Cr(I,X)3"
],
[
"Abstract Magnetic skyrmions are nano-scale spin structures that are promising for ultra-dense memory and logic devices.",
"Recent progresses in two-dimensional magnets encourage the idea to realize skyrmionic states in freestanding monolayers.",
"However, monolayers such as CrI3 lack Dzyaloshinskii-Moriya interactions (DMI) and thus do not naturally exhibit skyrmions but rather a ferromagnetic state.",
"Here we propose the fabrication of Cr(I,X)3 Janus monolayers, in which the Cr atoms are covalently bonded to the underlying I ions and top-layer Br or Cl atoms.",
"By performing first-principles calculations and Monte-Carlo simulations, we identify strong enough DMI, which leads to not only helical cycloid phases, but also to intrinsic skyrmionic states in Cr(I,Br)3 and magnetic-field-induced skyrmions in Cr(I,Cl)3."
],
[
"Magnetic Skyrmion States in Janus Monolayers of Chromium Trihalides Cr(I,X)$_3$ Changsong Xu$^{1}$ , Junsheng Feng$^{2}$ , Hongjun Xiang$^{3,4,*}$ & L. Bellaiche$^{1,*}$ Physics Department and Institute for Nanoscience and Engineering, University of Arkansas, Fayetteville, Arkansas 72701, USA School of Physics and Materials Engineering, Hefei Normal University, Hefei 230601, P. R. China Key Laboratory of Computational Physical Sciences (Ministry of Education), State Key Laboratory of Surface Physics, and Department of Physics, Fudan University, Shanghai, 200433, China Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, P. R. China Magnetic skyrmions are nano-scale spin structures that are promising for ultra-dense memory and logic devices.",
"Recent progresses in two-dimensional magnets encourage the idea to realize skyrmionic states in freestanding monolayers.",
"However, monolayers such as CrI$_3$ lack Dzyaloshinskii-Moriya interactions (DMI) and thus do not naturally exhibit skyrmions but rather a ferromagnetic state.",
"Here we propose the fabrication of Cr(I,X)$_3$ Janus monolayers, in which the Cr atoms are covalently bonded to the underlying I ions and top-layer Br or Cl atoms.",
"By performing first-principles calculations and Monte-Carlo simulations, we identify strong enough DMI, which leads to not only helical cycloid phases, but also to intrinsic skyrmionic states in Cr(I,Br)$_3$ and magnetic-field-induced skyrmions in Cr(I,Cl)$_3$ .",
"Magnetic skyrmions are nano-scale spin clusters with topological stability, and are promising for advanced spintronics [1], [2].",
"One requirement toward such applications is that the hosting materials should be thin films, so that the nano size of skyrmions can be taken full advantage of.",
"Besides previous studies on bulk MnSi [3], [4], [5], [6], recent works focused on ultrathin films, such as FeGe [7], [8] and rare-earth ion garnet [9], [10], which both take advantage of the Dzyaloshinskii-Moriya interaction (DMI) arising from the heavy metal substrate.",
"However, no skyrmionic state has ever been reported to intrinsically exist in free-standing monolayers, to the best of our knowledge, while two-dimensional (2D) semiconducting magnets, such as monolayer CrI$_3$ [11], are recently attracting much attention due to their novel physics and rich applications [12].",
"The ferromagnetic monolayer CrI$_3$ crystalizes in honeycomb lattice made of edge-sharing octahedra.",
"Its ferromagnetic order is stabilized by an out-of-plane anisotropy [11], which arises from single ion anisotropy (SIA) and Kitaev-type exchange coupling that both result from the SOC of its heavy ligands [13], [14].",
"However, the ingredient DMI is absent between the most strongly coupled first nearest neighbor (1st NN) Cr-Cr pairs, because the inversion center between the two Cr atoms prevents its existence [15].",
"Interestingly, very recent theoretical study proposed the application of electric field to break the inversion center and induce DMI in monolayer CrI$_3$ [16].",
"Although this clever method leads to CrI$_3$ monolayers becoming closer to adopt a skyrmion phase, the weak effects of electric field in generating DMI, as well as the rather strong out-of-plane anisotropy, hinders the actual creation of skyrmions in this system.",
"Here we propose a more effective approach that consists in fabricating Janus monolayers of chromium trihalides Cr(I,X)$_3$ (X = Br, Cl).",
"One example of Janus monolayers is the transition metal dichalcogenides MoSSe, which originates from the well-known monolayer MoS$_2$ but with one layer of S being substituted by Se.",
"Such Janus monolayer is experimentally achievable, since MoSSe has been reproducibly obtained using different methods [17], [18], which strongly suggests the feasibility of also creating Cr(I,X)$_3$ Janus monolayers.",
"In this manuscript, we apply density functional theory (DFT) and parallel tempering Monte Carlo (PTMC) simulations to study the magnetic interactions and to investigate skyrmionic states of two Janus monolayers, namely Cr(I,Br)$_3$ and Cr(I,Cl)$_3$ , as well as the prototype CrI$_3$ (see structural schematics in Figs.",
"1a and 1b).",
"As we will show, these Janus monolayers exhibit not only strong enough DMI, but also a decrease in magnetic anisotropy energy (MAE), which both contribute to stabilizing skyrmionic states.",
"We will also demonstrate that the presence of Kitaev interaction and the application of magnetic field both benefit to the emergence of skyrmions in these Cr(I,X)$_3$ Janus monolayers.",
"Results Hamiltonian and magnetic parameters We consider the following Hamiltonian for describing the magnetic interactions of the Cr(I,Br)$_3$ and Cr(I,Cl)$_3$ Janus monolayers, as well as the prototype CrI$_3$ monolayer, using the generalized matrix form up to the second nearest neighbors (2nd NN) for the spins, $\\mathbf {{\\rm S}}$ : $\\mathcal {H} = \\frac{1}{2} \\sum _{n=1,2}~ \\sum _{(i,j)_n} \\mathbf {{\\rm S}}_i {\\cdot } \\mathcal {J}_{n,ij} {\\cdot } \\mathbf {{\\rm S}}_j+\\sum _{i} \\mathbf {{\\rm S}}_i {\\cdot } \\mathcal {A}_{ii} {\\cdot } \\mathbf {{\\rm S}}_i ~,$ where the first term is the exchange coupling that runs over all first $(i,j)_1$ and second $(i,j)_2$ NN Cr pairs, respectively, and the second term represents SIA that runs over all Cr sites.",
"$\\mathcal {J}$ and $\\mathcal {A}$ are 3$\\times $ 3 matrices, for which the elements can be extracted using the four-state energy mapping method and density-functional theory (DFT) (see Method section and Ref.",
"[13], [19], [20], [21] for details).",
"The $\\mathcal {J}$ matrices can be further decomposed into an isotropic parameter $J^{\\prime }$ (to be defined later), the symmetric Kitaev term $K$ [13] and the antisymmetric DMI vector $\\mathbf {{\\rm D}}$ [19]; and the $\\mathcal {A}$ matrix simply reduces to its $A_{zz}$ components by symmetry.",
"The Hamiltonian can then be rewritten as $\\mathcal {H} = \\frac{1}{2} \\sum _{n=1,2}~ \\sum _{(i,j)_n}\\lbrace J^{\\prime }_n \\mathbf {{\\rm S}}_i {\\cdot } \\mathbf {{\\rm S}}_j+ K_n S_i^{\\gamma } S_j^{\\gamma }+ \\mathbf {{\\rm D}}_{n,ij} {\\cdot } (\\mathbf {{\\rm S}}_i {\\times } \\mathbf {{\\rm S}}_j)\\rbrace + \\sum _{i} A_{zz} (S_i^{z})^2 ~.$ where $S^{\\gamma }$ is the $\\gamma $ component of S in a local basis {$\\alpha ,\\beta ,\\gamma $ } that is related to the Kitaev interaction.",
"Such local basis, as well as the direction of DMI vector, are schematized in Figs.",
"1c and 1d, and will be discussed in detail in the following paragraphs.",
"Note that the Kitaev interaction is a bond-wised anisotropic exchange coupling that was first proposed by A. Kitaev [22] and then found in Na$_2$ IrO$_3$ by G. Jackeli and G. Khaliullin [23].",
"A non-negligible Kitaev coefficient has also been previously identified in CrI$_3$ and CrGeTe$_3$ and was found to be crucial for determining and understanding the different magnetic anisotropies of these two latter compounds [13].",
"A Kitaev term was also needed in order to reproduce the temperature evolution of magnetization in CrBr$_3$ few layers [24] and is the key ingredient to create quantum spin liquid states (when the $J^{\\prime }$ parameter is vanishing [25]).",
"In the present case, for the 1st NN of Cr(I,X)$_3$ , as well as that of CrI$_3$ , diagonalization of the symmetric part of $\\mathcal {J}_1$ matrix leads to (i) nearly degenerate lower-in-energy $J^{\\prime }_{1,\\alpha }$ and $J^{\\prime }_{1,\\beta }$ , where the eigenvectors $\\alpha _1$ and $\\beta _1$ lie in the Cr$_2$ L$_2$ (L for ligand) plane; and (ii) higher-in-energy $J^{\\prime }_{1,\\gamma }$ being associated with the perpendicular $\\gamma _1$ axis (see Figs.",
"1c and 1d).",
"Similarly for the 2nd NN, the low-energy easy plane is spanned by the perpendicular 3rd NN Cr pair ($\\alpha _2$ axis) and the two most related ligands ($\\beta _2$ axis); the hard axis ($\\gamma _2$ axis) is thus perpendicular to the easy plane (see schematics in Figs.",
"1c and 1d).",
"For both the 1st and the 2nd NN, the aforementioned isotropic exchange parameter is thus defined as $J^{\\prime }=(J^{\\prime }_{\\alpha }+J^{\\prime }_{\\beta })/2$ and the Kitaev coefficient as $K=J^{\\prime }_{\\gamma }-J^{\\prime }$ .",
"Note that we further define another isotropic parameter as $J=(J_{xx}+J_{yy}+J_{zz})/3$ , which is equivalent to the $J$ coefficient involved in the commonly used $J\\mathbf {{\\rm S}}_i {\\cdot } \\mathbf {{\\rm S}}_j$ term.",
"Such isotropic $J$ is thus different from $J^{\\prime }$ and its ratio with the DMI parameter ($|D/J|$ ) can provide information about the existence of magnetic skyrmions or not.",
"Furthermore, the directions of the DMI vectors are also distinct between 1st and 2nd NN.",
"Specifically and as further illustrated in Figs.",
"1c and 1d, the $\\mathbf {{\\rm D}}_1$ vectors of all Cr(I,X)$_3$ are basically parallel to $\\gamma _1$ , which satisfy Moriya's rule [15] and thus testify the accuracy of our calculations.",
"On the other hand, the $\\mathbf {{\\rm D}}_2$ vectors all stay in the ($\\alpha _2,\\beta _2$ ) easy plane and are close to the $\\beta _2$ axis.",
"Let us now look at the precise values of the magnetic parameters obtained from DFT and four-state method [13], [19], [20], [21].",
"As shown in Table I, both $J_1$ and $J_2$ yield negative values, which imply ferromagnetism (FM) for all investigated systems, which is consistent with the measured ferromagnetism of CrX$_3$ , with X = Cl, Br and I [26], [24], [11].",
"The $J_1$ , $K_1$ and $A_{zz}$ parameters all decrease in magnitude when the X ion of Cr(I,X)$_3$ varies from I to Cl, via Br.",
"This decrease in $J_1$ results from the shrinking of lattice constants (leading to shortened Cr-Cr distance and enhanced direct antiferromagnetic exchange), while that of $K_1$ and $A_{zz}$ root in the weakening of the SOC strength (as consistent with the location of I, Br and Cl in the periodic Table).",
"On the other hand, the $J_2$ and $K_2$ coefficients show no significant changes with the X ion.",
"Moreover, the Janus monolayers Cr(I,Br)$_3$ and Cr(I,Cl)$_3$ exhibit remarkable $D_1$ values of 0.270 meV and 0.194 meV, respectively, while the prototype CrI$_3$ has no finite $D_1$ , as aforementioned.",
"Such values, altogether with $J_1$ , yield large $|D_1/J_1|$ ratios of 0.150 for Cr(I,Br)$_3$ and 0.194 for Cr(I,Cl)$_3$ , which are within the typical range of 0.1-0.2 known to generate skyrmionic phases [2].",
"Such $|D_1/J_1|$ ratios are much larger than the reported value 0.071, obtained when applying an extremely large 2 V/nm electric field to CrI$_3$ monolayer [16].",
"Notably, our three investigated systems also show non-negligible $D_2$ values, which leads to $|D_2/J_2|$ ratios comparable to their 1st NN counterparts, as detailed in Table I.",
"The significant $|D_1/J_1|$ and $|D_2/J_2|$ ratios of Cr(I,Br)$_3$ and Cr(I,Cl)$_3$ demonstrate that the fabrication of Janus monolayers is an effective way to induce considerable DMI and is thus promising to create magnetic skyrmions.",
"Another important factor that affects the formation of skyrmions is the magnetic anisotropy energy (MAE), which is defined as the energy difference between out-of-plane FM (zFM) and in-plane FM (xFM) states, $\\epsilon =E_{zFM}-E_{xFM}$ , where $\\epsilon < 0$ ($> 0$ , respectively) favors out-of-plane (in-plane, respectively) FM.",
"Here, for Cr(I,X)$_3$ and CrI$_3$ , the total anisotropy is the result of the interplay between Kitaev interaction and SIA [13] and is calculated using the model energy of Eq.",
"(2) with the parameters of Table 1.",
"As shown in Fig.",
"2a, CrI$_3$ , which has an $\\epsilon $ = -0.693 meV/Cr, favors strong out-of-plane anisotropy and Cr(I,Br)$_3$ , with $\\epsilon $ = -0.218 meV/Cr, exhibits mild out-of-plane anisotropy.",
"In contrast, Cr(I,Cl)$_3$ shows weak in-plane anisotropy since $\\epsilon $ = 0.111 meV/Cr (note that Fig.",
"2a also displays the $\\epsilon $ directly calculated from DFT, that agree rather well with those obtained by the model – which thus further testifies the accuracy of our model Hamiltonian).",
"As we will see latter, such differences in anisotropy between Cr(I,Br)$_3$ and Cr(I,Cl)$_3$ plays a role on the morphology of the skyrmions.",
"Intrinsic skyrmionic states in Cr(I,Br)$_3$ Janus monolayer As detailed in the Method section, parallel tempering Monte Carlo (PTMC) simulations using the Hamiltonian of Eq.",
"(1) are performed over a 50$\\times $ 50$\\times $ 1 supercell to find spin structures with low energies.",
"These latter spin structures are then further relaxed with the conjugate gradient (CG) method, until arriving at energy minima.",
"Such optimization scheme guarantees the converged spin structures to be either metastable states or the ground state at the temperature of zero Kelvin.",
"Practically, the ground state of Cr(I,Br)$_3$ is determined to be out-of-plane ferromagnetism (zFM), of which the energy is set to be zero.",
"The first metastable state (the convention of terminology here is that the $n$ th metastable state has the $(n+1)$ th lowest energy) of Cr(I,Br)$_3$ is an out-of-plane cycloid structure, as shown in Fig.",
"3a, with an energy of 0.056 meV/Cr.",
"Its propagation direction is symmetrically equivalent along $\\mathbf {{\\rm (a+2b)}}$ , $\\mathbf {{\\rm (2a+b)}}$ and $\\mathbf {{\\rm (b-a)}}$ .",
"Such cycloid structure yields no finite topological charge $Q$ that is defined as $Q=\\frac{1}{4\\pi }\\int \\mathbf {{\\rm m}} \\cdot (\\frac{\\partial \\mathbf {{\\rm m}}}{\\partial x})\\times (\\frac{\\partial \\mathbf {{\\rm m}}}{\\partial y})dxdy~,$ where $\\mathbf {{\\rm m}}$ is the unit vector lying along the local magnetic moment's direction.",
"On the other hand, skyrmionic states with finite $Q$ are found as the second and even higher metastable states.",
"As shown in Fig.",
"3b, anomalous spin patterns occur at the boundary of dark and bright zones, which are actually the domains with spins having “in” and “out” out-of-plane components , respectively, i.e., z-components that are negative and positive, respectively, along the z-axis (note that, on the other hand, the arrows displayed in Fig.3 represent the in-plane components of the magnetic dipoles).",
"We name such boundary as domain wall.",
"The novel spin structure of Fig.",
"3b renders $Q=1$ and is thus topologically identical to the common bubble-like skyrmion [3], [4], [5], [6].",
"Such skyrmionic state has an energy of 0.069 meV/Cr (with respect to the ground state) and is also 0.013 meV/Cr higher than the cycloid state in our used 50$\\times $ 50$\\times $ 1 supercell.",
"Moreover, it is found that either the upper or the lower domain wall can host multiple skyrmions.",
"For example, state possessing two skyrmions ($Q=2$ ) in the supercell can have (i) one skyrmion at each domain wall (Fig.",
"3c) or (ii) two skyrmions at the same domain wall (Fig.",
"3d).",
"Interestingly, such two cases possess degenerate energy of 0.082 meV/Cr.",
"In fact, the energy is a linearly increasing function with simply the number of skyrmions, as shown in Fig.",
"2c, with the maximum number of skyrmions existing within the 50$\\times $ 50$\\times $ 1 supercell being four (see Fig.",
"3e for that four-skyrmion state).",
"Furthermore, it is found that the domain wall can host not only skyrmions, but also anti-skyrmions.",
"As a matter of fact and as shown in Fig.",
"3f, (i) the spins rotate clockwise from left to right along the lower domain wall, which renders $Q=2$ and characterizes two skyrmions, as similar to the case of Fig.",
"3d; (ii) but, in contrast, the spins rotate anticlockwise from right to left along the upper domain wall, which yields $Q=-2$ and indicates two antiskyrmions.",
"The total topological charge of Fig.",
"3f is thus $Q=0$ .",
"Comparing Fig.",
"3e and 3f, we find that either domain wall (upper or lower) can host either skyrmions or anti-skyrmions.",
"The energies of the two spin configurations in Figs.",
"3e and 3f are further found to be exactly the same, indicating that skyrmion and anti-skyrmion are energetically degenerate, which has never been reported, to the best of our knowledge.",
"Besides, the existence of antisyrmions is actually rather rare in surface or interface systems [27].",
"It is also found here that skyrmions and anti-skyrmions can not exist at the same domain wall, since they tend to annihilate with each other which would in fact lead to the transformation to the cycloidal state (resulting in $Q=0$ ).",
"Magnetic field induced skyrmion states in Cr(I,Cl)$_3$ Janus monolayer.",
"Let us now turn our attention to the Cr(I,Cl)$_3$ system.",
"As consistent with its positive MAE, the ground state of Cr(I,Cl)$_3$ is determined to be in-plane zigzag-canted FM, as shown in Fig.",
"4a.",
"The spin vectors at the A and B sites of the honeycomb lattice make an angle of 6.7$^\\circ $ .",
"Such spin canting is further evidenced by DFT calculations on a unit cell, as a moment of 0.3 $\\mu _B$ along the $y$ -direction emerges from the initial FM state fully lying along the x-direction (xFM), with an energy lowering of 0.024 meV/Cr with respect to that xFM state (note that the resulting angle between spins is equal to 6.2$^\\circ $ in DFT calculations, which compares rather well with the aforementioned 6.7$^\\circ $ given by numerical simulations using the magnetic Hamiltonian of Eq.",
"(2)).",
"Such zigzag-canted FM state is degenerate when the main component of the magnetization is along the $\\mathbf {{\\rm a}}$ direction (which is precisely our x-direction), or along $\\mathbf {{\\rm b}}$ and $\\mathbf {{\\rm (a+b)}}$ directions.",
"The first metastable state is a mostly in-plane cycloidal structure superimposed with small out-of-plane components forming a spin wave, as shown in Fig.",
"4b.",
"Note that such spin arrangement is reminiscent of the well-known complex spin organization of BiFeO$_3$ that consists in the coexistence of an in-plane cycloidal structure and an out-of-plane spin density wave [28].",
"Such cycloidal structure in Cr(I,Cl)$_3$ has an energy higher by 0.026 meV/Cr from the ground state and is degenerate for propagation directions being along $\\mathbf {{\\rm (a+2b)}}$ , $\\mathbf {{\\rm (2a+b)}}$ or $\\mathbf {{\\rm (b-a)}}$ directions.",
"Interestingly, skyrmionic states are not “naturally” stable in Cr(I,Cl)$_3$ within our Hamiltonian using the original DFT-derived magnetic parameters.",
"On the other hand, one can stabilize skyrmionic states there by applying an out-of-plane magnetic field $B$ in order to compensate the in-plane anisotropy.",
"It is numerically found that a critical field of 0.65 T tunes the MAE from positive to negative.",
"However, the magnetic field ($<$ 2 T) does not polarize the spins to the fully out-of-plane direction, but rather renders the in-plane zigzag-canted FM now possessing an out-of-plane component, as shown in Fig.",
"4c.",
"The previously metastable cycloid state can not survive and transform to such FM state when $B >$ 0.2 T. Practically, skyrmions are numerically found to be stabilized as metastable states when the magnetic field ranges between 0.5 T and 1.3 T. For instance and as shown in Fig.",
"4d for a magnetic field of 0.8 T, one skyrmion, with a very low energy of 0.015 meV/Cr, is created as a metastable state, rendering $Q = 1$ .",
"Such skyrmion is quite similar to the common bubble-like skyrmions [3], [4], [5], [6], but also possesses some unique characteristics.",
"For instance, it is not isotropic in the plane and there is always a bright (spin-out) zone being at one side of the dark (spin-in) skyrmion center, as shown in Figs.",
"4d-f.",
"In fact, up to three skymions can be stabilized within the 50$\\times $ 50$\\times $ 1 supercell and the energy also linearly increases with the skymion numbers in Cr(I,Cl)$_3$ , as shown in Fig.",
"2d.",
"Interestingly, we further find that, wherever the skyrmions are initialized in the supercell, after optimization, their core tend to be aligned equivalently along $\\mathbf {{\\rm a}}$ , $\\mathbf {{\\rm b}}$ and $\\mathbf {{\\rm (a+b)}}$ directions, which are the degenerate directions for the in-plane zigzag-canted FM state, as shown in Figs.",
"4e and 4f.",
"Such well organized structure may be of benefit for potential application in memory devices.",
"Discussion and Conclusion Note that we also studied the effects of Kitaev interactions in our investigated systems.",
"It is found that, if the Kitaev terms are turned off in our considered Hamiltonian, the skyrmionic state can no longer be stabilized in Cr(I,Br)$_3$ Janus monolayers.",
"As shown in Fig.",
"2b, $K_1$ (respectively, $K_2$ ) results in negative (respectively, positive) MAE and thus contributes to out-of-plane (respectively, in-plane) anisotropy.",
"The interplay between Kitaev terms and SIA is responsible for the total anisotropy, which plays an important role in the morphology of skyrmions.",
"For instance, we numerically found that arbitrarily modifying the MAE gives the following results: (i) a mild out-of-plane anisotropy benefits the formation of skyrmions, while (ii) a too small anisotropy results in skyrmions with large diameter, as shown in Fig.",
"S1.",
"Moreover, the frustration arising from Kitaev interaction [13] adds to the disorder of the system, which also facilitate the formation of skyrmions.",
"We also studied an analogous system, that is Cr(Br,Cl)$_3$ .",
"It exhibits in-plane zigzag-canted FM ground state and a metastable in-plane cycloid, as well as magnetic field induced skyrmion states, which makes it very much similar to Cr(I,Cl)$_3$ .",
"In summary, we proposed the fabrication of Cr(I,X)$_3$ (X = Br, Cl) Janus monolayers to induce large DMI and subsequent magnetic skyrmion states.",
"By combining DFT and MC simulations, we find that Cr(I,Br)$_3$ can intrinsically host metastable skyrmionic phases, while a skyrmionic state of Cr(I,Cl)$_3$ can be stabilized by applying an out-of-plane magnetic field.",
"Our study thus suggests a feasible approach to create skyrmions in semiconducting magnets consisting of chromium trihalides Janus monolayers.",
"Such presently predicted skyrmionic phases are not only useful for memory and logic devices, but can also be promising for energy storage using topological spin textures [29].",
"DFT calculations are performed using the Vienna ab-initio simulation package (VASP) [30].",
"The projector augmented wave (PAW) method [31] is employed with the following electrons being treated as valence electrons: Cr 3$p$ , 4$s$ and 3$d$ , I 5$s$ and 5$p$ , Br 4$s$ and 4$p$ , and Cl 3$s$ and 3$p$ .",
"The plane wave energy cut off is chosen to be 350 eV.",
"The length of the $c$ axis of all investigated systems is set to be 19.807 Å , which yields thick enough vacuum layers of more than 16.5 Å.",
"The local density approximation (LDA) [32] is used, with an effective Hubbard $U$ parameter chosen to be 0.5 eV for the localized 3$d$ electrons of Cr ions [13].",
"$k$ -point meshes are chosen such as they are commensurate with the choice of 4$\\times $ 4$\\times $ 1 for the unit cell (that contains 10 atoms).",
"The choices on LDA, Hubbard $U$ and $k$ -point meshes were previously tested to be accurate [13].",
"The Hellman-Feynman forces are taken to be converged when they become smaller than 0.001 eV/Å on each ion.",
"The four-state energy mapping method is applied to obtain the elements of the $\\mathcal {J}$ and $A_{zz}$ matrices [20], [21], [13], [19].",
"The four-state method considers one specific magnetic pair (or site) at a time.",
"Without loss of generality, the total energy can be written as: $E = E_{spin} + E_{nonspin} = E_0 + \\mathbf {{\\rm S}}_1{\\cdot }\\mathcal {J}_{12}{\\cdot }\\mathbf {{\\rm S}}_2 + \\mathbf {{\\rm S}}_1 {\\cdot } \\mathbf {{\\rm K}}_1+ \\mathbf {{\\rm S}}_2 {\\cdot } \\mathbf {{\\rm K}}_2$ where $\\mathbf {{\\rm S}}_1{\\cdot }\\mathcal {J}_{12}{\\cdot }\\mathbf {{\\rm S}}_2$ is the exchange coupling between the magnetic site 1 and site 2; $\\mathbf {{\\rm S}}_1 {\\cdot } \\mathbf {{\\rm K}}_1$ ($\\mathbf {{\\rm S}}_2 {\\cdot } \\mathbf {{\\rm K}}_2$ , respectively) represents the coupling between site 1 (site 2, respectively) and all the other magnetic sites that are different from sites 1 and 2; and the energy from interactions among all these other magnetic sites, together with non-magnetic energy $E_{nonspin}$ , is gathered in $E_0$ .",
"To calculate, e.g., the $J_{xy}$ component of $\\mathcal {J}_{12}$ , four spin states are considered for the {Cr1,Cr2} pair: {$(S~0~0),(0~S~0)$ }, {$(S~0~0),(0~-S~0)$ }, {$(-S~0~0),(0~S~0)$ } and {$(-S~0~0),(0~-S~0)$ }; while other spins are perpendicular to those of Cr1 and Cr2.",
"The $J_{xy}$ can then be constructed with the energies obtained from DFT calculations using the following equation: $J_{xy} = \\frac{E_1 - E_2 -E_3 +E_4}{4S^2}$ Such four-state method is accurate when using a large enough supercell (to prevent the coupling between some sites present in the supercell and periodically repeated sites from happening).",
"For example, SIA and the 1st NN parameters are calculated using a 2$\\times $ 2$\\times $ 1 supercell and the 2nd NN coefficients using a 3$\\times $ 3$\\times $ 1 supercell.",
"See Ref.",
"[20], [21], [13], [19] for more details.",
"Parallel tempering Monte Carlo (PTMC) simulations and conjugate gradient (CG) method.",
"PTMC simulations with heat bath algorithm [33] are performed using the Hamiltonian of Eq.",
"(1) to update the spin structures.",
"The results shown in the manuscript are based on a 50$\\times $ 50$\\times $ 1 supercell, while a 54$\\times $ 54$\\times $ 1 supercell is also adopted, which leads to similar skyrmion states.",
"200,000 MC sweeps are performed at each temperature.",
"After the MC simulations, a CG method [34] is applied to further optimize the spin configuration.",
"Specifically, the directions of spins are described with $(\\theta _i,\\phi _i)$ , which are independent variables.",
"Such $(\\theta _i,\\phi _i)$ can thus be optimized locally by the CG method to minimize the force on each spin.",
"The energy convergence criteria is set to be 1E-6 eV.",
"The combination of PTMC simulations and the CG method guarantees that our reported magnetic configurations are all located at global/local energy minima.",
"Calculation of topological charge $Q$ .",
"Although Eq.",
"(3) is commonly used to define topological charge $Q$ , it is not convenient to employ it to calculate $Q$ for discrete lattice of spins $\\mathbf {n}_i$ $(n_{i,x},n_{i,y},n_{i,z})$ .",
"Consequently, we alternatively adopt the definition of Berg and L$\\rm \\ddot{u}$ stcher [35], $Q=\\frac{1}{4\\pi } \\sum _l A_l$ ${\\rm cos}(\\frac{A_l}{2})=\\frac{1 + \\mathbf {n}_i\\cdot \\mathbf {n}_j + \\mathbf {n}_j\\cdot \\mathbf {n}_k + \\mathbf {n}_k\\cdot \\mathbf {n}_i}{\\sqrt{2(1+\\mathbf {n}_i\\cdot \\mathbf {n}_j)(1+\\mathbf {n}_j\\cdot \\mathbf {n}_k)(1+\\mathbf {n}_k\\cdot \\mathbf {n}_i)}}$ ${\\rm sign}(A_l)={\\rm sign}[\\mathbf {n}_i\\cdot (\\mathbf {n}_j\\times \\mathbf {n}_k)]$ where $l$ runs over all elementary triangles made of neighboring spin sites and $A_l$ is the solid angle formed by the three spin vectors $\\mathbf {n}_i$ , $\\mathbf {n}_j$ and $\\mathbf {n}_k$ of the $l$ th triangle.",
"Note that the triangles should cover all lattice area with no overlap and $\\mathbf {n}_i$ , $\\mathbf {n}_j$ and $\\mathbf {n}_k$ should be anticlockwisely ordered.",
"See details for exceptional cases in Refs.",
"[35], [36].",
"All data generated or analysed during this study are included in this published article (and its supplementary information files).",
"This work is supported by the Office of Basic Energy Sciences under contract ER-46612.",
"H.X.",
"is supported by NSFC (11374056), the Special Funds for Major State Basic Research (2015CB921700), Program for Professor of Special Appointment (Eastern Scholar), Qing Nian Ba Jian Program, and Fok Ying Tung Education Foundation.",
"J.F.",
"acknowledges the support from Anhui Provincial Natural Science Foundation (1908085MA10).",
"The Arkansas High Performance Computing Center (AHPCC) is also acknowledged.",
"C.X.",
"and J.F.",
"contribute equally to this work.",
"The idea of this paper is conceived during the discussion of all authors.",
"C.X.",
"performed the calculations.",
"H.X.",
"and L.B.",
"supervised this work.",
"C.X.",
"wrote the first draft of the manuscript.",
"All authors contributed to the discussion of the results and comments on the first draft of the manuscript.",
"See supplementary information for details about our predictions.",
"The authors declare that they have no competing financial interests.",
"Correspondence should be addressed to H.X.",
"(email: [email protected]) or L.B.",
"(email: [email protected])."
]
] | 1906.04336 |
[
[
"Linear recurrences indexed by $\\mathbb{Z}$"
],
[
"Abstract This note considers linear recurrences (also called linear difference equations) in unknowns indexed by the integers.",
"We characterize a unique \\emph{reduced} linear recurrence with the same solutions as a given linear recurrence, and construct a \\emph{solution matrix} which parametrizes the space of solutions.",
"Several properties of solution matrices are shown, including a combinatorial characterization of bases and dimension of the space of solutions."
],
[
"The problem",
"We consider the following version of a linear recurrenceSometimes called a `system of linear recurrence relations' or a `system of linear difference equations'.",
": a system of linear equations in the sequence of variables $...,x_{-1},x_0,x_1,x_2,...$ which equate each variable to a linear combination of the preceding variables.",
"This includes the (bi-infinite) Fibonacci recurrence: $x_i = x_{i-1}+x_{i-2},\\;\\;\\; \\forall i\\in \\mathbb {Z}$ but it also includes linear recurrences where the equations (and their length) may vary: $\\begin{array}{ll}x_i = x_{i-1} - x_{i-2} & \\forall i\\in 2 \\mathbb {Z} \\\\x_i = 2x_{i-1} - x_{i-2} + x_{i-4} & \\forall i\\in 2 \\mathbb {Z}+1\\end{array}$ The coefficients of the equations are taken from an arbitrary field $\\mathsf {k}$ , which we fix throughout.",
"A solution of a linear recurrence is a sequence of numbers in $\\mathsf {k}$ satisfying the equations.",
"For example, the linear recurrences (REF ) and (REF ) are respectively solved by the sequences: $...,-21,13,-8,5,-3, 2, -1, 1, &0, 1, 1, 2,3,5,8,13,21,... \\\\..., 1, 0, -1, 0, 1, 2, 1, 0, -1, 0, &1, 2, 1, 0, -1, 0, 1, 2,...$ The motivating problem of this note is to construct all solutions to a given linear recurrence.",
"We also consider affine recurrences, in which the equations may have a constant term.",
"The solutions to these systems are constructed in Section REF .",
"Sections 2-4 summarize the results, and Sections 5-8 contain the proofs and details."
],
[
"Simplifying the problem",
"We first reformulate the problem using ideas from finite dimensional linear algebra."
],
[
"Recurrence matrices",
"Like finite systems of linear equations, linear recurrences can be reformulated as matrix equations.",
"By moving the variables to the left-hand side of each equation and collecting the coefficients into a $\\mathbb {Z}\\times \\mathbb {Z}$ -matrix $\\mathsf {C}$ , we may rewrite the system of linear equations as $ \\mathsf {C}\\mathsf {x} =\\mathsf {0} $ where $\\mathsf {x}$ is a $\\mathbb {Z}$ -vector of variables.",
"For example, the Fibonacci recurrence may be rewritten as: $\\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][matrix of math nodes,nodes in empty cells,inner sep=0pt,nodes={anchor=center},column sep={.65cm,between origins},row sep={.65cm,between origins},left delimiter={[},right delimiter={]},] (M) at (0,0) {\\rotatebox {-45}{\\cdots }\\& \\& \\& \\& \\& \\& \\\\\\& 1 \\& 0 \\& 0 \\& 0 \\& 0 \\& \\\\\\& -1 \\& 1 \\& 0 \\& 0 \\& 0 \\& \\\\\\& -1 \\& -1 \\& 1 \\& 0 \\& 0 \\& \\\\\\& 0 \\& -1 \\& -1 \\& 1 \\& 0 \\& \\\\\\& 0 \\& 0 \\& -1 \\& -1 \\& 1 \\& \\\\\\& \\& \\& \\& \\& \\& \\rotatebox {-45}{\\cdots }\\\\};\\end{tikzpicture}\\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][matrix of math nodes,nodes in empty cells,inner sep=0pt,nodes={anchor=center},column sep={.65cm,between origins},row sep={.65cm,between origins},left delimiter={[},right delimiter={]},] (M) at (0,0) {\\rotatebox {90}{\\cdots }\\\\x_{-2} \\\\x_{-1} \\\\x_0 \\\\x_1 \\\\x_2 \\\\\\rotatebox {90}{\\cdots }\\\\};\\end{tikzpicture}=\\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][matrix of math nodes,nodes in empty cells,inner sep=0pt,nodes={anchor=center},column sep={.65cm,between origins},row sep={.65cm,between origins},left delimiter={[},right delimiter={]},] (M) at (0,0) {\\rotatebox {90}{\\cdots }\\\\0 \\\\0 \\\\0 \\\\0 \\\\0 \\\\\\rotatebox {90}{\\cdots }\\\\};\\end{tikzpicture}$ The $\\mathbb {Z}\\times \\mathbb {Z}$ -matrices $\\mathsf {C}$ which arise this way are precisely those which are: lower unitriangular; that is, $\\mathsf {C}_{a,b}=0$ if $a<b$ and $\\mathsf {C}_{a,a}=1$ for all $a$ , and horizontally bounded; that is, $\\mathsf {C}_{a,b}=0$ if $b\\ll a$ or $b \\gg a$ for all $a$ .The bounds need not be uniform; i.e.",
"they may depend on $a$ .",
"These are also called row finite matrices.",
"We define a recurrence matrix to be a $\\mathbb {Z}\\times \\mathbb {Z}$ -matrix with these two properties.",
"Hence, the original problem is equivalent to describing the kernel of a given recurrence matrix.",
"As can be seen above, recurrence matrices make inefficient use of space: their non-zero coefficients are concentrated below and sufficiently near the main diagonal.",
"To remedy this, we adopt the atypical convention of rotating $\\mathbb {Z}\\times \\mathbb {Z}$ -matrices $45^\\circ $ counterclockwise.This alignment is also chosen to match the existing literature on friezes; see Section REF .",
"So, a lower unitriangular matrix $\\mathsf {C}$ would be formatted as follows: $\\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][use as bounding box] (-5.25-.5,.5) rectangle (6+.5,-4);\\begin{scope}(-5.25-.5,.5) rectangle (6+.5,-3-.5);[matrix of math nodes,matrix anchor = M-4-8.center,throw/.style={},origin/.style={dark green,draw,circle,inner sep=0.25mm,minimum size=2mm},pivot/.style={draw,circle,inner sep=0.25mm,minimum size=2mm}, nodes in empty cells,inner sep=0pt,nodes={anchor=center,rotate=45},column sep={.75cm,between origins},row sep={.75cm,between origins},] (M) at (0,0) {\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& |[throw]| 1 \\& \\& |[throw]| 1 \\& \\& |[throw]| 1 \\& \\& |[throw]| 1 \\& \\&|[throw]| 1 \\& \\& |[throw]| 1 \\& \\& |[throw]| 1 \\& \\& \\cdots \\\\\\cdots \\& \\& \\mathsf {C}_{1,0} \\& \\& \\mathsf {C}_{2,1} \\& \\& \\mathsf {C}_{3,2} \\& \\&\\mathsf {C}_{4,3} \\& \\&\\mathsf {C}_{5,4} \\& \\&\\mathsf {C}_{6,5} \\& \\&\\mathsf {C}_{7,6} \\& \\\\\\&\\mathsf {C}_{1,-1} \\& \\&\\mathsf {C}_{2,0} \\& \\&\\mathsf {C}_{3,1} \\& \\&\\mathsf {C}_{4,2} \\& \\&\\mathsf {C}_{5,3} \\& \\&\\mathsf {C}_{6,4} \\& \\&\\mathsf {C}_{7,5} \\& \\& \\cdots \\\\\\cdots \\& \\&\\mathsf {C}_{2,-1} \\& \\&\\mathsf {C}_{3,0} \\& \\&\\mathsf {C}_{4,1} \\& \\&\\mathsf {C}_{5,2} \\& \\&\\mathsf {C}_{6,3} \\& \\&\\mathsf {C}_{7,4} \\& \\&\\mathsf {C}_{8,5} \\& \\\\\\& \\rotatebox {90}{\\cdots }\\& \\& \\rotatebox {90}{\\cdots }\\& \\& \\rotatebox {90}{\\cdots }\\& \\& \\rotatebox {90}{\\cdots }\\& \\& \\rotatebox {90}{\\cdots }\\& \\& \\rotatebox {90}{\\cdots }\\& \\& \\rotatebox {90}{\\cdots }\\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\};\\end{scope}[dark red, fill= dark red!50,opacity=.25,rounded corners] (M-1-2.center) -- (M-9-10.center) -- (M-9-12.center) -- (M-1-4.center) -- cycle;[dark blue, fill= dark blue!50,opacity=.25,rounded corners] (M-1-14.center) -- (M-9-6.center) -- (M-9-4.center) -- (M-1-12.center) -- cycle;[dark green, fill= dark green!50,opacity=.25,rounded corners] (-8,.35) rectangle (8.75,-.35);\\end{tikzpicture}\\node [dark red] at (M-9-11) {\\scriptsize 2nd column};\\node [dark blue] at (M-9-5) {\\scriptsize 4th row};\\node [dark green,right] at (6.5,0) {\\scriptsize Main diagonal};$ $Any omitted entries (including those above the main diagonal) are interpreted as $ $.$ The Fibonacci recurrence (REF ) corresponds to the recurrence matrix: $\\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][matrix of math nodes,matrix anchor = M-4-8.center,throw/.style={},origin/.style={},nodes in empty cells,inner sep=0pt,nodes={anchor=center,rotate=45},column sep={.5cm,between origins},row sep={.5cm,between origins},] (M) at (0,0) {\\& |[throw]| 1 \\& \\& |[throw]| 1 \\& \\& |[throw]| 1 \\& \\& |[origin]| 1 \\& \\&|[throw]| 1 \\& \\& |[throw]| 1 \\& \\& |[throw]| 1 \\& \\& \\cdots \\\\\\cdots \\& \\& -1 \\& \\& -1 \\& \\& -1 \\& \\& -1 \\& \\& -1 \\& \\& -1 \\& \\& -1 \\& \\\\\\& -1 \\& \\& -1 \\& \\& -1 \\& \\& -1 \\& \\& -1 \\& \\& -1 \\& \\& -1 \\& \\& \\cdots \\\\};\\end{tikzpicture}$ The linear recurrence (REF ) corresponds to the recurrence matrix: $\\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][matrix of math nodes,matrix anchor = M-4-8.center,throw/.style={},origin/.style={},faded/.style={black!25},nodes in empty cells,inner sep=0pt,nodes={anchor=center,rotate=45},column sep={.5cm,between origins},row sep={.5cm,between origins},] (M) at (0,0) {\\& |[throw]| 1 \\& \\& |[throw]| 1 \\& \\& |[throw]| 1 \\& \\& |[origin]| 1 \\& \\&|[throw]| 1 \\& \\& |[throw]| 1 \\& \\& |[throw]| 1 \\& \\& \\cdots \\\\\\cdots \\& \\& -1 \\& \\& -2 \\& \\& -1 \\& \\& -2 \\& \\& -1 \\& \\& -2 \\& \\& -1 \\& \\\\\\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& \\cdots \\\\\\cdots \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\\\\\& -1 \\& \\& |[faded]| 0 \\& \\& -1 \\& \\& |[faded]| 0 \\& \\& -1 \\& \\& |[faded]| 0 \\& \\& -1 \\& \\& \\cdots \\\\};\\end{tikzpicture}$ The zeroes in grey could have been suppressed.",
"[.05cm] The reader should keep the following two pathologies in mind when working with infinite matrices and vectors.",
"The product of an arbitrary $\\mathbb {Z}\\times \\mathbb {Z}$ -matrix with a $\\mathbb {Z}$ -vector or another $\\mathbb {Z}\\times \\mathbb {Z}$ -matrix may involve summing an infinite number of non-zero terms in $\\mathsf {k}$ .",
"When this occurs, we simply say the product does not exist.While one might allow infinite sums that absolutely converge in some topology on $\\mathsf {k}$ , we won't consider this.",
"Even when all constituent products exist, multiplication may not be associative.",
"The corresponding entries of $(\\mathsf {AB})\\mathsf {C}$ and $\\mathsf {A}(\\mathsf {BC})$ are defined by double infinite sums which coincide except for the order of summation, which cannot be exchanged in general.",
"Mercifully, the product $\\mathsf {AC}$ exists and the equality $(\\mathsf {AB})\\mathsf {C}=\\mathsf {A}(\\mathsf {BC})$ holds whenever (a) $\\mathsf {A}$ , $\\mathsf {B}$ , and $\\mathsf {C}$ are lower unitriangular, or (b) $\\mathsf {A}$ and $\\mathsf {B}$ are horizontally bounded and $\\mathsf {C}$ is any matrix or vector.",
"We will make extensive use of both of these cases.",
"Reduced recurrence matrices As we are primarily interested in the kernel of a recurrence matrix, let us say... ...a recurrence matrix is trivial if its kernel is trivial (i.e.",
"only the zero vector), and... ...two recurrence matrices are equivalent if their kernels are equal.",
"For example, a recurrence matrix is trivial if and only if it is equivalent to the $\\mathbb {Z}\\times \\mathbb {Z}$ identity matrix, whose kernel is clearly $\\lbrace 0\\rbrace $ .",
"This example can be extended to the following theorem.All theorems stated in Sections 2-4 are proven in a later section; e.g.",
"Theorem REF is proven in Section 5. thm: triv-equivTheorem REF Two recurrence matrices $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ are equivalent if and only $\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for some trivial recurrence matrix $\\mathsf {D}$ .",
"We would like to characterize a representative of each equivalence class which is `minimal' in some sense.",
"In finite linear algebra, this is accomplished by row reducing a matrix, but translating row reduction into our current setting runs into several issues.",
"Only one type of row operation can send recurrence matrices to recurrence matrices; specifically, adding a multiple of one row to a lower row.",
"There is no `upper left corner' in which to start a deterministic algorithm.",
"Sequences of row operations need not terminate, and two infinite sequences of row operations may have different limits (in an appropriate topology, see Section REF ).",
"Nevertheless, we may characterize those recurrence matrices which cannot be row reduced any further.",
"A recurrence matrix is reduced if the first non-zero entry in each row (called the pivot entry in that row) is the last non-zero entry in its column.",
"The following recurrence matrix is reduced.",
"$\\begin{tikzpicture}[nodes={anchor=center},ascending/.style={blue,opacity=.5},descending/.style={red,opacity=.5},][use as bounding box] (-6.30,-1.8) rectangle (5.80,.35);[matrix of math nodes,throw/.style={},pivot/.style={blue,draw,circle,inner sep=0mm,minimum size=5mm},matrix anchor = M-1-9.west,nodes in empty cells,nodes={anchor=center,rotate=45},column sep={.5cm,between origins},row sep={.5cm,between origins},] (M) at (0,0) {& |[throw]| 1 & & |[throw]| 1 & & |[throw]| 1 & & |[pivot]| 1 & &|[throw]| 1 & & |[throw]| 1 & & |[throw]| 1 & & \\cdots \\\\\\cdots & & 1 & & 1 & & & & & & |[pivot]| 1 & & |[pivot]| 1 & & |[pivot]| 1 & \\\\& |[pivot]| 1 & & |[pivot]| 1 & & & & |[pivot]| 1 & & & & & & & & \\cdots \\\\\\cdots & & & & & & & & & & & & & & & \\\\};[blue,dashed,->] (M-3-2) to (M-4-3.center);[blue,dashed,->] (M-3-4) to (M-4-5.center);[blue,dashed,->] (M-3-8) to (M-4-9.center);[blue,dashed,->] (M-1-8) to (M-4-11.center);[blue,dashed,->] (M-2-11) to (M-4-13.center);[blue,dashed,->] (M-2-13) to (M-4-15.center);\\end{tikzpicture}$ Below each pivot (in blueblue circles) there are only zeroes (along the dashed arrows).",
"Out first fundamental result is the following.",
"thm: reduced1Theorem REF Every recurrence matrix is equivalent to a unique reduced recurrence matrix.",
"We provide two independent proofs of this fact.",
"The proof in Section uses Zorn's Lemma to show that the the reduced matrix the limit of all `generalized row reductions'.",
"The proof in Section (Theorem REF ) constructs the reduced matrix directly from the space of solutions.",
"Constructing the solutions In this section, we construct all elements of the kernel of a reduced recurrence matrix, and thus all solutions to a reduced linear recurrence.",
"For $a\\le b\\in \\mathbb {Z}$ , let $[a,b]\\lbrace a,a+1,...,b\\rbrace \\subset \\mathbb {Z}$ .",
"For a $\\mathbb {Z}\\times \\mathbb {Z}$ -matrix $\\mathsf {C}$ and two sets $I,J\\subset \\mathbb {Z}$ , let $\\mathsf {C}_{I,J}$ denote the submatrix on row set $I$ and column set $J$ .",
"Adjugates Given a lower unitriangular $\\mathbb {Z}\\times \\mathbb {Z}$ -matrix $\\mathsf {C}$ , define the adjugate $\\mathsf {Adj}(\\mathsf {C})$ of $\\mathsf {C}$ to be the lower unitriangular $\\mathbb {Z}\\times \\mathbb {Z}$ -matrix whose subdiagonal entries are defined by $ \\mathsf {Adj}(\\mathsf {C})_{a,b} (-1)^{a+b}\\mathrm {det}(\\mathsf {C}_{[a+1,b],[a,b-1]}) $ Three examples of adjugates are given in Figure REF .",
"The determinant of the finite matrix $\\mathsf {C}_{[a+1,b],[a,b-1]}$ coincides with the determinantWhile general $\\mathbb {Z}\\times \\mathbb {Z}$ -matrices may not have well-defined determinants, the cofactor matrices of lower unitriangular matrices are lower unitriangular outside of a finite square, and thus have a well-defined determinant.",
"of the infinite cofactor matrix $\\mathsf {C}_{\\mathbb {Z}\\setminus \\lbrace a\\rbrace ,\\mathbb {Z}\\setminus \\lbrace b\\rbrace }$ , justifying the name `adjugate'.",
"Like its finite matrix counterpart, the adjugate matrix has many useful properties.",
"Proposition 3.3 Let $\\mathsf {C}$ and $\\mathsf {D}$ be lower unitriangular $\\mathbb {Z}\\times \\mathbb {Z}$ -matrices.",
"$\\mathsf {Adj}(\\mathsf {C})\\mathsf {C}=\\mathsf {C}\\mathsf {Adj}(\\mathsf {C}) = \\mathrm {Id}$ .",
"$\\mathsf {Adj}(\\mathsf {Adj}(\\mathsf {C}))=\\mathsf {C}$ .",
"$\\mathsf {Adj}(\\mathsf {C}\\mathsf {D}) = \\mathsf {Adj}(\\mathsf {D})\\mathsf {Adj}(\\mathsf {C})$ .",
"For $I,J\\subset [a,b]$ with $|I|=|J|$ , $\\det (\\mathsf {Adj}(\\mathsf {C})_{I,J}) = (-1)^{\\sum I+\\sum J}\\det ({\\mathsf {C}}_{[a,b]\\setminus J,[a,b]\\setminus I}) $ .",
"Since each entry of $\\mathsf {Adj}(\\mathsf {C})$ only depends on a finite submatrix of $\\mathsf {C}$ , these follow from their finite matrix counterparts.",
"E.g.",
"(4) follows by considering the submatrix $\\mathsf {C}_{[a,b],[a,b]}$ .",
"The first three parts of the prior proposition can be rephrased as follows.",
"Proposition 3.4 The lower unitriangular $\\mathbb {Z}\\times \\mathbb {Z}$ -matrices form a group with inverse $\\mathsf {Adj}$ .",
"[.05cm] Since multiplication is not always associative, inverses may not be unique in the larger set of $\\mathbb {Z}\\times \\mathbb {Z}$ -matrices (see Remark REF ), and so we write $\\mathsf {Adj}(\\mathsf {C})$ instead of $\\mathsf {C}^{-1}$ .",
"The recurrence matrices do not form a subgroup of the lower unitriangular matrices, as they are not closed under $\\mathsf {Adj}$ .",
"In fact, the recurrence matrices whose adjugate is also a recurrence matrix are precisely the trivial ones.",
"thm: triv-invTheorem REF A recurrence matrix $\\mathsf {C}$ is trivial if and only if $\\mathsf {Adj}(\\mathsf {C})$ is a recurrence matrix.",
"Since $\\mathsf {Adj}(\\mathsf {C})$ is lower unitriangular by construction, $\\mathsf {C}$ is trivial if and only if $\\mathsf {Adj}(\\mathsf {C})$ is horizontally bounded.",
"The recurrence matrices form a semigroup whose subgroup of invertible elements is the set of trivial recurrence matrices (by Theorem REF ).",
"The left orbits of this subgroup on the set of recurrence matrices are the equivalence classes (by Theorem REF ), and the reduced recurrence matrices are a transverse of these orbits (by Theorem REF ).",
"Figure: Examples of adjugate and solution matrices The solution matrix In this section, we construct a matrix $\\mathsf {Sol}(\\mathsf {C})$ whose image is the kernel of a given reduced recurrence matrix $\\mathsf {C}$ .",
"The shape of a recurrence matrix $\\mathsf {C}$ is the non-increasing function $S:\\mathbb {Z}\\rightarrow \\mathbb {Z}$ defined by $ S(a) := \\min \\lbrace b \\in \\mathbb {Z}\\mid \\mathsf {C}_{a,b}\\ne 0\\rbrace $ The pivot entry in the $a$ th row is then $\\mathsf {C}_{a,S(a)}$ , and so $\\mathsf {C}$ is reduced if and only if $\\mathsf {C}_{b,S(a)}=0$ for all $b>a$ .",
"In particular, the shape of a reduced recurrence matrix must be injective.",
"The shape of the Fibonacci recurrence matrix is $S(a)=a-2$ .",
"The shape of the recurrence matrix depicted in Example REF is $ S(a) = \\left\\lbrace \\begin{array}{cc}a-2 & \\text{if $a<0$} \\\\a & \\text{if $a=0$} \\\\a-2 & \\text{if $a=1$} \\\\a-1 & \\text{if $a>1$}\\end{array}\\right\\rbrace $ Given a recurrence matrix $\\mathsf {C}$ with shape $S$ , define a $\\mathbb {Z}\\times \\mathbb {Z}$ -matrix $\\mathsf {P}$ by $ \\mathsf {P}_{a,b} \\left\\lbrace \\begin{array}{cc}(\\mathsf {C}_{b,S(b)})^{-1} & \\text{if }a=S(b) \\\\0 & \\text{otherwise}\\end{array}\\right\\rbrace $ This is the generalized permutation matrix such that $\\mathsf {CP}$ moves the pivots to the main diagonal and rescales them to 1.",
"Thus, $\\mathsf {C}$ is reduced if and only if $\\mathsf {C}\\mathsf {P}$ is upper unitriangular.A $\\mathbb {Z}\\times \\mathbb {Z}$ -matrix $\\mathsf {C}$ is upper unitriangular if its transpose $\\mathsf {C}^\\top $ is lower unitriangular.",
"Such matrices also have an adjugate, which may be defined as $\\mathsf {Adj}(\\mathsf {C}^\\top )^\\top $ , for which the analog of Proposition REF holds.",
"Given a reduced linear recurrence $\\mathsf {C}$ , define the solution matrix $\\mathsf {Sol}(\\mathsf {C})$ of $\\mathsf {C}$ by $ \\mathsf {Sol}(\\mathsf {C}) \\mathsf {Adj}(\\mathsf {C}) - \\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P}) $ Examples are given in Figure REF .",
"The solution matrix is named for the following property.",
"Proposition 3.8 Each column of $\\mathsf {Sol}(\\mathsf {C})$ is a solution to $\\mathsf {C}\\mathsf {x}=\\mathsf {0}$ .",
"$\\displaystyle \\mathsf {C}\\mathsf {Sol}(\\mathsf {C}) = \\mathsf {C}[\\mathsf {Adj}(\\mathsf {C}) - \\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})]= \\mathsf {C}\\mathsf {Adj}(\\mathsf {C}) - (\\mathsf {C}\\mathsf {P})\\mathsf {Adj}(\\mathsf {C}\\mathsf {P}) = \\mathrm {Id} - \\mathrm {Id} = \\mathsf {0}$ .",
"If $\\mathsf {C}$ is the Fibonacci recurrence matrix, the $a$ th column of $\\mathsf {Sol}(\\mathsf {C})$ is given by $x_b = \\left\\lbrace \\begin{array}{cc}F_{b-a+1} & \\text{if $b-a+1\\ge 0$} \\\\(-1)^{a-b}F_{-b+a-1} & \\text{if $b-a+1<0$}\\end{array}\\right\\rbrace $ where $F_i$ is the $i$ th Fibonacci number (indexed so that $F_0=0$ and $F_1=1$ ).",
"This proposition can be extended to a complete characterization of solutions to $\\mathsf {C}\\mathsf {x}=\\mathsf {0}$ , and thus an answer to our original problem.",
"Given a $\\mathbb {Z}\\times \\mathbb {Z}$ matrix $\\mathsf {A}$ , define the image of $\\mathsf {A}$ to be the set of all sequences $\\mathsf {v}\\in \\mathsf {k}^\\mathbb {Z}$ equal to $\\mathsf {Aw}$ for some $\\mathsf {w\\in \\mathsf {k}^\\mathbb {Z}}$ .This is the usual definition of `image', with the explicit caveat that $\\mathsf {Aw}$ may not exist for all $\\mathsf {w}$ .",
"thm: solTheorem REF If $\\mathsf {C}$ is reduced, the kernel of $\\mathsf {C}$ equals the image of $\\mathsf {Sol}(\\mathsf {C})$ .",
"Any solution to the Fibonacci linear recurrence can be written as a linear combination of the solutions of the form (REF ).In fact, it suffices to only use two adjacent columns of the solution matrix here; see Example REF .",
"If $\\mathsf {C}$ is not reduced, the adjugate $\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})$ may not be defined, and so $\\mathsf {Sol}(\\mathsf {C})$ may not be constructed.",
"Balls and juggling We now connect the shape of a reduced recurrence matrix to the geometry of its kernel.",
"Fix a reduced recurrence matrix $\\mathsf {C}$ of shape $S$ for the remainder of the section.",
"An $S$ -ball is an equivalence class of integers under the equivalence relation $a\\sim S(a)$ , which does not contain a single element.",
"The significance of this notion is the following.",
"thm: ballsTheorem REF The dimension of $\\mathrm {ker}(\\mathsf {C})$ is equal to the number of $S$ -balls.",
"For the Fibonacci recurrence matrix, the shape $S$ is given by $S(a)=a-2$ .",
"There are two $S$ -balls: the set of even numbers, and the set of odd numbers; and the space of solutions to the Fibonacci recurrence is two dimensional (see Example REF ).",
"The recurrence matrix in Example REF has two $S$ -balls: $ \\lbrace ...,-6,-4,-2\\rbrace \\text{ and } \\lbrace ...,-5,-3,-1,1,2,3,...\\rbrace $ Since $S(0)=0$ , the singleton set $\\lbrace 0\\rbrace $ is also an equivalence class, but it is not an $S$ -ball.",
"As a non-increasing injection from $\\mathbb {Z}\\rightarrow \\mathbb {Z}$ , the shape $S$ can be thought of as a juggling pattern: instructions for how a juggler catches and throws balls over time.",
"At each moment $a$ , the juggler catches the ball they threw at moment $S(a)$ and immediately throws it again...unless $S(a)=a$ , in which case they neither catch nor throw a ball.At any moment $a$ which is not in the image of $S$ , the juggler throws the ball so high it never returns.",
"The ability to throw at escape velocity is a small stretch of the imagination for a juggler who is also immortal.",
"Each $S$ -ball lists those moments when a given ball is caught and thrown, and so the number of $S$ -balls equals the number of physical balls the juggler needs for the pattern.",
"The $S$ -balls of $\\mathsf {C}$ can be visualized as follows.",
"Circle each pivot entry and each entry on the main diagonal, and connect pairs of circles in the same column or in the same row.",
"Excluding any unconnected circles, the connected components of the resulting graph are the $S$ -balls.",
"There are four $S$ -balls below, each drawn in a different color.",
"$\\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][use as bounding box] (-9.5,-2.7) rectangle (6.3,0.3);[matrix of math nodes,matrix anchor = M-2-24.center,nodes in empty cells,inner sep=0pt,gthrow/.style={dark green,draw,circle,inner sep=0mm,minimum size=5mm},rthrow/.style={dark red,draw,circle,inner sep=0mm,minimum size=5mm},bthrow/.style={dark blue,draw,circle,inner sep=0mm,minimum size=5mm},pthrow/.style={dark purple,draw,circle,inner sep=0mm,minimum size=5mm},nodes={anchor=center,node font=\\scriptsize ,rotate=45},column sep={0.4cm,between origins},row sep={0.4cm,between origins},] (M) at (0,0) {\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& |[gthrow]| 1 \\& \\& |[rthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\& |[pthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[rthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[pthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\& |[rthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[pthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[rthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\\\\\cdots \\& \\& 5 \\& \\& 1 \\& \\& |[bthrow]| 1 \\& \\& 3 \\& \\& 2 \\& \\& 1 \\& \\& 5 \\& \\& 1 \\& \\& 5 \\& \\& 1 \\& \\& |[bthrow]| 1 \\& \\& 3 \\& \\& 2 \\& \\& 1 \\& \\& 5 \\& \\& 1 \\& \\& 5 \\& \\& 1 \\& \\& \\cdots \\\\\\& 3 \\& \\& 2 \\& \\& \\& \\& \\& \\& 5 \\& \\& 1 \\& \\& 3 \\& \\& 2 \\& \\& 3 \\& \\& 2 \\& \\& \\& \\& \\& \\& 5 \\& \\& 1 \\& \\& 3 \\& \\& 2 \\& \\& 3 \\& \\& 2 \\& \\& \\& \\\\\\cdots \\& \\& |[bthrow]| 1 \\& \\& \\& \\& -2 \\& \\& \\& \\& 2 \\& \\& 1 \\& \\& |[gthrow]| 1 \\& \\& 1 \\& \\& |[bthrow]| 1 \\& \\& \\& \\& -2 \\& \\& \\& \\& 2 \\& \\& 1 \\& \\& |[gthrow]| 1 \\& \\& 1 \\& \\& |[bthrow]| 1 \\& \\& \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& -1 \\& \\& -3 \\& \\& \\& \\& |[bthrow]| 1 \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& -1 \\& \\& -3 \\& \\& \\& \\& |[bthrow]| 1 \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& -1 \\& \\\\\\& \\& \\& \\& \\& \\& |[gthrow]| -1 \\& \\& |[rthrow]| -1 \\& \\& \\& \\& \\& \\& |[pthrow]| -1 \\& \\& \\& \\& \\& \\& \\& \\& |[gthrow]| -1 \\& \\& |[pthrow]| -1 \\& \\& \\& \\& \\& \\& |[rthrow]| -1 \\& \\& \\& \\& \\& \\& \\& \\& \\cdots \\\\\\& \\& \\& |[pthrow]| 1 \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[rthrow]| 1 \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[pthrow]| 1 \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\};\\end{tikzpicture}[dark green] (M-3-1) to (M-2-2) to (M-7-7) to (M-2-12) to (M-5-15) to (M-2-18) to (M-7-23) to (M-2-28) to (M-5-31) to (M-2-34) to (M-7-39);[dark red] (M-5-1) to (M-2-4) to (M-7-9) to (M-2-14) to (M-8-20) to (M-2-26) to (M-7-31) to (M-2-36) to (M-5-39);[dark blue] (M-3-1) to (M-5-3) to (M-2-6) to (M-3-7) to (M-2-8) to (M-6-12) to (M-2-16) to (M-5-19) to (M-2-22) to (M-3-23) to (M-2-24) to (M-6-28) to (M-2-32) to (M-5-35) to (M-2-38) to (M-3-39);[dark purple] (M-5-1) to (M-8-4) to (M-2-10) to (M-7-15) to (M-2-20) to (M-7-25) to (M-2-30) to (M-8-36) to (M-5-39);$ $Note that reducedness implies that non-zero entries of $ C$ only occur in circles and where two lines cross (See Remark \\ref {rem: soljugs} for a contrasting property of the solution matrix $ Sol(C)$).$ The connection to juggling is justified by considering the transpose of the above picture.",
"$\\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][use as bounding box] (-9.5,2.7) rectangle (6.3,-0.3);[matrix of math nodes,matrix anchor = M-2-24.center,nodes in empty cells,inner sep=0pt,gthrow/.style={dark green,draw,circle,inner sep=0mm,minimum size=5mm},rthrow/.style={dark red,draw,circle,inner sep=0mm,minimum size=5mm},bthrow/.style={dark blue,draw,circle,inner sep=0mm,minimum size=5mm},pthrow/.style={dark purple,draw,circle,inner sep=0mm,minimum size=5mm},nodes={anchor=center,node font=\\scriptsize ,rotate=45},column sep={0.4cm,between origins},row sep={-0.4cm,between origins},] (M) at (0,0) {\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& |[gthrow]| \\& \\& |[rthrow]| \\& \\& |[bthrow]| \\& \\& |[bthrow]| \\& \\& |[pthrow]| \\& \\& |[gthrow]| \\& \\& |[rthrow]| \\& \\& |[bthrow]| \\& \\& |[gthrow]| \\& \\& |[pthrow]| \\& \\& |[bthrow]| \\& \\& |[bthrow]| \\& \\& |[rthrow]| \\& \\& |[gthrow]| \\& \\& |[pthrow]| \\& \\& |[bthrow]| \\& \\& |[gthrow]| \\& \\& |[rthrow]| \\& \\& |[bthrow]| \\& \\\\\\cdots \\& \\& \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\cdots \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[gthrow]| \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[gthrow]| \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& |[gthrow]| \\& \\& |[rthrow]| \\& \\& \\& \\& \\& \\& |[pthrow]| \\& \\& \\& \\& \\& \\& \\& \\& |[gthrow]| \\& \\& |[pthrow]| \\& \\& \\& \\& \\& \\& |[rthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\cdots \\\\\\& \\& \\& |[pthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[rthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[pthrow]| \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\};\\end{tikzpicture}[dark green] (M-3-1) to (M-2-2) to (M-7-7) to (M-2-12) to (M-5-15) to (M-2-18) to (M-7-23) to (M-2-28) to (M-5-31) to (M-2-34) to (M-7-39);[dark red] (M-5-1) to (M-2-4) to (M-7-9) to (M-2-14) to (M-8-20) to (M-2-26) to (M-7-31) to (M-2-36) to (M-5-39);[dark blue] (M-3-1) to (M-5-3) to (M-2-6) to (M-3-7) to (M-2-8) to (M-6-12) to (M-2-16) to (M-5-19) to (M-2-22) to (M-3-23) to (M-2-24) to (M-6-28) to (M-2-32) to (M-5-35) to (M-2-38) to (M-3-39);[dark purple] (M-5-1) to (M-8-4) to (M-2-10) to (M-7-15) to (M-2-20) to (M-7-25) to (M-2-30) to (M-8-36) to (M-5-39);$ $This can be viewed as an idealized plot of the height of the `balls^{\\prime } over time.$ Balls and juggling patterns can also be used to parametrize the space of solutions to a linear recurrence.",
"Given a shape $S$ and an integer $b$ , let $T_b$ consist of the largest element in each $S$ -ball less than or equal to $b$ ; that is, $ T_b := \\lbrace a \\text{ such that $a\\le b$ but there is no $c\\le b$ with $S(c)=a$} \\rbrace $ In juggling terms, the set $T_b$ considers the balls in the air just after moment $b$ and records when each ball was thrown.",
"thm: solextendTheorem REF Restricting to the entries indexed by $T_b$ gives an isomorphism $\\mathrm {ker}(\\mathsf {C})\\rightarrow \\mathsf {k}^{T_b}$ .",
"That is, any choice of values for the variables $x_a$ for all $a\\in T_a$ can be uniquely extended to a solution to $\\mathsf {C}\\mathsf {x=0}$ .",
"We may also use $T_b$ to parametrize the space of solutions.",
"Let $\\mathsf {Sol}(\\mathsf {C})_{\\mathbb {Z},T_b}$ denote the $\\mathbb {Z}\\times T_b$ -matrix consisting of the columns of $\\mathsf {Sol}(\\mathsf {C})$ indexed by $T_b$ .",
"thm: solbasisTheorem REF Multiplication by $\\mathsf {Sol}(\\mathsf {C})_{\\mathbb {Z}\\times T_b}$ gives an isomorphism $\\mathsf {k}^{T_b}\\rightarrow \\mathrm {ker}(\\mathsf {C})$ .",
"Equivalently, every solution to $\\mathsf {C}\\mathsf {x=0}$ can be written uniquely as a (possibly infinite) linear combination of the columns of $\\mathsf {Sol}(\\mathsf {C})$ indexed by $T_b$ , and all such linear combinations exist (i.e.",
"no infinite non-zero sums).In the literature on linear recurrences (e.g.",
"[1]), this is called a a fundamental system of solutions.",
"When $T_b$ is finite, this is merely the definition of a basis.",
"Corollary 3.17 If $\\dim (\\mathrm {ker}(\\mathsf {C}))\\!<\\!\\infty $ , the columns of $\\mathsf {Sol}(\\mathsf {C})$ indexed by $T_b$ are a basis for $\\mathrm {ker}(\\mathsf {C})$ .",
"Let us consider the Fibonacci recurrence matrix $\\mathsf {C}$ once more.",
"The two $S$ -balls are the sets of even and odd numbers, and so an $S$ -schedule is a pair of consecutive integers.",
"Theorems REF and REF state that any pair of adjacent columns of $\\mathsf {Sol}(\\mathsf {C})$ form a basis of $\\mathrm {ker}(\\mathsf {C})$ , and any pair of values $x_{a-1},x_a\\in \\mathsf {k}$ may be extended uniquely to an element of $\\mathrm {ker}(\\mathsf {C})$ .",
"Definition REF generalizes $T_b$ to a broader class of sets, called schedules, for which analogs of Theorems REF and REF hold.",
"[.05cm] The isomorphisms in Theorems REF and REF are not mutually inverse.",
"Generalizations and connections We consider a few variations of this problem and applications of these ideas.",
"Affine recurrences Let us briefly consider the affine case.",
"An affine recurrence is a system of equations in the sequence of variables $...,x_{-1},x_0,x_1,x_2,...$ which equates each variable to an affine combination of the previous variables (i.e.",
"a degree 1 polynomial).",
"For example, we could add a constant terms $b_i\\in \\mathsf {k}$ to each equation in the Fibonacci recurrence: $x_i = x_{i-1}+x_{i-2}+b_i,\\;\\;\\; \\forall i\\in \\mathbb {Z}$ As before, we may move the variables to the left and factor the coefficients into a matrix $\\mathsf {C}$ : $ \\mathsf {C}\\mathsf {x} = \\mathsf {b}$ Here, $\\mathsf {C}$ is a recurrence matrix, and $\\mathsf {b}$ collects the constant terms from each equation.",
"If $\\mathsf {C}$ is reduced, define $\\mathsf {P}$ as in Section REF , and define the splitting matrix $\\mathsf {Spl}(\\mathsf {C})$ of $\\mathsf {C}$ by $ \\mathsf {Spl}(\\mathsf {C})_{a,b}:= \\left\\lbrace \\begin{array}{cc}\\mathsf {Adj}(\\mathsf {C})_{a,b} & \\text{if }b\\ge 0\\\\(\\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P}))_{a,b} & \\text{if }b<0\\end{array}\\right\\rbrace $ The right half of this matrix is lower unitriangular, and the left half of this matrix is upper triangular, resulting in non-zero entries concentrated into two antipodal wedges.",
"The splitting matrix for the Fibonacci recurrence matrix is below.",
"$\\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,]\\node [right, dark blue] at (3,-1.5) {\\scriptsize Entries from \\mathsf {Adj}(\\mathsf {C})};\\node [left, dark red] at (-3,1.2) {\\scriptsize Entries from \\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})};[use as bounding box] (-2.925,-2.925) rectangle (2.925,2.325);[matrix of math nodes,matrix anchor = M-10-10.center,nodes in empty cells,inner sep=0pt,nodes={anchor=center,node font=\\scriptsize ,rotate=45},column sep={0.3cm,between origins},row sep={0.3cm,between origins},] (M) at (0,0) {\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\cdots \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& -5 \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\cdots \\& \\& 3 \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& -2 \\& \\& -2 \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\cdots \\& \\& 1 \\& \\& 1 \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& -1 \\& \\& -1 \\& \\& -1 \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& 2 \\& \\& 2 \\& \\& 2 \\& \\& 2 \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& 3 \\& \\& 3 \\& \\& 3 \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& 5 \\& \\& 5 \\& \\& 5 \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& 8 \\& \\& 8 \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& 13 \\& \\& 13 \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& 21 \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& 34 \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\cdots \\\\};\\end{tikzpicture}[dark blue,rounded corners,fill=dark blue, opacity=.1] (M-19-19)+(.2,-.5) to ($ M-10-10)+(-.3,0)$) to ($ (M-10-10)+(-.2,.2)$) to ($ (M-10-19)+(.2,.2)$);[dark red,rounded corners,fill=dark red, opacity=.1] (M-3-1)+(-.2,.5) to ($ (M-10-8)+(.3,0)$) to ($ (M-10-8)+(.2,-.2)$) to ($ (M-10-1)+(-.2,-.2)$);$ $The lower/upper triangular conditions imply the non-zero entries coming from $ Adj(C)$ and $ PAdj(CP)$ must be contained in the blue and red cones, respectively.$ Proposition 4.2 Let $\\mathsf {C}$ be a reduced recurrence matrix.",
"$\\mathsf {Spl}(\\mathsf {C})$ is horizontally bounded.",
"$\\mathsf {C}\\mathsf {Spl}(\\mathsf {C})=\\mathsf {Id}$ ; that is, $\\mathsf {Spl}(\\mathsf {C})$ is a right inverse to $\\mathsf {C}$ .",
"For all $\\mathsf {b}\\in \\mathsf {k}^\\mathbb {Z}$ , the product $\\mathsf {x}=\\mathsf {Spl}(\\mathsf {C})\\mathsf {b}$ exists and is a solution to $\\mathsf {C}\\mathsf {x}=\\mathsf {b}$ .",
"In particular, $\\mathsf {C}\\mathsf {x}=\\mathsf {b}$ has a solution for all $\\mathsf {b}$ .",
"(1) If $a\\ge 0$ , then the $a$ th row of $\\mathsf {Spl}(\\mathsf {C})$ is zero outside the interval $[0,a]$ .",
"If $a<0$ , then the $a$ th row of $\\mathsf {Spl}(\\mathsf {C})$ is zero outside $[a,0]$ .This bound can be sharpened, though we won't need a sharp bound.",
"When $a<0$ , the $a$ th row of $\\mathsf {Spl}(\\mathsf {C})$ is zero outside the interval $[b,0]$ when $S(b)=a$ , and the row is entirely zero if there is no such $b$ .",
"Thus, $\\mathsf {Spl}(\\mathsf {C})$ is horizontally bounded.",
"(2) If $a\\ge 0$ , then the $a$ th column of $\\mathsf {C}\\mathsf {Spl}(\\mathsf {C})$ equals the $a$ th column of $\\mathsf {C}\\mathsf {Adj}(\\mathsf {C})=\\mathsf {Id}$ .",
"If $a<0$ , then the $a$ th column of $\\mathsf {C}\\mathsf {Spl}(\\mathsf {C})$ equals the $a$ th column of $\\mathsf {C}(\\mathsf {P}(\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})))=\\mathsf {Id}$ .",
"Therefore, $\\mathsf {C}\\mathsf {Spl}(\\mathsf {C})=\\mathsf {Id}$ .",
"(3) Since $\\mathsf {C}$ and $\\mathsf {Spl}(\\mathsf {C})$ are horizontally bounded, $\\mathsf {C}(\\mathsf {Spl}(\\mathsf {C})\\mathsf {b}) = (\\mathsf {C}\\mathsf {Spl}(\\mathsf {C}) ) \\mathsf {b} = \\mathsf {Idb}= \\mathsf {b} $ .",
"The existence of solutions to every affine recurrence is equivalent saying that, for any recurrence matrix $\\mathsf {C}$ , the associated multiplication map $\\mathsf {k}^\\mathbb {Z}\\rightarrow \\mathsf {k}^\\mathbb {Z}$ is surjective.",
"Given a solution to $\\mathsf {C}\\mathsf {x}=\\mathsf {b}$ , all other solutions are obtained by adding solutions to $\\mathsf {C}\\mathsf {x}=\\mathsf {0}$ .",
"Proposition 4.4 If $\\mathsf {C}$ is reduced, the solutions to $\\mathsf {C}\\mathsf {x} = \\mathsf {b}$ consist of sequences of the form $ \\mathsf {Spl}(\\mathsf {C}) \\mathsf {b} + \\mathsf {Sol}(\\mathsf {C}) \\mathsf {v}, $ running over all $\\mathsf {v}\\in \\mathsf {k}^\\mathbb {Z}$ such that the product $\\mathsf {Sol}(\\mathsf {C}) \\mathsf {v}$ exists.",
"This follows immediately from Proposition REF and Theorem REF .",
"We have now constructed three right inverses to a reduced recurrence matrix $\\mathsf {C}$ , each possessing an additional property: $\\mathsf {Adj}(\\mathsf {C})$ is lower unitriangular, $\\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})$ is upper unitriangular, and $\\mathsf {Spl}(\\mathsf {C})$ is horizontally bounded.",
"If $\\mathsf {C}\\ne \\mathsf {Id}$ , these are all distinct.",
"Linear recurrences indexed by $\\mathbb {N}$ Variations of `linear recurrences' have been studied for centuries.",
"Most often, one considers a system with variables indexed by $\\mathbb {N}$ (rather than $\\mathbb {Z}$ ) and relations defining each variable except at finitely many `initial variables'.",
"For example, the one-sided Fibonacci recurrence has initial variables $x_0$ and $x_1$ and equations $x_i = x_{i-1}+x_{i-2},\\;\\;\\; \\forall i\\ge 2$ The study of $\\mathbb {N}$ -indexed linear recurrences differs fundamentally from $\\mathbb {Z}$ -indexed linear recurrences.",
"Solutions to an $\\mathbb {N}$ -indexed system are determined by the values of the initial variables, which trivializes the kinds of questions we have considered (e.g.",
"existence and parametrization of solutions).",
"Rather, most work in the $\\mathbb {N}$ -indexed context has focused on finding simple formulas for the terms in a solution.",
"We review a few of these approaches.",
"When the equations in a linear recurrence are the same (a `constant' linear recurrence), shifting the indices of a sequence $ x_0,x_1,x_2,... \\longmapsto x_1,x_2,x_3,... $ defines a linear transformation from the space of solutions to itself.",
"Standard tools from linear algebra (e.g.",
"the characteristic polynomial) can then construct a basis of eigenvectors or generalized eigenvectors for the space of solutions.",
"Since the eigenvectors are geometric sequences, an eigenbasis expresses any solution as a linear combination of geometric sequences.",
"This is covered in textbooks like [10].",
"A sequence $x_0, x_1, x_2,...$ can be translated into formal series in several ways, such as $ F_\\mathsf {x}(t) := x_0 + x_1 t + x_2 t^2 + x_3t^3 + \\cdots $ Some linear recurrences (such as constant ones) translate into functional equations involving these generating functions.",
"Clever manipulation of these equations can then yield simple formulas for solutions.",
"This is covered in textbooks like [14].",
"The asymptotics of solutions, that is, the behavior of $x_i$ for sufficiently large $i$ , can be studied analytically.",
"Poincare [13] and othersA curiosity: [4] is the dissertation of Robert Carmichael, of Carmichael numbers in number theory.",
"[4], [2] construct integrals which coincide with the generating function $F_\\mathsf {x}(t)$ in an `infinitesmal neighborhood of infinity'.",
"See [1] for further details.",
"The techniques of the current work can be adapted to this setting.",
"We first add equations fixing the initial values and rewrite the system as a matrix equation $\\mathsf {C}\\mathsf {x}=\\mathsf {b}$ .",
"For example, the (one-sided) Fibonacci recurrence with initial values $x_0=a$ and $x_1=b$ is rewritten as $ \\begin{bmatrix}1 & 0 & 0 & 0 & \\\\0 & 1 & 0 & 0 & \\\\-1 & -1 & 1 & 0 & \\\\0 & -1 & -1 & 1 & \\\\& & & & \\rotatebox {-45}{\\cdots }\\\\\\end{bmatrix}\\begin{bmatrix}x_0 \\\\ x_1 \\\\ x_2 \\\\ x_3 \\\\ \\vdots \\end{bmatrix}=\\begin{bmatrix}a \\\\ b \\\\ 0 \\\\ 0 \\\\ \\vdots \\end{bmatrix}$ The recurrence matrix $\\mathsf {C}$ is $\\mathbb {N}\\times \\mathbb {N}$ , lower unitriangular, and horizontally bounded.",
"The adjugate $\\mathsf {Adj}(\\mathsf {C})$ is defined as before, and the identity $\\mathsf {Adj}(\\mathsf {C})\\mathsf {C}=\\mathsf {C}\\mathsf {Adj}(\\mathsf {C})=\\mathsf {Id}$ still holds.",
"However, there is a crucial difference.",
"In the $\\mathbb {N}\\times \\mathbb {N}$ case, the adjugate matrix $\\mathsf {Adj}(\\mathsf {C})$ is horizontally bounded, and so $\\mathsf {Adj}(\\mathsf {C}) (\\mathsf {C}\\mathsf {x}) = (\\mathsf {Adj}(\\mathsf {C}) \\mathsf {C}) \\mathsf {x}$ for all $\\mathbb {Z}$ -vectors $\\mathsf {x}$ .",
"If $\\mathsf {C}\\mathsf {x=b}$ , then $ \\mathsf {Adj}(\\mathsf {C}) \\mathsf {b} = \\mathsf {Adj}(\\mathsf {C}) (\\mathsf {C}\\mathsf {x})= (\\mathsf {Adj}(\\mathsf {C}) \\mathsf {C}) \\mathsf {x} =\\mathsf {x}$ Consequently, the unique solution to $\\mathsf {C}\\mathsf {x=b}$ can be computed as a linear combination of the columns of $\\mathsf {Adj}(\\mathsf {C})$ indexed by the initial variables.",
"We can restate this as follows.",
"Proposition 4.6 If $x_i$ is an initial variable in an $\\mathbb {N}$ -indexed linear recurrence with recurrence matrix $\\mathsf {C}$ , then the $i$ th column of $\\mathsf {Adj}(\\mathsf {C})$ is the solution for which $x_i=1$ and all other initial variables are 0.",
"Columns of this form are a basis for the space of all solutions.",
"While Proposition REF gives a basis of solutions, it is unclear how useful this is in general.",
"Computationally, the entries of $\\mathsf {Adj}(\\mathsf {C})$ are determinant of submatrices of $\\mathsf {C}$ , which are (naively) no simpler than recursively computing $x_0,x_1,....,x_j$ directly.",
"Friezes The author's original motivation for the work in this paper is a connection and forthcoming application to the following curious objects.",
"A tame $SL(k)$ -frieze consists of finitely many rows of integers (offset in a diamond pattern) such that: the top and bottom rows consist entirely of 1s, every $k\\times k$ diamond has determinant 1, and every $(k+1)\\times (k+1)$ diamond has determinant 0.",
"An example of an $SL(2)$ -frieze is given below.",
"$\\begin{tikzpicture}[baseline=(current bounding box.south),ampersand replacement=\\&,][matrix of math nodes,matrix anchor = M-1-8.center,origin/.style={},throw/.style={},pivot/.style={draw,circle,inner sep=0.25mm,minimum size=2mm}, nodes in empty cells,inner sep=0pt,nodes={anchor=center,node font=\\scriptsize },column sep={.35cm,between origins},row sep={.35cm,between origins},] (M) at (0,0) {\\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& \\cdots \\\\\\cdots \\& \\& 3 \\& \\& 2 \\& \\& 2 \\& \\& 1 \\& \\& 4 \\& \\& 3 \\& \\& 1 \\& \\& 2 \\& \\& 3 \\& \\& 2 \\& \\& 2 \\& \\& 1 \\& \\& 4 \\& \\& 3 \\& \\& 1 \\& \\& 2 \\& \\& 3 \\& \\\\\\& 5 \\& \\& 5 \\& \\& 3 \\& \\& 1 \\& \\& 3 \\& \\& 11 \\& \\& 2 \\& \\& 1 \\& \\& 5 \\& \\& 5 \\& \\& 3 \\& \\& 1 \\& \\& 3 \\& \\& 11 \\& \\& 2 \\& \\& 1 \\& \\& 5 \\& \\& \\cdots \\\\\\cdots \\& \\& 8 \\& \\& 7 \\& \\& 1 \\& \\& 2 \\& \\& 8 \\& \\& 7 \\& \\& 1 \\& \\& 2 \\& \\& 8 \\& \\& 7 \\& \\& 1 \\& \\& 2 \\& \\& 8 \\& \\& 7 \\& \\& 1 \\& \\& 2 \\& \\& 8 \\& \\\\\\& 3 \\& \\& 11 \\& \\& 2 \\& \\& 1 \\& \\& 5 \\& \\& 5 \\& \\& 3\\& \\& 1 \\& \\& 3 \\& \\& 11 \\& \\& 2 \\& \\& 1 \\& \\& 5 \\& \\& 5 \\& \\& 3\\& \\& 1 \\& \\& 3 \\& \\& \\cdots \\\\\\cdots \\& \\& 4 \\& \\& 3 \\& \\& 1 \\& \\& 2 \\& \\& 3 \\& \\& 2 \\& \\& 2 \\& \\& 1 \\& \\& 4 \\& \\& 3 \\& \\& 1 \\& \\& 2 \\& \\& 3 \\& \\& 2 \\& \\& 2 \\& \\& 1 \\& \\& 4 \\& \\\\\\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& \\cdots \\\\};\\end{tikzpicture}$ The study of friezes was initiated in [7], [5], [6] for $k=2$ , and generalized to arbitrary $k$ in [8], [3].",
"Friezes enjoy many remarkable properties; for example, the rows of a tame $SL(k)$ -frieze must be periodic.",
"An excellent overview is given in [11].",
"A frieze may be converted into a recurrence matrix, by rotating $45^\\circ $ clockwise and using the top row as the main diagonal.",
"[12] and others also use an alternating sign when translating a frieze into a linear recurrence.",
"Remarkably, the solutions have a periodicity condition.",
"Theorem 4.9 [12] If $\\mathsf {C}$ is the recurrence matrix associated to a tame $SL(k)$ -frieze, then every solution to $\\mathsf {C}\\mathsf {x=0}$ is superperiodic: $x_{i+n}=(-1)^sx_i$ for some $n$ and $s$ and all $i$ .",
"In fact, [12] proves a stronger result.",
"For each frieze $\\mathsf {C}$ , they construct a Gale dual frieze $\\mathsf {C}^\\dagger $ whose diagonals encode distinguished solutions to $\\mathsf {C}\\mathsf {x=0}$ .",
"In a sequel [9] to the current work, we will extend Theorem REF to an equivalence.",
"Specifically, if $\\mathsf {C}$ is a reduced recurrence matrix of shape $S$ , then the following are equivalent.",
"$\\mathsf {C}$ satisfies a family of determinantal identities generalizing the tame frieze conditions.",
"Every solution to $\\mathsf {C}\\mathsf {x=0}$ is $n$ -quasiperiodic; that is, $x_{i+n}=\\lambda x_i$ for some $\\lambda $ and all $i$ .",
"The Gale dual $\\mathsf {C}^\\dagger $ , a truncation of $\\mathsf {Sol}(\\mathsf {C})$ , has shape $S^\\dagger $ , where $S^\\dagger (i):=S^{-1}(i)+n$ .",
"The space of such linear recurrences (of fixed shape $S$ ) is the cluster $\\mathcal {X}$ -variety dual to the positroid variety corresponding to $S$ ; this will be explained in [9].",
"The rest of this note proves the promised results.",
"Kernel containment and factorization In this section, we prove a useful equivalence between containments of kernels and factorizations in the semigroup of recurrence matrices.",
"Let $\\mathsf {k}^\\mathbb {Z}_b\\subset \\mathsf {k}^\\mathbb {Z}$ denote the subspace of bounded sequences (i.e.",
"non-zero in finitely many terms).",
"If $\\mathsf {v}\\in \\mathsf {k}^\\mathbb {Z}_b$ and $\\mathsf {w}\\in \\mathsf {k}^\\mathbb {Z}$ , then the dot product $\\mathsf {v}\\cdot \\mathsf {w}$ is well-defined.",
"Lemma 5.2 Let $\\mathsf {C}$ be a recurrence matrix and let $\\mathsf {v}\\in \\mathsf {k}^\\mathbb {Z}_b$ .",
"If $\\mathsf {v}\\cdot \\mathsf {w}=0$ for all $\\mathsf {w}\\in \\mathrm {ker}(\\mathsf {C})$ , then $\\mathsf {v}$ is in the span of the rows of $\\mathsf {C}$ .",
"Let $V\\subset \\mathsf {k}^\\mathbb {Z}_b$ denote the span of the rows of $\\mathsf {C}$ , and assume for contradiction that $\\mathsf {v}\\notin V$ .",
"We may therefore choose a linear map $f:\\mathsf {k}^\\mathbb {Z}_b\\rightarrow \\mathsf {k}$ such that $f(V)=0$ and $f(\\mathsf {v})=1$ .The existence of such a map may depend on the Axiom of Choice, which we therefore assume.",
"Let $\\mathsf {e}_a\\in \\mathsf {k}^\\mathbb {Z}_b$ denote the standard basis vector which is 1 in the $a$ th term and 0 everywhere else, and set $\\mathsf {w}:= ( f(\\mathsf {e}_a)) _{a\\in \\mathbb {Z}}\\in \\mathsf {k}^\\mathbb {Z}$ .",
"By linearity, $f(\\mathsf {u})=\\mathsf {u}\\cdot \\mathsf {w}$ for all $\\mathsf {u}\\in \\mathsf {k}^\\mathbb {Z}_b$ .",
"Since $f$ kills each row of $\\mathsf {C}$ , $\\mathsf {C}\\mathsf {w}=0$ and so $\\mathsf {w}\\in \\mathrm {ker}(\\mathsf {C})$ .",
"However, $\\mathsf {v}\\cdot \\mathsf {w}=1$ , contradicting the hypothesis.",
"Lemma 5.3 Let $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ be recurrence matrices.",
"Then the following are equivalent.",
"$\\mathrm {ker}(\\mathsf {C})\\subseteq \\mathrm {ker}(\\mathsf {C}^{\\prime })$ .",
"$\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for some horizontally bounded matrix $\\mathsf {D}$ .",
"$\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for some recurrence matrix $\\mathsf {D}$ .",
"$\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C})$ is a recurrence matrix.",
"Furthermore, the matrix $\\mathsf {D}$ in (2) and (3) must equal $\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C})$ and is therefore unique.",
"($1\\Rightarrow 2$ ) If $\\mathrm {ker}(\\mathsf {C})\\subseteq \\mathrm {ker}(\\mathsf {C}^{\\prime })$ , then each row of $\\mathsf {C}^{\\prime }$ kills $\\mathrm {ker}(\\mathsf {C})$ .",
"By Lemma REF , the $a$ th row of $\\mathsf {C}^{\\prime }$ is equal to $\\mathsf {D}_a\\mathsf {C}$ for some bounded sequence $\\mathsf {D}_a\\in \\mathsf {k}^\\mathbb {Z}_b$ .",
"The vectors $\\mathsf {D}_a$ may be combined into the rows of a matrix $\\mathsf {D}$ which is horizontally bounded and satisfies $\\mathsf {D}\\mathsf {C}=\\mathsf {C}^{\\prime }$ .",
"($2\\Rightarrow 3+4+$ Uniqueness) Assume that $\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for a horizontally bounded $\\mathsf {D}$ .",
"Then $\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C}) = (\\mathsf {D}\\mathsf {C})\\mathsf {Adj}(\\mathsf {C}) \\stackrel{*}{=} \\mathsf {D}(\\mathsf {C}\\mathsf {Adj}(\\mathsf {C})) = \\mathsf {D}$ Equality ($*$ ) holds because $\\mathsf {D}$ and $\\mathsf {C}$ are horizontally bounded.",
"Since $\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C})$ is lower unitriangular and $\\mathsf {D}$ is horizontally bounded, they are the same recurrence matrix.",
"($3\\Rightarrow 1$ ) Assume $\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for some recurrence matrix $\\mathsf {D}$ .",
"If $\\mathsf {v}\\in \\mathrm {ker}(\\mathsf {C})$ , then $ \\mathsf {C}^{\\prime }\\mathsf {v} = (\\mathsf {D}\\mathsf {C})\\mathsf {v} \\stackrel{*}{=} \\mathsf {D}(\\mathsf {C}\\mathsf {v}) = \\mathsf {D}\\mathsf {0} = \\mathsf {0} $ Equality ($*$ ) holds because $\\mathsf {D}$ and $\\mathsf {C}$ are horizontally bounded.",
"Therefore, $\\mathrm {ker}(\\mathsf {C})\\subseteq \\mathrm {ker}(\\mathsf {C}^{\\prime })$ .",
"($4\\Rightarrow 3$ ) Setting $\\mathsf {D}\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C})$ , we check that $ \\mathsf {D}\\mathsf {C}= (\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C}) ) \\mathsf {C}\\stackrel{*}{=} \\mathsf {C}^{\\prime }(\\mathsf {Adj}(\\mathsf {C})\\mathsf {C}) = \\mathsf {C}^{\\prime }$ Equality ($*$ ) holds because $\\mathsf {C}^{\\prime }$ , $\\mathsf {Adj}(\\mathsf {C})$ , and $\\mathsf {C}$ are lower unitriangular.",
"Theorem 5.4 A recurrence matrix $\\mathsf {C}$ is trivial if and only if $\\mathsf {Adj}(\\mathsf {C})$ is a recurrence matrix.",
"The recurrence matrix $\\mathsf {C}$ is trivial when $\\mathrm {ker}(\\mathsf {C})\\subseteq \\mathrm {ker}(\\mathsf {Id})$ .",
"Applying Lemma REF with $\\mathsf {C}^{\\prime }=\\mathsf {Id}$ , this holds if and only if $\\mathsf {Adj}(\\mathsf {C})$ is a recurrence matrix.",
"Theorem 5.5 Two recurrence matrices $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ are equivalent if and only if $\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for a trivial recurrence matrix $\\mathsf {D}$ .",
"Two recurrence matrices $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ are equivalent if and only if $\\mathrm {ker}(\\mathsf {C})=\\mathrm {ker}(\\mathsf {C}^{\\prime })$ .",
"By Lemma REF , this holds if and only if $\\mathsf {D}=\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C})$ and $\\mathsf {Adj}(\\mathsf {D})=\\mathsf {C}\\mathsf {Adj}(\\mathsf {C}^{\\prime })$ are recurrence matrices.",
"By Theorem REF , this is equivalent to $\\mathsf {D}$ being a trivial linear recurrence.",
"Lemma REF also allows us to make a connection between kernel containment and shapes.",
"Lemma 5.6 Let $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ be recurrence matrices with shape $S$ and $S^{\\prime }$ , respectively.",
"If $\\mathrm {ker}(\\mathsf {C})\\subseteq \\mathrm {ker}(\\mathsf {C}^{\\prime })$ and $S$ is injective (e.g.",
"if $\\mathsf {C}$ is reduced), then $S(a)\\ge S^{\\prime }(a)$ for all $a$ .",
"By Lemma REF , $\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for some recurrence matrix $\\mathsf {D}$ .",
"Fix $a\\in \\mathbb {Z}$ , and consider $B:=\\lbrace b \\in \\mathbb {Z}\\mid \\mathsf {D}_{a,b} \\ne 0\\rbrace $ .",
"This set is bounded and contains $a$ .",
"Let $b_{0}$ be the element of $B$ on which $S$ is minimal; this is unique because $S$ is injective.",
"$ (\\mathsf {D}\\mathsf {C})_{a,S(b_{0})} = \\sum _{b\\in \\mathbb {Z}} \\mathsf {D}_{a,b} \\mathsf {C}_{b,S(b_{0})} = \\mathsf {D}_{a,b_{0}}\\mathsf {C}_{b_{0},S(b_{0})} \\ne 0 $ Therefore, $S^{\\prime }(a)\\le S(b_0)$ .In fact, this is equality, but we won't need this stronger statement.",
"Since $a\\in B$ , $S(b_0)\\le S(a)$ , and so $S^{\\prime }(a)\\le S(a)$ .",
"Proposition 5.7 Reduced recurrence matrices that are equivalent must be equal.",
"Let $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ be reduced and equivalent.",
"Lemma REF implies that $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ have the same shape; call it $S$ .",
"By Lemma REF , $\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for some recurrence matrix $\\mathsf {D}$ .",
"Let $T$ denote the shape of $\\mathsf {D}$ , so that $\\mathsf {D}_{a,b}=0$ whenever $b<T(a)$ .",
"Since $\\mathsf {C}$ is reduced of shape $S$ , $\\mathsf {C}_{b,S(T(a))}=0$ whenever $b>T(a)$ .",
"Therefore, $ \\mathsf {C}^{\\prime }_{a,S(T(a))} = \\sum _b \\mathsf {D}_{a,b} \\mathsf {C}_{b,S(T(a))} = \\mathsf {D}_{a,T(a)}\\mathsf {C}_{T(a),S(T(a))} \\ne 0 $ Since $\\mathsf {C}^{\\prime }$ is also reduced of shape $S$ , this is only possible if $a= T(a)$ .",
"Since this holds for all $a$ , the only non-zero entries of $\\mathsf {D}$ are on the main diagonal.",
"Thus, $\\mathsf {D}=\\mathsf {Id}$ and $\\mathsf {C}^{\\prime }=\\mathsf {C}$ .",
"Gauss-Zordan Elimination Because we are working with $\\mathbb {Z}\\times \\mathbb {Z}$ -matrices, we must consider infinite sequences of row reductions that may be chosen in an arbitrary order.",
"We furthermore consider generalized row reductions: limits of such row reductions (in an appropriate topology).",
"Row reduction Given a recurrence matrix $\\mathsf {C}$ of shape $S$ , a row reduction of $\\mathsf {C}$ is a matrix $\\mathsf {C}^{\\prime }$ obtained by adding $\\mathsf {C}_{a,S(b)} / \\mathsf {C}_{b,S(b)}$ times the $b$ th row to the $a$ th row, for some $b> a$ with $\\mathsf {C}_{a,S(b)}\\ne 0$ .",
"By design, the resulting matrix $\\mathsf {C}^{\\prime }$ has a zero in the $(a,S(b))$ entry.",
"Proposition 6.1 A recurrence matrix is reduced if and only if it has no row reductions.",
"A row reduction of $\\mathsf {C}$ can be reformulated as a factorization $\\mathsf {C}=\\mathsf {D}\\mathsf {C}^{\\prime }$ such that $\\mathsf {D}$ differs from the identity matrix in a single entry $\\mathsf {D}_{a,b}$ , and such that $\\mathsf {C}_{a,S(b)}\\ne 0$ and $\\mathsf {C}^{\\prime }_{a,S(b)}=0$ .",
"This perspective leads to the following generalization.",
"A generalized row reduction of $\\mathsf {C}$ is a recurrence matrix $\\mathsf {C}^{\\prime }$ such that $\\mathsf {C}=\\mathsf {D}\\mathsf {C}^{\\prime }$ for a trivial recurrence matrix $\\mathsf {D}$ with the property that, for each $a$ such that $\\lbrace b <a \\mid \\mathsf {D}_{a,b}\\ne 0\\rbrace $ is non-empty, we have $ \\mathsf {C}_{a,b_a} \\ne 0\\text{ and } \\mathsf {C}^{\\prime }_{a,b_a} = 0 $ where $b_a:= \\min \\lbrace S(b) \\mid b< a \\text{ s.t.",
"}\\mathsf {D}_{a,b}\\ne 0\\rbrace $ .",
"We write $\\mathsf {C}\\succeq \\mathsf {C}^{\\prime }$ to denote that $\\mathsf {C}^{\\prime }$ is a generalized row reduction of $\\mathsf {C}$ .",
"The index $b_a$ may be defined as the leftmost entry of the $a$ th row that multiplication by $\\mathsf {D}$ is `big enough' to change, and so $(\\mathsf {D}\\mathsf {C})_{a,b}=\\mathsf {C}_{a,b}$ whenever $b<b_a$ .",
"Thus, if $\\mathsf {C}\\succeq \\mathsf {C}^{\\prime }$ , then $\\mathsf {C}^{\\prime }$ must vanish in the leftmost entry in which the $a$ th rows of $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ differ.",
"Proposition 6.3 The relation $\\succeq $ defines a partial order on the set of recurrence matrices.",
"As a consequence, an iterated sequence of row reductions is a generalized row reduction.",
"(Antisymmetry) Assume $\\mathsf {C}\\succeq \\mathsf {C}^{\\prime }$ and $\\mathsf {C}\\preceq \\mathsf {C}^{\\prime }$ .",
"By Remark REF , both $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ vanish in the leftmost entry in which the $a$ th rows of $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ differ.",
"However, two entries cannot both vanish and be different, so the $a$ th rows of $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ coincide for all $a$ .",
"Thus, $\\mathsf {C}=\\mathsf {C}^{\\prime }$ .",
"(Transitivity) Let $\\mathsf {C}\\preceq \\mathsf {D}\\mathsf {C}\\preceq \\mathsf {D}^{\\prime }\\mathsf {D}\\mathsf {C}$ , and let $S$ and $S^{\\prime }$ denote the shapes of $\\mathsf {C}$ and $\\mathsf {D}\\mathsf {C}$ , respectively.",
"Fix some $a$ .",
"If $\\lbrace b< a\\mid \\mathsf {D}_{a,b}\\ne 0\\rbrace = \\varnothing $ or $\\lbrace b< a\\mid \\mathsf {D}^{\\prime }_{a,b}\\ne 0\\rbrace = \\varnothing $ , the generalized row reduction condition is easy to check.",
"Assume neither set is empty and let $b_0 &:= \\min \\lbrace S(b) \\mid b<a\\text{ s.t.",
"}\\mathsf {D}_{a,b}\\ne 0\\rbrace \\\\b_0^{\\prime } &:= \\min \\lbrace S^{\\prime }(b) \\mid b<a\\text{ s.t.",
"}\\mathsf {D}^{\\prime }_{a,b}\\ne 0\\rbrace $ By the definition of generalized row reductions, $\\mathsf {C}_{a,b_0}=0,\\;\\;\\; (\\mathsf {D}\\mathsf {C})_{a,b_0}\\ne 0,\\;\\;\\; (\\mathsf {D}\\mathsf {C})_{a,b_0^{\\prime }}=0 ,\\;\\;\\; (\\mathsf {D}^{\\prime }\\mathsf {D}\\mathsf {C})_{a,b_0^{\\prime }}\\ne 0$ This ensures that $\\mathsf {C}_{a,\\min (b_0,b_0^{\\prime })}=0$ and $(\\mathsf {D}^{\\prime }\\mathsf {D}\\mathsf {C})_{a,\\min (b_0,b_0^{\\prime })}\\ne 0$ .",
"Since these entries differ, $ \\min \\lbrace S(b) \\mid b<a\\text{ s.t.",
"}(\\mathsf {D}^{\\prime }\\mathsf {D})_{a,b}\\ne 0\\rbrace \\le \\min (b_0,b_0^{\\prime }) $ To show this is equality, consider some $b<a$ such that $(\\mathsf {D}^{\\prime }\\mathsf {D})_{a,b}\\ne 0$ .",
"We split into cases.",
"Assume $\\mathsf {D}^{\\prime }_{a,c}\\mathsf {D}_{c,b}\\ne 0$ for some $c<a$ .",
"Since $\\mathsf {D}_{c,b}\\ne 0$ and $\\mathsf {C}\\preceq \\mathsf {D}\\mathsf {C}$ , $S^{\\prime }(c)\\le S(b)$ .",
"Since $\\mathsf {D}^{\\prime }_{a,c}\\ne 0$ and $\\mathsf {D}\\mathsf {C}\\preceq \\mathsf {D}^{\\prime }\\mathsf {D}\\mathsf {C}$ , $S^{\\prime }(c)\\ge b_0^{\\prime }$ .",
"Therefore, $S(b)\\ge b^{\\prime }_0$ .",
"Otherwise, $\\mathsf {D}^{\\prime }_{a,c}\\mathsf {D}_{c,b}=0$ for all $c<a$ , and so $(\\mathsf {D}^{\\prime }\\mathsf {D})_{a,b}= \\mathsf {D}^{\\prime }_{a,a}\\mathsf {D}_{a,b}=\\mathsf {D}_{a,b}$ .",
"Since $\\mathsf {D}_{a,b}\\ne 0$ and $\\mathsf {C}\\preceq \\mathsf {D}\\mathsf {C}$ , we know that $S(b)\\ge b_0$ .",
"Therefore, $\\mathsf {C}_{a,\\min (b_0,b_0^{\\prime })}=0$ and $(\\mathsf {D}^{\\prime }\\mathsf {D}\\mathsf {C})_{a,\\min (b_0,b_0^{\\prime })}\\ne 0$ and $ \\min \\lbrace S(b) \\mid b<a\\text{ s.t.",
"}(\\mathsf {D}^{\\prime }\\mathsf {D})_{a,b}\\ne 0\\rbrace = \\min (b_0,b_0^{\\prime }) $ Since this holds for all $a$ , $\\mathsf {C}\\preceq \\mathsf {D}^{\\prime }\\mathsf {D}\\mathsf {C}$ .",
"Limits To define limits of generalized row reductions, we endow the set of recurrence matrices with the topology of row-wise stabilization: a sequence of recurrence matrices converges if each row stabilizes after finitely many steps.",
"We next show that sequences of generalized row reductions must stabilize row-wise to another generalized row reduction, via the following more general result.",
"Lemma 6.4 Let $\\mathcal {C}$ be a set of recurrence matrices in which every pair is comparable in the row reduction partial order.Sometimes called a `chain' in the literature on partially ordered sets.",
"Then the closure of $\\mathcal {C}$ in the space of recurrence matrices contains a lower bound of $\\mathcal {C}$ .",
"Equivalently, there is a descending sequence of recurrence matrices in $\\mathcal {C}$ (i.e.",
"generalized row reductions of the initial matrix in the sequence) which converges (i.e.",
"stabilizes row-wise) to a lower bound of $\\mathcal {C}$ (i.e.",
"a generalized row reduction of every matrix in $\\mathcal {C}$ ).",
"Given a recurrence matrix $\\mathsf {C}$ and an integer $a$ , define $ n_a(\\mathsf {C}) := \\sum _{b \\text{ s.t. }",
"\\mathsf {C}_{(a,b)}\\ne 0} (a-b)^2 $ If $\\mathsf {C}\\preceq \\mathsf {C}^{\\prime }$ , then $n_a(\\mathsf {C})\\le n_a(\\mathsf {C}^{\\prime })$ and equality implies the $a$ th rows of $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ coincide.",
"For each $a$ , let $\\mathcal {C}_a:= \\lbrace \\mathsf {C}\\in \\mathcal {C} \\mid \\forall \\mathsf {C}^{\\prime }\\in \\mathcal {C},\\;n_a (\\mathsf {C}) \\le n_a(\\mathsf {C}^{\\prime })\\rbrace $ ; that is, $\\mathcal {C}_a$ is the set of matrices in $\\mathcal {C}$ which attain the minimum value of $n_a$ .",
"This set is non-empty and the $a$ th row of each matrix in $\\mathcal {C}_a$ is the same, since $n_a$ has the same value and the matrices are comparable.",
"Consider $a,a^{\\prime }\\in \\mathbb {Z}$ and assume, for contradiction, that there exist $\\mathsf {C}\\in \\mathcal {C}_a\\setminus \\mathcal {C}_{a^{\\prime }}$ and $\\mathsf {C}^{\\prime } \\in \\mathcal {C}_{a^{\\prime }}\\setminus \\mathcal {C}_a$ .",
"If $\\mathsf {C}^{\\prime }\\preceq \\mathsf {C}$ , then $n_a(\\mathsf {C}^{\\prime })\\le n_a(\\mathsf {C})$ .",
"By the minimality of $n_a(\\mathsf {C})$ , this is an equality and so $\\mathsf {C}^{\\prime }\\in \\mathcal {C}_a$ ; a contradiction.",
"By a symmetric argument, $\\mathsf {C}\\preceq \\mathsf {C}^{\\prime }$ forces a contradiction.",
"Therefore, $\\mathcal {C}_a\\cap \\mathcal {C}_{a^{\\prime }}$ is either equal to $\\mathcal {C}_a$ or equal to $\\mathcal {C}_{a^{\\prime }}$ .",
"Applying this repeatedly, for any $i\\in \\mathbb {N}$ , there is some $a_i\\in [-i,i]$ such that $ \\bigcap _{a\\in [-i,i]} \\mathcal {C}_a =\\mathcal {C}_{a_i} \\ne \\varnothing $ Choose a matrix $\\mathsf {C}^i$ in $\\mathcal {C}_{a_i}$ for each $i$ .",
"The $a$ th rows in the sequence $\\mathsf {C}^1,\\mathsf {C}^2,\\mathsf {C}^3,...$ , stabilize after the $a$ th term, and so this sequence converges to the recurrence matrix $\\mathsf {C}$ whose $a$ th row coincides with the $a$ th row in each matrix in $\\mathcal {C}_a$ .",
"Let $S$ be the shape of $\\mathsf {C}^1$ .",
"Define a sequence $\\mathsf {D}^1,\\mathsf {D}^2,\\mathsf {D}^3,...$ of trivial recurrence matrices by $\\mathsf {C}^1 = \\mathsf {D}^n\\mathsf {C}^n$ for all $n$ .",
"Since $\\mathsf {C}^1\\succeq \\mathsf {C}^n$ , if $\\mathsf {D}^n_{a,b}\\ne 0$ , then $S(a)\\le S(b)\\le b$ ; that is, the $a$ th row $\\mathsf {D}$ can be non-zero only on the interval $[S(a),a]$ .",
"When $n>|S(a)|$ , the $a$ th row of the product $\\mathsf {D}^n\\mathsf {C}^n$ only depends on rows in $\\mathsf {C}^n$ that coincide with rows in $\\overline{\\mathsf {C}}$ .",
"Therefore, the $a$ th row of $\\mathsf {D}^n\\overline{\\mathsf {C}}$ is equal to $\\mathsf {C}^1$ .",
"Therefore, the sequence $\\mathsf {D}^1,\\mathsf {D}^2,\\mathsf {D}^3,...$ stabilizes row-wise to a matrix $\\overline{\\mathsf {D}}$ such that $\\overline{\\mathsf {D}}\\overline{\\mathsf {C}}=\\mathsf {C}^1$ .",
"As $\\overline{\\mathsf {D}}_{a,b}=\\mathsf {D}^n_{a,b}$ for large enough $n$ , this shows that $\\mathsf {C}\\preceq \\mathsf {C}^1$ .",
"Since the sequence $\\mathsf {C}^\\bullet $ could have started at any matrix in $\\mathcal {C}$ , this shows $\\overline{\\mathsf {C}}$ is a lower bound for $\\mathcal {C}$ .",
"Theorem 6.5 Every recurrence matrix is equivalent to a unique reduced recurrence matrix.",
"Let $\\mathcal {C}$ be an equivalence class of recurrence matrices, with the row reduction partial order.",
"Every non-empty chain in $\\mathcal {C}$ has a lower bound (by Lemma REF ).",
"By Zorn's Lemma, $\\mathcal {C}$ contains a minimal element $\\overline{\\mathsf {C}}$ .",
"If $\\overline{C}$ was not reduced, then there would be a row operation which would strictly decrease it in the reduction partial order; contradicting minimality.",
"Therefore, $\\overline{\\mathsf {C}}$ is reduced.",
"By Proposition REF , this reduced recurrence matrix is unique.",
"This provides a transfinite, non-deterministic analog of Gauss-Jordan elimination, which we humorously dub Gauss-Zordan elimination (both for `Zorn' and the integers $\\mathbb {Z}$ ).",
"Given a recurrence matrix $\\mathsf {C}$ , an arbitrary sequence of row reductions will stabilize row-wise to a matrix equivalent to $\\mathsf {C}$ .",
"While this limit may not be reduced, further arbitrary row reductions generate another convergent sequence.",
"Zorn's Lemma guarantees that some transfinite iteration of this process will eventually converge to the reduced representative of $\\mathsf {C}$ .",
"Constructing recurrences from spaces of solutions In this section, we consider the inverse problem to the motivating problem of this note: Given a subspace $V\\subset \\mathsf {k}^\\mathbb {Z}$ , how can we construct a linear recurrence $\\mathsf {C}\\mathsf {x}=\\mathsf {0}$ whose solutions are $V$ ?",
"We give a characterization of when this is possible in Theorem REF .",
"For any $I\\subset \\mathbb {Z}$ , let $\\pi _{I}:\\mathsf {k}^\\mathbb {Z}\\rightarrow \\mathsf {k}^{I}$ restrict a sequence to the indices in $I$ , and let $\\iota _I:\\mathsf {k}^I\\rightarrow \\mathsf {k}^\\mathbb {Z}$ extend a sequence by 0.",
"Given a subspace $V\\subset \\mathsf {k}^\\mathbb {Z}$ , let $V_{I}\\pi _{I}(V)\\subset \\mathsf {k}^{I}$ .",
"Rank matrices The rank matrix of a subspace $V\\subset \\mathsf {k}^\\mathbb {Z}$ is the $\\mathbb {Z}\\times \\mathbb {Z}$ -matrix withThe entries below the diagonal are unimportant; we set them to $b-a+1$ to avoid special cases later.",
"$ \\mathsf {R}_{a,b}:= \\left\\lbrace \\begin{array}{cc}\\dim _\\mathsf {k}(V_{[a,b]}) & \\text{if }a\\le b \\\\b-a+1 & \\text{otherwise}\\end{array} \\right\\rbrace $ Let $V$ be the space of sequences such that (a) the $-1$ st term is 0, (b) the $-2$ nd and 0th term are equal, and (c) the 0th, 1st, and 2nd terms sum to 0.",
"The rank matrix is $\\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][matrix of math nodes,matrix anchor = M-1-8.center,origin/.style={},throw/.style={},defect/.style={dark red,draw,circle,inner sep=0.25mm,minimum size=2mm},pivot/.style={draw,circle,inner sep=0.25mm,minimum size=2mm}, nodes in empty cells,inner sep=0pt,nodes={anchor=center,rotate=45},column sep={.5cm,between origins},row sep={.5cm,between origins},] (M) at (0,0) {\\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\\\\\& 5 \\& \\& 5 \\& \\& 4 \\& \\& 4 \\& \\& 4 \\& \\& 5 \\& \\& 6 \\& \\& \\cdots \\\\\\cdots \\& \\& 4 \\& \\& 4 \\& \\& 3 \\& \\& 3 \\& \\& 4 \\& \\& 5 \\& \\& 6 \\& \\\\\\& 4 \\& \\& 3 \\& \\& 3 \\& \\& 2 \\& \\& 3 \\& \\& 4 \\& \\& 5 \\& \\& \\cdots \\\\\\cdots \\& \\& 3 \\& \\& 2 \\& \\& 2 \\& \\& 2 \\& \\& 3 \\& \\& 4 \\& \\& 4 \\& \\\\\\& 3 \\& \\& 2 \\& \\& |[defect]| 1 \\& \\& 2 \\& \\& |[defect]| 2 \\& \\& 3 \\& \\& 3 \\& \\& \\cdots \\\\\\cdots \\& \\& 2 \\& \\& 1 \\& \\& 1 \\& \\& 2 \\& \\& 2 \\& \\& 2 \\& \\& 2 \\& \\\\\\& |[throw]| 1 \\& \\& |[throw]| 1 \\& \\& |[defect]| 0 \\& \\& |[origin]| 1 \\& \\&|[throw]| 1 \\& \\& |[throw]| 1 \\& \\& |[throw]| 1 \\& \\& \\cdots \\\\};\\end{tikzpicture}$ The subdiagonal entries have been omitted.",
"The dark redred circles are the defects (see below).",
"Proposition 7.3 If $\\mathsf {R}$ is the rank matrix of $V$ , then the following hold for any $a,b\\in \\mathbb {Z}$ .",
"$\\mathsf {R}_{a,b}-\\mathsf {R}_{a+1,b}$ must be 0 or 1.",
"$\\mathsf {R}_{a,b}-\\mathsf {R}_{a,b-1}$ must be 0 or 1.",
"$\\mathsf {R}_{a,b}-\\mathsf {R}_{a+1,b}-\\mathsf {R}_{a,b-1}+\\mathsf {R}_{a+1,b-1}$ must be 0 or $-1$ .",
"The projection $V_{[a,b]}\\rightarrow V_{[a+1,b]}$ is surjective with at most 1-dimensional kernel.",
"This proves the first result; the second is proven similarly.",
"The first result implies that $-1\\le (\\mathsf {R}_{a,b}-\\mathsf {R}_{a+1,b})-(\\mathsf {R}_{a,b-1} - \\mathsf {R}_{a+1,b-1}) \\le 1$ .",
"The map $V_{[a+1,b]} \\oplus V_{[a,b-1]}\\rightarrow V_{[a,b]}$ which sends $(\\mathsf {v},\\mathsf {w})$ to $\\mathsf {v+w}$ is a surjection whose kernel is the image of the map $V_{[a+1,b-1]} \\rightarrow V_{[a+1,b]} \\oplus V_{[a,b-1]}$ which sends $\\mathsf {v}$ to $(\\mathsf {v},-\\mathsf {v})$ .",
"Therefore, $ \\dim (V_{[a,b]}) \\le \\dim (V_{[a+1,b]} \\oplus V_{[a,b-1]}) - \\dim (V_{[a+1,b-1]}) $ This proves that $\\mathsf {R}_{a,b}-\\mathsf {R}_{a+1,b}-\\mathsf {R}_{a,b-1} + \\mathsf {R}_{a+1,b-1} \\le 0$ .",
"Let us say the pair $(a,b)\\in \\mathbb {Z}\\times \\mathbb {Z}$ is a defect of a rank matrix $\\mathsf {R}$ if $ \\mathsf {R}_{a,b} - \\mathsf {R}_{a+1,b}-\\mathsf {R}_{a,b-1}+\\mathsf {R}_{a+1,b-1} =-1 $ Proposition 7.4 The defects of a rank matrix $\\mathsf {R}$ have the following properties.",
"$\\mathsf {R}_{a,b} = (b-a+1) - \\#(\\text{defects in the box }[a,b]\\times [a,b]) $ .",
"Each row and column of a rank matrix can contain at most one defect.",
"If $[a,b]\\times \\lbrace b\\rbrace $ does not contain any defects, then $V_{[a,b]}$ contains the vector $(0,0,...,0,1)$ .",
"If $\\lbrace a\\rbrace \\times [a,b]$ does not contain any defects, then $V_{[a,b]}$ contains the vector $(1,0,...,0,0)$ .",
"Fix $a$ and consider the sequence $(\\mathsf {R}_{a,b}-\\mathsf {R}_{a+1,b})$ for all $b$ .",
"This sequence starts at 1 for sufficiently negative $b$ , switches from 1 to 0 whenever $(a,b)$ is a defect, and must remain 0 once it does (by Proposition REF .3).",
"Since there are no defects when $a<b$ , this implies that $ \\mathsf {R}_{a,b}=\\mathsf {R}_{a+1,b}+1-\\#(\\text{defects in the line }\\lbrace a\\rbrace \\times [a,b])$ In particular, there can be at most one defect in each row, and inductively implies that $ \\mathsf {R}_{a,b} = (b-a+1) - \\#(\\text{defects in the box }[a,b]\\times [a,b]) $ If $\\lbrace a\\rbrace \\times [a,b]$ does not contain any defect, then $\\mathsf {R}_{a,b}=\\mathsf {R}_{a+1,b}+1$ and the map $V_{[a,b]}\\rightarrow V_{[a+1,b]}$ has 1-dimensional kernel.",
"This kernel must be spanned by the vector $(1,0,...,0,0)$ .",
"The remaining results follow by a dual argument on the sequence $(\\mathsf {R}_{a,b}-\\mathsf {R}_{a,b-1})$ .",
"Given a rank matrix $\\mathsf {R}$ and a consecutive subset $I\\subset \\mathbb {Z}$ , an $\\mathsf {R}$ -schedule for $I$ is a subset $J\\subset I$ for which there is a sequence of subintervals $[a_0,b_0] \\subset [a_1,b_1] \\subset [a_2,b_2] \\subset \\cdots \\subset I$ such that $b_i-a_i=i$ , $\\bigcup [a_i,b_i] = I$ , and $|J\\cap [a_i,b_i]| = \\mathsf {R}_{a_i,b_i}$ .",
"Note that the sequence of intervals determines the $\\mathsf {R}$ -schedule, and, for all $i$ , $J\\cap [a_i,b_i]$ is an $\\mathsf {R}$ -schedule for $[a_i,b_i]$ .",
"Lemma 7.5 Given a subspace $V\\subset \\mathsf {k}^\\mathbb {Z}$ with rank matrix $\\mathsf {R}$ , and an $\\mathsf {R}$ -schedule $J$ for a subset $I$ , the restriction map $V_{I}\\rightarrow \\mathsf {k}^J$ is an isomorphism.",
"In particular, if $J$ is an $\\mathsf {R}$ -schedule for $\\mathbb {Z}$ , then $V\\rightarrow \\mathsf {k}^J$ is an isomorphism.",
"We prove the case when $I=[a,b]$ by induction on $n:=b-a$ .",
"If $n<0$ , the lemma holds vacuously.",
"Assume that the lemma holds for all intervals shorter than $n$ .",
"Choose a sequence of subintervals as in (REF ), and set $[a^{\\prime },b^{\\prime }]:=[a_{n-1},b_{n-1}]$ .",
"The restriction maps fit into a commutative diagram.",
"$\\begin{tikzpicture}[baseline=(current bounding box.center)]\\node (V2) at (0,0) {V_{[a,b]}};\\node (V3) at (3,0) {V_{[a^{\\prime },b^{\\prime }]}};\\node (k2) at (0,-1.5) {\\mathsf {k}^{J}};\\node (k3) at (3,-1.5) {\\mathsf {k}^{J\\cap [a^{\\prime },b^{\\prime }]}};[->>] (V2) to (V3);[->] (V2) to node[left] {\\pi _{J}} (k2);[->] (V3) to node[right] {\\pi _{J\\cap [a^{\\prime },b^{\\prime }]}} (k3);[->>] (k2) to (k3);\\end{tikzpicture}$ By the inductive hypothesis, $\\pi _{J\\cap [a^{\\prime },b^{\\prime }]}$ is an isomorphism, and so $V_{[a,b]}\\rightarrow \\mathsf {k}^{T_{[a^{\\prime },b^{\\prime }]}}$ is surjective.",
"Since $b^{\\prime }-a^{\\prime }=n-1$ , either $[a^{\\prime },b^{\\prime }]=[a+1,b]$ or $[a^{\\prime },b^{\\prime }]=[a,b-1]$ .",
"We have three cases.",
"If $\\mathsf {R}_{a^{\\prime },b^{\\prime }}=\\mathsf {R}_{a,b}$ , then $J\\cap [a^{\\prime },b^{\\prime }]=J$ and so the bottom arrow is an isomorphism.",
"Therefore, $\\pi _J:V_{[a,b]}\\rightarrow \\mathsf {k}^J$ is surjective.",
"If $\\mathsf {R}_{a^{\\prime },b^{\\prime }}=\\mathsf {R}_{a,b}-1$ and $[a^{\\prime },b^{\\prime }]=[a+1,b]$ , then there are no defects in $\\lbrace a\\rbrace \\times [a,b]$ , and so $V_{[a,b]}$ contains $(1,0,...,0,0)$ (by Proposition REF .4).",
"Since $|J| =|J\\cap [a^{\\prime },b^{\\prime }]|+1$ , $a\\in J$ and so the image of $(1,0,...,0,0)$ under $\\pi _{J}$ is non-zero and spans the kernel of $\\mathsf {k}^{J}\\rightarrow \\mathsf {k}^{J\\cap [a^{\\prime },b^{\\prime }]}$ .",
"Thus, $\\pi _J:V_{[a,b]}\\rightarrow \\mathsf {k}^{J}$ is surjective.",
"If $\\mathsf {R}_{a^{\\prime },b^{\\prime }}=\\mathsf {R}_{a,b}-1$ and $[a^{\\prime },b^{\\prime }]=[a,b-1]$ , then there are no defects in $[a,b] \\times \\lbrace b\\rbrace $ , and so $V_{[a,b]}$ contains $(0,0,...,0,1)$ (by Proposition REF .3).",
"By an analogous argument to the previous case, $\\pi _J:V_{[a,b]}\\rightarrow \\mathsf {k}^{J}$ is surjective.",
"The map $\\pi _J:V_{[a,b]}\\rightarrow \\mathsf {k}^{J}$ is surjective in all cases.",
"Since $ \\dim (V_{[a,b]}) = \\mathsf {R}_{[a,b]}= J = \\dim (\\mathsf {k}^{J}) $ this map is an isomorphism, completing the induction.",
"For infinite $I$ , the lemma follows since $I=\\bigcup [a_i,b_i]$ and the lemma holds on each $[a_i,b_i]$ .",
"Recurrence matrices from rank matrices We can now characterize when a subspace of $\\mathsf {k}^\\mathbb {Z}$ is the kernel of a reduced recurrence matrix.",
"Theorem 7.6 Given a subspace $V$ of $\\mathsf {k}^\\mathbb {Z}$ , the following are equivalent.",
"$V$ is the space of solutions to a linear recurrence $\\mathsf {C}\\mathsf {x}=\\mathsf {0}$ .",
"The only left-bounded sequence in $V$ is the zero sequence; that is, if $\\mathsf {v}\\in V$ and $\\mathsf {v}_i=0$ for all $i\\ll 0$ , then $\\mathsf {v}_i=0$ for all $i$ .",
"Every column of the rank matrix $\\mathsf {R}$ of $V$ contains a defect.",
"$V$ is the space of solutions to a reduced linear recurrence $\\overline{\\mathsf {C}}\\mathsf {x} =\\mathsf {0}$ .",
"The shape of $\\overline{\\mathsf {C}}$ is the function $S:\\mathbb {Z}\\rightarrow \\mathbb {Z}$ such that $(S(b),b)$ is a defect of $\\mathsf {R}$ .",
"$(4 \\Rightarrow 1)$ is automatic.",
"$(1 \\Rightarrow 2)$ If a sequence $\\mathsf {v}$ solves a linear recurrence, then every term in $\\mathsf {v}$ is equal to a linear combination of previous terms in the sequence.",
"If every term in $\\mathsf {v}$ of sufficiently negative index is 0, then recursively every term must be 0.",
"$(\\text{not } 3 \\Rightarrow \\text{not }2)$ Assume that the $b$ th column of the rank matrix of $V$ does not contain a defect.",
"By Proposition REF .3, $V_{[a,b]}$ contains the vector $(0,0,...,0,1)$ for all $a\\le b$ .",
"It follows that $V_{(-\\infty ,b]}$ contains the vector $(...,0,0,1)$ .",
"This implies that $V$ contains a sequence $\\mathsf {v}$ with $\\mathsf {v}_b=1$ and $\\mathsf {v}_a=0$ whenever $a<b$ .",
"$(3 \\Rightarrow 4)$ Assume that there is a function $S:\\mathbb {Z}\\rightarrow \\mathbb {Z}$ such that $(S(b),b)$ is a defect of $\\mathsf {R}$ for each $b$ .",
"For each interval $[a,b]$ , define the $\\mathsf {R}$ -schedule $T_{[a,b]} := [a,b] \\setminus S([a,b])$ .",
"We note that $T_{[S(b),b-1]} \\cup \\lbrace b\\rbrace = T_{[S(b)+1,b]}\\cup \\lbrace S(b) \\rbrace $ and consider the following commutative diagram.",
"$ \\begin{tikzpicture}\\node (V1) at (-3,0) {V_{[S(b),b-1]}};\\node (V2) at (0,0) {V_{[S(b),b]}};\\node (V3) at (3,0) {V_{[S(b)+1,b]}};\\node (k1) at (-3,-1.5) {\\mathsf {k}^{T_{[S(b),b-1]}}};\\node (k2) at (0,-1.5) {\\mathsf {k}^{T_{[S(b),b-1]}\\cup \\lbrace b\\rbrace }};\\node (k3) at (3,-1.5) {\\mathsf {k}^{T_{[S(b)+1,b]}}};[->] (V2) to (V1);[->] (V2) to (V3);[->] (V1) to (k1);[->] (V2) to (k2);[->] (V3) to (k3);[->] (k2) to (k1);[->] (k2) to (k3);\\end{tikzpicture}$ Since $(S(b),b)$ is a defect, the maps in the top row are isomorphisms.",
"Since $T_{[S(b),b-1]}$ and $T_{[S(b)+1,b]}$ are $\\mathsf {R}$ -schedules, the maps on the left and right are isomorphisms (by Lemma REF ).",
"Therefore, the map $V_{[S(b),b]} \\longrightarrow \\mathsf {k} ^{J }$ is an embedding of codimension 1.",
"Its image is defined by a relation (unique up to scaling) of the form $\\sum _{a\\in T_{[S(b),b-1]}\\cup \\lbrace b\\rbrace } \\mathsf {C}_{b,a}x_{a} = 0$ Because the left and right maps are isomorphisms, $\\mathsf {C}_{b,b}\\ne 0$ and $\\mathsf {C}_{S(b),b}\\ne 0$ .",
"Rescaling the relation as necessary, we assume that $\\mathsf {C}_{b,b}=1$ .",
"Construct a recurrence matrix $\\mathsf {C}$ such that, for each $b$ , the $b$ th row collects the coefficients of the corresponding equation (REF ).",
"For any pair $b<a$ , $S(b)\\notin T_{[S(a)+1,a]}$ and so $\\mathsf {C}_{a,S(b)}=0$ .",
"Therefore, $\\mathsf {C}$ is a reduced linear recurrence of shape $S$ , such that $V\\subseteq \\mathrm {ker}(\\mathsf {C})$ .",
"Consider any interval $[a,b]$ .",
"For each $b^{\\prime }\\in [a,b] \\setminus T_{[a,b]}$ , the corresponding relation (REF ) only involves terms with index in $[a,b]$ .",
"Since these relations are linearly independent, the codimension of $\\mathrm {ker}(\\mathsf {C})_{[a,b]}$ in $\\mathsf {k}^{[a,b]}$ is at least the cardinality of $[a,b]\\setminus T_{[a,b]}$ .",
"Therefore, $ \\dim (\\mathrm {ker}(\\mathsf {C})_{[a,b]})\\le |T_{[a,b]}| = \\mathsf {R}_{a,b} = \\dim (V_{[a,b]}) $ Since $V_{[a,b]}\\subseteq \\mathrm {ker}(\\mathsf {C})_{[a,b]}$ , $V_{[a,b]}=\\mathrm {ker}(\\mathsf {C})_{[a,b]}$ .",
"Since this holds for all intervals, $V=\\mathrm {ker}(\\mathsf {C})$ .",
"From rank matrices to shapes The theorem relates the shape of a reduced recurrence matrix $\\mathsf {C}$ to the defects of the rank matrix $\\mathsf {R}$ of $\\mathrm {ker}(\\mathsf {C})$ , as follows.",
"Corollary 7.7 If $\\mathsf {C}$ is a reduced recurrence matrix, then $(a,b)$ is a pivot of $\\mathsf {C}$ if and only if $(b,a)$ is a defect of the rank matrix of $\\mathrm {ker}(\\mathsf {C})$ .",
"Therefore, we may translate several earlier results into the language of shapes.",
"Definition 7.8 Given a non-increasing injection $S$ , an $S$ -schedule for a subset $I\\subset \\mathbb {Z}$ is a subset $J\\subset I$ for which there is a subsequence of subintervals $[a_0,b_0] \\subset [a_1,b_1] \\subset [a_2,b_2] \\subset \\cdots \\subset I$ such that $b_i-a_i=i$ , $\\bigcup [a_i,b_i] = I$ , and $|J\\cap [a_i,b_i]|$ equals the number of $S$ -balls in $[a_i,b_i]$ .",
"If $S$ is the shape of a reduced recurrence matrix $\\mathsf {C}$ and $\\mathsf {R}$ is the rank matrix of $\\mathrm {ker}(\\mathsf {C})$ , then $\\mathsf {R}$ -schedules and $S$ -schedules coincide.",
"Note that $I$ only admits an $S$ -schedule if $I$ consists of consecutive elements.",
"The following is a direct translation of Lemma REF .",
"Proposition 7.9 Let $\\mathsf {C}$ be a reduced recurrence matrix with shape $S$ , and let $J$ be an $S$ -schedule for $I$ .",
"Then the restriction map $\\mathrm {ker}(\\mathsf {C})_{I}\\rightarrow \\mathsf {k}^J$ is an isomorphism.",
"In particular, if $J$ is an $S$ -schedule for $\\mathbb {Z}$ , then $\\mathrm {ker}(\\mathsf {C})\\rightarrow \\mathsf {k}^J$ is an isomorphism.",
"As a special case, for any $b\\in \\mathbb {Z}$ , the sequence of intervals $[b,b] \\subset [b-1,b] \\subset [b-2,b] \\subset \\cdots $ determines the following $S$ -schedule for $\\mathbb {Z}$ : $ T_b := \\bigcup _{a\\le b} T_{[a,b]} = \\lbrace a\\le b \\mid \\forall c\\le b, S(c) \\ne a \\rbrace = (-\\infty ,b] \\setminus S\\left( ( -\\infty ,b]\\right) $ Therefore, the proposition specializes to the following.",
"Theorem 7.10 The restriction map $\\pi _{T_b}:\\mathrm {ker}(\\mathsf {C})\\rightarrow \\mathsf {k}^{T_b}$ is an isomorphism.",
"Since $T_b$ contains a unique representative of each $S$ -ball, this implies the following.",
"Theorem 7.11 Then dimension of $\\mathrm {ker}(\\mathsf {C})$ equals the number of $S$ -balls.",
"Constructing $S$ -schedules is easy and intuitive using the juggling pattern of $S$ .",
"Consider any zigzagging path in the juggling pattern which starts on the main diagonal, only travels up (northwest) or right (northeast), and ends above the $(a,b)$ th entry.",
"$\\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][use as bounding box] (-9.5,3.3) rectangle (6.3,-0.3);[matrix of math nodes,matrix anchor = M-2-24.center,nodes in empty cells,inner sep=0pt,gthrow/.style={dark green,draw,circle,inner sep=0mm,minimum size=5mm},rthrow/.style={dark red,draw,circle,inner sep=0mm,minimum size=5mm},bthrow/.style={dark blue,draw,circle,inner sep=0mm,minimum size=5mm},pthrow/.style={dark purple,draw,circle,inner sep=0mm,minimum size=5mm},nodes={anchor=center,node font=\\scriptsize ,rotate=45},column sep={0.4cm,between origins},row sep={-0.4cm,between origins},] (M) at (0,0) {\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& |[gthrow]| \\& \\& |[rthrow]| \\& \\& |[bthrow]| \\& \\& |[bthrow]| \\& \\& |[pthrow]| \\& \\& |[gthrow]| \\& \\& |[rthrow,rotate=-45]| 1 \\& \\& |[bthrow,rotate=-45]| 2 \\&\\& |[gthrow,rotate=-45]| 3 \\& |[inner sep=8pt]| \\& |[pthrow,rotate=-45]| 4 \\& \\& |[bthrow,rotate=-45]| 5 \\& \\& |[bthrow,rotate=-45]| 6 \\& \\& |[rthrow,rotate=-45]| 7 \\& \\& |[gthrow,rotate=-45]| 8 \\& \\& |[pthrow]| \\& \\& |[bthrow]| \\&\\& |[gthrow]| \\& \\& |[rthrow]| \\& \\& |[bthrow]| \\& \\\\\\cdots \\& \\& \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\cdots \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[gthrow]| \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[gthrow]| \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& |[gthrow]| \\& \\& |[rthrow]| \\& \\& \\& \\& \\& \\& |[pthrow]| \\& \\& \\& \\& \\& \\& \\& \\& |[gthrow]| \\& \\& |[pthrow]| \\& \\& \\& \\& \\& \\& |[rthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\cdots \\\\\\& \\& \\& |[pthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[rthrow]| \\& |[inner sep=8pt]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[pthrow]| \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\};\\end{tikzpicture}[dark green] (M-3-1) to (M-2-2) to (M-7-7) to (M-2-12) to (M-5-15) to (M-2-18) to (M-7-23) to (M-2-28) to (M-5-31) to (M-2-34) to (M-7-39);[dark red] (M-5-1) to (M-2-4) to (M-7-9) to (M-2-14) to (M-8-20) to (M-2-26) to (M-7-31) to (M-2-36) to (M-5-39);[dark blue] (M-3-1) to (M-5-3) to (M-2-6) to (M-3-7) to (M-2-8) to (M-6-12) to (M-2-16) to (M-5-19) to (M-2-22) to (M-3-23) to (M-2-24) to (M-6-28) to (M-2-32) to (M-5-35) to (M-2-38) to (M-3-39);[dark purple] (M-5-1) to (M-8-4) to (M-2-10) to (M-7-15) to (M-2-20) to (M-7-25) to (M-2-30) to (M-8-36) to (M-5-39);$ [dashed] (M-2-13.center) to (M-10-21.center) to (M-2-29.center); [thick] (M-2-19.center) to (M-3-18.center) to (M-4-19.center) to (M-5-18.center) to (M-9-22.center)to (M-10-21.center); $Each time the path crosses a colored line, record the row (if it is ascending) or the column (if it is descending).The resulting subset is an $ $-schedule for the interval $ [a,b]$, and every $ S$-schedule can be constructed this way.In the picture above, the path in black determines the $ S$-schedule $ {3,4,2,7}$ for the interval $ [1,8]$.",
"The set $ Tb$ comes from the path which starts to the right of $ (b,b)$ and only travels up (northeast).$ Properties of the solution matrix Fix a reduced recurrence matrix $\\mathsf {C}$ of shape $S$ for the rest of the section.",
"Vanishing The unitriangularity of $\\mathsf {Adj}(\\mathsf {C})$ and $\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})$ mean that the solution matrix $ \\mathsf {Sol}(\\mathsf {C}) := \\mathsf {Adj}(\\mathsf {C}) - \\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})$ has zeroes between the support of $\\mathsf {Adj}(\\mathsf {C})$ and $\\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})$ , which we make precise as follows.",
"Proposition 8.1 $\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ whenever $a<b$ and there is no $c\\le b$ with $S(c)=a$ .",
"If $S$ is bijective, this can be restated as $\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ whenever $a<b<S^{-1}(a)$ .",
"By unitriangularity, $\\mathsf {Adj}(\\mathsf {C})_{a,b}=0$ whenever $a<b$ .",
"Dually, $(\\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P}))_{a,b}$ is only non-zero if there is a $c$ with $\\mathsf {P}_{a,c}\\ne 0$ and $\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})_{c,b}\\ne 0$ ; that is, if $S(c)=a$ and $c\\ge b$ .",
"For fixed $b$ , the proposition determines the value of the $b$ th column of $\\mathsf {Sol}(\\mathsf {C})$ on the set $T_b$ .",
"Since this column solves $\\mathsf {C}\\mathsf {x}=0$ , these entries determine the column (Theorem REF ).",
"Corollary 8.2 The $b$ th column of $\\mathsf {Sol}(\\mathsf {C})$ is the unique solution to $\\mathsf {C}\\mathsf {x}=\\mathsf {0}$ for which $x_a=0$ whenever $a<b$ but there is no $c\\le b$ with $S(c)=a$ , and $x_b=1$ unless $S(b)=b$ , in which case $x_b=0$ .",
"We also have a vanishing condition which guarantees consecutive zeros in each column.",
"Proposition 8.3 $\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ whenever $S(b) < a < b$ .",
"We proceed by induction on $b-a>0$ .",
"Assume that $\\mathsf {Sol}(\\mathsf {C})_{a+1,b}=...=\\mathsf {Sol}(\\mathsf {C})_{b-1,b}=0$ (which is vacuous for the base case $b-a=1$ ).",
"We split into two cases.",
"If there is no $c\\le b$ with $S(c)=a$ , then $\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ by Proposition REF .",
"Otherwise, there is a $c\\le b$ with $S(c)=a$ .",
"Since $S(b)<a$ , we know $c\\ne b$ and so $c<b$ .",
"The $(c,b)$ th entry of the equality $\\mathsf {C}\\mathsf {Sol}(\\mathsf {C})=0$ is $ \\mathsf {C}_{c,a}\\mathsf {Sol}(\\mathsf {C})_{a,b} + \\mathsf {C}_{c,a+1}\\mathsf {Sol}(\\mathsf {C})_{a+1,b} + ... + \\mathsf {C}_{c,c}\\mathsf {Sol}(\\mathsf {C})_{c,b} =0 $ By assumption, $\\mathsf {Sol}(\\mathsf {C})_{a+1,b}=...=\\mathsf {Sol}(\\mathsf {C})_{c,b}=0$ , and so $\\mathsf {C}_{c,a}\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ .",
"Since $\\mathsf {C}_{c,a}=\\mathsf {C}_{c,S(c)}$ is invertible, $\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ .",
"This completes the induction.",
"Propositions REF and REF can be visualized in terms of juggling patterns.",
"As in Remark REF , the $S$ -balls can be visualized by connecting each pivot entry with a line to the diagonal entries in the same row or column.",
"The dashed circle is not in an $S$ -ball.",
"$ \\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][matrix of math nodes,matrix anchor = M-4-8.center,gthrow/.style={dark green,draw,circle,inner sep=0mm,minimum size=5mm},bthrow/.style={dark blue,draw,circle,inner sep=0mm,minimum size=5mm},dashedthrow/.style={dashed,draw,circle,inner sep=0mm,minimum size=5mm},nodes in empty cells,inner sep=0pt,nodes={anchor=center,node font=\\footnotesize ,rotate=45},column sep={.5cm,between origins},row sep={.5cm,between origins},] (M) at (0,0) {\\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\&|[dashedthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& \\cdots \\\\\\cdots \\& \\& -1 \\& \\& -1 \\& \\& -1 \\& \\& \\& \\& \\& \\& |[gthrow]| -1 \\& \\& |[gthrow]| -1 \\& \\\\\\& |[gthrow]| -1 \\& \\& |[bthrow]| -1 \\& \\& |[gthrow]| -1 \\& \\& \\& \\& |[gthrow]| -1 \\& \\& \\& \\& \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\};\\end{tikzpicture}[dark blue,->] (M-2-1) to (M-1-2) to (M-3-4) to (M-1-6) to (M-4-9);[dark green] (M-2-1) to (M-3-2) to (M-1-4) to (M-3-6) to (M-1-8) to (M-3-10) to (M-1-12) to (M-2-13) to (M-1-14) to (M-2-15) to (M-1-16);$ $If we transpose and superimpose the juggling pattern onto the solution matrix...$ [baseline=(current bounding box.center), ampersand replacement=&, ] [matrix of math nodes, matrix anchor = M-8-8.center, gthrow/.style=dark green,draw,circle,inner sep=0mm,minimum size=5mm, bthrow/.style=dark blue,draw,circle,inner sep=0mm,minimum size=5mm, dashedthrow/.style=dashed,draw,circle,inner sep=0mm,minimum size=5mm, faded/.style=black!25, nodes in empty cells, inner sep=0pt, nodes=anchor=center,node font=,rotate=45, column sep=.5cm,between origins, row sep=.5cm,between origins, ] (M) at (0,0) & & & & & & & & & & & & & & & & 5 & & |[faded]| 0 & & 2 & & -1 & & 1 & & |[faded]| 0 & & 1 & & & & -3 & & |[faded]| 0 & & -1 & & 1 & & |[faded]| 0 & & 1 & & |[faded]| 0 & & 2 & & 2 & & |[faded]| 0 & & 1 & & |[faded]| 0 & & 1 & & |[faded]| 0 & & & & -1 & & -1 & & |[faded]| 0 & & |[faded]| 0 & & 1 & & |[faded]| 0 & & 1 & & |[gthrow]| 1 & & |[bthrow]| 1 & & |[gthrow]| 1 & & |[faded]| 0 & & |[gthrow]| 1 & & |[faded]| 0 & & 1 & & & & |[faded]| 0 & & |[faded]| 0 & & |[faded]| 0 & & |[faded]| 0 & & |[faded]| 0 & & |[gthrow]| 1 & & |[gthrow]| 1 & & |[bthrow]| 1 & & |[gthrow]| 1 & & |[bthrow]| 1 & & |[gthrow]| 1 & & |[dashedthrow,faded]| 0 & &|[gthrow]| 1 & & |[gthrow]| 1 & & & & 1 & & 1 & & 1 & & |[faded]| 0 & & |[faded]| 0 & & 1 & & 1 & & 2 & & 2 & & 2 & & |[faded]| 0 & & 1 & & |[faded]| 0 & & 1 & & & & 3 & & 3 & & |[faded]| 0 & & 1 & & 1 & & |[faded]| 0 & & 1 & & 5 & & 5 & & |[faded]| 0 & & 2 & & 1 & & 1 & & |[faded]| 0 & & & & 8 & & |[faded]| 0 & & 3 & & 2 & & 1 & & 1 & & |[faded]| 0 & & 13 & & |[faded]| 0 & & 5 & & 3 & & 2 & & 1 & & 1 & & & & & & & & & & & & & & & & & ; [dark blue,->] (M-7-1) to (M-8-2) to (M-6-4) to (M-8-6) to (M-1-13); [dark green] (M-7-1) to (M-6-2) to (M-8-4) to (M-6-6) to (M-8-8) to (M-6-10) to (M-8-12) to (M-7-13) to (M-8-14) to (M-7-15) to (M-8-16); $Propositions \\ref {prop: colvan} and \\ref {prop: rowvan} state that the lines do not cross any non-zero entries.",
"The circles must be non-zero, except the dashed circle, whose entire row and column must vanish.\\footnote {We note that all of these observations will be combined into and generalized by Lemma \\ref {lemma: boxballs}.",
"}$ Rank conditions The prior vanishing results can be generalized to a formula for ranks of certain rectangular submatrices of $\\mathsf {Sol}(\\mathsf {C})$ .",
"Given integers $a\\le b$ and $c\\le d$ , the box $[a,b]\\times [c,d]\\subset \\mathbb {Z}\\times \\mathbb {Z}$ indexes a rectangular submatrix of $\\mathsf {Sol}(\\mathsf {C})$ .",
"Define an $S$ -ball in the box $[a,b]\\times [c,d]$ to be an equivalence class in the set $ \\left(\\lbrace (i,i) \\mid S(i) \\ne i \\rbrace \\cup \\lbrace (S(j),j) \\mid S(j)\\ne j\\rbrace \\right) \\subset [a,b] \\times [c,d] $ under the equivalence relation generated by $(i,i) \\sim (S(i),i)\\sim (S(i),S(i))$ .",
"This is a 2-dimensional analog of $S$ -balls as defined in Section REF .",
"The relation is that $S$ -balls in an interval $[a,b]$ are in bijection with $S$ -balls in the box $[a,b]\\times [a,b]$ , and also with the $S$ -balls in $[a,b]\\times [c,d]$ and in $[c,d]\\times [a,b]$ for any $[c,d]\\supset [a,b]$ .",
"In terms of the juggling pattern, this counts the number of different colors of circles inside the rectangular submatrix indexed by $[a,b]\\times [c,d]$ .",
"$ \\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][matrix of math nodes,matrix anchor = M-8-8.center,gthrow/.style={dark green,draw,circle,inner sep=0mm,minimum size=5mm},bthrow/.style={dark blue,draw,circle,inner sep=0mm,minimum size=5mm},dashedthrow/.style={dashed,draw,circle,inner sep=0mm,minimum size=5mm},faded/.style={black!25},nodes in empty cells,inner sep=0pt,nodes={anchor=center,node font=\\footnotesize ,rotate=45},column sep={.5cm,between origins},row sep={.5cm,between origins},] (M) at (0,0) {\\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\\\\\& 5 \\& \\& |[faded]| 0 \\& \\& 2 \\& \\& -1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& \\cdots \\\\\\cdots \\& \\& -3 \\& \\& |[faded]| 0 \\& \\& -1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\\\\\& 2 \\& \\& 2 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& \\cdots \\\\\\cdots \\& \\& -1 \\& \\& -1 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\\\\\& |[gthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[faded]| 0 \\& \\& |[gthrow]| 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& \\cdots \\\\\\cdots \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& |[gthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\\\\\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[dashedthrow,faded]| 0 \\& \\&|[gthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& \\cdots \\\\\\cdots \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& 1 \\& \\\\\\& 2 \\& \\& 2 \\& \\& 2 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& \\cdots \\\\\\cdots \\& \\& 3 \\& \\& 3 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\\\\\& 5 \\& \\& 5 \\& \\& |[faded]| 0 \\& \\& 2 \\& \\& 1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& \\cdots \\\\\\cdots \\& \\& 8 \\& \\& |[faded]| 0 \\& \\& 3 \\& \\& 2 \\& \\& 1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\\\\\& 13 \\& \\& |[faded]| 0 \\& \\& 5 \\& \\& 3 \\& \\& 2 \\& \\& 1 \\& \\& 1 \\& \\& \\cdots \\\\\\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\\\};\\end{tikzpicture}[dark blue,->] (M-7-1) to (M-8-2) to (M-6-4) to (M-8-6) to (M-1-13);[dark green] (M-7-1) to (M-6-2) to (M-8-4) to (M-6-6) to (M-8-8) to (M-6-10) to (M-8-12) to (M-7-13) to (M-8-14) to (M-7-15) to (M-8-16);$ [dark red, fill=dark red!50,opacity=.25,rounded corners] (M-3-4.center) – (M-2-5.center) – (M-10-13.center) – (M-11-12.center) – cycle; [dark blue, fill=dark blue!50,opacity=.25,rounded corners] (M-6-9.center) – (M-3-12.center) – (M-7-16.center) – (M-10-13.center) – cycle; [dark purple, fill=dark purple!50,opacity=.25,rounded corners] (M-7-2.center) – (M-4-5.center) – (M-7-8.center) – (M-10-5.center) – cycle; $The {dark purple}{purple}, {dark red}{red}, and {dark blue}{blue} boxes above contain $ dark purple2$, $ dark red0$, and $ dark blue1$ $ S$-balls, respectively.$ The following lemma relates the rank of a submatrix to the number of $S$ -balls in the box.",
"Lemma 8.7 Let $\\mathsf {C}$ be a reduced recurrence matrix of shape $S$ .",
"For any integers $a\\le b$ and $c\\le d$ , the rank of $\\mathsf {Sol}(\\mathsf {C})_{[a,b]\\times [c,d]}$ is at least the number of $S$ -balls in the box $[a,b]\\times [c,d]$ , with equality when $b-c\\ge -1 \\text{ and }\\min (a-c,b-d)\\le 0$ Conceptually, Condition (REF ) implies the box has at least one corner on or below the first superdiagonal, and at least two corners on or above the main diagonal.",
"Condition (REF ) holds for the three boxes in Example REF , and one may check that their ranks coincide with the number of $S$ -balls they contain.",
"First, we bound the rank below by finding a full-rank submatrix of the appropriate size.",
"Fix $[a,b]\\times [c,d]$ .",
"Index $I:=[c,d]\\cap [a,b]\\setminus S([c,d])$ by $i_1<i_2<\\cdots < i_k$ , and index $ J:= \\lbrace j\\in [c,d] \\mid j\\notin [a,b] \\text{ and }S(j) \\in [a,b] \\rbrace $ by $j_1<j_2<\\cdots < j_\\ell $ .",
"Note that $i_k<j_1$ and $S(j_h)<i_1$ for all $h$ .",
"Each $S$ -ball in the box $[a,b]\\times [c,d]$ contains a unique final circle, by which we mean one of the two types of pair: $(i,i)\\in [a,b]\\times [c,d]$ such that there is no $j\\in [c,d]$ with $S(j)=i$ .",
"$(S(i),i) \\in [a,b]\\times [c,d]$ such that $(i,i)\\notin [a,b]\\times [c,d]$ .",
"Furthermore, each column in the box can contain at most one such circle.",
"The columns containing the first kind of final circle are indexed by $I$ , and the columns containing the second kind are indexed by $J$ .",
"Therefore, the number of $S$ -balls in the box is $|I|+|J|$ .",
"Consider the submatrixWe are stretching the definition of `submatrix' to allow for rearranging the order of the rows and columns.",
"of $\\mathsf {Sol}(\\mathsf {C})$ with row set $ \\lbrace i_k,i_{k-1},...,i_1,S(j_1),S(j_2),...,S(J_\\ell )\\rbrace $ and column set $ \\lbrace i_k,i_{k-1},...,i_1,j_1,j_2,...,j_\\ell \\rbrace $ .",
"This matrix is upper triangular with non-zero entries on the diagonal, and so it has rank $|I|+|J|$ .",
"Consequently, $\\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]})\\ge \\text{the number of $S$-balls in the box $[a,b]\\times [c,d]$}$ Note that this inequality did not assume Conditions (REF ).",
"Next, assume that $b-c\\ge -1$ and that $b-d\\le 0$ , and define disjoint sets $K&:=\\lbrace k \\in [a,b] \\text{ such that there is an $m\\in [a,b]$ with $S(m)=k$} \\rbrace \\\\L &:= \\lbrace \\ell \\in [a,c-1] \\text{ such that there is no $m\\le d$ with $S(m)=\\ell $} \\rbrace $ Construct a $(K \\cup L)\\times [a,b]$ -matrix $\\mathsf {M}$ , such that for each $k\\in K$ , the corresponding row of $\\mathsf {M}$ is $\\mathsf {C}_{S^{-1}(k),[a,b]}$ and for each $\\ell \\in L$ , the corresponding row of $\\mathsf {M}$ is $e_\\ell $ (the row vector with a 1 in the $\\ell $ th place).",
"Since $\\mathsf {C}\\mathsf {Sol}(\\mathsf {C})=0$ (Proposition REF ), $\\mathsf {M}_{k,[a,b]}\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]} =0$ for all $k\\in K$ .",
"Since each row of $\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]}$ indexed by $\\ell \\in L$ vanishes (Proposition REF )Explicitly: Since $\\ell \\in [a,b]$ but not in $[c,d]$ , and $b-d\\le 0$ , $\\ell <c$ .",
"Since $\\ell \\notin S( (-\\infty ,d])$ , either $S^{-1}(\\ell )>d$ or $S^{-1}(\\ell )$ is empty.",
"In either case, $\\mathsf {Sol}(\\mathsf {C})_{\\ell ,[c,d]}$ consists of zeroes by Proposition REF ., $\\mathsf {M}_{\\ell , [a,b]}\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]} =0$ for all $\\ell \\in L$ .",
"Therefore, the matrix product vanishes: $ \\mathsf {M}\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]} =0$ Since the first non-zero entry in the $k$ th row of $\\mathsf {M}$ is in the $k$ th column and the first non-zero entry in the $\\ell $ th row of $\\mathsf {M}$ is in the $\\ell $ th column, $\\mathsf {M}$ is in row-echelon form and has rank $|K \\cup L|$ .",
"Therefore, $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]})\\le \\dim (\\mathrm {ker}(\\mathsf {M})) = |[a,b]\\setminus (K\\cup L) |$ Next, we note that $K\\cup L$ indexes the rows of the box $[a,b]\\times [c,d]$ which do not contain the final circle of an $S$ -ball in the box.",
"Combined with the lower bound (REF ), $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]})= \\text{the number of $S$-balls in the box $[a,b]\\times [c,d]$}$ This establishes the theorem in the case when $b-d\\le 0$ .",
"For the final case, we need a source of relations among the columns of $\\mathsf {Sol}(\\mathsf {C})$ .",
"Proposition 8.9 $\\mathsf {Sol}(\\mathsf {C})(\\mathsf {C}\\mathsf {P})=0$ .",
"We compute directly.",
"$\\mathsf {Sol}(\\mathsf {C})(\\mathsf {C}\\mathsf {P})&= \\mathsf {Adj}(\\mathsf {C})(\\mathsf {C}\\mathsf {P}) - (\\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P}) (\\mathsf {C}\\mathsf {P}) \\\\&\\stackrel{*}{=} (\\mathsf {Adj}(\\mathsf {C})\\mathsf {C})\\mathsf {P}- \\mathsf {P}(\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})(\\mathsf {C}\\mathsf {P}))= \\mathsf {P}- \\mathsf {P}= \\mathsf {0}$ Equality ($*$ ) holds because $\\mathsf {P}$ is a generalized permutation matrix.",
"Finally, assume that $b-c\\ge -1$ and that $a-c\\le 0$ , and define disjoint sets $K&:=\\lbrace k \\in [c,d] \\text{ such that $S(k)\\in [c,d]$} \\rbrace \\\\L &:= \\lbrace \\ell \\in [b+1,d] \\text{ such that $S(\\ell )<a$} \\rbrace $ Construct a $[c,d]\\times (K \\cup L)$ -matrix $\\mathsf {M}$ , such that for each $k\\in K$ , the corresponding column of $\\mathsf {M}$ is $\\mathsf {C}_{[a,b],k}$ and for each $\\ell \\in L$ , the corresponding column of $\\mathsf {M}$ is $e_\\ell $ (the column vector with a 1 in the $\\ell $ th place).",
"Since $\\mathsf {Sol}(\\mathsf {C})\\mathsf {C}\\mathsf {P}=0$ (Proposition REF ), $\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]} \\mathsf {M}_{[c,d],k}=0$ for all $k\\in K$ .",
"Since each column of $\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]}$ indexed by $\\ell \\in L$ vanishes (Proposition REF ), $\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]}\\mathsf {M}_{[c,d],\\ell } =0$ for all $\\ell \\in L$ .",
"Therefore, the matrix product vanishes: $ \\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]}\\mathsf {M}=0$ The transpose $\\mathsf {M}^\\top $ is in row echelon form and has rank $|K \\cup L|$ .",
"Therefore, $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]})\\le \\dim (\\mathrm {ker}(\\mathsf {M}^\\top )) = |[c,d]\\setminus (K\\cup L) |$ Next, we note that $K\\cup L$ indexes the columns of the box $[a,b]\\times [c,d]$ which do not contain the `initial circle' of an $S$ -ball in the box.",
"Combined with the lower bound (REF ), $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]})= \\text{the number of $S$-balls in the box $[a,b]\\times [c,d]$} $ Lemma REF extends to improper intervals such as $(-\\infty , \\infty ) $ and $ (-\\infty , d]$ by considering appropriate limits, as the number of $S$ -balls in a box is monotonic in nested intervals.",
"For example, the rank of the entire matrix $\\mathsf {Sol}(\\mathsf {C})$ equals the number of $S$ -balls in the unbounded `box' $(-\\infty ,\\infty )\\times ( -\\infty ,\\infty )$ ; that is, the number of $S$ -balls.",
"The lemma allows us to prove the following fundamental result.",
"Theorem 8.10 If $\\mathsf {C}$ is reduced, the kernel of $\\mathsf {C}$ equals the image of $\\mathsf {Sol}(\\mathsf {C})$ .",
"By Proposition REF , $\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C})) \\subseteq \\mathrm {ker}(\\mathsf {C})$ .",
"For any interval $[a,b]$ and any $S$ -schedule $J$ of $[a,b]$ , Proposition REF implies that $ \\dim (\\mathrm {ker}(\\mathsf {C})_{[a,b]})=|J|$ , and Lemma REF implies that $\\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b], \\mathbb {Z}}) = \\text{\\# of $S$-balls in the box $[a,b]\\times \\mathbb {Z}$} $ The number of $S$ -balls in $[a,b]\\times \\mathbb {Z}$ equals the number of $S$ -balls in $[a,b]$ (Remark REF ), which in turn equals $|J|$ .",
"Thus, $\\mathrm {ker}(\\mathsf {C})_{[a,b]}=\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C}))_{[a,b]}$ for all intervals $[a,b]$ , and so $\\mathrm {ker}(\\mathsf {C})=\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C}))$ .",
"Bases for solutions We can minimally parametrize the kernel using $S$ -schedules.See Definition REF ; note that $I$ only admits an $S$ -schedule if $I$ is a set of consecutive elements.",
"Proposition 8.11 Let $\\mathsf {C}$ be a reduced recurrence matrix with shape $S$ , and let $J$ be an $S$ -schedule for $I$ .",
"Then multiplication by $\\mathsf {Sol}(\\mathsf {C})_{I,J}$ gives an isomorphism $\\mathsf {k}^J\\rightarrow \\mathrm {ker}(\\mathsf {C})_I$ .",
"If $J$ is an $S$ -schedule for $\\mathbb {Z}$ , then multiplication by $\\mathsf {Sol}(\\mathsf {C})_{\\mathbb {Z},J}$ is an isomorphism $\\mathsf {k}^J\\rightarrow \\mathrm {ker}(\\mathsf {C})$ .",
"We first consider the the case when $I$ is finite.",
"Consider any interval $[c,d]$ such that $I\\subset [c,d]$ .",
"By Remark REF , the number of $S$ -balls in $[c,d]\\times I$ equals the number of $S$ -balls in $I$ , which is equal to $J$ by Definition REF .",
"By Lemma REF , this implies that $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],I}) = |J| $ .",
"Since $J\\subset I$ , this implies that $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],J}) \\le |J| $ .",
"We will prove that $\\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],J}) = |J|$ by induction on the number of elements in $I$ .",
"If $I=\\varnothing $ , the claim holds vacuously.",
"Assume that the claim holds for all $S$ -schedules of intervals shorter than $I$ , and choose a sequence of subintervals as in Definition REF .",
"Since $I$ is finite, the sequence terminates at $[a_n,b_n]=I$ .",
"Since $J^{\\prime } := J\\cap [a_{n-1},b_{n-1}]$ is an $S$ -schedule for $[a_{n-1},b_{n-1}]$ , by the inductive hypothesis, $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],J^{\\prime }}) = |J^{\\prime }| $ .",
"If $J^{\\prime }=J$ , this immediately implies (REF ).",
"If $J^{\\prime }\\ne J$ , then $|J| = |J^{\\prime }|+1$ , and so $\\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],[a_{n},b_{n}]} ) = \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],[a_{n-1},b_{n-1}]} ) +1 $ Hence, the column indexed by the unique element of $[a_n,b_n]\\setminus [a_{n-1},b_{n-1}]$ is linearly independent from the other columns.",
"Since this is also the unique element in $J\\setminus J^{\\prime }$ , $\\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],J} ) = \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],J^{\\prime }} ) +1 $ This proves (REF ) and completes the induction.",
"Since $\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C}))= \\mathrm {ker}(\\mathsf {C})$ , we know that $\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C})_{I,J}) \\subseteq \\mathrm {ker}(\\mathsf {C})_I$ .",
"Equation REF for $I=[c,d]$ and Proposition REF imply these both have dimension $|J|$ , so $\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C})_{I,J}) = \\mathrm {ker}(\\mathsf {C})_I$ .",
"When $I$ is infinite, Definition REF and the preceding argument guarantee it is a union of finite intervals on which the proposition holds, so $\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C})_{I,J}) = \\mathrm {ker}(\\mathsf {C})_I$ .",
"The special case of the $S$ -schedule $T_b$ for $\\mathbb {Z}$ yields the following.",
"Theorem 8.12 Multiplication by $\\mathsf {Sol}(\\mathsf {C})_{\\mathbb {Z}\\times T_b}$ gives an isomorphism $\\mathsf {k}^{T_b}\\rightarrow \\mathrm {ker}(\\mathsf {C})$ .",
"appendix"
],
[
"Constructing the solutions",
"In this section, we construct all elements of the kernel of a reduced recurrence matrix, and thus all solutions to a reduced linear recurrence.",
"For $a\\le b\\in \\mathbb {Z}$ , let $[a,b]\\lbrace a,a+1,...,b\\rbrace \\subset \\mathbb {Z}$ .",
"For a $\\mathbb {Z}\\times \\mathbb {Z}$ -matrix $\\mathsf {C}$ and two sets $I,J\\subset \\mathbb {Z}$ , let $\\mathsf {C}_{I,J}$ denote the submatrix on row set $I$ and column set $J$ ."
],
[
"Adjugates",
"Given a lower unitriangular $\\mathbb {Z}\\times \\mathbb {Z}$ -matrix $\\mathsf {C}$ , define the adjugate $\\mathsf {Adj}(\\mathsf {C})$ of $\\mathsf {C}$ to be the lower unitriangular $\\mathbb {Z}\\times \\mathbb {Z}$ -matrix whose subdiagonal entries are defined by $ \\mathsf {Adj}(\\mathsf {C})_{a,b} (-1)^{a+b}\\mathrm {det}(\\mathsf {C}_{[a+1,b],[a,b-1]}) $ Three examples of adjugates are given in Figure REF .",
"The determinant of the finite matrix $\\mathsf {C}_{[a+1,b],[a,b-1]}$ coincides with the determinantWhile general $\\mathbb {Z}\\times \\mathbb {Z}$ -matrices may not have well-defined determinants, the cofactor matrices of lower unitriangular matrices are lower unitriangular outside of a finite square, and thus have a well-defined determinant.",
"of the infinite cofactor matrix $\\mathsf {C}_{\\mathbb {Z}\\setminus \\lbrace a\\rbrace ,\\mathbb {Z}\\setminus \\lbrace b\\rbrace }$ , justifying the name `adjugate'.",
"Like its finite matrix counterpart, the adjugate matrix has many useful properties.",
"Proposition 3.3 Let $\\mathsf {C}$ and $\\mathsf {D}$ be lower unitriangular $\\mathbb {Z}\\times \\mathbb {Z}$ -matrices.",
"$\\mathsf {Adj}(\\mathsf {C})\\mathsf {C}=\\mathsf {C}\\mathsf {Adj}(\\mathsf {C}) = \\mathrm {Id}$ .",
"$\\mathsf {Adj}(\\mathsf {Adj}(\\mathsf {C}))=\\mathsf {C}$ .",
"$\\mathsf {Adj}(\\mathsf {C}\\mathsf {D}) = \\mathsf {Adj}(\\mathsf {D})\\mathsf {Adj}(\\mathsf {C})$ .",
"For $I,J\\subset [a,b]$ with $|I|=|J|$ , $\\det (\\mathsf {Adj}(\\mathsf {C})_{I,J}) = (-1)^{\\sum I+\\sum J}\\det ({\\mathsf {C}}_{[a,b]\\setminus J,[a,b]\\setminus I}) $ .",
"Since each entry of $\\mathsf {Adj}(\\mathsf {C})$ only depends on a finite submatrix of $\\mathsf {C}$ , these follow from their finite matrix counterparts.",
"E.g.",
"(4) follows by considering the submatrix $\\mathsf {C}_{[a,b],[a,b]}$ .",
"The first three parts of the prior proposition can be rephrased as follows.",
"Proposition 3.4 The lower unitriangular $\\mathbb {Z}\\times \\mathbb {Z}$ -matrices form a group with inverse $\\mathsf {Adj}$ .",
"[.05cm] Since multiplication is not always associative, inverses may not be unique in the larger set of $\\mathbb {Z}\\times \\mathbb {Z}$ -matrices (see Remark REF ), and so we write $\\mathsf {Adj}(\\mathsf {C})$ instead of $\\mathsf {C}^{-1}$ .",
"The recurrence matrices do not form a subgroup of the lower unitriangular matrices, as they are not closed under $\\mathsf {Adj}$ .",
"In fact, the recurrence matrices whose adjugate is also a recurrence matrix are precisely the trivial ones.",
"thm: triv-invTheorem REF A recurrence matrix $\\mathsf {C}$ is trivial if and only if $\\mathsf {Adj}(\\mathsf {C})$ is a recurrence matrix.",
"Since $\\mathsf {Adj}(\\mathsf {C})$ is lower unitriangular by construction, $\\mathsf {C}$ is trivial if and only if $\\mathsf {Adj}(\\mathsf {C})$ is horizontally bounded.",
"The recurrence matrices form a semigroup whose subgroup of invertible elements is the set of trivial recurrence matrices (by Theorem REF ).",
"The left orbits of this subgroup on the set of recurrence matrices are the equivalence classes (by Theorem REF ), and the reduced recurrence matrices are a transverse of these orbits (by Theorem REF ).",
"Figure: Examples of adjugate and solution matrices"
],
[
"The solution matrix",
"In this section, we construct a matrix $\\mathsf {Sol}(\\mathsf {C})$ whose image is the kernel of a given reduced recurrence matrix $\\mathsf {C}$ .",
"The shape of a recurrence matrix $\\mathsf {C}$ is the non-increasing function $S:\\mathbb {Z}\\rightarrow \\mathbb {Z}$ defined by $ S(a) := \\min \\lbrace b \\in \\mathbb {Z}\\mid \\mathsf {C}_{a,b}\\ne 0\\rbrace $ The pivot entry in the $a$ th row is then $\\mathsf {C}_{a,S(a)}$ , and so $\\mathsf {C}$ is reduced if and only if $\\mathsf {C}_{b,S(a)}=0$ for all $b>a$ .",
"In particular, the shape of a reduced recurrence matrix must be injective.",
"The shape of the Fibonacci recurrence matrix is $S(a)=a-2$ .",
"The shape of the recurrence matrix depicted in Example REF is $ S(a) = \\left\\lbrace \\begin{array}{cc}a-2 & \\text{if $a<0$} \\\\a & \\text{if $a=0$} \\\\a-2 & \\text{if $a=1$} \\\\a-1 & \\text{if $a>1$}\\end{array}\\right\\rbrace $ Given a recurrence matrix $\\mathsf {C}$ with shape $S$ , define a $\\mathbb {Z}\\times \\mathbb {Z}$ -matrix $\\mathsf {P}$ by $ \\mathsf {P}_{a,b} \\left\\lbrace \\begin{array}{cc}(\\mathsf {C}_{b,S(b)})^{-1} & \\text{if }a=S(b) \\\\0 & \\text{otherwise}\\end{array}\\right\\rbrace $ This is the generalized permutation matrix such that $\\mathsf {CP}$ moves the pivots to the main diagonal and rescales them to 1.",
"Thus, $\\mathsf {C}$ is reduced if and only if $\\mathsf {C}\\mathsf {P}$ is upper unitriangular.A $\\mathbb {Z}\\times \\mathbb {Z}$ -matrix $\\mathsf {C}$ is upper unitriangular if its transpose $\\mathsf {C}^\\top $ is lower unitriangular.",
"Such matrices also have an adjugate, which may be defined as $\\mathsf {Adj}(\\mathsf {C}^\\top )^\\top $ , for which the analog of Proposition REF holds.",
"Given a reduced linear recurrence $\\mathsf {C}$ , define the solution matrix $\\mathsf {Sol}(\\mathsf {C})$ of $\\mathsf {C}$ by $ \\mathsf {Sol}(\\mathsf {C}) \\mathsf {Adj}(\\mathsf {C}) - \\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P}) $ Examples are given in Figure REF .",
"The solution matrix is named for the following property.",
"Proposition 3.8 Each column of $\\mathsf {Sol}(\\mathsf {C})$ is a solution to $\\mathsf {C}\\mathsf {x}=\\mathsf {0}$ .",
"$\\displaystyle \\mathsf {C}\\mathsf {Sol}(\\mathsf {C}) = \\mathsf {C}[\\mathsf {Adj}(\\mathsf {C}) - \\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})]= \\mathsf {C}\\mathsf {Adj}(\\mathsf {C}) - (\\mathsf {C}\\mathsf {P})\\mathsf {Adj}(\\mathsf {C}\\mathsf {P}) = \\mathrm {Id} - \\mathrm {Id} = \\mathsf {0}$ .",
"If $\\mathsf {C}$ is the Fibonacci recurrence matrix, the $a$ th column of $\\mathsf {Sol}(\\mathsf {C})$ is given by $x_b = \\left\\lbrace \\begin{array}{cc}F_{b-a+1} & \\text{if $b-a+1\\ge 0$} \\\\(-1)^{a-b}F_{-b+a-1} & \\text{if $b-a+1<0$}\\end{array}\\right\\rbrace $ where $F_i$ is the $i$ th Fibonacci number (indexed so that $F_0=0$ and $F_1=1$ ).",
"This proposition can be extended to a complete characterization of solutions to $\\mathsf {C}\\mathsf {x}=\\mathsf {0}$ , and thus an answer to our original problem.",
"Given a $\\mathbb {Z}\\times \\mathbb {Z}$ matrix $\\mathsf {A}$ , define the image of $\\mathsf {A}$ to be the set of all sequences $\\mathsf {v}\\in \\mathsf {k}^\\mathbb {Z}$ equal to $\\mathsf {Aw}$ for some $\\mathsf {w\\in \\mathsf {k}^\\mathbb {Z}}$ .This is the usual definition of `image', with the explicit caveat that $\\mathsf {Aw}$ may not exist for all $\\mathsf {w}$ .",
"thm: solTheorem REF If $\\mathsf {C}$ is reduced, the kernel of $\\mathsf {C}$ equals the image of $\\mathsf {Sol}(\\mathsf {C})$ .",
"Any solution to the Fibonacci linear recurrence can be written as a linear combination of the solutions of the form (REF ).In fact, it suffices to only use two adjacent columns of the solution matrix here; see Example REF .",
"If $\\mathsf {C}$ is not reduced, the adjugate $\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})$ may not be defined, and so $\\mathsf {Sol}(\\mathsf {C})$ may not be constructed."
],
[
"Balls and juggling",
"We now connect the shape of a reduced recurrence matrix to the geometry of its kernel.",
"Fix a reduced recurrence matrix $\\mathsf {C}$ of shape $S$ for the remainder of the section.",
"An $S$ -ball is an equivalence class of integers under the equivalence relation $a\\sim S(a)$ , which does not contain a single element.",
"The significance of this notion is the following.",
"thm: ballsTheorem REF The dimension of $\\mathrm {ker}(\\mathsf {C})$ is equal to the number of $S$ -balls.",
"For the Fibonacci recurrence matrix, the shape $S$ is given by $S(a)=a-2$ .",
"There are two $S$ -balls: the set of even numbers, and the set of odd numbers; and the space of solutions to the Fibonacci recurrence is two dimensional (see Example REF ).",
"The recurrence matrix in Example REF has two $S$ -balls: $ \\lbrace ...,-6,-4,-2\\rbrace \\text{ and } \\lbrace ...,-5,-3,-1,1,2,3,...\\rbrace $ Since $S(0)=0$ , the singleton set $\\lbrace 0\\rbrace $ is also an equivalence class, but it is not an $S$ -ball.",
"As a non-increasing injection from $\\mathbb {Z}\\rightarrow \\mathbb {Z}$ , the shape $S$ can be thought of as a juggling pattern: instructions for how a juggler catches and throws balls over time.",
"At each moment $a$ , the juggler catches the ball they threw at moment $S(a)$ and immediately throws it again...unless $S(a)=a$ , in which case they neither catch nor throw a ball.At any moment $a$ which is not in the image of $S$ , the juggler throws the ball so high it never returns.",
"The ability to throw at escape velocity is a small stretch of the imagination for a juggler who is also immortal.",
"Each $S$ -ball lists those moments when a given ball is caught and thrown, and so the number of $S$ -balls equals the number of physical balls the juggler needs for the pattern.",
"The $S$ -balls of $\\mathsf {C}$ can be visualized as follows.",
"Circle each pivot entry and each entry on the main diagonal, and connect pairs of circles in the same column or in the same row.",
"Excluding any unconnected circles, the connected components of the resulting graph are the $S$ -balls.",
"There are four $S$ -balls below, each drawn in a different color.",
"$\\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][use as bounding box] (-9.5,-2.7) rectangle (6.3,0.3);[matrix of math nodes,matrix anchor = M-2-24.center,nodes in empty cells,inner sep=0pt,gthrow/.style={dark green,draw,circle,inner sep=0mm,minimum size=5mm},rthrow/.style={dark red,draw,circle,inner sep=0mm,minimum size=5mm},bthrow/.style={dark blue,draw,circle,inner sep=0mm,minimum size=5mm},pthrow/.style={dark purple,draw,circle,inner sep=0mm,minimum size=5mm},nodes={anchor=center,node font=\\scriptsize ,rotate=45},column sep={0.4cm,between origins},row sep={0.4cm,between origins},] (M) at (0,0) {\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& |[gthrow]| 1 \\& \\& |[rthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\& |[pthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[rthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[pthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\& |[rthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[pthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[rthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\\\\\cdots \\& \\& 5 \\& \\& 1 \\& \\& |[bthrow]| 1 \\& \\& 3 \\& \\& 2 \\& \\& 1 \\& \\& 5 \\& \\& 1 \\& \\& 5 \\& \\& 1 \\& \\& |[bthrow]| 1 \\& \\& 3 \\& \\& 2 \\& \\& 1 \\& \\& 5 \\& \\& 1 \\& \\& 5 \\& \\& 1 \\& \\& \\cdots \\\\\\& 3 \\& \\& 2 \\& \\& \\& \\& \\& \\& 5 \\& \\& 1 \\& \\& 3 \\& \\& 2 \\& \\& 3 \\& \\& 2 \\& \\& \\& \\& \\& \\& 5 \\& \\& 1 \\& \\& 3 \\& \\& 2 \\& \\& 3 \\& \\& 2 \\& \\& \\& \\\\\\cdots \\& \\& |[bthrow]| 1 \\& \\& \\& \\& -2 \\& \\& \\& \\& 2 \\& \\& 1 \\& \\& |[gthrow]| 1 \\& \\& 1 \\& \\& |[bthrow]| 1 \\& \\& \\& \\& -2 \\& \\& \\& \\& 2 \\& \\& 1 \\& \\& |[gthrow]| 1 \\& \\& 1 \\& \\& |[bthrow]| 1 \\& \\& \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& -1 \\& \\& -3 \\& \\& \\& \\& |[bthrow]| 1 \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& -1 \\& \\& -3 \\& \\& \\& \\& |[bthrow]| 1 \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& -1 \\& \\\\\\& \\& \\& \\& \\& \\& |[gthrow]| -1 \\& \\& |[rthrow]| -1 \\& \\& \\& \\& \\& \\& |[pthrow]| -1 \\& \\& \\& \\& \\& \\& \\& \\& |[gthrow]| -1 \\& \\& |[pthrow]| -1 \\& \\& \\& \\& \\& \\& |[rthrow]| -1 \\& \\& \\& \\& \\& \\& \\& \\& \\cdots \\\\\\& \\& \\& |[pthrow]| 1 \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[rthrow]| 1 \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[pthrow]| 1 \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\};\\end{tikzpicture}[dark green] (M-3-1) to (M-2-2) to (M-7-7) to (M-2-12) to (M-5-15) to (M-2-18) to (M-7-23) to (M-2-28) to (M-5-31) to (M-2-34) to (M-7-39);[dark red] (M-5-1) to (M-2-4) to (M-7-9) to (M-2-14) to (M-8-20) to (M-2-26) to (M-7-31) to (M-2-36) to (M-5-39);[dark blue] (M-3-1) to (M-5-3) to (M-2-6) to (M-3-7) to (M-2-8) to (M-6-12) to (M-2-16) to (M-5-19) to (M-2-22) to (M-3-23) to (M-2-24) to (M-6-28) to (M-2-32) to (M-5-35) to (M-2-38) to (M-3-39);[dark purple] (M-5-1) to (M-8-4) to (M-2-10) to (M-7-15) to (M-2-20) to (M-7-25) to (M-2-30) to (M-8-36) to (M-5-39);$ $Note that reducedness implies that non-zero entries of $ C$ only occur in circles and where two lines cross (See Remark \\ref {rem: soljugs} for a contrasting property of the solution matrix $ Sol(C)$).$ The connection to juggling is justified by considering the transpose of the above picture.",
"$\\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][use as bounding box] (-9.5,2.7) rectangle (6.3,-0.3);[matrix of math nodes,matrix anchor = M-2-24.center,nodes in empty cells,inner sep=0pt,gthrow/.style={dark green,draw,circle,inner sep=0mm,minimum size=5mm},rthrow/.style={dark red,draw,circle,inner sep=0mm,minimum size=5mm},bthrow/.style={dark blue,draw,circle,inner sep=0mm,minimum size=5mm},pthrow/.style={dark purple,draw,circle,inner sep=0mm,minimum size=5mm},nodes={anchor=center,node font=\\scriptsize ,rotate=45},column sep={0.4cm,between origins},row sep={-0.4cm,between origins},] (M) at (0,0) {\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& |[gthrow]| \\& \\& |[rthrow]| \\& \\& |[bthrow]| \\& \\& |[bthrow]| \\& \\& |[pthrow]| \\& \\& |[gthrow]| \\& \\& |[rthrow]| \\& \\& |[bthrow]| \\& \\& |[gthrow]| \\& \\& |[pthrow]| \\& \\& |[bthrow]| \\& \\& |[bthrow]| \\& \\& |[rthrow]| \\& \\& |[gthrow]| \\& \\& |[pthrow]| \\& \\& |[bthrow]| \\& \\& |[gthrow]| \\& \\& |[rthrow]| \\& \\& |[bthrow]| \\& \\\\\\cdots \\& \\& \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\cdots \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[gthrow]| \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[gthrow]| \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& |[gthrow]| \\& \\& |[rthrow]| \\& \\& \\& \\& \\& \\& |[pthrow]| \\& \\& \\& \\& \\& \\& \\& \\& |[gthrow]| \\& \\& |[pthrow]| \\& \\& \\& \\& \\& \\& |[rthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\cdots \\\\\\& \\& \\& |[pthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[rthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[pthrow]| \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\};\\end{tikzpicture}[dark green] (M-3-1) to (M-2-2) to (M-7-7) to (M-2-12) to (M-5-15) to (M-2-18) to (M-7-23) to (M-2-28) to (M-5-31) to (M-2-34) to (M-7-39);[dark red] (M-5-1) to (M-2-4) to (M-7-9) to (M-2-14) to (M-8-20) to (M-2-26) to (M-7-31) to (M-2-36) to (M-5-39);[dark blue] (M-3-1) to (M-5-3) to (M-2-6) to (M-3-7) to (M-2-8) to (M-6-12) to (M-2-16) to (M-5-19) to (M-2-22) to (M-3-23) to (M-2-24) to (M-6-28) to (M-2-32) to (M-5-35) to (M-2-38) to (M-3-39);[dark purple] (M-5-1) to (M-8-4) to (M-2-10) to (M-7-15) to (M-2-20) to (M-7-25) to (M-2-30) to (M-8-36) to (M-5-39);$ $This can be viewed as an idealized plot of the height of the `balls^{\\prime } over time.$ Balls and juggling patterns can also be used to parametrize the space of solutions to a linear recurrence.",
"Given a shape $S$ and an integer $b$ , let $T_b$ consist of the largest element in each $S$ -ball less than or equal to $b$ ; that is, $ T_b := \\lbrace a \\text{ such that $a\\le b$ but there is no $c\\le b$ with $S(c)=a$} \\rbrace $ In juggling terms, the set $T_b$ considers the balls in the air just after moment $b$ and records when each ball was thrown.",
"thm: solextendTheorem REF Restricting to the entries indexed by $T_b$ gives an isomorphism $\\mathrm {ker}(\\mathsf {C})\\rightarrow \\mathsf {k}^{T_b}$ .",
"That is, any choice of values for the variables $x_a$ for all $a\\in T_a$ can be uniquely extended to a solution to $\\mathsf {C}\\mathsf {x=0}$ .",
"We may also use $T_b$ to parametrize the space of solutions.",
"Let $\\mathsf {Sol}(\\mathsf {C})_{\\mathbb {Z},T_b}$ denote the $\\mathbb {Z}\\times T_b$ -matrix consisting of the columns of $\\mathsf {Sol}(\\mathsf {C})$ indexed by $T_b$ .",
"thm: solbasisTheorem REF Multiplication by $\\mathsf {Sol}(\\mathsf {C})_{\\mathbb {Z}\\times T_b}$ gives an isomorphism $\\mathsf {k}^{T_b}\\rightarrow \\mathrm {ker}(\\mathsf {C})$ .",
"Equivalently, every solution to $\\mathsf {C}\\mathsf {x=0}$ can be written uniquely as a (possibly infinite) linear combination of the columns of $\\mathsf {Sol}(\\mathsf {C})$ indexed by $T_b$ , and all such linear combinations exist (i.e.",
"no infinite non-zero sums).In the literature on linear recurrences (e.g.",
"[1]), this is called a a fundamental system of solutions.",
"When $T_b$ is finite, this is merely the definition of a basis.",
"Corollary 3.17 If $\\dim (\\mathrm {ker}(\\mathsf {C}))\\!<\\!\\infty $ , the columns of $\\mathsf {Sol}(\\mathsf {C})$ indexed by $T_b$ are a basis for $\\mathrm {ker}(\\mathsf {C})$ .",
"Let us consider the Fibonacci recurrence matrix $\\mathsf {C}$ once more.",
"The two $S$ -balls are the sets of even and odd numbers, and so an $S$ -schedule is a pair of consecutive integers.",
"Theorems REF and REF state that any pair of adjacent columns of $\\mathsf {Sol}(\\mathsf {C})$ form a basis of $\\mathrm {ker}(\\mathsf {C})$ , and any pair of values $x_{a-1},x_a\\in \\mathsf {k}$ may be extended uniquely to an element of $\\mathrm {ker}(\\mathsf {C})$ .",
"Definition REF generalizes $T_b$ to a broader class of sets, called schedules, for which analogs of Theorems REF and REF hold.",
"[.05cm] The isomorphisms in Theorems REF and REF are not mutually inverse.",
"Generalizations and connections We consider a few variations of this problem and applications of these ideas.",
"Affine recurrences Let us briefly consider the affine case.",
"An affine recurrence is a system of equations in the sequence of variables $...,x_{-1},x_0,x_1,x_2,...$ which equates each variable to an affine combination of the previous variables (i.e.",
"a degree 1 polynomial).",
"For example, we could add a constant terms $b_i\\in \\mathsf {k}$ to each equation in the Fibonacci recurrence: $x_i = x_{i-1}+x_{i-2}+b_i,\\;\\;\\; \\forall i\\in \\mathbb {Z}$ As before, we may move the variables to the left and factor the coefficients into a matrix $\\mathsf {C}$ : $ \\mathsf {C}\\mathsf {x} = \\mathsf {b}$ Here, $\\mathsf {C}$ is a recurrence matrix, and $\\mathsf {b}$ collects the constant terms from each equation.",
"If $\\mathsf {C}$ is reduced, define $\\mathsf {P}$ as in Section REF , and define the splitting matrix $\\mathsf {Spl}(\\mathsf {C})$ of $\\mathsf {C}$ by $ \\mathsf {Spl}(\\mathsf {C})_{a,b}:= \\left\\lbrace \\begin{array}{cc}\\mathsf {Adj}(\\mathsf {C})_{a,b} & \\text{if }b\\ge 0\\\\(\\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P}))_{a,b} & \\text{if }b<0\\end{array}\\right\\rbrace $ The right half of this matrix is lower unitriangular, and the left half of this matrix is upper triangular, resulting in non-zero entries concentrated into two antipodal wedges.",
"The splitting matrix for the Fibonacci recurrence matrix is below.",
"$\\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,]\\node [right, dark blue] at (3,-1.5) {\\scriptsize Entries from \\mathsf {Adj}(\\mathsf {C})};\\node [left, dark red] at (-3,1.2) {\\scriptsize Entries from \\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})};[use as bounding box] (-2.925,-2.925) rectangle (2.925,2.325);[matrix of math nodes,matrix anchor = M-10-10.center,nodes in empty cells,inner sep=0pt,nodes={anchor=center,node font=\\scriptsize ,rotate=45},column sep={0.3cm,between origins},row sep={0.3cm,between origins},] (M) at (0,0) {\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\cdots \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& -5 \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\cdots \\& \\& 3 \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& -2 \\& \\& -2 \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\cdots \\& \\& 1 \\& \\& 1 \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& -1 \\& \\& -1 \\& \\& -1 \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& 2 \\& \\& 2 \\& \\& 2 \\& \\& 2 \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& 3 \\& \\& 3 \\& \\& 3 \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& 5 \\& \\& 5 \\& \\& 5 \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& 8 \\& \\& 8 \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& 13 \\& \\& 13 \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& 21 \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& 34 \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\cdots \\\\};\\end{tikzpicture}[dark blue,rounded corners,fill=dark blue, opacity=.1] (M-19-19)+(.2,-.5) to ($ M-10-10)+(-.3,0)$) to ($ (M-10-10)+(-.2,.2)$) to ($ (M-10-19)+(.2,.2)$);[dark red,rounded corners,fill=dark red, opacity=.1] (M-3-1)+(-.2,.5) to ($ (M-10-8)+(.3,0)$) to ($ (M-10-8)+(.2,-.2)$) to ($ (M-10-1)+(-.2,-.2)$);$ $The lower/upper triangular conditions imply the non-zero entries coming from $ Adj(C)$ and $ PAdj(CP)$ must be contained in the blue and red cones, respectively.$ Proposition 4.2 Let $\\mathsf {C}$ be a reduced recurrence matrix.",
"$\\mathsf {Spl}(\\mathsf {C})$ is horizontally bounded.",
"$\\mathsf {C}\\mathsf {Spl}(\\mathsf {C})=\\mathsf {Id}$ ; that is, $\\mathsf {Spl}(\\mathsf {C})$ is a right inverse to $\\mathsf {C}$ .",
"For all $\\mathsf {b}\\in \\mathsf {k}^\\mathbb {Z}$ , the product $\\mathsf {x}=\\mathsf {Spl}(\\mathsf {C})\\mathsf {b}$ exists and is a solution to $\\mathsf {C}\\mathsf {x}=\\mathsf {b}$ .",
"In particular, $\\mathsf {C}\\mathsf {x}=\\mathsf {b}$ has a solution for all $\\mathsf {b}$ .",
"(1) If $a\\ge 0$ , then the $a$ th row of $\\mathsf {Spl}(\\mathsf {C})$ is zero outside the interval $[0,a]$ .",
"If $a<0$ , then the $a$ th row of $\\mathsf {Spl}(\\mathsf {C})$ is zero outside $[a,0]$ .This bound can be sharpened, though we won't need a sharp bound.",
"When $a<0$ , the $a$ th row of $\\mathsf {Spl}(\\mathsf {C})$ is zero outside the interval $[b,0]$ when $S(b)=a$ , and the row is entirely zero if there is no such $b$ .",
"Thus, $\\mathsf {Spl}(\\mathsf {C})$ is horizontally bounded.",
"(2) If $a\\ge 0$ , then the $a$ th column of $\\mathsf {C}\\mathsf {Spl}(\\mathsf {C})$ equals the $a$ th column of $\\mathsf {C}\\mathsf {Adj}(\\mathsf {C})=\\mathsf {Id}$ .",
"If $a<0$ , then the $a$ th column of $\\mathsf {C}\\mathsf {Spl}(\\mathsf {C})$ equals the $a$ th column of $\\mathsf {C}(\\mathsf {P}(\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})))=\\mathsf {Id}$ .",
"Therefore, $\\mathsf {C}\\mathsf {Spl}(\\mathsf {C})=\\mathsf {Id}$ .",
"(3) Since $\\mathsf {C}$ and $\\mathsf {Spl}(\\mathsf {C})$ are horizontally bounded, $\\mathsf {C}(\\mathsf {Spl}(\\mathsf {C})\\mathsf {b}) = (\\mathsf {C}\\mathsf {Spl}(\\mathsf {C}) ) \\mathsf {b} = \\mathsf {Idb}= \\mathsf {b} $ .",
"The existence of solutions to every affine recurrence is equivalent saying that, for any recurrence matrix $\\mathsf {C}$ , the associated multiplication map $\\mathsf {k}^\\mathbb {Z}\\rightarrow \\mathsf {k}^\\mathbb {Z}$ is surjective.",
"Given a solution to $\\mathsf {C}\\mathsf {x}=\\mathsf {b}$ , all other solutions are obtained by adding solutions to $\\mathsf {C}\\mathsf {x}=\\mathsf {0}$ .",
"Proposition 4.4 If $\\mathsf {C}$ is reduced, the solutions to $\\mathsf {C}\\mathsf {x} = \\mathsf {b}$ consist of sequences of the form $ \\mathsf {Spl}(\\mathsf {C}) \\mathsf {b} + \\mathsf {Sol}(\\mathsf {C}) \\mathsf {v}, $ running over all $\\mathsf {v}\\in \\mathsf {k}^\\mathbb {Z}$ such that the product $\\mathsf {Sol}(\\mathsf {C}) \\mathsf {v}$ exists.",
"This follows immediately from Proposition REF and Theorem REF .",
"We have now constructed three right inverses to a reduced recurrence matrix $\\mathsf {C}$ , each possessing an additional property: $\\mathsf {Adj}(\\mathsf {C})$ is lower unitriangular, $\\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})$ is upper unitriangular, and $\\mathsf {Spl}(\\mathsf {C})$ is horizontally bounded.",
"If $\\mathsf {C}\\ne \\mathsf {Id}$ , these are all distinct.",
"Linear recurrences indexed by $\\mathbb {N}$ Variations of `linear recurrences' have been studied for centuries.",
"Most often, one considers a system with variables indexed by $\\mathbb {N}$ (rather than $\\mathbb {Z}$ ) and relations defining each variable except at finitely many `initial variables'.",
"For example, the one-sided Fibonacci recurrence has initial variables $x_0$ and $x_1$ and equations $x_i = x_{i-1}+x_{i-2},\\;\\;\\; \\forall i\\ge 2$ The study of $\\mathbb {N}$ -indexed linear recurrences differs fundamentally from $\\mathbb {Z}$ -indexed linear recurrences.",
"Solutions to an $\\mathbb {N}$ -indexed system are determined by the values of the initial variables, which trivializes the kinds of questions we have considered (e.g.",
"existence and parametrization of solutions).",
"Rather, most work in the $\\mathbb {N}$ -indexed context has focused on finding simple formulas for the terms in a solution.",
"We review a few of these approaches.",
"When the equations in a linear recurrence are the same (a `constant' linear recurrence), shifting the indices of a sequence $ x_0,x_1,x_2,... \\longmapsto x_1,x_2,x_3,... $ defines a linear transformation from the space of solutions to itself.",
"Standard tools from linear algebra (e.g.",
"the characteristic polynomial) can then construct a basis of eigenvectors or generalized eigenvectors for the space of solutions.",
"Since the eigenvectors are geometric sequences, an eigenbasis expresses any solution as a linear combination of geometric sequences.",
"This is covered in textbooks like [10].",
"A sequence $x_0, x_1, x_2,...$ can be translated into formal series in several ways, such as $ F_\\mathsf {x}(t) := x_0 + x_1 t + x_2 t^2 + x_3t^3 + \\cdots $ Some linear recurrences (such as constant ones) translate into functional equations involving these generating functions.",
"Clever manipulation of these equations can then yield simple formulas for solutions.",
"This is covered in textbooks like [14].",
"The asymptotics of solutions, that is, the behavior of $x_i$ for sufficiently large $i$ , can be studied analytically.",
"Poincare [13] and othersA curiosity: [4] is the dissertation of Robert Carmichael, of Carmichael numbers in number theory.",
"[4], [2] construct integrals which coincide with the generating function $F_\\mathsf {x}(t)$ in an `infinitesmal neighborhood of infinity'.",
"See [1] for further details.",
"The techniques of the current work can be adapted to this setting.",
"We first add equations fixing the initial values and rewrite the system as a matrix equation $\\mathsf {C}\\mathsf {x}=\\mathsf {b}$ .",
"For example, the (one-sided) Fibonacci recurrence with initial values $x_0=a$ and $x_1=b$ is rewritten as $ \\begin{bmatrix}1 & 0 & 0 & 0 & \\\\0 & 1 & 0 & 0 & \\\\-1 & -1 & 1 & 0 & \\\\0 & -1 & -1 & 1 & \\\\& & & & \\rotatebox {-45}{\\cdots }\\\\\\end{bmatrix}\\begin{bmatrix}x_0 \\\\ x_1 \\\\ x_2 \\\\ x_3 \\\\ \\vdots \\end{bmatrix}=\\begin{bmatrix}a \\\\ b \\\\ 0 \\\\ 0 \\\\ \\vdots \\end{bmatrix}$ The recurrence matrix $\\mathsf {C}$ is $\\mathbb {N}\\times \\mathbb {N}$ , lower unitriangular, and horizontally bounded.",
"The adjugate $\\mathsf {Adj}(\\mathsf {C})$ is defined as before, and the identity $\\mathsf {Adj}(\\mathsf {C})\\mathsf {C}=\\mathsf {C}\\mathsf {Adj}(\\mathsf {C})=\\mathsf {Id}$ still holds.",
"However, there is a crucial difference.",
"In the $\\mathbb {N}\\times \\mathbb {N}$ case, the adjugate matrix $\\mathsf {Adj}(\\mathsf {C})$ is horizontally bounded, and so $\\mathsf {Adj}(\\mathsf {C}) (\\mathsf {C}\\mathsf {x}) = (\\mathsf {Adj}(\\mathsf {C}) \\mathsf {C}) \\mathsf {x}$ for all $\\mathbb {Z}$ -vectors $\\mathsf {x}$ .",
"If $\\mathsf {C}\\mathsf {x=b}$ , then $ \\mathsf {Adj}(\\mathsf {C}) \\mathsf {b} = \\mathsf {Adj}(\\mathsf {C}) (\\mathsf {C}\\mathsf {x})= (\\mathsf {Adj}(\\mathsf {C}) \\mathsf {C}) \\mathsf {x} =\\mathsf {x}$ Consequently, the unique solution to $\\mathsf {C}\\mathsf {x=b}$ can be computed as a linear combination of the columns of $\\mathsf {Adj}(\\mathsf {C})$ indexed by the initial variables.",
"We can restate this as follows.",
"Proposition 4.6 If $x_i$ is an initial variable in an $\\mathbb {N}$ -indexed linear recurrence with recurrence matrix $\\mathsf {C}$ , then the $i$ th column of $\\mathsf {Adj}(\\mathsf {C})$ is the solution for which $x_i=1$ and all other initial variables are 0.",
"Columns of this form are a basis for the space of all solutions.",
"While Proposition REF gives a basis of solutions, it is unclear how useful this is in general.",
"Computationally, the entries of $\\mathsf {Adj}(\\mathsf {C})$ are determinant of submatrices of $\\mathsf {C}$ , which are (naively) no simpler than recursively computing $x_0,x_1,....,x_j$ directly.",
"Friezes The author's original motivation for the work in this paper is a connection and forthcoming application to the following curious objects.",
"A tame $SL(k)$ -frieze consists of finitely many rows of integers (offset in a diamond pattern) such that: the top and bottom rows consist entirely of 1s, every $k\\times k$ diamond has determinant 1, and every $(k+1)\\times (k+1)$ diamond has determinant 0.",
"An example of an $SL(2)$ -frieze is given below.",
"$\\begin{tikzpicture}[baseline=(current bounding box.south),ampersand replacement=\\&,][matrix of math nodes,matrix anchor = M-1-8.center,origin/.style={},throw/.style={},pivot/.style={draw,circle,inner sep=0.25mm,minimum size=2mm}, nodes in empty cells,inner sep=0pt,nodes={anchor=center,node font=\\scriptsize },column sep={.35cm,between origins},row sep={.35cm,between origins},] (M) at (0,0) {\\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& \\cdots \\\\\\cdots \\& \\& 3 \\& \\& 2 \\& \\& 2 \\& \\& 1 \\& \\& 4 \\& \\& 3 \\& \\& 1 \\& \\& 2 \\& \\& 3 \\& \\& 2 \\& \\& 2 \\& \\& 1 \\& \\& 4 \\& \\& 3 \\& \\& 1 \\& \\& 2 \\& \\& 3 \\& \\\\\\& 5 \\& \\& 5 \\& \\& 3 \\& \\& 1 \\& \\& 3 \\& \\& 11 \\& \\& 2 \\& \\& 1 \\& \\& 5 \\& \\& 5 \\& \\& 3 \\& \\& 1 \\& \\& 3 \\& \\& 11 \\& \\& 2 \\& \\& 1 \\& \\& 5 \\& \\& \\cdots \\\\\\cdots \\& \\& 8 \\& \\& 7 \\& \\& 1 \\& \\& 2 \\& \\& 8 \\& \\& 7 \\& \\& 1 \\& \\& 2 \\& \\& 8 \\& \\& 7 \\& \\& 1 \\& \\& 2 \\& \\& 8 \\& \\& 7 \\& \\& 1 \\& \\& 2 \\& \\& 8 \\& \\\\\\& 3 \\& \\& 11 \\& \\& 2 \\& \\& 1 \\& \\& 5 \\& \\& 5 \\& \\& 3\\& \\& 1 \\& \\& 3 \\& \\& 11 \\& \\& 2 \\& \\& 1 \\& \\& 5 \\& \\& 5 \\& \\& 3\\& \\& 1 \\& \\& 3 \\& \\& \\cdots \\\\\\cdots \\& \\& 4 \\& \\& 3 \\& \\& 1 \\& \\& 2 \\& \\& 3 \\& \\& 2 \\& \\& 2 \\& \\& 1 \\& \\& 4 \\& \\& 3 \\& \\& 1 \\& \\& 2 \\& \\& 3 \\& \\& 2 \\& \\& 2 \\& \\& 1 \\& \\& 4 \\& \\\\\\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& \\cdots \\\\};\\end{tikzpicture}$ The study of friezes was initiated in [7], [5], [6] for $k=2$ , and generalized to arbitrary $k$ in [8], [3].",
"Friezes enjoy many remarkable properties; for example, the rows of a tame $SL(k)$ -frieze must be periodic.",
"An excellent overview is given in [11].",
"A frieze may be converted into a recurrence matrix, by rotating $45^\\circ $ clockwise and using the top row as the main diagonal.",
"[12] and others also use an alternating sign when translating a frieze into a linear recurrence.",
"Remarkably, the solutions have a periodicity condition.",
"Theorem 4.9 [12] If $\\mathsf {C}$ is the recurrence matrix associated to a tame $SL(k)$ -frieze, then every solution to $\\mathsf {C}\\mathsf {x=0}$ is superperiodic: $x_{i+n}=(-1)^sx_i$ for some $n$ and $s$ and all $i$ .",
"In fact, [12] proves a stronger result.",
"For each frieze $\\mathsf {C}$ , they construct a Gale dual frieze $\\mathsf {C}^\\dagger $ whose diagonals encode distinguished solutions to $\\mathsf {C}\\mathsf {x=0}$ .",
"In a sequel [9] to the current work, we will extend Theorem REF to an equivalence.",
"Specifically, if $\\mathsf {C}$ is a reduced recurrence matrix of shape $S$ , then the following are equivalent.",
"$\\mathsf {C}$ satisfies a family of determinantal identities generalizing the tame frieze conditions.",
"Every solution to $\\mathsf {C}\\mathsf {x=0}$ is $n$ -quasiperiodic; that is, $x_{i+n}=\\lambda x_i$ for some $\\lambda $ and all $i$ .",
"The Gale dual $\\mathsf {C}^\\dagger $ , a truncation of $\\mathsf {Sol}(\\mathsf {C})$ , has shape $S^\\dagger $ , where $S^\\dagger (i):=S^{-1}(i)+n$ .",
"The space of such linear recurrences (of fixed shape $S$ ) is the cluster $\\mathcal {X}$ -variety dual to the positroid variety corresponding to $S$ ; this will be explained in [9].",
"The rest of this note proves the promised results.",
"Kernel containment and factorization In this section, we prove a useful equivalence between containments of kernels and factorizations in the semigroup of recurrence matrices.",
"Let $\\mathsf {k}^\\mathbb {Z}_b\\subset \\mathsf {k}^\\mathbb {Z}$ denote the subspace of bounded sequences (i.e.",
"non-zero in finitely many terms).",
"If $\\mathsf {v}\\in \\mathsf {k}^\\mathbb {Z}_b$ and $\\mathsf {w}\\in \\mathsf {k}^\\mathbb {Z}$ , then the dot product $\\mathsf {v}\\cdot \\mathsf {w}$ is well-defined.",
"Lemma 5.2 Let $\\mathsf {C}$ be a recurrence matrix and let $\\mathsf {v}\\in \\mathsf {k}^\\mathbb {Z}_b$ .",
"If $\\mathsf {v}\\cdot \\mathsf {w}=0$ for all $\\mathsf {w}\\in \\mathrm {ker}(\\mathsf {C})$ , then $\\mathsf {v}$ is in the span of the rows of $\\mathsf {C}$ .",
"Let $V\\subset \\mathsf {k}^\\mathbb {Z}_b$ denote the span of the rows of $\\mathsf {C}$ , and assume for contradiction that $\\mathsf {v}\\notin V$ .",
"We may therefore choose a linear map $f:\\mathsf {k}^\\mathbb {Z}_b\\rightarrow \\mathsf {k}$ such that $f(V)=0$ and $f(\\mathsf {v})=1$ .The existence of such a map may depend on the Axiom of Choice, which we therefore assume.",
"Let $\\mathsf {e}_a\\in \\mathsf {k}^\\mathbb {Z}_b$ denote the standard basis vector which is 1 in the $a$ th term and 0 everywhere else, and set $\\mathsf {w}:= ( f(\\mathsf {e}_a)) _{a\\in \\mathbb {Z}}\\in \\mathsf {k}^\\mathbb {Z}$ .",
"By linearity, $f(\\mathsf {u})=\\mathsf {u}\\cdot \\mathsf {w}$ for all $\\mathsf {u}\\in \\mathsf {k}^\\mathbb {Z}_b$ .",
"Since $f$ kills each row of $\\mathsf {C}$ , $\\mathsf {C}\\mathsf {w}=0$ and so $\\mathsf {w}\\in \\mathrm {ker}(\\mathsf {C})$ .",
"However, $\\mathsf {v}\\cdot \\mathsf {w}=1$ , contradicting the hypothesis.",
"Lemma 5.3 Let $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ be recurrence matrices.",
"Then the following are equivalent.",
"$\\mathrm {ker}(\\mathsf {C})\\subseteq \\mathrm {ker}(\\mathsf {C}^{\\prime })$ .",
"$\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for some horizontally bounded matrix $\\mathsf {D}$ .",
"$\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for some recurrence matrix $\\mathsf {D}$ .",
"$\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C})$ is a recurrence matrix.",
"Furthermore, the matrix $\\mathsf {D}$ in (2) and (3) must equal $\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C})$ and is therefore unique.",
"($1\\Rightarrow 2$ ) If $\\mathrm {ker}(\\mathsf {C})\\subseteq \\mathrm {ker}(\\mathsf {C}^{\\prime })$ , then each row of $\\mathsf {C}^{\\prime }$ kills $\\mathrm {ker}(\\mathsf {C})$ .",
"By Lemma REF , the $a$ th row of $\\mathsf {C}^{\\prime }$ is equal to $\\mathsf {D}_a\\mathsf {C}$ for some bounded sequence $\\mathsf {D}_a\\in \\mathsf {k}^\\mathbb {Z}_b$ .",
"The vectors $\\mathsf {D}_a$ may be combined into the rows of a matrix $\\mathsf {D}$ which is horizontally bounded and satisfies $\\mathsf {D}\\mathsf {C}=\\mathsf {C}^{\\prime }$ .",
"($2\\Rightarrow 3+4+$ Uniqueness) Assume that $\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for a horizontally bounded $\\mathsf {D}$ .",
"Then $\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C}) = (\\mathsf {D}\\mathsf {C})\\mathsf {Adj}(\\mathsf {C}) \\stackrel{*}{=} \\mathsf {D}(\\mathsf {C}\\mathsf {Adj}(\\mathsf {C})) = \\mathsf {D}$ Equality ($*$ ) holds because $\\mathsf {D}$ and $\\mathsf {C}$ are horizontally bounded.",
"Since $\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C})$ is lower unitriangular and $\\mathsf {D}$ is horizontally bounded, they are the same recurrence matrix.",
"($3\\Rightarrow 1$ ) Assume $\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for some recurrence matrix $\\mathsf {D}$ .",
"If $\\mathsf {v}\\in \\mathrm {ker}(\\mathsf {C})$ , then $ \\mathsf {C}^{\\prime }\\mathsf {v} = (\\mathsf {D}\\mathsf {C})\\mathsf {v} \\stackrel{*}{=} \\mathsf {D}(\\mathsf {C}\\mathsf {v}) = \\mathsf {D}\\mathsf {0} = \\mathsf {0} $ Equality ($*$ ) holds because $\\mathsf {D}$ and $\\mathsf {C}$ are horizontally bounded.",
"Therefore, $\\mathrm {ker}(\\mathsf {C})\\subseteq \\mathrm {ker}(\\mathsf {C}^{\\prime })$ .",
"($4\\Rightarrow 3$ ) Setting $\\mathsf {D}\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C})$ , we check that $ \\mathsf {D}\\mathsf {C}= (\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C}) ) \\mathsf {C}\\stackrel{*}{=} \\mathsf {C}^{\\prime }(\\mathsf {Adj}(\\mathsf {C})\\mathsf {C}) = \\mathsf {C}^{\\prime }$ Equality ($*$ ) holds because $\\mathsf {C}^{\\prime }$ , $\\mathsf {Adj}(\\mathsf {C})$ , and $\\mathsf {C}$ are lower unitriangular.",
"Theorem 5.4 A recurrence matrix $\\mathsf {C}$ is trivial if and only if $\\mathsf {Adj}(\\mathsf {C})$ is a recurrence matrix.",
"The recurrence matrix $\\mathsf {C}$ is trivial when $\\mathrm {ker}(\\mathsf {C})\\subseteq \\mathrm {ker}(\\mathsf {Id})$ .",
"Applying Lemma REF with $\\mathsf {C}^{\\prime }=\\mathsf {Id}$ , this holds if and only if $\\mathsf {Adj}(\\mathsf {C})$ is a recurrence matrix.",
"Theorem 5.5 Two recurrence matrices $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ are equivalent if and only if $\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for a trivial recurrence matrix $\\mathsf {D}$ .",
"Two recurrence matrices $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ are equivalent if and only if $\\mathrm {ker}(\\mathsf {C})=\\mathrm {ker}(\\mathsf {C}^{\\prime })$ .",
"By Lemma REF , this holds if and only if $\\mathsf {D}=\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C})$ and $\\mathsf {Adj}(\\mathsf {D})=\\mathsf {C}\\mathsf {Adj}(\\mathsf {C}^{\\prime })$ are recurrence matrices.",
"By Theorem REF , this is equivalent to $\\mathsf {D}$ being a trivial linear recurrence.",
"Lemma REF also allows us to make a connection between kernel containment and shapes.",
"Lemma 5.6 Let $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ be recurrence matrices with shape $S$ and $S^{\\prime }$ , respectively.",
"If $\\mathrm {ker}(\\mathsf {C})\\subseteq \\mathrm {ker}(\\mathsf {C}^{\\prime })$ and $S$ is injective (e.g.",
"if $\\mathsf {C}$ is reduced), then $S(a)\\ge S^{\\prime }(a)$ for all $a$ .",
"By Lemma REF , $\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for some recurrence matrix $\\mathsf {D}$ .",
"Fix $a\\in \\mathbb {Z}$ , and consider $B:=\\lbrace b \\in \\mathbb {Z}\\mid \\mathsf {D}_{a,b} \\ne 0\\rbrace $ .",
"This set is bounded and contains $a$ .",
"Let $b_{0}$ be the element of $B$ on which $S$ is minimal; this is unique because $S$ is injective.",
"$ (\\mathsf {D}\\mathsf {C})_{a,S(b_{0})} = \\sum _{b\\in \\mathbb {Z}} \\mathsf {D}_{a,b} \\mathsf {C}_{b,S(b_{0})} = \\mathsf {D}_{a,b_{0}}\\mathsf {C}_{b_{0},S(b_{0})} \\ne 0 $ Therefore, $S^{\\prime }(a)\\le S(b_0)$ .In fact, this is equality, but we won't need this stronger statement.",
"Since $a\\in B$ , $S(b_0)\\le S(a)$ , and so $S^{\\prime }(a)\\le S(a)$ .",
"Proposition 5.7 Reduced recurrence matrices that are equivalent must be equal.",
"Let $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ be reduced and equivalent.",
"Lemma REF implies that $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ have the same shape; call it $S$ .",
"By Lemma REF , $\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for some recurrence matrix $\\mathsf {D}$ .",
"Let $T$ denote the shape of $\\mathsf {D}$ , so that $\\mathsf {D}_{a,b}=0$ whenever $b<T(a)$ .",
"Since $\\mathsf {C}$ is reduced of shape $S$ , $\\mathsf {C}_{b,S(T(a))}=0$ whenever $b>T(a)$ .",
"Therefore, $ \\mathsf {C}^{\\prime }_{a,S(T(a))} = \\sum _b \\mathsf {D}_{a,b} \\mathsf {C}_{b,S(T(a))} = \\mathsf {D}_{a,T(a)}\\mathsf {C}_{T(a),S(T(a))} \\ne 0 $ Since $\\mathsf {C}^{\\prime }$ is also reduced of shape $S$ , this is only possible if $a= T(a)$ .",
"Since this holds for all $a$ , the only non-zero entries of $\\mathsf {D}$ are on the main diagonal.",
"Thus, $\\mathsf {D}=\\mathsf {Id}$ and $\\mathsf {C}^{\\prime }=\\mathsf {C}$ .",
"Gauss-Zordan Elimination Because we are working with $\\mathbb {Z}\\times \\mathbb {Z}$ -matrices, we must consider infinite sequences of row reductions that may be chosen in an arbitrary order.",
"We furthermore consider generalized row reductions: limits of such row reductions (in an appropriate topology).",
"Row reduction Given a recurrence matrix $\\mathsf {C}$ of shape $S$ , a row reduction of $\\mathsf {C}$ is a matrix $\\mathsf {C}^{\\prime }$ obtained by adding $\\mathsf {C}_{a,S(b)} / \\mathsf {C}_{b,S(b)}$ times the $b$ th row to the $a$ th row, for some $b> a$ with $\\mathsf {C}_{a,S(b)}\\ne 0$ .",
"By design, the resulting matrix $\\mathsf {C}^{\\prime }$ has a zero in the $(a,S(b))$ entry.",
"Proposition 6.1 A recurrence matrix is reduced if and only if it has no row reductions.",
"A row reduction of $\\mathsf {C}$ can be reformulated as a factorization $\\mathsf {C}=\\mathsf {D}\\mathsf {C}^{\\prime }$ such that $\\mathsf {D}$ differs from the identity matrix in a single entry $\\mathsf {D}_{a,b}$ , and such that $\\mathsf {C}_{a,S(b)}\\ne 0$ and $\\mathsf {C}^{\\prime }_{a,S(b)}=0$ .",
"This perspective leads to the following generalization.",
"A generalized row reduction of $\\mathsf {C}$ is a recurrence matrix $\\mathsf {C}^{\\prime }$ such that $\\mathsf {C}=\\mathsf {D}\\mathsf {C}^{\\prime }$ for a trivial recurrence matrix $\\mathsf {D}$ with the property that, for each $a$ such that $\\lbrace b <a \\mid \\mathsf {D}_{a,b}\\ne 0\\rbrace $ is non-empty, we have $ \\mathsf {C}_{a,b_a} \\ne 0\\text{ and } \\mathsf {C}^{\\prime }_{a,b_a} = 0 $ where $b_a:= \\min \\lbrace S(b) \\mid b< a \\text{ s.t.",
"}\\mathsf {D}_{a,b}\\ne 0\\rbrace $ .",
"We write $\\mathsf {C}\\succeq \\mathsf {C}^{\\prime }$ to denote that $\\mathsf {C}^{\\prime }$ is a generalized row reduction of $\\mathsf {C}$ .",
"The index $b_a$ may be defined as the leftmost entry of the $a$ th row that multiplication by $\\mathsf {D}$ is `big enough' to change, and so $(\\mathsf {D}\\mathsf {C})_{a,b}=\\mathsf {C}_{a,b}$ whenever $b<b_a$ .",
"Thus, if $\\mathsf {C}\\succeq \\mathsf {C}^{\\prime }$ , then $\\mathsf {C}^{\\prime }$ must vanish in the leftmost entry in which the $a$ th rows of $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ differ.",
"Proposition 6.3 The relation $\\succeq $ defines a partial order on the set of recurrence matrices.",
"As a consequence, an iterated sequence of row reductions is a generalized row reduction.",
"(Antisymmetry) Assume $\\mathsf {C}\\succeq \\mathsf {C}^{\\prime }$ and $\\mathsf {C}\\preceq \\mathsf {C}^{\\prime }$ .",
"By Remark REF , both $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ vanish in the leftmost entry in which the $a$ th rows of $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ differ.",
"However, two entries cannot both vanish and be different, so the $a$ th rows of $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ coincide for all $a$ .",
"Thus, $\\mathsf {C}=\\mathsf {C}^{\\prime }$ .",
"(Transitivity) Let $\\mathsf {C}\\preceq \\mathsf {D}\\mathsf {C}\\preceq \\mathsf {D}^{\\prime }\\mathsf {D}\\mathsf {C}$ , and let $S$ and $S^{\\prime }$ denote the shapes of $\\mathsf {C}$ and $\\mathsf {D}\\mathsf {C}$ , respectively.",
"Fix some $a$ .",
"If $\\lbrace b< a\\mid \\mathsf {D}_{a,b}\\ne 0\\rbrace = \\varnothing $ or $\\lbrace b< a\\mid \\mathsf {D}^{\\prime }_{a,b}\\ne 0\\rbrace = \\varnothing $ , the generalized row reduction condition is easy to check.",
"Assume neither set is empty and let $b_0 &:= \\min \\lbrace S(b) \\mid b<a\\text{ s.t.",
"}\\mathsf {D}_{a,b}\\ne 0\\rbrace \\\\b_0^{\\prime } &:= \\min \\lbrace S^{\\prime }(b) \\mid b<a\\text{ s.t.",
"}\\mathsf {D}^{\\prime }_{a,b}\\ne 0\\rbrace $ By the definition of generalized row reductions, $\\mathsf {C}_{a,b_0}=0,\\;\\;\\; (\\mathsf {D}\\mathsf {C})_{a,b_0}\\ne 0,\\;\\;\\; (\\mathsf {D}\\mathsf {C})_{a,b_0^{\\prime }}=0 ,\\;\\;\\; (\\mathsf {D}^{\\prime }\\mathsf {D}\\mathsf {C})_{a,b_0^{\\prime }}\\ne 0$ This ensures that $\\mathsf {C}_{a,\\min (b_0,b_0^{\\prime })}=0$ and $(\\mathsf {D}^{\\prime }\\mathsf {D}\\mathsf {C})_{a,\\min (b_0,b_0^{\\prime })}\\ne 0$ .",
"Since these entries differ, $ \\min \\lbrace S(b) \\mid b<a\\text{ s.t.",
"}(\\mathsf {D}^{\\prime }\\mathsf {D})_{a,b}\\ne 0\\rbrace \\le \\min (b_0,b_0^{\\prime }) $ To show this is equality, consider some $b<a$ such that $(\\mathsf {D}^{\\prime }\\mathsf {D})_{a,b}\\ne 0$ .",
"We split into cases.",
"Assume $\\mathsf {D}^{\\prime }_{a,c}\\mathsf {D}_{c,b}\\ne 0$ for some $c<a$ .",
"Since $\\mathsf {D}_{c,b}\\ne 0$ and $\\mathsf {C}\\preceq \\mathsf {D}\\mathsf {C}$ , $S^{\\prime }(c)\\le S(b)$ .",
"Since $\\mathsf {D}^{\\prime }_{a,c}\\ne 0$ and $\\mathsf {D}\\mathsf {C}\\preceq \\mathsf {D}^{\\prime }\\mathsf {D}\\mathsf {C}$ , $S^{\\prime }(c)\\ge b_0^{\\prime }$ .",
"Therefore, $S(b)\\ge b^{\\prime }_0$ .",
"Otherwise, $\\mathsf {D}^{\\prime }_{a,c}\\mathsf {D}_{c,b}=0$ for all $c<a$ , and so $(\\mathsf {D}^{\\prime }\\mathsf {D})_{a,b}= \\mathsf {D}^{\\prime }_{a,a}\\mathsf {D}_{a,b}=\\mathsf {D}_{a,b}$ .",
"Since $\\mathsf {D}_{a,b}\\ne 0$ and $\\mathsf {C}\\preceq \\mathsf {D}\\mathsf {C}$ , we know that $S(b)\\ge b_0$ .",
"Therefore, $\\mathsf {C}_{a,\\min (b_0,b_0^{\\prime })}=0$ and $(\\mathsf {D}^{\\prime }\\mathsf {D}\\mathsf {C})_{a,\\min (b_0,b_0^{\\prime })}\\ne 0$ and $ \\min \\lbrace S(b) \\mid b<a\\text{ s.t.",
"}(\\mathsf {D}^{\\prime }\\mathsf {D})_{a,b}\\ne 0\\rbrace = \\min (b_0,b_0^{\\prime }) $ Since this holds for all $a$ , $\\mathsf {C}\\preceq \\mathsf {D}^{\\prime }\\mathsf {D}\\mathsf {C}$ .",
"Limits To define limits of generalized row reductions, we endow the set of recurrence matrices with the topology of row-wise stabilization: a sequence of recurrence matrices converges if each row stabilizes after finitely many steps.",
"We next show that sequences of generalized row reductions must stabilize row-wise to another generalized row reduction, via the following more general result.",
"Lemma 6.4 Let $\\mathcal {C}$ be a set of recurrence matrices in which every pair is comparable in the row reduction partial order.Sometimes called a `chain' in the literature on partially ordered sets.",
"Then the closure of $\\mathcal {C}$ in the space of recurrence matrices contains a lower bound of $\\mathcal {C}$ .",
"Equivalently, there is a descending sequence of recurrence matrices in $\\mathcal {C}$ (i.e.",
"generalized row reductions of the initial matrix in the sequence) which converges (i.e.",
"stabilizes row-wise) to a lower bound of $\\mathcal {C}$ (i.e.",
"a generalized row reduction of every matrix in $\\mathcal {C}$ ).",
"Given a recurrence matrix $\\mathsf {C}$ and an integer $a$ , define $ n_a(\\mathsf {C}) := \\sum _{b \\text{ s.t. }",
"\\mathsf {C}_{(a,b)}\\ne 0} (a-b)^2 $ If $\\mathsf {C}\\preceq \\mathsf {C}^{\\prime }$ , then $n_a(\\mathsf {C})\\le n_a(\\mathsf {C}^{\\prime })$ and equality implies the $a$ th rows of $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ coincide.",
"For each $a$ , let $\\mathcal {C}_a:= \\lbrace \\mathsf {C}\\in \\mathcal {C} \\mid \\forall \\mathsf {C}^{\\prime }\\in \\mathcal {C},\\;n_a (\\mathsf {C}) \\le n_a(\\mathsf {C}^{\\prime })\\rbrace $ ; that is, $\\mathcal {C}_a$ is the set of matrices in $\\mathcal {C}$ which attain the minimum value of $n_a$ .",
"This set is non-empty and the $a$ th row of each matrix in $\\mathcal {C}_a$ is the same, since $n_a$ has the same value and the matrices are comparable.",
"Consider $a,a^{\\prime }\\in \\mathbb {Z}$ and assume, for contradiction, that there exist $\\mathsf {C}\\in \\mathcal {C}_a\\setminus \\mathcal {C}_{a^{\\prime }}$ and $\\mathsf {C}^{\\prime } \\in \\mathcal {C}_{a^{\\prime }}\\setminus \\mathcal {C}_a$ .",
"If $\\mathsf {C}^{\\prime }\\preceq \\mathsf {C}$ , then $n_a(\\mathsf {C}^{\\prime })\\le n_a(\\mathsf {C})$ .",
"By the minimality of $n_a(\\mathsf {C})$ , this is an equality and so $\\mathsf {C}^{\\prime }\\in \\mathcal {C}_a$ ; a contradiction.",
"By a symmetric argument, $\\mathsf {C}\\preceq \\mathsf {C}^{\\prime }$ forces a contradiction.",
"Therefore, $\\mathcal {C}_a\\cap \\mathcal {C}_{a^{\\prime }}$ is either equal to $\\mathcal {C}_a$ or equal to $\\mathcal {C}_{a^{\\prime }}$ .",
"Applying this repeatedly, for any $i\\in \\mathbb {N}$ , there is some $a_i\\in [-i,i]$ such that $ \\bigcap _{a\\in [-i,i]} \\mathcal {C}_a =\\mathcal {C}_{a_i} \\ne \\varnothing $ Choose a matrix $\\mathsf {C}^i$ in $\\mathcal {C}_{a_i}$ for each $i$ .",
"The $a$ th rows in the sequence $\\mathsf {C}^1,\\mathsf {C}^2,\\mathsf {C}^3,...$ , stabilize after the $a$ th term, and so this sequence converges to the recurrence matrix $\\mathsf {C}$ whose $a$ th row coincides with the $a$ th row in each matrix in $\\mathcal {C}_a$ .",
"Let $S$ be the shape of $\\mathsf {C}^1$ .",
"Define a sequence $\\mathsf {D}^1,\\mathsf {D}^2,\\mathsf {D}^3,...$ of trivial recurrence matrices by $\\mathsf {C}^1 = \\mathsf {D}^n\\mathsf {C}^n$ for all $n$ .",
"Since $\\mathsf {C}^1\\succeq \\mathsf {C}^n$ , if $\\mathsf {D}^n_{a,b}\\ne 0$ , then $S(a)\\le S(b)\\le b$ ; that is, the $a$ th row $\\mathsf {D}$ can be non-zero only on the interval $[S(a),a]$ .",
"When $n>|S(a)|$ , the $a$ th row of the product $\\mathsf {D}^n\\mathsf {C}^n$ only depends on rows in $\\mathsf {C}^n$ that coincide with rows in $\\overline{\\mathsf {C}}$ .",
"Therefore, the $a$ th row of $\\mathsf {D}^n\\overline{\\mathsf {C}}$ is equal to $\\mathsf {C}^1$ .",
"Therefore, the sequence $\\mathsf {D}^1,\\mathsf {D}^2,\\mathsf {D}^3,...$ stabilizes row-wise to a matrix $\\overline{\\mathsf {D}}$ such that $\\overline{\\mathsf {D}}\\overline{\\mathsf {C}}=\\mathsf {C}^1$ .",
"As $\\overline{\\mathsf {D}}_{a,b}=\\mathsf {D}^n_{a,b}$ for large enough $n$ , this shows that $\\mathsf {C}\\preceq \\mathsf {C}^1$ .",
"Since the sequence $\\mathsf {C}^\\bullet $ could have started at any matrix in $\\mathcal {C}$ , this shows $\\overline{\\mathsf {C}}$ is a lower bound for $\\mathcal {C}$ .",
"Theorem 6.5 Every recurrence matrix is equivalent to a unique reduced recurrence matrix.",
"Let $\\mathcal {C}$ be an equivalence class of recurrence matrices, with the row reduction partial order.",
"Every non-empty chain in $\\mathcal {C}$ has a lower bound (by Lemma REF ).",
"By Zorn's Lemma, $\\mathcal {C}$ contains a minimal element $\\overline{\\mathsf {C}}$ .",
"If $\\overline{C}$ was not reduced, then there would be a row operation which would strictly decrease it in the reduction partial order; contradicting minimality.",
"Therefore, $\\overline{\\mathsf {C}}$ is reduced.",
"By Proposition REF , this reduced recurrence matrix is unique.",
"This provides a transfinite, non-deterministic analog of Gauss-Jordan elimination, which we humorously dub Gauss-Zordan elimination (both for `Zorn' and the integers $\\mathbb {Z}$ ).",
"Given a recurrence matrix $\\mathsf {C}$ , an arbitrary sequence of row reductions will stabilize row-wise to a matrix equivalent to $\\mathsf {C}$ .",
"While this limit may not be reduced, further arbitrary row reductions generate another convergent sequence.",
"Zorn's Lemma guarantees that some transfinite iteration of this process will eventually converge to the reduced representative of $\\mathsf {C}$ .",
"Constructing recurrences from spaces of solutions In this section, we consider the inverse problem to the motivating problem of this note: Given a subspace $V\\subset \\mathsf {k}^\\mathbb {Z}$ , how can we construct a linear recurrence $\\mathsf {C}\\mathsf {x}=\\mathsf {0}$ whose solutions are $V$ ?",
"We give a characterization of when this is possible in Theorem REF .",
"For any $I\\subset \\mathbb {Z}$ , let $\\pi _{I}:\\mathsf {k}^\\mathbb {Z}\\rightarrow \\mathsf {k}^{I}$ restrict a sequence to the indices in $I$ , and let $\\iota _I:\\mathsf {k}^I\\rightarrow \\mathsf {k}^\\mathbb {Z}$ extend a sequence by 0.",
"Given a subspace $V\\subset \\mathsf {k}^\\mathbb {Z}$ , let $V_{I}\\pi _{I}(V)\\subset \\mathsf {k}^{I}$ .",
"Rank matrices The rank matrix of a subspace $V\\subset \\mathsf {k}^\\mathbb {Z}$ is the $\\mathbb {Z}\\times \\mathbb {Z}$ -matrix withThe entries below the diagonal are unimportant; we set them to $b-a+1$ to avoid special cases later.",
"$ \\mathsf {R}_{a,b}:= \\left\\lbrace \\begin{array}{cc}\\dim _\\mathsf {k}(V_{[a,b]}) & \\text{if }a\\le b \\\\b-a+1 & \\text{otherwise}\\end{array} \\right\\rbrace $ Let $V$ be the space of sequences such that (a) the $-1$ st term is 0, (b) the $-2$ nd and 0th term are equal, and (c) the 0th, 1st, and 2nd terms sum to 0.",
"The rank matrix is $\\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][matrix of math nodes,matrix anchor = M-1-8.center,origin/.style={},throw/.style={},defect/.style={dark red,draw,circle,inner sep=0.25mm,minimum size=2mm},pivot/.style={draw,circle,inner sep=0.25mm,minimum size=2mm}, nodes in empty cells,inner sep=0pt,nodes={anchor=center,rotate=45},column sep={.5cm,between origins},row sep={.5cm,between origins},] (M) at (0,0) {\\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\\\\\& 5 \\& \\& 5 \\& \\& 4 \\& \\& 4 \\& \\& 4 \\& \\& 5 \\& \\& 6 \\& \\& \\cdots \\\\\\cdots \\& \\& 4 \\& \\& 4 \\& \\& 3 \\& \\& 3 \\& \\& 4 \\& \\& 5 \\& \\& 6 \\& \\\\\\& 4 \\& \\& 3 \\& \\& 3 \\& \\& 2 \\& \\& 3 \\& \\& 4 \\& \\& 5 \\& \\& \\cdots \\\\\\cdots \\& \\& 3 \\& \\& 2 \\& \\& 2 \\& \\& 2 \\& \\& 3 \\& \\& 4 \\& \\& 4 \\& \\\\\\& 3 \\& \\& 2 \\& \\& |[defect]| 1 \\& \\& 2 \\& \\& |[defect]| 2 \\& \\& 3 \\& \\& 3 \\& \\& \\cdots \\\\\\cdots \\& \\& 2 \\& \\& 1 \\& \\& 1 \\& \\& 2 \\& \\& 2 \\& \\& 2 \\& \\& 2 \\& \\\\\\& |[throw]| 1 \\& \\& |[throw]| 1 \\& \\& |[defect]| 0 \\& \\& |[origin]| 1 \\& \\&|[throw]| 1 \\& \\& |[throw]| 1 \\& \\& |[throw]| 1 \\& \\& \\cdots \\\\};\\end{tikzpicture}$ The subdiagonal entries have been omitted.",
"The dark redred circles are the defects (see below).",
"Proposition 7.3 If $\\mathsf {R}$ is the rank matrix of $V$ , then the following hold for any $a,b\\in \\mathbb {Z}$ .",
"$\\mathsf {R}_{a,b}-\\mathsf {R}_{a+1,b}$ must be 0 or 1.",
"$\\mathsf {R}_{a,b}-\\mathsf {R}_{a,b-1}$ must be 0 or 1.",
"$\\mathsf {R}_{a,b}-\\mathsf {R}_{a+1,b}-\\mathsf {R}_{a,b-1}+\\mathsf {R}_{a+1,b-1}$ must be 0 or $-1$ .",
"The projection $V_{[a,b]}\\rightarrow V_{[a+1,b]}$ is surjective with at most 1-dimensional kernel.",
"This proves the first result; the second is proven similarly.",
"The first result implies that $-1\\le (\\mathsf {R}_{a,b}-\\mathsf {R}_{a+1,b})-(\\mathsf {R}_{a,b-1} - \\mathsf {R}_{a+1,b-1}) \\le 1$ .",
"The map $V_{[a+1,b]} \\oplus V_{[a,b-1]}\\rightarrow V_{[a,b]}$ which sends $(\\mathsf {v},\\mathsf {w})$ to $\\mathsf {v+w}$ is a surjection whose kernel is the image of the map $V_{[a+1,b-1]} \\rightarrow V_{[a+1,b]} \\oplus V_{[a,b-1]}$ which sends $\\mathsf {v}$ to $(\\mathsf {v},-\\mathsf {v})$ .",
"Therefore, $ \\dim (V_{[a,b]}) \\le \\dim (V_{[a+1,b]} \\oplus V_{[a,b-1]}) - \\dim (V_{[a+1,b-1]}) $ This proves that $\\mathsf {R}_{a,b}-\\mathsf {R}_{a+1,b}-\\mathsf {R}_{a,b-1} + \\mathsf {R}_{a+1,b-1} \\le 0$ .",
"Let us say the pair $(a,b)\\in \\mathbb {Z}\\times \\mathbb {Z}$ is a defect of a rank matrix $\\mathsf {R}$ if $ \\mathsf {R}_{a,b} - \\mathsf {R}_{a+1,b}-\\mathsf {R}_{a,b-1}+\\mathsf {R}_{a+1,b-1} =-1 $ Proposition 7.4 The defects of a rank matrix $\\mathsf {R}$ have the following properties.",
"$\\mathsf {R}_{a,b} = (b-a+1) - \\#(\\text{defects in the box }[a,b]\\times [a,b]) $ .",
"Each row and column of a rank matrix can contain at most one defect.",
"If $[a,b]\\times \\lbrace b\\rbrace $ does not contain any defects, then $V_{[a,b]}$ contains the vector $(0,0,...,0,1)$ .",
"If $\\lbrace a\\rbrace \\times [a,b]$ does not contain any defects, then $V_{[a,b]}$ contains the vector $(1,0,...,0,0)$ .",
"Fix $a$ and consider the sequence $(\\mathsf {R}_{a,b}-\\mathsf {R}_{a+1,b})$ for all $b$ .",
"This sequence starts at 1 for sufficiently negative $b$ , switches from 1 to 0 whenever $(a,b)$ is a defect, and must remain 0 once it does (by Proposition REF .3).",
"Since there are no defects when $a<b$ , this implies that $ \\mathsf {R}_{a,b}=\\mathsf {R}_{a+1,b}+1-\\#(\\text{defects in the line }\\lbrace a\\rbrace \\times [a,b])$ In particular, there can be at most one defect in each row, and inductively implies that $ \\mathsf {R}_{a,b} = (b-a+1) - \\#(\\text{defects in the box }[a,b]\\times [a,b]) $ If $\\lbrace a\\rbrace \\times [a,b]$ does not contain any defect, then $\\mathsf {R}_{a,b}=\\mathsf {R}_{a+1,b}+1$ and the map $V_{[a,b]}\\rightarrow V_{[a+1,b]}$ has 1-dimensional kernel.",
"This kernel must be spanned by the vector $(1,0,...,0,0)$ .",
"The remaining results follow by a dual argument on the sequence $(\\mathsf {R}_{a,b}-\\mathsf {R}_{a,b-1})$ .",
"Given a rank matrix $\\mathsf {R}$ and a consecutive subset $I\\subset \\mathbb {Z}$ , an $\\mathsf {R}$ -schedule for $I$ is a subset $J\\subset I$ for which there is a sequence of subintervals $[a_0,b_0] \\subset [a_1,b_1] \\subset [a_2,b_2] \\subset \\cdots \\subset I$ such that $b_i-a_i=i$ , $\\bigcup [a_i,b_i] = I$ , and $|J\\cap [a_i,b_i]| = \\mathsf {R}_{a_i,b_i}$ .",
"Note that the sequence of intervals determines the $\\mathsf {R}$ -schedule, and, for all $i$ , $J\\cap [a_i,b_i]$ is an $\\mathsf {R}$ -schedule for $[a_i,b_i]$ .",
"Lemma 7.5 Given a subspace $V\\subset \\mathsf {k}^\\mathbb {Z}$ with rank matrix $\\mathsf {R}$ , and an $\\mathsf {R}$ -schedule $J$ for a subset $I$ , the restriction map $V_{I}\\rightarrow \\mathsf {k}^J$ is an isomorphism.",
"In particular, if $J$ is an $\\mathsf {R}$ -schedule for $\\mathbb {Z}$ , then $V\\rightarrow \\mathsf {k}^J$ is an isomorphism.",
"We prove the case when $I=[a,b]$ by induction on $n:=b-a$ .",
"If $n<0$ , the lemma holds vacuously.",
"Assume that the lemma holds for all intervals shorter than $n$ .",
"Choose a sequence of subintervals as in (REF ), and set $[a^{\\prime },b^{\\prime }]:=[a_{n-1},b_{n-1}]$ .",
"The restriction maps fit into a commutative diagram.",
"$\\begin{tikzpicture}[baseline=(current bounding box.center)]\\node (V2) at (0,0) {V_{[a,b]}};\\node (V3) at (3,0) {V_{[a^{\\prime },b^{\\prime }]}};\\node (k2) at (0,-1.5) {\\mathsf {k}^{J}};\\node (k3) at (3,-1.5) {\\mathsf {k}^{J\\cap [a^{\\prime },b^{\\prime }]}};[->>] (V2) to (V3);[->] (V2) to node[left] {\\pi _{J}} (k2);[->] (V3) to node[right] {\\pi _{J\\cap [a^{\\prime },b^{\\prime }]}} (k3);[->>] (k2) to (k3);\\end{tikzpicture}$ By the inductive hypothesis, $\\pi _{J\\cap [a^{\\prime },b^{\\prime }]}$ is an isomorphism, and so $V_{[a,b]}\\rightarrow \\mathsf {k}^{T_{[a^{\\prime },b^{\\prime }]}}$ is surjective.",
"Since $b^{\\prime }-a^{\\prime }=n-1$ , either $[a^{\\prime },b^{\\prime }]=[a+1,b]$ or $[a^{\\prime },b^{\\prime }]=[a,b-1]$ .",
"We have three cases.",
"If $\\mathsf {R}_{a^{\\prime },b^{\\prime }}=\\mathsf {R}_{a,b}$ , then $J\\cap [a^{\\prime },b^{\\prime }]=J$ and so the bottom arrow is an isomorphism.",
"Therefore, $\\pi _J:V_{[a,b]}\\rightarrow \\mathsf {k}^J$ is surjective.",
"If $\\mathsf {R}_{a^{\\prime },b^{\\prime }}=\\mathsf {R}_{a,b}-1$ and $[a^{\\prime },b^{\\prime }]=[a+1,b]$ , then there are no defects in $\\lbrace a\\rbrace \\times [a,b]$ , and so $V_{[a,b]}$ contains $(1,0,...,0,0)$ (by Proposition REF .4).",
"Since $|J| =|J\\cap [a^{\\prime },b^{\\prime }]|+1$ , $a\\in J$ and so the image of $(1,0,...,0,0)$ under $\\pi _{J}$ is non-zero and spans the kernel of $\\mathsf {k}^{J}\\rightarrow \\mathsf {k}^{J\\cap [a^{\\prime },b^{\\prime }]}$ .",
"Thus, $\\pi _J:V_{[a,b]}\\rightarrow \\mathsf {k}^{J}$ is surjective.",
"If $\\mathsf {R}_{a^{\\prime },b^{\\prime }}=\\mathsf {R}_{a,b}-1$ and $[a^{\\prime },b^{\\prime }]=[a,b-1]$ , then there are no defects in $[a,b] \\times \\lbrace b\\rbrace $ , and so $V_{[a,b]}$ contains $(0,0,...,0,1)$ (by Proposition REF .3).",
"By an analogous argument to the previous case, $\\pi _J:V_{[a,b]}\\rightarrow \\mathsf {k}^{J}$ is surjective.",
"The map $\\pi _J:V_{[a,b]}\\rightarrow \\mathsf {k}^{J}$ is surjective in all cases.",
"Since $ \\dim (V_{[a,b]}) = \\mathsf {R}_{[a,b]}= J = \\dim (\\mathsf {k}^{J}) $ this map is an isomorphism, completing the induction.",
"For infinite $I$ , the lemma follows since $I=\\bigcup [a_i,b_i]$ and the lemma holds on each $[a_i,b_i]$ .",
"Recurrence matrices from rank matrices We can now characterize when a subspace of $\\mathsf {k}^\\mathbb {Z}$ is the kernel of a reduced recurrence matrix.",
"Theorem 7.6 Given a subspace $V$ of $\\mathsf {k}^\\mathbb {Z}$ , the following are equivalent.",
"$V$ is the space of solutions to a linear recurrence $\\mathsf {C}\\mathsf {x}=\\mathsf {0}$ .",
"The only left-bounded sequence in $V$ is the zero sequence; that is, if $\\mathsf {v}\\in V$ and $\\mathsf {v}_i=0$ for all $i\\ll 0$ , then $\\mathsf {v}_i=0$ for all $i$ .",
"Every column of the rank matrix $\\mathsf {R}$ of $V$ contains a defect.",
"$V$ is the space of solutions to a reduced linear recurrence $\\overline{\\mathsf {C}}\\mathsf {x} =\\mathsf {0}$ .",
"The shape of $\\overline{\\mathsf {C}}$ is the function $S:\\mathbb {Z}\\rightarrow \\mathbb {Z}$ such that $(S(b),b)$ is a defect of $\\mathsf {R}$ .",
"$(4 \\Rightarrow 1)$ is automatic.",
"$(1 \\Rightarrow 2)$ If a sequence $\\mathsf {v}$ solves a linear recurrence, then every term in $\\mathsf {v}$ is equal to a linear combination of previous terms in the sequence.",
"If every term in $\\mathsf {v}$ of sufficiently negative index is 0, then recursively every term must be 0.",
"$(\\text{not } 3 \\Rightarrow \\text{not }2)$ Assume that the $b$ th column of the rank matrix of $V$ does not contain a defect.",
"By Proposition REF .3, $V_{[a,b]}$ contains the vector $(0,0,...,0,1)$ for all $a\\le b$ .",
"It follows that $V_{(-\\infty ,b]}$ contains the vector $(...,0,0,1)$ .",
"This implies that $V$ contains a sequence $\\mathsf {v}$ with $\\mathsf {v}_b=1$ and $\\mathsf {v}_a=0$ whenever $a<b$ .",
"$(3 \\Rightarrow 4)$ Assume that there is a function $S:\\mathbb {Z}\\rightarrow \\mathbb {Z}$ such that $(S(b),b)$ is a defect of $\\mathsf {R}$ for each $b$ .",
"For each interval $[a,b]$ , define the $\\mathsf {R}$ -schedule $T_{[a,b]} := [a,b] \\setminus S([a,b])$ .",
"We note that $T_{[S(b),b-1]} \\cup \\lbrace b\\rbrace = T_{[S(b)+1,b]}\\cup \\lbrace S(b) \\rbrace $ and consider the following commutative diagram.",
"$ \\begin{tikzpicture}\\node (V1) at (-3,0) {V_{[S(b),b-1]}};\\node (V2) at (0,0) {V_{[S(b),b]}};\\node (V3) at (3,0) {V_{[S(b)+1,b]}};\\node (k1) at (-3,-1.5) {\\mathsf {k}^{T_{[S(b),b-1]}}};\\node (k2) at (0,-1.5) {\\mathsf {k}^{T_{[S(b),b-1]}\\cup \\lbrace b\\rbrace }};\\node (k3) at (3,-1.5) {\\mathsf {k}^{T_{[S(b)+1,b]}}};[->] (V2) to (V1);[->] (V2) to (V3);[->] (V1) to (k1);[->] (V2) to (k2);[->] (V3) to (k3);[->] (k2) to (k1);[->] (k2) to (k3);\\end{tikzpicture}$ Since $(S(b),b)$ is a defect, the maps in the top row are isomorphisms.",
"Since $T_{[S(b),b-1]}$ and $T_{[S(b)+1,b]}$ are $\\mathsf {R}$ -schedules, the maps on the left and right are isomorphisms (by Lemma REF ).",
"Therefore, the map $V_{[S(b),b]} \\longrightarrow \\mathsf {k} ^{J }$ is an embedding of codimension 1.",
"Its image is defined by a relation (unique up to scaling) of the form $\\sum _{a\\in T_{[S(b),b-1]}\\cup \\lbrace b\\rbrace } \\mathsf {C}_{b,a}x_{a} = 0$ Because the left and right maps are isomorphisms, $\\mathsf {C}_{b,b}\\ne 0$ and $\\mathsf {C}_{S(b),b}\\ne 0$ .",
"Rescaling the relation as necessary, we assume that $\\mathsf {C}_{b,b}=1$ .",
"Construct a recurrence matrix $\\mathsf {C}$ such that, for each $b$ , the $b$ th row collects the coefficients of the corresponding equation (REF ).",
"For any pair $b<a$ , $S(b)\\notin T_{[S(a)+1,a]}$ and so $\\mathsf {C}_{a,S(b)}=0$ .",
"Therefore, $\\mathsf {C}$ is a reduced linear recurrence of shape $S$ , such that $V\\subseteq \\mathrm {ker}(\\mathsf {C})$ .",
"Consider any interval $[a,b]$ .",
"For each $b^{\\prime }\\in [a,b] \\setminus T_{[a,b]}$ , the corresponding relation (REF ) only involves terms with index in $[a,b]$ .",
"Since these relations are linearly independent, the codimension of $\\mathrm {ker}(\\mathsf {C})_{[a,b]}$ in $\\mathsf {k}^{[a,b]}$ is at least the cardinality of $[a,b]\\setminus T_{[a,b]}$ .",
"Therefore, $ \\dim (\\mathrm {ker}(\\mathsf {C})_{[a,b]})\\le |T_{[a,b]}| = \\mathsf {R}_{a,b} = \\dim (V_{[a,b]}) $ Since $V_{[a,b]}\\subseteq \\mathrm {ker}(\\mathsf {C})_{[a,b]}$ , $V_{[a,b]}=\\mathrm {ker}(\\mathsf {C})_{[a,b]}$ .",
"Since this holds for all intervals, $V=\\mathrm {ker}(\\mathsf {C})$ .",
"From rank matrices to shapes The theorem relates the shape of a reduced recurrence matrix $\\mathsf {C}$ to the defects of the rank matrix $\\mathsf {R}$ of $\\mathrm {ker}(\\mathsf {C})$ , as follows.",
"Corollary 7.7 If $\\mathsf {C}$ is a reduced recurrence matrix, then $(a,b)$ is a pivot of $\\mathsf {C}$ if and only if $(b,a)$ is a defect of the rank matrix of $\\mathrm {ker}(\\mathsf {C})$ .",
"Therefore, we may translate several earlier results into the language of shapes.",
"Definition 7.8 Given a non-increasing injection $S$ , an $S$ -schedule for a subset $I\\subset \\mathbb {Z}$ is a subset $J\\subset I$ for which there is a subsequence of subintervals $[a_0,b_0] \\subset [a_1,b_1] \\subset [a_2,b_2] \\subset \\cdots \\subset I$ such that $b_i-a_i=i$ , $\\bigcup [a_i,b_i] = I$ , and $|J\\cap [a_i,b_i]|$ equals the number of $S$ -balls in $[a_i,b_i]$ .",
"If $S$ is the shape of a reduced recurrence matrix $\\mathsf {C}$ and $\\mathsf {R}$ is the rank matrix of $\\mathrm {ker}(\\mathsf {C})$ , then $\\mathsf {R}$ -schedules and $S$ -schedules coincide.",
"Note that $I$ only admits an $S$ -schedule if $I$ consists of consecutive elements.",
"The following is a direct translation of Lemma REF .",
"Proposition 7.9 Let $\\mathsf {C}$ be a reduced recurrence matrix with shape $S$ , and let $J$ be an $S$ -schedule for $I$ .",
"Then the restriction map $\\mathrm {ker}(\\mathsf {C})_{I}\\rightarrow \\mathsf {k}^J$ is an isomorphism.",
"In particular, if $J$ is an $S$ -schedule for $\\mathbb {Z}$ , then $\\mathrm {ker}(\\mathsf {C})\\rightarrow \\mathsf {k}^J$ is an isomorphism.",
"As a special case, for any $b\\in \\mathbb {Z}$ , the sequence of intervals $[b,b] \\subset [b-1,b] \\subset [b-2,b] \\subset \\cdots $ determines the following $S$ -schedule for $\\mathbb {Z}$ : $ T_b := \\bigcup _{a\\le b} T_{[a,b]} = \\lbrace a\\le b \\mid \\forall c\\le b, S(c) \\ne a \\rbrace = (-\\infty ,b] \\setminus S\\left( ( -\\infty ,b]\\right) $ Therefore, the proposition specializes to the following.",
"Theorem 7.10 The restriction map $\\pi _{T_b}:\\mathrm {ker}(\\mathsf {C})\\rightarrow \\mathsf {k}^{T_b}$ is an isomorphism.",
"Since $T_b$ contains a unique representative of each $S$ -ball, this implies the following.",
"Theorem 7.11 Then dimension of $\\mathrm {ker}(\\mathsf {C})$ equals the number of $S$ -balls.",
"Constructing $S$ -schedules is easy and intuitive using the juggling pattern of $S$ .",
"Consider any zigzagging path in the juggling pattern which starts on the main diagonal, only travels up (northwest) or right (northeast), and ends above the $(a,b)$ th entry.",
"$\\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][use as bounding box] (-9.5,3.3) rectangle (6.3,-0.3);[matrix of math nodes,matrix anchor = M-2-24.center,nodes in empty cells,inner sep=0pt,gthrow/.style={dark green,draw,circle,inner sep=0mm,minimum size=5mm},rthrow/.style={dark red,draw,circle,inner sep=0mm,minimum size=5mm},bthrow/.style={dark blue,draw,circle,inner sep=0mm,minimum size=5mm},pthrow/.style={dark purple,draw,circle,inner sep=0mm,minimum size=5mm},nodes={anchor=center,node font=\\scriptsize ,rotate=45},column sep={0.4cm,between origins},row sep={-0.4cm,between origins},] (M) at (0,0) {\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& |[gthrow]| \\& \\& |[rthrow]| \\& \\& |[bthrow]| \\& \\& |[bthrow]| \\& \\& |[pthrow]| \\& \\& |[gthrow]| \\& \\& |[rthrow,rotate=-45]| 1 \\& \\& |[bthrow,rotate=-45]| 2 \\&\\& |[gthrow,rotate=-45]| 3 \\& |[inner sep=8pt]| \\& |[pthrow,rotate=-45]| 4 \\& \\& |[bthrow,rotate=-45]| 5 \\& \\& |[bthrow,rotate=-45]| 6 \\& \\& |[rthrow,rotate=-45]| 7 \\& \\& |[gthrow,rotate=-45]| 8 \\& \\& |[pthrow]| \\& \\& |[bthrow]| \\&\\& |[gthrow]| \\& \\& |[rthrow]| \\& \\& |[bthrow]| \\& \\\\\\cdots \\& \\& \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\cdots \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[gthrow]| \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[gthrow]| \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& |[gthrow]| \\& \\& |[rthrow]| \\& \\& \\& \\& \\& \\& |[pthrow]| \\& \\& \\& \\& \\& \\& \\& \\& |[gthrow]| \\& \\& |[pthrow]| \\& \\& \\& \\& \\& \\& |[rthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\cdots \\\\\\& \\& \\& |[pthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[rthrow]| \\& |[inner sep=8pt]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[pthrow]| \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\};\\end{tikzpicture}[dark green] (M-3-1) to (M-2-2) to (M-7-7) to (M-2-12) to (M-5-15) to (M-2-18) to (M-7-23) to (M-2-28) to (M-5-31) to (M-2-34) to (M-7-39);[dark red] (M-5-1) to (M-2-4) to (M-7-9) to (M-2-14) to (M-8-20) to (M-2-26) to (M-7-31) to (M-2-36) to (M-5-39);[dark blue] (M-3-1) to (M-5-3) to (M-2-6) to (M-3-7) to (M-2-8) to (M-6-12) to (M-2-16) to (M-5-19) to (M-2-22) to (M-3-23) to (M-2-24) to (M-6-28) to (M-2-32) to (M-5-35) to (M-2-38) to (M-3-39);[dark purple] (M-5-1) to (M-8-4) to (M-2-10) to (M-7-15) to (M-2-20) to (M-7-25) to (M-2-30) to (M-8-36) to (M-5-39);$ [dashed] (M-2-13.center) to (M-10-21.center) to (M-2-29.center); [thick] (M-2-19.center) to (M-3-18.center) to (M-4-19.center) to (M-5-18.center) to (M-9-22.center)to (M-10-21.center); $Each time the path crosses a colored line, record the row (if it is ascending) or the column (if it is descending).The resulting subset is an $ $-schedule for the interval $ [a,b]$, and every $ S$-schedule can be constructed this way.In the picture above, the path in black determines the $ S$-schedule $ {3,4,2,7}$ for the interval $ [1,8]$.",
"The set $ Tb$ comes from the path which starts to the right of $ (b,b)$ and only travels up (northeast).$ Properties of the solution matrix Fix a reduced recurrence matrix $\\mathsf {C}$ of shape $S$ for the rest of the section.",
"Vanishing The unitriangularity of $\\mathsf {Adj}(\\mathsf {C})$ and $\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})$ mean that the solution matrix $ \\mathsf {Sol}(\\mathsf {C}) := \\mathsf {Adj}(\\mathsf {C}) - \\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})$ has zeroes between the support of $\\mathsf {Adj}(\\mathsf {C})$ and $\\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})$ , which we make precise as follows.",
"Proposition 8.1 $\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ whenever $a<b$ and there is no $c\\le b$ with $S(c)=a$ .",
"If $S$ is bijective, this can be restated as $\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ whenever $a<b<S^{-1}(a)$ .",
"By unitriangularity, $\\mathsf {Adj}(\\mathsf {C})_{a,b}=0$ whenever $a<b$ .",
"Dually, $(\\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P}))_{a,b}$ is only non-zero if there is a $c$ with $\\mathsf {P}_{a,c}\\ne 0$ and $\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})_{c,b}\\ne 0$ ; that is, if $S(c)=a$ and $c\\ge b$ .",
"For fixed $b$ , the proposition determines the value of the $b$ th column of $\\mathsf {Sol}(\\mathsf {C})$ on the set $T_b$ .",
"Since this column solves $\\mathsf {C}\\mathsf {x}=0$ , these entries determine the column (Theorem REF ).",
"Corollary 8.2 The $b$ th column of $\\mathsf {Sol}(\\mathsf {C})$ is the unique solution to $\\mathsf {C}\\mathsf {x}=\\mathsf {0}$ for which $x_a=0$ whenever $a<b$ but there is no $c\\le b$ with $S(c)=a$ , and $x_b=1$ unless $S(b)=b$ , in which case $x_b=0$ .",
"We also have a vanishing condition which guarantees consecutive zeros in each column.",
"Proposition 8.3 $\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ whenever $S(b) < a < b$ .",
"We proceed by induction on $b-a>0$ .",
"Assume that $\\mathsf {Sol}(\\mathsf {C})_{a+1,b}=...=\\mathsf {Sol}(\\mathsf {C})_{b-1,b}=0$ (which is vacuous for the base case $b-a=1$ ).",
"We split into two cases.",
"If there is no $c\\le b$ with $S(c)=a$ , then $\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ by Proposition REF .",
"Otherwise, there is a $c\\le b$ with $S(c)=a$ .",
"Since $S(b)<a$ , we know $c\\ne b$ and so $c<b$ .",
"The $(c,b)$ th entry of the equality $\\mathsf {C}\\mathsf {Sol}(\\mathsf {C})=0$ is $ \\mathsf {C}_{c,a}\\mathsf {Sol}(\\mathsf {C})_{a,b} + \\mathsf {C}_{c,a+1}\\mathsf {Sol}(\\mathsf {C})_{a+1,b} + ... + \\mathsf {C}_{c,c}\\mathsf {Sol}(\\mathsf {C})_{c,b} =0 $ By assumption, $\\mathsf {Sol}(\\mathsf {C})_{a+1,b}=...=\\mathsf {Sol}(\\mathsf {C})_{c,b}=0$ , and so $\\mathsf {C}_{c,a}\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ .",
"Since $\\mathsf {C}_{c,a}=\\mathsf {C}_{c,S(c)}$ is invertible, $\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ .",
"This completes the induction.",
"Propositions REF and REF can be visualized in terms of juggling patterns.",
"As in Remark REF , the $S$ -balls can be visualized by connecting each pivot entry with a line to the diagonal entries in the same row or column.",
"The dashed circle is not in an $S$ -ball.",
"$ \\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][matrix of math nodes,matrix anchor = M-4-8.center,gthrow/.style={dark green,draw,circle,inner sep=0mm,minimum size=5mm},bthrow/.style={dark blue,draw,circle,inner sep=0mm,minimum size=5mm},dashedthrow/.style={dashed,draw,circle,inner sep=0mm,minimum size=5mm},nodes in empty cells,inner sep=0pt,nodes={anchor=center,node font=\\footnotesize ,rotate=45},column sep={.5cm,between origins},row sep={.5cm,between origins},] (M) at (0,0) {\\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\&|[dashedthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& \\cdots \\\\\\cdots \\& \\& -1 \\& \\& -1 \\& \\& -1 \\& \\& \\& \\& \\& \\& |[gthrow]| -1 \\& \\& |[gthrow]| -1 \\& \\\\\\& |[gthrow]| -1 \\& \\& |[bthrow]| -1 \\& \\& |[gthrow]| -1 \\& \\& \\& \\& |[gthrow]| -1 \\& \\& \\& \\& \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\};\\end{tikzpicture}[dark blue,->] (M-2-1) to (M-1-2) to (M-3-4) to (M-1-6) to (M-4-9);[dark green] (M-2-1) to (M-3-2) to (M-1-4) to (M-3-6) to (M-1-8) to (M-3-10) to (M-1-12) to (M-2-13) to (M-1-14) to (M-2-15) to (M-1-16);$ $If we transpose and superimpose the juggling pattern onto the solution matrix...$ [baseline=(current bounding box.center), ampersand replacement=&, ] [matrix of math nodes, matrix anchor = M-8-8.center, gthrow/.style=dark green,draw,circle,inner sep=0mm,minimum size=5mm, bthrow/.style=dark blue,draw,circle,inner sep=0mm,minimum size=5mm, dashedthrow/.style=dashed,draw,circle,inner sep=0mm,minimum size=5mm, faded/.style=black!25, nodes in empty cells, inner sep=0pt, nodes=anchor=center,node font=,rotate=45, column sep=.5cm,between origins, row sep=.5cm,between origins, ] (M) at (0,0) & & & & & & & & & & & & & & & & 5 & & |[faded]| 0 & & 2 & & -1 & & 1 & & |[faded]| 0 & & 1 & & & & -3 & & |[faded]| 0 & & -1 & & 1 & & |[faded]| 0 & & 1 & & |[faded]| 0 & & 2 & & 2 & & |[faded]| 0 & & 1 & & |[faded]| 0 & & 1 & & |[faded]| 0 & & & & -1 & & -1 & & |[faded]| 0 & & |[faded]| 0 & & 1 & & |[faded]| 0 & & 1 & & |[gthrow]| 1 & & |[bthrow]| 1 & & |[gthrow]| 1 & & |[faded]| 0 & & |[gthrow]| 1 & & |[faded]| 0 & & 1 & & & & |[faded]| 0 & & |[faded]| 0 & & |[faded]| 0 & & |[faded]| 0 & & |[faded]| 0 & & |[gthrow]| 1 & & |[gthrow]| 1 & & |[bthrow]| 1 & & |[gthrow]| 1 & & |[bthrow]| 1 & & |[gthrow]| 1 & & |[dashedthrow,faded]| 0 & &|[gthrow]| 1 & & |[gthrow]| 1 & & & & 1 & & 1 & & 1 & & |[faded]| 0 & & |[faded]| 0 & & 1 & & 1 & & 2 & & 2 & & 2 & & |[faded]| 0 & & 1 & & |[faded]| 0 & & 1 & & & & 3 & & 3 & & |[faded]| 0 & & 1 & & 1 & & |[faded]| 0 & & 1 & & 5 & & 5 & & |[faded]| 0 & & 2 & & 1 & & 1 & & |[faded]| 0 & & & & 8 & & |[faded]| 0 & & 3 & & 2 & & 1 & & 1 & & |[faded]| 0 & & 13 & & |[faded]| 0 & & 5 & & 3 & & 2 & & 1 & & 1 & & & & & & & & & & & & & & & & & ; [dark blue,->] (M-7-1) to (M-8-2) to (M-6-4) to (M-8-6) to (M-1-13); [dark green] (M-7-1) to (M-6-2) to (M-8-4) to (M-6-6) to (M-8-8) to (M-6-10) to (M-8-12) to (M-7-13) to (M-8-14) to (M-7-15) to (M-8-16); $Propositions \\ref {prop: colvan} and \\ref {prop: rowvan} state that the lines do not cross any non-zero entries.",
"The circles must be non-zero, except the dashed circle, whose entire row and column must vanish.\\footnote {We note that all of these observations will be combined into and generalized by Lemma \\ref {lemma: boxballs}.",
"}$ Rank conditions The prior vanishing results can be generalized to a formula for ranks of certain rectangular submatrices of $\\mathsf {Sol}(\\mathsf {C})$ .",
"Given integers $a\\le b$ and $c\\le d$ , the box $[a,b]\\times [c,d]\\subset \\mathbb {Z}\\times \\mathbb {Z}$ indexes a rectangular submatrix of $\\mathsf {Sol}(\\mathsf {C})$ .",
"Define an $S$ -ball in the box $[a,b]\\times [c,d]$ to be an equivalence class in the set $ \\left(\\lbrace (i,i) \\mid S(i) \\ne i \\rbrace \\cup \\lbrace (S(j),j) \\mid S(j)\\ne j\\rbrace \\right) \\subset [a,b] \\times [c,d] $ under the equivalence relation generated by $(i,i) \\sim (S(i),i)\\sim (S(i),S(i))$ .",
"This is a 2-dimensional analog of $S$ -balls as defined in Section REF .",
"The relation is that $S$ -balls in an interval $[a,b]$ are in bijection with $S$ -balls in the box $[a,b]\\times [a,b]$ , and also with the $S$ -balls in $[a,b]\\times [c,d]$ and in $[c,d]\\times [a,b]$ for any $[c,d]\\supset [a,b]$ .",
"In terms of the juggling pattern, this counts the number of different colors of circles inside the rectangular submatrix indexed by $[a,b]\\times [c,d]$ .",
"$ \\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][matrix of math nodes,matrix anchor = M-8-8.center,gthrow/.style={dark green,draw,circle,inner sep=0mm,minimum size=5mm},bthrow/.style={dark blue,draw,circle,inner sep=0mm,minimum size=5mm},dashedthrow/.style={dashed,draw,circle,inner sep=0mm,minimum size=5mm},faded/.style={black!25},nodes in empty cells,inner sep=0pt,nodes={anchor=center,node font=\\footnotesize ,rotate=45},column sep={.5cm,between origins},row sep={.5cm,between origins},] (M) at (0,0) {\\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\\\\\& 5 \\& \\& |[faded]| 0 \\& \\& 2 \\& \\& -1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& \\cdots \\\\\\cdots \\& \\& -3 \\& \\& |[faded]| 0 \\& \\& -1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\\\\\& 2 \\& \\& 2 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& \\cdots \\\\\\cdots \\& \\& -1 \\& \\& -1 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\\\\\& |[gthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[faded]| 0 \\& \\& |[gthrow]| 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& \\cdots \\\\\\cdots \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& |[gthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\\\\\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[dashedthrow,faded]| 0 \\& \\&|[gthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& \\cdots \\\\\\cdots \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& 1 \\& \\\\\\& 2 \\& \\& 2 \\& \\& 2 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& \\cdots \\\\\\cdots \\& \\& 3 \\& \\& 3 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\\\\\& 5 \\& \\& 5 \\& \\& |[faded]| 0 \\& \\& 2 \\& \\& 1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& \\cdots \\\\\\cdots \\& \\& 8 \\& \\& |[faded]| 0 \\& \\& 3 \\& \\& 2 \\& \\& 1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\\\\\& 13 \\& \\& |[faded]| 0 \\& \\& 5 \\& \\& 3 \\& \\& 2 \\& \\& 1 \\& \\& 1 \\& \\& \\cdots \\\\\\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\\\};\\end{tikzpicture}[dark blue,->] (M-7-1) to (M-8-2) to (M-6-4) to (M-8-6) to (M-1-13);[dark green] (M-7-1) to (M-6-2) to (M-8-4) to (M-6-6) to (M-8-8) to (M-6-10) to (M-8-12) to (M-7-13) to (M-8-14) to (M-7-15) to (M-8-16);$ [dark red, fill=dark red!50,opacity=.25,rounded corners] (M-3-4.center) – (M-2-5.center) – (M-10-13.center) – (M-11-12.center) – cycle; [dark blue, fill=dark blue!50,opacity=.25,rounded corners] (M-6-9.center) – (M-3-12.center) – (M-7-16.center) – (M-10-13.center) – cycle; [dark purple, fill=dark purple!50,opacity=.25,rounded corners] (M-7-2.center) – (M-4-5.center) – (M-7-8.center) – (M-10-5.center) – cycle; $The {dark purple}{purple}, {dark red}{red}, and {dark blue}{blue} boxes above contain $ dark purple2$, $ dark red0$, and $ dark blue1$ $ S$-balls, respectively.$ The following lemma relates the rank of a submatrix to the number of $S$ -balls in the box.",
"Lemma 8.7 Let $\\mathsf {C}$ be a reduced recurrence matrix of shape $S$ .",
"For any integers $a\\le b$ and $c\\le d$ , the rank of $\\mathsf {Sol}(\\mathsf {C})_{[a,b]\\times [c,d]}$ is at least the number of $S$ -balls in the box $[a,b]\\times [c,d]$ , with equality when $b-c\\ge -1 \\text{ and }\\min (a-c,b-d)\\le 0$ Conceptually, Condition (REF ) implies the box has at least one corner on or below the first superdiagonal, and at least two corners on or above the main diagonal.",
"Condition (REF ) holds for the three boxes in Example REF , and one may check that their ranks coincide with the number of $S$ -balls they contain.",
"First, we bound the rank below by finding a full-rank submatrix of the appropriate size.",
"Fix $[a,b]\\times [c,d]$ .",
"Index $I:=[c,d]\\cap [a,b]\\setminus S([c,d])$ by $i_1<i_2<\\cdots < i_k$ , and index $ J:= \\lbrace j\\in [c,d] \\mid j\\notin [a,b] \\text{ and }S(j) \\in [a,b] \\rbrace $ by $j_1<j_2<\\cdots < j_\\ell $ .",
"Note that $i_k<j_1$ and $S(j_h)<i_1$ for all $h$ .",
"Each $S$ -ball in the box $[a,b]\\times [c,d]$ contains a unique final circle, by which we mean one of the two types of pair: $(i,i)\\in [a,b]\\times [c,d]$ such that there is no $j\\in [c,d]$ with $S(j)=i$ .",
"$(S(i),i) \\in [a,b]\\times [c,d]$ such that $(i,i)\\notin [a,b]\\times [c,d]$ .",
"Furthermore, each column in the box can contain at most one such circle.",
"The columns containing the first kind of final circle are indexed by $I$ , and the columns containing the second kind are indexed by $J$ .",
"Therefore, the number of $S$ -balls in the box is $|I|+|J|$ .",
"Consider the submatrixWe are stretching the definition of `submatrix' to allow for rearranging the order of the rows and columns.",
"of $\\mathsf {Sol}(\\mathsf {C})$ with row set $ \\lbrace i_k,i_{k-1},...,i_1,S(j_1),S(j_2),...,S(J_\\ell )\\rbrace $ and column set $ \\lbrace i_k,i_{k-1},...,i_1,j_1,j_2,...,j_\\ell \\rbrace $ .",
"This matrix is upper triangular with non-zero entries on the diagonal, and so it has rank $|I|+|J|$ .",
"Consequently, $\\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]})\\ge \\text{the number of $S$-balls in the box $[a,b]\\times [c,d]$}$ Note that this inequality did not assume Conditions (REF ).",
"Next, assume that $b-c\\ge -1$ and that $b-d\\le 0$ , and define disjoint sets $K&:=\\lbrace k \\in [a,b] \\text{ such that there is an $m\\in [a,b]$ with $S(m)=k$} \\rbrace \\\\L &:= \\lbrace \\ell \\in [a,c-1] \\text{ such that there is no $m\\le d$ with $S(m)=\\ell $} \\rbrace $ Construct a $(K \\cup L)\\times [a,b]$ -matrix $\\mathsf {M}$ , such that for each $k\\in K$ , the corresponding row of $\\mathsf {M}$ is $\\mathsf {C}_{S^{-1}(k),[a,b]}$ and for each $\\ell \\in L$ , the corresponding row of $\\mathsf {M}$ is $e_\\ell $ (the row vector with a 1 in the $\\ell $ th place).",
"Since $\\mathsf {C}\\mathsf {Sol}(\\mathsf {C})=0$ (Proposition REF ), $\\mathsf {M}_{k,[a,b]}\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]} =0$ for all $k\\in K$ .",
"Since each row of $\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]}$ indexed by $\\ell \\in L$ vanishes (Proposition REF )Explicitly: Since $\\ell \\in [a,b]$ but not in $[c,d]$ , and $b-d\\le 0$ , $\\ell <c$ .",
"Since $\\ell \\notin S( (-\\infty ,d])$ , either $S^{-1}(\\ell )>d$ or $S^{-1}(\\ell )$ is empty.",
"In either case, $\\mathsf {Sol}(\\mathsf {C})_{\\ell ,[c,d]}$ consists of zeroes by Proposition REF ., $\\mathsf {M}_{\\ell , [a,b]}\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]} =0$ for all $\\ell \\in L$ .",
"Therefore, the matrix product vanishes: $ \\mathsf {M}\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]} =0$ Since the first non-zero entry in the $k$ th row of $\\mathsf {M}$ is in the $k$ th column and the first non-zero entry in the $\\ell $ th row of $\\mathsf {M}$ is in the $\\ell $ th column, $\\mathsf {M}$ is in row-echelon form and has rank $|K \\cup L|$ .",
"Therefore, $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]})\\le \\dim (\\mathrm {ker}(\\mathsf {M})) = |[a,b]\\setminus (K\\cup L) |$ Next, we note that $K\\cup L$ indexes the rows of the box $[a,b]\\times [c,d]$ which do not contain the final circle of an $S$ -ball in the box.",
"Combined with the lower bound (REF ), $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]})= \\text{the number of $S$-balls in the box $[a,b]\\times [c,d]$}$ This establishes the theorem in the case when $b-d\\le 0$ .",
"For the final case, we need a source of relations among the columns of $\\mathsf {Sol}(\\mathsf {C})$ .",
"Proposition 8.9 $\\mathsf {Sol}(\\mathsf {C})(\\mathsf {C}\\mathsf {P})=0$ .",
"We compute directly.",
"$\\mathsf {Sol}(\\mathsf {C})(\\mathsf {C}\\mathsf {P})&= \\mathsf {Adj}(\\mathsf {C})(\\mathsf {C}\\mathsf {P}) - (\\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P}) (\\mathsf {C}\\mathsf {P}) \\\\&\\stackrel{*}{=} (\\mathsf {Adj}(\\mathsf {C})\\mathsf {C})\\mathsf {P}- \\mathsf {P}(\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})(\\mathsf {C}\\mathsf {P}))= \\mathsf {P}- \\mathsf {P}= \\mathsf {0}$ Equality ($*$ ) holds because $\\mathsf {P}$ is a generalized permutation matrix.",
"Finally, assume that $b-c\\ge -1$ and that $a-c\\le 0$ , and define disjoint sets $K&:=\\lbrace k \\in [c,d] \\text{ such that $S(k)\\in [c,d]$} \\rbrace \\\\L &:= \\lbrace \\ell \\in [b+1,d] \\text{ such that $S(\\ell )<a$} \\rbrace $ Construct a $[c,d]\\times (K \\cup L)$ -matrix $\\mathsf {M}$ , such that for each $k\\in K$ , the corresponding column of $\\mathsf {M}$ is $\\mathsf {C}_{[a,b],k}$ and for each $\\ell \\in L$ , the corresponding column of $\\mathsf {M}$ is $e_\\ell $ (the column vector with a 1 in the $\\ell $ th place).",
"Since $\\mathsf {Sol}(\\mathsf {C})\\mathsf {C}\\mathsf {P}=0$ (Proposition REF ), $\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]} \\mathsf {M}_{[c,d],k}=0$ for all $k\\in K$ .",
"Since each column of $\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]}$ indexed by $\\ell \\in L$ vanishes (Proposition REF ), $\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]}\\mathsf {M}_{[c,d],\\ell } =0$ for all $\\ell \\in L$ .",
"Therefore, the matrix product vanishes: $ \\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]}\\mathsf {M}=0$ The transpose $\\mathsf {M}^\\top $ is in row echelon form and has rank $|K \\cup L|$ .",
"Therefore, $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]})\\le \\dim (\\mathrm {ker}(\\mathsf {M}^\\top )) = |[c,d]\\setminus (K\\cup L) |$ Next, we note that $K\\cup L$ indexes the columns of the box $[a,b]\\times [c,d]$ which do not contain the `initial circle' of an $S$ -ball in the box.",
"Combined with the lower bound (REF ), $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]})= \\text{the number of $S$-balls in the box $[a,b]\\times [c,d]$} $ Lemma REF extends to improper intervals such as $(-\\infty , \\infty ) $ and $ (-\\infty , d]$ by considering appropriate limits, as the number of $S$ -balls in a box is monotonic in nested intervals.",
"For example, the rank of the entire matrix $\\mathsf {Sol}(\\mathsf {C})$ equals the number of $S$ -balls in the unbounded `box' $(-\\infty ,\\infty )\\times ( -\\infty ,\\infty )$ ; that is, the number of $S$ -balls.",
"The lemma allows us to prove the following fundamental result.",
"Theorem 8.10 If $\\mathsf {C}$ is reduced, the kernel of $\\mathsf {C}$ equals the image of $\\mathsf {Sol}(\\mathsf {C})$ .",
"By Proposition REF , $\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C})) \\subseteq \\mathrm {ker}(\\mathsf {C})$ .",
"For any interval $[a,b]$ and any $S$ -schedule $J$ of $[a,b]$ , Proposition REF implies that $ \\dim (\\mathrm {ker}(\\mathsf {C})_{[a,b]})=|J|$ , and Lemma REF implies that $\\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b], \\mathbb {Z}}) = \\text{\\# of $S$-balls in the box $[a,b]\\times \\mathbb {Z}$} $ The number of $S$ -balls in $[a,b]\\times \\mathbb {Z}$ equals the number of $S$ -balls in $[a,b]$ (Remark REF ), which in turn equals $|J|$ .",
"Thus, $\\mathrm {ker}(\\mathsf {C})_{[a,b]}=\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C}))_{[a,b]}$ for all intervals $[a,b]$ , and so $\\mathrm {ker}(\\mathsf {C})=\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C}))$ .",
"Bases for solutions We can minimally parametrize the kernel using $S$ -schedules.See Definition REF ; note that $I$ only admits an $S$ -schedule if $I$ is a set of consecutive elements.",
"Proposition 8.11 Let $\\mathsf {C}$ be a reduced recurrence matrix with shape $S$ , and let $J$ be an $S$ -schedule for $I$ .",
"Then multiplication by $\\mathsf {Sol}(\\mathsf {C})_{I,J}$ gives an isomorphism $\\mathsf {k}^J\\rightarrow \\mathrm {ker}(\\mathsf {C})_I$ .",
"If $J$ is an $S$ -schedule for $\\mathbb {Z}$ , then multiplication by $\\mathsf {Sol}(\\mathsf {C})_{\\mathbb {Z},J}$ is an isomorphism $\\mathsf {k}^J\\rightarrow \\mathrm {ker}(\\mathsf {C})$ .",
"We first consider the the case when $I$ is finite.",
"Consider any interval $[c,d]$ such that $I\\subset [c,d]$ .",
"By Remark REF , the number of $S$ -balls in $[c,d]\\times I$ equals the number of $S$ -balls in $I$ , which is equal to $J$ by Definition REF .",
"By Lemma REF , this implies that $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],I}) = |J| $ .",
"Since $J\\subset I$ , this implies that $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],J}) \\le |J| $ .",
"We will prove that $\\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],J}) = |J|$ by induction on the number of elements in $I$ .",
"If $I=\\varnothing $ , the claim holds vacuously.",
"Assume that the claim holds for all $S$ -schedules of intervals shorter than $I$ , and choose a sequence of subintervals as in Definition REF .",
"Since $I$ is finite, the sequence terminates at $[a_n,b_n]=I$ .",
"Since $J^{\\prime } := J\\cap [a_{n-1},b_{n-1}]$ is an $S$ -schedule for $[a_{n-1},b_{n-1}]$ , by the inductive hypothesis, $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],J^{\\prime }}) = |J^{\\prime }| $ .",
"If $J^{\\prime }=J$ , this immediately implies (REF ).",
"If $J^{\\prime }\\ne J$ , then $|J| = |J^{\\prime }|+1$ , and so $\\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],[a_{n},b_{n}]} ) = \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],[a_{n-1},b_{n-1}]} ) +1 $ Hence, the column indexed by the unique element of $[a_n,b_n]\\setminus [a_{n-1},b_{n-1}]$ is linearly independent from the other columns.",
"Since this is also the unique element in $J\\setminus J^{\\prime }$ , $\\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],J} ) = \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],J^{\\prime }} ) +1 $ This proves (REF ) and completes the induction.",
"Since $\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C}))= \\mathrm {ker}(\\mathsf {C})$ , we know that $\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C})_{I,J}) \\subseteq \\mathrm {ker}(\\mathsf {C})_I$ .",
"Equation REF for $I=[c,d]$ and Proposition REF imply these both have dimension $|J|$ , so $\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C})_{I,J}) = \\mathrm {ker}(\\mathsf {C})_I$ .",
"When $I$ is infinite, Definition REF and the preceding argument guarantee it is a union of finite intervals on which the proposition holds, so $\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C})_{I,J}) = \\mathrm {ker}(\\mathsf {C})_I$ .",
"The special case of the $S$ -schedule $T_b$ for $\\mathbb {Z}$ yields the following.",
"Theorem 8.12 Multiplication by $\\mathsf {Sol}(\\mathsf {C})_{\\mathbb {Z}\\times T_b}$ gives an isomorphism $\\mathsf {k}^{T_b}\\rightarrow \\mathrm {ker}(\\mathsf {C})$ .",
"appendix"
],
[
"Generalizations and connections",
"We consider a few variations of this problem and applications of these ideas."
],
[
"Affine recurrences",
"Let us briefly consider the affine case.",
"An affine recurrence is a system of equations in the sequence of variables $...,x_{-1},x_0,x_1,x_2,...$ which equates each variable to an affine combination of the previous variables (i.e.",
"a degree 1 polynomial).",
"For example, we could add a constant terms $b_i\\in \\mathsf {k}$ to each equation in the Fibonacci recurrence: $x_i = x_{i-1}+x_{i-2}+b_i,\\;\\;\\; \\forall i\\in \\mathbb {Z}$ As before, we may move the variables to the left and factor the coefficients into a matrix $\\mathsf {C}$ : $ \\mathsf {C}\\mathsf {x} = \\mathsf {b}$ Here, $\\mathsf {C}$ is a recurrence matrix, and $\\mathsf {b}$ collects the constant terms from each equation.",
"If $\\mathsf {C}$ is reduced, define $\\mathsf {P}$ as in Section REF , and define the splitting matrix $\\mathsf {Spl}(\\mathsf {C})$ of $\\mathsf {C}$ by $ \\mathsf {Spl}(\\mathsf {C})_{a,b}:= \\left\\lbrace \\begin{array}{cc}\\mathsf {Adj}(\\mathsf {C})_{a,b} & \\text{if }b\\ge 0\\\\(\\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P}))_{a,b} & \\text{if }b<0\\end{array}\\right\\rbrace $ The right half of this matrix is lower unitriangular, and the left half of this matrix is upper triangular, resulting in non-zero entries concentrated into two antipodal wedges.",
"The splitting matrix for the Fibonacci recurrence matrix is below.",
"$\\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,]\\node [right, dark blue] at (3,-1.5) {\\scriptsize Entries from \\mathsf {Adj}(\\mathsf {C})};\\node [left, dark red] at (-3,1.2) {\\scriptsize Entries from \\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})};[use as bounding box] (-2.925,-2.925) rectangle (2.925,2.325);[matrix of math nodes,matrix anchor = M-10-10.center,nodes in empty cells,inner sep=0pt,nodes={anchor=center,node font=\\scriptsize ,rotate=45},column sep={0.3cm,between origins},row sep={0.3cm,between origins},] (M) at (0,0) {\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\cdots \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& -5 \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\cdots \\& \\& 3 \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& -2 \\& \\& -2 \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\cdots \\& \\& 1 \\& \\& 1 \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& -1 \\& \\& -1 \\& \\& -1 \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& 2 \\& \\& 2 \\& \\& 2 \\& \\& 2 \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& 3 \\& \\& 3 \\& \\& 3 \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& 5 \\& \\& 5 \\& \\& 5 \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& 8 \\& \\& 8 \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& 13 \\& \\& 13 \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& 21 \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& 34 \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\cdots \\\\};\\end{tikzpicture}[dark blue,rounded corners,fill=dark blue, opacity=.1] (M-19-19)+(.2,-.5) to ($ M-10-10)+(-.3,0)$) to ($ (M-10-10)+(-.2,.2)$) to ($ (M-10-19)+(.2,.2)$);[dark red,rounded corners,fill=dark red, opacity=.1] (M-3-1)+(-.2,.5) to ($ (M-10-8)+(.3,0)$) to ($ (M-10-8)+(.2,-.2)$) to ($ (M-10-1)+(-.2,-.2)$);$ $The lower/upper triangular conditions imply the non-zero entries coming from $ Adj(C)$ and $ PAdj(CP)$ must be contained in the blue and red cones, respectively.$ Proposition 4.2 Let $\\mathsf {C}$ be a reduced recurrence matrix.",
"$\\mathsf {Spl}(\\mathsf {C})$ is horizontally bounded.",
"$\\mathsf {C}\\mathsf {Spl}(\\mathsf {C})=\\mathsf {Id}$ ; that is, $\\mathsf {Spl}(\\mathsf {C})$ is a right inverse to $\\mathsf {C}$ .",
"For all $\\mathsf {b}\\in \\mathsf {k}^\\mathbb {Z}$ , the product $\\mathsf {x}=\\mathsf {Spl}(\\mathsf {C})\\mathsf {b}$ exists and is a solution to $\\mathsf {C}\\mathsf {x}=\\mathsf {b}$ .",
"In particular, $\\mathsf {C}\\mathsf {x}=\\mathsf {b}$ has a solution for all $\\mathsf {b}$ .",
"(1) If $a\\ge 0$ , then the $a$ th row of $\\mathsf {Spl}(\\mathsf {C})$ is zero outside the interval $[0,a]$ .",
"If $a<0$ , then the $a$ th row of $\\mathsf {Spl}(\\mathsf {C})$ is zero outside $[a,0]$ .This bound can be sharpened, though we won't need a sharp bound.",
"When $a<0$ , the $a$ th row of $\\mathsf {Spl}(\\mathsf {C})$ is zero outside the interval $[b,0]$ when $S(b)=a$ , and the row is entirely zero if there is no such $b$ .",
"Thus, $\\mathsf {Spl}(\\mathsf {C})$ is horizontally bounded.",
"(2) If $a\\ge 0$ , then the $a$ th column of $\\mathsf {C}\\mathsf {Spl}(\\mathsf {C})$ equals the $a$ th column of $\\mathsf {C}\\mathsf {Adj}(\\mathsf {C})=\\mathsf {Id}$ .",
"If $a<0$ , then the $a$ th column of $\\mathsf {C}\\mathsf {Spl}(\\mathsf {C})$ equals the $a$ th column of $\\mathsf {C}(\\mathsf {P}(\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})))=\\mathsf {Id}$ .",
"Therefore, $\\mathsf {C}\\mathsf {Spl}(\\mathsf {C})=\\mathsf {Id}$ .",
"(3) Since $\\mathsf {C}$ and $\\mathsf {Spl}(\\mathsf {C})$ are horizontally bounded, $\\mathsf {C}(\\mathsf {Spl}(\\mathsf {C})\\mathsf {b}) = (\\mathsf {C}\\mathsf {Spl}(\\mathsf {C}) ) \\mathsf {b} = \\mathsf {Idb}= \\mathsf {b} $ .",
"The existence of solutions to every affine recurrence is equivalent saying that, for any recurrence matrix $\\mathsf {C}$ , the associated multiplication map $\\mathsf {k}^\\mathbb {Z}\\rightarrow \\mathsf {k}^\\mathbb {Z}$ is surjective.",
"Given a solution to $\\mathsf {C}\\mathsf {x}=\\mathsf {b}$ , all other solutions are obtained by adding solutions to $\\mathsf {C}\\mathsf {x}=\\mathsf {0}$ .",
"Proposition 4.4 If $\\mathsf {C}$ is reduced, the solutions to $\\mathsf {C}\\mathsf {x} = \\mathsf {b}$ consist of sequences of the form $ \\mathsf {Spl}(\\mathsf {C}) \\mathsf {b} + \\mathsf {Sol}(\\mathsf {C}) \\mathsf {v}, $ running over all $\\mathsf {v}\\in \\mathsf {k}^\\mathbb {Z}$ such that the product $\\mathsf {Sol}(\\mathsf {C}) \\mathsf {v}$ exists.",
"This follows immediately from Proposition REF and Theorem REF .",
"We have now constructed three right inverses to a reduced recurrence matrix $\\mathsf {C}$ , each possessing an additional property: $\\mathsf {Adj}(\\mathsf {C})$ is lower unitriangular, $\\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})$ is upper unitriangular, and $\\mathsf {Spl}(\\mathsf {C})$ is horizontally bounded.",
"If $\\mathsf {C}\\ne \\mathsf {Id}$ , these are all distinct.",
"Linear recurrences indexed by $\\mathbb {N}$ Variations of `linear recurrences' have been studied for centuries.",
"Most often, one considers a system with variables indexed by $\\mathbb {N}$ (rather than $\\mathbb {Z}$ ) and relations defining each variable except at finitely many `initial variables'.",
"For example, the one-sided Fibonacci recurrence has initial variables $x_0$ and $x_1$ and equations $x_i = x_{i-1}+x_{i-2},\\;\\;\\; \\forall i\\ge 2$ The study of $\\mathbb {N}$ -indexed linear recurrences differs fundamentally from $\\mathbb {Z}$ -indexed linear recurrences.",
"Solutions to an $\\mathbb {N}$ -indexed system are determined by the values of the initial variables, which trivializes the kinds of questions we have considered (e.g.",
"existence and parametrization of solutions).",
"Rather, most work in the $\\mathbb {N}$ -indexed context has focused on finding simple formulas for the terms in a solution.",
"We review a few of these approaches.",
"When the equations in a linear recurrence are the same (a `constant' linear recurrence), shifting the indices of a sequence $ x_0,x_1,x_2,... \\longmapsto x_1,x_2,x_3,... $ defines a linear transformation from the space of solutions to itself.",
"Standard tools from linear algebra (e.g.",
"the characteristic polynomial) can then construct a basis of eigenvectors or generalized eigenvectors for the space of solutions.",
"Since the eigenvectors are geometric sequences, an eigenbasis expresses any solution as a linear combination of geometric sequences.",
"This is covered in textbooks like [10].",
"A sequence $x_0, x_1, x_2,...$ can be translated into formal series in several ways, such as $ F_\\mathsf {x}(t) := x_0 + x_1 t + x_2 t^2 + x_3t^3 + \\cdots $ Some linear recurrences (such as constant ones) translate into functional equations involving these generating functions.",
"Clever manipulation of these equations can then yield simple formulas for solutions.",
"This is covered in textbooks like [14].",
"The asymptotics of solutions, that is, the behavior of $x_i$ for sufficiently large $i$ , can be studied analytically.",
"Poincare [13] and othersA curiosity: [4] is the dissertation of Robert Carmichael, of Carmichael numbers in number theory.",
"[4], [2] construct integrals which coincide with the generating function $F_\\mathsf {x}(t)$ in an `infinitesmal neighborhood of infinity'.",
"See [1] for further details.",
"The techniques of the current work can be adapted to this setting.",
"We first add equations fixing the initial values and rewrite the system as a matrix equation $\\mathsf {C}\\mathsf {x}=\\mathsf {b}$ .",
"For example, the (one-sided) Fibonacci recurrence with initial values $x_0=a$ and $x_1=b$ is rewritten as $ \\begin{bmatrix}1 & 0 & 0 & 0 & \\\\0 & 1 & 0 & 0 & \\\\-1 & -1 & 1 & 0 & \\\\0 & -1 & -1 & 1 & \\\\& & & & \\rotatebox {-45}{\\cdots }\\\\\\end{bmatrix}\\begin{bmatrix}x_0 \\\\ x_1 \\\\ x_2 \\\\ x_3 \\\\ \\vdots \\end{bmatrix}=\\begin{bmatrix}a \\\\ b \\\\ 0 \\\\ 0 \\\\ \\vdots \\end{bmatrix}$ The recurrence matrix $\\mathsf {C}$ is $\\mathbb {N}\\times \\mathbb {N}$ , lower unitriangular, and horizontally bounded.",
"The adjugate $\\mathsf {Adj}(\\mathsf {C})$ is defined as before, and the identity $\\mathsf {Adj}(\\mathsf {C})\\mathsf {C}=\\mathsf {C}\\mathsf {Adj}(\\mathsf {C})=\\mathsf {Id}$ still holds.",
"However, there is a crucial difference.",
"In the $\\mathbb {N}\\times \\mathbb {N}$ case, the adjugate matrix $\\mathsf {Adj}(\\mathsf {C})$ is horizontally bounded, and so $\\mathsf {Adj}(\\mathsf {C}) (\\mathsf {C}\\mathsf {x}) = (\\mathsf {Adj}(\\mathsf {C}) \\mathsf {C}) \\mathsf {x}$ for all $\\mathbb {Z}$ -vectors $\\mathsf {x}$ .",
"If $\\mathsf {C}\\mathsf {x=b}$ , then $ \\mathsf {Adj}(\\mathsf {C}) \\mathsf {b} = \\mathsf {Adj}(\\mathsf {C}) (\\mathsf {C}\\mathsf {x})= (\\mathsf {Adj}(\\mathsf {C}) \\mathsf {C}) \\mathsf {x} =\\mathsf {x}$ Consequently, the unique solution to $\\mathsf {C}\\mathsf {x=b}$ can be computed as a linear combination of the columns of $\\mathsf {Adj}(\\mathsf {C})$ indexed by the initial variables.",
"We can restate this as follows.",
"Proposition 4.6 If $x_i$ is an initial variable in an $\\mathbb {N}$ -indexed linear recurrence with recurrence matrix $\\mathsf {C}$ , then the $i$ th column of $\\mathsf {Adj}(\\mathsf {C})$ is the solution for which $x_i=1$ and all other initial variables are 0.",
"Columns of this form are a basis for the space of all solutions.",
"While Proposition REF gives a basis of solutions, it is unclear how useful this is in general.",
"Computationally, the entries of $\\mathsf {Adj}(\\mathsf {C})$ are determinant of submatrices of $\\mathsf {C}$ , which are (naively) no simpler than recursively computing $x_0,x_1,....,x_j$ directly.",
"Friezes The author's original motivation for the work in this paper is a connection and forthcoming application to the following curious objects.",
"A tame $SL(k)$ -frieze consists of finitely many rows of integers (offset in a diamond pattern) such that: the top and bottom rows consist entirely of 1s, every $k\\times k$ diamond has determinant 1, and every $(k+1)\\times (k+1)$ diamond has determinant 0.",
"An example of an $SL(2)$ -frieze is given below.",
"$\\begin{tikzpicture}[baseline=(current bounding box.south),ampersand replacement=\\&,][matrix of math nodes,matrix anchor = M-1-8.center,origin/.style={},throw/.style={},pivot/.style={draw,circle,inner sep=0.25mm,minimum size=2mm}, nodes in empty cells,inner sep=0pt,nodes={anchor=center,node font=\\scriptsize },column sep={.35cm,between origins},row sep={.35cm,between origins},] (M) at (0,0) {\\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& \\cdots \\\\\\cdots \\& \\& 3 \\& \\& 2 \\& \\& 2 \\& \\& 1 \\& \\& 4 \\& \\& 3 \\& \\& 1 \\& \\& 2 \\& \\& 3 \\& \\& 2 \\& \\& 2 \\& \\& 1 \\& \\& 4 \\& \\& 3 \\& \\& 1 \\& \\& 2 \\& \\& 3 \\& \\\\\\& 5 \\& \\& 5 \\& \\& 3 \\& \\& 1 \\& \\& 3 \\& \\& 11 \\& \\& 2 \\& \\& 1 \\& \\& 5 \\& \\& 5 \\& \\& 3 \\& \\& 1 \\& \\& 3 \\& \\& 11 \\& \\& 2 \\& \\& 1 \\& \\& 5 \\& \\& \\cdots \\\\\\cdots \\& \\& 8 \\& \\& 7 \\& \\& 1 \\& \\& 2 \\& \\& 8 \\& \\& 7 \\& \\& 1 \\& \\& 2 \\& \\& 8 \\& \\& 7 \\& \\& 1 \\& \\& 2 \\& \\& 8 \\& \\& 7 \\& \\& 1 \\& \\& 2 \\& \\& 8 \\& \\\\\\& 3 \\& \\& 11 \\& \\& 2 \\& \\& 1 \\& \\& 5 \\& \\& 5 \\& \\& 3\\& \\& 1 \\& \\& 3 \\& \\& 11 \\& \\& 2 \\& \\& 1 \\& \\& 5 \\& \\& 5 \\& \\& 3\\& \\& 1 \\& \\& 3 \\& \\& \\cdots \\\\\\cdots \\& \\& 4 \\& \\& 3 \\& \\& 1 \\& \\& 2 \\& \\& 3 \\& \\& 2 \\& \\& 2 \\& \\& 1 \\& \\& 4 \\& \\& 3 \\& \\& 1 \\& \\& 2 \\& \\& 3 \\& \\& 2 \\& \\& 2 \\& \\& 1 \\& \\& 4 \\& \\\\\\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& \\cdots \\\\};\\end{tikzpicture}$ The study of friezes was initiated in [7], [5], [6] for $k=2$ , and generalized to arbitrary $k$ in [8], [3].",
"Friezes enjoy many remarkable properties; for example, the rows of a tame $SL(k)$ -frieze must be periodic.",
"An excellent overview is given in [11].",
"A frieze may be converted into a recurrence matrix, by rotating $45^\\circ $ clockwise and using the top row as the main diagonal.",
"[12] and others also use an alternating sign when translating a frieze into a linear recurrence.",
"Remarkably, the solutions have a periodicity condition.",
"Theorem 4.9 [12] If $\\mathsf {C}$ is the recurrence matrix associated to a tame $SL(k)$ -frieze, then every solution to $\\mathsf {C}\\mathsf {x=0}$ is superperiodic: $x_{i+n}=(-1)^sx_i$ for some $n$ and $s$ and all $i$ .",
"In fact, [12] proves a stronger result.",
"For each frieze $\\mathsf {C}$ , they construct a Gale dual frieze $\\mathsf {C}^\\dagger $ whose diagonals encode distinguished solutions to $\\mathsf {C}\\mathsf {x=0}$ .",
"In a sequel [9] to the current work, we will extend Theorem REF to an equivalence.",
"Specifically, if $\\mathsf {C}$ is a reduced recurrence matrix of shape $S$ , then the following are equivalent.",
"$\\mathsf {C}$ satisfies a family of determinantal identities generalizing the tame frieze conditions.",
"Every solution to $\\mathsf {C}\\mathsf {x=0}$ is $n$ -quasiperiodic; that is, $x_{i+n}=\\lambda x_i$ for some $\\lambda $ and all $i$ .",
"The Gale dual $\\mathsf {C}^\\dagger $ , a truncation of $\\mathsf {Sol}(\\mathsf {C})$ , has shape $S^\\dagger $ , where $S^\\dagger (i):=S^{-1}(i)+n$ .",
"The space of such linear recurrences (of fixed shape $S$ ) is the cluster $\\mathcal {X}$ -variety dual to the positroid variety corresponding to $S$ ; this will be explained in [9].",
"The rest of this note proves the promised results.",
"Kernel containment and factorization In this section, we prove a useful equivalence between containments of kernels and factorizations in the semigroup of recurrence matrices.",
"Let $\\mathsf {k}^\\mathbb {Z}_b\\subset \\mathsf {k}^\\mathbb {Z}$ denote the subspace of bounded sequences (i.e.",
"non-zero in finitely many terms).",
"If $\\mathsf {v}\\in \\mathsf {k}^\\mathbb {Z}_b$ and $\\mathsf {w}\\in \\mathsf {k}^\\mathbb {Z}$ , then the dot product $\\mathsf {v}\\cdot \\mathsf {w}$ is well-defined.",
"Lemma 5.2 Let $\\mathsf {C}$ be a recurrence matrix and let $\\mathsf {v}\\in \\mathsf {k}^\\mathbb {Z}_b$ .",
"If $\\mathsf {v}\\cdot \\mathsf {w}=0$ for all $\\mathsf {w}\\in \\mathrm {ker}(\\mathsf {C})$ , then $\\mathsf {v}$ is in the span of the rows of $\\mathsf {C}$ .",
"Let $V\\subset \\mathsf {k}^\\mathbb {Z}_b$ denote the span of the rows of $\\mathsf {C}$ , and assume for contradiction that $\\mathsf {v}\\notin V$ .",
"We may therefore choose a linear map $f:\\mathsf {k}^\\mathbb {Z}_b\\rightarrow \\mathsf {k}$ such that $f(V)=0$ and $f(\\mathsf {v})=1$ .The existence of such a map may depend on the Axiom of Choice, which we therefore assume.",
"Let $\\mathsf {e}_a\\in \\mathsf {k}^\\mathbb {Z}_b$ denote the standard basis vector which is 1 in the $a$ th term and 0 everywhere else, and set $\\mathsf {w}:= ( f(\\mathsf {e}_a)) _{a\\in \\mathbb {Z}}\\in \\mathsf {k}^\\mathbb {Z}$ .",
"By linearity, $f(\\mathsf {u})=\\mathsf {u}\\cdot \\mathsf {w}$ for all $\\mathsf {u}\\in \\mathsf {k}^\\mathbb {Z}_b$ .",
"Since $f$ kills each row of $\\mathsf {C}$ , $\\mathsf {C}\\mathsf {w}=0$ and so $\\mathsf {w}\\in \\mathrm {ker}(\\mathsf {C})$ .",
"However, $\\mathsf {v}\\cdot \\mathsf {w}=1$ , contradicting the hypothesis.",
"Lemma 5.3 Let $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ be recurrence matrices.",
"Then the following are equivalent.",
"$\\mathrm {ker}(\\mathsf {C})\\subseteq \\mathrm {ker}(\\mathsf {C}^{\\prime })$ .",
"$\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for some horizontally bounded matrix $\\mathsf {D}$ .",
"$\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for some recurrence matrix $\\mathsf {D}$ .",
"$\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C})$ is a recurrence matrix.",
"Furthermore, the matrix $\\mathsf {D}$ in (2) and (3) must equal $\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C})$ and is therefore unique.",
"($1\\Rightarrow 2$ ) If $\\mathrm {ker}(\\mathsf {C})\\subseteq \\mathrm {ker}(\\mathsf {C}^{\\prime })$ , then each row of $\\mathsf {C}^{\\prime }$ kills $\\mathrm {ker}(\\mathsf {C})$ .",
"By Lemma REF , the $a$ th row of $\\mathsf {C}^{\\prime }$ is equal to $\\mathsf {D}_a\\mathsf {C}$ for some bounded sequence $\\mathsf {D}_a\\in \\mathsf {k}^\\mathbb {Z}_b$ .",
"The vectors $\\mathsf {D}_a$ may be combined into the rows of a matrix $\\mathsf {D}$ which is horizontally bounded and satisfies $\\mathsf {D}\\mathsf {C}=\\mathsf {C}^{\\prime }$ .",
"($2\\Rightarrow 3+4+$ Uniqueness) Assume that $\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for a horizontally bounded $\\mathsf {D}$ .",
"Then $\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C}) = (\\mathsf {D}\\mathsf {C})\\mathsf {Adj}(\\mathsf {C}) \\stackrel{*}{=} \\mathsf {D}(\\mathsf {C}\\mathsf {Adj}(\\mathsf {C})) = \\mathsf {D}$ Equality ($*$ ) holds because $\\mathsf {D}$ and $\\mathsf {C}$ are horizontally bounded.",
"Since $\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C})$ is lower unitriangular and $\\mathsf {D}$ is horizontally bounded, they are the same recurrence matrix.",
"($3\\Rightarrow 1$ ) Assume $\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for some recurrence matrix $\\mathsf {D}$ .",
"If $\\mathsf {v}\\in \\mathrm {ker}(\\mathsf {C})$ , then $ \\mathsf {C}^{\\prime }\\mathsf {v} = (\\mathsf {D}\\mathsf {C})\\mathsf {v} \\stackrel{*}{=} \\mathsf {D}(\\mathsf {C}\\mathsf {v}) = \\mathsf {D}\\mathsf {0} = \\mathsf {0} $ Equality ($*$ ) holds because $\\mathsf {D}$ and $\\mathsf {C}$ are horizontally bounded.",
"Therefore, $\\mathrm {ker}(\\mathsf {C})\\subseteq \\mathrm {ker}(\\mathsf {C}^{\\prime })$ .",
"($4\\Rightarrow 3$ ) Setting $\\mathsf {D}\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C})$ , we check that $ \\mathsf {D}\\mathsf {C}= (\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C}) ) \\mathsf {C}\\stackrel{*}{=} \\mathsf {C}^{\\prime }(\\mathsf {Adj}(\\mathsf {C})\\mathsf {C}) = \\mathsf {C}^{\\prime }$ Equality ($*$ ) holds because $\\mathsf {C}^{\\prime }$ , $\\mathsf {Adj}(\\mathsf {C})$ , and $\\mathsf {C}$ are lower unitriangular.",
"Theorem 5.4 A recurrence matrix $\\mathsf {C}$ is trivial if and only if $\\mathsf {Adj}(\\mathsf {C})$ is a recurrence matrix.",
"The recurrence matrix $\\mathsf {C}$ is trivial when $\\mathrm {ker}(\\mathsf {C})\\subseteq \\mathrm {ker}(\\mathsf {Id})$ .",
"Applying Lemma REF with $\\mathsf {C}^{\\prime }=\\mathsf {Id}$ , this holds if and only if $\\mathsf {Adj}(\\mathsf {C})$ is a recurrence matrix.",
"Theorem 5.5 Two recurrence matrices $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ are equivalent if and only if $\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for a trivial recurrence matrix $\\mathsf {D}$ .",
"Two recurrence matrices $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ are equivalent if and only if $\\mathrm {ker}(\\mathsf {C})=\\mathrm {ker}(\\mathsf {C}^{\\prime })$ .",
"By Lemma REF , this holds if and only if $\\mathsf {D}=\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C})$ and $\\mathsf {Adj}(\\mathsf {D})=\\mathsf {C}\\mathsf {Adj}(\\mathsf {C}^{\\prime })$ are recurrence matrices.",
"By Theorem REF , this is equivalent to $\\mathsf {D}$ being a trivial linear recurrence.",
"Lemma REF also allows us to make a connection between kernel containment and shapes.",
"Lemma 5.6 Let $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ be recurrence matrices with shape $S$ and $S^{\\prime }$ , respectively.",
"If $\\mathrm {ker}(\\mathsf {C})\\subseteq \\mathrm {ker}(\\mathsf {C}^{\\prime })$ and $S$ is injective (e.g.",
"if $\\mathsf {C}$ is reduced), then $S(a)\\ge S^{\\prime }(a)$ for all $a$ .",
"By Lemma REF , $\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for some recurrence matrix $\\mathsf {D}$ .",
"Fix $a\\in \\mathbb {Z}$ , and consider $B:=\\lbrace b \\in \\mathbb {Z}\\mid \\mathsf {D}_{a,b} \\ne 0\\rbrace $ .",
"This set is bounded and contains $a$ .",
"Let $b_{0}$ be the element of $B$ on which $S$ is minimal; this is unique because $S$ is injective.",
"$ (\\mathsf {D}\\mathsf {C})_{a,S(b_{0})} = \\sum _{b\\in \\mathbb {Z}} \\mathsf {D}_{a,b} \\mathsf {C}_{b,S(b_{0})} = \\mathsf {D}_{a,b_{0}}\\mathsf {C}_{b_{0},S(b_{0})} \\ne 0 $ Therefore, $S^{\\prime }(a)\\le S(b_0)$ .In fact, this is equality, but we won't need this stronger statement.",
"Since $a\\in B$ , $S(b_0)\\le S(a)$ , and so $S^{\\prime }(a)\\le S(a)$ .",
"Proposition 5.7 Reduced recurrence matrices that are equivalent must be equal.",
"Let $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ be reduced and equivalent.",
"Lemma REF implies that $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ have the same shape; call it $S$ .",
"By Lemma REF , $\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for some recurrence matrix $\\mathsf {D}$ .",
"Let $T$ denote the shape of $\\mathsf {D}$ , so that $\\mathsf {D}_{a,b}=0$ whenever $b<T(a)$ .",
"Since $\\mathsf {C}$ is reduced of shape $S$ , $\\mathsf {C}_{b,S(T(a))}=0$ whenever $b>T(a)$ .",
"Therefore, $ \\mathsf {C}^{\\prime }_{a,S(T(a))} = \\sum _b \\mathsf {D}_{a,b} \\mathsf {C}_{b,S(T(a))} = \\mathsf {D}_{a,T(a)}\\mathsf {C}_{T(a),S(T(a))} \\ne 0 $ Since $\\mathsf {C}^{\\prime }$ is also reduced of shape $S$ , this is only possible if $a= T(a)$ .",
"Since this holds for all $a$ , the only non-zero entries of $\\mathsf {D}$ are on the main diagonal.",
"Thus, $\\mathsf {D}=\\mathsf {Id}$ and $\\mathsf {C}^{\\prime }=\\mathsf {C}$ .",
"Gauss-Zordan Elimination Because we are working with $\\mathbb {Z}\\times \\mathbb {Z}$ -matrices, we must consider infinite sequences of row reductions that may be chosen in an arbitrary order.",
"We furthermore consider generalized row reductions: limits of such row reductions (in an appropriate topology).",
"Row reduction Given a recurrence matrix $\\mathsf {C}$ of shape $S$ , a row reduction of $\\mathsf {C}$ is a matrix $\\mathsf {C}^{\\prime }$ obtained by adding $\\mathsf {C}_{a,S(b)} / \\mathsf {C}_{b,S(b)}$ times the $b$ th row to the $a$ th row, for some $b> a$ with $\\mathsf {C}_{a,S(b)}\\ne 0$ .",
"By design, the resulting matrix $\\mathsf {C}^{\\prime }$ has a zero in the $(a,S(b))$ entry.",
"Proposition 6.1 A recurrence matrix is reduced if and only if it has no row reductions.",
"A row reduction of $\\mathsf {C}$ can be reformulated as a factorization $\\mathsf {C}=\\mathsf {D}\\mathsf {C}^{\\prime }$ such that $\\mathsf {D}$ differs from the identity matrix in a single entry $\\mathsf {D}_{a,b}$ , and such that $\\mathsf {C}_{a,S(b)}\\ne 0$ and $\\mathsf {C}^{\\prime }_{a,S(b)}=0$ .",
"This perspective leads to the following generalization.",
"A generalized row reduction of $\\mathsf {C}$ is a recurrence matrix $\\mathsf {C}^{\\prime }$ such that $\\mathsf {C}=\\mathsf {D}\\mathsf {C}^{\\prime }$ for a trivial recurrence matrix $\\mathsf {D}$ with the property that, for each $a$ such that $\\lbrace b <a \\mid \\mathsf {D}_{a,b}\\ne 0\\rbrace $ is non-empty, we have $ \\mathsf {C}_{a,b_a} \\ne 0\\text{ and } \\mathsf {C}^{\\prime }_{a,b_a} = 0 $ where $b_a:= \\min \\lbrace S(b) \\mid b< a \\text{ s.t.",
"}\\mathsf {D}_{a,b}\\ne 0\\rbrace $ .",
"We write $\\mathsf {C}\\succeq \\mathsf {C}^{\\prime }$ to denote that $\\mathsf {C}^{\\prime }$ is a generalized row reduction of $\\mathsf {C}$ .",
"The index $b_a$ may be defined as the leftmost entry of the $a$ th row that multiplication by $\\mathsf {D}$ is `big enough' to change, and so $(\\mathsf {D}\\mathsf {C})_{a,b}=\\mathsf {C}_{a,b}$ whenever $b<b_a$ .",
"Thus, if $\\mathsf {C}\\succeq \\mathsf {C}^{\\prime }$ , then $\\mathsf {C}^{\\prime }$ must vanish in the leftmost entry in which the $a$ th rows of $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ differ.",
"Proposition 6.3 The relation $\\succeq $ defines a partial order on the set of recurrence matrices.",
"As a consequence, an iterated sequence of row reductions is a generalized row reduction.",
"(Antisymmetry) Assume $\\mathsf {C}\\succeq \\mathsf {C}^{\\prime }$ and $\\mathsf {C}\\preceq \\mathsf {C}^{\\prime }$ .",
"By Remark REF , both $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ vanish in the leftmost entry in which the $a$ th rows of $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ differ.",
"However, two entries cannot both vanish and be different, so the $a$ th rows of $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ coincide for all $a$ .",
"Thus, $\\mathsf {C}=\\mathsf {C}^{\\prime }$ .",
"(Transitivity) Let $\\mathsf {C}\\preceq \\mathsf {D}\\mathsf {C}\\preceq \\mathsf {D}^{\\prime }\\mathsf {D}\\mathsf {C}$ , and let $S$ and $S^{\\prime }$ denote the shapes of $\\mathsf {C}$ and $\\mathsf {D}\\mathsf {C}$ , respectively.",
"Fix some $a$ .",
"If $\\lbrace b< a\\mid \\mathsf {D}_{a,b}\\ne 0\\rbrace = \\varnothing $ or $\\lbrace b< a\\mid \\mathsf {D}^{\\prime }_{a,b}\\ne 0\\rbrace = \\varnothing $ , the generalized row reduction condition is easy to check.",
"Assume neither set is empty and let $b_0 &:= \\min \\lbrace S(b) \\mid b<a\\text{ s.t.",
"}\\mathsf {D}_{a,b}\\ne 0\\rbrace \\\\b_0^{\\prime } &:= \\min \\lbrace S^{\\prime }(b) \\mid b<a\\text{ s.t.",
"}\\mathsf {D}^{\\prime }_{a,b}\\ne 0\\rbrace $ By the definition of generalized row reductions, $\\mathsf {C}_{a,b_0}=0,\\;\\;\\; (\\mathsf {D}\\mathsf {C})_{a,b_0}\\ne 0,\\;\\;\\; (\\mathsf {D}\\mathsf {C})_{a,b_0^{\\prime }}=0 ,\\;\\;\\; (\\mathsf {D}^{\\prime }\\mathsf {D}\\mathsf {C})_{a,b_0^{\\prime }}\\ne 0$ This ensures that $\\mathsf {C}_{a,\\min (b_0,b_0^{\\prime })}=0$ and $(\\mathsf {D}^{\\prime }\\mathsf {D}\\mathsf {C})_{a,\\min (b_0,b_0^{\\prime })}\\ne 0$ .",
"Since these entries differ, $ \\min \\lbrace S(b) \\mid b<a\\text{ s.t.",
"}(\\mathsf {D}^{\\prime }\\mathsf {D})_{a,b}\\ne 0\\rbrace \\le \\min (b_0,b_0^{\\prime }) $ To show this is equality, consider some $b<a$ such that $(\\mathsf {D}^{\\prime }\\mathsf {D})_{a,b}\\ne 0$ .",
"We split into cases.",
"Assume $\\mathsf {D}^{\\prime }_{a,c}\\mathsf {D}_{c,b}\\ne 0$ for some $c<a$ .",
"Since $\\mathsf {D}_{c,b}\\ne 0$ and $\\mathsf {C}\\preceq \\mathsf {D}\\mathsf {C}$ , $S^{\\prime }(c)\\le S(b)$ .",
"Since $\\mathsf {D}^{\\prime }_{a,c}\\ne 0$ and $\\mathsf {D}\\mathsf {C}\\preceq \\mathsf {D}^{\\prime }\\mathsf {D}\\mathsf {C}$ , $S^{\\prime }(c)\\ge b_0^{\\prime }$ .",
"Therefore, $S(b)\\ge b^{\\prime }_0$ .",
"Otherwise, $\\mathsf {D}^{\\prime }_{a,c}\\mathsf {D}_{c,b}=0$ for all $c<a$ , and so $(\\mathsf {D}^{\\prime }\\mathsf {D})_{a,b}= \\mathsf {D}^{\\prime }_{a,a}\\mathsf {D}_{a,b}=\\mathsf {D}_{a,b}$ .",
"Since $\\mathsf {D}_{a,b}\\ne 0$ and $\\mathsf {C}\\preceq \\mathsf {D}\\mathsf {C}$ , we know that $S(b)\\ge b_0$ .",
"Therefore, $\\mathsf {C}_{a,\\min (b_0,b_0^{\\prime })}=0$ and $(\\mathsf {D}^{\\prime }\\mathsf {D}\\mathsf {C})_{a,\\min (b_0,b_0^{\\prime })}\\ne 0$ and $ \\min \\lbrace S(b) \\mid b<a\\text{ s.t.",
"}(\\mathsf {D}^{\\prime }\\mathsf {D})_{a,b}\\ne 0\\rbrace = \\min (b_0,b_0^{\\prime }) $ Since this holds for all $a$ , $\\mathsf {C}\\preceq \\mathsf {D}^{\\prime }\\mathsf {D}\\mathsf {C}$ .",
"Limits To define limits of generalized row reductions, we endow the set of recurrence matrices with the topology of row-wise stabilization: a sequence of recurrence matrices converges if each row stabilizes after finitely many steps.",
"We next show that sequences of generalized row reductions must stabilize row-wise to another generalized row reduction, via the following more general result.",
"Lemma 6.4 Let $\\mathcal {C}$ be a set of recurrence matrices in which every pair is comparable in the row reduction partial order.Sometimes called a `chain' in the literature on partially ordered sets.",
"Then the closure of $\\mathcal {C}$ in the space of recurrence matrices contains a lower bound of $\\mathcal {C}$ .",
"Equivalently, there is a descending sequence of recurrence matrices in $\\mathcal {C}$ (i.e.",
"generalized row reductions of the initial matrix in the sequence) which converges (i.e.",
"stabilizes row-wise) to a lower bound of $\\mathcal {C}$ (i.e.",
"a generalized row reduction of every matrix in $\\mathcal {C}$ ).",
"Given a recurrence matrix $\\mathsf {C}$ and an integer $a$ , define $ n_a(\\mathsf {C}) := \\sum _{b \\text{ s.t. }",
"\\mathsf {C}_{(a,b)}\\ne 0} (a-b)^2 $ If $\\mathsf {C}\\preceq \\mathsf {C}^{\\prime }$ , then $n_a(\\mathsf {C})\\le n_a(\\mathsf {C}^{\\prime })$ and equality implies the $a$ th rows of $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ coincide.",
"For each $a$ , let $\\mathcal {C}_a:= \\lbrace \\mathsf {C}\\in \\mathcal {C} \\mid \\forall \\mathsf {C}^{\\prime }\\in \\mathcal {C},\\;n_a (\\mathsf {C}) \\le n_a(\\mathsf {C}^{\\prime })\\rbrace $ ; that is, $\\mathcal {C}_a$ is the set of matrices in $\\mathcal {C}$ which attain the minimum value of $n_a$ .",
"This set is non-empty and the $a$ th row of each matrix in $\\mathcal {C}_a$ is the same, since $n_a$ has the same value and the matrices are comparable.",
"Consider $a,a^{\\prime }\\in \\mathbb {Z}$ and assume, for contradiction, that there exist $\\mathsf {C}\\in \\mathcal {C}_a\\setminus \\mathcal {C}_{a^{\\prime }}$ and $\\mathsf {C}^{\\prime } \\in \\mathcal {C}_{a^{\\prime }}\\setminus \\mathcal {C}_a$ .",
"If $\\mathsf {C}^{\\prime }\\preceq \\mathsf {C}$ , then $n_a(\\mathsf {C}^{\\prime })\\le n_a(\\mathsf {C})$ .",
"By the minimality of $n_a(\\mathsf {C})$ , this is an equality and so $\\mathsf {C}^{\\prime }\\in \\mathcal {C}_a$ ; a contradiction.",
"By a symmetric argument, $\\mathsf {C}\\preceq \\mathsf {C}^{\\prime }$ forces a contradiction.",
"Therefore, $\\mathcal {C}_a\\cap \\mathcal {C}_{a^{\\prime }}$ is either equal to $\\mathcal {C}_a$ or equal to $\\mathcal {C}_{a^{\\prime }}$ .",
"Applying this repeatedly, for any $i\\in \\mathbb {N}$ , there is some $a_i\\in [-i,i]$ such that $ \\bigcap _{a\\in [-i,i]} \\mathcal {C}_a =\\mathcal {C}_{a_i} \\ne \\varnothing $ Choose a matrix $\\mathsf {C}^i$ in $\\mathcal {C}_{a_i}$ for each $i$ .",
"The $a$ th rows in the sequence $\\mathsf {C}^1,\\mathsf {C}^2,\\mathsf {C}^3,...$ , stabilize after the $a$ th term, and so this sequence converges to the recurrence matrix $\\mathsf {C}$ whose $a$ th row coincides with the $a$ th row in each matrix in $\\mathcal {C}_a$ .",
"Let $S$ be the shape of $\\mathsf {C}^1$ .",
"Define a sequence $\\mathsf {D}^1,\\mathsf {D}^2,\\mathsf {D}^3,...$ of trivial recurrence matrices by $\\mathsf {C}^1 = \\mathsf {D}^n\\mathsf {C}^n$ for all $n$ .",
"Since $\\mathsf {C}^1\\succeq \\mathsf {C}^n$ , if $\\mathsf {D}^n_{a,b}\\ne 0$ , then $S(a)\\le S(b)\\le b$ ; that is, the $a$ th row $\\mathsf {D}$ can be non-zero only on the interval $[S(a),a]$ .",
"When $n>|S(a)|$ , the $a$ th row of the product $\\mathsf {D}^n\\mathsf {C}^n$ only depends on rows in $\\mathsf {C}^n$ that coincide with rows in $\\overline{\\mathsf {C}}$ .",
"Therefore, the $a$ th row of $\\mathsf {D}^n\\overline{\\mathsf {C}}$ is equal to $\\mathsf {C}^1$ .",
"Therefore, the sequence $\\mathsf {D}^1,\\mathsf {D}^2,\\mathsf {D}^3,...$ stabilizes row-wise to a matrix $\\overline{\\mathsf {D}}$ such that $\\overline{\\mathsf {D}}\\overline{\\mathsf {C}}=\\mathsf {C}^1$ .",
"As $\\overline{\\mathsf {D}}_{a,b}=\\mathsf {D}^n_{a,b}$ for large enough $n$ , this shows that $\\mathsf {C}\\preceq \\mathsf {C}^1$ .",
"Since the sequence $\\mathsf {C}^\\bullet $ could have started at any matrix in $\\mathcal {C}$ , this shows $\\overline{\\mathsf {C}}$ is a lower bound for $\\mathcal {C}$ .",
"Theorem 6.5 Every recurrence matrix is equivalent to a unique reduced recurrence matrix.",
"Let $\\mathcal {C}$ be an equivalence class of recurrence matrices, with the row reduction partial order.",
"Every non-empty chain in $\\mathcal {C}$ has a lower bound (by Lemma REF ).",
"By Zorn's Lemma, $\\mathcal {C}$ contains a minimal element $\\overline{\\mathsf {C}}$ .",
"If $\\overline{C}$ was not reduced, then there would be a row operation which would strictly decrease it in the reduction partial order; contradicting minimality.",
"Therefore, $\\overline{\\mathsf {C}}$ is reduced.",
"By Proposition REF , this reduced recurrence matrix is unique.",
"This provides a transfinite, non-deterministic analog of Gauss-Jordan elimination, which we humorously dub Gauss-Zordan elimination (both for `Zorn' and the integers $\\mathbb {Z}$ ).",
"Given a recurrence matrix $\\mathsf {C}$ , an arbitrary sequence of row reductions will stabilize row-wise to a matrix equivalent to $\\mathsf {C}$ .",
"While this limit may not be reduced, further arbitrary row reductions generate another convergent sequence.",
"Zorn's Lemma guarantees that some transfinite iteration of this process will eventually converge to the reduced representative of $\\mathsf {C}$ .",
"Constructing recurrences from spaces of solutions In this section, we consider the inverse problem to the motivating problem of this note: Given a subspace $V\\subset \\mathsf {k}^\\mathbb {Z}$ , how can we construct a linear recurrence $\\mathsf {C}\\mathsf {x}=\\mathsf {0}$ whose solutions are $V$ ?",
"We give a characterization of when this is possible in Theorem REF .",
"For any $I\\subset \\mathbb {Z}$ , let $\\pi _{I}:\\mathsf {k}^\\mathbb {Z}\\rightarrow \\mathsf {k}^{I}$ restrict a sequence to the indices in $I$ , and let $\\iota _I:\\mathsf {k}^I\\rightarrow \\mathsf {k}^\\mathbb {Z}$ extend a sequence by 0.",
"Given a subspace $V\\subset \\mathsf {k}^\\mathbb {Z}$ , let $V_{I}\\pi _{I}(V)\\subset \\mathsf {k}^{I}$ .",
"Rank matrices The rank matrix of a subspace $V\\subset \\mathsf {k}^\\mathbb {Z}$ is the $\\mathbb {Z}\\times \\mathbb {Z}$ -matrix withThe entries below the diagonal are unimportant; we set them to $b-a+1$ to avoid special cases later.",
"$ \\mathsf {R}_{a,b}:= \\left\\lbrace \\begin{array}{cc}\\dim _\\mathsf {k}(V_{[a,b]}) & \\text{if }a\\le b \\\\b-a+1 & \\text{otherwise}\\end{array} \\right\\rbrace $ Let $V$ be the space of sequences such that (a) the $-1$ st term is 0, (b) the $-2$ nd and 0th term are equal, and (c) the 0th, 1st, and 2nd terms sum to 0.",
"The rank matrix is $\\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][matrix of math nodes,matrix anchor = M-1-8.center,origin/.style={},throw/.style={},defect/.style={dark red,draw,circle,inner sep=0.25mm,minimum size=2mm},pivot/.style={draw,circle,inner sep=0.25mm,minimum size=2mm}, nodes in empty cells,inner sep=0pt,nodes={anchor=center,rotate=45},column sep={.5cm,between origins},row sep={.5cm,between origins},] (M) at (0,0) {\\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\\\\\& 5 \\& \\& 5 \\& \\& 4 \\& \\& 4 \\& \\& 4 \\& \\& 5 \\& \\& 6 \\& \\& \\cdots \\\\\\cdots \\& \\& 4 \\& \\& 4 \\& \\& 3 \\& \\& 3 \\& \\& 4 \\& \\& 5 \\& \\& 6 \\& \\\\\\& 4 \\& \\& 3 \\& \\& 3 \\& \\& 2 \\& \\& 3 \\& \\& 4 \\& \\& 5 \\& \\& \\cdots \\\\\\cdots \\& \\& 3 \\& \\& 2 \\& \\& 2 \\& \\& 2 \\& \\& 3 \\& \\& 4 \\& \\& 4 \\& \\\\\\& 3 \\& \\& 2 \\& \\& |[defect]| 1 \\& \\& 2 \\& \\& |[defect]| 2 \\& \\& 3 \\& \\& 3 \\& \\& \\cdots \\\\\\cdots \\& \\& 2 \\& \\& 1 \\& \\& 1 \\& \\& 2 \\& \\& 2 \\& \\& 2 \\& \\& 2 \\& \\\\\\& |[throw]| 1 \\& \\& |[throw]| 1 \\& \\& |[defect]| 0 \\& \\& |[origin]| 1 \\& \\&|[throw]| 1 \\& \\& |[throw]| 1 \\& \\& |[throw]| 1 \\& \\& \\cdots \\\\};\\end{tikzpicture}$ The subdiagonal entries have been omitted.",
"The dark redred circles are the defects (see below).",
"Proposition 7.3 If $\\mathsf {R}$ is the rank matrix of $V$ , then the following hold for any $a,b\\in \\mathbb {Z}$ .",
"$\\mathsf {R}_{a,b}-\\mathsf {R}_{a+1,b}$ must be 0 or 1.",
"$\\mathsf {R}_{a,b}-\\mathsf {R}_{a,b-1}$ must be 0 or 1.",
"$\\mathsf {R}_{a,b}-\\mathsf {R}_{a+1,b}-\\mathsf {R}_{a,b-1}+\\mathsf {R}_{a+1,b-1}$ must be 0 or $-1$ .",
"The projection $V_{[a,b]}\\rightarrow V_{[a+1,b]}$ is surjective with at most 1-dimensional kernel.",
"This proves the first result; the second is proven similarly.",
"The first result implies that $-1\\le (\\mathsf {R}_{a,b}-\\mathsf {R}_{a+1,b})-(\\mathsf {R}_{a,b-1} - \\mathsf {R}_{a+1,b-1}) \\le 1$ .",
"The map $V_{[a+1,b]} \\oplus V_{[a,b-1]}\\rightarrow V_{[a,b]}$ which sends $(\\mathsf {v},\\mathsf {w})$ to $\\mathsf {v+w}$ is a surjection whose kernel is the image of the map $V_{[a+1,b-1]} \\rightarrow V_{[a+1,b]} \\oplus V_{[a,b-1]}$ which sends $\\mathsf {v}$ to $(\\mathsf {v},-\\mathsf {v})$ .",
"Therefore, $ \\dim (V_{[a,b]}) \\le \\dim (V_{[a+1,b]} \\oplus V_{[a,b-1]}) - \\dim (V_{[a+1,b-1]}) $ This proves that $\\mathsf {R}_{a,b}-\\mathsf {R}_{a+1,b}-\\mathsf {R}_{a,b-1} + \\mathsf {R}_{a+1,b-1} \\le 0$ .",
"Let us say the pair $(a,b)\\in \\mathbb {Z}\\times \\mathbb {Z}$ is a defect of a rank matrix $\\mathsf {R}$ if $ \\mathsf {R}_{a,b} - \\mathsf {R}_{a+1,b}-\\mathsf {R}_{a,b-1}+\\mathsf {R}_{a+1,b-1} =-1 $ Proposition 7.4 The defects of a rank matrix $\\mathsf {R}$ have the following properties.",
"$\\mathsf {R}_{a,b} = (b-a+1) - \\#(\\text{defects in the box }[a,b]\\times [a,b]) $ .",
"Each row and column of a rank matrix can contain at most one defect.",
"If $[a,b]\\times \\lbrace b\\rbrace $ does not contain any defects, then $V_{[a,b]}$ contains the vector $(0,0,...,0,1)$ .",
"If $\\lbrace a\\rbrace \\times [a,b]$ does not contain any defects, then $V_{[a,b]}$ contains the vector $(1,0,...,0,0)$ .",
"Fix $a$ and consider the sequence $(\\mathsf {R}_{a,b}-\\mathsf {R}_{a+1,b})$ for all $b$ .",
"This sequence starts at 1 for sufficiently negative $b$ , switches from 1 to 0 whenever $(a,b)$ is a defect, and must remain 0 once it does (by Proposition REF .3).",
"Since there are no defects when $a<b$ , this implies that $ \\mathsf {R}_{a,b}=\\mathsf {R}_{a+1,b}+1-\\#(\\text{defects in the line }\\lbrace a\\rbrace \\times [a,b])$ In particular, there can be at most one defect in each row, and inductively implies that $ \\mathsf {R}_{a,b} = (b-a+1) - \\#(\\text{defects in the box }[a,b]\\times [a,b]) $ If $\\lbrace a\\rbrace \\times [a,b]$ does not contain any defect, then $\\mathsf {R}_{a,b}=\\mathsf {R}_{a+1,b}+1$ and the map $V_{[a,b]}\\rightarrow V_{[a+1,b]}$ has 1-dimensional kernel.",
"This kernel must be spanned by the vector $(1,0,...,0,0)$ .",
"The remaining results follow by a dual argument on the sequence $(\\mathsf {R}_{a,b}-\\mathsf {R}_{a,b-1})$ .",
"Given a rank matrix $\\mathsf {R}$ and a consecutive subset $I\\subset \\mathbb {Z}$ , an $\\mathsf {R}$ -schedule for $I$ is a subset $J\\subset I$ for which there is a sequence of subintervals $[a_0,b_0] \\subset [a_1,b_1] \\subset [a_2,b_2] \\subset \\cdots \\subset I$ such that $b_i-a_i=i$ , $\\bigcup [a_i,b_i] = I$ , and $|J\\cap [a_i,b_i]| = \\mathsf {R}_{a_i,b_i}$ .",
"Note that the sequence of intervals determines the $\\mathsf {R}$ -schedule, and, for all $i$ , $J\\cap [a_i,b_i]$ is an $\\mathsf {R}$ -schedule for $[a_i,b_i]$ .",
"Lemma 7.5 Given a subspace $V\\subset \\mathsf {k}^\\mathbb {Z}$ with rank matrix $\\mathsf {R}$ , and an $\\mathsf {R}$ -schedule $J$ for a subset $I$ , the restriction map $V_{I}\\rightarrow \\mathsf {k}^J$ is an isomorphism.",
"In particular, if $J$ is an $\\mathsf {R}$ -schedule for $\\mathbb {Z}$ , then $V\\rightarrow \\mathsf {k}^J$ is an isomorphism.",
"We prove the case when $I=[a,b]$ by induction on $n:=b-a$ .",
"If $n<0$ , the lemma holds vacuously.",
"Assume that the lemma holds for all intervals shorter than $n$ .",
"Choose a sequence of subintervals as in (REF ), and set $[a^{\\prime },b^{\\prime }]:=[a_{n-1},b_{n-1}]$ .",
"The restriction maps fit into a commutative diagram.",
"$\\begin{tikzpicture}[baseline=(current bounding box.center)]\\node (V2) at (0,0) {V_{[a,b]}};\\node (V3) at (3,0) {V_{[a^{\\prime },b^{\\prime }]}};\\node (k2) at (0,-1.5) {\\mathsf {k}^{J}};\\node (k3) at (3,-1.5) {\\mathsf {k}^{J\\cap [a^{\\prime },b^{\\prime }]}};[->>] (V2) to (V3);[->] (V2) to node[left] {\\pi _{J}} (k2);[->] (V3) to node[right] {\\pi _{J\\cap [a^{\\prime },b^{\\prime }]}} (k3);[->>] (k2) to (k3);\\end{tikzpicture}$ By the inductive hypothesis, $\\pi _{J\\cap [a^{\\prime },b^{\\prime }]}$ is an isomorphism, and so $V_{[a,b]}\\rightarrow \\mathsf {k}^{T_{[a^{\\prime },b^{\\prime }]}}$ is surjective.",
"Since $b^{\\prime }-a^{\\prime }=n-1$ , either $[a^{\\prime },b^{\\prime }]=[a+1,b]$ or $[a^{\\prime },b^{\\prime }]=[a,b-1]$ .",
"We have three cases.",
"If $\\mathsf {R}_{a^{\\prime },b^{\\prime }}=\\mathsf {R}_{a,b}$ , then $J\\cap [a^{\\prime },b^{\\prime }]=J$ and so the bottom arrow is an isomorphism.",
"Therefore, $\\pi _J:V_{[a,b]}\\rightarrow \\mathsf {k}^J$ is surjective.",
"If $\\mathsf {R}_{a^{\\prime },b^{\\prime }}=\\mathsf {R}_{a,b}-1$ and $[a^{\\prime },b^{\\prime }]=[a+1,b]$ , then there are no defects in $\\lbrace a\\rbrace \\times [a,b]$ , and so $V_{[a,b]}$ contains $(1,0,...,0,0)$ (by Proposition REF .4).",
"Since $|J| =|J\\cap [a^{\\prime },b^{\\prime }]|+1$ , $a\\in J$ and so the image of $(1,0,...,0,0)$ under $\\pi _{J}$ is non-zero and spans the kernel of $\\mathsf {k}^{J}\\rightarrow \\mathsf {k}^{J\\cap [a^{\\prime },b^{\\prime }]}$ .",
"Thus, $\\pi _J:V_{[a,b]}\\rightarrow \\mathsf {k}^{J}$ is surjective.",
"If $\\mathsf {R}_{a^{\\prime },b^{\\prime }}=\\mathsf {R}_{a,b}-1$ and $[a^{\\prime },b^{\\prime }]=[a,b-1]$ , then there are no defects in $[a,b] \\times \\lbrace b\\rbrace $ , and so $V_{[a,b]}$ contains $(0,0,...,0,1)$ (by Proposition REF .3).",
"By an analogous argument to the previous case, $\\pi _J:V_{[a,b]}\\rightarrow \\mathsf {k}^{J}$ is surjective.",
"The map $\\pi _J:V_{[a,b]}\\rightarrow \\mathsf {k}^{J}$ is surjective in all cases.",
"Since $ \\dim (V_{[a,b]}) = \\mathsf {R}_{[a,b]}= J = \\dim (\\mathsf {k}^{J}) $ this map is an isomorphism, completing the induction.",
"For infinite $I$ , the lemma follows since $I=\\bigcup [a_i,b_i]$ and the lemma holds on each $[a_i,b_i]$ .",
"Recurrence matrices from rank matrices We can now characterize when a subspace of $\\mathsf {k}^\\mathbb {Z}$ is the kernel of a reduced recurrence matrix.",
"Theorem 7.6 Given a subspace $V$ of $\\mathsf {k}^\\mathbb {Z}$ , the following are equivalent.",
"$V$ is the space of solutions to a linear recurrence $\\mathsf {C}\\mathsf {x}=\\mathsf {0}$ .",
"The only left-bounded sequence in $V$ is the zero sequence; that is, if $\\mathsf {v}\\in V$ and $\\mathsf {v}_i=0$ for all $i\\ll 0$ , then $\\mathsf {v}_i=0$ for all $i$ .",
"Every column of the rank matrix $\\mathsf {R}$ of $V$ contains a defect.",
"$V$ is the space of solutions to a reduced linear recurrence $\\overline{\\mathsf {C}}\\mathsf {x} =\\mathsf {0}$ .",
"The shape of $\\overline{\\mathsf {C}}$ is the function $S:\\mathbb {Z}\\rightarrow \\mathbb {Z}$ such that $(S(b),b)$ is a defect of $\\mathsf {R}$ .",
"$(4 \\Rightarrow 1)$ is automatic.",
"$(1 \\Rightarrow 2)$ If a sequence $\\mathsf {v}$ solves a linear recurrence, then every term in $\\mathsf {v}$ is equal to a linear combination of previous terms in the sequence.",
"If every term in $\\mathsf {v}$ of sufficiently negative index is 0, then recursively every term must be 0.",
"$(\\text{not } 3 \\Rightarrow \\text{not }2)$ Assume that the $b$ th column of the rank matrix of $V$ does not contain a defect.",
"By Proposition REF .3, $V_{[a,b]}$ contains the vector $(0,0,...,0,1)$ for all $a\\le b$ .",
"It follows that $V_{(-\\infty ,b]}$ contains the vector $(...,0,0,1)$ .",
"This implies that $V$ contains a sequence $\\mathsf {v}$ with $\\mathsf {v}_b=1$ and $\\mathsf {v}_a=0$ whenever $a<b$ .",
"$(3 \\Rightarrow 4)$ Assume that there is a function $S:\\mathbb {Z}\\rightarrow \\mathbb {Z}$ such that $(S(b),b)$ is a defect of $\\mathsf {R}$ for each $b$ .",
"For each interval $[a,b]$ , define the $\\mathsf {R}$ -schedule $T_{[a,b]} := [a,b] \\setminus S([a,b])$ .",
"We note that $T_{[S(b),b-1]} \\cup \\lbrace b\\rbrace = T_{[S(b)+1,b]}\\cup \\lbrace S(b) \\rbrace $ and consider the following commutative diagram.",
"$ \\begin{tikzpicture}\\node (V1) at (-3,0) {V_{[S(b),b-1]}};\\node (V2) at (0,0) {V_{[S(b),b]}};\\node (V3) at (3,0) {V_{[S(b)+1,b]}};\\node (k1) at (-3,-1.5) {\\mathsf {k}^{T_{[S(b),b-1]}}};\\node (k2) at (0,-1.5) {\\mathsf {k}^{T_{[S(b),b-1]}\\cup \\lbrace b\\rbrace }};\\node (k3) at (3,-1.5) {\\mathsf {k}^{T_{[S(b)+1,b]}}};[->] (V2) to (V1);[->] (V2) to (V3);[->] (V1) to (k1);[->] (V2) to (k2);[->] (V3) to (k3);[->] (k2) to (k1);[->] (k2) to (k3);\\end{tikzpicture}$ Since $(S(b),b)$ is a defect, the maps in the top row are isomorphisms.",
"Since $T_{[S(b),b-1]}$ and $T_{[S(b)+1,b]}$ are $\\mathsf {R}$ -schedules, the maps on the left and right are isomorphisms (by Lemma REF ).",
"Therefore, the map $V_{[S(b),b]} \\longrightarrow \\mathsf {k} ^{J }$ is an embedding of codimension 1.",
"Its image is defined by a relation (unique up to scaling) of the form $\\sum _{a\\in T_{[S(b),b-1]}\\cup \\lbrace b\\rbrace } \\mathsf {C}_{b,a}x_{a} = 0$ Because the left and right maps are isomorphisms, $\\mathsf {C}_{b,b}\\ne 0$ and $\\mathsf {C}_{S(b),b}\\ne 0$ .",
"Rescaling the relation as necessary, we assume that $\\mathsf {C}_{b,b}=1$ .",
"Construct a recurrence matrix $\\mathsf {C}$ such that, for each $b$ , the $b$ th row collects the coefficients of the corresponding equation (REF ).",
"For any pair $b<a$ , $S(b)\\notin T_{[S(a)+1,a]}$ and so $\\mathsf {C}_{a,S(b)}=0$ .",
"Therefore, $\\mathsf {C}$ is a reduced linear recurrence of shape $S$ , such that $V\\subseteq \\mathrm {ker}(\\mathsf {C})$ .",
"Consider any interval $[a,b]$ .",
"For each $b^{\\prime }\\in [a,b] \\setminus T_{[a,b]}$ , the corresponding relation (REF ) only involves terms with index in $[a,b]$ .",
"Since these relations are linearly independent, the codimension of $\\mathrm {ker}(\\mathsf {C})_{[a,b]}$ in $\\mathsf {k}^{[a,b]}$ is at least the cardinality of $[a,b]\\setminus T_{[a,b]}$ .",
"Therefore, $ \\dim (\\mathrm {ker}(\\mathsf {C})_{[a,b]})\\le |T_{[a,b]}| = \\mathsf {R}_{a,b} = \\dim (V_{[a,b]}) $ Since $V_{[a,b]}\\subseteq \\mathrm {ker}(\\mathsf {C})_{[a,b]}$ , $V_{[a,b]}=\\mathrm {ker}(\\mathsf {C})_{[a,b]}$ .",
"Since this holds for all intervals, $V=\\mathrm {ker}(\\mathsf {C})$ .",
"From rank matrices to shapes The theorem relates the shape of a reduced recurrence matrix $\\mathsf {C}$ to the defects of the rank matrix $\\mathsf {R}$ of $\\mathrm {ker}(\\mathsf {C})$ , as follows.",
"Corollary 7.7 If $\\mathsf {C}$ is a reduced recurrence matrix, then $(a,b)$ is a pivot of $\\mathsf {C}$ if and only if $(b,a)$ is a defect of the rank matrix of $\\mathrm {ker}(\\mathsf {C})$ .",
"Therefore, we may translate several earlier results into the language of shapes.",
"Definition 7.8 Given a non-increasing injection $S$ , an $S$ -schedule for a subset $I\\subset \\mathbb {Z}$ is a subset $J\\subset I$ for which there is a subsequence of subintervals $[a_0,b_0] \\subset [a_1,b_1] \\subset [a_2,b_2] \\subset \\cdots \\subset I$ such that $b_i-a_i=i$ , $\\bigcup [a_i,b_i] = I$ , and $|J\\cap [a_i,b_i]|$ equals the number of $S$ -balls in $[a_i,b_i]$ .",
"If $S$ is the shape of a reduced recurrence matrix $\\mathsf {C}$ and $\\mathsf {R}$ is the rank matrix of $\\mathrm {ker}(\\mathsf {C})$ , then $\\mathsf {R}$ -schedules and $S$ -schedules coincide.",
"Note that $I$ only admits an $S$ -schedule if $I$ consists of consecutive elements.",
"The following is a direct translation of Lemma REF .",
"Proposition 7.9 Let $\\mathsf {C}$ be a reduced recurrence matrix with shape $S$ , and let $J$ be an $S$ -schedule for $I$ .",
"Then the restriction map $\\mathrm {ker}(\\mathsf {C})_{I}\\rightarrow \\mathsf {k}^J$ is an isomorphism.",
"In particular, if $J$ is an $S$ -schedule for $\\mathbb {Z}$ , then $\\mathrm {ker}(\\mathsf {C})\\rightarrow \\mathsf {k}^J$ is an isomorphism.",
"As a special case, for any $b\\in \\mathbb {Z}$ , the sequence of intervals $[b,b] \\subset [b-1,b] \\subset [b-2,b] \\subset \\cdots $ determines the following $S$ -schedule for $\\mathbb {Z}$ : $ T_b := \\bigcup _{a\\le b} T_{[a,b]} = \\lbrace a\\le b \\mid \\forall c\\le b, S(c) \\ne a \\rbrace = (-\\infty ,b] \\setminus S\\left( ( -\\infty ,b]\\right) $ Therefore, the proposition specializes to the following.",
"Theorem 7.10 The restriction map $\\pi _{T_b}:\\mathrm {ker}(\\mathsf {C})\\rightarrow \\mathsf {k}^{T_b}$ is an isomorphism.",
"Since $T_b$ contains a unique representative of each $S$ -ball, this implies the following.",
"Theorem 7.11 Then dimension of $\\mathrm {ker}(\\mathsf {C})$ equals the number of $S$ -balls.",
"Constructing $S$ -schedules is easy and intuitive using the juggling pattern of $S$ .",
"Consider any zigzagging path in the juggling pattern which starts on the main diagonal, only travels up (northwest) or right (northeast), and ends above the $(a,b)$ th entry.",
"$\\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][use as bounding box] (-9.5,3.3) rectangle (6.3,-0.3);[matrix of math nodes,matrix anchor = M-2-24.center,nodes in empty cells,inner sep=0pt,gthrow/.style={dark green,draw,circle,inner sep=0mm,minimum size=5mm},rthrow/.style={dark red,draw,circle,inner sep=0mm,minimum size=5mm},bthrow/.style={dark blue,draw,circle,inner sep=0mm,minimum size=5mm},pthrow/.style={dark purple,draw,circle,inner sep=0mm,minimum size=5mm},nodes={anchor=center,node font=\\scriptsize ,rotate=45},column sep={0.4cm,between origins},row sep={-0.4cm,between origins},] (M) at (0,0) {\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& |[gthrow]| \\& \\& |[rthrow]| \\& \\& |[bthrow]| \\& \\& |[bthrow]| \\& \\& |[pthrow]| \\& \\& |[gthrow]| \\& \\& |[rthrow,rotate=-45]| 1 \\& \\& |[bthrow,rotate=-45]| 2 \\&\\& |[gthrow,rotate=-45]| 3 \\& |[inner sep=8pt]| \\& |[pthrow,rotate=-45]| 4 \\& \\& |[bthrow,rotate=-45]| 5 \\& \\& |[bthrow,rotate=-45]| 6 \\& \\& |[rthrow,rotate=-45]| 7 \\& \\& |[gthrow,rotate=-45]| 8 \\& \\& |[pthrow]| \\& \\& |[bthrow]| \\&\\& |[gthrow]| \\& \\& |[rthrow]| \\& \\& |[bthrow]| \\& \\\\\\cdots \\& \\& \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\cdots \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[gthrow]| \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[gthrow]| \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& |[gthrow]| \\& \\& |[rthrow]| \\& \\& \\& \\& \\& \\& |[pthrow]| \\& \\& \\& \\& \\& \\& \\& \\& |[gthrow]| \\& \\& |[pthrow]| \\& \\& \\& \\& \\& \\& |[rthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\cdots \\\\\\& \\& \\& |[pthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[rthrow]| \\& |[inner sep=8pt]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[pthrow]| \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\};\\end{tikzpicture}[dark green] (M-3-1) to (M-2-2) to (M-7-7) to (M-2-12) to (M-5-15) to (M-2-18) to (M-7-23) to (M-2-28) to (M-5-31) to (M-2-34) to (M-7-39);[dark red] (M-5-1) to (M-2-4) to (M-7-9) to (M-2-14) to (M-8-20) to (M-2-26) to (M-7-31) to (M-2-36) to (M-5-39);[dark blue] (M-3-1) to (M-5-3) to (M-2-6) to (M-3-7) to (M-2-8) to (M-6-12) to (M-2-16) to (M-5-19) to (M-2-22) to (M-3-23) to (M-2-24) to (M-6-28) to (M-2-32) to (M-5-35) to (M-2-38) to (M-3-39);[dark purple] (M-5-1) to (M-8-4) to (M-2-10) to (M-7-15) to (M-2-20) to (M-7-25) to (M-2-30) to (M-8-36) to (M-5-39);$ [dashed] (M-2-13.center) to (M-10-21.center) to (M-2-29.center); [thick] (M-2-19.center) to (M-3-18.center) to (M-4-19.center) to (M-5-18.center) to (M-9-22.center)to (M-10-21.center); $Each time the path crosses a colored line, record the row (if it is ascending) or the column (if it is descending).The resulting subset is an $ $-schedule for the interval $ [a,b]$, and every $ S$-schedule can be constructed this way.In the picture above, the path in black determines the $ S$-schedule $ {3,4,2,7}$ for the interval $ [1,8]$.",
"The set $ Tb$ comes from the path which starts to the right of $ (b,b)$ and only travels up (northeast).$ Properties of the solution matrix Fix a reduced recurrence matrix $\\mathsf {C}$ of shape $S$ for the rest of the section.",
"Vanishing The unitriangularity of $\\mathsf {Adj}(\\mathsf {C})$ and $\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})$ mean that the solution matrix $ \\mathsf {Sol}(\\mathsf {C}) := \\mathsf {Adj}(\\mathsf {C}) - \\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})$ has zeroes between the support of $\\mathsf {Adj}(\\mathsf {C})$ and $\\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})$ , which we make precise as follows.",
"Proposition 8.1 $\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ whenever $a<b$ and there is no $c\\le b$ with $S(c)=a$ .",
"If $S$ is bijective, this can be restated as $\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ whenever $a<b<S^{-1}(a)$ .",
"By unitriangularity, $\\mathsf {Adj}(\\mathsf {C})_{a,b}=0$ whenever $a<b$ .",
"Dually, $(\\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P}))_{a,b}$ is only non-zero if there is a $c$ with $\\mathsf {P}_{a,c}\\ne 0$ and $\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})_{c,b}\\ne 0$ ; that is, if $S(c)=a$ and $c\\ge b$ .",
"For fixed $b$ , the proposition determines the value of the $b$ th column of $\\mathsf {Sol}(\\mathsf {C})$ on the set $T_b$ .",
"Since this column solves $\\mathsf {C}\\mathsf {x}=0$ , these entries determine the column (Theorem REF ).",
"Corollary 8.2 The $b$ th column of $\\mathsf {Sol}(\\mathsf {C})$ is the unique solution to $\\mathsf {C}\\mathsf {x}=\\mathsf {0}$ for which $x_a=0$ whenever $a<b$ but there is no $c\\le b$ with $S(c)=a$ , and $x_b=1$ unless $S(b)=b$ , in which case $x_b=0$ .",
"We also have a vanishing condition which guarantees consecutive zeros in each column.",
"Proposition 8.3 $\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ whenever $S(b) < a < b$ .",
"We proceed by induction on $b-a>0$ .",
"Assume that $\\mathsf {Sol}(\\mathsf {C})_{a+1,b}=...=\\mathsf {Sol}(\\mathsf {C})_{b-1,b}=0$ (which is vacuous for the base case $b-a=1$ ).",
"We split into two cases.",
"If there is no $c\\le b$ with $S(c)=a$ , then $\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ by Proposition REF .",
"Otherwise, there is a $c\\le b$ with $S(c)=a$ .",
"Since $S(b)<a$ , we know $c\\ne b$ and so $c<b$ .",
"The $(c,b)$ th entry of the equality $\\mathsf {C}\\mathsf {Sol}(\\mathsf {C})=0$ is $ \\mathsf {C}_{c,a}\\mathsf {Sol}(\\mathsf {C})_{a,b} + \\mathsf {C}_{c,a+1}\\mathsf {Sol}(\\mathsf {C})_{a+1,b} + ... + \\mathsf {C}_{c,c}\\mathsf {Sol}(\\mathsf {C})_{c,b} =0 $ By assumption, $\\mathsf {Sol}(\\mathsf {C})_{a+1,b}=...=\\mathsf {Sol}(\\mathsf {C})_{c,b}=0$ , and so $\\mathsf {C}_{c,a}\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ .",
"Since $\\mathsf {C}_{c,a}=\\mathsf {C}_{c,S(c)}$ is invertible, $\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ .",
"This completes the induction.",
"Propositions REF and REF can be visualized in terms of juggling patterns.",
"As in Remark REF , the $S$ -balls can be visualized by connecting each pivot entry with a line to the diagonal entries in the same row or column.",
"The dashed circle is not in an $S$ -ball.",
"$ \\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][matrix of math nodes,matrix anchor = M-4-8.center,gthrow/.style={dark green,draw,circle,inner sep=0mm,minimum size=5mm},bthrow/.style={dark blue,draw,circle,inner sep=0mm,minimum size=5mm},dashedthrow/.style={dashed,draw,circle,inner sep=0mm,minimum size=5mm},nodes in empty cells,inner sep=0pt,nodes={anchor=center,node font=\\footnotesize ,rotate=45},column sep={.5cm,between origins},row sep={.5cm,between origins},] (M) at (0,0) {\\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\&|[dashedthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& \\cdots \\\\\\cdots \\& \\& -1 \\& \\& -1 \\& \\& -1 \\& \\& \\& \\& \\& \\& |[gthrow]| -1 \\& \\& |[gthrow]| -1 \\& \\\\\\& |[gthrow]| -1 \\& \\& |[bthrow]| -1 \\& \\& |[gthrow]| -1 \\& \\& \\& \\& |[gthrow]| -1 \\& \\& \\& \\& \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\};\\end{tikzpicture}[dark blue,->] (M-2-1) to (M-1-2) to (M-3-4) to (M-1-6) to (M-4-9);[dark green] (M-2-1) to (M-3-2) to (M-1-4) to (M-3-6) to (M-1-8) to (M-3-10) to (M-1-12) to (M-2-13) to (M-1-14) to (M-2-15) to (M-1-16);$ $If we transpose and superimpose the juggling pattern onto the solution matrix...$ [baseline=(current bounding box.center), ampersand replacement=&, ] [matrix of math nodes, matrix anchor = M-8-8.center, gthrow/.style=dark green,draw,circle,inner sep=0mm,minimum size=5mm, bthrow/.style=dark blue,draw,circle,inner sep=0mm,minimum size=5mm, dashedthrow/.style=dashed,draw,circle,inner sep=0mm,minimum size=5mm, faded/.style=black!25, nodes in empty cells, inner sep=0pt, nodes=anchor=center,node font=,rotate=45, column sep=.5cm,between origins, row sep=.5cm,between origins, ] (M) at (0,0) & & & & & & & & & & & & & & & & 5 & & |[faded]| 0 & & 2 & & -1 & & 1 & & |[faded]| 0 & & 1 & & & & -3 & & |[faded]| 0 & & -1 & & 1 & & |[faded]| 0 & & 1 & & |[faded]| 0 & & 2 & & 2 & & |[faded]| 0 & & 1 & & |[faded]| 0 & & 1 & & |[faded]| 0 & & & & -1 & & -1 & & |[faded]| 0 & & |[faded]| 0 & & 1 & & |[faded]| 0 & & 1 & & |[gthrow]| 1 & & |[bthrow]| 1 & & |[gthrow]| 1 & & |[faded]| 0 & & |[gthrow]| 1 & & |[faded]| 0 & & 1 & & & & |[faded]| 0 & & |[faded]| 0 & & |[faded]| 0 & & |[faded]| 0 & & |[faded]| 0 & & |[gthrow]| 1 & & |[gthrow]| 1 & & |[bthrow]| 1 & & |[gthrow]| 1 & & |[bthrow]| 1 & & |[gthrow]| 1 & & |[dashedthrow,faded]| 0 & &|[gthrow]| 1 & & |[gthrow]| 1 & & & & 1 & & 1 & & 1 & & |[faded]| 0 & & |[faded]| 0 & & 1 & & 1 & & 2 & & 2 & & 2 & & |[faded]| 0 & & 1 & & |[faded]| 0 & & 1 & & & & 3 & & 3 & & |[faded]| 0 & & 1 & & 1 & & |[faded]| 0 & & 1 & & 5 & & 5 & & |[faded]| 0 & & 2 & & 1 & & 1 & & |[faded]| 0 & & & & 8 & & |[faded]| 0 & & 3 & & 2 & & 1 & & 1 & & |[faded]| 0 & & 13 & & |[faded]| 0 & & 5 & & 3 & & 2 & & 1 & & 1 & & & & & & & & & & & & & & & & & ; [dark blue,->] (M-7-1) to (M-8-2) to (M-6-4) to (M-8-6) to (M-1-13); [dark green] (M-7-1) to (M-6-2) to (M-8-4) to (M-6-6) to (M-8-8) to (M-6-10) to (M-8-12) to (M-7-13) to (M-8-14) to (M-7-15) to (M-8-16); $Propositions \\ref {prop: colvan} and \\ref {prop: rowvan} state that the lines do not cross any non-zero entries.",
"The circles must be non-zero, except the dashed circle, whose entire row and column must vanish.\\footnote {We note that all of these observations will be combined into and generalized by Lemma \\ref {lemma: boxballs}.",
"}$ Rank conditions The prior vanishing results can be generalized to a formula for ranks of certain rectangular submatrices of $\\mathsf {Sol}(\\mathsf {C})$ .",
"Given integers $a\\le b$ and $c\\le d$ , the box $[a,b]\\times [c,d]\\subset \\mathbb {Z}\\times \\mathbb {Z}$ indexes a rectangular submatrix of $\\mathsf {Sol}(\\mathsf {C})$ .",
"Define an $S$ -ball in the box $[a,b]\\times [c,d]$ to be an equivalence class in the set $ \\left(\\lbrace (i,i) \\mid S(i) \\ne i \\rbrace \\cup \\lbrace (S(j),j) \\mid S(j)\\ne j\\rbrace \\right) \\subset [a,b] \\times [c,d] $ under the equivalence relation generated by $(i,i) \\sim (S(i),i)\\sim (S(i),S(i))$ .",
"This is a 2-dimensional analog of $S$ -balls as defined in Section REF .",
"The relation is that $S$ -balls in an interval $[a,b]$ are in bijection with $S$ -balls in the box $[a,b]\\times [a,b]$ , and also with the $S$ -balls in $[a,b]\\times [c,d]$ and in $[c,d]\\times [a,b]$ for any $[c,d]\\supset [a,b]$ .",
"In terms of the juggling pattern, this counts the number of different colors of circles inside the rectangular submatrix indexed by $[a,b]\\times [c,d]$ .",
"$ \\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][matrix of math nodes,matrix anchor = M-8-8.center,gthrow/.style={dark green,draw,circle,inner sep=0mm,minimum size=5mm},bthrow/.style={dark blue,draw,circle,inner sep=0mm,minimum size=5mm},dashedthrow/.style={dashed,draw,circle,inner sep=0mm,minimum size=5mm},faded/.style={black!25},nodes in empty cells,inner sep=0pt,nodes={anchor=center,node font=\\footnotesize ,rotate=45},column sep={.5cm,between origins},row sep={.5cm,between origins},] (M) at (0,0) {\\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\\\\\& 5 \\& \\& |[faded]| 0 \\& \\& 2 \\& \\& -1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& \\cdots \\\\\\cdots \\& \\& -3 \\& \\& |[faded]| 0 \\& \\& -1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\\\\\& 2 \\& \\& 2 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& \\cdots \\\\\\cdots \\& \\& -1 \\& \\& -1 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\\\\\& |[gthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[faded]| 0 \\& \\& |[gthrow]| 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& \\cdots \\\\\\cdots \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& |[gthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\\\\\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[dashedthrow,faded]| 0 \\& \\&|[gthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& \\cdots \\\\\\cdots \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& 1 \\& \\\\\\& 2 \\& \\& 2 \\& \\& 2 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& \\cdots \\\\\\cdots \\& \\& 3 \\& \\& 3 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\\\\\& 5 \\& \\& 5 \\& \\& |[faded]| 0 \\& \\& 2 \\& \\& 1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& \\cdots \\\\\\cdots \\& \\& 8 \\& \\& |[faded]| 0 \\& \\& 3 \\& \\& 2 \\& \\& 1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\\\\\& 13 \\& \\& |[faded]| 0 \\& \\& 5 \\& \\& 3 \\& \\& 2 \\& \\& 1 \\& \\& 1 \\& \\& \\cdots \\\\\\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\\\};\\end{tikzpicture}[dark blue,->] (M-7-1) to (M-8-2) to (M-6-4) to (M-8-6) to (M-1-13);[dark green] (M-7-1) to (M-6-2) to (M-8-4) to (M-6-6) to (M-8-8) to (M-6-10) to (M-8-12) to (M-7-13) to (M-8-14) to (M-7-15) to (M-8-16);$ [dark red, fill=dark red!50,opacity=.25,rounded corners] (M-3-4.center) – (M-2-5.center) – (M-10-13.center) – (M-11-12.center) – cycle; [dark blue, fill=dark blue!50,opacity=.25,rounded corners] (M-6-9.center) – (M-3-12.center) – (M-7-16.center) – (M-10-13.center) – cycle; [dark purple, fill=dark purple!50,opacity=.25,rounded corners] (M-7-2.center) – (M-4-5.center) – (M-7-8.center) – (M-10-5.center) – cycle; $The {dark purple}{purple}, {dark red}{red}, and {dark blue}{blue} boxes above contain $ dark purple2$, $ dark red0$, and $ dark blue1$ $ S$-balls, respectively.$ The following lemma relates the rank of a submatrix to the number of $S$ -balls in the box.",
"Lemma 8.7 Let $\\mathsf {C}$ be a reduced recurrence matrix of shape $S$ .",
"For any integers $a\\le b$ and $c\\le d$ , the rank of $\\mathsf {Sol}(\\mathsf {C})_{[a,b]\\times [c,d]}$ is at least the number of $S$ -balls in the box $[a,b]\\times [c,d]$ , with equality when $b-c\\ge -1 \\text{ and }\\min (a-c,b-d)\\le 0$ Conceptually, Condition (REF ) implies the box has at least one corner on or below the first superdiagonal, and at least two corners on or above the main diagonal.",
"Condition (REF ) holds for the three boxes in Example REF , and one may check that their ranks coincide with the number of $S$ -balls they contain.",
"First, we bound the rank below by finding a full-rank submatrix of the appropriate size.",
"Fix $[a,b]\\times [c,d]$ .",
"Index $I:=[c,d]\\cap [a,b]\\setminus S([c,d])$ by $i_1<i_2<\\cdots < i_k$ , and index $ J:= \\lbrace j\\in [c,d] \\mid j\\notin [a,b] \\text{ and }S(j) \\in [a,b] \\rbrace $ by $j_1<j_2<\\cdots < j_\\ell $ .",
"Note that $i_k<j_1$ and $S(j_h)<i_1$ for all $h$ .",
"Each $S$ -ball in the box $[a,b]\\times [c,d]$ contains a unique final circle, by which we mean one of the two types of pair: $(i,i)\\in [a,b]\\times [c,d]$ such that there is no $j\\in [c,d]$ with $S(j)=i$ .",
"$(S(i),i) \\in [a,b]\\times [c,d]$ such that $(i,i)\\notin [a,b]\\times [c,d]$ .",
"Furthermore, each column in the box can contain at most one such circle.",
"The columns containing the first kind of final circle are indexed by $I$ , and the columns containing the second kind are indexed by $J$ .",
"Therefore, the number of $S$ -balls in the box is $|I|+|J|$ .",
"Consider the submatrixWe are stretching the definition of `submatrix' to allow for rearranging the order of the rows and columns.",
"of $\\mathsf {Sol}(\\mathsf {C})$ with row set $ \\lbrace i_k,i_{k-1},...,i_1,S(j_1),S(j_2),...,S(J_\\ell )\\rbrace $ and column set $ \\lbrace i_k,i_{k-1},...,i_1,j_1,j_2,...,j_\\ell \\rbrace $ .",
"This matrix is upper triangular with non-zero entries on the diagonal, and so it has rank $|I|+|J|$ .",
"Consequently, $\\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]})\\ge \\text{the number of $S$-balls in the box $[a,b]\\times [c,d]$}$ Note that this inequality did not assume Conditions (REF ).",
"Next, assume that $b-c\\ge -1$ and that $b-d\\le 0$ , and define disjoint sets $K&:=\\lbrace k \\in [a,b] \\text{ such that there is an $m\\in [a,b]$ with $S(m)=k$} \\rbrace \\\\L &:= \\lbrace \\ell \\in [a,c-1] \\text{ such that there is no $m\\le d$ with $S(m)=\\ell $} \\rbrace $ Construct a $(K \\cup L)\\times [a,b]$ -matrix $\\mathsf {M}$ , such that for each $k\\in K$ , the corresponding row of $\\mathsf {M}$ is $\\mathsf {C}_{S^{-1}(k),[a,b]}$ and for each $\\ell \\in L$ , the corresponding row of $\\mathsf {M}$ is $e_\\ell $ (the row vector with a 1 in the $\\ell $ th place).",
"Since $\\mathsf {C}\\mathsf {Sol}(\\mathsf {C})=0$ (Proposition REF ), $\\mathsf {M}_{k,[a,b]}\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]} =0$ for all $k\\in K$ .",
"Since each row of $\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]}$ indexed by $\\ell \\in L$ vanishes (Proposition REF )Explicitly: Since $\\ell \\in [a,b]$ but not in $[c,d]$ , and $b-d\\le 0$ , $\\ell <c$ .",
"Since $\\ell \\notin S( (-\\infty ,d])$ , either $S^{-1}(\\ell )>d$ or $S^{-1}(\\ell )$ is empty.",
"In either case, $\\mathsf {Sol}(\\mathsf {C})_{\\ell ,[c,d]}$ consists of zeroes by Proposition REF ., $\\mathsf {M}_{\\ell , [a,b]}\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]} =0$ for all $\\ell \\in L$ .",
"Therefore, the matrix product vanishes: $ \\mathsf {M}\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]} =0$ Since the first non-zero entry in the $k$ th row of $\\mathsf {M}$ is in the $k$ th column and the first non-zero entry in the $\\ell $ th row of $\\mathsf {M}$ is in the $\\ell $ th column, $\\mathsf {M}$ is in row-echelon form and has rank $|K \\cup L|$ .",
"Therefore, $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]})\\le \\dim (\\mathrm {ker}(\\mathsf {M})) = |[a,b]\\setminus (K\\cup L) |$ Next, we note that $K\\cup L$ indexes the rows of the box $[a,b]\\times [c,d]$ which do not contain the final circle of an $S$ -ball in the box.",
"Combined with the lower bound (REF ), $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]})= \\text{the number of $S$-balls in the box $[a,b]\\times [c,d]$}$ This establishes the theorem in the case when $b-d\\le 0$ .",
"For the final case, we need a source of relations among the columns of $\\mathsf {Sol}(\\mathsf {C})$ .",
"Proposition 8.9 $\\mathsf {Sol}(\\mathsf {C})(\\mathsf {C}\\mathsf {P})=0$ .",
"We compute directly.",
"$\\mathsf {Sol}(\\mathsf {C})(\\mathsf {C}\\mathsf {P})&= \\mathsf {Adj}(\\mathsf {C})(\\mathsf {C}\\mathsf {P}) - (\\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P}) (\\mathsf {C}\\mathsf {P}) \\\\&\\stackrel{*}{=} (\\mathsf {Adj}(\\mathsf {C})\\mathsf {C})\\mathsf {P}- \\mathsf {P}(\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})(\\mathsf {C}\\mathsf {P}))= \\mathsf {P}- \\mathsf {P}= \\mathsf {0}$ Equality ($*$ ) holds because $\\mathsf {P}$ is a generalized permutation matrix.",
"Finally, assume that $b-c\\ge -1$ and that $a-c\\le 0$ , and define disjoint sets $K&:=\\lbrace k \\in [c,d] \\text{ such that $S(k)\\in [c,d]$} \\rbrace \\\\L &:= \\lbrace \\ell \\in [b+1,d] \\text{ such that $S(\\ell )<a$} \\rbrace $ Construct a $[c,d]\\times (K \\cup L)$ -matrix $\\mathsf {M}$ , such that for each $k\\in K$ , the corresponding column of $\\mathsf {M}$ is $\\mathsf {C}_{[a,b],k}$ and for each $\\ell \\in L$ , the corresponding column of $\\mathsf {M}$ is $e_\\ell $ (the column vector with a 1 in the $\\ell $ th place).",
"Since $\\mathsf {Sol}(\\mathsf {C})\\mathsf {C}\\mathsf {P}=0$ (Proposition REF ), $\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]} \\mathsf {M}_{[c,d],k}=0$ for all $k\\in K$ .",
"Since each column of $\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]}$ indexed by $\\ell \\in L$ vanishes (Proposition REF ), $\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]}\\mathsf {M}_{[c,d],\\ell } =0$ for all $\\ell \\in L$ .",
"Therefore, the matrix product vanishes: $ \\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]}\\mathsf {M}=0$ The transpose $\\mathsf {M}^\\top $ is in row echelon form and has rank $|K \\cup L|$ .",
"Therefore, $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]})\\le \\dim (\\mathrm {ker}(\\mathsf {M}^\\top )) = |[c,d]\\setminus (K\\cup L) |$ Next, we note that $K\\cup L$ indexes the columns of the box $[a,b]\\times [c,d]$ which do not contain the `initial circle' of an $S$ -ball in the box.",
"Combined with the lower bound (REF ), $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]})= \\text{the number of $S$-balls in the box $[a,b]\\times [c,d]$} $ Lemma REF extends to improper intervals such as $(-\\infty , \\infty ) $ and $ (-\\infty , d]$ by considering appropriate limits, as the number of $S$ -balls in a box is monotonic in nested intervals.",
"For example, the rank of the entire matrix $\\mathsf {Sol}(\\mathsf {C})$ equals the number of $S$ -balls in the unbounded `box' $(-\\infty ,\\infty )\\times ( -\\infty ,\\infty )$ ; that is, the number of $S$ -balls.",
"The lemma allows us to prove the following fundamental result.",
"Theorem 8.10 If $\\mathsf {C}$ is reduced, the kernel of $\\mathsf {C}$ equals the image of $\\mathsf {Sol}(\\mathsf {C})$ .",
"By Proposition REF , $\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C})) \\subseteq \\mathrm {ker}(\\mathsf {C})$ .",
"For any interval $[a,b]$ and any $S$ -schedule $J$ of $[a,b]$ , Proposition REF implies that $ \\dim (\\mathrm {ker}(\\mathsf {C})_{[a,b]})=|J|$ , and Lemma REF implies that $\\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b], \\mathbb {Z}}) = \\text{\\# of $S$-balls in the box $[a,b]\\times \\mathbb {Z}$} $ The number of $S$ -balls in $[a,b]\\times \\mathbb {Z}$ equals the number of $S$ -balls in $[a,b]$ (Remark REF ), which in turn equals $|J|$ .",
"Thus, $\\mathrm {ker}(\\mathsf {C})_{[a,b]}=\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C}))_{[a,b]}$ for all intervals $[a,b]$ , and so $\\mathrm {ker}(\\mathsf {C})=\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C}))$ .",
"Bases for solutions We can minimally parametrize the kernel using $S$ -schedules.See Definition REF ; note that $I$ only admits an $S$ -schedule if $I$ is a set of consecutive elements.",
"Proposition 8.11 Let $\\mathsf {C}$ be a reduced recurrence matrix with shape $S$ , and let $J$ be an $S$ -schedule for $I$ .",
"Then multiplication by $\\mathsf {Sol}(\\mathsf {C})_{I,J}$ gives an isomorphism $\\mathsf {k}^J\\rightarrow \\mathrm {ker}(\\mathsf {C})_I$ .",
"If $J$ is an $S$ -schedule for $\\mathbb {Z}$ , then multiplication by $\\mathsf {Sol}(\\mathsf {C})_{\\mathbb {Z},J}$ is an isomorphism $\\mathsf {k}^J\\rightarrow \\mathrm {ker}(\\mathsf {C})$ .",
"We first consider the the case when $I$ is finite.",
"Consider any interval $[c,d]$ such that $I\\subset [c,d]$ .",
"By Remark REF , the number of $S$ -balls in $[c,d]\\times I$ equals the number of $S$ -balls in $I$ , which is equal to $J$ by Definition REF .",
"By Lemma REF , this implies that $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],I}) = |J| $ .",
"Since $J\\subset I$ , this implies that $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],J}) \\le |J| $ .",
"We will prove that $\\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],J}) = |J|$ by induction on the number of elements in $I$ .",
"If $I=\\varnothing $ , the claim holds vacuously.",
"Assume that the claim holds for all $S$ -schedules of intervals shorter than $I$ , and choose a sequence of subintervals as in Definition REF .",
"Since $I$ is finite, the sequence terminates at $[a_n,b_n]=I$ .",
"Since $J^{\\prime } := J\\cap [a_{n-1},b_{n-1}]$ is an $S$ -schedule for $[a_{n-1},b_{n-1}]$ , by the inductive hypothesis, $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],J^{\\prime }}) = |J^{\\prime }| $ .",
"If $J^{\\prime }=J$ , this immediately implies (REF ).",
"If $J^{\\prime }\\ne J$ , then $|J| = |J^{\\prime }|+1$ , and so $\\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],[a_{n},b_{n}]} ) = \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],[a_{n-1},b_{n-1}]} ) +1 $ Hence, the column indexed by the unique element of $[a_n,b_n]\\setminus [a_{n-1},b_{n-1}]$ is linearly independent from the other columns.",
"Since this is also the unique element in $J\\setminus J^{\\prime }$ , $\\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],J} ) = \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],J^{\\prime }} ) +1 $ This proves (REF ) and completes the induction.",
"Since $\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C}))= \\mathrm {ker}(\\mathsf {C})$ , we know that $\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C})_{I,J}) \\subseteq \\mathrm {ker}(\\mathsf {C})_I$ .",
"Equation REF for $I=[c,d]$ and Proposition REF imply these both have dimension $|J|$ , so $\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C})_{I,J}) = \\mathrm {ker}(\\mathsf {C})_I$ .",
"When $I$ is infinite, Definition REF and the preceding argument guarantee it is a union of finite intervals on which the proposition holds, so $\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C})_{I,J}) = \\mathrm {ker}(\\mathsf {C})_I$ .",
"The special case of the $S$ -schedule $T_b$ for $\\mathbb {Z}$ yields the following.",
"Theorem 8.12 Multiplication by $\\mathsf {Sol}(\\mathsf {C})_{\\mathbb {Z}\\times T_b}$ gives an isomorphism $\\mathsf {k}^{T_b}\\rightarrow \\mathrm {ker}(\\mathsf {C})$ .",
"appendix"
],
[
"Kernel containment and factorization",
"In this section, we prove a useful equivalence between containments of kernels and factorizations in the semigroup of recurrence matrices.",
"Let $\\mathsf {k}^\\mathbb {Z}_b\\subset \\mathsf {k}^\\mathbb {Z}$ denote the subspace of bounded sequences (i.e.",
"non-zero in finitely many terms).",
"If $\\mathsf {v}\\in \\mathsf {k}^\\mathbb {Z}_b$ and $\\mathsf {w}\\in \\mathsf {k}^\\mathbb {Z}$ , then the dot product $\\mathsf {v}\\cdot \\mathsf {w}$ is well-defined.",
"Lemma 5.2 Let $\\mathsf {C}$ be a recurrence matrix and let $\\mathsf {v}\\in \\mathsf {k}^\\mathbb {Z}_b$ .",
"If $\\mathsf {v}\\cdot \\mathsf {w}=0$ for all $\\mathsf {w}\\in \\mathrm {ker}(\\mathsf {C})$ , then $\\mathsf {v}$ is in the span of the rows of $\\mathsf {C}$ .",
"Let $V\\subset \\mathsf {k}^\\mathbb {Z}_b$ denote the span of the rows of $\\mathsf {C}$ , and assume for contradiction that $\\mathsf {v}\\notin V$ .",
"We may therefore choose a linear map $f:\\mathsf {k}^\\mathbb {Z}_b\\rightarrow \\mathsf {k}$ such that $f(V)=0$ and $f(\\mathsf {v})=1$ .The existence of such a map may depend on the Axiom of Choice, which we therefore assume.",
"Let $\\mathsf {e}_a\\in \\mathsf {k}^\\mathbb {Z}_b$ denote the standard basis vector which is 1 in the $a$ th term and 0 everywhere else, and set $\\mathsf {w}:= ( f(\\mathsf {e}_a)) _{a\\in \\mathbb {Z}}\\in \\mathsf {k}^\\mathbb {Z}$ .",
"By linearity, $f(\\mathsf {u})=\\mathsf {u}\\cdot \\mathsf {w}$ for all $\\mathsf {u}\\in \\mathsf {k}^\\mathbb {Z}_b$ .",
"Since $f$ kills each row of $\\mathsf {C}$ , $\\mathsf {C}\\mathsf {w}=0$ and so $\\mathsf {w}\\in \\mathrm {ker}(\\mathsf {C})$ .",
"However, $\\mathsf {v}\\cdot \\mathsf {w}=1$ , contradicting the hypothesis.",
"Lemma 5.3 Let $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ be recurrence matrices.",
"Then the following are equivalent.",
"$\\mathrm {ker}(\\mathsf {C})\\subseteq \\mathrm {ker}(\\mathsf {C}^{\\prime })$ .",
"$\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for some horizontally bounded matrix $\\mathsf {D}$ .",
"$\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for some recurrence matrix $\\mathsf {D}$ .",
"$\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C})$ is a recurrence matrix.",
"Furthermore, the matrix $\\mathsf {D}$ in (2) and (3) must equal $\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C})$ and is therefore unique.",
"($1\\Rightarrow 2$ ) If $\\mathrm {ker}(\\mathsf {C})\\subseteq \\mathrm {ker}(\\mathsf {C}^{\\prime })$ , then each row of $\\mathsf {C}^{\\prime }$ kills $\\mathrm {ker}(\\mathsf {C})$ .",
"By Lemma REF , the $a$ th row of $\\mathsf {C}^{\\prime }$ is equal to $\\mathsf {D}_a\\mathsf {C}$ for some bounded sequence $\\mathsf {D}_a\\in \\mathsf {k}^\\mathbb {Z}_b$ .",
"The vectors $\\mathsf {D}_a$ may be combined into the rows of a matrix $\\mathsf {D}$ which is horizontally bounded and satisfies $\\mathsf {D}\\mathsf {C}=\\mathsf {C}^{\\prime }$ .",
"($2\\Rightarrow 3+4+$ Uniqueness) Assume that $\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for a horizontally bounded $\\mathsf {D}$ .",
"Then $\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C}) = (\\mathsf {D}\\mathsf {C})\\mathsf {Adj}(\\mathsf {C}) \\stackrel{*}{=} \\mathsf {D}(\\mathsf {C}\\mathsf {Adj}(\\mathsf {C})) = \\mathsf {D}$ Equality ($*$ ) holds because $\\mathsf {D}$ and $\\mathsf {C}$ are horizontally bounded.",
"Since $\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C})$ is lower unitriangular and $\\mathsf {D}$ is horizontally bounded, they are the same recurrence matrix.",
"($3\\Rightarrow 1$ ) Assume $\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for some recurrence matrix $\\mathsf {D}$ .",
"If $\\mathsf {v}\\in \\mathrm {ker}(\\mathsf {C})$ , then $ \\mathsf {C}^{\\prime }\\mathsf {v} = (\\mathsf {D}\\mathsf {C})\\mathsf {v} \\stackrel{*}{=} \\mathsf {D}(\\mathsf {C}\\mathsf {v}) = \\mathsf {D}\\mathsf {0} = \\mathsf {0} $ Equality ($*$ ) holds because $\\mathsf {D}$ and $\\mathsf {C}$ are horizontally bounded.",
"Therefore, $\\mathrm {ker}(\\mathsf {C})\\subseteq \\mathrm {ker}(\\mathsf {C}^{\\prime })$ .",
"($4\\Rightarrow 3$ ) Setting $\\mathsf {D}\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C})$ , we check that $ \\mathsf {D}\\mathsf {C}= (\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C}) ) \\mathsf {C}\\stackrel{*}{=} \\mathsf {C}^{\\prime }(\\mathsf {Adj}(\\mathsf {C})\\mathsf {C}) = \\mathsf {C}^{\\prime }$ Equality ($*$ ) holds because $\\mathsf {C}^{\\prime }$ , $\\mathsf {Adj}(\\mathsf {C})$ , and $\\mathsf {C}$ are lower unitriangular.",
"Theorem 5.4 A recurrence matrix $\\mathsf {C}$ is trivial if and only if $\\mathsf {Adj}(\\mathsf {C})$ is a recurrence matrix.",
"The recurrence matrix $\\mathsf {C}$ is trivial when $\\mathrm {ker}(\\mathsf {C})\\subseteq \\mathrm {ker}(\\mathsf {Id})$ .",
"Applying Lemma REF with $\\mathsf {C}^{\\prime }=\\mathsf {Id}$ , this holds if and only if $\\mathsf {Adj}(\\mathsf {C})$ is a recurrence matrix.",
"Theorem 5.5 Two recurrence matrices $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ are equivalent if and only if $\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for a trivial recurrence matrix $\\mathsf {D}$ .",
"Two recurrence matrices $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ are equivalent if and only if $\\mathrm {ker}(\\mathsf {C})=\\mathrm {ker}(\\mathsf {C}^{\\prime })$ .",
"By Lemma REF , this holds if and only if $\\mathsf {D}=\\mathsf {C}^{\\prime }\\mathsf {Adj}(\\mathsf {C})$ and $\\mathsf {Adj}(\\mathsf {D})=\\mathsf {C}\\mathsf {Adj}(\\mathsf {C}^{\\prime })$ are recurrence matrices.",
"By Theorem REF , this is equivalent to $\\mathsf {D}$ being a trivial linear recurrence.",
"Lemma REF also allows us to make a connection between kernel containment and shapes.",
"Lemma 5.6 Let $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ be recurrence matrices with shape $S$ and $S^{\\prime }$ , respectively.",
"If $\\mathrm {ker}(\\mathsf {C})\\subseteq \\mathrm {ker}(\\mathsf {C}^{\\prime })$ and $S$ is injective (e.g.",
"if $\\mathsf {C}$ is reduced), then $S(a)\\ge S^{\\prime }(a)$ for all $a$ .",
"By Lemma REF , $\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for some recurrence matrix $\\mathsf {D}$ .",
"Fix $a\\in \\mathbb {Z}$ , and consider $B:=\\lbrace b \\in \\mathbb {Z}\\mid \\mathsf {D}_{a,b} \\ne 0\\rbrace $ .",
"This set is bounded and contains $a$ .",
"Let $b_{0}$ be the element of $B$ on which $S$ is minimal; this is unique because $S$ is injective.",
"$ (\\mathsf {D}\\mathsf {C})_{a,S(b_{0})} = \\sum _{b\\in \\mathbb {Z}} \\mathsf {D}_{a,b} \\mathsf {C}_{b,S(b_{0})} = \\mathsf {D}_{a,b_{0}}\\mathsf {C}_{b_{0},S(b_{0})} \\ne 0 $ Therefore, $S^{\\prime }(a)\\le S(b_0)$ .In fact, this is equality, but we won't need this stronger statement.",
"Since $a\\in B$ , $S(b_0)\\le S(a)$ , and so $S^{\\prime }(a)\\le S(a)$ .",
"Proposition 5.7 Reduced recurrence matrices that are equivalent must be equal.",
"Let $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ be reduced and equivalent.",
"Lemma REF implies that $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ have the same shape; call it $S$ .",
"By Lemma REF , $\\mathsf {C}^{\\prime }=\\mathsf {D}\\mathsf {C}$ for some recurrence matrix $\\mathsf {D}$ .",
"Let $T$ denote the shape of $\\mathsf {D}$ , so that $\\mathsf {D}_{a,b}=0$ whenever $b<T(a)$ .",
"Since $\\mathsf {C}$ is reduced of shape $S$ , $\\mathsf {C}_{b,S(T(a))}=0$ whenever $b>T(a)$ .",
"Therefore, $ \\mathsf {C}^{\\prime }_{a,S(T(a))} = \\sum _b \\mathsf {D}_{a,b} \\mathsf {C}_{b,S(T(a))} = \\mathsf {D}_{a,T(a)}\\mathsf {C}_{T(a),S(T(a))} \\ne 0 $ Since $\\mathsf {C}^{\\prime }$ is also reduced of shape $S$ , this is only possible if $a= T(a)$ .",
"Since this holds for all $a$ , the only non-zero entries of $\\mathsf {D}$ are on the main diagonal.",
"Thus, $\\mathsf {D}=\\mathsf {Id}$ and $\\mathsf {C}^{\\prime }=\\mathsf {C}$ ."
],
[
"Gauss-Zordan Elimination",
"Because we are working with $\\mathbb {Z}\\times \\mathbb {Z}$ -matrices, we must consider infinite sequences of row reductions that may be chosen in an arbitrary order.",
"We furthermore consider generalized row reductions: limits of such row reductions (in an appropriate topology)."
],
[
"Row reduction",
"Given a recurrence matrix $\\mathsf {C}$ of shape $S$ , a row reduction of $\\mathsf {C}$ is a matrix $\\mathsf {C}^{\\prime }$ obtained by adding $\\mathsf {C}_{a,S(b)} / \\mathsf {C}_{b,S(b)}$ times the $b$ th row to the $a$ th row, for some $b> a$ with $\\mathsf {C}_{a,S(b)}\\ne 0$ .",
"By design, the resulting matrix $\\mathsf {C}^{\\prime }$ has a zero in the $(a,S(b))$ entry.",
"Proposition 6.1 A recurrence matrix is reduced if and only if it has no row reductions.",
"A row reduction of $\\mathsf {C}$ can be reformulated as a factorization $\\mathsf {C}=\\mathsf {D}\\mathsf {C}^{\\prime }$ such that $\\mathsf {D}$ differs from the identity matrix in a single entry $\\mathsf {D}_{a,b}$ , and such that $\\mathsf {C}_{a,S(b)}\\ne 0$ and $\\mathsf {C}^{\\prime }_{a,S(b)}=0$ .",
"This perspective leads to the following generalization.",
"A generalized row reduction of $\\mathsf {C}$ is a recurrence matrix $\\mathsf {C}^{\\prime }$ such that $\\mathsf {C}=\\mathsf {D}\\mathsf {C}^{\\prime }$ for a trivial recurrence matrix $\\mathsf {D}$ with the property that, for each $a$ such that $\\lbrace b <a \\mid \\mathsf {D}_{a,b}\\ne 0\\rbrace $ is non-empty, we have $ \\mathsf {C}_{a,b_a} \\ne 0\\text{ and } \\mathsf {C}^{\\prime }_{a,b_a} = 0 $ where $b_a:= \\min \\lbrace S(b) \\mid b< a \\text{ s.t.",
"}\\mathsf {D}_{a,b}\\ne 0\\rbrace $ .",
"We write $\\mathsf {C}\\succeq \\mathsf {C}^{\\prime }$ to denote that $\\mathsf {C}^{\\prime }$ is a generalized row reduction of $\\mathsf {C}$ .",
"The index $b_a$ may be defined as the leftmost entry of the $a$ th row that multiplication by $\\mathsf {D}$ is `big enough' to change, and so $(\\mathsf {D}\\mathsf {C})_{a,b}=\\mathsf {C}_{a,b}$ whenever $b<b_a$ .",
"Thus, if $\\mathsf {C}\\succeq \\mathsf {C}^{\\prime }$ , then $\\mathsf {C}^{\\prime }$ must vanish in the leftmost entry in which the $a$ th rows of $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ differ.",
"Proposition 6.3 The relation $\\succeq $ defines a partial order on the set of recurrence matrices.",
"As a consequence, an iterated sequence of row reductions is a generalized row reduction.",
"(Antisymmetry) Assume $\\mathsf {C}\\succeq \\mathsf {C}^{\\prime }$ and $\\mathsf {C}\\preceq \\mathsf {C}^{\\prime }$ .",
"By Remark REF , both $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ vanish in the leftmost entry in which the $a$ th rows of $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ differ.",
"However, two entries cannot both vanish and be different, so the $a$ th rows of $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ coincide for all $a$ .",
"Thus, $\\mathsf {C}=\\mathsf {C}^{\\prime }$ .",
"(Transitivity) Let $\\mathsf {C}\\preceq \\mathsf {D}\\mathsf {C}\\preceq \\mathsf {D}^{\\prime }\\mathsf {D}\\mathsf {C}$ , and let $S$ and $S^{\\prime }$ denote the shapes of $\\mathsf {C}$ and $\\mathsf {D}\\mathsf {C}$ , respectively.",
"Fix some $a$ .",
"If $\\lbrace b< a\\mid \\mathsf {D}_{a,b}\\ne 0\\rbrace = \\varnothing $ or $\\lbrace b< a\\mid \\mathsf {D}^{\\prime }_{a,b}\\ne 0\\rbrace = \\varnothing $ , the generalized row reduction condition is easy to check.",
"Assume neither set is empty and let $b_0 &:= \\min \\lbrace S(b) \\mid b<a\\text{ s.t.",
"}\\mathsf {D}_{a,b}\\ne 0\\rbrace \\\\b_0^{\\prime } &:= \\min \\lbrace S^{\\prime }(b) \\mid b<a\\text{ s.t.",
"}\\mathsf {D}^{\\prime }_{a,b}\\ne 0\\rbrace $ By the definition of generalized row reductions, $\\mathsf {C}_{a,b_0}=0,\\;\\;\\; (\\mathsf {D}\\mathsf {C})_{a,b_0}\\ne 0,\\;\\;\\; (\\mathsf {D}\\mathsf {C})_{a,b_0^{\\prime }}=0 ,\\;\\;\\; (\\mathsf {D}^{\\prime }\\mathsf {D}\\mathsf {C})_{a,b_0^{\\prime }}\\ne 0$ This ensures that $\\mathsf {C}_{a,\\min (b_0,b_0^{\\prime })}=0$ and $(\\mathsf {D}^{\\prime }\\mathsf {D}\\mathsf {C})_{a,\\min (b_0,b_0^{\\prime })}\\ne 0$ .",
"Since these entries differ, $ \\min \\lbrace S(b) \\mid b<a\\text{ s.t.",
"}(\\mathsf {D}^{\\prime }\\mathsf {D})_{a,b}\\ne 0\\rbrace \\le \\min (b_0,b_0^{\\prime }) $ To show this is equality, consider some $b<a$ such that $(\\mathsf {D}^{\\prime }\\mathsf {D})_{a,b}\\ne 0$ .",
"We split into cases.",
"Assume $\\mathsf {D}^{\\prime }_{a,c}\\mathsf {D}_{c,b}\\ne 0$ for some $c<a$ .",
"Since $\\mathsf {D}_{c,b}\\ne 0$ and $\\mathsf {C}\\preceq \\mathsf {D}\\mathsf {C}$ , $S^{\\prime }(c)\\le S(b)$ .",
"Since $\\mathsf {D}^{\\prime }_{a,c}\\ne 0$ and $\\mathsf {D}\\mathsf {C}\\preceq \\mathsf {D}^{\\prime }\\mathsf {D}\\mathsf {C}$ , $S^{\\prime }(c)\\ge b_0^{\\prime }$ .",
"Therefore, $S(b)\\ge b^{\\prime }_0$ .",
"Otherwise, $\\mathsf {D}^{\\prime }_{a,c}\\mathsf {D}_{c,b}=0$ for all $c<a$ , and so $(\\mathsf {D}^{\\prime }\\mathsf {D})_{a,b}= \\mathsf {D}^{\\prime }_{a,a}\\mathsf {D}_{a,b}=\\mathsf {D}_{a,b}$ .",
"Since $\\mathsf {D}_{a,b}\\ne 0$ and $\\mathsf {C}\\preceq \\mathsf {D}\\mathsf {C}$ , we know that $S(b)\\ge b_0$ .",
"Therefore, $\\mathsf {C}_{a,\\min (b_0,b_0^{\\prime })}=0$ and $(\\mathsf {D}^{\\prime }\\mathsf {D}\\mathsf {C})_{a,\\min (b_0,b_0^{\\prime })}\\ne 0$ and $ \\min \\lbrace S(b) \\mid b<a\\text{ s.t.",
"}(\\mathsf {D}^{\\prime }\\mathsf {D})_{a,b}\\ne 0\\rbrace = \\min (b_0,b_0^{\\prime }) $ Since this holds for all $a$ , $\\mathsf {C}\\preceq \\mathsf {D}^{\\prime }\\mathsf {D}\\mathsf {C}$ ."
],
[
"Limits",
"To define limits of generalized row reductions, we endow the set of recurrence matrices with the topology of row-wise stabilization: a sequence of recurrence matrices converges if each row stabilizes after finitely many steps.",
"We next show that sequences of generalized row reductions must stabilize row-wise to another generalized row reduction, via the following more general result.",
"Lemma 6.4 Let $\\mathcal {C}$ be a set of recurrence matrices in which every pair is comparable in the row reduction partial order.Sometimes called a `chain' in the literature on partially ordered sets.",
"Then the closure of $\\mathcal {C}$ in the space of recurrence matrices contains a lower bound of $\\mathcal {C}$ .",
"Equivalently, there is a descending sequence of recurrence matrices in $\\mathcal {C}$ (i.e.",
"generalized row reductions of the initial matrix in the sequence) which converges (i.e.",
"stabilizes row-wise) to a lower bound of $\\mathcal {C}$ (i.e.",
"a generalized row reduction of every matrix in $\\mathcal {C}$ ).",
"Given a recurrence matrix $\\mathsf {C}$ and an integer $a$ , define $ n_a(\\mathsf {C}) := \\sum _{b \\text{ s.t. }",
"\\mathsf {C}_{(a,b)}\\ne 0} (a-b)^2 $ If $\\mathsf {C}\\preceq \\mathsf {C}^{\\prime }$ , then $n_a(\\mathsf {C})\\le n_a(\\mathsf {C}^{\\prime })$ and equality implies the $a$ th rows of $\\mathsf {C}$ and $\\mathsf {C}^{\\prime }$ coincide.",
"For each $a$ , let $\\mathcal {C}_a:= \\lbrace \\mathsf {C}\\in \\mathcal {C} \\mid \\forall \\mathsf {C}^{\\prime }\\in \\mathcal {C},\\;n_a (\\mathsf {C}) \\le n_a(\\mathsf {C}^{\\prime })\\rbrace $ ; that is, $\\mathcal {C}_a$ is the set of matrices in $\\mathcal {C}$ which attain the minimum value of $n_a$ .",
"This set is non-empty and the $a$ th row of each matrix in $\\mathcal {C}_a$ is the same, since $n_a$ has the same value and the matrices are comparable.",
"Consider $a,a^{\\prime }\\in \\mathbb {Z}$ and assume, for contradiction, that there exist $\\mathsf {C}\\in \\mathcal {C}_a\\setminus \\mathcal {C}_{a^{\\prime }}$ and $\\mathsf {C}^{\\prime } \\in \\mathcal {C}_{a^{\\prime }}\\setminus \\mathcal {C}_a$ .",
"If $\\mathsf {C}^{\\prime }\\preceq \\mathsf {C}$ , then $n_a(\\mathsf {C}^{\\prime })\\le n_a(\\mathsf {C})$ .",
"By the minimality of $n_a(\\mathsf {C})$ , this is an equality and so $\\mathsf {C}^{\\prime }\\in \\mathcal {C}_a$ ; a contradiction.",
"By a symmetric argument, $\\mathsf {C}\\preceq \\mathsf {C}^{\\prime }$ forces a contradiction.",
"Therefore, $\\mathcal {C}_a\\cap \\mathcal {C}_{a^{\\prime }}$ is either equal to $\\mathcal {C}_a$ or equal to $\\mathcal {C}_{a^{\\prime }}$ .",
"Applying this repeatedly, for any $i\\in \\mathbb {N}$ , there is some $a_i\\in [-i,i]$ such that $ \\bigcap _{a\\in [-i,i]} \\mathcal {C}_a =\\mathcal {C}_{a_i} \\ne \\varnothing $ Choose a matrix $\\mathsf {C}^i$ in $\\mathcal {C}_{a_i}$ for each $i$ .",
"The $a$ th rows in the sequence $\\mathsf {C}^1,\\mathsf {C}^2,\\mathsf {C}^3,...$ , stabilize after the $a$ th term, and so this sequence converges to the recurrence matrix $\\mathsf {C}$ whose $a$ th row coincides with the $a$ th row in each matrix in $\\mathcal {C}_a$ .",
"Let $S$ be the shape of $\\mathsf {C}^1$ .",
"Define a sequence $\\mathsf {D}^1,\\mathsf {D}^2,\\mathsf {D}^3,...$ of trivial recurrence matrices by $\\mathsf {C}^1 = \\mathsf {D}^n\\mathsf {C}^n$ for all $n$ .",
"Since $\\mathsf {C}^1\\succeq \\mathsf {C}^n$ , if $\\mathsf {D}^n_{a,b}\\ne 0$ , then $S(a)\\le S(b)\\le b$ ; that is, the $a$ th row $\\mathsf {D}$ can be non-zero only on the interval $[S(a),a]$ .",
"When $n>|S(a)|$ , the $a$ th row of the product $\\mathsf {D}^n\\mathsf {C}^n$ only depends on rows in $\\mathsf {C}^n$ that coincide with rows in $\\overline{\\mathsf {C}}$ .",
"Therefore, the $a$ th row of $\\mathsf {D}^n\\overline{\\mathsf {C}}$ is equal to $\\mathsf {C}^1$ .",
"Therefore, the sequence $\\mathsf {D}^1,\\mathsf {D}^2,\\mathsf {D}^3,...$ stabilizes row-wise to a matrix $\\overline{\\mathsf {D}}$ such that $\\overline{\\mathsf {D}}\\overline{\\mathsf {C}}=\\mathsf {C}^1$ .",
"As $\\overline{\\mathsf {D}}_{a,b}=\\mathsf {D}^n_{a,b}$ for large enough $n$ , this shows that $\\mathsf {C}\\preceq \\mathsf {C}^1$ .",
"Since the sequence $\\mathsf {C}^\\bullet $ could have started at any matrix in $\\mathcal {C}$ , this shows $\\overline{\\mathsf {C}}$ is a lower bound for $\\mathcal {C}$ .",
"Theorem 6.5 Every recurrence matrix is equivalent to a unique reduced recurrence matrix.",
"Let $\\mathcal {C}$ be an equivalence class of recurrence matrices, with the row reduction partial order.",
"Every non-empty chain in $\\mathcal {C}$ has a lower bound (by Lemma REF ).",
"By Zorn's Lemma, $\\mathcal {C}$ contains a minimal element $\\overline{\\mathsf {C}}$ .",
"If $\\overline{C}$ was not reduced, then there would be a row operation which would strictly decrease it in the reduction partial order; contradicting minimality.",
"Therefore, $\\overline{\\mathsf {C}}$ is reduced.",
"By Proposition REF , this reduced recurrence matrix is unique.",
"This provides a transfinite, non-deterministic analog of Gauss-Jordan elimination, which we humorously dub Gauss-Zordan elimination (both for `Zorn' and the integers $\\mathbb {Z}$ ).",
"Given a recurrence matrix $\\mathsf {C}$ , an arbitrary sequence of row reductions will stabilize row-wise to a matrix equivalent to $\\mathsf {C}$ .",
"While this limit may not be reduced, further arbitrary row reductions generate another convergent sequence.",
"Zorn's Lemma guarantees that some transfinite iteration of this process will eventually converge to the reduced representative of $\\mathsf {C}$ ."
],
[
"Constructing recurrences from spaces of solutions",
"In this section, we consider the inverse problem to the motivating problem of this note: Given a subspace $V\\subset \\mathsf {k}^\\mathbb {Z}$ , how can we construct a linear recurrence $\\mathsf {C}\\mathsf {x}=\\mathsf {0}$ whose solutions are $V$ ?",
"We give a characterization of when this is possible in Theorem REF .",
"For any $I\\subset \\mathbb {Z}$ , let $\\pi _{I}:\\mathsf {k}^\\mathbb {Z}\\rightarrow \\mathsf {k}^{I}$ restrict a sequence to the indices in $I$ , and let $\\iota _I:\\mathsf {k}^I\\rightarrow \\mathsf {k}^\\mathbb {Z}$ extend a sequence by 0.",
"Given a subspace $V\\subset \\mathsf {k}^\\mathbb {Z}$ , let $V_{I}\\pi _{I}(V)\\subset \\mathsf {k}^{I}$ ."
],
[
"Rank matrices",
"The rank matrix of a subspace $V\\subset \\mathsf {k}^\\mathbb {Z}$ is the $\\mathbb {Z}\\times \\mathbb {Z}$ -matrix withThe entries below the diagonal are unimportant; we set them to $b-a+1$ to avoid special cases later.",
"$ \\mathsf {R}_{a,b}:= \\left\\lbrace \\begin{array}{cc}\\dim _\\mathsf {k}(V_{[a,b]}) & \\text{if }a\\le b \\\\b-a+1 & \\text{otherwise}\\end{array} \\right\\rbrace $ Let $V$ be the space of sequences such that (a) the $-1$ st term is 0, (b) the $-2$ nd and 0th term are equal, and (c) the 0th, 1st, and 2nd terms sum to 0.",
"The rank matrix is $\\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][matrix of math nodes,matrix anchor = M-1-8.center,origin/.style={},throw/.style={},defect/.style={dark red,draw,circle,inner sep=0.25mm,minimum size=2mm},pivot/.style={draw,circle,inner sep=0.25mm,minimum size=2mm}, nodes in empty cells,inner sep=0pt,nodes={anchor=center,rotate=45},column sep={.5cm,between origins},row sep={.5cm,between origins},] (M) at (0,0) {\\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\\\\\& 5 \\& \\& 5 \\& \\& 4 \\& \\& 4 \\& \\& 4 \\& \\& 5 \\& \\& 6 \\& \\& \\cdots \\\\\\cdots \\& \\& 4 \\& \\& 4 \\& \\& 3 \\& \\& 3 \\& \\& 4 \\& \\& 5 \\& \\& 6 \\& \\\\\\& 4 \\& \\& 3 \\& \\& 3 \\& \\& 2 \\& \\& 3 \\& \\& 4 \\& \\& 5 \\& \\& \\cdots \\\\\\cdots \\& \\& 3 \\& \\& 2 \\& \\& 2 \\& \\& 2 \\& \\& 3 \\& \\& 4 \\& \\& 4 \\& \\\\\\& 3 \\& \\& 2 \\& \\& |[defect]| 1 \\& \\& 2 \\& \\& |[defect]| 2 \\& \\& 3 \\& \\& 3 \\& \\& \\cdots \\\\\\cdots \\& \\& 2 \\& \\& 1 \\& \\& 1 \\& \\& 2 \\& \\& 2 \\& \\& 2 \\& \\& 2 \\& \\\\\\& |[throw]| 1 \\& \\& |[throw]| 1 \\& \\& |[defect]| 0 \\& \\& |[origin]| 1 \\& \\&|[throw]| 1 \\& \\& |[throw]| 1 \\& \\& |[throw]| 1 \\& \\& \\cdots \\\\};\\end{tikzpicture}$ The subdiagonal entries have been omitted.",
"The dark redred circles are the defects (see below).",
"Proposition 7.3 If $\\mathsf {R}$ is the rank matrix of $V$ , then the following hold for any $a,b\\in \\mathbb {Z}$ .",
"$\\mathsf {R}_{a,b}-\\mathsf {R}_{a+1,b}$ must be 0 or 1.",
"$\\mathsf {R}_{a,b}-\\mathsf {R}_{a,b-1}$ must be 0 or 1.",
"$\\mathsf {R}_{a,b}-\\mathsf {R}_{a+1,b}-\\mathsf {R}_{a,b-1}+\\mathsf {R}_{a+1,b-1}$ must be 0 or $-1$ .",
"The projection $V_{[a,b]}\\rightarrow V_{[a+1,b]}$ is surjective with at most 1-dimensional kernel.",
"This proves the first result; the second is proven similarly.",
"The first result implies that $-1\\le (\\mathsf {R}_{a,b}-\\mathsf {R}_{a+1,b})-(\\mathsf {R}_{a,b-1} - \\mathsf {R}_{a+1,b-1}) \\le 1$ .",
"The map $V_{[a+1,b]} \\oplus V_{[a,b-1]}\\rightarrow V_{[a,b]}$ which sends $(\\mathsf {v},\\mathsf {w})$ to $\\mathsf {v+w}$ is a surjection whose kernel is the image of the map $V_{[a+1,b-1]} \\rightarrow V_{[a+1,b]} \\oplus V_{[a,b-1]}$ which sends $\\mathsf {v}$ to $(\\mathsf {v},-\\mathsf {v})$ .",
"Therefore, $ \\dim (V_{[a,b]}) \\le \\dim (V_{[a+1,b]} \\oplus V_{[a,b-1]}) - \\dim (V_{[a+1,b-1]}) $ This proves that $\\mathsf {R}_{a,b}-\\mathsf {R}_{a+1,b}-\\mathsf {R}_{a,b-1} + \\mathsf {R}_{a+1,b-1} \\le 0$ .",
"Let us say the pair $(a,b)\\in \\mathbb {Z}\\times \\mathbb {Z}$ is a defect of a rank matrix $\\mathsf {R}$ if $ \\mathsf {R}_{a,b} - \\mathsf {R}_{a+1,b}-\\mathsf {R}_{a,b-1}+\\mathsf {R}_{a+1,b-1} =-1 $ Proposition 7.4 The defects of a rank matrix $\\mathsf {R}$ have the following properties.",
"$\\mathsf {R}_{a,b} = (b-a+1) - \\#(\\text{defects in the box }[a,b]\\times [a,b]) $ .",
"Each row and column of a rank matrix can contain at most one defect.",
"If $[a,b]\\times \\lbrace b\\rbrace $ does not contain any defects, then $V_{[a,b]}$ contains the vector $(0,0,...,0,1)$ .",
"If $\\lbrace a\\rbrace \\times [a,b]$ does not contain any defects, then $V_{[a,b]}$ contains the vector $(1,0,...,0,0)$ .",
"Fix $a$ and consider the sequence $(\\mathsf {R}_{a,b}-\\mathsf {R}_{a+1,b})$ for all $b$ .",
"This sequence starts at 1 for sufficiently negative $b$ , switches from 1 to 0 whenever $(a,b)$ is a defect, and must remain 0 once it does (by Proposition REF .3).",
"Since there are no defects when $a<b$ , this implies that $ \\mathsf {R}_{a,b}=\\mathsf {R}_{a+1,b}+1-\\#(\\text{defects in the line }\\lbrace a\\rbrace \\times [a,b])$ In particular, there can be at most one defect in each row, and inductively implies that $ \\mathsf {R}_{a,b} = (b-a+1) - \\#(\\text{defects in the box }[a,b]\\times [a,b]) $ If $\\lbrace a\\rbrace \\times [a,b]$ does not contain any defect, then $\\mathsf {R}_{a,b}=\\mathsf {R}_{a+1,b}+1$ and the map $V_{[a,b]}\\rightarrow V_{[a+1,b]}$ has 1-dimensional kernel.",
"This kernel must be spanned by the vector $(1,0,...,0,0)$ .",
"The remaining results follow by a dual argument on the sequence $(\\mathsf {R}_{a,b}-\\mathsf {R}_{a,b-1})$ .",
"Given a rank matrix $\\mathsf {R}$ and a consecutive subset $I\\subset \\mathbb {Z}$ , an $\\mathsf {R}$ -schedule for $I$ is a subset $J\\subset I$ for which there is a sequence of subintervals $[a_0,b_0] \\subset [a_1,b_1] \\subset [a_2,b_2] \\subset \\cdots \\subset I$ such that $b_i-a_i=i$ , $\\bigcup [a_i,b_i] = I$ , and $|J\\cap [a_i,b_i]| = \\mathsf {R}_{a_i,b_i}$ .",
"Note that the sequence of intervals determines the $\\mathsf {R}$ -schedule, and, for all $i$ , $J\\cap [a_i,b_i]$ is an $\\mathsf {R}$ -schedule for $[a_i,b_i]$ .",
"Lemma 7.5 Given a subspace $V\\subset \\mathsf {k}^\\mathbb {Z}$ with rank matrix $\\mathsf {R}$ , and an $\\mathsf {R}$ -schedule $J$ for a subset $I$ , the restriction map $V_{I}\\rightarrow \\mathsf {k}^J$ is an isomorphism.",
"In particular, if $J$ is an $\\mathsf {R}$ -schedule for $\\mathbb {Z}$ , then $V\\rightarrow \\mathsf {k}^J$ is an isomorphism.",
"We prove the case when $I=[a,b]$ by induction on $n:=b-a$ .",
"If $n<0$ , the lemma holds vacuously.",
"Assume that the lemma holds for all intervals shorter than $n$ .",
"Choose a sequence of subintervals as in (REF ), and set $[a^{\\prime },b^{\\prime }]:=[a_{n-1},b_{n-1}]$ .",
"The restriction maps fit into a commutative diagram.",
"$\\begin{tikzpicture}[baseline=(current bounding box.center)]\\node (V2) at (0,0) {V_{[a,b]}};\\node (V3) at (3,0) {V_{[a^{\\prime },b^{\\prime }]}};\\node (k2) at (0,-1.5) {\\mathsf {k}^{J}};\\node (k3) at (3,-1.5) {\\mathsf {k}^{J\\cap [a^{\\prime },b^{\\prime }]}};[->>] (V2) to (V3);[->] (V2) to node[left] {\\pi _{J}} (k2);[->] (V3) to node[right] {\\pi _{J\\cap [a^{\\prime },b^{\\prime }]}} (k3);[->>] (k2) to (k3);\\end{tikzpicture}$ By the inductive hypothesis, $\\pi _{J\\cap [a^{\\prime },b^{\\prime }]}$ is an isomorphism, and so $V_{[a,b]}\\rightarrow \\mathsf {k}^{T_{[a^{\\prime },b^{\\prime }]}}$ is surjective.",
"Since $b^{\\prime }-a^{\\prime }=n-1$ , either $[a^{\\prime },b^{\\prime }]=[a+1,b]$ or $[a^{\\prime },b^{\\prime }]=[a,b-1]$ .",
"We have three cases.",
"If $\\mathsf {R}_{a^{\\prime },b^{\\prime }}=\\mathsf {R}_{a,b}$ , then $J\\cap [a^{\\prime },b^{\\prime }]=J$ and so the bottom arrow is an isomorphism.",
"Therefore, $\\pi _J:V_{[a,b]}\\rightarrow \\mathsf {k}^J$ is surjective.",
"If $\\mathsf {R}_{a^{\\prime },b^{\\prime }}=\\mathsf {R}_{a,b}-1$ and $[a^{\\prime },b^{\\prime }]=[a+1,b]$ , then there are no defects in $\\lbrace a\\rbrace \\times [a,b]$ , and so $V_{[a,b]}$ contains $(1,0,...,0,0)$ (by Proposition REF .4).",
"Since $|J| =|J\\cap [a^{\\prime },b^{\\prime }]|+1$ , $a\\in J$ and so the image of $(1,0,...,0,0)$ under $\\pi _{J}$ is non-zero and spans the kernel of $\\mathsf {k}^{J}\\rightarrow \\mathsf {k}^{J\\cap [a^{\\prime },b^{\\prime }]}$ .",
"Thus, $\\pi _J:V_{[a,b]}\\rightarrow \\mathsf {k}^{J}$ is surjective.",
"If $\\mathsf {R}_{a^{\\prime },b^{\\prime }}=\\mathsf {R}_{a,b}-1$ and $[a^{\\prime },b^{\\prime }]=[a,b-1]$ , then there are no defects in $[a,b] \\times \\lbrace b\\rbrace $ , and so $V_{[a,b]}$ contains $(0,0,...,0,1)$ (by Proposition REF .3).",
"By an analogous argument to the previous case, $\\pi _J:V_{[a,b]}\\rightarrow \\mathsf {k}^{J}$ is surjective.",
"The map $\\pi _J:V_{[a,b]}\\rightarrow \\mathsf {k}^{J}$ is surjective in all cases.",
"Since $ \\dim (V_{[a,b]}) = \\mathsf {R}_{[a,b]}= J = \\dim (\\mathsf {k}^{J}) $ this map is an isomorphism, completing the induction.",
"For infinite $I$ , the lemma follows since $I=\\bigcup [a_i,b_i]$ and the lemma holds on each $[a_i,b_i]$ ."
],
[
"Recurrence matrices from rank matrices",
"We can now characterize when a subspace of $\\mathsf {k}^\\mathbb {Z}$ is the kernel of a reduced recurrence matrix.",
"Theorem 7.6 Given a subspace $V$ of $\\mathsf {k}^\\mathbb {Z}$ , the following are equivalent.",
"$V$ is the space of solutions to a linear recurrence $\\mathsf {C}\\mathsf {x}=\\mathsf {0}$ .",
"The only left-bounded sequence in $V$ is the zero sequence; that is, if $\\mathsf {v}\\in V$ and $\\mathsf {v}_i=0$ for all $i\\ll 0$ , then $\\mathsf {v}_i=0$ for all $i$ .",
"Every column of the rank matrix $\\mathsf {R}$ of $V$ contains a defect.",
"$V$ is the space of solutions to a reduced linear recurrence $\\overline{\\mathsf {C}}\\mathsf {x} =\\mathsf {0}$ .",
"The shape of $\\overline{\\mathsf {C}}$ is the function $S:\\mathbb {Z}\\rightarrow \\mathbb {Z}$ such that $(S(b),b)$ is a defect of $\\mathsf {R}$ .",
"$(4 \\Rightarrow 1)$ is automatic.",
"$(1 \\Rightarrow 2)$ If a sequence $\\mathsf {v}$ solves a linear recurrence, then every term in $\\mathsf {v}$ is equal to a linear combination of previous terms in the sequence.",
"If every term in $\\mathsf {v}$ of sufficiently negative index is 0, then recursively every term must be 0.",
"$(\\text{not } 3 \\Rightarrow \\text{not }2)$ Assume that the $b$ th column of the rank matrix of $V$ does not contain a defect.",
"By Proposition REF .3, $V_{[a,b]}$ contains the vector $(0,0,...,0,1)$ for all $a\\le b$ .",
"It follows that $V_{(-\\infty ,b]}$ contains the vector $(...,0,0,1)$ .",
"This implies that $V$ contains a sequence $\\mathsf {v}$ with $\\mathsf {v}_b=1$ and $\\mathsf {v}_a=0$ whenever $a<b$ .",
"$(3 \\Rightarrow 4)$ Assume that there is a function $S:\\mathbb {Z}\\rightarrow \\mathbb {Z}$ such that $(S(b),b)$ is a defect of $\\mathsf {R}$ for each $b$ .",
"For each interval $[a,b]$ , define the $\\mathsf {R}$ -schedule $T_{[a,b]} := [a,b] \\setminus S([a,b])$ .",
"We note that $T_{[S(b),b-1]} \\cup \\lbrace b\\rbrace = T_{[S(b)+1,b]}\\cup \\lbrace S(b) \\rbrace $ and consider the following commutative diagram.",
"$ \\begin{tikzpicture}\\node (V1) at (-3,0) {V_{[S(b),b-1]}};\\node (V2) at (0,0) {V_{[S(b),b]}};\\node (V3) at (3,0) {V_{[S(b)+1,b]}};\\node (k1) at (-3,-1.5) {\\mathsf {k}^{T_{[S(b),b-1]}}};\\node (k2) at (0,-1.5) {\\mathsf {k}^{T_{[S(b),b-1]}\\cup \\lbrace b\\rbrace }};\\node (k3) at (3,-1.5) {\\mathsf {k}^{T_{[S(b)+1,b]}}};[->] (V2) to (V1);[->] (V2) to (V3);[->] (V1) to (k1);[->] (V2) to (k2);[->] (V3) to (k3);[->] (k2) to (k1);[->] (k2) to (k3);\\end{tikzpicture}$ Since $(S(b),b)$ is a defect, the maps in the top row are isomorphisms.",
"Since $T_{[S(b),b-1]}$ and $T_{[S(b)+1,b]}$ are $\\mathsf {R}$ -schedules, the maps on the left and right are isomorphisms (by Lemma REF ).",
"Therefore, the map $V_{[S(b),b]} \\longrightarrow \\mathsf {k} ^{J }$ is an embedding of codimension 1.",
"Its image is defined by a relation (unique up to scaling) of the form $\\sum _{a\\in T_{[S(b),b-1]}\\cup \\lbrace b\\rbrace } \\mathsf {C}_{b,a}x_{a} = 0$ Because the left and right maps are isomorphisms, $\\mathsf {C}_{b,b}\\ne 0$ and $\\mathsf {C}_{S(b),b}\\ne 0$ .",
"Rescaling the relation as necessary, we assume that $\\mathsf {C}_{b,b}=1$ .",
"Construct a recurrence matrix $\\mathsf {C}$ such that, for each $b$ , the $b$ th row collects the coefficients of the corresponding equation (REF ).",
"For any pair $b<a$ , $S(b)\\notin T_{[S(a)+1,a]}$ and so $\\mathsf {C}_{a,S(b)}=0$ .",
"Therefore, $\\mathsf {C}$ is a reduced linear recurrence of shape $S$ , such that $V\\subseteq \\mathrm {ker}(\\mathsf {C})$ .",
"Consider any interval $[a,b]$ .",
"For each $b^{\\prime }\\in [a,b] \\setminus T_{[a,b]}$ , the corresponding relation (REF ) only involves terms with index in $[a,b]$ .",
"Since these relations are linearly independent, the codimension of $\\mathrm {ker}(\\mathsf {C})_{[a,b]}$ in $\\mathsf {k}^{[a,b]}$ is at least the cardinality of $[a,b]\\setminus T_{[a,b]}$ .",
"Therefore, $ \\dim (\\mathrm {ker}(\\mathsf {C})_{[a,b]})\\le |T_{[a,b]}| = \\mathsf {R}_{a,b} = \\dim (V_{[a,b]}) $ Since $V_{[a,b]}\\subseteq \\mathrm {ker}(\\mathsf {C})_{[a,b]}$ , $V_{[a,b]}=\\mathrm {ker}(\\mathsf {C})_{[a,b]}$ .",
"Since this holds for all intervals, $V=\\mathrm {ker}(\\mathsf {C})$ ."
],
[
"From rank matrices to shapes",
"The theorem relates the shape of a reduced recurrence matrix $\\mathsf {C}$ to the defects of the rank matrix $\\mathsf {R}$ of $\\mathrm {ker}(\\mathsf {C})$ , as follows.",
"Corollary 7.7 If $\\mathsf {C}$ is a reduced recurrence matrix, then $(a,b)$ is a pivot of $\\mathsf {C}$ if and only if $(b,a)$ is a defect of the rank matrix of $\\mathrm {ker}(\\mathsf {C})$ .",
"Therefore, we may translate several earlier results into the language of shapes.",
"Definition 7.8 Given a non-increasing injection $S$ , an $S$ -schedule for a subset $I\\subset \\mathbb {Z}$ is a subset $J\\subset I$ for which there is a subsequence of subintervals $[a_0,b_0] \\subset [a_1,b_1] \\subset [a_2,b_2] \\subset \\cdots \\subset I$ such that $b_i-a_i=i$ , $\\bigcup [a_i,b_i] = I$ , and $|J\\cap [a_i,b_i]|$ equals the number of $S$ -balls in $[a_i,b_i]$ .",
"If $S$ is the shape of a reduced recurrence matrix $\\mathsf {C}$ and $\\mathsf {R}$ is the rank matrix of $\\mathrm {ker}(\\mathsf {C})$ , then $\\mathsf {R}$ -schedules and $S$ -schedules coincide.",
"Note that $I$ only admits an $S$ -schedule if $I$ consists of consecutive elements.",
"The following is a direct translation of Lemma REF .",
"Proposition 7.9 Let $\\mathsf {C}$ be a reduced recurrence matrix with shape $S$ , and let $J$ be an $S$ -schedule for $I$ .",
"Then the restriction map $\\mathrm {ker}(\\mathsf {C})_{I}\\rightarrow \\mathsf {k}^J$ is an isomorphism.",
"In particular, if $J$ is an $S$ -schedule for $\\mathbb {Z}$ , then $\\mathrm {ker}(\\mathsf {C})\\rightarrow \\mathsf {k}^J$ is an isomorphism.",
"As a special case, for any $b\\in \\mathbb {Z}$ , the sequence of intervals $[b,b] \\subset [b-1,b] \\subset [b-2,b] \\subset \\cdots $ determines the following $S$ -schedule for $\\mathbb {Z}$ : $ T_b := \\bigcup _{a\\le b} T_{[a,b]} = \\lbrace a\\le b \\mid \\forall c\\le b, S(c) \\ne a \\rbrace = (-\\infty ,b] \\setminus S\\left( ( -\\infty ,b]\\right) $ Therefore, the proposition specializes to the following.",
"Theorem 7.10 The restriction map $\\pi _{T_b}:\\mathrm {ker}(\\mathsf {C})\\rightarrow \\mathsf {k}^{T_b}$ is an isomorphism.",
"Since $T_b$ contains a unique representative of each $S$ -ball, this implies the following.",
"Theorem 7.11 Then dimension of $\\mathrm {ker}(\\mathsf {C})$ equals the number of $S$ -balls.",
"Constructing $S$ -schedules is easy and intuitive using the juggling pattern of $S$ .",
"Consider any zigzagging path in the juggling pattern which starts on the main diagonal, only travels up (northwest) or right (northeast), and ends above the $(a,b)$ th entry.",
"$\\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][use as bounding box] (-9.5,3.3) rectangle (6.3,-0.3);[matrix of math nodes,matrix anchor = M-2-24.center,nodes in empty cells,inner sep=0pt,gthrow/.style={dark green,draw,circle,inner sep=0mm,minimum size=5mm},rthrow/.style={dark red,draw,circle,inner sep=0mm,minimum size=5mm},bthrow/.style={dark blue,draw,circle,inner sep=0mm,minimum size=5mm},pthrow/.style={dark purple,draw,circle,inner sep=0mm,minimum size=5mm},nodes={anchor=center,node font=\\scriptsize ,rotate=45},column sep={0.4cm,between origins},row sep={-0.4cm,between origins},] (M) at (0,0) {\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& |[gthrow]| \\& \\& |[rthrow]| \\& \\& |[bthrow]| \\& \\& |[bthrow]| \\& \\& |[pthrow]| \\& \\& |[gthrow]| \\& \\& |[rthrow,rotate=-45]| 1 \\& \\& |[bthrow,rotate=-45]| 2 \\&\\& |[gthrow,rotate=-45]| 3 \\& |[inner sep=8pt]| \\& |[pthrow,rotate=-45]| 4 \\& \\& |[bthrow,rotate=-45]| 5 \\& \\& |[bthrow,rotate=-45]| 6 \\& \\& |[rthrow,rotate=-45]| 7 \\& \\& |[gthrow,rotate=-45]| 8 \\& \\& |[pthrow]| \\& \\& |[bthrow]| \\&\\& |[gthrow]| \\& \\& |[rthrow]| \\& \\& |[bthrow]| \\& \\\\\\cdots \\& \\& \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\cdots \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[gthrow]| \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[gthrow]| \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[bthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& |[gthrow]| \\& \\& |[rthrow]| \\& \\& \\& \\& \\& \\& |[pthrow]| \\& \\& \\& \\& \\& \\& \\& \\& |[gthrow]| \\& \\& |[pthrow]| \\& \\& \\& \\& \\& \\& |[rthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\cdots \\\\\\& \\& \\& |[pthrow]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[rthrow]| \\& |[inner sep=8pt]| \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& |[pthrow]| \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\};\\end{tikzpicture}[dark green] (M-3-1) to (M-2-2) to (M-7-7) to (M-2-12) to (M-5-15) to (M-2-18) to (M-7-23) to (M-2-28) to (M-5-31) to (M-2-34) to (M-7-39);[dark red] (M-5-1) to (M-2-4) to (M-7-9) to (M-2-14) to (M-8-20) to (M-2-26) to (M-7-31) to (M-2-36) to (M-5-39);[dark blue] (M-3-1) to (M-5-3) to (M-2-6) to (M-3-7) to (M-2-8) to (M-6-12) to (M-2-16) to (M-5-19) to (M-2-22) to (M-3-23) to (M-2-24) to (M-6-28) to (M-2-32) to (M-5-35) to (M-2-38) to (M-3-39);[dark purple] (M-5-1) to (M-8-4) to (M-2-10) to (M-7-15) to (M-2-20) to (M-7-25) to (M-2-30) to (M-8-36) to (M-5-39);$ [dashed] (M-2-13.center) to (M-10-21.center) to (M-2-29.center); [thick] (M-2-19.center) to (M-3-18.center) to (M-4-19.center) to (M-5-18.center) to (M-9-22.center)to (M-10-21.center); $Each time the path crosses a colored line, record the row (if it is ascending) or the column (if it is descending).The resulting subset is an $ $-schedule for the interval $ [a,b]$, and every $ S$-schedule can be constructed this way.In the picture above, the path in black determines the $ S$-schedule $ {3,4,2,7}$ for the interval $ [1,8]$.",
"The set $ Tb$ comes from the path which starts to the right of $ (b,b)$ and only travels up (northeast).$ Properties of the solution matrix Fix a reduced recurrence matrix $\\mathsf {C}$ of shape $S$ for the rest of the section.",
"Vanishing The unitriangularity of $\\mathsf {Adj}(\\mathsf {C})$ and $\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})$ mean that the solution matrix $ \\mathsf {Sol}(\\mathsf {C}) := \\mathsf {Adj}(\\mathsf {C}) - \\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})$ has zeroes between the support of $\\mathsf {Adj}(\\mathsf {C})$ and $\\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})$ , which we make precise as follows.",
"Proposition 8.1 $\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ whenever $a<b$ and there is no $c\\le b$ with $S(c)=a$ .",
"If $S$ is bijective, this can be restated as $\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ whenever $a<b<S^{-1}(a)$ .",
"By unitriangularity, $\\mathsf {Adj}(\\mathsf {C})_{a,b}=0$ whenever $a<b$ .",
"Dually, $(\\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P}))_{a,b}$ is only non-zero if there is a $c$ with $\\mathsf {P}_{a,c}\\ne 0$ and $\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})_{c,b}\\ne 0$ ; that is, if $S(c)=a$ and $c\\ge b$ .",
"For fixed $b$ , the proposition determines the value of the $b$ th column of $\\mathsf {Sol}(\\mathsf {C})$ on the set $T_b$ .",
"Since this column solves $\\mathsf {C}\\mathsf {x}=0$ , these entries determine the column (Theorem REF ).",
"Corollary 8.2 The $b$ th column of $\\mathsf {Sol}(\\mathsf {C})$ is the unique solution to $\\mathsf {C}\\mathsf {x}=\\mathsf {0}$ for which $x_a=0$ whenever $a<b$ but there is no $c\\le b$ with $S(c)=a$ , and $x_b=1$ unless $S(b)=b$ , in which case $x_b=0$ .",
"We also have a vanishing condition which guarantees consecutive zeros in each column.",
"Proposition 8.3 $\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ whenever $S(b) < a < b$ .",
"We proceed by induction on $b-a>0$ .",
"Assume that $\\mathsf {Sol}(\\mathsf {C})_{a+1,b}=...=\\mathsf {Sol}(\\mathsf {C})_{b-1,b}=0$ (which is vacuous for the base case $b-a=1$ ).",
"We split into two cases.",
"If there is no $c\\le b$ with $S(c)=a$ , then $\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ by Proposition REF .",
"Otherwise, there is a $c\\le b$ with $S(c)=a$ .",
"Since $S(b)<a$ , we know $c\\ne b$ and so $c<b$ .",
"The $(c,b)$ th entry of the equality $\\mathsf {C}\\mathsf {Sol}(\\mathsf {C})=0$ is $ \\mathsf {C}_{c,a}\\mathsf {Sol}(\\mathsf {C})_{a,b} + \\mathsf {C}_{c,a+1}\\mathsf {Sol}(\\mathsf {C})_{a+1,b} + ... + \\mathsf {C}_{c,c}\\mathsf {Sol}(\\mathsf {C})_{c,b} =0 $ By assumption, $\\mathsf {Sol}(\\mathsf {C})_{a+1,b}=...=\\mathsf {Sol}(\\mathsf {C})_{c,b}=0$ , and so $\\mathsf {C}_{c,a}\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ .",
"Since $\\mathsf {C}_{c,a}=\\mathsf {C}_{c,S(c)}$ is invertible, $\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ .",
"This completes the induction.",
"Propositions REF and REF can be visualized in terms of juggling patterns.",
"As in Remark REF , the $S$ -balls can be visualized by connecting each pivot entry with a line to the diagonal entries in the same row or column.",
"The dashed circle is not in an $S$ -ball.",
"$ \\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][matrix of math nodes,matrix anchor = M-4-8.center,gthrow/.style={dark green,draw,circle,inner sep=0mm,minimum size=5mm},bthrow/.style={dark blue,draw,circle,inner sep=0mm,minimum size=5mm},dashedthrow/.style={dashed,draw,circle,inner sep=0mm,minimum size=5mm},nodes in empty cells,inner sep=0pt,nodes={anchor=center,node font=\\footnotesize ,rotate=45},column sep={.5cm,between origins},row sep={.5cm,between origins},] (M) at (0,0) {\\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\&|[dashedthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& \\cdots \\\\\\cdots \\& \\& -1 \\& \\& -1 \\& \\& -1 \\& \\& \\& \\& \\& \\& |[gthrow]| -1 \\& \\& |[gthrow]| -1 \\& \\\\\\& |[gthrow]| -1 \\& \\& |[bthrow]| -1 \\& \\& |[gthrow]| -1 \\& \\& \\& \\& |[gthrow]| -1 \\& \\& \\& \\& \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\};\\end{tikzpicture}[dark blue,->] (M-2-1) to (M-1-2) to (M-3-4) to (M-1-6) to (M-4-9);[dark green] (M-2-1) to (M-3-2) to (M-1-4) to (M-3-6) to (M-1-8) to (M-3-10) to (M-1-12) to (M-2-13) to (M-1-14) to (M-2-15) to (M-1-16);$ $If we transpose and superimpose the juggling pattern onto the solution matrix...$ [baseline=(current bounding box.center), ampersand replacement=&, ] [matrix of math nodes, matrix anchor = M-8-8.center, gthrow/.style=dark green,draw,circle,inner sep=0mm,minimum size=5mm, bthrow/.style=dark blue,draw,circle,inner sep=0mm,minimum size=5mm, dashedthrow/.style=dashed,draw,circle,inner sep=0mm,minimum size=5mm, faded/.style=black!25, nodes in empty cells, inner sep=0pt, nodes=anchor=center,node font=,rotate=45, column sep=.5cm,between origins, row sep=.5cm,between origins, ] (M) at (0,0) & & & & & & & & & & & & & & & & 5 & & |[faded]| 0 & & 2 & & -1 & & 1 & & |[faded]| 0 & & 1 & & & & -3 & & |[faded]| 0 & & -1 & & 1 & & |[faded]| 0 & & 1 & & |[faded]| 0 & & 2 & & 2 & & |[faded]| 0 & & 1 & & |[faded]| 0 & & 1 & & |[faded]| 0 & & & & -1 & & -1 & & |[faded]| 0 & & |[faded]| 0 & & 1 & & |[faded]| 0 & & 1 & & |[gthrow]| 1 & & |[bthrow]| 1 & & |[gthrow]| 1 & & |[faded]| 0 & & |[gthrow]| 1 & & |[faded]| 0 & & 1 & & & & |[faded]| 0 & & |[faded]| 0 & & |[faded]| 0 & & |[faded]| 0 & & |[faded]| 0 & & |[gthrow]| 1 & & |[gthrow]| 1 & & |[bthrow]| 1 & & |[gthrow]| 1 & & |[bthrow]| 1 & & |[gthrow]| 1 & & |[dashedthrow,faded]| 0 & &|[gthrow]| 1 & & |[gthrow]| 1 & & & & 1 & & 1 & & 1 & & |[faded]| 0 & & |[faded]| 0 & & 1 & & 1 & & 2 & & 2 & & 2 & & |[faded]| 0 & & 1 & & |[faded]| 0 & & 1 & & & & 3 & & 3 & & |[faded]| 0 & & 1 & & 1 & & |[faded]| 0 & & 1 & & 5 & & 5 & & |[faded]| 0 & & 2 & & 1 & & 1 & & |[faded]| 0 & & & & 8 & & |[faded]| 0 & & 3 & & 2 & & 1 & & 1 & & |[faded]| 0 & & 13 & & |[faded]| 0 & & 5 & & 3 & & 2 & & 1 & & 1 & & & & & & & & & & & & & & & & & ; [dark blue,->] (M-7-1) to (M-8-2) to (M-6-4) to (M-8-6) to (M-1-13); [dark green] (M-7-1) to (M-6-2) to (M-8-4) to (M-6-6) to (M-8-8) to (M-6-10) to (M-8-12) to (M-7-13) to (M-8-14) to (M-7-15) to (M-8-16); $Propositions \\ref {prop: colvan} and \\ref {prop: rowvan} state that the lines do not cross any non-zero entries.",
"The circles must be non-zero, except the dashed circle, whose entire row and column must vanish.\\footnote {We note that all of these observations will be combined into and generalized by Lemma \\ref {lemma: boxballs}.",
"}$ Rank conditions The prior vanishing results can be generalized to a formula for ranks of certain rectangular submatrices of $\\mathsf {Sol}(\\mathsf {C})$ .",
"Given integers $a\\le b$ and $c\\le d$ , the box $[a,b]\\times [c,d]\\subset \\mathbb {Z}\\times \\mathbb {Z}$ indexes a rectangular submatrix of $\\mathsf {Sol}(\\mathsf {C})$ .",
"Define an $S$ -ball in the box $[a,b]\\times [c,d]$ to be an equivalence class in the set $ \\left(\\lbrace (i,i) \\mid S(i) \\ne i \\rbrace \\cup \\lbrace (S(j),j) \\mid S(j)\\ne j\\rbrace \\right) \\subset [a,b] \\times [c,d] $ under the equivalence relation generated by $(i,i) \\sim (S(i),i)\\sim (S(i),S(i))$ .",
"This is a 2-dimensional analog of $S$ -balls as defined in Section REF .",
"The relation is that $S$ -balls in an interval $[a,b]$ are in bijection with $S$ -balls in the box $[a,b]\\times [a,b]$ , and also with the $S$ -balls in $[a,b]\\times [c,d]$ and in $[c,d]\\times [a,b]$ for any $[c,d]\\supset [a,b]$ .",
"In terms of the juggling pattern, this counts the number of different colors of circles inside the rectangular submatrix indexed by $[a,b]\\times [c,d]$ .",
"$ \\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][matrix of math nodes,matrix anchor = M-8-8.center,gthrow/.style={dark green,draw,circle,inner sep=0mm,minimum size=5mm},bthrow/.style={dark blue,draw,circle,inner sep=0mm,minimum size=5mm},dashedthrow/.style={dashed,draw,circle,inner sep=0mm,minimum size=5mm},faded/.style={black!25},nodes in empty cells,inner sep=0pt,nodes={anchor=center,node font=\\footnotesize ,rotate=45},column sep={.5cm,between origins},row sep={.5cm,between origins},] (M) at (0,0) {\\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\\\\\& 5 \\& \\& |[faded]| 0 \\& \\& 2 \\& \\& -1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& \\cdots \\\\\\cdots \\& \\& -3 \\& \\& |[faded]| 0 \\& \\& -1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\\\\\& 2 \\& \\& 2 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& \\cdots \\\\\\cdots \\& \\& -1 \\& \\& -1 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\\\\\& |[gthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[faded]| 0 \\& \\& |[gthrow]| 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& \\cdots \\\\\\cdots \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& |[gthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\\\\\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[dashedthrow,faded]| 0 \\& \\&|[gthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& \\cdots \\\\\\cdots \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& 1 \\& \\\\\\& 2 \\& \\& 2 \\& \\& 2 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& \\cdots \\\\\\cdots \\& \\& 3 \\& \\& 3 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\\\\\& 5 \\& \\& 5 \\& \\& |[faded]| 0 \\& \\& 2 \\& \\& 1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& \\cdots \\\\\\cdots \\& \\& 8 \\& \\& |[faded]| 0 \\& \\& 3 \\& \\& 2 \\& \\& 1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\\\\\& 13 \\& \\& |[faded]| 0 \\& \\& 5 \\& \\& 3 \\& \\& 2 \\& \\& 1 \\& \\& 1 \\& \\& \\cdots \\\\\\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\\\};\\end{tikzpicture}[dark blue,->] (M-7-1) to (M-8-2) to (M-6-4) to (M-8-6) to (M-1-13);[dark green] (M-7-1) to (M-6-2) to (M-8-4) to (M-6-6) to (M-8-8) to (M-6-10) to (M-8-12) to (M-7-13) to (M-8-14) to (M-7-15) to (M-8-16);$ [dark red, fill=dark red!50,opacity=.25,rounded corners] (M-3-4.center) – (M-2-5.center) – (M-10-13.center) – (M-11-12.center) – cycle; [dark blue, fill=dark blue!50,opacity=.25,rounded corners] (M-6-9.center) – (M-3-12.center) – (M-7-16.center) – (M-10-13.center) – cycle; [dark purple, fill=dark purple!50,opacity=.25,rounded corners] (M-7-2.center) – (M-4-5.center) – (M-7-8.center) – (M-10-5.center) – cycle; $The {dark purple}{purple}, {dark red}{red}, and {dark blue}{blue} boxes above contain $ dark purple2$, $ dark red0$, and $ dark blue1$ $ S$-balls, respectively.$ The following lemma relates the rank of a submatrix to the number of $S$ -balls in the box.",
"Lemma 8.7 Let $\\mathsf {C}$ be a reduced recurrence matrix of shape $S$ .",
"For any integers $a\\le b$ and $c\\le d$ , the rank of $\\mathsf {Sol}(\\mathsf {C})_{[a,b]\\times [c,d]}$ is at least the number of $S$ -balls in the box $[a,b]\\times [c,d]$ , with equality when $b-c\\ge -1 \\text{ and }\\min (a-c,b-d)\\le 0$ Conceptually, Condition (REF ) implies the box has at least one corner on or below the first superdiagonal, and at least two corners on or above the main diagonal.",
"Condition (REF ) holds for the three boxes in Example REF , and one may check that their ranks coincide with the number of $S$ -balls they contain.",
"First, we bound the rank below by finding a full-rank submatrix of the appropriate size.",
"Fix $[a,b]\\times [c,d]$ .",
"Index $I:=[c,d]\\cap [a,b]\\setminus S([c,d])$ by $i_1<i_2<\\cdots < i_k$ , and index $ J:= \\lbrace j\\in [c,d] \\mid j\\notin [a,b] \\text{ and }S(j) \\in [a,b] \\rbrace $ by $j_1<j_2<\\cdots < j_\\ell $ .",
"Note that $i_k<j_1$ and $S(j_h)<i_1$ for all $h$ .",
"Each $S$ -ball in the box $[a,b]\\times [c,d]$ contains a unique final circle, by which we mean one of the two types of pair: $(i,i)\\in [a,b]\\times [c,d]$ such that there is no $j\\in [c,d]$ with $S(j)=i$ .",
"$(S(i),i) \\in [a,b]\\times [c,d]$ such that $(i,i)\\notin [a,b]\\times [c,d]$ .",
"Furthermore, each column in the box can contain at most one such circle.",
"The columns containing the first kind of final circle are indexed by $I$ , and the columns containing the second kind are indexed by $J$ .",
"Therefore, the number of $S$ -balls in the box is $|I|+|J|$ .",
"Consider the submatrixWe are stretching the definition of `submatrix' to allow for rearranging the order of the rows and columns.",
"of $\\mathsf {Sol}(\\mathsf {C})$ with row set $ \\lbrace i_k,i_{k-1},...,i_1,S(j_1),S(j_2),...,S(J_\\ell )\\rbrace $ and column set $ \\lbrace i_k,i_{k-1},...,i_1,j_1,j_2,...,j_\\ell \\rbrace $ .",
"This matrix is upper triangular with non-zero entries on the diagonal, and so it has rank $|I|+|J|$ .",
"Consequently, $\\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]})\\ge \\text{the number of $S$-balls in the box $[a,b]\\times [c,d]$}$ Note that this inequality did not assume Conditions (REF ).",
"Next, assume that $b-c\\ge -1$ and that $b-d\\le 0$ , and define disjoint sets $K&:=\\lbrace k \\in [a,b] \\text{ such that there is an $m\\in [a,b]$ with $S(m)=k$} \\rbrace \\\\L &:= \\lbrace \\ell \\in [a,c-1] \\text{ such that there is no $m\\le d$ with $S(m)=\\ell $} \\rbrace $ Construct a $(K \\cup L)\\times [a,b]$ -matrix $\\mathsf {M}$ , such that for each $k\\in K$ , the corresponding row of $\\mathsf {M}$ is $\\mathsf {C}_{S^{-1}(k),[a,b]}$ and for each $\\ell \\in L$ , the corresponding row of $\\mathsf {M}$ is $e_\\ell $ (the row vector with a 1 in the $\\ell $ th place).",
"Since $\\mathsf {C}\\mathsf {Sol}(\\mathsf {C})=0$ (Proposition REF ), $\\mathsf {M}_{k,[a,b]}\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]} =0$ for all $k\\in K$ .",
"Since each row of $\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]}$ indexed by $\\ell \\in L$ vanishes (Proposition REF )Explicitly: Since $\\ell \\in [a,b]$ but not in $[c,d]$ , and $b-d\\le 0$ , $\\ell <c$ .",
"Since $\\ell \\notin S( (-\\infty ,d])$ , either $S^{-1}(\\ell )>d$ or $S^{-1}(\\ell )$ is empty.",
"In either case, $\\mathsf {Sol}(\\mathsf {C})_{\\ell ,[c,d]}$ consists of zeroes by Proposition REF ., $\\mathsf {M}_{\\ell , [a,b]}\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]} =0$ for all $\\ell \\in L$ .",
"Therefore, the matrix product vanishes: $ \\mathsf {M}\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]} =0$ Since the first non-zero entry in the $k$ th row of $\\mathsf {M}$ is in the $k$ th column and the first non-zero entry in the $\\ell $ th row of $\\mathsf {M}$ is in the $\\ell $ th column, $\\mathsf {M}$ is in row-echelon form and has rank $|K \\cup L|$ .",
"Therefore, $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]})\\le \\dim (\\mathrm {ker}(\\mathsf {M})) = |[a,b]\\setminus (K\\cup L) |$ Next, we note that $K\\cup L$ indexes the rows of the box $[a,b]\\times [c,d]$ which do not contain the final circle of an $S$ -ball in the box.",
"Combined with the lower bound (REF ), $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]})= \\text{the number of $S$-balls in the box $[a,b]\\times [c,d]$}$ This establishes the theorem in the case when $b-d\\le 0$ .",
"For the final case, we need a source of relations among the columns of $\\mathsf {Sol}(\\mathsf {C})$ .",
"Proposition 8.9 $\\mathsf {Sol}(\\mathsf {C})(\\mathsf {C}\\mathsf {P})=0$ .",
"We compute directly.",
"$\\mathsf {Sol}(\\mathsf {C})(\\mathsf {C}\\mathsf {P})&= \\mathsf {Adj}(\\mathsf {C})(\\mathsf {C}\\mathsf {P}) - (\\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P}) (\\mathsf {C}\\mathsf {P}) \\\\&\\stackrel{*}{=} (\\mathsf {Adj}(\\mathsf {C})\\mathsf {C})\\mathsf {P}- \\mathsf {P}(\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})(\\mathsf {C}\\mathsf {P}))= \\mathsf {P}- \\mathsf {P}= \\mathsf {0}$ Equality ($*$ ) holds because $\\mathsf {P}$ is a generalized permutation matrix.",
"Finally, assume that $b-c\\ge -1$ and that $a-c\\le 0$ , and define disjoint sets $K&:=\\lbrace k \\in [c,d] \\text{ such that $S(k)\\in [c,d]$} \\rbrace \\\\L &:= \\lbrace \\ell \\in [b+1,d] \\text{ such that $S(\\ell )<a$} \\rbrace $ Construct a $[c,d]\\times (K \\cup L)$ -matrix $\\mathsf {M}$ , such that for each $k\\in K$ , the corresponding column of $\\mathsf {M}$ is $\\mathsf {C}_{[a,b],k}$ and for each $\\ell \\in L$ , the corresponding column of $\\mathsf {M}$ is $e_\\ell $ (the column vector with a 1 in the $\\ell $ th place).",
"Since $\\mathsf {Sol}(\\mathsf {C})\\mathsf {C}\\mathsf {P}=0$ (Proposition REF ), $\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]} \\mathsf {M}_{[c,d],k}=0$ for all $k\\in K$ .",
"Since each column of $\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]}$ indexed by $\\ell \\in L$ vanishes (Proposition REF ), $\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]}\\mathsf {M}_{[c,d],\\ell } =0$ for all $\\ell \\in L$ .",
"Therefore, the matrix product vanishes: $ \\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]}\\mathsf {M}=0$ The transpose $\\mathsf {M}^\\top $ is in row echelon form and has rank $|K \\cup L|$ .",
"Therefore, $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]})\\le \\dim (\\mathrm {ker}(\\mathsf {M}^\\top )) = |[c,d]\\setminus (K\\cup L) |$ Next, we note that $K\\cup L$ indexes the columns of the box $[a,b]\\times [c,d]$ which do not contain the `initial circle' of an $S$ -ball in the box.",
"Combined with the lower bound (REF ), $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]})= \\text{the number of $S$-balls in the box $[a,b]\\times [c,d]$} $ Lemma REF extends to improper intervals such as $(-\\infty , \\infty ) $ and $ (-\\infty , d]$ by considering appropriate limits, as the number of $S$ -balls in a box is monotonic in nested intervals.",
"For example, the rank of the entire matrix $\\mathsf {Sol}(\\mathsf {C})$ equals the number of $S$ -balls in the unbounded `box' $(-\\infty ,\\infty )\\times ( -\\infty ,\\infty )$ ; that is, the number of $S$ -balls.",
"The lemma allows us to prove the following fundamental result.",
"Theorem 8.10 If $\\mathsf {C}$ is reduced, the kernel of $\\mathsf {C}$ equals the image of $\\mathsf {Sol}(\\mathsf {C})$ .",
"By Proposition REF , $\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C})) \\subseteq \\mathrm {ker}(\\mathsf {C})$ .",
"For any interval $[a,b]$ and any $S$ -schedule $J$ of $[a,b]$ , Proposition REF implies that $ \\dim (\\mathrm {ker}(\\mathsf {C})_{[a,b]})=|J|$ , and Lemma REF implies that $\\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b], \\mathbb {Z}}) = \\text{\\# of $S$-balls in the box $[a,b]\\times \\mathbb {Z}$} $ The number of $S$ -balls in $[a,b]\\times \\mathbb {Z}$ equals the number of $S$ -balls in $[a,b]$ (Remark REF ), which in turn equals $|J|$ .",
"Thus, $\\mathrm {ker}(\\mathsf {C})_{[a,b]}=\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C}))_{[a,b]}$ for all intervals $[a,b]$ , and so $\\mathrm {ker}(\\mathsf {C})=\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C}))$ .",
"Bases for solutions We can minimally parametrize the kernel using $S$ -schedules.See Definition REF ; note that $I$ only admits an $S$ -schedule if $I$ is a set of consecutive elements.",
"Proposition 8.11 Let $\\mathsf {C}$ be a reduced recurrence matrix with shape $S$ , and let $J$ be an $S$ -schedule for $I$ .",
"Then multiplication by $\\mathsf {Sol}(\\mathsf {C})_{I,J}$ gives an isomorphism $\\mathsf {k}^J\\rightarrow \\mathrm {ker}(\\mathsf {C})_I$ .",
"If $J$ is an $S$ -schedule for $\\mathbb {Z}$ , then multiplication by $\\mathsf {Sol}(\\mathsf {C})_{\\mathbb {Z},J}$ is an isomorphism $\\mathsf {k}^J\\rightarrow \\mathrm {ker}(\\mathsf {C})$ .",
"We first consider the the case when $I$ is finite.",
"Consider any interval $[c,d]$ such that $I\\subset [c,d]$ .",
"By Remark REF , the number of $S$ -balls in $[c,d]\\times I$ equals the number of $S$ -balls in $I$ , which is equal to $J$ by Definition REF .",
"By Lemma REF , this implies that $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],I}) = |J| $ .",
"Since $J\\subset I$ , this implies that $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],J}) \\le |J| $ .",
"We will prove that $\\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],J}) = |J|$ by induction on the number of elements in $I$ .",
"If $I=\\varnothing $ , the claim holds vacuously.",
"Assume that the claim holds for all $S$ -schedules of intervals shorter than $I$ , and choose a sequence of subintervals as in Definition REF .",
"Since $I$ is finite, the sequence terminates at $[a_n,b_n]=I$ .",
"Since $J^{\\prime } := J\\cap [a_{n-1},b_{n-1}]$ is an $S$ -schedule for $[a_{n-1},b_{n-1}]$ , by the inductive hypothesis, $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],J^{\\prime }}) = |J^{\\prime }| $ .",
"If $J^{\\prime }=J$ , this immediately implies (REF ).",
"If $J^{\\prime }\\ne J$ , then $|J| = |J^{\\prime }|+1$ , and so $\\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],[a_{n},b_{n}]} ) = \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],[a_{n-1},b_{n-1}]} ) +1 $ Hence, the column indexed by the unique element of $[a_n,b_n]\\setminus [a_{n-1},b_{n-1}]$ is linearly independent from the other columns.",
"Since this is also the unique element in $J\\setminus J^{\\prime }$ , $\\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],J} ) = \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],J^{\\prime }} ) +1 $ This proves (REF ) and completes the induction.",
"Since $\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C}))= \\mathrm {ker}(\\mathsf {C})$ , we know that $\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C})_{I,J}) \\subseteq \\mathrm {ker}(\\mathsf {C})_I$ .",
"Equation REF for $I=[c,d]$ and Proposition REF imply these both have dimension $|J|$ , so $\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C})_{I,J}) = \\mathrm {ker}(\\mathsf {C})_I$ .",
"When $I$ is infinite, Definition REF and the preceding argument guarantee it is a union of finite intervals on which the proposition holds, so $\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C})_{I,J}) = \\mathrm {ker}(\\mathsf {C})_I$ .",
"The special case of the $S$ -schedule $T_b$ for $\\mathbb {Z}$ yields the following.",
"Theorem 8.12 Multiplication by $\\mathsf {Sol}(\\mathsf {C})_{\\mathbb {Z}\\times T_b}$ gives an isomorphism $\\mathsf {k}^{T_b}\\rightarrow \\mathrm {ker}(\\mathsf {C})$ .",
"appendix"
],
[
"Properties of the solution matrix",
"Fix a reduced recurrence matrix $\\mathsf {C}$ of shape $S$ for the rest of the section."
],
[
"Vanishing",
"The unitriangularity of $\\mathsf {Adj}(\\mathsf {C})$ and $\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})$ mean that the solution matrix $ \\mathsf {Sol}(\\mathsf {C}) := \\mathsf {Adj}(\\mathsf {C}) - \\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})$ has zeroes between the support of $\\mathsf {Adj}(\\mathsf {C})$ and $\\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})$ , which we make precise as follows.",
"Proposition 8.1 $\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ whenever $a<b$ and there is no $c\\le b$ with $S(c)=a$ .",
"If $S$ is bijective, this can be restated as $\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ whenever $a<b<S^{-1}(a)$ .",
"By unitriangularity, $\\mathsf {Adj}(\\mathsf {C})_{a,b}=0$ whenever $a<b$ .",
"Dually, $(\\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P}))_{a,b}$ is only non-zero if there is a $c$ with $\\mathsf {P}_{a,c}\\ne 0$ and $\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})_{c,b}\\ne 0$ ; that is, if $S(c)=a$ and $c\\ge b$ .",
"For fixed $b$ , the proposition determines the value of the $b$ th column of $\\mathsf {Sol}(\\mathsf {C})$ on the set $T_b$ .",
"Since this column solves $\\mathsf {C}\\mathsf {x}=0$ , these entries determine the column (Theorem REF ).",
"Corollary 8.2 The $b$ th column of $\\mathsf {Sol}(\\mathsf {C})$ is the unique solution to $\\mathsf {C}\\mathsf {x}=\\mathsf {0}$ for which $x_a=0$ whenever $a<b$ but there is no $c\\le b$ with $S(c)=a$ , and $x_b=1$ unless $S(b)=b$ , in which case $x_b=0$ .",
"We also have a vanishing condition which guarantees consecutive zeros in each column.",
"Proposition 8.3 $\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ whenever $S(b) < a < b$ .",
"We proceed by induction on $b-a>0$ .",
"Assume that $\\mathsf {Sol}(\\mathsf {C})_{a+1,b}=...=\\mathsf {Sol}(\\mathsf {C})_{b-1,b}=0$ (which is vacuous for the base case $b-a=1$ ).",
"We split into two cases.",
"If there is no $c\\le b$ with $S(c)=a$ , then $\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ by Proposition REF .",
"Otherwise, there is a $c\\le b$ with $S(c)=a$ .",
"Since $S(b)<a$ , we know $c\\ne b$ and so $c<b$ .",
"The $(c,b)$ th entry of the equality $\\mathsf {C}\\mathsf {Sol}(\\mathsf {C})=0$ is $ \\mathsf {C}_{c,a}\\mathsf {Sol}(\\mathsf {C})_{a,b} + \\mathsf {C}_{c,a+1}\\mathsf {Sol}(\\mathsf {C})_{a+1,b} + ... + \\mathsf {C}_{c,c}\\mathsf {Sol}(\\mathsf {C})_{c,b} =0 $ By assumption, $\\mathsf {Sol}(\\mathsf {C})_{a+1,b}=...=\\mathsf {Sol}(\\mathsf {C})_{c,b}=0$ , and so $\\mathsf {C}_{c,a}\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ .",
"Since $\\mathsf {C}_{c,a}=\\mathsf {C}_{c,S(c)}$ is invertible, $\\mathsf {Sol}(\\mathsf {C})_{a,b}=0$ .",
"This completes the induction.",
"Propositions REF and REF can be visualized in terms of juggling patterns.",
"As in Remark REF , the $S$ -balls can be visualized by connecting each pivot entry with a line to the diagonal entries in the same row or column.",
"The dashed circle is not in an $S$ -ball.",
"$ \\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][matrix of math nodes,matrix anchor = M-4-8.center,gthrow/.style={dark green,draw,circle,inner sep=0mm,minimum size=5mm},bthrow/.style={dark blue,draw,circle,inner sep=0mm,minimum size=5mm},dashedthrow/.style={dashed,draw,circle,inner sep=0mm,minimum size=5mm},nodes in empty cells,inner sep=0pt,nodes={anchor=center,node font=\\footnotesize ,rotate=45},column sep={.5cm,between origins},row sep={.5cm,between origins},] (M) at (0,0) {\\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\&|[dashedthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& \\cdots \\\\\\cdots \\& \\& -1 \\& \\& -1 \\& \\& -1 \\& \\& \\& \\& \\& \\& |[gthrow]| -1 \\& \\& |[gthrow]| -1 \\& \\\\\\& |[gthrow]| -1 \\& \\& |[bthrow]| -1 \\& \\& |[gthrow]| -1 \\& \\& \\& \\& |[gthrow]| -1 \\& \\& \\& \\& \\& \\& \\cdots \\\\\\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\& \\\\};\\end{tikzpicture}[dark blue,->] (M-2-1) to (M-1-2) to (M-3-4) to (M-1-6) to (M-4-9);[dark green] (M-2-1) to (M-3-2) to (M-1-4) to (M-3-6) to (M-1-8) to (M-3-10) to (M-1-12) to (M-2-13) to (M-1-14) to (M-2-15) to (M-1-16);$ $If we transpose and superimpose the juggling pattern onto the solution matrix...$ [baseline=(current bounding box.center), ampersand replacement=&, ] [matrix of math nodes, matrix anchor = M-8-8.center, gthrow/.style=dark green,draw,circle,inner sep=0mm,minimum size=5mm, bthrow/.style=dark blue,draw,circle,inner sep=0mm,minimum size=5mm, dashedthrow/.style=dashed,draw,circle,inner sep=0mm,minimum size=5mm, faded/.style=black!25, nodes in empty cells, inner sep=0pt, nodes=anchor=center,node font=,rotate=45, column sep=.5cm,between origins, row sep=.5cm,between origins, ] (M) at (0,0) & & & & & & & & & & & & & & & & 5 & & |[faded]| 0 & & 2 & & -1 & & 1 & & |[faded]| 0 & & 1 & & & & -3 & & |[faded]| 0 & & -1 & & 1 & & |[faded]| 0 & & 1 & & |[faded]| 0 & & 2 & & 2 & & |[faded]| 0 & & 1 & & |[faded]| 0 & & 1 & & |[faded]| 0 & & & & -1 & & -1 & & |[faded]| 0 & & |[faded]| 0 & & 1 & & |[faded]| 0 & & 1 & & |[gthrow]| 1 & & |[bthrow]| 1 & & |[gthrow]| 1 & & |[faded]| 0 & & |[gthrow]| 1 & & |[faded]| 0 & & 1 & & & & |[faded]| 0 & & |[faded]| 0 & & |[faded]| 0 & & |[faded]| 0 & & |[faded]| 0 & & |[gthrow]| 1 & & |[gthrow]| 1 & & |[bthrow]| 1 & & |[gthrow]| 1 & & |[bthrow]| 1 & & |[gthrow]| 1 & & |[dashedthrow,faded]| 0 & &|[gthrow]| 1 & & |[gthrow]| 1 & & & & 1 & & 1 & & 1 & & |[faded]| 0 & & |[faded]| 0 & & 1 & & 1 & & 2 & & 2 & & 2 & & |[faded]| 0 & & 1 & & |[faded]| 0 & & 1 & & & & 3 & & 3 & & |[faded]| 0 & & 1 & & 1 & & |[faded]| 0 & & 1 & & 5 & & 5 & & |[faded]| 0 & & 2 & & 1 & & 1 & & |[faded]| 0 & & & & 8 & & |[faded]| 0 & & 3 & & 2 & & 1 & & 1 & & |[faded]| 0 & & 13 & & |[faded]| 0 & & 5 & & 3 & & 2 & & 1 & & 1 & & & & & & & & & & & & & & & & & ; [dark blue,->] (M-7-1) to (M-8-2) to (M-6-4) to (M-8-6) to (M-1-13); [dark green] (M-7-1) to (M-6-2) to (M-8-4) to (M-6-6) to (M-8-8) to (M-6-10) to (M-8-12) to (M-7-13) to (M-8-14) to (M-7-15) to (M-8-16); $Propositions \\ref {prop: colvan} and \\ref {prop: rowvan} state that the lines do not cross any non-zero entries.",
"The circles must be non-zero, except the dashed circle, whose entire row and column must vanish.\\footnote {We note that all of these observations will be combined into and generalized by Lemma \\ref {lemma: boxballs}.",
"}$ Rank conditions The prior vanishing results can be generalized to a formula for ranks of certain rectangular submatrices of $\\mathsf {Sol}(\\mathsf {C})$ .",
"Given integers $a\\le b$ and $c\\le d$ , the box $[a,b]\\times [c,d]\\subset \\mathbb {Z}\\times \\mathbb {Z}$ indexes a rectangular submatrix of $\\mathsf {Sol}(\\mathsf {C})$ .",
"Define an $S$ -ball in the box $[a,b]\\times [c,d]$ to be an equivalence class in the set $ \\left(\\lbrace (i,i) \\mid S(i) \\ne i \\rbrace \\cup \\lbrace (S(j),j) \\mid S(j)\\ne j\\rbrace \\right) \\subset [a,b] \\times [c,d] $ under the equivalence relation generated by $(i,i) \\sim (S(i),i)\\sim (S(i),S(i))$ .",
"This is a 2-dimensional analog of $S$ -balls as defined in Section REF .",
"The relation is that $S$ -balls in an interval $[a,b]$ are in bijection with $S$ -balls in the box $[a,b]\\times [a,b]$ , and also with the $S$ -balls in $[a,b]\\times [c,d]$ and in $[c,d]\\times [a,b]$ for any $[c,d]\\supset [a,b]$ .",
"In terms of the juggling pattern, this counts the number of different colors of circles inside the rectangular submatrix indexed by $[a,b]\\times [c,d]$ .",
"$ \\begin{tikzpicture}[baseline=(current bounding box.center),ampersand replacement=\\&,][matrix of math nodes,matrix anchor = M-8-8.center,gthrow/.style={dark green,draw,circle,inner sep=0mm,minimum size=5mm},bthrow/.style={dark blue,draw,circle,inner sep=0mm,minimum size=5mm},dashedthrow/.style={dashed,draw,circle,inner sep=0mm,minimum size=5mm},faded/.style={black!25},nodes in empty cells,inner sep=0pt,nodes={anchor=center,node font=\\footnotesize ,rotate=45},column sep={.5cm,between origins},row sep={.5cm,between origins},] (M) at (0,0) {\\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\\\\\& 5 \\& \\& |[faded]| 0 \\& \\& 2 \\& \\& -1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& \\cdots \\\\\\cdots \\& \\& -3 \\& \\& |[faded]| 0 \\& \\& -1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\\\\\& 2 \\& \\& 2 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& \\cdots \\\\\\cdots \\& \\& -1 \\& \\& -1 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\\\\\& |[gthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[faded]| 0 \\& \\& |[gthrow]| 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& \\cdots \\\\\\cdots \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& |[gthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\\\\\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[bthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& |[dashedthrow,faded]| 0 \\& \\&|[gthrow]| 1 \\& \\& |[gthrow]| 1 \\& \\& \\cdots \\\\\\cdots \\& \\& 1 \\& \\& 1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& 1 \\& \\\\\\& 2 \\& \\& 2 \\& \\& 2 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& \\cdots \\\\\\cdots \\& \\& 3 \\& \\& 3 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& 1 \\& \\\\\\& 5 \\& \\& 5 \\& \\& |[faded]| 0 \\& \\& 2 \\& \\& 1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\& \\cdots \\\\\\cdots \\& \\& 8 \\& \\& |[faded]| 0 \\& \\& 3 \\& \\& 2 \\& \\& 1 \\& \\& 1 \\& \\& |[faded]| 0 \\& \\\\\\& 13 \\& \\& |[faded]| 0 \\& \\& 5 \\& \\& 3 \\& \\& 2 \\& \\& 1 \\& \\& 1 \\& \\& \\cdots \\\\\\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\& \\cdots \\& \\\\};\\end{tikzpicture}[dark blue,->] (M-7-1) to (M-8-2) to (M-6-4) to (M-8-6) to (M-1-13);[dark green] (M-7-1) to (M-6-2) to (M-8-4) to (M-6-6) to (M-8-8) to (M-6-10) to (M-8-12) to (M-7-13) to (M-8-14) to (M-7-15) to (M-8-16);$ [dark red, fill=dark red!50,opacity=.25,rounded corners] (M-3-4.center) – (M-2-5.center) – (M-10-13.center) – (M-11-12.center) – cycle; [dark blue, fill=dark blue!50,opacity=.25,rounded corners] (M-6-9.center) – (M-3-12.center) – (M-7-16.center) – (M-10-13.center) – cycle; [dark purple, fill=dark purple!50,opacity=.25,rounded corners] (M-7-2.center) – (M-4-5.center) – (M-7-8.center) – (M-10-5.center) – cycle; $The {dark purple}{purple}, {dark red}{red}, and {dark blue}{blue} boxes above contain $ dark purple2$, $ dark red0$, and $ dark blue1$ $ S$-balls, respectively.$ The following lemma relates the rank of a submatrix to the number of $S$ -balls in the box.",
"Lemma 8.7 Let $\\mathsf {C}$ be a reduced recurrence matrix of shape $S$ .",
"For any integers $a\\le b$ and $c\\le d$ , the rank of $\\mathsf {Sol}(\\mathsf {C})_{[a,b]\\times [c,d]}$ is at least the number of $S$ -balls in the box $[a,b]\\times [c,d]$ , with equality when $b-c\\ge -1 \\text{ and }\\min (a-c,b-d)\\le 0$ Conceptually, Condition (REF ) implies the box has at least one corner on or below the first superdiagonal, and at least two corners on or above the main diagonal.",
"Condition (REF ) holds for the three boxes in Example REF , and one may check that their ranks coincide with the number of $S$ -balls they contain.",
"First, we bound the rank below by finding a full-rank submatrix of the appropriate size.",
"Fix $[a,b]\\times [c,d]$ .",
"Index $I:=[c,d]\\cap [a,b]\\setminus S([c,d])$ by $i_1<i_2<\\cdots < i_k$ , and index $ J:= \\lbrace j\\in [c,d] \\mid j\\notin [a,b] \\text{ and }S(j) \\in [a,b] \\rbrace $ by $j_1<j_2<\\cdots < j_\\ell $ .",
"Note that $i_k<j_1$ and $S(j_h)<i_1$ for all $h$ .",
"Each $S$ -ball in the box $[a,b]\\times [c,d]$ contains a unique final circle, by which we mean one of the two types of pair: $(i,i)\\in [a,b]\\times [c,d]$ such that there is no $j\\in [c,d]$ with $S(j)=i$ .",
"$(S(i),i) \\in [a,b]\\times [c,d]$ such that $(i,i)\\notin [a,b]\\times [c,d]$ .",
"Furthermore, each column in the box can contain at most one such circle.",
"The columns containing the first kind of final circle are indexed by $I$ , and the columns containing the second kind are indexed by $J$ .",
"Therefore, the number of $S$ -balls in the box is $|I|+|J|$ .",
"Consider the submatrixWe are stretching the definition of `submatrix' to allow for rearranging the order of the rows and columns.",
"of $\\mathsf {Sol}(\\mathsf {C})$ with row set $ \\lbrace i_k,i_{k-1},...,i_1,S(j_1),S(j_2),...,S(J_\\ell )\\rbrace $ and column set $ \\lbrace i_k,i_{k-1},...,i_1,j_1,j_2,...,j_\\ell \\rbrace $ .",
"This matrix is upper triangular with non-zero entries on the diagonal, and so it has rank $|I|+|J|$ .",
"Consequently, $\\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]})\\ge \\text{the number of $S$-balls in the box $[a,b]\\times [c,d]$}$ Note that this inequality did not assume Conditions (REF ).",
"Next, assume that $b-c\\ge -1$ and that $b-d\\le 0$ , and define disjoint sets $K&:=\\lbrace k \\in [a,b] \\text{ such that there is an $m\\in [a,b]$ with $S(m)=k$} \\rbrace \\\\L &:= \\lbrace \\ell \\in [a,c-1] \\text{ such that there is no $m\\le d$ with $S(m)=\\ell $} \\rbrace $ Construct a $(K \\cup L)\\times [a,b]$ -matrix $\\mathsf {M}$ , such that for each $k\\in K$ , the corresponding row of $\\mathsf {M}$ is $\\mathsf {C}_{S^{-1}(k),[a,b]}$ and for each $\\ell \\in L$ , the corresponding row of $\\mathsf {M}$ is $e_\\ell $ (the row vector with a 1 in the $\\ell $ th place).",
"Since $\\mathsf {C}\\mathsf {Sol}(\\mathsf {C})=0$ (Proposition REF ), $\\mathsf {M}_{k,[a,b]}\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]} =0$ for all $k\\in K$ .",
"Since each row of $\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]}$ indexed by $\\ell \\in L$ vanishes (Proposition REF )Explicitly: Since $\\ell \\in [a,b]$ but not in $[c,d]$ , and $b-d\\le 0$ , $\\ell <c$ .",
"Since $\\ell \\notin S( (-\\infty ,d])$ , either $S^{-1}(\\ell )>d$ or $S^{-1}(\\ell )$ is empty.",
"In either case, $\\mathsf {Sol}(\\mathsf {C})_{\\ell ,[c,d]}$ consists of zeroes by Proposition REF ., $\\mathsf {M}_{\\ell , [a,b]}\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]} =0$ for all $\\ell \\in L$ .",
"Therefore, the matrix product vanishes: $ \\mathsf {M}\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]} =0$ Since the first non-zero entry in the $k$ th row of $\\mathsf {M}$ is in the $k$ th column and the first non-zero entry in the $\\ell $ th row of $\\mathsf {M}$ is in the $\\ell $ th column, $\\mathsf {M}$ is in row-echelon form and has rank $|K \\cup L|$ .",
"Therefore, $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]})\\le \\dim (\\mathrm {ker}(\\mathsf {M})) = |[a,b]\\setminus (K\\cup L) |$ Next, we note that $K\\cup L$ indexes the rows of the box $[a,b]\\times [c,d]$ which do not contain the final circle of an $S$ -ball in the box.",
"Combined with the lower bound (REF ), $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]})= \\text{the number of $S$-balls in the box $[a,b]\\times [c,d]$}$ This establishes the theorem in the case when $b-d\\le 0$ .",
"For the final case, we need a source of relations among the columns of $\\mathsf {Sol}(\\mathsf {C})$ .",
"Proposition 8.9 $\\mathsf {Sol}(\\mathsf {C})(\\mathsf {C}\\mathsf {P})=0$ .",
"We compute directly.",
"$\\mathsf {Sol}(\\mathsf {C})(\\mathsf {C}\\mathsf {P})&= \\mathsf {Adj}(\\mathsf {C})(\\mathsf {C}\\mathsf {P}) - (\\mathsf {P}\\mathsf {Adj}(\\mathsf {C}\\mathsf {P}) (\\mathsf {C}\\mathsf {P}) \\\\&\\stackrel{*}{=} (\\mathsf {Adj}(\\mathsf {C})\\mathsf {C})\\mathsf {P}- \\mathsf {P}(\\mathsf {Adj}(\\mathsf {C}\\mathsf {P})(\\mathsf {C}\\mathsf {P}))= \\mathsf {P}- \\mathsf {P}= \\mathsf {0}$ Equality ($*$ ) holds because $\\mathsf {P}$ is a generalized permutation matrix.",
"Finally, assume that $b-c\\ge -1$ and that $a-c\\le 0$ , and define disjoint sets $K&:=\\lbrace k \\in [c,d] \\text{ such that $S(k)\\in [c,d]$} \\rbrace \\\\L &:= \\lbrace \\ell \\in [b+1,d] \\text{ such that $S(\\ell )<a$} \\rbrace $ Construct a $[c,d]\\times (K \\cup L)$ -matrix $\\mathsf {M}$ , such that for each $k\\in K$ , the corresponding column of $\\mathsf {M}$ is $\\mathsf {C}_{[a,b],k}$ and for each $\\ell \\in L$ , the corresponding column of $\\mathsf {M}$ is $e_\\ell $ (the column vector with a 1 in the $\\ell $ th place).",
"Since $\\mathsf {Sol}(\\mathsf {C})\\mathsf {C}\\mathsf {P}=0$ (Proposition REF ), $\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]} \\mathsf {M}_{[c,d],k}=0$ for all $k\\in K$ .",
"Since each column of $\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]}$ indexed by $\\ell \\in L$ vanishes (Proposition REF ), $\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]}\\mathsf {M}_{[c,d],\\ell } =0$ for all $\\ell \\in L$ .",
"Therefore, the matrix product vanishes: $ \\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]}\\mathsf {M}=0$ The transpose $\\mathsf {M}^\\top $ is in row echelon form and has rank $|K \\cup L|$ .",
"Therefore, $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]})\\le \\dim (\\mathrm {ker}(\\mathsf {M}^\\top )) = |[c,d]\\setminus (K\\cup L) |$ Next, we note that $K\\cup L$ indexes the columns of the box $[a,b]\\times [c,d]$ which do not contain the `initial circle' of an $S$ -ball in the box.",
"Combined with the lower bound (REF ), $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b],[c,d]})= \\text{the number of $S$-balls in the box $[a,b]\\times [c,d]$} $ Lemma REF extends to improper intervals such as $(-\\infty , \\infty ) $ and $ (-\\infty , d]$ by considering appropriate limits, as the number of $S$ -balls in a box is monotonic in nested intervals.",
"For example, the rank of the entire matrix $\\mathsf {Sol}(\\mathsf {C})$ equals the number of $S$ -balls in the unbounded `box' $(-\\infty ,\\infty )\\times ( -\\infty ,\\infty )$ ; that is, the number of $S$ -balls.",
"The lemma allows us to prove the following fundamental result.",
"Theorem 8.10 If $\\mathsf {C}$ is reduced, the kernel of $\\mathsf {C}$ equals the image of $\\mathsf {Sol}(\\mathsf {C})$ .",
"By Proposition REF , $\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C})) \\subseteq \\mathrm {ker}(\\mathsf {C})$ .",
"For any interval $[a,b]$ and any $S$ -schedule $J$ of $[a,b]$ , Proposition REF implies that $ \\dim (\\mathrm {ker}(\\mathsf {C})_{[a,b]})=|J|$ , and Lemma REF implies that $\\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[a,b], \\mathbb {Z}}) = \\text{\\# of $S$-balls in the box $[a,b]\\times \\mathbb {Z}$} $ The number of $S$ -balls in $[a,b]\\times \\mathbb {Z}$ equals the number of $S$ -balls in $[a,b]$ (Remark REF ), which in turn equals $|J|$ .",
"Thus, $\\mathrm {ker}(\\mathsf {C})_{[a,b]}=\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C}))_{[a,b]}$ for all intervals $[a,b]$ , and so $\\mathrm {ker}(\\mathsf {C})=\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C}))$ .",
"Bases for solutions We can minimally parametrize the kernel using $S$ -schedules.See Definition REF ; note that $I$ only admits an $S$ -schedule if $I$ is a set of consecutive elements.",
"Proposition 8.11 Let $\\mathsf {C}$ be a reduced recurrence matrix with shape $S$ , and let $J$ be an $S$ -schedule for $I$ .",
"Then multiplication by $\\mathsf {Sol}(\\mathsf {C})_{I,J}$ gives an isomorphism $\\mathsf {k}^J\\rightarrow \\mathrm {ker}(\\mathsf {C})_I$ .",
"If $J$ is an $S$ -schedule for $\\mathbb {Z}$ , then multiplication by $\\mathsf {Sol}(\\mathsf {C})_{\\mathbb {Z},J}$ is an isomorphism $\\mathsf {k}^J\\rightarrow \\mathrm {ker}(\\mathsf {C})$ .",
"We first consider the the case when $I$ is finite.",
"Consider any interval $[c,d]$ such that $I\\subset [c,d]$ .",
"By Remark REF , the number of $S$ -balls in $[c,d]\\times I$ equals the number of $S$ -balls in $I$ , which is equal to $J$ by Definition REF .",
"By Lemma REF , this implies that $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],I}) = |J| $ .",
"Since $J\\subset I$ , this implies that $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],J}) \\le |J| $ .",
"We will prove that $\\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],J}) = |J|$ by induction on the number of elements in $I$ .",
"If $I=\\varnothing $ , the claim holds vacuously.",
"Assume that the claim holds for all $S$ -schedules of intervals shorter than $I$ , and choose a sequence of subintervals as in Definition REF .",
"Since $I$ is finite, the sequence terminates at $[a_n,b_n]=I$ .",
"Since $J^{\\prime } := J\\cap [a_{n-1},b_{n-1}]$ is an $S$ -schedule for $[a_{n-1},b_{n-1}]$ , by the inductive hypothesis, $ \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],J^{\\prime }}) = |J^{\\prime }| $ .",
"If $J^{\\prime }=J$ , this immediately implies (REF ).",
"If $J^{\\prime }\\ne J$ , then $|J| = |J^{\\prime }|+1$ , and so $\\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],[a_{n},b_{n}]} ) = \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],[a_{n-1},b_{n-1}]} ) +1 $ Hence, the column indexed by the unique element of $[a_n,b_n]\\setminus [a_{n-1},b_{n-1}]$ is linearly independent from the other columns.",
"Since this is also the unique element in $J\\setminus J^{\\prime }$ , $\\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],J} ) = \\mathrm {rank}(\\mathsf {Sol}(\\mathsf {C})_{[c,d],J^{\\prime }} ) +1 $ This proves (REF ) and completes the induction.",
"Since $\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C}))= \\mathrm {ker}(\\mathsf {C})$ , we know that $\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C})_{I,J}) \\subseteq \\mathrm {ker}(\\mathsf {C})_I$ .",
"Equation REF for $I=[c,d]$ and Proposition REF imply these both have dimension $|J|$ , so $\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C})_{I,J}) = \\mathrm {ker}(\\mathsf {C})_I$ .",
"When $I$ is infinite, Definition REF and the preceding argument guarantee it is a union of finite intervals on which the proposition holds, so $\\mathrm {im}(\\mathsf {Sol}(\\mathsf {C})_{I,J}) = \\mathrm {ker}(\\mathsf {C})_I$ .",
"The special case of the $S$ -schedule $T_b$ for $\\mathbb {Z}$ yields the following.",
"Theorem 8.12 Multiplication by $\\mathsf {Sol}(\\mathsf {C})_{\\mathbb {Z}\\times T_b}$ gives an isomorphism $\\mathsf {k}^{T_b}\\rightarrow \\mathrm {ker}(\\mathsf {C})$ .",
"appendix"
]
] | 1906.04311 |
[
[
"Linking dissipation, anisotropy and intermittency in rotating stratified\n turbulence"
],
[
"Abstract Analyzing a large data base of high-resolution three-dimensional direct numerical simulations of decaying rotating stratified flows, we show that anomalous mixing and dissipation, marked anisotropy, and strong intermittency are all observed simultaneously in an intermediate regime of parameters in which both waves and eddies interact nonlinearly.",
"A critical behavior governed by the stratification occurs at Richardson numbers of order unity, and with the flow close to being in a state of instability.",
"This confirms the central dynamical role, in rotating stratified turbulence, of large-scale intermittency, which occurs in the vertical velocity and temperature fluctuations, as an adjustment mechanism of the energy transfer in the presence of strong waves."
],
[
"Introduction, equations and diagnostics",
"A signature of fully developed turbulence (FDT) is its intermittency, i.e.",
"the occurrence of intense and sparse small-scale structures such as vortex sheets, filaments and fronts.",
"This translates into non-Gaussian Probability Distribution Functions (PDFs) of velocity and temperature gradients, as well as chemical tracer gradients.",
"On the other hand, the atmosphere and the ocean are also known for their intermittency in the large scales, with strong wings in the PDFs of the vertical velocity and temperature fields themselves.",
"Such high vertical velocities are observed in the nocturnal Planetary Boundary Layer[1]; this leads to strong spatial and temporal variations of the rate of kinetic energy dissipation, as measured in oceanic ridges[2], [3].",
"Similarly, micro-structures, observed in the frontal Antarctic Circumpolar Current are formed by quasi-geostrophic eddies flowing over bottom topography[4].",
"These structures are due to the bathymetry which has recently been assessed with increased accuracy [5].",
"Such interactions between turbulence and stratification can affect many processes in the atmosphere and the ocean, such as for example rain formation [6], or the lifetime of melting ice shelves[7].",
"Large-scale intermittency is also found in high-resolution Direct Numerical Simulations (DNS) of stratified flows, with or without rotation[8], [9], with a direct correlation to high levels of dissipation.",
"However, isotropy is classically assumed when estimating energy dissipation of turbulent flows, from laboratory experiments to oceanic measurements, even though it has been known for a long time that small-scale isotropy recovers slowly in terms of the controlling parameter, such as in wakes, boundary layers, and pipe or shear flows.",
"A lack of isotropy can be associated with intermittency, as well as with the long-range interactions between large-scale coherent structures and small-scale dissipative eddies[10].",
"In the purely rotating case, vertical Taylor columns form and, using particle image velocimetry, space-time dependent anisotropy has been shown to be important[11].",
"Spectra follow weak turbulence arguments for strong rotation[12], and pressure acts on nonlocal interactions between strong vortices at small scales and large-scale fluctuations [13].",
"In the case of pure stratification, its role on small-scale anisotropy was studied experimentally in detail[14].",
"Spectral data and dissipation data are mostly stream-wise anisotropic because of the shear, on top of the anisotropy induced by the vertical direction of stratification[15].",
"The vertical integral length scale does not grow, contrary to its horizontal counterpart[16], and vertical scales are strongly intermittent.",
"Different components of the energy dissipation tensor have been evaluated, for purely stably stratified flows or wall turbulence, as a function of governing parameters[17], [18], [19], [20], and a slow return to isotropy is found only for rather high buoyancy Reynolds number [17], of the order of $R_B\\approx 10^3$ (see next section for definitions of parameters).",
"With strong imposed shear and using anisotropic boxes, anisotropy is found to be strongest when turbulence is weakest, as expected, and anisotropic eddies in the small scales depend on the effective scale-separation between the large-scale containing eddies, or the buoyancy scale for stratified flows, and the dissipative scale of the turbulence[17].",
"Part of the difficulty in assessing the return to isotropy in either the large or the small scales, however, is that there is a strong coupling between scales, through the interactions of gravity waves and fine-structure shear layers[21], as well as in fronts.",
"In this context, we evaluate quantitatively the link between mixing and dissipation, anisotropy and intermittency in the presence of both rotation and stratification, and as a function of the intensity of the turbulence.",
"This is accomplished in the framework of a large series of unforced DNS runs for the Boussinesq equations.",
"With ${\\cal P}$ the total pressure, ${\\bf u}={\\bf u}_\\perp +w\\hat{e}_z$ the velocity, $\\theta $ the temperature fluctuation (normalized to have dimensions of a velocity), and $\\nabla \\cdot {\\bf u} =0$ because of incompressibility, we have in the unforced case: $\\frac{\\partial {\\bf u}}{\\partial t} + \\mbox{$\\omega $} \\times {\\bf u} + 2 \\mbox{$\\Omega $} \\times {\\bf u} &=& -N \\theta \\hat{e}_z - \\nabla {\\cal P} + \\nu \\nabla ^2 {\\bf u}, \\\\\\frac{\\partial \\theta }{\\partial t} + {\\bf u} \\cdot \\nabla \\theta &=& Nw + \\kappa \\nabla ^2 \\theta \\ ;$ $\\nu $ is the viscosity, $\\kappa $ the diffusivity, $\\mbox{$\\omega $}=\\nabla \\times {\\bf u}$ the vorticity and $N$ the Brunt-Väisälä frequency.",
"Rotation, of intensity $\\Omega =f/2$ , and stratification are in the vertical ($z$ ) direction.",
"We use the pseudo-spectral Geophysical High Order Suite for Turbulence (GHOST) code with hybrid MPI/OpenMP/CUDA parallelization and linear scaling up to at least 130,000 cores[22].",
"The GHOST-generated database considered here consists of fifty-six simulations on grids of $1024^3$ points, as well as three at $512^3$ , twelve at $256^3$ , and two at $128^3$ resolutions, all in a triply periodic box[23], [24].",
"Initial conditions for most runs are isotropic in the velocity; thus at t=0, $w/u_\\perp \\lesssim 1$ , and we take zero temperature fluctuations, so that $\\theta $ develops in a dynamically consistent way.",
"Initial conditions in quasi-geostrophic (QG) equilibrium have also been considered[23], in that case with $N/f\\approx 5$ , $w(t=0)=0$ and $\\theta (t=0)\\ne 0$ .",
"The analysis of the QG set of runs, indicated in the figures by star symbols, has not introduced any major change in the conclusions[24], although it displays more intermittency and anisotropy (see Figs.",
"REF and REF below).",
"Finally, with $\\perp $ referring to the horizontal direction, $k=\\sqrt{|{\\bf k}_\\perp |^2 + k_z^2}$ is the isotropic wavenumber.",
"The dimensionless parameters of the problem are the Reynolds, Froude, Rossby and Prandtl numbers: $Re=\\frac{U_0L_{int}}{\\nu }\\ , \\ \\ Fr=\\frac{U_0}{L_{int}N}\\ , \\ \\ Ro=\\frac{U_0}{L_{int}f}\\ , \\ \\ Pr=\\frac{\\nu }{\\kappa }, $ where $U_0$ is the rms velocity and $L_{int}=\\int [E_v(k)/k]dk/E_v$ is the isotropic integral scale, both evaluated at the peak of dissipation.",
"For all runs, we set $Pr=1$ .",
"The kinetic, potential and total energies $E_v, E_p$ and $E_T=E_v+E_p$ , of respective isotropic Fourier spectra $E_{v,p,T}(k)$ , and their dissipation rates ${\\bar{\\epsilon }}_{v,p,T}$ are: $E_v= \\left<|{\\bf u}|^2/2\\right>, {\\bar{\\epsilon }}_v= DE_v/Dt=\\nu {Z_V, \\ Z_V=\\left<|\\mbox{$\\omega $}|^2\\right>,} $ $E_p= \\left< \\theta ^2/2\\right>, {\\bar{\\epsilon }}_p=DE_p/Dt=\\kappa {Z_P, \\ Z_P=\\left<|\\nabla \\theta |^2\\right>,} $ with ${\\bar{\\epsilon }}_T={\\bar{\\epsilon }}_v+{\\bar{\\epsilon }}_p$ , and where $Z_{V,P}$ are the kinetic and potential enstrophies.",
"Spectra can also be expressed in terms of $k_\\perp $ or $k_z$ (as in equation (REF ) below).",
"It may also be useful to define other derived parameters.",
"For example, the Richardson number $Ri$ , buoyancy Reynolds number $R_B$ , buoyancy interaction parameter $R_{IB}$ and gradient Richardson number $Ri_g$ are written as: $Ri&=& (N/S)^2 \\ \\ , \\ \\ R_B=ReFr^2 , \\\\R_{IB}&=& {\\bar{\\epsilon }}_v/(\\nu N^2)\\ , \\ Ri_g= N(N-\\partial _z\\theta )/S({\\bf x})^2, $ with $S=\\left<\\partial _z u_\\perp \\right>$ representing the internal shear that develops in a dynamically consistent way, and in the absence of imposed external shear.",
"$Ri_g$ is a point-wise measure of instability; it can be negative when the vertical temperature gradient is locally larger than the (constant) Brunt-Väisälä frequency, indicative of strong local overturning.",
"We should note here that different definitions can be found in the literature.",
"In particular, the buoyancy Reynolds number is often expressed as[25] ${\\bar{\\epsilon }}_v/[\\nu N^2]$ , corresponding to $R_{IB}$ here.",
"The distinction between $R_B$ and $R_{IB}$ is physically important.",
"Indeed, we can also define $\\beta $ as a global measure of the efficiency of kinetic energy dissipation, with respect to its dimensional evaluation $\\epsilon _D {\\sim U_0^3/L_{int}}$ : $\\beta &\\equiv &{\\bar{\\epsilon }}_v/\\epsilon _D= \\tau _{NL}/T_v, \\\\\\tau _{NL}&=&L_{int}/U_0, \\ T_v=E_v/{\\bar{\\epsilon }}_v\\ .",
"$ $\\tau _{NL}$ and $T_v$ are the two characteristic times defining nonlinear transfer and energy dissipation; one can also define the waves periods as $\\tau _{BV}=2\\pi /N$ and $\\tau _f=2\\pi /f$ .",
"It follows that, in the intermediate regime of wave-eddy interactions, one has the following simple relationship : $R_{IB}=\\beta R_B.",
"$ This can be justified through a dimensional argument corroborated by numerical results[24].",
"Note that, in fully developed turbulence, one has $T_v=\\tau _{NL}$ and $\\beta =1$ .",
"We also showed that the characteristic times associated with the velocity and temperature and based on their respective dissipation rates, $T_v$ and $T_p=E_p/{\\bar{\\epsilon }}_p$ , vary substantially with governing parameters, being comparable in a narrow range of Froude numbers when large-scale shear layers destabilize[26].",
"The direct numerical simulations cover a wide range of parameters[27], [26], [24]: $10^{-3} \\le Fr \\le 5.5 \\ , \\ 2.4\\le N/f \\le 312 \\ , \\ 1600\\le Re \\le 18590.",
"$ $R_B$ and $R_{IB}$ vary roughly from $10^{-2}$ to $10^5$ , values which, at the upper end, are relevant to the ocean and atmosphere.",
"A few purely stratified runs are considered as well.",
"Figure: Left: Variation with Richardson number of the kinetic energy dissipation efficiency β\\beta .",
"The Roman numerals at the bottom delineate the three regimes of rotating stratified turbulence.Right: Variation with buoyancy interaction parameterR IB R_{IB}of the mixing efficiency Γ f \\Gamma _fdefined in equ.",
"().Colored symbols indicate Rossby number ranges (see inset).Anisotropy has been studied extensively for a variety of flows[28], and many diagnostics have been devised.",
"Here, we concentrate on the following set, starting with the integral scale, the subscript $\\mu $ representing $z,\\perp $ : $\\frac{L_{int,\\mu }}{2\\pi }=\\frac{\\Sigma k^{-1}_{\\mu } E_{v}(k_\\mu )}{\\Sigma E_{v}(k_\\mu )}, $ with $L_{int}$ representing the integral scale for the isotropic case, that is in terms of isotropic wavenumber.",
"Integral scales are known to increase with time in FDT, and it has been shown to do the same in rotating and/or stratified turbulence.",
"This is a manifestation of the interactions between widely separated scale that feed the large-scale flow through what is known as eddy noise together with, in the rotating case in the presence of forcing, the occurrence of an inverse cascade of energy.",
"Figure: (a)-(c): Temporal variations, in units of turn-over time,of kinetic energy (left, blue axis in each plot) and of kinetic enstrophy Z V =|ω| 2 Z_V=\\left< |\\mbox{$\\omega $}|^2 \\right> (right, red axis in each plot).",
"All runs are performed on grids of 1024 3 1024^3 points and are in one of the three regimes identified in Fig.",
"(see text for parameters): I (a), II (b) and III (c).",
"(d)-(f): Joint PDFs of point-wise kinetic energy dissipation ϵ v (𝐱)\\epsilon _v({\\bf x}) and gradient Richardson number Ri g (𝐱)Ri_g({\\bf x})for the same three runs as above.Ri g =1/4Ri_g=1/4is indicated by the thin vertical lines.",
"All plots use the same color bar given at left.Other signatures of anisotropy can be obtained through the properties of the following tensors: $b_{ij}= \\frac{\\left< u_iu_j \\right>}{\\left< u_k u_k \\right>} - \\frac{\\delta _{ij}}{3}, \\ d_{ij}=\\frac{\\left< \\partial _k u_i \\partial _k u_j \\right>}{\\left< \\partial _k u_m \\partial _k u_m \\right>} - \\frac{\\delta _{ij}}{3}, $ $g_{ij}= \\frac{\\left< \\partial _i \\theta \\ \\partial _j \\theta \\right>}{\\left< \\partial _k \\theta \\ \\partial _k \\theta \\right>}- \\frac{\\delta _{ij}}{3}, \\ v_{ij}= \\frac{\\left<\\omega _i \\omega _j \\right>}{\\left<\\omega _k \\omega _k \\right>} -\\frac{\\delta _{ij}}{3} \\ .",
"$ These tensors are equal to zero in the isotropic case.",
"For reference, we also write the point-wise dissipation, $\\epsilon _v ({\\bf x})= 2\\nu s_{ij} s^{ij}$ , where $s_{ij}({\\bf x}) = \\frac{1}{2}(\\partial _i u_j + \\partial _j u_i)$ is the strain rate tensor.",
"We define as usual the second and third-order invariants of a tensor $T_{ij}$ as $T_{II}=T_{ij}T_{ji}$ and $T_{III}= T_{ij}T_{jk}T_{ki}$ .",
"For the tensors above, they are denoted respectively $b_{II,III}$ , $d_{II,III}$ , $g_{II,III}$ , and $v_{II,III}$ (see for example Refs.",
"[10], [29], [17], [28] for details and interpretation).",
"They refer in particular to the geometry of the fields (one-dimensional or 1D vs. 2D, 3D, and axisymmetric, oblate or prolate).",
"In what follows, all anisotropy tensors and their invariants are computed from a snapshot of the data cube at the peak of total enstrophy $Z_T=\\left< |\\mbox{$\\omega $}|^2 + |\\nabla \\theta |^2 \\right>$ (and thus, at the peak of dissipation ${\\bar{\\epsilon _T}}$ ) for each run, as are all PDFs and quantities associated with buoyancy flux, e.g., $\\Gamma _f$ defined in the next section.",
"All other quantities that are plotted are computed based on spectra that are averaged in time over the peak in enstrophy.",
"Specifically, the chosen time intervals, different for different runs, are taken so that the variation of the total enstrophy in each case is no more than 2.5% from its peak value when the turbulence is fully developed.",
"This ensures a lack of correlation between data points within the parametric study.",
"Note that most of the symbols used throughout the paper, together with their definitions, are provided for convenience in Appendix Table REF ."
],
[
"At the threshold of shear instabilities",
"Rotating stratified turbulence (RST) consists of an ensemble of interacting inertia-gravity waves and (nonlinear) eddies.",
"It can be classified into three regimes, I, II, and III, with dominance of waves in I for small Froude number (and small $R_{IB}$ ), and dominance of eddies in III for high $R_{IB}$ : then, the waves play a secondary role and dissipation recovers its fully developed turbulence isotropic limit ${\\bar{\\epsilon }}_D$ , within a factor of order unity[30].",
"In the intermediate regime(regime II), one finds (i) $\\beta \\sim Fr$ , as required by weak turbulence arguments; this is the first central result in Ref.",
"[24], together with the following two other laws: (ii) kinetic and potential energies are proportional (but not equal), with no dependence on governing parameters in regime II where waves and nonlinear eddies strongly interact; and (iii) similarly for the ratio of vertical to total kinetic energy, $E_z/E_v$ .",
"With these three constitutive laws, (i)-(iii), one can recover and establish a large number of scaling relationships, such as the ratio of characteristic scales[24], or for the mixing efficiency defined as: $\\Gamma _f\\equiv B_f/{\\bar{\\epsilon }}_v,\\ B_f=N\\left<w\\theta \\right>, $ $B_f$ being the buoyancy flux.",
"One finds $\\Gamma _f \\sim R_B^{-1}\\sim Fr^{-2}$ in regimes I and II, and $\\sim R_{IB}^{-1/2}\\sim Fr^{-1}$ in regime III.",
"Such scalings, predicted from simple physical arguments in [24] have been observed at high $R_{IB}$ , for example in oceanic data[31].",
"Defining $\\Gamma _\\ast \\equiv {\\bar{\\epsilon }}_p/{\\bar{\\epsilon }}_v$ as the reduced mixing efficiency provides another simple measure of irreversible mixing by looking at how much dissipation occurs in the potential and kinetic energy respectively.",
"It is easily shown using the laws given above that, for the saturated regime III, $\\Gamma _\\ast \\sim Fr^{-2}$ since the Ellison scale $L_{Ell}=2\\pi \\theta _{rms}/N$ becomes comparable to $L_{int}$ in that case (see Fig.",
"6 in that paper).",
"These scaling laws extend smoothly to the purely stratified flows we have analyzed, where, for regime II, the reduced mixing efficiency was found to vary linearly with Froude number [9].",
"These results are also compatible with other results obtained for that case[32], [25], [33], [19], [20].",
"We thus begin our investigation by examining mixing and dissipation.",
"We show in Fig.",
"REF the dissipation efficiency $\\beta ={\\bar{\\epsilon }}_v/\\epsilon _D$ as a function of Richardson number.",
"Unless specified otherwise, data is binned in Rossby number (refer to the legend in Fig.",
"REF (left)), as in most subsequent scatter plots, with roughly the same number of runs in each bin.",
"For runs initialized with random isotropic conditions, the color and symbol of a given data point both indicate which Rossby number bin it resides in.",
"Star symbols are used for quasi-geostrophic initial conditions, with a balance between pressure gradient, Coriolis force and gravity, and the color alone indicates the bin range it belongs to.",
"For all scatter plots, the size of a symbol is proportional to the viscosity of the run, with the smallest symbols denoting runs on grids of $1024^3$ points and higher Reynolds numbers, and the two largest symbols denoting the two runs on grids of $128^3$ points at the lowest $Re$ .",
"Note in the plot of $\\beta (Ri)$ the presence of an inflection point for $Ri\\lesssim 1/4$ , and the two plateaux starting at $Ri\\approx 10^{-2}$ and $\\approx 10$ with an approximate scaling $\\beta \\sim Ri^{-1/2}$ in the intermediate regime, consistent with $\\beta \\sim Fr$ , as found before[24].",
"As stated earlier, this defines the three regimes of rotating stratified turbulence, I, II and III, in a similar fashion as for the case of purely stratified turbulence[25].",
"The mixing efficiency $\\Gamma _f$ is plotted in Fig.",
"REF (right) as a function of buoyancy interaction parameter $R_{IB}$ .",
"The three data points with $R_{IB} \\gtrsim 10^4$ have Froude numbers above unity.",
"It also follows approximately two scaling laws.",
"It can become singular in the quasi-absence of kinetic energy dissipation (when measured in terms of buoyancy flux), and indeed $\\Gamma _f$ takes high values for the runs at low $Fr$ .",
"Its slower decay with $R_{IB}$ for strongly turbulent flows starts at a pivotal value of $R_{IB}\\approx 1$ , a threshold which will be present in most of the data analyzed herein.",
"The decay of $\\Gamma _f$ to low values is inexorable in the absence of forcing and with zero initial conditions in the temperature field which, at high $R_{IB}$ , becomes decoupled from the velocity and evolves in time in a way close to that of a passive scalar.",
"Figure: As a function of buoyancy interaction parameter R IB =ϵ ¯ v /[νN 2 ]R_{IB}={\\bar{\\epsilon }}_v/[\\nu N^2], we plot:(a) Ratio of vertical to horizontal integral scales (see equation ();(b) d II 1/2 d_{II}^{1/2}; and(c) v II 1/2 v_{II}^{1/2}(see equations () for definitions of second tensor invariants for the velocity and vorticity,d II 1/2 d_{II}^{1/2} and v II 1/2 v_{II}^{1/2}).",
"Binning is in Rossby number.Figure REF (a)-(c) displays the variation with time of the kinetic energy and enstrophy, $E_v$ and $Z_v$ , where the time is expressed in units of turn-over time, $\\tau _{NL}$ defined in equation (REF ).",
"The specific runs are computed on grids of $1024^3$ points.",
"There is roughly a factor of 10 in Froude number from regimes I to II, and from II to III; specifically, we have: Run 5, with $Fr\\approx 0.007$ , $N/f\\approx 31$ , $Re \\approx 14000$ , ${\\cal R}_B\\approx 0.75$ in regime I; Run 32, with $Fr\\approx 0.07$ , $N/f\\approx 42$ , $Re \\approx 12200$ , ${\\cal R}_B\\approx 65$ in regime II; and Run 58, with $Fr\\approx 0.89$ , $N/f\\approx 2.5$ , $Re \\approx 4700$ , ${\\cal R}_B\\approx 3760$ in the third regime[23], [24].",
"Note the different scales on both axes, and the different ranges of values for enstrophy in the three regimes: there is more enstrophy (and hence more dissipation) as we move from regime I to regime III.",
"There are fast oscillations in the first regime (Fig.",
"REF (a)).",
"They are a signature of the fast exchanges (compared to the turn-over time) of energy between the kinetic and potential modes.",
"In regime II (Fig, REF (b)), the oscillations are slower and become more complex once the maximum of enstrophy is reached and the flow is a superposition of nonlinearly interactive modes.",
"Finally, the higher enstrophy values in the last regime (Fig.",
"REF (c)) is related to strong small-scale dissipative structures.",
"The maximum of kinetic enstrophy (and thus of kinetic energy dissipation) is reached at a later time in regime I than in the other two regimes, corresponding to a slower development of small scales through weak nonlinear mode coupling.",
"Also, as expected, the energy decays faster as we approach the fully turbulent regime.",
"Similar results hold for the potential energy and its dissipation (not shown).",
"Joint PDFs of the point-wise gradient Richardson number and kinetic energy dissipation, for the same runs as in the top panels, are shown in Fig.",
"REF (d)-(f).",
"The threshold of shear instability, $Ri_g=1/4$ , is indicated in all three plots by a thin vertical line.",
"For regime II (Fig.",
"REF (e)), most points in the flow are close to (but still slightly above) the threshold, $Ri_g\\gtrsim 1$ .",
"The point-wise kinetic energy dissipation is centered on $\\approx 3 \\times 10^{-3}$ but covers locally a range of values almost two orders of magnitude wide for $Ri_g\\approx 1$ .",
"For runs in regime I (Fig.",
"REF (d)), no data point reaches $Ri_g=1$ , and rather dissipation values are found in a narrow band extending to high $Ri_g$ .",
"On the other hand, in the opposite case of strongly turbulent flows (regime III in Fig.",
"REF (f)), the bulge of points around $Ri_g\\approx 1/4$ is much narrower with a flow almost everywhere at the brink of linear instability, reaching higher values of local dissipation, and with a larger extension in its local values, over 2.5 orders of magnitude; this can be seen as being indicative of intermittent behavior, as we shall analyze below in Fig.",
"REF .",
"This high number of data points, for a given set-up, with local gradient Richardson numbers close to $1/4$ has been noted before by several authors.",
"For example, an equivalent result based on an earlier analysis of oceanic data[34] (see Figure 15) shows that in that case, most of the points are centered at $Ri_g\\approx 0.4$ , with also roughly four orders of magnitude in the variation of the local energy dissipation rate.",
"Such a high density of values for $Ri_g\\approx 1/4$ has recently been interpreted as a manifestation of self-organized criticality[35], with flow destabilization occurring in a wide range of intensity displaying power-law behavior, as analyzed on observations of oceanic microstructures.",
"We now focus on the anisotropy of these flows.",
"Large-scale anisotropy can be measured by the ratio $L_{int,z}/L_{int,\\perp }$ .",
"As shown in Fig.",
"REF (a), it increases with $R_{IB}$ at a slow rate, starting at $R_{IB}\\approx 1$ before settling sharply to a value close to unity for high $R_{IB}\\approx 10^3$ .",
"At small $R_{IB}$ , the larger vertical integral scale (with respect to its horizontal counterpart), indicative of a lesser anisotropy for strong rotation and stratification (blue triangles), can be attributed to initial conditions that are isotropic together with, in that range, weak nonlinear coupling.",
"Note that, at a given $R_{IB}$ , vertical scales are almost a factor of 2 larger for stronger rotation, with a clear clustering of points with $Ro\\le 0.3$ (blue triangles) at intermediate values of $R_{IB}$ .",
"This can be associated with a stronger inverse energy transfer due to rotation, although an inverse energy cascade is not directly observed in the absence of forcing, but can appear, for long times, as an envelope to the temporal decay behavior of a turbulent flow[36].",
"In Fig.",
"REF (b)-(c) are shown the second invariants, $d_{II}^{1/2}$ and $v_{II}^{1/2}$ of the velocity gradient and vorticity tensors (see equation (REF ) for definitions), again as functions of $R_{IB}$ .",
"While anisotropy expressed in terms of $d_{II}^{1/2}$ seems to show an approximate power law decrease towards isotropy (with power law index -1/3), in $v_{II}^{1/2}$ the three regimes of mixing are again visible.",
"In the latter, a sharp transition is observed at $R_{IB} \\gtrsim 100$ .",
"In terms of Froude number, the intermediate regime is bounded by $Fr \\in [0.03, 0.2]$ , and in terms of $R_B$ it is bounded by $R_B \\in [10,300]$ .",
"Note that the $Fr$ bounds encompass that for which the large-scale intermittency is strongest in the case of purely stratified forced flows, as measured by the kurtosis of the vertical Lagrangian velocity[9] (see also Fig.",
"REF ).",
"Note also that, for the highest values of the buoyancy interaction parameter, we have a small $d_{II}^{1/2}\\approx 10^{-3}$ , whereas in terms of the vorticity tensor, the tendency toward isotropy is much slower, with a lowest value of order $10^{-1}$ , indicative of vorticity structures that retain a signature of the imposed anisotropy.",
"This variable anisotropy associated with strong mixing properties is also accompanied by marked intermittency, at small scales and also in the large scales.",
"We first analyze in Fig.",
"REF the PDFs of the vertical temperature gradients, either normalized (a, left), or without normalization (b, middle).",
"Both are binned in $N/f=Ro/Fr$ , and at left the dotted line represents the corresponding Gaussian distribution.",
"As expected, the PDFs are non-Gaussian for all parameters, with wings the intensity of which varies somewhat with $N/f$ .",
"We note however that such wings are present from purely stratified flows to strongly rotating (and stratified) flows with $Ro\\le 0.3$ .",
"Gradients favor small scales, but large scales are intermittent as well, as found already for purely stratified flows[8], at least for an interval of parameters[9].",
"As an example of such large-scale intermittency, we plot at right (c) the kurtosis of the vertical component of the Eulerian velocity, $w$ , at the peak of dissipation, with the kurtosis defined as $K_w=\\left<w^4\\right>/\\left<w^2\\right>^2$ .",
"When considering only the runs with isotropic initial conditions, the increase in $K_w$ is rather smooth and with a peak at $R_{IB} \\approx \\mathcal {O}(10)$ of $K_w\\lesssim 4$ .",
"What is particularly striking, however, is the “bursty” behavior seen in the runs with QG initial conditions (indicated by stars) with a peak of $K_w \\approx 7.5$ at $R_{IB} \\gtrsim 1$ , or at $Fr \\approx 0.07$ , in good agreement with what is found for forced flows[9].",
"The high values we see in $K_w$ are comparable to those observed in the atmosphere[37], [1].",
"Note however that the peaks in $K_w, K_\\theta $ are intermittent in time[9], whereas our analysis is done at a fixed time close to the maximum dissipation of the flows, in order to maximize the effective Reynolds number of each run.",
"The behavior of the QG runs with significantly higher kurtosis is probably due to the fact that their initial conditions are two-dimensional, and with $w=0$ .",
"In such a case, for small Froude number and at least for small times, the advection term leads to smooth fields, and the flow has to develop strong vertical excitation characteristic of stratified turbulence, through local instabilities, in order to catch up with energy dissipation and with emerging tendencies towards isotropy in the small scales.",
"The temperature (not shown) displays for most runs a relatively flat kurtosis at close to its Gaussian value, $K_\\theta ^{(G)} \\approx 3$ , but still exhibits a rather sharp increase to $K_\\theta \\gtrsim 4.2$ in the QG-initialized runs at $R_{IB} \\approx 1$ , as well as for smaller values of Froude number and buoyancy interaction parameter $R_{IB}$ .",
"We provide in Fig.",
"REF the parametric variations for some of the velocity- and temperature-related anisotropy tensor invariants defined in equ.",
"(REF ).",
"Fig.",
"REF (a)-(b) show $b_{II}$ as a function of $R_{IB}$ , and $g_{II}$ as a function of $Ri$ , respectively.",
"Both have a peak at $R_{IB}\\approx 1$ , $Ri \\approx 1$ (corresponding also to $Fr\\approx 0.075$ , $R_B\\approx 10$ , not shown); however, we note that $g_{II,III}$ have a maxima for slightly smaller values of $Fr$ .",
"The final transition to a plateau approaching isotropic values, seen in Fig.",
"REF (a), occurs for high $R_{IB}$ $\\approx 10^3$ , as advocated on the basis of oceanic and estuary measurements[38], or from DNS[17], [20].",
"Figure: (a):NormalizedPDFs of ∂ z θ\\partial _z \\theta with binning in N/fN/f (see legend).The dotted black line is the corresponding Gaussian distribution.",
"(b): The same PDFs without the normalization.",
"(c): Kurtosis of vertical velocity as a function of R IB R_{IB}, with binning in Froude number as indicated in the legend.Having scaled nonlinearly both the second and third invariants of tensors in order for them to have the same physical dimension, we find that third invariants have similar scaling with control parameters, except that they can and do become negative, in ways comparable to what is found for purely stratified flows[17].",
"We illustrate this in Fig.",
"REF (c) in a scatter plot of the second and third invariants of $b_{ij}$ that, to a large degree, fills in Fig.",
"6 of Ref.",
"[17] for $b_{II}^{1/2} < 0.2$ ; it highlights the fact that at the peak of enstrophy, the majority of our runs are dominated by oblate axisymmetric structures, in the form of sheets.",
"This is complementary to what is performed by using many temporal snapshots, when one can probe more of the permissible $b_{II} - b_{III}$ domain[17].",
"Figure: Velocity and temperature invariants defined in eq.",
"():(a) b II 1/2 b_{II}^{1/2} versus R IB R_{IB};(b) g II 1/2 g_{II}^{1/2} vs. RiRi;(c) b II 1/2 b_{II}^{1/2} vs. b III 1/3 b_{III}^{1/3}; and(d) β\\beta vs. b II 1/2 b_{II}^{1/2}, showing the three regimes asin Fig.",
".In (e) is given themixing efficiency Γ f \\Gamma _f vs. b II 1/2 b_{II}^{1/2}, with a best-fitreference line for Fr>0.05Fr > 0.05.In (a,b,c), color binning is done in terms of Rossby number (see inset in Fig.",
"), whereas in (d,e) it is in terms of FrFr.We do note that there are two straggler points at high $b_{II}, g_{II}$ , and low negative $b_{III}$ , and in $v_{II}$ as seen in Fig.",
"REF (c).",
"These runs, indicated by blue stars, have quasi-geostrophic initial conditions and are at low Froude, Rossby and buoyancy Reynolds number ($R_B\\lesssim 1$ ); specifically, they are[24] runs Q9 and Q10 (see their Table 2).",
"Again, the quasi two-dimensional nature of such flows at the peak of enstrophy is confirmed in Fig.",
"REF (c), which places these QG-initialized runs on the upper left branch.",
"This indicates that these flows are dominated by quasi two-dimensional sheets[17] (see, e.g., their Fig.",
"6).",
"Indeed, the high anisotropy observed in the vicinity of $R_B \\approx 1, R_{IB}\\approx 1, Fr\\approx 0.07$ in Fig.",
"REF (c) corresponds to two-dimensional structures in the form of shear layers with strong quasi-vertical gradients at low $Fr$ , and which eventually roll-up as they become unstable.",
"Fig.",
"REF (d) shows the dependence of the kinetic energy dissipation efficiency, $\\beta $ , on the second invariant of the velocity anisotropy tensor, this time with binning in $Fr$ .",
"The figure serves to complement both Fig.",
"1(c) in Ref.",
"[24] and Fig.",
"REF (right), illustrating behavior in the three RST regimes, where $\\beta $ is low at reasonably high measures of (large-scale) anisotropy (as measured by $b_{ij}$ ) in regime I, approaches its highest value at largely constant $b_{II}^{1/2}$ in regime II, and as anisotropy begins to diminish at the end of regime II, remains essentially constant in regime III, as the anisotropy continues to decrease with decreasing stratification.",
"Finally, in order to render more explicit the correlation between mixing and anisotropy, we show in Fig.",
"REF (e) the mixing efficiency $\\Gamma _f$ , displayed against $b_{II}^{1/2}$ , again with binning in Froude number.",
"One observes an approximate power law increase in mixing efficiency as anisotropy grows with stratification, from large to moderate $Fr$ , with a best fit slope of $\\approx 1$ .",
"Using the definitions for $\\beta $ and $\\Gamma _f$ in terms of the buoyancy flux, we can write $\\Gamma _f =[\\beta Fr]^{-1} \\left< w \\theta \\right>/\\left<u^2\\right>$ .",
"Noting again that in regime III, $\\beta $ is independent of anisotropy (Fig.",
"REF (d)), and that $b_{zz}$ is remarkably linear in (indeed, nearly equal to) $b_{II}^{1/2}$ for all runs (not shown), the power law dependence of $\\Gamma _f$ on $b_{II}^{1/2}$ mainly results from the increasingly passive nature of the scalar in transitioning from regime II to regime III, and continuing to larger $Fr$ .",
"There is also an abrupt increase in $\\Gamma _f$ in the smallest $Fr$ range, corresponding to regime I with negligible kinetic energy dissipation.",
"The transitory regime (green diamonds) in the vicinity of the peak of vertical velocity kurtosis also corresponds to maximum $b_{II}$ , i.e.",
"maximum anisotropy, together with mixing efficiency of order unity.",
"The accumulation of points for Froude numbers in the intermediate range of values has large $b_{II}$ and a mixing efficiency around unity, with quasi-balanced vertical buoyancy flux and kinetic energy dissipation."
],
[
"Conclusion and discussion",
"We have shown in this paper that, in rotating stratified turbulence, a sharp increase in dissipation and mixing efficiency is associated, in an intermediate regime of parameters, with large-scale anisotropy, as seen in the velocity tensor $b_{ij}$ , and large-scale intermittency, as observed in the vertical velocity through its kurtosis.",
"The return to isotropy is slow and takes place mostly for buoyancy interaction parameters $R_{IB}$ larger than $\\approx 10^3$ .",
"The temporal persistence of anisotropic structures at a given Froude number could be related to the slow decay of energy in the presence of waves, particularly when helicity is strong[39].",
"Furthermore, rotation plays a role in the large scales, with a larger vertical integral scale at a given Froude number for small Rossby numbers (see Fig.",
"REF (a)).",
"The return to large-scale isotropy, as measured by ${L_z}/{L_\\perp }\\approx 1$ , is very sharp.",
"These results evoke threshold behavior and avalanche dynamics, as analyzed for numerous physical systems[40], [41], [42], [43] in the context of the solar wind, and as found as well recently in observational oceanic data[35].",
"In order to determine whether a given system is undergoing self-organized criticality (SOC) in the form of so-called avalanches, and if so what SOC class the system belongs to, one needs to resort to spatio-temporal analysis, although proxies are possible, such as using static snapshots of dissipative structures and applying the law of probability conservation[41].",
"Furthermore, different conclusions may be drawn whether one examines structures in the inertial range of turbulent flows, or whether one is in the dissipative range[41].",
"Perhaps localized Kelvin-Helmoltz overturning vortices merge into larger regions, as a reflection of non-locality of interactions in these flows, together with sweeping of small eddies by large-scale ones, close to the linear instability for $Ri_g=1/4$ , and leading to rare large-amplitude dissipative (avalanche) events.",
"In that context, long-time dynamics, in the presence of forcing, should be investigated to see whether correlations emerge.",
"A threshold analysis could be performed in these flows in terms of the number of excited sites, say above a local dissipation rate ${\\bar{\\epsilon }}_C$ , as a function of a control parameter, likely the local gradient Richardson number.",
"Temporal dynamics should also be analyzed in terms of life-time of overturning structures, as performed classically for example for pipe flows[44], [45].",
"The burstiness of these rotating stratified flows is accompanied by a turbulence collapse.",
"This takes place once the energy has been dissipated at a rate close to that of homogeneous isotropic turbulence.",
"However, we note that this rate has been found to be dependent on the ratio of the wave period controlling the waves to the turn-over time (in other words, the Froude number), in an intermediate regime of parameters[24].",
"This type of behavior has been studied e.g.",
"for shear flows, emphasizing both the inter-scale interactions between large and small eddies with rather similar statistics[46], and the importance of sharp edges in frontal dynamics[47].",
"This has been analyzed in the laboratory at the onset of instabilities including for Taylor-Couette flows or for pipe flows [45], [48], and it may be related to frontal dynamics observed in the atmosphere and ocean[49], [50], given the tendencies of such flows to be, at least in the idealized dynamical setting studied herein, at the margin of such instabilities.",
"Recent observations[35], modeling[49] and numerous DNS indicate that indeed the gradient Richardson number resides mainly around its classical threshold for linear instability ($\\approx 1/4$ ), as also observed in our results, exhibiting a strong correlation with dissipation.",
"In that light, it may be noted that the range of parameters for the mixing efficiency to be comparable to its canonical value observed in oceanic data is close to the instability threshold: $\\Gamma _f\\approx 0.2$ for $0.02 \\lesssim Fr \\lesssim 0.1$ .",
"Similarly, the kurtosis of the temperature and vertical velocity $K_{\\theta ,w}$ are high[9] in a narrow window around $0.07 \\le Fr \\le 0.1$ .",
"As a specific example of marginal instability behavior in the framework of a classical model of turbulence[51], [52] extended to the stratified case, it can be shown[53] that the flow remains close to the stable manifold of a reduced system of equations governing the temporal evolution of specific field gradients, involving in particular the vertically sheared horizontal flows through the second and third invariants of the velocity gradient matrix, and a cross-correlation velocity-temperature gradient tensor.",
"The link between localized intermittency, anisotropy and dissipation is also found in fully developed turbulence, in the form of strong vortex filaments, non-Gaussianity of velocity gradients and localized dissipative events.",
"It has also been shown that a Kolmogorov spectrum $E_v(k)\\sim k^{-5/3}$ can still be observed when the small scales are anisotropic[54].",
"The new element in rotating stratified flows is what the wave dynamics brings about, namely a fluid in a state of marginal instability, almost everywhere close to the threshold of linear instability in terms of $Ri_g \\approx 1/4$ .",
"It is already known that in magnetohydrodynamics (MHD), when coupling the velocity to a magnetic field leading to the propagation of Alfvén waves, there is stronger small-scale intermittency than for FDT, as found in models of MHD[55], [56], in DNS[57], [58] as well as in observations of the solar wind[59].",
"In RST, the added feature is having intermittency in the vertical component of the velocity and temperature fluctuations themselves, thus at large scale, as found in many observations in the atmosphere and in climatology as well[60], [61], and limited to a narrow range of parameters[9] centered on the marginal instability threshold.",
"Thus, not only does this interplay between waves and nonlinear eddies not destroy these characteristic features of turbulent flows, but in fact it acts in concert with them and can rather enhance them as well.",
"The large data base we use is at a relatively constant Reynolds number, $Re\\approx 10^4$ , and thus an analysis of the variation of anisotropy with $Re$ , for fixed rotation and stratification remains to be done, in the spirit of earlier pioneering studies[62], [63] for fluids.",
"Also, scale by scale anisotropy might be best studied with Fourier spectra.",
"This will be accomplished in the future, together with a study of the role of forcing.",
"This paper is centered on a large parametric study of rotating stratified turbulence.",
"Each flow taken individually is strongly intermittent in space, and thus presents zones that are active as well as zones that are quiescent.",
"It was proposed recently to partition a given flow in such zones, with strong layers delimiting such patches[64], depending on the buoyancy interaction parameter $R_{IB}$ , and with threshold values, of roughly 1, 10 and 100.",
"The intermediate range corresponds, in our DNS runs, to the peak of anisotropy and intermittency together with mixing efficiency being close to its canonical value, $\\Gamma _f\\approx 0.2$ .",
"In this regard, it will be of interest to perform such a local study for a few given runs of our data base in the three regimes.",
"Many other extensions of this work can be envisaged.",
"For example, one could perform a wavelet decomposition to examine the scale-by scale anisotropy and intermittency in such flows[65].",
"Moreover, kinetic helicity, the correlation between velocity and vorticity, is created by turbulence in rotating stratified flows[66], [67].",
"It is the first breaker of isotropy, since flow statistics depend only on the modulus of wavenumbers, but two defining functions (energy and helicity density) are necessary to fully describe the dynamics.",
"In FDT, helicity is slaved to the energy in the sense that $H_v(k)/E_v(k)\\sim 1/k$ , i.e.",
"isotropy is recovered in the small scales at the rate $1/k$ .",
"In the stratified case, its scale distribution changes with Brunt-Väisälä frequency[39], as measured for example in the PBL [68], and it undergoes a direct cascade to small scales while energy goes to large scales in the presence of strong rotation and forcing[69].",
"What role helicity and the nonlinear part of potential vorticity, namely $\\mbox{$\\omega $}\\cdot \\nabla \\theta $ , will play in the fast destabilization of shear layers, their intermittency, anisotropy and criticality are topics for future work.",
"We conclude by noting that a deeper understanding of the structure of small-scale rotating stratified turbulence, and of the nonlocal interactions between small scales and large scales, will allow for better modeling in weather and climate codes.",
"Many models of anisotropic flows have been proposed[70], [71], [72], [73] including artificial neural networks.",
"They extend isotropic formulations for kinetic energy dissipation by adding several off-diagonal terms, and assuming (or not) isotropy in the orthogonal plane[14], [15], including for two-point closures[28].",
"This modeling strategy has already been found useful in models of turbulent mixing in oceanic simulations[74], [75].",
"The runs analyzed in this paper have been performed using an ASD allocation at NCAR, supplemented by a large amount of background time, for both of which we are thankful.",
"NCAR is funded by the National Science Foundation.",
"RM acknowledges support from the program PALSE (Programme Avenir Lyon Saint-Etienne) of the University of Lyon, in the framework of the program Investissements d'Avenir (ANR-11-IDEX-0007), and from the project \"DisET\" (ANR-17-CE30-0003).",
"Support for AP from LASP, and in particular from Bob Ergun, is gratefully acknowledged."
],
[
"Appendix",
"We put together in the following Table, for convenience, the various symbols which are used in the paper, with their definitions and names.",
"Table: List of most symbols used in the paper, with definitions and names (refer as well to the main text).One can also define perpendicular and vertical integral scales, when using the spectra based on perpendicular and vertical wavevectors (see main text).",
"Also note that, in the third regime of rotating stratified turbulence, R B =R IB R_B=R_{IB} because now ϵ ¯ v =ϵ D {\\bar{\\epsilon }}_v=\\epsilon _D."
]
] | 1906.04302 |
[
[
"Ultraviolet to Infrared Evolution and Nonperturbative Behavior of ${\\rm\n SU}(N) \\otimes {\\rm SU}(N-4) \\otimes {\\rm U}(1)$ Chiral Gauge Theories"
],
[
"Abstract We analyze the ultraviolet to infrared evolution and nonperturbative properties of asymptotically free ${\\rm SU}(N) \\otimes {\\rm SU}(N-4) \\otimes {\\rm U}(1)$ chiral gauge theories with $N_f$ copies of chiral fermions transforming according to $([2]_N,1)_{N-4} + ([\\bar 1]_N,[\\bar 1]_{N-4})_{-(N-2)} + (1,(2)_{N-4})_N$, where $[k]_N$ and $(k)_N$ denote the antisymmetric and symmetric rank-$k$ tensor representations of SU($N$) and the rightmost subscript is the U(1) charge.",
"We give a detailed discussion for the lowest nondegenerate case, $N=6$.",
"These theories can exhibit both self-breaking of a strongly coupled gauge symmetry and induced dynamical breaking of a weakly coupled gauge interaction symmetry due to fermion condensates produced by a strongly coupled gauge interaction.",
"A connection with the dynamical breaking of ${\\rm SU}(2)_L \\otimes {\\rm U}(1)_Y$ electroweak gauge symmetry by the quark condensates $\\langle \\bar q q\\rangle$ due to color SU(3)$_c$ interactions is discussed.",
"We also remark on direct-product chiral gauge theories with fermions in higher-rank tensor representations."
],
[
"Introduction",
"A problem of basic field-theoretic interest concerns the behavior of strongly coupled chiral gauge theories.",
"In general, there are two types of chiral gauge theories, namely those based on a single gauge group and those with a direct-product ($dp$ ) gauge group of the form $G_{dp} = \\bigotimes _{i=1}^{N_G} G_i$ with $N_G \\ge 2$ .",
"Strongly coupled direct-product chiral gauge theories are of particular interest because they can exhibit a phenomenon that cannot occur in a chiral gauge theory with a single gauge group, namely the induced dynamical breaking of a weakly coupled gauge symmetry by a different, strongly coupled, gauge interaction.",
"This phenomenon is important not only from the point of view of abstract quantum field theory, but also because it actually occurs in nature.",
"In the Standard Model (SM), with the gauge group $G_{SM}={\\rm SU}(3)_c\\otimes G_{EW}$ , where the electroweak gauge group is $G_{EW} = {\\rm SU}(2)_L\\otimes {\\rm U}(1)_Y$ , the bilinear quark condensates $\\langle \\bar{q} q\\rangle $ produced by the strongly coupled SU(3)$_c$ color gauge interaction dynamically break $G_{EW}$ to the elctromagnetic gauge symmetry, U(1)$_{em}$ .",
"This breaking contributes terms of the form $g^2 f_\\pi ^2/4$ and $(g^2+g^{\\prime 2})f_\\pi ^2/4$ to the squared masses of the $W$ and $Z$ bosons, $m_W^2$ and $m_Z^2$ , respectively, where $g$ and $g^{\\prime }$ are the SU(2)$_L$ and U(1)$_Y$ gauge couplings, and $f_\\pi = 93$ MeV is the pion decay constant.",
"Thus, although textbook discussions usually mention only the vacuum expectation value (VEV) $\\langle \\phi \\rangle _0 = {0 \\frac{v}{\\sqrt{2}}}$ of the Higgs field $\\phi = {\\phi ^+ \\phi ^0}$ as the source of electroweak symmetry breaking in the SM, this breaking really arises from two different sources, one of which is the Higgs VEV (REF ), yielding $m_W^2 = g^2 v^2/4$ and $m_Z^2 = (g^2+g^{\\prime 2})v^2/4$ , where $v=246$ GeV, and the other of which is the above-mentioned dynamical contribution due to the formation of bilinear quark condensates in quantum chromodynamics (QCD).",
"Although this dynamical breaking of electroweak gauge symmetry by the SU(3)$_c$ color gauge interaction is very small compared with the contribution due to the VEV of the Higgs field, it is important as a physical example of how, in a direct-product chiral gauge theory, one strongly coupled gauge interaction can induce the breaking of a weakly coupled one [1], [2].",
"Indeed, a gedanken modification of the Standard Model in which the Higgs field is removed is a perfectly well-defined theory in which the $W$ and $Z$ masses are entirely due to the dynamical breaking of the electroweak gauge symmetry by the SU(3)$_c$ interaction [2], [3].",
"In the Standard Model, the SU(3)$_c$ gauge interaction is vectorial, while $G_{EW}$ is chiral, but this mechanism can also break a vectorial gauge symmetry; in Ref.",
"[3] it was shown that in this gedanken modification of the SM without any Higgs field, if one reversed the order of the coupling strengths of the non-Abelian gauge interactions so that the SU(2)$_L$ coupling were much stronger than the SU(3)$_c$ coupling, then the SU(2)$_L$ gauge interaction would produce bilinear fermion condensates of quarks and leptons that would break the vectorial SU(3)$_c$ , as well as U(1)$_Y$ and U(1)$_{em}$ , [3], while preserving SU(2)$_L$ .",
"Since dynamical symmetry breaking of a weakly coupled gauge symmetry occurs in nature, as shown by the breaking of electroweak gauge symmetry $G_{EW}$ by the $\\langle \\bar{q} q \\rangle $ quark condensates produced by SU(3)$_c$ gauge interaction, there is a motivation to investigate chiral gauge theories that can exhibit this phenomenon of the dynamical breaking of a weakly coupled gauge symmetry by a different, strongly coupled gauge interaction.",
"As noted above, this requires that one consider theories with direct-product chiral gauge symmetries.",
"Some previous studies of strongly coupled chiral gauge theories with direct-product gauge groups (and without any fundamental scalar fields) include [3]-[13], [14], [15].",
"In this paper we shall analyze chiral gauge theories with the direct-product gauge group $G = {\\rm SU}(N) \\otimes {\\rm SU}(N-4) \\otimes {\\rm U}(1) \\ .$ This group is of the form (REF ) with $N_G=3$ , $G_1 = {\\rm SU}(N)$ , $G_2 ={\\rm SU}(N-4)$ , and $G_3 = {\\rm U}(1)$ .",
"The group (REF ) has order $o(G)$ and rank $rk(G)$ given by $o(G)=2N^2-8N+15, \\quad rk(G)=2N-5 \\ .$ The fermion content of the theory consists of $N_f$ copies (“flavors”) of chiral fermions transforming as $([2]_N,1)_{N-4} + ([\\bar{1}]_N, [\\bar{1}]_{N-4})_{-(N-2)} +(1,(2)_{N-4})_N \\ ,$ where the meaning of the notation $(R_1,R_2)_q$ is as follows: the first and second entries refer to the representation $R_1$ of $G_1={\\rm SU}(N)$ and $R_2$ of $G_2={\\rm SU}(N-4)$ , and the subscript $q$ is the U(1) charge of the given fermion.",
"The symbols $[k]_N$ and $(k)_N$ denote the $k$ -fold antisymmetric and symmetric tensor representations of SU($N$ ), respectively, and $R_i=1$ denotes a singlet of $G_i$ , where $i=1$ or $i=2$ .",
"The fermion fields are denoted explicitly as $&& ([2]_6,1)_2: \\ \\psi ^{ij}_{p,L} \\ , \\cr \\cr && ([\\bar{1}]_6: \\ [\\bar{1}]_2)_{-4}: \\ \\chi _{i,\\alpha ,p,L} \\ , \\cr \\cr && (1,(2)_2)_6: \\ \\omega ^{\\alpha \\beta }_{p,L} \\ ,$ where $i,j$ are SU($N$ ) group indices, $\\alpha ,\\beta $ are SU($N-4$ ) group indices, and $p$ is a copy (flavor) index, running from 1 to $N_f$ .",
"We exclude the trivial value $N_f=0$ , because it does not produce a chiral gauge theory, but instead just a set of three decoupled pure gauge theories.",
"There are no bare fermion masses in the theory, since they are forbidden by the chiral gauge symmetry.",
"Without loss of generality, we write the fermions as left-handed.",
"This theory is free of anomalies in gauged currents, as is necessary for renormalizability, and is also free of global anomalies and mixed gauge-gravitational anomalies [12], [16].",
"We note two equivalent theories with the same gauge group, (REF ).",
"The first of these has all of the representations of the left-handed chiral fermions in (REF ) conjugated.",
"The second has the representations of SU($M$ ) conjugated relative to those of SU($N$ ), i.e., its fermion content consists of $N_f$ copies of the set $([2]_N,1)_{N-4} + ([\\bar{1}]_N, [1]_{N-4})_{-(N-2)} +(1,(\\bar{2})_{N-4})_N \\ ,$ Since these theories are equivalent to (REF ) with (REF ), it will suffice to study only the latter.",
"This model is of particular interest for the following reason.",
"A natural construction of a chiral gauge theory with a non-Abelian gauge group uses (left-handed chiral) fermions transforming according to an antisymmetric or symmetric rank-$k$ tensor representation of the gauge group, together with the requisite number of fermions transforming according to the conjugate fundamental representation, so as to yield zero gauge anomaly.",
"The simplest of these uses $k=2$ , so let us focus on these theories with $k=2$ .",
"With a special unitary gauge group, there are two such constructions: (i) $G={\\rm SU}(N)$ and chiral fermion content consisting of $N_f$ copies of the set $[2]_N + (N-4) \\, [\\bar{1}]_N$ and (ii) $G={\\rm SU}(M)$ and chiral fermion content consisting of $N_f$ copies of the set $(2)_M + (M+4) \\, [\\bar{1}]_M \\ .$ A basic question in the analysis of chiral gauge theories is whether one can combine these two separate single-gauge-group theories (i) and (ii) into a single chiral gauge theory with a direct-product gauge group that contains ${\\rm SU}(N) \\otimes {\\rm SU}(M)$ such that it is again anomaly-free.",
"The answer is yes, if we set $M=N-4$ , and the theory (REF ) with (REF ) provides an explicit realization of this combination.",
"Indeed, not only does this theory successfully combine the two separate chiral gauge symmetries SU($N$ ) and SU($M$ ) in an anomaly-free manner; it also incorporates a third gauge symmetry, namely the U(1).",
"A general classification of chiral gauge theories with direct-product gauge groups was given in Ref.",
"[13].",
"In this classification, a factor group $G_i$ is labelled as $G_c$ if it has complex representations and $G_r$ if it has (only) real or pseudoreal representations.",
"If a group $G_c$ has no gauge anomaly from any of its representations, then it was denoted as $G_{cs}$ , where the subscript $s$ stands for “safe”.",
"In this classification, if $N \\ge 7$ , then the gauge group (REF ) is of the form $(G_c,G_c,G_c)$ .",
"In contrast, if $N=6$ , then the second factor group is SU(2), which has (pseudo)real representations, so that the $N=6$ special case of (REF ) is of the form $(G_c,G_r,G_c)$ in this classification.",
"In accordance with the order of labelling of the $G_i$ factor groups, we denote the corresponding running gauge couplings as $g_1(\\mu )$ for $G_1 = {\\rm SU}(N)$ , $g_2(\\mu )$ for $G_2 = {\\rm SU}(N-4)$ , and $g_3(\\mu )$ for $G_3 = {\\rm U}(1)$ , where $\\mu $ is the Eucidean energy/momentum reference scale where $g_i(\\mu )$ is measured.",
"We further define $\\alpha _i(\\mu ) = g_i(\\mu )^2/(4\\pi )$ and $a_i(\\mu ) = g_i(\\mu )^2/(16\\pi ^2)$ , with $i=1,2,3$ .",
"(The argument $\\mu $ will sometimes be suppressed in the notation.)",
"As usual with a U(1) gauge interaction, the U(1) charge assignments in (REF ) involve an implicit normalization convention; the physics is unchanged if one redefines $q_f \\rightarrow \\lambda q_f$ for each fermion $f$ and $g_3 \\rightarrow \\lambda ^{-1}g_3$ , since only the product $q_f g_3$ appears in the U(1) covariant derivative.",
"Each of the two non-Abelian gauge interactions is required to be asymptotically free (AF), because this enables us to calculate the corresponding beta functions self-consistently at a high scale $\\mu =\\mu _{_{UV}}$ in the deep ultraviolet (UV) region, where they are weakly coupled.",
"These beta functions then describe the running of the non-Abelian couplings toward the infrared (IR) at small $\\mu $ , where these couplings become larger.",
"Since we are interested in the nonperturbative behavior of the non-Abelian gauge interactions, we will assume the U(1) gauge interaction to be weakly coupled at the initial reference scale $\\mu _{_{UV}}$ ; owing to the property that the beta function for this U(1) interaction is non-asymptotically free, the U(1) coupling $\\alpha _3(\\mu )$ becomes even weaker as $\\mu $ decreases below $\\mu _{_{UV}}$ and hence can be treated perturbatively in the full range $\\mu < \\mu _{_{UV}}$ under consideration here.",
"In addition to the phenomenon of a strongly coupled gauge interaction inducing the dynamical breaking of a different gauge symmetry, a chiral gauge theory can also exhibit a different phenomenon in which a strongly coupled gauge interaction corresponding to a given gauge symmetry produces fermion condensates that break this gauge symmetry itself [1], [17].",
"In particular, for a given gauge interaction corresponding to the non-Abelian gauge group $G_i$ , as $\\mu $ decreases from $\\mu _{_{UV}}$ and $\\alpha _i(\\mu )$ grows, it may become large enough at a certain scale, which we will denote as $\\mu =\\Lambda _1$ , to produce a fermion condensate that breaks the gauge symmetry $G_i$ to a subgroup $H_i\\subset G_i$ .",
"The fermions involved in this condensate gain dynamical masses of order $\\Lambda _1$ and are integrated out of the low-energy effective field theory (EFT) that describes the physics as $\\mu $ decreases below $\\Lambda _1$ .",
"The gauge bosons in the coset space $G_i/H_i$ pick up dynamical masses of order $g_i(\\Lambda _1) \\, \\Lambda _1$ and are also integrated out of the low-energy effective theory.",
"This low-energy theory has a gauge coupling inherited from the UV theory, but since the fermion and gauge boson content is different, this gauge coupling runs according to a different beta function.",
"Then this process of self-breaking of a gauge can repeat at one or more lower scales.",
"The final low-energy effective field theory may be a vectorial theory that confines and produces fermion condensates with associated spontaneous chiral symmetry breaking (S$\\chi $ SB) but no further gauge self-breaking.",
"Besides being of abstract field-theoretic interest, this mechanism of gauge self-breaking has been used in constructions and studies of reasonably ultraviolet-complete models of dynamical electroweak symmetry breaking (EWSB) and fermion mass generation [4]-[8], [11].",
"In these constructions, one starts with an asymptotically free chiral gauge theory that undergoes either self-breaking or a combination of self-breaking and induced symmetry breaking in a sequence of three different scales, $\\Lambda _1 >\\Lambda _2 > \\Lambda _3$ , with an associated breaking of the UV chiral gauge symmetry $G_{UV} \\rightarrow H_1 \\rightarrow H_2 \\rightarrow H_3$ , where the $H_3$ symmetry is vectorial.",
"At a lower scale $\\Lambda _T$ of order 1 TeV, the $H_3$ gauge interaction confines and produces condensates that break $G_{EW}$ .",
"It also produces a spectrum of $H_3$ -singlet bound states.",
"Gauge bosons in the coset space $G_{UV}/H_1$ gain dynamical masses of order $\\Lambda _1$ , while gauge bosons in the coset spaces $H_1/H_2$ and $H_2/H_3$ gain dynamical masses of order $\\Lambda _2$ and $\\Lambda _3$ , respectively.",
"Exchanges of these three different types of massive vector bosons produce the three generations of quark and lepton masses.",
"More complicated exchanges can also produce light neutrino masses via an appropriate seesaw mechanism [5].",
"This scenario has the potential to naturally explain the generational hierarchy in fermion masses, which reflects the hierarchy of self-breaking scales $\\Lambda _i$ , $i=1,2,3$ .",
"This construction is also an ultraviolet completion of low-energy effective Lagrangians for dynamical EWSB that use four-fermion operators [18] and predicts the coefficients of these four-fermion operators.",
"Our theory does not include any fundamental scalar fields.",
"Thus, the pattern of possible dynamical gauge symmetry breaking depends only on the gauge and fermion content, and the initial values of the gauge couplings at the reference scale $\\mu _{_{UV}}$ .",
"This is in contrast with theories in which gauge symmetry breaking is produced by VEVs of Higgs fields, because in these latter theories, the nature of the symmetry breaking depends on various parameters in the Higgs potential, which can be chosen at will, subject to the constraint that this Higgs potential should be bounded from below [19], [20].",
"An alternate application of strongly coupled chiral gauge theories was to efforts at modelling the quarks and leptons as composites of more fundamental fermions, commonly called preons.",
"This involved a scenario in which it was envisioned that the strongly coupled gauge interaction would produce confinement of the preons in gauge-singlet composite fermions, but no spontaneous chiral symmetry breaking.",
"The presumed absence of S$\\chi $ SB was necessary in order for the composite fermions to be very light compared to the inverse of the spatial compositeness scale $\\Lambda _{comp.}",
"= 1/r_{comp.",
"}$ .",
"For this purpose, theories were constructed that satisfied certain matching conditions of chiral symmetries between preons and the composite fermions [21], [22].",
"In the present paper we will focus on studying possible patterns of bilinear fermion condensate formation and resultant dynamical gauge symmetry breaking in the strongly coupled gauge theory (REF ) with (REF ) and (REF )-(REF ) rather than on possible scenarios with light composite fermions.",
"In addition to our analysis of the general theory (REF ) with (REF ), we will study the $N=6$ special case in detail.",
"This $N=6$ theory, with the gauge group $G_{N=6} = {\\rm SU}(6) \\otimes {\\rm SU}(2) \\otimes {\\rm U}(1) \\ ,$ is of particular interest because it is the lowest non-degenerate member of this family.",
"(If $N=5$ , then the SU($N-4$ ) group is trivial).",
"It is also special in two related aspects, namely that (i) as mentioned above, the resultant second factor group is SU(2), with (pseudo)real representations, in contrast to the situation for $N \\ge 7$ , where the SU($N-4$ ) group has complex representations; and (ii) the symmetric rank-2 tensor representation $(2)_2$ of SU(2) is the adjoint representation.",
"The fermion content for this $N=6$ theory, comprised of $N_f$ copies of $([2]_6,1)_2+([\\bar{1}]_6, [\\bar{1}]_2)_{-4}+(1,(2)_2)_6 \\ ,$ can also be conveniently expressed in terms of the dimensionalities of the representations as $(15,1)_2 + (\\bar{6},2)_{-4} + (1,3)_6 \\ .$ Owing to property (ii) above, we will often use the equivalent isovector notation ${\\vec{\\omega }}_{p,L}$ for the $\\omega ^{\\alpha \\beta }_{p,L}$ fermion.",
"Thus, the theory (REF ) with (REF ) and, in particular, the $N=6$ special case, provide valuable theoretical laboratories for the study of nonperturbative properties of chiral gauge theories, including self-breaking of a strongly coupled chiral gauge symmetry, induced breaking of a weakly coupled gauge symmetry by a strongly coupled gauge interaction, and the sequential construction of low-energy effective field theories.",
"This paper is organized as follows.",
"The general methods used in our analysis are described in Section .",
"In Section we analyze the UV to IR evolution, possible fermion condensation channels, and corresponding gauge symmetry breaking patterns of the theory (REF ) with (REF ).",
"In Sections - we present a detailed analysis of the $N=6$ theory.",
"Some remarks on related constructions of direct-product chiral gauge theories with fermions in higher-rank tensor representations are given in Section .",
"Our conclusions are contained in Section .",
"In this section we discuss the general methods that are used for our analysis.",
"We first explain our application of the renormalization group (RG).",
"Recall our labelling conventions given above for the gauge couplings, namely $g_1(\\mu )$ for SU($N$ ), $g_2(\\mu )$ for SU($N-4$ ), and $g_3(\\mu )$ for U(1).",
"The evolution of the three gauge couplings $g_i(\\mu )$ , or equivalently, the corresponding $\\alpha _i(\\mu )$ with $i=1,2,3$ , is determined by the RG beta functions $\\beta _{G_i} = \\frac{d\\alpha _i(\\mu )}{d\\ln \\mu } \\ .$ These have the series expansions $\\beta _{G_i} & = &-8\\pi a_i \\bigg [ b^{(G_i)}_{1\\ell ,i}a_i +\\sum _{j=1}^3 b^{(G_i)}_{2\\ell ;ij} a_i a_j \\cr \\cr & + & \\sum _{j,k=1}^3 b^{(G_i)}_{3\\ell ;ijk}a_ia_ja_k + ... \\bigg ] \\ ,$ where an overall minus sign is extracted, the dots $...$ indicate higher-loop terms, and there is no sum on repeated $i$ indices in the square bracket.",
"Here, $b^{(G_i)}_{1\\ell ,i}$ is the one-loop ($1\\ell $ ) coefficient, multiplying $a_i$ inside the square bracket in (REF ); $b^{(G_i)}_{2\\ell ;ij}$ is the two-loop coefficient, multiplying $a_ia_j$ in the square bracket, and so forth for higher-loop terms.",
"The one-loop coefficients $b^{(G_i)}_{1\\ell ,i}$ are scheme-independent.",
"We focus on the beta functions for the two non-Abelian gauge interactions, since these determine the upper bound on $N_f$ and are relevant for the formation of various possible fermion condensates as $\\alpha _1(\\mu )$ or $\\alpha _2(\\mu )$ become large in the infrared.",
"The one-loop coefficients in Eq.",
"(REF ) are $b^{({\\rm SU}(N))}_{1\\ell ,1} = \\frac{1}{3}\\Big [ 11N - 2N_f(N-3) \\Big ] \\ ,$ $b^{({\\rm SU}(N-4))}_{1\\ell ,2} = \\frac{1}{3}\\Big [ 11(N-4) - 2N_f(N-1) \\Big ] \\ ,$ and $b^{({\\rm U}(1))}_{1\\ell ,3} = -\\frac{4}{3} \\, N_f \\, N(N-1)(N-3)(N-4) \\ .$ As mentioned before, we assume that the U(1) gauge interaction is weakly coupled at the UV reference scale $\\mu _{_{UV}}$ ; then its gauge coupling decreases as $\\mu $ decreases from the UV to the IR, and hence can be treated perturbatively.",
"The requirements that the SU($N$ ) and SU($N-4$ ) gauge interactions must be asymptotically free are that $b^{({\\rm SU}(N))}_{1\\ell ,1} > 0$ and $b^{({\\rm SU}(N-4))}_{1\\ell ,2} > 0$ .",
"These impose the respective upper limits $N_f < N_{f,b1z}$ and $N_f < N_{f,b1z}^{\\prime }$ , where $N_{f,b1z} = \\frac{11N}{2(N-3)}$ and $N_{f,b1z}^{\\prime } = \\frac{11(N-4)}{2(N-1)} \\ ,$ where we use a prime to indicate the upper limit on $N_f$ from the condition $b^{({\\rm SU}(N-4))}_{1\\ell } > 0$ .",
"The upper bound (REF ), is more restrictive than the upper bound (REF ), as is clear, since the difference $N_{f,b1z} - N_{f,b1z}^{\\prime } = \\frac{33(N-2)}{(N-1)(N-3)}$ is positive for all of the relevant values of $N$ under consideration here.",
"Hence, we restrict $N_f < \\frac{11(N-4)}{2(N-1)} \\ .$ The (nonzero) values of $N_f$ that are allowed by the inequality (REF ) depend on $N$ and are as follows: $1 \\le N_f \\le 2$ if $6 \\le N \\le 7$ $1 \\le N_f \\le 3$ if $8 \\le N \\le 12$ $1 \\le N_f \\le 4$ if $13 \\le N \\le 34$ $1 \\le N_f \\le 5$ if $N_f \\ge 35$ .",
"As $N \\rightarrow \\infty $ , the upper limit on $N_f$ (formally generalized to a non-negative real number) approaches 11/2, thus allowing physical integral values up to 5, inclusive, as indicated above.",
"In general, the set of equations (REF ) is comprised of three coupled nonlinear first-order ordinary differential equations for the quantities $\\alpha _i$ , $i=1,2,3$ .",
"The solutions for the three $\\alpha _i(\\mu )$ depend on $N_f$ and the three initial values $\\alpha _i(\\mu _{_{UV}})$ at the UV reference scale $\\mu _{_{UV}}$ .",
"Since we do not assume that the group (REF ) is embedded in a single gauge group higher in the UV, we may choose these initial values $\\alpha _i(\\mu _{_{UV}})$ arbitrarily, subject to the constraint that for $\\mu =\\mu _{_{UV}}$ , the values are sufficiently small that the perturbative calculation of the beta functions $\\beta _{\\alpha _i}$ are self-consistent.",
"To leading order, i.e., to one-loop order, the differential equations making up this set decouple from each other, and one has the simple solution for each $i=1,2,3$ : $\\alpha _i(\\mu _1)^{-1} = \\alpha _i(\\mu _2)^{-1}-\\frac{b^{(G_i)}_{1\\ell ,i}}{2\\pi } \\,\\ln \\Big ( \\frac{\\mu _2}{\\mu _1} \\Big ) \\ ,$ where we take $\\mu _1 < \\mu _2$ .",
"At the level of two loops and higher, due to the fact that each of the fermions has nonzero U(1) charge and one of the fermions, $\\chi _{i,\\alpha ,p,L}$ , is a nonsinglet under both of the non-Abelian gauge groups, there are mixed terms $a_ia_j$ , $a_ia_ja_k$ , etc., that involve different gauge interactions, in the three beta functions $\\beta _{\\alpha _i}$ , so that the three beta functions become coupled differential equations.",
"In view of the mixing terms in (REF ) at the two-loop level, it is natural to focus first on two special cases, namely those in which one of the non-Abelian gauge interactions is much stronger than the other.",
"This can be arranged by specifying appropriate initial values of $\\alpha _1(\\mu _{_{UV}})$ and $\\alpha _2(\\mu _{_{UV}})$ at the UV scale $\\mu _{_{UV}}$ .",
"In these two cases, one can neglect the two-loop term that mixes these two non-Abelian gauge interactions in Eq.",
"(REF ), so that, to two-loop level, these interactions decouple, and the corresponding beta functions have the form, to this level, $\\beta _{\\alpha _1}= \\frac{d\\alpha _1}{d\\ln \\mu } =-8\\pi a_1 \\, \\Big [b^{({\\rm SU}(N))}_{1\\ell ,1}a_1 +b^{({\\rm SU}(N))}_{2\\ell ;11} a_1^2 \\Big ]$ and $\\beta _{\\alpha _2}= \\frac{d\\alpha _2}{d\\ln \\mu } =-8\\pi a_2 \\, \\Big [b^{({\\rm SU}(N-4))}_{1\\ell ,2}a_2 +b^{({\\rm SU}(N-4))}_{2\\ell ;22} a_2^2 \\Big ] \\ ,$ where the one-loop coefficients $b^{({\\rm SU}(N))}_{1\\ell ,1}$ and $b^{({\\rm SU}(N-4))}_{1\\ell ,2}$ were given above in Eqs.",
"(REF )-(REF ), and the two-loop coefficients are $&& b^{({\\rm SU}(N))}_{2\\ell ;11} = \\frac{1}{6N}\\bigg [ 68N^3-N_f(N-3)(29N^2-3N-12) \\bigg ] \\cr \\cr &&$ and $b^{({\\rm SU}(N-4))}_{2\\ell ;22} = \\frac{1}{6(N-4)}\\bigg [ 68(N-4)^3 -N_f(N-1)(29N^2-229N+440) \\bigg ] \\ .$ Both of these two-loop coefficients for the non-Abelian gauge couplings are positive for small $N_f$ and decrease with increasing $N_f$ , eventually passing through zero to negative values.",
"We denote the values of $N$ (formally generalized from positive integers $N \\ge 6$ to positive real numbers) at which $b^{({\\rm SU}(N))}_{2\\ell ;11}$ and $b^{({\\rm SU}(N-4))}_{2\\ell ;22}$ pass through zero as $N_{f,b2z}^{({\\rm SU}(N))}$ and $N_{f,b2z}^{({\\rm SU}(N-4))}$ .",
"These are $N_{f,b2z}^{({\\rm SU}(N))} = \\frac{68N^3}{(N-3)(29N^2-3N-12)}$ and $N_{f,b2z}^{({\\rm SU}(N-4))} = \\frac{68(N-4)^3}{(N-1)(29N^2-229N+440)} \\ .$ As $N \\rightarrow \\infty $ , $N_{f,b2z}^{({\\rm SU}(N))}$ approaches $68/29 = 2.34483$ from above, while $N_{f,b2z}^{({\\rm SU}(N-4))}$ approaches the same value from below.",
"With these inputs, we can investigate the presence or absence of an IR zero in the respective two-loop beta functions for the SU($N$ ) and SU($N-4$ ) theories.",
"The two-loop beta function for SU($N$ ) has no IR zero for $N_f=1$ or $N_f=2$ ; it does have an IR zero for higher values of $N_f$ , as allowed by the asymptotic freedom requirement for a fixed $N$ .",
"With a given $N$ , for the range of $N_f$ such that $b^{({\\rm SU}(N))}_{1\\ell ,1} > 0$ and $b^{({\\rm SU}(N))}_{2\\ell ;11} < 0$ , the IR zero of the two-loop SU($N$ ) beta function occurs at $\\alpha _{1,IR,2\\ell } &=& \\frac{8\\pi N[11N-2N_f(N-3)]}{N_f(N-3)(29N^2-3N-12)-68N^3} \\ .",
"\\cr \\cr &&$ Similarly, given a value of $N$ , for the range of $N_f$ such that $b^{({\\rm SU}(N-4))}_{1\\ell ,2} > 0$ , while $b^{({\\rm SU}(N-4))}_{2\\ell ;22} < 0$ , the two-loop SU($N-4$ ) beta function has an IR zero at $\\alpha _{2,IR,2\\ell } &=& \\frac{8\\pi (N-4)[11(N-4)-2N_f(N-1)]}{N_f(N-1)(29N^2-229N+440)-68(N-4)^3} \\ .",
"\\cr \\cr &&$ As $N \\rightarrow \\infty $ , the rescaled IRFP values of the SU($N$ ) and SU($N-4$ ) gauge interactions have the same limit: $\\lim _{N \\rightarrow \\infty } \\alpha _{1,IR,2\\ell }N &=&\\lim _{N \\rightarrow \\infty } \\alpha _{2,IR,2\\ell }N \\cr \\cr \\ &=& \\frac{8\\pi (11-2N_f)}{29N_f-68} \\ .$ We will analyze the UV to IR evolution using these beta functions below."
],
[
"Global Flavor Symmetries",
"The theory (REF ) with (REF ) has the classical global flavor (cgb) symmetry $G_{cgb} = {\\rm U}(N_f)_{\\psi } \\otimes {\\rm U}(N_f)_{\\chi }\\otimes {\\rm U}(N_f)_{\\omega } \\ ,$ where, for each fermion $f=\\psi ^{ij}_{p,L}$ , $\\chi _{i,\\alpha ,p,L}$ , and ${\\vec{\\omega }}_{p,L}$ , the elements of the group ${\\rm U}(N_f)_f$ act on the flavor indices $p$ , leaving all gauge indices unchanged.",
"Each ${\\rm U}(N_f)_f$ factor group in (REF ) can equivalently be written as ${\\rm SU}(N_f)_f \\otimes {\\rm U}(1)_f$ .",
"The instantons present in the SU($N$ ) gauge sector break both of the global abelian symmetries U(1)$_\\psi $ and U(1)$_\\chi $ .",
"Separately, the instantons in the SU($N-4$ ) gauge sector break both the U(1)$_\\chi $ and U(1)$_{\\omega }$ symmetries.",
"There are two special cases that will be of particular interest, namely the respective cases in which one non-Abelian gauge interaction is much stronger than the other.",
"First, let us consider the case in which the SU($N$ ) gauge interaction is much stronger than the SU($N-4$ ) gauge interaction, which, like the U(1) interaction, is weakly coupled.",
"In this theory, the effects of instantons in the SU($N-4$ ) gauge sector are exponentially suppressed and can be neglected [23].",
"Although the SU($N$ ) instantons break the global U(1)$_\\psi $ and U(1)$_\\chi $ flavor symmetries, one can construct a current which is a linear combination of the U(1)$_\\psi $ and U(1)$\\chi $ currents and is conserved in the presence of the SU($N$ ) instantons (see, e.g., Section V of [24]), which we denote as U(1)$_{\\psi \\chi }$ .",
"The effective non-anomalous global flavor (gb) symmetry of this theory is thus $G_{gb}={\\rm SU}(N_f)_\\psi \\otimes {\\rm SU}(N_f)_\\chi \\otimes {\\rm U}(1)_{\\psi \\chi } \\otimes {\\rm U}(N_f)_\\omega $ .",
"Similarly, in the other case, in which the SU($N$ ) and U(1) gauge interactions are weak, and the SU($N-4$ ) gauge interaction is strong, the effects of SU($N$ ) instantons are exponentially suppressed and are negligible.",
"Although the SU($N-4$ ) instantons break the global U(1)$_\\omega $ and U(1)$_\\chi $ flavor symmetries, one can construct a current which is a linear combination of the U(1)$_\\omega $ and U(1)$\\chi $ currents and is conserved in the presence of the SU($N$ ) instantons, which we denote as U(1)$_{\\omega \\chi }$ .",
"The effective non-anomalous global flavor symmetry of this theory is thus $G_{gb} = {\\rm SU}(N_f)_\\omega \\otimes {\\rm SU}(N_f)_\\chi \\times {\\rm U}(1)_{\\omega \\chi } \\otimes {\\rm U}(N_f)_\\psi $ ."
],
[
"UV to IR Evolution and Fermion Condensates",
"We next discuss the UV to IR evolution of this theory and the general analysis of possible fermion condensate formation in various channels.",
"We begin with the two respective cases in which one of the two non-Abelian gauge interactions is much stronger than the other and then remark on the case where both are present with comparable strength.",
"Let us denote the dominant coupling as $\\alpha _i(\\mu )$ .",
"As the reference scale $\\mu $ decreases below $\\mu _{_{UV}}$ , the coupling $\\alpha _i(\\mu )$ for this interaction increases.",
"There are two general possibilities for the associated beta function, $\\beta _{\\alpha _i}$ : (i) it does not have an IR zero or (ii) it has an IR zero.",
"In the first case, (i), the coupling continues to increase with decreasing $\\mu $ until it eventually exceeds the range where it can be calculated with the perturbative beta function.",
"This can then lead to the formation of (bilinear) fermion condensates.",
"In the second case, let us denote the value of $\\alpha _i$ at this IR zero as $\\alpha _{IR}$ , and consider a possible condensation channel, $R \\times R^{\\prime } \\rightarrow R_c \\ ,$ where $R$ and $R^{\\prime }$ denote fermion representations under the strongly coupled gauge symmetry $G_i$ , and $R_c$ denotes the representation of the condensate under $G_i$ .",
"Assuming that this is an attractive channel, we denote the minimal critical coupling for condensation in this channel as $\\alpha _{cr}$ .",
"If the beta function does not have an IR zero, then $\\alpha _i$ will certainly exceed $\\alpha _{cr}$ as $\\mu $ decreases to some scale.",
"If the beta function $\\beta _{\\alpha _i}$ does have an IR zero, then there are two subcases: (iia) $\\alpha _{IR} \\ge \\alpha _{cr}$ and (iib) $\\alpha _{IR} < \\alpha _{cr}$ .",
"In case (iia), the condensate can form, similarly to case (i), while in case (iib), this condensate will not form.",
"For the possible condensation channel (REF ), an approximate measure of its attractiveness (motivated by iterated one-gluon exchange) is $\\Delta C_2 = C(R) + C(R^{\\prime }) - C(R_c) \\ ,$ where $C_2(R)$ is the quadratic Casimir invariant for the representation $R$ [25].",
"Among several possible fermion condensation channels, the one with the largest (positive) value of $\\Delta C_2$ is commonly termed the most attractive channel (MAC) and is the one that is expected to occur.",
"Approximate solutions of Schwinger-Dyson equations for the fermion propagator in a vectorial theory have shown that if one starts with a massless fermion, it follows that if $\\alpha > \\alpha _{cr}$ , where $3 \\alpha _{cr} C_2(R)/\\pi =1$ , then the Schwinger-Dyson equation has a solution with a dynamically generated mass, indicating spontaneous chiral symmetry breaking and associated bilinear fermion condensate formation [26].",
"In a vectorial gauge theory such as quantum chromodynamics, the condensate is a gauge-singlet, so $\\Delta C_2 = 2C_2(R)$ .",
"Hence, one can write the condition for the critical coupling in the form that can be taken over for a chiral gauge theory, namely $3 \\alpha _{cr}\\Delta C_2/(2\\pi )=1$ , so that $\\alpha _{cr} = \\frac{2\\pi }{3\\Delta C_2 } \\ .$ Because this is based on a rough approximation (an iterated one-gluon exchange approximation to the Schwinger-Dyson equation), it is used only as a rough estimate.",
"Since without loss of generality we write all fermions as left-handed, the Lorentz-invariant bilinears involving two fermion fields $f_L$ and $f^{\\prime }_L$ are of the form $f^T_L C f^{\\prime }_L$ , where $C$ is the Dirac charge-conjugation matrix satisfying $C\\gamma _\\mu C^{-1} = -(\\gamma _\\mu )^T$ .",
"If $f_L$ and $f^{\\prime }_L$ transform according to the same representation $R_1$ of a symmetry group $G_1$ and $R_2$ of a symmetry group $G_2$ , then we may write the bilinear fermion operator product abstractly as $f^T_{{\\cal R},p,L} C f_{{\\cal R},p^{\\prime },L} \\ ,$ where gauge group indices are suppressed in the notation, ${\\cal R}$ denotes the representations under the gauge groups, and, as before, $p$ and $p^{\\prime }$ are flavor indices.",
"From the property $C^T = -C$ together with the anticommutativity of fermion fields, it follows that the bilinear fermion operator product (REF ) is symmetric under interchange of the order of fermion fields and therefore is symmetric in the overall product $\\prod _{i} (R_i \\times R_i)] \\, R_{fl} \\ ,$ where $R_{fl}$ abstractly denotes the symmetry property under interchange of flavors [13].",
"For our theory, with its two non-Abelian groups, this means that the fermion bilinears are of the form $(s,s,s), \\quad (s,a,a), \\quad (a,s,a), \\ \\ {\\rm or} \\ \\ (a,a,s) \\ ,$ where here $s$ and $a$ indicate symmetric and antisymmetric, and the three entries refer to the representations $R_1$ of $G_1$ , $R_2$ of $G_2$ , and $R_{fl}$ .",
"If, as $\\mu $ decreases through a scale $\\Lambda _1$ and the coupling $\\alpha _i(\\mu )$ of the strongly coupled gauge interaction corresponding to the factor group $G_i$ increases beyond $\\alpha _{cr}$ for the condensation channel (REF ) and the condensate forms, then the fermions involved in the condensate gain dynamical masses of order $\\Lambda _1$ and are integrated out of the low-energy effective field theory that describes the physics as $\\mu $ decreases below $\\Lambda _1$ .",
"If this condensate either self-breaks the $G_i$ symmetry or produces induced breaking of a weakly coupled gauge symmetry $G_j$ to a respective subgroup $H_i \\subset G_i$ or $H_j \\subset G_j$ , then the gauge bosons in the respective coset spaces $G_i/H_i$ or $G_j/H_j$ pick up dynamical masses of order $g_i(\\Lambda _1) \\Lambda _1$ or $g_j(\\Lambda _1) \\, \\Lambda _1$ , respectively.",
"Hence, like the fermions with dynamically generated masses, these now massive vector bosons are integrated out of the low-energy effective field theory applicable as $\\mu $ decreases below $\\Lambda $ .",
"In this section we analyze possible fermion condensation channels in the general-$N$ theory (REF ) with fermion content (REF )."
],
[
"${\\rm SU}(N)$ Gauge Interaction Dominant ",
"We begin by focusing on the case where the SU($N$ ) gauge interaction is much stronger than the SU($N-4$ ) (and U(1)) gauge interactions.",
"This theory is labelled SUND, standing for “SU($N$ )-dominant”.",
"Although we keep $\\alpha _2$ and $\\alpha _3$ nonzero, we note parenthetically that if one were to set $\\alpha _2=\\alpha _3=0$ , then the resultant theory would be the $k=2$ special case of a family of chiral gauge theories analyzed in Ref.",
"[24] with a single gauge group $G={\\rm SU}(N)$ and an anomaly-free content of chiral fermions transforming as $[k]_N$ and $n_{\\bar{F}}$ copies of $[\\bar{1}]_N$ , where $n_{\\bar{F}} = (N-3)!(N-2k)/[(N-k-1)!(k-1)!",
"]$ (plus SU($N$ )-singlet fermions).",
"Since the SU($N$ ) gauge interaction is asymptotically free, $\\alpha _1(\\mu )$ increases as $\\mu $ decreases from the initial reference scale $\\mu _{_{UV}}$ in the UV.",
"We focus on the subset of values of $N_f$ such that the beta function $\\beta _{\\alpha _1}$ either has no IR zero at the two-loop level or has an IRFP at a sufficiently large value that fermion condensation can take place.",
"There are three possible (bilinear) fermion condensation channels.",
"We give shorthand names to these based on the fermions involved in each condensate.",
"The first is the $\\psi \\chi $ channel, $\\psi \\chi : \\quad ([2]_N,1)_{N-4} \\times ([\\bar{1}]_N,[\\bar{1}]_{N-4})_{-(N-2)}\\rightarrow ([1]_N,[\\bar{1}]_{N-4})_{-2} \\ ,$ with associated condensate $\\langle \\psi ^{ij \\ T}_{p,L} C \\chi _{j, \\beta ,p^{\\prime },L} \\rangle \\ ,$ where $i,j$ are SU($N$ ) group indices and $\\beta $ is an SU($N-4$ ) group index.",
"The condensate (REF ) transforms as the fundamental ($[1]_N$ ) representation of SU($N$ ) and the conjugate fundamental ($[\\bar{1}]_{N-4}$ ) representation of SU($N-4$ ), so it self-breaks SU($N$ ) to SU($N-1$ ) and produces an induced breaking of the weakly coupled SU($N-4$ ) to SU($N-5$ ).",
"Since the condensate (REF ) has a nonzero U(1) charge, $q_{\\psi \\chi }=-2$ , it also breaks U(1).",
"Thus, here the residual gauge symmetry in the effective field theory that is applicable as $\\mu $ decreases below $\\Lambda _1$ is ${\\rm SU}(N-1) \\otimes {\\rm SU}(N-5) \\ .$ If $N=6$ , then the residual gauge symmetry is just SU(5).",
"For this channel we calculate $(\\Delta C_2)_{\\psi \\chi } = C_2([2]_N) = \\frac{(N-2)(N+1)}{N} \\ .$ For this and other possible fermion condensation channels, we record these properties in Table .",
"This table refers to the possible initial condensation patterns at the highest condensation scale; subsequent evolution further into the infrared is discussed below.",
"The second possible channel is the $\\psi \\psi $ channel, $\\psi \\psi : \\quad ([2]_N,1)_{N-4} \\times ([2]_N,1)_{N-4} \\rightarrow ([4]_N,1)_{2(N-4)} \\ ,$ Note that $[4]_N \\approx [\\overline{N-4}]_N$ , where $R \\approx R^{\\prime }$ means that the representations $R$ and $R^{\\prime }$ are equivalent.",
"The associated condensate is $\\epsilon _{...k\\ell mn} \\langle \\psi ^{k \\ell \\ T}_{p,L} C\\psi ^{m n}_{p^{\\prime },L} \\rangle \\ ,$ where the antisymmetric tensor $\\epsilon _{...k\\ell mn}$ has $N$ indices, four of which are indicated explicitly, with the rest implicit.",
"From the general group-theoretic analysis in [27], [24], it follows that since the condensate (REF ) transforms as a $[4]_N$ of SU($N$ ), it breaks SU($N$ ) to ${\\rm SU}(N-4) \\otimes {\\rm SU}(4)$ .",
"Since the $\\psi ^{k\\ell }_{p,L}$ are singlets under the original SU($N-4$ ) group in (REF ), this condensate is obviously invariant under this SU($N-4$ ).",
"Furthermore, since this condensate has a nonzero U(1) charge (namely, $q_{\\psi \\psi }=2(N-4)$ ), it breaks the U(1) gauge symmetry.",
"Hence, the condensate (REF ) breaks $G$ to $[{\\rm SU}(N-4) \\otimes {\\rm SU}(4)] \\otimes {\\rm SU}(N-4) \\ .$ where we have inserted brackets to distinguish the two different SU($N-4$ ) groups.",
"The measure of attractiveness for this condensation channel is $(\\Delta C_2)_{\\psi \\psi } = 2C_2([2]_N) - C_2([4]_N) = \\frac{4(N+1)}{N} \\ .$ The third possible channel is the $\\chi \\chi $ channel, $\\chi \\chi : \\quad ([\\bar{1}]_N, [\\bar{1}]_{N-4})_{-(N-2)} \\times ([\\bar{1}]_N, [\\bar{1}]_{N-4})_{-(N-2)} \\rightarrow ([\\bar{2}]_N,[\\bar{2}]_{N-4})_{-2(N-2)} \\ ,$ with associated condensate $\\epsilon ^{...mn} \\epsilon ^{...\\alpha \\beta }\\langle \\chi _{m \\alpha ,p,L}^T C \\chi _{n,\\beta ,p^{\\prime },L} \\rangle \\ ,$ where $\\epsilon ^{...mn}$ and $\\epsilon ^{...\\alpha \\beta }$ are antisymmetric tensors under SU($N$ ) and SU($N-4$ ), respectively, with two indices shown explicitly and the rest understood implicitly.",
"From the general group-theoretic analysis [27], [24], it follows that since the condensate (REF ) transforms as a $[\\bar{2}]_N$ representation of SU($N$ ), it breaks SU($N$ ) to ${\\rm SU}(N-2) \\otimes {\\rm SU}(2)^{\\prime }$ , and similarly, since it transforms as a $[\\bar{2}]_{N-4}$ representation of SU($N-4$ ), it breaks SU($N-4$ ) to ${\\rm SU}(N-6) \\otimes {\\rm SU}(2)^{\\prime \\prime }$ .",
"Here we append a single prime to the first SU(2) and a double prime to the second SU(2) to distinguish them and also to disguish them from the SU(2) group of the $N=6$ theory (REF ).",
"Since the condensate (REF ) carries a nonzero U(1) charge (namely, $q_{\\chi \\chi }=-2(N-2)$ ), it breaks the U(1) gauge symmetry.",
"Thus, this condensate (REF ) breaks $G$ to the group $[{\\rm SU}(N-2) \\otimes {\\rm SU}(2)^{\\prime }] \\otimes [{\\rm SU}(N-6) \\otimes {\\rm SU}(2)^{\\prime \\prime }] \\ ,$ where the square brackets here are inserted to indicate the origin of the different factor groups from the original SU($N$ ) and SU($N-4$ ) factor groups in (REF ).",
"We find $(\\Delta C_2)_{\\chi \\chi } = 2C_2([\\bar{1}]_N) - C_2([\\bar{2}]_N) = \\frac{N+1}{N} \\ .$ From these results we calculate the relative attractiveness of these three possible fermion condensation channels in this SU($N$ )-dominant case.",
"We compute the differences $(\\Delta C_2)_{\\psi \\chi }-(\\Delta C_2)_{\\psi \\psi } = \\frac{(N-6)(N+1)}{N}$ and $(\\Delta C_2)_{\\psi \\psi }-(\\Delta C_2)_{\\chi \\chi } = \\frac{3(N+1)}{N} \\ ,$ whence $(\\Delta C_2)_{\\psi \\chi }-(\\Delta C_2)_{\\chi \\chi } = \\frac{(N-3)(N+1)}{N} \\ ,$ and the ratios $\\frac{(\\Delta C_2)_{\\psi \\chi }}{(\\Delta C_2)_{\\psi \\psi }}= \\frac{N-2}{4}$ and $\\frac{(\\Delta C_2)_{\\psi \\psi }}{(\\Delta C_2)_{\\chi \\chi }} = 4 \\ ,$ whence $\\frac{(\\Delta C_2)_{\\psi \\chi }}{(\\Delta C_2)_{\\chi \\chi }} = N-2 \\ .$ Therefore, in this SU($N$ )-dominant case with $N \\ge 7$ , the $\\psi \\chi $ channel is the MAC, with greater attractiveness than the $\\psi \\psi $ channel, which, in turn, is more attractive than the $\\chi \\chi $ channel.",
"Summarizing: ${\\rm SU}(N)-{\\rm dominant \\ with} \\ N \\ge 7 \\ \\Longrightarrow \\ \\psi \\chi \\ {\\rm channel \\ is \\ the \\ MAC}.$ One interesting feature of these comparisons is that the ratio $(\\Delta C_2)_{\\psi \\psi }/(\\Delta C_2)_{\\chi \\chi }$ is independent of $N$ .",
"As is evident from these results, in the lowest non-degenerate case, namely $N=6$ , the $\\psi \\chi $ and $\\psi \\psi $ channels are equally attractive, and are a factor 4 more attractive than the $\\chi \\chi $ channel.",
"Thus, ${\\rm SU}(N)-{\\rm dominant \\ with} \\ N =6 \\ \\Longrightarrow \\ \\psi \\chi \\ {\\rm and} \\ \\psi \\psi \\ {\\rm channels \\ are \\ the \\ MACs}.$ We focus here on the range $N \\ge 7$ ; a detailed analysis of the $N=6$ case will be given below.",
"Since the $\\psi \\chi $ channel is the MAC, it is expected that as the Euclidean reference scale $\\mu $ decreases below a value that we denote as $\\Lambda _1$ , the coupling $\\alpha _1(\\mu )$ increases sufficiently to cause condensation in this channel.",
"This condensation self-breaks SU($N$ ) to SU($N-1$ ) and breaks the weakly coupled gauge symmetry SU($N-4$ ) to SU($N-5$ ) and also the abelian symmetry U(1).",
"Without loss of generality, we may choose the SU($N$ ) group index $i$ in the condensate (REF ) to be $i=N$ and the SU($N-4$ ) group index to be $\\alpha =N-4$ .",
"The condensate (REF ) also spontaneously breaks the global flavor group $G_{gb}$ for this theory, producing a set of Nambu-Goldstone bosons (NGBs).",
"Earlier works in related chiral gauge theories have studied the resultant change in counts of the UV versus IR degrees of freedom [24], [28]-[31].",
"Here we focus on the dynamical self-breaking and induced breaking of gauge symmetries, together with the construction of resultant low-energy effective field theories.",
"The fermions involved in the condensate (REF ), namely $\\psi ^{Nj}_{p,L} = -\\psi ^{jN}_{p,L}$ with $1 \\le j \\le N-1$ and $\\chi _{j,N-4,p^{\\prime },L}$ with $1 \\le j \\le N-1$ , gain dynamical masses of order $\\Lambda _1$ .",
"The $2N-1$ SU($N$ ) gauge bosons in the coset space ${\\rm SU}(N)/{\\rm SU}(N-1)$ gain dynamical masses of order $g_1(\\Lambda _1)\\Lambda _1$ , while the $(2N-9)$ SU($N-4$ ) gauge bosons in the coset space ${\\rm SU}(N-4)/{\\rm SU}(N-5)$ gain dynamical masses of order $g_2(\\Lambda _1)\\Lambda _1$ .",
"Finally, the U(1) gauge boson picks up a dynamical mass of order $g_3(\\Lambda _1) \\Lambda _1$ .",
"These massive fields are integrated out of the low-energy effective field theory that describes the physics as the reference scale $\\mu $ decreases below $\\Lambda _1$ .",
"This low-energy effective field theory that is applicable as $\\mu $ decreases below $\\Lambda _1$ is invariant under the gauge symmetry (REF ).",
"The massless gauge-nonsinglet fermion content of this EFT consists of $\\psi ^{ij}_{p,L}$ with $1 \\le i,j \\le N-1$ , $1 \\le p \\le N_f$ , forming $N_f$ copies of a $([2]_{N-1},1)$ representation under the group (REF ), $\\chi _{j,\\beta ,p^{\\prime },L}$ with $1 \\le j \\le N-1$ , $1 \\le \\beta \\le N-5$ , and $1 \\le p^{\\prime } \\le N_f$ , comprising $N_f$ copies of the $([\\bar{1}]_{N-1},[\\bar{1}]_{N-5})$ representation of (REF ), and $\\omega ^{\\alpha \\beta }_{p,L}$ with $1 \\le \\alpha , \\beta \\le N-5$ and $1 \\le p \\le N_f$ , comprising $N_f$ copies of $(1,(2)_{N-5})$ .",
"(We do not list the U(1) charges, since there is no U(1) gauge symmetry in this low-energy effective theory.)",
"The condensation process then repeats, with the $\\psi \\chi $ condensation channel again being the MAC in this ${\\rm SU}(N-1)\\otimes {\\rm SU}(N-5)$ theory.",
"One can treat the successive self-breakings and induced dynamical breakings iteratively at the various steps."
],
[
"${\\rm SU}(N-4)$ Gauge Interaction Dominant, {{formula:41c52770-7b07-4879-9779-43c549615db3}}",
"Here we analyze the case in which the SU($N-4$ ) gauge interaction is strongly coupled and dominates over the SU($N$ ) gauge interaction (as well as the weakly coupled U(1) gauge interaction).",
"We restrict to the range $N \\ge 7$ here and will consider the $N=6$ in detail below.",
"It will sometimes be convenient to use the quantity $M=N-4$ as defined before.",
"We will denote this theory as SUMD, standing for “SU($M$ )-dominant”.",
"If we were to completely turn off the SU($N$ ) and U(1) gauge interactions, then this theory would be equivalent to a chiral gauge theory with an SU($M$ ) gauge group, and $N_f$ flavors of chiral fermions transforming according to the anomaly-free set $(2)_M$ + $M+4$ copies of $[\\bar{1}]_M$ , which has been studied in [28]-[31].",
"However, here we do not completely turn off the SU($N$ ) or U(1) gauge interactions.",
"There are two possible (bilinear) fermion condensation channels.",
"The first is $\\chi \\omega : \\quad && ([\\bar{1}]_N, [\\bar{1}]_{N-4})_{-(N-2)} \\times (1,(2)_{N-4})_N \\rightarrow \\cr \\cr &&\\rightarrow ([\\bar{1}]_N,[1]_{N-4})_2 \\ ,$ with associated condensate $\\langle \\chi _{i,\\alpha ,p,L}^T C \\omega ^{\\alpha \\beta }_{p^{\\prime },L} \\rangle \\ ,$ where $i$ is an SU($N$ ) group index and $\\alpha , \\ \\beta $ are ${\\rm SU}(N-4)$ group indices.",
"The value of $\\Delta C_2$ for this condensation, as produced by the SU($N-4$ ) gauge interaction, is $(\\Delta C_2)_{\\chi \\omega ,SUMD} &=&C_2((2)_{N-4}) = \\frac{(N-2)(N-5)}{N-4} \\cr \\cr && {\\rm in} \\ {\\rm SU}(N-4) \\ .$ This condensate transforms as $([\\bar{1}]_N,[1]_{N-4})_2$ and hence self-breaks SU($N-4$ ) to SU($N-5$ ) and produces induced breaking of the weakly coupled symmetries SU($N$ ) to SU($N-1$ ) and of U(1).",
"It leaves invariant the same residual gauge symmetry, (REF ), as the $\\psi \\chi $ condensate (REF ), which is the MAC for the SU($N$ )-dominant case (REF ).",
"By convention, one may choose the SU($N-4$ ) index $\\beta $ in the condensate (REF ) to be $\\beta =N-4$ and the SU($N$ ) index $i$ to be $i=N$ .",
"Then the fermions $\\chi _{N,\\alpha ,p,L}$ and $\\omega ^{\\alpha ,N-4}_{p^{\\prime },L}$ with $1 \\le \\alpha \\le N-4$ , $1 \\le p,p^{\\prime } \\le N_f$ involved in the condensate pick up dynamical masses of order $\\Lambda _1$ .",
"The dynamical mass generation for the SU($N$ ) and SU($N-4$ ) gauge bosons in the respective coset spaces ${\\rm SU}(N)/{\\rm SU}(N-1)$ and ${\\rm SU}(N-4)/{\\rm SU}(N-5)$ is the same as described above in the SU($N$ )-dominant scenario, as is the dynamical mass generation for the U(1) gauge boson.",
"A second possible condensation channel is the $\\chi \\chi $ channel (REF ), with associated condensate (REF ).",
"This condensate breaks $G$ to the group given above in Eq.",
"(REF ).",
"The measure of attractiveness of this condensation channel, as produced by the ${\\rm SU}(M)={\\rm SU}(N-4)$ gauge interaction, is $(\\Delta C_2)_{\\chi \\chi } &=& 2C_2([\\bar{1}]_M) - C_2([\\bar{2}]_M) = \\frac{M+1}{M}\\cr \\cr &=& \\frac{N-3}{N-4} \\ .$ Comparing the attractiveness measure of the channels (REF ) and (REF ), we calculate the difference $(\\Delta C_2)_{\\chi \\omega } - (\\Delta C_2)_{\\chi \\chi } = \\frac{N^2-8N+13}{N-4}$ and the ratio $\\frac{(\\Delta C_2)_{\\chi \\omega }}{(\\Delta C_2)_{\\chi \\chi }} =\\frac{(N-2)(N-5)}{N-3} \\ .$ For the range $N \\ge 6$ , the difference $(\\Delta C_2)_{\\chi \\omega } - (\\Delta C_2)_{\\chi \\chi }$ is positive and, equivalently, the ratio $(\\Delta C_2)_{\\chi \\omega }/(\\Delta C_2)_{\\chi \\chi } > 1$ .",
"Hence, the $\\chi \\omega $ channel is always more attractive than the $\\chi \\chi $ channel in this SU($N-4$ )-dominant case.",
"Thus, ${\\rm SU(N-4)-dominant \\ with} \\ N \\ge 7: \\ \\Longrightarrow \\ \\chi \\omega \\ {\\rm channel \\ is \\ the \\ MAC}.$ In addition to breaking gauge symmetries, the MAC condensate (REF ) spontaneously breaks the global symmetry $G_{gb}$ for this theory, yielding a set of NGBs.",
"Here we focus on the gauge symmetry breaking.",
"We have restricted to the range $N \\ge 7$ ; as will be discussed below, the MAC is different in the special case $N=6$ , where it is the $\\omega \\omega $ channel.",
"Although the $\\chi \\chi $ channel is not the MAC, we comment on its symmetry properties.",
"It breaks SU($N-4$ ) to ${\\rm SU}(N-6) \\otimes {\\rm SU}(2)$ and also breaks U(1), since the condensate has nonzero U(1) charge $q_{\\chi \\chi }=-2(N-2)$ .",
"In terms of SU($N-4$ ), the associated condensate has the form $\\epsilon ^{...\\alpha \\beta }\\langle \\chi _{i,\\alpha ,p,L}^T C \\chi _{j,\\beta ,p^{\\prime },L}\\rangle \\ ,$ where $\\epsilon ^{...\\alpha \\beta }$ is an antisymmetric SU($N-4$ ) tensor and we have indicated $N-6$ of the indices implicitly with dots.",
"For this $\\chi \\chi $ channel, as regards the SU($N$ ) and flavor symmetry, there are two channels and corresponding condensates.",
"The (REF ) channel that involves an antisymmetric structure for SU($N$ ) group indices is $&& ([\\bar{1}]_N, [\\bar{1}]_{N-4})_{-(N-2)} \\times ([\\bar{1}]_N,[\\bar{1}]_{N-4})_{-(N-2)} \\rightarrow \\cr \\cr && \\rightarrow ([\\bar{2}]_N,[\\bar{2}]_{N-4})_{-2(N-2)} \\ ,$ with corresponding condensate $\\epsilon ^{...mn} \\epsilon ^{...\\alpha \\beta }\\langle \\chi _{m,\\alpha ,p,L}^T C \\chi _{n,\\beta ,p^{\\prime },L}\\rangle \\ .$ Here $\\epsilon ^{...mn}$ is an antisymmetric tensor under SU($N$ ), $\\epsilon ^{...\\alpha \\beta }$ was defined, and we indicate the rest of the indices in each tensor implicitly with dots.",
"This condensate is automatically symmetrized in the flavor indices $p$ and $p^{\\prime }$ and is of the form $(a,a,s)$ in the classification of Ref.",
"[13].",
"The (REF ) channel that involves a symmetric structure for SU($N$ ) group indices is $&& ([\\bar{1}]_N, [\\bar{1}]_{N-4})_{-(N-2)} \\times ([\\bar{1}]_N,[\\bar{1}]_{N-4})_{-(N-2)} \\rightarrow \\cr \\cr && \\rightarrow ((\\bar{2})_N,[\\bar{2}]_{N-4})_{-2(N-2)} \\ ,$ with corresponding condensate $\\epsilon ^{...\\alpha \\beta }\\langle \\chi _{i,\\alpha ,p,L}^T C \\chi _{j,\\beta ,p^{\\prime },L}\\rangle - (p \\leftrightarrow p^{\\prime }) \\ .$ Because this condensate is antisymmetrized in flavor indices, it is automatically symmetric in SU($N$ ) group indices and is thus of the form $(s,a,a)$ in the classification of Ref.",
"[13]."
],
[
"SU($N$ ) and SU({{formula:7b1e2f36-bdd2-466d-872e-9f9ec6672887}} ) Gauge Interactions of Comparable Strength",
"Finally, we analyze the situation in which the SU($N$ ) and SU($N-4$ ) gauge interactions are of comparable strength at the scale relevant for the initial condensation.",
"We restrict to $N \\ge 7$ here and will discuss the $N=6$ theory below.",
"The value of $\\Delta C_2$ for the most attractive channel, $\\psi \\chi $ , in the SU($N$ )-dominant case was given in Eq.",
"(REF ), and the value of $\\Delta C_2$ for the MAC $\\chi \\omega $ in the SU($N-4$ )-dominant case was given in Eq.",
"(REF ) above.",
"The difference is $&& (\\Delta C_2)_{\\psi \\chi , SUND} - (\\Delta C_2)_{\\chi \\omega , SUMD} =\\frac{4(N-2)}{N(N-4)} \\ .",
"\\cr \\cr &&$ Since this is positive for the relevant range of $N$ considered here, it follows that, as the reference scale decreases and the SU($N$ ) and SU($N-4$ ) couplings increase, the minimal value of $\\alpha $ for condensation is reached first for the SU($N$ )-induced $\\psi \\chi $ condensate, at a scale $\\mu $ that we may again denote $\\Lambda _1$ , where $\\alpha _1(\\Lambda _1)$ exceeds $\\alpha _{cr}$ for the $\\psi \\chi $ condensation.",
"At a slightly lower scale, $\\Lambda _1^{\\prime } \\mathrel {\\raisebox {-.6ex}{\\stackrel{\\textstyle <}{\\sim }}}\\Lambda _1$ , the SU($N-4$ ) gauge interaction, of comparable strength, increases through the slightly larger critical value for condensation in the $\\chi \\omega $ channel.",
"These condensates both break the gauge symmetry in the same way, to the residual subgroup ${\\rm SU}(N-1) \\otimes {\\rm SU}(N-5)$ , as given in Eq.",
"(REF ).",
"We have described the fermions and gauge bosons that gain dynamical masses from the $\\psi \\chi $ and $\\chi \\omega $ condensations above, and we combine these results here By convention, one may choose the SU($N$ ) index $i$ and the SU($N-4$ ) index $\\alpha $ in the $\\psi \\chi $ condensate $\\langle \\psi ^{ij \\ T}_{p,L} C \\chi _{j,\\alpha ,p^{\\prime },L}\\rangle $ in Eq.",
"(REF ) to be $i=N$ and $\\alpha =N-4$ , respectively.",
"The fermions involved in this condensate are then $\\psi ^{Nj}_{p,L}$ and $\\chi _{j,N-4,p^{\\prime },L}$ with $1 \\le j \\le N-1$ .",
"These gain dynamical masses of order $\\Lambda _1$ .",
"The $2N-1$ SU($N$ ) gauge bosons in the coset ${\\rm SU}(N)/{\\rm SU}(N-1)$ and the $2M-1=2N-9$ SU($M$ ) gauge bosons in the coset ${\\rm SU}(M)/{\\rm SU}(M-1)={\\rm SU}(N-4)/{\\rm SU}(N-5)$ gain dynamical masses of order $\\simeq g_1(\\Lambda _1)\\Lambda _1$ and $\\simeq g_2(\\Lambda _1)\\Lambda _1$ , while the U(1) gauge boson gains a dynamical mass $\\simeq g_3(\\Lambda _1)\\Lambda _1$ .",
"A vacuum alignment argument [1], [32] suggests that the condensation process would be such as to preserve the maximal residual gauge symmetry, with gauge group of the largest order, thereby minimizing the number of gauge bosons that pick up masses.",
"In the present case, one can use this argument to infer that in the condensate $\\langle \\chi _{i,\\alpha ,p,L}^T C\\omega ^{\\alpha \\beta }_{p^{\\prime },L}\\rangle $ in Eq.",
"(REF ), the SU($N$ ) index is the same as the unmatched index in the $\\langle \\psi ^{ij \\ T}_{p,L} C \\chi _{j,\\alpha ,p^{\\prime },L}\\rangle $ condensate, namely $i=N$ , and the $\\beta $ index is the same as the unmatched SU($N-4$ ) index $\\alpha $ in the $\\psi \\chi $ condensate, namely $N-4$ , so that these two condensates break the initial UV gauge group $G$ in the same way, to the subgroup ${\\rm SU}(N-1)\\otimes {\\rm SU}(N-5)$ in Eq.",
"(REF ).",
"Then the fermions involved in the $\\chi \\omega $ condensate, $\\chi _{N,\\alpha ,p,L}$ and $\\omega ^{\\alpha \\beta }_{p^{\\prime },L}$ with $1 \\le \\alpha \\le N-4$ and $\\beta =N-4$ , gain dynamical masses of order $\\Lambda _1$ and $\\Lambda _1^{\\prime }$ .",
"The resultant low-energy effective field theory that describes the physics as the reference scale $\\mu $ decreases below $\\Lambda _1^{\\prime }$ contains the following massless fermions that are nonsinglets under the residual gauge group ${\\rm SU}(N-1) \\otimes {\\rm SU}(N-5)$ : $\\psi ^{ij}_{p,L}$ with $1 \\le i,j \\le N-1$ , $1 \\le p \\le N_f$ , forming $N_f$ copies of a $([2]_{N-1},1)$ representation under the group (REF ), $\\chi _{j,\\alpha ,p^{\\prime },L}$ with $1 \\le j \\le N-1$ , $1 \\le \\beta \\le N-5$ , and $1 \\le p^{\\prime } \\le N_f$ , forming $N_f$ copies of the $([\\bar{1}]_{N-1},[\\bar{1}]_{N-5})$ representation of (REF ), and $\\omega ^{\\alpha \\beta }_{p^{\\prime },L}$ with $1 \\le \\alpha , \\ \\beta \\le N-5$ , forming $N_f$ copies of the $(1,(2)_{N-5})$ representation of the group (REF ) This theory also includes certain massless fermions that are singlets under the gauge group (REF ), e.g., $\\chi _{N,N-4,p,L}$ ."
],
[
"Beta Function and Constraints on $N_f$",
"In this section we study the lowest nondegenerate case of the chiral gauge theory (REF ) with the fermions (REF ), namely the $N=6$ theory, for which the fermion content was given in Eq.",
"(REF ).",
"From the general formulas (REF ) and (REF ), it follows that the one-loop coefficients for the SU(6) and SU(2) gauge interactions in this theory are $b^{({\\rm SU}(6))}_{1\\ell ,1} =2(11-N_f)$ and $b^{({\\rm SU}(2))}_{1\\ell ,2} = \\frac{2}{3}\\Big (11-5N_f \\Big ) \\ .$ Substituting $N=6$ into the upper bound on $N_f$ in Eq.",
"(REF ), we find that $N_f < 11/5$ , i.e., for physical integral values, $N=6 \\ \\Longrightarrow \\ N_f=1, \\ 2 \\ ,$ in accord with the general result given in Section .",
"When discussing the $N_f=1$ case, we will suppress the flavor indices in the notation, since they are all the same.",
"For the study of the UV to IR evolution of this theory, we substitute $N=6$ into the general formulas (REF ) and (REF ) to obtain the two-loop coefficients in the SU(6) and SU(2) beta functions, which are $b^{({\\rm SU}(6))}_{2\\ell ;11} = \\frac{1}{2}\\Big ( 816-169N_f \\Big )$ and $b^{({\\rm SU}(2))}_{2\\ell ;22} = \\frac{1}{6}\\Big ( 272 - 275N_f \\Big ) \\ .$ As before, it is natural to begin by analyzing the UV to IR evolution in the case where one non-Abelian gauge interaction is much stronger than the other.",
"We start with the situation in which the SU(2) gauge interaction is much stronger than the SU(6) interaction, so that, to first approximation, we may treat the SU(6) (as well as U(1)) gauge interaction perturbatively.",
"By analogy with our notation above, this will be denoted as the SU2D case, where again, D stands for “dominant”.",
"Then, since $b^{({\\rm SU}(2))}_{1\\ell ,2} > 0$ while $b^{({\\rm SU}(2))}_{2\\ell ,22} < 0$ , the two-loop beta function $\\beta _{\\alpha _2}$ for the SU(2) gauge interaction has an IR zero at $\\alpha _{2,IR,2\\ell } &=& -\\frac{4\\pi b^{({\\rm SU}(2))}_{1\\ell ,2}}{b^{({\\rm SU}(2))}_{2\\ell ;22}} \\cr \\cr &=& \\frac{16\\pi (11-5N_f)}{275N_f-272} \\ .$ For $N_f=1$ , $\\alpha _{2,IR,2\\ell }=32\\pi = 100.5$ , while for $N_f=2$ , $\\alpha _{2,IR,2\\ell }=8\\pi /139 = 0.181$ .",
"The IRFP value for $N_f=1$ is too large for the two-loop calculation to be considered to be quantitatively accurate, but it does indicate that the theory becomes strongly coupled in the IR.",
"The IRFP value for $N_f=2$ is considerably smaller than the estimates of the critical values $\\alpha _{cr}$ for any of the three attractive condensation channels (which will be given below).",
"Hence, this theory with $N_f=2$ is expected to evolve in the IR limit to an exact IR fixed point (IRFP) in a scale-invariant and conformally-invariant non-Abelian Coulomb phase (NACP), without any spontaneous chiral symmetry breaking or associated fermion condensate formation.",
"We therefore focus on the $N_f=1$ case.",
"Since the flavor subscripts $p, p^{\\prime }$ are always equal to 1, they are suppressed in the notation."
],
[
"Condensation at Scale $\\Lambda _1$ ",
"We proceed to determine the most attractive channel for the formation of bilinear condensates of SU(2)-nonsinglet fermions in this $N_f=1$ case.",
"There are, a priori, several possible channels.",
"The first is $\\omega \\omega : \\quad (1,Adj)_6 \\times (1,Adj)_6 \\rightarrow (1,1)_{12} \\ ,$ where $Adj$ is the adjoint (triplet) representation of SU(2) and the notation follows Eq.",
"(REF ).",
"The shorthand name for this channel, $\\omega \\omega $ , follows from the condensate, which is $\\langle {\\vec{\\omega }}^T_{L} C \\, \\cdot {\\vec{\\omega }}_{L}\\rangle \\ .$ In terms of dimensions of the SU(2) representations, this channel has the form $3 \\times 3 \\rightarrow 1$ .",
"The measure of attractiveness of this channel due to the strongly coupled SU(2) gauge interaction is $\\Delta C_2 = 2C_2(Adj) = 4 \\quad {\\rm for} \\ 3 \\times 3 \\rightarrow 1 \\ {\\rm in} \\ {\\rm SU}(2) \\ .$ This is the most attractive channel: ${\\rm SU}(2)-{\\rm dominant} \\ \\Longrightarrow \\ \\omega \\omega \\ {\\rm channel \\ is \\ the \\ MAC}.$ The rough estimate of the minimal (critical) coupling $\\alpha _2(\\mu )=\\alpha _{cr}$ for this channel is given by Eq.",
"(REF ) as $\\alpha _{cr} \\simeq \\pi /6 = 0.5$ .",
"Since the condensate involves the SU(6)-singlet fermion ${\\vec{\\omega }}_L$ , it obviously preserves the SU(6) gauge symmetry.",
"As a scalar product of the isovector ${\\vec{\\omega }}_L$ with itself, this condensate is also invariant under the strongly coupled SU(2) gauge symmetry.",
"Because the condensate has a nonzero U(1) charge (namely, $q=12$ ), it breaks the U(1) gauge symmetry.",
"The continuous gauge symmetry under which the condensate (REF ) is invariant is therefore ${\\rm SU}(4) \\otimes {\\rm SU}(2) \\ .$ This residual symmetry group has order 38 and rank 6.",
"For this and other possible fermion condensation channels, we record these properties in Table .",
"This table refers to the possible initial condensation patterns at the highest condensation scale; subsequent evolution further into the infrared is discussed below A second possible condensation channel is $\\chi \\chi : \\quad ([\\bar{1}]_6, [\\bar{1}]_2)_{-4} \\times ([\\bar{1}]_6, [\\bar{1}]_2)_{-4} \\rightarrow ([\\bar{2}]_6, 1)_{-8} \\ ,$ where the shorthand name $\\chi \\chi $ reflects the associated condensate, $\\epsilon ^{\\alpha \\beta }\\langle \\chi _{i,\\alpha ,L}^T C \\chi _{j,\\beta ,L}\\rangle $ .",
"Since SU(2) has only pseudoreal representations, this channel has the form $2\\times 2 \\rightarrow 1$ with respect to SU(2).",
"The measure of attractiveness of this channel due to the strongly coupled SU(2) gauge interaction is $\\Delta C_2 = 2C_2([\\bar{1}]_2) = \\frac{3}{2} \\quad {\\rm for} \\ 2 \\times 2 \\rightarrow 1 \\ {\\rm in} \\ {\\rm SU}(2) \\ .$ From (REF ), we find that the minimal critical coupling for condensation in this channel is $\\alpha _{cr} \\simeq 4\\pi /9 = 1.4$ .",
"From the general structural analysis of fermion condensates given above, it follows that, since the SU(2) tensor $\\epsilon ^{\\alpha \\beta }$ is antisymmetric, the condensate must be of the form $(a,a,s)$ .",
"(It cannot be of the form $(s,a,a)$ because with $N_f=1$ , this would vanish identically.)",
"Hence, under SU(6), it transforms as $[4]_6$ , or equivalently, as $[\\bar{2}]_6$ , as indicated in Eq.",
"(REF ).",
"Consequently, it is proportional to $\\epsilon ^{ijk\\ell mn} \\epsilon ^{\\alpha \\beta }\\langle \\chi _{m,\\alpha ,L}^T C \\chi _{n,\\beta ,L} \\rangle \\ ,$ where $i,j,k,\\ell ,m,n$ are SU(6) indices and $\\alpha ,\\beta $ are SU(2) indices.",
"This condensation channel thus preserves SU(2) while breaking U(1).",
"As regards SU(6), from a general group-theoretic analysis [27], [24], one infers that the condensate (REF ) breaks this SU(6) gauge symmetry to the subgroup ${\\rm SU}(4) \\otimes {\\rm SU}(2)^{\\prime }$ , where we mark the ${\\rm SU}(2)^{\\prime }$ with a prime to distinguish it from the SU(2) in the original gauge group (REF ).",
"Hence, the full continuous gauge symmetry under which the condensate (REF ) is invariant is $[{\\rm SU}(4) \\otimes {\\rm SU}(2)^{\\prime }] \\otimes {\\rm SU}(2) \\ ,$ where we insert the brackets to indicate the origin of the $[{\\rm SU}(4) \\otimes {\\rm SU}(2)^{\\prime }]$ group from the breaking of the original SU(6) in (REF ).",
"This residual symmetry group has order 21 and rank 5.",
"A third type of condensation channel is $\\chi \\omega : \\quad ([\\bar{1}]_6, [\\bar{1}]_2)_{-4} \\times (1,Adj)_6 \\rightarrow ([\\bar{1}]_6,[1]_2)_2 \\ ,$ where the shorthand name $\\chi \\omega $ reflects the condensate $\\langle \\chi _{i,\\alpha ,L}^T C \\omega ^{\\alpha \\beta }_{L} \\rangle \\ .$ With respect to SU(2), this channel is $2 \\times 3 \\rightarrow 2$ .",
"The measure of attractiveness for this channel due to SU(2) gauge interactions is $\\Delta C_2 = C_2(Adj) = 2 \\quad {\\rm for} \\ 2 \\times 3 \\rightarrow 2 \\ {\\rm in} \\ {\\rm SU}(2) \\ .$ The corresponding estimate of the critical coupling from Eq.",
"(REF ) is $\\alpha _{cr}=\\pi /3$ .",
"Evidently, this channel is more attractive than the $\\chi \\chi $ channel (REF ), but less attractive than the $\\omega \\omega $ channel (REF ).",
"All of these three types of fermion condensation exhibit the phenomenon of a strongly coupled gauge interaction producing condensate(s) that dynamically break a more weakly coupled gauge interaction, namely U(1).",
"Furthermore, the condensate in the $\\chi \\chi $ channel (REF ) dynamically breaks not only the U(1) gauge symmetry, but also the more weakly coupled SU(6) gauge symmetry.",
"If a condensate were to form in the $\\chi \\omega $ channel (REF ), it would self-break the strongly coupled SU(2) symmetry, as well as breaking the weakly coupled SU(6) symmetry.",
"However, as will be shown below, a condensate is not likely to form in the $\\chi \\omega $ channel.",
"Since the $\\omega \\omega $ channel (REF ) is the MAC, one expects that, as this theory evolves from the UV to the IR, at a scale that we denote $\\mu =\\Lambda _1$ where the running coupling $\\alpha _2(\\mu )$ increases above the critical value for condensation in this $\\omega \\omega $ channel, the condensate (REF ) forms, breaking the U(1) gauge symmetry, but leaving the SU(2) and SU(6) symmetries intact.",
"As the condensate $\\langle {\\vec{\\omega }}^T_{L} C \\, \\cdot {\\vec{\\omega }}_{L}\\rangle $ in Eq.",
"(REF ) maintains the SU(2) symmetry, all of the three components of the fermion ${\\vec{\\omega }}_L$ involved in this condensate gain equal dynamical masses $\\sim \\Lambda _1$ and are integrated out of the low-energy effective field theory that describes the physics as the reference scale $\\mu $ decreases below $\\Lambda _1$ .",
"The U(1) gauge field gains a mass $\\sim g_3(\\Lambda _1) \\, \\Lambda _1$ .",
"With these fermion and vector boson fields integrated out, the one-loop and two-loop coefficients in the SU(2) beta function in the low-energy effective theory have the same sign, so as the reference momentum scale $\\mu $ decreases below $\\Lambda _1$ , the coupling $\\alpha _2(\\mu )$ continues to increase.",
"Because the ${\\vec{\\omega }}_L$ fermions have been integrated out at the scale $\\Lambda _1$ , they are no longer available to form a condensate in the $\\chi \\omega $ channel (REF ) in the low-energy effective theory below $\\Lambda _1$ ."
],
[
"EFT Below $\\Lambda _1$ and Condensation at\nScale {{formula:4991d14b-247d-49ad-a1ea-5a348e60daca}} ",
"In Ref.",
"[31] it was proved that if one starts with a chiral gauge theory with gauge group $G$ that is free of gauge and global anomalies, and it is broken dynamically to a theory with gauge group $H \\subseteq G$ , with some set of fermions gaining dynamical masses and being integrated out, then the low-energy theory with the gauge group $H$ is also free of gauge and global anomalies.",
"As a special case of this theorem, the low-energy theory that is operative here as $\\mu $ decreases below $\\Lambda _1$ is also an anomaly-free theory.",
"One easily checks that it is chiral.",
"As $\\mu $ decreases below a lower scale that we denote as $\\Lambda _2$ , $\\alpha _2(\\mu )$ increases past the critical value for the attractive $\\chi \\chi $ condensation channel (REF ), which is the MAC in this low-energy effective theory, and the condensate (REF ) is expected to form.",
"As noted above, this leaves SU(2) invariant and breaks SU(6) to ${\\rm SU}(4) \\otimes {\\rm SU}(2)^{\\prime }$ .",
"By convention, one may label the SU(6) indices $i,j$ of the fermions in the condensate (REF ) as $m=5$ and $n=6$ .",
"Then the fermions $\\chi _{5\\alpha ,L}$ and $\\chi _{6\\beta ,L}$ that are involved in this condensate gain dynamical masses of order $\\Lambda _2$ and are integrated out of the low-energy effective theory applicable for $\\mu < \\Lambda _2$ .",
"Furthermore, the gauge fields in the coset space ${\\rm SU}(6)/[{\\rm SU}(4) \\otimes {\\rm SU}(2)]$ gain dynamical masses of order $g_1(\\Lambda ) \\Lambda _2$ ."
],
[
"EFT Below $\\Lambda _2$ and Further Condensation",
"By the same theorem as before, this low-energy theory is anomaly-free and one can again check that it is chiral.",
"The low-energy effective theory below $\\Lambda _2$ thus has a gauge symmetry $[{\\rm SU}(4) \\otimes {\\rm SU}(2)^{\\prime }] \\otimes {\\rm SU}(2)$ , where the ${\\rm SU}(2)^{\\prime }$ arises from the breaking of the SU(6) and the second SU(2) was present in the original theory.",
"The fermions that have gained masses and have been integrated out are no longer dynamical.",
"The elements of the residual SU(4) subgroup of SU(6) operate on the indices $1 \\le i \\le 4$ , while the elements of ${\\rm SU}(2)^{\\prime }$ operate on the indices $i=5,6$ .",
"Thus, the massless fermions in this effective field theory below $\\Lambda _2$ are as follows, where we categorize them with a three-component vector, indicating the representations with respect to the group (REF ) in the indicated order: $\\psi ^{ij}_L$ with $1 \\le i, \\ j \\le 4$ , forming a (self-conjugate) $([2]_4,1,1)$ representation of the group ${\\rm SU}(4) \\otimes {\\rm SU}(2)^{\\prime } \\otimes {\\rm SU}(2)$ in (REF ), $\\psi ^{i5}_L$ and $\\psi ^{i6}_L$ , forming a $([1]_4,[1]_{2^{\\prime }}1,1)$ representation of (REF ), $\\chi _{i,\\alpha ,L}$ with $1 \\le i \\le 4$ , forming a $([\\bar{1}]_4,1,[\\bar{1}]_2)$ representation of (REF ).",
"In this low-energy EFT below $\\Lambda _2$ , the MAC for SU(4)-induced condensate formations is $[2]_4 \\times [2]_4 \\rightarrow 1$ with the self-conjugate $\\psi ^{ij}_L$ transforming as $[2]_4$ of SU4), producing the condensate $\\sum _{i,j,k,\\ell = 1}^4\\epsilon _{ijk\\ell }\\langle \\psi ^{ij \\ T}_L C \\psi ^{k\\ell }_L\\rangle \\ .$ This is a singlet under the SU(4) gauge symmetry and is obviously invariant under the two other gauge symmetries, ${\\rm SU}(2)^{\\prime } \\otimes {\\rm SU}(2)$ , since the fermions in (REF ) are singlets under these groups.",
"Let us denote the scale at which this condensate forms as $\\Lambda _3$ .",
"The SU(4)-induced condensation producing this condensate (REF ) has $\\Delta C_2 = 5$ .",
"The $\\psi ^{ij}_L$ with $1 \\le i,j \\le 4$ involved in this condensate pick up dynamical masses of order $\\Lambda _3$ and are integrated out of the low-energy EFT that is operative below $\\Lambda _3$ .",
"The SU(4) gauge interaction can also produce the condensate $\\sum _{i=1}^4 \\langle \\psi ^{ir \\ T}_L C \\chi _{i,\\alpha }\\rangle \\ ,$ where $r=5, \\ 6$ .",
"This condensate is invariant under SU(4) and breaks ${\\rm SU}(2)^{\\prime } \\otimes {\\rm SU}(2)$ (since it involves the uncontracted ${\\rm SU}(2)^{\\prime }$ index $r$ and the uncontracted SU(2) index $\\alpha $ ).",
"For this condensation, $\\Delta C_2 = 15/4$ .",
"With these condensates, only the SU(4) gauge symmetry remains, and all SU(4)-nonsinglet fermions have picked up dynamical masses.",
"This vectorial SU(4) theory confines and produces a spectrum of SU(4)-singlet bound state hadrons."
],
[
"RG Evolution from UV",
"In this section we analyze the $N=6$ theory for the case in which the SU(6) gauge interaction becomes strongly coupled and is dominant over the weakly coupled SU(2) (and U(1)) gauge interactions.",
"We denote this as the SU6D case.",
"The one-loop and two-loop terms in the beta function were given above in Eqs.",
"(REF ) and (REF ).",
"For both of the cases allowed by the requirement of asymptotic freedom for the SU(6) and SU(2) gauge interactions, namely $N_f=1$ and $N_f=2$ , these coefficients have the same sign, so that the two-loop beta function of this SU(6) theory has no IR zero.",
"Hence, as the scale $\\mu $ decreases from $\\mu _{_{UV}}$ to the IR, $\\alpha _1(\\mu )$ increases until it eventually exceeds the range of values where it can be calculated perturbatively."
],
[
"Highest-Scale Condensation Channels",
"We examine the various possible fermion condensation channels produced by the strongly coupled SU(6) gauge interaction.",
"The first is the $\\psi \\psi $ channel $\\psi \\psi : \\quad ([2]_6,1)_2 \\times ([2]_6,1)_2 \\rightarrow ([4]_6,1)_4 \\approx ([\\bar{2}]_6,1)_4 \\ ,$ with associated condensate $\\epsilon _{ijk\\ell mn} \\langle \\psi ^{k\\ell \\ T}_{p,L} C\\psi ^{mn}_{p^{\\prime },L} \\rangle \\ .$ This condensate is automatically symmetrized in the flavor indices.",
"Since it transforms as a $[\\bar{2}]_6$ representation of SU(6), it breaks SU(6) to ${\\rm SU}(4) \\otimes {\\rm SU}(2)^{\\prime }$ .",
"Because the constituent fermion fields in (REF ) are singlets under SU(2), this condensate is obviously SU(2)-invariant.",
"Finally, owing to the property that the condensate (REF ) has nonzero U(1) charge, it also breaks U(1).",
"The residual subgroup of the original group (REF ) that is left invariant by the condensate (REF ) is thus $[{\\rm SU}(4) \\otimes {\\rm SU}(2)^{\\prime }] \\otimes {\\rm SU}(2)$ (see Eq.",
"(REF )), as in the condensation process (REF ).",
"The condensation (REF ) thus provides another example of an induced, dynamical breaking of one gauge symmetry, namely U(1), by a different, strongly coupled, gauge interaction in a direct-product chiral gauge theory.",
"The measure of attractiveness of this condensation channel involving the SU(6) gauge interaction is $\\Delta C_2 &=& C_2([\\bar{2}]_6) = \\frac{14}{3} \\cr \\cr && {\\rm for} \\ [2]_6 \\times [2]_6 \\rightarrow [\\bar{2}]_6 \\ {\\rm in} \\ SU(6) .$ From the rough estimate for the minimal critical coupling strength to produce this condensate, (REF ), one has $\\alpha _{cr} \\simeq \\pi /7 = 0.45$ .",
"A second possible condensation channel is $\\psi \\chi : \\quad ([2]_6,1)_{2} \\times ([\\bar{1}]_6,[\\bar{1}]_2)_{-4} \\rightarrow ([1]_6,[\\bar{1}]_2)_{-2}$ with associated condensate $\\langle \\psi ^{ij \\ T}_{p,L} C \\chi _{j, \\beta ,p^{\\prime },L} \\rangle \\ .$ This condensation breaks SU(6) to SU(5) and also breaks SU(2) and U(1), so that the residual invariance group is SU(5), with order 24 and rank 4.",
"The total number of broken generators is thus 15 and the reduction in rank is by 3.",
"Again, this illustrates the dynamical breaking of more weakly coupled gauge symmetries by a strongly coupled gauge interaction in a direct-product gauge theory.",
"The measure of attractiveness of this channel (REF ) is $\\Delta C_2 &=& C_2([2]_6) = \\frac{14}{3} \\cr \\cr &&{\\rm for} \\ [2]_6 \\times [\\bar{1}]_6 \\rightarrow [1]_6 \\ {\\rm in} \\ SU(6) .$ Evidently, this is the same as the attractiveness for the channel (REF ), so the critical coupling $\\alpha _{cr}$ is also the same as for that channel.",
"This $\\Delta C_2=14/3$ is also larger than the $\\Delta C_2$ for the third channel (to be discussed below), so that, as was stated above in (REF ), for this $N=6$ theory, with SU(6) being the dominant gauge interaction, the $\\psi \\psi $ and $\\psi \\chi $ channels are the MACs.",
"A third condensation channel produced by the dominant SU(6) gauge interaction is $\\chi \\chi : \\quad [\\bar{1}]_6 \\times [\\bar{1}]_6 \\rightarrow [\\bar{2}]_6 \\approx [4]_6 \\ \\ {\\rm in } \\ \\ {\\rm SU}(6)$ with condensate $\\epsilon ^{ijk\\ell mn} \\langle \\chi _{m,\\alpha ,p,L}^T C\\chi _{n,\\beta ,p^{\\prime },L}\\rangle \\ .$ Although we use the same shorthand name, $\\chi \\chi $ , for this channel as in Eq.",
"(REF ), it is understood that here it is the SU(6) gauge interaction that is responsible for the formation of this condensate, rather than the SU(2) gauge interaction in (REF ).",
"The measure of attractiveness for this condensation, as produced by the SU(6) gauge interaction, is $\\Delta C_2 &=& 2C_2([\\bar{1}]_6) - C_2([2]_6) = \\frac{7}{6} \\cr \\cr && {\\rm for} \\ [\\bar{1}]_6 \\times [\\bar{1}]_6 \\rightarrow [\\bar{2}]_6 \\ {\\rm in} \\ {\\rm SU}(6)\\ .$ This $\\Delta C_2$ is a factor of 4 smaller than the common value $\\Delta C_2=14/3$ for the condensation channels (REF ) and (REF ) and hence is predicted not to occur in this SU(6)-dominant case.",
"We proceed to discuss in greater detail the two different patterns of UV to IR evolution for the most attractive condensation channels in this SU(6)-dominant case."
],
[
"$\\psi \\psi $ Condensation Channel ",
"Here we consider the $\\psi \\psi $ condensation channel (REF ), i.e., $([2]_6,1)_2 \\times ([2]_6,1)_2 \\rightarrow ([\\bar{2}]_6,1)_4$ .",
"We denote the scale at which the condensate (REF ) forms as $\\Lambda _1$ .",
"(To avoid cumbersome notation, we use the same symbol for this highest-level condensation as we did in the subsection dealing with the case where the SU(2) gauge interaction is dominant, but it is understood implicitly that this scale has generically different values for these different cases.)",
"Without loss of generality, one may choose the SU(6) group indices of the $\\psi $ fermions involved in the condensate (REF ) to be $k, \\ \\ell , \\ m, \\ n \\in \\lbrace 1,2,3,4 \\rbrace $ and the uncontracted indices in (REF ) to be $i,j \\in \\lbrace 5,6\\rbrace $ .",
"The $\\psi $ fermions involved in the condensate (REF ) gain dynamical masses of order $\\Lambda _1$ .",
"The gauge bosons in the coset ${\\rm SU}(6)/[{\\rm SU}(4) \\otimes {\\rm SU}(2)^{\\prime }]$ pick up dynamical masses of order $g_1(\\Lambda _1) \\, \\Lambda _1$ , and the U(1) gauge boson picks up a dynamical mass $\\simeq g_3(\\Lambda _1) \\, \\Lambda _1$ .",
"These massive fermion and vector boson fields are integrated out of the low-energy effective field theory that describes the physics as the reference scale $\\mu $ decreases below $\\Lambda _1$ .",
"The resultant low-energy effective theory contains the following massless fermions: (1) ${\\rm SU}(4) \\otimes {\\rm SU}(2)^{\\prime }$ -nonsinglets $\\psi ^{ia}_{p,L}$ with $1 \\le i \\le 4$ , $a \\in \\lbrace 5,6\\rbrace $ , and $1 \\le p \\le N_f$ , which are singlets under SU(2); (2) ${\\rm SU}(4) \\otimes {\\rm SU}(2)$ -nonsinglets $\\chi _{i \\alpha ,p,L}$ with $1 \\le i \\le 4$ , $\\alpha =1,2$ , and $1 \\le p \\le N_f$ , which are singlets under ${\\rm SU}(2)^{\\prime }$ ; and (3) ${\\rm SU}(2)^{\\prime } \\otimes {\\rm SU}(2)$ -nonsinglets $\\chi _{i,\\alpha ,p,L}$ with $i=5,6$ , which are singlets under SU(4).",
"There are also the massless fermions $\\psi ^{56}_{p,L}$ with $1 \\le p \\le N_f$ , which are singlets under all three factor groups in (REF ).",
"The fermions (1) transform as $2N_f$ fundamental representations $F=[1]_4$ of SU(4), while the fermions (2) transform as $2N_f$ conjugate fundamental representations $\\bar{F}=[\\bar{1}]_4$ of SU(4), so that the SU(4) gauge symmetry is vectorial.",
"Combining this property with the fact that the ${\\rm SU}(2)^{\\prime }$ and SU(2) groups have only real representations, it follows that this low-energy theory is vectorial.",
"The action of an element $U \\in {\\rm SU}(4)$ is $\\psi ^{ia}_{p,L} &=& U^i_j \\psi ^{ja}_{p,L} \\cr \\cr \\chi _{ia,p,L} &=& (U^\\dagger )^j_i \\chi _{ja,p,L} \\ ,$ with fixed $a=5,6$ and $1 \\le p \\le N_f$ .",
"The elements of ${\\rm SU}(2)^{\\prime }$ operate on the indices $a=5,6$ of the fermions (1) and (3).",
"(The operation of the elements of SU(2) on the $\\alpha ,\\beta $ indices has already been discussed.)",
"The couplings of the SU(4) and ${\\rm SU}(2)^{\\prime }$ gauge interactions start out equal at $\\mu =\\Lambda _1$ , as descendents of the gauge coupling $\\alpha _1$ of the UV gauge coupling for the SU(6) gauge interaction.",
"As the theory evolves further into the IR, several possible patterns of gauge symmetry breaking are possible.",
"The SU(4) gauge interaction can produce a condensate in the $[1]_4 \\times [\\bar{1}]_4 \\rightarrow 1$ , i.e., $F \\times \\bar{F} \\rightarrow 1$ channel: $\\langle \\sum _{i=1}^4 \\psi ^{ia \\ T}_{p,L} C \\chi _{i,\\alpha ,p^{\\prime },L} \\rangle \\ ,$ where, as indicated, the sum on $i$ is over the active SU(4) gauge indices, while the other indices take on the values $a=5, \\ 6$ , $\\alpha =1, \\ 2$ , and $1\\le p,p^{\\prime } \\le N_f$ .",
"The measure of attractive of this condensation, as produced by the SU(4) gauge interaction, is $\\Delta C_2 = 2C_2([1]_4) = 15/4 = 3.75$ .",
"This condensate preserves the SU(4) gauge symmetry and breaks the ${\\rm SU}(2)^{\\prime }$ gauge symmetry operating on the indices $a=5,6$ and the SU(2) gauge symmetry operating on the indices $\\alpha =1,2$ .",
"In contrast, the ${\\rm SU}(2)^{\\prime }$ gauge interaction could produce the condensate $\\sum _{a,b=5}^6 \\epsilon _{ab}\\langle \\psi ^{ia \\ T}_{p,L} C \\psi ^{jb}_{p^{\\prime },L}\\rangle \\ .$ The measure of attractiveness for this condensation, as produced by the ${\\rm SU}(2)^{\\prime }$ interaction, is $\\Delta C_2 = 3/2$ .",
"Since the fermions involved in this condensate are SU(2)-singlets, it obviously preserves SU(2).",
"With the contraction on the ${\\rm SU}(2)^{\\prime }$ indices $a,b \\in \\lbrace 5,6\\rbrace $ , it also preserves ${\\rm SU}(2)^{\\prime }$ .",
"If $N_f=1$ , then the condensate is automatically symmetric in the single flavor index, so it has the form $(a,a,s)$ in the notation of Eq.",
"(REF ) and hence transforms like the $[2]_4$ representation of SU(4).",
"This breaks SU(4) to $SU(2)^{\\prime \\prime } \\otimes SU(2)^{\\prime \\prime \\prime }$ , where we use repeated primes to indicate that these SU(2) subgroups of SU(4) are distinct from both the original UV SU(2) symmetry and the ${\\rm SU}(2)^{\\prime }$ symmetry.",
"If $N_f=2$ , then there are two possibilities; $(a,a,s)$ if one constructs a linear combination that is symmetrized in flavor indices, and $(a,s,a)$ , if one antisymmetrizes over flavor indices.",
"For each of these possibilities, one can track the evolution further into the IR using the same methods as above."
],
[
"$\\psi \\chi $ Condensation Channel ",
"Here we consider the $\\psi \\chi $ condensation channel (REF ), i.e., $([2]_6,1)_2 \\times ([\\bar{1}]_6,[\\bar{1}]_2)_{-4} \\rightarrow ([1]_6,[\\bar{1}]_2)_{-2}$ , with the associated condensate $\\langle \\psi ^{ij \\ T}_{p,L} C \\chi _{j,\\beta ,p^{\\prime },L} \\rangle $ in Eq.",
"(REF ).",
"By convention, we may choose the SU(6) index $i=6$ and the SU(2) index $\\beta =2$ in this condensate.",
"Then the fermions involved in the condensate, namely $\\psi ^{6j}_{p,L}$ and $\\chi _{j,2,p^{\\prime },L}$ with $1 \\le j \\le 5$ gain dynamical masses of order $\\Lambda _1$ and are integrated out of the low-energy effective theory applicable for $\\mu < \\Lambda _1$ .",
"The 11 SU(6) gauge bosons in the coset ${\\rm SU}(6)/{\\rm SU}(5)$ gain dynamical masses of order $g_1(\\Lambda _1) \\Lambda _1$ , while the SU(2) and U(1) gauge bosons gain masses of order $g_i(\\Lambda _1) \\Lambda _1$ with $i=2,3$ , respectively.",
"These fields are integrated out of the low-energy effective theory applicable for $\\mu < \\Lambda _1$ .",
"For this channel, the low-energy effective theory that describes the physics as $\\mu $ decreases below $\\Lambda _1$ has an SU(5) gauge symmetry with (massless) SU(5)-nonsinglet fermions $\\psi ^{ij}_{p,L}$ and $\\chi _{i,1,p^{\\prime },L}$ , where $1 \\le i,j \\le 5$ and $1 \\le p,p^{\\prime } \\le N_f$ .",
"In addition, there are massless SU(5)-singlet fermions $\\chi _{6,\\beta ,p^{\\prime },L}$ and $\\omega ^{\\alpha \\beta }_{p,L}$ with $1 \\le \\alpha , \\beta \\le 2$ and $1 \\le p,p^{\\prime } \\le N_f$ remaining from the UV theory.",
"In this low-energy theory, the SU(5) gauge coupling inherited from the SU(6) UV theory continues to increase as $\\mu $ decreases below $\\Lambda _1$ , and is expected to trigger a further fermion condensation $[2]_5 \\times [2]_5 \\rightarrow [\\bar{1}]_5$ with $\\Delta C_2 = 24/5$ and associated condensate $\\sum _{i,j,k,\\ell ,m=1}^5 \\,\\epsilon _{ijk\\ell m} \\langle \\psi ^{jk \\ T}_{p,L} C \\psi ^{\\ell m}_{p^{\\prime },L}\\rangle \\ ,$ where the indices $i, \\ j, \\ k, \\ell , m$ are SU(5) group indices.",
"By convention, we may choose the uncontracted SU(5) group index in (REF ) to be $i=5$ .",
"This condensate breaks SU(5) to SU(4).",
"The fermions $\\psi ^{jk}_{p,L}$ with $j,k \\in \\lbrace 1,2,3,4\\rbrace $ and $1 \\le p\\le N_f$ gain dynamical masses of order $\\Lambda _2$ .",
"The 9 gauge bosons in the coset SU(5)/SU(4) gain dynamical masses of order $g_1(\\Lambda _2)\\Lambda _2$ .",
"All of these fields are integrated out of the low-energy effective theory that describes the physics at scales $\\mu < \\Lambda _2$ .",
"The low-energy theory that is operative for $\\mu < \\Lambda _2$ has a gauge group SU(4) and (massless) SU(4)-nonsinglet fermion content consisting of $\\psi ^{ij}_{p,L}$ with $1 \\le i,j \\le 4$ and $1 \\le p \\le N_f$ .",
"However, this representation, $[2]_4$ , in SU(4) is self-conjugate, i.e., $[2]_4 \\approx [\\bar{2}]_4$ , so this theory is vectorial.",
"The two-loop beta function for this theory has no IR zero and as $\\mu $ continues to decrease, the SU(4) coupling inherited from the SU(5) theory continues to increase.",
"Because of the vectorial nature of this descendent SU(4) theory, the condensate that forms is in the channel $[2]_4 \\times [2]_4 \\rightarrow 1$ , with condensate $\\sum _{i,j,k,\\ell =1}^4 \\,\\epsilon _{ijk\\ell } \\langle \\psi ^{ij \\ T}_{p,L} C \\psi ^{k\\ell }_{p^{\\prime },L}\\rangle \\ ,$ with $\\Delta C_2=5$ , where here, $i, \\ j, \\ k, \\ \\ell $ are SU(4) group indices.",
"This condensate preserves the SU(4) gauge symmetry, while breaking global chiral symmetries spontaneously.",
"The fermions involved in this condensate pick up dynamical masses of order the condensation scale.",
"This theory confines and produces a spectrum of SU(4)-singlet bound state hadrons."
],
[
"General Discussion",
"In this section we consider the situation in which both the SU(6) and SU(2) gauge interactions are of comparable strength and hence must be treated together (with the U(1) gauge interaction still being weak).",
"In this case, one cannot neglect the mixing terms at the two-loop and higher-loop level in the beta functions $\\beta _{\\alpha _i}$ , Eq.",
"(REF ), so the calculation the evolution of the gauge couplings down from the initial reference point $\\mu =\\mu _{_{UV}}$ in the UV is more complicated.",
"For our present purposes, it will suffice to consider a case in which $\\alpha _1(\\mu ) \\simeq \\alpha _2(\\mu ) \\simeq O(1)$ at a lower scale $\\mu $ .",
"Since the SU(2) interaction by itself would evolve to a relatively weakly coupled IRFP if $N_f=2$ , expected to be in the non-Abelian Coulomb phase, we will assume $N_f=1$ here, to guarantee that not just the SU(6) interaction, but also the SU(2) interaction become strongly coupled in the infrared.",
"We have shown above that the most attractive condensation channels are different in the simple situations where either the SU(6) or the SU(2) gauge interactions are dominant.",
"Specifically, in the SU(2)-dominant case, the MAC is the $\\omega \\omega $ channel, with $\\Delta C_2=4$ , while in the SU(6)-dominant case, the MACs are the $\\psi \\psi $ and $\\psi \\chi $ channels, with the same measure of attractiveness, $\\Delta C_2=14/3 = 4.7$ .",
"One would thus expect that as the reference scale decreases, the first condensate(s) to form would be in the $\\psi \\psi $ and/or $\\psi \\chi $ channels, as produced by the SU(6) gauge interaction.",
"Since the $\\psi \\psi $ channels involves SU(2)-singlet fermions, it would not be affected by the fact that the SU(2) gauge interaction is also strongly coupled.",
"The other SU(6) MAC, namely the $\\psi \\chi $ channel involves the SU(2)-singlet fermion $\\psi $ and the SU(2)-nonsinglet fermion $\\chi $ , so the binding is only caused by the SU(6) interaction.",
"Since the $\\psi \\chi $ condensation leaves the residual gauge symmetry group SU(5), of order 24, while the $\\chi \\chi $ condensation would leave the residual gauge symmetry (REF ), of order 21, a vacuum alignment argument suggests that the $\\psi \\chi $ condensation channel is preferred over the $\\chi \\chi $ channel.",
"Thus, the $\\psi \\chi $ condensate (REF ) is expected to form at a scale that we will denote as $\\Lambda _1$ , self-breaking SU(6) to SU(5) and also producing induced dynamical breaking of U(1).",
"The 11 gauge bosons in the coset ${\\rm SU}(6)/{\\rm SU}(5)$ gain dynamical masses of order $g_1(\\Lambda _1) \\, \\Lambda _1$ , and the U(1) gauge boson gains a mass of order $g_3(\\Lambda _1) \\, \\Lambda _1$ ."
],
[
"EFT Below $\\Lambda _1$",
"Next, one would expect that condensation would occur in the $\\omega \\omega $ channel, as produced by the strong SU(2) gauge interaction.",
"Owing to the fact that the value of $\\Delta C_2$ for this condensation is equal to 4, slightly less than the value of 4.7 for the $\\psi \\psi $ condensation, one expects that this occurs at a slightly lower scale.",
"Because this second condensation would give dynamical masses to the $\\omega $ fermions, which would thus be integrated out of the low-energy theory applicable below this condensation scale, it would preclude the formation of an SU(2)-induced condensate in the $\\chi \\omega $ channel.",
"There remains the $\\chi \\chi $ condensation channel.",
"Although the value of $\\Delta C_2$ for this condensation, as produced by the SU(6) interaction, is 7/6, which is a factor of 4 smaller than the value of 14/3 for the MACs, and although the value of $\\Delta C_2$ for this condensation, as produced by the SU(2) interaction, is 3/2, considerably smaller than the value $\\Delta C_2 = 4$ for the SU(2)-induced MAC channel, $\\omega \\omega $ , the $\\chi \\chi $ channel has the special property that it involves both the SU(2) and SU(6) gauge interactions, in contrast to all of the other possible condensation channels ($\\psi \\psi $ , $\\psi \\chi $ , $\\omega \\omega $ , and $\\chi \\omega $ ), each of which only involves one of these two non-Abelian gauge interactions.",
"If $\\alpha _1(\\mu )=\\alpha _2(\\mu )$ and one were simply to add the two terms $(7/6)\\alpha _1(\\mu )+(3/2)\\alpha _2(\\mu ) =(8/3)\\alpha _1(\\mu )$ , the effective $\\Delta C_2$ would be 8/3 = 3.7, which is still less than values for the MACs for both the SU(6)-induced condensates and the SU(2)-induced condensates."
],
[
"Related Constructions",
"At the beginning of this paper we remarked on how the theory (REF ) with (REF ) successfully combines two different (anomaly-free) chiral gauge theories, SU($N$ ) with $N_f$ copies of (REF ), and SU($M$ ) with $N_f$ copies of (REF ), where $M=N-4$ .",
"A natural question concerns related constructions of direct-product chiral gauge theories with fermions in higher-rank tensor representations of the factor groups.",
"The next step up in complexity involves rank-3 antisymmetric and symmetric tensor representations for the fermions.",
"Two theories with these rank-3 representations use a gauge group of the form ${\\rm SU}(N) \\otimes {\\rm SU}(M) \\otimes {\\rm U}(1) \\ ,$ where now $M$ can take on two different values as a function of $N$ , namely $M=N-3$ or $N=N-6$ .",
"In both cases, the fermion content consists of $N_f$ copies of the set [12], [16] $([3]_N,1)_{q_{30}} + ([\\bar{2}]_N,(\\bar{1})_M)_{q_{21}} + ([1]_N,(2)_M)_{q_{12}}+ (1,(\\bar{3})_M)_{q_{03}}$ with $M=N-3 \\ & \\Longrightarrow & (q_{30},q_{21},q_{12},q_{03}) = \\cr \\cr &=& \\Big ( -(N-3),(N-2),-(N-1),N \\Big ) \\cr \\cr &&$ and $M=N-6 \\ & \\Longrightarrow & (q_{30},q_{21},q_{12},q_{03})= \\cr \\cr &=& \\Big (-(N-6),(N-4),-(N-2),N \\Big ) \\ .",
"\\cr \\cr &&$ Owing to the presence of the factor group SU($M$ ) in (REF ), the lowest nondegenerate cases are $N=5$ if $M=N-3$ and $N=8$ if $M=N-6$ .",
"As before, there are equivalent theories.",
"One has all of the representations of the (left-handed chiral) fermions conjugated.",
"The second has the SU($M$ ) representations conjugated relative to the SU($N$ ) representations, i.e., it has a fermion content comprised of $N_f$ copies of the set $([3]_N,1)_{q_{30}} + ([\\bar{2}]_N,(1)_M)_{q_{21}} + ([1]_N,(\\bar{2})_M)_{q_{12}}+ (1,(3)_M)_{q_{03}} \\ .$ Since these are equivalent to the theory with gauge group (REF ) and fermions (REF ), with the indicated U(1) charges for $M=N-3$ and $M=N-6$ , it suffices to discuss only the latter theories.",
"However, none of these theories satisfies the requisite condition for our analysis, that both the SU($N$ ) and SU($M$ ) gauge interactions are asymptotically free (AF).",
"The reason for this is as follows.",
"In the theory (REF ) with (REF ), the one-loop term in the SU($N$ ) and SU($N-4$ ) beta functions involves the trace invariants for the fundamental and symmetric or antisymmetric rank-2 representations.",
"While $T([2]_N)=(N-2)/2$ and $T((2)_N)$ are linear functions of $N$ and hence enter the one-loop coefficients in the beta functions with the same polynomial degree as the pure gauge contribution, $T([3]_N)$ and $T((3)_M)$ are quadratic functions of $N$ and $M$ , respectively, namely $T([3]_N=(N-3)(N-4)/4)$ and $T((3)_M)=(M+3)(M+4)/4$ .",
"The most stringent restriction arises from the constraint that the SU($M$ ) beta function be negative.",
"The one-loop coefficient in this beta function is $b^{({\\rm SU}(M))}_{1\\ell ;11} &=& \\frac{1}{3}\\bigg [ 11M - N_f \\bigg \\lbrace \\frac{N(N-1)}{2} + N(M+2) + \\cr \\cr &+& \\frac{(M+2)(M+3)}{2} \\bigg \\rbrace \\bigg ] \\ .$ For the theory with $M=N-3$ , this is $b^{({\\rm SU}(N-3))}_{1\\ell ;11} = \\frac{1}{3}\\Big [11(N-3)-2N_fN(N-1)\\Big ] \\ .$ This one-loop coefficient is negative if $N_f > N_{f,b1z,k3a}$ , where $N_{f,b1z,k3a} = \\frac{11(N-3)}{2N(N-1)} \\ .$ We find that $N_{f,b1z,k3a} < 1$ for all $N$ in the relevant range $N \\ge 5$ .",
"Hence, the AF constraint does not allow any nonzero value of $N_f$ .",
"Similarly, for the theory with $M=N-6$ , $b^{({\\rm SU}(N-6))}_{1\\ell ;11} = \\frac{1}{3}\\Big [11(N-6)-2N_f(N^2-4N+3)\\Big ] \\ .$ This one-loop coefficient is negative if $N_f > N_{f,b1z,k3b}$ , where $N_{f,b1z,k3b} = \\frac{11(N-6)}{2N(N^2-4N+3)} \\ .$ The value of $N_{f,b1z,k3b}$ is less than 1 for all $N$ in the relevant range, $N \\ge 8$ .",
"Therefore, the AF constraint does not allow any nonzero value of $N_f$ .",
"We recall that $N_f$ must be nonzero in order for the theory to be a chiral gauge theory, since if $N_f=0$ , then the theory degenerates into decoupled purely gluonic sectors.",
"Thus, in neither of these theories with rank-3 fermion representations and $M=N-3$ or $M=N-6$ is the SU($M$ ) gauge interaction asymptotically free.",
"Similar comments apply to ${\\rm SU}(N)\\otimes {\\rm SU}(M) \\otimes {\\rm U}(1)$ theories with fermions in sets of representations containing antisymmetric and symmetric rank-$k$ tensor representations of the non-Abelian gauge groups with $k \\ge 4$ .",
"As was discussed above, the requirement of asymptotic freedom of both of the non-Abelian gauge interactions was imposed because of (i) the purpose of studying the strong-coupling behavior of one or both of these interactions as the theory evolves from the UV to the IR and (ii) the necessity to be able to carry out a self-consistent perturbative calculation of the beta functions for these interactions at a reference scale, $\\mu _{_{UV}}$ ."
],
[
"Conclusions",
"In nature, the ${\\rm SU}(2)_L \\otimes {\\rm U}(1)_Y$ electroweak symmetry is broken not only by the vacuum expectation value of the Higgs field, but also dynamically, by the $\\langle \\bar{q} q \\rangle $ quark condensates produced by the color SU(3)$_c$ gauge interaction.",
"Moreover, sequential self-breakings of strongly coupled chiral gauge symmetries have also been used in models of dynamical generation of fermion masses.",
"In this paper we have investigated a chiral gauge theory that serves as a theoretical laboratory that exhibits both induced breaking of a weakly coupled gauge symmetry via condensates formed by a different, strongly coupled gauge interaction, and also self-breaking of strongly coupled chiral gauge symmetries.",
"We have studied an asymptotically free chiral gauge theory with the direct-product gauge group ${\\rm SU}(N)\\otimes {\\rm SU}(N-4) \\otimes {\\rm U}(1)$ and chiral fermion content consisting of $N_f$ flavors of fermions transforming according to the representations $([2]_N,1)_{N-4} + ([\\bar{1}]_N,[\\bar{1}]_{N-4})_{-(N-2)} + (1,(2)_{N-4})_N$ .",
"One of the reasons for interest in this theory is that it may be viewed as a combination of two separate (anomaly-free) chiral gauge theories, namely (i) an SU($N$ ) theory with fermion content consisting of $N_f$ flavors of fermions in the $[2]_N$ and $N-4$ copies of $[\\bar{1}]_N$ , and (ii) an SU($M$ ) theory with fermions consisting of $N_f$ flavors of fermions in the $(2)_M$ and $M+4$ copies of $[\\bar{1}]_M$ , with $M=N-4$ , which also incorporates a U(1) gauge symmetry.",
"We have analyzed the UV to IR evolution of this theory and have investigated patterns of possible bilinear condensate formation.",
"A detailed discussion of the lowest nondegenerate case, $N=6$ was given.",
"This analysis involved a sequential construction and analysis of low-energy effective field theories that describe the physics as the theory evolves through various condensation scales and certain fermions and gauge bosons pick up dynamically generated masses.",
"Our findings provide new insights into the phenomenon of induced breaking of a weakly coupled gauge symmetry by a different, strongly coupled gauge interaction, and self-breaking of a strongly coupled chiral gauge symmetry.",
"This research was supported in part by the Danish National Research Foundation grant DNRF90 to CP$^3$ -Origins at SDU (T.A.R.)",
"and by the U.S. NSF Grant NSF-PHY-16-1620628 (R.S.)",
"Properties of possible initial (highest-scale) bilinear fermion condensates in the UV theory (REF ) with (REF ) for $N \\ge 7$ .",
"The shorthand name of the condensation channel is listed in the first column, and the corresponding condensate is displayed in the the second column.",
"The third and fourth columns list the values of $\\Delta C_2$ with respect to the SU($N$ ) and SU($N-4$ ) gauge interactions.",
"The entries in the fifth, sixth, and seventh columns indicate whether a given condensate is invariant (inv.)",
"under the SU($N$ ), SU($N-4$ ), and U(1) gauge symmetries, respectively, or breaks (bk.)",
"one or more of these symmetries.",
"The entry in the eighth column gives the representation $(R_1,R_2)_q$ of the condensate under the group (REF ), following the notation of Eq.",
"(REF ).",
"The ninth column lists the continuous gauge symmetry group under which a given condensate is invariant.",
"The tensors $\\epsilon _{...k\\ell mn}$ and $\\epsilon ^{...mn}$ are antisymmetric SU($N$ ) tensors, while $\\epsilon ^{...\\alpha \\beta }$ is an antisymmetric SU($N-4$ ) tensor.",
"These results are for the case $N_f=1$ and for $N_f \\ge 2$ with condensates symmetrized over the flavor indices, which are suppressed in the notation.",
"The $\\psi \\chi $ channel is the MAC for the SU($N$ )-dominant case, while the $\\chi \\omega $ channel is the MAC for the SU($N-4$ )-dominant case in this $N \\ge 7$ range.",
"See text for further discussion.",
"Table: NO_CAPTIONProperties of possible initial bilinear fermion condensates in the UV theory (REF ) with (REF ).",
"The shorthand name of the condensation channel and the condensate in this channel are displayed in the the first and second columns.",
"The third and fourth columns list the values of $\\Delta C_2$ with respect to the SU(6) and SU(2) gauge interactions.",
"The entries in the fifth, sixth, and seventh columns indicate whether a given condensate is invariant (inv.)",
"under the SU(6), SU(2), and U(1) gauge symmetries, respectively, or breaks (bk.)",
"one or more of these symmetries.",
"The entry in the eighth column gives the representation $(R_1,R_2)_q$ of the condensate under the group (REF ), following the notation of Eq.",
"(REF ).",
"The ninth column lists the continuous gauge symmetry group under which a given condensate is invariant.",
"These results are for the case $N_f=1$ and for $N_f=2$ with condensates symmetrized over the flavor indices, which are suppressed in the notation.",
"The $\\psi \\psi $ and $\\psi \\chi $ channels are the MACs for the SU(6)-dominant case, while the $\\omega \\omega $ is the MAC for the SU(2)-dominant case.",
"See text for further discussion.",
"Table: NO_CAPTION"
]
] | 1906.04255 |
[
[
"Combining the small-x evolution and DGLAP for description of inclusive\n photon induced processes"
],
[
"Abstract Interest in studying the inclusive photon induced processes of Deep-Inelastic Scattering (DIS) and Diffractive DIS (DDIS) at high energies includes their experimental investigation and thorough theoretical description.",
"The conventional instrument for theoretical investigation of DIS at large x is DGLAP.",
"It describes the Q^2-evolution.",
"Description of DIS in the small-x region requires accounting for the both Q^2 and x -evolution at the same time.",
"Combining DGLAP with total resummation of double logarithms of x and Q^2, we present a description of structure function F_1 at large Q^2 and arbitrary x.",
"Making use of the necessity of the shift of Q^2 in order to regulate infrared singularities, we obtain an expression for F_1 valid at arbitrary x and Q^2.",
"The obtained expressions coincide with the DLA expressions at small x and at the same time coincides with the DGLAP result at large x and large Q^2.",
"Accounting for virtualities k^2 of the external partons allows us to obtain expressions for F_1 in K_T-Factorization, which are valid at arbitrary Q^2 and arbitrary relations between Q^2 and k^2.",
"Expressions for F_1 in all types of QCD factorization exhibit the small-x asymptotics of the Pomeron type.",
"This Pomeron is generated by the high-energy asymptotics of amplitudes of the 2 -> 2 parton-parton forward scattering.",
"These amplitudes are calculated in DLA, so their asymptotics have nothing to do with the BFKL Pomeron.",
"We demonstrate that the parton-parton amplitudes can be used in the DDIS models instead of BFKL Pomeron or combinations of hard and soft Pomerons at non-asymptotic energies.",
"On the other hand, this modification automatically ensures the Pomerons asymptotics, which simplifies and makes more consistent theory of DDIS."
],
[
"Introduction",
"Photon induced inclusive processes such as Deep-Inelastic Scattering (DIS), Photoproduction and Diffractive Deep-Inelastic Scattering (DDIS) are the objects of intensive experimental and theoretical investigation.",
"Theoretical studying these processes is based on the concept of QCD factorization which makes possible to separate perturbative and non-perturbative contributions.",
"There are several forms/types of QCD factorization in the literature.",
"Firstly, there is Collinear Factorization[1].",
"Secondly, there is the more general $K_T$ Factorization suggested in Refs.",
"[2], [3].",
"The third, the most general, type of QCD factorization is Basic Factorization suggested in Ref. [4].",
"In any of those factorizations the approximation of the single-parton photon-hadron interactions is used and as a result, the subjects of consideration are represented through convolutions of perturbative and non-perturbative objects connected by two-parton intermediate states.",
"For instance, representation of the amplitude $T$ of the elastic forward Compton scattering off a hadron in any type of QCD factorization is depicted in Fig.",
"1 .",
"Figure: QCD factorization of amplitude T. The dashed lines denote virtual photons with momentum qq, the horizontal straight linescorrespond to the incoming hadron with momentum pp, the vertical straight lines standfor intermediate quarks and the wavy lines denote intermediate gluons with momentum kk.",
"The upper blobs denote DIS off partons,they are described with Perturbative QCD while the lowestblobs are non-perturbative.In the analytical form, QCD factorization of $T$ is generically written as follows: $T = T_q \\otimes \\Phi _q + T_g \\otimes \\Phi _g,$ where $T_q$ and $T_g$ are the perturbative amplitudes of the elastic Compton scattering off a quark and gluon respectively.",
"Non-perturbative blobs $\\Phi _q$ and $\\Phi _g$ denote distributions of those partons in the hadrons.",
"The notations $\\otimes $ refer to integrations over momentum $k$ of any parton in the two-quark and two-gluon intermediate states.",
"The specific form of the integrations as well as a parametrization of momentum $k$ of the intermediate partons depend on the type of chosen factorizationSee for detail Ref.",
"[4] and refs therein.. Amplitudes $T_{q,g}$ as well as amplitudes $\\Phi _{q,g}$ are also given by expressions depending on the type of factorization.",
"In the present paper we focus on the perturbative amplitudes $T_{q,g}$ .",
"In any specific type of factorization, there are kinematic regions where amplitudes $T_{q,g}$ are given by essentially different expressions.",
"Such regions are : Region A.",
"Large $x$ and large $Q^2$ : $(x \\sim 1) \\bigotimes (Q^2 \\gg \\mu ^2)$ .",
"Region B.",
"Small $x$ and large $Q^2$ : $(x \\ll 1)\\bigotimes (Q^2 \\gg \\mu ^2)$ .",
"Region C. Small $x$ and small $Q^2$ : $(x \\ll 1)\\bigotimes (Q^2 < \\mu ^2)$ .",
"Region D. Large $x$ and small $Q^2$ : $(x \\sim 1)\\bigotimes (Q^2 < \\mu ^2)$ .",
"We have used above the conventional notations: $Q^2 = - q^2$ , with $q$ being the external photon momentum, and $x = Q^2/w$ , where $w = 2pq$ , with $p$ being the momentum of the incoming/outgoing parton.",
"The parameter $\\mu $ is associated with the factorization scale.",
"Besides, it plays the role of the infrared (IR) cut-off in Collinear, $K_T$ and Basic Factorizations, when the double-logarithmic (DL) contributions to amplitudes $T_{q,g}$ are accounted for.",
"The physical meaning of $\\mu $ for the parton distributions $\\Phi _{q,g}$ in $K_T$ Factorization and its role in transition from $K_T$ Factorization to Collinear Factorization were considered in Ref. [4].",
"Now let us consider the status of knowledge of amplitudes $T_{q,g}$ in the Regions A-D.",
"The region A is the DGLAP applicability region, so expressions for $T_{q,g}$ in this region are provided by the DGLAP evolution equations[5].",
"Such expressions can easily be found in the literature.",
"They account for logarithms of $Q^2$ but neglect total resummations of logarithms of $x$ because such logarithms are unessential in Region A.",
"Description of $T_{q,g}$ in the small-$x$ region B in the double-logarithmic approximation (DLA) in the framework of Collinear Factorization was obtained in Ref. [6].",
"In particular, it was shown in Ref.",
"[6] that the small-$x$ asymptotics of $T_{q,g}$ exhibits a new, DL contribution to Pomeron.",
"In contrast, double-logarithmic expressions for $T_{q,g}$ in region B in $K_T$ Factorization have not been obtained.",
"We do it in the present paper, using the same method as in Ref.",
"[6]: constructing and solving Infrared Evolution Equations (IREE)This method was suggested by L.N. Lipatov.",
"History of this method and its development are discussed in detail in Ref. [12]..",
"Then we obtain expressions for $T_{q,g}$ which combine DL and non-DL contributions available in DGLAP, so these expressions can universally be used at large $Q^2$ and arbitrary $x$ (i.e.",
"in the region $\\textbf {A}\\oplus \\textbf {B}$ ) in Collinear, $K_T$ and Basic Factorizations.",
"Extending expressions for $T_{qg}$ to low $Q^2$ was suggested in Ref.",
"[11] for Collinear Factorization.",
"In the present paper we generalize this extension to the cases of $K_T$ and Basic Factorizations, obtaining thereby expressions which can universally be used at arbitrary $x$ and $Q^2$ in any form of QCD factorization.",
"Putting $q^2 = 0$ in Eq.",
"(REF ), one arrives at the photoproduction amplitudes $A_{\\gamma }$ : $A_{\\gamma } \\equiv T|_{q^2 = 0}.$ Combining it with Eq.",
"(REF ) leads to the factorized form of $A_{\\gamma q}$ : $A_{\\gamma } = A_{\\gamma q} \\otimes \\Phi _q + A_{\\gamma g} \\otimes \\Phi _g.$ The Optical theorem relates the amplitude $A_{\\gamma }$ to the total cross section of photoproduction.",
"Using the expressions for $T_{qg}$ at low $Q^2$ allows us to obtain expressions for the perturbative amplitudes $A_{\\gamma q}, A_{\\gamma g}$ first in Collinear and then in the other forms of QCD factorization.",
"Our paper is organized as follows: in Sect.",
"II we remind results of Ref.",
"[6] for amplitudes $T_{q,g}$ in Region B in Collinear Factorization and extend $T_{q,g}$ to Regions A,C,D, obtaining explicit expressions for $T_{q,g}$ which can be used at any $x$ and $Q^2$ .",
"Then we briefly discuss the small-$x$ asymptotics of $T_{q,g}$ , their applicability region the power $Q^2$ -corrections to $T_{q,g}$ in the low-$Q^2$ region C. In Sect.",
"III we study amplitudes $T_{q,g}$ in $K_T$ -Factorization.",
"This type of factorization involves dealing with essentially off-shell external partons, which brings a new technical problem which is absent in Collinear Factorization: there is no universal description of $T_{q,g}$ in Region B at at arbitrary virtualities of the external partons.",
"We deal with this problem also by constructing and solving appropriate IREEs.",
"In Sect.",
"IV we consider the photoproduction amplitudes in Collinear, $K_T$ and Basic Factorizations.",
"Finally, Sect.",
"V is for concluding remarks."
],
[
"Compton amplitudes in Collinear Factorization",
"Eq.",
"(REF ) for Collinear Factorization takes the following form: $T^{(col)} = \\int _x^1 \\frac{d \\beta }{\\beta }\\left[ T^{(col)}_q(x/\\beta ,Q^2/\\mu ^2) \\Phi ^{(col)}_q (\\beta ,\\mu ^2)+ T^{(col)}_g(x/\\beta ,Q^2/\\mu ^2) \\Phi ^{(col)}_g (\\beta ,\\mu ^2)\\right],$ where the superscript $\"col\"$ refers to Collinear Factorization and $\\beta $ is the longitudinal fraction of momentum $k$ in Fig. 1.",
"We start with considering amplitudes $T^{(col)}_{q,g}$ in region B and then extend these expressions to regions A,C,D."
],
[
"Amplitudes $T^{(col)}_{q.g}$ in region B",
"Perturbative amplitudes $T^{(col)}_{q,g}$ in region B were calculated in Ref.",
"[6] in the double-logarithmic approximation (DLA) and the cases of fixed and running $\\alpha _s$ were investigated separately.",
"Amplitudes $T^{(col)}_{q,g}$ were in Ref.",
"[6] represented as follows: $T^{(col)}_q (x/\\beta , Q^2/\\mu ^2)&=& \\int _{-\\imath \\infty }^{\\imath \\infty } \\frac{d \\omega }{2 \\imath \\pi } \\left(x/\\beta \\right)^{-\\omega }\\xi ^{(+)} (\\omega ) F^{(col)}_q (\\omega , y),\\\\ \\nonumber T^{(col)}_g (x/\\beta , Q^2/\\mu ^2)&=& \\int _{-\\imath \\infty }^{\\imath \\infty } \\frac{d \\omega }{2 \\imath \\pi } \\left(x/\\beta \\right)^{-\\omega }\\xi ^{(+)} (\\omega ) F^{(col)}_g (\\omega ,y),$ with $\\mu $ being the factorization scale.",
"The logarithmic variable $y$ in Eq.",
"(REF ) is related to $Q^2$ : $y = \\ln (Q^2/\\mu ^2)$ and $\\xi ^{(+)} (\\omega )$ is the positive signature factor: $\\xi ^{(+)} (\\omega ) = - \\left(e^{-\\imath \\pi \\omega } + 1\\right)/2.$ The signature factor $\\xi ^{(+)} (\\omega )$ guarantees that $T_{q,g}$ are invariant with respect to permutation of the incoming and outgoing external photons.",
"The integral representation (REF ) is the asymptotic form of the Sommerfeld-Watson transform but it is frequently addressed as the Mellin transform.",
"Throughout the paper we will name $T_{q,g}$ the Mellin amplitudes.",
"Explicit expressions for the Mellin amplitudes $F^{(col)}_{q,g}$ in the region B are: $F^{(col)}_q = C_q^{(+)} e^{\\Omega _{(+)} y} + C_q^{(-)} e^{\\Omega _{(-)} y},\\\\ \\nonumber F^{(col)}_g = C_g^{(+)} e^{\\Omega _{(+)} y} +C_g^{(-)} e^{\\Omega _{(-)} y}.$ Obtained in Ref.",
"[6] expressions for $\\Omega _{(\\pm )}, C_q^{(\\pm )}$ and $C_g^{(\\pm )}$ can be found in Appendix B where they are expressed through the on-shell parton-parton amplitudes $A_{rr^{\\prime }}$ ($r,r^{\\prime } = q,g$ ).",
"Graphs for amplitudes $A_{rr^{\\prime }}$ are depicted in Fig. 2.",
"Figure: Graphs for the parton-parton amplitudes.The Mellin transform for the parton-parton amplitudes is similar to the one in Eq.",
"(REF ): $A_{rr^{\\prime }} (s/\\mu ^2) = \\int _{-\\imath \\infty }^{\\imath \\infty } \\frac{d \\omega }{2 \\imath \\pi } \\left(s/\\mu ^2\\right)^{\\omega }\\xi ^{(+)} (\\omega ) f_{rr^{\\prime }} (\\omega ).$ Technically, it is more convenient to use the Mellin amplitudes $h_{rr^{\\prime }}$ which are proportional to $f_{rr^{\\prime }}$ : $h_{rr^{\\prime }}(\\omega ) = f_{rr^{\\prime }}(\\omega )/(8 \\pi ^2).$ Explicit expression for amplitudes $h_{rr^{\\prime }}$ can be found in Appendix A.",
"Let us discuss Eq.",
"(REF ).",
"It is easy to see that its structure exhibits a distinct similarity to the structure of the DGLAP description of $F^{(col)}_{q,g}$ , which is especially obvious when the approximation of fixed $\\alpha _s$ is used: $F^{DGLAP}_q &=& \\hat{C}_q^{(+)} e^{\\gamma _{(+)}^{DGLAP} y} + \\hat{C}_q^{(-)} e^{\\gamma _{(-)}^{DGLAP} y},\\\\ \\nonumber F^{DGLAP}_g &=& \\hat{C}_g^{(+)} e^{\\gamma _{(+)}^{DGLAP} y} +\\hat{C}_g^{(-)} e^{\\gamma _{(-)}^{DGLAP} y}.$ Indeed, the factors $\\Omega _{(\\pm )}$ in the exponents of Eq.",
"(REF ) are new anomalous dimensions instead of $\\gamma ^{DGLAP}_{(\\pm )}$ in Eq.",
"(REF ) while the factors $C_g^{(\\pm )}, C_q^{(\\pm )}$ are new coefficient functions instead of the DGLAP coefficient functions $\\hat{C}^{(\\pm )}_{q,g}$ .",
"However in contrast to DGLAP, the coefficient functions and the anomalous dimensions in (REF ) are calculated with the same means and they include the total resummations of the DL contributions.",
"The both coefficient functions and anomalous dimensions in (REF ) are made of amplitudes $h_{rr^{\\prime }}$ .",
"When the partonic amplitudes $h_{rr^{\\prime }}$ are replaced by their Born values, the integrands in Eq.",
"(REF ) coincide with the integrands of the LO DGLAP expressions in which the most singular in $\\omega $ terms are retained.",
"Further expansions of $h_{rr^{\\prime }}$ and substituting them in Eq.",
"(REF ) lead to the NLO (as well as to NNLO DGLAP, etc), where the most singular terms only are accounted for in each order in $\\alpha _s$ ."
],
[
"Extending the small-x Eqs. (",
"Expressions in Eqs.",
"(REF ,REF ) are defined in the region B.",
"Now we are going to obtain an interpolation formulae for amplitudes $T^{(col)}_{q,g}$ , which would reproduce the DGLAP expressions at large $x$ and at the same time would coincide with Eqs.",
"(REF ,REF ) at small $x$ .",
"Such extension can be obtained with the four steps: Step 1: Subtract the most singular in $\\omega $ terms (i.e.",
"the terms $\\sim \\alpha _s/\\omega $ in the case of LO DGLAP; $\\alpha ^2_s/\\omega ^3$ in the case of NLO DGLAP, etc.)",
"from the DGLAP anomalous dimensions $\\gamma _{(\\pm )}^{(DGLAP)} (\\omega , \\alpha _s)$ .",
"We denote $\\widetilde{\\gamma }^{DGLAP}_{(\\pm )}$ the result of such amputation.",
"Step 2: Add $\\widetilde{\\gamma }^{DGLAP}_{(\\pm )}$ to $\\Omega _{(\\pm )}$ .",
"The new anomalous dimensions $\\widetilde{\\Omega }_{(\\pm )} = \\Omega _{(\\pm )} + \\widetilde{\\gamma }^{DGLAP}_{(\\pm )}$ contain the total resummations of the double-logarithmic contributions.",
"They are essential at small $x$ and unimportant at large $x$ .",
"At the same time, (REF ) contains less singular in $1/\\omega ^n$ DGLAP contributions which are dominant at large $x$ .",
"Step 3: Subtract the LO DGLAP term $=1$ and the most singular in $\\omega $ terms (i.e.",
"the term 1 $\\sim \\alpha _s/\\omega ^2$ in the case of NLO DGLAP, $\\alpha ^2_s/\\omega ^4$ in the case of NNLO DGLAP, etc.)",
"from the DGLAP expressions for the coefficient functions.",
"We denote $\\Delta \\hat{C}^{(\\pm )}_{q,g}$ the result of such amputation.",
"Step 4: Add the results obtained in Step 3 to the DL expressions $C^{(\\pm )}_{q,g}$ , arriving thereby at new coefficient functions $\\widetilde{C}^{(\\pm )}_{q,g}$ : $\\widetilde{C}^{(\\pm )}_{q,g} = C^{(\\pm )}_{q,g} + \\Delta \\hat{C}^{(\\pm )}_{q,g}.$ The coefficient functions $\\widetilde{C}^{(\\pm )}_{q,g}$ include both DL contributions and the less singular DGLAP terms.",
"The subtractions in Steps 1,3 are necessary to avoid the double counting.",
"Replacing $\\Omega _{(\\pm )}$ and $C^{(\\pm )}_{q,g}$ in Eq.",
"(REF ) by $\\widetilde{\\Omega }_{(\\pm )}$ and $\\widetilde{C}^{(\\pm )}_{q,g}$ , we obtain the interpolation expressions $\\widetilde{T}^{(col)}_{q,g}$ for the Compton amplitudes : $\\widetilde{T}^{(col)}_q (x/\\beta , Q^2/\\mu ^2)&=& \\int _{-\\imath \\infty }^{\\imath \\infty } \\frac{d \\omega }{2 \\imath \\pi } \\left(x/\\beta \\right)^{-\\omega }\\xi ^{(+)} (\\omega ) \\left[\\widetilde{C}_q^{(+)} e^{\\widetilde{\\Omega }_{(+)} y} + \\widetilde{C}_q^{(-)} e^{\\widetilde{\\Omega }_{(-)} y}\\right],\\\\ \\nonumber T^{(col)}_g (x/\\beta , Q^2/\\mu ^2)&=& \\int _{-\\imath \\infty }^{\\imath \\infty } \\frac{d \\omega }{2 \\imath \\pi } \\left(x/\\beta \\right)^{-\\omega }\\xi ^{(+)} (\\omega )\\left[ \\widetilde{C}_g^{(+)} e^{\\widetilde{\\Omega }_{(+)} y} +\\widetilde{C}_g^{(-)} e^{\\widetilde{\\Omega }_{(-)} y} \\right].$ Eq.",
"(REF ) combines the small-$x$ evolution in DLA with the DGLAP results for the coefficient functions and anomalous dimensions, which are important at large $x$ .",
"These expressions can universally be used as the interpolation formulae for $T^{(col)}_{q,g}$ in the region $\\textbf {A} \\oplus \\textbf {B}$ ."
],
[
"Extending Eq. (",
"Extension of the expressions (REF ) to describe the amplitudes $\\widetilde{T}^{(col)}_{q,g}$ to the region $\\textbf {C} \\oplus \\textbf {D}$ is also not rigorous.",
"The standard approach suggested in Ref.",
"[9] and used since that in many papers (see e.g.",
"Ref.",
"[10]) to describe DIS at low $Q^2$ is to make a shift of $Q^2$ : $Q^2 \\rightarrow Q^2 + m^2,$ where the mass scale $m$ was fixed on basis of certain physical considerations, depending on specific situation.",
"In contrast to preceding papers, we proved in Ref.",
"[11] (see also the overview[12]) that the scale $m$ with DL accuracy can be unambiguously specified: the IR cut-off $\\mu $ plays the role of the scale $m$ .",
"Our proof was based on the well-known fact that DL contributions arise from integrations $\\sim d k^2_{i \\perp }/k^2_{i \\perp }$ over the transverse momenta $k_{i \\perp }$ of virtual partons, which requires introducing an IR cut-off $\\mu $ .",
"It can be introduced through the shift $k^2_{i \\perp } \\rightarrow k^2_{i \\perp } + \\mu ^2$ , which eventually leads to the shift (REF ), with $m = \\mu .$ The proof of (REF ) in Refs.",
"[11], [12] was done in the context of the spin-dependent structure function $g_1$ but it holds for amplitudes $\\widetilde{T}^{(col)}_{q,g}$ either.",
"Applying the Principle of Minimal Sensitivity[13], we estimated in Ref.",
"[6] the value of $\\mu $ of Eq.",
"(REF ): $\\mu = 2.3 \\Lambda _{QCD} \\approx 1 GeV.$ So, the universal expression for the Compton amplitudes $\\widetilde{T}^{(col)}_{q,g}$ valid in the region $\\textbf {A} \\oplus \\textbf {B}\\oplus \\textbf {C} \\oplus \\textbf {D}$ (i.e.",
"at arbitrary values of $x$ and $Q^2$ ) in the framework of Collinear Factorization is $\\widetilde{T}^{(col)}_q (x/\\beta , Q^2/\\mu ^2)&=& \\int _{-\\imath \\infty }^{\\imath \\infty } \\frac{d \\omega }{2 \\imath \\pi }\\left(\\widetilde{x}/\\beta \\right)^{-\\omega }\\xi ^{(+)} (\\omega ) \\left[\\widetilde{C}_q^{(+)} e^{\\widetilde{\\Omega }_{(+)} \\widetilde{y}} + \\widetilde{C}_q^{(-)} e^{\\widetilde{\\Omega }_{(-)} \\widetilde{y}}\\right],\\\\ \\nonumber \\widetilde{T}^{(col)}_g (x/\\beta , Q^2/\\mu ^2)&=& \\int _{-\\imath \\infty }^{\\imath \\infty } \\frac{d \\omega }{2 \\imath \\pi } \\left(\\widetilde{x}/\\beta \\right)^{-\\omega }\\xi ^{(+)} (\\omega )\\left[ \\widetilde{C}_g^{(+)} e^{\\widetilde{\\Omega }_{(+)} \\widetilde{y}} +\\widetilde{C}_g^{(-)} e^{\\widetilde{\\Omega }_{(-)} \\widetilde{y}} \\right],$ where we have denoted $\\widetilde{x} = x + \\mu ^2/w \\equiv x + x_0, ~~~~ \\widetilde{y} = \\ln \\left((Q^2 + \\mu ^2)/\\mu ^2\\right).$"
],
[
"Small-$x$ asymptotics of the Compton amplitudes and comparison with the DGLAP asymptotics",
"Asymptotics of $T^{(col)}_{q,g}$ at $x \\rightarrow 0$ and $Q^2 > \\mu ^2$ were considered in detail in Ref. [6].",
"In the present paper we briefly remind results of Ref.",
"[6] and compare these asymptotics with the asymptotics of the same amplitudes obtained in the DGLAP approach.",
"The standard mathematical tool to calculate the small-$x$ asymptotics of $T^{(col)}_{q,g}$ is Saddle Point method.",
"Applying this method to $T^{(col)}_{q,g}$ , we obtain that the small-$x$ asymptotics of amplitudes $T^{(col)}_{q,g}$ are of the Regge type.",
"They both have the same intercepts $\\omega _0$ : $T^{(col)}_q &\\sim & \\frac{\\Pi _q(\\omega _0)}{(\\ln (1/x))^{3/2}} x^{- \\omega _0} (Q^2/\\mu ^2)^{\\omega _0/2},~~T^{(col)}_g \\sim \\frac{\\Pi _g(\\omega _0)}{(\\ln (1/x))^{3/2}} x^{- \\omega _0} (Q^2/\\mu ^2)^{\\omega _0/2}.$ The factors $\\Pi _{q,g}(\\omega _0)$ in Eq.",
"(REF ) are called the impact factorsExplicit expressions for the impact factors can be found un Ref. [6].",
"They are the only difference between the asymptotics of $T^{(col)}_q $ and $T^{(col)}_g$ .",
"The value of the intercept $\\omega _0$ in Eq.",
"(REF ) proved to be dependent on accuracy of the calculations.",
"The maximal intercept $\\omega ^{DL}_H$ corresponded to the roughest approximation where the quark contributions were neglected and $\\alpha _s$ was fixed while the minimal intercept $\\omega ^{DL}_S$ corresponded to the most accurate calculation where both gluon and quark contributions were accounted for and $\\alpha _s$ was running: $\\omega ^{DL}_H = 1.35,~~\\omega ^{DL}_S = 1.07.$ As the both intercepts $> 1$ , the Reggeons of Eq.",
"(REF ) with the intercepts (REF ) are the DL contributions to Pomeron, or DL Pomerons.",
"In accordance with the conventional Pomeron terminology we call them hard (with the subscript H) and soft (with the subscript S) DL Pomerons respectively.",
"It is interesting fact that $\\omega ^{DL}_H$ is close to the LO BFLK Pomeron intercept and $\\omega ^{DL}_S$ almost coincides with the NLO BFLK Pomeron intercept despite that BFKL have nothing in common with resummation of DL contributions.",
"The asymptotics in Eq.",
"(REF ) are given by much simpler expressions than the parent amplitudes in Eq.",
"(REF ).",
"However, the asymptotics should not be used outside their applicability region.",
"It was shown in Ref.",
"[6] that the asymptotics should be used at $x < x_{max}$ and estimates of $x_{max}$ at various values of $Q^2$ were obtained.",
"In particular, at $Q^2 = 10$ GeV$^2$ $x_{max} = 10^{-6}.$ It was shown in Ref.",
"[6] that the greater is $Q^2$ , the less is $x_{max}$ .",
"When $x \\ge x_{max}$ the parent amplitudes $T_{q,g}$ should be used instead of the asymptotics."
],
[
"Remark on Asymptotic Scaling",
"Eq.",
"(REF ) can be written in such a way: $T^{(col)}_q \\sim \\frac{\\Pi _q(\\omega _0)}{\\ln ^{3/2} (1/x)}\\left(\\zeta /\\mu ^2\\right)^{\\omega _0/2},~~T^{(col)}_g \\sim \\frac{\\Pi _g(\\omega _0)}{\\ln ^{3/2} (1/x)}\\left(\\zeta /\\mu ^2\\right)^{\\omega _0/2},$ with $\\zeta = Q^2/x^2.$ Although both $T^{(col)}_q$ and $T^{(col)}_g$ by definition depend on two independent variables $x$ and $Q^2$ , Eq.",
"(REF ) manifests that $T^{(col)}_{q,g} \\ln ^{3/2} (1/x)$ asymptotically depend on the single variable $\\zeta $ only.",
"We name this remarkable property the Asymptotic Scaling.",
"This property is absent in the DGLAP approach.",
"Indeed, DGLAP predicts that the $x$ - and $Q^2$ - dependence of $T^{(col)}_{q,g}$ at $x \\rightarrow 0$ are unrelated: $T^{(DGLAP)}_{q,g} \\sim x^{-a} (Q^2/\\mu ^2)^{\\gamma _{DGLAP}/b},$ where $\\gamma _{DGLAP}$ is the anomalous dimension and $b$ is the first coefficient of the $\\beta $ -function.",
"The intercept $a$ has the phenomenological origin: it is generated by the terms $x^{-a}$ in the fits for the initial parton densities."
],
[
"Perturbative power $Q^2$ -contributions",
"We begin studying the power $Q^2$ -corrections with considering $T^{(col)}_q$ at the large-$Q^2$ region, where $Q^2 \\gg \\mu ^2$ and where the logarithmic variable $\\widetilde{y}$ defined in (REF ) can be expanded in the series as follows: $\\widetilde{y} = y + \\sum _{n=1}^{\\infty } c_n \\left(\\frac{\\mu ^2}{Q^2}\\right)^n.$ Being substituted in Eq.",
"(REF ), it allows us to write $T^{(col)}_q$ as $\\widetilde{T}^{(col)}_q (x/\\beta , \\widetilde{Q}^2) = \\widetilde{T}^{(col)}_q (x/\\beta ,Q^2) +\\left(\\frac{\\mu ^2}{Q^2}\\right) C_1 +\\left(\\frac{\\mu ^2}{Q^2}\\right)^2 C_2 +...$ The first term in the r.h.s.",
"of Eq.",
"(REF ) describes the Compton amplitude in the region $\\textbf {A}\\otimes \\textbf {B}$ .",
"At large $x$ it is given by the DGLAP formulae whereas the second and third terms can wrongly be interpreted as contributions of the higher twists, probably of the non-perturbative origin though actually they are altogether perturbative: $C_1 = \\frac{\\partial \\widetilde{T}^{(col)}_q (x/\\beta ,y)}{\\partial y},~~C_2 = \\frac{1}{2} \\frac{\\partial ^2 \\widetilde{T}^{(col)}_q (x/\\beta ,y)}{\\partial y^2}.$ This example demonstrates that ignoring the shift (REF ) and interpretating all terms inversely proportional to $Q^2$ as an impact of higher twist could be misleading.",
"On the other, at $Q^2 < \\mu ^2$ the expansion (REF ) is replaced by another one: $\\widetilde{y} = \\sum _{n=1}^{\\infty } c^{\\prime }_n \\left(\\frac{Q^2}{\\mu ^2}\\right)^n.$ It leads to the small-$Q^2$ expression for $\\widetilde{T}^{(col)}_q$ : $\\widetilde{T}^{(col)}_q (x/\\beta , \\widetilde{Q}^2)= \\sum _{n-1}^{\\infty } \\left(\\frac{Q^2}{\\mu ^2}\\right)^n C^{\\prime }_n$ which decreases down to zero when $Q^2 \\rightarrow 0$ .",
"It explains very well why the terms $\\sim 1/(Q^2)^n$ are never seen at small $Q^2$ where, naively, their impact could be great.",
"We remind that our estimate of $\\mu $ is $\\mu \\approx 1$ GeV (see Eq.",
"(REF )).",
"Disappearing of the terms $\\sim 1/(Q^2)^n$ at $Q^2 \\lesssim 1$ GeV obtained from analysis of experimental data could confirm correctness of our reasoning."
],
[
" Calculating perturbative Compton amplitudes in $K_T$ Factorization",
"Originally, $K_T$ Factorization was introduced in Refs.",
"[2], [3] to make possible applying BFKL for description of hadronic reactions.",
"It implied the kinematic region of very high, or asymptotic, energies, i.e.",
"at $x \\ll 1$ .",
"However, in the cases when BFKL is not involved one can use $K_T$ Factorization at $x \\sim 1$ as well.",
"In the present Sect.",
"we generalize DL expressions for $T^{(col)}_{q,g}$ to $K_T$ Factorization.",
"In the framework of $K_T$ Factorization Eq.",
"(REF ) takes the following form: $T^{(KT)} = \\int _x^1 \\frac{d \\beta }{\\beta }\\int _{\\mu ^2}^w \\frac{d k^2_{\\perp }}{k^2_{\\perp }}\\left[ T^{(KT)}_q(x/\\beta ,Q^2,k^2_{\\perp },\\mu ^2) \\Phi ^{(KT)}_q (\\beta ,k^2_{\\perp },\\mu ^2)+ T^{(KT)}_g(x/\\beta ,Q^2,k^2_{\\perp },\\mu ^2) \\Phi ^{(KT)}_g (\\beta ,k^2_{\\perp },\\mu ^2)\\right],$ where the superscript $\"KT\"$ refers to $K_T$ -Factorization, $\\mu $ is the factorization scale, $\\beta $ is the longitudinal fraction of the external partons (see Fig.",
"1) and $k^2_{\\perp }$ stands for their transverse momentum.",
"$\\Phi ^{(KT)}_{q,g}$ denote initial parton distributions and amplitudes $T^{(KT)}_{q,g}$ are perturbative.",
"The subscripts $\"q,g\"$ refer to quarks and gluons in the same way as in Sect. II.",
"We are going to calculate $T^{(KT)}_{q,g}$ in DLA.",
"DL contributions to them arrive from the kinematic region $w \\gg Q^2, k^2_{\\perp } \\gg \\mu ^2.$ However, amplitudes $T^{(KT)}_{q,g}$ in DLA cannot be described by a single universal expression at different virtualities $k^2_{\\perp }$ in (REF ).",
"There are two regions, where they are given by different expressions: Region E: Moderate-virtual $k^2_{\\perp }$ , where $k^2_{\\perp } \\ll \\mu ^2/x.$ Region F: Deeply-virtual $k^2_{\\perp }$ , where $k^2_{\\perp } \\gg \\mu ^2/x.$ In order to avoid confusions, we will denote $T^{(\\textbf {E})}_{q,g}$ and $T^{(\\textbf {F})}_{q,g}$ the amplitudes $T^{(KT)}_{q,g}$ in the regions E and F respectively.",
"We will use the Mellin transform for $T^{(\\textbf {E,F)}}_{q,g}$ in the following form: $T^{(\\textbf {E})}_{q,g} = \\int _{-\\imath \\infty }^{\\imath \\infty } \\frac{d \\omega }{2 \\imath \\pi } \\left(w/\\mu ^2\\right)^{\\omega }\\xi ^{(+)} (\\omega ) \\varphi ^{(\\textbf {E})}_{q,g}(\\omega , y, z),\\\\ \\nonumber T^{(\\textbf {F})}_{q,g} = \\int _{-\\imath \\infty }^{\\imath \\infty } \\frac{d \\omega }{2 \\imath \\pi } \\left(w/\\mu ^2\\right)^{\\omega }\\xi ^{(+)} (\\omega ) \\varphi ^{(\\textbf {F})}_{q,g}(\\omega , y, z),$ where $y$ is given by Eq.",
"(REF ) and the new variable $z = \\ln (k^2_{\\perp }/\\mu ^2)$ describes dependence of $T^{(KT)}_{q,g}$ on $k_{\\perp }$ in the kinematics (REF ).",
"This dependence differs $T^{(KT)}_{q,g}$ from amplitudes $T^{(col)}_{q,g}$ considered in Sect.",
"II in Collinear Factorization.",
"In addition to $y$ and $z$ , it is convenient to introduce one more logarithmic variable: $\\rho = \\ln (w/\\mu ^2).$ In terms of the logarithmic variables regions E and F defined in Eqs.",
"(REF ,REF ) look as follows: Region E: $\\rho > y + z$ , Region F: $\\rho < y + z$ ."
],
[
"Calculating the off-shell Compton amplitudes in region E",
"IREEs for $\\varphi ^{(\\textbf {E})}_{q,g}$ have the following form: $\\frac{\\partial \\varphi ^{(\\textbf {E})}_q }{\\partial y} + \\frac{\\partial \\varphi ^{(\\textbf {E})}_q }{\\partial z} +\\omega \\varphi ^{(\\textbf {E})}_q = F^{(col)}_q (\\omega ,y) H_{qq} (\\omega ,z)+ F^{(col)}_g (\\omega ,y) H_{gq} (\\omega ,z),\\\\ \\nonumber \\frac{\\partial \\varphi ^{(\\textbf {E})}_g}{\\partial y} +\\frac{\\partial \\varphi ^{(\\textbf {E})}_g}{\\partial z} +\\omega \\varphi ^{(\\textbf {E})}_g= F^{(col)}_q (\\omega , y) H_{qg} (\\omega ,z)+ F^{(col)}_g (\\omega ,y) H_{gg} (\\omega ,z).$ The l.h.s.",
"of IREEs in (REF ) are obtained with applying the differential operator $-\\mu ^2 d/d \\mu ^2 = \\partial /\\partial \\rho + \\partial /\\partial y + \\partial /\\partial z$ to Eq.",
"(REF ).",
"As a result, the l.h.s.of Eq.",
"(REF ) contain explicitly the derivatives over $y$ and $z$ while the factor $\\omega $ corresponds to differentiation of the Mellin factor $(w/\\mu ^2)^{\\omega }$ .",
"The convolutions in r.h.s.",
"of (REF ) are similar in structure to the convolutions in the DGLAP evolution equations.",
"They involve amplitudes $F^{(col)}_q$ which were considered in Sect.",
"II and amplitudes $H_{rr^{\\prime }}$ which are still unknown.",
"We will calculate amplitudes $H_{rr^{\\prime }}$ in the next Sect.",
"In order to specify general solutions to Eq.",
"(REF ), we will use the matching $T^{(\\textbf {E})}_q (w,y,z)|_{z = 0} = T^{(col)}_q (\\omega , y),~~~T^{(\\textbf {E})}_g(w,y,z)|_{z = 0} = T^{(col)}_g (\\omega , y),$ with $T^{(col)}_{q,g}$ defined in Eqs.",
"(REF ).",
"The matching (REF ) implies the following hierarchy between $y$ and $z$ : $y > z.$ The requirement (REF ) is temporary.",
"The final expressions for $T^{(\\textbf {E})}_{q,g}$ will be written in the form free of this hierarchy.",
"Solving Eq.",
"(REF ) goes easier when $y$ and $z$ are replaced by new variables $\\xi , \\eta $ : $\\xi = \\frac{1}{2} (y + z),~~~\\eta = \\frac{1}{2} (y - z).$ Obviously, the inverse transform is $y = \\xi + \\eta ,~~~z = \\xi - \\eta .$ In terms of $\\xi ,\\eta $ Eq.",
"(REF ) looks simpler: $\\frac{\\partial \\varphi ^{(\\textbf {E})}_q }{\\partial \\xi } +\\omega \\varphi ^{(\\textbf {E})}_q = F^{(col)}_q (\\omega ,y) H_{qq} (\\omega ,z)+ F^{(col)}_g (\\omega ,y) H_{gq} (\\omega ,z),\\\\ \\nonumber \\frac{\\partial \\varphi ^{(\\textbf {E})}_g}{\\partial \\xi } +\\omega \\varphi ^{(\\textbf {E})}_g= F^{(col)}_q (\\omega , y) H_{qg} (\\omega ,z)+ F^{(col)}_g (\\omega ,y) H_{gg} (\\omega ,z).$ Solution to Eq.",
"(REF ) is: $\\varphi ^{(\\textbf {E})}_q = e^{-\\omega \\xi } \\left[T^{(col)}_q (\\omega , \\eta ) +\\int _{\\eta }^{\\xi } d v e^{\\omega v}\\left(F^{(col)}_q (\\omega ,y^{\\prime }) H_{qq} (\\omega ,z^{\\prime })+ F^{(col)}_g (\\omega ,y^{\\prime }) H_{gq} (\\omega ,z^{\\prime })\\right)\\right],\\\\ \\nonumber \\varphi ^{(\\textbf {E})}_g = e^{-\\omega \\xi } \\left[T^{(col)}_g (\\omega , \\eta ) +\\int _{\\eta }^{\\xi } d v e^{\\omega v}\\left(F^{(col)}_q (\\omega ,y^{\\prime }) H_{qg} (\\omega ,z^{\\prime })+ F^{(col)}_g (\\omega ,y^{\\prime }) H_{gg} (\\omega ,z^{\\prime })\\right)\\right],$ with the variables $y^{\\prime },z^{\\prime }$ defined as follows: $y^{\\prime } = v + \\eta ,~~z^{\\prime } = v - \\eta .$ In order to lift the relation between $y$ and $z$ of Eq.",
"(REF ) we replace $\\eta $ with $|\\eta |$ everywhere save Eq.",
"(REF ).",
"Doing so and substituting Eq.",
"(REF ) in (REF ), we arrive at expressions for amplitudes $T^{(\\textbf {E})}_{q,g}$ in region E: $T^{(\\textbf {E})}_q = \\int _{-\\imath \\infty }^{\\imath \\infty } \\frac{d \\omega }{2 \\imath \\pi }\\xi ^{(+)} (\\omega ) \\left(\\frac{s}{\\sqrt{Q^2 k^2_{\\perp }}}\\right)^{\\omega }\\left[T^{(col)}_q (\\omega , |\\eta |) +\\right.\\\\ \\nonumber \\left.\\int _{|\\eta |}^{\\xi } d v e^{\\omega v}\\left(F^{(col)}_q (\\omega ,y^{\\prime }) H_{qq} (\\omega ,z^{\\prime })+ F^{(col)}_g (\\omega ,y^{\\prime }) H_{gq} (\\omega ,z^{\\prime })\\right)\\right],\\\\ \\nonumber T^{(\\textbf {E})}_g = \\int _{-\\imath \\infty }^{\\imath \\infty } \\frac{d \\omega }{2 \\imath \\pi }\\xi ^{(+)} (\\omega ) \\left(\\frac{s}{\\sqrt{Q^2 k^2_{\\perp }}}\\right)^{\\omega }\\left[T^{(col)}_g (\\omega , |\\eta |) +\\right.\\\\ \\nonumber \\left.\\int _{|\\eta |}^{\\xi } d v e^{\\omega v}\\left(F^{(col)}_g (\\omega ,y^{\\prime }) H_{qg} (\\omega ,z^{\\prime })+ F^{(col)}_g (\\omega ,y^{\\prime }) H_{gg} (\\omega ,z^{\\prime })\\right)\\right].$"
],
[
"Calculating the off-shell Compton amplitudes in region F",
"IREEs for $T^{(\\textbf {F})}_{q,g}$ are very simple: $\\frac{\\partial T^{(\\textbf {F})}_q}{\\partial \\rho } + \\frac{\\partial T^{(\\textbf {F})}_q}{\\partial y}+ \\frac{\\partial T^{(\\textbf {F})}_q}{\\partial z} = 0,\\\\ \\nonumber \\frac{\\partial T^{(\\textbf {F})}_g}{\\partial \\rho } + \\frac{\\partial T^{(\\textbf {F})}_g}{\\partial y}+ \\frac{\\partial T^{(\\textbf {F})}_g}{\\partial z} = 0.$ We do not use the Mellin transform for $T^{(\\textbf {F})}_{q,g}$ and because of it the l.h.s.",
"of (REF ) contain derivatives over $\\rho $ , $y$ and $z$ .",
"Integration over momenta of all virtual partons in the deeply-virtual region F do not bring any dependenceA detailed derivation of IREEs in the case of Deeply-Virtual kinematics can be found in Ref. [14].",
"of $T^{(\\textbf {F})}_{q,g}$ on $\\mu $ .",
"As a result, the r.h.s.",
"of Eq.",
"(REF ) are zeros.",
"A general solution to Eq.",
"(REF ) is $T^{(\\textbf {F})}_q = \\Psi _q \\left((\\rho - y),(\\rho - z)\\right),~~T^{(\\textbf {F})}_g = \\Psi _g \\left((\\rho - y),(\\rho - z)\\right),$ with $\\Psi _q$ and $\\Psi _g$ being arbitrary functions.",
"In order to specify them we use the following matching: Amplitudes $T^{(\\textbf {F})}_{q,g}$ coincide with amplitudes $T^{(\\textbf {E})}_{q,g}$ on the border surface between regions E and F, where $\\rho = y + z$ : $T^{(\\textbf {F})}_q (y,z) = \\bar{T}^{(\\textbf {E})}_q (y,z), ~~T^{(\\textbf {F})}_g (y,z) = \\bar{T}^{(\\textbf {E})}_g (y,z),$ with the new amplitudes $\\bar{T}^{(\\textbf {E})}_{q,g}$ defined as follows: $\\bar{T}^{(\\textbf {E})}_q (y,z) = T^{(\\textbf {E})}_q (\\rho ,y,z)|_{\\rho = y + z},~~\\bar{T}^{(\\textbf {E})}_g (y,z) = T^{(\\textbf {E})}_g (\\rho ,y,z)|_{\\rho = y + z}.$ According to Eq.",
"(REF ) the expression for $T^{(\\textbf {F})}_q, T^{(\\textbf {F})}_g$ in the whole region F can be obtained from amplitudes $\\bar{T}^{(\\textbf {E})}_{q,g}$ by the simple change of their arguments: $T^{(\\textbf {F})}_q (\\rho , y, z) = \\bar{T}^{(\\textbf {E})}_q (\\rho -y, \\rho - z),~~~T^{(\\textbf {F})}_g (\\rho , y, z) = \\bar{T}^{(\\textbf {E})}_g (\\rho -y, \\rho - z).$ Both $\\rho - y$ and $\\rho - z$ do not depend on the IR cut-off $\\mu $ , so amplitudes $T^{(\\textbf {F})}_{q,g}$ are IR stable."
],
[
"Extension of amplitudes $T^{(\\textbf {E,F})}_{q,g}$ to the region of small {{formula:faef11c1-996a-4f78-a44e-22c9867e40d4}}",
"Extension of $T^{(\\textbf {E})}_{q,g}$ to the small-$Q^2$ region can be done with the shift of $Q^2$ like it was done in Collinear Factorization.",
"As a result, amplitudes $T^{(\\textbf {E})}_{q,g}$ can be extended to the small-$Q^2$ region with replacement of $x, y$ in Eq.",
"(REF ) by $\\widetilde{x},\\widetilde{y}$ defined in Eq.",
"(REF ).",
"A similar extension of $T^{(\\textbf {F})}_{q,g}$ is impossible because it would involve partons with virtualities $k^2_{\\perp } > w$ which contradicts to Eq.",
"(REF )."
],
[
"Extension of amplitudes $T^{(\\textbf {E,F})}_{q,g}$ to Basic Factorization",
"It follows from Ref.",
"[4] that extension of Compton amplitudes $T^{(\\textbf {E})}_{q,g}$ and $T^{(\\textbf {F})}_{q,p}$ defined in Eqs.",
"(REF ,REF ) to Basic factorization can be done with the very simple replacement: it is necessary and sufficient to replace $k^2_{\\perp }$ by $|k^2|$ in Eqs.",
"(REF ,REF )."
],
[
"Off-shell parton-parton amplitudes $H_{rr^{\\prime }}$",
"Expressions for Compton amplitudes $T^{(KT)}_{q,g}$ in Eqs.",
"(REF ,REF ) include off-shell Mellin amplitudes $H_{rr^{\\prime }}$ .",
"Below we calculate them in DLA.",
"They stem from the Mellin transform for off-shell parton-parton amplitudes $\\widetilde{A}_{rr^{\\prime }}$ : $\\widetilde{A}_{rr^{\\prime }} (w,z) = \\int _{- \\imath \\infty }^{\\imath \\infty } \\frac{d \\omega }{2 \\pi \\imath } e^{\\omega \\rho }\\xi ^{(\\omega )} \\widetilde{f}_{rr^{\\prime }} (\\omega ,z)$ and the following definition (cf.",
"(REF )): $H_{rr^{\\prime }} (\\omega ,z) = \\frac{1}{8 \\pi ^2} \\widetilde{f}_{rr^{\\prime }} (\\omega ,z).$ Amplitudes $H_{rr^{\\prime }} (\\omega ,z)$ can also be found with constructing and solving appropriate IREEs.",
"The IREEs for the off-shell $H_{rr^{\\prime }}$ are quite similar to Eq.",
"(REF ): $\\frac{\\partial }{\\partial z} H_{qq} + \\omega H_{qq} &=& h_{qq} H_{qq} + h_{qg} H_{gq},\\\\ \\nonumber \\frac{\\partial }{\\partial z} H_{gq} + \\omega H_{gq} &=& h_{gq} H_{qq} + h_{gg} H_{gq},\\\\ \\nonumber \\frac{\\partial }{\\partial z} H_{qg} + \\omega H_{qg} &=& h_{qq} H_{qg} + h_{qg} H_{gg},\\\\ \\nonumber \\frac{\\partial }{\\partial z} H_{gg} + \\omega H_{gg} &=& h_{gq} H_{qg} + h_{gg} H_{gg},$ where l.h.s.",
"of each equation corresponds to applying operator $-\\mu ^2 d/d \\mu ^2$ to Eq.",
"(REF ) while each r.h.s involves convolutions of $H_{rr^{\\prime }}$ and $h_{rr^{\\prime }}$ .",
"Specifying the general solution to Eq.",
"(REF ) should be done with using the matching: $H_{rr^{\\prime }}|_{z = 0} = h_{rr^{\\prime }}.$ Solving (REF ) and using the matching (REF ), we obtain the following expressions for $H_{rr^{\\prime }}$ : $H_{qq} &=& e^{- \\omega z} \\left[C_1 e^{\\Omega _{(+)} z} + C_2 e^{\\Omega _{(-)} z}\\right],\\\\ \\nonumber H_{gq} &=& e^{- \\omega z} \\left[\\frac{h_{qq} - \\Omega _{(+)}}{h_{qg}}~ C_1 e^{\\Omega _{(+)} z} +\\frac{h_{qq} - \\Omega _{(-)}}{h_{qg}}~C_2 e^{\\Omega _{(-)} z}\\right],\\\\ \\nonumber H_{gg} &=& e^{- \\omega z} \\left[C^{\\prime }_1 e^{\\Omega _{(+)} z} + C^{\\prime }_2 e^{\\Omega _{(-)} z}\\right],\\\\ \\nonumber H_{qg} &=& e^{- \\omega z} \\left[\\frac{h_{gg} - \\Omega _{(+)}}{h_{gq}}~ C^{\\prime }_1 e^{\\Omega _{(+)} z} +\\frac{h_{gg} - \\Omega _{(-)}}{h_{gq}}~C^{\\prime }_2 e^{\\Omega _{(-)} z}\\right].$ Explicit expressions for the terms $\\Omega _{(\\pm )}$ are presented in Eq.",
"(REF ) while $C_{1,2}$ and $C^{\\prime }_{1,2}$ are defined in Eq.",
"(REF ).",
"The overall factor $e^{- \\omega z}$ in Eq.",
"(REF ) converts the IR-dependent Mellin factor $(w/\\mu ^2)^{\\omega }$ of Eq.",
"(REF ) in the IR-stable factor $\\left(w/k^2_{\\perp }\\right)^{\\omega }$ ."
],
[
"Photoproduction amplitudes",
"It follows from Eqs.",
"(REF ,REF ) that the perturbative components $A_{\\gamma q}, A_{\\gamma g}$ of the photoproduction are related to the perturbative Compton amplitudes $T_{q,g}$ in a simple manner: $A_{\\gamma q} = T_q|_{q^2 = 0},~~~A_{\\gamma g} = T_g|_{q^2 = 0}.$ Eq.",
"(REF ) holds in any form of QCD factorization but expressions for the photoproduction amplitudes are different in different forms of factorizations.",
"We start with obtaining them in Collinear Factorization."
],
[
"Photoproduction amplitudes in Collinear kinematics",
"According to Eq.",
"(REF ), putting $q^2 = 0$ in Eqs.",
"(REF , REF ) should yield $A^{(col)}_{\\gamma q}$ and $A^{(col)}_{\\gamma q}$ in DLA.",
"However, such procedure cannot be done in the straightforward way because the Mellin amplitudes $F^{(col)}_{q,g}$ in (REF ) were obtained in Ref.",
"[6] for $Q^2 \\gg \\mu ^2$ only and they cannot be used at $Q^2 < \\mu ^2$ .",
"In order to describe $A^{(col)}_{q,g}$ at low $Q^2$ , including $Q^2 = 0$ , we use the prescription in the Sect.",
"II and replace $x,y$ in Eqs.",
"(REF , REF ) by $\\widetilde{x},\\widetilde{y}$ defined in Eq.",
"(REF ).",
"Then, putting $Q^2 = 0$ in $\\widetilde{x},\\widetilde{y}$ , we obtain $\\widetilde{x}|_{q^2 = 0} \\equiv x_0 = \\mu ^2/w,~~~\\widetilde{y}|_{q^2 = 0} \\equiv y_0 = 0.$ Replacement of $\\widetilde{x},\\widetilde{y}$ by $x_0, y_0$ in Eqs.",
"(REF , REF ) allows us to obtain photoproduction amplitudes in DLA in Collinear Factorization: Combining Eqs.",
"(REF ,REF ) and (REF ), we obtain: $A^{(col)}_{\\gamma q} &=& a_{\\gamma q} \\int _{-\\imath \\infty }^{\\imath \\infty } \\frac{d \\omega }{2 \\imath \\pi } \\left(x_0/ \\beta \\right)^{-\\omega }\\xi ^{(+)} (\\omega ) f^{(col)}_{\\gamma q} (\\omega ),\\\\ \\nonumber A^{(col)}_{\\gamma g} &=& a_{\\gamma q} \\int _{-\\imath \\infty }^{\\imath \\infty } \\frac{d \\omega }{2 \\imath \\pi } \\left(x_0/ \\beta \\right)^{\\omega } \\xi ^{(+)} (\\omega ) f^{(col)}_{\\gamma g} (\\omega ),$ with $f^{(col)}_{\\gamma q} &=& a_{\\gamma q} \\frac{ (\\omega - h_{gg})}{G(\\omega )},\\\\ \\nonumber f^{(col)}_{\\gamma g} &=& a_{\\gamma q} \\frac{ h_{qg}}{G(\\omega )},$ where $a_{\\gamma q}$ is the averaged photon-quark coupling $a_{\\gamma q} = \\bar{e_q^2}$ , $x_0 = \\mu ^2/w$ and $G = (\\omega - h_{qq})(\\omega - h_{gg})- h_{gg}h_{qg}.$ Accounting for non-DL radiative correction in $A^{(col)}_{\\gamma q}, A^{(col)}_{\\gamma g}$ can be done by the same way as it was done in Sec.",
"IIB for the Compton amplitudes."
],
[
"Photoproduction amplitudes in $K_T$ - Factorization ",
"DL contributions to the perturbative photoproduction amplitudes $A^{(KT)}_{\\gamma q}(s, k^2_{\\perp })$ and $A^{(KT)}_{\\gamma g}(s, k^2_{\\perp })$ come from the region $w \\gg k^2_{\\perp }.$ We will use the Mellin transform for $A^{(KT)}_{\\gamma q}$ and $A^{(KT)}_{\\gamma g}$ in the following form: $A^{(KT)}_{\\gamma q} &=& \\int _{-\\imath \\infty }^{\\imath \\infty } \\frac{d \\omega }{2 \\imath \\pi } \\left( w/\\mu ^2\\right)^{\\omega }\\xi ^{(+)} (\\omega ) F^{(KT)}_{\\gamma q} (\\omega , z),\\\\ \\nonumber A^{(KT)}_{\\gamma g} &=& \\int _{-\\imath \\infty }^{\\imath \\infty } \\frac{d \\omega }{2 \\imath \\pi } \\left(w/\\mu ^2\\right)^{\\omega }\\xi ^{(+)} (\\omega ) F^{(KT)}_{\\gamma g} (\\omega , z).$ IREE for the Mellin amplitudes $F_{\\gamma q} (\\omega , z),F_{\\gamma g} (\\omega , z)$ are similar to Eq.",
"(REF ): $\\frac{\\partial }{\\partial z} F^{(KT)}_{\\gamma q} (\\omega ,z) + \\omega F^{(KT)}_{\\gamma q} (\\omega ,z) = f^{(col)}_{\\gamma q} (\\omega ) H_{qq} (\\omega ,z)+ f^{(col)}_{\\gamma g} (\\omega ) H_{gq} (\\omega ,z),\\\\ \\nonumber \\frac{\\partial }{\\partial z} F^{(KT)}_{\\gamma g} (\\omega ,z) + \\omega F^{(KT)}_{\\gamma g} (\\omega ,z) = f^{(col)}_{\\gamma q} (\\omega ) H_{qg} (\\omega ,z)+ f^{(col)}_{\\gamma g} (\\omega ) H_{gg} (\\omega ,z).$ Expressions for on-shell amplitudes $f^{(col)}_{\\gamma q} $ and $f^{(col)}_{\\gamma g} $ in the r.h.s.",
"of Eq.",
"(REF ) are given by Eq.",
"(REF ) while new off-shell parton-parton amplitudes $H_{rr^{\\prime }} (\\omega ,z)$ are unknown and should be specified.",
"The general solution to Eq.",
"(REF ) should be specified by the matching: $A^{(KT)}_{\\gamma q}(w,z) |_{z = 0} = A^{(col)}_{\\gamma q} (w).$ Solving Eq.",
"(REF ) and using the matching of Eq.",
"(REF ) yields expressions for $A^{(KT)}_{\\gamma q}$ and $A^{(KT)}_{\\gamma g}$ : $A^{(KT)}_{\\gamma q} &=& \\int _{-\\imath \\infty }^{\\imath \\infty } \\frac{d \\omega }{2 \\imath \\pi } \\left( w/k^2_{\\perp }\\right)^{\\omega }\\xi ^{(+)} (\\omega )\\left[f^{(col)}_{\\gamma q}\\left(1 +\\int _0^z d u e^{ \\omega u} H_{qq} (\\omega ,z^{\\prime })\\right) +f^{(col)}_{\\gamma g} \\int _0^z d u e^{ \\omega u} H_{gq} (\\omega ,z^{\\prime })\\right],\\\\ \\nonumber A^{(KT)}_{\\gamma g} &=& \\int _{-\\imath \\infty }^{\\imath \\infty } \\frac{d \\omega }{2 \\imath \\pi } \\left( w/k^2_{\\perp }\\right)^{\\omega }\\xi ^{(+)} (\\omega )\\left[f^{(col)}_{\\gamma q} \\int _0^z d u e^{ \\omega u} H_{qg} (\\omega ,z^{\\prime }) +f^{(col)}_{\\gamma g}\\left(1 + \\int _0^z d u e^{ \\omega u} H_{gg} (\\omega ,z^{\\prime })\\right)\\right].$ We remind that expressions for amplitudes $f^{(col)}_{\\gamma q}$ and $f^{(col)}_{\\gamma g}$ can are obtained in Eq.",
"(REF ) while $H_{rr^{\\prime }}$ are defined in Eq.",
"(REF ).",
"To conclude this Sect.",
"we notice that non-DL corrections can be incorporated in Eqs.",
"(REF , REF ) absolutely similarly to the prescription of Sect. IIB.",
"Transition from $K_T$ Factorization to Basic one is done with replacement of $k^2_{\\perp }$ by $|k^2|$ ."
],
[
"Conclusion",
"In this paper we have studied first the perturbative amplitudes $T_{q,g}$ of elastic Compton scattering off quarks and gluons and then perturbative components $A_{\\gamma q},A_{\\gamma g}$ of the photoproduction amplitudes.",
"$T_{q,g}$ were calculated in Ref.",
"[6] in DLA at small $x$ and large photon virtualities $Q^2$ (i.e.",
"in kinematic region B) in the framework of Collinear Factorization .",
"We converted these results in expressions which can be used at arbitrary $x$ and $Q^2$ in Collinear, $K_T$ and Basic Factorizations.",
"Extension to Region A was done with combining the total resummation of DL contributions and DGLAP description of $T_{q,g}$ .",
"By doing so we obtained in Eq.",
"(REF ) the interpolation expressions reproducing $T_{q,g}$ in DLA at small $x$ and coinciding with the DGLAP description of $T_{q,g}$ when $x$ are not small.",
"Using the shift of Eq.",
"(REF ), we extended description $T_{q,g}$ to small $Q^2$ .",
"As a result, we arrived at Eq.",
"(REF ) to the expressions for $T_{q,g}$ in Collinear Factorization, which can be used at arbitrary $x$ and $Q^2$ .",
"In contrast to Collinear Factorization, $T_{q,g}$ are the essentially off-shell in $K_T$ Factorization and they cannot be described in DLA by a single expression valid at arbitrary values of virtualities $k^2_{\\perp }$ of the external partons.",
"It made us consider separately regions of moderate (Eq.",
"(REF )) and deep (Eq.",
"REF ) virtualities and obtain expressions Eqs.",
"(REF ) and (REF ) for $T_{q,g}$ in those regions.",
"We obtained them by constructing appropriate IREEs and solving them.",
"After transition from Region B to the to low $Q^2$ Region C has been studied, we became able to obtain explicit expressions (REF ) and (REF ) for the photoproduction amplitudes $A_{\\gamma q}, A_{\\gamma g}$ in Collinear and $K_T$ Factorizations respectively.",
"Small-$x$ asymptotics of amplitudes $T_{q,g}$ are of the Regge type with the same intercept in any form of QCD factorization.",
"Because of it we considered such asymptotics in Collinear Factorization (see Eq.",
"(REF )) and discussed dependence of its intercept on accuracy of the calculations: the higher is the accuracy, the lesser is the intercept.",
"In other words, the hard Pomeron becomes the soft one, when the accuracy grows.",
"We think that further increasing the accuracy can lead to vanishing supercritical Pomeron(s) and restoration of the Unitarity.",
"The next interesting point is that when $T_{q,g}$ are calculated in DLA, their asymptotics depends on the single variable $\\zeta = Q^2/x^2$ as shown in Eq.",
"(REF ).",
"Neither DGLAP nor BFKL cause such dependence.",
"It is important to use the Regge asymptotics within their applicability region, i.e.",
"at $x < x_{max}$ , with $x_{max}$ given by Eq.",
"(REF ).",
"When $x > x_{max}$ the asymptotic is considerably less than the parent amplitudes, so such amplitudes should be used instead of the asymptotics.",
"Ignoring this point leads to various misconceptions: for instance, appearance of artificial/model (hard) Pomerons and spin-dependent Pomerons.",
"To conclude, let us notice that DL Pomeron can play an important role for description of various hadronic reactions where Pomerons are used and the diffractive DIS in the first place, see e.g.",
"[15]."
],
[
"Acknowledgement",
"We are grateful to S.I.",
"Alekhin, D.Yu.",
"Ivanov, G.I.",
"Lykasov, F. Olness and O.V.",
"Teryaev for useful communications."
],
[
"Expressions for the parton-parton amplitudes",
"where $Z = \\frac{1}{\\sqrt{2}}\\sqrt{ Y + W}~,$ with $Y = \\omega ^2 - 2(b_{qq} + b_{gg})$ and $W = \\sqrt{(\\omega ^2 - 2(b_{qq} + b_{gg}))^2 - 4 (b_{qq} - b_{gg})^2 -16b_{gq} b_{qg} }.$ Eqs.",
"(REF ,REF ,REF ) express $h_{rr^{\\prime }}$ through terms $b_{rr^{\\prime }}$ .",
"The terms $b_{rr^{\\prime }}$ include the Born factors $a_{rr^{\\prime }}$ and contributions of non-ladder graphs $V_{rr^{\\prime }}$ : $b_{rr^{\\prime }} = a_{rr^{\\prime }} + V_{rr^{\\prime }}.$ The Born factors are (see Ref.",
"[12] for detail): $a_{qq} = \\frac{A(\\omega )C_F}{2\\pi },~a_{qg} = \\frac{A^{\\prime }(\\omega )C_F}{\\pi },~a_{gq} = -\\frac{A^{\\prime }(\\omega )n_f}{2 \\pi }.~a_{gg} = \\frac{2N A(\\omega )}{\\pi },$ where $A$ and $A^{\\prime }$ stand for the running QCD couplings: $A = \\frac{1}{b} \\left[\\frac{\\eta }{\\eta ^2 + \\pi ^2} - \\int _0^{\\infty } \\frac{d z e^{- \\omega z}}{(z + \\eta )^2 + \\pi ^2}\\right],A^{\\prime } = \\frac{1}{b} \\left[\\frac{1}{\\eta } - \\int _0^{\\infty } \\frac{d z e^{- \\omega z}}{(z + \\eta )^2}\\right],$ with $\\eta = \\ln \\left(\\mu ^2/\\Lambda ^2_{QCD}\\right)$ and $b$ being the first coefficient of the Gell-Mann- Low function.",
"When the running effects for the QCD coupling are neglected, $A(\\omega )$ and $A^{\\prime }(\\omega )$ are replaced by $\\alpha _s$ .",
"The terms $V_{rr^{\\prime }}$ are represented in a similar albeit more involved way (see Ref.",
"[12] for detail): $ V_{rr^{\\prime }} = \\frac{m_{rr^{\\prime }}}{\\pi ^2} D(\\omega )~,$ with $ m_{qq} = \\frac{C_F}{2 N}~,\\quad m_{gg} = - 2N^2~,\\quad m_{gq} = n_f \\frac{N}{2}~,\\quad m_{qg} = - N C_F~,$ and $ D(\\omega ) = \\frac{1}{2 b^2} \\int _{0}^{\\infty } d ze^{- \\omega z} \\ln \\big ( (z + \\eta )/\\eta \\big ) \\Big [\\frac{z + \\eta }{(z + \\eta )^2 + \\pi ^2} - \\frac{1}{z +\\eta }\\Big ]~.$"
],
[
"Explicit expressions for ingredients of Eq. (",
"The factors $C^{(\\pm )}_{q,g}$ and $\\Omega _{(\\pm )}$ of Eq.",
"(REF were obtained in Ref. [6].",
"We list them below.",
"All of them are expressed though the parton-parton amplitudes $h_{rr^{\\prime }}$ of Appendix A.",
"The factors $\\Omega _{(\\pm )}$ are: $\\Omega _{(\\pm )} = \\frac{1}{2} \\left[ h_{gg} + h_{qq} \\pm \\sqrt{R}\\right]$ and $R = (h_{gg} + h_{qq})^2 - 4(h_{qq}h_{gg} - h_{qg}h_{gq}) = (h_{gg} - h_{qq})^2 + 4 h_{qg}h_{gq} .$ The factors $C_{(\\pm )}$ are also expressed through the parton-parton amplitudes: $C_q^{(+)} &=& a_{\\gamma q}\\frac{h_{qg}h_{gq} - (\\omega - h_{gg})\\left(h_{gg} - h_{qq} - \\sqrt{R}\\right)}{2 G \\sqrt{R}},\\\\ \\nonumber C_q^{(-)} &=& a_{\\gamma q}\\frac{ -h_{qg}h_{gq} + (\\omega - h_{gg})\\left(h_{gg} - h_{qq} + \\sqrt{R}\\right)}{2 G \\sqrt{R}}.$ $C_g^{(+)} &=& C_q^{(+)} \\frac{h_{gg} - h_{qq} + \\sqrt{R}}{2h_{qg}},\\\\ \\nonumber C_g^{(-)} &=& C_q^{(-)} \\frac{h_{gg} - h_{qq} - \\sqrt{R}}{2h_{qg}}.$"
],
[
"Expressions for the factors $C_{1,2}$ and {{formula:f4eef51c-fb53-4554-a684-6d6e2d822a4d}} of Eq. (",
"The terms $C_1 = \\frac{2 h_{qg}h_{gq} - h^2_{qq} + h_{qq}h_{gg} + h_{qq}\\sqrt{R}}{2 \\sqrt{R}},\\\\ \\nonumber C_2 = \\frac{h^2_{qq} - h_{qq}h_{gg} - 2 h_{qg}h_{gq} + h_{qq}\\sqrt{R}}{2 \\sqrt{R}},\\\\ \\nonumber C^{\\prime }_1 = \\frac{2 h_{qg}h_{gq} - h^2_{gg} + h_{qq}h_{gg} + h_{gg}\\sqrt{R}}{2 \\sqrt{R}},\\\\ \\nonumber C^{\\prime }_2 = \\frac{h^2_{gg} - h_{qq}h_{gg} - 2 h_{qg}h_{gq} + h_{gg}\\sqrt{R}}{2 \\sqrt{R}},$ where $R$ is given by Eq.",
"(REF )."
]
] | 1906.04506 |
[
[
"Nilpotent elements in the cohomology of the classifying space of a\n connected Lie group"
],
[
"Abstract We give an example of a compact connected Lie group of the lowest rank such that the mod 2 cohomology ring of its classifying space has a nonzero nilpotent element."
],
[
"Introduction",
"Let $p$ be a prime number and $G$ a compact Lie group.",
"In , Quillen defined a homomorphism, $q_G \\colon H^{*}(BG;\\mathbb {Z}/p)\\rightarrow \\lim _{A\\in \\mathfrak {A}} H^{*}(BA;\\mathbb {Z}/p),$ where $\\mathfrak {A}$ is a category of elementary abelian $p$ -subgroups of $G$ , and proved that $q_G$ is an $F$ -isomorphism.",
"In particular, an element in its kernel is nilpotent.",
"For $p=2$ , an element in the image of $q_G$ is not nilpotent, so the nilradical of $H^{*}(BG;\\mathbb {Z}/2)$ is exactly the kernel of $q_G$ .",
"In , Kono and Yagita showed that $q_G$ is not injective for $p=2$ , $G=\\mathrm {Spin}(11), E_7$ by showing the existence of a nonzero nilpotent element in $H^{*}(BG;\\mathbb {Z}/2)$ .",
"For an odd prime number $p$ , Adams conjectured that $q_G$ is injective for all compact connected Lie groups.",
"Adams' conjecture still remains as an open problem as of today.",
"On the other hand, for a compact connected Lie group $G$ , there exists a maximal torus $T$ .",
"Let $W$ be the Weyl group $N(T)/T$ .",
"We denote by $H^{*}(BT;\\mathbb {Z})^W$ the ring of invariants of $W$ .",
"We denote by $\\mathrm {Tor}$ the torsion part of $H^{*}(BG;\\mathbb {Z})$ .",
"Then, the inclusion map of $T$ induces a homomorphism, $\\iota _T^{*}\\colon H^{*}(BG;\\mathbb {Z})/\\mathrm {Tor}\\rightarrow H^{*}(BT;\\mathbb {Z})^W.$ Borel showed that $\\iota _T^*$ is injective.",
"In , Feshbach gave a criterion for $\\iota _T^*$ to be surjective, hence an isomorphism.",
"If $H_{*}(G;\\mathbb {Z})$ has no odd torsion, $\\iota _T^*$ is surjective if and only if the $E_\\infty $ -term of the mod 2 Bockstein spectral sequence of $BG$ , $H^{*}(BG;\\mathbb {Z})/\\mathrm {Tor}\\otimes \\mathbb {Z}/2,$ has no nonzero nilpotent element.",
"Feshbach also showed that for $G=\\mathrm {Spin}(12)$ , the $E_\\infty $ -term of the mod 2 Bockstein spectral sequence of $BG$ has a nonzero nilpotent element.",
"As for spin groups $\\mathrm {Spin}(n)$ , Benson and Wood computed the ring of invariants of the Weyl group and they showed that $\\iota _T^*$ is not surjective if and only if $n\\ge 11$ and $n \\equiv 3, 4, 5 \\mod {8}$ .",
"However, as in the case of Adams' conjecture, for an odd prime $p$ , no example of a compact connected Lie group $G$ such that the $E_\\infty $ -term of the mod $p$ Bockstein spectral sequence of $BG$ has a nonzero nilpotent element is known.",
"So, nilpotent elements in the cohomology of classifying spaces of compact connected Lie groups are interesting subject for study.",
"However, no example of a compact connected Lie group $G$ such that $H^{*}(BG;\\mathbb {Z}/2)$ has a nonzero nilpotent element is known except for spin groups and the exceptional Lie group $E_7$ .",
"The purpose of this paper is to give a simpler example.",
"First, we define a compact connected Lie group $G$ .",
"Let us consider the three fold product $SU(2)^3$ of the special unitary groups $SU(2)$ .",
"Its center is an elementary abelian 2-group $(\\mathbb {Z}/2)^3$ .",
"Let $\\Gamma $ be the kernel of the group homomorphism $\\det \\colon (\\mathbb {Z}.2)^3 \\rightarrow \\mathbb {Z}/2$ defined by $\\det (a_1, a_2, a_3)=a_1a_2a_3$ .",
"We define $G$ to be $SU(2)^3/\\Gamma $ .",
"Next, we state our results saying that $G=SU(2)^3/\\Gamma $ satisfies the required conditions.",
"Since $SU(2)^3/(\\mathbb {Z}/2)^3=SO(3)^3$ , we have the following fibre sequence: $B\\mathbb {Z}/2 \\rightarrow BG \\stackrel{\\pi }{\\longrightarrow } BSO(3)^3.$ Let $\\pi _i\\colon BSO(3)^3 \\rightarrow BSO(3)$ be the projection onto the $i^{\\mathrm {th}}$ factor.",
"Let $w_2$ , $w_3$ be the generators of $H^2(BSO(3);\\mathbb {Z}/2)$ , $H^3(BSO(3);\\mathbb {Z}/2)$ , respectively.",
"Let $w_k^{\\prime }=\\pi ^*(\\pi _1^{*}(w_k))$ and $w_k^{\\prime \\prime }=\\pi ^*(\\pi _2^{*}(w_k))$ .",
"Let $w_{16}$ be the Stiefel-Whitney class $w_{16}(\\rho )$ of a real representation $\\rho \\colon G\\rightarrow O(16)$ .",
"We will give the definition of $\\rho $ in Section .",
"Let $f_5$ , $f_9$ , $g_4$ , $g_7$ , $g_8$ be polynomials defined by $f_5&=w_2^{\\prime }w_3^{\\prime \\prime }+w_2^{\\prime \\prime }w_3^{\\prime },\\\\f_9& =w_3^{\\prime 2} w_3^{\\prime \\prime }+w_3^{\\prime \\prime 2}w_3^{\\prime },\\\\g_4&=w_2^{\\prime }w_2^{\\prime \\prime },\\\\g_7&=w_2^{\\prime }w_2^{\\prime \\prime }(w_3^{\\prime }+w_3^{\\prime \\prime }),\\\\g_8 &=w_3^{\\prime }w_3^{\\prime \\prime }(w_2^{\\prime }+w_2^{\\prime \\prime }),$ respectively.",
"Then, our results are stated as follows: Theorem 1.1 The mod 2 cohomology ring of $BG$ is $\\mathbb {Z}/2[ w_2^{\\prime }, w_2^{\\prime \\prime }, w_3^{\\prime }, w_3^{\\prime \\prime }, w_{16}]/(f_5, f_9)$ and its nilradical is generated by $g_7$ , $g_8$ .",
"Theorem 1.2 The $E_\\infty $ -term of the mod 2 Bockstein spectral sequence of $BG$ is $\\mathbb {Z}/2[w_2^{\\prime 2}, w_2^{\\prime \\prime 2}, w_{16}] \\otimes \\Delta (g_4, g_8),$ where $\\Delta (g_4, g_8)$ is the vector space over $\\mathbb {Z}/2$ spanned by 1, $g_4$ , $g_8$ and $g_4g_8$ .",
"Its nilradical is generated by $g_8$ , The rank of $SU(2)^3/\\Gamma $ is 3.",
"If the rank of a compact connected Lie group is lower than 3, then it is homotopy equivalent to one of $T$ , $SU(2)$ , $T^2$ , $T \\times SU(2)$ , $SU(2)\\times SU(2)$ , $SU(3)$ , $G_2$ or their quotient groups by their central subgroups.",
"For such a compact connected Lie group, the mod 2 cohomology ring of its classifying space is a polynomial ring, so that it has no nonzero nilpotent element.",
"Thus our example is an example of the lowest rank.",
"We hope our results shed some light on Adams' conjecture since on the contrary to spin groups we have odd primary analogue of the group $SU(2)^3/\\Gamma $ .",
"Let $\\Gamma _2$ is the kernel of the determinant homomorphism $\\det \\colon (S^1)^3 \\rightarrow S^1$ .",
"Consider the quotient group $U(p)^3/\\Gamma _2.$ It is the odd primary counter part as the group $U(2)^3/\\Gamma _2$ is the central extension of the group $SU(2)^3/\\Gamma $ by $S^1$ .",
"But that's another story and we wish to deal with the group $U(p)^3/\\Gamma _2$ in another paper.",
"In what follows, we assume that $G$ is the compact connected Lie group $SU(2)^3/\\Gamma $ .",
"We also denote the mod 2 cohomology ring of $X$ by $H^{*}(X)$ rather than $H^{*}(X;\\mathbb {Z}/2)$ .",
"This paper is organized as follows: In Section , we compute the Leray-Serre spectral sequence associated with the fibre sequence $B\\mathbb {Z}/2 \\stackrel{\\iota }{\\longrightarrow } BG \\stackrel{\\pi }{\\longrightarrow } BSO(3)^3$ to compute the mod 2 cohomology ring $H^{*}(BG)$ and prove Theorem REF .",
"In Section , we compute the $Q_0$ -cohomology of $H^{*}(BG)$ to complete the proof of Theorem REF ."
],
[
"The mod 2 cohomology ring",
"In this section, we compute the mod 2 cohomology ring of $BG$ by computing the Leray-Serre spectral sequence associated with the fibre sequence $B\\mathbb {Z}/2 \\stackrel{\\iota }{\\longrightarrow } BG \\stackrel{\\pi }{\\longrightarrow } BSO(3)^3.$ First, we recall the mod 2 cohomology rings of $BSO(3)$ and $BSO(3)^3$ .",
"Let $Q_i$ be the Milnor operation $Q_i\\colon H^k(X) \\rightarrow H^{k+2^{i+1}-1}(X)$ defined inductively by $Q_0=\\mathrm {Sq}^1, \\quad Q_{i+1}= \\mathrm {Sq}^{2^{i+1}} Q_i + Q_i \\mathrm {Sq}^{2^{i+1}}$ for $i\\ge 0$ .",
"The mod 2 cohomology ring of $BSO(3)$ is a polynomial ring generated by two elements $w_2$ , $w_3$ of degree 2, 3, respectively, i.e.",
"$H^{*}(BSO(3))=\\mathbb {Z}/2[w_2, w_3].$ The action of Steenrod squares is given by the Wu formula.",
"However, what we use in this paper is the action of $Q_0$ , $Q_1$ and $Q_2$ only.",
"It is given by $Q_0(w_2)&=w_3,\\\\Q_1(w_2)&=w_2w_3,\\\\Q_2(w_2)&= w_2^3 w_3+w_3^3.$ Recall that $\\pi _i\\colon BSO(3)^3\\rightarrow BSO(3)$ ($i=1,2,3$ ) is the projection onto the $i^{\\mathrm {th}}$ factor.",
"By abuse of notation, we define elements $w_k^{\\prime }$ , $w_k^{\\prime \\prime }$ , $w_k^{\\prime \\prime \\prime }$ ($k=2, 3$ ) in $H^{*}(BSO(3)^3)$ by $w_k^{\\prime }=\\pi _1^*(w_k)$ , $w_k^{\\prime \\prime }=\\pi _2^*(w_k)$ , $w_k^{\\prime \\prime \\prime }=\\pi _3^*(w_k).$ Let use define elements $v_2$ , $v_3$ by $v_2&=w_2^{\\prime }+w_2^{\\prime \\prime }+w_2^{\\prime \\prime \\prime },\\\\v_3&=w_3^{\\prime }+w_3^{\\prime \\prime }+w_3^{\\prime \\prime \\prime },$ and ideals $I_1$ , $I_2$ by $I_1&=(v_2, v_3),\\\\I_2&=(v_2, v_3, Q_1(v_2)).$ Again, by abuse of notation, we consider $f_5$ , $f_9$ as polynomials $w_2^{\\prime }w_3^{\\prime \\prime }+w_2^{\\prime \\prime }w_3^{\\prime }$ , $w_3^{\\prime 2}w_3^{\\prime \\prime }+w_3^{\\prime \\prime }w_3^{\\prime }$ in $H^{*}(BSO(3)^3)$ , respectively.",
"Then, by direct calculations, we have $Q_0 v_2&=v_3,\\\\Q_1 v_2&\\equiv f_5 \\quad \\mod {I}_1,\\\\Q_2v_2&\\equiv f_9 \\quad \\mod {I}_2.$ Now, we compute the Leray-Serre spectral sequence.",
"The $E_2$ -term is given by $E_2^{p,q}=H^p(BSO(3)^3) \\otimes H^q(B\\mathbb {Z}/2),$ so that $E_2=\\mathbb {Z}/2[w_2^{\\prime }, w_2^{\\prime \\prime }, v_2, w_3^{\\prime }, w_3^{\\prime \\prime }, v_3, u_1],$ where $u_1$ is the generator of $H^{1}(B\\mathbb {Z}/2)$ .",
"The first nontrivial differential is $d_2$ .",
"Let $\\iota _i\\colon SU(2) \\rightarrow SU(2)^3$ be the inclusion map to the $i^{\\mathrm {th}}$ factor, that is, $\\iota _{1}(g)=(g, 1, 1),\\quad \\iota _{2}(g)=(1, g, 1), \\quad \\iota _{3}(g)=(1, 1, g).$ Then, they induce the following commutative diagrams.",
"$\\begin{diagram}\\node {B\\mathbb {Z}/2} {e,t}{=}{s} \\node {B\\mathbb {Z}/2}{s}\\\\\\node {BSU(2)} {e,t}{\\iota _i} {s} \\node {BG} {s}\\\\\\node {BSO(3)} {e,t}{\\iota _i} \\node {BSO(3)^3.",
"}\\end{diagram}$ Since the differential in the Leray-Serre spectral sequence associated with the fibre sequence $B\\mathbb {Z}/2\\rightarrow BSU(2) \\rightarrow BSO(3)$ is $d_2(u_1)=w_2,$ we have $d_2(u_1)=v_2$ in the Leray-Serre spectral sequence for $H^{*}(BG)$ .",
"To compute the higher differentials, we use the transgression theorem.",
"In order to use the transgression theorem, we use the following lemma on Milnor operations and Steenrod squares.",
"Lemma 2.1 For $x\\in H^{2}(X)$ and $k\\ge 1$ , we have $Q_k(x)= \\mathrm {Sq}^{2^k}\\cdots \\mathrm {Sq}^{2^0} (x)$ For $k \\ge 2$ , by the definition of $Q_{i+1}$ and the unstable condition, we have $Q_{k}(x)&=\\mathrm {Sq}^{2^{k}} Q_{k-1} (x)+ Q_{k-1} \\mathrm {Sq}^{2^{k}} (x)\\\\&=\\mathrm {Sq}^{2^{k}} Q_{k-1} (x).$ Suppose $k=1$ .",
"By the unstable condition, we have $\\mathrm {Sq}^2(x)=x^2$ .",
"By the Cartan formula, we have $\\mathrm {Sq}^1 (x^2)=0$ .",
"Hence, we have $Q_{1}(x)&=\\mathrm {Sq}^{2} Q_{0} (x)+ Q_{0}{\\mathrm {Sq}^2}(x)\\\\&=\\mathrm {Sq}^{2} Q_0 (x).",
"$ From $d_2(u_1)=v_2$ and the action of $Q_0$ , $Q_1$ , $Q_2$ on $v_2$ , by Lemma REF and the transgression theorem, we have $d_3(u_1^2)&= v_3, \\\\d_5(u_1^4)&=f_5, \\\\d_9(u_1^8)&=f_9.$ It is easy to see that $E_3&=\\mathbb {Z}/2[w_2^{\\prime }, w_2^{\\prime \\prime }, w_3^{\\prime }, w_3^{\\prime \\prime }, v_3, u_1^2],\\\\E_4&=\\mathbb {Z}/2[w_2^{\\prime }, w_2^{\\prime \\prime }, w_3^{\\prime }, w_3^{\\prime \\prime }, u_1^4],\\\\E_6&=\\mathbb {Z}/2[w_2^{\\prime }, w_2^{\\prime \\prime }, w_3^{\\prime }, w_3^{\\prime \\prime }, u_1^8]/(f_5),$ In $\\mathbb {Z}/2[w_2^{\\prime }, w_2^{\\prime \\prime }, w_3^{\\prime }, w_3^{\\prime \\prime }]$ , the sequence $f_5$ , $f_9$ is a regular sequence since their greatest common divisor is 1.",
"Therefore, we have $E_{10}=\\mathbb {Z}/2[w_2^{\\prime }, w_2^{\\prime \\prime }, w_3^{\\prime }, w_3^{\\prime \\prime }, u_1^{16}]/(f_5,f_9).$ In order to prove that the spectral sequence collapses at the $E_{10}$ -level, we consider the Stiefel-Whitney class of a real representation $\\rho \\colon G \\rightarrow O(16).$ Let us define the representation $\\rho $ .",
"Since $SO(4)=SU(2)\\times _{\\mathbb {Z}/2} SU(2),$ we may regard $G=SU(2) \\times _{\\mathbb {Z}/2} (SU(2)\\times _{\\mathbb {Z}/2} SU(2)) = SU(2) \\times _{\\mathbb {Z}/2} SO(4)$ as a subgroup of $SO(4) \\times _{\\mathbb {Z}/2} SO(4)$ by regarding the first factor $SU(2)$ as the subgroup of $SO(4)$ in the usual manner.",
"Let $\\varphi \\colon SO(4) \\times SO(4) \\rightarrow O(16)$ be the real representation given by $(g_1, g_2) m= g_1 m g_2^{-1}$ where $(g_1, g_2) \\in SO(4) \\times SO(4)$ and $m$ is a $4\\times 4$ matrix with real coefficients.",
"Then, $\\varphi $ induced a 16-dimensional real representation $\\varphi ^{\\prime }\\colon SO(4)\\times _{\\mathbb {Z}/2} SO(4) \\rightarrow O(16)$ We define the representation $\\rho $ to be the restriction of $\\varphi ^{\\prime }$ to $G$ .",
"Proposition 2.2 The Stiefel-Whitney class $w_{16}(\\rho )$ of the real representation $\\rho $ is not decomposable in $H^{*}(BG)$ .",
"It is represented by $u_1^{16}$ in the Leray-Serre spectral sequence.",
"Let $\\iota \\colon \\mathbb {Z}/2\\rightarrow G$ be the inclusion map of the center $\\mathbb {Z}/2$ .",
"The restriction of $\\rho $ to the center of $G$ is $16\\lambda $ where $\\lambda $ is the nontrivial 1 dimensional real representation of $\\mathbb {Z}/2$ .",
"So, the Stiefel-Whitney class $w_{16}(\\rho \\circ \\iota )$ is nonzero.",
"If $u_1^{16}$ supports a nontrivial differential in the Leray-Serre spectral sequence then, up to degree $\\le 16$ , $H^{*}(BG)$ is generated by $w_2^{\\prime }, w_2^{\\prime \\prime }, w_3^{\\prime }, w_3^{\\prime \\prime }$ .",
"However, since $\\iota $ factors through $BSU(2)^2$ , the induced homomorphism sends $w_2^{\\prime }, w_2^{\\prime \\prime }, w_3^{\\prime }, w_3^{\\prime \\prime }$ to zero.",
"So, $w_{16}(\\rho \\circ \\iota )$ is zero.",
"It is a contradiction.",
"Therefore, $u_1^{16}$ is a permanent cycle in the Leray-Serre spectral sequence and it is represented by $w_{16}(\\rho )$ .",
"Hence, the spectral sequence collapses at the $E_{10}$ -level, that is, $E_\\infty =E_{10}$ and we obtain the first half of Theorem REF .",
"Proposition 2.3 We have $ H^{*}(BG)=\\mathbb {Z}/2[w_2^{\\prime }, w_2^{\\prime \\prime }, w_3^{\\prime }, w_3^{\\prime \\prime }, w_{16}]/(f_5, f_{9}),$ where $w_{16}$ is the Stiefel-Whitney class $w_{16}(\\rho )$ .",
"In order to prove the second half of Theorem REF , let us define a ring homomorphism $\\eta \\colon \\mathbb {Z}/2[w_2^{\\prime }, w_2^{\\prime \\prime }, w_3^{\\prime }, w_3^{\\prime \\prime }, w_{16}]\\rightarrow \\mathbb {Z}/2[w_2^{\\prime }, w_2^{\\prime \\prime }, u, w_{16}]$ by $\\eta (w_2^{\\prime })=w_2^{\\prime }$ , $\\eta (w_2^{\\prime \\prime })=w_2^{\\prime \\prime }$ , $\\eta (w_3^{\\prime })=w_2^{\\prime } u$ , $\\eta (w_3^{\\prime \\prime })=w_2^{\\prime \\prime }u$ , $\\eta (w_{16})=w_{16}$ .",
"It induces the following ring homomorphism $\\eta ^{\\prime }\\colon H^{*}(BG)\\rightarrow \\mathbb {Z}/2[w_2^{\\prime }, w_2^{\\prime \\prime }, u , w_{16}]/(u^3 w_2^{\\prime }w_2^{\\prime \\prime }(w_2^{\\prime }+w_2^{\\prime \\prime })).$ Let $R_0=\\mathbb {Z}/2[w_2^{\\prime }, w_2^{\\prime \\prime }, w_3^{\\prime }, w_3^{\\prime \\prime }, w_{16}].$ From Proposition REF , using the fact that $f_5$ , $f_9$ is a regular sequence in $R_0$ , we have the Poincaré series of $H^{*}(BG)$ , $PS(H^{*}(BG), t)=\\dfrac{(1-t^5)(1-t^9)}{(1-t^2)^2(1-t^3)^2(1-t^{16})}.$ On the other hand, it is also easy to see that the image of $\\eta ^{\\prime }$ is spanned by monomials $u^\\ell w_2^{\\prime m} w_2^{\\prime \\prime n}w_{16}^k,$ where $k$ ranges over all non-negative integers, for $\\ell =0,1,2$ , $(m, n)$ satisfies the condition $m+n\\ge \\ell $ , and for $\\ell \\ge 3$ , $(m,n)$ satisfies one of the following conditions: $m\\ge \\ell $ , $n=0$ or $m=1$ , $n\\ge \\ell -1$ or $m=0$ , $n\\ge \\ell $ .",
"Thus, the Poincaré series $PS(\\mathrm {Im}\\, \\eta ^{\\prime }, t)$ is $\\dfrac{1}{1-t^{16}}\\left( \\dfrac{1}{(1-t^2)^2}+ t\\left( \\dfrac{1}{(1-t^2)^2}-1\\right)+t^2 \\left( \\dfrac{1}{(1-t^2)^2}-1-2t^2\\right)+\\sum _{\\ell =3}^\\infty \\dfrac{3t^{3\\ell }}{1-t^2}\\right).$ By computing this, we have $PS(H^{*}(BG),t)=PS(\\mathrm {Im}\\, \\eta ^{\\prime }, t).$ Thus, $\\eta ^{\\prime }$ is injective.",
"In view of this injective homomorphism $\\eta ^{\\prime }$ , it is easy to see that elements $g_7$ , $g_8$ corresponding to $t w_2^{\\prime }w_2^{\\prime \\prime }(w_2^{\\prime }+w_2^{\\prime \\prime })$ , $t^2 w_2^{\\prime }w_2^{\\prime \\prime }(w_2^{\\prime }+w_2^{\\prime \\prime })$ , respectively, are nilpotent.",
"So we obtain the following second half of Theorem REF .",
"Proposition 2.4 The nilradical of $H^{*}(BG)$ is the ideal generated by two elements $g_7$ and $g_8$ ."
],
[
"The mod 2 Bockstein spectral sequence",
"In this section, in order to show that the mod 2 Bockstein spectral sequence of $BG$ collapses at the $E_2$ -level and to compute its $E_\\infty $ -term, we compute the $Q_0$ -cohomology, i.e.",
"$H^{*}(H^{*}(BG),Q_0)=\\mathrm {Ker}\\, Q_0/\\mathrm {Im}\\, Q_0,$ of $H^{*}(BG)$ .",
"First, we recall the action of $Q_0$ on $H^{*}(BG)$ .",
"The action of $Q_0$ on $w_2^{\\prime }, w_2^{\\prime \\prime }, w_3^{\\prime }, w_3^{\\prime \\prime }$ is clear from that on $H^{*}(BSO(3))$ .",
"We need to determine the action of $Q_0$ on $w_{16}$ .",
"Proposition 3.1 In $H^{*}(BG)$ , we have $Q_0(w_{16})=0$ .",
"The generator $w_{16}$ is defined as the Stiefel-Whitney class $w_{16}(\\rho )$ of the 16-dimensional real representation $\\rho \\colon G\\rightarrow O(16)$ .",
"Hence, $w_{17}(\\rho )=0$ .",
"Since $BG$ is simply-connected, we also have $w_1(\\rho )=0$ .",
"By the Wu formula $\\mathrm {Sq}^1 w_{16}(\\rho )=w_{17}(\\rho )+w_1(\\rho )w_{16}(\\rho )$ , we have the desired result.",
"Let $R_0=\\mathbb {Z}/2[w_2^{\\prime }, w_2^{\\prime \\prime }, w_3^{\\prime }, w_3^{\\prime \\prime }, w_{16}].$ We consider the action of $Q_0$ on $w_2^{\\prime }$ , $w_2^{\\prime \\prime }$ , $w_3^{\\prime }$ , $w_3^{\\prime \\prime }$ , $w_{16}$ in $R_0$ by $Q_0(w_2^{\\prime })=w_3^{\\prime }, \\quad Q_0(w_2^{\\prime \\prime })=w_3^{\\prime \\prime }, \\quad Q_0(w_3^{\\prime })=0, \\quad Q_0(w_3^{\\prime \\prime })=0, \\quad Q_0(w_{16})= 0.$ Let $R_1=R_0/(f_5),\\quad R_2=R_0/(f_5, f_9).$ It is clear that $R_2=H^{*}(BG)$ and $H^*(H^{*}(BG), Q_0)=H^*(R_2, Q_0)$ .",
"We will prove the following Proposition REF at the end of this section.",
"Proposition 3.2 We have $H^{*}(R_2, Q_0)=\\mathbb {Z}/2[w_2^{\\prime 2}, w_2^{\\prime \\prime 2}, w_{16}]\\otimes \\Delta (g_4, g_8).$ The $E_1$ -term of the mod 2 Bockstein spectral sequence of $BG$ is the mod 2 cohomology ring of $BG$ and $d_1$ is $Q_0$ .",
"Since, by Proposition REF , the $E_2$ -term has no nonzero odd degree element, the spectral sequence collapses at the $E_2$ -level.",
"It is also clear that $g_4^2=w_2^{\\prime 2}w_2^{\\prime \\prime 2}\\ne 0$ , $g_8^2=0$ from Theorem REF .",
"Now, we complete the proof of Theorem REF by proving Proposition REF .",
"We start with $H^*(R_0, Q_0)$ .",
"It is clear that $H^*(R_0,Q_0)=\\mathbb {Z}/2[w_2^{\\prime 2}, w_2^{\\prime \\prime 2}, w_{16}].$ We denote by $(-)\\times a$ the multiplication by $a$ , Consider a short exact sequence $0 \\rightarrow R_0 \\stackrel{(-)\\times f_5}{\\longrightarrow } R_0 \\rightarrow R_1 \\rightarrow 0.$ Since $Q_0$ commutes with $f_5$ , this short exact sequence induces the long exact sequences in $Q_0$ -cohomology: $\\cdots \\rightarrow H^{i}(R_0,Q_0) \\rightarrow H^{i}(R_1, Q_0) \\stackrel{\\delta _4}{\\longrightarrow } H^{i-4}(R_0,Q_0) \\rightarrow \\cdots $ Since $H^{odd}(R_0,Q_0)=0$ , this long exact sequence splits into short exact sequences: $0 \\rightarrow H^{2i}(R_0;Q_0) \\rightarrow H^{2i}(R_1, Q_0) \\stackrel{\\delta _4}{\\longrightarrow } H^{2i-4}(R_0,Q_0) \\rightarrow 0$ and $H^{odd}(R_1, Q_0)=0$ .",
"Since $Q_0 g_4=f_5$ in $R_0$ , $g_4$ is nonzero in $R_1$ and $\\delta _4(g_4)=1$ .",
"Therefore, we have $H^{*}(R_1,Q_0)=\\mathbb {Z}/2[w_2^{\\prime 2}, w_2^{\\prime \\prime 2}, w_{16}]\\otimes \\Delta (g_4).$ Next, let us consider a short exact sequence $0 \\rightarrow R_1 \\stackrel{(-)\\times f_9}{\\longrightarrow } R_1 \\rightarrow R_2 \\rightarrow 0.$ As above, since $H^{odd}(R_1, Q_0)=\\lbrace 0\\rbrace $ , we have short exact sequences $0 \\rightarrow H^{2i}(R_1;Q_0) \\rightarrow H^{2i}(R_2, Q_0) \\stackrel{\\delta _8}{\\longrightarrow } H^{2i-8}(R_1,Q_0) \\rightarrow 0$ and $H^{odd}(R_2, Q_0)=\\lbrace 0\\rbrace $ .",
"Since $Q_0g_8=f_9$ , we obtain the desired result $H^{*}(R_2, Q_0)=\\mathbb {Z}/2[w_2^{\\prime 2}, w_2^{\\prime \\prime 2}, w_{16}]\\otimes \\Delta (g_4, g_8).",
"$ benson-wood-1995article author=Benson, D. J., author=Wood, Jay A., title=Integral invariants and cohomology of $B{\\rm Spin}(n)$ , journal=Topology, volume=34, date=1995, number=1, pages=13–28, issn=0040-9383, doi=10.1016/0040-9383(94)E0019-G, feshbach-1981article author=Feshbach, Mark, title=The image of $H^{\\ast } (BG,\\,{\\bf Z})$ in $H^{\\ast } (BT,\\,{\\bf Z})$ for $G$ a compact Lie group with maximal torus $T$ , journal=Topology, volume=20, date=1981, number=1, pages=93–95, issn=0040-9383, doi=10.1016/0040-9383(81)90015-X, kono-yagita-1993article author=Kono, Akira, author=Yagita, Nobuaki, title=Brown-Peterson and ordinary cohomology theories of classifying spaces for compact Lie groups, journal=Trans.",
"Amer.",
"Math.",
"Soc., volume=339, date=1993, number=2, pages=781–798, issn=0002-9947, doi=10.2307/2154298, quillen-1971article author=Quillen, Daniel, title=The spectrum of an equivariant cohomology ring.",
"I, II, journal=Ann.",
"of Math.",
"(2), volume=94, date=1971, pages=549–572; ibid.",
"(2) 94 (1971), 573–602, issn=0003-486X, doi=10.2307/1970770,"
]
] | 1906.04499 |
[
[
"Establishing a relativistic model for atomic gravimeters"
],
[
"Abstract This work establishes a high-precision relativistic theoretical model: start from studying finite speed of light effect based on a coordinate transformation, and further extend the research methods to analyze the overall relativistic effects.",
"This model promotes the development of testing General Relativity with atomic interferometry."
],
[
"Introduction",
"Since a high-precision gravimeter helps to accurately build a tide model, determine the geoid, test the General Relativity (GR) and so on, it is very necessary to develop high-precision atomic gravimeters and the corresponding theoretical model.",
"Therefore, in the gravity measurements, except for some Newtonian effects, we should also consider some special and general relativistic effects to establish a high-precision theoretical model [1].",
"In general, researches for the relativistic effects in atomic gravimeters can be divided into two aspects: the finite speed of light (FSL) effect [3], [2], [4], and the GR effects [5], [6].",
"Since current researches about them exist some disagreements [3], [2], [4], [5], [6] or being incomplete [5], [6], we recalculate these effects, and derive a more complete and general expression for them."
],
[
"Calculation idea for the interferometric phase shift",
"According to the working principle of atomic gravimeters, the three-Raman-pulse sequence is usually used to interact with the moving atoms.",
"These pulses split, reflect and recombine the atomic wave packets, respectively.",
"As the evolutions of atoms are space separated, the interferometric signal carries the information of the gravitational field.",
"Thus, the gravitational acceleration can be derived from the measured interferometric phase shift.",
"A complete atomic interferometry system consists of two parts: atoms and laser lights, which are the main contribution sources to the total interferometric phase.",
"Since the phase is a scalar, it does't depend on the selection of coordinate systems, and we can equivalently observe this atom-laser interacting system in different coordinate systems.",
"In the freely falling system attached to the atoms, all the relativistic effects are reflected in the laser lights, while in the laser-platform coordinate system fixed on the laboratory, all the relativistic effects are reflected in the atoms.",
"To a large extent, we here apply the latter idea.",
"Figure: A spacetime diagram of a light-pulse atom interferometry.",
"The solid and dashed lines respectively represent the motions of the atoms in the ground and excited states, and the dotted lines stand for the lights manipulating the atoms.For a typical atomic gravimeter, one of the two Raman beams is reflected by a mirror (see $\\vec{k}_2$ beam in Fig.",
"1).",
"As $\\vec{k}_2$ beam reaches atoms latter than $\\vec{k}_1$ beam, and the stimulated Raman transitions occur only when both of the two beams interact with atoms, $\\vec{k}_1$ beam can be considered as a “background light\", and the $\\vec{k}_2$ beam can be considered as a “control light\".",
"Based on Fig.",
"1, the total interferometric phase shift can be calculated, which contains three parts: the atomic propagation phase shift, laser phase shift, and the separation phase shift.",
"Conventionally, to calculate the relativistic phase shift, one should first solve the geodesic equations of atoms and photons in the general relativistic frame to derive the trajectories of them, solve the five intersections $A$ , $B$ , $C$ , $D_1$ , $D_2 $ , and then calculate the propagation phase shift with the path integral method, and finally obtain the total interferometric phase shift as well as the gravitational acceleration.",
"However, as the integral intervals for the two paths are temporally different due to the FSL, the calculation is not easy, and usually uses a computer to give a scalar expression.",
"The main idea of this work is to make a coordinate transformation to transfer FSL effect of the “control light\" into the atomic Lagrangian, based on which the integral intervals of the classical Lagrangians for the two paths are temporally same in new coordinate system.",
"The velocity of $\\vec{k}_2 $ light becomes $dr^{\\prime }/dt^{\\prime } = \\infty $ , and the atomic Lagrangian contains the FSL disturbance in the new system.",
"Then, the total phase shift can be calculated in Bord$\\acute{}e$ ABCD matrix and perturbation methods."
],
[
"Results",
"In the special relativistic frame, we calculated the FSL effect, and derived a more complete vectorial expression [7], [8].",
"We find that, except for the results given by Kasevich's group [5], [6] and Steven Chu's group [2], [3], the FSL correction also includes ${ - 2\\frac{{\\vec{v}(T) \\cdot {{\\vec{e}}_k}}}{c}\\frac{{{\\alpha _1} - {\\alpha _2}}}{{{{\\vec{k}}_{{\\rm {eff}}}} \\cdot {{\\vec{g}}_0}}}}$ , which has been missed before but at the same magnitude order with the other terms.",
"We further make clear the physical roots of these corrective terms.",
"The main interferometric phase shift arises from the atomic absorbing or emitting laser phase shifts, which can be simply described by ${\\phi _{{\\rm {laser}}}} = {\\vec{k}_{{\\rm {eff}}}} \\cdot \\vec{r} - {\\omega _{{\\rm {eff}}}}t+\\phi _0 & \\rightarrow \\left( {\\frac{{{\\omega _1}{{\\vec{n}}_1} - {\\omega _2}{{\\vec{n}}_2}}}{c} + \\frac{{{\\alpha _1}{{\\vec{n}}_1} - {\\alpha _2}{{\\vec{n}}_2}}}{c}T} \\right) \\cdot \\vec{r}(T,\\delta T) \\nonumber \\\\&- \\left[ {{\\omega _1} - {\\omega _2} + ({\\alpha _1} - {\\alpha _2})T} \\right]t(T,\\delta T)+\\phi _0.$ Here, $\\delta T$ is the time delay due to finite propagating speed of light, and $\\alpha _1$ , $\\alpha _2$ are the frequency chirps, which should be introduced to compensate the doppler shift due to atomic motion.",
"We defined the $1/c$ terms related as the FSL effect.",
"Since $\\delta T$ is $1/c$ related, the FSL effect includes three parts: the pure FSL time delay, the coupling of the frequency chirp and the time delay, and the chirp-dependence changes of the wave vector.",
"In addition, we find the FSL correction depends on the propagating directions of the lights involved in the measurement process.",
"Therefore, the subterms of FSL correction may be experimentally tested by adjusting the experimental configuration.",
"That's why Cheng et al.",
"[4] reported they only experimentally verified the FSL effect associated with the coupling of the frequency chirp and the time delay.",
"Based on the calculation for FSL correction, and derived a more complete relativistic expression of the interferometric phase shift, which is suitable to analyze the atoms moving in three dimensions.",
"In addition, this result first considered the relativistic effects related to Earth's rotation in atomic gravimeters, and also completed the effects related to gravity gradient."
],
[
"conclusion and prospect",
"We mainly developed an analytical study method, based on which the FSL effect is clearly studied, and further a more complete relativistic model for atom gravimeters is established.",
"This work will help to test GR with atomic interferometry.",
"In the near future, on one hand, we will consider exploring the error-elimination schemes, such as the frequency-shift gravity-gradient compensation technique [10], and in fact we have started this related work [11]; on the other hand, we want to explore the GR-test scheme, such as the test of Lorentz violation and gravitational wave."
],
[
"Acknowledgments",
"This work is supported by the Post-doctoral Science Foundation of China (Grant Nos.",
"2017M620308 and 2018T110750)."
]
] | 1906.04348 |
[
[
"Convergence of D\\\"umbgen's Algorithm for Estimation of Tail Inflation"
],
[
"Abstract Given a density $f$ on the non-negative real line, D\\\"umbgen's algorithm is a routine for finding the (unique) log-convex, non-decreasing function $\\hat\\phi$ such that $\\int\\hat\\phi(x)f(x)dx=1$ and such that the likelihood $\\prod_{i=1}^{n}f(x_i)\\hat\\phi(x_i)$ of given data $x_1,\\ldots,x_n$ under density $x\\mapsto \\hat\\phi(x)f(x)$ is maximized.",
"We summarize D\\\"umbgen's algorithm for finding this MLE $\\hat\\phi$, and we present a novel guarantee of the algorithm's termination and convergence."
],
[
"Introduction",
"Motivated by the study of statistical sparsity, we suppose that $f$ is a density on the set $\\mathbb {R}_{\\ge 0}$ of non-negative real numbers, and we consider the class $\\Phi _1:=\\left\\lbrace \\phi :\\mathbb {R}_{\\ge 0}\\rightarrow \\mathbb {R}_{>0}~|~\\text{$\\phi $ log-convex,non-decreasing s.t.",
"}\\int \\phi (x)f(x)dx=1\\right\\rbrace $ of positive, log-convex, non-decreasing functions $\\phi $ such that $x\\mapsto \\phi (x)f(x)$ defines a probability density.",
"The interpretation is that, for any given $\\phi $ in $\\Phi _1$ , the product $\\phi \\cdot f$ represents the modification of $f$ by the tail-inflation function $\\phi $ ; the non-decreasing, log-convex nature of $\\phi $ means that the tails of the density $\\phi \\cdot f$ contain relatively more probability mass than do the tails of $f$ .",
"Treating $\\lbrace \\phi \\cdot f~:~\\phi \\in \\Phi _1\\rbrace $ as a family of densities parameterized by $\\Phi _1$ , one defines a maximum likelihood estimate $\\hat{\\phi }$ for given data $x_1,\\ldots ,x_n\\in \\mathbb {R}_{\\ge 0}$ as $\\hat{\\phi }:=\\operatornamewithlimits{arg\\,sup}_{\\phi \\in \\Phi _1}\\prod _{i=1}^n \\phi (x_i)f(x_i).$ Dümbgen [1] has provided an iterative active set algorithm for finding this maximum-likelihood estimate $\\hat{\\phi }$ .",
"Despite the fact that the space $\\Phi _1$ of tail inflation functions is infinite-dimensional, Dümbgen's algorithm is able to produce a sequence $\\phi _0,\\phi _1,\\ldots ,\\phi _k,\\ldots \\in \\Phi _1$ of functions that converges (in likelihood) to the MLE.",
"Making use of the fact that the logarithm $\\hat{\\theta }:=\\log (\\hat{\\phi })$ of $\\hat{\\phi }$ is piecewise linear and has finitely many breakpoints (which is proved in Section 5.1 of [1]), Dumbgen's algorithm iteratively updates an active set $D\\subset \\mathbb {R}_{\\ge 0}$ of breakpoints for a convex, piecewise linear candidate function $\\theta $ satisfying $\\int e^{\\theta (x)}f(x)dx=1$ .",
"After the $k$ th iteration of the algorithm we obtain $\\phi _k$ by exponentiating the $k$ th candidate function $\\theta _k$ : $\\phi _k(x):=e^{\\theta _k(x)}.$ The main aim of this paper is to establish a guarantee of convergence for Dümbgen's algorithm.",
"The proof presented relies on the following three assumptions: Assumption 1 The density $f$ is continuous and has full support on $\\mathbb {R}_{>0}$ , that is, $f(x)>0$ for all positive $x$ .",
"Assumption 2 The density $f$ has an exponential tail, that is, there exists a constant $\\beta \\in \\mathbb {R}$ such that $(\\forall \\lambda \\in \\mathbb {R})\\quad \\quad \\quad \\lambda <\\beta \\iff \\int e^{\\lambda x}f(x)dx<\\infty $ and $\\lim _{\\lambda \\rightarrow \\beta ^-}\\int e^{\\lambda x}f(x)dx = \\infty .$ Assumption 3 Exponential tilting of $f$ results in a density with a finite second moment: for $\\lambda \\in \\mathbb {R}$ , $\\int _0^{\\infty }e^{\\lambda x}f(x)dx<\\infty \\Rightarrow \\int _0^{\\infty }x^2e^{\\lambda x}f(x)dx<\\infty .$ The family of Gamma distributions is a prototypical example satisfying the above three requirements.",
"It should be noted that Dümbgen's paper [1] provides an algorithm that works in the enlarged setting where $f$ is defined on $\\mathbb {R}$ and $\\phi $ is not required to be monotone, and that the setting where $\\phi $ is log-concave (rather than log-convex) is also addressed in [1].",
"The focus of the present paper is restricted to what Dumbgen calls “Setting 2B”, where $f$ is defined on $\\mathbb {R}_{\\ge 0}$ and where the tail-inflation functions $\\phi $ are required to be log-convex and non-decreasing.",
"This being said, the results presented in this paper generalize well to the other settings considered in [1].",
"In Section we give the statement of Dümbgen's algorithm for estimation of $\\hat{\\phi }$ in the setting where $f$ is defined on the non-negative real half-line and $\\hat{\\phi }$ is required to be log-convex.",
"Additionally, we state several key results from Dümbgen's paper [1] that are important in demonstrating convergence.",
"In section we give an overview of the proof of convergence.",
"In section we calculate a bound on the suboptimality of $\\theta _k$ in terms of the directional derivatives of our objective function $L$ (which is a modified log-likelihood function).",
"In section we build on a key result from Dümbgen's paper to show that the maximal slope $\\sup _{k,x}\\theta _k^{\\prime }(x)$ attained by any candidate function $\\theta _k$ is bounded above by some number strictly less than $\\beta $ .",
"Additionaly, we give a bound on the values ${\\theta _k(0)}$ taken by the candidate functions at zero.",
"The results from Section are used in section to give a lower bound on the change in $L$ resulting from each step taken by Dümbgen's algorithm.",
"In section we finish the proof guaranteeing convergence of the algorithm.",
"In section we conclude."
],
[
"Statement of Dumbgen's Algorithm and Results from\n{{cite:200c21b394f915a4c2aca6db5900638d4911584e}}",
"The aim of this section is to summarize the derivation of Dümbgen's algorithm for estimation of a log-convex tail-inflation factor.",
"We also summarise the results from Dümbgen's paper that are used later in proving convergence of the algorithm, glossing over proofs when convenient.",
"As mentioned in the introduction, Dümbgen's paper also considers settings where the ambient density$f$ is defined on the whole real line (as opposed to on $\\mathbb {R}_{\\ge 0}$ ), and where the inflation factor $\\phi $ is log-concave instead of log-convex; these settings are not considered here.",
"See Dümbgen's paper for full discussion.",
"We are given data $x_1,\\ldots ,x_n\\in \\mathbb {R}_{\\ge 0}$ and a density $f$ on $\\mathbb {R}_{\\ge 0}$ with full support.",
"To find the log-convex, non-decreasing function $\\hat{\\theta }$ maximizing the likelihood of $x_1,\\ldots ,x_n$ , an active-set strategy is used.",
"First, we define the set $\\Theta _1=\\left\\lbrace \\theta :\\mathbb {R}_{\\ge 0}\\rightarrow \\mathbb {R}~|~\\text{$\\theta $ convex,non-decreasing s.t.",
"}\\int e^{\\theta (x)}f(x)dx=1\\right\\rbrace $ of candidate functions $\\theta $ .",
"We note the set bijection $\\Theta _1\\cong \\Phi _1$ defined by $\\theta \\mapsto e^\\theta $ .",
"The log-likelihood of a given candidate $\\theta \\in \\Theta _1$ is given by $\\log \\left[\\prod _{i=1}^n e^{\\theta (x_i)}f(x_i)\\right] =\\sum _{i=1}^n\\left[\\theta (x_i)+\\log f(x_i)\\right].$ Seeing as $f$ and $x_1,\\ldots ,x_n$ are fixed, we take as our objective for optimiziation the simplification $l(\\theta ):=\\sum _{i=1}^n \\theta (x_i).$ of the log-likelihood (REF ).",
"Here $l$ defines a function from $\\Theta _1$ to $\\mathbb {R}$ .",
"In order to employ techniques from convex optimization, we consider the superset $\\Theta =\\left\\lbrace \\theta :\\mathbb {R}_{\\ge 0}\\rightarrow \\mathbb {R}~|~\\text{$\\theta $ convex,non-decreasing}\\right\\rbrace $ of $\\Theta _1$ .",
"This set $\\Theta $ is closed under convex combinations, that is if $\\theta _a$ and $\\theta _b$ are elements of $\\Theta $ then so is $\\lambda \\theta _a+(1-\\lambda )\\theta _b$ , so long as $\\lambda $ satisfies $0\\le \\lambda \\le 1$ .",
"Dumbgen defines the function $L:\\Theta \\rightarrow \\bar{\\mathbb {R}}$ by $L(\\theta ):=\\sum _{i=1}^n\\theta (x_i)-\\int e^{\\theta (x)}f(x)dx+1$ We note that $L(\\theta )$ is finite if and only if $\\int e^{\\theta (x)}f(x)$ is finite.",
"Following the notation used in [1], we use $\\hat{P}$ to denote the empirical distribution $\\frac{1}{n}\\sum _{i=1}^n\\delta _{x_n}$ of the observed data $x_1,\\ldots ,x_n$ , so that the log-likelihood function (REF ) can be written as $l(\\theta )=\\int \\theta d\\hat{P}.$ We write $M$ for the measure on $\\mathbb {R}_{\\ge 0}$ having density $f$ , so that the objective (REF ) can be written as $L(\\theta )=\\int \\theta d\\hat{P}-\\int e^\\theta dM+1.$ The following four properties demonstrate that $L$ is a suitable objective function for finding the MLE $\\hat{\\theta }$ .",
"Property 2.1 For any $\\theta $ in $\\Theta $ , we have $L(\\theta )=\\sum _{i=1}^n\\theta (x_i)$ if and only if $\\int e^{\\theta (x)}f(x)dx=1$ .",
"It follows that we have functional equality $l=L|_{\\Theta _1}$ between the log-likelihood function $l$ and the restriction of $L$ to $\\Theta _1$ .",
"For any $\\theta $ in $\\Theta $ , we have $\\int e^\\theta dM=1\\iff l(\\theta )=\\sum _{i=1}^n\\theta (x_i)=\\sum _{i=1}^n\\theta (x_i)-\\int e^\\theta dM+1=L(\\theta )$ Property 2.2 The function $L$ is strictly concave on the set $\\Theta $ .",
"To see that $L$ is strictly concave, observe that the first term $\\int \\theta d\\hat{P}$ of (REF ) is affine, and that the exponential function appearing in the second term is strictly convex.",
"Lemma 2.1 $L$ attains its maximum at a unique point $\\hat{\\theta }:=\\operatornamewithlimits{arg\\,sup}_{\\theta \\in \\Theta }L(\\theta ).$ By the previous property, $\\hat{\\theta }$ is an element of $\\Theta _1$ .",
"For a proof of the above result, see Lemma 2.7 and Section 5.1 from [1].",
"We remark that Dümbgen's proof of this fact does not rely on Assumptions REF or REF , and that the result can still be proved even with weakend versions of Assumption REF .",
"Property 2.3 The maximum $\\hat{\\theta }:=\\operatornamewithlimits{arg\\,sup}_{\\theta \\in \\Theta }L(\\theta )$ is an element of $\\Theta _1$ .",
"This can be seen by letting $c\\in \\mathbb {R}$ and taking the derivative of $L(\\theta +c)$ with respect to $c$ .",
"We find that $\\frac{\\partial }{\\partial c}L(\\theta +c)=0$ only if $c=-\\log \\int e^\\theta dM$ which implies $\\theta +c\\in \\Theta _1$ .",
"It follows from the above four properties that maximizing $L$ over $\\Theta $ is equivalent to maximizing $l$ over $\\Theta _1$ , in the sense that $\\operatornamewithlimits{arg\\,sup}_{\\theta \\in \\Theta }L(\\theta )=\\hat{\\theta }=\\operatornamewithlimits{arg\\,sup}_{\\theta \\in \\Theta _1}l(\\theta ).$ Dümbgen's algorithm relies on the following crucial lemma, which is listed as Lemma 2.7 in Dümbgen's paper [1] and is proved in section 5.1 of the same.",
"Lemma 2.2 The MLE $\\hat{\\theta }$ is a piecewise linear function having finitely many breakpoints.",
"Writing $x_1\\le \\cdots \\le x_n$ without loss of generality, there is at most one breakpoint in the open inverval $(x_i,x_{i+1})$ between each pair of adjacent observations $x_i,x_{i+1}$ .",
"Moreover, every breakpoint of $\\hat{\\theta }$ is an element of the set $\\left(\\lbrace 0\\rbrace \\cup [x_1,x_n]\\right)\\setminus \\lbrace x_1,\\ldots ,x_n\\rbrace $ , where a breakpoint at 0 is interpreted as saying that the right derivative of $\\hat{\\theta }$ at 0 is nonzero.",
"It follows that $\\hat{\\theta }$ has at most $n$ breakpoints.",
"Thus we consider the set $\\mathbb {V}:=\\left\\lbrace v:\\mathbb {R}_{\\ge 0}\\rightarrow \\mathbb {R}~|~\\text{$v$ piecewise linear with finitelymany breakpoints}\\right\\rbrace $ of piecewise linear functions with finitely many breakpoints.",
"In particular, we know that $\\hat{\\theta }$ belongs to the set $\\Theta _1\\cap \\mathbb {V}$ .",
"Notation 2.1 Given an element $v$ of $\\mathbb {V}$ , we let $D(v)$ denote the set $D(v):=\\lbrace \\tau \\in \\mathbb {R}_{\\ge 0}~:~v^{\\prime }(\\tau -)\\ne v^{\\prime }(\\tau +)\\rbrace $ of breakpoints the given function $v$ .",
"In the above display equation, $v^{\\prime }(\\tau -)$ denotes the left derivative of $v$ at $\\tau $ , and $v^{\\prime }(\\tau +)$ denotes the right derivative at $\\tau $ .",
"By the definition of $\\mathbb {V}$ , the set $D(v)$ is a finite subset of $\\mathbb {R}_{\\ge 0}$ for any $v$ in $\\mathbb {V}$ .",
"Notation 2.2 (The subset $\\mathbb {V}_S$ of $\\mathbb {V}$ ) Given a finite subset $S$ of $\\mathbb {R}_{\\ge 0}$ , we let $\\mathbb {V}_S$ denote the subset $V_S:=\\lbrace v\\in \\mathbb {V}~:~D(v)\\subseteq S\\rbrace $ of $\\mathbb {V}$ consisting of functions $v$ with breakpoints in $S$ .",
"Notation 2.3 (The function $V_\\tau $ ) For a given non-negative real number $\\tau $ , let $V_\\tau :\\mathbb {R}_{\\ge 0}\\rightarrow \\mathbb {R}$ denote the function $V_\\tau (x):= (x-\\tau )^+={\\left\\lbrace \\begin{array}{ll}0&x\\le \\tau \\\\x-\\tau &\\tau \\le x\\end{array}\\right.",
"}$ that is constantly zero on the interval $[0,\\tau ]$ , and that is increasing with unit slope on the interval $[\\tau ,\\infty )$ .",
"We note that, for any non-negative $\\tau $ , the function $V_\\tau $ is an element of the set $\\Theta \\cap \\mathbb {V}$ of convex non-decreasing functions with finitely many breakpoints.",
"Also, it is worth mentioning that $ \\Theta \\cap \\mathbb {V}= \\text{ span } ^+\\lbrace ( x-\\tau ) ^+\\rbrace $ is the convex cone consisting precisely of the set of finite linear combinations of functions $ \\lbrace V_\\tau ~:~\\tau \\in \\mathbb {R}^+\\rbrace $ , where the coefficients of the terms in the linear combinations are non-negative.",
"Dumbgen's algorithm works by maintaining a set $S$ of breakpoints.",
"The algorithm alternates between a “Local Search”, which finds $\\hat{\\theta }_S:=\\operatornamewithlimits{arg\\,sup}_{\\theta \\in \\Theta \\cap \\mathbb {V}_S}L(v)$ and replaces $S$ with the subset $D(\\hat{\\theta }_S)$ of $S$ , and a “Global Search” which replaces the set $S$ with $S\\cup \\lbrace \\tau \\rbrace $ , where $\\tau $ is chosen as to maximize the directional derivative $\\lim _{t\\rightarrow 0^+}\\frac{L(\\hat{\\theta }_S+tV_\\tau )-L(\\hat{\\theta }_S)}{t}$ in the direction of $V_\\tau $ .",
"In general, for $\\theta $ in $\\Theta $ and $v$ in $\\mathbb {V}$ , we write $DL(\\theta ,v):=\\lim _{t\\rightarrow 0^+}\\frac{L(\\theta +tv)-L(\\theta )}{t}.$ Dümbgen's Algorithm for Finding of Log-Convex Tail Inflation MLE [1] Dümbgen'sAlgorithm$\\theta _0,\\delta _0,\\delta _1,\\epsilon $ $k\\leftarrow 0$ $\\tau _k\\leftarrow \\operatornamewithlimits{arg\\,sup}_\\tau DL(\\theta _k,V_\\tau )$ to within $\\delta _0$ suboptimality $h_k\\leftarrow DL(\\theta _k, V_{\\tau _k})$ $h_k\\le \\epsilon $ $h_k\\le \\epsilon $ is the termination criterion $\\theta _k$ $k\\leftarrow k+1$ $S_k\\leftarrow D(\\theta _{k-1})\\cup \\lbrace \\tau _{k-1}\\rbrace $ Note that $D(\\theta _{k-1})\\subseteq S_{k-1}$ $\\theta _k\\leftarrow \\operatornamewithlimits{arg\\,sup}_{\\theta \\in \\Theta \\cap \\mathbb {V}_{S_k}}L(\\theta )$ to within $\\delta _1$ suboptimality It is possible to efficiently find $\\operatornamewithlimits{arg\\,sup}_\\tau DL(\\theta _k,V_\\tau )$ on line of Algorithm because of the following crucial lemma, proved in Section 3.2 of Dümbgen's paper [1].",
"Lemma 2.3 For any given $\\theta \\in \\Theta $ , the function $\\tau \\mapsto DL(\\theta ,V_\\tau )$ is strictly concave on each of the intervals $[x_i,x_{i+1}]$ , where $x_1,\\ldots ,x_n$ are the observed data sorted in increasing order.",
"In particular, we can use a concave optimization routine on the intervals $[x_i,x_{i+1}]$ to find the point $\\tau _k$ such that $DL(\\theta _k,\\tau _k)\\ge \\operatornamewithlimits{arg\\,sup}_\\tau DL(\\theta _k,V_\\tau )-\\delta _0$ .",
"Thus, if the termination criterion $h_k\\le \\epsilon $ on line on Algorithm is met, then we can guarantee $\\sup _\\tau DL(\\theta _k,V_\\tau )\\le DL(\\theta _k,\\tau _k)+\\delta _0= h_k+\\delta _0\\le \\epsilon +\\delta _0.$ We prove in Section that if $\\left(\\sup _{\\theta \\in {\\Theta \\cap \\mathbb {V}_{S_{k}}}}L(\\theta )\\right)-L(\\theta _k)\\le \\delta _1$ and if $\\sup _\\tau DL(\\theta _k,V_\\tau )\\le \\epsilon +\\delta _0$ then $L(\\hat{\\theta })-L(\\theta _k)\\le \\beta (\\epsilon +\\delta _0)+\\delta _1$ for some constant $\\beta $ .",
"Thus the termination criterion $h_k\\le \\epsilon $ corresponds directly to suboptimality of the candidate function $\\theta _k$ produced by step $k$ of the algorithm.",
"The set $\\mathbb {V}_{S_k}$ consists of piecewise linear functions having breakpoints in the set $S_k$ .",
"Finally, we should mention that, because the space $V_{S_k}$ is finite-dimensional (having dimension ${S_k}$ ), it is possible to find $\\operatornamewithlimits{arg\\,sup}_{\\theta \\in \\Theta \\cap \\mathbb {V}_{S_k}}L(\\theta )$ , as on line of Algorithm , using standard convex optimization procedure.",
"In Dümbgen's implementation of the algorithm, Newton's method is used with a Goldstein-Armijo stepsize correction.",
"In Appendix , we sketch a proof that the cardinality of $S_k$ is bounded above by $2n-1$ ."
],
[
"Other useful results from {{cite:200c21b394f915a4c2aca6db5900638d4911584e}}",
"Lemma 2.4 (Dümbgen) Let $\\mathbb {1}:\\mathbb {R}_{\\ge 0}\\rightarrow \\mathbb {R}$ denote the constant function $x\\mapsto 1$ .",
"An element $\\theta $ of $\\Theta $ belongs to $\\Theta _1$ if and only if $DL(\\theta ,\\mathbb {1})=0$ , i.e.",
"$\\Theta _1=\\lbrace \\theta \\in \\Theta ~:~DL(\\theta ,\\mathbb {1})=0\\rbrace .$ This proof is from Section 1 of [1].",
"We consider the derivative $\\frac{\\partial }{\\partial c}L(\\theta +c\\mathbb {1})$ .",
"We have $DL(\\theta ,\\mathbb {1})&\\equiv \\frac{\\partial }{\\partial t}L(\\theta +t\\mathbb {1})\\big |_{t=0^+}\\\\&=\\frac{\\partial }{\\partial t}\\left[\\int (\\theta +t\\mathbb {1})d\\hat{P} -\\int e^{\\theta +t\\mathbb {1}}dM+1\\right|_{t=0^+}\\\\&=\\left[\\int \\mathbb {1} d\\hat{P} -\\int \\mathbb {1} e^{\\theta +t\\mathbb {1}}dM\\right|_{t=0^+}\\\\&=1 -\\int \\mathbb {1} e^{\\theta }dM=1-\\int e^\\theta dM$ which shows that $DL(\\theta ,\\mathbb {1})=0$ if and only if $1=\\int e^\\theta dM$ , which (by definition) holds if and only if $\\theta $ is an element of $\\Theta _1$ .",
"Definition 2.1 We say that a function $\\theta \\in \\Theta \\cap \\mathbb {V}$ is locally optimal if $\\theta $ maximizes $L$ over the set $\\Theta \\cap \\mathbb {V}_{D(\\theta )}$ , that is, if $\\theta =\\operatornamewithlimits{arg\\,sup}_{v\\in \\Theta \\cap \\mathbb {V}_{D(\\theta )}}L(v).$ Lemma 2.5 (Dümbgen) An element $\\theta $ of $\\Theta \\cap \\mathbb {V}$ is locally optiomal if and only if $DL(\\theta ,\\mathbb {1})=0$ and $DL(\\theta ,V_\\tau )=0$ for all $\\tau $ in $D(\\theta )$ , that is, $\\left(\\theta =\\operatornamewithlimits{arg\\,sup}_{v\\in \\Theta \\cap \\mathbb {V}_{D(\\theta )}}L(v)\\right)\\iff \\biggl (DL(\\theta ,\\mathbb {1})=0\\quad \\text{and}\\quad DL(\\theta ,V_\\tau )=0~ \\forall \\tau \\in D(\\theta )\\biggr ).$ Given an element $\\theta $ of $\\Theta \\cap \\mathbb {V}$ , the set $\\mathbb {V}_{D(\\theta )}$ is a finite-dimensional vector space with basis $\\lbrace \\mathbb {1}\\rbrace \\cup \\lbrace V_\\tau ~:~\\tau \\in D(\\theta )\\rbrace $ .",
"The set $\\Theta \\cap \\mathbb {V}_{D(\\theta )}$ is the convex cone $\\Theta \\cap \\mathbb {V}_{D(\\theta )}=\\left\\lbrace \\alpha \\mathbb {1}+\\sum _{\\tau \\in D(\\theta )}\\beta _\\tau V_\\tau ~|~\\alpha \\in \\mathbb {R},~\\beta _\\tau \\ge 0\\forall \\tau \\in D(\\theta )\\right\\rbrace .$ in the vector space $\\mathbb {V}_{D(\\theta )}$ .",
"We note that since $\\Theta \\cap \\mathbb {V}_{D(\\theta )}$ is a convex set, and is a subset of $\\Theta $ , the restriction of $L$ to $\\Theta \\cap \\mathbb {V}_{D(\\theta )}$ is a convex function.",
"We use the notation $\\beta _{\\tau ,\\theta }$ to denote the change in slope of $\\theta $ at breakpoint $\\tau $ , so that $\\theta =\\theta (0)\\mathbb {1}+\\sum _{\\tau \\in D(\\theta )}\\beta _{\\tau ,\\theta }V_\\tau $ .",
"By the definition of $D(\\theta )$ , we have $\\beta _{\\tau ,\\theta }>0$ for all $\\tau \\in D(\\theta )$ .",
"This is to say that $\\theta $ lies in the interior of the convex cone $\\Theta \\cap \\mathbb {V}_{D(\\theta )}$ .",
"Since $\\theta $ is not on the boundary of this set, the local optimality of $\\theta $ is equivalent to the condition $\\frac{\\partial L(\\theta )}{\\partial \\mathbb {1}}=0$ and $\\frac{\\partial L(\\theta )}{\\partial V_\\tau }=0$ for each $\\tau $ .",
"Property 2.4 If an element $\\theta $ of $\\Theta \\cap \\mathbb {V}$ is locally optimal, then $\\theta $ is an element of $\\Theta _1$ .",
"This follows directly from Lemmas REF and REF : If $\\theta $ is locally optimal then $DL(\\theta ,\\mathbb {1})=0$ , which is equivalent to $\\theta $ 's membership in the subset $\\Theta _1$ of $\\Theta $ ."
],
[
"Proof of Convergence: Overview",
"To simplify our analysis, we suppose that the search steps $\\tau _k\\leftarrow \\operatornamewithlimits{arg\\,sup}_\\tau DL(\\theta _k,V_\\tau ) \\quad \\quad \\text{and}\\quad \\quad \\theta _k\\leftarrow \\operatornamewithlimits{arg\\,sup}_{\\theta \\in \\Theta \\cap \\mathbb {V}_{S_{k-1}\\cup \\lbrace \\tau _{k-1}\\rbrace }}L(\\theta )$ in Algorithm are exact.",
"As mentioned at the end of the previous section, if the termination criterion $h_k\\le \\epsilon $ from line of Algorithm is satisfied then the bound $L(\\hat{\\theta })-L(\\theta _k)<\\beta \\epsilon $ follows.",
"This is proved in Section .",
"Now, supposing that the termination criterion $h_k\\le \\epsilon $ is not met, we show that, for any fixed positive $\\epsilon $ and any real number $R$ , if $L(\\theta _k)\\ge R$ then there exists some constant $C_{R,\\epsilon }>0$ such that $h_k > \\epsilon \\quad \\Rightarrow \\quad L(\\theta _{k+1})-L(\\theta _k)\\ge C_{R,\\epsilon }$ for each $k$ .",
"This will be proved in Section , with help from lemmas proved in Section .",
"Because $\\theta _{k}$ is defined as the $\\operatornamewithlimits{arg\\,sup}$ of $L$ over the class $\\Theta \\cup \\mathbb {V}_{S_{k-1}\\cup \\lbrace \\tau _{k-1}\\rbrace }$ of convex piecewise linear functions having breakpoints in the set $S_{k-1}\\cup \\lbrace \\tau _{k-1}\\rbrace $ , and because $\\theta _{k-1}$ belongs to this same class, we are guaranteed that $L(\\theta _k)\\ge L(\\theta _{k-1})$ for each $k$ .",
"Therefore $L(\\theta _k)\\ge L(\\theta _0)$ for each $k$ , hence $ h_k > \\epsilon \\quad \\Rightarrow \\quad L(\\theta _{k+1})-L(\\theta _k)\\ge C_{L(\\theta _0),\\epsilon }$ for each $k$ .",
"Therefore, we can guarantee that after finitely many steps (bounded above in number by the ratio $(L(\\hat{\\theta })-L(\\theta _0))/{C_{L(\\theta _0),\\epsilon }}$ ) the bound (REF ) is reached."
],
[
"$\\theta $ locally optimal and {{formula:c68ee7dd-0864-4886-abf1-da6fe269a1f8}} implies\n{{formula:10860421-c023-47af-890f-43812afbfb02}}",
"The goal of this section is to show that suboptimality $L(\\hat{\\theta })-L(\\theta _k)<const\\cdot \\epsilon $ is implied by the termination condition $\\operatornamewithlimits{arg\\,sup}_{\\tau }DL(\\theta _k,V_\\tau )<\\epsilon $ .",
"In other words, if there is no $\\tau $ satisfying $DL(\\theta _k,V_\\tau )>\\epsilon $ then the suboptimality $L(\\hat{\\theta })-L(\\theta _k)$ of $\\theta _k$ must be small.",
"As mentioned in the previous section, we simplify out analysis by assuming that the local search step $\\theta \\leftarrow \\operatornamewithlimits{arg\\,sup}_{v\\in \\Theta \\cap \\mathbb {V}_{S}}L(v)$ is exact, that is, that $\\theta _k$ is locally optimal.",
"Not making this assumption, we would instead obtain a bound $L(\\hat{\\theta })-L(\\theta _k)<\\delta _1+(const)\\cdot \\epsilon $ on the suboptimality of $\\theta _k$ , where $\\delta _1$ is the tolerance parameter for local suboptimality of $\\theta _k$ : $\\delta _1\\ge \\left(\\sup _{v\\in {\\Theta \\cap {V_{D(\\theta _k)}}}}L(v)\\right) - L(\\theta _k).$ We have stated in Assumption REF that the density $f$ has an exponential tail, that is, there exists some constant $\\beta \\in \\mathbb {R}$ satisfying $\\int _0^\\infty e^{\\kappa x}M(dx)<\\infty \\iff \\kappa <\\beta .$ Before we can bound the suboptimality of $\\theta _k$ directly need a lemma formalizing the relationship between $\\beta $ and the maximal slope obtained by a convex function $\\theta \\in \\Theta $ .",
"Lemma 4.1 Suppose that $\\theta \\in \\Theta $ .",
"Define $m(\\theta ):=\\sup _x\\theta ^{\\prime }(x+)$ to be the maximal slope attained by $\\theta $ .",
"We have $m(\\theta )<\\beta \\iff \\int e^{theta(x)}M(dx)<\\infty .$ Note that $ \\theta (x_n)+m(\\theta )(x-x_n)\\le \\theta (x)\\le \\theta (0)+m(\\theta )x.",
"$ Therefore $\\int e^\\theta dM\\le \\int e^{\\theta (0)+m(\\theta )x}M(dx)=e^{\\theta (0)}\\int e^{m(\\theta )x}M(dx)$ which gives $m(\\theta )<\\beta \\Rightarrow \\int e^\\theta dM<\\infty ,$ and $\\int e^\\theta dM\\ge \\int e^{\\theta (x_n)+m(\\theta )(x-x_n)}M(dx)=e^{\\theta (x_n)-m(\\theta )x_n}\\int e^{m(\\theta )x}M(dx)$ which gives $m(\\theta )\\ge \\beta \\Rightarrow \\int e^\\theta dM=\\infty .$ In particular, we note that $\\beta $ is is an upper bound for the maximal slope $m(\\hat{\\theta })=\\hat{\\theta }^{\\prime }(x_n)$ attained by $\\hat{\\theta }$ .",
"Note that by Lemma REF , the MLE $\\hat{\\theta }$ does not have a breakpoint at $x_n$ , so we may write $\\hat{\\theta }^{\\prime }(x_n)$ to refer unambiguously to the derivative $\\hat{\\theta }^{\\prime }(x_n-)=\\hat{\\theta }^{\\prime }(x_n)=\\hat{\\theta }^{\\prime }(x_n+)$ of $\\hat{\\theta }$ at $x_n$ .",
"We now state and prove the main result of this section.",
"Lemma 4.2 Let $\\theta \\in \\Theta _1$ .",
"Suppose that $\\theta $ is locally optimal, that is, $\\theta =\\operatornamewithlimits{arg\\,sup}_{v\\in \\Theta \\cap \\mathbb {V}_{D(\\theta )}}L(\\theta ).$ If $\\sup _{\\tau }DL(\\theta ,V_\\tau )<\\epsilon $ then $L(\\hat{\\theta })-L(\\theta )\\le \\hat{\\theta }^{\\prime }(x_n)\\epsilon <\\beta \\epsilon .$ Define ${v:=\\hat{\\theta }-\\theta }$ .",
"Although $v$ might not be convex, we do have ${v\\in \\mathbb {V}}$ , that is, $v$ is a piecewise linear function with finitely many breakpoints.",
"For each breakpoint ${\\tau \\in D(v)}$ , let ${\\beta _{\\tau ,v}:=v^{\\prime }(\\tau +)-v^{\\prime }(\\tau -)}$ denote the change in slope of $v$ at ${\\tau }$ .",
"Similarly, we write ${\\beta _{\\tau ,\\theta }}$ and ${\\beta _{\\tau ,\\hat{\\theta }}}$ , respectively, for the changes in slope ${\\beta _{\\tau ,\\theta }:=\\theta ^{\\prime }(\\tau +)-\\theta ^{\\prime }(\\tau -)}$ and ${\\beta _{\\tau ,\\hat{\\theta }}:=\\hat{\\theta }^{\\prime }(\\tau +)-\\hat{\\theta }^{\\prime }(\\tau -)}$ of $\\theta $ and $\\hat{\\theta }$ .",
"The proof proceeds as follows: first, we shall show that, for any $\\theta \\in \\Theta $ , $L(\\hat{\\theta })-L(\\theta )\\le DL(\\theta ,v).$ Next, we show that if $\\theta \\in \\Theta _1$ then ${DL(\\theta ,v)=\\sum _{\\tau \\in D(v)}\\beta _{\\tau ,v}DL(\\theta ,V_\\tau )}.$ Finally, we show that if $\\sup _\\tau D(\\theta ,V_\\tau )<\\epsilon $ then $\\sum _{\\tau \\in D(v)}\\beta _{\\tau ,v}DL(\\theta ,V_\\tau )<\\epsilon \\hat{\\theta }^{\\prime }(x_n+)<\\epsilon \\beta .$ Combining (REF ), (REF ) and (REF ) gives the desired result.",
"Validitiy of inequality (REF ) follows from concavity of $L$ on $\\Theta $ , which gives $L(\\hat{\\theta })-L(\\theta )&=\\frac{L(\\hat{\\theta })t-L(\\theta )t}{t}\\\\&= \\frac{\\left(L(\\theta )(1-t)+L(\\hat{\\theta })t\\right)- L(\\theta )}{t}\\\\&\\le \\frac{L\\left(\\theta (1-t)+\\hat{\\theta }t\\right)-L(\\theta )}{t}\\\\&=\\frac{L\\left(\\theta + tv\\right)- L(\\theta )}{t}$ for all $t$ in the interval $(0,1)$ .",
"Taking the limit as $t$ approaches 0 from above, we have $L(\\hat{\\theta })-L(\\theta )\\le \\lim _{t\\rightarrow -^+}\\frac{L\\left(\\theta + tv\\right)- L(\\theta )}{t}\\equiv DL(\\theta ,v)$ as required.",
"To demonstrate validity of equation (REF ), first re-write $v$ as $v =v(0)\\mathbb {1}+\\sum _{\\tau \\in D(v)}\\beta _{\\tau ,v}V_\\tau $ where $\\mathbb {1}$ denotes the constant function $(x\\mapsto 1)$ .",
"Note that the changes in slope $\\beta _{\\tau ,v}$ can be negative, if $\\beta _{\\tau ,\\theta }>\\beta _{\\tau ,\\hat{\\theta }}$ .",
"In Theorem REF from Appendix we show that the operator $v\\mapsto DL(\\theta ,v)$ is linear (with a caveat regarding the domain on which $L$ is finite); using this linearity, we have $DL(\\theta ,v)&=DL\\left(\\theta ,~v(0)\\mathbb {1}+\\sum _{\\tau \\in D(v)}\\beta _{\\tau ,v}V_\\tau \\right)\\\\&=v(0)DL\\left(\\theta ,~\\mathbb {1}\\right)+\\sum _{\\tau \\in D(v)}\\beta _{\\tau ,v}DL\\left(\\theta ,V_\\tau \\right).$ Because $\\theta $ is assumed to be locally optimal, we have $DL(\\theta ,\\mathbb {1})=0$ and thus equation (REF ) follows.",
"Finally, assuming that $\\sup _\\tau D(\\theta ,V_\\tau )<\\epsilon $ , and using the fact that $DL(\\theta ,V_\\tau )=0\\quad \\forall \\tau \\in D(\\theta )$ follows from local optimality of $\\theta $ (c.f.",
"Lemma REF ), we have $\\sum _{\\tau \\in D(v)}\\beta _{\\tau ,v}DL(\\theta ,V_\\tau )&=\\sum _{\\tau \\in D(\\hat{\\theta })\\setminus D(\\theta )}\\beta _{\\tau ,v}DL(\\theta ,V_\\tau )+\\sum _{\\tau \\in D(\\theta )}\\beta _{\\tau ,v}DL(\\theta ,V_\\tau )\\\\&=\\sum _{\\tau \\in D(\\hat{\\theta })\\setminus D(\\theta )}\\beta _{\\tau ,v}DL(\\theta ,V_\\tau )&\\left(\\text{since}~DL(\\theta ,V_\\tau )=0\\quad \\forall \\tau \\in D(\\theta )\\right)\\\\&=\\sum _{\\tau \\in D(\\hat{\\theta })\\setminus D(\\theta )}\\left[\\beta _{\\tau ,\\hat{\\theta }}-\\beta _{\\tau ,\\theta }\\right]DL(\\theta ,V_\\tau )\\\\&=\\sum _{\\tau \\in D(\\hat{\\theta })\\setminus D(\\theta )}\\beta _{\\tau ,\\hat{\\theta }}DL(\\theta ,V_\\tau )&(\\text{since}~\\tau \\notin D(\\theta )\\Rightarrow \\beta _{\\tau ,\\theta }=0)\\\\&\\le \\sum _{\\tau \\in D(\\hat{\\theta })}\\beta _{\\tau ,\\hat{\\theta }}\\cdot \\epsilon \\\\&=\\epsilon \\cdot \\hat{\\theta }^{\\prime }(x_n+)<\\epsilon \\beta .$ It follows from the above lemma that, if the termination criterion $h_k\\le \\epsilon $ in Algorithm is met, then $L(\\hat{\\theta })-L(\\theta _k)\\le \\hat{\\theta }^{\\prime }(x_n+)\\cdot \\epsilon <\\beta \\cdot \\epsilon $ if we are using exact searches on lines and of Algorithm , or $L(\\hat{\\theta })-L(\\theta _k)\\le \\delta _1+\\hat{\\theta }^{\\prime }(x_n+)\\cdot (\\epsilon +\\delta _0)<\\delta _1+\\beta \\cdot (\\epsilon +\\delta _0)$ when using inexact searches with tolerances $\\delta _0$ and $\\delta _1$ ."
],
[
"Upper Bounds on the Slope and Intercept of $\\theta _k$",
"In this section we show that, for any constant $R\\in \\mathbb {R}$ , there exist real numbers numbers $s_R\\in (0,\\infty )$ and $m_R\\in (0,\\beta )$ such that, for any $\\theta \\in \\Theta $ , $R\\le L(\\theta )\\Rightarrow \\left({\\theta (0)}\\le s_R\\quad \\text{and}\\quad \\sup _{x}\\theta ^{\\prime }(x)\\le m_R\\right).$ As in the previous section, we will write $m(\\theta ):=\\sup _x\\theta (x+)$ for the maximal slope obtained by a function $\\theta $ in $\\Theta $ .",
"Lemma 5.1 (Dumbgen) For any $\\theta $ in $\\Theta $ , $L(\\theta )\\le -\\log \\int e^{\\theta (x)-\\theta (x_n)}M(dx).$ This result is proved in section 5.1 of Dümbgen's paper [1].",
"We reproduce the proof here: Seeing as $\\theta $ is non-decreasing on $[0,x_n]$ and $\\hat{P}$ is the empirical distribution of the data $x_1,\\ldots ,x_n$ , we have $\\int \\theta d\\hat{P}\\le \\theta (x_n)$ .",
"Thus $L(\\theta )&=\\int \\theta d\\hat{P}-\\int e^\\theta dM+1 \\\\&\\le \\theta (x_n)-\\int e^{\\theta (x)+\\theta (x_n)-\\theta (x_n)}M(dx)+1 \\\\&=\\theta (x_n)-e^{\\theta (x_n)}\\int e^{\\theta (x)-\\theta (x_n)} M(dx)+1 \\\\&\\le \\sup _{p\\in \\mathbb {R}}\\left[p-e^p\\int e^{\\theta (x)-\\theta (x_n)} M(dx)\\right]+1\\\\&=-\\log \\int e^{\\theta (x)-\\theta (x_n)}M(dx).$ The result below follows directly from the preceding lemma.",
"Proposition 5.1 (Dumbgen) For any $\\theta $ in $\\Theta $ , $L(\\theta )\\le -\\log \\int e^{m(\\theta )\\cdot (x-x_n)}M(dx).$ Because $\\theta $ is convex, $\\theta (x)-\\theta (x_n)\\ge m(\\theta )\\cdot (x-x_n)$ for all $x$ .",
"Therefore, $L(\\theta )\\le -\\log \\int e^{\\theta (x)-\\theta (x_n)}M(dx)\\le -\\log \\int e^{m(\\theta )\\cdot (x-x_n)}M(dx)$ .",
"We note that $\\lim _{m(\\theta )\\rightarrow \\infty }-\\log \\int e^{m(\\theta )\\cdot (x-x_n)}M(dx)=-\\infty .$ Using this fact, we are able to prove the following: Theorem 5.1 For any $R\\in \\mathbb {R}$ there exists a positive real number $m_R<\\beta $ such that $ R\\le L(\\theta )\\Rightarrow m(\\theta )\\le m_R $ for all $\\theta $ in $\\Theta $ .",
"Fix $R$ in $\\mathbb {R}$ .",
"By (REF ) there exists a constant $m_R$ such that $m> m_R\\Rightarrow -\\log \\int e^{m\\cdot (x-x_n)}M(dx)< R$ for all $m$ in $\\mathbb {R}$ .",
"This is equivalent to the statement that there exists $m_R$ satisfying $R\\le -\\log \\int e^{m\\cdot (x-x_n)}M(dx)\\Rightarrow m\\le m_R$ for all $m$ in $\\mathbb {R}$ .",
"By Proposition REF , if $R\\le L(\\theta )$ then $R\\le -\\log \\int e^{m(\\theta )\\cdot (x-x_n)}M(dx)$ , implying $ R\\le L(\\theta )\\Rightarrow m(\\theta )\\le m_R$ .",
"To guarantee that such a value $m_R$ can be found satisfying $m_R<\\beta $ , we take a closer look at (REF ).",
"The statement $-\\log \\int e^{m\\cdot (x-x_n)}M(dx)< R$ is equivalent to $e^{-R}< e^{-mx_n}\\int e^{mx}M(dx).$ Therefore, (REF ) can be restated as $m> m_R\\Rightarrow e^{-R}<e^{-mx_n}\\int e^{mx}M(dx).$ By Assumption REF we have $\\lim _{\\lambda \\rightarrow \\beta ^-}e^{-\\lambda x_n}\\int e^{\\lambda x}f(x)dx = \\infty ,$ which goes to show that, for any $R$ , we can find a number $m_R<\\beta $ such that (REF ) is satisfied.",
"We now go through a similar argument to obtain a bound on ${\\theta (0)}$ .",
"Theorem 5.2 (Dumbgen) For any $R\\in \\mathbb {R}$ , there exists a non-negative real number $s_R$ such that $ R\\le L(\\theta )\\Rightarrow {\\theta (0)}\\le s_R $ for all $\\theta $ in $\\Theta $ .",
"We will first obtain a bound on $\\theta (x_n)$ of the form $ R\\le L(\\theta )\\Rightarrow {\\theta (x_n)}\\le q_R .",
"$ We will then combine the bounds $q_R$ and $m_R$ to obtain a bound $s_R$ on $\\theta (0)$ .",
"To begin, fix $R$ in $\\mathbb {R}$ .",
"From () we have $L(\\theta ) \\le \\theta (x_n)-e^{\\theta (x_n)}\\int _0^\\infty e^{\\theta (x)-\\theta (x_n)}M(dx)+1.$ Since $\\theta (x)-\\theta (x_n)\\ge 0$ for $x$ larger than $x_n$ , we have $\\int _0^\\infty e^{\\theta (x)-\\theta (x_n)}M(dx)\\ge \\int _{x_n}^\\infty M(dx)$ which, combined with (REF ), gives the following inequality: $L(\\theta )\\le \\theta (x_n)-e^{\\theta (x_n)}\\int _{x_n}^\\infty M(dx)+1.$ We note that $\\lim _{{\\theta (x_n)}\\rightarrow \\infty }L(\\theta )\\le \\theta (x_n)-e^{\\theta (x_n)}\\int _{x_n}^\\infty M(dx)+1=-\\infty $ so that, by the same argument as in the proof of Theorem REF , there exists a constant $q_R$ satisfying $ R\\le L(\\theta )\\Rightarrow {\\theta (x_n)}\\le q_R $ for all $\\theta $ in $\\Theta $ .",
"Suppose now that $R\\le L(\\theta )$ , so that ${\\theta (x_n)}\\le q_R$ and $m(\\theta )\\le m_R$ .",
"Because $\\theta $ is non-decreasing, we have $\\theta (0)\\le \\theta (x_n)\\le q_R$ .",
"Because $\\theta $ is convex, we have $-q_R-m_R\\cdot x_n\\le \\theta (x_n)-m_R\\cdot x_n\\le \\theta (0)$ .",
"Combining these two inequalities $ -q_R-m_R\\cdot x_n\\le \\theta (0)\\le q_R$ we obtain $ {\\theta (0)}\\le q_R+m_R\\cdot x_n.$ Thus the required bound on ${\\theta (0)}$ is given by $s_R:=q_R+m_R\\cdot x_n$ .",
"Recall that in Lemma REF we cited Dümbgen's proof that there exists a unique maximizer $\\hat{\\theta }$ for the function $L$ .",
"Although we do not attempt to prove this in the present paper, we remark that the bounds $m_R$ and $s_R$ derived in this section can be used given an upper bound for $L$ : Remark For any $ R\\in \\mathbb {R}$ there is a constant $ U_R $ such that $ R\\le L ( \\theta ) \\Rightarrow L ( \\theta ) \\le U_R.",
"$ It follows that for $\\theta \\in \\Theta $ and for any $ R\\in \\mathbb {R}$ $ L ( \\theta ) \\le \\max ( R,U_R ) $ so that the objective function $ L$ is bounded above.",
"Suppose $ R\\le L ( \\theta ) $ .",
"Then $L(\\theta )&\\le -\\log \\int _{0}^\\infty e^{m(\\theta )(x-x_n)}M(dx)\\\\&=-\\log \\left[\\int _{0}^{x_n}e^{m(\\theta )(x-x_n)}M(dx)+\\int _{x_n}^{\\infty }e^{m(\\theta )(x-x_n)}M(dx)\\right]\\\\&=-\\log \\left[\\int _{0}^{x_n}e^{m_R(x-x_n)}M(dx)+\\int _{x_n}^{\\infty }e^{0(x-x_n)}M(dx)\\right]\\\\&=const_R.$"
],
[
"A local bound on the second directional derivative of $L$",
"In this section we show that, supposing $DL(\\theta _{k-1},V_{\\tau _{k-1}})\\ge \\epsilon $ , the improvement $L(\\theta _{k-1})-L(\\theta _{k})$ at step $k$ is bounded below by a constant.",
"Define $R:=L(\\theta _{k-1})$ so that by Theorems REF and REF there are numbers $s_R$ and $m_R$ satisfying ${\\theta _{k-1}(0)}\\le s_R\\quad \\text{and}\\quad \\sup _x\\theta _{k-1}^{\\prime }(x+)\\le m_R<\\beta .$ Note that $L(\\theta _k)-L(\\theta _{k-1})\\ge L(\\theta _{k-1}+tV_{\\tau _{k-1}})-L(\\theta _{k-1})$ for every real number $t$ .",
"This follows from the fact that $\\theta _k$ is defined as $\\theta _k \\equiv \\operatornamewithlimits{arg\\,sup}_{\\theta \\in \\Theta \\cap \\mathbb {V}_{D(\\theta _{k-1})\\cup \\lbrace \\tau _{k-1}\\rbrace }} L(\\theta )$ and that, for each $t$ , the function $\\theta _{k-1}+tV_{\\tau _{k-1}}$ is also a member of the set $\\Theta \\cap \\mathbb {V}_{D(\\theta _{k-1})\\cup \\lbrace \\tau _{k-1}\\rbrace }$ .",
"Given inequality (REF ), we can bound $L(\\theta _k)-L(\\theta _{k-1})$ below by finding a lower bound for $\\sup _{t\\ge 0}L(\\theta _{k-1}+tV_{\\tau _{k-1}})-L(\\theta _{k-1})$ .",
"For notational convenience, we define the function $g(t):=L(\\theta _{k-1}+tV_{\\tau _{k-1}})-L(\\theta _{k-1}).$ To find a lower bound for $\\sup _{t\\ge 0}g(t)$ , we first note that $g^{\\prime }(t)=\\frac{\\partial }{\\partial t}\\left[L(\\theta _{k-1}+tV_{\\tau _{k-1}})\\right]$ so that $g^{\\prime }(0)\\equiv DL(\\theta _{k-1},V_{\\tau _{k-1}})$ .",
"Moreover, by strict concavity of $L$ , our function $g$ is strictly concave in $t$ , that is, $g^{\\prime \\prime }(t)<0$ so long as $t$ satisfies $g(t)>-\\infty $ .",
"Below, assuming $g^{\\prime }(0)=DL(\\theta _{k-1},V_{\\tau _{k-1}})>\\epsilon $ , we argue that there exists $T>0$ such that $g(T)\\le 0$ .",
"This, together with strict concavity of $g$ and the facts $g(0)=0$ and $g^{\\prime }(0)>0$ , go to show that $g$ obtains its supremum at a unique point $t^*\\in (0,T)$ .",
"The details of this argument rely on the fact that the maximal slope operator $m:\\Theta \\cap \\mathbb {V}\\rightarrow \\mathbb {R}$ is additive: for any $t\\ge 0$ we have $ m(\\theta _{k-1}+tV_{\\tau _{k-1}})=m(\\theta _{k-1})+t\\cdot m(V_{\\tau _{k-1}})=m(\\theta _{k-1})+t.",
"$ Defining $R=L(\\theta _{k-1})$ , we have by Theorem REF that there exists a constant $m_R$ such that $m(\\theta _{k-1})+t>m_R\\Rightarrow L(\\theta _{k-1}+tV_{\\tau _{k-1}})<L(\\theta _{k-1}).$ Writing $T:=m_R-m(\\theta _{k-1})$ gives the desired property $g(T)=L(\\theta _{k-1}+TV_{\\tau _{k-1}})-L(\\theta _{k-1})\\le 0$ , and so the supremum $t^*:=\\operatornamewithlimits{arg\\,sup}_tg(t)$ must exist.",
"The following lemma allows us to find a quadratic function $y(t)$ that bounds $g(t)$ below on the set $[0,T]=\\lbrace t~:~g(t)\\ge 0\\rbrace $ , giving us a lower bound on $g(t^*)$ .",
"Lemma 6.1 Suppose $\\theta \\in \\Theta $ and $\\tau \\in \\mathbb {R}_{\\ge 0}$ satisfy $DL(\\theta ,V_\\tau )>\\epsilon $ , and that $R$ is a parameter satisfying $R\\le L(\\theta )$ so that ${\\theta (0)}\\le s_R\\quad \\text{and}\\quad \\sup _x\\theta ^{\\prime }(x+)\\le m_R<\\beta .$ Let $g:\\mathbb {R}^+\\rightarrow \\mathbb {R}$ denote the function $t\\mapsto L(\\theta +tV_\\tau )-L(\\theta )$ .",
"For $t$ in the set $\\lbrace t~:~g(t)\\ge 0\\rbrace $ , the magnitude ${g^{\\prime \\prime }(t)}$ of the second derivative of $g^{\\prime \\prime }$ is bounded above by a constant depending only on $s_R$ and $m_R$ .",
"If $t$ satisfies $0\\le g(t)$ then $R\\le L(\\theta +tV_{\\tau })$ because $0\\le g(t)=L(\\theta +tV_\\tau )-L(\\theta )=L(\\theta +tV_\\tau )-R.$ Therefore, for all such $t$ we have $\\theta (0)+tV_{\\tau }(0)\\le s_R\\quad \\text{and}\\quad \\sup _x\\left(\\theta ^{\\prime }(x+)+tV_{\\tau }^{\\prime }(x+)\\right)\\le m_R.$ This gives $\\theta (x)+tV_\\tau (x)\\le s_R+m_Rx$ for all $x$ , so that $\\nonumber {g^{\\prime \\prime }(t)}&=\\left|\\frac{\\partial ^2}{\\partial t^2} \\left[L(\\theta +tV_\\tau )-L(\\theta )\\right]\\right|=\\left|\\frac{\\partial ^2}{\\partial t^2} \\left[L(\\theta +tV_\\tau )\\right]\\right|\\\\\\nonumber &=\\left|\\frac{\\partial ^2}{\\partial t^2} \\left[\\int \\left(\\theta +tV_\\tau \\right)d\\hat{P} -\\int e^{\\theta +tV_\\tau }dM+1\\right]\\right|\\\\\\nonumber &=\\left|\\frac{\\partial }{\\partial t} \\left[\\int V_\\tau d\\hat{P} -\\int V_\\tau e^{\\theta +tV_\\tau }dM\\right]\\right|\\\\\\nonumber &=\\left|-\\int V_\\tau ^2 e^{\\theta +tV_\\tau }dM\\right|=\\int _0^\\infty V_\\tau ^2 e^{\\theta +tV_\\tau }dM\\\\\\nonumber &\\le \\int _0^\\infty V_\\tau ^2 e^{s_R+m_Rx}dM\\\\\\nonumber &=\\int _\\tau ^\\infty (x-\\tau )^2 e^{s_R+m_Rx}M(dx)\\\\\\nonumber &=\\int _\\tau ^\\infty (x^2-2x\\tau +\\tau ^2) e^{s_R+m_Rx}M(dx)\\\\\\nonumber &\\le \\int _\\tau ^\\infty (x^2+\\tau ^2) e^{s_R+m_Rx}M(dx)\\\\\\nonumber &\\le \\int _\\tau ^\\infty (x^2+x_n^2) e^{s_R+m_Rx}M(dx)\\\\\\nonumber &\\le \\int _0^\\infty (x^2+x_n^2) e^{s_R+m_Rx}M(dx)\\\\\\nonumber &\\le \\int _0^\\infty (x^2+x_n^2) e^{s_R+m_Rx}M(dx)\\\\&= e^{s_R}\\int _0^\\infty x^2e^{m_Rx}M(dx)+x_n^2e^{s_R}\\int _0^\\infty e^{m_Rx}M(dx)$ for all $t$ satisfying the hypothesis $g(t)\\ge 0$ .",
"Since $m_R<\\beta $ , Assumption REF gives that the last line (REF ) above is finite.",
"By the lemma above, $L(\\theta _k)-L(\\theta _{k-1})$ is greater than or equal to the supremum attained by the parabola $y(t)=\\epsilon x-\\gamma x^2/2$ , where $\\gamma =\\int _0^\\infty (x^2+x_n^2)e^{s_R+m_Rx}M(dx)$ .",
"The peak of the parabola $y$ is attained at $t=\\frac{\\epsilon }{\\gamma }$ , and so $\\sup _{t}y(t)=\\frac{\\epsilon ^2}{2\\gamma }$ .",
"Thus we have: Theorem 6.1 $L(\\theta _k)-L(\\theta _{k-1})\\ge \\frac{\\epsilon ^2}{2\\gamma }$ if $ \\operatornamewithlimits{arg\\,sup}_\\tau DL ( \\theta _{k-1},V_\\tau ) \\ge \\epsilon $ .",
"We emphasize that this bound holds for all $k$ such that the termination criterion $\\sup _\\tau DL(\\theta _k,V_\\tau )<\\epsilon $ for Dümbgen's algorithm has not been met."
],
[
"Proof of Convergence",
"We produce a bound $K_\\epsilon \\in \\mathbb {N}$ such that, starting with a guess $\\theta _0\\in \\Theta _1\\cap \\mathbb {V}$ , Dümbgen's Algorithm is guaranteed to converge in likelihood within $K_\\epsilon $ steps: $ L(\\hat{\\theta })-L(\\theta _{K_\\epsilon })<\\beta \\cdot \\epsilon .",
"$ In the following theorem, we let $R:=L(\\theta _0)$ denote the objective value of $\\theta _0$ and let $s_R\\in (0,\\infty )$ and $m_R\\in (0,\\beta )$ denote constants satisfying $ {\\theta _k(0)}\\le s_R\\quad \\text{and}\\quad \\sup _x\\theta _k^{\\prime }(x)\\le m_R $ for all $k$ in $\\mathbb {N}$ .",
"Such bounds $m_R$ and $s_R$ are guaranteed to exist by Theorems REF and REF , respectively.",
"We let $h_0$ denote the maximal directional derivative $ h_0:=\\sup _\\tau DL(\\theta _0,V_\\tau ) $ , and we assume that the search procedures on lines and of Algorithm are exact.",
"Theorem 7.1 Let $\\gamma _R$ denote the constant $ \\gamma _R:= \\int _0^\\infty (x^2+x_n^2) e^{s_R+m_Rx}M(dx) $ derived in Lemma REF .",
"Dümbgen's Algorithm reaches suboptimality within $K_\\epsilon $ steps, $ L(\\hat{\\theta })-L(\\theta _{K_\\epsilon })<\\beta \\cdot \\epsilon , $ where $K_\\epsilon =*{\\frac{L(\\hat{\\theta })-L(\\theta _0)-\\beta \\epsilon }{\\epsilon ^2/(2\\gamma _R)}}\\le *{\\frac{\\beta (h_0-\\epsilon )}{\\epsilon ^2/(2\\gamma _R)}}$ .",
"By Lemma REF , if there is $k$ such that $\\operatornamewithlimits{arg\\,sup}_\\tau DL(\\theta _k,V_\\tau )\\le \\epsilon $ then $L(\\hat{\\theta })-L(\\theta _k)\\le \\beta \\epsilon $ as required.",
"Defining $K_\\epsilon $ as in the statement of the Theorem above, suppose that the suboptimality criterion (REF ) has not been met for any of the first $K_\\epsilon $ steps taken by the algorithm, that is, suppose $\\operatornamewithlimits{arg\\,sup}_\\tau DL(\\theta ,V_\\tau )\\ge \\epsilon \\quad \\quad \\quad (\\forall k<K_\\epsilon ).$ Then by Theorem REF , $L(\\theta _{k+1})-L(\\theta _{k})\\ge \\frac{\\epsilon ^2}{2\\gamma _R}\\quad \\quad \\quad (\\forall k<K_\\epsilon ).$ Thus we may derive $L(\\hat{\\theta })-L(\\theta _{K_\\epsilon })&=L(\\hat{\\theta })-L(\\theta _0)-\\sum _{k=1}^{K_\\epsilon }\\bigl [L(\\theta _k)-L(\\theta _{k-1})\\bigr ]\\\\&\\le L(\\hat{\\theta })-L(\\theta _0)-\\sum _{k=1}^{K_\\epsilon }\\frac{\\epsilon ^2}{2\\gamma _R}\\\\&=L(\\hat{\\theta })-L(\\theta _0)-*{\\frac{L(\\hat{\\theta })-L(\\theta _0)-\\beta \\epsilon }{\\epsilon ^2/(2\\gamma _R)}}\\frac{\\epsilon ^2}{2\\gamma _R}\\\\&\\le \\beta \\cdot \\epsilon $ to complete the proof.",
"We note here that in practice, the algorithm appears to converge quite quickly."
],
[
"Conclusion",
"A key result proved by Dümbgen in deriving this algorithm is that the logarithm $\\hat{\\theta }(x):=\\log (\\hat{\\phi }(x))$ is necessary piecewise linear, with at most one breakpoint between each pair $x_i,x_j\\in \\lbrace x_1,\\ldots ,x_n\\rbrace $ of adjacent observations.",
"This motivates the decision to optimize over the space $\\Theta $ of convex functions with finitely many breakpoints.",
"This space $\\Theta $ is infinite-dimensional.",
"Despite this, we have been able to prove that Dümbgen's algorithm produces a sequence $\\theta _0,\\ldots ,\\theta _k,\\theta _{k+1},\\ldots $ of functions that converges to the optimum $\\hat{\\theta }$ .",
"The proof of convergence is made possible by the following characteristics of the problem: The optimum $\\hat{\\theta }$ is piecewise linear and has finitely many breakpoints.",
"For any given set $D$ of breakpoints, we can use a finite-dimensional convex optimization routine to find $\\operatornamewithlimits{arg\\,sup}_{v\\in \\mathbb {V}_D}L(v)$ .",
"Thus, this problem of finding $\\hat{\\theta }$ lends itself to an active-set approach.",
"It is possible to efficiently find a good candidate $\\tau \\in \\mathcal {D}$ for addition to the active set of breakpoints.",
"Given $\\epsilon >0$ , if $L(\\hat{\\theta })-L(\\theta _k)\\ge \\epsilon \\beta $ then the directional derivative ${\\frac{\\partial }{\\partial t}L(\\theta +tV_\\tau )\\vert _{t=0+}}$ must be larger than $\\epsilon $ (c.f.",
"Lemma REF ).",
"Strict concavity of $L$ guarantees that the second directional derivative ${\\frac{\\partial ^2}{\\partial t^2}L(\\theta +tV_\\tau )}$ is negative for all $t$ .",
"Although this second derivative is unbounded below, restriction to the level set $\\lbrace t ~:~ L(\\theta +tV_\\tau )>R\\rbrace $ , where $R$ is an arbitrary real number, will allow us to produce a bound on ${\\frac{\\partial ^2}{\\partial t^2}L(\\theta +tV_\\tau )}$ that is uniform for different values of $\\theta $ and $\\tau $ .",
"This is be the key to producing a lower bound on the improvement $L(\\theta _{k+1})-L(\\theta _k)$ in objective value (as in Theorem REF )."
],
[
"Sketch of proof that each $\\theta _k$ has at most {{formula:ab5f702a-5eb7-4e66-95fc-6542a82b4a90}} breakpoints",
"We will show that the locally-optimal parameter $\\theta $ returned by the LocalSearch procedure as defined in [1] will have at most $2n$ breakpoints.",
"It will follow that, since the global parameter search results in the addition of just one breakpoint, candidate functions considered by the algorithm are limited to at most $2n+1$ breakpoints.",
"In Dümbgen's paper it is also proved that the optimal $\\hat{\\theta }$ has at most $n$ breakpoints.",
"The proof of the claim that LocalSearch results in at most $2n$ breakpoints is a consequence of the fact that a locally-optimal parameter can have at most 2 breakpoints between any two adjacent observations.",
"Formally, we have the following Theorem 1.1 (Boundedness of number of breakpoints) Let $x_i,x_j\\in \\lbrace x_1,\\ldots ,x_n\\rbrace $ be adjacent observations, so that $x_i<x_j$ and $\\nexists x^{\\prime }\\in \\lbrace x_1,\\ldots ,x_n\\rbrace $ such that $x_i<x^{\\prime }<x_j$ .",
"For any element $\\theta \\in \\mathbb {V}\\cap \\Theta $ , the result $\\theta ^{\\prime }:=\\arg \\max _{v\\in \\mathbb {V}_{D(\\theta )}\\cap \\Theta _1}L(v)$ of a local search over the set $\\mathbb {V}_{D(\\theta )}\\cap \\Theta _1$ has at most two breakpoints between $x_i$ and $x_j$ , i.e.",
"$\\left|D(\\theta ^{\\prime })\\cap (x_i,x_j)\\right|\\le 2.$ Suppose that $\\theta $ has more than two breakpoints between $x_i$ an $x_j$ , i.e.",
"$\\left|D(\\theta )\\cap (x_i,x_j)\\right|\\ge 3.$ It will suffice to show that $\\theta $ is not locally optimal, i.e.",
"that there exists some element $v\\in \\mathbb {V}_{D(theta)}$ satisfying $DL(\\theta ,v)>0.$ Let $\\tau _1,\\tau _2,\\tau _3\\in D(\\theta )\\cap (x_i,x_j)$ be three distinct breakpoints of $\\theta $ on the interval $(x_i,x_j)$ .",
"Without loss of generality we suppose that $\\tau _1<\\tau _2<\\tau _3$ .",
"Of course, we must have $\\beta _{\\tau _1},\\beta _{\\tau _2},\\beta _{\\tau _3}>0$ .",
"We claim that there is an element $v$ of $\\mathbb {V}_{\\lbrace \\tau _1,\\tau _2,\\tau _3\\rbrace }$ satisfying the condition (REF ) above.",
"Indeed, we can define $V_{\\tau _1,\\tau _2,\\tau _3}\\in \\mathbb {V}{\\lbrace \\tau _1,\\tau _2,\\tau _3\\rbrace }$ by $V_{\\tau _1,\\tau _2,\\tau _3}(x):= {\\left\\lbrace \\begin{array}{ll}0 & x\\le \\tau _1 \\\\-\\frac{x-\\tau _1}{\\tau _2-\\tau _1} & \\tau _1\\le x\\le \\tau _2\\\\-\\frac{\\tau _3-x}{\\tau _3-\\tau _2} & \\tau _2\\le x\\le \\tau _3\\\\0 & x\\ge \\tau _3\\end{array}\\right.}.",
"$ Let $\\gamma $ be any positive number.",
"We find that because $V_{\\tau _1,\\tau _2,\\tau _3}(x_k)=0$ for all observations $x_k\\in \\lbrace x_1,\\ldots ,x_n\\rbrace $ , we have $\\int \\theta d\\hat{P}=\\int (\\theta +\\gamma V_{\\tau _1,\\tau _2,\\tau _3})d\\hat{P}$ .",
"Moreover, $\\theta (x)+\\gamma V_{\\tau _1,\\tau _2,\\tau _3}(x)<\\theta (x)~~~\\forall x\\in (x_i,x_j)$ because $V_{\\tau _1,\\tau _2,\\tau _3}$ is negative everywhere on $(x_i,x_j)$ .",
"Therefore, we have $\\int e^{\\theta +\\gamma V_{\\tau _1,\\tau _2,\\tau _3}}dM\\le \\int e^\\theta dM.$ We conclude that (REF ) does indeed hold for $v=V_{\\tau _1,\\tau _2,\\tau _3}$ ."
],
[
"Linearity of the operator $DL(\\theta ,-)$",
"The goal of this section is to show that the map $v\\mapsto DL(\\theta ,v)$ is linear, provided that $L$ is finite.",
"First, we state a Lemma.",
"Lemma 2.1 Suppose that $y:\\mathbb {R}\\rightarrow \\bar{\\mathbb {R}}$ is a concave function and that $r\\in \\operatorname{dom}y$ , that is, $y(r)$ is finite.",
"Suppose that $s\\in \\mathbb {R}$ and that there exists $\\epsilon >0$ such that $y(r+\\epsilon s)$ is finite.",
"Then the one-sided derivative $\\frac{\\partial }{\\partial t}y(r+ts)\\Bigr |_{t=0^+}\\equiv \\lim _{t\\downarrow 0}\\frac{y(r+ts)-y(r)}{t}$ exists and is finite.",
"See the proof of Theorem 23.1 from Rockafellar's book [3].",
"Now we can state the main result of this appendix: Theorem 2.1 Take $\\theta \\in \\Theta $ such that $L(\\theta )>-\\infty $ .",
"Let $v_1,\\ldots ,v_K\\in \\mathbb {V}$ such that there exists $\\epsilon >0$ satisfying $\\theta +\\epsilon v_k\\in \\Theta \\quad \\bigl (\\forall k\\in \\lbrace 1,\\ldots ,K\\rbrace \\bigr ).$ Then for any non-negative real coefficients $h_1,\\ldots ,h_K\\ge 0$ we have: $DL(\\theta ,v_k)$ exists and is finite for each $k$ in $\\lbrace 1,\\ldots ,K\\rbrace $ , $DL(\\theta ,\\sum _{k=1}^K h_kv_k)$ exists and is finite, and there is equality $\\sum _{k=1}^K h_k DL(\\theta ,v_k)=DL(\\theta ,\\sum _{k=1}^K h_kv_k)$ .",
"First we establish that there exists a number $\\tilde{\\epsilon }>0$ such that, for each $k$ , $L(\\theta +\\tilde{\\epsilon }v_k)>-\\infty .$ This fact follows from continuity of $L$ together with (REF ) and the assumption that $L(\\theta )>-\\infty $ .",
"The first statement (REF ) in the theorem above then follows from Lemma REF .",
"Next, we establish that for any $h_1,\\ldots ,h_K\\ge 0$ , there exists $\\epsilon _h>0$ such that $L(\\theta +\\epsilon _h\\sum _{k=1}^{K}h_kv_k)>-\\infty .$ Indeed, defining $\\epsilon _h:=\\frac{\\tilde{\\epsilon }}{\\sum _{k=1}^{K}h_k}$ we see that $\\theta +\\epsilon _h\\sum _{k=1}^{K}h_kv_k=\\theta +\\frac{\\sum _{k=1}^{K}h_k\\tilde{\\epsilon }v_k}{\\sum _{k=1}^{K}h_k}$ is a convex combination of the points $\\theta +\\tilde{\\epsilon }v_1,\\ldots ,\\theta +\\tilde{\\epsilon }v_k$ .",
"Finiteness (REF ) then follows from concavity of $L$ and finiteness (REF ) of $L(\\theta +\\tilde{\\epsilon }v_k)$ for each $k$ .",
"Thus, Lemma REF gives that $DL(\\theta , \\sum _{k=1}^{K}h_kv_k)$ exists and is finite, verifying statement (REF ) above.",
"Finally, we confirm statement (REF ) by an application of the Leibnitz Integral Rule: we have equality $\\nonumber DL(\\theta ,\\sum _{k=1}^{K}h_kv_k)&= \\frac{\\partial }{\\partial t}\\left[L\\left(\\theta +t\\sum _{k=1}^{K}h_kv_k\\right)\\right|_{t=0^+} \\\\\\nonumber &= \\frac{\\partial }{\\partial t} \\left[\\int \\left(\\theta +t\\sum _{k=1}^{K}h_kv_k\\right)d\\hat{P}- \\int e^{\\theta +t\\sum _{k=1}^{K}h_kv_k}dM+1\\right|_{t=0^+} \\\\\\nonumber &= \\sum _{k=1}^{K}h_k\\int v_kd\\hat{P}-\\frac{\\partial }{\\partial t}\\int e^{\\theta +t\\sum _{k=1}^{K}h_kv_k}dM\\biggr |_{t=0^+} \\\\&= \\sum _{k=1}^{K}h_k\\int v_kd\\hat{P}-\\int \\frac{\\partial }{\\partial t}e^{\\theta +t\\sum _{k=1}^{K}h_kv_k}dM\\biggr |_{t=0^+} \\\\\\nonumber &= \\sum _{k=1}^{K}h_k\\int v_kd\\hat{P}-\\int \\sum _{k=1}^{K}h_kv_ke^{\\theta +t\\sum _{k=1}^{K}h_kv_k}dM\\biggr |_{t=0^+} \\\\\\nonumber &= \\sum _{k=1}^{K}h_k\\int v_kd\\hat{P}-\\sum _{k=1}^{K}h_k\\int v_ke^{\\theta }dM \\\\\\nonumber &= \\sum _{k=1}^{K}h_k\\frac{\\partial }{\\partial t} \\left[\\int \\left(\\theta +tv_k\\right)d\\hat{P}- \\int e^{\\theta +tv_k}dM+1\\right|_{t=0^+} \\\\\\nonumber &= \\sum _{k=1}^{K}h_k \\frac{\\partial }{\\partial t}\\left[L\\left(\\theta +tv_k\\right)\\right|_{t=0^+}= \\sum _{k=1}^{K}h_kDL(\\theta ,v_k)$ where the interchange (REF ) of differentiation and integration is allowed because, for $t\\in [0,\\epsilon _h]$ , the function $e^{\\theta (x)+t\\sum _{k=1}^{K}h_kv_k(x)}f(x)$ is continuous and its partial derivative $\\frac{\\partial }{\\partial t}e^{\\theta (x)+t\\sum _{k=1}^{K}h_kv_k(x)}f(x)=\\sum _{k=1}^{K}h_kv_k(x)e^{\\theta (x)+t\\sum _{k=1}^{K}h_kv_k(x)}f(x)$ is also continuous."
]
] | 1906.04544 |
[
[
"Parabolic Hitchin Maps and Their Generic Fibers"
],
[
"Abstract We set up a BNR correspondence for moduli spaces of Higgs bundles over a curve with a parabolic structure over any algebraically closed field.",
"This leads to a concrete description of generic fibers of the associated strongly parabolic Hitchin map.",
"We also show that the global nilpotent cone is equi-dimensional with half dimension of the total space.",
"As a result, we prove the flatness and surjectivity of this map and the existence of very stable parabolic vector bundles."
],
[
"Introduction",
"Hitchin [9] introduced the map now named after him, and showed that it defines a completely integrable system in the complex-algebraic sense.",
"Subsequently Beauville, Narasimhan and Ramanan [4] constructed a correspondence — indeed nowadays refered to as the BNR correspondence — which among other things characterizes the generic fiber of a Hitchin map as a compactified Jacobian.",
"Our paper is concerned with a parabolic version of these results in the setting of algebraic geometry.",
"By this we mean that we work over an arbitrary algebraically closed field $k$ .",
"To be concrete, let us fix a smooth projective curve $X$ over $k$ of genus $g(X)\\ge 2$ and a finite subset $D\\subset X$ , which we shall also regard as a reduced effective divisor on $X$ .",
"We also fix a positive integer $r$ which will be the rank of vector bundles on $X$ that we shall consider (but if $k$ has characteristic 2, we shall assume $r\\ge 3$ in order to avoid issues involving ampleness) and we specify for each $x\\in D$ a finite sequence $m^{\\scriptscriptstyle \\bullet }(x)=(m^1(x), m^2(x), \\dots , m^{\\sigma _x}(x))$ of positive integers summing up to $r$ .",
"We refer to these data as a quasi-parabolic structure; let us denote this simply by $P$ .",
"A quasi-parabolic vector bundle of type $P$ is then a rank $r$ vector bundle $E$ on $X$ which for every $x\\in D$ is endowed with a filtration $E|_{x}=F^0(x)\\supset F^1(x)\\supset \\cdots \\supset F^{\\sigma _x}(x)=0$ such that $\\dim F^{j-1}(x)/ F^{j}(x)=m^j(x)$ .",
"A parabolic Higgs field on a such a bundle is a ${\\mathcal {O}}_X$ -homomorphism $\\theta : E\\rightarrow E\\otimes _{{\\mathcal {O}}_X} \\omega _X(D)$ with the property that it takes each $F^j(x)$ to $F^{j+1}(x)\\otimes _{{\\mathcal {O}}_X}T^*_x(X)$ .",
"We call it a weak parabolic Higgs field, if it only takes $F^j(x)$ to $F^{j}(x)\\otimes _{{\\mathcal {O}}_X}T^*_x(X)$ .",
"A weak parabolic Higgs field $\\theta $ has a characteristic polynomial with coefficients as an element of $\\mathcal {H}:=\\prod _{j=1}^rH^0 (X, (\\omega (D))^{\\otimes j})$ and the characteristic polynomial itself defines the spectral curve in the cotangent bundle of $X$ that is finite over $X$ .",
"With the help of Geometric Invariant Theory one can construct moduli spaces of such objects, but this requires “polarization data”, which in the present context take the form of a weight function $\\alpha $ which assigns to every $x\\in D$ a set of real numbers $0= \\alpha _0(x)<\\alpha _1(x)<\\cdots <\\alpha _{\\sigma _x}(x)=1$ .",
"As we will recall later, this then gives rise to notions of parabolic structures and corresponding stability conditions.",
"And leads to quasi-projective varieties parametrizing the classes of $\\alpha $ -stable objects of type $P$ : for the parabolic vector bundles we get ${\\textbf {M}}_{P,\\alpha }$ , for weak parabolic Higgs bundles we get $\\mathbf {Higgs}^W_{P,\\alpha }$ and for ordinary parabolic Higgs bundles we get $\\mathbf {Higgs}_{P,\\alpha }$ , the latter being contained in $\\mathbf {Higgs}^W_{P,\\alpha }$ as a closed subset.",
"If we choose $\\alpha $ generic, then the notions of semistability and stability coincide, so that these have an interpretation as coarse moduli spaces, and the varieties in question will be nonsingular.",
"By assigning to a Higgs field the coefficients of its characteristic polynomial we obtain a (Hitchin) map $h^W_{P,\\alpha }: \\mathbf {Higgs}^W_{P,\\alpha }\\rightarrow \\mathcal {H}$ .",
"We prove that $h^W_{P,\\alpha }$ is flat, show that each connected component of the generic fiber of $h^{W}_{P,\\alpha }$ is a torsor of the Picard variety of the corresponding spectral curve and compute the number of connected components.",
"But our main results concern the image $\\mathcal {H}_{P}$ of $\\mathbf {Higgs}_{P,\\alpha }$ and the resulting morphism $h_{P,\\alpha }: \\mathbf {Higgs}_{P,\\alpha }\\rightarrow \\mathcal {H}_{P}$ .",
"We characterize $\\mathcal {H}_{P}$ as an affine subspace of $\\mathbf {Higgs}_{P,\\alpha }$ (this was obtained earlier by Baraglia and Kamgarpour [2]) and prove essentially that $h_{P,\\alpha }$ has all the properties that one would hope for.",
"We have a commutative diagram $\\begin{diagram}\\mathbf {Higgs}_{P,\\alpha }&^{h_{P,\\alpha }}& \\mathcal {H}_{P}\\\\& &\\\\\\mathbf {Higgs}^W_{P,\\alpha }&^{h^W_{P,\\alpha }}& \\mathcal {H}\\end{diagram}$ but beware that this is not Cartesian unless all the $m^j(x)$ are equal to 1.",
"We give a concrete description of generic fibers of $h_{P,\\alpha }$ and we also obtain the parabolic BNR correspondence in this setting, which roughly speaking amounts to (see Theorem REF ): Theorem 1.1 (Parabolic BNR Correspondence) There is a one to one correspondence between: $\\left\\lbrace \\begin{array}{c}\\text{isomorphism classes of parabolic Higgs bundles}\\\\ \\text{with prescibed characteristic polynomial} \\end{array} \\right\\rbrace $ and $ \\left\\lbrace \\begin{array}{c}\\text{line bundles over the normalized spectral curve}\\\\ \\text{with a fixed degree determined by the parabolic data}\\end{array} \\right\\rbrace .$ In particular, generic fibers of $h_{P,\\alpha }$ are connected.",
"Furthermore, we compute the dimension of the parabolic nilpotent cones and derive from this (see Theorem REF ): Theorem 1.2 When $\\mathbf {Higgs}_{P,\\alpha }$ is smooth, the parabolic Hitchin map $h_{P,\\alpha }$ is flat and surjective.",
"Let us now indicate how this relates to previous work.",
"After the fundamental work of Hitchin and Beauville-Narasimhan-Ramanan mentioned above, several papers investigated various properties of the Hitchin map over the complex field, for example in [10], [7], [8].",
"Niture [15] constructed the moduli space of (semi-)stable Higgs bundles over an algebraically closed field and showed the properness of Hitchin maps.",
"In the parabolic setting, Yokogawa [21], [22] constructed the moduli space of (semi-)stable parabolic Higgs bundles and the weak version of this notion and proved that a weak parabolic Hitchin map is proper.",
"His construction works over any algebraically closed field.",
"Logares and Martens [11], working over the complex field, studied the generic fibers and constructed a Poisson structure on $\\mathbf {Higgs}^W_{P,\\alpha }$ and proved that $h^W_{P,\\alpha }$ is an integrable system in the Poisson sense.",
"Scheinost and Schottenloher [19], also working over $\\mathbb {C}$ , defined the parabolic Hitchin map $h_{P,\\alpha }$ and proved by means of a non-abelian Hodge correspondence that $h_{P,\\alpha }$ is an algebraically completely integrable system.",
"Baraglia, Kamgarpour and Varma [20], [3], [2] generalized this to a $G$ -parahoric Hitchin system, here $G$ can be a simple simply connected algebraic group over $\\mathbb {C}$ .",
"We close this section by describing how this paper is organized.",
"In section 2, we recall the parabolic setting and review the properties of ${\\textbf {M}}_{P,\\alpha }$ , $\\mathbf {Higgs}_{P,\\alpha }$ and $\\mathbf {Higgs}^W_{P,\\alpha }$ .",
"In section 3, we recall the construction of the Hitchin maps $h^W_{P,\\alpha }$ and $h_{P,\\alpha }$ and determine the corresponding parabolic Hitchin base space ${\\mathcal {H}}_P$ as in [2].",
"In section 4, we set up the parabolic BNR correspondence (Theorem REF ) and determine the generic fibers of a parabolic Hitchin map.",
"In section 5, we do the same for a weak parabolic Hitchin map.",
"And finally, in section 6, we compute the dimension of parabolic nilpotent cones and prove Theorem REF .",
"We also prove the existence of very stable parabolic vector bundle.",
"As an application, we use co-dimension estimate to give an embedding of conformal blocks into theta functions.",
"Acknowledgements The authors thank Eduard Looijenga.",
"Discussions with Eduard motivated our proof of the first main theorem and significantly affected the organization of this paper.",
"The authors also thank Bingyi Chen, Yifei Chen, Hélène Esnault, Yi Gu, Peigen Li, Yichen Qin and Junchao Shentu, Xiaotao Sun and Xiaokui Yang for helpful discussions."
],
[
"Parabolic vector bundles",
"We use the notions and the notation introduced above.",
"In particular, we fix $X$ and a set of quasi-parabolic data $P=(D, \\lbrace m^{\\scriptscriptstyle \\bullet }(x)\\rbrace _{x\\in D})$ .",
"We denote by $P_x\\subseteq {\\operatorname{GL}}_r=G$ to be the standard parabolic subgroup with Levi type $\\lbrace m^j(x)\\rbrace $ .",
"We also fix a weight function $\\alpha =\\lbrace \\alpha _{{\\scriptscriptstyle \\bullet }}(x)\\rbrace _{x\\in D}$ and call $(P, \\alpha )$ a parabolic structure.",
"We fix a positive integer $r$ and let $E$ be a rank $r$ vector bundle over $X$ endowed with a quasi-parabolic structure of type $P$ .",
"Remark 2.1 From now on, we will use calligraphic letters ${\\mathcal {E}}, {\\mathcal {F}}, \\ldots $ to denote parabolic bundles of a given type (with certain quasi-parabolic structure), and use the normal upright Roman letters $E, F,\\ldots $ to denote underlying vector bundles.",
"We will also consider a local version (where $X$ is replaced by the spectrum of a DVR).",
"Then $D$ will be the closed point, and we will write $\\sigma $ , $\\lbrace m^{j}\\rbrace _{j=1}^{\\sigma }$ and $\\lbrace \\alpha _{j}\\rbrace _{j=1}^{\\sigma }$ instead.",
"Let be given a parabolic vector bundle ${\\mathcal {E}}$ on $X$ .",
"Then every coherent ${\\mathcal {O}}_X$ -submodule $F$ of $E$ inherits from $E$ a quasi-parabolic structure so that it may be regarded as a parabolic vector bundle ${\\mathcal {F}}$ .",
"Note that the weight function $\\alpha $ for ${\\mathcal {E}}$ determines one for ${\\mathcal {F}}$ .",
"Similarly, for any line bundle $L$ on $X$ we have a natural parabolic structure on $E\\otimes _{{\\mathcal {O}}_X}L$ , which we then denote by ${\\mathcal {E}}\\otimes _{{\\mathcal {O}}_X}L$ .",
"For more details, please refer to [21].",
"An endomorphism of ${\\mathcal {E}}$ is of course a vector bundle endomorphism of $E$ which preserves the filtrations $F^{\\scriptscriptstyle \\bullet }(x)$ .",
"We call this a strongly parabolic endomorphism if it takes $F^i(x))$ to $F^{i+1}(x)$ for all $x\\in D$ and $i$ .",
"We denote the subspaces of $\\operatorname{End}_{{\\mathcal {O}}_X}(E)$ defined by these properties $ParEnd({\\mathcal {E}}) \\text{ resp.\\ } SParEnd({\\mathcal {E}}).$ Similarly we can define the sheaf of parabolic endomorphisms and sheaf of strongly parabolic endomorphisms, denoted by ${\\mathcal {P}}ar{\\mathcal {E}}nd({\\mathcal {E}})$ and ${\\mathcal {S}}{\\mathcal {P}}ar{\\mathcal {E}}nd({\\mathcal {E}})$ respectively.",
"Remark 2.2 Following [22], we have $ {\\mathcal {P}}ar {\\mathcal {E}}nd({\\mathcal {E}})^{\\vee }={\\mathcal {S}}{\\mathcal {P}}ar{\\mathcal {E}}nd({\\mathcal {E}})\\otimes _{{\\mathcal {O}}_X}{\\mathcal {O}}_X(D).$ We now define the parabolic degree (or $\\alpha $ -degree) of ${\\mathcal {E}}$ to be $\\text{par-}deg({\\mathcal {E}}):=deg(E)+\\sum _{x\\in D}\\sum _{j=1}^{\\sigma _x}\\alpha _{j}(x)m^j(x).$ And the parabolic slope or $\\alpha $ -slope of ${\\mathcal {E}}$ is given by $\\text{par-}\\mu ({\\mathcal {E}})=\\frac{\\text{par-}deg({\\mathcal {E}})}{r}$ Definition 2.3 A parabolic vector bundle ${\\mathcal {E}}$ is said to be (semi-)stable if for every proper coherent ${\\mathcal {O}}_X$ -submodule $F\\subsetneq E$ , we have $\\text{par-}\\mu ({\\mathcal {F}})<\\text{par-}\\mu ({\\mathcal {E}}) \\text{ resp.",
"}(\\le ),$ where the parabolic structure on ${\\mathcal {F}}$ is inherited from ${\\mathcal {E}}$ .",
"There exists a coarse moduli space for semistable parabolic vector bundles of rank $r$ with fixed quasi-parabolic type $P$ and weights $\\alpha $ .",
"For the constructions and properties, we refer the interested readers to [13], [21], [22].",
"Denote the moduli space by ${\\textbf {M}}_{P,\\alpha }$ (the stable locus is denoted by ${\\textbf {M}}_{P,\\alpha }^s$ ).",
"${\\textbf {M}}_{P,\\alpha }$ is a normal projective variety of dimension $\\dim ({\\textbf {M}}_{P,\\alpha })&=(g-1)r^2+1+\\sum \\limits _{x\\in D}\\frac{1}{2}(r^2-\\sum _{j=1}^{\\sigma _x}(m^{j}(x))^2)\\\\&=(g-1)r^2+1+\\sum \\limits _{x\\in D}\\dim (G/P_{x}),$"
],
[
"Parabolic Higgs bundles",
"Let us define the parabolic Higgs bundles.",
"It is reasonable that a general Higgs bundle should be a cotangent vector of a stable parabolic vector bundle in its moduli space.",
"Recall in (REF ) that ${\\mathcal {P}}ar{\\mathcal {E}}nd({\\mathcal {E}})$ is naturally dual to ${\\mathcal {S}}{\\mathcal {P}}ar{\\mathcal {E}}nd({\\mathcal {E}})(D)$ .",
"Yokogawa [22] showed: $ T^*_{\\left[{\\mathcal {E}}\\right]}{\\textbf {M}}_{P,\\alpha }^{s}=(H^1(X,{\\mathcal {P}}ar{\\mathcal {E}}nd({\\mathcal {E}})))^*\\cong H^0(X,{\\mathcal {S}}{\\mathcal {P}}ar {\\mathcal {E}}nd({\\mathcal {E}})\\otimes _{{\\mathcal {O}}_X}\\omega _X(D)).$ So we define the parabolic Higgs bundles as follows: Definition 2.4 A parabolic Higgs bundle on $X$ with fixed parabolic data $(P,\\alpha )$ is a parabolic vector bundle ${\\mathcal {E}}$ together with a Higgs field $\\theta $ , $\\theta :{\\mathcal {E}}\\rightarrow {\\mathcal {E}}\\otimes _{{\\mathcal {O}}_X}\\omega _X(D)$ such that $\\theta $ is a strongly parabolic map between ${\\mathcal {E}}$ and ${\\mathcal {E}}\\otimes _{{\\mathcal {O}}_X}\\omega _X(D)$ .",
"If $\\theta $ is merely parabolic, we say that $({\\mathcal {E}},\\theta )$ is a weak parabolic Higgs bundle.",
"Remark 2.5 The category of (weak) parabolic filtered Higgs sheaves is an abelian category with enough injectives which contains the category of (weak) parabolic Higgs bundles as a full subcategory.",
"See [22].",
"One can similarly define the stability condition for a (weak) parabolic Higgs bundles.",
"A (weak) parabolic Higgs bundle $({\\mathcal {E}},\\theta )$ is called $\\alpha $ -semi-stable (resp.",
"stable) if for all proper sub-Higgs bundle $(F,\\theta )\\subsetneq (E,\\theta )$ , one has $\\text{par-}\\mu ({\\mathcal {F}})\\le \\text{par-}\\mu ({\\mathcal {E}})$ (resp.",
"$<$ ).",
"Similar to the vector bundle case, an $\\alpha $ -stable parabolic Higgs bundle $({\\mathcal {E}},\\theta )$ is simple, i.e.",
"$ParEnd({\\mathcal {E}},\\theta )\\cong k$ .",
"As mentioned in the introduction, Geometric Invariant Theory shows that the $\\alpha $ -stable objects define a moduli spaces $\\mathbf {Higgs}^W_{P,\\alpha }$ and $\\mathbf {Higgs}_{P,\\alpha }$ that are normal quasi-projective varieties (see[13], [22] and [21]).",
"We have $\\dim (\\mathbf {Higgs}^W_{P,\\alpha } )= (2g-2+\\deg (D))r^2 + 1.$ and $\\mathbf {Higgs}_{P,\\alpha }$ is a closed subvariety of $\\mathbf {Higgs}^W_{P,\\alpha }$ (see [22]) and $\\dim (\\mathbf {Higgs}_{P,\\alpha }) =2(g-1)r^2 +2+ \\sum _{x\\in D}2\\dim (G/P_{x})=2\\dim ({\\textbf {M}}_{P,\\alpha }).$ For generic $\\alpha $ , a bundle (or pair) is $\\alpha $ -semistable if and only if it is $\\alpha $ -stable.",
"In these cases, the moduli spaces ${\\textbf {M}}_{P,\\alpha }$ , $\\mathbf {Higgs}_{P,\\alpha }$ and $\\mathbf {Higgs}^W_{P,\\alpha }$ are smooth.",
"In what follows, we will always assume that $\\alpha $ is generic in this sense.",
"For simplicity, we will always drop the weight $\\alpha $ in the subscripts and abbreviate the parabolic structure $(P,\\alpha )$ as $P$ ."
],
[
"The (weak) parabolic Hitchin Maps",
"Weak parabolic Hitchin maps are defined by Yokogawa [21].",
"According to [21] and [22], $\\mathbf {Higgs}^W_P$ is a geometric quotient by an algebraic group ${\\operatorname{PGL}}(V)$ of some ${\\operatorname{PGL}}(V)$ -scheme $\\mathcal {Q}$ .",
"On $X_{{\\mathcal {Q}}}=X\\times {\\mathcal {Q}}$ one has a universal family of stable weak parabolic Higgs bundles $(\\widetilde{{\\mathcal {E}}},\\widetilde{\\theta })$ and a surjection $V\\otimes _k {\\mathcal {O}}_{X_{\\mathcal {Q}}}\\twoheadrightarrow \\widetilde{{\\mathcal {E}}}$ .",
"Thus the coefficients of the characteristic polynomial of $\\widetilde{\\theta }$ $(a_1(\\widetilde{\\theta }),\\cdots , a_n(\\widetilde{\\theta })):=({\\operatorname{tr}}_{{\\mathcal {O}}_{X_{\\mathcal {Q}}}}(\\widetilde{\\theta }),{\\operatorname{tr}}_{{\\mathcal {O}}_{X_{\\mathcal {Q}}}}(\\wedge ^2_{{\\mathcal {O}}_{X_{\\mathcal {Q}}}}\\widetilde{\\theta }), \\cdots ,\\wedge ^r_{{\\mathcal {O}}_{X_{\\mathcal {Q}}}}\\widetilde{\\theta }))$ determine a section of $\\bigoplus _{i=1}^r(\\pi _X^*\\omega _X(D))^{\\otimes i}$ over $X_{{\\mathcal {Q}}}$ .",
"We write $\\mathbf {H}^0(X,(\\omega _X(D))^{\\otimes i})$ for the affine variety underlying $H^0(X,(\\omega _X(D))^{\\otimes i})$ .",
"Since $H^{0}(X_{\\mathcal {Q}},\\bigoplus _{i=1}^r(\\pi _X^*\\omega _X(D))^{\\otimes i}) =\\operatorname{Hom}_{\\operatorname{\\mathbf {Sch}}}({\\mathcal {Q}},\\prod _{i=1}^r \\mathbf {H}^0(X,(\\omega _X(D))^{\\otimes i})),$ the characteristic polynomial of $\\widetilde{\\theta }$ defines a morphism of schemes ${\\mathcal {Q}}\\rightarrow \\prod _{i=1}^r \\mathbf {H}^0(X,(\\omega _X(D))^{\\otimes i}).$ This map is equivariant under the ${\\operatorname{PGL}}(V)$ -action [21] and hence factors through the moduli space $\\mathbf {Higgs}^W_P$ .",
"Definition 3.1 The Hitchin base space for the pair $(X,D)$ is ${\\mathcal {H}}:=\\prod _{i=1}^r \\mathbf {H}^0(X,(\\omega _X(D))^{\\otimes i})$ and $h_P^W: \\mathbf {Higgs}^W_P\\rightarrow {\\mathcal {H}}$ is called the weak parabolic Hitchin map.",
"Note that $h_P^W$ is pointwise defined as $({\\mathcal {E}},\\theta )\\mapsto (a_1(\\theta ),\\cdots ,a_r(\\theta ))\\in {\\mathcal {H}}$ .",
"It is easy to see $\\dim ({\\mathcal {H}}) =r^2(g-1)+\\frac{r(r+1)\\deg (D)}{2},$ and in general a generic fiber of $h^W_{P}$ has smaller dimension than ${\\mathcal {H}}$ .",
"We shall now define an affine subspace ${\\mathcal {H}}_P$ of ${\\mathcal {H}}$ (which as the notation indicates depends on $P$ ) such that $h_{P}^W(\\mathbf {Higgs}_P)\\subset {\\mathcal {H}}_P$ .",
"Baraglia and Kamgarpour [2] have already determined parabolic Hitchin base spaces for all classical groupsTheir notation for ${\\mathcal {H}}_P$ is ${\\mathcal {A}}_{{\\mathcal {G}},P}$ ..",
"Moreover when $k=\\mathbb {C}$ , they show in [3] that $h_{P}$ is surjective by symplectic methods.",
"We here do the calculation for $G={\\operatorname{GL}}_{r}$ , not just for completeness, but also because it involves some facts of Young tableaux which will we need later.",
"Our proof is simple and direct.",
"In Section , we will give a proof of surjectivity over general $k$ ."
],
[
"Intermezzo on partitions",
"A partition of $r$ is a sequence of integers $n_{1}\\ge n_{2}\\ge \\cdots \\ge n_{\\sigma }> 0$ with sum $r$ .",
"Its conjugate partition is the sequence of integers $\\mu _{1}\\ge \\mu _{2}\\ge \\cdots \\ge \\mu _{n_{1}}>0$ (also with sum $r$ ) given by $\\mu _{j}=\\#\\lbrace \\ell :n_{\\ell }\\ge j, 1\\le \\ell \\le \\sigma \\rbrace .$ It is customary to depict this as a Young diagram: For example for $(n_{1},n_{2},n_{3})=(5,4,2)$ , we have the Young diagram: $\\large (5,4,2)$ We can read the conjugate partition from the diagram: $(\\mu _{1},\\mu _{2},\\mu _{3},\\mu _{4},\\mu _{5})=(3,3,2,2,1).$ Number the boxes as indicated: 147911 25810 36 For each partition of $r$ , we assign a level function: $j\\rightarrow \\gamma _{j}, 1\\le j\\le r$ , such that $\\gamma _{j}=l$ if and only if $\\sum _{t\\le l-1}\\mu _{t}< j\\le \\sum _{t\\le l}\\mu _{t}.$ For example, combined with the former numbered Young Tableau, $\\gamma _{j}$ is illustrated as following: 12345 1234 12 It is clear that: $&\\sum _{j}\\gamma _{j}=\\sum _{t}t\\mu _{t}=\\sum _{i}\\sum _{j\\le n_{i}}j=\\sum _{i}\\frac{1}{2}n_{i}(n_{i}+1).\\\\&\\sum _{i=1}^{\\sigma }(n_i)^{2}=\\sum _{t=1}^{n_{\\sigma }}t^{2}(\\mu _{t}-\\mu _{t+1})=\\sum _{t=1}^{n_{\\sigma }}(2t-1)\\mu _{t}.$ In the following, we reorder the Levi type $\\lbrace m^{j}(x)\\rbrace _{j=1}^{\\sigma _{x}}$ from large to small as $\\lbrace n_{j}(x)\\rbrace _{j=1}^{\\sigma _{x}}$ , so that $n_1(x)\\ge n_2(x)\\ge \\cdots \\ge n_{\\sigma _x}(x)>0$ .",
"This is a partition of $r$ .",
"Definition 3.2 The parabolic Hitchin base for the parabolic data $P$ is $\\mathcal {H}_{P}:=\\prod _{j=1}^r\\mathbf {H}^{0}\\Big (X,\\omega _X^{\\otimes j}\\otimes {\\mathcal {O}}_X\\big (\\sum _{x\\in D}(j-\\gamma _{j}(x))\\cdot x\\big ) \\Big )\\subset {\\mathcal {H}},$ where the right hand side is regarded as an affine space.",
"Lemma 3.3 $\\dim \\mathcal {H}_{P}=\\frac{1}{2}\\dim \\mathbf {Higgs}_P$ Recall that $\\dim \\mathbf {Higgs}_P=\\dim T^{*}{\\textbf {M}}_{P}=2\\dim {\\textbf {M}}_{P}$ .",
"By Riemann-Roch theorem, we have $\\dim \\mathcal {H}_{P}&=\\sum _{j=1}^{r}\\dim H^{0}\\Big (X,\\omega _X^{\\otimes j}\\otimes {\\mathcal {O}}_X\\big (\\sum \\limits _{x\\in D}(j-\\gamma _j(x))\\cdot x\\big )\\Big )\\\\&=1+r(1-g)+\\frac{r(r+1)}{2}(2g-2)+\\sum _{j=1}^{r}\\sum _{x\\in D}(j-\\gamma _j(x))\\\\&=1+r^{2}(g-1)+\\frac{r(r+1)\\deg D}{2}-\\sum _{x\\in D}\\sum _{j=1}^{r}\\gamma _j(x)\\\\&=\\dim ({\\textbf {M}}_P)+\\frac{1}{2}\\sum \\limits _{x\\in D}\\Big (r+\\sum \\limits _{l=1}^{\\sigma _x}m^l(x)^{2}-2\\sum \\limits _{j=1}^{r}\\gamma _j(x)\\Big )\\\\&=\\frac{1}{2}\\dim \\mathbf {Higgs}_P $ The last equality follows from (REF ), ().",
"Theorem 3.4 For $({\\mathcal {E}},\\theta )\\in \\mathbf {Higgs}_P$ , $h_{P}^W({\\mathcal {E}},\\theta )\\in {\\mathcal {H}}_P$ .",
"i.e., we have $a_{j}(\\theta )\\in H^{0}\\Big (X,\\omega _{X}^{\\otimes j}\\otimes {\\mathcal {O}}_X\\big (\\sum _{x\\in D}(j-\\gamma _{j}(x))\\cdot x\\big )\\Big ).$ Without loss of generality, we may assume $D=x$ .",
"We denote the characteristic polynomial of $\\theta $ as $\\lambda ^{r}+a_{1}\\lambda ^{r-1}+\\cdots a_{r}$ where $a_{i}={\\operatorname{tr}}(\\wedge ^{i}\\theta )$ .",
"We denote the formal local ring at $x$ by ${\\mathcal {O}}$ with natural valuation denoted by $v$ .",
"We denote its fraction field by $\\mathcal {K}$ .",
"We fix a local coordinate $t$ in a formal neighborhood of $x$ and choose local section $\\frac{dt}{t}$ to get a trivialization of $\\omega _{X}(x)$ near $x$ .",
"Then the characteristic polynomial around $x$ becomes $f(t,\\lambda ):=\\lambda ^{r}+b_{1}\\lambda ^{r-1}+\\cdots b_{r},$ where $b_i\\in {\\mathcal {O}}$ .",
"Following the above argument, we only need to show that: $v(b_{i})\\ge \\gamma _{i}, \\quad 1\\le i\\le r$ It amounts to prove the following statement: Claim 3.5 Let ${\\mathcal {E}}$ be a free ${\\mathcal {O}}$ -module of rank $r$ .",
"$F^{\\bullet }$ is a filtration ${\\mathcal {E}}\\otimes _{{\\mathcal {O}}} k$ .",
"Denote by $n_{1}\\ge n_{2}\\ge \\cdots n_{\\sigma }>0$ a partition of $r$ , with $n_{i}=\\dim _{k}\\frac{F^{i-1}}{F^{i}}, 1\\le i\\le \\sigma $ .",
"Then for $\\theta \\in \\operatorname{End}_{{\\mathcal {O}}}({\\mathcal {E}})$ such that $\\theta $ respects $F^{\\bullet }$ , $ v({\\operatorname{tr}}(\\wedge _{{\\mathcal {O}}}^{i}\\theta ^{i}))\\ge \\gamma _{i}$ Now we prove the claim.",
"Lift $F^{\\bullet }$ to a filtration ${\\mathcal {F}}^{\\bullet }$ on ${\\mathcal {E}}$ .",
"This induces a filtration of $\\wedge _{{\\mathcal {O}}}^{i}{\\mathcal {E}}$ with associated graded ${\\mathcal {O}}$ -module: $\\bigoplus _{\\delta _{1}+\\cdots +\\delta _{\\sigma }=i}\\wedge _{{\\mathcal {O}}}^{\\delta _{1}}({\\mathcal {F}}^{0}/{\\mathcal {F}}^{1})\\otimes \\cdots \\otimes \\wedge _{{\\mathcal {O}}}^{\\delta _{\\sigma }}({\\mathcal {F}}^{\\sigma -1}/{\\mathcal {F}}^{\\sigma }).$ Any $\\theta $ as above induces a map in each summand, this map has trace has valuation no less than $\\min \\lbrace \\delta _{1},\\cdots ,\\delta _{\\sigma }\\rbrace $ .",
"Since ${\\operatorname{tr}}(\\wedge _{{\\mathcal {O}}}^{i}\\theta ^{i})$ is the sum of these traces, then our claim follows from intermezzo above.",
"Yokogawa [21] showed that $h^{W}_{P}$ is projective and $\\mathbf {Higgs}_P\\subset \\mathbf {Higgs}^{W}_{P}$ is a closed sub-variety.",
"By Theorem REF , the image of $\\mathbf {Higgs}_P$ under $h^W_P$ is contained in ${\\mathcal {H}}_{P}\\subset {\\mathcal {H}}$ .",
"We denote this restriction $h_P=h_P^W|_{\\mathbf {Higgs}_P}:\\mathbf {Higgs}_P\\rightarrow {\\mathcal {H}}_P$ and refer to it as the parabolic Hitchin map.",
"We conclude that: Proposition-definition 3.6 The parabolic Hitchin map for the parabolic structure $P$ is the morphism $h_P=h_P^W|_{\\mathbf {Higgs}_P}:\\mathbf {Higgs}_P\\rightarrow {\\mathcal {H}}_P$ This morphism is proper.",
"In the next two sections, we determine generic fibers of the (weak) parabolic Hitchin map.",
"As in [4], we introduce the spectral curve to realize the Hitchin fibers as a particular kind of sheaves on the spectral curve.",
"One observe that ${\\mathcal {H}}$ is also the Hitchin base of $\\omega _{X}(D)$ -valued Higgs bundles.",
"So for $a\\in {\\mathcal {H}}$ , one has the spectral curve $X_a\\subset {\\mathbb {P}}({\\mathcal {O}}_X\\oplus \\omega _X(D))$ for $\\omega _X(D)$ -valued Higgs bundles, cut out by the characteristic polynomial $a\\in {\\mathcal {H}}$ .",
"Denote the projection by $\\pi _a:X_a\\rightarrow X$ .",
"One can compute the arithmetic genus as: $P_{a}(X_{a})=1-\\chi (X,\\pi _{*}{\\mathcal {O}}_{X_{a}})=1+r^{2}(g-1)+\\frac{r(r-1)}{2}\\deg (D).$ When we work in the weak parabolic case, $X_a$ is smooth for generic $a\\in {\\mathcal {H}}$ .",
"On the other hand, for any $a\\in {\\mathcal {H}}_P$ , the spectral curve $X_a$ is singular (except for the Borel case).",
"Yet for a generic $a\\in {\\mathcal {H}}_{P}$ , $X_{a}$ is integral, totally ramified at $x\\in D$ and smooth elsewhere.",
"Please refer to the appendix."
],
[
"Generic Fiber of Parabolic Hitchin Map",
"In this section, we determine generic fibers of parabolic Hitchin map.",
"We will start from a local analysis, and then derive from it the parabolic correspondence as stated in Theorem REF .",
"The analysis of local case is also of its own interest."
],
[
"Local case",
"Suppose we're given the triple $(V,F^{\\bullet },\\theta )$ as following, (a) $V$ is a free ${\\mathcal {O}}=k[[t]]$ -module of rank $r$ , with filtration $F^{\\bullet }V$ : $V=V^{0}\\supset V^1\\supset \\cdots \\supset V^{\\sigma }=t\\cdot V.$ with $\\dim V^{i}/V^{i+1}=m_{i+1}$ .",
"As before we rearrange $(m_{i})$ as $(n_{i})$ to give a partition of $r$ .",
"(b) $\\theta :V\\rightarrow V$ be a $k[[t]]$ module morphism and $\\theta (V^{i})\\subset V^{i+1}$ .",
"(c) $\\text{char}_\\theta =f(\\lambda ,t)=\\prod _{i=1}^{n_{1}}f_{i}$ , each $f_{i}$ is an Eisenstein polynomial with $\\deg (f_{i})=\\mu _{i}$ , here $(\\cdots ,\\mu _{i},\\cdots )$ as before is conjugate partition.",
"Besides the constant term of $f_{i}$ are different in $t\\cdot k[[t]]$ .",
"Let $A:={\\mathcal {O}}[\\lambda ]/(f)$ , and $A_i=k[[t]][\\lambda ]/(f_i(t,\\lambda ))$ , then each $A_{i}$ is a DVR and let $\\widetilde{A}=\\prod _{i=1}^{n_{1}}A_i$ .",
"Then we have a natural injection $A\\hookrightarrow \\widetilde{A}$ .",
"$\\widetilde{A}$ can be treated as normalization of $A$ .",
"Then: Claim 4.1 $V$ is a principal $\\widetilde{A}$ -module.",
"From the Intermezzo, we know $\\sigma =\\mu _{1}$ .",
"It is easy to see $\\theta ^{\\sigma }(v)\\in tV$ for $\\forall v\\in V$ .",
"We define $\\operatorname{Ker}f_{i}:=\\lbrace v\\in V|f_{i}(\\theta )(v)=0\\rbrace $ , it's easy to see $\\operatorname{Ker}f_{i}$ is a direct summand of $V$ .",
"Proposition 4.2 The image of $\\operatorname{Ker}f_{1}\\rightarrow V\\rightarrow V/\\operatorname{Ker}f_{i}$ is $\\operatorname{Ker}\\overline{f}_{1}$ , for $1< i\\le n_{1}$ .",
"Here $\\overline{f}_{1}$ is the induced map of $f_{1}$ on $V/\\operatorname{Ker}f_{i}$ .",
"In particular, $\\operatorname{Ker}f_{1}\\oplus \\operatorname{Ker}f_{i}$ is a direct summand of $V$ .",
"We denote the natural quotient map $V\\rightarrow V/\\operatorname{Ker}f_{i}$ by $q_{i}$ .Then $q_{i}^{-1}(\\operatorname{Ker}\\overline{f}_{1})=\\lbrace v\\in V|f_{1}(v)\\in \\operatorname{Ker}f_{i}\\rbrace $ For simplicity we write $f_{1}$ as $\\lambda ^{\\mu _{1}}+\\alpha _{1}$ , here $\\alpha _{1}\\in tk[[t]]\\backslash t^{2}k[[t]]$ by generic condition.",
"We denote $f_{1}(v)=w\\in \\operatorname{Ker}f_{i}$ .",
"By definition, to show $q_{i}(\\operatorname{Ker}f_{1})=\\operatorname{Ker}\\overline{f}_{1}$ , it suffices to show $\\exists w^{^{\\prime }}\\in \\operatorname{Ker}f_{i}, f_{1}(w^{^{\\prime }})=w$ This amounts to solve the following linear equations: $\\left\\lbrace \\begin{array}{rcl}(\\theta ^{\\mu _{1}}+\\alpha _{1})w^{^{\\prime }}&=&w\\\\(\\theta ^{\\mu _{i}}+\\alpha _{i})w^{^{\\prime }}&=&0\\end{array}\\right.$ i.e.",
"$ (-\\alpha _{i}\\theta ^{\\mu _{1}-\\mu _{i}}+\\alpha _{1})w^{^{\\prime }}=w$ .",
"It is easy to see that $\\theta $ -action on $V$ is continuous with respect to $t$ -adic topology on $V$ , thus $V$ can be treated as $ k[[t]][[\\lambda ]]$ -module.",
"We rewrite: $(-\\alpha _{i}\\theta ^{\\mu _{1}-\\mu _{i}}+\\alpha _{1})=t\\varphi _{t}(\\theta ),$ then $\\varphi _{t}(\\theta )^{-1}$ is a well-defined map on $V$ , since we assume $f_1$ and $f_i$ have different constant terms $t^{2}\\lnot |(\\alpha _{i}-\\alpha _{1})$ .",
"Notice that $f_{1}(v)=w\\Rightarrow \\theta ^{\\mu _{1}}(v)\\equiv w (\\text{mod } t)$ .",
"Since $\\theta ^{\\mu _{1}}v\\in tV$ , we know that $w\\in tV$ , then we can find a (unique) $w^{^{\\prime }}=\\varphi _{t}^{-1}(w/t),$ such that $q_{i}(v-w^{^{\\prime }})=q_{i}(v)$ and $v-w^{^{\\prime }}\\in \\operatorname{Ker}f_{1}$ .",
"Thus $q_{i}:\\operatorname{Ker}f_{1}\\rightarrow \\operatorname{Ker}\\overline{f}_{1}$ is surjective.",
"Since $\\operatorname{Ker}\\overline{f}_{1}$ is a direct summand of $V/\\ker f_{i}$ , $\\operatorname{Ker}f_{1}\\oplus \\operatorname{Ker}f_{i}$ is a direct summand of $V$ .",
"Proposition 4.3 We have the following decomposition: $V\\simeq \\bigoplus _{i=1,\\ldots ,n_{1}}\\operatorname{Ker}f_{i}$ That is to say $V$ is a principal $\\widetilde{A}$ -module.",
"Remark 4.4 It is obvious that we Can Not lift a principal $A$ -module structure to $\\widetilde{A}$ -module structure.",
"The reason lies in that over a principal $A$ -module, we do not have a filtration $F^{\\bullet }$ of type $(m_1, m_2,\\ldots ,m_{\\sigma })$ , and a $\\theta $ strongly preserves it.",
"This proposition actually shows the effect of parabolic condition on local structure of Higgs bundles.",
"We prove this by induction both on rank of $V$ and the number of irreducible factors of $\\text{char}_\\theta $ .",
"From Proposition REF , we know that $\\operatorname{Ker}f_{1}\\oplus \\operatorname{Ker}f_{i}$ is direct summand of $V$ for $\\forall i, 2\\le i\\le n_{1}$ .",
"Consider the map: $q_{1}:V\\rightarrow V/\\operatorname{Ker}f_{1}$ Since $\\operatorname{Ker}f_1\\oplus \\operatorname{Ker}f_i$ is a direct summand of $V$ by Proposition REF , $q_{1}(\\operatorname{Ker}f_{i})$ is a direct summand and is contained in $\\operatorname{Ker}\\overline{f_{i}}\\subset V/\\operatorname{Ker}f_{1}$ .",
"Because $\\operatorname{Ker}f_{i}\\cap \\operatorname{Ker}f_{1}=0$ , $q_{1}$ is injective when restricted to $\\operatorname{Ker}f_{i}$ .",
"By passing to $V\\otimes _{{\\mathcal {O}}} K$ , and the obvious decomposition: $V\\otimes _{{\\mathcal {O}}} K=\\bigoplus _{i=1}^{n_{1}}\\operatorname{Ker}f_{i}\\otimes _{{\\mathcal {O}}}K$ We know that $rk(\\operatorname{Ker}f_{i})=rk(\\operatorname{Ker}\\overline{f}_{i})$ , then: $q_{1}(\\operatorname{Ker}f_{i})=\\operatorname{Ker}\\overline{f}_{i}$ Thus we only need to prove that: $V/\\operatorname{Ker}f_{1}=\\bigoplus _{i=2}^{n_{1}}\\operatorname{Ker}\\overline{f}_{i}$ $\\theta $ acts on $V/\\operatorname{Ker}f_{1}$ with characteristic polynomial $\\prod _{i=2}^{\\sigma }f_{i}$ .",
"The filtration on $V$ , actually induces a filtration on $V/\\operatorname{Ker}f_{1}$ .because $\\operatorname{Ker}f_{1}\\cap V_{j}$ is a direct summand of $V_{j}$ To use induction, we only need to show that the length of this filtration is $\\mu _{2}$ .",
"This follows from that $\\operatorname{Ker}f_{1}$ is rank one module over $A_{1}$ which is a DVR.",
"Then by induction, we have decomposition on $V/\\operatorname{Ker}f_{1}$ ,i.e: $V/\\operatorname{Ker}f_{1}\\simeq \\bigoplus _{i=2,\\ldots ,n_{1}}\\operatorname{Ker}\\overline{f}_i$ As $q_{1}:\\operatorname{Ker}f_{i}\\rightarrow \\operatorname{Ker}\\overline{f}_i$ is surjective, we have the decomposition.",
"In the following, we fix a generic $a\\in {\\mathcal {H}}_{P}$ .",
"For simplicity, we assume $D=x$ .",
"Our first goal is normalize the singular spectral curve $X_{a}$ and analyse local property of its normalization."
],
[
"Normalization of spectral curves",
"We denote $N:\\widetilde{X}_{a}\\rightarrow X_a$ as normalization of $X_{a}$ .",
"And we denote by $\\widetilde{\\pi }$ the composition map: $\\widetilde{\\pi }: \\widetilde{X}_{a}\\xrightarrow{} X_{a}\\xrightarrow{} X.$ As in Theorem REF , $f\\in {\\mathcal {O}}[\\lambda ]\\cong k[[t]][\\lambda ]$ define the spcetral curve locally, so that the formal completion of the local ring of $X_a$ at $x$ is $A:={\\mathcal {O}}[\\lambda ]/(f)$ .",
"Notice that $\\operatorname{Spec}(A)$ and $X_a-\\lbrace \\pi ^{-1}(x)\\rbrace $ form an fpqc covering of $X_a$ .",
"Since $X_a-\\lbrace \\pi ^{-1}(x)\\rbrace $ is smooth, we only need to construct the normalization of $\\operatorname{Spec}(A)$ .",
"For generic choose of $a\\in {\\mathcal {H}}_P$ , we may assume the coefficient $b_i\\in {\\mathcal {O}}$ has valuation $\\gamma _i$ .",
"We denote the Newton polygon of $f$ by $\\Gamma $ .",
"Figure: Newton Polygon of Characteristic PolynomialFigure REF is a Newton Polygon of characteristic polynomial corresponding to the example in the Intermezzo.",
"We define $C=\\Gamma +{\\mathbb {R}}_{\\ge 0}^2$ , so that $\\Gamma $ determines $C$ which is a closed convex subset of ${\\mathbb {R}}_{\\ge 0}^2$ .",
"Let $p_0, p_1, \\dots ,p_s$ , be the `singular' points of $\\partial C$ : points where it has an angle $<\\pi $ (so that $p_s$ lies on the $x$ -axis).",
"The standard theory of toric modifications was developed in the 1970's and is due to several people.",
"For the construction we refer the readers to the introduction paper [16] and references there in.",
"It assigns to $\\Gamma $ a toric modification $\\pi : T_\\Gamma \\rightarrow \\mathbb {A}^2$ of $\\mathbb {A}^2$ , where $T_\\Gamma $ is a normal variety.",
"The morphism $\\pi $ is proper and is an isomorphism over $\\mathbb {A}^2\\backslash \\lbrace (0,0)\\rbrace $ .",
"The exceptional locus $\\pi ^{-1}(0,0)$ is the union of irreducible components $\\lbrace D_e\\rbrace $ , where $e$ runs over the edges of $\\Gamma $ .",
"For every edge $e$ of $\\Gamma $ , denote by $f_e$ the subsum of $f$ over $e\\cap {\\mathbb {Z}}^2$ .",
"Assumption 4.5 We impose the genericity condition that all these roots of $f_{e}$ are nonzero and pairwise distinct for all $e\\in \\partial \\Gamma $ .",
"This is the concept `non-degeneracy' in [16].",
"Under this assumption, according to [16].",
"The strict transform $\\widehat{Z}(f)$ of $Z(f)\\subset \\mathbb {A}^{2}$ in $T_\\Gamma $ is the normalization of $Z(f)$ , and meets $D_e$ transversally in a set that can be effectively indexed by the connected components of $e\\backslash {\\mathbb {Z}}^2$ .",
"In particular, $Z(f)$ has as many branches at the origin as connected components of $\\Gamma \\backslash \\mathbb {Z}^2$ .",
"In our case, the slope of $e$ is $-1/\\mu _{e}$ for some $\\mu _{e}\\in \\lbrace \\mu _{1},\\ldots ,\\mu _{n_1}\\rbrace $ , then one can check that each branch of $Z(f)$ whose strict transform meets $D_e$ is formally given by an Eisenstein equation of degree $\\mu _{e}$ .",
"To conclude: Proposition 4.6 Under Assumption REF , $f$ decomposes in $k[[t,\\lambda ]]$ into a product of Eisenstein polynomials $f=\\prod _{i=1}^{n_1} f_i$ .",
"Exactly $\\#\\lbrace i|\\mu _{i}=\\mu _{e}\\rbrace $ of them are of degree $\\mu _{e}$ (but the difference of two such have their constant terms not divisible by $t^2$ ), and $\\prod _i k[[t]][\\lambda ]/(f_i)$ is the normalization of $k[[t]][\\lambda ]/(f)$ .",
"Remark 4.7 This is a stronger conclusion than that in [14], because of our Assumption REF .",
"Corollary 4.8 For generic $a\\in {\\mathcal {H}}_{P}$ , there are $n_{1}$ (the length of conjugate partition) points in $\\widetilde{X}_{a}$ over $x\\in X$ .",
"Ramification degrees are $(\\mu _{1},\\mu _{2},\\ldots ,\\mu _{n_{1}})$ .",
"The geometric genus of $\\widetilde{X}_a$ is $ P_g(\\widetilde{X}_a)=r^2(g-1)+1+\\dim (G/P_x).$ The ramification degree is due to degree of Eisenstein polynomials defining strict transform of local branches.",
"The geometric genus $P_g(\\widetilde{X}_a)$ then follows from the ramification degrees."
],
[
"Parabolic BNR correspondence",
"This subsection is devoted to build the parabolic BNR correspondence(also stated as Theorem REF ): Theorem 4.9 (Parabolic BNR Correspondence for ${\\operatorname{GL}}_{r}$ ) For generic $a\\in {\\mathcal {H}}_{P}$ , there is a one to one correspondence between: $\\left\\lbrace \\begin{array}{c}\\text{Parabolic Higgs bundle } (\\mathcal {E},\\theta )\\\\\\text{ with }\\ \\text{char}_\\theta =a,\\ \\deg (E)=d\\end{array} \\right\\rbrace \\leftrightarrow \\lbrace \\text{degree } \\delta \\text{ line bundles over } \\widetilde{X}_a\\rbrace $ where $\\delta =(r^2-r)(g-1)+\\dim (G/P_x)+d$ .",
"By the classical BNR correspondence, a parabolic Higgs bundle $({\\mathcal {E}},\\theta )$ corresponds to a torsion free rank one ${\\mathcal {O}}_{X_a}$ -module $V$ with a filtration on $V_{\\pi ^{-1}(x)}$ .",
"Then to prove this theorem, we only need to check that $V$ with filtration induces a $N_*{\\mathcal {O}}_{\\widetilde{X}_a}$ module structure.",
"Since the normalization map is finite and isomorphism over $X_a-\\pi ^{-1}(x)$ , we reduce to consider the local problem near $x$ .",
"By our argument in former subsection, when we specialize $({\\mathcal {E}},\\theta )\\in h^{-1}(a)$ to the marked point $x$ , we exactly get the triple as in Local Case.",
"Then $E$ has a locally free rank 1 $\\widetilde{\\pi }_*{\\mathcal {O}}_{\\widetilde{X}_a}$ -module structure induced by $({\\mathcal {E}},\\theta )$ .",
"Now we can prove our Theorem REF : Firstly, given a parabolic Higgs bundle $(\\mathcal {E},\\theta )$ with $\\text{char}_\\theta =a$ , $\\deg (E)=d$ by proposition REF and discussion before, we have a line bundle $L$ of degree $\\delta $ over $\\widetilde{X}_a$ such that $\\widetilde{\\pi }_*L=E$ .",
"There is an action of $\\theta $ on $\\widetilde{\\pi }_{*}L$ induced by the $\\widetilde{\\pi }_{*}{\\mathcal {O}}_{\\widetilde{X}_a}$ -module structure on $\\widetilde{\\pi }_{*}L$ , and $\\text{char}_\\theta =a$ since $X_a$ is integral.",
"Hence $(\\widetilde{\\pi }_{*}L, \\theta )=({\\mathcal {E}},\\theta )$ .",
"Conversely, given a degree $\\delta $ line bundle $L$ over $\\widetilde{X}_a$ , a Young tableaux argument shows that there exists a unique filtration $L=L_0\\supset L_1\\supset \\cdots \\supset L_{\\sigma }=L(-\\widetilde{\\pi }^{-1}(x))$ such that the graded terms have the same dimension as the Levi type of $P_x$ .",
"The push forward filtration on $\\widetilde{\\pi }_*L$ and $\\widetilde{\\pi }_*{\\mathcal {O}}_{\\widetilde{X}_a}$ -module structure induce a parabolic Higgs bundles structure on $(\\widetilde{\\pi }_{*}L, \\theta )$ with $\\text{char}_\\theta =a$ and $\\deg =d$ .",
"Again, as before, this gives us the correspondence.",
"The degree $\\delta $ can be calculated using Riemann-Roch theorem, as $P_g(\\widetilde{X}_a)=r^2(g-1)+1+\\dim (G/P_x)$ in Corollary REF .",
"Corollary 4.10 Under the same assumption of Theorem REF , for generic $a\\in \\mathcal {H}_{P}$ , the parabolic Hitchin fiber $h_{P}^{-1}(a)$ is isomorphic to $\\operatorname{Pic}^{\\delta }(\\widetilde{X}_{a})$ .",
"By Theorem REF , we only need to check the stability of $(\\widetilde{\\pi }_*L,\\theta )$ for line bundle $L$ over $\\widetilde{X}_a$ .",
"However, the spectral curve $X_a$ is integral, which tells that there is no proper sub-Higgs bundle of $(\\widetilde{\\pi }_*L,\\theta )$ , hence it is a stable parabolic Higgs bundle.",
"Remark 4.11 Scheinost and Schottenloher [19] proved a similar result over $\\mathbb {C}$ by uniquely extending the eigen line bundle on $X_{a}-\\pi ^{-1}(D)$ to $\\widetilde{X}_{a}$ .",
"This extension is announced there.",
"We use a different strategy here which is similar to that in [4] to prove the correspondence.",
"Notice that we only put generic condition on the characteristic polynomial of Higgs field $\\theta $ , but due to the decomposition, for all $({\\mathcal {E}},\\theta )\\in h^{-1}(a)$ we have: Corollary 4.12 The Jordan blocks of $\\theta \\mod {t}$ is of size $(\\mu _{1},\\mu _{2},\\ldots ,\\mu _{n_1})$ By Theorem REF , $V$ has a natural $\\widetilde{A}$ module structure.",
"Since each $A_{i}$ is a DVR, we may find $e_{i}\\in \\operatorname{Ker}f_{i}$ , such that it is free module over $k[[t]]$ , with basis: $\\lbrace v,\\theta v,\\ldots ,\\theta ^{\\mu _{i}-1}v\\rbrace .$ After mod out $t$ , then the matrix of $\\theta $ on $\\operatorname{Ker}f_{i}$ with respect to this basis is a Jordan block of size $\\mu _{i}$ .",
"Remark 4.13 It means that when given a sufficiently general characteristic polynomial, each global Higgs field $\\theta $ with this prescribed characteristic polynomial has same Jordan normal form after reduction at marked point $x\\in D$ .",
"Actually, they are the so-called Richardson elements.",
"We refer the readers to [1] for more details.",
"If we replace ${\\operatorname{GL}}_{r}$ by ${\\operatorname{SL}}_{r}$ , we also have coarse moduli spaces ${\\textbf {M}}_{P,\\alpha }^{\\circ }$ , $\\mathbf {Higgs}_{P,\\alpha }^{\\circ }$ .",
"And the parabolic Hitchin space is: ${\\mathcal {H}}_{P}^{\\circ }:=\\prod _{j=2}^r\\mathbf {H}^{0}\\left(X,\\omega _X^{\\otimes j}\\otimes {\\mathcal {O}}_X(\\sum _{x\\in D}(j-\\gamma _{j}(x))\\cdot x)\\right)$ We use '$\\circ $ ' to emphasize trace zero.",
"We denote corresponding parabolic Hitchin map as $h_{P,\\alpha }^{\\circ }$ .",
"Considering the following commutative diagram: $\\begin{diagram}\\mathbf {Higgs}^{\\circ }_{P,\\alpha }&^{h^{\\circ }_{P,\\alpha }}& \\mathcal {H}^{\\circ }_{P}\\\\& &\\\\\\mathbf {Higgs}_{P,\\alpha }&^{h_{P,\\alpha }}& \\mathcal {H}_{P}\\end{diagram}$ It follows that $h^{\\circ }_{P,\\alpha }$ is proper.",
"Then by our parabolic BNR correspondence Theorem REF , generic fibers of $h^{\\circ }_{P,\\alpha }$ is Prym variety of $\\operatorname{Pic}(\\widetilde{X}_{a})$ .",
"Then by dimension argument and properness, $h^{\\circ }_{P,\\alpha }$ is surjective."
],
[
"Generic fiber of weak parabolic Hitchin maps",
"In this section, we give a concrete description of generic fibers of the weak parabolic Hitchin map.",
"In what follows, we fix $a\\in {\\mathcal {H}}$ , such that $X_a$ is smooth and $\\pi _a$ is unramified over $x$ .",
"And abbreviate $\\pi _a$ as $\\pi $ .",
"To simplify notation, we omit $\\delta $ using $Pic(X_{a})$ to denote some connected component of its Picard variety.",
"Choose a marked point $q\\in X_{a}$ , thus we have an embedding $\\tau :X_{a}\\rightarrow Pic(X_{a})$ .",
"Then we have a universal line bundle over $Pic(X_{a})\\times X_{a}$ which is the pull back of a Poincaré line bundle.",
"We denote the universal line bundle by ${P}$ .",
"Consider the following projection: $ Pic(X_a)\\times X_a \\xrightarrow{} Pic(X_a)\\times X$ We denote $V:=(id\\times \\pi )_{*}{P}$ , which is a rank $r$ vector bundle over $Pic(X_{a})\\times X$ .",
"Thus the $(id\\times \\pi )_*{\\mathcal {O}}_{Pic(X_a)\\times X}$ -module structure induces a Higgs field $\\theta _{Pic}:V\\rightarrow V\\otimes _{{\\mathcal {O}}_X}\\pi _X^*\\omega _X(x).$ And $(V,\\theta _{Pic})$ can be viewed as the universal family of Higgs bundles on $X$ with characteristic polynomial $a$ .",
"For simplicity, we use $V|_{x}$ to denote the restriction of $V$ to $\\lbrace x\\rbrace \\times Pic(X_{a})$ .",
"Proposition 5.1 We have a group scheme ${\\mathfrak {T}}$ over $Pic(X_{a})$ , for any point $z\\in Pic(X_a)$ , the fiber ${\\mathfrak {T}}(z)$ is the centralizer $T$ of $\\theta |_x$ .",
"Restricting $\\theta _{Pic}$ to $\\lbrace x\\rbrace \\times Pic(X_{a})$ gives $\\theta _{Pic}|_x:V|_x\\rightarrow V\\otimes _{{\\mathcal {O}}_X}\\pi _x^*\\omega _X(x)|_x\\cong V|_x$ which is regular semi-simple everywhere because $\\pi $ is unramified over $x$ .",
"Denote by ${\\mathcal {A}}ut(V|_x)$ the group scheme of local automorphisms of vector bundle $V|_x$ .",
"Then we consider the centralizer of $\\theta _{Pic}|_x:V|_x\\rightarrow V|_x$ in ${\\mathcal {A}}ut(V|_x)$ over $Pic(X_{a})\\times \\lbrace x\\rbrace $ .",
"This gives us a group scheme ${\\mathfrak {T}}$ over $Pic(X_{a})$ , fiber-wise it is a maximal torus in $G$ .",
"In the following we construct a flag bundle $\\mathfrak {Fl}$ on $Pic(X_a)$ classifies all the possible filtrations at $x$ .",
"Fiber-wise this is isomorphic to $G/P_x$ .",
"And show that ${\\mathfrak {T}}$ acts on it naturally.",
"Definition 5.2 Denote by $\\text{Fr}(V|_x)$ the frame bundle given by the vector bundle $V|_x$ .",
"We define the (partial) flag bundle $\\mathfrak {Fl}$ over $Pic(X_{a})$ as the associate bundle $\\text{Fr}(V|_x)\\times _G G/P_x.$ Here $P_x$ is the parabolic subgroup given by the parabolic structure at $x$ .",
"By definition, $\\mathfrak {Fl}$ parametrize all the vector bundle filtrations with type given by $P_x$ on $V|_x$ .",
"Denote by $W_x$ the Coxeter subgroup corresponding to the parabolic subgroup $P_x$ , we have Lemma 5.3 $\\mathfrak {T}$ acts on $\\mathfrak {Fl}$ over $Pic(X_a)$ , and the fixed points $\\mathfrak {Fl}^{\\mathfrak {T}}$ is a $W_x$ torsor on $Pic(X_a)$ .",
"We know that ${\\mathfrak {T}}\\subset {\\mathcal {A}}ut(V|_{x})$ as a sub-group scheme, thus ${\\mathfrak {T}}$ acts on $\\mathfrak {Fl}$ .",
"Since fiber-wise we know that the invariant point of $G/P_x$ under the action of $T\\subset P_x$ is bijective to $W_{x}$ , we finish the proof.",
"Now we can give a description of the weak parabolic Hitchin fiber $(h^{W}_P)^{-1}(a)$ .",
"Theorem 5.4 For general $a\\in {\\mathcal {H}}$ , we have $(h^{W}_P)^{-1}(a)\\cong \\mathfrak {Fl}^{\\mathfrak {T}}$ .",
"Remark 5.5 The intuition of this theorem is that: Filtrations coming from a parabolic structure should be compatible with the Higgs field at $x$ , thus they corresponds to the fixed points of ${\\mathfrak {T}}$ action on $\\mathfrak {Fl}$ .",
"Fiber-wise, fixed points are those $\\lbrace P_{i}\\rbrace \\subset G/P_{x}$ , such that $P_{i}\\supset Z_{G}(\\theta _{x})=T$ .",
"Since then $\\theta _{x}\\in \\mathfrak {p}_{i}$ , filtration determined by $P_{i}$ is compatible with $\\theta _{x}$ .",
"Conversely, $({\\mathcal {E}},\\theta )$ lies in $(h^{W}_{P})^{-1}(a)$ , meaning that $E$ is a line bundle over $X_{a}$ , and has a filtration at $x$ compatible with $\\theta _{x}$ .",
"A sub-bundle of parabolic sub-groups, $P^{\\prime }\\subset {\\mathcal {A}}ut(V|_{x})$ , determines a filtration of $V|_x$ .",
"This filtration is compatible with $\\theta _{x}$ if and only if $\\theta _{x}\\in \\mathfrak {p}^{\\prime }$ .",
"Since $\\theta _{x}$ is regular semi-simple, this implies that $P^{\\prime }\\supset \\mathfrak {T}$ .",
"Thus it is a fixed point of $\\mathfrak {T}$ -action in $\\mathfrak {Fl}$ .",
"Since in our case $G={\\operatorname{GL}}_{r}$ , we can give a more explicit description.",
"First, we denote $\\pi ^{-1}(x)$ by $\\lbrace y_{1},\\ldots ,y_{r}\\rbrace \\subset X_{a}$ .",
"Then we restrict the universal line bundle ${P}$ to each $Pic(X_a)\\times {y_i}$ , and denote it by ${P}|_{y_i}$ .",
"One has $ V|_x\\cong \\oplus _{i=1}^r {P}|_{y_i}$ since $\\pi ^{-1}(x)$ are $r$ -distinct reduced points.",
"Indeed, factors in the decomposition (REF ) are eigenspaces of $\\theta _{Pic}|_x$ .",
"So under this decomposition, $\\theta _{Pic}|_x$ is a direct sum of $\\theta _{y_i}:{P}_{y_i}\\rightarrow {P}_{y_i}$ and ${\\mathfrak {T}}$ preserve the decomposition.",
"To conclude: Corollary 5.6 The connected components $\\pi _{0}((h^{W}_P)^{-1}(a))$ is bijective to the Coxeter group $W_{x}$ associated with $P_x$ ."
],
[
"Global Nilpotent Cone of the Parabolic Hitchin Maps",
"In this section, we study global properties of (weak) parabolic Hitchin maps, i.e.",
"flatness and surjectivity.",
"Definition 6.1 We call $h_{P}^{-1}(0)$ (resp.",
"$(h^{W}_{P})^{-1}(0)$ ) the parabolic global nilpotent cone (resp.",
"the weak parabolic global nilpotent cone).",
"We denote $h_{P}^{-1}(0)$ (resp.",
"$(h^{W}_{P})^{-1}(0)$ ) by ${\\mathcal {N}}il_{P}$ (resp.",
"${\\mathcal {N}}il_{P}^{W}$ ).",
"By Lemma REF , we have $& \\dim (\\text{fiber of }h_{P}) \\ge \\dim ({\\textbf {M}}_P)\\\\& \\dim (\\text{fiber of }h_P^W) \\ge r^2(g-1)+1+\\frac{r(r-1)\\deg (D)}{2}.$"
],
[
"$\\mathbb {G}_m$ -actions on {{formula:9c820d0f-a8ef-421b-ac21-18b2c48b6e6e}} and {{formula:dd05dcd8-1d12-403e-956a-6ccbf5affb54}}",
"There is a natural $\\mathbb {G}_m$ -action on $({\\mathcal {E}},\\theta )$ given by $({\\mathcal {E}},\\theta )\\mapsto ({\\mathcal {E}},t\\theta ), t\\in \\mathbb {G}_{m}$ .",
"It preserves stability and leaves Hilbert polynomials invariant.",
"Thus it can be defined on moduli spaces, i.e $\\mathbf {Higgs}_{P}$ and $\\mathbf {Higgs}^{W}_{P}$ .",
"This action was first studied by Simpson in [17] and [18].",
"It contains a lot of information of both moduli spaces and Hitchin maps.",
"There is also a natural $\\mathbb {G}_m$ -action on ${\\mathcal {H}}$ and ${\\mathcal {H}}_{P}$ : $(a_1,a_2,\\cdots ,a_r)\\mapsto (ta_1,t^2a_2,\\cdots ,t^ra_r),$ and $h_{P}$ , $h^{W}_{P}$ are equivariant under this $\\mathbb {G}_m$ -action.",
"This can be used to show the flatness of Hitchin map [8] if one has the dimension estimate of the global nilpotent cones.",
"We will use deformation theory to estimate the dimension of the global nilpotent cones in the next sub-section."
],
[
"The dimension of the global nilpotent cone",
"The study of infinitesimal deformations of parabolic Higgs bundles was done in [22] and [5]."
],
[
"Parabolic global nilpotent cone",
"In this sub-subsection, we will use infinitesimal method to calculate the dimension of the parabolic global nilpotent cone.",
"Recall that in [21],[22], $\\mathbf {Higgs}_P$ is a geometric quotient by an algebraic group ${\\operatorname{PGL}}(V)$ of some ${\\operatorname{PGL}}(V)$ -scheme ${\\mathcal {Q}}$ .",
"Moreover, one has a universal family of framed stable parabolic Higgs bundles $(\\widetilde{{\\mathcal {E}}},\\widetilde{\\theta }) \\text{ with surjection }V\\otimes _k {\\mathcal {O}}_{X_{\\mathcal {Q}}}\\twoheadrightarrow \\widetilde{{\\mathcal {E}}}.$ Denote the quotient map by $q:{\\mathcal {Q}}\\rightarrow \\mathbf {Higgs}_P$ .",
"Restricting the universal family $(V\\otimes _k {\\mathcal {O}}_{X_{{\\mathcal {Q}}}}\\twoheadrightarrow \\widetilde{{\\mathcal {E}}},\\widetilde{\\theta })$ to $q^{-1}({\\mathcal {N}}il_{P})$ , we get: $\\mathbb {U}_{{\\mathcal {N}}il_P}:=(V\\otimes _k {\\mathcal {O}}_{X_{q^{-1}({\\mathcal {N}}il_{P})}}\\twoheadrightarrow \\widetilde{{\\mathcal {E}}},\\widetilde{\\theta }).$ For any scheme $S$ and flat family $(V\\otimes _k {\\mathcal {O}}_S\\twoheadrightarrow {\\mathcal {E}}_{S},\\theta _S)$ of parabolic Higgs bundles with $\\text{char}_{\\theta _{S}}=0$ on $S$ , there is a map $\\varphi :S\\rightarrow q^{-1}({\\mathcal {N}}il_{P})$ such that $({ id}_X\\times \\varphi )^*\\mathbb {U}_{{\\mathcal {N}}il_P}\\cong (V\\otimes _k {\\mathcal {O}}_S\\twoheadrightarrow {\\mathcal {E}}_{S},\\theta _S).$ To determine the dimension of ${\\mathcal {N}}il_P$ , it is sufficient to calculate the dimension of each irreducible component with reduced structure.",
"Restrict to any generic point $\\eta $ of ${\\mathcal {N}}il_P$ , $\\theta _\\eta :=\\widetilde{\\theta }|_{q^{-1}(\\eta )^\\text{red}}$ gives a filtration $\\lbrace \\operatorname{Ker}(\\theta _\\eta ^i)\\rbrace $ of vector bundles of $\\widetilde{E}|_{q^{-1}(\\eta )^\\text{red}}$ (i.e.",
"the graded terms are all vector bundles), because $X\\times \\eta $ is a curve.",
"Spread out this.",
"Lemma 6.2 There exists an irreducible open subset $W\\subset {\\mathcal {N}}il^{\\text{red}}_P$ with generic point $\\eta $ , such that $\\theta _W:=\\widetilde{\\theta }|_{q^{-1}(W)^\\text{red}}$ gives a filtration $\\operatorname{Ker}(\\theta _W^i)$ of vector bundles of $E_{W}:=\\widetilde{E}|_{q^{-1}(W)^\\text{red}}$ over $X\\times W$ .",
"We fix some notations for filtered bundle maps.",
"Let $E^1,E^2$ and $E$ be vector bundles on $X$ with decreasing filtrations by subbundles $K_i^\\bullet $ , $i=1,2,\\emptyset $ on each of them .",
"We denote by ${\\mathcal {H}}om^{\\text{fil}}(E^1,E^2)\\text{ and }{\\mathcal {E}}nd^{\\text{fil}}(E)$ the coherent subsheaf of ${\\mathcal {H}}om(E_1,E_2)$ (resp.",
"${\\mathcal {E}}nd(E)$ ) consisting of those local homomorphisms preserving filtrations.",
"And ${\\mathcal {H}}om^{\\text{s-fil}}(E_1,E_2)$ (resp.",
"${\\mathcal {E}}nd^{\\text{s-fil}}(E)$ ) consists of the local homomorphisms $\\varphi $ such that $\\varphi (K^{j}_1|_{U})\\subset K^{j+1}_2|_U$ (resp.",
"$\\varphi (K^{j}|_{U})\\subset K^{j+1}|_U$ ).",
"Let us denote the decreasing filtration: $K^\\bullet _W:E_{W}=K^0_W\\supset K^1_W\\supset \\cdots \\supset K^{r^{\\prime }}_W=0.$ induced by $\\lbrace \\operatorname{Ker}(\\theta _W^i)\\rbrace $ on $E_{W}$ .",
"For $x\\in D$ we also use $x$ to denote the closed immersion $\\lbrace x\\rbrace \\times W\\rightarrow X\\times W$ .",
"Lemma 6.3 At each punctured point $x:\\lbrace x\\rbrace \\times W\\rightarrow X\\times W$ , $K^{\\bullet }_W|_{x}$ is a coarser flag than the parabolic structure ${\\mathcal {E}}_W|_{x}$ , so $\\begin{array}{r}SParHom({\\mathcal {E}}_{W},{\\mathcal {E}}_{W}\\otimes \\omega _X(D))\\cap Hom^{\\text{s-fil}} (E_W,E_W\\otimes \\omega _X(D))\\\\=Hom^{\\text{s-fil}} (E_W,E_W\\otimes \\omega _X(D))\\end{array}$ and $\\theta _W\\in Hom^{\\text{s-fil}} (E_W,E_W\\otimes \\omega _X(D)).$ $\\theta _W$ is a strongly parabolic map thus $F^{i}(x)\\subset K_W^i$ .",
"In other words, ${\\mathcal {E}}_{W}|_{x}$ is a finer flag than $K^{\\bullet }_W|_{x}$ .",
"Besides, $\\theta _W\\in SParHom({\\mathcal {E}}_{W},{\\mathcal {E}}_{W}\\otimes \\omega _X(D))\\cap Hom^{\\text{s-fil}} (E_W,E_W\\otimes \\omega _X(D))$ by definition.",
"Thus the nilpotent parabolic bundle ${\\mathcal {E}}_{W}$ has a filtration of vector bundles which do not depend on the surjection $V\\otimes _k {\\mathcal {O}}_{q^{-1}(W)^{\\text{red}}}\\twoheadrightarrow E_{W}$ .",
"Theorem 6.4 The space of infinitesimal deformations in $W$ of a nilpotent parabolic Higgs bundle $({\\mathcal {E}},\\theta )$ , is canonically isomorphic to ${\\mathbb {H}}^1(X,{\\mathcal {A}}^\\bullet )$ .",
"Here ${\\mathcal {A}}^\\bullet $ is the following complex of sheaves on $X$ : $0\\rightarrow {\\mathcal {P}}ar{\\mathcal {E}}nd({\\mathcal {E}})\\cap {\\mathcal {E}}nd^{\\text{fil}}(E)\\xrightarrow{}({\\mathcal {S}}{\\mathcal {P}}ar{\\mathcal {E}}nd({\\mathcal {E}})\\cap {\\mathcal {E}}nd^{\\text{s-fil}}(E) )\\otimes \\omega _X(D)\\rightarrow 0$ which is isomorphic to $0\\rightarrow {\\mathcal {P}}ar{\\mathcal {E}}nd({\\mathcal {E}})\\cap {\\mathcal {E}}nd^{\\text{fil}}(E)\\xrightarrow{}{\\mathcal {E}}nd^{\\text{s-fil}}(E) \\otimes \\omega _X(D)\\rightarrow 0.$ An infinitesimal deformation of a parabolic pair $({\\mathcal {E}},\\theta )=u\\in W$ is a flat family $({\\mathcal {E}},\\theta )$ with $\\text{char}_{\\theta }=0$ parametrized by $\\operatorname{Spec}(k[\\epsilon ]/\\epsilon ^2)$ together with a given isomorphism of $({\\mathcal {E}}, \\theta )$ with the specialization of $({\\mathcal {E}},\\theta )$ .",
"By the local universal property of $\\mathbb {U}_{{\\mathcal {N}}il_P}$ , $({\\mathcal {E}},\\theta )$ is the pull back of $(\\widetilde{{\\mathcal {E}}},\\widetilde{\\theta })$ by a map $\\varphi :\\operatorname{Spec}(k[\\epsilon ]/\\epsilon ^2)\\rightarrow {\\mathcal {N}}il_P$ .",
"Moreover, if the deformation is inside $W$ , then $\\varphi $ factor through $q^{-1}(W)^{\\text{red}}$ and $({\\mathcal {E}},\\theta )$ is a pull back of $({\\mathcal {E}}_W,\\theta _W)$ .",
"Thus $K^{\\bullet }:=(id_X\\times \\varphi )^*K^{\\bullet }_W$ is a filtration on $E$ such that $\\theta \\in SParHom({\\mathcal {E}},{\\mathcal {E}}\\otimes \\omega _X(D))\\cap Hom^{\\text{s-fil}} (E,E\\otimes \\omega _X(D)).$ $K^{\\bullet }_W$ do not depend on the surjection $V\\otimes _k {\\mathcal {O}}_{q^{-1}(W)^{red}}\\twoheadrightarrow E_{W}$ , so $K^{\\bullet }$ is uniquely determined by $({\\mathcal {E}},\\theta )$ .",
"Let us denote the projection by $\\pi :X_{\\epsilon }=X\\times \\operatorname{Spec}(k[\\epsilon ]/\\epsilon ^2)\\rightarrow X$ .",
"Tensoring $({\\mathcal {E}},K^{\\bullet },\\theta )$ with $0\\rightarrow (\\epsilon )\\rightarrow k[\\epsilon ]/\\epsilon ^2 \\rightarrow k\\rightarrow 0 ,$ we have an extension of filtered parabolic Higgs ${\\mathcal {O}}_{X_\\epsilon }$ -modules $ 0\\rightarrow ({\\mathcal {E}},K^{\\bullet },\\theta )(\\epsilon )\\rightarrow ({\\mathcal {E}},K^{\\bullet },\\theta ) \\rightarrow ({\\mathcal {E}},K^{\\bullet },\\theta )\\rightarrow 0 .$ Pushing forward (REF ) by $\\pi $ , we have an extension $0\\rightarrow ({\\mathcal {E}},K^{\\bullet },\\theta )\\rightarrow \\pi _*({\\mathcal {E}},K^{\\bullet },\\theta ) \\rightarrow ({\\mathcal {E}},K^{\\bullet },\\theta )\\rightarrow 0$ of locally free filtered parabolic Higgs ${\\mathcal {O}}_X$ -modules.",
"The left inclusion will recover the ${\\mathcal {O}}_{X_\\epsilon }$ -module structure of $\\pi _*({\\mathcal {E}},K^{\\bullet },\\theta )$ .",
"Thus $({\\mathcal {E}},K^{\\bullet },\\theta )$ is formally determined by an element in $\\operatorname{Ext}_{\\text{fil}-par-Higgs-{\\mathcal {O}}_X}(({\\mathcal {E}},K^{\\bullet },\\theta ),({\\mathcal {E}},K^{\\bullet },\\theta )).$ One can reinterpret the extension class using Čech cohomology.",
"Let ${\\mathcal {U}}=\\lbrace U_i\\rbrace _{i}$ be an affine finite covering of $X$ , trivializing $E$ and all $K^j$ .",
"Then on each $U_i$ , there is a splitting $\\varphi _i:({\\mathcal {E}},K^{\\bullet })|_{U_i}\\rightarrow ({\\mathcal {E}},K^{\\bullet })|_{U_i}$ preserving the two compatible filtrations.",
"The Higgs fields induce a filtered map $\\psi _i=\\theta \\varphi _i-\\varphi _i\\theta :({\\mathcal {E}},K^{\\bullet })|_{U_i}\\rightarrow ({\\mathcal {E}},K^{\\bullet })|_{U_i}.$ Thus the extension $({\\mathcal {E}},K^{\\bullet })$ is given by a Čech one cycle $(\\varphi _{ij}:=\\varphi _i-\\varphi _j)$ with value in ${\\mathcal {P}}ar{\\mathcal {E}}nd({\\mathcal {E}})\\cap {\\mathcal {E}}nd^{\\text{fil}}(E)$ and a Ĉech 0-cycle $(\\psi _i:=\\theta \\varphi _i-\\varphi _i\\theta )$ with value in $ {\\mathcal {S}}{\\mathcal {P}}ar{\\mathcal {E}}nd({\\mathcal {E}})\\otimes \\omega _X(D)\\cap {\\mathcal {E}}nd^{\\text{s-fil}}(E)\\otimes \\omega _X(D).$ One has $\\delta (\\varphi _{ij})_{abc}&=\\varphi _{bc}-\\varphi _{ac}+\\varphi _{ab}\\\\&=\\varphi _b-\\varphi _c-\\varphi _a+\\varphi _c+\\varphi _a-\\varphi _b=0,$ and $\\delta (\\psi _i)_{ab}&=\\theta \\varphi _{ab}-\\varphi _{ab}\\theta \\\\&=\\theta \\varphi _{ab}-\\varphi _{ab}\\theta =ad(\\theta )(\\varphi _{ab}).$ It means that $((\\varphi _{ij}),(\\psi _i))$ is a Čech 1-cocycle of the following complex of sheaves ${\\mathcal {A}}^\\bullet $ on $X$ : $0\\rightarrow {\\mathcal {P}}ar{\\mathcal {E}}nd({\\mathcal {E}})\\cap {\\mathcal {E}}nd^{\\text{fil}}(E)\\xrightarrow{}({\\mathcal {S}}{\\mathcal {P}}ar{\\mathcal {E}}nd({\\mathcal {E}})\\cap {\\mathcal {E}}nd^{\\text{s-fil}}(E) )\\otimes \\omega _X(D)\\rightarrow 0$ If the extension is trivial, then $\\varphi _i=( 1, \\varphi _i^{\\prime })$ where $\\varphi _i^{\\prime }\\in {\\mathcal {P}}ar{\\mathcal {E}}nd({\\mathcal {E}})\\cap {\\mathcal {E}}nd^{\\text{fil}}(E)$ and $\\psi _i=\\theta \\varphi ^{\\prime }-\\varphi ^{\\prime }\\theta =ad(\\theta )(\\varphi ^{\\prime }_i).$ Thus trivial extensions correspond to Čech 1-coboundary of ${\\mathcal {A}}^\\bullet $ .",
"On the other hand, if we have a Čech 1-cocycle $((\\varphi _{ij}),(\\psi _i))$ , then use $\\begin{bmatrix}I&\\varphi _{ij}\\\\ 0&I\\end{bmatrix}$ to glue $\\lbrace ({\\mathcal {E}},K^\\bullet )|_{U_i}\\oplus ({\\mathcal {E}},K^\\bullet )|_{U_i}\\rbrace $ with the local Higgs field $\\begin{bmatrix}\\theta &\\psi _i\\\\0&\\theta \\end{bmatrix}$ .",
"One can check that the gluing condition of the local Higgs fields: $\\begin{bmatrix}\\theta &\\psi _i\\\\0&\\theta \\end{bmatrix}\\begin{bmatrix}I&\\varphi _{ij}\\\\ 0&I\\end{bmatrix}=\\begin{bmatrix}\\theta &\\psi _j\\\\0&\\theta \\end{bmatrix}\\begin{bmatrix}I&\\varphi _{ij}\\\\ 0&I\\end{bmatrix}$ is equivalent to the cocycle condition $\\delta (\\psi _i)_{ab}=ad(\\theta )(\\varphi _{ab}).$ If $((\\varphi _{ij}),(\\psi _i))$ is a coboundary, i.e.",
"$((\\varphi _{ij}),(\\psi _i))=\\left( (\\varphi _i^{\\prime }-\\varphi _j^{\\prime }),(ad(\\theta )(\\varphi _i^{\\prime })) \\right).$ One can check : $\\begin{bmatrix}-\\varphi ^{\\prime }_i\\\\I\\end{bmatrix}:({\\mathcal {E}},K^\\bullet )|_{U_i}\\rightarrow ({\\mathcal {E}},K^\\bullet )|_{U_i}\\oplus ({\\mathcal {E}},K^\\bullet )|_{U_i}$ can be glued to a global splitting of filtered Higgs bundle.",
"The filtration $K^\\bullet $ of bundles on $E$ is equivalent as a ${\\mathcal {P}}$ reduction, where ${\\mathcal {P}}\\subset {\\operatorname{GL}}_r$ is the corresponding parabolic subgroup.",
"We denote the principal ${\\mathcal {P}}$ -bundle by $^{\\mathcal {P}}E$ , and let $U\\subset {\\mathcal {P}}$ be the unipotent radical.",
"We denote their Lie algebras by $\\mathfrak {n}$ , $\\mathfrak {p}$ .",
"Thus ${\\mathcal {E}}nd^{\\text{fil}}(E)\\cong ad_{^{\\mathcal {P}}E}$ , ${\\mathcal {E}}nd^{\\text{s-fil}}({\\mathcal {E}})\\cong ad_{^{\\mathcal {P}}E}(\\mathfrak {n})$ .",
"By Lemma REF , we have $P_x\\subset {\\mathcal {P}}$ for all $x\\in D$ .",
"Denoting by $\\mathfrak {p}_x$ the Lie algebra of $P_x$ , we have $ 0\\rightarrow {\\mathcal {P}}ar{\\mathcal {E}}nd({\\mathcal {E}})\\cap {\\mathcal {E}}nd^{\\text{fil}}(E)\\rightarrow {\\mathcal {E}}nd^{\\text{fil}}(E)\\rightarrow \\bigoplus _{x\\in D}i_{x*}\\text{ }\\mathfrak {p}_{/\\mathfrak {p}_x}\\rightarrow 0.$ According to (REF ) we have $0\\rightarrow {\\mathcal {A}}^\\bullet \\rightarrow {\\mathcal {A}}^{\\prime \\bullet }\\rightarrow \\bigoplus _{x}i_{x*}\\text{ }\\mathfrak {p}_{/\\mathfrak {p}_{x}}\\rightarrow 0,$ where ${\\mathcal {A}}^{\\prime \\bullet }$ is $0\\rightarrow {\\mathcal {E}}nd^{\\text{fil}}(E)\\xrightarrow{}{\\mathcal {E}}nd^{\\text{s-fil}}(E) \\otimes \\omega _X(D)\\rightarrow 0.$ We also need the following Lemmas.",
"Lemma 6.5 Assume ${\\mathcal {P}}$ be a parabolic subgroup of ${\\operatorname{GL}}_r$ , $U$ be its unipotent radical.",
"Denote by $\\mathfrak {p}$ , ${\\mathfrak {g}}$ and $\\mathfrak {n}$ their Lie algebras.",
"${\\mathcal {P}}$ act on them by conjugation.",
"One have $\\mathfrak {n}^\\vee \\cong \\mathfrak {g}/\\mathfrak {p}$ as ${\\mathcal {P}}$ -linear representations.",
"For ${\\mathfrak {g}}=\\mathfrak {gl}_r$ , the form $\\beta :{\\mathfrak {g}}\\times {\\mathfrak {g}}\\rightarrow k\\quad (A,B)\\mapsto {\\operatorname{tr}}(AB)$ is a non-degenerate ${\\operatorname{GL}}_r$ -equivariant bilinear form.",
"Thus the isomorphism holds.",
"Lemma 6.6 Let $E$ be a finite dimensional vector space over a field.",
"$\\theta :E\\rightarrow E$ is a nilpotent endomorphism.",
"$\\mathfrak {p}$ is the parabolic algebra preserving the decreasing filtration given by $\\lbrace \\operatorname{Ker}(\\theta ^{i})\\rbrace $ .",
"Let $\\mathfrak {n}$ be the nilpotent radical.",
"Then $ad(\\theta ):\\mathfrak {p}\\rightarrow \\mathfrak {n}$ is surjective.",
"Do induction on the number of Levi factors of $\\mathfrak {p}$ .",
"Proposition 6.7 we get the dimension estimate $\\dim _k(T_{u}W)=\\dim _k({\\mathbb {H}}^1(X,{\\mathcal {A}}^\\bullet )) =\\dim ({\\textbf {M}}_P).$ Thus any irreducible component of ${\\mathcal {N}}il_P$ has same dimension as ${\\textbf {M}}_P$ .",
"In particular, ${\\mathcal {N}}il_P$ is equi-dimensional.",
"One have $\\chi (X,{\\mathcal {A}}^{\\prime \\bullet })&=\\chi ({\\mathcal {E}}nd^{\\text{fil}}(E))-\\chi ({\\mathcal {E}}nd^{\\text{s-fil}}(E) \\otimes \\omega _X(D))\\\\&=\\chi (ad_{^{\\mathcal {P}}E}(\\mathfrak {p}))-\\chi (ad_{^{\\mathcal {P}}E}(\\mathfrak {n}) \\otimes \\omega _X)-\\deg (D)\\cdot \\dim _k(\\mathfrak {n})\\\\&=\\chi (ad_{^{\\mathcal {P}}E}(\\mathfrak {p}))+\\chi (ad_{^{\\mathcal {P}}E}(\\mathfrak {g}/\\mathfrak {p}))-\\deg (D)\\cdot \\dim _k(\\mathfrak {n})\\\\&= r^2(1-g)-\\deg (D)\\cdot \\dim _k(\\mathfrak {n})$ Thus $\\chi (X,{\\mathcal {A}}^\\bullet )= \\chi (X,{\\mathcal {A}}^{\\prime \\bullet })-\\sum _{x\\in D} \\dim _k(\\mathfrak {p}_{/\\mathfrak {p}_{x}})= r^2(1-g)-\\sum _{x} \\dim (G/P_x).$ ${\\mathbb {H}}^0(X,{\\mathcal {A}}^\\bullet )$ are those endomorphism of ${\\mathcal {E}}$ commuting with $\\theta $ , then by stability of $({\\mathcal {E}},\\theta )$ , we have $h^0(X,{\\mathcal {A}}^\\bullet )=1$ .",
"Base change ${\\mathcal {A}}^\\bullet $ to the generic point $\\xi $ of $X$ , we have ${\\mathcal {A}}^\\bullet _\\xi :{\\mathcal {E}}nd^{\\text{fil}}(E_\\xi )\\xrightarrow{} {\\mathcal {E}}nd^{\\text{s-fil}}(E) \\otimes \\omega _X(D)_{\\xi }\\cong {\\mathcal {E}}nd^{\\text{s-fil}}(E_{\\xi }).$ This map is surjective by Lemma REF .",
"Thus $\\tau ^{\\ge 1}{\\mathcal {A}}^{\\bullet }$ is supported on finitely many closed points of $X$ and ${\\mathbb {H}}^2(X,\\tau ^{\\ge 1}{\\mathcal {A}}^\\bullet )=0$ .",
"By $\\tau ^{\\le 0}{\\mathcal {A}}^\\bullet \\rightarrow {\\mathcal {A}}^\\bullet \\rightarrow \\tau ^{\\ge 1}{\\mathcal {A}}^\\bullet \\xrightarrow{},$ we have $h^2(X,{\\mathcal {A}}^\\bullet )=0$ , thus $\\dim _k(T_{u}W)=\\dim _k({\\mathbb {H}}^1(X,{\\mathcal {A}}^\\bullet ))=1-\\dim _k(\\chi (X,{\\mathcal {A}}^\\bullet )) =\\dim ({\\textbf {M}}_P).$ Theorem 6.8 If $\\mathbf {Higgs}_P$ is smooth, then the parabolic Hitchin map $h_P$ is flat and surjective.",
"The proof is similar as in [8].",
"For $\\forall $ $s\\in {\\mathcal {H}}_P-\\lbrace 0\\rbrace $ , $\\overline{\\mathbb {G}_m\\cdot s}$ contains 0.",
"Since $h_P$ is equivariant under the $\\mathbb {G}_m$ -action, for each point $t\\in \\mathbb {G}_m\\cdot s$ , $h_P^{-1}(t)\\cong h_P^{-1}(s)$ .",
"Thus $\\dim (h_P^{-1}(s))=\\dim (\\text{generic fiber of }h_P|_{\\overline{\\mathbb {G}_m\\cdot s}})\\le \\dim h_P^{-1}(0).$ By (REF ), we have $\\dim (h_P^{-1}(s))=\\dim ({\\textbf {M}}_P)$ for any $s\\in {\\mathcal {H}}_P$ .",
"Since $\\mathbf {Higgs}^{W}_{P}$ and ${\\mathcal {H}}$ are both smooth, $h_P$ is flat.",
"Because all fibers are of dimension $\\frac{1}{2}\\mathbf {Higgs}_P=\\dim \\mathcal {H}_{P}$ , $h_{P}$ is dominant.",
"Since $h_P$ is proper by Proposition REF , it is surjective."
],
[
"Weak Parabolic global nilpotent cone",
"Let us compute the dimension of the weak parabolic global nilpotent cone to show the flatness of $h^{W}_{P}$ .",
"However for a weak parabolic Higgs bundles $({\\mathcal {E}},\\theta )$ in ${\\mathcal {N}}il_{P}^{W}$ , the filtration $\\lbrace \\ker (\\theta ^{i})\\rbrace $ and the parabolic filtration are not compatible, it is not obvious to construct a complex dominating deformation within ${\\mathcal {N}}il^{W}_{P}$ as before.",
"We still can calculate $\\dim {\\mathcal {N}}il^{W}_{P}$ by dominating ${\\mathcal {N}}il^W_{P,\\alpha }$ by finite union of ${\\mathcal {N}}il^{W}_{B_i,\\beta _i}$ .",
"Here $B_i$ is a Borel quasi-parabolic structure refining $P$ .",
"More precisely, for a Borel parabolic structure $(B,\\beta )$ , the weak parabolic nilpotent cone coincide with the parabolic nilpotent cone, thus by Theorem REF , ${\\mathcal {N}}il^W_{B,\\beta }$ has the expected dimension $r^2(g-1)+1+\\frac{r(r-1)\\deg (D)}{2}$ .",
"For any generic point $\\eta $ of ${\\mathcal {N}}il^W_{P,\\alpha }$ , by restricting the universal family on $\\lbrace D\\rbrace \\times \\eta $ , it is not difficult to see there exist a Borel refinement $B_\\eta $ of $P$ , such that for general $({\\mathcal {E}},\\theta )$ in the $\\eta $ -irreducible component of ${\\mathcal {N}}il^W_{P,\\alpha }$ , $\\theta $ preserve the filtration given by $B_\\eta $ .",
"One can choose a parabolic weight $\\beta _\\eta $ for each $B_\\eta $ , such that stability is preserved after the forget the parabolic structure from $B_\\eta ,\\beta _\\eta $ to $(P,\\alpha )$ .",
"In other words, the forgetful map is well defined on the moduli spaces and restrict to $f_{\\eta }:{\\mathcal {N}}il_{B_\\eta ,\\beta _\\eta }^W \\rightarrow {\\mathcal {N}}il^W_{P,\\alpha }$ which dominate the generic point $\\eta $ .",
"Thus $\\sqcup _{\\eta }{\\mathcal {N}}il_{B_\\eta ,\\beta _\\eta }^W$ dominate ${\\mathcal {N}}il^W_{P,\\alpha }$ and we conclude: Theorem 6.9 The weak parabolic nilpotent cone has dimension $r^2(g-1)+1+\\frac{r(r-1)\\deg (D)}{2}.$ Indeed, if $\\mathbf {Higgs}_{P,\\alpha }^W$ is smooth, the weak parabolic Hitchin map $h_{P,\\alpha }^W:\\mathbf {Higgs}_{P,\\alpha }^W \\rightarrow {\\mathcal {H}}$ is flat."
],
[
"Existence of very stable parabolic bundles",
"Definition 6.10 We recall that a system of parabolic Hodge bundles is a parabolic Higgs bundle $({\\mathcal {E}},\\theta )$ with following decomposition: ${\\mathcal {E}}\\cong \\oplus {\\mathcal {E}}^i$ such that $\\theta $ is decomposed as a direct sum of $\\theta _i:{\\mathcal {E}}^i\\rightarrow {\\mathcal {E}}^{i+1}$ .",
"Here $\\lbrace {\\mathcal {E}}^{i}\\rbrace $ are subbundles with induced parabolic structures.",
"If a parabolic Higgs bundle $({\\mathcal {E}},\\theta )$ is a fixed point of $\\mathbb {G}_m$ - action, then it has a structure of system of parabolic Hodge bundles.",
"We have the following lemma similar to [18], [17] and [22]: Lemma 6.11 If the parabolic Higgs bundle $({\\mathcal {E}},\\theta )$ satisfies $({\\mathcal {E}},\\theta )\\cong ({\\mathcal {E}},t\\cdot \\theta )$ for some $t\\in \\mathbb {G}_m(k)$ which is not a root of unity, then ${\\mathcal {E}}$ has a structure of system of parabolic Hodge bundles.",
"In particular, if $\\theta \\ne 0$ , then the decomposition ${\\mathcal {E}}\\cong \\oplus {\\mathcal {E}}^i $ given by the system of parabolic Hodge bundles is non-trivial.",
"Remarks 6.12 One conclude that given a parabolic Higgs bundle $({\\mathcal {E}},\\theta )$ , if ${\\mathcal {E}}$ is stable and $\\theta \\ne 0$ , it can not be fixed by the $\\mathbb {G}_m$ -action.",
"A section $s$ of ${\\mathcal {S}}{\\mathcal {P}}ar {\\mathcal {E}}nd({\\mathcal {E}})\\otimes \\omega _X(D)$ is nilpotent if $({\\mathcal {E}},s)\\in {\\mathcal {N}}il_P$ .",
"Definition 6.13 A stable parabolic bundle ${\\mathcal {E}}$ is said to be very stable if there is no non-zero nilpotent section in $H^0(X,{\\mathcal {S}}{\\mathcal {P}}ar {\\mathcal {E}}nd({\\mathcal {E}})\\otimes \\omega _X(D))$ .",
"Theorem 6.14 The set of very stable parabolic bundles contains a non-empty Zariski open set in the moduli of stable parabolic bundles ${\\textbf {M}}_P$ .",
"Denote by $N^0$ the open dense subset of $\\mathbf {Higgs}_P$ consists of $({\\mathcal {E}},\\theta )$ such that ${\\mathcal {E}}$ is a stable parabolic vector bundle.",
"Then $\\pi : N^0\\rightarrow {\\textbf {M}}_P$ by forgetting the Higgs fields is a well defined projection.",
"$N^0$ is $\\mathbb {G}_m$ -equivariant in $\\mathbf {Higgs}_P$ , and $\\pi $ is also $\\mathbb {G}_m$ -equivariant.",
"Denote by $Z_1$ the set of $({\\mathcal {E}},\\theta )$ with ${\\mathcal {E}}$ stable, $\\theta $ nilpotent and nonzero.",
"One observe that $Z_1\\subset {\\mathcal {N}}il_P$ , $Z_1$ is $\\mathbb {G}_m$ -equivariant and all the stable parabolic bundle which is not very stable is contained in $\\pi (Z_1)$ .",
"Because ${\\mathcal {E}}$ is stable(can not be decomposed), $\\theta $ is non-zero, then by Lemma REF , $\\mathbb {G}_m$ acts freely on $Z_1$ .",
"Thus $Z_1/\\mathbb {G}_m\\twoheadrightarrow \\pi (Z_1)$ .",
"One have $\\dim (Z_1)\\le \\dim ({\\mathcal {N}}il_P)=\\dim ({\\textbf {M}}_P)$ , so $\\dim (\\pi (Z_1))\\le \\dim (Z_1/\\mathbb {G}_m)=\\dim ({\\textbf {M}}_P)-1<\\dim ({\\textbf {M}}_P).$ Thus the set of very stable parabolic bundles contains a non-empty Zariski open set in ${\\textbf {M}}_P$ .",
"Corollary 6.15 For a generic choice of $a\\in \\mathcal {H}_P$ , the natural forgetful map $h_{P}^{-1}(a)\\dashrightarrow {\\textbf {M}}_P$ is a dominant rational map.",
"By Theorem REF , we know that the image of $\\mathbf {Higgs}_P$ is contained in $\\mathcal {H}_P$ .",
"Consider the following rational map: $\\rho :\\mathbf {Higgs}_P\\dashrightarrow {\\mathcal {H}}_P\\times {\\textbf {M}}_P\\quad u\\mapsto (h_P(u),\\pi (u)).$ By the existence of very stable parabolic vector bundle, i.e.",
"there exist $(0,{\\mathcal {E}})\\in \\mathcal {H}_{P}\\times {\\textbf {M}}_P$ whose pre-image is $(0,{\\mathcal {E}})\\in \\mathbf {Higgs}_P$ .",
"By Corollary REF , we know that $\\dim \\mathbf {Higgs}_P=\\dim \\mathcal {H}_P+\\dim {\\textbf {M}}_P$ , it means that $\\rho $ is generically finite.",
"Thus $h_{P}^{-1}(a)\\rightarrow {\\textbf {M}}_P$ is dominant, for generic $a\\in \\mathcal {H}_P$ .",
"As an application, we can also show that the rational forgetful map $F:h^{-1}(a)\\dashrightarrow {\\textbf {M}}_{P}$ is defined on an open sub-variety $U\\subset h^{-1}(a)$ and $h^{-1}(a)\\backslash U$ is of co-dimension $\\ge 2$ .",
"This can be proved using similar method in [6].",
"It is well-known that there is a parabolic theta line bundle ${\\mathcal {L}}_{P}$ (which is not canonically defined) over ${\\textbf {M}}_{P}$ .",
"Then $F^{*}{\\mathcal {L}}_{P}$ can be extended to a line bundle over $h^{-1}(a)$ , we still denote it by $F^{*}{\\mathcal {L}}_{P}$ .",
"To conclude: Corollary 6.16 For an $\\ell \\in \\mathbb {Z}$ , there is an embedding: $H^{0}({\\textbf {M}}_{P},{\\mathcal {L}}_{P}^{\\otimes \\ell })\\hookrightarrow H^{0}(h^{-1}_P(a),F^{*}{\\mathcal {L}}_{P}^{\\otimes \\ell })$ This is a generalization of that in [4] to parabolic case.",
"It is interesting to see that the left hand side vector space is also know as generalized parabolic theta functions of level $\\ell $ (also referred to as conformal blocks) as in [12]."
],
[
"Appendix",
"In this appendix, we discuss singularities of generic spectral curves, along with ramification.",
"Since we may work over positive characteristics, it needs a little bit more work to use Jacobian criterion.",
"We assume $D=x$ and if $\\text{char}(k)=2$ , rank $r\\ge 3$ .",
"Lemma 7.1 For a generic choice of $a\\in \\mathcal {H}_{P}$ , the corresponding spectral curve $X_{a}$ is integral, totally ramified over $x$ , and smooth elsewhere.",
"Since being integral is an open condition, similar as in [4], we only need to show there exist $ a\\in \\mathcal {H}_P$ , such that $X_{a}$ is integral.",
"Take $\\text{char}_{\\theta }=\\lambda ^{r}+a_{r}=0$ with $a_{r}\\in H^{0}(X, \\omega ^{\\otimes r}((r-\\gamma _{r})\\cdot x))$ .",
"The spectral $X_a$ is integral if $a_{r}$ is not $r$ -th power of an element in $H^{0}(X,\\omega _X(x))$ , this is true for generic $a_{r}$ .",
"Since smoothness outside $x$ is an open condition, it is sufficient to find such a spectral curve.",
"When $\\text{char}(k)\\nmid r$ , we take $\\text{char}_{\\theta }=\\lambda ^{r}+a_{r}=0$ .",
"Due to the weak Bertini theorem, we can choose $a_{r}$ with only simple roots outside $x$ .",
"Applying Jacobian criterion, $X_{a}$ is what we want.",
"When $\\text{char}(k)\\mid r$ , we take following equation: $\\text{char}_{\\theta }=\\lambda ^{r}+a_{r-1}\\lambda +a_{r}=0$ Then consider following equations: $\\left\\lbrace \\begin{array}{l}\\lambda ^{r}+a_{r-1}\\lambda +a_{r}=0\\\\a_{r-1}=0\\\\a^{\\prime }_{r-1}\\lambda +a^{\\prime }_{r}=0.\\end{array}\\right.$ Since rank $r\\ge 3$ , by weak Bertini theorem, we can choose $a_{r-1}$ with only simple roots outside $x$ .",
"Take $s\\in H^0(X,\\omega ((1+\\gamma _r-\\gamma _{r-1})x))$ with zeros outside $zero(a_{r-1})$ , we can find $a_{r}=a_{r-1}\\otimes s$ such that $zero(a_{r-1})\\supset zero(a_{r})$ , and $zero(a_{r-1})$ are simple zeros of $a_r$ , then $X_{a}$ is smooth outside $x$ .",
"At $x$ , $a_{r}$ will always has multiple zeros except Borel type Similarly, we have Lemma 7.2 For generic $a\\in \\mathcal {H}$ , the corresponding spectral curve $X_{a}$ is smooth.",
"What's more, in this case we can also say something about ramification: Lemma 7.3 For generic choice $a\\in {\\mathcal {H}}$ , we have $\\pi _a:X_{a}\\rightarrow X$ is unramified over $x$ .",
"The ramification divisor of $\\pi _a$ is defined by the resultant.",
"It is a divisor in the linear system of the line bundle $R:=\\omega _X(x)^{\\otimes r(r-1)}$ .",
"Considering the following morphism given by the resultant: $Res: {\\mathcal {H}}\\rightarrow H^{0}(X,R),\\quad a\\mapsto \\text{Res}(a).$ We have the codimension 1 sub-space $W:=H^{0}(X,R(-x))\\subset H^{0}(X,R),$ such that $Res(a)\\in W$ if and only if $\\pi _a$ is ramified over $x$ .",
"$Res$ is an polynomial map so the image is a sub-variety.",
"To prove our statement, we only need to find one particular $a$ so that $\\pi _a$ is unramified over $x$ .",
"Consider the characteristic polynomial of the form $\\lambda ^r+a_r$ .",
"In the neighbourhood of $x$ , we can write it as $\\lambda ^r+b_r\\cdot (\\frac{dt}{t})^{\\otimes r}$ .",
"Here $\\frac{dt}{t}$ is the trivialization of $\\omega (x)$ near $x$ .",
"By Jacobian criterion, $\\pi _a$ is unramified over $x$ if $b_r\\in {\\mathcal {O}}_{X,x}$ is indecomposable.",
"Take $b_r=t$ and extend $t\\cdot (\\frac{dt}{t})^{\\otimes r}$ to a global section $s$ , we find $a=\\lambda ^r+s$ such that $\\pi _a$ is unramified at $x$ ."
]
] | 1906.04475 |
[
[
"A volume of fluid framework for interface-resolved simulations of\n vaporizing liquid-gas flows"
],
[
"Abstract This work demonstrates a computational framework for simulating vaporizing, liquid-gas flows.",
"It is developed for the general vaporization problem which solves the vaporization rate based as from the local thermodynamic equilibrium of the liquid-gas system.",
"This includes the commonly studied vaporization regimes of film boiling and isothermal evaporation.",
"The framework is built upon a Cartesian grid solver for low-Mach, turbulent flows which has been modified to handle multiphase flows with large density ratios.",
"Interface transport is performed using an unsplit volume of fluid solver.",
"A novel, divergence-free extrapolation technique is used to create a velocity field that is suitable for interface transport.",
"Sharp treatments are used for the vapor mass fractions and temperature fields.",
"The pressure Poisson equation is treated using the Ghost Fluid Method.",
"Interface equilibrium at the interface is computed using the Clausius-Clapeyron relation, and is coupled to the flow solver using a monotone, unconditionally stable scheme.",
"It will be shown that correct prediction of the interface properties is fundamental to accurate simulations of the vaporization process.",
"The convergence and accuracy of the proposed numerical framework is verified against solutions in one, two, and three dimensions.",
"The simulations recover first order convergence under temporal and spatial refinement for the general vaporization problem.",
"The work is concluded with a demonstration of unsteady vaporization of a droplet at intermediate Reynolds number."
],
[
"Introduction",
"Several common engineering systems involve the flow of liquids and gases.",
"For many of these flows, it is often the case that the primary objective of these systems is to exchange heat or mass between the two phases.",
"Some common systems include heat exchangers, spray coolers, bubble column reactors, and spray combustors.",
"In these systems heat and mass transfer occurs primarily through the process of phase change.",
"As such, understanding phase change is an important step in better understanding the dynamics of these engineering systems.",
"This paper focuses on liquid-to-gas phase change, here called vaporization.",
"For many of these problems, it is appropriate to conceptualize the flow as consisting of a continuous carrier phase with dispersed droplets.",
"Several reviews have been written on the theory of the vaporization of droplets [7], [8], [9].",
"These studies were focused primarily on either vaporization for idealized flows.",
"The study of the interaction of vaporization and turbulent flows remains an open problem.",
"Recently, experiments have been performed studying vaporization and combustion of droplets in turbulent flows (e.g., the review by Birouk and Gokalp [10]), however a fundamental understanding of the mechanism of vaporization-turbulence interaction is lacking.",
"In particular, the question of which scales of turbulence are most important to the interaction is yet to be answered.",
"Direct numerical simulation (DNS) can provide an alternative avenue to study this problem.",
"In this context, DNS are taken to be those simulations that solve the dynamics of the flow from first principles, i.e., directly from the laws for conservation of mass, momentum, and energy.",
"Vaporization is represented as a set of coupled matching conditions at the liquid-gas interface which ensure the conservative transfer of mass, momentum, and energy across the interface.",
"According to Sirignano [11], a multiphase flow simulation can be considered a true DNS when it resolves both the gas film around the droplet and the internal droplet flow.",
"Unfortunately, due to the sharp discontinuities in pressure, velocity, and density at the interface, the stable numerical simulation of this problem is difficult.",
"Only recently have numerical methods capable of performing resolved simulations of vaporization have been discussed [12], [13], [14], [15], [16], [17], [5], [1], [18].",
"The aforementioned numerical frameworks have been used to study film boiling, vaporizing bubbles rising in liquid, and the classical $d^2$ law.",
"However, the studies are performed under several assumptions regarding the regime of vaporization and the flow conditions.",
"First, simulations are typically performed within one of two limiting cases of vaporization: an evaporation limit wherein the vaporization rate is assumed to be limited by the local values in the vapor species [14], [16]; and a boiling limit wherein the vaporization rate is assumed to be limited by the local values in temperature [12], [13], [15], [17], [18].",
"Few authors have studied vaporization without the imposition of these constraints [16], [5], [1].",
"All of the frameworks mentioned so far used incompressible solvers, and assumed constant thermophysical properties.",
"Furthermore, most of the studies were performed in two dimensions which inherently neglects many important physical processes in turbulent flows.",
"The literature also contains very little discussion of the convergence and stability of the numerical methods used in the simulations.",
"The current work demonstrates a numerical framework for vaporization in three dimensions, and develops a suite of tests to study the stability and convergence of the flow solver.",
"In developing the framework, it will be shown that traditional semi-implicit treatments of scalar transport can lead to stability issues for large timesteps.",
"The work is organized as follows.",
"sec:physics gives the physical description of the problem and outlines the mathematical model of the flow.",
"sec:numerics details the algorithms used in the work.",
"sec:results verifies the numerical algorithm against known analytical solutions in one dimension.",
"sec:2D performs a verification study in two dimensions, and it demonstrates the temporal accuracy and stability of the solver.",
"sec:3D verifies the solver against the well know $d^2$ solution in three dimensions.",
"Finally, demonstrations of unsteady droplet vaporization (sec:cross) are shown."
],
[
"Physical Description",
"This section details the mathematical equations used to describe the vaporization process.",
"The process is described by the conservation equations for low-Mach flow, i.e., conservation of mass, momentum, energy, and chemical species in each phase.",
"Thermodynamic equilibrium at the interface is resolved using the Clausius-Clapeyron relation.",
"As an initial assumption, a single component fuel is used, and the fuel vapor does not react with the surrounding gas.",
"Both the liquid and the gas are assumed to be of constant density.",
"The exposition of the governing equations follows.",
"The motion of a Newtonian fluid is governed by the Navier-Stokes equations, $\\frac{\\partial \\left(\\rho \\mathbf {u}\\right)}{\\partial t}+\\nabla \\cdot \\left(\\rho \\mathbf {u}\\otimes \\mathbf {u}\\right)=-\\nabla p+\\nabla \\cdot \\left(\\mu \\mathbf {\\mathcal {S}}\\right), \\text{ where} \\\\\\mathbf {\\mathcal {S}}=\\left(\\nabla \\mathbf {u}+{\\nabla \\mathbf {u}}^\\top -\\frac{2}{3}\\left(\\nabla \\cdot \\mathbf {u}\\right)\\mathbf {\\mathcal {I}}\\right), $ which is statement of the conservation of momentum.",
"Here, $\\rho $ and $\\mu $ are the fluid density and dynamic viscosity; $\\mathbf {u}$ is the velocity; and $p$ is the pressure.",
"The identity tensor is represented with $\\mathbf {\\mathcal {I}}$ .",
"eq:NS must be coupled with the continuity equation, here expressed as $\\frac{\\partial \\rho }{\\partial t}+\\nabla \\cdot \\left(\\rho \\mathbf {u}\\right)=0,$ which is an expression of the conservation of mass.",
"In the gas phase, conservation of mass must be supplemented with an equation for conservation of chemical species, $\\frac{\\partial \\left(\\rho Y\\right)}{\\partial t}+\\nabla \\cdot \\left(\\rho Y\\mathbf {u}\\right)=\\nabla \\cdot \\left(\\rho D\\nabla Y\\right),$ where the vapor mass fraction and its diffusivity are represented using $Y$ and $D$ , respectively.",
"As the problem is nonreactive, remaining inert gas can be expressed as $1-Y$ .",
"The equation for conservation of energy can be expressed as $\\frac{\\partial \\left(\\rho C_p T\\right)}{\\partial t}+\\nabla \\cdot \\left(\\rho C_p T\\mathbf {u}\\right)=\\nabla \\cdot \\left(k\\nabla T\\right)+\\frac{Dp}{Dt}+\\frac{\\mu }{2}\\mathbf {\\mathcal {S}}:\\mathbf {\\mathcal {S}},$ where $T$ is the fluid temperature, $C_p$ is the specific heat under constant pressure, and Fourier's law is used for conduction ($k$ is the thermal conductivity).",
"The notation, $D/Dt$ represents the Lagrangian derivative, i.e., $\\frac{Dp}{Dt}=\\frac{\\partial p}{\\partial t}+\\mathbf {u}\\cdot \\nabla p,$ and double tensor contraction expands as $\\mathbf {\\mathcal {S}}:\\mathbf {\\mathcal {S}}=S_{ij}S_{ij},$ using Einstein summation convention.",
"However, the full energy equation is not used in this work.",
"Consistent with the low-Mach assumption, we neglect those terms of the energy equation that are quadratic in the Mach number (or equivalently the velocity), yielding $\\frac{\\partial \\left(\\rho C_p T\\right)}{\\partial t}+\\nabla \\cdot \\left(\\rho C_p T\\mathbf {u}\\right)=\\nabla \\cdot \\left(k\\nabla T\\right).$ In deriving eq:energy it was assumed that the pressure scales as the dynamic pressure, $\\frac{1}{2}\\rho |\\mathbf {u}|^2$ .",
"The resulting equation neglects all pressure and viscous work on the system and, accordingly, acoustic effects."
],
[
"Phase Change Strategy",
"The conservation equations eq:NS,,eq:cont,eq:vaporfrac,eq:energy do not offer a complete description of the flow for multiphase problems.",
"A sample domain is shown in fig:interfacegeom.",
"To completely describe the system, the interface location and interface matching conditions must be known.",
"Label the liquid and gas portions of the domain using $L$ and $G$ , respectively.",
"The liquid-gas interface is then defined as the intersection of these two regions, and is referred to with $\\Gamma $ .",
"On the interface, an outward facing normal, $\\mathbf {n}_\\Gamma $ , is defined pointing away from the liquid.",
"An vaporization mass flux, $\\dot{m}$ , forms normal to the interface, and removes an amount of mass $\\Delta M$ from the liquid (fig:interfacemdot).",
"The notation $\\cap $ will be used to refer to the intersection of two sets, so that $\\Gamma =L\\cap G$ .",
"The notation $\\cup $ will be used for the union of two sets, so that the entire domain is given by $L\\cup G$ .",
"In fig:interface, the liquid region is represented using a dark color, the gas is represented using white, and the interface is represented as a thick line.",
"This convention is used throughout the remainder of this work.",
"Figure: Geometry near the liquid gas interface.",
"fig:interfacegeom: The liquid and gas regions are labeled as LL and GG, respectively, whereas the interface is Γ\\Gamma .",
"The local interface normal vector, 𝐧 Γ \\mathbf {n}_\\Gamma is shown as well.",
"fig:interfacemdot: Over a time, Δt\\Delta t, vaporization at the rate, m ˙\\dot{m}, causes a change in liquid mass of ΔM\\Delta MBecause only two phases exist in this problem, $\\Gamma $ can be identified using the liquid indicator function, $f(\\mathbf {x},t)=\\left\\lbrace \\begin{array}{cc}1 & \\text{if }\\mathbf {x} \\in L,\\\\0 & \\text{otherwise.",
"}\\end{array}\\right.$ The interface lies where $f$ passes between 0 and 1, or equivalently when $\\nabla f$ is nonzero.",
"In practice, numerical schemes transport the liquid volume fraction, $\\alpha $ , which is the volume average of $f$ in each computational cell."
],
[
"Interfacial Conditions",
"Consider the local velocity of interface motion, $\\mathbf {u_S}$ .",
"It follows from the principle of conservation of mass that, $\\mathbf {n}_\\Gamma \\cdot \\rho _G\\left(\\mathbf {u}_G-\\mathbf {u}_S\\right)=\\mathbf {n}_\\Gamma \\cdot \\rho _L\\left(\\mathbf {u}_L-\\mathbf {u}_S\\right)=\\dot{m}.$ This expresses the fact that the flux of mass into the gas from the interface is the same as the flux from the liquid into the interface.",
"This flux is denoted $\\dot{m}$ .",
"It satisfies $\\frac{d}{dt}\\int \\limits _L\\rho _LdV=-\\int \\limits _{\\Gamma }\\dot{m}dS,$ hence, $\\dot{m}$ is the local vaporization rate per unit area.",
"(By convention, integrals of type $dS$ refer to surface integrals and of type $dV$ refer to volume integrals.)",
"This also implies a transport equation for the liquid volume fraction in the form, $\\frac{\\partial \\rho _L\\alpha }{\\partial t}+\\nabla \\cdot \\left(\\rho _L\\alpha \\mathbf {u}\\right)=-\\dot{m}\\delta _\\Gamma ,$ which is derived from substituting the definitions of $f$ and $\\alpha $ into eq:masscont and using the Reynolds transport theorem.",
"Here $\\delta _\\Gamma $ is the interfacial surface area density which can be defined for a region $\\Omega $ as, $\\delta _\\Gamma =\\int \\limits _{\\Gamma \\cap {\\Omega }}dS\\left\\bad.\\int \\limits _{\\Omega }dV\\right..$ eq:interfacecont can also be expressed as an interfacial compatibility condition between the velocities of the two phases.",
"Defining a jump notation as $\\left[\\phi \\right]_\\Gamma =\\phi _G-\\phi _L,$ the interface condition, eq:interfacecont can be written as $\\left[\\mathbf {n}_\\Gamma \\cdot \\mathbf {u}\\right]_\\Gamma =\\dot{m}\\left[\\frac{1}{\\rho }\\right]_\\Gamma .$ For the momentum equation, the interfacial condition is the jump in the pressure across the interface given by $\\left[p\\right]_\\Gamma =-\\sigma \\kappa -{\\dot{m}}^2\\left[\\frac{1}{\\rho }\\right]_\\Gamma .$ The term $\\sigma \\kappa $ is the pressure caused by surface tension, where $\\sigma $ is the surface tension coefficient and $\\kappa =\\nabla \\cdot \\mathbf {n}_\\Gamma $ is twice the mean curvature.",
"The interfacial conditions for the remaining equations are more complex.",
"The gradients of the vapor mass fraction and temperature are related implicitly through the definition of $\\dot{m}$ .",
"The value of $\\dot{m}$ varies locally on the interface, and its value reflects the local imbalance in thermal energy and species mass across the interface, $\\dot{m} = \\frac{\\left[\\mathbf {n}_\\Gamma \\cdot k\\nabla T\\right]_\\Gamma }{\\left[h\\right]},$ and, $\\dot{m} = \\frac{\\mathbf {n}_\\Gamma \\cdot \\rho _GD\\nabla Y}{Y^\\Gamma -1},$ which are derived form eq:energy,eq:vaporfrac, respectively.",
"Since both eq:mdotT,eq:mdotY must be satisfied simultaneously, it follows that the gradients of vapor mass fraction and temperature are not independent at the interface.",
"In eq:mdotT, the specific enthalpy, $h$ , in each phase is approximated as $h=h_{sat}+C_p\\left(T-T_{sat}\\right)$ .",
"This leads to $\\left[h\\right]=L_V+\\left[C_p\\right]\\left(T_\\Gamma -T_{sat}\\right)$ , where $L_V$ is the specific latent heat of boiling.",
"Accordingly, $\\left[h\\right]$ is an effective latent heat, which varies as a function of temperature.",
"Because eq:mdotT is derived from the thermal energy equation, eq:energy, it implicitly presumes that thermal energy transfer at the interface dominates mechanical energy transfer.",
"However, this presumption is consistent with the thermodynamic assumptions used in the work.",
"The relationship between $T$ and $Y$ is further complicated by the requirement of the thermodynamic equilibrium at the interface between the fuel and its vapor.",
"This implies a functional relationship between the vapor mass fraction and the temperature fields.",
"This relationship can be described by the Clausius-Clapeyron relations, $X &=\\exp \\left(-\\frac{L_VW_V}{R}\\left(\\frac{1}{T_\\Gamma }-\\frac{1}{T_{sat}}\\right)\\right), \\text{ where} \\\\Y &= \\frac{XW_V}{XW_V+(1-X)W_A}.",
"$ Here $X$ is the vapor mole fraction and $W_V$ and $W_A$ are the vapor and ambient gas molar masses, respectively.",
"eq:CC is derived assuming the gas behaves ideally, hence the use of the gas constant, $R$ .",
"The saturation temperature, $T_{sat}$ are assumed to be known and are fixed in time and space.",
"The temperature across the interface is assumed to be continuous, i.e., $\\left.T_L\\right|_\\Gamma =\\left.T_G\\right|_\\Gamma =T_\\Gamma $ .",
"eq:mdotT,eq:mdotY,eq:CCx,eq:CCy must be satisfied simultaneously to correctly resolve the vaporization."
],
[
"Numerical Approach",
"This section details the algorithms used to perform direct numerical simulations of vaporizing liquid-gas flows.",
"The algorithms are embedded into NGA, a computational code for solving low-Mach, multiphase flows [2].",
"NGA uses a mixture of finite volume and finite difference schemes to solve the conservation equations (sec:physics) on a staggered grid.",
"Two time integration schemes are used which are detailed in sec:temps.",
"A detailed exposition of the methods used to solve the vaporization problem follows.",
"The numerical methods used in this work will be detailed in semi-discrete form.",
"For time evolving fields, superscripts are used.",
"When necessary, secondary superscripts are used to indicate sub-iterations, e.g., $f^{n,k}$ occurs at time level $n$ at the $k$ sub-iteration.",
"By convention, $f^{n+1,0}=f^{n,m}$ where $m$ is the total number of sub-iteration used in each timestep.",
"The elapsed time between time levels is represented as $\\Delta t$ .",
"For spatially varying fields, subscripts are used to identify the position, e.g., $f_{i,j,k}^n\\equiv f(\\mathbf {x},t)$ , where $\\mathbf {x}$ is the centroid of the grid cell indexed with $(i,j,k)$ and $t=n\\Delta t$ .",
"The distance between the grid cell centroids $(i,j,k)$ and $(i+1,j,k)$ is given by $\\Delta x$ .",
"The distances $\\Delta y$ and $\\Delta z$ are defined analogously.",
"For brevity, superscript and subscript notations will be omitted when the reference is unambiguous."
],
[
"Sharp Representation of Liquid-Gas Flows",
"Before discussing the numerical methods themselves, it is worthwhile to discuss the strategy used to represent physical quantities and field variables in each grid cell.",
"Following, Anumolu and Trujillo [18], a strategy is said to be sharp when the interface position is known, the thermophysical properties at the interface are discontinuous, and fluxes are evaluated using only information from one phase.",
"enum:1 requires the use of an interface tracking scheme with explicit interface reconstruction.",
"Such a scheme is used in this work (detailed in sec:transport).",
"With precise knowledge of the interface position, enum:2,enum:3 reduce to a statement on how information is averaged between the two phases.",
"This work uses a mixture of both sharp and non-sharp representations.",
"A two-phase momentum equation is formed by taking the volume average of the Navier-Stokes equations, eq:NS, evaluated in each phase.",
"Using $\\alpha $ , the effective density and viscosity in a computational cell can be defined as $\\rho =\\rho _L\\alpha +\\rho _G(1-\\alpha ),$ and $\\mu =\\mu _L\\alpha +\\mu _G(1-\\alpha ).$ The form of the momentum equations remains the same, simply replacing the effective density and viscosity for their single phase counterparts.",
"A two-phase continuity equation is constructed similarly.",
"Throughout the remainder of this work, the effective density and viscosity will be referred to using $\\rho $ and $\\mu $ while the phase-specific values will always use subscripts $L$ and $G$ .",
"The resulting representation of $\\rho $ , $\\mu $ , and $\\mathbf {u}$ is not sharp, since values from both sides of the interface are used in the computation of these quantities.",
"Even so, this approach is preferred as it is consistent with the finite volume methodology used in this work.",
"In contrast, the temperature is represented using two fields, $T_L$ and $T_G$ , which are defined within the liquid and gas, respectively.",
"This approach follows Ma and Bothe [5] who demonstrated that such a representation produced more accurate temperature profiles for vaporization simulations.",
"This representation of the temperature field is sharp, since it uses only information from one side of the interface.",
"The vapor mass fraction field is defined only in the gas, hence it is also treated sharply.",
"For grid cells that contain both liquid and gas, data are computed at the phase barycenters $\\mathbf {x}_L$ and $\\mathbf {x}_G$ , respectively.",
"For example, in the cell $(i)$ the value of $T_L$ is computed at $\\mathbf {x}_{L,i}$ while the value of $T_G$ is computed at $\\mathbf {x}_{G,i}$ ."
],
[
"Time Integration",
"In general, the transport equations used in this work can be rewritten for a variable $\\phi $ in the form $\\frac{\\partial \\left(\\rho \\phi \\right)}{\\partial t}=-\\nabla \\cdot \\left(\\rho \\phi \\mathbf {u}\\right)+\\nabla \\cdot \\left(\\rho \\xi \\nabla \\phi \\right)+S(\\mathbf {x}),$ where $\\xi $ is some appropriately defined diffusion coefficient and $S$ represents sources to the equation.",
"Selecting a time integration involves finding appropriate temporal discretization of these operators.",
"Two time integration schemes are used in this work.",
"The default scheme is an iterative semi-implicit scheme given in [19] which is second order accurate.",
"This time integration scheme corresponds to solving the semi-discrete equation $\\frac{\\left(\\rho \\phi \\right)^{n+1,k+1}-\\left(\\rho \\phi \\right)^n}{\\Delta t}&=-\\nabla \\cdot \\left(\\rho \\phi \\mathbf {u}\\right)^{n+1/2,k}+\\nabla \\cdot \\left(\\rho \\xi \\nabla \\phi \\right)^{n+1/2,k}+S^{n+1/2,k}, \\\\\\phi ^{n+1/2,k}&=\\frac{1}{2}\\left(\\phi ^{n+1,k}+\\phi ^n\\right),$ with an arbitrary discretization of the spatial operators.",
"It can be proven that this scheme is linearly stable [19].",
"The second discretization scheme used in this work is a $\\frac{\\left(\\rho \\phi \\right)^{n+1,k+1}-\\left(\\rho \\phi \\right)^n}{\\Delta t}&=-\\nabla \\cdot \\left(\\rho \\phi \\mathbf {u}\\right)^{n+1/2,k}+\\nabla \\cdot \\left(\\rho \\xi \\nabla \\phi \\right)^{n+1,k+1}+S^{n+1/2,k}, \\\\\\phi ^{n+1/2,k}&=\\frac{1}{2}\\left(\\phi ^{n+1,k}+\\phi ^n\\right).$ The major difference between the two scheme is the time discretization of the diffusive fluxes, which are handled fully implicitly in the later expression.",
"The reason for this is as follows.",
"Consider a simplified transport equation consisting only of the diffusive component of the original equation, $\\frac{\\partial \\phi }{\\partial t}=\\frac{\\partial }{\\partial x}\\left(\\xi \\frac{\\partial \\phi }{\\partial x}\\right),$ which has been taken in one dimension, for ease of exposition.",
"(eq:timeEase exploits the constant density assumption by dividing density from both sides of the equation.)",
"When discretized temporally using eq:timeMonotone and spatially using the standard second order finite volume discretization $\\frac{{\\partial }^2\\phi }{\\partial x^2}=\\frac{\\phi _{i+1}-2\\phi _i+\\phi _{i-1}}{\\Delta x^2},$ this yields, $-Cu_{i+1}^{n+1,k+1}+(1+2C)u_i^{n+1,k+1}-Cu_{i-1}^{n+1,k+1}=u_i^n,$ where $C=\\xi \\Delta t/{\\Delta x}^2$ .",
"This equation corresponds to a monotone scheme (see the proof in sec:monotoneProof) and has the property of being not only linearly stable, but also non-oscillatory.",
"Since vaporization is controlled by diffusive mechanisms via eq:mdotT,eq:mdotY, the non-oscillatory solution of diffusion is important.",
"This idea is further explored in sec:stability."
],
[
"Momentum Equation",
"The momentum equation is solved using a mass-momentum consistent advection strategy [3].",
"The scheme is known to be stable for flows with large density ratios such as those encountered in a fuel-air system.",
"The basic idea of the scheme is to update both $\\rho $ and $\\rho \\mathbf {u}$ in time such that their discrete solution implies the discrete transport equation for $\\mathbf {u}$ , via $\\mathbf {u}^{n+1}=\\left(\\rho \\mathbf {u}\\right)^{n+1}/\\rho ^{n+1}$ .",
"A overview of the solution is shown below for a one dimensional system.",
"Due to the staggered grid approach, velocities are solved on a mesh that is offset from the primary mesh (fig:offset).",
"A new set of effective densities and viscosities, $\\rho _u$ and $\\mu _u$ , must be defined using $\\alpha _u$ , the local fraction of liquid volume in the $u$ velocity cell.",
"This local volume fraction is computed by constructing the $u$ velocity cell, and computing the liquid volume contained therein (fig:offset).",
"An interface reconstruction scheme must be used to compute the exact volume inside of the $u$ velocity cell.",
"The scheme used in this work is discussed in sec:transport.",
"At each time step, $\\alpha ^n_u$ is computed from the cell centered field, $\\alpha ^n$ at the current time level.",
"The resulting algorithm to update $u$ is $u^{n+1/2,k} &=\\frac{1}{2}\\left(u^{n+1,k}+u^n\\right), \\\\\\frac{\\rho _u^{n+1,k+1}-\\rho _u^n}{\\Delta t}+\\frac{\\partial \\left(\\rho _u^nu^{n+1/2,k}\\right)}{\\partial x}&=0, \\\\\\frac{\\rho _u^{n+1,k+1}u^{n+1,k+1}-\\rho _u^nu^n}{\\Delta t}+\\frac{\\partial \\left(\\rho _u^nu^nu^{n+1/2,k}\\right)}{\\partial x}&=RHS, $ where $RHS$ represents all non-advective terms in the Navier-Stokes equations.",
"First eq:Mconst3 is solved for $\\rho ^{n+1,k+1}_u$ , and that value is used in eq:Mconst4 to update the velocity.",
"The key aspect of this treatment is that the same discrete representation of the flux operator $\\partial /{\\partial x}$ is used for advection in eq:Mconst3,eq:Mconst4.",
"Although any discretization may be used for this operator in general, care must be taken to ensure the stability of the scheme.",
"In this work the flux operator is discretized using first order upwind in interface containing cells, and a second order finite volume discretization[2] otherwise.",
"The specifics of the discretization including the definition of $RHS$ are given in [3].",
"Figure: Construction of volume fraction on velocity cell, i u i_u.",
"Data from cells i-1i-1 and ii are used to construct i u i_u."
],
[
"Pressure Poisson Equation",
"For low-Mach flows, the continuity equation is enforced through the pressure.",
"In this work, Chorin splitting [20] is used to couple the momentum and continuity equations.",
"First, an intermediate velocity is computed by solving the Navier-Stokes equations, eq:NS, with zero pressure.",
"This field is referred to as ${\\mathbf {u}}^{NS}$ , the Navier-Stokes velocity.",
"Next the velocity field is updated with a pressure projection according to ${\\mathbf {u}}^{n+1}={\\mathbf {u}}^{NS}-\\frac{\\Delta t}{\\rho }\\nabla p.$ Taking the divergence of eq:pressureSplit yields the Poisson equation for pressure, $\\nabla \\cdot \\left(\\frac{1}{\\rho }\\nabla p\\right)=-\\frac{1}{\\Delta t}\\nabla \\cdot \\left({\\mathbf {u}}^{n+1}-{\\mathbf {u}}^{NS}\\right).$ Conceptually, eq:pressureSplit,eq:PPE are similar to the projection technique used in purely incompressible flows.",
"In that context, the velocity is projected onto its divergence-free part using a Helmholtz decomposition.",
"However, in this work, it is not assumed that $\\mathbf {u}^{n+1}$ is incompressible, so the right hand side of eq:PPE is nontrivial.",
"By rearranging eq:cont, we see $\\nabla \\cdot \\mathbf {u}=-\\frac{1}{\\rho }\\frac{D\\rho }{Dt},$ and this value is used in eq:PPE.",
"For brevity, the notation $H=\\frac{1}{\\rho }\\frac{D\\rho }{Dt},$ is adopted.",
"Here it should be noted the abuse of notation used in writing operators such as $\\nabla \\cdot \\mathbf {u}$ in the preceding discussions.",
"The flow solver solves the equations of motion in the finite volume sense, meaning that the flow variables are updated in terms of fluxes through control volumes.",
"This approach is consistent with directly solving the integral form of the conservation equations.",
"However, in this work flow variables are not necessarily differentiable, requiring a careful definition of the various operators.",
"As such, operators must be understood in the following sense.",
"For some vector, $\\mathbf {f}$ , define the operator, $\\text{div}\\left(\\mathbf {f}\\right)=\\lim \\limits _{\\int \\limits _\\Omega dV\\rightarrow 0}\\frac{\\int \\limits _{\\partial \\Omega }\\mathbf {n}\\cdot \\mathbf {f} dS}{\\int \\limits _\\Omega dV}$ which is the net flux per unit volume out of $\\Omega $ .",
"For sufficiently smooth vector fields this limit yields $\\text{div}\\left(\\mathbf {f}\\right)=\\frac{\\partial f_i}{\\partial x_i}=\\nabla \\cdot \\mathbf {f},$ which follows from the Gauss-Ostrogradsky (Divergence) Theorem.",
"In this work, all flow variables are continuous except at the interface, so relation eq:GOD applies almost everywhere.",
"Throughout this work, the suggestive notation $\\nabla \\cdot \\mathbf {f}$ has been used with the meaning $\\text{div}\\left(f\\right)$ .",
"The form of eq:GOD as a surface integral divided by an volume integral suggests that the surface density, $\\delta _\\Gamma $ , may be important in evaluating the divergence operator, $\\text{div}$ , for discontinuous vectors.",
"This can be seen by directly computing $\\text{div}\\left(\\mathbf {u}\\right)$ near the interface.",
"Define a volume $\\Omega _\\Gamma $ by extending a small distance from the interface in both directions (i.e.",
"in directions of both $\\mathbf {n}_\\Gamma $ and $-\\mathbf {n}_\\Gamma $ ).",
"In such a volume $\\text{div}\\left(\\mathbf {u}\\right)_\\Gamma $ satisfies, $\\lim \\limits _{\\int \\limits _{\\Omega _\\Gamma }dV\\rightarrow 0}\\frac{\\int \\limits _{\\Gamma }\\left[\\mathbf {n}_\\Gamma \\cdot \\mathbf {u}\\right]_\\Gamma dS}{\\int \\limits _{\\Omega _\\Gamma }dV}=\\lim \\limits _{\\int \\limits _{\\Omega _\\Gamma }dV\\rightarrow 0}\\frac{\\int \\limits _{\\Gamma }\\left[\\mathbf {n}_\\Gamma \\cdot \\mathbf {u}\\right]_\\Gamma \\delta _\\Gamma dV}{\\int \\limits _{\\Omega _\\Gamma }dV}=\\left[\\mathbf {n}_\\Gamma \\cdot \\mathbf {u}\\right]_\\Gamma \\delta _\\Gamma .$ In deriving eq:discreteDIV it was important to assume that there is no jump in tangential velocity, such that the net flux through $\\partial \\Omega _{\\Gamma }$ equals the net flux through $\\Gamma $ .",
"The overall value of $\\text{div}\\left(\\mathbf {u}\\right)$ in a discrete computational cell contains three components: a component from the fully gas portion of the cell, a component from the fully liquid component of the cell, and an interfacial component as detailed in eq:discreteDIV.",
"Using this information, and expanding $\\left[\\mathbf {n}_\\Gamma \\cdot \\mathbf {u}\\right]_\\Gamma $ using eq:divU, the pressure Poisson equation can be written as $\\nabla \\cdot \\left(\\frac{1}{\\rho }\\nabla p\\right)=\\frac{1}{\\Delta t}\\left(H-\\dot{m}\\left[\\frac{1}{\\rho }\\right]_\\Gamma \\delta _\\Gamma +\\nabla \\cdot {\\mathbf {u}}^{NS}\\right),$ where eq:H is applied separately in each phase, so that $H=\\alpha H_L+(1-\\alpha )H_G.$ $H$ can change due to variations in the chemical composition, temperature, and thermodynamic pressure of the fluid.",
"As a first step, the current work explicitly neglects changes in density due to chemical composition and temperature.",
"This simplifies the exposition of the system thermodynamics considerably.",
"Consistent with the low-Mach assumption, the thermodynamic pressure is assumed to vary only temporally.",
"This implies that each phase's value of $H$ is also a function of time only.",
"Finally, we explicitly assume that the liquid is fully incompressible, implying that $H$ can be nonzero only in the gas.",
"By assumption, nonzero $H$ correspond to a build up of pressure in the computational domain.",
"Since sources due to chemical composition and temperature are neglected, this build up is due entirely to the evaporative flow.",
"In the presence of outflow conditions, this flow is able to freely exit the domain.",
"Accordingly, vaporization does not cause any pressure buildup, and $H=0$ .",
"For closed boundary conditions, such as periodic boundary conditions, the flow cannot exit the domain, and a gradual rise in pressure occurs.",
"For low-Mach flows, a consistent choice for this source is $H_G= \\frac{-1}{\\int \\limits _GdV}\\int \\limits _{L\\cup G}\\left(-\\dot{m}\\left[\\frac{1}{\\rho }\\right]_\\Gamma \\delta _\\Gamma +\\nabla \\cdot {\\mathbf {u}}^{NS}\\right)dV.$ This choice ensures that the compatibility condition of the pressure Poisson equation is satisfied, i.e, $\\int \\limits _{L\\cup G}\\nabla \\cdot \\left(\\frac{1}{\\rho }\\nabla p\\right)dV=\\int \\limits _{L\\cup G}\\frac{1}{\\Delta t}\\left(H-\\dot{m}\\left[\\frac{1}{\\rho }\\right]_\\Gamma \\delta _\\Gamma +\\nabla \\cdot {\\mathbf {u}}^{NS}\\right)dV=0.$ Satisfaction of this condition is sufficient to guarantee existence and uniqueness of the pressure correction, $\\nabla p$ [21].",
"The gas density is then updated as ${\\rho _G}^{n+1}={\\rho _G}^n(1+\\Delta tH_G),$ which can be considered to be a low-Mach correction to the Poisson equation that accounts for gas phase compressibility.",
"The pressure Poisson equation, eq:PPEtrue, is solved using a technique based upon the Ghost Fluid Method [6].",
"An illustrative discretization in one dimension is defined below.",
"The pressure Laplacian operator is discretized as, $\\frac{\\partial }{\\partial x}\\left(\\frac{1}{\\rho }\\frac{\\partial p}{\\partial x}\\right)\\approx \\frac{1}{\\Delta x}\\left(\\frac{1}{\\rho _{u,i+1}}\\frac{p_{i+1}-p_i}{\\Delta x}-\\frac{1}{\\rho _{u,i}}\\frac{p_i-p_{i-1}}{\\Delta x}\\right),$ where the assumption of constant grid spacing has been used.",
"The velocity divergence is written as $\\frac{\\partial u}{\\partial x}\\approx \\frac{u_{i+1}-u_i}{\\Delta x}.$ The singular source term involving $\\delta $ requires a special treatment.",
"Here this term is discretized as $\\dot{m}\\left[\\frac{1}{\\rho }\\right]_\\Gamma \\delta _\\Gamma \\approx Q\\dot{m}_i\\left[\\frac{1}{\\rho }\\right]_\\Gamma \\frac{1}{\\Delta x},$ the value $Q$ is one in interface containing cells and zero otherwise.",
"The pressure jump, eq:Pjump is discretized using the standard GFM technique.",
"As a single pressure field is solved for the entire domain, this discretization is not sharp, in the sense of [18].",
"However, this approach was chosen to be consistent with the one field approach to the velocity field.",
"Furthermore, in practice, this technique only distributes the pressure jump over a region of one or two grid cells around the interface."
],
[
"Scalar Transport Equation",
"Transport of the temperature and vapor mass fraction fields is handled sharply to avoid artificial smearing of the profiles.",
"One of the necessary aspects of a sharp representation is that data for one phase are not computed using values from the other phase.",
"For this purpose, several auxiliary variables are defined that will be necessary to discuss in the transport equations.",
"At each time step, a gas volume fraction $\\alpha _G^n=1-\\alpha ^n$ and secondary liquid volume fraction $\\alpha _L^n=\\alpha ^n$ are constructed.",
"To aid in the advection of scalars, a continuous velocity field is constructed for both the liquid and gas fields, labeled as ${\\mathbf {u}}_L$ and ${\\mathbf {u}}_G$ , respectively.",
"The fields are initialized using the true velocity field at the latest time level, ${\\mathbf {u}}^{n+1,k}$ , and the data are extrapolated using the zero normal gradient technique of Aslam [22].",
"In the technique, a PDE is solved in pseudotime, $t_{ps}$ , until steady state, and the steady state solution verifies the zero normal gradient property.",
"The equation to extend an arbitrary field, $\\phi $ , from the liquid into the gas can be expressed as $\\frac{\\partial \\phi }{\\partial t_{ps}}+A\\mathbf {n}_\\Gamma \\cdot \\nabla \\phi =0$ where $A=\\left\\lbrace \\begin{array}{cc}0 & \\text{if }\\alpha _L=1,\\\\1 & \\text{otherwise.",
"}\\end{array}\\right.$ A similar process is used to extrapolate from gas to liquid.",
"The technique results in an extrapolated field that is constant along the direction of $\\mathbf {n}_\\Gamma $ .",
"In [22], a general technique for extrapolation to arbitrary polynomial order is discussed.",
"However, for the current work it was found by experience that constant extrapolation yields superior numerical stability.",
"The transport of scalars uses a variation of the mass consistent scheme [3], outlined below for the vapor mass fraction $Y^{n+1/2,k} &=\\frac{1}{2}\\left(Y^{n+1,k}+Y^{n}\\right) \\\\\\frac{\\rho _G^{n+1}\\alpha _G^{n+1,k+1}-\\rho _G^{n+1}\\alpha _G^n}{\\Delta t}+\\frac{\\partial \\left(\\rho _G^{n+1}\\alpha _G^n{\\mathbf {u}}_G\\right)}{\\partial x}&=0 \\\\\\frac{\\rho _G^{n+1}\\alpha _G^{n+1,k+1}Y^{n+1,k}-\\rho _G^{n+1}\\alpha _G^nY^n}{\\Delta t}+\\frac{\\partial \\left(\\rho _G^{n+1}\\alpha _G^nY^n{\\mathbf {u}}_G\\right)}{\\partial x}&=\\frac{\\partial }{\\partial x}\\left(D\\frac{\\partial Y^{n+1/2,k}}{\\partial x}\\right)$ First, eq:Ytrue2 is solved for $\\rho _G^{n+1}\\alpha _G^{n+1,k+1}$ and then this value is used in eq:Ytrue3 to update $Y$ .",
"For advection, the BQUICK algorithm [23] is used.",
"The algorithm modifies the QUICK advection routine, by locally switching to an simple upwind scheme wherever the computed $Y^{n+1}$ becomes unbounded.",
"For the present work, a second switching condition is used.",
"Whenever the QUICK stencil crosses the interface, the simple upwind scheme is used.",
"To use the BQUICK algorithm, reasonable bounds must be chosen for each scalar.",
"For the mass fraction field, the maximum and minimum values of 1 and 0, respectively, are chosen.",
"For the temperature fields, the maximum of the initial domain temperature, $T_M$ , and 0 are chosen.",
"This choice is appropriate if there are no sources of heat, in which case $T_M$ will always be the highest domain temperature physically possible.",
"The overall accuracy of this scheme is first order, because the simple upwind scheme is first order accurate.",
"In practice, higher accuracy has been observed [23].",
"The treatment of diffusion in interface-containing cells requires special care.",
"fig:diff demonstrates a region near the interface in one dimension.",
"The interface does not occur inside either of cells $i$ and $i+1$ , so the flux between the cells is computed using the second order finite volume discretization, $\\frac{1}{2}(D_{i+1}+D_i)\\frac{Y_{i+1}-Y_i}{\\Delta x}.$ A different treatment is used to compute the flux whenever one (or both) of the two cells contains the interface, such is the case with cells $i$ and $i-1$ .",
"In this case, the flux is computed using the relative distances between the appropriate phase barycenters.",
"For example, the discretization of the flux for the vapor mass fraction is $\\beta _{i-1/2} D_{i-1/2}\\frac{x_i-x_{i-1}}{\\Vert \\mathbf {x}_{G,i}-\\mathbf {x}_{G,i-1}\\Vert }(Y_i-Y_{i-1}).",
"$ Here $x_i$ is the $x$ axis component of $\\mathbf {x}_{G,i}$ .",
"The value $\\beta _{i-1/2}$ is the fraction of gas area on the face between cells $i$ and $i-1$ .",
"The value $D_{i-1/2}$ is linearly interpolated between $D_i$ and $D_{i-1}$ .",
"Note that the VOF scheme does not represent the interface smoothly between cells, and accordingly, the gas area fraction may be different on either side of a cell face.",
"In such a case, $\\beta _{i-1/2}$ is taken to be the minimum of the values.",
"In cell $i-1$ , an additional diffusive flux from the interface into the cell must be computed which satisfies $\\int \\limits _{\\Gamma \\cup \\Omega _{i-1}} \\mathbf {n}_\\Gamma \\cdot D\\nabla Y dS,$ or in terms of the surface area density, $\\int \\limits _{\\Omega _{i-1}}\\left(\\mathbf {n}_\\Gamma \\cdot D\\nabla Y\\right)_\\Gamma \\delta _\\Gamma dV.$ The evaluation of the flux at the interface is the focus of sec:flux.",
"Although not detailed here $T_G$ and $T_L$ are solved in a manner analogous to the vapor mass fraction field.",
"Figure: Diffusion fluxes of vapor mass fraction near the interface.",
"Diffusion fluxes are computed on grid cell faces (white circles) using data from the gas phase barycenters (black circles).",
"Fluxes between cells are restricted to the gaseous fraction of the cell faces (circled in red) and the interface (thick red line)."
],
[
"Thermodynamic Equilibrium at the Interface",
"Vaporization involves the flux of mass, momentum, and energy across the interface.",
"The imbalance of these fluxes is directly tied to the definition of $\\dot{m}$ in eq:Pjump,eq:Ujump,eq:mdotT,eq:mdotY.",
"This relation is further complicated by the requirement of the system to satisfy thermodynamic equilibrium, i.e., eq:CC.",
"Because of this complexity, the solution of this nonlinear system is often neglected in the literature by assuming a priori a limiting vaporization regime such as boiling conditions.",
"There are, however, a few strategies in the literature capable of solving the general problem.",
"The first strategy used in this work is similar to a technique used in Ma and Bothe [5].",
"In their work, they assume an error function profile for the temperature field, and substitute the thermodynamic relations to yield a nonlinear system that can be solved for the interface temperature.",
"Such an assumption is appropriate for low $\\dot{m}$ vaporization (see sec:SPconv), but fails to be appropriate for situations with appreciable convection.",
"Instead, the current work constructs the system directly from the discretization of the governing equations.",
"Recalling that gas and liquid temperatures (as well as mass fraction) are stored in their respective phase barycenter, specialized computational stencils must be use to accurately compute the gradient if the stencil includes the interface.",
"Although several discretizations are possible, a general form of such a stencil is $\\mathbf {n}_\\Gamma \\cdot \\nabla T_G\\approx \\sum _i w_{G,i} T_{G,i},$ where the summation is taken over computational cells near the interface and is restricted to the gas side.",
"A liquid stencil can be defined similarly.",
"Substituting these expressions into eq:mdotT yields $\\dot{m}\\left[h\\right] = \\sum _i w_{G,i}T_{G,i}+\\sum _j w_{L,j} T_{L,j}+(w_{L,\\Gamma }+w_{G,\\Gamma })T_\\Gamma ,\\text{ or} \\\\T_\\Gamma = \\frac{\\dot{m}\\left[h\\right] -\\left(\\sum _i w_{G,i}T_{G,i}+\\sum _j w_{L,j} T_{L,j}\\right)}{w_{L,\\Gamma }+w_{G,\\Gamma }},$ where $w_{G,\\Gamma }$ and $w_{L,\\Gamma }$ refer to the gas-side and liquid-side coefficient to the interface-containing cell.",
"Notably, knowledge of $\\dot{m}$ and the current temperature field allows the expression of the interface temperature in the form $T_\\Gamma ={\\mathcal {T}}(\\dot{m})$ .",
"The system is closed by noting that discretizing eq:mdotY yields a relationship of the form $\\dot{m}=\\dot{\\mathcal {M}}(Y_\\Gamma )$ , and the Clausius-Clapeyron relation, eq:CC, requires $Y_\\Gamma =\\mathcal {Y}(T_\\Gamma )$ .",
"The entire system may thus be written as $T_\\Gamma -{\\mathcal {T}}({\\dot{\\mathcal {M}}}(\\mathcal {Y}(T_\\Gamma )))=0.$ The solution of eq:fixed can be solved numerically using any root finding routine.",
"A simple regula falsi solver was used in this work.",
"The method outlined above solves the system with discrete consistency in the sense that each equation is satisfied simultaneously in each grid cell.",
"This method for computing interface equilibrium will be henceforth referred to as MB method.",
"Rueda-Villegas et al.",
"[1] forgo the solution of the nonlinear problem, eq:fixed.",
"Instead, they use the approximation $Y_\\Gamma \\approx Y(\\mathbf {x})$ , and then invert the Clausius-Clapeyron relation to find $T_\\Gamma $ .",
"This method is not discretely consistent.",
"To correct for this, they do not directly compute $\\mathbf {n}_\\Gamma \\cdot \\nabla Y$ on the interface and instead substitute $\\frac{1-Y_\\Gamma }{\\rho _GD}\\dot{m}$ where this quantity is needed.",
"Note that the temperature field is used to compute $\\dot{m}$ through eq:mdotT.",
"This method is abbreviated as the REA method.",
"The use of the local value of $Y$ for the interface value introduces a first order error into the system.",
"In the current work, we introduce a modified version of the REA technique that instead extrapolates the $Y$ field to the interface with second order accuracy.",
"The importance of this higher order accuracy will be discussed in sec:interface.",
"This modified technique is referred to as REA2.",
"In this work, two stencils are used to discretize the vaporization fluxes at the interface.",
"The pressure Poisson equation, eq:PPE, and the VOF transport equation, eq:VOF, use $\\dot{m}$ but have no direct knowledge of the scalar fields.",
"The scalar transport equations, in turn, are effectively independent of each other, only being coupled implicitly through the boundary conditions, $T_\\Gamma $ and $Y_\\Gamma $ .",
"This leads to the following splitting.",
"Inside of scalar transport equations, the vaporization fluxes are implemented using a first order accurate gradient approximation.",
"For the gas temperature, this takes the form, $\\mathbf {n}_\\Gamma \\cdot \\nabla T_G=\\frac{T_G^{n+1,k+1}-T^{n+1}_{G,\\Gamma }}{\\Vert \\mathbf {x}_G-\\mathbf {x}_\\Gamma \\Vert },$ where $\\mathbf {x}_G$ and $\\mathbf {x}_\\Gamma $ are the gas phase barycenter and interface barycenter within the computational cell.",
"Note that eq:upwind is discretized using the $n+1$ time level, so that its use corresponds to a fully implicit time integration.",
"Furthermore, this discretization is monotone and therefore does not introduce oscillations to the solution of the scalar fields.",
"This discretization is important.",
"Since the interface moves freely through the domain, $\\Vert \\mathbf {x}_G-\\mathbf {x}_\\Gamma \\Vert $ can approach zero making the interfacial flux term be of arbitrary magnitude.",
"For interface containing cells, this term usually dominates the scalar transport phenomena, so its accurate resolution is paramount.",
"For the pressure Poisson equation and VOF transport equation it was found that a higher order treatment of $\\dot{m}$ was necessary to achieve a convergent solution.",
"First, each scalar is extended across the interface using linear extrapolation [22].",
"Then a least squares fit to the temperature profile in each phase is computed over each cell and its $5^{dim}-1$ neighbors, in $dim=3$ dimensions.",
"Gradients are calculated from the least square fit, and finally eq:mdotT is used to compute $\\dot{m}$ .",
"For example, in one dimension the cell $\\Omega _k$ is fit with the functional form $T=c_0+c_1(x-x_k),$ yielding the system $\\mathbf {\\mathcal {L}}\\mathbf {c}&=\\mathbf {T} \\\\{\\mathbf {\\mathcal {L}}}_{i,j} &= \\frac{1}{j!",
"}{(x_{G,k+i}-x_k)}^j \\\\\\mathbf {c} &= [c_0;c_1]^\\top \\\\{\\mathbf {T}}_i &= T_{G,k+i} \\\\i &\\in \\lbrace -2,-1,0,1,2\\rbrace \\\\j &\\in \\lbrace 0,1\\rbrace $ which is solved in the least squares sense, i.e., $\\mathbf {c}=({\\mathbf {\\mathcal {L}}}^\\top {\\mathbf {\\mathcal {L}}})^{-1}{\\mathbf {\\mathcal {L}}}^\\top \\mathbf {T}$ .",
"It follows from eq:LSQ that the coefficients $c_i$ are the least squares fits for the temperature profile at the cell centroid.",
"In practice, this technique does not lead to any issues with stability, due to the weak coupling of eq:PPE,eq:VOF with the scalar transport equations.",
"The technique provides another interesting property.",
"As noted in [18], knowledge of $\\dot{m}$ is needed not only in interface-containing cells, but also in a narrow band of cells around the interface.",
"The treatment of $\\dot{m}$ given in our work provides a natural framework to extend $\\dot{m}$ across into such a band, namely, by computing eq:mdotT locally in each cell using the extended temperature field."
],
[
"Interface Transport and Divergence-free Velocity Extrapolation",
"Interface transport is accomplished using a second order accurate, unsplit, geometric volume of fluid method [4].",
"The volume fraction is transported using the extrapolated liquid velocity, ${\\mathbf {u}}_L$ .",
"The VOF method used in this work ensures consistency between the discrete velocity divergence and the flux volumes used in the transport step.",
"An implication of this feature is that errors in divergence of the extrapolated velocity field lead to erroneous variations in the liquid volume.",
"To remove this source of error, the extrapolation is completed in two steps.",
"First, the velocity is extrapolated using constant extrapolation [22].",
"The resulting velocity is labeled ${\\mathbf {u}}_L^*$ .",
"Next, ${\\mathbf {u}}_L^*$ is projected onto its divergence-free part using ${\\mathbf {u}}_L^n={\\mathbf {u}}_L^*-\\nabla W,$ where W is a potential derived from the Helmholtz-type equation, $aW+{\\nabla }^2 W=\\nabla \\cdot {\\mathbf {u}}_L^*.$ The coefficient $a$ is zero for all liquid-containing cells and cells that are within three grid cells of the interface, which reduces it to a Poisson equation in this region.",
"For all other cells, $a$ is set to an arbitrary constant, here given by $\\frac{1}{{\\Delta t}^2}$ .",
"The advantage of solving eq:PPEL over a Poisson equation is that the given technique does not need to satisfy the compatibility relation $\\int \\limits _{L\\cup G}\\left(\\nabla \\cdot {{\\mathbf {u}}_L^*}\\right)dV=0,$ eq:compatuL will not hold, in general, for the extrapolated velocity field.",
"The volume of fluid advection equation, eq:VOF, can be solved in a modified form.",
"The interface recession term, $\\dot{m}\\left\\bad.\\rho _L\\delta _\\Gamma \\right.$ , is rewritten in terms of a recession velocity, $\\dot{m}/\\rho _L\\mathbf {n}_\\Gamma $ .",
"Furthermore, since $\\nabla \\cdot {\\mathbf {u}}_L=0$ everywhere that $\\alpha $ is nonzero, the resulting equation simplifies to $\\frac{\\partial \\alpha }{\\partial t}+\\left({\\mathbf {u}}_L-\\frac{\\dot{m}}{\\rho _L}\\mathbf {n}_\\Gamma \\right)\\cdot \\nabla \\alpha =0.$ In practice, the velocity $\\left({\\mathbf {u}}_L-\\frac{\\dot{m}}{\\rho _L}\\mathbf {n}_\\Gamma \\right)$ is used as the velocity seen by the VOF solver.",
"The volume fraction field is used to compute all relevant geometric information for the solver.",
"The interface is reconstructed from $\\alpha $ using the Piecewise Linear Interface Calculation (PLIC) method with ELVIRA [24] which also defines the interface normal $\\mathbf {n}_\\Gamma $ .",
"PLIC represents the interface in each cell as a simple polygon which separates a liquid polyhedron from a gas polyhedron.",
"The representation of the geometry in terms of polytopes allows for a simple algorithm to compute the area (volume) of each polygon (polyhedron) by subdividing it into triangles (tetrahedra) [4].",
"The curvature $\\kappa $ is computed using the mesh decoupled height function approach [25].",
"A diagram showing the flow of a single time step of the solver is shown in tikz:flow.",
"Figure: Outline of a single time step of the flow solver.",
"Sub-iterations are indexed with kk."
],
[
"Verification in One Dimension",
"A suite of numerical tests is performed to study the accuracy of the code.",
"First, spatial convergence is demonstrated against a well-studied heat transfer limited case.",
"Second, this case is modified to study vaporization without the constraint of heat transfer limitation.",
"All simulations are required to satisfy a CFL stability restraint, which is a numerical limitation on the possible time step that can be used in a simulation.",
"The CFL numbers depend on the physical problem being studied as well as the numerical discretization of the governing equation.",
"The CFL numbers for advection, viscous diffusion, mass diffusion, temperature diffusion, and surface tension are $C_u=&\\frac{\\Delta t}{\\Delta _m} \\max \\limits _{L\\cup G} u_M \\\\C_\\mu =& \\frac{4\\mu \\Delta t}{{\\Delta _m}^2} \\\\C_D =& \\frac{4D\\Delta t}{{\\Delta _m}^2} \\\\C_\\lambda =& \\frac{4\\lambda \\Delta t}{{\\Delta _m}^2} \\\\C_\\sigma =& \\Delta t\\sqrt{\\frac{\\sigma }{\\left(\\rho _L+\\rho _G\\right)\\left(\\Delta _m/{2\\pi }\\right)^{3}}}$ where the notations $u_M=\\sum \\limits _{i\\in 1,2,3}|u_i|$ and $\\Delta _m =\\min \\left(\\Delta x,\\Delta y,\\Delta z\\right)$ are used.",
"The thermal diffusivity, $\\lambda =k/(\\rho C_p)$ , has been introduced for convenience.",
"eq:cflm,eq:cfll should be understood to apply to both the liquid and the gas values of $\\mu $ and $\\lambda $ .",
"For the numerical schemes used in this work, the conditions $C_\\sigma <1$ and $C_u <1$ must be satisfied, whereas the relations eq:cflm,eq:cflD,eq:cfll are relaxed due to the use of implicit solvers."
],
[
"Spatial Convergence",
"Due to the inherent complexities of the vaporizing problem, there are very few analytical solutions to verify against.",
"One problem that does have such a solution is Neumann's problem, whose solution is given in the paper by Hardt and Wondra [17] The vaporization problem is solved in one dimension with an assumption of a slow moving interface and negligible convective effects.",
"A schematic is shown in fig:Neumann.",
"A region of hot gas heats the liquid, causing vaporization.",
"Dirichlet conditions are specified for the temperature and mass fraction of the gas at the left boundary whereas the liquid is able to flow freely from the right boundary.",
"The domain is 1 mm in length.",
"Figure: Geometry for Neumann's problem.",
"The gas is heated on the left by a wall of constant temperature, T w T_w.",
"The liquid vaporizes into the gas, and is free to flow from the right boundary.",
"Sample temperature profile shown.Under the assumptions given here, the energy equation, eq:energy, reduces to the unsteady diffusion equation with time varying boundary.",
"Define the initial and current interface location as $x_0$ and $x_\\Gamma $ , respectively.",
"The solution is given in terms of the error function as $T = T_\\infty +\\frac{T_\\Gamma -T_\\infty }{\\operatornamewithlimits{erf}(\\ell )}\\operatornamewithlimits{erf}\\left(\\frac{x-x_0}{2\\sqrt{\\lambda t}}\\right).",
"$ In eq:Neusol, the interface position has been assumed to vary as $x_\\Gamma = x_0+2\\ell \\sqrt{\\lambda t}$ .",
"The diffusion layer value, $\\ell $ , must be solved from thermodynamic variables as $\\ell \\exp (\\ell ^2)\\operatornamewithlimits{erf}(\\ell )=\\frac{C_{p,G}(T_\\Gamma -T_\\infty )}{\\sqrt{\\pi }L}.$ In their work, Hardt and Wondra solve only for the gas temperature field, however, the same analysis can be performed for the mass fraction field, yielding $Y = Y_\\infty +\\frac{Y_\\Gamma -Y_\\infty }{\\operatornamewithlimits{erf}(\\ell _Y)}\\operatornamewithlimits{erf}\\left(\\frac{x-x_0}{2\\sqrt{D t}}\\right) .$ In general, an equation similar to equation eq:Neudelta can be written for $\\ell _Y$ , however for the unity Lewis number case, the uniqueness of the interface position implies $\\ell _Y=\\ell $ .",
"By assumption, the liquid temperature field remains at the interface temperature, $T_\\Gamma $ .",
"eq:Neusol,eq:NeusolY can be combined with the condition of equality of the two expression for $\\dot{m}$ , eq:mdotT,eq:mdotY, to yield an analytical relation between the interface temperature and mass fraction, $T_\\Gamma =T_\\infty +\\frac{L}{C_p}\\left(\\frac{D}{\\lambda }\\right)^{1/2}\\frac{Y_\\Gamma -Y_\\infty }{Y_\\Gamma -1}.$ eq:Neutemp should be combined with the Clausius-Clapeyron relation, eq:CC to close the system.",
"The parameters used in this study mimic the properties of a vaporizing water-air system.",
"They are listed in tab:Neumann.",
"The fluid and thermal properties listed are the same as those used in the work of Hardt and Wondra [17].",
"A unity Lewis number is imposed, and for consistency with their work, the vapor was given the molar mass of water.",
"Two test cases are performed with these parameters by varying $T_\\infty $ and $Y_\\infty $ .",
"Table: Fluid properties for Neumann's problem."
],
[
"Case 1: Film Boiling",
"The first test, Case 1, corresponds exactly to that used in the paper of Hardt and Wondra, which uses $T_\\infty =383.15\\text{ K}$ and $Y_\\infty =1$ , leading to $T_\\Gamma =373.15\\text{ K}$ and $Y_\\Gamma =1$ .",
"To aid in the imposition of the boundary conditions, $x_0$ is chosen to be inside the computational domain at $x_0=0.02L=2\\times 10^{-5}\\text{ m}$ .",
"We chose to initialize the simulation with the analytical solution at time $t=0.1\\text{ s}$ .",
"Simulations are performed of vaporization using the given parameters between times $t=0.\\text{ s}$ and $t=0.2\\text{ s}$ .",
"We initialize the liquid field to the interface temperature, $T_\\Gamma $ .",
"Both the MB technique and REA technique are used to resolve interface equilibrium.",
"A grid convergence study is performed using 50, 75, 100, 150, and 200 grid cells in the domain.",
"All simulations are performed for a fixed $C_\\lambda =C_D=2$ .",
"The analytical solution at $t=0.3\\text{ s}$ is shown in fig:Neumann for the 50 cell grid.",
"It can be seen that both the MB and REA simulations match well with the expected solution.",
"We define error norms for the gas temperature as $\\varepsilon & = \\left|T-T_{exact}\\right|, \\\\L_1 & = \\int \\limits _G \\varepsilon dV\\left\\bad.\\int \\limits _GdV\\right., \\text{ and} \\\\L_\\infty & = \\max _G\\varepsilon .",
"$ The vapor mass fraction and liquid temperature fields are omitted, because they remain within machine precision of their initial values for all cases, as expected.",
"The $L_1$ and $L_\\infty $ errors in gas temperature are plotted in fig:Neu1:L1,fig:Neu1:Linf.",
"It can be seen that approximately first order convergence is noted, which is consistent with the overall scheme accuracy.",
"Furthermore, the two schemes produce almost identical results.",
"Figure: Case 1.",
"Errors computed at t=0.1st=0.1\\text{ s}.",
"MB (), REA ().",
"A first order trend line has been added.While valuable as a comparison to previous work, the standard Neumann's problem does not test the accuracy of the dynamic interface computation, in a situation with nontrivial mass fraction field.",
"A second case, Case 2, was performed using the same grid and fluid parameters with $T_\\infty =323.15\\text{ K}$ and $Y_\\infty =0.2$ .",
"This yields $T_\\Gamma =296.163\\text{ K}$ and $Y_\\Gamma =0.221022$ .",
"The simulation was performed between times $t=0.01$ and $0.1\\text{ s}$ .",
"A grid convergence study is performed with $C_\\lambda =C_D=2$ .",
"For a more complete analysis of this field, additional error measurements are taken.",
"Errors for the vapor mass fraction, the liquid temperature, and liquid velocity are defined in manner analogously to eq:L1,eq:Linf.",
"The gas temperature and vapor mass fraction are referenced to the analytical solution, eq:Neusol,eq:NeusolY.",
"The liquid temperature is referenced to initial value, which is predicted to be constant in time.",
"The velocity error is computed relative to the following solution.",
"Since the problem is one dimensional, the continuity equation predicts a piecewise constant velocity, with a discontinuity only at the interface.",
"The liquid velocity can then be predicted from the definition of $\\dot{m}$ $\\dot{m}=\\rho _L(u_S-u_L)=\\rho _G(u_S-u_G),$ where $u_L=-\\mathbf {n}_\\Gamma \\cdot \\mathbf {u}_L$ , $u_G=-\\mathbf {n}_\\Gamma \\cdot \\mathbf {u}_G$ , $u_S=-\\mathbf {n}_\\Gamma \\cdot \\mathbf {u}_S$ .",
"Note that eq:mdotNeumann has been expressed such that positive speeds corresponds to motion in the positive $x$ direction.",
"This expression rearranges into, $u_L=\\left(1-\\frac{\\rho _G}{\\rho _L}\\right)u_S+\\frac{\\rho _G}{\\rho _L}u_G.$ Since $x_\\Gamma = x_0+2\\ell \\sqrt{\\lambda t}$ , it follows $u_S=\\ell \\sqrt{\\frac{\\lambda }{t}}.$ With this information, the liquid velocity can be expressed as $u_L=\\ell \\sqrt{\\frac{\\lambda }{t}}\\left(1-\\frac{\\rho _G}{\\rho _L}\\right),$ where $u_G=0$ has been substituted.",
"The gas temperature converges with first order accuracy in both $L_1$ and $L_\\infty $ sense for all three techniques (fig:Neu2:L1,fig:Neu2:Linf).",
"However, it is interesting to note that errors using REA are consistently higher than either of the other two techniques, and errors using REA2 are consistently the lowest.",
"A similar trend is seen for the vapor mass fraction field in fig:Neu2:L1Vap,fig:Neu2:LinfVap, although greater variance about the trend is seen for the REA technique.",
"The liquid temperature field (fig:Neu2:L1Liq,fig:Neu2:LinfLiq) demonstrates a different trend.",
"While first order convergence is seen using the REA and REA2 techniques, no convergence is seen using MB.",
"Indeed, MB appears to divergence under grid refinement.",
"However, the overall liquid temperature error associated with the MB technique is orders of magnitude lower than that of the REA or REA2 techniques.",
"The liquid velocity error is demonstrated in fig:Neu2:LinfVelo.",
"Since the liquid velocity is spatially constant, there is no distinction between $L_1$ and $L_\\infty $ errors.",
"This property has been verified, but the redundant plot is not shown here.",
"A trend of approximately linear convergence is displayed, although the trend appears to have some variation.",
"Figure: Case 2.",
"Errors computed at t=0.1st=0.1\\text{ s}.",
"MB (), REA (), REA2 ().",
"A first order trend line has been added."
],
[
"Vaporization of a Curved Interface: Convergence and Stability",
"The solutions to the vaporization problems presented so far have been analytical solutions in one dimension.",
"These solutions, while interesting for verification, do not allow for a more full test of the capabilities of the solver.",
"One of the goals of this work was to develop a robust numerical solver for simulating vaporizing multiphase flows.",
"However solutions in 1D cannot fully test this.",
"Many of the most challenging aspects of multiphase flow simulation are due to non-alignment between the mesh and flow structures; however, in one dimension all flow structures are aligned with the mesh.",
"To understand what influence, if any, mesh alignment has on the flow solver simulations are performed in two dimensions with a complex, curved interface.",
"Since no analytical solution is known to this problem, errors are approximated by comparing simulation data to those of the same simulation run on a fine grid.",
"This section also serves to test the numerical stability of the solver, as a temporal convergence study is performed.",
"Results are compared for both the semi-implicit and fully implicit numerical time integrator.",
"The interface shape and initial conditions are depicted in fig:2Dinterface.",
"The interface shape is parameterized as a function of the $y$ position and is given by $x_\\Gamma =L_x\\left(0.04+w_h+w_a\\left(\\frac{2\\pi w_ny}{L_y}\\right)^{-1}\\sin \\left(\\frac{2\\pi w_ny}{L_y}\\right)\\right)$ where $w_h$ , $w_a$ , and $w_n$ are arbitrary constants here chosen as $\\frac{2}{3}$ , $\\frac{1}{6}$ , and 5, respectively.",
"The grid length in the $x$ and $y$ directions are represented by $L_x$ and $L_y$ , respectively and are equal $L_x=L_y=1e-3$ .",
"The initial conditions used in this problem are chosen arbitrarily, but have been inspired by those of the previous section.",
"The condition may be written as $T_G&=T_\\Gamma +(T_\\infty -T_\\Gamma )\\operatornamewithlimits{erf}\\left(5\\frac{x_\\Gamma -x}{L_x}\\right) \\\\Y_G&=Y_\\Gamma +(Y_\\infty -Y_\\Gamma )\\operatornamewithlimits{erf}\\left(5\\frac{x_\\Gamma -x}{L_x}\\right) \\\\T_L&=T_\\Gamma .$ The fluid properties are the same as the previous cases and are listed in tab:Neumann.",
"Figure: Initial field used in 2D curved interface case here shown on the 200 2 200^2 mesh.",
"The interface is shown as a gray line."
],
[
"Solution Convergence",
"Simulations are performed for grids of size $25\\times 25$ , $50\\times 50$ , $75\\times 75$ , and $100\\times 100$ , and the data are compared to a simulation with grid size $200\\times 200$ for accuracy.",
"Simulation data are taken at time $t=0.05s$ .",
"Both the MB and REA2 techniques have been used for this study, and both technique perform well for grid convergence.",
"Images of the convergence of the interface position are shown in fig:2DconvergeInt.",
"The profile seems to sharpen under grid refinement.",
"The center most hump appears taller, whereas the outer humps appear to decrease in height consistent with the principle of mass conservation.",
"The interface converges rapidly towards the fine mesh profile.",
"fig:2D depicts the convergence of the gas temperature, vapor mass fraction, liquid temperature, and liquid velocity in the domain.",
"A first order scaling is recovered for both $L_1$ and $L_\\infty $ errors in the gas fields.",
"For the liquid temperature field, first order convergence is seen in $L_1$ , but slower convergence is noted for $L_\\infty $ .",
"The liquid velocity field appears to converge at a very slow rate in $L_1$ , but diverges in $L_\\infty $ .",
"This is not entirely surprising.",
"From the previous section, we noted that the liquid velocity is directly proportional to $\\dot{m}$ and therefore directly proportional to the scalar gradient at the interface.",
"Since the scalar field is itself only first order accurate, its gradient would be expected to lose an order of accuracy, recovering only zeroth order accuracy.",
"This explains the lack of convergence in the liquid velocity field.",
"Since the velocity controls the liquid motion and the rate of heat advection into and out of the interface, it is reasonable that this would have an impact on the accuracy in the liquid temperature field as well.",
"However, since the gas is stationary, only the boundary effect is felt by the gas, decreasing the impact of the liquid velocity on its accuracy.",
"It is important to note the significance of the trends in fig:2D.",
"First, it can be seen that the effective accuracy of the simulation in two dimensions is somewhat lower than what was recovered in one dimension.",
"This is unsurprising because of the greater geometric complexity of the interface in two dimensions.",
"The resolution of this complex interface shape becomes a limiting factor in the resolution of the fluid flow.",
"Second, it can be seen that the error effect is particularly strong when measured in the $L_\\infty $ sense.",
"It is common within the literature to report errors in terms of smooth error norms such as $L_1$ .",
"This form of measurement can be misleading when numerical errors are not distributed uniformly, and that is the case for these simulations where errors are largest near the interface.",
"This calls into question the use of smooth error norms for numerical analysis of flows with sharp features or discontinuities.",
"While the current work fails to convergence in $L_\\infty $ , it does converge in the more traditional $L_1$ sense.",
"Consistent with the trends in the literature, the remainder of this paper will assume that the recovered $L_1$ convergence is sufficient to perform exploratory studies of vaporizing multiphase flows.",
"However, future work will focus on improving the accuracy of the flow solver, so that improved convergence will can be recovered.",
"Figure: Interface convergence using REA2.",
"The grids 25×2525\\times 25, 50×5050\\times 50, 75×7575\\times 75, 100×100100\\times 100, and 200×200200\\times 200 are represented using black, blue, green, red, and gray, respectively.Figure: Case 3.",
"Errors computed at t=0.1st=0.1\\text{ s}.",
"MB (), REA2 ().",
"A first order trend line has been added where appropriate."
],
[
"Temporal Stability and Convergence",
"The temporal convergence and stability of the solver is tested by performing simulations on the $100\\times 100$ grid using various time steps.",
"The time steps are chosen to yield $C_D=C_\\lambda \\in \\lbrace 10,25,50,100,200\\rbrace $ .",
"The two time integration schemes are compared using both the MB and REA2 techniques.",
"The data are summarized in tab:2Dsemi,tab:2Dfull.",
"Using the semi-implicit time integrator, similar trends are seen in errors for gas temperature and vapor mass fraction using both MB and REA2 techniques.",
"Both $L_1$ and $L_\\infty $ errors decrease with $C_\\lambda $ .",
"There is a noticeable difference in the error of the liquid temperature and velocity, as has been seen in the previous sections.",
"It is interesting to note that neither of these field appears to converge under temporal refinement (decreasing $C_\\lambda $ ) according to the $L_1$ metric.",
"Surprisingly, the liquid velocity converges in both cases under the $L_\\infty $ metric.",
"It should also be noted that for the REA2 technique, there is spike in solution error at $C_\\lambda =200$ .",
"It will later be demonstrated that this is due to an issue with solution monotonicity.",
"Using the fully implicit time integrator a few trends can be noted.",
"The MB technique seems to recover superior temporal convergence using the fully implicit solver than the semi-implicit solver.",
"Ignoring the $C_\\lambda =200$ case of REA2 technique, their errors are largely the same using both time integrators.",
"This is not a surprise.",
"While the semi-implicit solver has higher formal accuracy than the fully implicitly integrator, this property is no use when the leading error term in the flow is dominated by the first order treatment of the interfacial diffusion flux sec:flux.",
"Table: Errors using the semi-implicit time integrator.Table: Errors using the fully implicit time integrator."
],
[
"Monotonicity and Stability",
"The data suggested both MB and REA2 schemes recovered better stability and convergence properties using the fully implicit time integration scheme.",
"This follows from the fact that this scheme is a monotone scheme, and therefore disallows the development of numerical oscillation developing in the flow.",
"fig:wiggle demonstrates snapshots of the vapor mass fraction and $\\dot{m}$ fields at simulation end using the REA2 technique and $C_\\lambda =200$ .",
"Using the semi-implicit time integration a region of high vapor mass fraction appears near the interface leading to a non-monotonic variation of the field.",
"This is not seen in the fully implicit time integrator.",
"As a result, the $\\dot{m}$ fields look very different.",
"The fully implicit scheme demonstrates smooth variation of $\\dot{m}$ , whereas the semi-implicit scheme demonstrates chunking.",
"The effect can also be seen by monitoring the temporal behavior of $\\dot{m}$ in fig:monomdot.",
"While $\\dot{m}$ varies smoothly for the fully implicit scheme, it is oscillatory for the semi-implicit scheme.",
"This translates to rapid changes in the slope of the normalized volume (fig:monovol), and the eventual divergence of the solution.",
"Figure: Influence of time integration on field data using fully implicit and semi implicit solver.",
"Data taken at simulation end using C=200C=200.",
"The interface is shown as a gray line.Figure: Influence of time integration on temporal trace of data using fully implicit and semi implicit solver.",
"Semi implicit (), Fully implicit ()."
],
[
"Vaporization of an Isolated Spherical Droplet",
"The previous solutions, while interesting for verification, have little practical value.",
"A somewhat more physically relevant problem is the problem of vaporizing a spherical droplet in a quiescent flow.",
"Assuming the problem is quasi-steady (temporal derivatives of the scalars and momentum are negligible) and the gas obeys the continuity equation, $\\nabla \\cdot \\mathbf {u}=0$ , the solution to the temperature and mass fraction fields can be found analytically, and is given in any text on the subject [11].",
"The remarkable aspect of the solution is that the analytical result predicts that the squared diameter of the droplet ($d^2$ ) varies linearly in time.",
"This result is also observed experimentally for both vaporization and combustion [7].",
"We follow the derivation of Rueda-Villegas et al.",
"[1] who obtain the solution $\\frac{T-T_\\Gamma +L/C}{T_\\infty -T_\\Gamma +L/C}=&e^{-\\frac{QC_p}{2\\pi k}\\frac{1}{d}}, \\text{ and} \\\\\\frac{Y-1}{Y_\\infty -1}=&e^{-\\frac{Q}{2\\pi \\rho D}\\frac{1}{d}}.",
"$ In eq:D2T,eq:D2Y, $Q$ is a constant related to $\\dot{m}$ .",
"The values $\\dot{m}$ , $Y_\\Gamma $ , and $T_\\Gamma $ are related through $Q = -2\\pi d\\rho _GD\\ln \\left(\\frac{Y-1}{Y_\\infty -1}\\right), \\text{ and} \\\\T_\\Gamma =T_\\infty +\\frac{L}{C_p}\\left(1-{\\left(\\frac{Y_\\infty -1}{Y_\\Gamma -1}\\right)}^{D/\\lambda }\\right),$ which is derived form eq:D2T,eq:D2Y.",
"eq:D2mdot is more commonly written in the form $Q = 2\\pi d\\rho _GD\\ln \\left(1+B_M\\right)$ where $B_M=\\frac{Y_\\Gamma -Y_\\infty }{1-Y_\\Gamma }$ is the mass transfer number.",
"This relation can also be used to compute the vaporization rate per unit area, yielding $\\dot{m} \\equiv \\frac{Q}{\\pi d^2}= \\frac{2}{d}\\rho _GD\\ln \\left(1+B_M\\right)$ Note the explicit dependence of the solution on the current droplet diameter, $d$ .",
"eq:D2mdot,eq:D2link must be solved simultaneously with the Clausius-Clapeyron relation, eq:CC, to complete the system.",
"In this section, parameters are chosen to imitate the properties of acetone.",
"It was pointed out by Raessi et al.",
"[26], that for practical simulation of liquid vaporization using real physical properties, the condition $C_\\sigma <1$ often becomes the limiting CFL condition.",
"Since surface tension is not important to the mechanics of the $d^2$ law, the surface tension was reduced to zero, allowing for the use of a larger numerical time step.",
"Similarly, the use of the monotone, fully implicit scalar diffusion solver frees the simulation from time scales $C_D$ , $C_{\\lambda , L}$ , and $C_{\\lambda ,G}$ .",
"Only the advection CFL condition need be satisfied, here forced to be $C_u<1$ for all simulations.",
"Some discussion must be devoted to the implementation of the boundary conditions.",
"The simulation is run on a periodic, cubic domain using a Cartesian solver in three dimensions.",
"The spherical symmetry implied by the far field boundary condition is approximated by centering the domain on the droplet center of mass.",
"Tests are performed to characterize the influence of the boundary condition on the evaporation rate (see sec:far).",
"It was found that at sufficiently early times in the simulation, boundary effects were negligible.",
"A further modification is required to deal with the boundary conditions.",
"Experience has shown that small errors in the liquid velocity field can lead to motion of the droplet away from the center of the domain, breaking the implied symmetry of the boundary conditions.",
"To remove this source of error from this suite of tests, $-\\mathbf {n}_\\Gamma \\dot{m}$ is used as the velocity for VOF transport.",
"This effectively enforces the condition, $\\mathbf {u}_L=\\mathbf {0}$ .",
"However, the reconstructed $\\mathbf {u}_L$ is still used (rather than $\\mathbf {0}$ ) for the liquid temperature advection.",
"The chosen fluid properties are listed in tab:D2.",
"The actual fluid surface tension is listed, even though the value of $\\sigma =0$ is used in these simulations.",
"The far field temperature is chosen to be $700\\text{ K}$ and the far field vapor mass fraction is 0.",
"This results in interface temperature and vapor mass fraction of $294.92\\text{ K}$ and $0.43993$ , respectively.",
"The vapor mass fraction and gas temperature fields are initialized with the analytical solution, eq:D2T,eq:D2Y, and the liquid temperature field is initialized simply as $T_\\Gamma $ .",
"Note that in contrast to the simulations of previous sections, this section demonstrates a large range of values in both the temperature and vapor mass fraction fields.",
"The initialization is demonstrated in fig:D2t,fig:D2y.",
"Figure: Initial field used in d 2 d^2 law cases, here shown on the 128 3 128^3 mesh.",
"A planar cut has been taken at the center of the domain.",
"The interface is shown as a gray line.A grid convergence study was performed using a domain of size $8\\times 10^{-4}$ m which corresponds to exactly eight droplet diameters.",
"The convergence study was performed using $N=32$ , 64, 128, and 256 grid cells in each of the three grid directions.",
"This corresponds to resolutions of 4, 8, 16, and 32 cells per droplet diameter.",
"The study focuses on the prediction of the $d^2$ law vaporization time constant, because this parameter is one of the few values that is easily measured in both experimental and computational studies.",
"Following Sirignano [11] we write $\\frac{d^2}{d_0^2}=1-\\frac{t}{\\tau },$ where $d_0$ is the initial droplet diameter, and $\\tau $ is the vaporization time constant.",
"From eq:d2t it can be seen that $\\tau $ is the time necessary to fully vaporize a droplet.",
"This value can be computed as $\\tau = \\frac{\\rho _Ld_0^2}{8\\rho _GD\\ln (1+B)}.$ Using the parameters of this study, the time constant is computed to be $\\tau =0.029126\\text{ s}$ [11].",
"Simulations are run until time $t=1.5\\times 10^{-3}\\text{ s}$ when the ratio $d^2/d_0^2$ is computed, and eq:d2t is used to extract a time constant.",
"This choice of simulation end time corresponds to $t/\\tau \\approx 0.01$ , which is before boundary effects are important.",
"fig:D2trends shows the temporal variation of the effective droplet square diameter, computed as $\\frac{d^2}{d^2_0}={\\left(\\frac{V}{V_0}\\right)}^{2/3},$ where $V_0$ and $V$ are the initial and current droplet volumes, respectively.",
"It can be seen that a linear trend in $d^2$ is recovered for all cases, however, the slope of the line depends on the mesh resolution.",
"As the grid is refined, $\\tau $ appears to approach its expected value as shown in tab:D2data, however, the rate appears to be less than first order accurate.",
"This is not entirely surprising.",
"As pointed out in sec:2D, the evaporation rate is related to the derivative of the scalar fields, and therefore is expected to be less accurate than the scalar field which is only first order.",
"Table: Fluid properties for d 2 d^2 law problem.Figure: Time constants from d 2 d^2 law convergence study.",
"The theoretical value is τ=0.029126\\tau =0.029126 s."
],
[
"Vaporizing Droplet in Uniform Flow",
"A final test is performed of a vaporizing droplet in uniform flow.",
"There is no known analytical solution to this problem.",
"A droplet is placed at the center of a gaseous domain with inflow velocity $U=40$ m/s.",
"Unlike the simulation of the previous section, the assumption $\\mathbf {u}_L=0$ is not imposed, so the droplet is able to move and deform freely in space.",
"The same fluid properties are used as the $d^2$ law case, tab:D2.",
"The solution is initialized with a $d^2$ law initialization.",
"The parameters chosen lead to a flow Reynolds number $Re=\\rho _GUd/\\mu _G=400$ and gas Schmidt and Prandtl numbers as $Sc=\\mu _G/\\left(\\rho _GD\\right)=0.192\\text{ and }Pr=\\mu _G/\\left(\\rho _G\\lambda _G\\right)=0.192$ , respectively.",
"The surface tension is not assumed to be zero, and is given as $\\sigma = 0.0237$ N/m.",
"This leads to a Weber number $We=\\frac{\\rho _G U^2d}{\\sigma }\\approx 7$ .",
"The simulation used 128 grid cells in each direction, which was chosen to resolve the gas film thickness, $\\delta $ .",
"Following the paper by Sirignano and Abramzon [27], the following correlation for gas film thickness can be derived, $\\delta = \\frac{d}{0.552{Re}^{1/2}{\\max \\left(Sc,Pr\\right)}^{1/3}}.",
"$ For the parameters in this study, eq:thickness yields $\\delta \\approx 0.157 d$ or $\\delta /\\Delta \\approx 2.51$ .",
"eq:thickness is noted to underpredict the film thickness [27], so the use of eq:thickness can be considered a upper bound on the required resolution.",
"A series of visualizations is shown in fig:crossseries.",
"fig:cross1 demonstrates the initial condition.",
"As the flow progresses, aerodynamic forces cause the droplet to flatten and move slightly in the domain (fig:cross2).",
"fig:cross3 shows the wake becoming asymmetrical and the formation of a large vortex under the droplet.",
"After some time, the droplet wake becomes chaotic due to the interaction of several shed vortices, fig:cross4.",
"The temporal variation in vaporization rate is shown in fig:crossmdot.",
"The vaporization rate is greatly enhanced by the flow when compared to the $Re=0$ limit (the $d^2$ law).",
"The vaporization rate appears to increase in time until the attached vortices begin to shed, indicating strong coupling between vortex formation and vaporization.",
"Figure: Volume render of vapor mass fraction at representative times.",
"The rendering is truncated at the midplane in order to visualize the interface (shown in gray).Figure: m ˙\\dot{m} in time.",
"Re=400Re=400 (), Re=0Re=0 ()."
],
[
"Conclusion",
"A numerical solver capable of simulating droplet vaporization in a complex flow field is demonstrated.",
"A discretely conservative volume of fluid scheme is coupled with appropriate sources based on local thermodynamic equilibrium of the vapor mass fraction and temperature fields.",
"The evaporation sources are coupled to the scalar transport equation using an unconditionally stable, monotone scheme.",
"The convergence and stability properties of the solver are explored, and it shown that first order convergence is seen for gas temperature and vapor mass fraction, consistent with the schemes used.",
"Also consistent with the scheme accuracy is the fact that the liquid velocity seems to converge poorly.",
"Future work will require the use of higher order schemes for the scalar transport, to avoid this issue.",
"Finally, it is shown that when used with a fully implicit time integrator the solver is capable of running stably at high time steps (for example $C_\\lambda =200$ ) with non-oscillatory solutions.",
"This is an improvement over the common Crank-Nicolson time integrator, and the improvement comes without any sacrifice of accuracy.",
"As true test of the robustness of the numerical solver, three dimensional simulations at high Reynolds number are performed.",
"A vaporizing droplet in uniform flow at $Re=400$ is demonstrated.",
"Because of the high speed flow, there is significant deformation of the interface, so no analytical solution is known for this problem.",
"It is noted that vaporization is enhanced by the flow, which is consistent with the literature on non-deforming vaporizing droplets [27]."
],
[
"Acknowledgments",
"This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No.",
"DGE-1650441 and by the Alfred P. Sloan Foundation.",
"This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1548562.",
"The XSEDE resource used was Stampede2 at the Texas Advanced Computing Center (TACC) at The University of Texas at Austin.",
"The authors also acknowledge Advanced Research Computing at Virginia Tech for providing computational resources and technical support that have contributed to the results reported within this paper."
],
[
"Monotone Schemes",
"Harten [28] in the development of total variation diminishing (TVD) schemes defines three types of schemes.",
"Monotonicity preserving schemes, TVD schemes, and monotone schemes.",
"Monotonicity preserving schemes are those that do not introduce new extrema to the solution, nor do they change the value of the current extrema.",
"TVD schemes are defined in terms of a mathematical property called the total variation, and they are shown to be a class of monotonicity preserving schemes.",
"The final category, monotone schemes, are a robust class of TVD schemes, and therefore also preserve monotonicity.",
"Following Harten, define an update operator such that $\\mathbf {\\phi }^{n+1}=\\mathbf {\\mathcal {L}}\\mathbf {\\phi }^{n},$ where the vector $\\mathbf {\\phi }=[\\phi _1,\\phi _2,\\ldots ]^\\top $ .",
"A monotone scheme is one such that the coefficients of $\\mathbf {\\mathcal {L}}$ are all positive.",
"Comparing eq:monotonediscrete to eq:TVD, the chosen time-space integration scheme is monotone when the matrix, $\\mathbf {\\mathcal {A}}=\\left\\lbrace \\begin{array}{lc}A_{i,j}=1+2C & j=i \\\\A_{i,j}=-C & j=i-1 \\text{ or } j=i+1\\\\A_{i,j}=0 & \\text{otherwise,}\\end{array}\\right.$ is invertible, and the coefficients of ${\\mathbf {\\mathcal {A}}}^{-1}$ are all positive.",
"This property follows directly from the fact that $\\mathbf {\\mathcal {A}}$ is a nonsignular M matrix[29].",
"That is, $\\mathbf {\\mathcal {A}}$ is an M matrix, and it is strictly diagonally dominant with positive diagonal."
],
[
"Farfield Boundary",
"The $d^2$ law is the vaporization of an isolated droplet in a quiescent field, which calls for an infinite domain.",
"In this work, the domain is approximated using a triply periodic cube, with a fixed droplet in the center.",
"The domain is made to be much larger than the droplet so that the effects of the boundary will be minimal.",
"Still, as the finite sized domain fills with vapor, the effects of the boundary condition will be felt.",
"A study was performed to look at the impact of domain size (measured in droplet diameters) on the evaporation rate.",
"The droplet diameter, fluid properties, and initial conditions were made to be consistent with those in sec:3D.",
"For this study, the grid resolution is fixed so that the number of grid cells per droplet diameter is always 8.",
"The domain sizes were chosen to be integral multiples of the droplet diameter, viz., 4, 8, and 16 diameters.",
"The results of this study can be seen in fig:boundary.",
"At the beginning of the simulation, all domains give the same trend in $d^2$ .",
"As the simulation progresses, boundary effects become important.",
"This manifests itself in a gradual plateau of the droplet square diameter as a function of time.",
"Simulations with a greater ratio of domain size to droplet diameter are able to continue longer without this plateau.",
"As a reference, a straight line is shown by extrapolating the slope from the initial condition to the first time step.",
"This reference was chosen instead of the analytical result, $1/\\tau $ , since the effective vaporization rate, and therefore the slope, changes under mesh refinement.",
"For the given parameters, a domain eight times the droplet diameter leaves an error of only about $1.3\\%$ in the slope at $t=0.01\\tau $ , which is the standard used in sec:3D.",
"Figure: d 2 d^2 scaling as a function of domain size.",
"Domain size varies asfour diameter (), eight diameters (), and sixteen diameters ().",
"An extrapolated linear trend is shown as ()."
]
] | 1906.04565 |
[
[
"Alzheimer's Disease Brain MRI Classification: Challenges and Insights"
],
[
"Abstract In recent years, many papers have reported state-of-the-art performance on Alzheimer's Disease classification with MRI scans from the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset using convolutional neural networks.",
"However, we discover that when we split that data into training and testing sets at the subject level, we are not able to obtain similar performance, bringing the validity of many of the previous studies into question.",
"Furthermore, we point out that previous works use different subsets of the ADNI data, making comparison across similar works tricky.",
"In this study, we present the results of three splitting methods, discuss the motivations behind their validity, and report our results using all of the available subjects."
],
[
"Introduction",
"Alzheimer's Disease is a progressive neurodegenerative disease characterized by cognitive decline and memory loss.",
"It is one of the leading causes of death in the aging population, affecting millions of people around the world.",
"Automatic diagnosis and early detection can help identify high risk cohorts for better treatment planning and improved quality of life [2].",
"Initial works on brain MRI classification rely on domain-specific medical knowledge to compute volumetric segmentation of the brain in order to analyze key components that are strongly correlated with the disease [8], [16], [4].",
"With recent advances in deep learning and an increasing number of MRI scans being made public by initiatives such as Alzheimer's Disease Neuroimaging Initiative (adni.loni.usc.edu) and Open Access Series of Imaging Studies (www.oasis-brains.org), various convolutional neural network (CNN) architectures, including 2D and 3D variants of ResNet, DenseNet, InceptionNet, and VGG models, have been proposed to holistically learn feature representations of the brain [12], [13], [14], [19].",
"A patient's diagnosis in the dataset is typically categorized as Alzheimer's disease (AD); mild cognitive impairment (MCI); or cognitively normal (CN).",
"While many previous related works report high accuracy in three-class classification of brain MRI scans from the ADNI dataset, we point out a major issue that puts into question the validity of their results.",
"A number of papers [15], [6], [20], [12] perform classification with training and testing splits done randomly across the brain MRIs, in which the implicit assumption is that the MRIs are independently and identically distributed.",
"As a result, a given patient's scans from different visits could exist across both the training and testing sets.",
"We point out that transitions in disease stage occur at a very low frequency in the ADNI dataset, which means that a model can overfit to the brain structure corresponding to an individual patient and not learn the key differences in feature representation among the different disease stages.",
"The model can simply regurgitate patient-level information learned during training and rely on that to obtain good classification results on the test set.",
"We refer to generating the splits by sampling at random from the pool of images, ignoring the patient IDs, as splitting by MRI randomly.",
"Our main contribution in this study is exploring alternative ways to split the data into training and testing sets such that the evaluation gives valid insights into the performance of the proposed methods.",
"First, we look at splitting the data by patient, where all of the available visits of a patient are assigned to either the training or testing set, with no patient data split across the two sets.",
"This method is motivated by the use case for automatic diagnosis from MRI scans, where the assumption is that no prior or subsequent patient information is given.",
"Then, we look at splitting the data by visit history, meaning that the first $n-1$ visits of a patient's data are used for training, and the $n$th visit is used for testing.",
"This method is motivated by the use case wherein we incorporate MRI scans into personalized trajectory prediction.",
"Finally, we compare the accuracy of training and testing on sets split randomly by MRI, with the accuracies we obtain using the two aforementioned ways of splitting the training and testing set.",
"We run experiments with both popular 2D models pre-trained on ImageNet, as well as a 22-layer ResNet-based 3D model we built.",
"Unlike many other studies that perform experiments on a subset of the ADNI dataset, our experiments include all of the currently available data from ADNI.",
"Our results suggest that there is still much room for improvement in classifying MRI scans for Alzheimer's Disease detection."
],
[
"Related Work",
"Before the current wave of interest in deep learning, the field of neuroimaging relied on domain knowledge to compute some predefined sets of handcrafted metrics on brain MRI scans that reveal details on Alzheimer's disease progression, such as the amount of tissue shrinkage that has occurred.",
"A common method is to run MRI scans through brain segmentation pipelines, such as FreeSurfer [7], to compute values measuring the anatomical structure of the brain, and enter those values into a statistical or machine learning model to classify the state of the brain [8], [16], [4].",
"With the advent of big data and deep learning, convolutional neural network models have become increasingly popular for the task of image classification.",
"While a variety of architectures and training methods have been proposed, the advantages and disadvantages of these design choices are not clear.",
"[10] used the pre-trained 2D VGG model [17] and Inception V4 model [18] to train on a set of 6,400 slices from the axial view of 200 patients' MRI scans from the OASIS dataset to perform two-class (Normal vs. Alzheimer's) classification, where 32 slices are extracted from each patient's MRI scans based on the entropy of the image.",
"[12] followed a similar approach using a 2D VGG model [17] on the ADNI dataset to train on a set of 4,800 brain images augmented from 150 subjects' scans.",
"For each subject's scan, they selected 32 slices based on the entropy of the slices to compose the dataset.",
"They then proceed to shuffle the data and split it with a 4:1 training to testing ratio to perform the more complicated task of three-class classification for Normal, Mild Cognitive Impairment, and Alzheimer.",
"Other methods used 3D models, which are more computationally expensive but have more learning capacity for the three-dimensional MRI data.",
"[15] pretrained a sparse autoencoder on randomly generated $5\\times 5\\times 5$ patches and used the weights as part of a 3D CNN to perform classification on the ADNI dataset, splitting randomly by MRI.",
"[11] employed a similar approach of using sparse autoencoder to pretrain on scans, but instead of randomly selecting patches from the training dataset, they used scans from the CADDementia dataset [5], and performed ten-fold cross-validation on the ADNI dataset.",
"[13] extracted regions of interest around the lobes of the hippocampus using atlas Automated Anatomical Labeling (AAL), augmented their data by applying random shifts of their data by up to two pixels and random Gaussian blur with $\\sigma $ up to 1.2, and classified using a Siamese network on each of the regions of interest.",
"More recently, [19] trained a 3D DensetNet with ensemble voting on the ADNI dataset to achieve the state-of-the-art accuracy on three-class classification.",
"They split by patients but treated MRIs of the same patient that are over three years apart as different patients, giving away some information from the training to testing process.",
"Table REF summarizes information on the data and evaluation approaches used by a few recent papers.",
"Unlike most of the aforementioned papers that report high performance but do not explicitly mention their training and testing data split methodology, [3] pointed out the problem of potentially giving away information from the training set into the testing set when splitting randomly by MRI scans, and showed that splitting MRI data by patient led to worse model performance.",
"However, they only report two-class classification (Normal vs. Alzheimer's) in their results.",
"They also only used a subset that is less than half the size of the MRIs available in ADNI, even though they reported empirical studies showing that the same model performing well on a small dataset can experience a significantly reduced performance on a large dataset.",
"Our work differs in that we use as much of the data as available from ADNI and provide results in three-class classification (CN vs MCI vs AD).",
"In addition, we introduce a method of splitting the data by visit history motivated by real-world use case that is new to the application in Alzheimer's disease brain MRI classification, and we compare the effects of the different splitting approaches in model performance.",
"Table: Performance and methodology of some of the state-of-the-art studies on three-class Alzheimer's Disease classification.",
"The different splitting schemes and subsets of the ADNI dataset used in evaluation make it hard to interpret the results meaningfully."
],
[
"Methodology and Results",
"In this section, we will briefly discuss the dataset we worked with, the preprocessing steps we took to produce our results, the model architectures we chose and training hyperparameters we used, as well as the results we obtained from the various methods of splitting the dataset."
],
[
"Data Acquisition and Preprocessing",
"We use MRI data from ADNI because it is the largest publicly available dataset on Alzheimer's Disease, consisting of patients who experience mental decline getting their brain scanned and cognition assessed in longitudinal visits over the span of several years.",
"We collected all of the scans that underwent Multiplanar Reconstruction (MPR), Gradwarp, and B1 Correction pre-processing from the dataset, discarding the replicated scans of the brain for the same patient during the same visit.",
"We performed statistical parametric mapping (SPM) [1] using the T1–volume pipeline in the Clinica software platform (www.clinica.run), developed by the ARAMIS lab (www.aramislab.fr), as an additional pre-processing step on the images to spatially normalize them.",
"In total, 2,731 MRIs from 657 patients are selected for our experiments.",
"For the split by patient history, we used 2,074 scans for training and validation, and 657 scans for testing, where each scan in the testing set is from the last visit of the patient.",
"For splitting by MRI, we used the same number of scans for all sets as the previous split by patient history.",
"And finally, for splitting at the patient level, we had 2,074 scans from 484 patients in the training set and validation set, and 657 scans from 173 patients in the testing set.",
"For consistency, we used the same number of scans in all subsets for all three splits.",
"Our code repository is publicly available https://github.com/Information-Fusion-Lab-Umass/alzheimers-cnn-study, and it includes the patients ids and visits that we evaluated."
],
[
"Model Architecture and Training",
"We compared 2D and 3D CNN architecture performances.",
"For our 2D CNN architecture, we used ResNet18 [9] that was pretrained on ImageNet, allowing the model to learn how to better extract low-level features from images.",
"For our 3D CNN architecture, we followed a residual network-based architectural design as well.",
"We used the “bottleneck\" configuration, where the inner convolutional layer of each residual block contains half the number of filters, and we used the “full pre-activation\" layout for the residual blocks.",
"See Figure REF for more details.",
"For the 2D networks, we used a learning rate of 0.0001 and L2 regularization constant of 0.01.",
"For the 3D network, we used a learning rate of 0.001 and regularization constant of 0.0001.",
"Both networks were trained for 36 epochs, with early stopping.",
"Figure: Comparison of the spatially normalized MRI scans of 4 subjects in each of the CN, MCI, and AD categories.",
"To the human eye, distinguishing the difference across the disease stages is a difficult task.Figure: The architectures of our 3D CNN model and residual blocks.",
"With the exception of the first and last convolution layers, which have a stride of 1, all other layers have a stride of 2 for down-sampling.",
"The first convolution layers takes in a 1-channel image and outputs a 32-channel output."
],
[
"Splitting Methodology and Results",
"Our main goal is to investigate how differently the model performs under the following three scenarios: (1) random training and testing split across the brain MRIs, (2) training and testing split by patient ID, and (3) training and testing split based on visit history across the patients.",
"Table: Our classification result of CNN models by different train/test splitting scheme, averaged over five runs.In Table REF , we present the three-way classification result of the models described in Section REF .",
"In summary, when the training and testing sets were split by MRI scans randomly, the 2D and 3D models attained accuracy close to 84%.",
"When the training and test sets were split by visits, in which only the latest visit of each patient was set aside for testing, accuracy was slightly lower.",
"However, accuracy dropped to around 50% when the training and test set were split by patient.",
"We note that the other state-of-the-art 2D CNN architectures we tried (DenseNet, InceptionNet, VGGNet) performed similarly to ResNet, and the choice of view in the 2D slice (coronal, axial, sagittal) did not lead to significant differences in testing accuracy as long as the slices were chosen to be close to the center of the brain.",
"We report results on the 88th slice of the coronal view for our 2D model.",
"Our 3D model performs slightly better but is still limited in performance, suffering the same problem of over-fitting to information on individual patients instead of learning what generally differentiates a brain in the different stages of Alzheimer's Disease.",
"Table: Confusion matrix of the 3D CNN experiment on the test set, with data split by patient."
],
[
"Analysis",
"Analysis of the dataset shows that the frequency of disease stage transition for patients between any two consecutive visits is low in the ADNI dataset.",
"There are only 152 transitions in the entire dataset, which contains 2,731 scans from 657 patients.",
"We believe that due to the relatively few transition points in the dataset, the models are still able to achieve accuracy in the 80% range for the splitting by visit experiments by repeating the diagnostic label of the previous visits.",
"Out of the 152 total transitions we found across the whole dataset, only 52 of them happened for patients between the $n-1$th visit and $n$th visit.",
"This suggests that the model is able to encode the structure of a patient's brain from the training set, in turn aiding its performance on the testing set.",
"Furthermore, we set up additional experiments where we trained on visits $t_{1} ... t_{n-1}$ from all patients, but only tested on the 52 patients that had a transition from the $n-1$th to $n$th visit.",
"The classification accuracy on this experiment dropped to around 54%, which suggests that the network was repeating information about the patient's brain structure instead of learning to be discriminative among the different stages of Alzheimer's Disease exhibited by a particular MRI scan."
],
[
"Technical Challenges",
"We summarize the main challenges in working with the ADNI dataset as follows: 1.",
"Lack of transitions in a patient's health status between consecutive visits.",
"There are only 152 transitions total out of the entire dataset of 2,731 images collected from patient visits.",
"It is easy for the model to overfit and memorize the state of a patient at each visit instead of generalizing the key distinctions between the different stages of Alzheimer's Disease.",
"2.",
"Coarse-grained data labels.",
"The data labels are coarse-grained in nature so our classifier may become confused when trying to learn on cases when a patient's cognitive state may be borderline, such as being between MCI and AD.",
"The confusion matrix in Table REF demonstrates this.",
"3.",
"Difficulty in distinguishing the visual difference of a brain in the different Alzheimer's stages.",
"Human brains are distinct by nature, and the quality of MRI collections from different clinical settings add to the noise level of the data.",
"In Figure REF , we plotted out the brains of subjects in CN/MCI/AD, and show that the difference in anatomical structure from CN to MCI to AD is very subtle.",
"4.",
"No clear baseline.",
"Many studies evaluated the performance of their models on different subsets of the ADNI dataset, making fair comparison a tricky task.",
"In addition, studies that use a separate testing set do not report the subjects or scans that they used in their testing set, further complicating the comparison process.",
"We hope to keep these challenges in mind when designing future experiments and ultimately design models that can reliably classify brain MRIs with its true stage in Alzheimer's Disease progression, which is robust to visit number, lack of patient transitions, and minor fluctuations in scan quality."
],
[
"Insights and Future Work",
"Many studies in Alzheimer's disease brain MRI classification do not take into account how the data should be properly split, putting into question the ability of the proposed models to generalize on unseen data.",
"We fill this gap by providing detailed analysis of model performance across splitting schemes.",
"Additionally, to our knowledge, none of the previous studies use all of the MRIs available in the ADNI dataset and do not present a clear explanation for this decision.",
"To address the issue, we perform our experiments on all available data while also reporting the subjects used in the training and test split of all our experiments for reproducibility.",
"In the future, we would like to explore utilizing the covariate data collected from patients to aid image feature extraction.",
"Most of the studies we have come across do not use any covariate information collected from patients.",
"The covariates, such as patient demographics and cognitive test scores, may be helpful for the classification task since they correlate with the disease stage of the patient.",
"A scenario could be a multitask learning setup, where the model predicts the Mini-Mental State Examination (MMSE) and Alzheimer's Disease Assessment Scale (ADAS) cognitive scores in addition to the labels.",
"We think this may be helpful in training the model because the cognitive test scores can provide finer-grained signal for the model, making the prediction more robust."
]
] | 1906.04231 |
[
[
"Shape versus timing: linear responses of a limit cycle with hard\n boundaries under instantaneous and static perturbation"
],
[
"Abstract When dynamical systems that produce rhythmic behaviors operate within hard limits, they may exhibit limit cycles with sliding components, that is, closed isolated periodic orbits that make and break contact with a constraint surface.",
"Examples include heel-ground interaction in locomotion, firing rate rectification in neural networks and stick-slip oscillators.",
"In many rhythmic systems, robustness against external perturbations involves response of both the shape and the timing of the limit cycle trajectory.",
"The existing methods of infinitesimal phase response curve (iPRC) and variational analysis are well established for quantifying changes in timing and shape for smooth systems.",
"These tools have recently been extended to nonsmooth dynamics with transversal crossing boundaries.",
"In this work, we further extend the iPRC method to nonsmooth systems with sliding components, which enables us to make predictions about the synchronization properties of weakly coupled stick-slip oscillators.",
"We observe a new feature of the isochrons in a planar limit cycle with hard sliding boundaries.",
"Moreover, the classical variational analysis neglects timing information and is restricted to instantaneous perturbations.",
"By defining the \"infinitesimal shape response curve\" (iSRC), we incorporate timing sensitivity of an oscillator to describe the shape response of this oscillator to parametric perturbations.",
"In order to extract timing information, we also develop a \"local timing response curve\" (lTRC) that measures the timing sensitivity of a limit cycle within any given region.",
"We demonstrate in a specific example that taking into account local timing sensitivity in a nonsmooth system greatly improves the accuracy of the iSRC over global timing analysis given by the iPRC."
],
[
"Introduction",
"A limit cycle with sliding component (LCSC) is a closed, isolated, periodic orbit of an $n$ -dimensional, autonomous, deterministic nonsmooth dynamical system, in which the trajectory is constrained to move along a surface of dimension $k<n$ during some portion of the orbit.",
"The motion of a trajectory sliding along a constraint surface is called a sliding mode.",
"LCSCs appear naturally in dynamical systems models of physiological and robotic motor control systems [2], [22], [43], [29], [48], [39], [25] as well as mechanical stick-slip systems [20], [21], [11], [38].",
"In control theory, the sliding mode concept has been used to design controllers for nonlinear systems [54], [55], [37].",
"Both natural and engineered motor systems are robust to certain short and long term disturbances.",
"Studies of the robustness properties of these systems have relied on applying the infinitesimal phase response curve (iPRC) and variational analysis to quantify changes in timing and shape of the motor trajectory to weak perturbations.",
"For understanding the response to instantaneous perturbations, the two methods are ubiquitously used in the literature of smooth dynamical systems [57], [58], [45].",
"[19], [3], [38], [11] show that variational analysis can be applied to nonsmooth systems including LCSC systems.",
"Recently, the iPRC has also been generalized to nonsmooth systems, provided the flow is always transverse to any switching surfaces at which nonsmooth transitions occur [52], [46], [8], [62].",
"However, there have been fewer reported studies that analyze the model response (especially the shape response) to perturbations that are sustained over long times.",
"Even fewer works have analyzed the response of LCSCs, in which the transverse flow condition fails, to both instantaneous and sustained perturbations.",
"Our goal in this paper is to bridge such a knowledge gap by providing a first description of the infinitesimal shape response curve (iSRC) that can account for the shape response of an oscillator to sustained (e.g., parametric) perturbations and extending both iPRC and iSRC to LCSCs in nonsmooth systems.",
"In this paper we consider the case of continuous LCSC solutions to nonsmooth systems with degree of smoothness one or higher; that is, systems with continuous trajectories, also known as Filippov systems [19], [3].",
"The simplest model of such a system can be written as follows $ \\frac{d\\textbf {x}}{dt}=F(\\mathbf {x}):=\\left\\lbrace \\begin{array}{cccccccccc}F^{\\rm slide}(\\mathbf {x}), &\\quad \\quad \\mathbf {x}\\in {\\mathcal {R}}^{\\rm slide}\\subset \\Sigma \\\\F^{\\rm interior}(\\mathbf {x}), & \\text{otherwise}\\\\\\end{array}\\right.$ where $\\mathbf {x}$ denotes the state variable.",
"The trajectory is confined to travel within the closure of the domain ${\\mathcal {R}}$ , the boundary of which is defined to be the hard boundary $\\Sigma $ .",
"A trajectory entering the sliding region ${\\mathcal {R}}^{\\rm slide}\\subset \\Sigma $ will slide along it with the vector field $F^{\\rm slide}$ until it is allowed to reenter the interior.",
"For points not in ${\\mathcal {R}}^{\\rm slide}$ , the dynamics is determined by $F^{\\rm interior}$ .",
"A stick-slip oscillator is one example of a system that exhibits a LCSC [20].",
"A mass on a belt that moves at a constant velocity $u$ is connected to a fixed support by a linear elastic spring and by a linear dashpot.",
"Mechanical systems of this type are referred to as stick-slip since there are times when the mass and the belt are moving together (stick phase) and others in which the mass slips relative to the belt (slip phase).",
"Such a solution trajectory alternating between stick and slip phases is a LCSC, as illustrated in Figure REF , left.",
"Here the hard boundary is given by $v=u$ , where $v$ is the velocity of the mass and $u$ is the driving velocity of the belt.",
"Thus, for a stick-slip system, the “sliding component\" of the limit cycle corresponds to the “stick\" phase, during which the mass moves with fewer degrees of freedom than during the “slip\" phase.",
"Figure: Periodic LCSC solution trajectories of a stick-slip system.",
"Left: projection of the unperturbed trajectory γ(t)\\gamma (t) onto the (x,v)(x,v) phase space (black), where xx is the displacement of the mass from the position in which the spring assumes its natural length and vv is the velocity of the mass.",
"The trajectory enters and leaves the hard boundary (blue line) at 𝐱 land \\mathbf {x}_{\\rm land} (blue square) and 𝐱 liftoff \\mathbf {x}_{\\rm liftoff} (blue circle), respectively.",
"Right: Projections of the unperturbed trajectory γ(t)\\gamma (t)(black) and the trajectory under parametric perturbation γ ε (t)\\gamma _{\\varepsilon }(t)(red).",
"See § for further details.While the response of an oscillator to an instantaneous perturbation is well understood, relatively little consideration has been given to studying the model response to sustained (e.g., parametric) perturbations even in the context of smooth systems.",
"Here, we develop the mathematical framework required to analyze the changes in shape and timing wrought by parametric changes.",
"In general, a small fixed change in a parameter gives rise to a new limit cycle, with different shape and timing than the original.",
"See Figure REF , right.",
"In order to account for the shape response of an oscillator to parametric perturbations, we apply Lighthill's method [33] to derive an iSRC in §REF .",
"Later in §REF , we use the iSRC to study the shape response of the stick-slip oscillator to a small parametric perturbation.",
"In contrast to standard variational analysis, which neglects timing changes, the iSRC takes into account both timing and shape changes arising due to a parametric perturbation.",
"In many applications of LCSCs, the impact of a perturbation on local timing can be as important as the global effects.",
"For instance, any motor control or mechanical system that operates by making and breaking physical contact (e.g., walking, scratching, grasping or stick-slip) would only experience perturbations limited to a discrete component of the limit cycle (e.g., the friction of the ground acts as a perturbation during the stance phase of locomotion and is absent during the swing phase).",
"In these cases, one would need to compute the local timing changes of the trajectory during the phase when the perturbation exists to understand the robustness of this system.",
"It is well known that the global change in timing of an oscillator due to a parametric perturbation can be captured using the iPRC, which, however, cannot be used to capture a local timing change induced by sustained perturbations in many LCSC systems.",
"In this paper, we develop a local timing response curve (lTRC) that is analogous to the iPRC but measures the local timing sensitivity of a limit cycle within any given local region (e.g., the stance phase in locomotion).",
"Development of the lTRC allows us to compute local timing changes in an oscillator due to nonuniform sustained perturbations.",
"Moreover, we show that the lTRC can be used to greatly improve the accuracy of the iSRC of a nonsmooth system.",
"Recently the iPRC has been extended to certain nonsmooth systems, but the theory does not extend directly to LCSCs in which the transverse flow condition fails.",
"In this paper, we bridge such a knowledge gap by extending the iPRC to LCSC systems (see Theorem REF ).",
"In contrast to the variational dynamics that exhibits discontinuities when a sliding motion begins [19], we find that the iPRC in a LCSC experiences a discontinuous jump when the trajectory leaves a sliding region and is continuous when the sliding mode begins.",
"To our knowledge, ours is the first work considering iPRC for systems with sliding components.",
"For example, although the well-known monograph Hybrid Dynamical Systems [23] addresses periodic orbits and synchronization in nonsmooth systems with transverse boundary crossings, it avoids systems with sliding components, which have non-transverse flow out of the constraint surface, nor does it discuss phase response curve methods.",
"We illustrate the theory using a planar model consisting of four sliding components and a stick-slip system with one sliding component.",
"We also use the iPRC to predict the synchronization properties of two weakly coupled stick-slip oscillators.",
"The rest of the paper is organized as follows.",
"We consider smooth systems in § and Filippov systems in §.",
"In each of the two sections, we first review classical theory and methods to provide context, and then present our new results.",
"The variational and phase response curve analysis for the responses of smooth dynamical systems to instantaneous and parametric perturbations are reviewed in §REF .",
"To account for shape responses to sustained perturbations, we define the iSRC in §REF and define the lTRC in §REF .",
"We present the classical two-zone Filippov system with transversal crossing boundary and define a Filippov system that produces a LCSC solution (see (REF )) in §REF .",
"While the applicability of the classical perturbative methods from § is generally limited by the constraint that the dynamics of the system is smooth, some elements of the methods have already been generalized to nonsmooth systems, which are reviewed in §REF .",
"We extend the iPRC to the LCSC case in nonsmooth systems in §REF .",
"The main result is summarized in Theorem REF .",
"Appendix gives a proof of the theorem.",
"Numerical algorithms for implementing all the methods are presented in Appendix .",
"In §, we illustrate both the theory and algorithms using a planar model, comprising a limit cycle with a linear vector field in the interior of a simply connected convex domain with four hard boundaries.",
"In this example, we show that under certain circumstances (e.g.",
"non-uniform perturbation), the iSRC together with the lTRC provides a more accurate representation of the combined timing and shape responses to static perturbations than using the global iPRC alone.",
"Surprisingly, we discover nondifferentiable “kinks\" in the isochron function that propagate backwards in time along an osculating trajectory that encounters the hard boundary exactly at the liftoff point (the point where the limit cycle trajectory smoothly departs the boundary).",
"In §, we use the iSRC to understand the shape response of an actual mechanical system - a stick-slip oscillator - to a parametric perturbation and use the iPRC to study the synchronization of two weakly coupled stick-slip oscillators.",
"Lastly, we discuss limitations of our methods and possible future directions in §.",
"Appendix provides a table of symbols used in the paper."
],
[
"Linear responses of smooth systems",
"In this section we consider smooth dynamical systems.",
"We begin by reviewing the classical variational theory for limit cycles, and then derive new methods including the infinitesimal shape response curve (iSRC) and the local timing response curve (lTRC) for linear approximation of the effects of small perturbations on the timing and shape of a limit cycle trajectory in the smooth case.",
"Specifically, in §REF we review the classical variational and infinitesimal phase response curve analyses that capture the shape response to small instantaneous perturbations and the effects of both instantaneous and sustained perturbations on the timing of trajectories near a limit cycle (LC) trajectory.",
"However, little consideration has been given to the effect of sustained perturbations on the shape of the orbit.",
"In §REF , we derive the iSRC to account for the combined shape and timing response of a LC trajectory under sustained (e.g., parametric) perturbations.",
"To obtain a more accurate iSRC when the limit cycle experiences different timing sensitivities in different regions within the domain, in §REF , we introduce the lTRC.",
"In contrast to the iPRC, which measures the global shift in the period, the lTRC lets us compute the timing change of a LC trajectory within regions bounded between specified Poincaré sections.",
"Consider a one-parameter family of $n$ -dimensional dynamical systems $\\frac{d\\mathbf {x}}{dt}=F_\\varepsilon (\\mathbf {x}),$ indexed by a parameter $\\varepsilon $ representing a static perturbation of a reference system $\\frac{d\\textbf {x}}{dt}=F_0(\\textbf {x}).$ Assumption 2.1 Throughout this section, we make the following assumptions: The vector field $F_{\\varepsilon }(\\mathbf {x}): \\Omega \\times {\\mathcal {I}}\\rightarrow \\mathbb {R}^n$ is $C^1$ in both the coordinates $\\mathbf {x}$ in some open subset $\\Omega \\subset \\mathbb {R}^n$ and the perturbation $\\varepsilon \\in {\\mathcal {I}}\\subset \\mathbb {R}$ , where ${\\mathcal {I}}$ is a small open neighborhood of zero.",
"For $\\varepsilon \\in {\\mathcal {I}}$ , system (REF ) has a linearly asymptotically stable limit cycle $\\gamma _\\varepsilon (t)$ , with a finite period $T_\\varepsilon $ depending (at least $C^1$ ) on $\\varepsilon $ .",
"It follows from Assumption REF that when $\\varepsilon =0$ , $F_0(\\mathbf {x})$ is $C^1$ in $\\mathbf {x}\\in \\Omega $ and the unperturbed system (REF ) exhibits a $T_0$ -periodic asymptotically stable limit cycle solution $\\gamma _0(t) = \\gamma _0(t+T_0)$ with $0<T_0<\\infty $ .",
"To simplify notation, we will drop the subscript 0 and use $F(\\mathbf {x})$ , $\\gamma (t)$ and $T$ to denote the unperturbed vector field, limit cycle solution and period, except where required to avoid confusion.",
"Moreover, Assumption REF implies that we have the following approximations that will be needed for deriving the iSRC and lTRC: $F_{\\varepsilon }(\\mathbf {x}) = F_0(\\mathbf {x}) + \\varepsilon \\frac{\\partial F_\\varepsilon }{\\partial \\varepsilon }(\\mathbf {x})\\Big |_{\\varepsilon =0} + O(\\varepsilon ^2),$ $ T_\\varepsilon =T_0+\\varepsilon T_1+O(\\varepsilon ^2),$ $ \\gamma _\\varepsilon (\\tau _\\varepsilon (t)) = \\gamma _0(t)+\\varepsilon \\gamma _1(t)+O(\\varepsilon ^2)\\quad (\\text{uniformly in }t),$ where $T_1$ is the linear shift in the limit cycle period $T_0$ in response to the static perturbation of size $\\varepsilon $ .",
"This global timing sensitivity $T_1>0$ if increasing $\\varepsilon $ increases the period.",
"The perturbed time $\\tau _\\varepsilon (t)$ , which satisfies $\\tau _0(t)\\equiv t$ and $\\tau _\\varepsilon (t+T_0)-\\tau _\\varepsilon (t)=T_\\varepsilon ,$ will be described in detail later (see (REF )); it allows the approximation (REF ) to be uniform in time and permits us to compare perturbed and unperturbed trajectories at corresponding time points.",
"The vector function $\\gamma _1(t)$ is a representative belonging to an equivalence class that comprises the iSRC."
],
[
"Shape and timing response to ",
"Suppose a small, brief perturbation is applied to (REF ) at time $t_0$ such that there is a small abrupt perturbation in the state space.",
"We have $\\tilde{\\gamma }(t_0)= \\gamma (t_0)+\\varepsilon P,$ where $\\tilde{\\gamma }$ indicates the trajectory subsequent to the instantaneous perturbation, $\\varepsilon $ is the magnitude of the perturbation, and $P$ is the unit vector in the direction of the perturbation in the state space.",
"As is well known, the effects of the small brief perturbation $\\varepsilon P$ on the shape and timing of the limit cycle trajectory are given, respectively, by the solution of the variational equation (REF ), and the iPRC which solves the adjoint equation (REF ).",
"The evolution of a trajectory $\\tilde{\\gamma }(t)$ close to the limit cycle $\\gamma (t)$ may be approximated as $\\tilde{\\gamma }(t)=\\gamma (t)+\\mathbf {u}(t)+O(\\varepsilon ^2)$ , where $\\mathbf {u}(t)$ satisfies the variational equation $\\frac{d\\mathbf {u}}{dt}=DF(\\gamma (t))\\mathbf {u}$ with initial displacement $\\mathbf {u}(t_0)=\\varepsilon P$ given by (REF ), for small $\\varepsilon $ .",
"Here $DF(\\gamma (t))$ is the Jacobian matrix evaluated along $\\gamma (t)$ .",
"On the other hand, an iPRC of an oscillator measures the timing sensitivity of the limit cycle to infinitesimally small perturbations at every point along its cycle.",
"It is defined as the shift in the oscillator phase $\\theta \\in [0, T)$ per size of the perturbation, in the limit of small perturbation size.",
"The limit cycle solution takes each phase to a unique point on the limit cycle, $\\textbf {x}=\\gamma (\\theta )$ , and its inverse maps each point on the cycle to a unique phase, $\\theta =\\phi (\\textbf {x})$ .",
"One may extend the domain of $\\phi (\\textbf {x})$ to points in the basin of attraction $\\mathcal {B}$ of the limit cycle by defining the asymptotic phase: $\\phi (\\textbf {x}): \\mathcal {B}\\rightarrow [0,T)$ with $\\frac{d\\phi (\\mathbf {x}(t))}{dt}=1,\\quad \\phi (\\mathbf {x}(t))=\\phi (\\mathbf {x}(t+T)).$ If $\\textbf {x}_0\\in \\gamma (t)$ and $\\textbf {y}_0 \\in \\mathcal {B}$ , then we say that $\\mathbf {y}_0$ has the same asymptotic phase as $\\mathbf {x}_0$ if $\\left\\Vert \\mathbf {x}(t; \\mathbf {x}_0)-\\mathbf {y}(t;\\mathbf {y}_0) \\right\\Vert \\rightarrow 0$ , as $t\\rightarrow \\infty $ .",
"This means that $\\phi (\\mathbf {x}_0)=\\phi (\\mathbf {y}_0)$ .",
"The set of all points off the limit cycle that have the same asymptotic phase as the point $\\mathbf {x}_0$ on the limit cycle is the isochron with phase $\\phi (\\mathbf {x}_0)$ .",
"The asymptotic phase function $\\phi $ is defined up to an additive constant; this constant is of no consequence other than to define an arbitrary reference point as the “zero phase\" location on the limit cycle trajectory.",
"Suppose $\\varepsilon P$ applied at phase $\\theta $ results in a new state $\\gamma (\\theta )+\\varepsilon P \\in \\mathcal {B}$ , which corresponds to a new phase $\\tilde{\\theta }=\\phi (\\gamma (\\theta )+\\varepsilon P)$ .",
"The phase difference $\\tilde{\\theta }-\\theta $ defines the phase response curve (PRC) of the oscillator.",
"One defines the iPRC as the vector function ${\\mathbf {z}}:[0,T)\\rightarrow {\\mathbf {R}}^n$ satisfying $ {\\mathbf {z}}(\\theta )\\cdot {P}=\\lim _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }\\left(\\phi (\\gamma (\\theta )+\\varepsilon {P})-\\theta \\right) = \\nabla _{\\mathbf {x}}\\phi (\\gamma (\\theta ))\\cdot {P}$ for arbitrary unit perturbation ${P}$ .",
"The first equality serves as a definition, while the second follows from routine arguments [5], [18], [49], [45].",
"It follows directly that the vector iPRC is the gradient of the asymptotic phase and it captures the phase (or timing) response to perturbations in any direction $P$ in state space.",
"By assumption REF , the vector field $F$ is $C^1$ .",
"It follows that the iPRC is a continuous $T$ -periodic solution satisfying the adjoint equation [49], $\\frac{d{\\mathbf {z}}}{dt}=-DF(\\gamma (t))^\\intercal {\\mathbf {z}},$ with the normalization condition $F(\\gamma (\\theta ))\\cdot {\\mathbf {z}}(\\theta ) = 1.$ Remark 2.2 By direct calculation, one can show that the solutions to the variational equation and the adjoint equation satisfy $\\mathbf {u}^\\intercal {\\mathbf {z}}=\\text{constant}$ : $\\frac{d(\\mathbf {u}^\\intercal {\\mathbf {z}})}{dt}=\\frac{d\\mathbf {u}^\\intercal }{dt} {\\mathbf {z}}+\\mathbf {u}^\\intercal \\frac{d{\\mathbf {z}}}{dt}=\\mathbf {u}^\\intercal DF^\\intercal {\\mathbf {z}}+\\mathbf {u}^\\intercal (-DF^\\intercal {\\mathbf {z}})=0.$ This relation holds for both smooth and nonsmooth systems with transverse crossings [46].",
"For completeness, we note that differences between phase variables, as in (REF ), will be interpreted as the periodic difference, $d_T(\\phi (\\mathbf {x}),\\phi (\\mathbf {y}))$ .",
"That is, if two angular variables $\\theta $ and $\\psi $ are defined on the circle $\\mathbb {S}\\equiv [0,T)$ , then we set $d_T(\\theta ,\\psi )={\\left\\lbrace \\begin{array}{ll}\\theta -\\psi +T,&\\theta -\\psi <-\\frac{T}{2}\\\\\\theta -\\psi ,&-\\frac{T}{2}\\le \\theta -\\psi \\le \\frac{T}{2}\\\\\\theta -\\psi -T,&\\theta -\\psi >\\frac{T}{2},\\end{array}\\right.",
"}$ which maps $d_T(\\theta ,\\psi )$ to the range $[-T/2,T/2]$ .",
"In what follows we will simply write $\\theta -\\psi $ for clarity rather than $d_T(\\theta ,\\psi )$ ."
],
[
"Timing response to ",
"Next we review how the iPRC can be used to estimate the linear shift in the limit cycle period in response to a sustained perturbation (see (REF )).",
"Note that the perturbed periodic solution $\\gamma _{\\varepsilon }(t)$ to system (REF ) can be represented, to leading order, by the single variable system $\\frac{d\\theta }{dt}=1+{\\mathbf {z}}(\\theta )^\\intercal G(\\mathbf {x},t),$ where $G(\\mathbf {x},t)=\\varepsilon \\frac{\\partial F_\\varepsilon (\\gamma (t))}{\\partial \\varepsilon }|_{\\varepsilon =0}$ represents the $O(\\varepsilon )$ perturbation of the vector field, $\\theta \\in [0, T_0)$ is the asymptotic phase as defined above, and ${\\mathbf {z}}:\\theta \\in [0,T_0)\\rightarrow {\\mathbf {R}}^n$ is the iPRC.",
"Recall that for $0\\le \\varepsilon \\ll 1$ we can represent $T_\\varepsilon $ with $T_\\varepsilon =T_0+\\varepsilon T_1+O(\\varepsilon ^2)$ (see (REF )).",
"From (REF ), $T_1$ can be calculated using the iPRC as $T_1=-\\int _{0}^{T_0} {\\mathbf {z}}(\\theta )^\\intercal \\frac{\\partial F_\\varepsilon (\\gamma (\\theta ))}{\\partial \\varepsilon }\\Big |_{\\varepsilon =0}d\\theta .$"
],
[
"Shape response to sustained perturbations: iSRC",
"In this section, we develop new tools to analyze the effects of sustained perturbations on the shape of a limit cycle solution.",
"As the parameter $\\varepsilon $ in (REF ) varies, the family of limit cycles produced by the flow forms a 2-dimensional “ribbon” in the $(n+1)$ -dimensional space parametrized by $(\\mathbf {x},\\varepsilon )$ .",
"The smoothness of this ribbon, when the vector field $F_\\varepsilon $ is smooth, follows immediately from the persistence of hyperbolic over- and under-flowing invariant manifolds ([61], §6.2).",
"In contrast to the instantaneous perturbation considered in the previous section, changes in each aspect of shape and timing can now influence the other, and hence a variational analysis of the combined shape and timing response of limit cycles under constant perturbation is needed.",
"To this end, we develop a new method that we call the iSRC (see (REF )).",
"In our analysis, we adapt Lighthill's method of coordinate perturbation (“strained coordinates\") to simultaneously stretch the time coordinate so as to accommodate the effect of parameter changes on period.",
"Generically, introducing a change in a parameter will lead to a change in period, as well as a displacement of the set of points comprising the limit cycle's orbit.",
"In order to quantify the change in shape, we must first accommodate any change in period.",
"To this end we introduce a rescaled time coordinate $t\\rightarrow \\tau _\\varepsilon (t)$ , which satisfies the consistency and smoothness conditions $\\frac{d\\tau _\\varepsilon }{dt}>0,\\quad \\text{ and }\\quad \\frac{1}{\\varepsilon }\\left(\\int _{t=0}^{T_0}\\left(\\frac{d\\tau _\\varepsilon }{dt}\\right)dt - T_0\\right)=T_1+O(\\varepsilon ),\\text{ as }\\varepsilon \\rightarrow 0,$ where $T_0$ is the unperturbed period and $T_1$ is the linear shift in the period as given by (REF ).",
"These conditions do not determine the value of the derivative of $\\tau _\\varepsilon $ , which we write as $d\\tau _\\varepsilon /dt=1/\\nu _\\varepsilon (t).$ In general, the iSRC will depend on the choice of $\\nu _\\varepsilon (t)$ .",
"However, some natural choices are particularly well adapted to specific problems, as we will see.",
"Initially, we will make the simple ansatz $\\nu _\\varepsilon (t)=\\text{const};$ that is, we will impose uniform local timing sensitivity.",
"Later in §REF we will introduce the local timing response curve (lTRC) to exploit alternative time rescalings for greater accuracy.",
"As discussed above, to understand how the static perturbation changes the shape of the limit cycle $\\gamma (t)$ , we need to rescale the time coordinate of the perturbed solution so that $\\gamma (t)$ and $\\gamma _{\\varepsilon }(t)$ may be compared at corresponding time points.",
"That is, for $\\varepsilon >0$ we wish to introduce a rescaled perturbed time coordinate $\\tau _\\varepsilon (t)$ so that (REF ) holds, uniformly in time $-\\infty <t<\\infty $ , which we repeat below: $\\gamma _\\varepsilon (\\tau _\\varepsilon (t))=\\gamma _0(t)+\\varepsilon \\gamma _1(t)+O(\\varepsilon ^2).$ We define the $T_0$ -periodic function $\\gamma _1(t)$ to be the infinitesimal shape response curve (iSRC).",
"We show next that $\\gamma _1(t)$ obeys an inhomogeneous variational equation (REF ).",
"This equation resembles (REF ), but has two additional non-homogeneous terms arising, respectively, from time rescaling $t\\rightarrow \\tau _\\varepsilon (t)$ , and directly from the constant perturbation acting on the vector field.",
"It follows from (REF ) and (REF ) that the scaling factor is $\\nu _\\varepsilon =\\frac{T_0}{T_\\varepsilon }$ .",
"Moreover, by (REF ), $\\nu _\\varepsilon $ can be written as $\\nu _\\varepsilon =1-\\varepsilon \\nu _1+ O(\\varepsilon ^2),$ where $\\nu _1=\\frac{T_1}{T_0}$ represents the relative change in frequency.",
"In terms of $\\nu _\\varepsilon $ that is time-independent, the rescaled time for $\\gamma _{\\varepsilon }(\\tau _\\varepsilon (t))$ can be written as $\\tau _\\varepsilon (t)=t/\\nu _\\varepsilon \\in [0, T_\\varepsilon ]$ for $t\\in [0, T_0]$ (see (REF )).",
"Differentiating $\\gamma _\\varepsilon (\\tau _\\varepsilon (t))$ given in (REF ) with respect to $t$ ($\\frac{d\\gamma _\\varepsilon }{dt} = \\frac{d\\gamma _\\varepsilon }{d\\tau _\\varepsilon } \\frac{d\\tau _\\varepsilon }{dt}$ ), substituting the ansatz ($\\frac{d\\tau _\\varepsilon }{dt}=\\frac{1}{\\nu _\\varepsilon }$ ) and rearranging lead to $\\begin{split}\\begin{array}{cccccccccc}\\frac{d \\gamma _{\\varepsilon }}{d\\tau _\\varepsilon }&=& \\nu _\\varepsilon \\,(\\gamma ^{\\prime }(t)+\\varepsilon \\gamma _1^{\\prime }( t)+O(\\varepsilon ^2))\\\\&=& (1-\\varepsilon \\nu _1+ O(\\varepsilon ^2))\\,(\\gamma ^{\\prime }(t)+\\varepsilon \\gamma _1^{\\prime }( t)+O(\\varepsilon ^2))\\\\&=& \\gamma ^{\\prime }( t)-\\varepsilon \\nu _1 \\gamma ^{\\prime }( t) +\\varepsilon \\gamma _1^{\\prime }( t) + O(\\varepsilon ^2) \\\\&=& F_0(\\gamma (t)) + \\varepsilon (-\\nu _1 F_0(\\gamma (t)) +\\gamma _1^{\\prime }( t))+O(\\varepsilon ^2),\\end{array}\\end{split}$ where $^{\\prime }$ denotes the derivative with respect to $t$ .",
"On the other hand, expanding the right hand side of (REF ) gives $\\begin{split}\\begin{array}{cccccccccc}\\frac{d \\gamma _{\\varepsilon }}{d\\tau _\\varepsilon } &=& F_\\varepsilon (\\gamma _{\\varepsilon }(\\tau _\\varepsilon ))\\\\&=& F_0(\\gamma (t))+\\varepsilon \\Big (DF_0(\\gamma (t)) \\gamma _1(t) +\\frac{\\partial F_\\varepsilon (\\gamma (t))}{\\partial \\varepsilon }\\Big |_{\\varepsilon =0}\\Big )+O(\\varepsilon ^2).\\end{array}\\end{split}$ Equating (REF ) and (REF ) to first order, we find that the linear shift in shape produced by a static perturbation, i.e.",
"the iSRC, satisfies $\\frac{d \\gamma _1(t)}{dt}&=& DF_0(\\gamma (t)) \\gamma _1(t) +\\nu _1 F_0(\\gamma (t)) +\\frac{\\partial F_\\varepsilon (\\gamma (t))}{\\partial \\varepsilon }\\Big |_{\\varepsilon =0},$ with period $T_0$ , as claimed before.",
"It remains to establish an initial condition $\\gamma _1(0)$ .",
"For smooth systems, without loss of generality, we may choose the initial condition for (REF ) by taking a Poincaré section orthogonal to the limit cycle at a chosen reference point $p_0=\\gamma (0)$ .",
"Then the initial condition is given by $\\gamma _1(0)=(p_\\varepsilon -p_0)/\\varepsilon $ where $p_\\varepsilon $ is the intersection point where the perturbed limit cycle $\\gamma _\\varepsilon $ crosses the Poincaré section.",
"For nonsmooth systems discussed in the balance of the paper, we choose the reference section $\\Sigma $ to be one of the switching or contact boundaries.",
"That such an arbitrary choice of an initial condition using an orthogonal Poincaré section or a switching boundary does not compromise generality is a consequence of the following Lemma, the proof of which is in given in Appendix .",
"Lemma 2.3 Let $\\gamma ^\\textbf {a}_1(t)$ and $\\gamma ^\\textbf {b}_1(t)$ be two $T_0$ -periodic solutions to the iSRC equation (REF ) for a smooth vector field $F_0$ with a hyperbolically stable limit cycle $\\gamma _0(t)$ .",
"Then, their difference satisfies $\\gamma ^\\textbf {b}_1(t)-\\gamma ^\\textbf {a}_1(t)=\\varphi F_0(\\gamma _0(t))$ , where $\\varphi $ is a constant representing a fixed phase offset.",
"Thus, if $\\gamma ^\\textbf {a}_1(t)$ is a representative iSRC specified by taking the orthogonal Poincaré section, and $\\gamma ^\\textbf {b}_1(t)$ is another representative specified by a different transverse section, then the two solutions differ by a fixed offset – namely a vector in the direction of the flow along the limit cycle – indexed by an additive difference in phase.",
"Hence the differences between distinct periodic solutions to (REF ) have precisely the same degree of ambiguity – and for the same reason – as the familiar ambiguity of the phase of an oscillator.",
"The accuracy of the iSRC in approximating the linear change in the limit cycle shape evidently depends on its timing sensitivity, that is, the choice of the relative change in frequency $\\nu _1$ .",
"In the preceding derivation, we chose $\\nu _1$ to be the relative change in the full period by assuming the limit cycle has constant timing sensitivity.",
"It is natural to expect that different choices of $\\nu _1$ will be needed for systems with varying timing sensitivities along the limit cycle.",
"This possibility motivates us to consider local timing surfaces which divide the limit cycle into a number of segments, each distinguished by its own timing sensitivity properties.",
"For each segment, we show that the linear shift in the time that $\\gamma (t)$ spends in that segment can be estimated using a local timing response curve (lTRC) derived in §REF .",
"The lTRC is analogous to the iPRC in the sense that they obey the same adjoint equation, but with different boundary conditions."
],
[
"Local timing response to perturbations: lTRC",
"The iPRC captures the net effect on timing of an oscillation – the phase shift – due to a transient perturbation (REF ), as well as the net change in period due to a sustained perturbation (REF ).",
"In order to study the impact of a perturbation on local timing as opposed to global timing, we introduce the notion of local timing surfaces that separate the limit cycle trajectory into segments with different timing sensitivities.",
"Examples of local timing surfaces in smooth systems include the passage of neuronal voltage through its local maximum or through a predefined threshold voltage, and the point of maximal extension of reach by a limb.",
"In nonsmooth systems, switching surfaces at which dynamics changes can also serve as local timing surfaces.",
"For instance, in the feeding system of Aplysia californica [50], [39], the open-closed switching boundary of the grasper defines a local timing surface.It is worth noting that the idea of exploiting the presence of timing surfaces to specify the lTRC, by which we are taking advantage of corresponding features present in both the perturbed and unperturbed limit cycles, may be seen as an example of the general notion of bisimulation [27], [28].",
"Whatever the origin of the local timing surface or surfaces of interest, it is natural to consider the phase space of a limit cycle as divided into multiple regions.",
"Hence we may consider a smooth system $d\\mathbf {x}/dt=F(\\mathbf {x})$ with a limit cycle solution $\\gamma (t)$ passing through multiple regions in succession (see Figure REF ).",
"In each region, we assume that $\\gamma (t)$ has constant timing sensitivity.",
"To compute the relative change in time in any given region, we define a lTRC to measure the timing shift of $\\gamma (t)$ in response to perturbations delivered at different times in that region.",
"Below, we illustrate the derivation of the lTRC in region I and show how it can be used to compute the relative change in time in this region, denoted by $\\nu _1^{\\rm I}$ .",
"Figure: Schematic illustration of a limit cycle solution for a system consisting of a number of regions, each with distinct constant timing sensitivities.Σ in \\Sigma ^{\\rm in} and Σ out \\Sigma ^{\\rm out} denote the local timing surfaces for region I.𝐱 in \\mathbf {x}^{\\rm in} and 𝐱 out \\mathbf {x}^{\\rm out} denote the points where the limit cycle enters region I through Σ in \\Sigma ^{\\rm in} and exits region I through Σ out \\Sigma ^{\\rm out}, respectively.Suppose that at time $t^{\\rm in}$ , $\\gamma (t)$ enters region I upon crossing the surface $\\Sigma ^{\\rm in}$ at the point $\\mathbf {x}^{\\rm in}$ ; at time $t^{\\rm out}$ , $\\gamma (t)$ exits region I upon crossing the surface $\\Sigma ^{\\rm out}$ at the point $\\mathbf {x}^{\\rm out}$ (see Figure REF ).",
"Denote the vector field under a constant perturbation by $F_\\varepsilon (\\mathbf {x})$ and let $\\mathbf {x}_{\\varepsilon }$ denote the coordinate of the perturbed trajectory.",
"Let $T_0^{\\rm I} = t^{\\rm out}-t^{\\rm in}$ denote the time $\\gamma (t)$ spent in region I and let $T_{\\varepsilon }^{\\rm I}$ denote the time the perturbed trajectory spent in region I.",
"Assume we can write $T_{\\varepsilon }^{\\rm I}=T_0^{\\rm I}+\\varepsilon T^{\\rm I}_1+O(\\varepsilon ^2)$ as we did before.",
"It follows that the relative change in time of $\\gamma (t)$ in region I is given by $\\nu _1^{\\rm I}=\\frac{T^{\\rm I}_1}{T_0^{\\rm I}} = \\frac{T^{\\rm I}_1}{t^{\\rm out}-t^{\\rm in}} .$ The goal is to compute $\\nu _1^{\\rm I}$ , which requires an estimate of $T^{\\rm I}_1$ .",
"To this end, we define the local timing response curve $\\eta ^{\\rm I}(t)$ associated with region I.",
"We show that $\\eta ^{\\rm I}(t)$ satisfies the adjoint equation (REF ) and the boundary condition (REF ).",
"Let $\\mathcal {T}^{\\rm I}(\\mathbf {x})$ for $\\mathbf {x}$ in region I be the time remaining until exiting region I through $\\Sigma ^{\\rm out}$ , under the unperturbed vector field.",
"This function is at least defined in some open neighborhood around the reference limit cycle trajectory $\\gamma (t)$ if not throughout region I.",
"For the unperturbed system, $\\mathcal {T}^{\\rm I}$ satisfies $\\frac{d\\mathcal {T}^{\\rm I}(\\mathbf {x}(t))}{dt}=-1$ along the limit cycle orbit $\\gamma (t)$ .",
"Hence $F(\\mathbf {x})\\cdot \\nabla \\mathcal {T}^{\\rm I}(\\mathbf {x})=-1$ for all $\\mathbf {x}$ for which $\\mathcal {T}^{\\rm I}$ is defined.",
"We define $\\eta ^{\\rm I}(t):= \\nabla \\mathcal {T}^{\\rm I}(\\mathbf {x}(t))$ to be the local timing response curve (lTRC) for region I.",
"It is defined for $t\\in [t^{\\rm in}, t^{\\rm out}]$ .",
"We show in Appendix that $T^{\\rm I}_1$ can be estimated as $T^{\\rm I}_{1} = \\eta ^{\\rm I}(\\mathbf {x}^{\\rm in})\\cdot \\frac{\\partial \\mathbf {x}_{\\varepsilon }^{\\rm in}}{\\partial \\varepsilon }\\Big |_{\\varepsilon =0}+\\int _{t^{\\rm in}}^{t^{\\rm out}}\\eta ^{\\rm I}(\\gamma (t))\\cdot \\frac{\\partial F_\\varepsilon (\\gamma (t))}{\\partial \\varepsilon }\\Big |_{\\varepsilon =0}dt,$ where $\\mathbf {x}_\\varepsilon ^{\\rm in}$ denotes the coordinate of the perturbed entry point into region I.",
"We may naturally view $\\eta ^{\\rm I}$ as either a function of space, as in (REF ), or as a function of time, evaluated e.g.",
"along the limit cycle trajectory.",
"Comparing (REF ) with (REF ), the integral terms have the same form, albeit with opposite signs.",
"In addition, (REF ) has an additional term arising from the impact of the perturbation on the point of entry to region I.",
"On the other hand, the impact of the perturbation on the exit point, denoted by $\\eta ^{\\rm I}(\\mathbf {x}_\\varepsilon ^{\\rm out})\\cdot \\frac{\\partial \\mathbf {x}_\\varepsilon ^{\\rm out}}{\\partial \\varepsilon }\\big |_{\\varepsilon =0}$ , is always zero because the exit boundary $\\Sigma ^{\\rm out}$ is a level curve of $\\mathcal {T}^{\\rm I}$ ; in other words, $\\mathcal {T}^{\\rm I}\\equiv 0$ at $\\Sigma ^{\\rm out}$ .",
"This indicates that the lTRC vector $\\eta ^{\\rm I}$ associated with a given region is always perpendicular to the exit boundary of that region.",
"Similar to the iPRC, it follows from (REF ) that $\\eta ^{\\rm I}$ satisfies the adjoint equation $\\frac{d\\eta ^{\\rm I}}{dt}=-DF(\\gamma (t))^\\intercal \\eta ^{\\rm I}$ together with the boundary (normalization) condition at the exit point $\\eta ^{\\rm I}(\\mathbf {x}^{\\rm out})=\\frac{-n^{\\rm out}}{n^{\\rm out \\intercal } F(\\mathbf {x}^{\\rm out})}$ where $n^{\\rm out}$ is a normal vector of $\\Sigma ^{\\rm out}$ at the unperturbed exit point $\\mathbf {x}^{\\rm out}$ .",
"The reason $\\eta ^{\\rm I}$ at the exit point has the direction $n^{\\rm out}$ is because $\\eta ^{\\rm I}$ is normal to the exit boundary as discussed above.",
"To summarize, in order to compute $\\nu _1^{\\rm I}=T^{\\rm I}_{1}/({t^{\\rm out}-t^{\\rm in}})$ , we need numerically to find $t^{\\rm in},t^{\\rm out}$ and evaluate (REF ) to estimate $T^{\\rm I}_{1}$ , for which we need to solve the boundary problem of the adjoint equation (REF )-(REF ) for the lTRC $\\eta ^{\\rm I}$ .",
"The procedures to obtain the relative change in time in other regions $\\nu _1^{\\rm II}, \\nu _1^{\\rm III},\\cdots $ are similar to computing $\\nu _1^{\\rm I}$ in region I and hence are omitted.",
"The existence of different timing sensitivities of $\\gamma (t)$ in different regions therefore leads to a piecewise-specified version of the iSRC (REF ) with period $T_0$ , $\\frac{d \\gamma _1^j(t)}{dt}&=& DF_0^j(\\gamma (t)) \\gamma _1^j(t) +\\nu _1^{j} F_0^j(\\gamma (t)) +\\frac{\\partial F_\\varepsilon ^j(\\gamma (t))}{\\partial \\varepsilon }\\Big |_{\\varepsilon =0},$ where $\\gamma _1^j$ , $F_0^j$ , $F_{\\varepsilon }^j$ and $\\nu _1^j$ denote the iSRC, the unperturbed vector field, the perturbed vector field, and the relative change in time in region $j$ , respectively, with $j\\in \\lbrace \\rm I, II, III, \\cdots \\rbrace $ .",
"Note that in a smooth system as concerned in this section, $F_0^j\\equiv F_0$ for all $j$ .",
"In § we will show in a specific example that the iSRC with piecewise-specified timing rescaling has much greater accuracy in approximating the linear shape response of the limit cycle to static perturbations than the iSRC using a global uniform rescaling.",
"Remark 2.4 The derivation of the lTRC in a given region still holds as long as the system is smooth in that region.",
"Hence the assumption that $F(\\mathbf {x})$ is smooth everywhere can be relaxed to $F(\\mathbf {x})$ being piecewise smooth.",
"Remark 2.5 The lTRC is an intrinsic property of a limit cycle that is determined by the choice of local timing surfaces that are transverse to the limit cycle.",
"In this work, we only use the lTRC in the situation where there are naturally occurring surfaces such as those corresponding to switching surfaces inherent in the geometry of a problem (e.g., the stick-slip system).",
"In general, the lTRC construction could be used to analyze smooth systems as well, for instance systems with heteroclinic cycling [51]."
],
[
"Linear responses of nonsmooth systems with continuous solutions",
"Nonsmooth dynamical systems arise in many areas of biology and engineering.",
"However, methods developed for smooth systems (discussed in §) do not extend directly to understanding the changes of periodic limit cycle orbits in nonsmooth systems, because their Jacobian matrices are not well defined [8], [62].",
"Specifically, nonsmooth systems exhibit discontinuities in the time evolution of the solutions to the variational equations, $\\mathbf {u}$ (REF ) and $\\gamma _1$ (REF ), and the solutions to the adjoint equations, ${\\mathbf {z}}$ (REF ) and $\\eta $ (REF ).",
"Following the terminology of [46] and [38], we call the discontinuities in ${\\mathbf {z}}$ and $\\eta $ “jumps” and call the discontinuities in $\\mathbf {u}$ and $\\gamma _1$ “saltations”.",
"Qualitatively, we use “jumps\" to refer to discontinuities in the timing response of a trajectory, and “saltations\" in the shape response.",
"Since ${\\mathbf {z}}$ and $\\eta $ satisfy the same adjoint equation, they have the same discontinuities.",
"Similarly, $\\mathbf {u}$ and $\\gamma _1$ obey versions of the variational equation with the same homogeneous term and different nonhomogeneous terms; since the jump conditions arise from the homogeneous terms (involving the Jacobian matrix in (REF ), (REF )) we will presume that $\\mathbf {u}$ and $\\gamma _1$ satisfy the same saltation conditions at the transition boundaries.",
"In this section, we characterize the discontinuities in the solutions to the adjoint equation in terms of ${\\mathbf {z}}$ , and discuss nonsmoothness of the variational dynamics in terms of $\\mathbf {u}$ .",
"In this paper, we consider nonsmooth systems with degree of smoothness one or higher (Filippov systems); that is, systems with continuous solutions.",
"In such systems, the right-hand-side changes discontinuously as one or more switching surfaces are crossed.",
"A trajectory reaching a switching surface has two behaviors: it may cross the surface transversally, or it may slide along it, in which case the motion is called sliding mode.",
"In [52], [46], [8], [62], [3], [38], the solutions to the adjoint and variational equations have been studied in the case of solutions which cross the surface of discontinuity transversally.",
"The concept of saltations in the variational dynamics $\\mathbf {u}$ has also been adapted by [19], [3], [38], [11] to the case of sliding motion on surfaces.",
"However, in the case of sliding mode motions, discontinuous jumps exhibited by the iPRC have not yet been characterized.",
"Bridging that gap is a principal goal of this paper.",
"In the remainder of this section, we first define Filippov systems with both types of discontinuities.",
"Then we review the existing methods for computing the saltations in $\\mathbf {u}$ (and $\\gamma _1$ ) in both cases, and the jumps in ${\\mathbf {z}}$ (and $\\eta $ ) in the case of transversal crossing boundaries, following [3], [38] and [44], [46].",
"Lastly in §REF , we present our main results about the discontinuous behavior of the iPRC for nonsmooth systems with sliding motions.",
"For completeness, we include both the old and new results in the case of sliding motions in the statement of Theorem REF .",
"Figure: Examples of trajectories of nonsmooth systems with a boundary Σ\\Sigma .",
"(A) The trajectory of a two-zone nonsmooth system () intersects the boundary Σ\\Sigma transversely at 𝐱 p \\mathbf {x}_p (black dot).F I F^{\\rm I} and F II F^{\\rm II} denote the vector fields in the two regions ℛ I {\\mathcal {R}}^{\\rm I} and ℛ II {\\mathcal {R}}^{\\rm II}.Components of F I F^{\\rm I} and F II F^{\\rm II} normal to Σ\\Sigma at the crossing point have the same sign, allowing for the transversal crossing.",
"(B) The trajectory of a nonsmooth system () hits the hard boundary Σ\\Sigma at the landing point (red dot) and begins sliding along Σ\\Sigma under the vector field F slide F^{\\rm slide}.At the liftoff point (blue dot), the trajectory naturally reenters the interior.F interior F^{\\rm interior} denotes the vector field in the interior domain."
],
[
"Filippov systems",
"It is sufficient to consider a Filippov system with a single boundary (switching surface) to illustrate the discontinuities of ${\\mathbf {z}}$ and $\\mathbf {u}$ at any boundary crossing point.",
"Below we give the definition of a two-zone Filippov system with a transversal crossing boundary (see (REF )) and a local representation of a Filippov system that exhibits the sliding mode (see (REF ))."
],
[
"Transversal crossing boundary",
"Definition 3.1 A two-zone system with uniform degree of smoothness one (or higher) is described by $\\frac{d\\textbf {x}}{dt}=F(\\mathbf {x}):=\\left\\lbrace \\begin{array}{cccccccccc}F^{\\rm I}(\\mathbf {x}), & \\mathbf {x}\\in {\\mathcal {R}}^{\\rm I} \\\\F^{\\rm II}(\\mathbf {x}), & \\mathbf {x}\\in {\\mathcal {R}}^{\\rm II} \\\\\\end{array}\\right.$ where ${\\mathcal {R}}^{\\rm I}:=\\lbrace \\mathbf {x}| H(\\mathbf {x})<0\\rbrace $ and ${\\mathcal {R}}^{\\rm II}:=\\lbrace \\mathbf {x}| H(\\mathbf {x})>0\\rbrace $ for a smooth function $H$ have non-empty interiors, and the vector fields $F^{\\rm I,II}:\\overline{{\\mathcal {R}}}^{\\rm I,II}\\rightarrow \\mathbb {R}^n$ are at least $C^1$ , where $\\overline{{\\mathcal {R}}}$ denotes the closure of ${\\mathcal {R}}$ in $\\mathbb {R}^n$ .",
"Definition 3.2 The switching boundary for the Filippov system (REF ) is the $\\mathbb {R}^{n-1}$ -dimensional manifold $\\Sigma := \\overline{{\\mathcal {R}}}^{\\rm I}\\cap \\overline{{\\mathcal {R}}}^{\\rm II}=\\lbrace \\mathbf {x}|H(\\mathbf {x})=0\\rbrace $ .",
"$\\Sigma $ is called a transversal crossing boundary if at $\\mathbf {x}_p\\in \\Sigma $ the following holds $\\begin{array}{cccccccccc}(n_{p}\\cdot F^{\\rm I}(\\mathbf {x}_p)) (n_{p} \\cdot F^{\\rm II}(\\mathbf {x}_p))> 0\\end{array}$ where $n_p=\\nabla H(\\mathbf {x}_p)$ refers to the vector normal to $\\Sigma $ at $\\mathbf {x}_p$ .",
"Without loss of generality, we assume at time $t=t_p$ a limit cycle solution $\\gamma (t)$ of (REF ) crosses the boundary $\\Sigma $ from ${\\mathcal {R}}^{\\rm I}$ to ${\\mathcal {R}}^{\\rm II}$ (see Figure REF A).",
"In this case, we have $\\begin{array}{cccccccccc}n_{p}\\cdot F^{\\rm I}(\\mathbf {x}_p)> 0\\quad \\quad n_{p} \\cdot F^{\\rm II}(\\mathbf {x}_p)> 0, \\end{array}$ where the boundary crossing point can be defined as $\\mathbf {x}_p:=\\lim _{t\\rightarrow t_p^-}\\gamma (t)=\\lim _{t\\rightarrow t_p^+}\\gamma (t)$ ."
],
[
"Sliding motion on a hard boundary",
"Next we consider the second type of switching surface on which the transversal flow condition (REF ) does not hold.",
"That is, parts of the solution trajectory slide along a surface (e.g., Figure REF B).",
"As an example of a hard boundary at which the transverse flow condition would break down, consider the requirement that firing rates in a neural network model be nonnegative.",
"When a nerve cell ceases firing because of inhibition, its firing rate will be held at zero until the balance of inhibition and excitation allow spiking to resume.",
"At the point at which the firing rate first resumes positive values, the vector field describing the system lies tangent to the constraint surface rather than transverse to it.",
"Below we present a model of a Filippov system with sliding motion along a hard boundary - system (REF ).",
"We begin with precise definitions of hard boundary, sliding region, sliding vector field and the liftoff condition.",
"In such systems, non-transversal crossing points include the landing point at which a sliding motion begins, and the liftoff point at which the sliding terminates (see Fig.",
"REF B).",
"Definition 3.3 Consider a system with domain $\\mathcal {R}$ .",
"We call a surface $\\Sigma $ a hard boundary if it is part of the boundary of the closure of $\\mathcal {R}$ .",
"Definition 3.4 The sliding region (${\\mathcal {R}}^{\\rm slide}$ ) is defined as the portion of a hard boundary $\\Sigma $ for which $\\begin{array}{cccccccccc}{\\mathcal {R}}^{\\rm slide}=\\lbrace \\mathbf {x}\\in \\Sigma \\,|\\,n_\\mathbf {x}\\cdot F^{\\rm interior}(\\mathbf {x})>0\\rbrace ,\\end{array}$ where $n_\\mathbf {x}$ is a unit normal vector of $\\Sigma $ at $\\mathbf {x}$ that points away from the interior and $F^{\\rm interior}$ denotes the vector field defined on the closure of the domain ${\\mathcal {R}}$ .",
"Definition 3.5 The interior domain (${\\mathcal {R}}^{\\rm interior}$ ) is defined as the complement of ${\\mathcal {R}}^{\\rm slide}$ in the closure of ${\\mathcal {R}}$ .",
"When the trajectory enters the sliding region, the solution will continue along ${\\mathcal {R}}^{\\rm slide}$ with time derivative $F^\\text{slide}$ that is tangent to the hard boundary.",
"While any vector field with vanishing normal component could be considered for $F^\\text{slide}$ , in this paper we adopt the natural choice of setting $F^\\text{slide}$ to be the following.",
"Definition 3.6 The sliding vector field $F^{\\rm slide}$ , defined on ${\\mathcal {R}}^{\\rm slide}$ , is given by the continuation of $F^{\\rm interior}$ in the component tangential to ${\\mathcal {R}}^{\\rm slide}$ : $\\begin{array}{cccccccccc}F^{\\rm slide}(\\mathbf {x})=F^{\\rm interior}(\\mathbf {x})-\\big (n_\\mathbf {x}\\cdot F^{\\rm interior}(\\mathbf {x})\\big )n_\\mathbf {x}.\\end{array}$ During the sliding motion, the flow will slide along $\\Sigma $ with the sliding vector field until it is allowed to reenter the interior; that is Definition 3.7 The flow exits the sliding region ${\\mathcal {R}}^{\\rm slide}$ as the trajectory crosses the liftoff boundary $\\mathcal {L}$ defined as $\\begin{array}{cccccccccc}\\mathcal {L}=\\lbrace \\mathbf {x}\\in \\Sigma \\,| \\,n_\\mathbf {x}\\cdot F^{\\rm interior}(\\mathbf {x})=0\\rbrace .\\end{array}$ Thus the liftoff boundary constitutes the edge of the sliding region of the hard boundary.",
"To identify the liftoff point at which the trajectory reenters the interior of the domain, we further require the nondegeneracy condition that the trajectory crosses the liftoff boundary $\\mathcal {L}$ at a finite velocity.",
"Specifically, the outward normal component of the interior velocity should switch from positive (outward) to negative (inward) at the liftoff boundary, as one moves in the direction of the flow (see Fig.",
"REF ).",
"That is, $\\begin{array}{cccccccccc}\\left.\\left[\\nabla (n_\\mathbf {x}\\cdot F^{\\rm interior}(\\mathbf {x})) \\cdot F^{\\rm slide}(\\mathbf {x})\\right] \\right|_{\\mathbf {x}\\in \\mathcal {L}}< 0\\end{array}.$ Note that the liftoff condition (REF ) together with the nondegeneracy condition (REF ) uniquely defines a liftoff point for the trajectory that slides along $\\Sigma $ .",
"At the liftoff point, we have $F^{\\rm slide}=F^{\\rm interior}$ .",
"Remark 3.8 Our definition of the sliding region and sliding vector field is consistent with that in [3] §5.2.2, except that the system of interest in this paper is only defined on one side of the sliding region.",
"However, our main Theorem REF , below, holds in either case.",
"Hence our results also apply to Filippov systems with sliding regions bordered by vector fields on either side, as in the example the stick-slip oscillator ([38] §6.5).",
"The motion along a trajectory is specified differently depending on the location of a point.",
"For a point in the interior, the dynamics is determined by $F^\\text{interior}$ .",
"For a point on $\\Sigma $ , the velocity obeys either $F^\\text{interior}$ or else $F^\\text{slide}$ that is tangent to $\\Sigma $ , depending on whether $F^\\text{interior}$ is directed inwardly or outwardly at a given boundary point.",
"This dual definition of the vector field has the effect that points driven into the boundary do not exit through the hard boundary, but rather slide along the boundary until the trajectory crosses the liftoff boundary $\\mathcal {L}$ .",
"Using the preceding notation, in the neighborhood of a hard boundary $\\Sigma $ , a system with a limit cycle component confined to the sliding region takes the following form $\\frac{d\\textbf {x}}{dt}=F(\\mathbf {x}):=\\left\\lbrace \\begin{array}{cccccccccc}F^{\\rm interior}(\\mathbf {x}), & \\mathbf {x}\\in {\\mathcal {R}}^{\\rm interior}\\\\F^{\\rm slide}(\\mathbf {x}), & \\mathbf {x}\\in {\\mathcal {R}}^{\\rm slide}\\subset \\Sigma \\\\\\end{array}\\right.$ where $F^{\\rm interior},{\\mathcal {R}}^{\\rm interior}, F^{\\rm side}, {\\mathcal {R}}^{\\rm slide}, \\Sigma $ are given in Definitions REF -REF .",
"Definition 3.9 In a general Filippov system which locally at a hard boundary $\\Sigma $ has the form (REF ), we call a closed, isolated periodic orbit that passes through a sliding region a limit cycle with sliding component, denoted as LCSC.",
"Figure: Trajectory from the interior with vector field F interior F^{\\rm interior} making a transverse entry to a hard boundary Σ\\Sigma , followed by motion confined to the sliding region ℛ slide \\mathcal {R}^{\\rm slide}, then a smooth liftoff at ℒ\\mathcal {L} back into the interior of the domain.Red dot: landing point (point at which the trajectory exits the interior and enters the hard boundary surface).Blue dot: liftoff point (point at which the trajectory crosses the liftoff boundary ℒ\\mathcal {L} and reenters the interior).After a suitable change of coordinates, the geometry may be arranged as shown, with the hard boundary Σ\\Sigma coinciding with one coordinate plane.Downward vertical arrow: 𝐧\\mathbf {n}, the outward normal vector for Σ\\Sigma .The region 𝐧·F interior >0\\mathbf {n}\\cdot F^\\text{interior}>0 defines the sliding region within Σ\\Sigma ; the condition 𝐧·F interior =0\\mathbf {n}\\cdot F^\\text{interior}=0 defines the liftoff boundary ℒ\\mathcal {L}.We make the following assumptions about the vector field $F$ , the hard boundary $\\Sigma $ and the liftoff boundary ${\\mathcal {L}}\\subset \\Sigma $ throughout: Assumption 3.10 $F^{\\rm interior}: \\overline{{\\mathcal {R}}}^{\\rm interior}\\rightarrow \\mathbb {R}^n$ is at least $C^1$ .",
"Under an appropriate smooth change of coordinates, the hard boundary $\\Sigma $ can be transformed into a lower dimensional manifold with a constant normal vector $n$ (cf. Fig.",
"REF ).",
"When the trajectory crosses the liftoff boundary ${\\mathcal {L}}$ , the nondegeneracy condition (REF ) holds so a liftoff point can be uniquely defined."
],
[
"Review of variational dynamics and iPRC in Filippov systems",
"Below we review the behaviors of the variational dynamics $\\mathbf {u}$ and the iPRC ${\\mathbf {z}}$ in the case of transversal intersection, as well as the characterization of discontinuities in $\\mathbf {u}$ in the case of sliding motion on a hard boundary.",
"As discussed before, the iSRC $\\gamma _1$ and the lTRC $\\eta $ experience the same discontinuities as $\\mathbf {u}$ and ${\\mathbf {z}}$ , respectively."
],
[
"Variational dynamics: transversal crossing and sliding motion",
"For a sufficiently small instantaneous perturbation, the displacement $\\mathbf {u}(t)$ evolves continuously over the domain in which (REF ) is smooth, and can be obtained to first order in the initial displacement by solving the variational equation (REF ).",
"As $\\gamma $ crosses $\\Sigma $ at time $t_p$ , $\\mathbf {u}(t)$ exhibits discontinuities (or “saltations”) since the Jacobian evaluated at $\\mathbf {x}_p$ is not uniquely defined.",
"The discontinuity in $\\mathbf {u}$ at $\\mathbf {x}_p$ can be expressed with the saltation matrix $S_p$ as ${\\mathbf {u}_p^+=S_p\\,\\mathbf {u}_p^-}$ where $\\mathbf {u}_p^-=\\lim _{t\\rightarrow t_p^-} \\mathbf {u}(t)$ and $\\mathbf {u}_p^+=\\lim _{t\\rightarrow t_p^+} \\mathbf {u}(t)$ represent the displacements between perturbed and unperturbed solutions just before and just after the crossing, respectively.",
"It is straightforward to show (cf.",
"[38] §7.2 or [3] §2.5) that $S_p$ can be constructed using the vector fields in the neighborhood of the crossing point and the vector $n_p$ normal to the switching boundary at $\\mathbf {x}_p$ as $S_p=I+\\frac{(F_p^+-F_p^-)n_{p}^\\intercal }{n_{p}^\\intercal F_p^-}$ where $F_p^-=\\lim _{\\mathbf {x}\\rightarrow \\mathbf {x}_p^-}F(\\mathbf {x}),\\,F_p^+=\\lim _{\\mathbf {x}\\rightarrow \\mathbf {x}_p^+}F(\\mathbf {x})$ are the vector fields of (REF ) just before and just after the crossing at $\\mathbf {x}_p$ .",
"Throughout this paper, $I$ denotes the identity matrix with size $n\\times n$ .",
"Remark 3.11 If the vector field $F$ evaluated along the limit cycle is continuous when crossing the boundary $\\Sigma $ transversely, so that $F_p^-=F_p^+$ and $n_p^\\intercal F_p^-\\ne 0$ , then the saltation matrix $S_p$ at such a boundary crossing point is the identity matrix, and there is no discontinuity in $\\mathbf {u}$ or $\\gamma _1$ at time $t_p$ .",
"When the transversal flow condition $n_p^\\intercal F_p^-\\ne 0$ is violated, the expression (REF ) cannot be used.",
"Nevertheless, [19], [3] showed how to adapt the definition of the saltation matrix to capture the discontinuity of $\\mathbf {u}$ at non-transversal crossings including both landing and liftoff points.",
"Specifically, at a landing point, the saltation matrix is given by $S_p = I - n_{p}n_p^\\intercal ,$ whereas at a liftoff point, the saltation matrix is $S_p = I.$ The above results about $S_p$ at the non-transversal crossing points are also summarized as parts (a) and (b) in Theorem REF , together with our new results about the iPRC in the case of sliding motion along a hard boundary.",
"For the derivation of $S_p$ at the landing and liftoff points, we refer to [19], [3] or our proof of Theorem REF in Appendix ."
],
[
"iPRC: transversal crossing",
"Now we consider discontinuous jumps in the iPRC ${\\mathbf {z}}$ for (REF ) with transversal intersection.",
"This curve obeys the adjoint equation (REF ) and is continuous within the interior of each subdomain in which (REF ) is smooth.",
"When the limit cycle path crosses the switching boundary at the point $\\mathbf {x}_p$ , ${\\mathbf {z}}$ exhibits a discontinuous jump which can be characterized by the jump matrix $J_p$ ${\\mathbf {z}}_p^+=J_p{\\mathbf {z}}_p^-$ where ${\\mathbf {z}}_p^-=\\lim _{t\\rightarrow t_p^-} {\\mathbf {z}}(t)$ and ${\\mathbf {z}}_p^+=\\lim _{t\\rightarrow t_p^+} {\\mathbf {z}}(t)$ are the iPRC just before and just after crossing the switching boundary at time $t_p$ .",
"As discussed in [46], the relation (REF ) between $\\mathbf {u}$ and ${\\mathbf {z}}$ for smooth systems remains valid at any transversal boundary crossing point.",
"In other words, $\\mathbf {u}_p^{-\\intercal }{\\mathbf {z}}_p^-=\\mathbf {u}_p^{+\\intercal }{\\mathbf {z}}_p^+$ holds at the transversal crossing point $\\mathbf {x}_p$ .",
"This leads to a relation between the saltation and jump matrices at $\\mathbf {x}_p$ $J_p^\\intercal S_p=I.$ The saltation matrix $S_p$ given by (REF ) has full rank at any transverse crossing point.",
"It follows that $J_p$ can be written as $J_p = ({S_p}^{-1})^\\intercal .$ The definition of $J_p$ given by (REF ), however, does not hold at a landing or a liftoff point.",
"See Remarks REF and REF for more details.",
"This motivates us to characterize discontinuous jumps in ${\\mathbf {z}}$ at non-transversal crossing points, which will then allow us to compute iPRC in the context of LCSC."
],
[
"Jumps in iPRC: sliding motion on a hard boundary",
"In this section, we now present our new results on the iPRC for LCSC.",
"As discussed in the previous section, the existence of the jump matrix (REF ) is guaranteed by the transversal flow condition (REF ), which, however, will no longer hold when part of the limit cycle slides along a boundary (e.g., Figure REF B).",
"Next, we establish the conditions relating ${\\mathbf {z}}$ at non-transversal crossings, including the landing point and the liftoff point.",
"Before that, we need to impose the following assumption on the asymptotic phase function for LCSC: Assumption 3.12 Within the stable manifold of a LCSC there is a well defined asymptotic phase function $\\phi (\\mathbf {x})$ satisfying $d\\phi /dt=1$ along the trajectory, where $\\phi $ is Lipschitz continuous.",
"Moreover, on the hard boundary $\\Sigma $ , the directional derivatives of $\\phi $ with respect to directions tangential to the surface are also Lipschitz continuous, except (possibly) at the liftoff and landing points.",
"Our main result, Theorem REF as shown below, gathers together several conclusions about the variational and infinitesimal phase response curve dynamics of a LCSC local to a sliding boundary.",
"For completeness, we review the established behavior of variational dynamics local to a sliding boundary (parts (a) and (b)) and present our new results regarding the iPRC in parts (c) through (e) (see Appendix for the proof).",
"Theorem 3.13 Consider a general LCSC described locally by (REF ) in the neighborhood of a hard boundary $\\Sigma $ , satisfying Assumption REF and Assumption REF .",
"The following properties hold for the variational dynamics $\\mathbf {u}$ and the iPRC ${\\mathbf {z}}$ along $\\Sigma $ : At the landing point of $\\Sigma $ , the saltation matrix is $S=I-n n^\\intercal $ , where $I$ is the identity matrix.",
"At the liftoff point of $\\Sigma $ , the saltation matrix is $S=I$ .",
"Along the sliding region within $\\Sigma $ , the component of ${\\mathbf {z}}$ normal to $\\Sigma $ is zero.",
"The normal component of ${\\mathbf {z}}$ is continuous at the landing point.",
"The tangential components of ${\\mathbf {z}}$ are continuous at both landing and liftoff points.",
"We make the following additional observations about Theorem REF : Remark 3.14 The following two statements follow directly from Theorem REF : It follows from (a) in Theorem REF that the component of $\\mathbf {u}$ normal to $\\Sigma $ vanishes when the LCSC hits $\\Sigma $ .",
"Once on the sliding region, the Jacobian used in the variational equation switches from $DF^{\\rm interior}$ to $DF^{\\rm slide}$ where $F^{\\rm slide}$ has zero normal component by construction (REF ).",
"Hence, the normal component of $\\mathbf {u}$ is stationary over time, and remains zero on the sliding region.",
"It follows from parts (d) and (e) in Theorem REF that the jump matrix of ${\\mathbf {z}}$ at a landing point is trivial (identity matrix).",
"Remark 3.15 Theorem REF excludes discontinuities in ${\\mathbf {z}}$ except at the liftoff point, and then only in its normal component.",
"Since the normal component of ${\\mathbf {z}}$ along each sliding component of a LCSC is zero by Theorem REF , a discontinuous jump occurring at a liftoff point must be a nonzero instantaneous jump, which cannot be specified directly in terms of the value of ${\\mathbf {z}}$ prior to the jump.",
"However, a time-reversed version of the jump matrix at the liftoff point, denoted as $\\mathcal {J}$ , is well defined as follows: $\\begin{array}{cccccccccc}{{\\mathbf {z}}_{\\text{lift}}^{-}}=\\mathcal {J} {{\\mathbf {z}}_{\\text{lift}}^{+}}\\end{array}$ where ${{\\mathbf {z}}_{\\text{lift}}^{-}}$ and ${{\\mathbf {z}}_{\\text{lift}}^{+}}$ are the iPRC just before and just after the trajectory crosses the liftoff point in forwards time, and $\\mathcal {J}$ at the liftoff point has the same form as the saltation matrix $S$ at the corresponding landing point $\\mathcal {J}=I-n {n}^\\intercal .$ That is, the component of ${\\mathbf {z}}$ normal to $\\Sigma $ becomes 0 as the trajectory enters $\\Sigma $ in backwards time.",
"Remark 3.16 Combining Theorem REF , Remarks REF and REF , we summarize the behavior of the solutions of the variational and adjoint equations $\\mathbf {u}$ and ${\\mathbf {z}}$ in limit cycles with sliding components: Table: NO_CAPTIONwhere $S$ is the regular saltation matrix, $\\mathcal {J}$ is the time-reversed jump matrix and $^\\perp $ denotes the normal component.",
"Remark 3.17 It follows directly from Remark REF that the relation between the saltation and jump matrices at a transversal boundary crossing point $J^\\intercal S=I$ (see (REF )) is no longer true at a landing or a liftoff point.",
"Instead, the following condition holds $\\mathcal {J}^\\intercal _p S_p = I-n n^\\intercal $ where $\\mathcal {J}_p$ and $S_p$ denote the time-reversed jump matrix and the regular saltation matrix at a landing or a liftoff point.",
"Remark 3.18 Assumption REF is necessary for the proof of part (c) in Theorem REF .",
"A stable limit cycle arising in a $C^r$ -smooth vector field, for $r\\ge 1$ , will have $C^r$ isochrons [61], [34].",
"In [46] and [62] the authors assume differentiability of the phase function with respect to a basis of vectors spanning a switching surface.",
"The assumption we require here is similarly plausible; it appears to hold at least for the model systems we have considered.",
"Next we illustrate the behavior of a limit cycle with sliding component via an analytically tractable planar model in § and a stick-slip oscillator in §.",
"In these two examples, we will see that a nonzero instantaneous jump discussed in Remark REF can occur in the normal component of ${\\mathbf {z}}$ at the liftoff point, reflecting a “kink” or nonsmooth feature in the isochrons (cf.",
"Figure REF ).",
"In both systems, the discontinuity in the iPRC reflects a curve of nondifferentiability in the asymptotic phase function propagating backwards along a trajectory from the liftoff point to the interior of the domain (see Figure REF A).",
"The presence of a discontinuous jump from zero to a nonzero normal component in ${\\mathbf {z}}$ in forward time implies that numerical evaluation of the iPRC (presented in Appendix ) should be accomplished via backward integration along the limit cycle.",
"In §REF , we further apply the iPRC developed for the LCSC system to study the synchronization of two weakly coupled stick-slip oscillators, which together form a four-dimensional nonsmooth system with two sliding components.",
"In addition to the iPRC, we also provide numerical algorithms for calculating the lTRC, the variational dynamics and the iSRC for a LCSC in a general nonsmooth system with hard boundaries, in Appendix ."
],
[
"Applications to a planar limit cycle model with sliding components\n",
"In this section, we apply our methods to a two dimensional, analytically tractable model that has a single interior domain with purely linear flow and hard boundary constraints that create a limit cycle with sliding components (LCSC).",
"We find the surprising result that the isochrons exhibit a nonsmooth “kink” propagating into the interior of the domain from the locations of the liftoff points, i.e.",
"the points where the limit cycle smoothly departs the boundary.",
"In addition, we show that using local timing response curve analysis gives significantly greater accuracy of the shape response than using a single, global, phase response curve.",
"MATLAB source code for simulating the model and reproducing the figures is available: https://github.com/yangyang-wang/LC_in_square.",
"In the interior of the domain $[-1,1]\\times [-1,1]$ , we take the vector field of a simple spiral source to define the interior dynamics of the planar model $\\frac{d\\textbf {x}}{dt}=F(\\textbf {x})=\\begin{bmatrix} \\alpha x - y\\\\x+\\alpha y \\end{bmatrix}$ where $\\textbf {x}=\\begin{bmatrix} x\\\\y\\end{bmatrix}$ and $\\alpha $ is the expansion rate of the source at the origin.",
"The rotation rate is fixed at a constant value 1.",
"The Jacobian matrix $DF$ evaluated along the limit cycle solution in the interior of the domain is $DF=\\begin{bmatrix} \\alpha & -1 \\\\ 1 & \\alpha \\end{bmatrix} .$ In what follows we will require $0<\\alpha < 1$ , so we have a weakly expanding source.",
"For illustration, $\\alpha =0.2$ provides a convenient value.",
"Every trajectory starting from the interior, except the origin, will eventually collide with one of the walls at $x=\\pm 1$ or $y=\\pm 1$ (in time not exceeding $\\frac{1}{2\\alpha }\\ln (2/(x(0)^2+y(0)^2)$ ).",
"As in §REF , we set the sliding vector field when the trajectory is traveling along the wall to be equal to the continuation of the interior vector field in the component parallel to the wall, while the normal component is set to zero (except where it is oriented into the domain interior).",
"The resulting vector fields of the planar LCSC model $F(\\mathbf {x})$ on the interior and along the walls are given in Table REF , and illustrated in Fig.",
"REF B.",
"Table: Vector field of the planar LCSC model on the interior and along the boundaries.The trajectory will naturally lift off the wall and return to the interior when the normal component of the unconstrained vector field changes from outward to inward, i.e., $(F^{\\text{interior}}\\cdot n^{\\text{wall}})|_{\\text{wall}}=0$ (see (REF )).",
"For instance, on the wall $x=1$ with a normal vector $n=[1,0]^\\intercal $ , we compute $F^{\\text{interior}}|_{\\text{wall}}\\cdot n^{\\text{wall}}=(\\alpha x-y)|_{\\text{wall}}=\\alpha -y=0.$ It follows that $y=\\alpha $ defines the liftoff condition on the wall $x=1$ .",
"For this planar model, there are four lift-off points with coordinates $(1,\\alpha ), (-\\alpha ,1), (-1,-\\alpha ), (\\alpha ,-1)$ on the walls $x=1,\\,y=1,\\,x=-1, y=-1$ , respectively.",
"Denote the LCSC produced by the planar model by $\\gamma (t)$ , whose time series over $[0, T_0]$ is shown in Figure REF A, where $T_0$ is the period.",
"The projection of $\\gamma (t)$ onto the $(x,y)$ -plane is shown in the right panel, together with an osculating trajectory that starts near the center and ends up running into the wall $x=1$ at the lift-off point $(1,\\alpha )$ (black star).",
"Figure: Simulation result of the planar LCSC model with parameter α=0.2\\alpha =0.2.",
"(A): Time series of the limit cycle γ(t)\\gamma (t) generated by the planar LCSC model over one cycle with initial condition γ(0)=[1,α] ⊺ \\gamma (0)=[1,\\alpha ]^\\intercal .",
"(B): Projection of γ(t)\\gamma (t) onto (x,y)(x,y) phase space (solid black) and the osculating trajectory (dashed black), starting near the center that ends up running into the wall at the liftoff point [1,α] ⊺ [1,\\alpha ]^\\intercal (black star).Red arrows represent the vector field.Next we implement algorithms given in Appendix to find the timing and shape responses of the LCSC to both instantaneous perturbations and sustained perturbation.",
"We start by finding the iPRC for the LCSC to understand the timing response, and then solve the variational equation to find the linear shape response of the planar LCSC model to an instantaneous perturbation.",
"Lastly, we compute the iSRC when the applied sustained perturbations are both uniform and nonuniform, to understand the shape response of the planar LCSC model to sustained perturbations."
],
[
"Infinitesimal phase response analysis",
"In the case of weak coupling or small perturbations of a strongly stable limit cycle, a linearized analysis of the phase response curve – the iPRC – suffices to predict the behavior of the perturbed system.",
"When trajectories slide along a hard boundary, however, the linearized analysis breaks down.",
"For nonsmooth systems such as the LCSCs, the asymptotic phase function $\\phi (\\mathbf {x})$ may itself be nonsmooth at certain locations, even when it remains well defined; its gradient (i.e., the iPRC) may therefore be discontinuous at those locations.",
"Nevertheless, one may be able to derive a consistent first order approximation to the phase response curve notwithstanding that the directional derivative (REF ) may not be well defined, as discussed in §.",
"The dynamics of the planar LCSC model are smooth except for the discontinuities when crossing the switching boundaries, that is, entering or exiting the walls.",
"The iPRC, ${\\mathbf {z}}(t)$ , will be continuous in the interior domain as well as in the interior of the four boundaries.",
"As discussed in §REF , the discontinuity of iPRCs only occurs at the liftoff point.",
"By Remark REF , the time-reversed jump matrix at a liftoff point, which takes the iPRC just after crossing the liftoff point to the iPRC just before crossing the liftoff point in backwards time, is given by $\\mathcal {J}=\\left[\\begin{array}{rrrrrrrrrrrrrrrrrr} 0 & 0 \\\\0 & 1 \\end{array}\\right]$ when the trajectory leaves the walls $x=\\pm 1$ , and is given by $\\mathcal {J}=\\left[\\begin{array}{rrrrrrrrrrrrrrrrrr} 1 & 0 \\\\0 & 0 \\end{array}\\right]$ when the trajectory leaves the walls $y=\\pm 1$ .",
"Figure: iPRCs for the planar LCSC model with parameter α=0.2\\alpha =0.2.",
"(A): Trajectories and isochrons for the LCSC model.The solid black and dashed black curves are the same as in Figure B.The colored scalloped curves are isochrons of the LCSC γ(t)\\gamma (t) (black solid) corresponding to 50 evenly distributed phases nT 0 /50nT_0/50, n=1,⋯,50n=1,\\cdots , 50.We define the phase at the liftoff point (black star) to be zero.",
"(B): iPRCs for the planar LCSC model.The blue and red curves represent the iPRC for perturbations in the positive xx and yy directions, respectively.The intervals during which γ(t)\\gamma (t) slides along a wall are indicated by the shaded regions.While y=+1y=+1, the iPRC vector is parallel to the wall (𝐳 y ≡0{\\mathbf {z}}_y\\equiv 0) and oriented opposite to the direction of flow (𝐳 x <0{\\mathbf {z}}_x < 0).Similarly, on the remaining walls, the iPRC vector has zero normal component relative to the active constraint wall, and parallel component opposite the direction of motion.Figure REF A shows the limit cycle (solid black curve), the osculating trajectory (dashed black curve) corresponding to the liftoff point (black star, $\\theta =0$ ) and the isochrons computed from a direct method, starting from a grid of initial conditions and tracking the phase of final locations (colored scalloped curves).",
"There appears to be a “kink\" in the isochron function, propagating backwards in time along the trajectories that encounter the boundaries exactly at the liftoff points, such as the dashed curve.",
"This apparent discontinuity in the gradient of the isochron function in the interior of the domain exactly corresponds, at the boundary, with the point of discontinuity occurring in the iPRC along the limit cycle (cf.",
"Remark REF ).",
"According to Figure REF , the isochron curves are perpendicular to the sliding region of the wall at which the interior vector field is pointing outward.",
"That is, the normal component of the iPRC when the trajectory slides along a wall is equal to 0.",
"There is no jump in ${\\mathbf {z}}$ when the trajectory enters the wall, but instead a discontinuous jump from zero to nonzero occurs in the normal component of ${\\mathbf {z}}$ at the liftoff point.",
"All of these observations are consistent with iPRC ${\\mathbf {z}}$ (Figure REF , right) that is computed using Algorithm for ${\\mathbf {z}}$ in §REF based on Theorem REF .",
"After the trajectory lifts off the east wall ($x=1$ ) at the point marked $\\theta =0$ in Figure REF A (black star), a perturbation along the positive $x$ -direction (resp., positive $y$ -direction) causes a phase delay (resp., advance).",
"While the timing sensitivity of the LCSC to small perturbations in the $x$ -direction reaches a local maximum before reaching the next wall $y=1$ , the phase advance caused by the $y$ -direction perturbation decreases continuously to 0 as the trajectory approaches $y=1$ .",
"As the trajectory is sliding along the wall ($y=1$ ), the positive $y$ -direction perturbation that is normal to the wall has no effect on the LCSC and hence will not affect its phase.",
"Moreover, we showed in Theorem REF that a perturbation in the negative $y$ -direction also has no effect on the phase, since the perturbed trajectory returns to the wall within time $O(\\varepsilon )$ , with a net phase offset that is at most $O(\\varepsilon ^2)$ , where $\\varepsilon $ is the size of the perturbation.",
"As the trajectory lifts off the wall, there is a discontinuous jump in ${\\mathbf {z}}_y$ , so that a negative $y$ -direction perturbation applied immediately after the liftoff point leads to a phase advance.",
"On the other hand, on the sliding region of the wall $y=1$ , a perturbation along the positive $x$ -direction, against the direction of the flow, results in a phase delay, which decreases in size as the phase increases, and becomes 0 upon reaching the next wall, $x=-1$ .",
"The timing sensitivity of the LCSC to perturbations applied afterwards are similar to what are observed in the first quarter of the period due to the $\\mathbf {Z}_4$ -symmetry $\\sigma (x,y) = (-y,x)$ .",
"The linear change in the oscillation period of the LCSC in response to a static perturbation can then be estimated by taking the integral of the iPRC multiplying the given perturbation, as shown in the last step in Algorithm for ${\\mathbf {z}}$ (§ REF ).",
"As noted before, the change in period will be needed to solve (REF ) for the iSRC to understand how this perturbation affects the shape of the LCSC.",
"In this example, the interior vector field (REF ) is linear.",
"Therefore its Jacobian is constant, and the iPRC may be obtained analytically [46].",
"The resulting curves are indistinguishable from the numerically calculated curves shown in Fig.",
"REF B."
],
[
"Variational analysis",
"Suppose a small instantaneous perturbation, applied at time $t=0$ , leads to an initial displacement $\\mathbf {u}(0)=\\tilde{\\gamma }(0)-\\gamma (0)$ , where $\\gamma (0)=[1,\\alpha ]$ is the liftoff point (black star in Figs.",
"REF B and REF A) as in the previous section.",
"We use the variational analysis to study how this perturbation evolves over time.",
"Similar to the iPRC, $\\mathbf {u}(t)$ will be continuous everywhere in the domain except when entering or exiting the walls.",
"In contrast to the iPRC, $\\mathbf {u}$ is continuous at all liftoff points, but exhibits discontinuous saltations when the trajectory enters a wall.",
"According to Theorem REF , the saltation matrix $S$ , which takes $\\mathbf {u}$ just before entering a wall to $\\mathbf {u}$ just after entering the wall in forwards time, for the planar LCSC model is given by $S=\\left[\\begin{array}{rrrrrrrrrrrrrrrrrr} 0 & 0 \\\\0 & 1 \\end{array}\\right]$ when the trajectory enters the walls $x=\\pm 1$ , and is given by $S=\\left[\\begin{array}{rrrrrrrrrrrrrrrrrr} 1 & 0 \\\\0 & 0 \\end{array}\\right]$ when the trajectory enters the walls $y=\\pm 1$ .",
"Solutions to the variational equation of the planar LCSC model with the given initial condition $\\mathbf {u}(0)$ can be computed using Algorithm for $\\mathbf {u}$ in §REF .",
"As discussed in Remark REF , an alternative way to find the displacement $\\mathbf {u}(t)$ is to compute the fundamental solution matrix $\\Phi (t, 0)$ by running Algorithm for $\\mathbf {u}$ twice and then to evaluate $\\mathbf {u}(t) = \\Phi (t,0) \\mathbf {u}(0)$ .",
"The advantage of the latter approach is that once $\\Phi (t,0)$ is obtained, it can be used to compute $\\mathbf {u}(t)$ with any given initial value by evaluating a matrix multiplication instead of solving the variational equation.",
"Here, by taking $[1,0]$ and $[0, 1]$ as the initial conditions for $\\mathbf {u}$ at the liftoff point A, we apply Algorithm for $\\mathbf {u}$ to compute the time evolution of the two columns for the fundamental matrix $\\Phi (t,0)$ .",
"A simple calculation shows that the monodromy matrix $\\Phi (T_0,0)$ has an eigenvalue $+1$ , whose eigenvector $[0,1]$ is tangent to the limit cycle at the liftoff point, as expected (Remark REF ).",
"It follows that if the initial displacement at the liftoff point is along the limit cycle direction, then the displacement after a full period becomes the same as the initial one.",
"To see this, we take the initial displacement $\\mathbf {u}(0)=[0,\\varepsilon ]$ where $\\varepsilon =0.1$ to be the tangent vector of the limit cycle at the liftoff point, and compute $\\mathbf {u}(t)$ , $x$ and $y$ components of which are shown in red dotted curves in Figure REF B,D.",
"The saltations in $\\mathbf {u}$ at time when the trajectory hits the walls can be clearly distinguished in the plot.",
"Moreover, $\\mathbf {u}(T_0)=\\mathbf {u}(0)$ as we expect.",
"To further validate the accuracy of $\\mathbf {u}$ , we solve and plot $\\gamma (t)$ with $\\gamma (0)=[1,\\alpha ]$ (black, Figure REF A,C) and the perturbed trajectory $\\tilde{\\gamma }(t)$ with $\\tilde{\\gamma }(0)=[1,\\alpha ]+\\mathbf {u}(0)$ (red dotted, Figure REF A,C).",
"The differences between the two trajectories along the $x$ -direction and the $y$ -direction are indicated by the black lines in Figure REF B and D, both showing good agreements with the approximated displacements computed from the variational equation, indicated by the red dotted lines in Figure REF B and D. Such an approximation becomes better as the perturbation size $\\varepsilon $ gets smaller (simulation result not shown).",
"Figure: Linear shape response 𝐮(t)\\mathbf {u}(t) of the LCSC trajectory γ(t)\\gamma (t) to an instantaneous perturbation applied at γ(0)=[1,α]\\gamma (0)=[1,\\alpha ] when time =0=0 where α=0.2\\alpha =0.2.The initial displacement is 𝐮(0)=[0,ε]\\mathbf {u}(0)=[0,\\varepsilon ] where ε=0.1\\varepsilon =0.1.",
"(A, C) Time series of γ(t)\\gamma (t) (black solid) and γ ˜(t)\\tilde{\\gamma }(t) (red dotted) with a perturbed initial condition γ ˜(0)=[1,α+ε]\\tilde{\\gamma }(0)=[1,\\alpha +\\varepsilon ].",
"(B, D) The difference between γ ˜(t)\\tilde{\\gamma }(t) and γ(t)\\gamma (t) obtained by direct calculation from the left panels (black) and the displacement solution 𝐮(t)\\mathbf {u}(t) obtained using the Algorithm for 𝐮\\mathbf {u} (red dotted).",
"(A) and (B) show trajectories and the displacement along the xx-direction, while (C) and (D) show trajectories and displacements along the yy-direction.Shaded regions have the same meanings as in Figure .Next, we study the effects of static perturbations on the timing and shape using the iPRC and iSRC."
],
[
"Shape response analysis",
"In this section, we illustrate how to compute the iSRC $\\gamma _1$ , the linear shape responses of the LCSC to small static perturbations.",
"Recall that we use $\\gamma _0(t)$ with period $T_0$ and $\\gamma _{\\varepsilon }(t)$ with period $T_{\\varepsilon }$ to denote the original and the perturbed LCSC solutions.",
"We write $\\gamma _1$ for the linear shift in the limit cycle shape in response to the static perturbation as indicated by (REF ), which we also repeat here: $\\gamma _\\varepsilon (\\tau _\\varepsilon (t))=\\gamma _0(t)+\\varepsilon \\gamma _1(t)+O(\\varepsilon ^2),$ where the time for the perturbed LCSC is rescaled to be $\\tau _\\varepsilon (t)$ to match the unperturbed time points.",
"The iSRC $\\gamma _1$ satisfies the nonhomogeneous variational equation (REF ).",
"To solve this equation, an estimation of the timing scaling factor $\\nu _1$ , determined by the choice of time rescaling $\\tau _\\varepsilon (t)$ , is needed.",
"Here we consider two kinds of static perturbations on the planar LCSC model: global perturbation and piecewise perturbation."
],
[
"Global perturbation.",
"We apply a small static perturbation to the planar LCSC model by increasing the model parameter $\\alpha $ by $\\varepsilon $ globally: $\\alpha \\rightarrow \\alpha +\\varepsilon $ .",
"To compare the LCSCs before and after perturbation at corresponding time points, we rescale the perturbed trajectory uniformly in time so that $\\tau _\\varepsilon (t) = T_{\\varepsilon }t/T_0$ .",
"It follows that $\\nu _1 = T_1/T_0$ , where the linear shift $T_1:=\\lim _{\\varepsilon \\rightarrow 0}(T_{\\varepsilon }-T_0)/\\varepsilon $ can be estimated using the iPRC (see (REF )).",
"Using Algorithm for $\\gamma _1$ with uniform rescaling, we numerically compute the iSRC $\\gamma _1(t)$ for $\\varepsilon =0.01$ .",
"The $x$ and $y$ components of $\\varepsilon \\gamma _1(t)$ are shown by the red curves in Figure REF A, both of which show good agreement with the numerical displacement $\\gamma _\\varepsilon (\\tau _\\varepsilon (t))-\\gamma _0(t)$ (black solid), as expected from our theory.",
"For $\\varepsilon $ over a range $[0, 0.01]$ , we repeat the above procedure and compute the Euclidean norms of both the numerical displacement vector $\\gamma _{\\varepsilon }(\\tau _\\varepsilon (t))-\\gamma (t)$ (Figure REF B, black solid) and the approximated displacement vector $\\varepsilon \\gamma _1(t)$ (Figure REF B, red dotted) over one cycle.",
"From the plot, we can see that the iSRC with uniform rescaling of time gives a good first-order $\\varepsilon $ approximation to the shape response of the planar LCSC model to a global static perturbation.",
"Figure: iSRC of the LCSC model to a small perturbation α→α+ε\\alpha \\rightarrow \\alpha +\\varepsilon with unperturbed parameter α=0.2\\alpha =0.2.",
"(A) Time series of the difference between the perturbed and unperturbed solutions along the xx-direction (top panel) and the yy-direction (bottom panel) with ε=0.01\\varepsilon =0.01.The black curve denotes the numerical displacement computed by subtracting the unperturbed solution trajectory from the perturbed trajectory, after globally rescaling time.The red dashed curve denotes the product of ε\\varepsilon and the shape response curve solution.Shaded regions have the same meanings as in Figure .",
"(B) The norm of the numerical difference (black) and the product of ε\\varepsilon and the iSRC (red dashed) grow linearly with respect to ε\\varepsilon with nearly identical slope, indicating that the iSRC is very good for approximating the numerical difference over a range of ε\\varepsilon and improves with smaller ε\\varepsilon ."
],
[
"Piecewise perturbation.",
"Uniform rescaling of time as used above is the simplest choice among many possible rescalings, and is shown to be adequate in the global perturbation case for computing an accurate iSRC.",
"As discussed in §, in certain cases we may instead need the technique of local timing response curves (lTRCs) to obtain nonuniform choices of rescaling for greater accuracy.",
"As an illustration, we add two local timing surfaces $\\Sigma ^{\\rm in}$ and $\\Sigma ^{\\rm out}$ to the planar LCSC model (see Figure REF A).",
"We denote the subdomain above $\\Sigma ^{\\rm in}$ and $\\Sigma ^{\\rm out}$ by region I (${\\mathcal {R}}^{\\rm I}$ ) and denote the remaining subdomain by region II (${\\mathcal {R}}^{\\rm II}$ ).",
"Moreover, we introduce a new parameter $\\omega $ , the rotation rate of the source at the origin, that has previously been fixed at 1, and rewrite the interior dynamics of the planar LCSC model as $\\frac{d\\textbf {x}}{dt}=F(\\textbf {x})=\\begin{bmatrix} \\alpha x -\\omega y\\\\\\omega x+\\alpha y \\end{bmatrix}.$ The vector fields on a given wall are obtained by replacing the coefficient of $y$ in $dx/dt$ (in Table REF ) by $-\\omega $ and replacing the coefficient of $x$ in $dy/dt$ (in Table REF ) by $\\omega $ on that wall.",
"We apply a static piecewise perturbation to the system by letting $(\\alpha ,\\omega )\\rightarrow (\\alpha +\\varepsilon ,\\omega -\\varepsilon )$ over region I but not region II.",
"Such a piecewise constant perturbation affects both the expansion and rotation rates of the source in region $\\rm I$ , and hence will lead to different timing sensitivities of $\\gamma (t)$ in the two regions.",
"It is therefore natural to use piecewise uniform rescaling when computing the shape response curve as opposed to using a uniform rescaling.",
"In the following, we first compute the lTRC (see Figure REF ) and use it to estimate the two time rescaling factors for ${\\mathcal {R}}^{\\rm I}$ and ${\\mathcal {R}}^{\\rm II}$ , which are denoted by $\\nu _1^{\\rm I}$ and $\\nu _1^{\\rm II}$ , respectively.",
"We then show the iSRC computed using the piecewise uniform rescaling factors provides a more accurate representation of the shape response to the piecewise static perturbation than using a uniform rescaling (see Figure REF and REF ).",
"Figure: lTRC of the planar LCSC model under perturbation (α,ω)→(α+ε,ω-ε)(\\alpha ,\\omega )\\rightarrow (\\alpha +\\varepsilon ,\\omega -\\varepsilon ) over region I with unperturbed parameters α=0.2\\alpha =0.2 and ω=1\\omega =1 held fixed in region II.",
"(A) Projection of the limit cycle solution to the planar model with two new added switching surfaces Σ in \\Sigma ^{\\rm in} (green dashed line) and Σ out \\Sigma ^{\\rm out} (blue dashed line) onto its phase plane.",
"(B) Time series of the lTRC η I \\eta ^{\\rm I} from t in t^{\\rm in} (the time of entry into region I at 𝐱 in \\mathbf {x}^{\\rm in}) to t out t^{\\rm out} (the time of exiting region I at 𝐱 out \\mathbf {x}^{\\rm out}).A discontinuous jump occurs when the trajectory exits the wall y=1y=1 indicated by the right boundary of the shaded region, which has the same meaning as in Figure .Although the lTRC $\\eta $ is defined throughout the domain, estimating the effect of the perturbation localized to region I only requires evaluating the lTRC in this region.",
"Figure REF B shows the time series of $\\eta ^{\\rm I}$ for the planar LCSC model in region I, obtained by numerically integrating the adjoint equation (REF ) backward in time with the initial condition of $\\eta ^{\\rm I}$ given by its value at the exit point of region I denoted by $\\mathbf {x}^{\\rm out}$ (see Algorithm for $\\eta ^j$).",
"Similar to the iPRC, the $y$ component of the lTRC $\\eta ^\\text{I}$ shown by the red curve in Figure REF B is zero along the wall $y=1$ , and the only discontinuous jump of $\\eta ^{\\rm I}$ occurs at the liftoff point.",
"Note that $\\eta ^{\\rm I}$ is defined as the gradient of the time remaining in ${\\mathcal {R}}^{\\rm I}$ until exiting through $\\Sigma ^{\\rm out}$ .",
"If the $x$ or $y$ component of $\\eta ^{\\rm I}$ is positive then the perturbation along the positive $x$ -direction or $y$ -direction increases the time remaining in ${\\mathcal {R}}^{\\rm I}$ , and the exit from ${\\mathcal {R}}^{\\rm I}$ will occur later.",
"On the other hand, if the $x$ or $y$ component $\\eta ^{\\rm I}$ is negative then the perturbation along the positive $x$ -direction or $y$ -direction decreases the time remaining in ${\\mathcal {R}}^{\\rm I}$ , and the exit from ${\\mathcal {R}}^{\\rm I}$ will occur sooner.",
"The relative shift in time spent in ${\\mathcal {R}}^{\\rm I}$ caused by a static perturbation can therefore be estimated using the lTRC (see (REF )) as illustrated in the last step of Algorithm for $\\eta ^j$.",
"Note that the first term in (REF ) implies that the timing change in a region generically depends on the shape change at the corresponding entry point, leading to the possibility of bidirectional coupling between timing and shape changes.",
"However, in this planar system, a perturbed trajectory with $\\varepsilon \\ll 1$ will converge back to the original trajectory, within region II (where the perturbation is absent), in finite time.",
"Under these circumstances, there is no shift between the perturbed and unperturbed trajectories in the entry location to region I.",
"Hence, in this case, the local timing shift does not depend on the shape change.",
"Let $T_{0}^{\\rm I}$ denote the time spent in region I, and let $T_{0}^{\\rm II}=T_{0}-T_{0}^{\\rm I}$ denote the time spent in region II (recall $T_{0}$ is the total period).",
"The linear shift in $T_0^{\\rm I}$ , denoted by $T_1^{\\rm I}$ , can be estimated using the lTRC $\\eta ^{\\rm I}$ as discussed above.",
"By definition the two time rescaling factors required to compute the iSRC are given by $\\nu _1^{\\rm I}=\\frac{T_1^{\\rm I}}{T_0^{\\rm I}}$ and $\\nu _1^{\\rm II}=\\frac{T_1-T_1^{\\rm I}}{T_0^{\\rm II}}$ where the global relative change in period, $T_1$ , can be estimated using the iPRC as discussed before.",
"With $\\nu _1^{\\rm I}$ and $\\nu _1^{\\rm II}$ known, we take $\\mathbf {x}^{\\rm in}$ , the coordinate of the entry point into ${\\mathcal {R}}^{\\rm I}$ , as the initial condition for $\\gamma (t)$ and apply Algorithm for $\\gamma _1$ with piecewise uniform rescaling to compute the iSRC $\\gamma _1$ for $\\varepsilon =0.1$ .",
"The $x$ and $y$ components of $\\varepsilon \\gamma _1$ are shown by the red dashed curves in Figure REF B, both of which show good agreement with the numerical displacement $\\gamma _{\\varepsilon }(\\tau _\\varepsilon (t)) - \\gamma (t)$ (black solid curves).",
"Here the rescaling $\\tau _\\varepsilon (t)$ is piecewise uniform: $\\tau _\\varepsilon (t)=\\left\\lbrace \\begin{array}{cccccccccc}t^{\\rm in}+T_{\\varepsilon }^{\\rm I}(t-t^{\\rm in})/T_0^{\\rm I}, & \\gamma (t) \\in {\\mathcal {R}}^{\\rm I}\\\\\\\\t^{\\rm in}+T_{\\varepsilon }^{\\rm I}+T_{\\varepsilon }^{\\rm II}(t-t^{\\rm out})/T_0^{\\rm II}, & \\gamma (t) \\in {\\mathcal {R}}^{\\rm II}\\\\\\end{array}\\right.$ where $T_{\\varepsilon }^i$ denotes the time $\\gamma _{\\varepsilon }$ spends in ${\\mathcal {R}}^i$ with $i\\in \\lbrace \\rm I, II\\rbrace $ .",
"It follows that the exit time of the trajectory from region I before (Figure REF B, vertical blue line) and after (Figure REF B, vertical magenta line) perturbation are the same.",
"As a comparison, for $\\varepsilon =0.1$ , we also compute the iSRC and the numerical displacement using the uniform rescaling of time as we did in the global perturbation case (see Figure REF A).",
"The difference between the vertical blue and magenta lines (the time when the unperturbed and perturbed trajectories leave region I) indicates region I and region II have different timing sensitivities.",
"As expected, the resulting $\\varepsilon \\gamma _1$ no longer shows good agreement with the numerical displacement obtained from subtracting the unperturbed solution from the rescaled perturbed solution.",
"Figure: A small perturbation is applied to the planar model over region I in which (α,ω)→(α+ε,ω-ε)(\\alpha ,\\omega )\\rightarrow (\\alpha +\\varepsilon ,\\omega -\\varepsilon ) with unperturbed parameters α=0.2\\alpha =0.2, ω=1\\omega =1 and perturbation ε=0.1\\varepsilon =0.1.Time series of the difference between the perturbed and unperturbed solutions along the xx-direction (top panel) and the yy-direction (lower panel) using (A) the global rescaling factor and (B) two different rescaling factors within regions I and II.The vertical blue dashed line denotes the exit time of the unperturbed trajectory from Region I, while the vertical magenta solid line denotes the exit time of the rescaled perturbed trajectory from Region I.Other color codings of lines are the same as in Figure A.Shaded regions have the same meanings as in Figure .Piecewise uniform rescaling, on the other hand, leads to a more accurate iSRC for the LCSC model (REF ) than uniform rescaling, when the LCSC $\\gamma (t)$ experiences distinct timing sensitivities for $\\varepsilon =0.1$ .",
"Fig.",
"REF contrasts the accuracy of the linearized shape response using global (A) versus local (B) timing response curves, for $\\varepsilon =0.1$ .",
"We also show the same conclusion holds for other $\\varepsilon $ values.",
"To this end, for $\\varepsilon $ over a range of $[0,0.1]$ we repeat the above procedure and compute the Euclidean norms of both the numerical displacement vector and the displacement vector approximated by the iSRC, as illustrated in Figure REF A.",
"The numerical and approximated norm curves using the uniform rescaling are shown in red solid and red dotted lines, while the numerical and approximated norm curves using the piecewise uniform rescaling are shown in blue solid and blue dotted lines.",
"Unsurprisingly, the norms of the displacements between the perturbed and original trajectory grow approximately linearly with respect to $\\varepsilon $ , and the displacement norms with piecewise uniform rescaling are smaller than that with uniform rescaling.",
"The fact that the difference between the lines in red is much bigger than the difference between the lines in blue suggests that the piecewise uniform rescaling gives a more accurate iSRC than using the uniform rescaling for $\\varepsilon \\in [0,0.1]$ , as we expect.",
"This heightened accuracy is further demonstrated in Figure REF B, where the relative difference between the numerical and approximated norms with uniform rescaling (red curve) is significantly larger than the relative difference when using piecewise uniform rescaling (blue curve).",
"Figure: A small parametric perturbation is applied to the planar model over region I in which (α,ω)→(α+ε,ω-ε)(\\alpha ,\\omega )\\rightarrow (\\alpha +\\varepsilon ,\\omega -\\varepsilon ) with unperturbed parameters α=0.2\\alpha =0.2, ω=1\\omega =1.",
"(A): Values of the Euclidean norm of (γ ε (τ ε (t))-γ(t))(\\gamma _\\varepsilon (\\tau _\\varepsilon (t))-\\gamma (t)) computed numerically (solid curve) versus those computed from the iSRC (dashed curve), as ε\\varepsilon varies.The norms grow approximately linearly with respect to ε\\varepsilon .",
"The approximation obtained by the iSRC when using piecewise uniform rescaling (blue) is closer to the actual simulation than using the uniform rescaling (red).",
"(B): The relative difference between the actual and approximated norms with a uniform rescaling (red) is larger than that when piecewise uniform rescaling is used (blue).The difference between the two curves expands as ε\\varepsilon increases."
],
[
"Applications to stick-slip oscillators",
"In this section, we return to our motivating example — the stick-slip oscillator — presented in §, which is a piecewise smooth system that exhibits a LCSC (see Figure REF ).",
"In §REF , we apply the analysis of lTRC and iSRC to compute the shape response of the single stick-slip oscillator to parametric perturbations as done in §.",
"In §REF we use the phase reduction method to show that two weakly coupled identical stick-slip oscillators exhibit anti-phase synchronization."
],
[
"Shape response of a stick-slip oscillator to parametric perturbations",
"We consider a stick-slip system consisting of a block of mass $m$ supported by a moving belt with constant velocity $u$ .",
"The block is connected to a fixed support by an elastic spring with stiffness $k$ and a linear dashpot with damping coefficient $c$ .",
"The surface between the block and the belt is rough so that the belt exerts a static friction force on the block which sticks to the belt during the stick phase until the elastic force due to the spring and the damping force generated by the dashpot build up to exceed the maximum static friction force.",
"At this point the slip phase begins and the slipping motion is described by the following equation [20], [11], $mx^{\\prime \\prime }+cx^{\\prime }+kx = f(x^{\\prime }-u),$ where $x(t)$ is the displacement of the oscillator from the position at which the spring assumes its natural length; $^{\\prime }$ and $^{\\prime \\prime }$ indicate first and second order differentiation with respect to $t$ .",
"The kinetic friction force of the block for $x^{\\prime }< u$ is given by $f(x^{\\prime }-u) = \\frac{1-\\delta }{1-\\gamma (x^{\\prime }-u)}+\\delta +\\eta (x^{\\prime }-u)^2$ where $\\delta \\in [0, 1],\\gamma >0$ , and $\\eta >0$ .",
"When $x^{\\prime }>u$ , the kinetic friction force is $\\frac{-(1-\\delta )}{1+\\gamma (x^{\\prime }-u)}-\\delta -\\eta (x^{\\prime }-u)^2$ .",
"The maximum static friction force is assumed to be $f_s=1$ when there is zero relative velocity (i.e., $x^{\\prime }=u$ ) and therefore the friction force is continuous in the stick-slip transition, whereas in general such a transition may be characterized by a finite jump in the friction force.",
"The slipping motion governed by (REF ) will continue to the point where there is no relative motion between the block and belt so that $x^{\\prime }=u$ , and the elastic and damping forces are balanced by the static friction force so that $x^{\\prime \\prime }=0$ .",
"Figure: Simulation result of the stick-slip system () when m=1,k=1,c=0.1,δ=0.5,γ=1,η=0.001,u=0.5m=1,k=1,c=0.1,\\delta =0.5,\\gamma =1,\\eta =0.001,u=0.5.",
"Left: Time series of the LCSC with initial condition [1.4127,0.0829] ⊺ [1.4127, 0.0829]^\\intercal .",
"The shaded region indicates the stick phase.",
"Right: Projection of the LCSC from the left onto (x,v)(x,v) phase space (solid black) and the hard boundary Σ:v≡u\\Sigma : v\\equiv u (solid blue).",
"The red star symbol denotes the starting point.The continuous repetition of sticking and slipping motions with appropriate parameters can lead to a stick-slip limit cycle with sliding components (LCSC) within one zone $x^{\\prime }\\le u$ .",
"See Figure REF .",
"The sliding region ${\\mathcal {R}}^{\\rm slide}$ when the system is constrained to one less degree of freedom by a hard boundary, corresponds physically to the stick phase of the stick-slip system, when the block is captured by the moving belt and is carried along with the velocity $x^{\\prime }\\equiv u$ until it escapes.",
"It follows that the hard boundary is $\\Sigma =\\lbrace x^{\\prime }=u\\rbrace $ with a unit normal vector $n=[0, 1]^\\intercal $ .",
"Letting $v = x^{\\prime }$ and $\\mathbf {x}=[x,v]^\\intercal $ , we rewrite the stick-slip system in the form of (REF ) where $\\frac{d\\mathbf {x}}{dt}=F(\\mathbf {x}):={\\left\\lbrace \\begin{array}{ll}\\begin{array}{lclr}F^{\\rm interior}(\\mathbf {x}) &=& \\left[\\begin{array}{cccccccccccc} v \\\\ -\\frac{k}{m}x-\\frac{c}{m}v+\\frac{f(v-u)}{m} \\end{array}\\right],& \\mathbf {x}\\in {\\mathcal {R}}^{\\rm interior} \\\\F^{\\rm slide}(\\mathbf {x}) &=& \\left[\\begin{array}{cccccccccccc} u \\\\ 0 \\end{array}\\right], &\\mathbf {x}\\in {\\mathcal {R}}^{\\rm slide}\\subset \\Sigma .\\end{array}\\end{array}\\right.",
"}$ where $F^{\\rm interior}$ is the vector field during the slip phase and $F^{\\rm slide}$ is the vector field during the stick phase.",
"It follows from Definition REF and Definition REF that ${\\mathcal {R}}^{\\rm slide} = \\lbrace \\mathbf {x}\\in \\Sigma \\, |\\, x < \\frac{1-cu}{k} \\rbrace $ and the flow exits the sliding region (i.e., the stick phase ends) when $x=\\frac{1-cu}{k}$ .",
"In other words, the stick phase terminates when the maximum static friction force $f_s=1$ acting on the block is exceeded by the other two forces; that is, $kx+cx^{\\prime }=f_s$ where $x^{\\prime }=u$ during the stick phase.",
"The interior domain is therefore ${\\mathcal {R}}^{\\rm interior} = \\lbrace \\mathbf {x}\\in \\Sigma \\, |\\, x \\ge \\frac{1-cu}{k} \\rbrace \\cup \\lbrace \\mathbf {x}\\in \\mathbb {R}^2\\,|\\, v< u\\rbrace .$ Remark 5.1 In contrast to our setup in which the stick-slip system (REF ) is restricted to the domain $v\\le u$ , [20], [11] used Filippov's Convex Method to write the stick-slip system in the full space $\\mathbb {R}^2$ $\\frac{d\\textbf {x}}{dt}=F(\\mathbf {x}):=\\left\\lbrace \\begin{array}{cccccccccc}F^{\\rm I}(\\mathbf {x}), & v<u \\\\F^{\\Sigma }(\\mathbf {x}), & v=u \\\\F^{\\rm II}(\\mathbf {x}), & v>u\\end{array}\\right.$ where $F^{\\Sigma }(\\mathbf {x})=(1-\\alpha (\\mathbf {x}))F^{\\rm I}+\\alpha (\\mathbf {x}) F^{\\rm II}$ and $\\alpha (\\mathbf {x})=\\frac{n^\\intercal F^{\\rm I}}{n^\\intercal (F^{\\rm I}-F^{\\rm II})}$ .",
"Nonetheless, the two systems (REF ) and (REF ) exhibit the same LCSC that has the same iPRC and iSRC in response to parametric perturbations.",
"This is because $F^{\\rm I}$ is identical to $F^{\\rm interior}$ in (REF ) and, moreover, it follows from direct calculation that $F^{\\Sigma }$ is identical to $F^{\\rm slide}$ in (REF ).",
"Since we are only interested in understanding the timing and shape response of the stick-slip LCSC to small parametric perturbations, it is enough for us to work with (REF ) that is defined in the domain $v\\le u$ where the LCSC exists.",
"Figure: Shape response of the stick-slip system () to a small parametric perturbation of the damping coefficient c→c+εc\\rightarrow c+\\varepsilon with ε=-0.05\\varepsilon =-0.05 and other unperturbed parameters the same as in Figure .",
"Time series of the displacement between the perturbed and unperturbed solutions along the xx-direction (resp., the vv-direction) are shown in the top panel (resp., the bottom panel).",
"The black curve denotes the numerical displacement computed by subtracting the unperturbed solution trajectory from the time-rescaled perturbed trajectory.",
"The red dashed curve denotes the product of the iSRC solution and the perturbation size ε\\varepsilon .",
"The white and shaded regions indicate the slip and stick phases, respectively, and the blue/magenta dashed line indicates the transition.When $m=1,k=1,c=0.1,\\delta =0.5,\\gamma =1,\\eta =0.001,u=0.5$ , the system (REF ) exhibits a LCSC, denoted by $\\gamma (t)$ , that slides along the hard boundary $\\Sigma $ (see Figure REF ).",
"In the following, we study the shape response of $\\gamma (t)$ to a small parametric perturbation using the iSRC and lTRC, as discussed in §.",
"Similar to the constructed planar model (REF ) with two local timing surfaces, variations of the model parameters in (REF ) (e.g., the damping coefficient $c$ ) only affect the vector field in the interior domain (i.e., during the slip phase), which naturally leads to a piecewise perturbation on the system.",
"The lTRC is therefore needed to compute rescaling factors in different regions (interior/boundary) or phases (slip/stick), which are required to compute the iSRC.",
"The iSRC obtained by using piecewise uniform rescaling factors is shown by red dashed lines in Figure REF , agreeing with the actual displacements (black solid line) between rescaled perturbed and unperturbed trajectories."
],
[
"Anti-synchrony of two weakly coupled stick-slip oscillators",
"The theory of weakly coupled oscillators has been used to predict the synchronization properties in networks of oscillators in smooth systems [49], [45] and nonsmooth systems with transversal crossing boundaries [46].",
"In this section, we consider a nonsmooth system with hard boundaries composed of two blocks connected by a spring on a moving belt and apply the weakly coupled oscillator theory to study the synchrony between the two coupled stick-slip oscillators.",
"Our results predict a non-intuitive result, namely, that the anti-synchrony solution is stable whereas in-phase synchronization is unstable for an identical pair of stick-slip oscillators.",
"Moreover, the anti-synchrony solution has an extremely slow rate of convergence.",
"A slip phase of the coupled stick-slip oscillators begins according to the following equations of motion [21], $\\begin{array}{lcl}m_1 x_1^{\\prime \\prime } &=&-k_1x_1-k_{3}(x_1-x_2)+f_{1}(x_1^{\\prime }-u), \\\\m_2 x_2^{\\prime \\prime } &=&-k_2x_2-k_{3}(x_2-x_1)+f_{2}(x_2^{\\prime }-u),\\end{array}$ where $x_i$ is the displacement, $m_i$ is the mass, $k_i$ for $i=1,2$ is the stiffness of the spring connecting block $i$ to the fixed support and $k_3$ is the stiffness of the coupling spring, $f_{i}(x_i^{\\prime }-u)$ is the kinetic friction force of the $i$ -th block and $u$ is the velocity of the moving belt.",
"The damping coefficient of the spring is assumed to be 0 for simplicity.",
"Each mass can undergo a stick phase, which leads to two hard boundaries: $\\Sigma ^1=\\lbrace x_1^{\\prime }=u\\rbrace , \\quad \\Sigma ^2=\\lbrace x_2^{\\prime }=u\\rbrace .$ Letting $X_1=[x_1,v_1]^\\intercal $ and $X_2 = [x_2,v_2]^\\intercal $ where $v_1 = x_1^{\\prime }$ and $v_2 = x_2^{\\prime }$ and assuming $m_1=m_2=m$ , $k_1=k_2=k$ and $f_1=f_2=f$ so the two uncoupled oscillators are identical, we rewrite the coupled stick-slip systems in the following form $\\begin{array}{lcl}X_1^{\\prime }&=& F(X_1)+k_3 G(X_2,X_1)\\\\X_2^{\\prime } &=& F(X_2)+k_3G(X_1,X_2)\\end{array}$ where $F(X_i):=\\left\\lbrace \\begin{array}{cccccccccc}\\left[\\begin{array}{cccccccccccc} v_i \\\\ -\\frac{k}{m}x_i +\\frac{f(v_i-u)}{m} \\end{array}\\right], & X_i \\in {\\mathcal {R}}^{{\\rm interior_i}}\\\\\\left[\\begin{array}{cccccccccccc} u \\\\ 0 \\end{array}\\right], & X_i \\in {\\mathcal {R}}^{{\\rm slide_i}}\\subset \\Sigma ^i\\end{array}\\right.$ $G(X_j,X_i):=\\left\\lbrace \\begin{array}{cccccccccc}\\left[\\begin{array}{cccccccccccc} 0\\\\-(x_i-x_j)/m \\end{array}\\right], & X_i \\in {\\mathcal {R}}^{{\\rm interior_i}}\\\\\\left[\\begin{array}{cccccccccccc} 0 \\\\ 0 \\end{array}\\right], & X_i \\in {\\mathcal {R}}^{{\\rm slide_i}}\\subset \\Sigma ^i\\end{array}\\right.$ and the kinetic force $f$ is given by (REF ).",
"The sliding regions (${\\mathcal {R}}^{{\\rm slide_1}}, {\\mathcal {R}}^{{\\rm slide_2}}$ ), confined to $\\Sigma ^1$ and $\\Sigma ^2$ , and the interior domains (${\\mathcal {R}}^{\\rm interior_1}$ , ${\\mathcal {R}}^{\\rm interior_2}$ ) can be found according to Definition REF and Definition REF , as we did for the one-mass stick-slip system (REF ).",
"When there is no coupling with $k_3=0$ , the two oscillators are identical and exhibit a $T_0$ -periodic LCSC solution denoted as $\\gamma (t)$ .",
"As discussed before, the periodic solutions of an asymptotically stable oscillator can be represented by a single variable phase model (see (REF )) and the coupled oscillators can then be converted to the following phase model $\\begin{array}{lcl}\\theta _1^{\\prime }&=& 1+k_3 {\\mathbf {z}}(\\theta _1)\\cdot G(X_2,X_1)\\\\\\theta _2^{\\prime } &=& 1+k_3{\\mathbf {z}}(\\theta _2)\\cdot G(X_1,X_2)\\end{array}$ where $\\theta _1\\in [0, T_0]$ and $\\theta _2\\in [0, T_0]$ are the phases of the two uncoupled oscillators.",
"${\\mathbf {z}}$ is the iPRC curve for the uncoupled stick-slip oscillator.",
"If the coupling strength $k_3$ is sufficiently small, the uncoupled oscillator is almost identical to the periodic solutions $X_j(t)$ .",
"The system (REF ) can then be approximated by $\\begin{array}{lcl}\\theta _1^{\\prime }&=& 1+k_3 {\\mathbf {z}}(\\theta _1)\\cdot G(\\gamma (\\theta _2),\\gamma (\\theta _1))\\\\\\theta _2^{\\prime } &=& 1+k_3{\\mathbf {z}}(\\theta _2)\\cdot G(\\gamma (\\theta _1),\\gamma (\\theta _2)).\\end{array}$ Averaging the right hand sides over one cycle $[0, T_0]$ and defining $\\psi =\\theta _2-\\theta _1$ , we can obtain the following scalar equation of the relative phase: $\\begin{array}{lcl}\\psi ^{\\prime } &=& k_3 (H(-\\psi ) - H(\\psi )) \\equiv k_3 {\\mathcal {H}}(\\psi ),\\end{array}$ where $H(\\psi ) = \\frac{1}{T_0}\\int _{0}^{T_0} {\\mathbf {z}}(t)\\cdot G(\\gamma (t+\\psi ),\\gamma (t)) \\,d t.$ The autonomous and scalar ODE (REF ) can then be used to predict the synchronization rates and stability using a standard stability analysis on the phase line.",
"Figure: iPRC for the uncoupled stick-slip system and the right hand side ℋ{\\mathcal {H}} of () when m=1,k=1,δ=0,γ=3,η=0m=1, k=1, \\delta =0, \\gamma =3, \\eta =0 and u=0.295u=0.295.",
"Left: iPRC for the unperturbed oscillator.",
"The shaded region indicates the stick phase.",
"Right: Stability analysis of the right hand side function ℋ{\\mathcal {H}} of ().",
"Filled (resp., hollow) circle denotes an asymptotically stable (resp., unstable) phase locked solution.When $m=1, k=1, \\delta =0, \\gamma =3, \\eta =0$ , and $u=0.295$ , the iPRC for the uncoupled stick-slip oscillator is shown in Figure REF , left panel.",
"As discussed before, the discontinuous jump in the iPRC occurs at the liftoff boundary, that is, when the stick phase ends.",
"The right panel shows the right hand side function ${\\mathcal {H}}$ of (REF ).",
"On the phase line, a filled circle at $\\psi =T_0/2$ (resp., open circle at $\\psi =0, T_0$ ) corresponds to an asymptotically stable (resp., unstable) phase locked solution.",
"Hence, the phase model predicts that coupled stick-slip oscillators will diverge from an in-phase synchrony and asymptotically converge to an anti-phase synchrony.",
"This prediction is supported by the numerical simulation in Figure REF , left panel, showing that oscillations beginning in-phase eventually converge to an anti-phase synchrony solution.",
"The accuracy of the synchronization rates is demonstrated in Figure REF , right panel, where the red dashed curve is the solution to (REF ) and the black curve is the numerical phase difference in the full model (REF ) for $k_3=0.001$ .",
"The two curves agree relatively well until $\\psi $ is close to the plateau region in ${\\mathcal {H}}(\\psi )$ shown in Figure REF .",
"During this region, ${\\mathcal {H}}(\\psi )$ is nearly zero so the convergence to the anti-phase synchrony is very slow.",
"Hence in the right panel of Figure REF , we only include the time evolution of $\\psi $ for $t\\in [0, 80000]$ which is not long enough for the phase difference to converge to the anti-synchrony state.",
"From the plot, we can see that $\\psi $ increases rapidly to about 4 over the first 2000 unit of time.",
"After that, the converging rate significantly slows down because $\\psi $ enters the plateau region of ${\\mathcal {H}}(\\psi )$ .",
"On the other hand, the shape of the ${\\mathcal {H}}(\\psi )$ curve also suggests that the anti-synchronous point is near-neutrally stable, which may explain why the prediction of the convergence rates near the plateau region is less accurate than the prediction in non-plateau regions.",
"Improving the accuracy needs further work, such as accounting for higher order effects that are neglected in our first-order phase reduction of coupled oscillators, which is beyond the scope of this paper.",
"Figure: Simulation result of the coupled stick-slip system () when the coupling strength k 3 =0.001k_3=0.001 and other parameters are the same as in Figure .",
"Left: Time series of velocities of the two oscillators: initial conditions at a phase difference of 0 (ψ=0\\psi =0) (top panel) approach a phase difference of 4.24494.2449 after duration time 80000 (bottom panel) and eventually converge to a phase-locked solution with a phase difference of T 0 /2T_0/2 where T 0 =10.02T_0=10.02 is the period of the uncoupled stick-slip oscillator (not shown here since the convergence rate of ψ\\psi near the plateau region of ℋ(ψ){\\mathcal {H}}(\\psi ) as shown in Figure is very slow).",
"Right: Time series of the predicted phase difference (red) and the actual phase difference (black) for time over [0,80000][0, 80000]."
],
[
"Discussion",
"Rhythmic motions making and breaking contact with a constraining boundary, and subject to external perturbations, arise in motor control systems such as walking, running, scratching, biting and swallowing, as well as other natural and engineered hybrid systems [4], [6].",
"Dynamical systems describing such rhythmic motions are therefore nonsmooth and often exhibit limit cycle trajectories with sliding components.",
"In smooth dynamical systems, classical analysis for understanding the change in periodic limit cycle orbits under weak perturbation relies on the Jacobian linearization of the flow near the limit cycle.",
"These methods do not apply directly to nonsmooth systems, for which the Jacobian matrices are not well defined.",
"In this work, we describe for the first time the infinitesimal phase response curves (iPRC) for limit cycles with sliding components (LCSC).",
"Moreover, we give a rigorous derivation of the jump matrix for the iPRC at the hard boundary crossing point.",
"We also report, for the first time, how the presence of a liftoff point, where a limit cycle leaves a constraint surface, can create a nondifferentiable “kink” in the asymptotic phase function, propagating backwards in time along an osculating trajectory (see Figure REF A).",
"Most significantly, we have developed the infinitesimal shape response curve (iSRC) to analyze the joint variation of both shape and timing of limit cycles with sliding components, under parametric perturbations.",
"We show that taking into account local timing sensitivity within a switching region improves the accuracy of the iSRC over global timing analysis alone.",
"This improvement in accuracy is facilitated by our introduction of a novel local timing response curve (lTRC) measuring the timing sensitivity of an oscillator within a given local region.",
"Our results clarify an important distinction between the effects of the boundary encounter on the timing and shape changes in limit cycles with sliding components.",
"We have extended the iPRC developed for smooth limit cycle systems to the LCSC case, presented here as Theorem REF .",
"In addition, our analysis yields an explicit expression for the iPRC jump matrix that characterizes the behavior of the iPRC at the landing and liftoff points.",
"Surprisingly, we find that the iPRC experiences no discontinuity when the trajectory first contacts a hard boundary, while the variational equation suffers a discontinuity, captured by the saltation matrix.",
"Even more interesting, at the liftoff point – where the saltation matrix for the variational problem is trivial – the iPRC does show a discontinuous change, captured by a nontrivial jump matrix.",
"Specifically, there is a discontinuous jump from zero to a nonzero normal component in the iPRC.",
"Consequently, numerical evaluation of the iPRC must be obtained by backward integration along the limit cycle, as discussed in §REF .",
"Finally, we find that both the iPRC and the variational dynamics have zero normal components during the sliding component of the limit cycle, due to dimensional compression at the hard boundary.",
"Limit cycles with sliding components can sometimes arise as the singular limits of smooth singularly perturbed systems [31], [32].",
"Specifically, [31] shows that a piecewise smooth system can be understood as a singular perturbation problem in the limiting situation by blowing up the discontinuity into a switching layer.",
"To the best of our knowledge, this literature does not address phase response curve and variational dynamics.",
"In principle, one might obtain results analogous to those we present here by first analyzing a smooth system and subsequently taking the singular limit.",
"While such an undertaking would be both interesting and challenging, our methods avoid the associated technical challenges by calculating the iSRC, iPRC and lTRC for the nonsmooth system directly.",
"Moreover, our results may provide some insights into phase response curve and variational dynamics of singularly perturbed systems that exhibit LCSC in the singular limit.",
"For example, [56], [30] analyze and predict synchronization properties of relaxation oscillators using the results from iPRC analysis in the singular relaxation limit.",
"Whether a similar relationship holds between singularly perturbed systems and LCSCs in singular limits has yet to be understood.",
"Standard variational and phase response curve analysis typically neglects changes in timing or shape, focusing instead on only one of the two aspects [35].",
"However, in many applications such as motor control systems, both the shape and timing of the trajectory are often affected under slow or parametric perturbations.",
"In this paper, we consider both timing and shape aspects using the iSRC, a first-order approximation to the change in shape of the limit cycle under a parameteric perturbation.",
"We have discussed two ways of incorporating timing changes into the iSRC: uniform timing rescaling based on the global timing analysis (iPRC) and piecewise uniform timing rescaling based on the local timing analysis (lTRC).",
"As demonstrated in the planar system example in §, when the trajectory exhibits approximately constant timing sensitivities, the iSRC with global timing rescaling is good enough for approximating the shape change (see Figure REF ); otherwise, we need take into account local timing changes to increase the accuracy of the iSRC (see Figure REF ).",
"LCSC with piecewise timing sensitivities naturally arise in many motor control systems due to nonuniform perturbations as well as the stick-slip mechanical system as studied in §.",
"Local timing analysis (lTRC) will then provide a better understanding of such systems compared with the global timing analysis (iPRC).",
"Ours is not the first work to characterize linear responses of limit cycles to parameteric perturbations.",
"[60] defines the “parametric impulse phase response curve (pIPRC)\" to capture the timing sensitivity of limit cycle systems to parametric perturbations, which is estimated in our paper through the iPRC as described at the end of §REF .",
"Other investigators have also considered variational [3], [38] and phase response analysis in nonsmooth systems [52], [46], [8], [62], but the studies on the iPRC were subject to transverse flow conditions.",
"Our work extends the iPRC analysis to the LCSC case in which the transversal crossing condition fails.",
"Combined timing and shape responses of limit cycles to perturbations have also been explored in other works.",
"[41] examined energy-optimal control of the timing of limit cycle systems including spiking neuron models and models of cardiac arrhythmia.",
"They showed that when one of the nontrivial Floquet multipliers of an unperturbed limit cycle system has magnitude close to unity, control inputs based solely on standard phase reduction, which neglects the effect on the shape of the controlled trajectory, can dramatically fail to achieve control objectives.",
"They and other authors have introduced augmented phase reduction techniques that use a system of coordinates (related to the Floquet coordinates) transverse to the limit cycle to improve the accuracy of phase reduction and control [7], [65], [67], [66], [42], [62], [63], [64], [47].",
"These methods require the underlying dynamics be smoothly differentiable, and rely on calculation of the Jacobian (first derivative) and in some cases the Hessian (second derivative) matrices [66].",
"For nonsmooth limit cycle systems with sliding components, our analysis is the first to address the combined effects of shape and timing, an essential element of improved control in biomedical applications as well as for understanding mechanisms of control in naturally occurring motor control systems.",
"For trajectories with different timing sensitivities in different regions, we rely on the local timing response curve (lTRC) to estimate the relative shift in time in each sub-region, in order to compute the full infinitesimal shape response curve (iSRC).",
"Conversely, solving for the lTRC in a given region may also require an understanding of the impact of the perturbation on the entry point associated with that region (see (REF )).",
"Thus, in general, the iSRC and the lTRC are interdependent.",
"While we have not derived a closed-form expression for the shape and timing response in the most general case, we have provided effective algorithms for solving each of them separately, which requires preliminary numerical work to find the trajectory shape shift at the entry point.",
"In the future, it may be possible to derive general closed-form expressions for the iSRC and lTRC in systems with distinct timing sensitivities.",
"While our methods are illustrated using a planar limit cycle system with hard boundaries and coupled stick-slip systems, they apply to higher dimensional and more realistic systems as well.",
"For instance, preliminary investigations suggest that the methods developed in this paper are applicable to analyzing the nonsmooth dynamics arising in the control system of feeding movements in the sea slug Aplysia [51], [50], [39].",
"More generally, limit cycles with discontinuous trajectories arise in neuroscience (e.g., integrate and fire neurons) and mechanics (e.g., ricochet dynamics).",
"If such systems manifest limit cycles with sliding components, our methods could be combined with variational methods adapted for piecewise continuous trajectories [10], [52].",
"It was observed heuristically by [39] that sensory feedback could in some circumstances lead to significant robustness against an increase in applied load, in the sense that although modest relative increases in external load (c. 20%) led to comparable changes in both the timing and shape of trajectories, the net effect on the performance (rate of intake of food) was an order of magnitude smaller (c. 1%).",
"Similarly, [12] showed that in a model for control of a central pattern generator regulating the breathing rhythm, mean arterial partial pressure of oxygen (PPO$_2$ ) remained approximately constant under changing metabolic loads when chemosensory feedback from the arterial PPO$_2$ to the central pattern generator was present, but varied widely otherwise [12].",
"Understanding how rhythmic biological control systems respond to such perturbations and maintain robust, adaptive performance is one of the fundamental problems within theoretical biology.",
"Solving these problems will then require variational analysis along the lines we develop here.",
"Nonsmooth dynamics arise naturally in many biological systems [1], [10], and thus, the approach in this paper is likely to have broad applicability to many other problems in biology."
],
[
"Acknowledgement",
"This work was made possible in part by grants from the National Science Foundation (DMS-1413770, DEB-1654989, IOS-174869 and IOS-1754869 to H.J.C).",
"P.J.T thanks the Oberlin College Department of Mathematics for research support.",
"This research has been supported in part by the National Science Foundation Grant DMS-1440386 to the Mathematical Biosciences Institute."
],
[
"Proof of Lemma ",
"In this section we prove Lemma REF , which we restate for the reader's convenience.",
"Lemma Let $\\gamma ^\\textbf {a}_1(t)$ and $\\gamma ^\\textbf {b}_1(t)$ be two $T_0$ -periodic solutions to the iSRC equation (REF ) for a smooth vector field $F_0$ with a hyperbolically stable limit cycle $\\gamma _0(t)$ .",
"Then, their difference satisfies $\\gamma ^\\textbf {b}_1(t)-\\gamma ^\\textbf {a}_1(t)=\\varphi F_0(\\gamma _0(t))$ , where $\\varphi $ is a constant representing a fixed phase offset.",
"Consider $\\mathbf {x}^{\\prime }=F_0(\\mathbf {x})$ with $\\mathbf {x}(0)=\\mathbf {x}_0\\in \\mathbb {R}^n$ .",
"Let $\\Phi (t,0)$ be the fundamental matrix solution.",
"Then $\\Phi (t,0)$ satisfies $\\Phi ^{\\prime }(t,0)= DF_0(\\mathbf {x}(t))\\Phi (t,0)$ and $\\Phi (0,0)=I$ , where $\\mathbf {x}=\\gamma _0(t)$ is the unperturbed limit cycle solution.",
"Suppose the monodromy matrix $M=\\Phi (T_0,0)$ is diagonalizable with eigenvalues $\\lbrace \\mu _i, i=1,\\cdots ,n\\rbrace $ associated with linearly independent eigenvectors $\\lbrace \\mathbf {v}_i, i=1,\\cdots ,n\\rbrace $ .",
"The eigenvalues $\\mu _i$ are often referred to as Floquet multipliers of the periodic orbit solution $\\gamma _0(t)$ of (REF ) [40].",
"Since $\\gamma _0(t)$ is hyperbolically stable, $M$ has a single trivial Floquet multiplier.",
"Without loss of generality, we assume $\\mu _1=1$ and hence $\\mathbf {v}_1=F_0(\\gamma _0(0))$ .",
"Let the vector $\\eta _i(t)$ be the solution to the variational equation $\\eta _i^{\\prime } = DF_0(\\gamma _0(t))\\eta _i$ that starts along the $i$ -th Floquet eigenvector direction $\\eta _i(0)=\\mathbf {v}_i$ .",
"It follows that $\\eta _i(T_0) = \\Phi (T_0,0)\\eta _i(0)=M \\mathbf {v}_i = \\mu _i \\mathbf {v}_i$ .",
"For simplicity, we denote $A(t) = DF_0(\\gamma _0(t))$ hereafter.",
"Let $\\rho _i = \\ln (\\mu _i)/T_0$ and let $\\mathbf {q}_i(t)=e^{-t\\rho _i}\\eta _i(t)$ be the rescaled version of the trajectory $\\eta _i(t)$ , which is the $i$ -th Floquet coordinate [40].",
"It follows from direct calculations that $\\mathbf {q}_i(t)$ is periodic so that $\\mathbf {q}_i(t) = \\mathbf {q}_i(t+T_0)$ and satisfies the initial value problem $\\mathbf {q}_i(t)^{\\prime } = A(t)\\mathbf {q}_i(t) - \\rho _i \\mathbf {q}_i(t)$ with $\\mathbf {q}_i(0)=\\mathbf {v}_i$ .",
"Since $\\rho _1=0$ , the first Floquet coordinate $\\mathbf {q}_1(t)$ satisfies the initial value problem $\\mathbf {q}_1^{\\prime }=A(t)\\mathbf {q}_1,\\quad \\mathbf {q}_1(0)=F_0(\\gamma _0(t)).$ Using the chain rule, upon differentiating $F_0(\\gamma _0(t))$ , one can show that setting $\\mathbf {q}_1(t)=F_0(\\gamma _0(t))$ for any $t$ , solves the initial value problem (REF ).",
"Note that (REF ) is similar to the equation satisfied by $\\mathcal {I}_i$ , the gradient of isostable coordinates that are related to Floquet coordinates $\\mathbf {q}_i$ [67], [47].",
"In fact, direct calculation implies that the relationship between the iPRC and the variational dynamics (see Remark REF ) also holds for $\\mathbf {q}_i$ and $\\mathcal {I}_i$ ; that is, $\\frac{d(\\mathbf {q}_i^\\intercal \\mathcal {I}_i)}{dt}=0$ for $i=1,\\cdots ,n$ .",
"Note that at each $t$ , the vector set $\\lbrace \\mathbf {q}_i(t), i=1,\\cdots ,n\\rbrace $ spans $\\mathbb {R}^n$ .",
"Therefore we can write $\\gamma _1(t)$ , the general solution to the iSRC equation (REF ), as a linear combination of $\\lbrace \\mathbf {q}_i(t), i=1,\\cdots ,n\\rbrace $ with coefficients $a_i(t)$ : $\\gamma _1(t) = \\sum _{i=1}^{n} a_i(t) \\mathbf {q}_i(t).$ Let $Q(t)$ be the $n\\times n$ matrix $Q(t)=\\left(\\mathbf {q}_1(t)|\\cdots | \\mathbf {q}_n(t)\\right)$ , and let $R=\\mathrm {diag}(\\rho _1,\\cdots ,\\rho _n)$ be the diagonal matrix with $\\lbrace \\rho _1,\\cdots , \\rho _n\\rbrace $ as the diagonal entries, and $\\mathbf {a}(t) =[a_1(t),\\cdots ,a_n(t)]^\\intercal $ .",
"Then $\\gamma _1(t)=Q(t)\\mathbf {a}(t)$ and (REF ) can be rewritten as $Q(t)^{\\prime } = A(t)Q(t)-Q(t)R.$ Differentiating both sides of (REF ) and substituting in (REF ) leads to $\\gamma _1^{\\prime }(t)&= Q^{\\prime }(t)\\mathbf {a}(t) +Q(t)\\mathbf {a}^{\\prime }(t)\\\\&=(A(t)Q(t)-Q(t)R )\\mathbf {a}(t) +Q(t)\\mathbf {a}^{\\prime }(t)\\\\&=(Q(t)\\mathbf {a}^{\\prime }(t)-Q(t)R \\mathbf {a}(t)) + A(t)Q(t)\\mathbf {a}(t)\\\\&=(Q(t)\\mathbf {a}^{\\prime }(t)-Q(t)R \\mathbf {a}(t)) + A(t)\\gamma _1(t).$ On the other hand, by (REF ) we have $\\gamma _1^{\\prime }(t)&=& A(t) \\gamma _1(t) +\\mathbf {c}(t)\\nonumber ,$ where $\\mathbf {c}(t)=\\nu _1 F_0(\\gamma _0(t)) +\\frac{\\partial F_\\varepsilon (\\gamma _0(t))}{\\partial \\varepsilon }\\Big |_{\\varepsilon =0}$ .",
"It follows that $Q(t)\\mathbf {a}^{\\prime }(t)= Q(t)R \\mathbf {a}(t)+ \\mathbf {c}(t).$ Since $\\lbrace \\mathbf {q}_i(t)\\rbrace _{i=1}^n$ spans $\\mathbb {R}^b$ for each time $t\\in [0, T_0]$ , the matrix $Q(t)$ is invertible at each $t$ .",
"Thus multiplying both sides of (REF ) by $Q(t)^{-1}$ gives $\\mathbf {a}^{\\prime }(t)= R \\mathbf {a}(t) + Q(t)^{-1}\\mathbf {c}(t).$ Suppose two different iSRC curves are given by $\\gamma _1^a(t)=\\sum _{i=1}^n a_i(t)\\mathbf {q}_i(t)=Q(t)\\mathbf {a}(t)$ and $\\gamma _1^b(t)=\\sum _{i=1}^n b_i(t)\\mathbf {q}_i(t)=Q(t)\\mathbf {b}(t)$ .",
"Then $\\mathbf {b}^{\\prime }(t)-\\mathbf {a}^{\\prime }(t) = R\\mathbf {b}(t)-R\\mathbf {a}(t)= \\left[\\begin{array}{cccccccccccc} \\rho _1&&\\\\ &\\ddots & \\\\ && \\rho _n \\end{array}\\right](\\mathbf {b}(t)-\\mathbf {a}(t)).$ It follows that for $i=1,\\cdots ,n$ we have $b_i(t)-a_i(t)=C_ie^{\\rho _i t}$ for some constant $C_i$ .",
"Note that $\\mathbf {a}(t)$ and $\\mathbf {b}(t)$ are both $T_0$ -periodic.",
"Therefore $C_i=C_i e^{\\rho _i T_0}$ .",
"So either $C_i=0$ , and hence $a_i(t)\\equiv b_i(t)$ , or else $\\rho _i=0$ .",
"However, recall there is only one trivial multiplier $\\mu _1=1$ , so that only $\\rho _1=0$ and $\\rho _i\\ne 0$ for $i=2,\\cdots ,n$ .",
"Hence, $a_i(t)\\equiv b_i(t)$ for $i=2,\\cdots ,n$ ; that is, there exists some constant $\\phi $ such that $\\mathbf {b}(t)-\\mathbf {a}(t)=\\phi \\mathbf {e}_1$ .",
"Thus, $\\gamma _1^b(t)-\\gamma _1^a(t) = Q(t) \\phi \\mathbf {e}_1 =\\phi \\mathbf {q}_1(t).$ Consequently, the two iSRC curves only differ along the first Floquet coordinate direction $\\mathbf {q}_1(t)=F_0(\\gamma _0(t))$ and hence only differ by a shift in phase along the direction of the limit cycle $\\gamma _0(t)$ : $\\gamma _1^b(t)-\\gamma _1^a(t) =\\phi F_0(\\gamma _0(t))$ where $\\phi $ is the constant phase shift introduced by the initial conditions $\\gamma _1^b(0)-\\gamma _1^a(0) =\\phi F_0(\\gamma _0(0))$ ."
],
[
"Derivation of Equation ",
"This section establishes equation (REF ), which specifies the first-order change in the transit time through region I, or $T_1^\\text{I}$ : $T^{\\rm I}_{1} = \\eta ^{\\rm I}(\\mathbf {x}^{\\rm in})\\cdot \\frac{\\partial \\mathbf {x}_{\\varepsilon }^{\\rm in}}{\\partial \\varepsilon }\\Big |_{\\varepsilon =0}+\\int _{t^{\\rm in}}^{t^{\\rm out}}\\eta ^{\\rm I}(\\gamma (t))\\cdot \\frac{\\partial F_\\varepsilon (\\gamma (t))}{\\partial \\varepsilon }\\Big |_{\\varepsilon =0}dt,$ Recall $\\mathcal {T}^{\\rm I}(\\mathbf {x})$ is the time remaining until exiting region I through $\\Sigma ^{\\rm out}$ , under the unperturbed vector field, starting from location $\\mathbf {x}$ ; $\\eta ^{\\rm I}:= \\nabla \\mathcal {T}^{\\rm I}(\\mathbf {x})$ is the local timing response curve (lTRC) for region I, defined for the component of the trajectory lying within region I, i.e.",
"for times $t\\in [t^{\\rm in}, t^{\\rm out}]$ ; and $\\mathbf {x}_\\varepsilon ^{\\rm in}$ is the coordinate of the perturbed entry point into region I.",
"We consider a single region $\\mathcal {R}$ with entry surface $\\Sigma ^{\\rm in}$ and exist surface $\\Sigma ^{\\rm out}$ .",
"We assume that these two surfaces are fixed, independent of static perturbation with size $\\varepsilon $ .",
"The limit cycle solution $\\mathbf {x}=\\gamma _\\varepsilon (\\tau )$ satisfies $\\frac{d\\mathbf {x}}{d\\tau }=F_{\\varepsilon }(\\mathbf {x})$ where $\\tau $ is the time coordinate of the perturbed trajectory.",
"Moreover, $\\gamma _\\varepsilon (\\tau )$ enters $\\mathcal {R}$ at $\\mathbf {x}^\\text{in}_\\varepsilon \\in \\Sigma ^{\\rm in}$ when $\\tau =t^\\text{in}_\\varepsilon $ and exits at $\\mathbf {x}^\\text{out}_\\varepsilon \\in \\Sigma ^{\\rm out}$ when $\\tau =t^\\text{out}_\\varepsilon $ .",
"Since the system is autonomous, we are free to choose the reference time along the limit cycle orbit.",
"For convenience of calculation, we set $t^\\text{out}_\\varepsilon \\equiv 0$ for all $\\varepsilon $ .",
"Denote the transit time that $\\gamma _\\varepsilon $ spends in $\\mathcal {R}$ by $T^{\\mathcal {R}}_\\varepsilon $ .",
"It follows that $t^\\text{in}_\\varepsilon = -T^{\\mathcal {R}}_{\\varepsilon }$ , where $\\varepsilon $ can be 0.",
"Assuming that the transit time has a well behaved expansion in $\\varepsilon ,$ we write $T^{\\mathcal {R}}_\\varepsilon &=T^{\\mathcal {R}}_0+\\varepsilon T^{\\mathcal {R}}_1+O(\\varepsilon ^2)$ where $T^{\\mathcal {R}}_0$ is the transit time for the unperturbed trajectory and $T^{\\mathcal {R}}_1$ is the linear shift in the transit time.",
"In the rest of this section, we drop the superscript ${\\mathcal {R}}$ on $T^{\\mathcal {R}}_\\varepsilon ,\\,T^{\\mathcal {R}}_0$ and $T^{\\mathcal {R}}_1$ for simplicity.",
"Our goal is to prove that $T_1$ is given by (REF ).",
"We do this in two steps.",
"First, we show that the transit time $T_{\\varepsilon }$ can be expressed in terms of the perturbed vector field and perturbed local timing response curve (see (REF )).",
"Second, we expand the expression for $T_{\\varepsilon }$ to first order in $\\varepsilon $ to obtain the expression for $T_1$ .",
"Since the time remaining to exit, denoted as $\\mathcal {T}_\\varepsilon $ , decreases at a constant rate along trajectories, for arbitrary $\\varepsilon $ we have $-1=\\frac{d\\mathcal {T}_\\varepsilon }{d\\tau }=F_\\varepsilon (\\gamma _\\varepsilon (\\tau ))\\cdot \\eta _\\varepsilon (\\gamma _\\varepsilon (\\tau )),$ where $\\eta _\\varepsilon (\\mathbf {x})=\\nabla \\mathcal {T}_\\varepsilon (\\mathbf {x})$ is defined as the local timing response curve under perturbation.",
"By (REF ), the transit time $T_\\varepsilon $ is therefore given by $T_\\varepsilon =\\int _{\\tau =t^\\text{out}_\\varepsilon }^{t^\\text{in}_\\varepsilon }F_\\varepsilon (\\gamma _\\varepsilon (\\tau ))\\cdot \\eta _\\varepsilon (\\gamma _\\varepsilon (\\tau ))\\,d\\tau .$ In this expression, we integrate backwards in time along the limit cycle trajectory, from the egress point $\\mathbf {x}^\\text{out}_\\varepsilon $ at time $t^\\text{out}_\\varepsilon $ , to the ingress point $\\mathbf {x}^\\text{in}_\\varepsilon $ at time $t^\\text{in}_\\varepsilon $ : For $\\varepsilon =0$ , and taking into account (REF ), this integral reduces to $T_0=\\int _{\\tau =t^\\text{out}_0}^{t^\\text{in}_0} F_0(\\gamma _0(\\tau ))\\cdot \\eta _0(\\gamma _0(\\tau ))\\,d\\tau =\\int _{\\tau =t^\\text{out}_0}^{t^\\text{in}_0}(-1)\\,d\\tau = t^\\text{out}_0-t^\\text{in}_0=0-(-T_0),$ since $t^\\text{in}_0= -T_0$ and $t_\\varepsilon ^\\text{out}\\equiv 0$ .",
"In order to derive an expression for $T_1$ , the first order shift in the transit time, we need to expand (REF ) to first order in $\\varepsilon $ .",
"To this end, we need to know the Taylor expansions for all terms in (REF ).",
"Suppose we can expand $F_\\varepsilon $ , $\\mathcal {T}_{\\varepsilon }$ , and $\\eta _\\varepsilon $ as follows: $\\begin{array}{cccccccccc}F_{\\varepsilon }(\\mathbf {x})&=&F_0(\\mathbf {x})+\\varepsilon F_1(\\mathbf {x})+O(\\varepsilon ^2),&\\text{ as }\\varepsilon \\rightarrow 0,\\\\\\mathcal {T}_\\varepsilon (\\mathbf {x})&=&\\mathcal {T}_0(\\mathbf {x})+\\varepsilon \\mathcal {T}_1(\\mathbf {x})+O(\\varepsilon ^2),&\\text{ as }\\varepsilon \\rightarrow 0,\\\\\\eta _\\varepsilon (\\mathbf {x})&=&\\eta _0(\\mathbf {x})+\\varepsilon \\eta _1(\\mathbf {x})+O(\\varepsilon ^2),&\\text{ as }\\varepsilon \\rightarrow 0, \\end{array}$ where $\\eta _0(\\mathbf {x}) = \\nabla \\mathcal {T}_0(\\mathbf {x})$ is the unperturbed local timing response curve.",
"Following the idea of deriving the infinitesimal shape response curve in §REF , we write the portion of the perturbed limit cycle trajectory within region $\\mathcal {R}$ in terms of the unperturbed limit cycle, plus a small correction, $\\gamma _\\varepsilon (\\tau )&=\\gamma \\left(\\nu _\\varepsilon \\tau \\right)+\\varepsilon \\gamma _1\\left(\\nu _\\varepsilon \\tau \\right)+O(\\varepsilon ^2)$ where $-T_\\varepsilon \\le \\tau \\le 0$ and $\\nu _\\varepsilon = \\frac{T_0}{T_\\varepsilon }$ .",
"Now we expand (REF ) to first order $T_\\varepsilon &=\\int _{\\tau =0}^{-T_{\\varepsilon }}\\Big [ F_0(\\gamma _0(\\nu _\\varepsilon \\tau ))+\\varepsilon DF_0(\\gamma _0(\\nu _\\varepsilon \\tau ))\\cdot \\gamma _1(\\nu _\\varepsilon \\tau )+\\varepsilon F_1(\\gamma _0(\\nu _\\varepsilon \\tau )) \\Big ]\\cdot \\\\ \\nonumber &\\quad \\quad \\quad \\Big [\\eta _0(\\gamma _0(\\nu _\\varepsilon \\tau ))+\\varepsilon D\\eta _0(\\gamma _0(\\nu _\\varepsilon \\tau )) \\cdot \\gamma _1(\\nu _\\varepsilon \\tau )+\\varepsilon \\eta _1(\\gamma _0(\\nu _\\varepsilon \\tau )) \\Big ]d\\tau +O(\\varepsilon ^2)\\\\ \\nonumber &=\\int _{\\tau =0}^{-T_{\\varepsilon }} F_0(\\gamma _0(\\nu _\\varepsilon \\tau ))\\cdot \\eta _0(\\gamma _0(\\nu _\\varepsilon \\tau )) d\\tau + \\varepsilon \\Big [ F_0(\\gamma _0(\\nu _\\varepsilon \\tau )) \\cdot \\eta _1(\\gamma _0(\\nu _\\varepsilon \\tau ))+ F_1(\\gamma _0(\\nu _\\varepsilon \\tau ))\\cdot \\eta _0(\\gamma _0(\\nu _\\varepsilon \\tau )) \\Big ] d\\tau + \\\\\\nonumber & \\quad \\quad \\quad \\varepsilon \\Big [ F_0(\\gamma _0(\\nu _\\varepsilon \\tau )) \\cdot D\\eta _0(\\gamma _0(\\nu _\\varepsilon \\tau )) \\cdot \\gamma _1(\\nu _\\varepsilon \\tau ) + DF_0(\\gamma _0(\\nu _\\varepsilon \\tau ))\\cdot \\gamma _1(\\nu _\\varepsilon \\tau ) \\cdot \\eta _0(\\gamma _0(\\nu _\\varepsilon \\tau )) \\Big ] d\\tau +O(\\varepsilon ^2)\\\\ \\nonumber &=\\frac{1}{\\nu _\\varepsilon }\\int _{t=0}^{-T_{0}} F_0(\\gamma _0(t))\\cdot \\eta _0(\\gamma _0(t)) dt + \\varepsilon \\Big [ F_0(\\gamma _0(t)) \\cdot \\eta _1(\\gamma _0(t))+ F_1(\\gamma _0(t))\\cdot \\eta _0(\\gamma _0(t)) \\Big ] dt + \\\\\\nonumber & \\quad \\quad \\quad \\varepsilon \\Big [ F_0(\\gamma _0(t)) \\cdot D\\eta _0(\\gamma _0(t)) \\cdot \\gamma _1(t) + DF_0(\\gamma _0(t))\\cdot \\gamma _1(t) \\cdot \\eta _0(\\gamma _0(t)) \\Big ] dt +O(\\varepsilon ^2)\\\\ \\nonumber $ To order $O(1)$ , we recover $T_0=\\int _{t=0}^{-T_0} F_0(\\gamma _0(t))\\cdot \\eta _0(\\gamma _0(t))\\,dt.$ This leads to $T_{\\varepsilon } = \\frac{1}{\\nu _{\\varepsilon }}T_0,$ as required for consistency.",
"We are therefore left with $0&=\\int _{t=0}^{-T_{0}} \\Big [ F_0(\\gamma _0(t)) \\cdot \\eta _1(\\gamma _0(t))+ F_1(\\gamma _0(t))\\cdot \\eta _0(\\gamma _0(t)) \\Big ] dt \\\\\\nonumber & + \\int _{t=0}^{-T_{0}} \\Big [ F_0(\\gamma _0(t)) \\cdot D\\eta _0(\\gamma _0(t)) \\cdot \\gamma _1(t) + DF_0(\\gamma _0(t))\\cdot \\gamma _1(t) \\cdot \\eta _0(\\gamma _0(t)) \\Big ] dt +O(\\varepsilon )\\\\ \\nonumber &=\\int _{t=0}^{-T_{0}} \\Big [ F_0(\\gamma _0(t)) \\cdot \\eta _1(\\gamma _0(t))+ F_1(\\gamma _0(t))\\cdot \\eta _0(\\gamma _0(t)) \\Big ] dt \\\\\\nonumber & + \\int _{t=0}^{-T_{0}} \\Big [F_0(\\gamma _0(t))^\\intercal D\\eta _0(\\gamma _0(t)) + \\eta _0(\\gamma _0(t))^\\intercal DF_0(\\gamma _0(t))\\Big ] \\cdot \\gamma _1(t) dt +O(\\varepsilon )\\\\ \\nonumber $ where the second equality follows from rearranging orders of factors in the second integral.",
"Note that since $F_0\\cdot \\eta _0\\equiv -1$ everywhere, we have the identity $0=\\frac{\\partial }{\\partial \\mathbf {x}_j}\\left(\\sum _i \\eta ^i F^i\\right)=\\sum _i\\frac{\\partial \\eta ^i}{\\partial \\mathbf {x}_j} F^i+\\sum _i\\eta ^i\\frac{\\partial F^i}{\\partial \\mathbf {x}_j}$ where $F^i$ and $\\eta ^i$ are the $i$ -th components for $F_0$ and $\\eta _0$ ; $\\mathbf {x}_j$ denotes the $j$ th component of $\\mathbf {x}$ for $j\\in \\lbrace 1,\\cdots , n\\rbrace $ .",
"It follows that $F_0^\\intercal (D\\eta _0)+\\eta _0^\\intercal (D F_0)=0$ in (REF ), leaving only $0=\\int _{t=0}^{-T_0} \\Big [F_0(\\gamma _0(t)) \\cdot \\eta _1(\\gamma _0(t))+ F_1(\\gamma _0(t))\\cdot \\eta _0(\\gamma _0(t))\\Big ] \\,dt.$ Since $F_0(\\gamma _0(t))=d\\gamma _0/dt$ and $\\eta _1(\\mathbf {x})=\\partial \\eta _\\varepsilon (\\mathbf {x})/\\partial \\varepsilon |_{\\varepsilon =0} = \\partial \\nabla \\mathcal {T}_\\varepsilon (\\mathbf {x})/\\partial \\varepsilon |_{\\varepsilon =0}$ , $\\int _{t=0}^{-T_0}F_0(\\gamma _0(t))\\cdot \\eta _1(\\gamma _0(t)) \\,dt \\nonumber &= \\int _{t=0}^{-T_0}\\left(\\frac{d\\gamma _0}{d t}\\right)\\cdot \\left.\\left(\\frac{\\partial }{\\partial \\varepsilon }\\left[\\nabla \\mathcal {T}_\\varepsilon (\\gamma _0(t)) \\right]\\right)\\right|_{\\varepsilon =0}\\,dt\\\\\\nonumber &=\\int _{t=0}^{-T_0}\\left(\\frac{d\\gamma _0}{dt}\\right)\\cdot \\nabla \\left.\\left(\\frac{\\partial }{\\partial \\varepsilon }\\left[ \\mathcal {T}_\\varepsilon (\\gamma _0(t)) \\right]\\right)\\right|_{\\varepsilon =0}\\,dt\\\\\\nonumber &=\\int _{t=0}^{-T_0}\\frac{d}{dt}\\left.\\left(\\frac{\\partial }{\\partial \\varepsilon }\\left[ \\mathcal {T}_\\varepsilon (\\gamma _0(t)) \\right]\\right)\\right|_{\\varepsilon =0}\\,dt\\\\\\nonumber &=\\left.\\left(\\frac{\\partial }{\\partial \\varepsilon }\\left[ \\mathcal {T}_\\varepsilon (\\mathbf {x}^\\text{in}_0) \\right]\\right)\\right|_{\\varepsilon =0} -\\left.\\left(\\frac{\\partial }{\\partial \\varepsilon }\\left[ \\mathcal {T}_\\varepsilon (\\mathbf {x}^\\text{out}_0) \\right]\\right)\\right|_{\\varepsilon =0} \\\\\\nonumber &=\\left.\\left(\\frac{\\partial }{\\partial \\varepsilon }\\left[ \\mathcal {T}_\\varepsilon (\\mathbf {x}^\\text{in}_0) \\right]\\right)\\right|_{\\varepsilon =0} -0\\\\\\nonumber &=\\mathcal {T}_1(\\mathbf {x}^\\text{in}_0).$ Therefore $\\mathcal {T}_1(\\mathbf {x}^\\text{in}_0) =\\int _{t=-T_0}^{0}F_1(\\gamma _0(t))\\cdot \\eta _0(\\gamma _0(t)) dt =\\int _{t=t_0^{\\rm in}}^{t_0^{\\rm out}}F_1(\\gamma _0(t))\\cdot \\eta _0(\\gamma _0(t)) dt .$ The second equality follows from our convention that $t^\\text{in}_0= -T_0$ and $t_\\varepsilon ^\\text{out}\\equiv 0$ .",
"We notice that $T_\\varepsilon &=\\mathcal {T}_\\varepsilon (\\mathbf {x}_\\varepsilon ^\\text{in})=T_0+ \\varepsilon \\left( \\mathcal {T}_1(\\mathbf {x}^\\text{in}_0)+\\nabla \\mathcal {T}_0(\\mathbf {x}^\\text{in}_0) \\cdot \\mathbf {x}^\\text{in}_1 \\right),$ where we have made use of the Taylor expansion $\\mathbf {x}^\\text{in}_\\varepsilon =\\mathbf {x}^\\text{in}_0+\\varepsilon \\mathbf {x}^\\text{in}_1+O(\\varepsilon ^2),\\text{ as }\\varepsilon \\rightarrow 0$ .",
"Equating the first order terms in (REF ) and (REF ) leads to $T_1=\\mathcal {T}_1(\\mathbf {x}^\\text{in}_0)+\\eta _0(\\mathbf {x}^\\text{in}_0)\\cdot \\mathbf {x}^\\text{in}_1.$ Substituting (REF ) into (REF ), we finally obtain $T_1= \\eta _0(\\mathbf {x}^\\text{in}_0)\\cdot \\mathbf {x}^\\text{in}_1+\\int _{t=t_0^{\\rm in}}^{t_0^{\\rm out}}F_1(\\gamma _0(t))\\cdot \\eta _0(\\gamma _0(t)) dt$ which is (REF ), as desired."
],
[
"Proof of Theorem ",
"In this section we present a proof of Theorem REF , which we restate for the reader's convenience.",
"As discussed before, parts (a) and (b) are already covered in [19], [3], whereas parts (c) through (d) are our new results.",
"For completeness, we still include parts (a) and (b) as well as our versions of proofs.",
"Theorem.",
"Consider a general LCSC described locally by (REF ) in the neighborhood of a hard boundary $\\Sigma $ , satisfying Assumption REF and Assumption REF .",
"The following properties hold for the variational dynamics $\\mathbf {u}$ and the iPRC ${\\mathbf {z}}$ along $\\Sigma $ : At the landing point of $\\Sigma $ , the saltation matrix is $S=I-n n^\\intercal $ , where $I$ is the identity matrix.",
"At the liftoff point of $\\Sigma $ , the saltation matrix is $S=I$ .",
"Along the sliding region within $\\Sigma $ , the component of ${\\mathbf {z}}$ normal to $\\Sigma $ is zero.",
"The normal component of ${\\mathbf {z}}$ is continuous at the landing point.",
"The tangential components of ${\\mathbf {z}}$ are continuous at both landing and liftoff points.",
"We choose coordinates $\\mathbf {x}=(\\mathbf {w},v)=(w_1,w_2,\\ldots ,w_{n-1},v)$ so that within a neighborhood containing both the landing and liftoff points, the hard boundary corresponds to $v=0$ , the interior of the domain coincides with $v>0$ , and the unit normal vector for the hard boundary is $\\mathbf {n}=(0,\\ldots ,0,1)$ .",
"Writing the velocity vector $\\mathbf {F}=(f_1,f_2,\\ldots ,f_{n-1},g)$ in these coordinates.",
"In addition, we use $\\mathbf {F}^{\\rm slide}$ to denote the vector field for points on the sliding region, whereas the dynamics of other points is governed by $\\mathbf {F}^{\\rm int}$ .",
"The transversal intersection condition for the trajectory entering the hard boundary is $g^\\text{int}(\\mathbf {x}_\\text{land},0)<0$ (cf. eq.",
"(REF ); note that $\\mathbf {n}$ defined here points in the opposite direction from the outward normal vector in (REF )).",
"At points $\\mathbf {x}\\in \\mathcal {L}$ on the liftoff boundary, $\\mathbf {F}^\\text{slide}$ and $\\mathbf {F}^\\text{int}$ coincide and we will use whichever notation seems clearer in a given instance.",
"Under the nondegeneracy condition at the liftoff point (REF ), we can further arrange the coordinates $(w_1,\\ldots ,w_{n-1})$ so that the unit vector normal to the liftoff boundary $\\mathcal {L}$ at the liftoff point is $\\ell =(0,\\ldots ,0,1,0)$ , and $g^\\text{int}\\gtrless 0 \\iff w_{n-1}\\gtrless 0$ .",
"With these coordinates, the nondegeneracy condition (REF ) is $\\mathbf {F}^{\\rm slide}(\\mathbf {x}_{\\rm lift})\\cdot \\ell =f^\\text{slide}_{n-1}(\\mathbf {x}_\\text{lift})>0$ ."
],
[
"(a) At the landing point, the saltation matrix is $S=I-\\mathbf {n}\\mathbf {n}^\\intercal $ , where {{formula:3b737c41-4590-4cb4-8eb8-e3106088df81}} is the identity matrix.",
"The saltation matrix at a transition from the interior to a sliding motion along a hard boundary is given in ([3], Example 2.14, p. 111) as $S=I+\\frac{(\\mathbf {F}^{\\rm slide}-\\mathbf {F}^{\\rm int})\\mathbf {n}^\\intercal }{\\mathbf {n}^\\intercal \\mathbf {F}^{\\rm int}},$ provided the trajectory approaches the hard boundary transversally.",
"It follows from the definition of the sliding vector field $F^{\\rm slide}$ given by (REF ) that $S=I - \\mathbf {n}\\mathbf {n}^\\intercal ,$ as claimed."
],
[
"(b) At the liftoff point, the saltation matrix is $S=I$ .",
"We adapt the argument in ([3], §2.5) to our hard boundary/liftoff construction.",
"The essential difference is that the trajectory is not transverse to the hard boundary at the liftoff point, indeed $\\mathbf {n}^\\intercal \\mathbf {F}=0$ at $\\mathbf {x}_\\text{lift}$ , so eq.",
"(REF ) does not give a well defined saltation matrix.",
"However, by replacing the vector $\\mathbf {n}$ normal to the hard boundary with the vector $\\ell $ normal to the liftoff boundary, we recover an equation analogous to (REF ), as we will show.",
"Since $\\mathbf {F}^\\text{slide}=\\mathbf {F}^\\text{int}$ at the liftoff point, we conclude that the saltation matrix at the liftoff point reduces to the identity matrix.",
"Let $\\Phi _\\text{I}$ and $\\Phi _\\text{II}$ denote the flow operators on the sliding region and in the domain complementary to the sliding region, respectively.",
"That is, $\\Phi _\\text{I}(\\mathbf {x},t)$ takes initial point $\\mathbf {x}\\in \\mathcal {R}^\\text{slide}$ at time zero to $\\Phi _\\text{I}(\\mathbf {x},t)$ at time $0\\le t \\le \\mathcal {T}(\\mathbf {x})$ .",
"So $\\Phi _\\text{I}$ is restricted to act for times up to the time $\\mathcal {T}(\\mathbf {x})$ at which the trajectory starting at $\\mathbf {x}$ reaches the liftoff point, $\\Phi _\\text{I}(\\mathbf {x},\\mathcal {T}(\\mathbf {x}))\\in \\mathcal {L}$ .",
"Such a trajectory necessarily has initial condition $\\mathbf {x}=(w_1,\\ldots ,w_{n-1},0)$ satisfying $w_{n-1}<0$ , by our coordinatization.",
"Let $\\mathbf {x}_\\text{a}\\in \\mathcal {R}^\\text{slide}$ be a point on the periodic limit cycle solution, so that $\\Phi _\\text{I}(\\mathbf {x}_\\text{a},\\mathcal {T}(\\mathbf {x}_\\text{a}))=\\mathbf {x}_\\text{lift}$ .",
"Write $\\tau =\\mathcal {T}(\\mathbf {x}_\\text{a})$ for the time it takes for the trajectory to reach the liftoff point after passing location $\\mathbf {x}_\\text{a}$ .",
"We require a first-order accurate estimate of the effect of the boundary on the displacement between the unperturbed trajectory and a nearby trajectory.",
"If we make a small (size $\\varepsilon $ ) perturbation into the domain interior, away from the constraint surface, the normal component of the perturbed trajectory will return to zero within a time interval of $O(\\varepsilon )$ duration, before the two trajectories reach the liftoff boundary.",
"Therefore we need only consider perturbations tangent to the constraint surface.",
"Let $\\mathbf {x}_\\text{a}^{\\prime }\\in \\mathcal {R}^\\text{slide}$ denote a point near $\\mathbf {x}_\\text{a}$ , and suppose it takes time $\\mathcal {T}(\\mathbf {x}_\\text{a}^{\\prime })=\\tau +\\delta $ for the trajectory through $\\mathbf {x}_\\text{a}^{\\prime }$ to liftoff, at some point $\\mathbf {x}_\\text{lift}^{\\prime }\\in \\mathcal {L}$ .",
"There are two cases to consider: either $\\delta \\ge 0$ or else $\\delta \\le 0$ .",
"The two cases are handled similarly; we focus on the first for brevity.",
"In case $\\delta >0$ , the original trajectory arrives at the liftoff boundary before the perturbed trajectory, and the point $\\mathbf {x}_\\text{b}^{\\prime }=\\Phi _\\text{I}(\\mathbf {x}_\\text{a}^{\\prime },\\tau )\\in \\mathcal {R}^\\text{slide}$ .",
"We write $\\mathbf {x}_\\text{b}^{\\prime }=\\mathbf {x}_\\text{lift}+\\Delta \\mathbf {x}_\\text{b}$ (see Fig.",
"REF B) and expand the flow operator as follows: $\\nonumber \\Phi _\\text{I}(\\mathbf {x}_\\text{b}^{\\prime },\\delta )=&\\mathbf {x}_\\text{b}^{\\prime }+\\delta \\, \\mathbf {F}^\\text{slide}(\\mathbf {x}_\\text{b}^{\\prime })+\\frac{\\delta ^2}{2}\\left(\\nabla ^\\text{slide}\\mathbf {F}^\\text{slide}(\\mathbf {x}_\\text{b}^{\\prime }) \\right)\\cdot \\mathbf {F}^\\text{slide}(\\mathbf {x}_\\text{b}^{\\prime })+O(\\delta ^3)\\\\=&\\mathbf {x}_\\text{lift}+\\Delta \\mathbf {x}_\\text{b} +\\delta \\, \\mathbf {F}^\\text{slide}(\\mathbf {x}_\\text{lift})+\\delta \\,\\left( \\nabla ^\\text{slide}\\mathbf {F}^\\text{slide}(\\mathbf {x}_\\text{lift} )\\right)\\cdot \\Delta \\mathbf {x}_\\text{b}\\\\ \\nonumber &+\\frac{\\delta ^2}{2}\\left(\\nabla ^\\text{slide}\\mathbf {F}^\\text{slide}(\\mathbf {x}_\\text{b}^{\\prime }) \\right)\\cdot \\mathbf {F}^\\text{slide}(\\mathbf {x}_\\text{b}^{\\prime })+O(3),$ where $\\nabla ^\\text{slide}$ is the gradient operator restricted to $\\mathbf {x}=(x_1,\\ldots ,x_{n-1})$ .",
"The Taylor expansion in (REF ) is justified in a neighborhood of $\\mathbf {x}_\\text{b}^{\\prime }$ contained in the sliding region of the hard boundary.",
"The transversality of the intersection of the reference trajectory with $\\mathcal {L}$ (that is, $\\mathbf {F}_{n-1}(\\mathbf {x}_\\text{lift})>0$ ) means that $\\delta $ and $|\\Delta \\mathbf {x}_\\text{b}|$ will be of the same order.",
"We write $O(n)$ to denote terms of order $\\left(|\\Delta \\mathbf {x}_\\text{b}|^p \\delta ^{n-p}\\right)$ for $0\\le p \\le n$ .",
"Next we estimate $\\delta $ and the location $\\mathbf {x}_\\text{lift}^{\\prime }$ at which the perturbed trajectory crosses $\\mathcal {L}$ .",
"To first order, $\\ell ^\\intercal \\mathbf {x}_\\text{b}^{\\prime }&=\\ell ^\\intercal \\mathbf {F}^\\text{slide}(\\mathbf {x}_\\text{b}^{\\prime })\\,\\delta \\\\\\ell ^\\intercal (\\mathbf {x}_\\text{lift}+\\Delta \\mathbf {x}_\\text{b})&=\\ell ^\\intercal \\left(\\mathbf {F}^\\text{slide}(\\mathbf {x}_\\text{lift}+\\Delta \\mathbf {x}_\\text{b})\\right)\\delta \\\\\\ell ^\\intercal \\Delta \\mathbf {x}_\\text{b} &=\\ell ^\\intercal \\left( \\mathbf {F}^\\text{slide}(\\mathbf {x}_\\text{lift})+\\left(\\nabla ^\\text{slide}\\mathbf {F}^\\text{slide}(\\mathbf {x}_\\text{lift})\\right)\\cdot \\Delta \\mathbf {x}_\\text{b} \\right)\\delta \\\\\\nonumber &=\\ell ^\\intercal \\mathbf {F}^\\text{slide}(\\mathbf {x}_\\text{lift})\\delta +O(2)\\\\\\delta &= \\frac{\\ell ^\\intercal \\Delta \\mathbf {x}_\\text{b}}{\\ell ^\\intercal \\mathbf {F}^\\text{slide}(\\mathbf {x}_\\text{lift})}+O(2).$ Combining this result with (REF ), the perturbed trajectory's liftoff location is $\\mathbf {x}_\\text{lift}^{\\prime }=\\mathbf {x}_\\text{lift}+\\Delta \\mathbf {x}_\\text{b}+\\mathbf {F}^\\text{slide}(\\mathbf {x}_\\text{lift})\\delta + O(2).$ Meanwhile, as the perturbed trajectory proceeds to $\\mathcal {L}$ , during a time interval of duration $\\delta $ , the unperturbed trajectory has reentered the interior and evolves according to $\\Phi _\\text{II}$ , the flow defined for all initial conditions not within the sliding region.",
"At a time $\\delta $ after reaching $\\mathcal {L}$ , the unperturbed trajectory is located, to first order, at a point $\\mathbf {x}_\\text{c}=\\mathbf {x}_\\text{lift}+\\mathbf {F}^\\text{int}(\\mathbf {x}_\\text{lift})\\,\\delta +O(2).$ Thus, combining (REF ) and (REF ) the displacement between the two trajectories immediately following liftoff of the perturbed trajectory, $\\Delta \\mathbf {x}_\\text{c}=\\mathbf {x}_\\text{lift}^{\\prime }-\\mathbf {x}_\\text{c}$ , is given (to first order) by $\\Delta \\mathbf {x}_\\text{c}&=\\mathbf {x}_\\text{lift}^{\\prime }-\\mathbf {x}_\\text{c}\\\\&=\\mathbf {x}_\\text{lift}+\\Delta \\mathbf {x}_\\text{b}+\\mathbf {F}^\\text{slide}(\\mathbf {x}_\\text{lift})\\delta -\\left( \\mathbf {x}_\\text{lift}+\\mathbf {F}^\\text{int}(\\mathbf {x}_\\text{lift})\\,\\delta \\right)\\\\&=\\Delta \\mathbf {x}_\\text{b} + \\left( \\mathbf {F}^\\text{slide}(\\mathbf {x}_\\text{lift})-\\mathbf {F}^\\text{int}(\\mathbf {x}_\\text{lift}) \\right)\\delta \\\\&=\\Delta \\mathbf {x}_\\text{b} + \\frac{\\left( \\mathbf {F}^\\text{slide}(\\mathbf {x}_\\text{lift})-\\mathbf {F}^\\text{int}(\\mathbf {x}_\\text{lift}) \\right)\\ell ^\\intercal \\Delta \\mathbf {x}_\\text{b}}{\\ell ^\\intercal \\mathbf {F}^\\text{slide}(\\mathbf {x}_\\text{lift})}\\\\&=S_\\text{lift}\\Delta \\mathbf {x}_\\text{b} + O(2).$ Therefore, the saltation matrix at the liftoff point is $S_\\text{lift}=I+\\frac{\\left( \\mathbf {F}^\\text{slide}(\\mathbf {x}_\\text{lift})-\\mathbf {F}^\\text{int}(\\mathbf {x}_\\text{lift}) \\right)\\ell ^\\intercal }{\\ell ^\\intercal \\mathbf {F}^\\text{slide}(\\mathbf {x}_\\text{lift})}.$ We take the vector field on the sliding region to be the projection of the vector field defined for the interior onto the boundary surface (cf.",
"(REF )).",
"Therefore for our construction $\\mathbf {F}^\\text{slide}(\\mathbf {x}_\\text{lift})=\\mathbf {F}^\\text{int}(\\mathbf {x}_\\text{lift})$ , and hence $S_\\text{lift}=I,$ as claimed.",
"We note that equation (REF ) will hold for more general constructions as well.",
"This concludes the proof of part (b).",
"In parts (c) and (d) of the proof, our goal is to show the normal component of the iPRC is zero along the sliding region on $\\Sigma $ and is continuous at the landing point.",
"To this end, we compute the normal component of the iPRC using its definition (REF ), which in $(\\mathbf {w},v)$ coordinates takes the form ${\\mathbf {z}}_v := {\\mathbf {z}}\\cdot \\mathbf {n}= \\lim _{\\varepsilon \\rightarrow 0}\\frac{\\phi (\\mathbf {x}+\\varepsilon \\mathbf {n}) -\\phi (\\mathbf {x})}{\\varepsilon },$ where $\\phi (\\mathbf {x})$ denotes the asymptotic phase at point $\\mathbf {x}$ on the limit cycle.",
"That is, we apply a small instantaneous perturbation to the limit cycle, either while it is sliding along $\\Sigma $ (part c) or else just before landing (part d), in the $\\mathbf {n}$ direction, and estimate the phase difference between the perturbed and unperturbed trajectories (cf. Fig.",
"REF ).",
"Figure: Unperturbed trajectory (black curve) and a perturbed trajectory (red curve) near the hard boundary Σ\\Sigma (horizontal plane) in the (𝐰,v)(\\mathbf {w}, v) phase space.Dashed line: intersection of liftoff boundary ℒ\\mathcal {L} and Σ\\Sigma .",
"(A) Trajectory moves downward towards the sliding region (the area in Σ\\Sigma where g<0g<0), hits Σ\\Sigma at the landing point 𝐱 land \\mathbf {x}_{\\rm land}, and exits Σ\\Sigma at the liftoff point 𝐱 lift \\mathbf {x}_{\\rm lift}.",
"(B) Construction for the proof of part (b).An instantaneous perturbation tangent to Σ\\Sigma is made to the point 𝐱 a \\mathbf {x}_a at t=0t=0, pushing it to a point 𝐱 a ' ∈Σ\\mathbf {x}_a^{\\prime }\\in \\Sigma .The trajectory starting at 𝐱 a \\mathbf {x}_a (resp., 𝐱 a ' \\mathbf {x}_a^{\\prime }) reaches the liftoff point 𝐱 lift \\mathbf {x}_{\\rm lift} (resp., 𝐱 b ' \\mathbf {x}_b^{\\prime }) after time τ\\tau , and reaches 𝐱 c \\mathbf {x}_c (resp., 𝐱 lift ' \\mathbf {x}_{\\rm lift}^{\\prime }) after additional time δ\\delta .The displacements Δ𝐱 b =𝐱 b ' -𝐱 lift \\Delta \\mathbf {x}_b=\\mathbf {x}_b^{\\prime }-\\mathbf {x}_{\\rm lift} and Δ𝐱 c =𝐱 lift ' -𝐱 c \\Delta \\mathbf {x}_c=\\mathbf {x}_{\\rm lift}^{\\prime }-\\mathbf {x}_{c} differ by an amount captured, to linear order, by the saltation matrix.",
"(C) Construction for the proof of part (c).An instantaneous perturbation with size ε\\varepsilon in the positive vv-direction (green arrow) is made to the point 𝐱 a ∈Σ\\mathbf {x}_a\\in \\Sigma , pushing it off the boundary to an interior point 𝐱 a ' \\mathbf {x}_a^{\\prime }.After time τ\\tau , the trajectory starting at 𝐱 a ' \\mathbf {x}_a^{\\prime } (resp., 𝐱 a \\mathbf {x}_a) reaches a landing point 𝐱 land ' \\mathbf {x}_{\\rm land}^{\\prime } (resp., 𝐱 b \\mathbf {x}_b).",
"(D) The same perturbation (green arrow) as in panel (C) is applied to the point 𝐱 a \\mathbf {x}_a located at a distance of hh above Σ\\Sigma , pushing it to a point 𝐱 a ' \\mathbf {x}_a^{\\prime }.The trajectory starting at 𝐱 a \\mathbf {x}_a lands on Σ\\Sigma at 𝐱 land \\mathbf {x}_{\\rm land}.After the same amount of time, the perturbed trajectory starting at 𝐱 a ' \\mathbf {x}_a^{\\prime } reaches 𝐱 b ' \\mathbf {x}_b^{\\prime }.After additional time τ\\tau , the two trajectories reach 𝐱 c \\mathbf {x}_c and 𝐱 land ' \\mathbf {x}_{\\rm land}^{\\prime }, respectively."
],
[
"(c) Along the sliding region, the component of ${\\mathbf {z}}$ normal to {{formula:502cb428-e9aa-46dd-8358-c85e2bde8669}} is zero.",
"By (REF ) the normal component of the iPRC for a point on the sliding component of the trajectory, denoted by $\\mathbf {x}_\\text{a} = (w_\\text{a},0)$ is given by ${\\mathbf {z}}_v(\\mathbf {x}_\\text{a})=\\lim _{\\varepsilon \\rightarrow 0}\\frac{\\phi (w_\\text{a},\\varepsilon )-\\phi (w_\\text{a},0)}{\\varepsilon }.$ By $\\mathbf {x}_\\text{a}^{\\prime }=(w_\\text{a},\\varepsilon )$ we denote a point that is located at a distance of $\\varepsilon $ above $\\mathbf {x}_\\text{a}$ .",
"Our goal is to show ${\\mathbf {z}}_v(\\mathbf {x}_\\text{a})=0$ .",
"The perturbed trajectory from $\\mathbf {x}_\\text{a}^{\\prime }$ is governed by the interior flow $\\Phi _\\text{II}$ until it reaches the sliding region at a point $\\mathbf {x}_\\text{b}^{\\prime }\\in \\Sigma $ , after some time $\\tau $ .",
"Meanwhile the unperturbed trajectory from $\\mathbf {x}_\\text{a}$ is governed by the sliding flow $\\Phi _\\text{I}$ until it crosses the liftoff point at $\\mathcal {L}$ (Fig.",
"REF , dotted line).",
"To first order in $\\varepsilon $ , the time for the perturbed trajectory $\\mathbf {x}^{\\prime }(t)$ to return to the constraint surface is $\\nonumber \\tau (\\varepsilon )&=-\\frac{\\varepsilon }{g^\\text{int}(\\mathbf {w}_\\text{a},\\varepsilon )}+O(\\varepsilon ^2)=-\\frac{\\varepsilon }{g^\\text{int}(\\mathbf {w}_\\text{a},0)+\\varepsilon D_v g^\\text{int}(\\mathbf {w}_\\text{a},0)+O(\\varepsilon ^2)}+O(\\varepsilon ^2)\\\\&=-\\frac{\\varepsilon }{g^\\text{int}(\\mathbf {w}_\\text{a},0)}+O(\\varepsilon ^2),\\text{ as }\\varepsilon \\rightarrow 0.$ Because $\\mathbf {x}_\\text{a}=(\\mathbf {w}_\\text{a},0)$ is in the sliding region, $g^\\text{int}(\\mathbf {w}_\\text{a},0)<0$ ; we conclude that $\\tau $ and $\\varepsilon $ are of the same order.",
"We use $(p)$ to denote terms of order $p$ in $\\varepsilon $ or $\\tau $ .",
"At time $\\tau $ following the perturbation, the location of the perturbed trajectory is $\\mathbf {x}_\\text{b}^{\\prime }&=\\Phi _\\text{II}(\\mathbf {x}_\\text{a}^{\\prime },\\tau )\\\\ \\nonumber &=\\mathbf {x}_\\text{a}^{\\prime }+\\tau \\mathbf {F}^\\text{int}(\\mathbf {x}_\\text{a}^{\\prime })+O(2)\\\\&=\\mathbf {x}_\\text{a}+\\varepsilon \\mathbf {n}+\\tau \\left( \\mathbf {F}^\\text{int}(\\mathbf {x}_\\text{a})+\\varepsilon \\mathbf {n}^\\intercal D\\mathbf {F}^\\text{int}(\\mathbf {x}_\\text{a})\\right)+O(2) \\nonumber \\\\ \\nonumber &=\\mathbf {x}_\\text{a}+(0,\\ldots ,0,\\varepsilon )-\\frac{\\varepsilon }{g^\\text{int}(\\mathbf {x}_\\text{a})}(f_1^\\text{int}(\\mathbf {x}_\\text{a}),\\ldots ,f_{n-1}^\\text{int}(\\mathbf {x}_\\text{a}),g^\\text{int}(\\mathbf {x}_\\text{a}))+O(2)\\\\ \\nonumber &=\\mathbf {x}_\\text{a}-\\frac{\\varepsilon }{g^\\text{int}(\\mathbf {x}_\\text{a})}(f_1^\\text{int}(\\mathbf {x}_\\text{a}),\\ldots ,f_{n-1}^\\text{int}(\\mathbf {x}_\\text{a}),0)+O(2).$ Simultaneously, the location of the unperturbed trajectory is $\\mathbf {x}_\\text{b}&=\\Phi _\\text{I}(\\mathbf {x}_\\text{a},\\tau )\\\\ \\nonumber &=\\mathbf {x}_\\text{a}+\\tau \\mathbf {F}^\\text{slide}(\\mathbf {x}_\\text{a})+O(2)\\\\ \\nonumber &=\\mathbf {x}_\\text{a}-\\frac{\\varepsilon }{g^\\text{int}(\\mathbf {x}_\\text{a})}(f_1^\\text{int}(\\mathbf {x}_\\text{a}),\\ldots ,f_{n-1}^\\text{int}(\\mathbf {x}_\\text{a}),0) +O(2),$ since for $\\mathbf {x}\\in \\Sigma $ , we have $f^\\text{slide}(\\mathbf {x})=f^\\text{int}(\\mathbf {x})$ by construction.",
"Comparing the difference in location of the two trajectories at time $\\tau $ after the perturbation, we see that $||\\mathbf {x}_b^{\\prime }-\\mathbf {x}_b||=O(\\varepsilon ^2).$ By assumption, the asymptotic phase function $\\phi (\\mathbf {x})$ is $C^1$ with respect to displacements tangent to the constraint surface.",
"Since both $\\mathbf {x}_\\text{b}$ and $\\mathbf {x}_\\text{b}^{\\prime }$ are on this surface, $\\mathbf {n}^\\intercal (\\mathbf {x}_\\text{b}^{\\prime }-\\mathbf {x}_\\text{b})=0,$ and $\\phi (\\mathbf {x}_\\text{b}^{\\prime })=\\phi (\\mathbf {x}_\\text{b})+O(\\varepsilon ^2)$ .",
"Therefore ${\\mathbf {z}}_v(\\mathbf {x}_\\text{a})=0$ for points $\\mathbf {x}_\\text{a}$ on the sliding component of the limit cycle.",
"This completes the proof of part (c)."
],
[
"(d) The normal component of ${\\mathbf {z}}$ is continuous at the landing point.",
"In order to show that the normal component of the iPRC (${\\mathbf {z}}_v$ ) is continuous at the landing point, we prove that ${\\mathbf {z}}_v$ has a well-defined limit at the landing point and moreover, this limit equals 0 which is the value of ${\\mathbf {z}}_v$ at the landing point as proved in (c).",
"To this end, consider a point on the limit cycle shortly ahead of the landing point, $\\mathbf {x}_\\text{a}=(\\mathbf {w}_\\text{a},h)$ with $0<h\\ll 1$ fixed, (cf. Fig.",
"REF D).",
"By (REF ) $\\begin{array}{cccccccccc}{\\mathbf {z}}_v(\\mathbf {x}_a)=\\lim _{\\varepsilon \\rightarrow 0}\\frac{\\phi (\\mathbf {w}_\\text{a},h+\\varepsilon )-\\phi (\\mathbf {w}_\\text{a},h)}{\\varepsilon }.\\end{array}$ Our goal is to show $\\lim _{h\\rightarrow 0}{\\mathbf {z}}_v(\\mathbf {x}_a) = {\\mathbf {z}}_v(\\mathbf {x}_\\text{land}) = 0$ .",
"We consider the case $\\varepsilon >0$ ; the treatment for $\\varepsilon <0$ is similar.",
"For $\\varepsilon >0$ , when the unperturbed trajectory arrives at the constraint surface (at landing point $\\mathbf {x}_\\text{land}$ ), the perturbed trajectory is at a point $\\mathbf {x}_\\text{b}^{\\prime }$ that is still in the interior of the domain.",
"Denote the unperturbed landing time $t=0$ ; denote the time of flight from initial point $\\mathbf {x}_\\text{a}$ to $\\mathbf {x}_\\text{land}$ by $s$ .",
"Through an estimate similar to that in part (c), to first order in $h$ , we have $s(h)=-\\frac{h}{g^\\text{int}(\\mathbf {x}_\\text{land})}+O(h^2).$ Between $t=-s$ and $t=0$ , the displacement between the perturbed trajectory ($\\mathbf {x}^{\\prime }(t)$ ) and the unperturbed trajectory ($\\mathbf {x}(t)$ ) satisfies $\\frac{d(\\mathbf {x}^{\\prime }-\\mathbf {x})}{dt}=D\\mathbf {F}^\\text{int}(\\mathbf {x}(t))\\cdot (\\mathbf {x}^{\\prime }-\\mathbf {x})+O(\\varepsilon ^2),$ with initial condition $\\mathbf {x}^{\\prime }(-s)-\\mathbf {x}(-s)=\\varepsilon \\mathbf {n}$ .",
"Because the interior vector field is presumed $C^1$ , for $h,s\\ll 1$ we have $\\mathbf {x}^{\\prime }_\\text{b}-\\mathbf {x}_\\text{land}&= \\mathbf {x}^{\\prime }_\\text{a}-\\mathbf {x}_\\text{a}+s\\left( \\varepsilon D_v \\mathbf {F}^\\text{int}(\\mathbf {x}_\\text{land}) + O(\\varepsilon ^2)\\right) + O(s^2)\\\\&= \\varepsilon \\mathbf {n}-h\\left( \\varepsilon \\frac{D_v \\mathbf {F}^\\text{int}(\\mathbf {x}_\\text{land})}{g^\\text{int}(\\mathbf {x}_\\text{land})} + O(\\varepsilon ^2)\\right)+O(h^2) \\\\&= (0,\\cdots ,0, \\varepsilon ) -\\frac{h\\varepsilon }{g^\\text{int}(\\mathbf {x}_\\text{land})} (f_{1,v}^\\text{int}(\\mathbf {x}_\\text{land}),\\ldots ,f_{n-1,v}^\\text{int}(\\mathbf {x}_\\text{land}),g^\\text{int}_v(\\mathbf {x}_\\text{land})) + O(2)\\\\&= \\left(-h\\varepsilon \\frac{\\mathbf {f}_v^\\text{int}(\\mathbf {x}_\\text{land})}{g^\\text{int}(\\mathbf {x}_\\text{land})}, \\varepsilon -h\\varepsilon \\frac{g^\\text{int}_v(\\mathbf {x}_\\text{land})}{g^\\text{int}(\\mathbf {x}_\\text{land})}\\right) + O(2).$ Here $\\mathbf {f}_v^\\text{int} = (f_{1,v}^\\text{int},\\ldots ,f_{n-1,v}^\\text{int})$ , where $f^{\\rm int}_{k,v}$ denotes $\\partial f^{\\rm int}_k/\\partial v$ , and $O(2)$ denotes terms of order 2 in $\\varepsilon $ or $h$ as in (c).",
"In the rest of this proof, we drop the dependence of the functions on $\\mathbf {x}_{\\rm land}$ for simplicity.",
"Since $\\mathbf {x}_\\text{land}$ is in the sliding region, it follows that $\\mathbf {x}_b^{\\prime }$ is $\\varepsilon -h\\varepsilon \\frac{g^\\text{int}_v}{g^\\text{int}} + O(2)$ above the sliding region.",
"Through a similar estimation as in part (c), to first order in $\\varepsilon $ and $h$ , the time for the perturbed trajectory to arrive at the sliding region is $\\tau (h,\\varepsilon ) = \\frac{\\varepsilon -h\\varepsilon \\frac{g^\\text{int}_v}{g^\\text{int}}}{-g^\\text{int}}+O(2) = -\\frac{\\varepsilon }{g^\\text{int}} + h\\varepsilon \\frac{g_v^\\text{int}}{(g^\\text{int})^2} +O(2).$ At time $\\tau $ , the location of the perturbed trajectory is $\\begin{array}{rcl}\\mathbf {x}_\\text{land}^{\\prime } &=& \\Phi _\\text{II}(\\mathbf {x}_b^{\\prime },\\tau )\\\\&=& \\mathbf {x}_b^{\\prime }+ \\tau \\mathbf {F}^\\text{int}(\\mathbf {x}_b^{\\prime }) + O(2)\\\\&=& \\mathbf {x}_\\text{land} + \\left(-h\\varepsilon \\frac{\\mathbf {f}_v^\\text{int}}{g^\\text{int}}, \\varepsilon -h\\varepsilon \\frac{g^\\text{int}_v}{g^\\text{int}}\\right) +\\tau \\mathbf {F}^\\text{int}(\\mathbf {x}_\\text{land})+O(2)\\\\&=& \\mathbf {x}_\\text{land} + \\left(-h\\varepsilon \\frac{\\mathbf {f}_v^\\text{int}}{g^\\text{int}}, \\varepsilon -h\\varepsilon \\frac{g^\\text{int}_v}{g^\\text{int}}\\right) +\\left(-\\frac{\\varepsilon }{g^\\text{int}} + h\\varepsilon \\frac{g_v^\\text{int}}{(g^\\text{int})^2} \\right) (\\mathbf {f}^\\text{int}, g^\\text{int})+O(2)\\\\&=& \\mathbf {x}_\\text{land} + \\left(-h\\varepsilon \\frac{\\mathbf {f}_v^\\text{int}}{g^\\text{int}}, 0) +(-\\frac{\\varepsilon }{g^\\text{int}} + h\\varepsilon \\frac{g_v^\\text{int}}{(g^\\text{int})^2} \\right) (\\mathbf {f}^\\text{int}, 0)+O(2).\\end{array}$ Simultaneously, the location of the unperturbed trajectory is $\\begin{array}{rcl}\\mathbf {x}_\\text{c}&=& \\Phi _\\text{I}(\\mathbf {x}_\\text{land},\\tau )\\\\&=& \\mathbf {x}_{\\rm land}+\\tau \\mathbf {F}^\\text{slide}(\\mathbf {x}_\\text{land}) +O(2)\\\\&=& \\mathbf {x}_\\text{land} + \\left(-\\frac{\\varepsilon }{g^\\text{int}} + h\\varepsilon \\frac{g_v^\\text{int}}{(g^\\text{int})^2} \\right)(\\mathbf {f}^\\text{int}, 0) + O(2).\\end{array}$ Comparing the difference between (REF ) and (REF ), we see that $\\left\\Vert \\mathbf {x}_{\\rm land}^{\\prime } - \\mathbf {x}_c\\right\\Vert = O(h\\varepsilon ).$ Recall that the asymptotic phase is assumed to be $C^1$ , with respect to displacements tangent to $\\Sigma $ .",
"Since $\\mathbf {x}_{\\rm land}^{\\prime }$ and $\\mathbf {x}_c$ are on $\\Sigma $ , it follows that $\\phi (\\mathbf {x}_{\\rm land}^{\\prime }) - \\phi (\\mathbf {x}_c) = O(h\\varepsilon ).$ Therefore, by (REF ), ${\\mathbf {z}}_v(\\mathbf {x}_a) = \\lim _{\\varepsilon \\rightarrow 0} \\frac{\\phi (\\mathbf {x}_a^{\\prime })-\\phi (\\mathbf {x}_a)}{\\varepsilon } = \\lim _{\\varepsilon \\rightarrow 0} \\frac{\\phi (\\mathbf {x}_{\\rm land}^{\\prime })-\\phi (\\mathbf {x}_c)}{\\varepsilon }=O(h)$ Consequently, $\\lim _{h\\rightarrow 0} {\\mathbf {z}}_v(\\mathbf {x}_a) = 0$ as required.",
"This completes the proof of part (d)."
],
[
"(e) The tangential components of ${\\mathbf {z}}$ are continuous at both landing and liftoff points.",
"We denote the tangential components of the iPRC by ${\\mathbf {z}}_\\mathbf {w}$ , where $\\mathbf {w}$ represents vectors in the $n-1$ dimensional tangent space of the hard boundary.",
"The $n-1$ dimensional iPRC vector ${\\mathbf {z}}_\\mathbf {w}$ obeys a restricted (i.e.",
"reduced dimension) adjoint equation given in terms of $f_\\mathbf {w}$ , the $(n-1)\\times (n-1)$ Jacobian derivative of $f$ with respect to the $n-1$ tangential coordinates ($\\mathbf {w}$ ), and $g_\\mathbf {w}$ , the $1\\times (n-1)$ Jacobian derivative of $g$ with respect to the tangential coordinates, and ${\\mathbf {z}}_v$ , the (scalar) component of ${\\mathbf {z}}$ in the normal direction $\\begin{array}{cccccccccc}\\frac{d{\\mathbf {z}}_\\mathbf {w}}{dt}=-f_\\mathbf {w}(\\mathbf {w},v)^\\intercal {\\mathbf {z}}_\\mathbf {w}-g_\\mathbf {w}(\\mathbf {w},v)^\\intercal {\\mathbf {z}}_v\\end{array}$ along the limit cycle in the interior domain.",
"On the other hand, along the sliding component of the limit cycle that is restricted to $\\lbrace \\Sigma : v=0\\rbrace $ , ${\\mathbf {z}}_u$ satisfies $\\begin{array}{cccccccccc}\\frac{d{\\mathbf {z}}_\\mathbf {w}}{dt}=-f_\\mathbf {w}(\\mathbf {w},0)^\\intercal {\\mathbf {z}}_\\mathbf {w}.\\end{array}$ By part (c), ${\\mathbf {z}}_v$ goes continuously to zero as the trajectory from the interior approaches the landing point.",
"Therefore ${\\mathbf {z}}_\\mathbf {w}$ is continuous at the landing point.",
"Figure: Unperturbed trajectory (black) leaves the hard boundary at the liftoff point 𝐱 lift \\mathbf {x}_{\\rm lift}, in the (𝐰,v)(\\mathbf {w}, v) phase space.An instantaneous perturbation tangent to Σ\\Sigma is made to the liftoff point at t=τt=\\tau , pushing it to 𝐱 a \\mathbf {x}_a on the sliding region or to 𝐱 b \\mathbf {x}_b that is outside the sliding region.The points 𝐱 c \\mathbf {x}_c and 𝐱 lift ' \\mathbf {x}_{\\rm lift}^{\\prime } denote the positions of the unperturbed trajectory and the perturbed trajectory at t=0t=0.Next we prove the continuity of ${\\mathbf {z}}_\\mathbf {w}$ at the liftoff point $\\mathbf {x}_{\\rm lift}=(\\mathbf {w}_{\\rm lift},0)$ .",
"Recall that in the coordinates employed, the unit vector tangent to $\\Sigma $ and normal to $\\mathcal {L}$ at $\\mathbf {x}_\\text{lift}$ is $\\ell =(0,\\ldots ,0,1,0)$ (cf. Fig.",
"REF ).",
"Fix an arbitrary tangential unit vector $\\hat{\\mathbf {w}}$ oriented away from the sliding region (such that $\\ell ^\\intercal \\hat{\\mathbf {w}}> 0$ ).",
"The left and right limits of ${\\mathbf {z}}_w$ at $\\mathbf {x}_{\\rm lift}$ are given by ${\\mathbf {z}}_{\\hat{\\mathbf {w}}}^-(\\mathbf {x}_{\\rm lift}) = \\lim _{\\varepsilon \\rightarrow 0^+} \\frac{\\phi (\\mathbf {w}_{\\rm lift}-\\varepsilon \\hat{\\mathbf {w}},0)-\\phi (\\mathbf {w}_{\\rm lift},0)}{-\\varepsilon }$ and ${\\mathbf {z}}_{\\hat{\\mathbf {w}}}^+(\\mathbf {x}_{\\rm lift}) = \\lim _{\\varepsilon \\rightarrow 0} \\frac{\\phi (\\mathbf {w}_{\\rm lift}+\\varepsilon \\hat{\\mathbf {w}},0)-\\phi (\\mathbf {w}_{\\rm lift},0^+)}{\\varepsilon }.$ By $\\mathbf {x}_a=(\\mathbf {w}_{\\rm lift}-\\varepsilon \\hat{\\mathbf {w}},0)$ and $\\mathbf {x}_b=(\\mathbf {w}_{\\rm lift}+\\varepsilon \\hat{\\mathbf {w}},0)$ we denote the two points that are located at a distance of $\\varepsilon $ away from $\\mathbf {x}_{\\rm lift}$ along the $-\\hat{\\mathbf {w}}$ and $\\hat{\\mathbf {w}}$ directions, respectively (cf. Fig.",
"REF ).",
"We will show that ${\\mathbf {z}}_{\\hat{\\mathbf {w}}}^-(\\mathbf {x}_{\\rm lift}) = {\\mathbf {z}}_{\\hat{\\mathbf {w}}}^+(\\mathbf {x}_{\\rm lift}).$ The equality of these limits will establish that ${\\mathbf {z}}_{\\mathbf {w}}$ is continuous at the liftoff point.",
"First, we consider ${\\mathbf {z}}_{\\hat{\\mathbf {w}}}^+(\\mathbf {x}_\\text{lift})$ .",
"Given $\\hat{\\mathbf {w}}$ , there exists a unique point $\\mathbf {x}_{\\rm lift}^{\\prime }$ at the liftoff boundary $\\mathcal {L}\\cap \\Sigma $ , and a time $\\tau >0$ , such that the trajectory beginning from $\\mathbf {x}_{\\rm lift}^{\\prime }$ at time 0 passes directly over $\\mathbf {x}_b^{\\prime }$ at time $\\tau $ , in the sense that $\\Phi _\\text{II}(\\mathbf {x}_\\text{lift}^{\\prime },\\tau )=(\\mathbf {w}_b, h)$ , where $\\Phi _\\text{II}$ is the flow operator in the complement of the sliding region, $h>0$ is the “height” of $\\mathbf {x}_b^{\\prime }$ above $\\mathbf {x}_b$ , and $\\mathbf {w}_b$ is the coordinate vector along the tangent space of the hard boundary.",
"Let $\\mathbf {x}_{\\rm lift} = (\\mathbf {w}_{\\rm lift},0)$ and $\\mathbf {x}_{\\rm lift}^{\\prime } = (\\mathbf {w}_{\\rm lift}^{\\prime },0)$ .",
"By our construction, $\\mathbf {w}_b = \\mathbf {w}_{\\rm lift} + \\varepsilon \\hat{\\mathbf {w}}$ .",
"Hence, the location of the perturbed trajectory at time $\\tau $ is $\\begin{array}{rcl}(\\mathbf {w}_b,h)= (\\mathbf {w}_{\\rm lift}+\\varepsilon \\hat{\\mathbf {w}},h) &=&\\Phi _{\\rm II}(\\mathbf {x}_{\\rm lift}^{\\prime },\\tau )\\\\ &=& (\\mathbf {w}_{\\rm lift}^{\\prime },0) + (f^{\\rm int}(\\mathbf {x}_{\\rm lift}^{\\prime }), 0) \\tau + O(\\tau ^2)\\\\ &=& (\\mathbf {w}_{\\rm lift}^{\\prime }+f^{\\rm int}(\\mathbf {x}_{\\rm lift}^{\\prime })\\tau +O(\\tau ^2), O(\\tau ^2)).\\end{array}$ Hence $\\mathbf {w}_{\\rm lift}-\\mathbf {w}_{\\rm lift}^{\\prime } = f^{\\rm int}(\\mathbf {x}_{\\rm lift}^{\\prime })\\tau -\\varepsilon \\hat{\\mathbf {w}}+O(\\tau ^2),$ $h = O(\\tau ^2),$ and $\\mathbf {w}_b - \\mathbf {w}_{\\rm lift}^{\\prime }= f^{\\rm int}(\\mathbf {x}_{\\rm lift}^{\\prime })\\tau +O(\\tau ^2)$ On the other hand, $\\varepsilon \\hat{\\mathbf {w}} + (\\mathbf {w}_{\\rm lift}^{\\prime } - \\mathbf {w}_{b}) = \\mathbf {w}_{\\rm lift}- \\mathbf {w}_{\\rm lift}^{\\prime }.$ By (REF ) and (REF ), the above equation becomes $\\varepsilon \\hat{\\mathbf {w}} - f^{\\rm int}(\\mathbf {x}_{\\rm lift}^{\\prime })\\tau =f^{\\rm int}(\\mathbf {x}_{\\rm lift}^{\\prime })\\tau -\\varepsilon \\hat{\\mathbf {w}} + O(\\tau ^2).$ That is, $\\varepsilon \\hat{\\mathbf {w}} = f^{\\rm int}(\\mathbf {x}_{\\rm lift}^{\\prime })\\tau + O(\\tau ^2).$ Taking the inner product of both sides with the unit vector $\\ell $ (normal to $\\mathcal {L}$ ), and noting that for sufficiently small $\\varepsilon $ , $\\ell ^\\intercal f^\\text{int}(x_\\text{lift}^{\\prime })>0$ (our nondegeneracy condition), we have $\\tau =\\varepsilon \\frac{\\ell ^\\intercal \\hat{\\mathbf {w}}}{\\ell ^\\intercal f^\\text{int}(\\mathbf {x}_\\text{lift}^{\\prime })}+O(\\tau ^2),$ and hence $\\tau =O(\\varepsilon )$ .",
"Therefore, (REF ) becomes $h=O(\\varepsilon ^2)$ and hence the phase difference between $\\mathbf {x}_b^{\\prime }$ and $\\mathbf {x}_b$ is $\\phi (\\mathbf {x}_b)-\\phi (\\mathbf {x}_b^{\\prime })=O(\\varepsilon ^2)$ due to the assumption that $\\phi $ is Lipschitz continuous.",
"Next we show (REF ) holds using (REF ) and (REF ).",
"Let the unperturbed trajectory pass through $\\mathbf {x}_\\text{lift}$ at time $\\tau $ , and let $\\mathbf {x}_c$ be the location of the unperturbed trajectory at time $t=0$ (see Fig.",
"REF ).",
"Let $\\Delta \\mathbf {x}_c=\\mathbf {x}_{\\rm lift}^{\\prime }-\\mathbf {x}_c$ and $\\Delta \\mathbf {x}_b=\\mathbf {x}_b^{\\prime } -\\mathbf {x}_{\\rm lift}$ .",
"Then by part (b), $\\Delta \\mathbf {x}_b-\\Delta \\mathbf {x}_c= O(|\\Delta \\mathbf {x}_b|^2);$ since the saltation matrix is equal to the identity matrix at the liftoff boundary.",
"Since $\\mathbf {x}_{\\rm lift},\\mathbf {x}_b,\\mathbf {x}_b^{\\prime }$ form a right triangle, $|\\Delta \\mathbf {x}_b|^2 = \\varepsilon ^2+ h^2 = \\varepsilon ^2+ O(\\varepsilon ^4),$ which implies that $\\Delta \\mathbf {x}_b-\\Delta \\mathbf {x}_c= O(\\varepsilon ^2).$ Direct computation shows $\\begin{array}{cccccccccc}\\phi (\\mathbf {x}_b)-\\phi (\\mathbf {x}_{\\rm lift}) &=&(\\phi (\\mathbf {x}_b)-\\phi (\\mathbf {x}_b^{\\prime })) + (\\phi (\\mathbf {x}_b^{\\prime })-\\phi (\\mathbf {x}_{\\rm lift}))\\\\&=& (\\phi (\\mathbf {x}_{\\rm lift}^{\\prime })-\\phi (\\mathbf {x}_c))+O(\\varepsilon ^2) \\\\&=& D_\\mathbf {w}\\phi (\\mathbf {x}_c)\\cdot \\Delta \\mathbf {x}_c+O(\\varepsilon ^2)\\\\&=& D_\\mathbf {w}\\phi (\\mathbf {x}_c)\\cdot \\Delta \\mathbf {x}_b+O(\\varepsilon ^2)\\\\&=& D_\\mathbf {w}\\phi (\\mathbf {x}_c)\\cdot (\\mathbf {x}_b^{\\prime } - \\mathbf {x}_{\\rm lift})+O(\\varepsilon ^2)\\\\&=& D_\\mathbf {w}\\phi (\\mathbf {x}_c)\\cdot (\\mathbf {x}_b - \\mathbf {x}_{\\rm lift})+O(\\varepsilon ^2)\\\\&=& D_\\mathbf {w}\\phi (\\mathbf {x}_c)\\cdot \\varepsilon \\hat{\\mathbf {w}}+O(\\varepsilon ^2)\\end{array}.$ To obtain the second equality, we translate the trajectories backward in time by $\\tau $ beginning from $\\mathbf {x}_b^{\\prime }$ and $\\mathbf {x}_\\text{lift}$ , respectively; shifting both trajectories by an equal time interval does not change their phase relationship.",
"The $O(\\varepsilon ^2)$ difference arises from (REF ).",
"The third equality follows from the assumption that $\\phi $ is differentiable with respect to displacements tangent to the sliding region.",
"The fourth equality uses (REF ); the fifth and seventh follow from the definitions; the sixth uses (REF ).",
"Recall the we assume $\\phi $ to have Lipschitz continuous derivatives in the tangential directions at the boundary surface (except possibly at the landing and liftoff points).",
"Under this assumption, taking the limit $\\varepsilon \\rightarrow 0^+$ leads to $\\mathbf {x}_c\\rightarrow \\mathbf {x}_{\\rm lift}^-$ and hence ${\\mathbf {z}}^+_{\\hat{\\mathbf {w}}}(\\mathbf {x}_{\\rm lift}) =D_\\mathbf {w}\\phi (\\mathbf {x}_{\\rm lift}^-)\\cdot \\hat{\\mathbf {w}}$ by (REF ).",
"On the other hand, $\\begin{array}{cccccccccc}\\phi (\\mathbf {x}_a)-\\phi (\\mathbf {x}_{\\rm lift}) &=&D_\\mathbf {w}\\phi (\\mathbf {x}_a)\\cdot (\\mathbf {x}_a-\\mathbf {x}_{\\rm lift}) +O(\\varepsilon ^2)\\\\&=& -D_\\mathbf {w}\\phi (\\mathbf {x}_a)\\cdot \\varepsilon \\hat{\\mathbf {w}}+O(\\varepsilon ^2).",
"\\end{array}$ Taking the limit $\\varepsilon \\rightarrow 0+$ results in $\\mathbf {x}_a\\rightarrow \\mathbf {x}_{\\rm lift}^-$ and hence (REF ) together with (REF ), implies ${\\mathbf {z}}^-_{\\hat{\\mathbf {w}}}(\\mathbf {x}_{\\rm lift}) =D_\\mathbf {w}\\phi (\\mathbf {x}_{\\rm lift}^-)\\cdot \\hat{\\mathbf {w}}.$ Hence, (REF ) holds."
],
[
"Numerical Algorithms",
"We will now describe how the results presented in § and § can be implemented as numerical algorithms.",
"MATLAB code that implements these algorithms for the example system described in § is available: https://github.com/yangyang-wang/LC_in_square.",
"Consider a multiple-zone Filippov system generalized from (REF ), $\\frac{d\\mathbf {x}}{dt}=F(\\mathbf {x}),$ that produces a $T_0$ -periodic limit cycle solution $\\gamma (t)\\subset {\\mathbf {R}}^n$ .",
"Suppose $\\gamma (t)$ includes $k$ sliding components confined to boundary surfaces denoted as $\\Sigma ^i\\subset {\\mathbf {R}}^{n-1},\\,i\\in \\lbrace 1,...,k\\rbrace $ .",
"$\\gamma (t)$ exits the $i$ -th boundary $\\Sigma ^i$ at a unique liftoff point $\\mathbf {x}_{\\text{lift}}^i$ given that the nondegeneracy condition (REF ) at $\\mathbf {x}_{\\text{lift}}^i$ is satisfied.",
"We denote the normal vector to $\\Sigma ^i$ at liftoff, landing, or boundary crossing points by $n^i$ .",
"We denote the interior domain by ${\\mathcal {R}}^{\\rm interior}$ , which can now consist of multiple subdomains separated by transversal crossing boundaries, and denote the piecewise smooth vector field in ${\\mathcal {R}}^{\\rm interior}$ by $F^{\\rm interior}$ .",
"By (REF ), the sliding vector field on the sliding region ${\\mathcal {R}}^{\\mathrm {slide}_i}\\subset \\Sigma ^i$ is therefore $F^{\\mathrm {slide}_i}(\\mathbf {x})= F^{\\rm interior}(\\mathbf {x}) - (n^i\\cdot F^{\\rm interior}(\\mathbf {x}))n^i$ Using this notation, the vector field (REF ) can be written as $F(\\mathbf {x}):=\\left\\lbrace \\begin{array}{cccccccccc}F^{\\rm interior}(\\mathbf {x}), & \\mathbf {x}\\in {\\mathcal {R}}^{\\rm interior}&\\\\F^{\\mathrm {slide}_i}(\\mathbf {x}), & \\mathbf {x}\\in {\\mathcal {R}}^{\\mathrm {slide}_i}&\\\\\\end{array}\\right.$ and we denote the vector field after a static perturbation by $F_{\\varepsilon }(\\mathbf {x}):=\\left\\lbrace \\begin{array}{cccccccccc}F^{\\rm interior}_\\varepsilon (\\mathbf {x}), & \\mathbf {x}\\in {\\mathcal {R}}^{\\rm interior}&\\\\F^{\\mathrm {slide}_i}_\\varepsilon (\\mathbf {x}), & \\mathbf {x}\\in {\\mathcal {R}}^{\\mathrm {slide}_i}&\\\\\\end{array}\\right.$ where $i\\in \\lbrace 1, ..., k\\rbrace $ .",
"Here we assume that the regions are independent of static perturbation with size $\\varepsilon $ .",
"Notice that the computation of the iSRC requires estimating the rescaling factors, for which we need to compute the iPRC or the lTRC depending on whether a global uniform rescaling (REF ) or a piecewise uniform rescaling (REF ) is needed.",
"We hence first present the numerical algorithms for obtaining the iPRC in §REF and the lTRC in §REF ; the algorithm for solving the homogeneous variational equation for the linear shape responses of $\\gamma (t)$ to instantaneous perturbations (the variational dynamics $\\mathbf {u}$ ) is presented in §REF ; lastly, in §REF we illustrate the algorithms for computing the linear shape responses of $\\gamma (t)$ to sustained perturbations (the iSRC $\\gamma _1$ ) with a uniform rescaling factor computed from the iPRC as well as with piecewise uniform rescaling factors computed from the lTRC.",
"For simplicity, we assume the initial time is $t_0=0$ ."
],
[
"Algorithm for Calculating the iPRC ${\\mathbf {z}}$ for LCSCs",
"It follows from Remark REF that the iPRC ${\\mathbf {z}}$ for the LCSCs need to be solved backward in time.",
"While there is no discontinuity of ${\\mathbf {z}}$ at a landing point, a time-reversed version of the jump matrix at the liftoff point on the hard boundary $\\Sigma ^i$ , denoted as $\\mathcal {J}^i_{\\rm lift}$ , is given by $\\mathcal {J}^i_{\\rm lift}=I-n^i {n^i}^\\intercal ,$ where $I$ is the identity matrix.",
"$\\mathcal {J}^i_{\\rm lift}$ updates ${\\mathbf {z}}$ local to the liftoff point as $\\begin{array}{cccccccccc}{{\\mathbf {z}}_{\\text{lift}}^{i^-}}=\\mathcal {J}^i_{\\rm lift} {{\\mathbf {z}}_{\\text{lift}}^{i^+}}\\end{array}$ where ${{\\mathbf {z}}_{\\text{lift}}^{i^-}}$ and ${{\\mathbf {z}}_{\\text{lift}}^{i^+}}$ are the iPRC just before and just after the trajectory crosses the liftoff point $x_{\\text{lift},i}$ in forwards time.",
"We now describe an algorithm for numerically obtaining the complete iPRC ${\\mathbf {z}}$ for $\\gamma (t)$ , a stable limit cycle with sliding components along hard boundaries and transversal crossing boundaries as described before."
],
[
"Algorithm for ${\\mathbf {z}}$",
" Fix an initial condition $\\mathbf {x}_0=\\gamma (0)$ on the limit cycle, and integrate (REF ) to compute $\\gamma (t)$ over $[0, T_0]$ .",
"Integrate the adjoint equation backward in time by defining $s=T_0-t$ and numerically solve for the fundamental matrix $\\Psi (s)$ over one period $0\\le s \\le T_0$ , where $\\Psi $ satisfies $\\Psi (0)=I$ , the identity matrix.",
"For $s$ such that $\\gamma (T_0-s)$ lies in the interior of the domain, $ \\frac{d\\Psi }{ds}=A^{\\rm interior}(T_0-s)\\Psi $ where $A^{\\rm interior}(t)=\\left(DF^{\\rm interior} (\\gamma (t))\\right)^\\intercal $ is the transpose of the Jacobian of the interior vector field $F^{\\rm interior}$ .",
"For $s$ such that $\\gamma (T_0-s)$ lies within a sliding component along boundary $\\Sigma ^i$ , $ \\frac{d\\Psi }{ds}= A^i(T_0-s)\\Psi $ where $A^i(t)=\\left(DF^{\\mathrm {slide}_i}(\\gamma (t))\\right)^\\intercal $ is the transpose of the Jacobian of the sliding vector field $F^{\\mathrm {slide}_i}$ , given in (REF ).",
"At any time $t_p$ when $\\gamma $ transversely crosses a switching surface with a normal vector $n_p$ , $ \\Psi ^- = \\mathcal {J} \\Psi ^+$ where $\\Psi ^-=\\lim _{s\\rightarrow (T_0-t_p)^+}\\Psi (s)$ and $\\Psi ^+=\\lim _{s\\rightarrow (T_0-t_p)^-}\\Psi (s)$ are the fundamental matrices just before and just after crossing the surface in forwards time.",
"$\\mathcal {J}=S^\\intercal $ since $J^\\intercal S=I$ as discussed in §REF , where the saltation matrix at any transversal crossing point is $S=I+\\frac{(F_p^+-F_p^-)n_p^\\intercal }{n_p^\\intercal F_p^-}$ where $F_p^-, F_p^+$ are the vector fields just before and just after the crossing in forwards time (see (REF )).",
"At a liftoff point on the $i$ -th hard boundary $\\Sigma ^i$ (in backwards time, a transition from the interior to $\\Sigma ^i$ ), update $\\Psi $ as $ \\Psi ^- = \\mathcal {J}^i \\Psi ^+$ where $\\mathcal {J}^i=I-n^i n^{i \\intercal }$ as defined in (REF ), and then switch the integration from the full Jacobian $A^{\\rm interior}$ to the restricted Jacobian $A^i$ .",
"At a landing point on the $i$ -th hard boundary $\\Sigma ^i$ (in backwards time, a transition from $\\Sigma ^i$ to the interior) switch integration from the restricted Jacobian $A^i$ to the full Jacobian $A^{\\rm interior}$ ; no other change in $\\Psi $ is needed.",
"Diagonalize the fundamental matrix at one period $\\Psi (T_0)$ ; it should have a single eigenvector $v$ with unit eigenvalue.",
"The initial value for ${\\mathbf {z}}_\\text{BW}$ (represented in backwards time) at the point $\\gamma (T_0)=\\gamma (0)=\\mathbf {x}_0$ is given by ${\\mathbf {z}}_\\text{BW}(0)=\\frac{v}{F(\\mathbf {x}_0)\\cdot v}$ The iPRC in backward time over $s\\in [0, T_0]$ is given by ${\\mathbf {z}}_\\text{BW}(s)=\\Psi (s){\\mathbf {z}}_{\\rm BW}(0)$ and is $T_0$ -periodic.",
"Equivalently, one may repeat step (2) by replacing $\\Psi (s)$ with ${\\mathbf {z}}_\\text{BW}(s)$ and replacing the initial condition $\\Psi (0)=I$ with ${\\mathbf {z}}_{\\rm BW}(0)$ to solve for the complete iPRC.",
"The iPRC in forward time is then given by ${\\mathbf {z}}(t)={\\mathbf {z}}_\\text{BW}(T_0-t)$ where $t\\in [0,T_0]$ .",
"The linear shift in period in response to the static perturbation can be calculated by evaluating the integral (see (REF )) $T_1=-\\int _{0}^{T_0} {\\mathbf {z}}^\\intercal (t)\\frac{\\partial F_\\varepsilon (\\gamma (t))}{\\partial \\varepsilon }\\Big |_{\\varepsilon =0}dt$ Remark E.1 An alternative way (in MATLAB) to do backward integration is reversing the time span in the numerical solver; that is, integrate the adjoint equation over $[T_0, 0]$ to compute ${\\mathbf {z}}(t)$ ."
],
[
"Algorithm for Calculating the lTRC for LCSCs",
"The lTRC satisfies the same adjoint equation, (REF ), as the iPRC, and hence exhibits the same jump matrix at each liftoff, landing and boundary crossing point.",
"It follows that the algorithm for the iPRC from §REF can mostly carry over to computing the lTRC.",
"Suppose the domain of $\\gamma (t)$ can be divided into $m$ regions ${\\mathcal {R}}^1, ..., {\\mathcal {R}}^m$ , each distinguished by its own timing sensitivity properties.",
"We denote the lTRC in ${\\mathcal {R}}^j$ by $\\eta ^j$ .",
"Below we describe the algorithm to compute $\\eta ^j$ in region ${\\mathcal {R}}^j$ bounded by the two local timing surfaces $\\Sigma ^{\\rm in}$ and $\\Sigma ^{\\rm out}$ .",
"Following the notations in §, $t^{\\rm in}$ and $t^{\\rm out}$ denote the time of entry into and exit out of ${\\mathcal {R}}^j$ , at locations $\\mathbf {x}^{\\rm in}$ and $\\mathbf {x}^{\\rm out}$ , respectively.",
"The algorithm for computing $\\eta ^j$ is described as follows."
],
[
"Algorithm for $\\eta ^j$",
" Compute $\\gamma $ , the unperturbed limit cycle, and $T_0$ , its period, by integrating (REF ).",
"Compute $t^{\\rm in}, t^{\\rm out}$ for region $j$ .",
"Evaluate $\\mathbf {x}^{\\rm in}=\\gamma (t^{\\rm in}),\\, \\mathbf {x}^{\\rm out}=\\gamma (t^{\\rm out})$ and $T_0^{j}=t^{\\rm out}-t^{\\rm in}$ .",
"Compute the boundary value for $\\eta ^j$ at the exit point $\\mathbf {x}^{\\rm out}$ (see (REF )) $\\eta ^{j}(\\mathbf {x}^{\\rm out})=\\frac{-n^{\\rm out}}{{n^{\\rm out}}^\\intercal F(\\mathbf {x}^{\\rm out})}$ where $n^{\\rm out}$ is a normal vector to $\\Sigma ^{\\rm out}$ .",
"Integrate the adjoint equation backward in time by defining $s=T_0-t$ and numerically solve for $\\eta ^j_{\\rm BW}(s)$ (represented in backwards time) over $[T_0-t^{\\rm out}, T_0-t^{\\rm in}]$ .",
"$\\eta ^j_{\\rm BW}(s)$ satisfies the initial condition $\\eta ^j_{\\rm BW}(T_0-t_{\\rm out})=\\eta ^{j}(t_{\\rm out})$ computed from step (3) as well as conditions (b) through (f) from step (2) of Algorithm for ${\\mathbf {z}}$ in §REF .",
"The lTRC in forward time is then given by $\\eta ^j(t) = \\eta ^j_{\\rm BW}(T_0-t)$ where $t\\in [t_{\\rm in}, t_{\\rm out}]$ .",
"Compute $\\gamma _\\varepsilon $ , the limit cycle under some small static perturbation $\\varepsilon \\ll 1$ , and find $\\mathbf {x}_{\\varepsilon }^{\\rm in}$ , the coordinate of the intersection point where $\\gamma _\\varepsilon (t)$ crosses $\\Sigma ^{\\rm in}$ .",
"The linear shift in time in region $j$ in response to the static perturbation can be calculated by evaluating the integral (see (REF )) $T^{j}_{1} = \\eta ^j(\\mathbf {x}^{\\rm in})\\cdot \\frac{ \\mathbf {x}_{\\varepsilon }^{\\rm in}-\\mathbf {x}^{\\rm in}}{\\varepsilon }+\\int _{t^{\\rm in}}^{t^{\\rm out}}\\eta ^j(\\gamma (t))\\cdot \\frac{\\partial F_\\varepsilon (\\gamma (t))}{\\partial \\varepsilon }\\Big |_{\\varepsilon =0}dt.$ Remark E.2 All the local linear shifts in time sum up to the global linear shift in period, that is, $T_1 = \\sum _{j=1}^{j=m}T_1^j$ ."
],
[
"Algorithm for Solving the Homogeneous Variational Equation for LCSCs",
"Here we describe the algorithm for solving the homogeneous variational equation for linear displacement $\\mathbf {u}$ , the shape response to an instantaneous perturbation.",
"This makes use of Theorem REF , which describes different jumping behaviors of $\\mathbf {u}$ at liftoff, landing, and boundary crossing points.",
"Unlike the iPRC and lTRC which require integration backwards in time, the variational dynamics can be solved with forward integration.",
"This makes the algorithm comparatively simpler by allowing $\\gamma (t)$ and $\\mathbf {u}(t)$ to be solved simultaneously."
],
[
"Algorithm for $\\mathbf {u}$ :",
" Fix an initial condition $\\mathbf {x}_0=\\gamma (0)$ on the limit cycle and an initial condition $\\mathbf {u}_0=\\mathbf {u}(0)$ for the displacement at $\\gamma (0)$ of the limit cycle.",
"Integrate the original differential equation (REF ) and the homogeneous variational equation (REF ) simultaneously forward in time and numerically solve for $\\mathbf {u}(t)$ over one period $0\\le t \\le T_0$ , where $\\mathbf {u}$ satisfies $\\mathbf {u}(0)=\\mathbf {u}_0$ .",
"For $t$ such that $\\gamma (t)$ lies in the interior of the domain, $ \\frac{d\\mathbf {u}}{dt}=DF^{\\rm interior} (\\gamma (t)) \\mathbf {u}$ For $t$ such that $\\gamma (t)$ lies within a sliding component along boundary $\\Sigma ^i$ , $ \\frac{d\\mathbf {u}}{dt}= DF^{\\mathrm {slide}_i}(\\gamma (t))\\mathbf {u}$ where $DF^{\\mathrm {slide}_i}$ is the Jacobian of the sliding vector field $F^{\\mathrm {slide}_i}$ given in (REF ).",
"At any time $t_p$ when $\\gamma $ transversely crosses a switching surface with a normal vector $n_p$ separating vector field $F_p^-$ on the incoming side from vector field $F_p^+$ on the outgoing side, $ \\mathbf {u}^+ = S \\mathbf {u}^-$ where $\\mathbf {u}^-=\\lim _{t\\rightarrow t_p^-}\\mathbf {u}(t)$ and $\\mathbf {u}^+=\\lim _{t\\rightarrow t_p^+}\\mathbf {u}(t)$ are the displacements just before and just after crossing the surface.",
"By the definition for the saltation matrix at transversal crossing point (REF ), we have $S=I+\\frac{(F_p^+-F_p^-)n_p^\\intercal }{n_p^\\intercal F_p^-}.$ At a landing point on the $i$ -th hard boundary $\\Sigma ^i$ , update $\\mathbf {u}$ as $ \\mathbf {u}^+ = S^i \\mathbf {u}^-$ where $S^i=I-n^i{n^i}^\\intercal $ (recall $n^i$ is the normal vector to $\\Sigma ^i$ ) and switch integration from the full Jacobian $DF^{\\rm interior}$ to the restricted Jacobian $DF^{\\mathrm {slide}_i}$ .",
"At a liftoff point on the $i$ -th hard boundary $\\Sigma ^i$ , switch integration from the restricted Jacobian $DF^{\\mathrm {slide}_i}$ to the full Jacobian $DF^{\\rm interior}$ ; no other change in $\\mathbf {u}$ is needed.",
"Remark E.3 The fundamental solution matrix satisfies $\\frac{d\\Phi (t,0)}{dt}=DF\\Phi (t,0),\\, \\text{with}\\quad \\Phi (0,0)=I$ and takes the initial perturbation $\\mathbf {u}(0)$ to the perturbation $\\mathbf {u}(t)$ at time $t$ , that is, $\\mathbf {u}(t)=\\Phi (t,0)\\mathbf {u}(0).$ Computing $\\Phi $ therefore requires applying Algorithm for $\\mathbf {u}$ $n$ times, once for each dimension of the state space.",
"Specifically, let $\\Phi (t,0)=[\\phi _1(t,0)\\, ...,\\,\\phi _n(t,0)]$ .",
"The $i$ -th column $\\phi _i(t,0)$ is the solution of the variational equation (REF ) with the initial condition $\\phi _i(0,0)=e_i$ , a unit column vector with zeros everywhere except at the $i$ -th row where the entry equals 1.",
"Remark E.4 Once $\\Phi $ is obtained, we can obtain the monodromy matrix, $M= \\Phi (T_0,0)$ .",
"It follows from the periodicity of $\\gamma (t)$ that $M$ has $+1$ as an eigenvalue with eigenvector $v$ tangent to the limit cycle at $\\mathbf {x}_0$ ; this condition provides a partial consistency check for the algorithm."
],
[
"Algorithms for computing iSRC, the response to sustained perturbation",
"Now we discuss the calculation of iSRC $\\gamma _1$ , the linear shape response to a sustained perturbation.",
"While $\\gamma _1$ shares the same saltation as $\\mathbf {u}$ at each liftoff, landing and boundary crossing point, $\\gamma _1$ satisfies the nonhomogeneous version of the variational equation, (REF ) or (REF ), where one of the nonhomogeneous terms depends on the time scaling factor, $\\nu _1$ or $\\nu ^j_1$ .",
"Moreover, the initial condition for $\\gamma _1$ depends on the given perturbation and hence needs to be computed in the algorithm whereas the initial value for $\\mathbf {u}$ is arbitrarily preassigned.",
"In the following, we first describe the algorithm for computing $\\gamma _1$ using the global uniform rescaling and then consider using piecewise uniform rescaling."
],
[
"Algorithm for $\\gamma _1$ with uniform rescaling",
" Fix an initial condition $\\mathbf {x}_0=\\gamma (0)$ on the limit cycle.",
"Compute the linear shift in period $T_1$ using Algorithm for ${\\mathbf {z}}$, then evaluate $\\nu _1=T_1/T_0$ .",
"Choose an arbitrary Poincaré section $\\Sigma $ (this can be one of the switching boundaries for appropriate $\\mathbf {x}_0$ ) that is transverse to $\\gamma $ at $\\mathbf {x}_0$ .",
"Compute $\\gamma _\\varepsilon $ , the limit cycle under some fixed small static perturbation, and find ${\\mathbf {x}_0}_\\varepsilon $ , the coordinate of the intersection point where $\\gamma _\\varepsilon (t)$ crosses $\\Sigma $ .",
"The initial value for $\\gamma _1$ at the initial point $\\mathbf {x}_0$ is then given by $\\gamma _1(0) = \\frac{{\\mathbf {x}_{0}}_\\varepsilon - \\mathbf {x}_0}{\\varepsilon }$ Integrate the original differential equation (REF ) with the initial condition $\\mathbf {x}_0$ and the nonhomogeneous variational equation (REF ) simultaneously forward in time and numerically solve for $\\gamma _1$ over one period $0\\le t \\le T_0$ , where $\\gamma _1$ satisfies $\\gamma _1(0)=({\\mathbf {x}_{0}}_\\varepsilon - \\mathbf {x}_0)/\\varepsilon $ .",
"For $t$ such that $\\gamma (t)$ lies in the interior of the domain, $ \\frac{d\\gamma _1}{dt}=DF^{\\rm interior}(\\gamma (t)) \\gamma _1 + \\nu _1 F^{\\rm interior}(\\gamma ( t)) +\\frac{\\partial F^{\\rm interior}_\\varepsilon (\\gamma (t))}{\\partial \\varepsilon }\\Big |_{\\varepsilon =0} $ For $t$ such that $\\gamma (t)$ lies within a sliding component along boundary $\\Sigma ^i$ , $ \\frac{d\\gamma _1}{dt}= DF^{\\mathrm {slide}_i}(\\gamma (t)) \\gamma _1 + \\nu _1 F^{\\mathrm {slide}_i}(\\gamma ( t)) + \\frac{\\partial F_\\varepsilon ^{\\mathrm {slide}_i}(\\gamma ( t))}{\\partial \\varepsilon }\\Big |_{\\varepsilon =0} $ where $DF^{\\mathrm {slide}_i}$ is the Jacobian of the sliding vector field $F^{\\mathrm {slide}_i}$ given in (REF ).",
"For transversal crossings, landing points, and liftoff points, apply (d), (e) and (f), respectively, from step 2) in Algorithm for $\\mathbf {u}$ in §REF , by replacing $\\mathbf {u}$ with $\\gamma _1$ .",
"Next we consider the case when $\\gamma (t)$ exhibits $m$ different uniform timing sensitivities at regions ${\\mathcal {R}}^1, ..., {\\mathcal {R}}^m$ , each bounded by two local timing surfaces, as discussed in §REF .",
"Piecewise uniform rescaling is therefore needed to compute the shape response curve.",
"The procedure for obtaining $\\gamma _1$ in this case is nearly the same as described in Algorithm for $\\gamma _1$ with uniform rescaling, except we now need to compute various rescaling factors using the lTRC.",
"This hence leads to different variational equations that need to be solved.",
"On the other hand, the local timing surfaces naturally serve as the Poincaré sections that are required to compute the initial values for $\\gamma _1$ in the uniform rescaling case."
],
[
"Algorithm for $\\gamma _1$ with piecewise uniform rescaling",
" Take the initial condition for $\\gamma (t)$ to be $\\gamma (0)=\\mathbf {x}_0\\in \\Sigma $ , where $\\Sigma $ is one of the local timing surfaces.",
"Compute $\\gamma (t)$ , the unperturbed trajectory, and $\\gamma _\\varepsilon (t)$ , the trajectory under some static perturbation $0<\\varepsilon \\ll 1$ , by integrating (REF ).",
"For $j\\in \\lbrace 1,...,m\\rbrace $ , compute $T^j_0$ , the time that $\\gamma (t)$ spends in region $j$ and $T^j_1$ , the linear shift in time in region $j$ using Algorithm for $\\eta ^j$, and then evaluate $\\nu ^j_1=T^j_1/T^j_0$ .",
"Compute ${\\mathbf {x}_0}_\\varepsilon $ , the coordinate of the intersection point where $\\gamma _\\varepsilon (t)$ crosses $\\Sigma $ .",
"The initial value for $\\gamma _1$ at the initial point $\\mathbf {x}_0$ is given by $\\gamma _1(0) = \\frac{{\\mathbf {x}_0}_\\varepsilon - \\mathbf {x}_0}{\\varepsilon }$ Integrate the original differential equation (REF ) with the initial condition $\\mathbf {x}_0$ and the piecewise nonhomogeneous variational equation (REF ) simultaneously forward in time and numerically solve for $\\gamma _1$ over one period $0\\le t \\le T_0$ , where $\\gamma _1$ satisfies $\\gamma _1(0)=({\\mathbf {x}_{0}}_\\varepsilon - \\mathbf {x}_0)/\\varepsilon $ .",
"For $t$ such that $\\gamma (t)$ lies in the intersection of the interior of the domain and region ${\\mathcal {R}}^j$ , $ \\frac{d\\gamma _1}{dt}=DF^{\\mathrm {interior}_j}(\\gamma (t)) \\gamma _1 + \\nu ^j_1 F^{\\mathrm {interior}_j}(\\gamma ( t)) +\\frac{\\partial F_\\varepsilon ^{\\mathrm {interior}_j}(\\gamma ( t))}{\\partial \\varepsilon }\\Big |_{\\varepsilon =0} $ where $DF^{\\mathrm {interior}_j}$ is the Jacobian of the interior vector field $F^{\\mathrm {interior}_j}$ in ${\\mathcal {R}}^j$ .",
"For $t$ such that $\\gamma (t)$ lies within the intersection of a hard boundary $\\Sigma ^i$ and region ${\\mathcal {R}}^j$ , $ \\frac{d\\gamma _1}{dt}= DF^{\\mathrm {slide}_i}(\\gamma (t)) \\gamma _1 + \\nu ^j_1 F^{\\mathrm {slide}_i}(\\gamma ( t)) + \\frac{\\partial F_\\varepsilon ^{\\mathrm {slide}_i}(\\gamma ( t))}{\\partial \\varepsilon }\\Big |_{\\varepsilon =0} $ where $DF^{\\mathrm {slide}_i}$ is the Jacobian of the sliding vector field $F^{\\mathrm {slide}_i}(\\mathbf {x})=F^{\\mathrm {interior}_j}(\\mathbf {x})-(n^i\\cdot F^{\\mathrm {interior}_j}(\\mathbf {x}))n^i$ given in (REF ).",
"For transversal crossings, landing points, and liftoff points, apply (d), (e) and (f), respectively, from step 2) in Algorithm for $\\mathbf {u}$ in §REF , replacing $\\mathbf {u}$ with $\\gamma _1$ ."
]
] | 1906.04387 |
[
[
"TW-SMNet: Deep Multitask Learning of Tele-Wide Stereo Matching"
],
[
"Abstract In this paper, we introduce the problem of estimating the real world depth of elements in a scene captured by two cameras with different field of views, where the first field of view (FOV) is a Wide FOV (WFOV) captured by a wide angle lens, and the second FOV is contained in the first FOV and is captured by a tele zoom lens.",
"We refer to the problem of estimating the inverse depth for the union of FOVs, while leveraging the stereo information in the overlapping FOV, as Tele-Wide Stereo Matching (TW-SM).",
"We propose different deep learning solutions to the TW-SM problem.",
"Since the disparity is proportional to the inverse depth, we train stereo matching disparity estimation (SMDE) networks to estimate the disparity for the union WFOV.",
"We further propose an end-to-end deep multitask tele-wide stereo matching neural network (MT-TW-SMNet), which simultaneously learns the SMDE task for the overlapped Tele FOV and the single image inverse depth estimation (SIDE) task for the WFOV.",
"Moreover, we design multiple methods for the fusion of the SMDE and SIDE networks.",
"We evaluate the performance of TW-SM on the popular KITTI and SceneFlow stereo datasets, and demonstrate its practicality by synthesizing the Bokeh effect on the WFOV from a tele-wide stereo image pair."
],
[
"Introduction",
"Depth estimation of real world elements in a scene has many applications in computer vision, scene understanding, image and video enhancement, autonomous driving, simultaneous localization and mapping, and 3D object reconstruction.",
"For example, accurate depth estimation allows accurate foreground-background segmentation, and hence the separation of the foreground (close) objects of interest from the background (far) objects in a scene.",
"Foreground-background segmentation can be used in object detection, tracking, and image Bokeh.",
"Bokeh is the soft out-of-focus blur of the background which can be mastered by using the right settings on expensive cameras with fast lens and wide apertures.",
"Moreover, achieving the Bokeh effect by capturing with a shallow depth-of-field often requires the camera to be physically closer to the subject of interest, and the subject to be further away from the background.",
"However, accurate depth estimation also allows one to synthesize the images captured by non-professional photographers or cameras with smaller lenses (such as mobile phone cameras) to obtain more aesthetically pleasant effects such as Bokeh (see Fig.",
"REF ).",
"Other applications of accurate depth estimation include 3D object reconstruction and virtual reality applications, where it may be desired to alter the background regions or the subject of interest to render them according to the desired virtual reality.",
"For autonomous driving applications, where safety is most important, the depth estimated from the images captured by the cameras can be fused with those obtained from other sensors to improve the accuracy of estimating the distance of the detected objects from the camera.",
"Figure: Demonstration of the Bokeh effect on the central Tele FOV only (left) and on the full Wide FOV (right).Figure: Tele-wide stereo matching on a KITTI example, to generate Wide FOV Bokeh)The problem of estimating depth in an image from two stereo cameras with an identical field of view (FOV) has been well studied [1], [2], [3].",
"Depth estimation from two stereo rectified images can be obtained by calculating the disparity (which is the horizontal displacement) between matching pixels in both images.",
"This is often done by stereo matching techniques that find the corresponding points in both images.",
"Stereo matching involves the extraction of features from the stereo images and the computation of a cost volume to match between the features along the same horizontal line in the left and right rectified images using different similarity measures.",
"This is then followed by aggregation of the cost metrics to optimize the disparity estimation.",
"Multi-camera systems, where more than two cameras with the same focal lengths are deployed at different baseline lengths and directions, have been proposed [4].",
"In this multi-baseline system, the cameras are assumed to have the same focal length.",
"Hence, multiple disparity measurements for the common (overlapping) field of view are estimated by stereo pairs with multiple baselines, and then fused to obtain a statistically more accurate depth map.",
"Recently, there has been a renewed interest in multi-camera system deployment in the autonomous driving systems, as well as in mobile handsets.",
"To make the best benefit of the extra cameras, the camera lenses are chosen to have different focal lengths in order for the device to have good resolution at both near and far objects, which results in different field of views (FOVs).",
"For example, recent cell phones are equipped with two or more cameras.",
"Example specifications of the cameras are ($f/1.7$ , $26 mm$ , $1 \\times $ optical zoom) and ($f/2.4$ , $52 mm$ , $2 \\times $ optical zoom), respectively.",
"One reason for having cameras with different focal lengths is for diversity, where the first camera has a wider aperture for better light sensitivity, and the second camera has a longer focal length for capturing a higher image resolution at twice the optical zoom.",
"We consider in this work a two camera system, where the first camera has $1 \\times $ the optical zoom, and the second camera has $2 \\times $ the optical zoom.",
"We call the field of view of the first camera with $1 \\times $ the zoom, as the Wide FOV (WFOV).",
"The field of view of the second camera with twice the optical zoom is the central part of the WFOV, and we call it the Tele FOV (TFOV).",
"We refer to the problem of the estimation of the inverse depth for the region described by the union of two different FOVs, while leveraging the stereo information from the overlapping FOV, as tele-wide stereo matching (TW-SM).",
"To the best of our knowledge, this work is the first to consider the TW-SM problem.",
"Since the disparity is directly proportional to the inverse depth, we also refer to this problem as disparity estimation of the union of stereo FOVs.",
"One application of stereo matching is to produce a Bokeh effect in the image, by blurring the background, while keeping the object of interest in focus.",
"With TW-SM, the Bokeh effect can be synthesized for the full WFOV image, rather than for the central TFOV region that is achievable by conventional stereo matching.",
"Fig.",
"REF demonstrates the big difference between the Bokeh effect on the full WFOV and on the central TFOV only.",
"Fig.",
"REF demonstrates the result achievable by this paper on an example from the KITTI stereo dataset, where the disparity map is estimated using TW-SM and used to synthesize the Bokeh effect on the WFOV.",
"Figure: The proposed solutions for tele-wide stereo matching, and their input and output features.We consider multiple deep-learning based approaches to the tele-wide stereo matching problem.",
"First, we develop tele-wide stereo matching networks (TW-SMNets) that attempt to do stereo matching between the tele and wide images to estimate the disparity.",
"Hence, we train TW-SMNet(T) to estimate a disparity map for the overlapped TFOV, and train TW-SMNet(W) to estimate a disparity map for the union WFOV.",
"Second, we pose the problem as a single-image inverse-depth estimation (SIDE) problem by formulating the disparity as the scaled inverse depth.",
"Hence, we train deep SIDE neural networks to estimate the inverse depth from the wide FOV image only.",
"Third, we design a deep multitask tele-wide stereo matching neural network (MT-TW-SMNet) which takes the stereo input images with different FOVs, and estimates the disparity for the overlapped FOV using the stereo input, as well as the inverse depth for the union FOV using the WFOV input image Our preliminary multitask tele-wide stereo matching results have been accepted for publication at the IEEE Conference on Image Processing, ICIP 2019 [5].. Fourth, we design multiple methods for the fusion of these two approaches, SIDE and TW-SM.",
"For example, we consider input feature fusion such as RGBD SIDE, where the SIDE network uses the disparity estimates from the multitask stereo matching network as additional input features, to guide its estimation of inverse depth.",
"We also consider output feature (decision) fusion, where we design a fusion network to fuse the disparity estimates from the tele-wide stereo matching network and the WFOV single image inverse-depth estimation network.",
"Examples of the inputs and outputs of the proposed networks are shown in Fig.",
"REF .",
"The rest of this paper is organized as follows.",
"Section explains the preliminaries and the related works to stereo disparity estimation and single image depth estimation.",
"In Section , our different proposals for tele-wide stereo matching are explained.",
"Section gives the experimental results of tele-wide networks on KITTI and SceneFlow datasets.",
"Section discusses our proposed input feature fusion and output feature fusion between the different tele-wide disparity estimation methods.",
"Section concludes the paper.",
"Stereo disparity estimation techniques involve feature extraction, matching cost computation, disparity aggregation and computation, and disparity refinement [1].",
"Matching cost computation at a given disparity is based on measuring the similarity between pixels in the left and right images at this disparity shift.",
"The cost computation can simply be the sum of absolute differences of pixel intensities at the given disparity [6].",
"Disparity aggregation can be done by simple aggregation of the matching cost over local box windows, or by guided-image cost volume filtering [3].",
"Estimation of the disparity calculation can be done by local, global, or semiglobal methods.",
"Semiglobal matching (SGM) [2] methods are less complex than global methods such as graph cut algorithms [7].",
"Semiglobal matching is also more robust than local window-based methods.",
"SGM performs cost aggregation by approximate minimization of a two dimensional energy function towards each pixel along eight one dimensional paths.",
"Disparity refinement is classically done by further checking for left and right consistencies, invalidating occlusions and mismatches, and filling such invalid segments by propagating neighboring disparity values.",
"Deep learning approaches to solve the disparity estimation problem surged after the significant efforts in collecting datasets with stereo input images and their ground truth disparity maps, e.g.",
"SceneFlow [8], KITTI 2012 [9], KITTI 2015 [10], and the Middlebury [11] stereo benchmark datasets.",
"The existence of such datasets enabled supervised training of deep neural networks for the task of stereo matching, as well as the transparent testing and benchmarking of different algorithms on their hosting servers.",
"Convolutional neural networks (CNNs) are now widely investigated to solve problems in computer vision and image processing.",
"Similar to the classical disparity estimation techniques, CNN-based disparity estimation involves feature extraction, matching cost estimation, disparity aggregation and computation, and disparity refinement.",
"First, deep features are extracted from the rectified left and right images using deep convolutional networks such as ResNet-50 [12] or VGG-16 [13].",
"The cost volume is formed by measuring the matching cost between the extracted left and right deep feature maps at different disparity shifts.",
"Typical choices for the matching cost are simple feature concatenation, or calculation of metrics such as absolute distance or correlation [8], [14], [15], [16] or by extending the cost volume to compute multiple metrics [17].",
"The cost volume is further processed and refined by a disparity computation module that regresses to the estimated disparity.",
"Refinement networks can then be used to further refine the initial coarse depth or disparity estimates.",
"Zbontar et.",
"al [18] designed a deep Siamese network for stereo matching, where the network was trained to predict the similarity between image patches.",
"Luo et al.",
"[19] improved this work [18] and made it more efficient, where they formulated the matching cost computation as a multi-label classification problem.",
"Shaked et al.",
"[20] proposed a highway network for the matching cost computation and a global disparity network for the prediction of disparity confidence scores.",
"Other studies focused on the post-processing of the disparity map for disparity refinement.",
"Seki et al.",
"further extended SGM to SGM-Net which deployed SGM penalties, instead of manually-tuned penalties, for regularization.",
"Post-processing networks were proposed to detect incorrect disparity and replace them by others estimated from the local regions, and further refined by a post-refining module [21].",
"More complicated convolutional neural networks are utilized in later approaches, including SharpMask [22], RefineNet [23], the label refinement network [24], the stacked hourglass (SHG) architecture [25][26], and the atrous multiscale architecture which does dense multiscale contextual aggregation [17].",
"Different neural network architectures were proposed for stereo matching.",
"There are two main approaches, the encoder-decoder architectures and the spatial pyramid pooling architecture.",
"The encoder-decoder utilizes a cascade architecture, which integrates top-down and bottom-up information through skip connections.",
"The fully convolutional network (FCN) [27] was one of the pioneering works following this architecture to aggregate coarse-to-fine predictions to improve the segmentation quality.",
"Mayer et al.",
"[28] designed an end-to-end network (DispNet) for both the disparity estimation and optical flow estimation.",
"Pang et al.",
"[29] developed a two-stage network based on DispNet, where the disparity map is calculated at the first stage, its multiscale residual map is extracted at the second stage, and the outputs of these two stages are further combined.",
"Kendall et al.",
"[30] introduced GC-Net, where 3D convolutional layers are designed for cost volume regularization using 3D convolutions.",
"These end-to-end networks exploit multiscale features and hierarchical relationships between the earlier and later neural network layers for disparity estimation.",
"The contextual information is utilized to reduce the mismatch at ambiguous regions, and improve the depth estimation.",
"Spatial pyramid pooling (SPP) integrates deep receptive fields with multiscale context information.",
"Global pooling with FCN [31] has been shown to enlarge the empirical receptive field to extract information from the whole image.",
"Later works improved the SPP by introducing the atrous (dilated) convolutions.",
"Zhao et al.",
"designed PSPNet [32], where a pyramid pooling module is adopted to generate a multiscale contextual prior.",
"Chen et al.",
"proposed the atrous spatial pyramid pooling (ASPP) module in DeepLab v2 [33] for multiscale feature embedding.",
"DeepLab v3 [34] introduced a newly designed ASPP module, where global feature pooling and parallel dilated convolutions with different dilation factors are utilized to aggregate features from different receptive fields.",
"Chang et al.",
"proposed PSMNet [35], a pyramid stereo matching network consisting of two main modules: spatial pyramid pooling and 3D CNN.",
"In PSMNet, the spatial pyramid pooling module takes advantage of the capacity of global context information by aggregating contexts at different scales and locations to form a cost volume.",
"The 3D CNN learns to regularize the cost volume using stacked multiple hourglass networks in conjunction with intermediate supervision.",
"In this paper, our proposed tele-wide stereo matching network uses PSMNet as a baseline of the stereo matching network TW-SMNet, and its architecture is demonstrated in Fig.",
"REF .",
"Figure: Architecture of the tele-wide stereo matching network (TW-SMNet)."
],
[
"Single image depth estimation",
"Previous approaches for depth estimation from single images can be categorized into three main groups: (i) methods operating on hand crafted features, (ii) methods based on graphical models and (iii) methods adopting deep neural networks.",
"Earlier works addressing the depth prediction task belong to the first category.",
"Hoiem et al.",
"[36] introduced photo pop-up, a fully automatic method for creating a basic 3D model from a single photograph.",
"Karsch et al.",
"[37] developed Depth Transfer, a non parametric approach where the depth of an input image is reconstructed by transferring the depth of multiple similar images and then applying some warping and optimizing procedures.",
"Ladicky [38] demonstrated the benefit of combining semantic object labels with depth features.",
"Other works exploited the flexibility of graphical models to reconstruct depth information.",
"One of the most commonly-used technologies is the Conditional Random Field (CRF).",
"By defining different unary and pairwise potentials, we can have different CRFs.",
"For instance, Delage et al.",
"[39] proposed a dynamic Bayesian framework for recovering 3D information from indoor scenes.",
"A discriminatively-trained multiscale Markov Random Field (MRF) was introduced in [40].",
"More recent approaches for depth estimation are based on CNNs.",
"The pioneering work by Eigen [41] introduced a two-scale architecture consisting of the coarse-scale network and the fine-scale network.",
"The coarse-scale network is a convolutional neural network that identifies the global scene context.",
"This coarse depth map, along with the original input image are then fed to the fine-scale fully convolutional network to refine the depth result.",
"The scale-invariant loss is utilized in this work, which further improves the robustness of the estimated depth map.",
"This work is further extended in [42], where the depth estimation, surface normal estimation, and semantic segmentation are integrated into one unified network.",
"Li et al.",
"[43] considered using a loss function with components in both the depth domain and the gradient of depth domain, yielding a two-channel network, to learn both the depth map and its gradient which are fused together to generate the final output.",
"There are a few works who considered combining the CNN and CRF together.",
"Conventional CRF-based depth estimation used the RGB image as the observation.",
"Mousavian et al.",
"[44] proposed to combine depth estimation and semantic segmentation together.",
"First, a CNN is utilized to extract feature maps of the depth and semantic labels at the same time.",
"Then, these feature maps are fused as the input of a CRF.",
"Xu et al.",
"[45] proposed a cascade structure for CNN-CRF depth estimation, where a side output of the CNN is used as the input to the CRF for depth estimation at a certain scale, and then estimates are refined at subsequent levels.",
"Two CRF architectures, multiscale CRF (one CRF uses all the feature maps in the potential function), and cascade CRF (the depth estimated by one CRF is the input to the next CRF) are proposed, where the CRFs are implemented as neural network modules to enable end-to-end training.",
"One disadvantage of the above works is that they are not fully convolutional since a fully connected layer is still utilized for depth regression.",
"This significantly increases the model complexity and the computational burden, which prompted Laina et al.",
"[46] to propose a fully convolutional network (FCN).",
"FCN is based on ResNet-50 with additional up-sampling layers to recover the loss in resolution from the pooling layers, and is trained with the BerHu loss.",
"Laina's work was extended so the input constitutes of a randomly sampled sparse depth map, in addition to the input RGB features of an image [47].",
"As expected, the additional prior information about the sparse depth samples improves the SIDE accuracy significantly.",
"Li et al.",
"[48] reformulated the SIDE problem as a classification problem by quantizing the depth in the log space and using a soft-weighted-sum inference.",
"DORN [49] further formulated the depth estimation problem as an ordinal regression problem and achieved state-of-art performance using atrous spatial pyramid pooling (ASPP) modules to capture multiscale information."
],
[
"Tele-wide Stereo Matching ",
"Given two stereo cameras, the stereo matching techniques described in Section can be utilized to estimate the disparity map for the overlapping FOVs.",
"However, to our knowledge disparity estimation for the union of FOVs has not been well studied or benchmarked previously in the literature.",
"Here, we consider the special case when the FOV of one camera is contained inside that of the other camera.",
"The practical need for this case has surfaced recently due to the equipment of recent mobile devices with stereo cameras (tele camera and wide camera) which have different focal lengths, and where the tele camera is specified to capture the center FOV of the other camera at twice the optical zoom.",
"In this section, we propose different deep learning solutions to solve the tele-wide disparity estimation problem.",
"We deploy a tele-wide stereo-matching network (TW-SMNet) to work on the left wide-image and the right tele-image to generate a disparity map for the wide FOV.",
"This network is expected to perform well on the TFOV.",
"We also benchmark the performance of this tele-wide network when trained to learn the disparity map for the full WFOV.",
"Next, we modify the TW-SMNet to be a single-image inverse depth estimation network (SIDENet), which works only on the left wide-image to generate the inverse depth prediction for the full WFOV.",
"Based on observations from these networks, we propose a multitask tele-wide stereo matching network (MT-TW-SMNet) which concurrently learns to do stereo matching and single image inverse depth estimation, to get better disparity estimates for the full WFOV.",
"Figure: Architecture of the stacked hourglass single image depth estimation network (SHG-SIDENet)."
],
[
"Tele-Wide Disparity Estimation ",
"We establish a baseline solution to the tele-wide disparity estimation problem by casting the problem as a standard stereo matching problem.",
"We implement a tele-wide stereo matching network (TW-SMNet) that takes as input a tele-wide stereo pair, representing the left WFOV and the right TFOV, cf.",
"Fig.",
"REF .",
"We train two different TW-SMNets; TW-SMNet(T) which is trained to output an estimated disparity map for the overlapping TFOV only, and TW-SMNet(W) which is trained to estimate the disparity for the full WFOV.",
"TW-SMNet(T) and TW-SMNet(W) modify the tele-wide stereo inputs differently according to the different training objectives, as demonstrated in Fig.",
"REF .",
"The input features of TW-SMNet(W) uses a wide stereo image pair, where the right tele image is zero-padded after down-sampling to match the resolution and size of the left wide image.",
"TW-SMNet(T) uses a tele stereo image pair, where its input features are the right tele image and the left wide image after it is center cropped for the TFOV and up-sampled to match the size and resolution of the right tele image.",
"The network architecture of the TW-SMNets is shown in Fig.",
"REF .",
"The TW-SMNet uses a similar architecture to PSMNet [35], and consists of four main parts: feature extractor, cost volume for left/right feature map matching, stacked hourglass module, and disparity regression.",
"Feature extraction is done by 3 $3\\times 3$ convolutional layers followed by a 50-layer residual network (ResNet-50) constituting of multiplicities of four different residual blocks with atrous convolutional layers.",
"This is followed by a spatial pyramid pooling (SPP) module that extracts hierarchical context information at multiple scales.",
"The SPP module is composed of four average pooling blocks of fixed-sizes $[64\\times 64, 32\\times 32, 16\\times 16, 8\\times 8]$ .",
"After feature extraction, a cost volume is utilized to learn the matching costs between the left and right feature maps, where the left feature maps are aligned and concatenated with their corresponding right feature maps at each disparity shift.",
"This results in a four dimensional feature map of dimension (number of channels$\\times $ maximum disparity$\\times $ height$\\times $ width).",
"Hence a 3D CNN is used for cost aggregation, where three hourglass modules are stacked to learn high level local contextual information while keeping the low level global contextual information.",
"The TW-SMNet is trained with a classification-based regression loss, where the network regresses the feature maps at the output of the stacked hourglass to a continuous disparity map.",
"The disparity is classified into $D$ bins whose values are the integers from 1 to $D$ , where $D$ is the maximum disparity.",
"The classification probabilities at the $D$ bins are calculated for the disparity at each pixel.",
"The expected disparity is then calculated from the estimated disparity probabilities.",
"The predicted disparity for each pixel location is given by (REF ), $d_i=\\sum _{j=1}^D{j\\times p_i^j},$ where $d_i$ is the estimated disparity of pixel $i$ , $p_i^j$ represents the soft probability of the disparity $d_i$ falling in the bin with value $j$ .",
"For robust regression, the Huber loss ($H_{1}$ ) is used to measure the difference between the predicted disparity $d_i$ and the ground truth disparity $d_i^{gt}$ , as shown in (REF ), $L(d_i,d_i^{gt})=\\frac{1}{N}\\sum _i {H_1} (d_i-d_i^{gt}),$ where $N$ is the total number of labeled pixels, and the Huber loss at any $\\delta $ is defined by [50] $ {H_\\delta }(x)= {\\left\\lbrace \\begin{array}{ll}0.5 x^2, & \\mbox{if}~|x| \\le \\delta , \\\\\\delta |x|-0.5 \\delta ^2, & \\mbox{otherwise}.",
"\\end{array}\\right.}",
"$ On the one hand, TW-SMNet(T) only provides disparity estimates for the overlapping TFOV based on stereo matching at the tele regions only, and it should be the most accurate within the TFOV.",
"On the other hand, TW-SMNet(W) estimates the disparity for the full WFOV including the region surrounding the TFOV.",
"Since TW-SMNet(W) has the same input information as TW-SMNet(T) in the tele region, the accuracy of TW-SMNet(W) should be close to that of TW-SMNet(T) in the TFOV.",
"However, due to the lack of stereo matching information at the surrounding region in the wide stereo input, TW-SMNet(W) suffers a relatively larger error when estimating the disparity for the pixels in the surrounding region, which are the pixels in the Wide FOV but not in the TFOV."
],
[
"Wide FOV Inverse Depth Estimation",
"To improve the performance on surrounding region, we propose a stacked hourglass single image inverse depth estimation network (SHG-SIDE), which estimates an inverse depth map from the left wide RGB image only.",
"For each pixel, the computed disparity $d$ is proportional to the `inverse depth' $\\zeta $ , where the depth $z$ (distance to the scene point) is related to the disparity by the camera baseline $B$ (distance between the two camera centers), and the camera focal length $F$ , as demonstrated by Fig.",
"REF , and $\\zeta = \\frac{1}{z}=\\frac{d}{FB}.$ Figure: Calculation of depth zz from stereo disparity dd, focal length FF, and camera baseline BB.Since, an inverse depth map can be converted to a disparity map by knowledge of the camera baseline and focal length which are predetermined constants for a specific camera setup, the SHG-SIDE network is trained to regress to the disparity (i.e.",
"a scaled inverse depth) directly.",
"The last convolutional layer of SHG-SIDENet is set to have $D$ output neurons for classification, and the same classification-based robust regression as TW-SMNet is used for training of the SHG-SIDE network and the prediction of the inverse-depth maps.",
"The network architecture of SHG-SIDENet is modified from that of TW-SMNet.",
"We remove the feature extraction channel for the right tele image and the cost volume.",
"All the three dimensional convolutional layers after the cost volume are hence shrunk to a two dimensional form.",
"The network structure is shown in Fig.",
"REF .",
"Since the SHG-SIDENet estimates the inverse depth by scene understanding it can provide more accurate depth estimates than TW-SMNet in the surrounding region where the stereo matching information is incomplete.",
"Figure: The network architecture of the multitask tele-wide stereo matching network (MT-TW-SMNet).Table: Error rate of tele-wide disparity estimation networks on the KITTI stereo 2015 validation dataset; `cen' stands for the `center' TFOV region, and `sur' stands for the region in the WFOV `surrounding' the TFOV.Table: End-point error of tele-wide disparity estimation networks on the SceneFlow test set."
],
[
"Multitask Network for Tele-Wide Stereo-Matching",
"We investigated both single image inverse depth estimation (SIDE) and stereo matching (SM) solutions to the tele-wide disparity estimation problem.",
"Whereas the TW-SMNet has an advantage in the tele FOV, the SHG-SIDENet has an advantage in the surrounding region of the wide FOV.",
"This inspires us to propose an end-to-end multitask tele-wide stereo matching network (MT-TW-SMNet) which combines the TW-SMNet and the SIDENet together.",
"The proposed MT-TW-SMNet takes as input both the left wide image and the zero-padded right tele image.",
"The ResNet-50 followed by SPP, as explained in Sec.",
"REF , are used as a feature extractor which is shared between both the SM and SIDE tasks.",
"As shown in Fig.",
"REF , the MT-TW-SMNet is a two-branch network, where the SM branch constructs a cost volume from the features extracted from the left and wide images for disparity computation, and the SIDE branch uses the features extracted from the left wide image for scene understanding.",
"The loss function of MT-TW-SMNet is a linear combination of the classification-based robust regression losses used to train the SIDENet and TW-SMNet, as computed by (REF ) $L(d_i,d_i^{gt})=L_{SMDE}(d_i,d_i^{gt})+\\alpha L_{SIDE}(d_i,d_i^{gt}),$ where $\\alpha $ is a constant that is empirically chosen as $1.0$ in this work.",
"In MT-TW-SMNet, the learning process of SIDE branch will assist the TW-SM branch to make a better understanding of the global context.",
"Since both branches share the same feature extractor and are trained end-to-end, the accuracy of the stereo matching branch will be improved.",
"At inference time, the disparity estimates from both branches can be fused together.",
"For example, the final disparity estimate can be a linear combination of that estimated by both the SIDE and SMDE branches.",
"However, for efficient inference, we only used the output of the SMDE branch, where we found that the additional learning of the SIDE objective enabled the SMDE branch to make a better understanding of the global context and provide a more accurate disparity estimate than TW-SMNet(W) in the surrounding region, while preserving its stereo matching accuracy in the tele region."
],
[
"Datasets",
"We use two popular stereo datasets to evaluate our tele-wide disparity estimation methods, SceneFlow [28], and the KITTI Stereo 2015 [51].",
"The left image is used as the left-wide image, while the center-cropped region of the right image is used as the right tele image, which has half the width and half the height of the left wide image.",
"SceneFlow is a large scale synthetic dataset containing $35,454$ training and $4,370$ testing images of size $960\\times 540$ .",
"This dataset provides dense and elaborate ground truth disparity maps.",
"The end-point disparity error is calculated among all pixels with valid disparity labels when evaluating the performance of our tele-wide networks.",
"KITTI is a real-world dataset with street views from a driving car.",
"It contains training stereo image pairs with sparse ground-truth disparities obtained using LiDAR.",
"The image size in KITTI is around $1242\\times 375$ .",
"We use the KITTI stereo 2015 validation set (40 labeled image pairs), and test set (200 unlabeled image pairs) to evaluate the performance of our networks.",
"Since only 200 training images are provided, we fine-tune our SceneFlow networks on these training images to obtain the KITTI models.",
"In the evaluation, a pixel to be correctly estimated if the disparity or flow end-point error is less than 3 pixels or less than $<5\\%$ .",
"We note that the disparity is not labeled for the sky regions in the KITTI training and testing datasets, and all non-labeled regions are ignored for evaluation purposes."
],
[
"Performance comparisons between TW-SMNet, SHG-SIDENet, and MT-TW-SMNet",
"We evaluate the accuracy of the proposed tele-wide disparity estimation networks: the single image inverse depth estimation network SHG-SIDENet, the stereo matching network TW-SMNet, as well as the multitask network MT-TW-SMNet.",
"In Table REF , we give the error rates on the KITTI 2015 validation datasets, as well as the accuracy of state-of-art single image depth estimation network DORN [49].",
"Since the DORN provides an output depth map, we convert it to a disparity map when calculating the error.",
"The center error is labeled `cen' and is the error when calculated over the pixels in the TFOV only, while the surrounding error is labeled `surr' and is calculated over the pixels in the WFOV which are not in center TFOV region.",
"It can be seen that the tele-wide networks achieve better accuracy than DORN.",
"We can find that the best center accuracy is achieved by TW-SMNet(T) which is based on stereo matching of the left-right tele image pairs.",
"However, TW-SMNet(T) does not provide the surrounding disparity.",
"TW-SMNet(W) uses the left wide image and the zero-padded right tele-image as input, and estimates a disparity map for the full WFOV, including the surrounding region.",
"Moreover, the accuracy of TW-SMNet(W) is close to that of TW-SMNet(T) in the TFOV region.",
"In contrast, the single image inverse depth estimation network SHG-SIDENet gives a more accurate estimate for the surrounding disparity than TW-SMNet(W), but the central accuracy is worse.",
"These results make sense because the stereo disparity estimation is a matching problem.",
"If both the left RGB image and right RGB image are given, the accuracy will be good.",
"In case of missing information, as in the surrounding region, the cost-volume computation of of TW-SMNet(W) will struggle to find the left-right correspondence, so the estimation accuracy will not be as good as with single image inverse depth estimation which relies on scene understanding.",
"We also observe that the center accuracy of the multitask network MT-TW-SMNet is a bit lower than that of TW-SMNet(W) and TW-SMNet(T) in the central TFOV region, but the surrounding accuracy and the overall-pixel accuracy are better.",
"Similar observations can be found by testing on the SceneFlow dataset, as shown in Table REF .",
"The end-point-error of our tele-wide disparity estimation networks on SceneFlow confirm that the MT-TW-SMNet acheives the the lowest error of $5.6$ pixels.",
"It has better accuracy than SIDENet in the overlapping Tele FOV, and better accuracy than TW-SMNet(W), which is only trained for stereo matching, in the surrounding non-overlapping region.",
"This shows the effectiveness of our multitask learning strategy, where learning SIDE as an auxiliary task helped the main SM branch provide a better estimate in the surrounding region.",
"We also submitted our results to the KITTI evaluation server for testing on the KITTI Stereo 2015 test set.",
"KITTI evaluates the disparity error for all the pixels in the wide FOV, and also breaks down the error rates separately for the pixels belonging to the foreground objects or to the background regions, as shown in Table REF .",
"The MT-TW-SMNet achieved better overall tele-wide disparity estimation accuracy than the SIDENet.",
"Our multitask tele-wide disparity estimation also ranks on the KITTI leader board better than other methods which use full wide left and right images for stereo disparity estimation, such as [3], [52], [53].",
"The table also shows that with fusion between the different methods (MT-TW Fusion), the result can be further improved so the overall disparity error is only $11.96\\%$Detailed analyses and visualizations of the MT-TW-SMNet disparity maps on KITTI Stereo 2015 test set can be found at [54]..",
"The different fusion methods are explained in the next section.",
"Table: Full FOV disparity error rate (%) on KITTI Stereo 2015 Test set, for different pixel types in the wide image."
],
[
"Tele-Wide Fusion ",
"In Section , we have introduced multiple ways to solve the tele-wide disparity estimation problem, including TW-SMNet, SHG-SIDENet, and MT-TW-SMNet.",
"In this section, we introduce two network fusion methods, which are input feature fusion and output feature fusion.",
"Moreover, we investigate architectures with both input and output fusion.",
"The SIDE network is modified to take the estimated disparity from MT-TW-SMNet as an input feature in addition to the RGB wide input image (RGBD input) which guides the estimation of the inverse depth, and the output of this RGBD SHG-SIDENet is further fused with the output MT-TW-SMNet for more accurate disparity estimation.",
"Table: Tele-wide disparity estimation error rates of the input feature fusion network RGBD-SHG-SIDENet using different input sparse disparity maps at the central (cen) and surrounding (sur) regions of the WFOV, on KITTI Stereo 2015 validation dataset."
],
[
"Input feature fusion: Single image inverse depth estimation with RGBD input",
"As a method of input feature fusion, we propose to use the estimated disparity from the stereo matching networks described above as an input disparity channel to SHG-SIDENet, in addition to the input RGB wide image, to guide the inverse depth estimation of SHG-SIDENet.",
"Since we will adopt the same stacked hourglass architecture for this network as the SHG-SIDENet, we will call this network RGBD-SHG-SIDENet.",
"However, the first layer of the RGBD-SHG-SIDENet is modified from that of the SHG-SIDENet to take the additional input disparity channel.",
"Previous works [47] have shown that using the knowledge of sparse depth can significantly improve the accuracy of depth estimation, where it is assumed that such sparse depths can be obtained by some computer vision systems such as simultaneous localization and mapping (SLAM).",
"Instead, here we utilize the disparity estimates obtained by tele-wide stereo matching.",
"As shown in Fig.",
"REF , we first use the TW stereo disparity estimation network to obtain reasonably accurate disparity estimates for the pixels in the tele region.",
"This tele disparity map is concatenated with the RGB wide image to make the RGBD feature that is input to the RGBD-SHG-SIDENet.",
"Whereas the single image inverse depth estimation network can easily learn the relative distance between objects by scene understanding, the input absolute values can help estimate more accurate inverse depth values.",
"The additional disparity input can be generated by any of the tele-wide disparity estimation networks introduced before.",
"RGBD-SHG-SIDE can also be used for input fusion of disparity results from the different SM or SIDE networks, where the disparity map can be constructed from the SM network in the Tele region and from the SIDE network in the surrounding network (by decision selection as in Sec.",
"REF ), or multiple disparity maps obtained by different algorithms can be concatenated with the RGB wide image at the RGBD SIDENet.",
"For complexity reduction, the input disparity map can also be estimated using classical stereo matching in the tele region.",
"Figure: Input feature fusion of SHG-SIDENet and TW-SMNet network."
],
[
"Experimental results of input feature fusion with RGBD-SHG-SIDENet",
"Next, we investigate the effectiveness of input feature fusion using the RGBD-SHG-SIDENet.",
"We evaluate the performance on the KITTI Stereo 2015 validation dataset.",
"We train RGBD-SHG-SIDENet by using the sparse ground-truth disparity map as the additional input disparity channel.",
"We conduct an ablation study and compare different options for the input disparity map.",
"At inference time, the additional input disparity channel is sampled from the previously estimated disparity maps by using the same sparsity rate of the ground truth maps used for training.",
"The RGBD-SHG-SIDENet networks could have been alternatively trained directly using the dense or sparse disparity outputs from the networks in Table REF as their additional input feature, but this has several drawbacks: the training convergence is worse, and this requires to train a different model for each possible input which also results in model overfitting and the inability to generalize to different input maps.",
"Table REF compares the accuracy of the disparity estimated by the RGBD-SHG-SIDENet for different options of construction of the input sparse disparity map.",
"We investigate the different cases where the sparse disparity is sampled from one estimated disparity map.",
"We also investigate the cases where the center disparity and the surrounding disparity are sampled from the outputs of two different networks using decision selection (cf.",
"Sec.",
"REF ).",
"The sparse input map is constructed by randomly sampling the estimated disparity maps at rates of $20\\%$ of the pixels at the central tele region, and $12\\%$ of the pixels at the surrounding region, if it exists.",
"From Table REF , it can be confirmed that input feature fusion clearly improves the accuracy.",
"Although the accuracy in the central tele region can degrade slightly, the surrounding accuracy improves significantly which improves the overall accuracy.",
"For example, comparing `RGBD-SHG-SIDENet + TW-SMNet(W)' to `TW-SMNet(W)', the overall error rate is reduced from $13.10\\%$ to $10.33\\%$ , which is even better than the $12.70\\%$ attainable from the MT-TW-SMNet.",
"We notice that sampling the central disparity from the stereo matching networks TW-SMNet(T), TW-SMNet(W), or MT-TW-SMNet results in better overall accuracy.",
"Sampling the input disparity map totally from that of the SHG-SIDENet shows the performance attainble using the left wide input image only, and can only slightly improve over the error rate of SHG-SIDENet.",
"However, it is noticed that the additional sampling of the disparity from the surrounding region estimated by SHG-SIDENet does not improve over only sampling from the central tele regions of the SM disparity maps.",
"The reason is that the estimated surrounding disparity of SHG-SIDENet is not as accurate as that of the central disparity estimated by stereo matching.",
"Since our training uses the ground truth disparity as the additional input disparity channel, the network expects to get an accurate sparse disparity map at its input.",
"Table: Error rate of output feature fusion based on decision selection (DS) and fast global smoothing (FGS) smoothing on the KITTI Stereo 2015 validation dataset.",
"The disparity input of RGBD-SHG-SIDENet comes from TW-SMNet(T)."
],
[
"Output feature fusion ",
"Fusion of the results from different classifiers has always been an important topic in machine learning [55], [56], [57].",
"Motivated by this, we investigate methods for the fusion of output features or disparity maps from two or more networks.",
"The general framework for the proposed output feature fusion is shown in Fig.",
"REF .",
"To obtain a more accurate disparity map, the output disparity maps from different networks can be merged together by using simple decision selection or by using deep fusion networks or a combination of them.",
"The merged disparity map can be further post-processed to improve its quality, and reduce visual artifacts for specific applications such as synthesizing the Bokeh effect.",
"Figure: Output feature fusion of SHG-SIDENet and TW-SMNet.Figure: Output feature fusion based on decision selection.",
"The border discontinuity can be observed."
],
[
"Decision selection:",
"A straightforward solution to obtain a more accurate disparity map from multiple disparity maps is by decision selection.",
"Decision selection attempts to generate a disparity map by selecting the more reliable disparity result at each pixel.",
"We know that the best disparity estimation for the Tele FOV comes from the networks having both the left view and right view as inputs (e.g., TW-SMNet or MT-TW-SMNet).",
"In contrast, since SIDE relies on scene understanding, the best estimation of the surrounding region comes from the SHG-SIDENet.",
"The results of Table REF and Table REF confirm that the SIDE networks are more accurate in the surrounding regions and that the SM networks give better disparity estimate in the overlapping tele region.",
"Hence, to get the best quantitative accuracy, we propose selecting the disparity estimates from one of the TW-SMNets in the central Tele FOV, and selecting it from the SHG-SIDENet in the surrounding region.",
"This will improve the overall disparity estimation accuracy with little additional computational cost.",
"Figure: Deep network fusion of disparity maps.Table REF and REF show that the best center accuracy is achieved by the TW-SMNet(T), and that the best surrounding accuracy is achieved by RGBD-SHG-SIDENet.",
"So we apply decision selection betwen these two networks, and test the performance on the KITTI Stereo 2015 validation set.",
"The results are shown in Table REF .",
"As expected, the overall error (`error-all') is reduced from $11.05\\%$ to $9.50\\%$ , due to using the most accurate disparity estimates in both the central and the surrounding regions."
],
[
"Post-processing of disparity maps:",
"Disparity selection is demonstrated in Fig.",
"REF , which shows that such operation may introduce a large disparity discontinuity along the tele FOV border.",
"Such a disparity discontinuity may be problematic for applications such as synthesizing the Bokeh effect on an image, where the degree of the applied blur depends on the estimated depth.",
"To solve this problem, we utilize a post-processing module to deliver perceptually pleasing disparity maps.",
"One potential way to reduce the effect of the abrupt change in disparity around the tele FOV border is to smooth the disparity map.",
"One solution is edge preserving smoothing using bilateral filters which are based on local averaging [58].",
"Another solution is to use the RGB images as a guidance to smoothing, so as to preserve the edges in the RGB image, which is called edge guided filtering.",
"Hence, for post processing we propose using the fast global smoother (FGS) [59], which optimizes a global objective function defined with data constraints and a smoothness prior.",
"Since the FGS filtered values around the tele FOV border depends on the whole disparity map, we calculate the filtered values around the border using the global filters by deploying FGS.",
"To smooth the transition in disparity values between the central and surrounding region, only the strip around the boundary in the merged disparity map is replaced with the filtered one.",
"As shown in the last row of Table REF , the error rate increases 0.04% by applying FGS due to the smoothing around the tele FOV border.",
"However, the perceptual quality improved significantly as the disparity discontinuity around the border is smoothed out, as shown the Fig.",
"REF .",
"Figure: Feature output fusion using decision selection fusion (left)and with additional post-processing using FGS smoothing (right).Figure: MT-TW Fusion: Tele-wide disparity estimation using both input feature fusion and output feature fusion, which has the best performance among the tested architectures.",
"The disparity merging module performs deep network fusion using a stacked hourglass network, followed by decision selection."
],
[
"Deep network fusion of disparity maps:",
"Above we observed that the fusion of multiple disparity maps with simple methods such as decision selection, often requires post processing with global smoothers to reduce the fusion artifacts.",
"That motivated us to design deep neural networks to fuse the disparity maps generated by different schemes.",
"Deep network fusion of multiple output features has been investigated before in different contexts, such as the fusion of the output images from different super-resolution networks [60].",
"Deep network fusion is characterized by its capability of taking into account both the global and local contexts, as defined by the deep network's receptive field.",
"The proposed disparity fusion network concatenates two estimated disparity maps at its input, and outputs a refined disparity map, as shown in Fig.",
"REF .",
"In our experiments, one of the input maps is obtained by SIDE and the other input disparity map is obtained by one of the stereo-matched disparity estimation (SMDE) techniques.",
"The fusion network is trained using the classification-based robust regression loss given by (REF ), as used for training our TW-SMNets.",
"For simplicity, we utilized the stacked hourglass head used in the TW-SMNet and SHG-SIDE as the fusion network.",
"Hence, the proposed fusion is called SHG-Fusion.Compared to decision selection, deep network fusion of the disparity maps achieves better accuracy.",
"Moreover, the deeply fused disparity maps do not need FGS post-processing as they do not suffer from disparity discontinuity around the tele FOV border.",
"Figure: Example outputs of our estimated tele-wide disparity from the TW-SM Fusion network.Table: Error rate of output feature fusion by disparity fusion network between RGBD-SHG-SIDENet and MT-TW-SMNet in KITTI validation set.",
"`cen' stands for `center', and `sur' stands for `surrounding'.Table: End-point error of output feature fusion by disparity fusion network between RGBD-SHG-SIDENet and MT-TW-SMNet in SceneFlow test set.",
"`cen' stands for `center', and `sur' stands for `surrounding'."
],
[
"Combining the different fusion methods:",
"As shown in Fig.",
"REF , decision selection can be further applied between the deeply fused disparity map at the output of the fusion network and the disparity map input to the fusion network which was obtained by stereo matching.",
"The crux of such additional merging is that the fusion network helps improve the overall disparity by especially improving the estimated disparity in the surrounding region, but the stereo matched disparity maps at the input may still be more accurate in the tele FOV region.",
"Next, we give some experimental results with output feature fusion.",
"In Table REF , we find that the error rate can be reduced from $10.37\\%$ to $9.85\\%$ by the stacked hourglass fusion network, and to $9.64\\%$ by further applying decision selection with MT-TW-SMNet.",
"Although the accuracy of stacked hourglass fusion is slightly lower than RGBD-SIDENet based on TW-SMNet (T) in Table REF , its output does not have boundary discontinuity due to the using of fusion network.",
"The accuracy of decision-level fusion of SHG-SIDENet and MT-TW-SMNet on the Sceneflow dataset is shown in Table REF .",
"From Table REF , we observe that by using SHG-fusion, the end-point-error is reduced from $5.61$ pixels to $5.50$ pixels, and further to $5.31$ pixels by decision selection.",
"These results confirm that our proposed fusion is effective in improving the accuracy of the tele-wide disparity estimation network.",
"Both output feature fusion and input feature fusion can also be combined together in the same scheme, as demonstrated in Fig.",
"REF .",
"For example, the SIDE map can be obtained by the RGBD-SHG-SIDENet which samples the stereo-matching output of the MT-TW-SMNet disparity map to construct a sparse disparity feature, that is fused at the input of the RGBD SIDE network.",
"The disparity maps from the same stereo-matching branch of the MT-TW-SMNet and from the output of the RGBD-SHG-SIDENet are then deeply fused using SHG-Fusion.",
"This architecture gave the best accuracy among the tele-wide disparity estimation methods, we tested.",
"We submitted our best results to the KITTI leaderboard server.",
"Our best performance on the KTTI test set on the leaderboard is by the fusion: RGBD-SHG-SIDENet (D comes from the estimated tele disparity from MT-TW-SMNet) + MT-TW-SMNet + SHG-fusion + decision selection.",
"KITTI leaderboard utilizes a different evaluation protocol as the validation set, where the error of foreground pixel (error-fg) and the error of background pixel (error-bg) are calculated.",
"This performance is shown in Table REF as `MT-TW Fusion'.",
"Although this accuracy is not as good as the top ranked submissions, it only utilizes the left wide image and the right tele image, and uses much less information than other submissions which use the full wide left and right images.",
"Our proposed tele-wide stereo matching fusion scheme still ranks better than recent schemes for wide-wide stereo matching that also use the interframe optical flow information [61].",
"Example disparity outputs from the MT-TW Fusion network on test examples are given in Fig.",
"REF , and more detailed visualizations can be found at [62].",
"Given a focused image of a scene and the depth map of the scene, the Bokeh effect can be synthesized by post-processing to render blurry out-of-focus areas in the image [63].",
"We use the estimated tele-wide depth map to synthesize the Bokeh effect on the full WFOV.",
"One problem is that there are some pixels which are not correctly estimated, especially at the top (sky) regions of KITTI images due to the lack of ground-truth labels.",
"So we propose an additional post-processing module to remove these incorrect pixels before using it for Bokeh.",
"We know that image Bokeh requires an input focus point to locate the desired foreground object.",
"A feasible way to remove the bad estimated pixels is to suppress the regions with high disparity that don't include the focus point.",
"So we design the following algorithm based on the connected components and distance transform, as shown in Fig.",
"REF .",
"After locating the foreground region by the Bokeh focus point, we suppress all other disparities relative to their distance from the foreground region.",
"Examples for the Bokeh effect synthesized from input tele-wide stereo image pairs are shown for the KITTI Stereo 2015 and the SceneFlow datasets in Fig.",
"REF and Fig.",
"REF , respectively.",
"Post-processing the estimated disparity for image Bokeh.",
"[1] Wide disparity $d$ , Focus point $(i,j)$ Post-processed disparity map $d^{\\prime }$ Define a pixel $(x,y)$ as foreground if and only if $0.7 d(i,j) \\le d(x,y) \\le 1.3 d(i,j)$ Extract a disparity map $d_f=d$ for all foreground pixels, and $d_f=0$ otherwise.",
"Generate the foreground mask $b_f=\\lbrace d_f \\ne 0\\rbrace $ .",
"Find $n$ connected components $\\lbrace C_1,C_2,…,C_n \\rbrace $ in the $b_f$ .",
"Find the connected component $C_t$ that includes the focus point $(i,j)$ .",
"$k = 1$ to $n$ , and $k \\ne t$ pixels $(x,y) \\in C_k$ , and $i \\ne 0$ Calculate the closet Euclidean distance $p(x,y)$ to any pixel not in $\\lbrace C_1,C_2,…,C_n \\rbrace $ $d^{\\prime }(x,y)=d(x,y)\\frac{1 - 2 p(x,y)}{p(x,y)}$ $d^{\\prime }$ Figure: Examples of the image Bokeh results using tele-wide disparity on SceneFlow dataset."
],
[
"Conclusion ",
"In this paper, we introduced the tele-wide stereo matching problem.",
"We established baseline solutions to estimate the inverse depth for the full wide field of view (FOV) from a tele-wide stereo image pair having a left wide FOV (WFOV) image and a right tele FOV (TFOV) image.",
"To improve the estimation accuracy, we further introduced a multitask tele-wide stereo matching network (MT-TW-SMNet) that is trained end-to-end to learn both a disparity estimation objective and an inverse depth estimation objective.",
"We showed that input feature fusion with RGBD inverse depth estimation can significantly improve the depth estimation quality in the regions surrounding the TFOV, where the stereo information is missing.",
"We further explored different output feature fusion methods, such as with decision selection between the proposed stereo matching and inverse depth estimation techniques, or with context-aware deep network fusion using stacked hour glass networks.",
"Experimental results on KITTI and SceneFlow datasets demonstrate that our proposed approaches achieve considerable performance in the tele-wide disparity estimation scheme, and perform better than other popular methods that perform stereo matching between two wide FOVs.",
"We demonstrate the usefulness of this approach, by synthesizing the Bokeh effect on the full WFOV, when the input is a tele-wide stereo image pair.",
"Although not considered in this paper, our work can also be generalized to systems with more than two cameras or multi-camera systems with different focal lengths, by estimating the union FOV for one stereo pair at a time, and recursively merging the results from the different stereo pairs to estimate the depth map for the union FOV of all cameras."
]
] | 1906.04463 |
[
[
"A Study of Berry Connection and Complex Analysis for Topological\n Characterization"
],
[
"Abstract We study and present the results of Berry connection for the topological states in quantum matter.",
"The Berry connection plays a central role in the geometric phase and topological phenomenon in quantum many-body system.",
"We present the necessary and sufficient conditions to characterize the topological nature of the system through the complex analysis.",
"We also present the different topological aspects of the system in the momentum space."
],
[
"Introduction",
"In nature, matter has different states such as solid, liquid and gaseous.",
"These states are characterized by their internal structures.",
"Other than internal structures, there exists symmetry.",
"With discovery of fractional quantum Hall effect, there arose a new type of matter, called topological state of matter [1].",
"These materials have a new type of internal order called topological order which makes them to be different from other materials.",
"They are materials which conduct along the edge [2] and are insulators at the bulk.",
"The conduction along the edge is protected by two major symmetries [3], [4], [5], time reversal and particle-conservation symmetry.",
"The physics of Landau Fermi liquid theory provides a major description of the symmetry breaking phenomenon of quantum many-particle system.",
"However the major limitations of Landau theory of phase transition is that it is related with the local order parameter.",
"But it is well known for the study of the topological state of quantum matter that they do not have any order parameter.",
"Berry phase is the main tool to characterize the topological state of the system [6], [7], [8], [9], [10], [11], [12].",
"When a time-dependent Hamiltonian with a state $|n\\rangle $ begins to evolve, it remains in the same state till the end.",
"But along with the dynamical phase, it may also acquire a geometric phase depending on the path of evolution [13], [14], [15], [16].",
"For a quantum adiabatic process, geometric phase is generally known as Berry phase.",
"The concept of Berry phase arises naturally from Berry connection [17], [18], [19], [20].",
"Berry connection is more fundamental than Berry phase and Berry curvature.",
"It is non-observable and non-vanishing quantity which shows the overlapping of the wavefunction in an evolving system.",
"It shows the connection from $\\mathbb {R}$ to $\\mathbb {R+dR}$ manifolds in the Hilbert space $\\mathbb {H}$ .",
"In other-words it describes the parallel transport of the Block state in momentum space.",
"There are observations of tensor Berry connection to explain the higher order gauge fields of quantum matter [21].",
"By the proper selection of gauge one can explain the Berry connection of tensor monopoles found in 4D-Weyl type systems, Berry phase with or without Krammer degeneracy, monopoles of Dirac and Yang [22], [23], [24].",
"To analyze the nature of topological state of matter, we need to understand the behavior of wavefunction.",
"Topological properties can also be determined by the complex analysis of the parameter space [25], [26].",
"Topological phase transition points are considered as the singularity points in the parameter space.",
"Topological phase transition results in the gapless condition of the energy spectrum.",
"Topological invariants like winding number, Chern numbers are ill defined at these points.",
"By using complex analysis method one can be define the topological properties both in gapless and gapped phases [27].",
"For the present study we show explicitly that how Berry connection is related to the topological quantum phase transition in the momentum space and we present a way of complex analysis to verify the method.",
"Motivation of the study: The main motivation of this work is to explain a few topological aspects of many-body system in an efficient manner.",
"Topological states of matter are characterized by the absence of local order parameters.",
"But there are some studies which considers Berry phase as the local topological order parameter [28].",
"So, one can calculate the local topological order parameter (Berry phase) as a integration of Berry connection over the all available energy states.",
"The Fourier transform of Berry connection is correlation function in 1D which gives the charge correlation function of Wannier states at different points [29], [30], [31], [32].",
"Recently there are studies to calculate correlation length, universality classes and scaling laws by using Berry connection [33], [34], [35], [34].",
"One can characterize the topological state of a quantum system by the physical entity called Berry phase [36].",
"Berry phase exists where there is a closed interval.",
"But Berry connection is a more fundamental quantity which reveals the path of evolution of ground-state wave function.",
"Berry connection is a gauge dependent quantity which is equivalent to vector potential in electromagnetic theory [17], [37].",
"The significance of non-trivial Berry connection can be found in the Aharanov-Bohm effect [38].",
"Here we are interested to show how one can directly study the topological behavior of the system just by studying the Berry connection and we argue that Berry connection is the fundamental tool to study the topological state of matter.",
"The study of geometric phase to characterize the topological state of matter has already been done in the literature [39].",
"It finally gives the parametric relation of the system to characterize the different topological aspects of the topological state of matter [40].",
"But we show explicitly how the different topological aspects manifests in the momentum space.",
"The other motivation of the study is to do the complex analysis to characterize the topological aspects of the problem.",
"Complex analysis is a effective way to describe the state of a system.",
"Depending on the on the position of poles and zeroes, topological state of the system is determined [41], [26].",
"Outline of the work: In section we introduce and explain our model Hamiltonian.",
"In section we present a detailed study of Berry connection and variation of topological angle.",
"Here we give a clear picture between Berry connection, geometric phase and energy spectrum of different topological phases.",
"In section , we explicitly show the existence of topological state from the perspective of complex analysis.",
"Thus we verify how we can verify the topological properties of the system."
],
[
"Introduction to Model Hamiltonian",
"In this section, we first introduce the Hamiltonian of the present study.",
"We consider the Kitaev chain as our model Hamiltonian [42], [40], [43].",
"It is a lattice model of p-wave superconductor in 1D [42].",
"Kitaev model has two phases.",
"Topological phase for $\\mu <2J$ , non-topological phase for $\\mu >2J$ .",
"Here the gapless condition occurs for the case $\\mu =2J$ at $k=0$ , where the gapless condition occurs.",
"Through energy dispersion study, we can understand gapless state formation [40].",
"The Hamiltonian can be written as $H_0 = [\\sum _{j}-{J} ({c_j}^{\\dagger } c_{j+1} + h.c )-{\\mu } {c_j}^{\\dagger } {c_j}+{|\\Delta |} ( {c_j} {c_{j+1}} + h.c ) ], $ where $ J$ is the hopping matrix element, $\\mu $ is the chemical potential and $|\\Delta |$ is the magnitude of the superconducting gap.",
"We write the Hamiltonian in the momentum space as H1 = k> 0 ( + 2J k) (k k + -k -k) + 2i k > 0 k (k -k + k -k), where $ {\\psi ^{\\dagger }} (k) (\\psi (k))$ is the creation (annihilation) operator of the spinless fermion of momentum $k$ .",
"We can write the Hamiltonian in the BdG format as $H_{BdG}(k)=\\left(\\begin{matrix}\\chi _1(k)&& i\\chi _2(k)\\\\-i\\chi _2(k)&& -\\chi _1(k)\\\\\\end{matrix}\\right).$ We can express the Hamiltonian by Anderson pseudo-spin approach [44], [45], [43].",
"One can write the BdG Hamiltonian in the pseudo-spin basis as $\\vec{H}(k)=\\chi _1(k)\\vec{\\tau _1}+\\chi _2(k)\\vec{\\tau _2}+\\chi _3(k) \\vec{\\tau _3}\\Rightarrow H_{BdG}(k)=\\Sigma _i\\vec{\\chi }_i(k).\\vec{\\tau }_i,$ where ${\\tau _i}=(\\tau _1,\\tau _2,\\tau _3)$ are the Pauli matrices, $\\chi _1(k)=0$ , $\\chi _2(k)=2\\Delta \\sin k$ and $\\chi _3(k)=-2J\\cos k -\\mu $ .",
"To characterize the topological trivial and non-trivial phases of the Kitaev chain one can construct the parametric space with axis, $X=\\frac{\\chi _1(k)}{|\\chi (k)|}$ and $Y=\\frac{\\chi _2(k)}{|\\chi (k)|}$ [41].",
"The unit vector $\\frac{\\chi (k)}{|\\chi (k)|}$ takes the closed path in the parametric space as $k$ varies across the Brillouin zone.",
"Therefore the number of windings that the unit vector takes around its origin in the parametric space defines the winding number.",
"The winding number $w=\\pm n$ with $n=0,1,2,...$ , characterizes the topological trivial ($w=0$ ) and non-trivial ($w=\\pm 1$ ) phases in Kitaev chain."
],
[
"Study of Berry connection and variation of topological angle for the model Hamiltonian",
"Here we present study of Berry connection and variation of topological angle within the Brillouin zone boundary.",
"The other important feature of the Berry connection is that it transforms as a vector gauge potential under gauge transformation.",
"Therefore in quantum many-body lattice system, Berry connection associated with a Bloch state acts as an effective electrostatic vector potential defined in quantum metric space.",
"Gauge invariant quantities can be defined from the Berry connection.",
"Here we mention very briefly how Berry connection is related with the Aharanov-Bohm effect to illustrate the importance of Berry connection.",
"In this experimental set-up magnetic field $(\\vec{B}=\\vec{\\nabla }\\times \\vec{A})$ outside the tube is zero [39].",
"But the vector potential is non-zero.",
"So for any closed trajectory around the flux tube, there is a global effect such that the line integral over the vector potential gives [38] $\\oint dr.\\vec{A}(\\vec{r})=n\\Phi ,$ where $\\Phi $ is total flux inside the tube and $n$ is integer winding number of the trajectory around the tube.",
"It has shown explicitly in [40] that the phase acquired in Aharanov-Bohm effect is the topological phase.",
"Therefore the non-local effect of $\\bar{A}$ causes the topological phase.",
"By using Stokes theorem, one can connect the Berry phase and Berry connection through the analytical relation, $\\gamma _n=\\oint _c A_n(r).dr$ , over the closed contour $c$ .",
"The resulting Berry phase will be $2\\pi $ or integral multiples of $2\\pi $ .",
"If the contour is not closed Berry phase won't exists, because gauge dependency is not present.",
"The geometric phase acquired by the the wavefunction in Aharanov-Bohm effect is a gauge-independent quantity.",
"The phase acquired by the wavefunction is $\\phi _k=\\int _{s}F(k).ds,$ where $F(k)$ which is nothing other than the magnetic flux density or Berry curvature.",
"It is a gauge independent quantity.",
"It can also be written as $F(k)=\\nabla _k\\times A(k)=\\nabla _k\\times \\langle \\psi (k)|i\\nabla _k|\\psi (k)\\rangle =i\\langle \\nabla _k\\psi (k)|\\times |\\nabla _k\\psi (k)\\rangle $ Where $A(k)$ is the Berry connection.",
"Berry curvature is a geometric property of the system.",
"It can be observes as magnetic field effects in electronic as well as optical Hall effects.",
"Sometimes there arises the singularities in the parameter space of the vector potential.",
"In such cases Stokes theorem becomes invalid and the geometric phase is ill defined.",
"Otherwise geometric phase is always the integral multiple of $2\\pi $ .",
"$\\phi _k=\\int _{s}F(k).ds=\\oint _cA(k).dl=\\gamma $ It is clear from the exclusive analysis that the Berry phase which is analogous to the magnetic flux may be zero but Berry connection which is analogous to the vector potential is always non-zero.",
"The important result of this study is that for the topological state of matter like topological insulator and topological superconductors, this relation between the Berry connection and Berry phase is verified.",
"Mathematically for our model Hamiltonian Berry connection is expressed as $A_k=\\left\\langle \\psi _{k}|i\\partial _k|\\psi _{k}\\right\\rangle .$ From eq.",
"REF , we get the eigenvalues, $E_k=\\pm \\sqrt{(2J\\cos k+\\mu )^2+(2\\Delta \\sin k)^2}=\\sqrt{2(J^2+\\Delta ^2)+\\mu ^2+2J\\mu \\cos k+2(j^2-\\Delta ^2)\\cos 2k}.$ Eigenvectors are given by |=-i2(+2Jk-Ek)k|+1|, where, $|\\uparrow \\rangle =(1,0)^T$ and $|\\downarrow \\rangle =(0,1)^T$ are the column matrices.",
"Berry connection is given by Ak=k|ik|k=i(-i2(+2Jk+Ekk)) *(i2(+2Jk+Ekk) -ik(-2Jk+-4Jk-4(J2-2)2kEk)).",
"After simplification we get Ak=-i(2J+k)(+2Jk+Ek)23 k42Ek.",
"Figure: The upper panel of the figure shows the variation of Berry connection for Kitaev Hamiltonian (H k H_k) with kk.",
"The red curve represents the topological case(μ<2J\\mu <2J), the blue curve represents transition state (μ=2J\\mu =2J) and the green curve represents non-topological state (μ>2J\\mu >2J).",
"The middle panel shows the geometric phase for corresponding parameter spaces.",
"The lower panel shows the energy dispersion curves for the corresponding parametric spaces.Fig.",
"REF contains three panels.",
"The upper panel shows the variation of $A(k)$ with momentum $k$ .",
"This figure presents three curves for different values of $J$ ($J=1,0.5,0.2$ ).",
"These different curves are for different topological states of system.",
"i.e.",
"$\\mu <2J$ , $\\mu =2J$ and $\\mu >2J$ , as one can find from the topological invariant number [45].",
"It is well known in the literature that at the topological quantum phase transition point $(\\mu =2J)$ winding number shows the discontinuous value.",
"In the present study we show explicitly at this particular point, at the first Brillouin zone boundary there is also a discontinuity in the Berry connection.",
"We show that at the topological quantum phase transition point, there is a jump of $A(k)$ .",
"For the topological states the $A(k)$ curve is continuous over the BZ boundary and for the non-topological states $A(k)$ is within the BZ boundary.",
"The middle panel shows the energy dispersion relations with $k$ for the corresponding three phases.",
"One can observe how the energy gap closes for the topological quantum phase transition point.",
"But for the topological and non-topological phases, there is a gap in BZ.",
"The system goes from one topological state to the other by closing the gap.",
"In the rotated basis we can write the Hamiltonian as $H(k) = (\\epsilon _{k}- \\mu ) \\sigma _{x} - 2\\Delta \\sin k \\sigma _{y},$ where $\\epsilon _{k} = -2t\\cos k$ .",
"We write the Hamiltonian in matrix form as $ H= \\begin{bmatrix}0 && r e^{i\\theta } \\\\r e^{-i\\theta } && 0\\end{bmatrix}, $ where $ r = \\sqrt{(2t\\cos k+\\mu )^2 + 4\\Delta ^2 \\sin ^2 k}$ and $ \\theta _k = -\\tan ^{-1}(\\frac{2\\Delta \\sin k}{2t\\cos k+\\mu }).$ $\\theta _k$ is topological angle, which is the angle made by wave-vector in the Brillouin zone.",
"The integration over the variation of topological angle with wave-vector gives the geometric phase.",
"i.e.",
"$\\gamma =\\int _{-\\pi }^{+\\pi }\\left(\\frac{d\\theta _k}{dk}\\right)dk.$ So, present model $\\left(\\frac{d\\theta _k}{dk}\\right)$ is given by $\\frac{d\\theta _k}{dk}=\\frac{2\\Delta (2J+\\mu \\cos k)}{(2J\\cos k+\\mu )^2+(2\\Delta \\sin k)^2}.$ Fig.",
"REF shows the variation of the topological angle $\\left(\\frac{d\\theta _k}{dk}\\right)$ with the momenta $k$ .",
"The figure consists of three panels.",
"Upper panel is for topological $(\\mu <2J)$ , middle is for critical $(\\mu =2J)$ , and lower is for non-topological $(\\mu >2J)$ case respectively.",
"Each figure consists of four curves for different values of $\\Delta $ ($\\Delta =0.2,0.5,0.8,1$ ).",
"The variation of $\\theta _k$ is almost absent for non-topological state, except at the very edge of the Brillouin zone boundary.",
"We observe that the variation of topological angle is maximum for topological state of matter.",
"We observe that variation is more prominent for lower values of $\\Delta $ .",
"We also observe that as the system goes from the topological state to the non-topological state, the variation of $\\theta _k$ is small.",
"This behavior of $\\theta _k$ in momentum space is physically consistent with the behavior of winding number.",
"Figure: Variation of topological angle (dθ k /dk)(d\\theta _k/dk) for Kitaev Hamiltonian (H) with kk.",
"The upper panel shows topological case(μ<2J\\mu <2J), middle panel shows transition case (μ=2J\\mu =2J) and the lower panel shows non-topological case (μ>2J\\mu >2J).",
"Red, blue, green and yellow curves represent Δ=0.2,0.5,0.8\\Delta =0.2,0.5,0.8 and 1 respectivelyThis study of $\\frac{d\\theta _k}{dk}$ show a reflection of topology in the momentum space.",
"i.e.",
"how the karnel of eq.",
"behaves in the momentum space.",
"It reveals for this study that the variation of topological angle for different topological states are different."
],
[
"An analysis of topology of the model Hamiltonian from the perspective of complex variable",
"The argument principle of complex analysis allows one to calculate winding number of simple closed contour about its origin.",
"Argument principle states that the winding number can be written as a contour integral of a meromorphic function $f(z)$ .",
"This integral can be expressed in terms of number of zeros and poles of $f(z)$ .",
"Therefore the winding number $w$ of a simple closed contour about its origin, can also be expressed as a difference between number of zeros and poles of $f(z)$ , i.e.",
"$w=Z-P$ [41], where $Z$ and $P$ are zeros and poles respectively.",
"The generic form of the Hamiltonian can be written as $H(k)=d(k).\\sigma = \\left( \\begin{matrix}0 && q^*(k)\\\\q(k) && 0\\end{matrix}\\right),$ where $q(k)=-2J\\cos k-\\mu +2i\\Delta \\sin k$ .",
"The winding number can be calculated using the definition [46] $w=\\frac{1}{2\\pi i}\\int \\limits _{-\\pi }^{\\pi } dk \\frac{\\partial _k q(k)}{q(k)}= \\frac{1}{2\\pi i}\\int \\limits _{-\\pi }^{\\pi } dk \\frac{2J\\sin k + 2i\\Delta \\cos k}{-2J \\cos k -\\mu + 2i\\Delta \\sin k}\\\\ \\hspace{8.5359pt}.$ Writing $\\sin k$ and $\\cos k$ in the exponential form and substituting $z=e^{ik}$ and $dz=ie^{ik}dk$ , one can rewrite the above equation [41], [47] as $w=\\frac{1}{2\\pi i}\\int \\limits _{-\\pi }^{\\pi } \\frac{dz}{z} \\frac{\\Delta (z^2+1)-t(z^2-1)}{(\\Delta -t)z^2 - \\mu z - (\\Delta +t)}\\hspace{8.5359pt}.$ Winding number can be calculated using the argument principle of complex analysis.",
"eq.",
"REF has a general form as $\\frac{1}{2\\pi i} \\oint \\frac{f^{\\prime }(z)}{f(z)} dz.$ This integral can be solved by calculating number of poles ($P$ ) and number of zeros ($Z$ ) of the function $f(z)$ as $\\frac{1}{2\\pi i} \\oint \\frac{f^{\\prime }(z)}{f(z)} dz = Z-P.$ In our case $f(z)$ has the form $f(z)= \\frac{1}{z}\\left( \\Delta (z^2-1)-J(z^2+1)-\\mu z \\right) $ .",
"Clearly it has a pole at $z=0$ .",
"It has two zeros, one at $z=+1$ if $\\mu =-2J$ , another at $z=-1$ if $\\mu =2J$ .",
"Therefore the winding number is $w=2-1=1$ if $-2J<\\mu <2J$ .",
"Thus, in parametric space it encircles the origin one time.",
"Either the winding number is $+1$ or $-1$ .",
"The sign indicates the direction of encircling, clockwise or counterclockwise.",
"This corresponds to the topological region of the Hamiltonian $H(k)$ .",
"If there are equal numbers of poles as well as zeros, there will be no winding number.",
"Here the curve will not encircle the origin.",
"This shows non-topological case.",
"There are some cases when the zeros and poles are present on the contour of the curve.",
"At this point we can not explain the winding number for the particular function.",
"Here the curve touches the origin and the function is ill defined.",
"This point is known as topological phase transition point (fig.REF ).",
"Figure: Parametric plot for the Hamiltonian H(k)H(k) for different values of μ\\mu .Fig REF contains three panels.",
"Left one indicates the topological state of the system where the origin is encircles by a closed curve.",
"Middle panel represents the topological phase-transition case where the curve touches the origin.",
"Right panel shows the non-topological state of the of the system where origin lies outside the closed curve.",
"This is a straight forward method to analyze the topological properties of the system.",
"The same results can be obtained in slightly different approach.",
"First we write the Hamiltonian (eq.1) in Majorana operators with $c_n^{\\dagger }=\\frac{a+ib}{2}$ and $c_n=\\frac{a-ib}{2}$ , $H=-i\\left[\\sum \\limits _{j=1}^{N-1} \\left( \\frac{J-\\Delta }{2}\\right)b_ja_{j-1} - \\sum \\limits _{j=1}^{N}\\left( \\frac{\\mu }{2}\\right)b_ja_j + \\sum \\limits _{j=1}^{N-1} \\left( \\frac{J+\\Delta }{2}\\right)b_ja_{j+1}\\right]$ Generic form of this Hamiltonian can be written as $H=\\sum _{\\alpha =-1,0,1} \\gamma _{\\alpha }H_{\\alpha },$ where $H_{\\alpha }=\\sum \\limits _{j=1}^{N}b_ja_{j+\\alpha }$ and $\\gamma _{\\alpha =-1,0,1}=\\left( \\frac{J-\\Delta }{2}\\right), \\mu , \\left( \\frac{J+\\Delta }{2}\\right)$ respectively.",
"The Fourier transform of the Hamiltonian is $f(k)=\\sum _{\\alpha =-1,0,1}\\gamma _{\\alpha }e^{ik\\alpha }$ .",
"With $z=e^{ik}$ we get the complex function associated with the Hamiltonian, $f(z)= \\sum _{\\alpha =-1,0,1}\\gamma _{\\alpha }z^{\\alpha }= \\frac{1}{z}\\gamma _{-1}-\\gamma _0+z\\gamma _1= \\frac{1}{z}\\left(\\gamma _{1}z^2-\\mu z+\\gamma _{-1} \\right)$ Thus we have the roots $z_{1,2}=\\frac{\\mu \\pm \\sqrt{\\mu ^2 - 4\\gamma _1\\gamma _{-1}}}{2\\gamma _1}=\\frac{\\mu \\pm \\sqrt{\\mu ^2-J^2+\\Delta ^2}}{J+\\Delta }$ .",
"The complex function $f(z)$ has a pole at $z=0$ and two zeros at $z=z_1$ and $z=z_2$ .",
"Topological, non-topological phases and the transition between them can be characterized by observing whether two zeros fall inside, outside or on the unit circle respectively.",
"Using the argument principle of complex analysis, topological invariant $w=Z-P$ (where $Z$ is number zeros that falls inside the unit circle and $P$ is number of pole) can be calculated.",
"In our case the pole is always at the origin of the unit circle.",
"But values of two zeros $z_{1,2}$ depends on the parameter values.",
"We already know for parameter condition $-J<\\mu <J$ the system is in the topological phase and for $\\mu <-J$ and $\\mu >J$ , system is in non-topological phase.",
"$\\mu =\\pm J$ is thus the gap closing point indicating the topological quantum phase transition.",
"This is shown more explicitly in the fig.REF in terms number of zeros and pole of $f(z)$ .",
"Left panel of fig REF shows topological phase for finite $\\mu $ satisfying $-J<\\mu <J$ .",
"We observe both the zeros lies inside the unit circle giving $w=2-1=1$ .",
"The middle and right panel of fig REF shows the system at the phase transition point and at non-topological phase respectively.",
"At $\\mu =\\pm J$ we see one of the zero lie on the unit circle while another lie inside, which gives $w=0$ .",
"For non-topological phase one of the zero lie inside and another lie outside the unit circle giving $w=0$ .",
"Figure REF consists of three panels.",
"Left panel represent the topological case.",
"Here we can observe both zeros lies inside the unit circle.",
"The corresponding pseudo-spin (eq. )",
"vector encircles the origin without any discontinuity.",
"The middle panel represents the topological phase-transition case.",
"Here the zeros lies on the unit circle.",
"The corresponding pseudo- spin vector encircles the origin but there is a discontinuity at the phase-transition point.",
"The right panel represents the non-topological case.",
"Here the zeros lies outside the unit circle.",
"The corresponding pseudo-spin vector does not encircles the origin.",
"Figure: Zeros and poles of the complex function f(z)f(z).",
"Left panel: topological phase, middle panel: phase transition case and right panel: non-topological phase respectively.Conclusion: We presented the detailed analysis of the Berry connection for the Kitaev model Hamiltonian and explained how the Berry connection can be helpful for the understanding of the topological properties of the system.",
"We showed that how the topological angle varies in the momentum-space as we vary the parameters of the system.",
"We gave the explicit explanation for the existence of topological state through the complex analysis technique.",
"Acknowledgments: The authors would like to acknowledge DST (EMR/2017/000898) for the funding and RRI library for the books and journals.",
"The authors would like to acknowledge Mr. N. A. Prakash for reading this manuscript critically and Mr. Amitava Banerjee for useful discussions.",
"Finally authors would like to acknowledge ICTS lectures/seminars/workshops/conferences/discussion meetings on different aspects of physics."
]
] | 1906.04400 |
[
[
"Optimizing Pipelined Computation and Communication for\n Latency-Constrained Edge Learning"
],
[
"Abstract Consider a device that is connected to an edge processor via a communication channel.",
"The device holds local data that is to be offloaded to the edge processor so as to train a machine learning model, e.g., for regression or classification.",
"Transmission of the data to the learning processor, as well as training based on Stochastic Gradient Descent (SGD), must be both completed within a time limit.",
"Assuming that communication and computation can be pipelined, this letter investigates the optimal choice for the packet payload size, given the overhead of each data packet transmission and the ratio between the computation and the communication rates.",
"This amounts to a tradeoff between bias and variance, since communicating the entire data set first reduces the bias of the training process but it may not leave sufficient time for learning.",
"Analytical bounds on the expected optimality gap are derived so as to enable an effective optimization, which is validated in numerical results."
],
[
"Introduction",
"Edge learning refers to the training of machine learning models on devices that are close to the end users [1].",
"The proximity to the user is instrumental in facilitating a low-latency response, in enhancing privacy, and in reducing backhaul congestion.",
"Edge learning processors include smart phones and other user-owned devices, as well as edge nodes of a wireless network that provide wireless access and computational resources [1].",
"As illustrated in Fig.",
"REF , the latter case hinges on the offloading of data from the data-bearing device to the edge processor, and can be seen as an instance of mobile edge computing [2].",
"Research on edge learning has so far instead focused mostly on scenarios in which training occurs locally at the data-bearing devices.",
"In these setups, devices can communicate either through a parameter server [3] or in a device-to-device manner [4].",
"The goal is to either learn a global model without exchanging directly the local data [5] or to train separate models while leveraging the correlation among the local data sets [6].",
"Devices can exchange either information about the local model parameters, as in federated learning [7], or gradient information, as in distributed Stochastic Gradient Descent (SGD) methods [8], [9].",
"In this work, we consider an edge learning scenario in which training takes place at an edge node of a wireless system as illustrated in Fig.",
"1.",
"The data is held by a device and has to be offloaded through a communication channel to the edge node.",
"The learning task has to be executed within a time limit, which might be insufficient to transmit the complete dataset.",
"Transmission of data blocks from device to edge node, and training at the edge node can be carried out simultaneously (see Fig.",
"REF ).",
"Each transmitted packet contains a fixed overhead, accounting e.g.",
"for meta-data and pilots.",
"Given the overhead of each data packet transmission, what is the optimal size of a communication block?",
"Communicating the entire data set first reduces the bias of the training process but it may not leave sufficient time for learning.",
"We investigate a more general strategy that communicates in blocks and pipelines communication and computation with an optimized block size, which is shown to be generally preferable.",
"Analysis and simulation results provide insights into the optimal duration of the communication block and on the performance gains attainable with an optimized communication and computation policy.",
"The rest of this letter is organized as follows.",
"In Sec.",
", we provide an overview of the model and the associated notations.",
"In Sec.",
", we examine the technical assumptions necessary for our work.",
"In Sec.",
", we provide our main result and discuss its implications.",
"Finally, in Sec.",
", we consider numerical experiments in the light of our result.",
"Figure: An edge computing system, in which training of a model parametrized by vector ww takes place at an edge processor based on data received from a device using a protocol with timeline illustrated in Fig.",
"(OH = overhead)."
],
[
"System model",
"As seen in Fig.",
"REF , we study an edge learning system in which a device communicates with an edge node, and associated server, over an error-free communication channel.",
"The device has access to a local training dataset $\\mathcal {X} = \\lbrace x_{1}, x_{2}, \\dots , x_{N}\\rbrace $ of $N$ data points $\\lbrace x_{n}\\rbrace _{n=1}^{N}$ , and training of a machine learning model is carried out at the edge node based on data received from the device.",
"As illustrated in Fig.",
"REF , communication and learning must be completed within a time limit $T$ .",
"To this end, the transmissions are organized into blocks, and transmission and computing at the edge node can be performed in parallel.",
"Training at the edge node aims at identifying a model parametrized by a vector $w \\in \\mathrm {R}^{d}$ within a given hypothesis class.",
"Training is carried out by (approximately) solving the Empirical Risk Minimization (ERM) problem (see, e.g, [10]).",
"This amounts to the minimization with respect to vector $w$ of the empirical average $\\mathcal {L}(w)$ of a loss function $\\ell (w,x)$ over all the data points $x$ in the training dataset, i.e., $\\mathcal {L}(w) = \\frac{1}{N} \\sum _{n=1}^{N}\\ell (w, x_{n}).$ As detailed below, the minimization of the function $\\mathcal {L}(w)$ is carried out at the edge node using SGD, based on the data points received from the device.",
"In order to elaborate on the communication and computation protocol illustrated in Fig.",
"REF , we normalize all time measures to the time required to transmit one data sample from the device to the edge node.",
"With this convention, we denote as $\\tau _{p}$ the time required to make one SGD update at the edge node.",
"As seen in Fig.",
"REF , transmission from the device to the edge node is organised into blocks.",
"In this study, we ignore the effect of channel errors, which is briefly discussed in Sec.",
".",
"In the $b$ -th block, the device transmits a subset $\\mathcal {X}_{b} \\subseteq \\mathcal {X}$ of $n_{c}$ new samples from its local dataset.",
"At the end of the block, the edge node adds these samples to the subset $\\tilde{\\mathcal {X}}_{b+1}$ of samples it has available for training in the $b+1$ -th block, i.e., $\\tilde{\\mathcal {X}}_{b+1} = \\tilde{\\mathcal {X}}_{b} \\cup \\mathcal {X}_{b}$ with $\\mathcal {X}_{0} = \\varnothing .$ The samples in $\\mathcal {X}_{b}$ are randomly and uniformly selected from the set $\\Delta \\mathcal {X}_{b} = \\mathcal {X} \\setminus \\tilde{\\mathcal {X}}_{b}$ of samples not yet transmitted to the edge node.",
"A packet sent in any block contains an overhead, e.g., for pilots and meta-data, of duration $n_{o}$ , irrespective of the number $n_{c}$ of transmitted samples.",
"It follows that the duration of a transmission block is $n_{c} + n_{o}$ .",
"There are at most $B_{d} = N/n_{c}$ transmission blocks, since $B_{d}$ blocks are sufficient to deliver the entire dataset to the edge node.",
"Therefore, we need to distinguish two cases.",
"As seen in Fig.",
"REF (a), when $T \\le B_{d}(n_{c} + n_{o})$ , the device is only able to deliver a fraction of the samples.",
"In particular, denoting as $B= T/(n_{c} + n_{o})$ the number of blocks, the fraction of data points delivered at the edge node at time $T$ equals $(B - 1)/B_{d}$ .",
"In contrast, if $T > B_{d}(n_{c} + n_{o})$ , as illustrated in Fig.",
"REF (b), the edge node has the entire dataset available after $B_{d}$ blocks, that is, for a duration equal to $ \\tau _{l} = T - B_{d}(n_{c} + n_{o})$ .",
"Henceforth, we refer to this last period as block $B_{l} = B_{d} + 1$ .",
"During each block $b \\le B_{d}$ , the edge node computes $n_{p} = (n_{c} + n_{o})/\\tau _{p}$ local SGD updates (REF ).",
"During block $B_{l}$ , the edge node computes $n_{l} = \\tau _{l}/\\tau _{p}$ SGD updates.",
"The $j$ -th local update at block $b$ , with $j = 1,\\dots , n_{p}$ , is given as $w_{b}^{j} = w_{b}^{j-1} - \\alpha \\nabla \\ell (w_{b}^{j-1}, \\xi _{b}^{j}),$ where $\\alpha $ is the learning rate, and $\\xi _{b}^{j}$ is a data point sampled i.i.d.",
"uniformly from the subset $\\tilde{\\mathcal {X}}_{b} = \\bigcup _{l=1}^{b-1} \\mathcal {X}_{l}$ of samples currently available at the edge node.",
"Note that we have $\\tilde{\\mathcal {X}}_{B_{l}} = \\mathcal {X}$ .",
"The goal of this work is to optimize the number of samples $n_{c}$ sent in each block with the aim of minimizing the empirical loss (REF ) at the edge node at the end of time $T$ .",
"In the next sections, we present an analysis of the empirical loss obtained at time $T$ that allows us to gain insights into the optimal choice of $n_{c}$ .",
"Figure: Transmission and training protocol: when (a) T≤B d (n c +n o )T \\le B_{d}(n_{c} + n_{o}); and (b) T>B d (n c +n o )T > B_{d}(n_{c} + n_{o})."
],
[
"Technical assumptions",
"In order to study the training loss achieved at the edge node at the end of the training process, we make the following standard assumptions, which apply, for instance, to linear models with quadratic or cross-entropy losses under suitable constraints (see the comprehensive review paper [9]): the sequence of iterates $w_{b}^{j}$ in (REF ) is contained in a bounded open set $\\mathcal {W} \\subseteq \\mathbf {R}^{d}$ with radius $D = \\max _{u, w \\in \\mathcal {W} \\times \\mathcal {W}} ||w - u||_{2}$ over which the function $\\ell (w, x)$ is bounded below by a scalar $\\ell _{\\text{inf}}$ for all $x$ ; the function $\\ell (w,x)$ is continuously differentiable in $w$ for any fixed value of $x$ and is $L$ -smooth in $w$ , i.e., $||\\nabla \\ell (w, x) - \\nabla \\ell (\\bar{w}, x)||_{2} \\le L||w - \\bar{w}||_{2}$ for all $(w, \\bar{w}) \\in \\mathcal {W}\\times \\mathcal {W}$ , and for all $x$ .",
"This implies $\\ell (w, x) \\le \\ell (\\bar{w}, x) + \\nabla \\ell (\\bar{w}, x)^{T}(w - \\bar{w}) + \\frac{L}{2}||w - \\bar{w}||_{2}^{2}$ for all $(w, \\bar{w}) \\in \\mathcal {W} \\times \\mathcal {W}$ , and for all $x$ ; the loss function $\\ell (w,x)$ is convex and satisties the Polyak-Lojasiewicz condition in $w$ , i.e., there exists a constant $c > 0$ such that $2c(\\ell (w, x) - \\ell (w_{\\ell }^{*}, x)) \\le ||\\nabla \\ell (w, x)||_{2}^{2}$ for all $(w, x) \\in \\mathcal {W} \\times \\mathbf {R}^{d}$ where $w_{\\ell }^{*}(x) = \\operatornamewithlimits{arg\\,min}_{w \\in \\mathcal {W}}\\ell (w,x)$ is a minimizer of $\\ell (w, x)$ .",
"The P-L condition is implied by, but does not imply, strong convexity [9].",
"We further need to make assumptions on the statistics of the gradient $\\nabla \\ell (w, \\xi _{b}^{j})$ used in the update (REF ).",
"To this end, for each block $b > 1$ , we define the empirical loss limited to the samples available at the edge node at block $b$ as $\\tilde{\\mathcal {L}}_{b}(w) = \\frac{1}{(b - 1) n_{c}} \\sum \\limits _{x_{i} \\in \\tilde{\\mathcal {X}}_{b}}\\ell (w, x_{i});$ the empirical loss over the samples transmitted at iteration $b \\ge 1$ as $\\mathcal {L}_{b}(w) = \\frac{1}{n_{c}}\\sum \\limits _{x_{i} \\in \\mathcal {X}_{b}}\\ell (w, x_{i});$ and the empirical loss over the samples not available at the edge at iteration $b > 1$ $\\Delta \\mathcal {L}_{b}(w) = \\frac{1}{N - (b - 1)n_{c}} \\sum \\limits _{x_{i} \\in \\Delta \\mathcal {X}_{b}} \\ell (w, x_{i}).$ Note that we have the identity $\\mathcal {L}(w) = \\big ((b - 1)n_{c}/N\\big )\\tilde{\\mathcal {L}}_{b}(w) + \\big ((N - (b - 1)n_{c})/N \\big )\\Delta \\mathcal {L}_{b}(w)$ .",
"First, we observe that given the previously transmitted data samples, the gradient $\\nabla \\ell (w_{b}^{j-1}, \\xi _{b}^{j})$ is an unbiased estimate of the gradient $\\nabla \\tilde{\\mathcal {L}}_{b}(w)$ of the empirical loss limited to the samples available at the edge node at block $b$ .",
"In formulas, $\\mathrm {E}_{\\xi _{b}^{j} | \\tilde{\\mathcal {X}}_{b}}[\\nabla \\ell (w, \\xi _{b}^{j})] = \\nabla \\tilde{\\mathcal {L}}_{b}(w) $ , where $\\mathrm {E}_{\\xi _{b}^{j} | \\tilde{\\mathcal {X}}_{b}}[\\ \\cdot \\ ]$ is the conditional expectation given the previously transmitted samples.",
"We finally make the following assumption (see, e.g., [9]): For any set $\\tilde{\\mathcal {X}}_{b}$ of samples available at the edge node, there exist scalars $M \\ge 0$ and $M_{V} \\ge 0$ such that $\\mathrm {V}_{\\xi _{b}^{j} | \\tilde{\\mathcal {X}}_{b}}[\\nabla \\ell (w, \\xi _{b}^{j})] \\le M + M_{V}||\\nabla \\tilde{\\mathcal {L}}_{b}(w)||_{2}^{2}$ where $\\mathrm {V}[\\ \\cdot \\ ] = \\mathrm {E}[||\\cdot ||^{2}] - ||\\mathrm {E}[\\ \\cdot \\ ]||^{2}$ is the variance."
],
[
"Convergence analysis",
"In this section, we present our main result and its implications on the optimal choice of the number $n_{c}$ of transmitted samples per block.",
"Henceforth, we use the notation $\\mathrm {E}_{b}[\\ \\cdot \\ ]$ to indicate the conditional expectation $\\mathrm {E}_{\\xi _{b}^{1}, \\dots \\xi _{b}^{n_{p}}|\\tilde{\\mathcal {X}}_{b}}[\\ \\cdot \\ ]$ on the samples selected for the SGD updates in the $b$ -th block given the set $\\tilde{\\mathcal {X}}_{b}$ of samples available at the edge node at $b$ .",
"We similarly define $\\mathrm {E}_{B_{l}}[\\ \\cdot \\ ] = \\mathrm {E}_{\\xi _{B_{l}}^{1}, \\dots , \\xi _{B_{l}}^{n_{l}}}[\\ \\cdot \\ ]$ as the conditional expectation on the samples selected for the SGD updates in block $B_{l}$ (see Fig.",
"REF (b)).",
"Theorem 1 Under assumptions REF -REF , assume that the SGD stepsize $\\alpha $ satisfies $0 < \\alpha \\le \\frac{2}{LM_{G}}$ and define $\\gamma = \\alpha \\Big (1 - \\frac{1}{2}\\alpha L M_{G}\\Big ).$ Then, for any sequence $\\tilde{\\mathcal {X}}_{1}, \\dots , \\tilde{\\mathcal {X}}_{B}$ the expected optimality gap at time $T$ is upper bounded as $& \\mathrm {E}_{B}[\\mathcal {L}(w_{B}^{n_{p}}) - \\mathcal {L}(w^{*})] & \\nonumber \\\\&\\le \\frac{\\alpha ^{2} L M}{2 \\gamma c} \\frac{(B - 1)}{B_{d}} + \\Big (1 - \\frac{(B - 1)}{B_{d}}\\Big ) \\mathrm {E}_{B}\\Big [\\Delta \\mathcal {L}_{B}(w_{B}^{n_{p}}) - \\Delta \\mathcal {L}_{B}(w^{*})\\Big ] + \\frac{1}{B_{d}}\\sum _{l=1}^{B - 1} (1 - \\gamma c)^{l n_{p}}\\mathrm {E}_{B - l}\\Big [\\mathcal {L}_{B - l}(w_{B - l}^{n_{p}}) - \\mathcal {L}_{B - l}(w^{*}) - \\frac{\\alpha ^{2} L M}{2 \\gamma c}\\Big ]&$ if $T \\le B_{d}(n_{c} + n_{o})$ ; and by $& \\mathrm {E}_{B_{l}}\\Big [\\mathcal {L}\\Big (w_{B_{l}}^{n_{l}}\\Big ) - \\mathcal {L}(w^{*})\\Big ] \\le \\frac{\\alpha ^{2} L M}{2 \\gamma c} + \\frac{1}{B_{d}}(1 - \\gamma c)^{n_{l}}\\sum _{l= 0}^{B_{d} - 1} (1 - \\gamma c)^{l n_{p}} \\mathrm {E}_{B_{d} - l}\\Big [\\mathcal {L}_{B_{d} -l}(w_{B_{d} - l}^{n_{p}}) - \\mathcal {L}_{B_{d} -l}(w^{*}) - \\frac{\\alpha ^{2} L M}{2 \\gamma c}\\Big ] &$ if $T > B_{d}(n_{c} + n_{o})$ .",
"Proof: See Appendix A.",
"The bound (REF )-(REF ) extends the classical analysis of the convergence of SGD for the case in which the entire dataset is available at the learner [9] to the set up under study.",
"The bound distinguishes the case in which the edge node has the entire data set by the last block, and the complementary case, as seen in Fig.",
"REF .",
"The first term in the bound (REF ) represents an asymptotic bias that does not vanish with the number of SGD updates, even when all the data points are available at the edge node.",
"It is due to the variance (REF ) of the stochastic gradient.",
"The bound (REF ) for smaller values of $T$ also comprises an additional bias term, that is the second term in (REF ), due to the lack of knowledge about samples not received at the edge node by the end of the training process.",
"In contrast, the last term in bound (REF )-(REF ) accounts for the standard geometric decrease of the initial error in gradient-based learning algorithms.",
"Here, the initial error for each block $b$ is given by $\\mathrm {E}_{b}\\big [\\mathcal {L}(w_{b - 1}^{n_{p}}) - \\mathcal {L}(w^{*})\\big ]$ .",
"Note that the additional factor with exponent $n_{l}$ in (REF ) accounts for the number of updates made after all the samples have been received at the edge node.",
"The bound (REF )-(REF ) can be in principle optimized numerically in order to find an optimal value to the block size $n_{c}$ .",
"However, in practice, doing so would require fixing the choice of the sequence $\\tilde{\\mathcal {X}}_{1}, \\dots , \\tilde{\\mathcal {X}}_{B}$ , and running Monte Carlo experiments for every randomly selected sample of the sequence of SGD updates (REF ), which is computationally intractable.",
"Therefore, in the following, we derive a generally looser bound that can be directly evaluated numerically without running any Monte Carlo simulations.",
"This bound will then be used in order to obtain an optimized value for $n_{c}$ .",
"Figure: Upper bound ()-() versus block size n c n_{c} for various values of the overhead n o n_{o}.",
"The full dots represent values of n c n_{c} at which we have T=B d (n c +n o )T = B_{d}(n_{c} + n_{o}) (see Fig.",
"), crosses represent the optimized value n ˜ c \\tilde{n}_{c}.Corollary 1 Under the conditions of Theorem REF , the expected optimality gap at time $T$ is upper bounded as $& \\mathrm {E}_{B}[\\mathcal {L}(w_{B}^{n_{p}}) - \\mathcal {L}(w^{*})] \\le \\frac{\\alpha ^{2} L M}{2 \\gamma c} \\frac{(B - 1)}{B_{d}} + \\Big (1 - \\frac{(B - 1)}{B_{d}}\\Big ) \\frac{LD^{2}}{2} + \\frac{1}{B_{d}}\\sum _{l=1}^{B - 1} (1 - \\gamma c)^{l n_{p}}\\Big [\\frac{LD^{2}}{2} - \\frac{\\alpha ^{2} L M}{2 \\gamma c}\\Big ], &$ if $T \\le B_{d}(n_{c} + n_{o})$ ; and by $& \\mathrm {E}_{B_{l}}[\\mathcal {L}(w_{B_{l}}^{n_{l}}) - \\mathcal {L}(w^{*})] \\le \\frac{\\alpha ^{2} L M}{2 \\gamma c} + \\frac{1}{B_{d}}(1 - \\gamma c)^{n_{l}}\\sum _{l= 0}^{B_{d} - 1} (1 - \\gamma c)^{l n_{p}}\\Big [\\frac{LD^{2}}{2} - \\frac{\\alpha ^{2} L M}{2 \\gamma c}\\Big ] &$ if $T > B_{d}(n_{c} + n_{o})$ .",
"Proof: See Appendix B.",
"We plot bound (REF )-(REF ) in Fig.",
"REF .",
"These results are obtained for $N = 18,576$ , $T = 1.5 N$ , $L = 1.908$ , $c= 0.061$ , $M = 1$ , $M_{G}=1$ , $\\tau _{p}=1$ , $\\alpha = 0.0001$ .",
"We note that $L$ and $c$ represent respectively the smallest and largest eigenvalues of the data Gramian matrix for the example studied in Sec.",
".",
"For each value of $n_{o}$ , we mark in the figure both the value of $n_{c}$ that minimizes the upper bound in Corollary REF and the value of $n_{c}$ at which we have the condition $T=B_{d}(n_{c} + n_{o})$ .",
"As seen in Fig.",
"REF , this is the minimum value of $n_{c}$ that allows the full transmission of the training set by the last training block.",
"A first observation is that the optimized value of $n_{c}$ , henceforth referred to as $\\tilde{n}_{c}$ , is generally smaller than the number $N$ of training points in $\\mathcal {X}$ , suggesting the advantages of pipelining communication and computation.",
"Furthermore, as the overhead $n_{o}$ increases, it becomes preferable, in terms of the bound (REF )-(REF ), to choose larger values $\\tilde{n}_{c}$ for the block size $n_{c}$ .",
"This is because a larger value of $n_{o}$ needs to be amortized by transmitting more data in each block, lest the transmission time is dominated by overhead transmission.",
"Finally, for smaller values of $n_{o}$ , the minimum $\\tilde{n}_{c}$ of the bound is obtained when the entire data set is eventually transferred to the edge node, i.e., $T > B_{d}(n_{c} + n_{o})$ , while the opposite is true for larger value of $n_{o}$ .",
"Interestingly, this suggests that it may be advantageous in terms of final training loss, to forego the transmission of some training points in exchange for more time to carry out training on a fraction of the data set."
],
[
"Numerical experiments",
"In this section, we validate the theoretical findings of the previous sections by means of a numerical example based on ridge regression on the California Housing dataset [11].",
"The dataset contains 20640 covariate vectors $x_{n} \\in \\mathrm {R}^{8}$ , each with a real label $y_{n}$ .",
"We randomly select $90\\%$ of the samples to define the set $\\mathcal {X}$ for training, i.e., we have $N = 18576$ .",
"As for Fig.",
"REF , we choose $\\tau _{p} = 1$ and $\\alpha =0.0001$ .",
"The parameter vector is initialized using i.i.d.",
"zero-mean Gaussian entries with unitary power.",
"The loss function is defined as $\\ell (w, x) = (w^{T}x - y)^{2} + \\frac{\\lambda }{N}||w||^{2}$ where $w \\in \\mathrm {R}^{8}$ and the regularization coefficient is chosen as $\\lambda = 0.05$ .",
"Figure: Training loss versus training time for different values of the block size n c n_{c}.",
"Solid line: experimental and theoretical optima.By computing the average final training loss for each value of $n_{c}$ , we can experimentally determine the optimal value $n_{c}^{*}$ of the block size.",
"We compare the performance using this experimental optimum with the performance obtained using the minimum $\\tilde{n}_{c}$ of the bound (REF )-(REF ).",
"To this end, in Fig.",
"REF , given a fixed overhead size $n_{o}$ , we plot the average training loss $\\mathcal {L}(w_{b}^{j})$ against the normalized training time $j$ for $n_{c}^{*}$ and for the value $\\tilde{n}_{c}$ obtained from the bound (REF )-(REF ).",
"As references, we also plot as dotted lines the losses obtained for selected values of $n_{c}$ .",
"The choice of the block size $n_{c}$ minimizing the average final loss is seen to be a trade-off between the rate of decrease of the loss and the final attained accuracy.",
"In particular, decreasing $n_{c}$ allows the edge node to reduce the loss more quickly, albeit with noisier updates and at the cost of a potentially larger final training loss due to the transmitted packet being dominated by the overhead.",
"Importantly, determining the optimum block size experimentally instead of using bound (REF )-(REF ) only provides a gain of $3.8\\%$ in terms of the final training loss, at the cost of a computationally burdensome parameter optimization."
],
[
"Conclusions",
"In this work, we considered an edge computing system in which an edge learner carries out training over a limited time period while receiving the training data from a device through a communication link.",
"Considering a strategy that allows communication and computation to be pipelined, we have analysed the optimal communication block size as a function of the packet overhead.",
"Among interesting directions for future work, we mention the inclusion of the effect of delays due to errors in the communication channel.",
"In this case, the optimization problem could be generalized to account for the selection of the data rate.",
"Other interesting extensions would be to consider online learning, where data sent in previous packets can be only partially stored at the server, and to investigate a scenario with multiple devices."
],
[
"Proof of Theorem 1",
"Using the same arguments as in the proof of [9], we can directly obtain the following inequality for each block $b$ : $& \\mathrm {E}_{b}[\\tilde{\\mathcal {L}}_{b}(w_{b}^{n_{p}}) - \\tilde{\\mathcal {L}}_{b}(w^{*})] \\le \\frac{\\alpha ^{2} L M}{2 \\gamma c} + (1 - \\gamma c)^{n_{p}} \\mathrm {E}_{b}\\Big [\\tilde{\\mathcal {L}}_{b}(w_{b}^{0}) - \\tilde{\\mathcal {L}}_{b}(w^{*}) - \\frac{\\alpha ^{2} L M}{2 \\gamma c}\\Big ].",
"&$ Note that we have $w_{b}^{0} = w_{b-1}^{n_{p}}$ , since the initial parameter at block $b$ is the final parameter obtained at block $b-1$ .",
"By definition of the local empirical losses (REF )-(REF ), we have the equality $\\tilde{\\mathcal {L}}_{b}(w_{b-1}^{n_{p}}) = \\frac{b - 2}{b - 1}\\tilde{\\mathcal {L}}_{b-1}(w_{b-1}^{n_{p}}) + \\frac{1}{b - 1}\\mathcal {L}_{b-1}(w_{b-1}^{n_{p}}).$ Plugging (REF ) into (REF ), we have $& \\mathrm {E}_{b}[\\tilde{\\mathcal {L}}_{b}(w_{b}^{n_{p}}) - \\tilde{\\mathcal {L}}_{b}(w^{*})] & \\nonumber \\\\& \\le \\frac{\\alpha ^{2} L M}{2 \\gamma c} + (1 - \\gamma c)^{n_{p}}\\mathrm {E}_{b}\\Big [\\Big (\\frac{b-2}{b-1}\\Big )\\Big (\\tilde{\\mathcal {L}}_{b-1}(w_{b-1}^{n_{p}})- \\tilde{\\mathcal {L}}_{b-1}(w^{*})\\Big ) + \\frac{1}{b-1}\\Big ( \\mathcal {L}_{b-1}(w_{b-1}^{n_{p}})- \\mathcal {L}_{b-1}(w^{*}) \\Big )- \\frac{\\alpha ^{2} L M}{2 \\gamma c}\\Big ].",
"&$ Iterating this substitution for all blocks $b-1, b-2, \\dots , 2$ , we obtain $& \\mathrm {E}_{b}[\\tilde{\\mathcal {L}}_{b}(w_{b}^{n_{p}}) - \\tilde{\\mathcal {L}}_{b}(w^{*})] \\le \\frac{\\alpha ^{2} L M}{2 \\gamma c} + \\sum _{l=1}^{b-1} (1 - \\gamma c)^{l n_{p}}\\frac{1}{b-1}\\mathrm {E}_{b}\\Big [\\mathcal {L}_{b-l}(w_{b-l}^{n_{p}})- \\mathcal {L}_{b-l}(w^{*})- \\frac{\\alpha ^{2} L M}{2 \\gamma c}\\Big ].",
"&$ While inequality (REF ) applies for any choice of $T$ , we now specialize the result to the case where the allocated amount of time $T$ is not sufficient to transmit the whole dataset, i.e., $T \\le B_{d}(n_{c} + n_{o})$ .",
"(see Fig.",
"REF (a)).",
"According to (REF )-(REF ), for this case, we have the equality $\\mathcal {L}(w) = \\frac{(b- 1)}{B_{d}}\\tilde{\\mathcal {L}}_{b}(w) + \\frac{N - (b- 1)}{B_{d}} \\Delta \\mathcal {L}_{b}(w).$ Plugging (REF ) into (REF ) for block $b=B$ , we then obtain $& \\mathrm {E}_{B}[\\mathcal {L}(w_{B}^{n_{p}}) - \\mathcal {L}(w^{*})] & \\nonumber \\\\&\\le \\frac{\\alpha ^{2} L M}{2 \\gamma c} \\frac{(B - 1)}{B_{d}} + \\Big (1 - \\frac{(B - 1)}{B_{d}} \\Big ) \\mathrm {E}_{b}\\Big [\\Delta \\mathcal {L}_{B}(w_{B}^{n_{p}}) - \\Delta \\mathcal {L}_{B}(w^{*})\\Big ] + \\frac{1}{B_{d}}\\sum _{l=1}^{B-1} (1 - \\gamma c)^{l n_{p}}\\mathrm {E}_{B}\\Big [\\mathcal {L}_{B - l}(w_{B - l}^{n_{p}})- \\mathcal {L}_{B - l}(w^{*})- \\frac{\\alpha ^{2} L M}{2 \\gamma c}\\Big ], &$ which is (REF ) in Theorem REF .",
"Finally, we consider the case where there is sufficient time to transmit the whole dataset, i.e., $T > B_{d}(n_{c} + n_{o})$ (see Fig.",
"REF (b)).",
"According to (REF ), we have $& \\mathrm {E}_{B_{l}}[\\mathcal {L}_{B_{l}}(w_{B_{l}}^{n_{l}}) - \\mathcal {L}_{B_{l}}(w^{*})] & \\nonumber \\\\& \\le \\frac{\\alpha ^{2} L M}{2 \\gamma c} + (1 - \\gamma c)^{n_{l}}\\mathrm {E}_{B_{l}}\\Big [\\mathcal {L}(w_{B_{l}}^{0})- \\mathcal {L}(w^{*}) - \\frac{\\alpha ^{2} L M}{2 \\gamma c}\\Big ] & \\nonumber \\\\& \\stackrel{\\text{(a)}}{\\le }\\frac{\\alpha ^{2} L M}{2 \\gamma c} + \\frac{1}{B_{d}}(1 - \\gamma c)^{n_{l}}\\sum _{l= 0}^{B_{d} - 1} (1 - \\gamma c)^{l n_{p}} \\mathrm {E}_{B_{l}}\\Big [\\mathcal {L}_{B_{d} -l}(w_{B_{d} - l}^{n_{p}}) - \\mathcal {L}_{B_{d} -l}(w^{*}) - \\frac{\\alpha ^{2} L M}{2 \\gamma c}\\Big ], &$ where (a) arises from plugging (REF ) in (REF ) with $B = B_{d}$ .",
"This is (REF ) in Theorem REF , concluding the proof."
],
[
"Proof of Corollary 1",
"Defining for all $t = 1,\\dots , B_{d}$ , the optimum solution $\\Delta w_{b}^{*} = \\operatornamewithlimits{arg\\,min}_{w} \\Delta \\mathcal {L}_{b}(w)$ , we can write $\\Delta \\mathcal {L}_{b}(\\Delta w_{b}^{*}) \\le \\Delta \\mathcal {L}_{b}(w^{*})$ , and hence also the inequality $&\\Delta \\mathcal {L}_{b}(w_{b}^{n_{p}}) - \\Delta \\mathcal {L}_{b}(w^{*}) \\le \\Delta \\mathcal {L}_{b}(w_{b}^{n_{p}}) - \\Delta \\mathcal {L}_{b}(\\Delta w_{b}^{*}).&$ Writing the Lipschitz continuity property of the gradients REF with $\\nabla (\\Delta \\mathcal {L}_{b}(\\Delta w_{b}^{*})) = 0$ and REF , we have $\\Delta \\mathcal {L}_{b}(w_{b}^{n_{p}}) - \\Delta \\mathcal {L}_{b}(\\Delta w_{b}^{*}) \\le \\frac{LD^{2}}{2}$ .",
"Using a similar argument, we can write $\\mathcal {L}_{b}(w_{b}^{n_{p}}) - \\mathcal {L}_{b}( w_{b}^{*}) \\le \\frac{LD^{2}}{2}$ , where $w_{b}^{*} = \\operatornamewithlimits{arg\\,min}_{w} \\mathcal {L}_{b}(w)$ .",
"Plugging this into (REF ), we obtain the inequality $& \\mathrm {E}_{B}[\\mathcal {L}(w_{B}^{n_{p}}) - \\mathcal {L}(w^{*})] \\le \\frac{\\alpha ^{2} L M}{2 \\gamma c} \\frac{(B- 1)}{B_{d}} + \\Big (1 - \\frac{(B - 1)}{B_{d}} \\Big )\\frac{LD^{2}}{2} + \\frac{1}{B_{d}}\\sum _{l=1}^{B-1} (1 - \\gamma c)^{l n_{p}}\\Big [\\frac{LD^{2}}{2}- \\frac{\\alpha ^{2} L M}{2 \\gamma c}\\Big ], &$ which is (REF ) in Corollary REF .",
"Following the same approach with (REF ), we obtain $& \\mathrm {E}_{B_{l}}[\\mathcal {L}(w_{B_{l}}^{n_{l}}) - \\mathcal {L}(w^{*})] \\le \\frac{\\alpha ^{2} L M}{2 \\gamma c} + \\frac{1}{B_{d}}(1 - \\gamma c)^{n_{l}}\\sum _{l= 0}^{B_{d} - 1} (1 - \\gamma c)^{l n_{p}} \\Big [\\frac{LD^{2}}{2} - \\frac{\\alpha ^{2} L M}{2 \\gamma c}\\Big ], &$ which is (REF ) in Corollary REF , completing the proof."
]
] | 1906.04488 |
[
[
"Interpreting OWL Complex Classes in AutomationML based on Bidirectional\n Translation"
],
[
"Abstract The World Wide Web Consortium (W3C) has published several recommendations for building and storing ontologies, including the most recent OWL 2 Web Ontology Language (OWL).",
"These initiatives have been followed by practical implementations that popularize OWL in various domains.",
"For example, OWL has been used for conceptual modeling in industrial engineering, and its reasoning facilities are used to provide a wealth of services, e.g.",
"model diagnosis, automated code generation, and semantic integration.",
"More specifically, recent studies have shown that OWL is well suited for harmonizing information of engineering tools stored as AutomationML (AML) files.",
"However, OWL and its tools can be cumbersome for direct use by engineers such that an ontology expert is often required in practice.",
"Although much attention has been paid in the literature to overcome this issue by transforming OWL ontologies from/to AML models automatically, dealing with OWL complex classes remains an open research question.",
"In this paper, we introduce the AML concept models for representing OWL complex classes in AutomationML, and present algorithms for the bidirectional translation between OWL complex classes and their corresponding AML concept models.",
"We show that this approach provides an efficient and intuitive interface for nonexperts to visualize, modify, and create OWL complex classes."
],
[
"Introduction",
"The World Wide Web Consortium (W3C) has published several recommendations for building ontologies, with the Resource Description Framework (RDF) and the Web Ontology Language (OWL) being the most popular ones.",
"OWL was designed as an extension of RDF with significant more expressivity and is preferred as a language for conceptual modeling in complex domains.",
"The reasoning facilities of OWL can, therefore, be used to support decision making in the domain of interest.",
"The Automation Markup Language (AutomationML, or AML) is a neutral, XML-based data format for data exchange between engineering tools [1].",
"AML is standardized as IEC 62714 and has its root in the data format CAEX (IEC 62424).",
"AML supports the modeling of plant topology, component structure, geometry and kinematics, logic behavior, and communication networks.",
"However, AML per se does not provide a formal semantics for automated data interpretation [2].",
"In practice, tools need to achieve a common understanding of the data and be responsible for the preservation of semantics.",
"Efforts have been made on adopting OWL and its reasoning facilities for processing AML data.",
"The typical approach comprises three steps: a) transform engineering data stored in an AML document to an AML ontology by explicitly define the semantics of AML notions; b) after communication with the domain experts, an ontology expert extends the AML ontology with additional knowledge for specific engineering purposes; c) utilizing the reasoner for providing advanced engineering services.",
"For example, with predefined ontological descriptions about error types in plant models, Abele et al.",
"were able to identify modeling errors in the plant topology [3].",
"Hua et al.",
"proposed a model-driven robot programming approach that is capable of inferring component capability and the associated programming interfaces from AML models [4].",
"In this paper, we use the term AML ontology to indicate an OWL ontology that is converted from an AML document.",
"It is evident that the approaches mentioned above are based on sophisticated domain knowledge that is modeled as OWL complex classes by nesting logic-based OWL constructors.",
"Therefore, a profound understanding of the domain and the language OWL is required.",
"Recently, Hildebrandt et al.",
"proposed the domain expert-centric approach for building ontologies of cyber-physical systems [5].",
"While this approach tackles the problem of incorporating domain expert's knowledge, it is unclear yet how to deal with OWL complex classes.",
"In the remainder of the paper, we use the term OWL complex class and the term OWL class interchangeably if the context is clear.",
"In this paper, we introduce the AML concept model for representing ontological semantics in native AML models.",
"Based on a bidirectional translation procedure between OWL and AML, OWL classes can be visualized as AML concept models for inspection and modification, and proper AML concept models can be transformed to OWL classes while preserving the ontological semantics.",
"We show that this approach demonstrates an efficient and intuitive interface for a non-expert to interact with OWL complex classes.",
"This paper is organized as follows.",
"Section discusses related work on model transformation between OWL and AML.",
"Section gives a brief overview of OWL and AML, and introduces the important notions used in this paper.",
"In section we present the AML concept model that is developed for preserving ontological semantics in AML.",
"In section we describe the bidirectional translation between OWL complex classes and AML concept models.",
"Finally, we demonstrate the utility of this approach with two typical use cases of ontology engineering in section and conclude the paper with future works in section ."
],
[
"Related Work",
"The first result about converting AML to OWL appeared in 2009 by Runde et al.",
"in their German paper [6].",
"Two approaches were proposed and discussed.",
"The abstract approach represents the CAEX vocabulary directly as OWL classes in the ontology and transforms CAEX classes, objects and attributes as individuals of these OWL classes.",
"The concrete approach generates an OWL class for each CAEX class with an annotation about its original type in the CAEX schema.",
"For example, an AML role class $\\mathsf {Robot}$ will be converted to an OWL class with the annotation $\\mathrm {RoleClass}$ .",
"Subsequent researches generally follow either the abstract or the concrete approach.",
"For example, Kovalenko et al.",
"proposed a lightweight ontology for covering core concepts of CAEX using the abstract approach [7], while Hua et al.",
"followed the concrete approach for learning unknown engineering concepts from AML data [2].",
"The backward transformation from OWL to AML is less studied, although the first approach was already published in 2010 in [8].",
"The transformation begins with mapping atomic OWL classes to appropriate CAEX classes using the CAEX type annotation of each OWL class.",
"It proceeds with OWL individuals of the top level OWL classes and transforms them into proper CAEX objects.",
"Then the transformation handles each property associated with the individuals until all information in OWL is processed.",
"It is evident that existing methods only target at \"simple\" knowledge types, that is, atomic classes, objects, and properties.",
"For handling complex ontological knowledge, e.g.",
"OWL complex classes, one challenge arises that no regular AML model can preserve complex ontological semantics.",
"In the remainder of the paper, we assume that we are given an AML ontology converted from an AML document following the approach proposed in [2].",
"Our goal is to develop a modeling approach that enhances native AML models with ontological semantics and a translation procedure between such native AML models and OWL complex classes."
],
[
"AutomationML",
"AML data is stored in an XML document which conforms to the underlying CAEX XML schema.",
"An AML document usually contains a set of class libraries and a structured collection of engineering objects that represents the plant topology.",
"We emphasize the following core concepts of CAEX that we consider in this paper: Role class ($\\mathbf {RC}$ ): a role class refers to a type of engineering objects, e.g.",
"$\\mathsf {Robot}$ .",
"As AML follows the object-oriented paradigm, role classes can be organized in inheritance hierarchies within so-called role class libraries.",
"Interface class ($\\mathbf {IC}$ ): an interface class represents a type of engineering interfaces, e.g.",
"$\\mathsf {SignalInterface}$ or $\\mathsf {AttachmentInterface}$ .",
"Similar to role classes, inheritance is allowed between interface classes and an interface class library stores a set of interface classes.",
"Internal element ($\\mathbf {IE}$ ): an internal element is the model of an engineering object, e.g.",
"a joint inside a robot or a real robot in the plant.",
"By referring to a role class, the meaning of an internal element is declared.",
"For describing the plant topology, internal elements are organized as tree structures in the instance hierarchy.",
"External interface ($\\mathbf {EI}$ ): an external interface is the model of an engineering interface, e.g.",
"an IO pin of a controller.",
"The type of an external interface is defined by referring to an interface class.",
"System unit class ($\\mathbf {SUC}$ ): a system unit class is a reusable engineering template that contains an internal structure, where internal elements are used to represent individual parts of the structure.",
"For all the concepts mentioned above, CAEX attributes can be defined to describe their properties.",
"In the rest of the paper, we use the notion AML model to refer to any XML model that can be generated according to the CAEX schema."
],
[
"OWL",
"OWLWhile OWL is the short name of the Web Ontology Language whose expressive power goes beyond the scope of description logics, we use this notion to refer to the specific sub-language OWL 2 DL.",
"belongs to the family of expressive Description Logics (DL) and is closely related to $\\mathcal {SROIQ}$ [9].",
"An OWL ontology defines a finite set of classes (e.g.",
"$\\mathsf {Robot}$ ), individuals (e.g.",
"a robot instance) and properties in a domain of discourse, and describes relations between these artifacts.",
"One further distinguishes between object properties and data properties.",
"The former one is used for relations between individuals (e.g.",
"a robot has a controller), and the latter one is for describing the concrete qualities of an individual (e.g.",
"weight of a robot).",
"Although an AML ontology generated by [2] has merely two object properties $\\mathsf {hasIE}/\\mathsf {hasEI}$ , we also consider the following inverse properties in this paper: $\\mathsf {isIEOf} \\equiv \\mathsf {hasIE^-}, \\mathsf {isEIOf} \\equiv \\mathsf {hasEI^-}$ An OWL class is either an atomic class or a complex one when it is generated by so-called concept constructors [9].",
"Table REF shows the concept constructors of OWL, their correspondences in the terminology of DL, their DL syntaxPlease refer to [10] for more details of the DL syntax., and their formal model-theoretic semantics in OWL.",
"We use conventional notions for the syntax: $A$ represents an atomic class, $C$ or $D$ stands for an OWL (complex) class, $R$ stands for an OWL property, $a$ or $b$ stands for an OWL individual, $DR$ is used for the data range of data properties, and $lt$ is used for a literal value.",
"The nested OWL class $C$ inside a restriction e.g.",
"$\\exists R.C$ is called the filler of the restriction.",
"Table: Syntax and semantics of OWL constructorsIn this paper, we consider an OWL complex class constructed by using arbitrarily many of the constructors in the covered part of Table REF , and assume that no constructors in the uncovered part would appear.",
"While it seems to be a strong assumption, we argue that: a) since CAEX attributes are mapped to data properties and one CAEX attribute is usually assigned to an object only once, the universal, at-least and at-most restrictions on data properties can be omitted; b) the local reflexivity cannot appear in an AML ontology, since we can not have an internal element (or external interface) which is the internal element (or external interface) of itself.",
"Formula REF shows some examples of OWL classes used in this paper.",
"Class A refers to $\\mathsf {Robot}$ s without any internal element of the type $\\mathsf {\\lnot IOController}$ .",
"Class B refers to internal elements of a $\\mathsf {Robot}$ from the manufacturer KUKA.",
"Class C refers to $\\mathsf {Robot}$ s with an $\\mathsf {IOController}$ that has at least three $\\mathsf {IOInterface}$ s. Class D refers to $\\mathsf {IOInterface}$ s from objects that have at least three $\\mathsf {IOInterface}$ s. Apparently, as the complexity grows, the intended meaning of an OWL complex class becomes more difficult to understand.",
"In the next section, we introduce the AML concept model that is able to represent OWL complex classes as native AML models.",
"$&\\mathsf {A} \\equiv \\mathsf {Robot \\sqcap \\lnot \\exists hasIE.",
"(\\lnot IOController)} \\nonumber \\\\&\\mathsf {B} \\equiv \\mathsf {\\exists isIEOf.",
"(Robot \\sqcap hasManufacturer.",
"\"KUKA\"}) \\nonumber \\\\&\\mathsf {C} \\equiv \\mathsf {Robot \\sqcap \\exists hasIE.",
"(IOController \\,\\, \\sqcap \\ge 3 hasEI.IOInterface}) \\nonumber \\\\&\\mathsf {D} \\equiv \\mathsf {IOInterface \\sqcap \\exists isEIOf.",
"(\\ge 3 hasEI.IOInterface})$"
],
[
"The AML Concept Model",
"Consider the OWL class constructors in Table REF .",
"It is evident that while atomic classes can be represented as AML classes directly [2], most of the features in OWL are not supported by AML.",
"Therefore, we propose the following approach to represent OWL class constructors as dedicated AML models: Atomic class: similar to the conventional translation procedure as proposed by [8], an atomic class is represented by a CAEX role or interface class.",
"A class reference in CAEX is therefore equivalent to a class assertion in OWL.",
"For example, an internal element $a$ of the role class $A$ is represented as $A(a)$ .",
"Thing: Thing is the most general concept in OWL and contains all individuals.",
"Therefore, it is represented by a CAEX object with no specific configurations.",
"Nothing: Nothing is the most specific concept in OWL and contains no individual.",
"Nothing is handled as the complement of Thing (see the complement case below).",
"Intersection: an intersection $C \\sqcap D$ contains individuals that are instances of all the operands $C$ and $D$ in the intersection.",
"Therefore, an intersection is represented by the composition of several AML models that correspond to each of the operands, including CAEX class references, attributes, and subordinate object structures.",
"Union: a union $C \\sqcup D$ contains individuals that are instances of at least one operand $C$ or $D$ of the union.",
"XML does not support unions in general.",
"In this paper, we handle each operand of a union separately and generate one AML model for each of them.",
"Nominal: a nominal $\\lbrace a, b, ...\\rbrace $ enumerates all individuals that an OWL class shall contain.",
"Similar to the union constructor, nominals cannot be directly represented in XML, and we generate one AML model for each element inside a nominal.",
"Existential restriction: an existential restriction $\\exists R.C$ or $\\exists R.(DR)$ states the existence of the relation $R$ with the filler $C$ or the data range $DR$ .",
"If $R$ is an object property, the existential restriction is represented by a child object (internal element or external interface) while the filler $C$ is represented by the model of the child object.",
"If $R$ is a data property, the existential restriction is represented by a CAEX attribute while the data range $DR$ is represented by the configuration of the CAEX attribute, e.g.",
"data type and value requirements.",
"Object cardinality restrictions: an object cardinality restriction, i.e.",
"an exact restriction $= n R.C$ , an at-least restriction $\\ge n R.C$ , or an at-most restriction $\\le n R.C$ , defines the number of child objects of the class $C$ w.r.t.",
"the relation $R$ .",
"The CAEX attributes minCardinality and maxCardinality are added to the child objects to represent the minimum and maximum number respectively.",
"The exact cardinality of $n$ is represented by $minCardinality = minCardinality = n$ .",
"Fills restriction: a fills restriction $\\exists R.\\lbrace a\\rbrace $ or $\\exists R.\\lbrace lt\\rbrace $ corresponds to an existential restriction with a Singleton filler.",
"If $R$ is an object property, the CAEX attribute isIdentifiedByID is used to restrict the ID of the child object, as ID is unique in AML.",
"If $R$ is a data property, $lt$ is set as the required value of the corresponding CAEX attribute.",
"Universal restriction: a universal restriction $\\forall R.C$ forces all child objects w.r.t.",
"the relation $R$ to be instances of the class $C$ .",
"For example, $\\mathsf {\\forall hasIE.C}$ describes things that have internal elements of type $\\mathsf {C}$ only.",
"While universal restrictions can not be directly represented in XML, it can be simulated by disallowing child objects that are instances of the class $\\lnot C$ [11] using the exact cardinality $= 0 \\, R.(\\lnot C)$ .",
"Complement: a complement $\\lnot C$ contains all individuals that are not instances of $C$ .",
"Since an OWL class can have arbitrarily nested complements, we first transform an OWL class to its negation normal form (NNF) so that complements are only bound to atomic classes [12].",
"For example, the NNF of the OWL class A in Formula REF is: $\\mathsf {NNF(A) \\equiv Robot \\sqcap \\forall hasIE.IOController}$ Obviously, $\\mathsf {NNF(A)}$ does not contain any complements.",
"In fact, complements can only appear in the following three cases in the NNF of an OWL class: (a) A complement can be bound to an atomic class as $\\lnot A$ or a data range as $\\lnot DR$ , and is not part of any restrictions.",
"In this case, a CAEX attribute negated=true is added to the AML model.",
"Note that intersections of a mixture of positive and negative atomic classes, e.g.",
"$\\lnot A_1 \\sqcap A_2$ , cannot be modeled in AML.",
"(b) A complement can be the filler of an existential restriction, i.e.",
"$\\exists R.(\\lnot A)$ or $\\exists R.(\\lnot DR)$ .",
"As with the existential restriction, a child CAEX object or CAEX attribute is first generated.",
"Then the CAEX attribute negated=true is added to the child model.",
"(c) A complement can be the filler of a universal restriction as $\\forall R.(\\lnot A)$ (recall that we ignore universal restrictions on data properties).",
"In this case, we disallow child objects of the class $A$ w.r.t.",
"the relation $R$ , which can be expressed using the exact cardinality $= 0 \\, R.A$ .",
"Table REF summarizes the introduced CAEX attributes that are used to capture the semantics of OWL constructors mentioned above.",
"We call them concept attributes.",
"The attribute primary is a helper flag to indicate which element in an AML model is described by the OWL class.",
"We call an AML model with concept attributes as an AML concept model and enumerate the values of concept attributes based on possible forms of NNF in Table REF .",
"Intuitively, AML concept models can be nested to represent nested OWL class expressions.",
"An AML concept model is proper if it has exactly one primary element.",
"Note that intersections, unions, and nominals are omitted in the mapping since we handle each element of them individually.",
"Table: The AML concept attributes for capturing ontological semantics.Table: Mapping between OWL constructors and AML concept attributes.Figure REF illustrates the AML concept models of the NNF of the OWL classes A, B, C and D in Formula REF as tree structures.",
"Internal elements (IE) and external interfaces (EI) are represented by tree nodes, and their class references and attributes are depicted as labels on the top right corner.",
"A negated object is marked as red.",
"The primary object is marked as bold with an underline.",
"Numbers in square brackets are the min and max cardinality of the object, while a value $-1$ means that it is unlimited.",
"Note that for the classes B and D, the primary object is not the root node since XML cannot depict \"part-of\" relations (i.e.",
"$\\mathsf {isIEOf, isEIOf}$ ).",
"Therefore, each inverse property is simulated as a predecessor node in the XML tree.",
"Figure: The AML concept models for the OWL classes in Formula ."
],
[
"Translation between OWL and AML",
"The core idea of the translation is to exploit the tree structure of OWL class expressions.",
"More concretely, we introduce AML concept trees that depict OWL complex classes in a tree structure similar to AML concept models.",
"Then we describe the forward translation $\\mathrm {TransF: OWL \\mapsto AML}$ via the AML concept trees.",
"Finally, we show that the backward translation $\\mathrm {TransB: AML \\mapsto OWL}$ can be directly carried out using the mappings in Table REF ."
],
[
"From OWL to AND-tree",
"We define a tree conventionally as a directed graph $\\mathcal {G = (V, E)}$ where $\\mathcal {V}$ is a finite set of nodes and $\\mathcal {E}$ is a finite set of edges, to which the following rules apply: A tree $\\mathcal {G}$ has a unique root node that has no predecessor.",
"Each node $n \\in \\mathcal {V}$ has a unique predecessor.",
"We call leaf nodes the tree nodes that have no successor, i.e.",
"at the bottom of the tree.",
"Furthermore, a branching node is an inner tree node that has a unique predecessor and arbitrarily many successors.",
"Based on these notions, an AND-tree is a tree with the following properties: The root of an AND-tree represents the expression of an OWL complex class.",
"Each branching node of an AND-tree represents either an intersection or a restriction (see the notions in Table REF ).",
"Each leaf node of an AND-tree represents either OWL Thing, OWL Nothing or an atomic class.",
"For each OWL complex class without unions and inverse properties, an AND-tree can be constructed by making a successor node for each operand of an intersection and the filler of a restriction, as shown in Algorithm REF .",
"[tbp] $\\mathrm {Construct}$ [1] The class expression $ce$ of an OWL class $C$ A tree node $root$ make a tree node $root$ for $ce$ $ce$ is an atomic class $root$ ($ce$ is an intersection) $operand \\in ce$ let $child = $ Construct($operand$ ) add $child$ as a successor to $root$ ($ce$ is a restriction) let $child = $ Construct($ce.filler$ ) add $child$ as a successor to $root$ $root$ Figure: The AND-tree constructed from the OWL class D in Formula .",
"The numbers in the tree nodes show the sequence of node construction.We illustrate the construction process in Figure REF .",
"Each box represents a tree node, and the number on the upper left corner of each box shows the sequence of node construction.",
"The root node of the AND-tree corresponds to the OWL class D in Formula REF .",
"Since the root is an intersection, the algorithm will handle each operand of it individually through line 4 to 6.",
"The atomic operand $\\mathsf {IOInterface}$ is returned directly and added as a child to the root in line 7.",
"For the complex operand $\\mathsf {\\exists isIEOf.",
"(\\ge 3 hasEI.IOInterface)}$ , the algorithm recursively generates sub-nodes until the final atomic filler $\\mathsf {IOInterface}$ is reached in line 10.",
"Note that all nodes are generated immediately in line 1 when $\\mathrm {Construct}$ is called.",
"It becomes more involved if the OWL class $C$ contains any disjunctions (unions or nominals) because XML does not support or statements generally.",
"The solution is to construct $m$ AND-trees for a disjunction with $m$ elements.",
"However, since disjunctions can appear in any nested part inside an OWL class expression, we need to traverse the logical structure of the class expression to produce a set of AND-trees that is logically equivalent to the OWL class.",
"[htbp] $\\mathrm {ConstructD}$ [1] The class expression $ce$ of an OWL class $C$ A set of tree nodes $roots$ initialize $roots = \\lbrace \\rbrace $ $ce$ is an union or a nominal each $element$ in $ce$ add ConstructD($element$ ) to $roots$ make a tree node $n$ for $ce$ , add $n$ to $roots$ $ce$ is an atomic class $roots$ ($ce$ is an intersection) $operand \\in ce$ let $nestedTrees = $ ConstructD($operand$ ) $root$ in $roots$ copy $root$ $nestedTrees.size-1$ times add the root of each $tree \\in nestedTrees$ as a successor to exactly one copy of $root$ ($ce$ is a restriction) let $nestedTrees = $ ConstructD($ce.filler$ ) $root$ in $roots$ copy $root$ $nestedTrees.size-1$ times add the root of each $tree \\in nestedTrees$ as a successor to exactly one copy of $root$ $roots$ Algorithm REF shows the AND-tree construction process for classes involving disjunctions.",
"If the input class expression $ce$ is a disjunction, then a set of tree nodes are generated for the elements of the disjunction (line 4).",
"In case the input is an intersection, the recursive call of $\\mathrm {ConstructD}$ in line 12 will handle possible nested disjunction in each element and produce a set of nested trees.",
"These nested trees need to be multiplexed with the existing trees in $roots$ through line 13 to 15.",
"The algorithm treats restrictions similarly to intersections despite that the filler of a restriction is used to produce nested trees in line 19.",
"It is worth noting that only $m-1$ copies of $root$ are made in line 14 and 21 since the original $root$ also counts during the construction.",
"Figure REF illustrates the tree construction process of the OWL class $\\mathsf {Robot \\sqcap \\exists hasIE.",
"(IOController \\sqcup IODevice)}$ .",
"In the first step, a root node is generated that contains the complete class expression (line 7).",
"Then, for each operand of the intersection, a child node is generated in step 2 and 3 (line 12).",
"Since the $\\mathsf {Robot}$ node is atomic, no further construction is required in the recursive call (line 9).",
"On the other hand, the restriction node $\\mathsf {\\exists hasIE.",
"(IOController \\sqcup IODevice)}$ is copied in step 4 (line 21), since its filler is a union and produces two atomic nodes $\\mathsf {IOController}$ and $\\mathsf {IODevice}$ (line 19).",
"In step 5 and 6, the atomic nodes are added to the original and copied restriction nodes (line 22).",
"Finally, the root node $\\mathsf {Robot \\sqcap \\exists hasIE.",
"(IOController \\sqcup IODevice)}$ is copied once to accept the two distinct restriction nodes in step 7 (line 14-15).",
"Figure: The tree construction process of the OWL complex class 𝖱𝗈𝖻𝗈𝗍⊓∃𝗁𝖺𝗌𝖨𝖤.",
"(𝖨𝖮𝖢𝗈𝗇𝗍𝗋𝗈𝗅𝗅𝖾𝗋⊔𝖨𝖮𝖣𝖾𝗏𝗂𝖼𝖾)\\mathsf {Robot \\sqcap \\exists hasIE.",
"(IOController \\sqcup IODevice)}.",
"The numbers in the tree nodes show the sequence of node construction."
],
[
"Working with Inverse Properties",
"For OWL classes that describe objects in the instance hierarchy, inverse properties might appear for gathering information about their ancestors or siblings (see the OWL classes B and D in Formula REF ).",
"Due to structural restrictions in AML, we assume that the following conditions hold when an inverse property $R^- \\in \\mathsf {\\lbrace isIEOf, isEIOf\\rbrace }$ appear: [itemindent=1em] C1: $R^-$ does not appear in the filler of any restriction that has $R$ as property, e.g.",
"$\\exists R.(\\exists R^-.C)$ .",
"C2: $R^-$ does not appear in the filler of cardinality restrictions, e.g.",
"$\\ge n \\,\\, R^-.C$ .",
"C3: $R^-$ does not appear in the filler of any restriction that has a different property $R^{\\prime } \\ne R$ , e.g.",
"$\\exists R^{\\prime }.",
"(\\exists R^-.C)$ .",
"C4: $\\mathsf {isEIOf}$ does not appear in the filler of any restrictions that has an inverse property, e.g.",
"$\\mathsf {\\exists R^-.",
"(\\exists isEIOf.C)}$ Figure: The construction of the AML concept tree of class D and the conversion to its AML concept model.",
"The numbers in the tree nodes show the sequence of node construction.",
"The orange dashed lines show the mapping between nodes in the AML concept tree and the AML concept model.The conditions C1 and C2 avoid modeling redundancies in OWL, since AML data has a tree structure, and each node in the tree has a unique predecessor.",
"A class expression $\\exists R.\\exists R^-.C$ is therefore logically equivalent to $C$ , and a cardinality restriction is redundant to an existential restriction.",
"The condition C3 avoids modeling errors in OWL since the set of internal elements is disjoint with the set of external interfaces.",
"The condition C4 holds since external interfaces have no child object in AML.",
"We call an OWL class that meets the conditions C1-C4 as a proper AML class.",
"The inverse properties of a proper AML class always appear continuously at the outermost layer of the class expression.",
"In other words, the AND-tree of a proper AML class has all inverse properties in the upper part of the tree.",
"Therefore, Algorithm REF iteratively removes the inverse properties from the root of an AND-tree.",
"We call an AND-tree that contains no disjunctions nor inverse properties as an AML concept tree.",
"[htbp] $\\mathrm {removeInverseProperty}$ [1] The root of an AND-tree $root$ The root of a new AND-tree $newRoot$ let $ce =$ class expression in $root$ $ce$ contains no (nested) inverse property $root$ $ce$ is a restriction construct a new node $newRoot$ for $ce.filler$ change the filler of $root$ to owl:Thing add $root$ as a successor of $newRoot$ move $root.children$ as sucessors of $newRoot$ $ce$ is an intersection construct a template node $newRoot$ let $inv$ be successors of $root$ with inverse property let $normal$ be other successors of $root$ construct a new node $normalChild$ as an existential restriction with $normal$ being its filler add $normalChild$ as a successor to $newRoot$ add the expression of $normalChild$ to $newRoot$ $node \\in inv$ add the filler of $node$ to $newRoot$ conjunctively move $node.child$ as a successor of $newRoot$ RemoveInverseProperty($newRoot$ ) Figure REF shows how the inverse property in the root of class D's AND-tree is removed.",
"Since the original root node is an intersection, the algorithm first constructs a template node for the new root (line 11).",
"Then a new child node is constructed for the previous child $\\mathsf {IOInterface}$ by formulating an existential restriction in step 2 (line 14 to 15).",
"To keep the consistency of the tree, the expression of the new child node is added to the new root node in the third step (line 16).",
"For the previous child $\\mathsf {\\exists isEIOf.",
"(\\ge 3 hasEI.IOInterface)}$ with the inverse property $\\mathsf {isEIOf}$ , the filler $\\mathsf {\\ge 3 hasEI.IOInterface}$ is added to the new root node as a conjunctive term in step 4 (line 18), and the corresponding grandchild with its sub-tree is added as a child to the new root in step 5 (line 19).",
"It is obvious that the inverse property $\\mathsf {isEIOf}$ is now removed.",
"Note that the OWL class expression of the new root node is different from the original one.",
"Informally, the original root describes the primary object in an arbitrary position of the CAEX instance hierarchy (marked as yellow), while the new root describes the predecessor of the primary object."
],
[
"The Forward Translation: from OWL to AML",
"Until now we have shown the algorithms to transform a proper AML class into an AML concept tree.",
"The forward translation $\\mathrm {TransF: OWL \\mapsto AML}$ can be implemented by traversing AML concept trees in a depth-first manner.",
"For every tree node, we generate a corresponding AML concept model whose concept attributes are configured based on the mappings in Table REF .",
"The CAEX type of the target AML concept model is determined either by the object property being used in case of a restriction or by the CAEX type annotation of the OWL class in case of an intersection in the root node.",
"The orange dashed lines in Figure REF show the translation from the AML concept tree of class D to its AML concept model illustrated in Figure REF .",
"Recall that OWL atomic classes are mapped to CAEX class references."
],
[
"The Backward Translation: from AML to OWL",
"If an AML concept model is proper, i.e.",
"it has exactly one primary element (see section ), then the backward translation $\\mathrm {TransB: AML \\mapsto OWL}$ can be directly carried out using the mappings in Table REF .",
"First, a traverse of the AML concept model is necessary to localize the primary object.",
"Afterwards, successors of the primary object are translated to restrictions with normal properties while the predecessors are translated to restrictions with inverse properties.",
"If Algorithm REF would have generated several AML concept models during the forward translation, they are translated independently to several OWL classes and combined disjunctively as a union.",
"In this case, an original OWL class with nested unions will be reproduced as a union of expressions, e.g.",
"$\\mathsf {\\exists r.(C \\sqcup D) \\rightarrow \\exists r.C \\sqcup \\exists r.D}$ .",
"It is worth noting that the mappings in Table REF are used for both $\\mathrm {TransF}$ and $\\mathrm {TransB}$ .",
"Therefore, the forward and backward translation are inverse functions of each other in terms of semantic equivalence.",
"That means, for an OWL class $C$ and an AML concept model $M$ , we have: $\\begin{split}&\\mathrm {TransB(TransF}(C)) \\equiv C \\\\&\\mathrm {TransF(TransB}(M)) \\equiv M\\end{split}$"
],
[
"Use Cases",
"We have implemented the AML concept model and the bidirectional translation in Java.",
"The source code and examples can be found in the GitHub repositoryhttps://github.com/kit-hua/ETFA2019.",
"To demonstrate the use cases of the proposed approach, we discuss two typical scenarios in ontology engineering (Figure REF ).",
"For editing AML models, we recommend the AML editorhttps://www.automationml.org/o.red.c/dateien.html.",
"Figure: The work flow for ontology engineering using bidirectional translation.In the first use case (orange arrows in Figure REF ), the required OWL class does not exist yet, and a user wants to create an AML concept model for the concept in mind: The user generates the primary AML concept model for the target concept, i.e.",
"a CAEX role class, system unit class, interface class, internal element or external interface with class reference and concept attributes.",
"The user adds CAEX attributes and sub-elements with sufficient constraints to the model.",
"This process repeats recursively for nested attributes and sub-elements.",
"If the primary AML concept model shall be further restricted by the properties of its predecessor or siblings, a parent AML concept model is generated.",
"This process repeats recursively for further predecessors and siblings.",
"The user generates the OWL class using the backward translation and adds it to the AML ontology.",
"The second use case (blue arrows in Figure REF ) refers to the ontology evolution procedure in which OWL complex classes already exist in an AML ontology.",
"These classes might be modeled by an ontology expert or created using the AML editor as described above.",
"Now the user might want to inspect a particular OWL class and modify it by demand.",
"First, the chosen OWL class is translated into AML concept models via its AML concept trees.",
"Then, the user can open the generated AML concept models in the AML editor and inspect them by browsing their XML structure.",
"If any modification is necessary, the user can edit the AML concept models as described above and export the new one to an OWL class.",
"In conclusion, AML concept models can be inspected and modified using the AML editor, while the forward and backward translation are transparent to the user.",
"By comparing the OWL complex classes in Formula REF and their corresponding AML concept models in Figure REF , we believe that the two use cases demonstrate an intuitive and efficient interaction with OWL complex classes.",
"Because the forward and backward translations are inverse functions of each other (see Formula REF in section REF ), Figure REF also illustrates that a round-trip engineering of OWL complex classes is possible by following the work flow of both use cases successively."
],
[
"Conclusion",
"In this paper, we studied the problem of interpreting OWL complex classes from an AML ontology.",
"We identified the inadequacy of existing approaches and introduced a native AML based approach for visualizing, editing and creating OWL complex classes.",
"More specifically, we presented the AML concept model that is capable of carrying ontological semantics, and a bidirectional translation procedure for the conversion between OWL complex classes and AML concept models.",
"With two typical use cases in ontology engineering, we demonstrated the utility of the proposed approach.",
"Future works are considered in two aspects.",
"First, the semantic expressivity of the AML concept model is restricted by the object properties $\\mathsf {hasIE, hasEI}$ and can be extended to cover further modeling facilities in AML, e.g.",
"connections between objects.",
"Second, the current implementation does not provide a friendly user interface and can be improved by integrating the translation procedure into the AML Editor."
],
[
"Acknowledgment",
"This work has been supported from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 688117 “Safe human-robot interaction in logistic applications for highly flexible warehouses (SafeLog)”."
]
] | 1906.04240 |
[
[
"Modeling Sentiment Dependencies with Graph Convolutional Networks for\n Aspect-level Sentiment Classification"
],
[
"Abstract Aspect-level sentiment classification aims to distinguish the sentiment polarities over one or more aspect terms in a sentence.",
"Existing approaches mostly model different aspects in one sentence independently, which ignore the sentiment dependencies between different aspects.",
"However, we find such dependency information between different aspects can bring additional valuable information.",
"In this paper, we propose a novel aspect-level sentiment classification model based on graph convolutional networks (GCN) which can effectively capture the sentiment dependencies between multi-aspects in one sentence.",
"Our model firstly introduces bidirectional attention mechanism with position encoding to model aspect-specific representations between each aspect and its context words, then employs GCN over the attention mechanism to capture the sentiment dependencies between different aspects in one sentence.",
"We evaluate the proposed approach on the SemEval 2014 datasets.",
"Experiments show that our model outperforms the state-of-the-art methods.",
"We also conduct experiments to evaluate the effectiveness of GCN module, which indicates that the dependencies between different aspects is highly helpful in aspect-level sentiment classification."
],
[
"Introduction",
"Aspect-level sentiment classification [1], [2] is a fundamental natural language processing task that gets lots of attention in recent years.",
"It is a fine-grained task in sentiment analysis, which aims to infer the sentiment polarities of aspects in their context.",
"For example, in the sentence “The price is reasonable although the service is poor\", the sentiment polarities for the two aspect terms, “price\" and “service\", are positive and negative respectively.",
"An aspect term (or simply aspect) is usually an entity or an entity aspect.",
"Aspect-level sentiment classification is much more complicated than sentence-level sentiment classification, because identifying the parts of sentence describing the corresponding aspects is difficult.",
"Traditional approaches [3], [4] mainly focus on statistical methods to design a set of handcrafted features to train a classifier (e.g., Support Vector Machine).",
"However, such kind of feature-based work is labor-intensive.",
"In recent years, neural network models [5], [6] are of growing interest for their capacity to automatically generate useful low dimensional representations from aspects and their contexts, and achieve great accuracy on the aspect-level sentiment classification without careful engineering of features.",
"Especially, by the ability to effectively identify which words in the sentence are more important on a given aspect, attention mechanisms [7], [8] implemented by neural networks are widely used in aspect-level sentiment classification [9], [10], [11], [12], [13], [14].",
"Chen et al.",
"[10] model a multiple attention mechanism with a gated recurrent unit network to capture the relevance between each context word and the aspect.",
"Ma et al.",
"[11] design a model which learns the representations of the aspect and context interactively with two attention mechanisms.",
"Song et al.",
"[13] propose an attentional encoder network, which employ multi-head attention for the modeling between context and aspect.",
"These attention-based models have proven to be successful and effective in learning aspect-specific representations.",
"Figure: An example to illustrate the usefulness of the sentiment dependencies between multiple aspects.",
"The dependencies can be inferred by some knowledge in the sentence, e.g., conjunction.",
"The evidence of the usefulness of the sentiment dependencies is that we can easily guess the true sentiment of “food\" even if we mask the word “horrible\".Despite these advances, the studies above still remain problems.",
"They all build models with each aspect individually ignoring the sentiment dependencies information between multiple aspects, which will lose some additional valuable information.",
"For example, as we can see from the example given in Fig.",
"REF , the sentiment polarity of the first aspect “setting\" is positive.",
"From the conjunction “but\", we are easy to know that the second aspect “food\" has opposite sentiment polarity with “setting\".",
"By this sentiment dependency relation, we can guess the polarity of aspect “food\" is negative.",
"Similarly, from the second comma, we conjecture that the sentiment polarity of the last aspect “service\" is likely the same as “food\".",
"Therefore, the sentiment dependencies are helpful to infer the sentiment polarities of aspects in one sentence.",
"In this paper, we propose a novel method to model Sentiment Dependencies with Graph Convolutional Networks (SDGCN) for aspect-level sentiment classification.",
"GCN is a simple and effective convolutional neural network operating on graphs, which can catch inter-dependent information from rich relational data [15].",
"For every node in graph, GCN encodes relevant information about its neighborhoods as a new feature representation vector.",
"In our case, an aspect is treated as a node, and an edge represents the sentiment dependency relation of two nodes.",
"Our model learns the sentiment dependencies of aspects via this graph structure.",
"As far as we know, our work is the first to consider the sentiment dependencies between aspects in one sentence for aspect-level sentiment classification task.",
"Furthermore, in order to capture the aspect-specific representations, our model applies bidirectional attention mechanism with position encoding before GCN.",
"We evaluate the proposed approach on the SemEval 2014 datasets.",
"Experiments show that our model outperforms the state-of-the-art methods.Source code is available at https://github.com/Pinlong-Zhao/SDGCN.",
"The main contributions of this paper are presented as follows: To the best of our knowledge, this is the first study to consider the sentiment dependencies between aspects in one sentence for aspect-level sentiment classification.",
"We design bidirectional attention mechanism with position encoding to capture the aspect-specific representations.",
"We propose a novel multi-aspects sentiment classification framework, which employs GCN to effectively capture the sentiment dependencies between different aspects in one sentence.",
"We evaluate our method on the SemEval 2014 datasets.",
"And experiments show that our model achieves superior performance over the state-of-the-art approaches."
],
[
"Related work",
"In this section, we will review related works on aspect-level sentiment classification and graph convolutional network briefly."
],
[
"Aspect-level sentiment classification",
"Sentiment analysis, also known as opinion mining [16], [17], is an important research topic in Natural Language Processing (NLP).",
"Aspect-level sentiment classification is a fine-grained task in sentiment analysis.",
"In aspect-level sentiment classification , early works mainly focus on extracting a set of features like bag-of-words features and sentiment lexicons features to train a sentiment classifier [18].",
"These methods including rule-based methods [19] and statistic-based methods [20] rely on feature-engineering which are labor intensive.",
"In recent years, deep neural network methods are getting more and more attention as they can generate the dense vectors of sentences without handcrafted features [21], [22].",
"And the vectors are low-dimensional word representations with rich semantic information remained.",
"Moreover, using the attention mechanism can enhance the sentence representation for concentrating on the key part of a sentence given an aspect [23], [24], [25].",
"Wang et al.",
"[9] propose ATAE-LSTM that combines LSTM and attention mechanism.",
"The model makes embeddings of aspects to participate in computing attention weights.",
"RAM is proposed by Chen et al.",
"[10] which adopts multiple-attention mechanism on the memory built with bidirectional LSTM.",
"Ma et al.",
"[11] design a model with the bidirectional attention mechanism, which interactively learns the attention weights on context and aspect words respectively.",
"Song et al.",
"[13] propose an attentional encoder network, which eschews recurrence and apply multi-head attention for the modeling between context and aspect.",
"However, these attention works model each aspect separately in one sentence, which may loss some sentiment dependency information on multiple aspects case."
],
[
"Graph convolutional network",
"Graph convolutional network [26] is effective at dealing with graph data which contains rich relation information.",
"Many works dedicate to extending GCN for image tasks [27], [28], [29], [30].",
"Chen et al.",
"[31] build the model via GCN for multi-label image recognition, which propagates information between multiple labels and consequently learns inter-dependent classifiers for each of image labels.",
"GCN has also received growing attention in NLP recently such as semantic role labeling [32], machine translation [33] and relation classification [34].",
"Some works explore graph neural networks for text classification [35], [36].",
"They view a document, a sentence or a word as a graph node and rely on the relation of nodes to construct the graph.",
"The studies above show that GCN can effectively capture relation between nodes.",
"Inspired by these, we adopt GCN to get the sentiment dependencies between multi-aspects."
],
[
"Methodology",
"Aspect-level sentiment classification can be formulated as follows.",
"Given an input context consists of $N$ words $W^c=\\lbrace w^c_1,w^c_2,\\ldots ,w^c_N\\rbrace $ , and $K$ aspect teams $W^a=\\lbrace W^{a_1},W^{a_2},\\ldots ,W^{a_K}\\rbrace $ .",
"Each aspect $W^{a_i}=\\lbrace w^{a_i}_1,w^{a_i}_2,\\ldots ,w^{a_i}_{M_i}\\rbrace $ is a subsequence of sentence $W^c$ , which contains $M_i\\in [1,N)$ words.",
"It is required to construct a sentiment classifier that predicts the sentiment polarities of the multiple aspect teams.",
"We present the overall architecture of the proposed SDGCN in Fig.",
"REF .",
"It consists of the input embedding layer, the Bi-LSTM, the position encoding, the bidirectional attention mechanism, the GCN and the output layer.",
"Next, we introduce all components sequentially from input to output."
],
[
"Input embedding layer ",
"Input embedding layer maps each word to a high dimensional vector space.",
"We employ the pretrained embedding matrix GloVe [37] and pretrained model BERT [38] to obtain the fixed word embedding of each word.",
"Then each word will be represented by an embedding vector $e_t\\in \\mathbb {R}^{d_{emb}\\times 1}$ , where $d_{emb}$ is the dimension of word vectors.",
"After embedding layer, the context embedding is denoted as a matrix $E^c\\in \\mathbb {R}^{d_{emb}\\times N}$ , and the $i$ -th aspect embedding is denoted as a matrix $E^{a_i}\\in \\mathbb {R}^{d_{emb}\\times M_i}$ ."
],
[
"Bidirectional Long Short-Term Memory (Bi-LSTM)",
"We employ Bi-LSTM on top of the embedding layer to capture the contextual information for each word.",
"After feeding word embedding to Bi-LSTM, the forward hidden state $\\overrightarrow{h_t}\\in \\mathbb {R}^{d_{hid}\\times 1}$ and the backward hidden state $\\overleftarrow{h_t}\\in \\mathbb {R}^{d_{hid}\\times 1}$ are obtained, where $d_{hid}$ is the number of hidden units.",
"We concatenate both the forward and the backward hidden state to form the final representation: $&h_t=[\\overrightarrow{h_t},\\overleftarrow{h_t}]\\in \\mathbb {R}^{2d_{hid}\\times 1}&$ In our model, we employ two Bi-LSTM separately to get the sentence contextual hidden output $H^c=[h^c_1,h^c_2,\\ldots ,h^c_N]\\in \\mathbb {R}^{2d_{hid}\\times N}$ and each aspect contextual hidden output $H^a_i=[h^{a_i}_1,h^{a_i}_2,\\ldots ,h^{a_i}_{M_i}]\\in \\mathbb {R}^{2d_{hid}\\times M_i}$ .",
"Note that, the Bi-LSTM for each different aspect shares the parameters."
],
[
"Position encoding",
"Based on the intuition that the polarity of a given aspect is easier to be influenced by the context words with closer distance to the aspect, we introduce position encoding to simulate this normal rules in natural language.",
"Formally, given an aspect $W^{a_i}$ that is one of the $K$ aspects, where $i\\in [1,K]$ is the index of aspects, the relative distance $d^{a_i}_t$ between the $t$ -th word and the $i$ -th aspect is defined as follows: $&d^{a_i}_t={\\left\\lbrace \\begin{array}{ll}1, & \\text{$dis=0$}\\\\1-\\frac{dis}{N}, & \\text{$1\\le dis\\le s$}\\\\0, & \\text{$dis> s$}\\end{array}\\right.",
"}&$ where $dis$ is the distance between a context word and the aspect (here we treat an aspect as a single unit, and $d = 0$ means that the context word is also the aspect word), $s$ is a pre-specified constant, and $N$ is the length of the context.",
"Finally, we can obtain the position-aware representation with position information: $&p^{a_i}_t=d^{a_i}_th^c_t&\\nonumber \\\\&P^{a_i}=P_i=[p^{a_i}_1,p^{a_i}_2,\\ldots ,p^{a_i}_N]&$"
],
[
"Bidirectional attention mechanism",
"In order to capture the interactive information between the context and the aspect, we employ a bidirectional attention mechanism in our model.",
"This mechanism consists of two modules: context to aspect attention module and aspect to context attention module.",
"Firstly, the former module is used to get new representations of aspects based on the context.",
"Secondly, based on the new representations, the later module is employed to obtain the aspect-specific context representations which will be fed into the downstream GCN."
],
[
"Context to aspect attention ",
"Context to aspect attention learns to assign attention weights to the aspect words according to a query vector, where the query vector is $\\overline{h^c}\\in \\mathbb {R}^{2d_{hid}\\times 1}$ which is obtained by average pooling operation over the context hidden output $H^c$ .",
"For each hidden word vector $h^{a_i}_t\\in \\mathbb {R}^{2d_{hid}\\times 1}$ in one aspect, the attention weight $\\beta ^{a_i}_t$ is computed as follows: $&f_{ca}(\\overline{h^c},h^{a_i}_t)=\\overline{h^c}^T\\cdot W_{ca}\\cdot h^{a_i}_t&$ $&\\beta ^{a_i}_t=\\frac{exp(f_{ca}(\\overline{h^c},h^{a_i}_t))}{\\sum ^{M_i}_{t=1}exp(f_{ca}(\\overline{h^c},h^{a_i}_t))}&$ where $W_{ca}\\in \\mathbb {R}^{2d_{hid}\\times 2d_{hid}}$ is the attention weight matrix.",
"After computing the word attention weights, we can get the weighted combination of the aspect hidden representation as a new aspect representation: $&m^{a_i}=\\sum ^{M_i}_{t=1}\\beta ^{a_i}_t\\cdot h^{a_i}_t&$"
],
[
"Aspect to context attention ",
"Aspect to context attention learns to capture the aspect-specific context representation, which is similar to context to aspect attention.",
"Specifically, the attention scores is calculated by the new aspect representation $m^{a_i}$ and the position-aware representation $p^{a_i}_t$ .",
"The process can be formulated as follows: $&f_{ac}(m^{a_i},p^{a_i}_t)={m^{a_i}}^T\\cdot W_{ac}\\cdot p^{a_i}_t&$ $&\\gamma ^{a_i}_t=\\frac{exp(f_{ac}(m^{a_i},p^{a_i}_t))}{\\sum ^{N}_{t=1}exp(f_{ac}(m^{a_i},p^{a_i}_t))}&$ $&x^{a_i}=x_i=\\sum ^{N}_{t=1}\\gamma ^{a_i}_t\\cdot h^c_t&$ where $W_{ac}\\in \\mathbb {R}^{2d_{hid}\\times 2d_{hid}}$ is the attention weight matrix.",
"By now, we get the aspect-specific representations $X=[x_1,x_2,\\ldots ,x_K]$ between each aspect and its context words, where K is the number of aspects in the context."
],
[
"Graph convolutional network",
"GCN is widely used to deal with data which contains rich relationships and interdependency between objects, because GCN can effectively capture the dependence of graphs via message passing between the nodes of graphs.",
"We also employ a graph to capture the sentiment dependencies between aspects.",
"The final output of each GCN node is designed to be the classifier of the corresponding aspect in our task.",
"Moreover, there are no explicit edges in our task.",
"Thus, we need to define the edges from scratch.",
"Figure: Illustration of our proposed sentiment graphs.",
"a 1 a_1, a 2 a_2, a 3 a_3, a 4 a_4 and a 5 a_5 denote five aspects in one context."
],
[
"Sentiment graph",
"We construct a graph, named sentiment graph, to capture the sentiment dependencies between multi-aspects in one sentence, where each node is regarded as an aspect and each edge is treated as the sentiment dependency relation.",
"As shown in Fig.",
"REF , we define two kinds of undirected sentiment graphs: adjacent-relation graph: An aspect is only connected to its nearby aspects.",
"global-relation graph: An aspect is connected to all other aspects.",
"If two nodes are connected by an edge, it means that the two nodes are neighboring to each other.",
"Formally, given a node $v$ , we use $N(v)$ to denote all neighbors of $v$ .",
"$u\\in N(v)$ means that $u$ and $v$ are connected with an edge.",
"Figure: Statistics of the number of aspects in one sentence on SemEval 2014 data set."
],
[
"Sentiment graph based GCN",
"GCN encodes relevant information about its neighborhood as a new representation vector, where each node in the graph indicates a representation of aspect.",
"In addition, as Kipf et al.",
"[15] do, we assume all nodes contain self-loops.",
"Then, the new node representation is computed as follows: $&x^1_v=relu(\\sum _{u\\in N(v)}W_{cross}x_u+b_{cross})+ReLU(W_{self}x_v+b_{self})&$ where $W_{cross}, W_{self}\\in \\mathbb {R}^{d_m\\times d_n}$ , $b_{cross}, b_{self}\\in \\mathbb {R}^{d_m\\times 1}$ , $x_u$ is the $u$ -th aspect-specific representation (see Eq.",
"(REF )), and $ReLU$ is the rectifier linear unit activation function.",
"In this work, we use $d_m=d_n=2h_{hid}$ .",
"By stacking multiple GCN layers, the final hidden representation of each node can receive messages from a further neighborhood.",
"Each GCN layer takes the node representations from previous layer as inputs and outputs new node representations: $&x^{l+1}_v=relu(\\sum _{u\\in N(v)}W^l_{cross}x^l_u+b^l_{cross})+relu(W^l_{self}x^l_v+b^l_{self})&$ where $l$ denotes the layer number and $1\\le l\\le L-1$ ."
],
[
"Output layer",
"The final output of each GCN node $x^L_i$ is treated as a classifier of the $i$ -th aspect.",
"At last, we use a fully-connected layer to map $x^L_i$ into the aspect space of $C$ classes: $&z_i=W_zx^L_i+b_z&$ where $W_z\\in \\mathbb {R}^{C\\times 2d_{hid}}$ is the weight matrix, and $b_z\\in \\mathbb {R}^{2d_{hid}\\times C}$ is the bias.",
"The predicted probability of the $i$ -th aspect with sentiment polarity $j\\in [1,C]$ is computed by: $&y^{\\prime }_{ij}=\\frac{exp(z_{ij})}{\\sum ^C_{k=1}exp(z_{ik})}&$"
],
[
"Model training",
"Our model is trained by minimizing the cross entropy with $L2$ -regularization term.",
"For a given sentence, the loss function is defined as: $&loss=\\sum ^K_{i=1}\\sum ^C_{j=1}y_{ij}log(y^{\\prime }_{ij})+\\lambda \\parallel \\theta \\parallel ^2&$ where $y_{ij}$ is a one-hot labels of the $i$ -th aspect for the $j$ -th class, $\\lambda $ is the coefficient for $L2$ -regularization, $\\theta $ is the parameters that need to be regularized.",
"Furthermore, we adopt the dropout strategy during training step to avoid over-fitting.",
"Table: The details of the experimental data sets."
],
[
"Data sets and experimental settings",
"To demonstrate the effectiveness of our proposed method, as most previous works [9], [11], [25], [13], we conduct experiments on two datasets from SemEval 2014 Task4The detailed introduction of this task can be found at http://alt.qcri.org/semeval2014/task4.",
"[39], which contains the reviews in laptop and restaurant.",
"The details of the SemEval 2014 datasets are shown in Table REF .",
"Each dataset consists of train and test set.",
"Each review (one sentence) contains one or more aspects and their corresponding sentiment polarities, i.e., positive, neutral and negative.",
"To be specific, the number in table means the number of aspects in each sentiment category.",
"To demonstrate the necessity of considering the sentiment dependencies between the aspects, we further calculate the number of aspects in each sentence, which is presented in Fig.",
"REF .",
"From the histogram in Fig.",
"REF , we can see that each sentence contains one to thirteen aspects.",
"The number of aspects in most reviews is 1 to 4.",
"The pie chart shows the proportion of only one aspect and more than one aspect in one sentence.",
"It can be seen that more than half of the aspects do not appear alone in a review.",
"According to these statistics, we can conclude that it is common to have multi-aspects within one sentence.",
"Our model mainly aims to model the sentiment dependencies between different aspects in one sentence.",
"In our implementation, we respectively use the GloVehttps://nlp.stanford.edu/projects/glove/ [37] word vector and the pre-trained language model word representation BERThttps://github.com/google-research/bert#pre-trained-models [38] to initialize the word embeddings.",
"The dimension of each word vector is 300 for GloVe and 768 for BERT.",
"The number of LSTM hidden units is set to 300, and the output dimension of GCN layer is set to 600.",
"The weight matrix of last fully connect layer is randomly initialized by a normal distribution $N(0, 1)$ .",
"Besides the last fully connect layer, all the weight matrices are randomly initialized by a uniform distribution $U(-0.01, 0.01)$ .",
"In addition, we add $L2$ -regularization to the last fully connect layer with a weight of 0.01.",
"During training, we set dropout to 0.5, the batch size is set to 32 and the optimizer is Adam Optimizer with a learning rate of 0.001.",
"We implement our proposed model using Tensorflowhttps://www.tensorflow.org/.",
"To evaluate performance of the model, we employ Accuracy and Macro-F1 metrics.",
"The Macro-F1 metric is more appropriate when the data set is not balanced."
],
[
"Comparative methods",
"To comprehensively evaluate the performance of proposed SDGAN, we compare our model with the following models.",
"TD-LSTM [6] constructs aspect-specific representation by the left context with aspect and the right context with aspect, then employs two LSTMs to model them respectively.",
"The last hidden states of the two LSTMs are finally concatenated for predicting the sentiment polarity of the aspect.",
"ATAE-LSTM [9] first attaches the aspect embedding to each word embedding to capture aspect-dependent information, and then employs attention mechanism to get the sentence representation for final classification.",
"MemNet [40] uses a deep memory network on the context word embeddings for sentence representation to capture the relevance between each context word and the aspect.",
"Finally, the output of the last attention layer is used to infer the polarity of the aspect.",
"IAN [11] generates the representations for aspect terms and contexts with two attention-based LSTM network separately.",
"Then the context representation and the aspect representation are concatenated for predicting the sentiment polarity of the aspect.",
"RAM [10] employs a gated recurrent unit network to model a multiple attention mechanism, and captures the relevance between each context word and the aspect.",
"Then the output of the gated recurrent unit network is obtained for final classification.",
"PBAN [41] appends the position embedding into each word embedding.",
"It then introduces a position-aware bidirectional attention network (PBAN) based on Bi-GRU to enhance the mutual relation between the aspect term and its corresponding sentence.",
"TSN [25] is a two-stage framework for aspect-level sentiment analysis.",
"The first stage, it uses a position attention to capture the aspect-dependent representation.",
"The second stage, it introduces penalization term to enhance the difference of the attention weights towards different aspects in one sentence.",
"AEN [13] mainly consists of an embedding layer, an attentional encoder layer, an aspect-specific attention layer, and an output layer.",
"In order to eschew the recurrence, it employs attention-based encoders for the modeling between the aspect and its corresponding context.",
"AEN-BERT [13] is AEN with BERT embedding."
],
[
"Overall results",
"Table REF shows the experimental results of competing models.",
"In order to remove the influence with different word representations and directly compare the performance of different models, we compare GloVe-based models and BERT-based models separately.",
"Our proposed model achieves the best performance on both GloVe-based models and BERT-based models, which demonstrates the effectiveness of our proposed model.",
"In particularly, SDGCN-BERT obtains new state-of-the-art results.",
"Among all the GloVe-based methods, the TD-LSTM approach performs worst because it takes the aspect information into consideration in a very coarse way.",
"ATAE-LSTM, MenNet and IAN are basic attention-based models.",
"After taking the importance of the aspect into account with attention mechanism, they achieve a stable improvement comparing to the TD-LSTM.",
"RAM achieves a better performance than other basic attention-based models, because it combines multiple attentions with a recurrent neural network to capture aspect-specific representations.",
"PBAN achieves a similar performance as RAM by employing a position embedding.",
"To be specific, PBAN is better than RAM on Restaurant dataset, but worse than RAN on Laptop dataset.",
"Compared with RAM and PBAN, the overall performance of TSN is not perform well on both Restaurant dataset and Laptop dataset, which might because the framework of TSN is too simple to model the representations of context and aspect effectively.",
"AEN is slightly better than TSN, but still worse than RAM and PBAN.",
"It indicates that the discard of the recurrent neural networks can reduce the size of model while lead to the loss of performance.",
"Comparing the results of SDGCN-A w/o position and SDGCN-G w/o position, SDGCN-A and SDGCN-G, respectively, we observe that the GCN built with global-relation is slightly higher than built with adjacent-relation in both accuracy and Macro-F1 measure.",
"This may indicate that the adjacent relationship is not sufficient to capture the interactive information among multiple aspects due to the neglect of the long-distance relation of aspects.",
"Moreover, the two models (SDGCN-A and SDGCN-G) with position information gain a significant improvement compared to the two models without position information.",
"It shows that the position encoding module is crucial for good performance.",
"Benefits from the power of pre-trained BERT, BERT-based models have shown huge superiority over GloVe-based models.",
"Furthermore, compared with AEN-BERT, on the Restaurant dataset, SDGCN-BERT achieves absolute increases of 1.09% and 1.86% in accuracy and Macro-F1 measure respectively, and gains absolute increases of 1.42% and 2.03% in accuracy and Macro-F1 measure respectively on the Laptop dataset.",
"The increments prove the effectiveness of our proposed SDGCN.",
"Figure: Comparisons with different depths of GCN in our model.Figure: Illustration of attention weights obtained by model with GCN and without GCN respectively.",
"(a) and (b) are two examples from the Laptop dataset.",
"(a) Aspects: keyboard, screen; (b) Aspects: resolution, fonts.Table: The effect of GCN."
],
[
"The effect of GCN module",
"In this section, we design a series of models to further verify the effectiveness of GCN module.",
"These models are: BiAtt+GCN is just another name of our proposed SDGCN model.",
"BiAtt is based on BiAtt -GCN, where we remove the GCN module.",
"Therefore, it predicts the sentiments of different aspects in one sentence independently.",
"Att+GCN is a simplified version of BiAtt+GCN.",
"The only difference between Att+GCN and BiAtt+GCN is that Att+GCN does not have context to aspect attention.",
"Att is the model of Att+GCN removing the GCN module.",
"Table REF shows the performances of all these models.",
"It is clear to see that, comparing with GCN-reduced models, the two models with GCN achieve higher performance, respectively.",
"The results verify that the modeling of the sentiment dependencies between different aspects with GCN plays a great role in predicting the sentiment polarities of aspects."
],
[
"Impact of GCN layer number",
"The number of GCN layers is one very important setting parameter that affects the performance of our model.",
"In order to investigate the impact of the GCN layer number, we conduct experiment with the different number of GCN layers from 1 to 8.",
"The performance results are shown in Fig.",
"REF .",
"As can be seen from the results, in general, when the number of GCN layers is 2, the model works best.",
"When the number of GCN layers is bigger than 2, the performance drops with the increase of the number of GCN layers on both the datasets.",
"The possible reason for the phenomenon of the performance drop may be that with the increase of the model parameters, the model becomes more difficult to train and over-fitting."
],
[
"Case study",
"In order to have an intuitive understanding of the difference between with-GCN model (our proposed model) and without-GCN model, we use two examples with multiple aspects from laptop dataset as a case study.",
"We draw heat maps to visualize the attention weights on the words computed by the two models, as shown in Fig.",
"REF .",
"The deeper the color, the more attention the model pays to it.",
"As we can see from the first example, i.e., “i love the keyboard and the screen.",
"\", with two aspects “keyboard\" and “screen\", without-GCN model mainly focuses on the word “love\" to predict the sentiment polarities of the two aspects.",
"While for with-GCN model, besides the word “love\", it also pays attention to the conjunction “and\".",
"This phenomenon indicates that with-GCN model captures the sentiment dependencies of the two aspects through the word “and\", and then predicts the sentiments of “keyboard\" and “screen\" simultaneously.",
"The second example is “air has higher resolution but the fonts are small.\"",
"with two aspects “resolution\" and “fonts\".",
"It is obvious that the sentiments of the two aspects “resolution\" and “fonts\" are opposite connected by the conjunction “but\".",
"Without-GCN model predicts the polarity of aspect “resolution\" by the word “higher and the polarity of aspect “fonts by the word “small\" in isolation, which ignores the relation between the two aspects.",
"In the contrary, with-GCN model enforces the model to pay attention on the word “but\" when predicting the sentiment polarity for aspect “fonts\".",
"From these examples, we can observe that our proposed model (with-GCN model) not only focuses the corresponding words which are useful for predicting the sentiment of each aspect, but also considers the textual information which is helpful for judging the relation between different aspects.",
"By using attention mechanism to focus on the textual words describing the interdependence between different aspects, the downstream GCN module can effectively further represent the sentiment dependencies between different aspects in one sentence.",
"With more useful information, our proposed model can predict aspect-level sentiment category more accurately."
],
[
"Conclusion",
"In this paper, we design a novel GCN based model (SDGCN) for aspect-level sentiment classification.",
"The key idea of our model is to employ GCN to model the sentiment dependencies between different aspects in one sentence.",
"Specifically, SDGCN first adopts bidirectional attention mechanism with position encoding to obtain aspect-specific representations, then captures the sentiment dependencies via message passing between aspects.",
"Thus, SDGCN benefits from such dependencies which are always ignored in previous studies.",
"Experiments on SemEval 2014 verify the effectiveness of the proposed mode, and SDGCN-BERT obtains new state-of-the-art results.",
"The case study shows that SDGCN can not only pay attention to those words which are important for predicting the sentiment polarities of aspects, but also pay attention to the words which are helpful for judging the sentiment dependencies between different aspects.",
"In our future work, we will explore how to build a more precise sentiment graph structure between aspects.",
"The two kinds of undirected sentiment graphs in this work are coarse.",
"We conjecture that making use of textual information to define a graph may create a better graph structure.",
"numbers,sortcompress"
]
] | 1906.04501 |
[
[
"Survey of Artificial Intelligence for Card Games and Its Application to\n the Swiss Game Jass"
],
[
"Abstract In the last decades we have witnessed the success of applications of Artificial Intelligence to playing games.",
"In this work we address the challenging field of games with hidden information and card games in particular.",
"Jass is a very popular card game in Switzerland and is closely connected with Swiss culture.",
"To the best of our knowledge, performances of Artificial Intelligence agents in the game of Jass do not outperform top players yet.",
"Our contribution to the community is two-fold.",
"First, we provide an overview of the current state-of-the-art of Artificial Intelligence methods for card games in general.",
"Second, we discuss their application to the use-case of the Swiss card game Jass.",
"This paper aims to be an entry point for both seasoned researchers and new practitioners who want to join in the Jass challenge."
],
[
"Introduction",
"The research field of AI applied to playing games has been subject to several breakthroughs in the last years.",
"In particular, the branch of PIG — where the entire game state is known to all players at all points in time — has seen machines triumph over human professional players in different occasions, such as for Chess, the Atari games or Go.",
"When it comes to IIG — where part of the information is unknown to the players, such as in card games — there is a thin line separating AI from humans, who still have the upper hand against state-of-the-art agents.",
"However, recent work shows that in constrained situations, the gap between humans and AI is becoming thinner.",
"This is particularly visible when considering advances on Texas hold'em no-limit poker [1] and the computer games Dota 2 and StarCraft II.",
"Hidden information is also present in many real world scenarios, like negotiations, surgical operations, business, physics and others.",
"Many of these situations can be formalized as games, which in turn can be solved using the methods refined in the test bed of card games.",
"Most card games involve hidden information, which makes them both a suitable and interesting domain for further research on AI.",
"There is a large variety of card games, where many use different cards and rules, which poses different challenges to the players.",
"To tackle these different issues, several methods have been proposed.",
"Unfortunately, these methods are often either very complex, or introduce only minor modifications to address a particular issue for a particular game.",
"Despite producing good empirical results, this practice leads to a more complex landscape of literature which is at times hard to navigate, especially for new practitioners in the field.",
"To combat this unwanted side effect, overviews of the current recent trends and methods are very helpful.",
"In this work, we aim to provide such an overview of AI methods applied to card games.",
"In the appendix there is a short description of the games we mention in this work.",
"To complement the overview, we chose to use the card game “Jass” as a use-case for a discussion of the methods present above.",
"Jass is a very popular card game in Switzerland and tightly linked to the Swiss culture.",
"From a research point of view, Jass is a challenging game because a) it is played by more than two players (specifically, four divided in two teams of two), b) it involves hidden information (the cards of the other players), c) it is difficult to master by humans and d) the number of information sets is much bigger than that of other popular card games such as Poker.",
"However to the best of our knowledge, a formal approach towards Jass has not been address in a scientific manner yet.",
"The Swiss Intercantonal Lottery and some Jass applications have deployed some AI agents, but these programs are not yet able to beat top human players."
],
[
"Main Contribution",
"In this work, we aim to address a gap in the literature regarding AI approaches towards card games with a particular emphasis on the popular Swiss card game Jass.",
"To the best of our knowledge, there has not been a formal scientific approach to Jass outlined in the literature.",
"To this end, we discuss the potential merits and demerits of the different methods outlined in the paper towards Jass."
],
[
"Related Work",
"In this section we review the relevant related work.",
"In their book, Yannakakis et al.",
"[2] gave a general overview of AI development in games, while Rubin et al.",
"[3] provided a more specific review on the methods used in computer Poker.",
"In his thesis, Burch [4] reviewed the state-of-the-art in CFR, a family of methods very heavily used in computer Poker.",
"Finally, Browne et al.",
"[5] surveyed the different variants of MCTS, a family of methods used for AI in many games of both perfect and imperfect information.",
"We are not aware of any work that specifically addresses the domain of card games."
],
[
"Theoretical Foundation",
"In this section we introduce terms necessary to understand AI for card games."
],
[
"Game Types",
"Games can be classified in many dimensions.",
"In this section we outline the ones most important for classifying card games.",
"Sequential games are normally formalized as extensive-form games.",
"These games are played on a decision tree, where a node represents a decision point for a player and an edge describes a possible action leading from one state to another.",
"For each node in this tree it is possible to define an information set.",
"An information set includes all the states a player could be in, given the information the player has observed so far.",
"In PIG, these information sets always only comprise exactly one state, because all information is known.",
"In an IIG like Poker, this information set contains all the card combinations the opponents could have, given the information the player has, i.e.",
"the cards on the table and the cards in the hand."
],
[
"Coordination Games",
"Unlike many strategic situations, collaboration is central in a coordination game, not conflict.",
"In a coordination game, the highest payoffs can only be achieved through team work.",
"Choosing the side of the road to drive on is a simple example of a coordination game.",
"It does not matter which side of the road you agree on, but to avoid crashes, an agreement is essential.",
"In card games, like Bridge or Jass, where there are two teams playing against each other, the interactions within the team can be seen as a coordination game.",
"When developing an AI, it is important to accurately measure its strength in comparison to other AI and humans.",
"The ultimate goal is to achieve optimal play.",
"When a player is playing optimally, s/he does not make any mistakes but plays the best possible move in every situation.",
"When an optimal strategy in a game is known, this game is considered solved."
],
[
"NE",
"A NE describes a combination of strategies in non-cooperative games.",
"When two or more players are playing their respective part of a NE, any unilateral deviation from the strategy leads to a negative relative outcome for the deviating player [6].",
"So when programming players for games, the goal is to get as close as possible to a NE.",
"When one is playing a NE strategy, the worst outcome that can happen is coming to a draw.",
"This means that a NE player wins against any player not playing a NE strategy.",
"In games involving chance (the cards dealt at the beginning in the case of Poker), the player may not win every single game.",
"Thus, many games may have to be played to evaluate the strategies.",
"A NE strategy is particularly beneficial against strong players.",
"Therefore, it does not make any mistakes the opponent could possibly exploit.",
"On the other hand, a NE strategy might not win over a sub-optimal player by a large margin because it does not actively try to exploit the opponent but rather tries not to commit any mistakes at all.",
"There exists a NE for every finite game [6]."
],
[
"Exploitability",
"Exploitability is a measure for this deviation from a NE [7].",
"The higher the exploitability, the greater the distance to a NE, and therefore, the weaker the player.",
"A NE strategy constitutes optimal play, since there is no possible better strategy.",
"However, there are different NE strategies which differ in their effectiveness of exploiting non-NE strategies [8].",
"If it is not possible to calculate such a strategy (for example, because the state space is too large), we want to estimate a strategy which minimizes the deviation from a NE."
],
[
"Comparison to Humans",
"When designing AI it is always interesting to evaluate how well they perform in comparison to humans.",
"Here we distinguish four categories: sub-human, par-human, high-human and super-human AI which respectively mean worse than, similar to, better than most and better than all humans.",
"The current best AI agents in Jass achieve par-human standards.",
"In Bridge, current computer programs achieve expert level, which constitutes high-human proficiency.",
"In many PIG like Go or Chess, current AI achieve super-human level."
],
[
"Rule-Based Systems",
"Rule-based systems leverage human knowledge to build an AI player [2].",
"Many simple AI for card games are rule-based and then used as baseline players.",
"This mostly entails a number of if-then-else statements which can be viewed as a man-made decision tree.",
"Ward et al.",
"[9] created a rule-based AI for Magic: The Gathering which was used as a baseline player.",
"Robilliard et al.",
"[10] developed a rule-based AI for 7 Wonders which was used as a baseline player.",
"Watanabe et al.",
"[11] implemented three rule-based players.",
"The greedy player behaves like a beginner player.",
"The other two follow more advanced strategies taken from strategy books and are behaving like expert players.",
"Osawa [12] presented several par-human rule-based strategies for Hanabi.",
"His results indicated that feedback-based strategies achieve higher scores than purely rational ones.",
"Van den Bergh et al.",
"[13] developed a strong par-human rule-based AI for Hanabi.",
"Whitehouse et al.",
"[14] evaluated the rule-based Spades player developed by AI Factory.",
"Based on player reviews they found it to decide weakly in certain situations but to be a strong par-human player overall."
],
[
"RL Methods",
"RL is a machine learning method which is frequently used to play games.",
"It consists of an agent performing actions in a given environment.",
"Based on its actions, the agent receives positive rewards which reinforce desirable behaviour and negative rewards which discourage unwanted behaviour.",
"Using a value function, the agent tries to find out which action is the most desirable in a given state."
],
[
"TDL",
"TDL updates the value function continuously after every iteration, as opposed to earlier strategies which waited until the episode's end [15].",
"Sturtevant et al.",
"[16] developed a sub-human AI for Hearts using Stochastic Linear Regression and TDL which outperforms players based on minimax search."
],
[
"PG",
"PG is an algorithm which directly learns a policy function mapping a state to an action [15].",
"PPO is an extension to the PG algorithm improving its stability and reducing the convergence time [17] Charlesworth [18] applied PPO to Big 2, reaching par-human level."
],
[
"CFR",
"CFR [19] is a self-playing method that works very well for IIG and has been used by the most successful poker AIs [1], [20].",
"“Counterfactual” denotes looking back and thinking “had I only known then...”.",
"“Regret” says how much better one would have done, if one had chosen a different action.",
"And “minimization” is used to minimize the total regret over all actions, so that the future regret is as small as possible.",
"Note that CFR only requires memory linear to the number of information sets and not to the number of states [3].",
"Additionally, CFR has been able to exploit non-NE strategies computed by UCT agents in simultaneous games [21]."
],
[
"CFR+",
"CFR+ is a re-engineered version of CFR, which drastically reduces convergence time.",
"It always iterates over the entire tree and only allows non-negative regrets.",
"[22] Bowling et al.",
"[22] used CFR+ to essentially solve heads-up limit Texas hold'em Poker in 2015.",
"Moravčík et al.",
"[20] developed a general algorithm for imperfect information settings, called DeepStack.",
"With statistical significance, it defeated professional poker players in a study over 44000 hands."
],
[
"Deep CFR",
"Deep CFR [23] combines CFR with deep ANN.",
"Brown et al.",
"[1] leverage deep CFR to decisively beat four top human poker players in 2017 with their program called Libratus."
],
[
"Discounted CFR",
"Discounted CFR [24] matches or outperforms the previous state-of-the-art variant CFR+ depending on the application by discounting prior iterations."
],
[
"NFSP",
"In NFSP, two players start with random strategies encoded in an ANN.",
"They play against each other knowing the other player's strategy improving the own strategy.",
"With an increasing number of iterations, the strategies typically approach a NE.",
"Since NFSP [25] has a slower convergence rate than CFR it is not widely used.",
"Heinrich et al.",
"[25] applied NFSP to Texas hold'em Poker and reported similar performance to the state-of-the-art super-human programs.",
"In Leduc Poker, a simplification of the former, they approached a NE.",
"Kawamura et al.",
"[26] calculated approximate NE strategies with NFSP in multiplayer IIG."
],
[
"FOM",
"FOM like EGT are, like CFR, methods which approximate NE strategies in IIG.",
"They have a better theoretical convergence rate than CFR because of lower computational and memory costs.",
"Note that, like CFR, EGT is only able to approach a NE in two-player games [27].",
"Kroer et al.",
"[27] applied a variant of EGT to Poker reporting faster convergence than some CFR variants.",
"They argue that, given more hyper parameter tuning, the performance of CFR+ can be reached."
],
[
"MC Methods",
"MC methods use randomness to solve problems that might be deterministic in principle."
],
[
"MC Simulation",
"MC Simulation uses a large number of random experiments to numerically solve large problems involving many random variables.",
"Mitsukami et al.",
"[28] developed a par-human AI for Japanese Mahjong using MC Simulation.",
"Kupferschmid et al.",
"[29] applied MC Simulation to Skat to obtain the game-theoretical value of a Skat hand.",
"Note that they converted the game to a PIG by making all the cards known.",
"Yan et al.",
"[30] report a 70% win rate using MC Simulation in a Klondike version, which has all cards revealed to the player.",
"Note that this converts the game to a PIG."
],
[
"Flat MC",
"Flat MC uses MC Simulation, with the actions in a given state being uniformly sampled [5].",
"Ginsberg [31] achieves world champion level play in Bridge using Flat MC in 2001."
],
[
"MCTS",
"MCTS consists of four stages: Selection, Expansion, Simulation and Backpropagation [5].",
"Selection: Starting from the root node, an expandable child node is selected.",
"A node is expandable if it is non-terminal (i.e.",
"it does have children) and has unvisited children.",
"Expansion: The tree is expanded by adding one or more child nodes to the previously selected node.",
"Simulation: From these new children nodes a simulation is run to acquire a reward at a terminal node.",
"Backpropagation: The simulation's result is used to update the information in the affected nodes (nodes in the selection path).",
"A tree policy is used for selecting and expanding a node and the simulation is run according to the default policy.",
"Browne et al.",
"[5] gives a detailed overview of the MCTS family .",
"In this section we outline the variants used on card games."
],
[
"UCT",
"UCT is the most common MCTS method, using upper confidence bounds as a tree policy, which is a formula that tries to balance the exploration/exploitation problem [32].",
"When the search explores too much, the optimal moves are not played frequently enough and therefore it may find a sub-optimal move.",
"When the search exploits too much, it may not find a path which promises much greater payoffs and it therefore also may find a sub-optimal move.",
"Minimax is a basic algorithm used for two-player zero-sum games, operating on the game tree.",
"When the entire tree is visited, minimax is optimal [2].",
"UCT converges to minimax given enough time and memory [32].",
"Sievers et al.",
"[33] applied UCT to Doppelkopf reaching par-human performance.",
"Schäfer [34] used UCT to build an AI for Skat, which is still sub-human but comparable to the MC Simulation based player proposed by Kupferschmid et al.",
"[29].",
"Swiechowski et al.",
"[35] combined an MCTS player with supervised learning on the logs of sample games, achieving par-human performance.",
"Santos et al.",
"[36] outperformed basic MCTS based AI by combining it with domain-specific knowledge.",
"Heinrich et al.",
"[37] combined UCT with self-play and apply it to Poker.",
"They reported convergence to a NE in a small Poker game and argue that, given enough training, convergence can also be reached in large limit Texas Hold'em Poker."
],
[
"Determinization",
"Determinization is a technique which allows solving an IIG with methods used for PIG.",
"Determinization samples many states from the information set and plays the game to a terminal state based on these states of perfect information.",
"Bjarnason et al.",
"[38] studied Klondike using UCT, hindsight optimization and sparse sampling.",
"Hindsight optimization uses determinization and hindsight knowledge to improve the strategy.",
"They developed a policy which wins at least 35% of games, which is a lower bound for an optimal Klondike policy.",
"Sturtevant [39] applied UCT with determinization to the multiplayer games Spades and Hearts.",
"He reported similar performance to the state-of-the-art at that time in Spades and slightly better performance in Hearts.",
"Cowling et al.",
"[40] applied MCTS with determinization approaches to the card game Magic: The Gathering achieving high-human performance and outperforming an expert-level rule-based player.",
"Robilliard et al.",
"[10] applied UCT with determinization to 7 Wonders outperforming rule-based AI.",
"The experiments against human players were promising but not statistically significant.",
"Solinas et al.",
"[41] used UCT and supervised learning to infer the cards of the other players, improving over the state-of-the-art in Skat card-play.",
"Edelkamp [42] combined distilled expert rules, winning probabilities aggregations and a fast tree exploration into an AI for the Misère variant of Skat significantly outperforming human experts."
],
[
"IS-MCTS",
"IS-MCTS tackles the problem of strategy fusion which includes the false assumption that different moves can be taken from different states in the information set [43].",
"However, because the player does not know of the different states in the information set, it cannot decide differently, based on different states.",
"IS-MCTS operates directly on a tree of information sets.",
"Whitehouse et al.",
"[44] used MCTS with determinization and information sets on Dou Di Zhu.",
"They did not report any significant differences in performance between the two proposed algorithms.",
"Watanabe et al.",
"[11] presented a high-human AI using IS-MCTS for the Italian card game Scopone which consistently beat strong rule-based players.",
"Walton-Rivers et al.",
"[45] applied IS-MCTS to Hanabi, but they measured inferior performance to rule-based players.",
"Whitehouse et al.",
"[14] found an MCTS player to be stronger than rule-based players in the card game Spades.",
"They integrated IS-MCTS with knowledge-based methods to create more engaging play.",
"Cowling et al.",
"[46] performed a statistical analysis over 27592 played games on a mobile platform to evaluate the player's difficulty for humans.",
"Devlin et al.",
"[47] combined insights from game play data with IS-MCTS to emulate human play."
],
[
"MCCFR",
"MCCFR drastically reduces the convergence time of CFR by using MC Sampling [48].",
"MCCFR samples blocks of paths from the root to a terminal node and then computes the immediate counterfactual regrets over these blocks.",
"Lanctot et al.",
"[48] showed this faster convergence rate in experiments on Goofspiel and One-Card-Poker.",
"Ponsen et al.",
"[49] evidences that MCCFR approaches a NE in Poker."
],
[
"OOS",
"OOS is an online variant to MCCFR which can decrease its exploitability with increasing search time [50].",
"Lisý et al.",
"[50] demonstrated that OOS can exploit IS-MCTS in Poker knowing the opponent's strategy and given enough computation time."
],
[
"EA",
"EA are inspired by evolutionary theory.",
"Strong individuals — strategies in the case of game AI — can survive and reproduce, whereas weaker ones eventually become extinct [2].",
"Mahlmann et al.",
"[51] compared three EA agents with different fitness functions in Dominion.",
"They argued that their method can be used for automatic game design and game balancing.",
"Noble [52] applied a EA evolving ANN to Poker in 2002 improving over the state-of-the-art at the time."
],
[
"Use-Case: Swiss Card Game Jass",
"Jass is a trick-taking traditional Swiss card game often played at social events.",
"It involves hidden information, is sequential, non-cooperative, finite and constant-sum, as there are always 157 points possible in each game.",
"The Swiss Intercantonal Lottery provide a guide for general Jass ruleswww.swisslos.ch/en/jass/informations/jass-rules/principles-of-jass.html and for the variant Schieber in particularwww.swisslos.ch/en/jass/informations/jass-rules/schieber-jass.html."
],
[
"Coordination Game Within Jass",
"Schieber is a non-cooperative game, since the two teams are opposing each other.",
"However, additionally, the activity within a team can be formulated as a coordination game.",
"This adds another dimension to the game as it enables cooperation between the players within the game to maximize the team's benefit.",
"Although the rules of the game forbid any communication during a game within the team, by playing specific cards in certain situations, the two players can convey information about the cards they have.",
"For this to work, of course they must have the same understanding of this communication by card play.",
"Humans have some existing “agreements” like a “discarding policy”.",
"Discarding tells the partner which suits the player is bad at.",
"It is interesting to investigate, whether AI are able to pick up these “agreements” or even come up with new ones."
],
[
"Suitable Methods for AI in Card Games by the Example of Jass",
"MCTS and CFR are the two families of algorithms that have most successfully been applied to card games.",
"In this section we are comparing these two methods' advantages and disadvantages in detail by the example of the trick-taking card game Jass.",
"To the best of our knowledge, CFR has almost exclusively been applied to Poker so far, although the authors claim that it can be applied to any IIG [23].",
"CFR provides theoretical guarantees for approaching a NE in two player IIG [19].",
"On the other hand, as we discussed in section REF , pure NE strategies may not be able to specifically exploit weak opponents.",
"Additionally, CFR needs a lot of time to converge, compared to MCTS [49].",
"MCTS has been applied to a plethora of complex card games including Bridge, Skat, Doppelkopf or Spades, as we have illustrated in the previous sections.",
"It finds good strategies fast but only converges to a NE in PIG and not necessarily in IIG [49].",
"As opposed to CFR, MCTS does not find the moves with lowest exploitability, but the ones with highest chance of winning [53].",
"MCTS eventually converges to minimax, but total convergence is infeasible for large problems [5].",
"So, if the goal is to find a good strategy relatively fast, MCTS should be chosen, whereas CFR should be selected, if the goal is to be minimally exploitable [49].",
"To put it simply, CFR is great at not losing, but not very good at destroying an opponent and MCTS is great at finding good strategies fast, but not very good at resisting against very strong opponents."
],
[
"Preliminary Results",
"Preliminary experiments not presented in this paper show that MCTS is a promising approach for a strong AI playing Jass."
],
[
"Conclusion",
"In this paper we first provided an overview of the methods used in AI development for card games.",
"Then we discussed the advantages and disadvantages of the two most promising families of algorithms (MCTS and CFR) in more detail.",
"Finally, we presented an analysis for how to apply these methods to the Swiss card game Jass.",
"Game Descriptions In this section we give the gist of the less well-known games discussed in the paper (in order of appearance).",
"Magic: The Gathering is a trading and digital collectible card game played by two or more players.",
"7 Wonders is a board game with strong elements of card games including hidden information for two to seven players.",
"Scopone is a variant of the Italian card game Scopa.",
"Hanabi is a French cooperative card game for two to five players.",
"Spades is a four player trick-taking card game mainly played in North America.",
"Big 2 is a Chinese card game for two to four players mainly played in East and South East Asia.",
"The goal is, to get rid of all of one's cards first.",
"Mahjong is a traditional Chinese tile-based game for four (or seldom three) players similar to the Western game Rummy.",
"Skat is a three player trick-taking card game mainly played in Germany.",
"Klondike is a single-player variant of the French card game Patience and shipped with Windows since version 3.",
"Bridge is a trick-taking card game for four players played world-wide in clubs, tournaments, online and socially at home.",
"It has often been used as a test bed for AI research and is still an active area of research, since super-human performance has not been achieved yet.",
"Doppelkopf is a trick-taking card game for four people, mainly played in Germany.",
"Hearthstone is an online collectible card video game, developed by Blizzard Entertainment.",
"Hearts is a four player trick-taking card game, mainly played in North America.",
"Dou Di Zhu is a Chinese card game for three players.",
"Goofspiel is a simple bidding card game for two or more players.",
"One-Card-Poker generalizes the minimal variant Kuhn-Poker.",
"Dominion is a modern deck-building card game similar to Magic: The Gathering."
],
[
"Game Descriptions",
"In this section we give the gist of the less well-known games discussed in the paper (in order of appearance).",
"Magic: The Gathering is a trading and digital collectible card game played by two or more players.",
"7 Wonders is a board game with strong elements of card games including hidden information for two to seven players.",
"Scopone is a variant of the Italian card game Scopa.",
"Hanabi is a French cooperative card game for two to five players.",
"Spades is a four player trick-taking card game mainly played in North America.",
"Big 2 is a Chinese card game for two to four players mainly played in East and South East Asia.",
"The goal is, to get rid of all of one's cards first.",
"Mahjong is a traditional Chinese tile-based game for four (or seldom three) players similar to the Western game Rummy.",
"Skat is a three player trick-taking card game mainly played in Germany.",
"Klondike is a single-player variant of the French card game Patience and shipped with Windows since version 3.",
"Bridge is a trick-taking card game for four players played world-wide in clubs, tournaments, online and socially at home.",
"It has often been used as a test bed for AI research and is still an active area of research, since super-human performance has not been achieved yet.",
"Doppelkopf is a trick-taking card game for four people, mainly played in Germany.",
"Hearthstone is an online collectible card video game, developed by Blizzard Entertainment.",
"Hearts is a four player trick-taking card game, mainly played in North America.",
"Dou Di Zhu is a Chinese card game for three players.",
"Goofspiel is a simple bidding card game for two or more players.",
"One-Card-Poker generalizes the minimal variant Kuhn-Poker.",
"Dominion is a modern deck-building card game similar to Magic: The Gathering."
]
] | 1906.04439 |
[
[
"A Linear Algorithm for Minimum Dominator Colorings of Orientations of\n Paths"
],
[
"Abstract In this paper we present an algorithm for finding a minimum dominator coloring of orientations of paths.",
"To date this is the first algorithm for dominator colorings of digraphs in any capacity.",
"We prove that the algorithm always provides a minimum dominator coloring of an oriented path and show that it runs in $\\mathcal{O}(n)$ time.",
"The algorithm is available at https://github.com/cat-astrophic/MDC-orientations_of_paths/."
],
[
"Introduction",
"Let $G=(V,E)$ be a graph.",
"A set $S\\subset V$ is called a dominating set if every vertex of $V$ is either in $S$ or adjacent to at least one member of $S$ .",
"The domination number of a graph, $\\gamma (G)$ , is the size of a smallest dominating set of $G$ .",
"A dominator coloring of a graph is a proper vertex coloring of the graph which additionally satisfies the property that every vertex dominates some color class in the dominator coloring.",
"Dominator colorings of graphs were first studied by Gera in [6], [7], [8].",
"Dominating sets and dominator colorings are useful for a myriad of problems and have been applied to studying electric power grids [9] and to sensors in networks [2].",
"Since the purpose of this paper is to introduce an algorithm for minimum dominator colorings of digraphs, it is important to know what algorithms exist for this problem in the undirected setting.",
"A linear algorithm which finds the domination number of trees was introduced in [5].",
"Building from this result, [11] established a polynomial time algorithm that provides a minimum dominator coloring of trees.",
"Additionally, it was proved in [1] that the dominator chromatic number, $\\chi _{d}(G)$ , cannot be found in polynomial time in general for many elementary families of graphs including bipartite and planar graphs.",
"The first results on dominator colorings of digraphs were the dominator chromatic number of paths and cycles [4], and that the dominator chromatic number of digraphs is that the dominator chromatic number of a tree is invariant under reversal of orientation [3].",
"Other interesting results established in [4] on dominator colorings of digraphs include the fact that the dominator chromatic number of a subgraph $H\\subset G$ can be larger than the dominator chromatic number of $G$ , as well as that $\\limsup \\limits _{n\\rightarrow \\infty }\\frac{\\chi _{d}(D)}{\\Delta (D)}=\\infty $ .",
"That these results differ from virtually every other vertex coloring problem makes dominator colorings of digraphs of particular interest in graph theory.",
"One of the major results from that paper was the following theorem which provides the minimum dominator chromatic number over all orientations of paths.",
"In the proof, many important structural characterizations relating orientations of paths and dominator colorings were established.",
"Theorem 1 The minimum dominator chromatic number over all orientations of the path $P_{n}$ is given by $\\chi _{d}(P_{n})={\\left\\lbrace \\begin{array}{ll}k+2 & \\mathrm {if}\\ n=4k\\\\k+2 & \\mathrm {if}\\ n=4k+1\\\\k+3 & \\mathrm {if}\\ n=4k+2\\\\k+3 & \\mathrm {if}\\ n=4k+3\\end{array}\\right.",
"}$ for $k\\ge 1$ with the exception $\\chi _{d}(P_{6})=3$ .",
"To conclude the introduction, an example of an oriented path of length five is presented below, and a minimum dominator coloring is provided in the caption.",
"The reader is referred to [10] as the standard reference for domination.",
"Figure: An orientation of the path P 5 P_{5} which has dominator chromatic number 3.",
"The color classes of P 5 P_{5} in a minimum dominator coloring may be given by C 0 ={v 2 ,v 4 }C_{0}=\\lbrace v_{2},v_{4}\\rbrace , C 1 ={v 1 ,v 3 }C_{1}=\\lbrace v_{1},v_{3}\\rbrace , and C 2 ={v 5 }C_{2}=\\lbrace v_{5}\\rbrace ."
],
[
"The Algorithm",
"The purpose of this paper is to present the first algorithm which provides a minimum dominator coloring of a directed graph.",
"In particular, we present an algorithm which provides a minimum dominator coloring of orientations of paths.",
"After providing the algorithm, we will prove that it runs in $\\mathcal {O}(n)$ time.",
"The algorithm works by sequentially through the vertex set, from $v_{1}$ through $v_{n}$ , and colors each vertex in such a a way as to minimize the number of colors used in a proper dominator coloring.",
"From Theorem REF in [4] we know that all vertices with in-degree equal to zero must belong to the same color class.",
"For this reason, the first thing the algorithm checks for is precisely this.",
"Assuming this is not the case, another immediately guaranteed coloring results is that any vertex that is dominated by a vertex of out-degree one must be uniquely colored.",
"Once these two cases are checked for, all that remains in an orientation of a path are vertices whose entire in-neighborhood consists of vertices with out-degree equal to two (notice that this may include vertices with out-degree zero or out-degree one).",
"It is easy to see that, for oriented paths, after removing all vertices with with at least one in-neighbor having out-degree one, what remains are subpaths in which every vertex has either out-degree zero or out-degree two, i.e., subpaths which have the following out-degree sequence pattern: $\\lbrace 0,2,0,\\dots ,0,2,0\\rbrace $ (it is possible that one or both of the end vertices of such a subpath do not have out-degree zero, but since all of the out-degree two vertices are assigned the same color, and since such an end vertex would have out-degree two in the full path, we may ignore end vertices of these subpaths that do not have out-degree zero).",
"From Theorem REF in [4] we know that such oriented paths are minimized in terms of dominator colorings precisely when the first vertex is assigned a color $C^{\\star }$ which is used for all vertices of out-degree zero that are not uniquely colored, and when this color is assigned to every other vertex with out-degree zero (every fourth vertex in the path).",
"For convenience, we refer to these paths as 2-chains since they are maximal subpaths with respect to the density of vertices with out-degree two.",
"Also, 2-chains of length three are an exception to this coloring scheme as they may have both vertices of out-degree zero belong to the same color class.",
"This case is addressed in the algorithm as well, but first we describe the process for handling all other 2-chains.",
"By paying attention to where we are within a 2-chain in an oriented path, we can proceed in order through the vertex set of an oriented path and provide a minimum dominator coloring.",
"The method used in the algorithm is to have a variable $\\alpha \\in \\mathbb {Z}_{2}$ indicate whether or not we should use the color $C^{\\star }$ or a new color when coloring vertices of out-degree zero.",
"The variable $\\beta $ indicates whether we have established the color $C^{\\star }$ yet.",
"Since it turns out that 2-chains of length three may use different colors for the two vertices of out-degree zero if one vertex is colored with $C^{\\star }$ and $C^{\\star }$ is present elsewhere in the path (this will be proven in the theorem below), the algorithm handles 2-chains of length three by coloring both vertices the same if and only if the color $C^{\\star }$ has not yet been established (i.e., if $\\beta $ still has a value of 0).",
"Lastly we address the exception of the path $P_{6}$ which has either the out-degree sequence $\\lbrace 0,2,0,2,0,1\\rbrace $ or its reversal.",
"As is stands, the algorithm would minimally color the reversal of this particular orientation of $P_{6}$ but not this particular orientation of $P_{6}$ .",
"Because this was the only exception listed in Theorem REF from [4], and because checking an input ($n$ ) will not alter the time complexity of the algorithm, we simply check for this occurrence in the same if statement as when we check to see if a 2-chain of length three exists.",
"Finally we are ready to present the algorithm.",
"After stating the algorithm, we provide its time complexity and prove that it always provides a minimum dominator coloring of an oriented path.",
"[h!]",
"[1] Minimum Dominator Coloring Algorithm for Oriented Paths input An orientated path $P_{n}$ of length $n$ initialize $\\mathcal {C}\\leftarrow \\lbrace C_{0}\\rbrace $ initialize $\\mathcal {F} \\leftarrow \\emptyset $ initialize $\\alpha \\leftarrow 0$ initialize $\\beta \\leftarrow 0$ $v_{i} \\in V(P_{n})$ $d^{-}(v_{i}) = 0$ Color $v_{i}$ with $C_{0}$ $\\mathcal {F} \\leftarrow \\lbrace \\mathcal {F}\\cup C_{0}\\rbrace $ $\\exists \\ v_{j} \\in N^{-}(v_{i})\\ \\mathrm {s.t.",
"}\\ d^{+}(v_{j}) = 1$ Color $v_{i}$ uniquely with a new color $C_{|\\mathcal {C}|}$ $\\mathcal {C} \\leftarrow \\lbrace \\mathcal {C}\\cup C_{|\\mathcal {C}|}\\rbrace $ $\\mathcal {F} \\leftarrow \\lbrace \\mathcal {F}\\cup C_{|\\mathcal {C}|}\\rbrace $ $\\alpha \\leftarrow 0$ This indicates the end of a 2-chain $\\alpha = 0$ $\\beta = 0$ Define new color $C^{\\star } = C_{|\\mathcal {C}|}$ Color $v_{i}$ with color $C^{\\star }$ $\\mathcal {C} \\leftarrow \\lbrace \\mathcal {C}\\cup C^{\\star }\\rbrace $ $\\mathcal {F} \\leftarrow \\lbrace \\mathcal {F}\\cup C^{\\star }\\rbrace $ $d^{+}(v_{i+1}) = 2 \\ne d^{+}(v_{i+3})$ or $n = 6$ $\\alpha \\leftarrow 0$ This indicates a 2-chain of length 3 or a $P_{6}$ $\\alpha \\leftarrow 1$ $\\beta \\leftarrow 1$ Color $v_{i}$ with existing color $C^{\\star }$ $\\mathcal {F} \\leftarrow \\lbrace \\mathcal {F}\\cup C^{\\star }\\rbrace $ $\\alpha \\leftarrow 1$ Color $v_{i}$ uniquely with color $C_{|\\mathcal {C}|}$ $\\mathcal {C} \\leftarrow \\lbrace \\mathcal {C}\\cup C_{|\\mathcal {C}|}\\rbrace $ $\\mathcal {F} \\leftarrow \\lbrace \\mathcal {F}\\cup C_{|\\mathcal {C}|}\\rbrace $ $\\alpha \\leftarrow 0$ return $|\\mathcal {C}|$ The Dominator Chromatic Number is: $\\chi _{d}(P_{n}) = |\\mathcal {C}|$ return $\\mathcal {F}$ The coloring of $V(P_{n})$ , i.e., $\\mathcal {F}_{i}=c(v_{i})\\ \\forall \\ v_{i}\\in V(P_{n})$ Now we show that the time complexity of the algorithm is $\\mathcal {O}(n)$ .",
"This result follows since we may count a constant number of things the algorithm needs to check for each vertex, hence the algorithm takes at most $cn$ steps to complete for some $c\\in \\mathbb {N}$ .",
"Next we show that the algorithm provides a minimum dominator coloring of any oriented path.",
"Theorem 2 The Minimum Dominator Coloring Algorithm results in a minimum dominator coloring of every oriented path.",
"Since any vertex with in-degree equal to zero is assigned to the same color class by this algorithm, no counterexample can come from these vertices.",
"Since every vertex with an in-neighbor of out-degree one is colored uniquely by this algorithm, no counterexample can come from these vertices either.",
"Thus any possible counterexample must come from the set of vertices whose entire in-neighborhood consists of vertices of out-degree two.",
"Since the algorithm colors each 2-chain optimally, it suffices to show that if each 2-chain is colored optimally, with the possible exception of 2-chains of length three whenever $C^{\\star }$ already has been used, and the color $C^{\\star }$ is common to all 2-chains which have non-uniquely colored vertices, all vertices of in-degree zero are colored similarly, and all vertices with an in-neighbor of out-degree one are colored uniquely (i.e., the conditions of the first paragraph of this proof are met), that the dominator coloring is minimum.",
"Clearly the color $C^{\\star }$ must be common to all 2-chains (with non-uniquely colored vertices), else there are at least two color classes that can be combined.",
"The MDC algorithm for orientations of paths does ensure that the color class $C^{\\star }$ is the unique color class shared by 2-chains.",
"With this established, it follows that each 2-chain must be (and in fact is) minimally dominator colored, except for possibly some 2-chains of length three which use the color $C^{\\star }$ which was established prior to the instance of that 2-chain, as the only remaining vertices in our path are those vertices of 2-chains which must be colored uniquely.",
"That the resulting dominator coloring is minimum immediately follows."
],
[
"Conclusion",
"In this paper we established the first algorithm which provides a minimum dominator coloring of directed graphs.",
"Specifically, this algorithm provides a minimum dominator coloring for orientations of paths.",
"We proved that this algorithm always results in a minimum dominator coloring of an oriented path, as well as that the algorithm runs in $\\mathcal {O}(n)$ time.",
"The most likely extensions of this algorithm are to orientations of trees and cycles.",
"While results on the dominator chromatic number of orientations of trees exist, specific results for this class of digraphs are not as established as they are for orientations of cycles, a class of graphs for which the dominator chromatic number is entirely determined.",
"However, the acyclic structure of trees (and theoretically of directed acyclic graphs as well) lend them to being possible candidates for easy extensions of this algorithm.",
"To conclude this paper we mention an application of this result.",
"For those interested, a python script, which uses this algorithm and provides a visual of a user selected orientation of a path, can be found online at the repository https://github.com/cat-astrophic/MDC-orientations_of_paths/.",
"Please feel free to use and modify this script to fit your needs!"
]
] | 1906.04523 |
[
[
"Six new rapidly oscillating Ap stars in the Kepler long-cadence data\n using super-Nyquist asteroseismology"
],
[
"Abstract We perform a search for rapidly oscillating Ap stars in the Kepler long-cadence data, where true oscillations above the Nyquist limit of 283.21 {\\mu}Hz can be reliably distinguished from aliases as a consequence of the barycentric time corrections applied to the Kepler data.",
"We find evidence for rapid oscillations in six stars: KIC 6631188, KIC 7018170, KIC 10685175, KIC 11031749, KIC 11296437 and KIC 11409673, and identify each star as chemically peculiar through either pre-existing classifications or spectroscopic measurements.",
"For each star, we identify the principal pulsation mode, and are able to observe several additional pulsation modes in KIC 7018170.",
"We find that KIC 7018170 and KIC 11409673 both oscillate above their theoretical acoustic cutoff frequency, whilst KIC 11031749 oscillates at the cutoff frequency within uncertainty.",
"All but KIC 11031749 exhibit strong amplitude modulation consistent with the oblique pulsator model, confirming their mode geometry and periods of rotation."
],
[
"Introduction",
"Since their discovery by [41], [42], only 70 rapidly oscillating Ap (roAp) stars have been found [80], [37], [17], [8].",
"Progress in understanding their pulsation mechanism, abundance, and the origin of their magnetic fields has been hindered by the relatively small number of known roAp stars.",
"A key difficulty in their detection lies in the rapid oscillations themselves, requiring dedicated observations at a short enough cadence to properly sample the oscillations.",
"In this paper, we show that the Kepler long-cadence data can be used to detect roAp stars, despite their pulsation frequencies being greater than the Nyquist frequency of the data.",
"As a class, the chemically peculiar A type (Ap) stars exhibit enhanced features of rare earth elements, such as Sr, Cr and Eu, in their spectra [54].",
"This enhancement is the result of a stable magnetic field on the order of a few to tens of kG [52], which typically allows for the formation of abundance `spots' on the surface, concentrated at the magnetic poles [69].",
"In most, but not all Ap stars, photometric and spectral variability over the rotation cycle can be observed [1].",
"Such characteristic spot-based modulation manifests as a low-frequency modulation of the light curve which is readily identified, allowing for the rotation period to be measured [19].",
"The roAp stars are a rare subclass of the Ap stars that exhibit rapid brightness and radial velocity variations with periods between 5 and 24 min and amplitudes up to 0.018 mag in Johnson $B$ [45], [39].",
"They oscillate in high-overtone, low-degree pressure (p) modes [70].",
"The excitation of high-overtone p-modes, as opposed to the low-overtones of other pulsators in the classical instability strip is suspected to be a consequence of the strong magnetic field – on the order of a few to tens of kG – which suppresses the convective envelope at the magnetic poles and increases the efficiency of the opacity mechanism in the region of hydrogen ionisation [4], [15].",
"Based on this, a theoretical instability strip for the roAp stars has been published by [15].",
"However, discrepancies between the observed and theoretical red and blue edges have been noted, with several roAp stars identified to be cooler than the theoretical red edge.",
"A further challenge to theoretical models of pulsations in magnetic stars are oscillations above the so-called acoustic cutoff frequency [71], [30].",
"In non-magnetic stars, oscillations above this frequency are not expected.",
"However, in roAp stars the strong magnetic field guarantees that part of the wave energy is kept inside the star in every pulsation cycle, for arbitrarily large frequencies [81].",
"For that reason, no theoretical limit exists to the frequency of the modes.",
"Nevertheless, for a mode to be observed, it has to be excited.",
"Models show that the opacity mechanism is capable of exciting modes of frequency close to, but below, the acoustic cutoff frequency.",
"The excitation mechanism for the oscillations above the acoustic cutoff is thought to be turbulent pressure in the envelope regions where convection is no longer suppressed [16].",
"The magnetic axis of roAp stars is closely aligned with the pulsation axis, with both being inclined to the rotation axis.",
"Observation of this phenomenon led to the development [42] and later refinement [20], [76], [77], [78], [83], [84], [11] of the oblique pulsator model.",
"The roAp stars present a unique testbed for models of magneto-acoustic interactions in stars, and have been widely sought with both ground and space-based photometry.",
"The launch of the Kepler Space Telescope allowed for the detection of oscillations well below the amplitude threshold for ground-based observations, even for stars fainter than 13 magnitude.",
"The vast majority of stars observed by Kepler were recorded in long-cadence (LC) mode, with exposures integrated over 29.43 min.",
"A further allocation of 512 stars at any given time were observed in the short-cadence (SC) mode, with an integration time of 58.85 s. These two modes correspond to Nyquist limits of 283.21 and 8496.18 $$ Hz respectively [12].",
"In its nominal mission, Kepler continuously observed around 150 000 stars in LC mode for 4 yr.",
"The Kepler SC data have been used to discover several roAp stars [46], [5], [7], [80], and to detect pulsation in previously known roAp stars with the extended K2 mission [28], [29].",
"Until now, only SC data have been used for identification of new roAp stars in the Kepler field.",
"However, with the limited availability of SC observation slots, a wide search for rapid oscillators has not been feasible.",
"Although ground-based photometric data have been used to search for roAp stars [48], [36], [62], [26], most previous work in using Kepler to identify such stars has relied solely on SC observations of targets already known to be chemically peculiar.",
"The number of targets in the Kepler field that possess LC data far outweigh those with SC data, but they have been largely ignored in the search for new roAp stars.",
"The key difficulty in searching for rapid oscillations in the LC data is that each pulsation frequency in the Fourier spectrum is accompanied by many aliases, reflected around integer multiples of the sampling frequency.",
"Despite this, it has previously been shown by [58] that the Nyquist ambiguity in the LC data can be resolved as a result of the barycentric corrections applied to Kepler time stamps, leading to a scenario where Nyquist aliases can be reliably distinguished from their true counterparts even if they are well above or below the nominal Nyquist limit.",
"The barycentric corrections modulate the cadence of the photometric observations, so that all aliases above the Nyquist limit appear as multiplets split by the orbital frequency of the Kepler telescope ($1/372.5$ d, 0.03 $$ Hz).",
"Furthermore, the distribution of power in Fourier space ensures that in the absence of errors, the highest peak of a set of aliases will always be the true one.",
"An example of distinguishing aliases is shown in Fig.",
"REF for the known roAp star KIC 10195926.",
"The true pulsation is evident as the highest peak in the LC data, and is not split by the Kepler orbital frequency.",
"Figure: Amplitude spectra of the a) long- and b) short-cadence Kepler data of KIC 10195926, a previously known roAp star .",
"The primary oscillation is detectable even though it lies far above the Nyquist frequency (shown in integer multiples of the Nyquist frequency as red dotted lines) for the LC data.",
"The green curves show the ratio of measured to intrinsic amplitudes in the data, showing the effects of apodization.",
"c) shows the aliased signal at 406.2 Hz of the true oscillation at d), 972.6 Hz, distinguishable by both the Kepler orbital separation frequency (dashed blue lines) and maximum amplitudes.",
"The Python code Pyquist has been used to plot the apodization .This technique, known as super-Nyquist asteroseismology, has previously been used with red giants and solar-like oscillators on a case-by-case basis [14], [88], [49], as well as in combinations of LC data with ground-based observations for compact pulsators [9].",
"Applications in the context of roAp stars have been limited only to frequency verification of SC or other data [27], [80].",
"Our approach makes no assumption about the spectroscopic nature or previous identification of the target, except that its effective temperature lies in the observed range for roAp stars.",
"We further note that super-Nyquist asteroseismology is applicable to the Transiting Exoplanet Survey Satellite [66] and future space-based missions [57], [75].",
"In this paper, we report six new roAp stars whose frequencies are identified solely from their LC data; KIC 6631188, KIC 7018170, KIC 10685175, KIC 11031749, KIC 11296437, and KIC 11409673.",
"These are all found to be chemically peculiar A/F stars with enhanced Sr, Cr, and/or Eu lines.",
"We selected Kepler targets with effective temperatures between 6000 and 10 000 K according to the `input' temperatures of [50].",
"We significantly extended the cooler edge of our search since few roAp stars are known to lie close to the red edge of the instability strip.",
"We used the Kepler LC light curves from Quarters 0 through 17, processed with the PDCSAP pipeline [82], yielding a total sample of 69 347 stars.",
"We applied a custom-written pipeline to all of these stars that have LC photometry for at least four quarters.",
"Nyquist aliases in stars with time-bases shorter than a full Kepler orbital period (4 quarters) have poorly defined multiplets and were discarded from the sample at run-time [58].",
"In addition to the automated search, we manually inspected the light curves of the 53 known magnetic chemically peculiar (mCP) stars from the list of [34].",
"These stars have pre-existing spectral classification, requiring only a super-Nyquist oscillation to be identified as a roAp star."
],
[
"Pipeline",
"The pipeline was designed to identify all oscillations between 580 and 3500 $$ Hz in the Kepler LC data, by first applying a high-pass filter to the light curve, removing both the long-period rotational modulation between 0 and 50 $$ Hz and low-frequency instrumental artefacts.",
"The high-pass filter reduced all power in this given range to noise level.",
"The skewness of the amplitude spectrum values, as measured between 0 and 3500 $$ Hz, was then used as a diagnostic for separating pulsators and non-pulsators, following [60].",
"Stars with no detectable pulsations, either aliased or otherwise, tend to have a skewness lower than unity, and were removed from the sample.",
"After filtering out non-pulsators, each frequency above 700 $$ Hz at a signal-to-noise ratio (SNR) greater than 5 was then checked automatically for sidelobes.",
"These sidelobes are caused by the uneven sampling of Kepler's data points once barycentric corrections to the time stamps have been made, as seen in Fig.",
"REF .",
"A simple peak-finding algorithm was used to determine the frequency of highest amplitude for each frequency in the set of aliases, which was further refined using a three-point parabolic interpolation to mitigate any potential frequency drift.",
"The frequencies above a SNR of 5 were then deemed aliases if their sidelobes were separated by the Kepler orbital frequency for a tolerance of $\\pm $ 0.002 $$ Hz.",
"Frequencies that did not display evidence of Nyquist aliasing were then flagged for manual inspection.",
"The general process for the pipeline can be summarised as follows: High-pass filter the light curve and calculate the skewness of the amplitude spectrum between 0 and 3500 $$ Hz; if skewness is less than unity, move to next star.",
"For each peak greater than 700 $$ Hz with a SNR above 5, identify all sidelobes and determine whether they are separated by the Kepler orbital frequency.",
"If at least one peak is not an alias, flag the star for manual inspection."
],
[
"Apodization",
"The high-pass filter was designed to remove all signals between 0 and 50 $$ Hz.",
"This had the additional effect of removing the reflected signals at integer multiples of the sampling frequency, regardless of whether they are aliased or genuine.",
"As a result, the pipeline presented here cannot reliably identify oscillations close to integer multiples of the sampling frequency (2$\\nu _{\\rm Nyq}$ ).",
"However, we note that if any of these stars are indeed oscillating in these regions, or even above the Nyquist frequency, the measured amplitude will be highly diminished as a result of the non-zero duration of Kepler integration times, a phenomenon referred to as apodization [55], [25] or phase smearing [9].",
"The amplitudes measured from the data ($A_{\\rm measured}$ ) are smaller than their intrinsic amplitudes in the Kepler filter by a factor of $\\eta $ , $\\eta = \\dfrac{A_{\\rm measured}}{A_{\\rm intrinsic}} = {\\rm sinc}\\Big [\\dfrac{\\nu }{2\\nu _{\\rm Nyq}} \\Big ],$ where $\\nu $ and $\\nu _{\\rm Nyq}$ are the observed and Nyquist frequencies, respectively.",
"This equation shows that frequencies lying near integer multiples of the sampling frequency are almost undetectable in Kepler and other photometric campaigns.",
"The factor $\\eta $ is shown as the green curves in Fig.",
"REF .",
"For the results in Sec.",
", both measured and intrinsic amplitudes are provided.",
"Each star found to have non-alias high-frequency pulsations by the pipeline was manually inspected.",
"Of the flagged candidates, 4 were previously identified roAp stars in the Kepler field, KIC 10195926 [46], KIC 10483436 [6], KIC 7582608 [27], and KIC 4768731 [80].",
"The fifth previously known Kepler roAp star KIC 8677585 [7], was not identified by the pipeline, due to the primary frequency of 1659.79 $$ Hz falling just within range of the filtered region.",
"We further identified one more high-frequency oscillator during manual inspection of the 53 stars in the mCP sample of [34].",
"For all six newly identified stars, we calculated an amplitude spectrum in the frequency range around the detected pulsation, following the method of [43].",
"The frequencies were then optimised by non-linear least-squares.",
"The signal-to-noise ratio (SNR) of the spectrum was calculated for the entire light curve by means of a box kernel convolution of frequency width 23.15 $$ Hz (2 d$^{-1}$), as implemented in the Lightkurve Python package [86]."
],
[
"Stellar properties",
"The properties of the six new high-frequency oscillators examined in this work are provided in Table REF .",
"Temperatures were obtained from LAMOST DR4 spectroscopy [89].",
"Since the temperatures of roAp stars are inherently difficult to measure as a result of their anomalous elemental distributions [53], we inflated the low uncertainties in the LAMOST catalogue ($\\sim $ 40 K) to a fixed 300 K. For the one star with an unusable spectrum in LAMOST, KIC 6631188, we took the temperature from the stellar properties catalogue of [50].",
"We derived apparent magnitudes in the SDSS $g-$ band by re-calibrating the KIC $g-$ and $r-$ bands following equation 1 of [63].",
"Distances were obtained from the Gaia DR2 parallaxes using the normalised posterior distribution and adopted length scale model of [3].",
"This produced a distribution of distances for each star, from which Monte Carlo draws could be sampled.",
"Unlike [3], no parallax zero-point correction has been applied to our sample, since it has previously been shown by [60] to not be appropriate for Kepler A stars.",
"Table: Properties of the 6 new roAp stars.Standard treatment of bolometric corrections [85] are unreliable for Ap stars, due to their anomalous flux distributions.",
"Working in SDSS $g$ minimises the bolometric correction, since the wavelength range is close to the peak of the spectral energy distribution of Ap stars.",
"We obtained $g$ -band bolometric corrections using the IsoClassify package [33], which interpolates over the mesa Isochrones & Stellar Tracks (MIST) tables [18] using stellar metallicities, effective temperatures, and surface gravities obtained from [50].",
"Extinction corrections of [23] as queried through the Dustmaps python package [22], were applied to the sample.",
"The corrections were re-scaled to SDSS $g$ following table A1 of [73].",
"To calculate luminosities, we followed the methodology of [60], using a Monte Carlo simulation to obtain uncertainties.",
"Masses were obtained via an interpolation over stellar tracks, and are discussed in more detail in Sec.",
"REF .",
"KIC 6631188 has previously been identified as a rotational variable with a period of 5.029 d [65], or 2.514 d [64].",
"The unfiltered light curve of KIC 6631188 shows a series of low-frequency harmonic signals beginning at multiples of 2.30 $$ Hz (Fig.",
"REF ).",
"Although the highest amplitude signal corresponds to a rotational period of 2.514 d, the true rotation period was confirmed by folding the light curve on the 2.30 $$ Hz frequency, yielding a period of 5.03117 $\\pm $ 0.00004 d. The folded light curve shows clear double-wave spot-based modulation, implying that both magnetic poles are observed.",
"Figure: a) Amplitude spectrum of KIC 6631188 out to the Nyquist frequency of 283.2 Hz.",
"The inset shows the low-frequency region with peaks due to rotation.",
"b) Light curve folded at the rotation period of 5.03 d and binned at a factor of 50:1. c) Amplitude spectrum of KIC 6631188 after a high-pass filter has removed the low-frequency signals – the true oscillation frequency of 1493.52 Hz has the highest amplitude (green).",
"All other peaks flagged as aliases above the 5 SNR are marked in blue.",
"The red dashed lines denote integer multiples of the Nyquist frequency.",
"d) Zoomed region of the primary frequency before, and e), after pre-whitening ν 1 \\nu _1.",
"The residual power in ν 1 \\nu _1 is due to frequency variability.",
"The four sidelobes are due to rotational modulation of the pulsation amplitude (see text).",
"The top x-axis, where shown, is the corresponding frequency in d -1 ^{-1}.After high-pass filtering the light curve, the primary pulsation frequency of 1493.52 $$ Hz is observable in the super-Nyquist regime.",
"We see evidence for rotational splitting through the detection of a quintuplet, indicating an $\\ell =2$ or distorted $\\ell =1$ mode.",
"It seems likely that the star is a pure quadrupole pulsator, unless an $\\ell =1$ mode is hidden at an integer multiple of the sampling frequency – where its amplitude would be highly diminished as a result of apodization.",
"It is also possible that other modes are of low intrinsic amplitude, making their detection in the super-Nyquist regime difficult.",
"We can measure the rotational period of KIC 6631188 from the sidelobe splitting as 5.0312 $\\pm $ 0.0003 d in good agreement with the low-frequency signal.",
"We list the pulsation and rotational frequencies in Table REF .",
"We are able to provide further constraints on the geometry of the star by assuming that the rotational sidelobes are split from the central peak by exactly the rotation frequency of the star.",
"We chose a zero-point in time such that the phases of the sidelobes were equal, and then applied a linear least squares fit to the data.",
"For a pure non-distorted mode, we expect the phases of all peaks in the multiplet to be the same.",
"We find that the phases are not identical, implying moderate distortion of the mode (Table REF ).",
"Table: Linear least squares fit to the pulsation and force-fitted sidelobes in KIC 6631188.",
"The zero-point for the fit is BJD 2455692.84871, and has been chosen as such to force the first pair of sidelobe phases to be equal.The oblique pulsator model can also be applied to obtain geometric constraints on the star's magnetic obliquity and inclination angles, $\\beta $ and $i$ , respectively.",
"The frequency quintuplet strongly suggests that the pulsation in KIC 6631188 is a quadrupole mode.",
"We therefore consider the axisymmetric quadrupole case, where $\\ell =2$ and $m=0$ and apply the relation of [44] for a non-distorted oblique quadrupole pulsation in the absence of limb-darkening and spots: $\\tan {i}\\tan {\\beta }= 4 \\dfrac{A_{+2}^{(2)}+A_{-2}^{(2)}}{A_{+1}^{(2)}+A_{-1}^{(2)}}.$ Here $i$ is the rotational inclination angle, $\\beta $ is the angle of obliquity between the rotation and magnetic axes, and $A_{\\pm 1,2}^{(1,2)}$ are the amplitudes of the first and second sidelobes of the quadrupole pulsation.",
"Using the values of Table REF , we find that $\\tan {i}\\tan {\\beta }=7.4\\,\\pm \\,0.7$ , and provide a summary of values satisfying this relation in Fig.",
"REF .",
"Since $i+\\beta \\ge 90^\\circ $ , both pulsation poles should be visible in the light curve over the rotation cycle of the star, a result consistent with observations of the double-wave light curve with spots at the the magnetic poles.",
"Figure: Possible i+βi + \\beta combinations for the roAp stars where analysis of the multiplets allows us to set constraints on their geometry.",
"The shaded region marks the uncertainty.",
"In stars for which i+β>90 ∘ i + \\beta > 90 ^\\circ , both magnetic poles are observed.",
"For KIC 7018170, only the primary ν 1 \\nu _1 solution has been shown.",
"KIC 10685175 has been omitted as its uncertainty dominates the figure."
],
[
"KIC 7018170",
"The low-frequency variability of KIC 7018170 exhibits no sign of rotational modulation, which is probably a result of the PDCSAP pipeline removing the long-period variability (Fig.",
"REF ).",
"It is therefore unsurprising that KIC 7018170 has not been detected as an Ap star in the Kepler data – the automatic removal of low-frequency modulation causes it to appear as an ordinary non-peculiar star in the LC photometry.",
"Figure: Same as in Fig.",
"for KIC 7018170.",
"In a) however, the long-period rotational modulation has been largely removed by the PDCSAP flux pipeline leading to a jagged light curve (b).",
"Panels f) and g) show the secondary frequencies ν 2 \\nu _2 and ν 3 \\nu _3 extracted after manual inspection of the filtered light curve.The rotational signal is clearly present in the sidelobe splitting of the primary and secondary pulsation frequencies.",
"The high-pass filtered light curve reveals the primary signal, $\\nu _1$ , at 1945.30 $$ Hz, with inspection of the amplitude spectrum revealing two more modes; $\\nu _2$ and $\\nu _3$ , at frequencies of 1920.28 and 1970.32 $$ Hz, respectively.",
"All three of these modes are split by 0.16 $$ Hz which we interpret as the rotational frequency.",
"KIC 7018170 exhibits significant frequency variability during the second half of the data, which destroys the clean peaks of the triplets.",
"To analyse them in detail, we analysed only the first half of the data where frequency variability is minimal.",
"This provided a good balance between frequency resolution and variability in the data.",
"To estimate the large frequency separation, $\\Delta \\nu $ , defined as the difference in frequency of modes of the same degree and consecutive radial order, we apply the general asteroseismic scaling relation, $\\dfrac{\\Delta \\nu }{\\Delta \\nu _{{\\odot }}} = \\sqrt{\\dfrac{\\rho }{\\rho _{\\odot }}} = \\dfrac{(M/{\\rm M}_{\\odot })^{0.5} (T_{\\rm eff} / {\\rm T}_{\\rm eff, \\odot })^3}{(L/{\\rm L}_{\\odot })^{0.75}}$ with adopted solar values $\\Delta \\nu _\\odot $ = 134.88 $\\pm $ 0.04 $$ Hz, and T$_{\\rm eff, \\odot } = 5777$ K [32].",
"Using the stellar properties in Table REF , we estimate the large separation as 55.19 $\\pm $ 7.27 $$ Hz.",
"The separation from the primary frequency $\\nu _1$ to $\\nu _2$ and $\\nu _3$ are 24.86 and 25.18 $$ Hz, respectively, indicating that the observed modes are likely of alternating even and odd degrees.",
"While we can not determine the degrees of the modes from the LC data alone, it suggests that the primary frequency $\\nu _1$ is actually a quintuplet with unobserved positive sidelobes.",
"If $\\nu _1$ is instead a triplet, it would have highly asymmetric rotational sidelobe peak amplitudes which the other modes do not exhibit.",
"Asymmetric sidelobe amplitudes are a signature of the Coriolis effect [10], but whether this is the sole explanation for such unequal distribution of power in the amplitude spectra can not be known without follow-up observations.",
"$\\nu _1$ is thus more likely to be a quintuplet as the large separation would then be 50.04 $$ Hz, a value much closer to the expected result.",
"The positive suspected sidelobes at frequencies of 1945.46 and 1945.62 $$ Hz have a SNR of 1.16 and 1.78 respectively, well below the minimum level required for confirmation.",
"To this end, we provide a fit to the full quintuplet in Table REF , but note that the frequencies should be treated with caution.",
"Assuming that $\\nu _2$ and $\\nu _3$ are triplets, while $\\nu _1$ is a quintuplet, we forced the rotational sidelobes to be equally separated from the pulsation mode frequency by the rotation frequency.",
"Lacking the rotational signal from the low-frequency amplitude spectrum, we instead obtained the rotational frequency by examining the variability in amplitudes of the modes themselves.",
"As the star rotates, the observed amplitudes of the oscillations will modulate in phase with the rotation.",
"We provide a full discussion of this phenomenon in Sec.",
"REF .",
"We obtained a rotation frequency of 0.160 $\\pm $ 0.005 $$ Hz, corresponding to a period of 72.7 $\\pm $ 2.5 d. We used this amplitude modulation frequency to fit the multiplets by linear least squares to test the oblique pulsator model.",
"By choosing the zero-point in time such that the phases of the $\\pm \\nu _{\\rm rot}$ sidelobes of $\\nu _1$ are equal, we found that $\\nu _1$ , does not appear to be distorted, and $\\nu _3$ only slightly.",
"$\\nu _2$ is heavily distorted, as shown by the unequal phases of the multiplet in Table REF .",
"Table: Linear least squares fit to the pulsation and force-fitted sidelobes in KIC 7018170.",
"The zero-point for the fit is BJD 2455755.69582, and has been chosen as such to force the sidelobe phases of ν 1 \\nu _1 to be equal.We can again constrain the inclination and magnetic obliquity angles for the modes.",
"In the case of a pure dipole triplet, $\\tan {i}\\tan {\\beta } = \\dfrac{A_{+1}^{(1)}+A_{-1}^{(1)}}{A_{0}^{(1)}},$ where again $A_{\\pm 1}^{(1)}$ are the dipole sidelobe amplitudes, and $A_{0}^{(1)}$ is the amplitude of the central peak.",
"Using Table REF , we find that $\\tan {i}\\tan {\\beta } = 1.0\\,\\pm \\,0.2$ , and $\\tan {i}\\tan {\\beta } = 1.7\\,\\pm \\,0.4$ for $\\nu _2$ , and $\\nu _3$ , respectively.",
"Using Eqn.",
"REF , we find $\\tan {i}\\tan {\\beta } = 2.4\\,\\pm \\,0.4$ for $\\nu _1$ , which agrees with $\\nu _3$ within the large errors, while disagreeing with $\\nu _2$ , which appears to be $\\pi $ rad out of phase.",
"We provide a summary of values satisfying these relations in Fig.",
"REF ."
],
[
"KIC 10685175",
"KIC 10685175 was detected by manual inspection of the mCP stars from [34], and was not flagged by the pipeline.",
"The star shows obvious rotational modulation in the low-frequency region of the amplitude spectrum (Fig.",
"REF ).",
"The period of rotation, 3.10198 $\\pm $ 0.00001 d was determined from the low-frequency signal at 0.322 $$ Hz.",
"Figure: a) Amplitude spectrum of KIC 10685175 out to the nominal Nyquist frequency.",
"The inset shows the low-frequency region of the spectrum corresponding to the rotation frequency.",
"b) Light curve folded at the rotation period of 5.03 d and binned at a ratio of 50:1. c) Amplitude spectrum of KIC 10685175 after pre-whitening – the true oscillation frequency of 2783.00 Hz can be observed as the signal of maximum amplitude (green).",
"All other peaks flagged as aliases above the 5 SNR are marked in blue.",
"The red dashed lines denote integer multiples of the Nyquist frequency.",
"d) Zoomed region of the primary frequency before, and e), after pre-whitening ν 1 \\nu _1.To study the roAp pulsations, we subtracted the rotational frequency and its first 30 harmonics.",
"Although the amplitude spectrum is too noisy to reveal Kepler orbital sidelobe splitting, the true peak is evident as the signal with the highest power: 2783.01 $$ Hz.",
"This frequency lies close to a multiple of the Kepler sampling rate, and thus has a highly diminished amplitude.",
"The primary frequency appears to be a quintuplet split by the rotational frequency.",
"However, the low SNR of the outermost rotational sidelobes necessitates careful consideration.",
"The outer sidelobes, at frequencies of 2775.55 and 2790.47 $$ Hz have a SNR of 1.28 and 0.49 respectively.",
"Similar to KIC 7018170, we provide a fit to the full suspected quintuplet in Table REF , but again note that the frequencies of the outermost sidelobes should be treated with caution.",
"If we are to assume that the pulsation is a triplet, while ignoring the outer sidelobes, then Eqn.",
"REF yields a value of $\\tan {i}\\tan {\\beta }=0.9\\,\\pm \\,0.4$ , implying that $i+\\beta <90^\\circ $ and that only one pulsation pole is observed.",
"The large uncertainty however indicates that either one or two poles can be observed if the pulsation is modelled as a triplet.",
"On the other hand, if we consider the star as a quadrupole pulsator, we obtain a value of $\\tan {i}\\tan {\\beta }=1.7\\,\\pm \\,1.6$ , a result almost completely dominated by its uncertainty.",
"We again investigate the distortion of the mode by assuming that the multiplet is split by the rotation frequency, and find that all phases agree within error, implying minimal distortion of the mode (Table REF ).",
"However, it should be noted that the low SNR of the spectrum greatly inflates the uncertainties on the amplitudes and phases of the fit.",
"Table: Linear least squares fit to the pulsation and force-fitted sidelobes in KIC 10685175.",
"The zero-point for the fit is BJD 2455689.78282, and has been chosen as such to force the first pair of sidelobe phases to be equal."
],
[
"KIC 11031749",
"KIC 11037149 does not appear to demonstrate spot-based amplitude modulation in its light curve (Fig.",
"REF ), nor does it show signs of rotational frequency splitting - consistent with a lack of rotational modulation.",
"Despite this, it is clear that it possesses unusual chemical abundances of Sr, Cr, and Eu from its spectrum (Sec.",
"REF ).",
"We theorise two possibilities behind the lack of observable modulation.",
"If the angles of inclination or magnetic obliquity are close to $0^\\circ $ , no modulation would be observed since the axis of rotation is pointing towards Kepler.",
"However, this assumes that the chemical abundance spots are aligned over the magnetic poles, which has been shown to not always be the case [40].",
"Another possibility is that the period of rotation could be much longer than the time-base of the Kepler LC data.",
"While the typical rotational period for A-type stars is rather short [68], [67], [56], the rotation for Ap types can exceed even 10 yr due to the effects of magnetic braking [47].",
"Indeed, a non-negligible fraction of Ap stars are known to have rotational periods exceeding several centuries [51], and so it is possible for the star to simply be an extremely slow rotator.",
"The aliased signal of the true pulsation is visible even in the unfiltered LC data (Fig.",
"REF ).",
"After filtering, we identified one pulsation frequency ($\\nu _1$ : 1372.72 $$ Hz), and provide a fit in Table REF .",
"With no clear multiplet structure around the primary frequency, we are unable to constrain the inclination and magnetic obliquity within the framework of the oblique pulsator model.",
"Indeed, without any apparent low-frequency modulation we are unable to even provide a rotation period.",
"This represents an interesting, although not unheard of challenge for determining the rotation period.",
"Since the photometric and spectral variability originate with the observed spot-based modulation, neither method can determine the rotation period without a longer time-base of observations.",
"Regardless, we include KIC 11031749 in our list of new roAp stars as it satisfies the main criterion of exhibiting both rapid oscillations and chemical abundance peculiarities."
],
[
"KIC 11296437",
"KIC 11296437 is a known rotationally variable star [65], whose period of rotation at 7.12433 $\\pm $ 0.00002 d is evident in the low-frequency region of the amplitude spectrum (Fig.",
"REF ).",
"Folding the light curve on this period shows only a single spot or set of spots, implying that $i$ +$\\beta < 90^\\circ $ .",
"The primary pulsation frequency was found to be 1409.78 $$ Hz in the high-pass filtered light curve.",
"This mode shows two sidelobes split by 1.625 $$ Hz which is in good agreement with the rotation frequency.",
"Two low-frequency modes, $\\nu _2$ and $\\nu _3$ , are also present below the Nyquist frequency, at 126.79 and 129.15 $$ Hz respectively.",
"These modes are clearly non-aliased pulsations, as they are not split by the Kepler orbital period.",
"However, neither of them are split by the rotational frequency.",
"We provide a fit to these frequencies in Table REF .",
"We apply the oblique pulsator model to $\\nu _1$ by assuming the mode is split by the rotation frequency, and find that all three phases agree within error, implying that $\\nu _1$ is not distorted.",
"The same test can not be applied to $\\nu _2$ and $\\nu _3$ due to their lack of rotational splitting.",
"We can further constrain the geometry of the star by again considering the sidelobe amplitude ratios (Eq.",
"REF ).",
"Using the values in Table REF , we find that $\\tan {i}\\tan {\\beta } = 0.11\\,\\pm \\,0.01$ , and provide a summary of angles satisfying these values in Fig.",
"REF , which demonstrates that only one pulsation pole should be observed over the rotation cycle ($i$ +$\\beta < 90^\\circ $ ).",
"Table: Linear least squares fit to the pulsation and force-fitted sidelobes in KIC 11296437.",
"The zero-point for the fit is BJD 2455690.64114.KIC 11296437 is highly unusual in that it displays both high-frequency roAp pulsations and low-frequency $p-$ mode pulsations that are typically associated with $\\delta $ Scuti stars.",
"A lack of rotational splitting in the low-frequency modes suggests that the star might be a binary composed of a $\\delta $ Scuti and roAp star component.",
"If KIC 11296437 is truly a single component system, then it would pose a major challenge to current theoretical models of roAp stars.",
"In particular, the low-frequency modes at 126.79 and 129.15 $$ Hz are expected to be damped by the magnetic field according to previous theoretical modelling [70].",
"KIC 11296437 would be the first exception to this theory amongst the roAp stars.",
"On the other hand, it would also be highly unusual if KIC 11296437 were a binary system.",
"It is rare for Ap stars to be observed in binaries, and much more so for roAp stars.",
"Currently, there is one known roAp star belonging to a spectroscopic binary [24], with several other suspected binaries [74].",
"Stellar multiplicity in roAp stars is important for understanding their evolutionary formation and whether tidal interactions may inhibit their pulsations."
],
[
"KIC 11409673",
"KIC 11409673 is a peculiar case, as it has previously been identified as an eclipsing binary, and later a heartbeat binary [38].",
"We note that a radial velocity survey of heartbeat stars has positively identified KIC 11409673 as a roAp star [79].",
"Here we provide independent confirmation of this result through super-Nyquist asteroseismology, as their result has been ignored in later catalogues of roAp stars.",
"KIC 11409673 has a clear low-frequency variation at 0.94 $$ Hz, corresponding to a rotational period of 12.3107 $\\pm $ 0.0003 d. Similar to KIC 6631188, the low-frequency region is dominated by a higher amplitude signal at 2$\\nu _{\\rm rot}$ consistent with observations of the double-wave nature, as seen in Fig.",
"REF .",
"The rotation period of 12.31 d is found by folding the light curve, and is confirmed after high-pass filtering the light curve and examining the triplet centred around the primary frequency of 2500.93 $$ Hz.",
"Similar to KIC 7018170, KIC 11409673 exhibits strong frequency variation which negatively affects the shape of the multiplets.",
"We thus split the LC data into four equally spaced sections and analysed the multiplet separately in each section.",
"This reduced the issues arising from frequency variation, despite leading to a decrease in frequency resolution.",
"The results of the least squares analysis for the first section of data are presented in Table REF .",
"Again, applying the oblique pulsator model by assuming that the sidelobes be separated from the primary frequency by the assumed rotation frequency, we fit each section of data via least squares.",
"By choosing the zero-point in time such that the phases of the sidelobes are equal, we are able to show that the mode is not distorted, as the three phases agree within error.",
"This is the case for all four separate fits.",
"The results of this test for the first section of the data are shown in Table REF , with the remaining sections in Appendix REF .",
"We find that $\\tan {i}\\tan {\\beta }=6.1\\,\\pm \\,1.1$ using Eqn.",
"REF for the first section of data, implying that both spots are observed in agreement with the light curve.",
"Table: Linear least squares fit to the pulsation and force-fitted sidelobes in KIC 11409673.",
"The zero-point for the fit is BJD 2455144.43981.",
"The data have been split into four equally spaced sets, with the sidelobes force-fitted in each set.",
"We show only the results of the first-set below and provide the rest in Appendix .",
"The results for each set are similar, and agree within the errors."
],
[
"Acoustic cutoff frequencies",
"As discussed in Sec.",
", several roAp stars are known to oscillate above their theoretical acoustic cutoff frequency ($\\nu _{\\rm ac}$ ).",
"The origin of the pulsation mechanism in these super-acoustic stars remains unknown, and presents a significant challenge to theoretical modelling.",
"We therefore calculate whether the stars presented here oscillate above their theoretical acoustic cutoff frequency following the relation $\\dfrac{\\nu _{\\rm ac}}{\\nu _{\\rm ac, \\odot }} = \\dfrac{{\\rm M}/{\\rm M}_\\odot ({\\rm T}_{\\rm eff}/{\\rm T}_{\\rm eff, \\odot })^{3.5}}{{\\rm L}/{\\rm L}_\\odot },$ where $\\nu _{\\rm ac, \\odot }$ = 5300 $$ Hz is the theoretical acoustic cutoff frequency of the Sun [35].",
"Using the values provided in Table REF , we find that KIC 7018170 and KIC 11409673 both oscillate above their theoretical limit (1739.5 and 2053.7 $$ Hz respectively).",
"The remaining stars do not oscillate above their theoretical acoustic cutoff frequency.",
"KIC 11031739, however, lies almost exactly on the border of the acoustic cutoff frequency within the errors."
],
[
"Spectral classification",
"The LAMOST [89] survey has collected low-resolution spectra between 3800-9000 Å for objects in the Kepler field.",
"We obtained LAMOST spectra from the 4th Data Release (DR4).",
"All stars presented here have at minimum one low-resolution spectrum available from LAMOST.",
"However, the spectrum of KIC 6631188 is of unusable SNR.",
"We thus obtained a high-resolution spectrum of KIC 6631188 on April 17 2019 using the HIRES spectrograph [87] at the Keck-I 10-m telescope on Maunakea observatory, Hawai`i.",
"The spectrum was obtained and reduced as part of the California Planet Search queue [31].",
"We obtained a 10-minute integration using the C5 decker, resulting in a S/N per pixel of 30 at $\\sim $ 6000 Å with a spectral resolving power of $R\\sim $ 60 000.",
"This spectrum has been down-sampled to match the MK standard spectrum.",
"Fig.",
"REF presents the spectra of KIC 6631188, KIC 7018170, KIC 11031749, KIC 11296437, and KIC 11409673, with MK standard stars down-sampled to match the resolution of either the HIRES or LAMOST via the SPECTRES package [13].",
"KIC 10685175 has a pre-existing spectral classification of A4 V Eu [34], and thus is not re-classified in this work.",
"In KIC 6631188, there is a strong enhancement of Sr ii at 4077 Å and 4215 Å.",
"The 4111 Å line of Cr ii is present, which is used to confirm a Cr peculiarity, but the strongest line of Cr ii 4172 Å line is not enhanced.",
"The hydrogen lines look narrow for a main-sequence star, but the metal lines are well bracketed by A9 V and F1 V. We place this star at F0 V Sr. KIC 7018170 shows evidence of chemical peculiarities which are only mild, but the spectrum is of low SNR.",
"The 4077 Å line is very strong, which is indicative of an over abundance of Sr or Cr or both, but matching peculiarities in other lines of these elements are less clear.",
"Sr ii at 4215 Å is marginally enhanced, and the 4111 and 4172 Å lines of Cr ii are also only marginally enhanced.",
"The Eu ii line at 4205 Å is significantly enhanced and is matched with a small enhancement at $4128-4132$ Å.",
"We place KIC 7018170 as a F2 V (SrCr)Eu type star.",
"Figure: Keck and LAMOST spectra of KIC 6631188, KIC 7018170, KIC 11031749, KIC 11296437, and KIC 11409673 from top to bottom.",
"MK standard spectra have been down-sampled to match the LAMOST resolution.",
"MK standard spectra have been obtained from https://web.archive.org/web/20140717065550/http://stellar.phys.appstate.edu/Standards/std1_8.htmlThe Ca ii K line in KIC 11031749 is broad and a little shallow, typical of magnetic Ap stars.",
"The 4077 Å line is very strong, suggesting enhancement of Sr and/or Cr.",
"A mild enhancement of other Sr and Cr lines suggests both are contributing to the enhancement of the 4077 Å line.",
"There is mild enhancement of the Eu ii 4205 Å line and the 4128-4132 Å doublet suggesting Eu is overabundant.",
"It is noteworthy that the Ca i 4226 Å line is a little deep.",
"We thus classify KIC 11031749 as F1 V SrCrEu.",
"In KIC 11296437, there is a strong enhancement of Eu ii at 4130 Å and 4205 Å.",
"There is no clear enhancement of Sr ii at 4216 Å but a slightly deeper line at 4077 Å which is also a line of Cr ii.",
"The 4111 Å line of Cr ii is present, which is used to confirm a Cr peculiarity, but the 4172 Å line does not look enhanced which is normally the strongest line.",
"The hydrogen lines look narrow for a main-sequence star, but the metal lines are well bracketed by A7 V and F0 V. We place this star at A9 V EuCr.",
"For KIC 11409673, there is enhanced absorption at 4216 Å which is a classic signature of a Sr overabundance in an Ap star.",
"This is usually met with an enhancement in the 4077 Å line, but that line is also a line of Cr.",
"The 4077 Å line is only moderately enhanced, but enough to support a classification of enhanced Sr.",
"Since the 4172 Å line is normal, it appears that Cr is not enhanced.",
"Other Cr lines cannot be relied upon at this SNR.",
"Since the Eu ii 4205 Å absorption line is strong, it appears that Eu is overabundant.",
"There is no other evidence for a Si enhancement.",
"The Ca ii K line is a little broad and shallow for A9, which suggests mild atmospheric stratification typical of magnetic Ap stars.",
"The hydrogen lines are a good fit intermediate to A7 and F0.",
"We thus classify this star as A9 V SrEu.",
"Figure: Left panel: Positions of the previously known (blue) and new roAp stars (red) discussed in this paper.",
"Uncertainties have only been shown on the new roAp stars for clarity.",
"Although stellar tracks are computed in 0.05-M ⊙ _\\odot intervals, only every second track has been displayed here.",
"Right panel: Interpolated and observed frequencies from pulsation modelling of the roAp stars, coloured by effective temperature.",
"The observed frequencies are taken as the signal of highest amplitude.",
"Circles mark previously known roAp stars, whereas outlined circles mark the six new stars.",
"Uncertainties on the interpolated frequencies are obtained through a Monte Carlo simulation."
],
[
"Positions in the H-R diagram",
"To place our sample on the HR diagram (Fig.",
"REF ), we derived new stellar tracks based on the models of [16].",
"We performed linear, non-adiabatic pulsation calculations for a grid of models covering the region of the HR diagram where roAp stars are typically found.",
"We considered models with masses between 1.4 and 2.5 M$_\\odot $ , in steps of 0.05 M$_\\odot $ , and fixed the interior chemical composition at $Y=0.278$ and $X=0.705$ .",
"The calculations followed closely those described for the polar models discussed in [16].",
"The polar models consider that envelope convection is suppressed by the magnetic field, a condition required for the excitation by the opacity mechanism of high radial order modes in roAp stars.",
"Four different cases were considered for each fixed effective temperature and luminosity in the grid.",
"The first case considered an equilibrium model with a surface helium abundance of $Y_{\\rm surf}=0.01$ and an atmosphere that extends to a minimum optical depth of $\\tau _{\\rm min}=3.5\\times 10^{-5}$ .",
"For this case the pulsations were computed with a fully reflective boundary condition.",
"The other three cases considered were in all similar to this one, except that the above options were modified one at a time to: $Y_{\\rm surf}=0.1$ ; $\\tau _{\\rm min}=3.5\\times 10^{-4}$ ; transmissive boundary condition [16].",
"We provide a summary of these models in Table REF .",
"We note that the impact of the choice of $Y$ and $X$ on the frequencies of the excited oscillations is negligible compared to the impact of changing the aspects of the physics described above.",
"Table: Model parameters of the non-adiabatic calculations.",
"Shown are the surface helium abundance Y surf Y_{\\rm surf}, minimum optical depth τ min \\tau _{\\rm min}, and outer boundary condition in the pulsation code.For each fixed point in the track we calculated the frequency of maximum growth rate as a comparison to our observed frequencies.",
"The observed frequency was assumed to be the mode of highest linear growth rate.",
"However, it should be noted that this may not necessarily be the case.",
"Using these tracks, we performed linear interpolation to obtain an estimate of the masses and frequencies derived from modelling.",
"Uncertainties on the masses have been artificially inflated to account for uncertainties in metallicity, temperature and luminosity (0.2, 0.1, and 0.05 $\\rm M/\\rm M_\\odot $ respectively), following [60].",
"An extra error component of 0.1 $\\rm M/\\rm M_\\odot $ is included to account for unknown parameters in the stellar modelling, such as overshooting and mixing length.",
"These four contributions are combined in quadrature, yielding a fixed uncertainty of 0.25 $\\rm M/\\rm M_\\odot $ .",
"Frequency interpolation is performed for each model (1 through 4), with the plotted value being the median of these results.",
"The uncertainty in the interpolation of both frequency and mass is obtained from a Monte Carlo simulation sampled from the uncertainty in the temperature and luminosity of the stars.",
"We s how the results of the positions in the H-R diagram and comparison of interpolated frequencies in Fig.",
"REF .",
"For frequencies below $\\sim $ 1800 $$ Hz the agreement between theory and observations is reasonable, albeit with discrepancies on a star-by-star case that may be due to an incomplete modelling of the physics of these complex stars.",
"However, for stars with higher characteristic frequencies there seems to be two distinct groups, one lying below and the other clearly above the 1:1 line.",
"Coloured by temperature, we note that the group lying above the 1:1 line tends to be cooler in general.",
"The suppression of envelope convection is key to the driving of roAp pulsations by the opacity mechanism.",
"As it is harder for suppression to take place in the coolest evolved stars, one may question whether an additional source of driving is at play in these stars.",
"In fact, it has been shown in a previous work that the driving of the very high frequency modes observed in some well known roAp stars cannot be attributed to the opacity mechanism.",
"It was argued that they may, instead, be driven by the turbulent pressure if envelope convection is not fully suppressed [16].",
"Whether that mechanism could contribute also to the driving of the modes observed in the stars laying clearly above the 1:1 line on the right panel of Fig.",
"10 is something that should be explored in future non-adiabatic modelling of roAp stars."
],
[
"Intrinsic amplitude and phase variability",
"Many of the known roAp stars have shown significant variation in the amplitudes and phases of their pulsation frequencies over the observation period.",
"To examine this variability in our sample, we conducted a time and frequency domain analysis where a continuous amplitude spectrum was generated by sliding a rectangular window across the light curve.",
"The Fourier amplitude and phase was then calculated within the window at each point.",
"For each star, the window length was chosen to be $100/\\nu _1$ d in width to minimise phase and amplitude uncertainty whilst correctly sampling the frequency.",
"Both rectangular and Gaussian windows were tested and found to have minimal difference in the resultant amplitudes.",
"Before calculating the variability, the rotational frequency and corresponding harmonics were pre-whitened manually to minimise any potential frequency beating.",
"Amplitude modulation of the normal modes provides a measure of the rotation period of these stars and confirms the nature of their oblique pulsations.",
"If the pulsation amplitudes goes to zero, then a node crosses the line-of-sight and is observed, and (for a dipole mode) the pole is at $90^\\circ $ to the line-of-sight.",
"However, if the geometry is such that the amplitude never goes to zero (e.g.",
"$\\alpha $ Cir), then we must always see a pole.",
"This leads to a periodic modulation of the pulsation amplitude, which is different to the spot-based rotational modulation seen in the light curves of Ap stars in general.",
"Similarly, a $\\pi $ rad phase change should be observed in the phase of the frequency whenever a node crosses the line-of-sight.",
"The continuous amplitude spectrum and corresponding rotational signal is shown in Fig.",
"REF .",
"The rotation signal was obtained by examining the amplitude spectrum of the modulation along the primary frequency of each star.",
"KIC 6631188 has obvious low-frequency modulation in its light curve, making identification of the rotation period straightforward (Sec. ).",
"Regardless, it makes for a useful test case for confirming modulation of its primary frequency.",
"The amplitude spectrum of the modulation signal in Fig.",
"REF has a curious peak at twice the rotational frequency, 4.60 $$ Hz, corresponding to a period of 2.52 d. This result implies that the spots are not necessarily aligned along the magnetic and pulsation axes, as was shown to be possible by [40].",
"The analysis of KIC 7018170 greatly benefits from amplitude modulation of its modes, as the PDCSAP flux pipeline removes any low-frequency content in the light curve.",
"The modulation is calculated for all three modes present in the LC data, with the amplitude spectrum taken on the weighted average signal, which is thus dominated by $\\nu _1$ .",
"The frequency components of each mode multiplet are in phase, in good agreement with the oblique pulsator model.",
"Indeed, it is quite remarkable that the secondary modes ($\\nu _2$ , $\\nu _3$ ) in KIC 7018170 exhibit such clear amplitude modulation despite being of low SNR (Fig.",
"REF ).",
"The amplitude spectrum of the variation in the signal is found to peak at 0.160 $$ Hz agreeing with the rotation frequency identified from sidelobe splitting.",
"No evidence of amplitude modulation can be found in KIC 11031749.",
"We can however speculate on the nature of the lack of amplitude modulation and attribute it to two possibilities; either the position of the pulsation pole does not move relative to the observer, or the rotation period is much longer than the 4-yr Kepler data.",
"Thus, no rotation period can be ascribed based on modulation of the principal frequency.",
"KIC 11296437 shows a periodic signal in its amplitude modulation corresponding to the rotational period for only the high-frequency ($\\nu _1$ ) mode, with a frequency of 1.63 $$ Hz.",
"The low-frequency modes ($\\nu _2$ , $\\nu _3$ ) show no evidence of amplitude modulation, suggesting that they could possibly belong to an orbital companion.",
"KIC 11409673 has a clear low-frequency signal and harmonic present in the light curve beginning at 0.940 $$ Hz.",
"Amplitude modulation of its primary oscillation frequency shows evidence of rotation in good agreement with the low-frequency signal.",
"The peak in the amplitude spectrum of the modulation confirms the rotation period of 12.310 d derived in Sec. .",
"Fig.",
"REF shows the variation of the pulsation amplitude and phase over the rotation period of the two stars in this work found to have observable phase crossings, KIC 7018170 and KIC 11409673.",
"The maximum amplitude coincides with light maximum, which is expected when the spots producing the light variations are closely aligned with the magnetic and pulsation poles.",
"The amplitude does not reach zero in KIC 7018170, but almost does in KIC 11409673.",
"Since the amplitude does not go to zero for KIC 7018170, we can see that the mode is distorted."
],
[
"Phase modulation from binarity",
"Both KIC 6631188 and KIC 11031749 show long term phase variations independent of their rotation.",
"If the frequency variability of KIC 11031749 is modelled as modulation of its phase due to binary reflex motion from an orbital companion, its orbital properties can be derived following the phase modulation technique [59].",
"To examine this, the light curve was separated into 5-day segments, where the phase of each segment at the primary frequency was calculated through a discrete Fourier transform.",
"We then converted the phases into light arrival times ($\\tau $ ) by dividing by the frequency, from which a map of the binary orbit was constructed.",
"A Hamiltonian Markov Chain Monte Carlo sampler was then applied to fit the time delay curve through the use of PyMC3 [72].",
"The sampler was run simultaneously over 4 chains for 5000 draws each, with 2000 tuning steps.",
"The resulting fit is shown in Fig.",
"REF , with extracted orbital parameters in Table REF .",
"These parameters give a binary mass function $f(m_1, m_2, sini)$ = 0.000 327 M$_\\odot $ .",
"Assuming a primary mass $M_1$ from Table REF of 1.78 M$_\\odot $ , we obtain a lower limit on the mass of the companion to be 0.105 $M_\\odot $ , placing its potential companion as a low-mass M-dwarf.",
"Figure: Left panel: Observed time delay for KIC 6631188.",
"The time delay, τ\\tau , is defined such that it is negative when the pulsating star is nearer to us than the barycenter of the system.",
"The blue line is not a orbital solution fit, but rather the time delay as obtained by binning the signal for clarity.",
"If the observed signal is truly periodic, it appears to be on a timescale longer than the LC data.",
"Right panel: Observed time delays (black dots) and the fitted model (green line) for KIC 11031749.",
"The phase modulation suggests a binary system of low eccentricity, whose orbital period is comparable to or longer than the time-base of the LC photometry.Table: Orbital parameters of KIC 11031749 obtained through phase modulation.",
"ϖ\\varpi is the angle from the nodal point to the periapsis, ii is the inclination angle.",
"aa is the semimajor axis, ee is the eccentricity, and φ p \\phi _p is the phase of periastron.KIC 6631188 also shows signs of frequency modulation (Fig.",
"REF ).",
"However, if this modulation is truly from a stellar companion then its orbital period must be much longer than the time base of the Kepler data.",
"The PM method can only be used when at least one full binary orbit is observed in the time delay curve.",
"Thus, no orbital solution can be presented here.",
"It is important to note the scarcity of Ap stars in binary systems.",
"Indeed, the much smaller subset of roAp stars have a low chance of being found in a binary [2], [61], [21], however, few techniques can adequately observe the low-mass companion presented here.",
"Although frequency modulation in roAp stars has been inferred in the past to be a consequence of binary motion, two out of the six stars presented in this work show evidence of coherent frequency/phase modulation.",
"Whether this modulation is a consequence of changes in the pulsation cavity, magnetic field, or externally caused by orbital perturbations of a companion remains to be seen, and requires spectroscopic follow-up to rule out orbital motion via a radial velocity analysis."
],
[
"Conclusions",
"We presented the results of a search for rapid oscillators in the Kepler long-cadence data using super-Nyquist asteroseismology to reliably distinguish between real and aliased pulsation frequencies.",
"We selected over 69 000 stars whose temperatures lie within the known range of roAp stars, and based on a search for high-frequency non-alias pulsations, have detected unambiguous oscillations in six stars - KIC 6631188, KIC 7018170, KIC 10685175, KIC 11031749, KIC 11296437, and KIC 11409673.",
"LAMOST or Keck spectra of five of these stars shows that they exhibit unusual abundances of rare earth elements, the signature of an Ap star, with the final target , KIC 10685175, already being confirmed as chemically peculiar in the literature.",
"This research marks a significant step in our search for roAp stars, and indeed, all high-frequency pulsators.",
"To the best of our knowledge, this is the first time super-Nyquist asteroseismology has been used solely for identification of oscillation modes to such a high frequency.",
"Although we expect many new roAp stars to be found in the TESS Data Releases, Kepler had the advantage of being able to observe stars of much fainter magnitude for a longer time-span, revealing pulsations of lower amplitude."
],
[
"Acknowledgements",
"We are thankful to the entire Kepler team for such incredible data.",
"DRH gratefully acknowledges the support of the Australian Government Research Training Program (AGRTP) and University of Sydney Merit Award scholarships.",
"This research has been supported by the Australian Government through the Australian Research Council DECRA grant number DE180101104.",
"DLH and DWK acknowledge financial support from the Science and Technology Facilities Council (STFC) via grant ST/M000877/1.",
"MC is supported in the form of work contract funded by national funds through Fundação para a Ciência e Tecnologia (FCT) and acknowledges the supported by FCT through national funds and by FEDER through COMPETE2020 by these grants: UID/FIS/04434/2019, PTDC/FIS-AST/30389/2017 & POCI-01-0145-FEDER-030389.",
"DH acknowledges support by the National Science Foundation (AST-1717000).",
"This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium).",
"Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.",
"The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawai`ian community.",
"We are most fortunate to have the opportunity to conduct observations from this mountain.",
"This research was partially conducted during the Exostar19 program at the Kavli Institute for Theoretical Physics at UC Santa Barbara, which was supported in part by the National Science Foundation under Grant No.",
"NSF PHY-1748958 Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope; LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences.",
"Funding for the project has been provided by the National Development and Reform Commission.",
"LAMOST is operated and managed by the National Astronomical Observatories, Chinese Academy of Sciences."
],
[
"least squares fit of pulsation modes for KIC 11409673",
"We provide here the force-fitted sidelobes for the other sections of data in KIC 11409673.",
"Table: Force-fitted pulsations in KIC 11409673 for sections 2 through 4.",
"The zero-points were chosen to force the first set of sidelobes to be equal, and are BJD 2455501.39754, BJD 2455870.64075, and BJD 2456240.07652 respectively."
]
] | 1906.04353 |
[
[
"Many-body effects in strongly-disordered III-nitride quantum wells:\n interplay between carrier localization and Coulomb interaction"
],
[
"Abstract The joint impact of Anderson localization and many-body interaction is observed in the optical properties of strongly-disordered III-nitride quantum wells, a system where the Coulomb interaction and the fluctuating potential are pronounced effects with similar magnitude.",
"A numerical method is introduced to solve the 6-dimensional coupled Schrodinger equation in the presence of disorder and Coulomb interaction, a challenging numerical task.",
"It accurately reproduces the measured absorption and luminescence dynamics of InGaN quantum wells at room-temperature: absorption spectra reveal the existence of a broadened excitonic peak, and carrier lifetime measurements show that luminescence departs from a conventional bimolecular behavior.",
"These results reveal that luminescence is governed by the interplay between localization and Coulomb interaction, and provide practical insight in the physics of modern light-emitting diodes."
],
[
" Many-body effects in strongly-disordered III-nitride quantum wells: interplay between carrier localization and Coulomb interaction Aurelien David [email protected] Soraa Inc., 6500 Kaiser Dr. Fremont CA 94555 Nathan G. Young Soraa Inc., 6500 Kaiser Dr. Fremont CA 94555 Michael D. Craven Soraa Inc., 6500 Kaiser Dr. Fremont CA 94555 The joint impact of Anderson localization and many-body interaction is observed in the optical properties of strongly-disordered III-nitride quantum wells, a system where the Coulomb interaction and the fluctuating potential are pronounced effects with similar magnitude.",
"A numerical method is introduced to solve the 6-dimensional coupled Schrodinger equation in the presence of disorder and Coulomb interaction, a challenging numerical task.",
"It accurately reproduces the measured absorption and luminescence dynamics of InGaN quantum wells at room-temperature: absorption spectra reveal the existence of a broadened excitonic peak, and carrier lifetime measurements show that luminescence departs from a conventional bimolecular behavior.",
"These results reveal that luminescence is governed by the interplay between localization and Coulomb interaction, and provide practical insight in the physics of modern light-emitting diodes.",
"The physics of compound semiconductors is influenced by many-body interaction as well as disorder.",
"In conventional semiconductors, disorder effects are weak compared to the Coulomb interaction, and disorder can be treated as a perturbation to the excitonic center-of-mass.",
"This has led to multiple signatures at cryogenic temperatures in the widely-studied system of GaAs/AlGaAs quantum wells (QWs) with disordered interfaces, using absorption and luminescence as probes for the quantum state of the system [1], [2], [3], [4], [5], [6], [7], [8], [9].",
"More recently, the mastery of III-nitride compounds –the material system of choice for modern light-emitting diodes (LEDs)– has renewed interest in the physics of luminescence of disordered semiconductors by giving access to a new physical regime.",
"Indeed, these materials constitute a remarkable system where the two effects are strongly pronounced: Coulomb interaction leads to an excitonic binding energy of tens of meVs, with an associated Bohr radius of a few nm; meanwhile, Anderson localization of carriers stems from inherent random alloy disorder, with fluctuations in potential energy of $\\sim 100$ meV and typical dimensions of a few nm.",
"Therefore, both phenomena occur with a similar scale of energy (large enough to be relevant at room temperature) and distance, and must be considered on equal footing rather than perturbatively.",
"III-nitride QWs thus offer an ideal testbed to study the unique regime of strong disorder with strong Coulomb interaction, with the practical perspective of better understanding the fundamental limits of this material for LED applications.",
"Recent experimental work has shown evidence of localization effects in III-nitride QWs [10], [11], [12], [13] with signatures reminiscent of that in conventional semiconductors [14], [2], [3], [4], [5], [6]; on the other hand, direct observation of Coulomb-interaction effects has been elusive.",
"From a theoretical standpoint, alloy disorder [15], [16], [17], [18], [19] and many-body effects [20], [21] have been investigated independently.",
"Their joint consideration is a more complex task; it was first tackled recently in Refs.",
"[16], [22], with the limitation that the Coulomb-interacting problem is solved perturbatively over a basis of a few free-carrier eigenstates.",
"Overall, these theoretical investigations only predict moderate corrections to disorder-free models, and have not been validated experimentally.",
"In this letter, we present direct experimental evidence of the effects of localization and many-body interaction on the room-temperature optical properties of high-quality InGaN QWs, and show that their magnitude well exceeds the aforementioned predictions.",
"We introduce an advanced numerical model which treats these on equal footing by solving the full 6-dimensional Schrodinger equation.",
"This powerful approach quantitatively reproduces experimental results, confirming that the interplay between the two effects is essential in understanding III-nitride luminescence and the resulting LED efficiency.",
"We begin by investigating the absorption coefficient ($\\alpha $ ) in InGaN QWs, since excitonic effects are commonly manifested by sharp absorption peaks.",
"Such peaks are indeed observed in bulk GaN samples at cryogenic temperature [23].",
"In contrast, measurements on QWs usually result in broad spectra which lack excitonic signatures.",
"We illustrate this using a sample, grown on a bulk GaN substrate by MOCVD, having a 3.5 nm-thick InGaN QW with [In]$=13\\%$ placed at the center of a p-i-n junction.",
"Its room-temperature absorption, measured by photocurrent spectroscopy [24], [11], lacks distinct features, as shown in Fig.",
"REF (a).",
"Figure: Absorption properties of a 3.5nm-thick InGaN QW.",
"(a) α\\alpha measured by photocurrent at various reverse biases, showing the progressive appearance of an excitonic resonance.",
"(b) Modeled α\\alpha (in log scale) in flat-band conditions.",
"Thin lines: individual configurations.",
"Thick line: average of 40 calculations.",
"Dashed line: average without Coulomb interaction, for comparison.",
"(c) Comparison of measured (solid line with dots) and modeled (solid line) α\\alpha , showing excellent agreement.",
"Dashed line: modeled α\\alpha without disorder.Such featureless absorption is attributed to the inhomogeneous broadening caused by the strong polarization field present across the QW, and to the reduction in Coulomb interaction due to electron-hole wavefunction separation.",
"To remove these effects, we perform photocurrent measurements under reverse bias to compensate for the polarization field.",
"As the bias is varied, the absorption spectrum progressively sharpens until the QW reaches flat-band conditions (Fig.",
"REF (a)), where an excitonic peak is observed [25].",
"We have systematically observed similar spectra in other samples of varying QW design (data not shown).",
"This excitonic peak constitutes a rare direct manifestation of many-body effects at room temperature in InGaN QWs, and offers an opportunity to validate theoretical models.",
"We now present a numerical model which predicts such optical features by taking into account alloy disorder and Coulomb interaction.",
"Since both effects are of similar magnitude, they cannot be treated perturbatively as has been done in other material systems.",
"We place ourselves in the two-band effective mass approximation; this is appropriate to describe the near-band-edge optical behavior.",
"The electron-hole Hamiltonian reads: $H_r = -\\frac{\\hbar ^2}{2m_e} \\Delta _e + V_e - \\frac{\\hbar ^2}{2m_h} \\Delta _h + V_h +V_{C}$ Here $\\Delta _{e,h}$ are the Laplacian operators on the electron and hole coordinates, $V_{e,h}$ the electrostatic potentials for the electron and hole (including alloy disorder), and $V_{C}$ the Coulomb interaction term.",
"Because it includes disorder and Coulomb interaction, Eq.",
"(REF ) is a 6-dimensional problem (3 dimensions per carrier, coupled together), making it very challenging numerically.",
"The standard approach (summing optical transitions over eigenstates of $H_r$ ) is prohibitive as several thousand eigenstates are required for a proper description of the optical joint density of states (JDOS).",
"Instead, we proceed by solution of the time-dependent Schrodinger equation in real space, following the general approach of Refs.",
"[26], [27].",
"In short, the optical polarization is propagated in time, which directly yields the JDOS and absorption without requiring eigenstates.",
"This makes the method computationally-efficient, although it has not previously been applied to problems with such complexity.",
"We mention important aspects of the model below; implementation details are in the Supplemental Material (SM).",
"The Hamiltonian is discretized in real space with a standard finite difference scheme.",
"To account for alloy disorder in $V_{e,h}$ , we create smoothed numerical maps of atomic potentials following the approach of Refs.",
"[15], [11].",
"The Coulomb term $V_{C}$ deserves caution.",
"Since $V_C \\sim 1/r$ , a naive discretization diverges at $r=0$ , causing well-known numerical difficulties.",
"The often-proposed simple regularization $V_C \\sim 1/(r^2+a^2)^{1/2}$ (with $a$ an empirical short-scale constant) suffers from poor convergence.",
"Instead, we have investigated two more sophisticated approaches: the so-called ground-state scheme [26], [27] and asymptotic-behavior-correspondence (ABC) scheme [28].",
"Both lead to near-identical results in our simulations.",
"As we will see hereafter, the model will ultimately need to account for carrier-screening effects.",
"Therefore, we select the ABC scheme because it can straightforwardly be generalized to arbitrary carrier populations In contrast, the ground-state scheme requires a preliminary calculation of the ground state for the 1-dimensional excitonic problem, which only exists below the Mott density.",
"The Coulomb term thus reads: $V_{C}=-e^2/4\\pi \\epsilon \\bar{r}$ , with $\\bar{r}$ the effective radius derived from the ABC scheme and $\\epsilon $ the dielectric constant of GaN.",
"With this model, we compute the optical absorption of a QW in flat-band conditions.",
"Fig.",
"REF (b) shows several calculations with different configurations of the alloy distribution, and the average of 40 calculations.",
"For each configuration, sharp excitonic peaks are observed.",
"Near the band edge these peaks are dense; each configuration also displays a few deeply-localized excitonic peaks: these correspond to excitonic states stemming from Anderson-localized holes at random In-rich locations in the QW.",
"These deep states produce an Urbach tail for the average absorption, with a characteristic energy $\\sim 20$ meV, similar to the measurements of Ref. [11].",
"Note that ignoring Coulomb interaction would lead to a narrower Urbach tail (7 meV, Fig.",
"REF (b)).",
"The Urbach tail is often considered as a manifestation of wavefunction localization[14], [30]; our results show that Coulomb interaction further affects its behavior.",
"We infer that the observed increase in Urbach energy is caused by a more pronounced Coulomb interaction among deeply-localized states.",
"Figure: Photoluminescence properties of InGaN QWs.",
"(a) Effective radiative coefficient versus carrier density.",
"Dots: experimental measurements, revealing a departure from bimolecular recombination.",
"Dashed lines: model including disorder but ignoring Coulomb interaction, resulting in a bimolecular rate, i.e.",
"a plateau at low density with an increase due to field-screening at high density (shown only for the thinnest and thickest QWs).",
"Solid lines: full model.",
"(b) Experimental luminescence spectra LL for a 3 nm QW, indicative of thermalized carrier distributions at all carrier densities (densities as labeled).",
"(c) Modeled carrier-density-dependent absorption for one configuration of a 3 nm QW (densities as labeled).",
"(d) Modeled absorption α\\alpha (lines) and luminescence LL (shaded shapes) for a 3 nm QW; blue/red: with/without Coulomb interaction.",
"(e) IQE of a 2.5 nm QW versus current density JJ.",
"Solid line: measured IQE; dashed line: predicted IQE in the absence of Coulomb enhancement.Fig.",
"REF (c) compares the average calculated absorption (smoothed with a conservative linewidth of 15 meV [31], [12], a value low-enough to smooth numerical noise without dominating the inhomogeneous broadening) to the experimental data.",
"An excellent agreement is obtained without adjusting any model parameter.",
"The amplitude and shape of the excitonic peak and of the low-energy edge are well-reproduced.",
"For comparison, Fig.",
"REF (c) also shows a calculation where alloy disorder was ignored: the corresponding excitonic peak is much too sharp.",
"This confirms that Anderson localization dominates the inhomogeneous broadening in flat-band absorption spectra.",
"Having addressed optical absorption, we turn to the more complex and important study of luminescence–more specifically luminescence dynamics, for which many-body effects can cause a departure from the conventional bimolecular radiative rate $G_r=Bn^2$ (with $B$ the radiative coefficient and $n$ the density of electrons and holes).",
"We study a series of QW samples similar to the previous sample, with varying QW thickness.",
"The epitaxial structure ensures the absence of artifacts from modulation doping [32] and carrier escape [33], thus enabling a proper measurement of carrier dynamics using an all-optical differential lifetime measurement [34].",
"In contrast to conventional large-signal measurements, this technique gives direct access to the lifetime and carrier density $n$ over a wide range of excitation levels, without requiring assumptions on a recombination model.",
"By combining this with a measurement of the sample's absolute internal quantum efficiency (IQE), we obtain the effective radiative coefficient $B(n)=G_r/n^2$ [35].",
"We measure $B$ for a series of five samples spanning QW thicknesses between 2.5 nm and 4.5 nm, as shown in Fig.",
"REF .",
"This data shows an intricate behavior.",
"At high carrier density, $B$ increases for most samples; as discussed in our previous work, this is simply due to screening of the polarization field by injected carriers [34].",
"Qualitatively, screening should be most pronounced for thicker samples (where the potential drop due to the polarization field is larger), as we do observe experimentally.",
"The behavior at low density, however, defies common expectation: instead of a plateau (characteristic of standard bimolecular recombination), $B$ shows a clear carrier dependence, increasing up to tenfold at low density–this is most pronounced in thin QWs.",
"The remainder of this Letter is dedicated to further investigating this remarkable trend.",
"One may first wonder if carrier localization can alone cause the observed increase of $B$ .",
"We have verified that this is not the case by computing the radiative rate for free carriers (i.e.",
"carriers without Coulomb interaction) in a QW with alloy disorder.",
"As shown as dashed lines in Fig.",
"REF (a), a bimolecular rate is still predicted.",
"This conclusion is unsurprising, as the bimolecular behavior is generally robust against the details of transition selection rules [36].",
"Furthermore, our calculations with disorder alone only lead to a small correction to $B$ from the disorder-free case (see SM): this is in line with other theoretical investigation [16], [17], [18], [19], but does not reproduce the experimental order-of-magnitude increase in $B$ , showing that disorder alone is insufficient to explain the data.",
"Instead, one must again consider many-body effects to account for the radiative dynamics.",
"Studies in various material systems have shown how Coulomb interaction increases the radiative rate at low density [37], [20], [38], [39], [40].",
"We therefore extend the model to encompass luminescence.",
"Highly-accurate approaches exist for this task [41]; however, these are prohibitively complex for the present 6-dimensional problem.",
"Instead, as will be detailed hereafter, we proceed at the lowest order by deriving a carrier-dependent absorption spectrum, then transforming it into a luminescence spectrum (see SM for further discussion).",
"The impact of carriers on absorption is included by computing a statically-screened Coulomb potential: $V_{Cs} = V_{C} \\exp {(-\\kappa \\bar{r})}$ , with $\\kappa $ the Thomas-Fermi screening length (see details in SM) [42].",
"This screened potential is used to compute the screened absorption spectrum, using the same procedure as before.",
"Fig.",
"REF (c) illustrates results for a 3 nm QW.",
"The Coulomb enhancement is maximal at low density, and is progressively screened until the free-carrier limit is reached at high density.",
"Note that the enhancement is comprised of excitonic peaks and the Sommerfeld factor [42], although the distinction between bound states and continuum is not well-defined in the presence of disorder.",
"To then obtain luminescence spectra, we make assumptions on the carrier populations.",
"First, we assume that all the carriers exist as an electrons-hole plasma, with no exciton population.",
"This is well-justified at room temperature considering the Saha equation [43]Specifically, assuming an exciton binding energy of 20meV, the Saha equation predicts less than 5% of carriers in an excitonic phase at any carrier density..",
"Thus in this model, the luminescence enhancement is not due to the presence of an excitonic population, but solely to an increase of the JDOS by the Coulomb interaction–a distinction discussed in detail in Refs.",
"[38], [39].",
"Second, we assume that the carriers are in quasi-thermal equilibrium, and described by Fermi-Dirac populations.",
"This assumption may not be obvious for holes in the presence of Anderson localization; however the following considerations justify it: (i) our free-carrier computations confirm that only the lowest-energy hole states are localized, while higher-energy hole states (50meV and above) extend laterally and enable population thermalization It can be calculated that about $25\\%$ of the populated hole states are fully delocalized.",
"These states are expected to undergo scattering with localized states and ensure thermalization.",
"; and (ii) experimental room-temperature luminescence spectra indeed display a thermalized tail, independent of carrier density (Fig.",
"REF (b)).",
"Under these assumptions, the densities of states for free electrons and holes are derived by computing eigenstates of the respective 3-dimensional Poisson-Schrodinger equations.",
"Several thousand states are computed, enough to generate accurate densities of states and the corresponding carrier distributions $f_{e,h}$ .",
"Absorption and luminescence are related by the Kubo-Martin-Schwinger (KMS) relationship [46], [47], [48], [49].",
"Here we use a generalization of this relationship, valid in disordered systems (see SM): $L(E)=\\frac{E^2n^2}{\\pi ^2c^2\\hbar ^3} \\alpha \\left< f_e(1-f_h) \\right>$ where $L$ is the luminescence spectrum, $E$ the emission energy, $n$ the refractive index, $\\alpha $ the absorption with Coulomb enhancement, and $\\left< \\cdot \\right>$ denotes an average over all pairs of electron-hole states with transition energy $E$ , weighed by their wavefunction overlap.",
"In the non-Coulomb-interacting case, Eq.",
"(REF ) exactly predicts $L$ .",
"To compute luminescence with many-body effects, we apply this relationship with carrier distributions given by their free densities of state, but replacing the free-carrier absorption with the Coulomb-interacting one [20].",
"This approximation is accurate at high enough temperature [38].",
"The resulting Coulomb-enhancement of $L$ is illustrated in Fig.",
"REF (d).",
"Finally, we obtain the radiative coefficient with many-body enhancement as $B=\\int {L(E)dE}/n^2$ .",
"The resulting $B$ coefficients are shown on Fig.",
"REF (a).",
"To best match the experimental data, the modeled QW thickness was adjusted by $+0.5$ nm for all samples (a correction within the experimental uncertainty on the QW thickness).",
"The resulting predictions are in excellent agreement with the experimental data, reproducing the relative variation of $B$ with QW thickness and its carrier-dependence.",
"Therefore, the behavior of the radiative rate at low current can be understood as follows.",
"By increasing $\\alpha $ , many-body interaction leads to an enhanced radiative rate; this interaction is most pronounced for thin QWs (where the electron and hole wavefunctions are closely confined) and weaker for thick QWs (where the wavefunctions are separated).",
"The enhancement is modulated by in-plane carrier localization, and is further screened by carriers, leading to a complex dependence on the structure design and carrier density.",
"Importantly, this enhancement can have a profound impact on the efficiency of real-world LED devices.",
"For instance, at a moderate carrier density $10^{17}$ cm$^{-3}$ , Coulomb enhancement increases the radiative rate by an order of magnitude in thin QWs.",
"As shown in Fig.",
"REF (e), this causes a substantial improvement in low-current efficiency – a regime of particular interest in applications such as micro-LEDs.",
"In summary, we have shown that strongly-disordered InGaN quantum wells are a remarkable system with pronounced localization and many-body effects directly observed in room-temperature optical properties.",
"We have introduced a state-of-the-art non-perturbative model which treats alloy disorder and Coulomb interaction on equal footing, accurately and efficiently tackling this challenging numerical problem.",
"The model confirms that the interplay between localization and Coulomb interaction dominates the optical properties of InGaN quantum emitters, making them an ideal testbed for future explorations of these complex optical effects, and shedding new insight into the efficiency of modern LEDs.",
"Supplemental Material: Many-body effects in strongly-disordered III-nitride quantum wells: interplay between carrier localization and Coulomb interaction We summarize the main steps of the numerical method to compute the absorption coefficient $\\alpha $ , as introduced in [27].",
"$\\alpha $ is related to the optical polarization $P$ (ignoring numerical prefactors): $\\alpha (\\omega ) \\propto \\text{Im} \\left(\\int _{0}^{\\infty }P(t) e^{i \\omega t} \\text{d}t \\right)$ For a semiconductor structure excited at $t=0$ by a dipole $\\mu \\propto \\delta (\\textbf {r}_e-\\textbf {r}_h)$ , the microscopic polarization $\\psi $ can be obtained by propagating the time-dependent Schrodinger equation: $i \\frac{\\partial }{\\partial t} \\psi (\\textbf {r},t) = H \\psi (\\textbf {r},t),$ with the boundary condition $\\psi (\\textbf {r},t=0) = i \\mu $ .",
"From this, $P$ is derived as: $P(t) = e^{-\\epsilon t / \\hbar } \\int \\mu ^* \\psi \\text{d}\\textbf {r},$ yielding $\\alpha $ .",
"Here, $\\epsilon $ is an empirical broadening energy which should be small enough that it does not dominate the inhomogeneous broadening.",
"The key step of the calculation is the time-propagation of Eq.",
"(REF ).",
"By discretizing $H$ and $\\psi $ with a finite-difference scheme, this step takes the form of a simple matrix multiplication.",
"The matrix is very large for a 6-dimensional problem, but efficient multiplication algorithms make the scheme tractable – in contrast to seeking numerous eigenstates of $H$ .",
"To construct $H$ , we employ standard material parameters for the band structure of GaN [50].",
"For the relative hole mass we adopt a value of 1.9, corresponding to the Luttinger parameter $A7$ far from the band edge.",
"This choice is suited to describe hole wave packets localized in space [51]; it leads to localized hole states on a scale of a few nm, consistent with ab-initio calculations [52], [53].",
"Note that the value of the hole mass is of importance to obtain proper localization results: using a lighter Gamma-point mass instead would lead to a weaker localization.",
"We assume a band offset of $70\\%$ in the conduction band; this is compatible with known values [54], although experimental uncertainty remains [55].",
"Different band offset values would slightly affect the electron localization.",
"Composition fluctuation maps are obtained following the approach of [51], [56]: namely, a random distribution of Ga and In atoms is generated on a fine grid, then smoothed out (by a Gaussian with 1 nm full-width at half maximum) to account for inherent averaging by carrier wavefunctions, and interpolated on the final computation grid to yield a spatial composition map (Fig.",
"REF ).",
"From this, the potential due to the local band gap is evaluated.",
"The electrostatic potential is obtained conventionally by computing the strain equations and solving the resulting Poisson equation, with the p-i-n junction and applied voltage being accounted-for as boundary conditions [57].",
"In this calculation, we assume that the piezoelectric constant $e_{15}$ is zero, since various works have found its effect to be small (compared to other piezoelectric terms) and uncertain, withe both positive and negative values reported [58].",
"The potential contributions from the band gap and electrostatics are summed to obtain $V_{e,h}$ .",
"The computation domain has $(xyz)$ dimensions of $16\\times 16\\times 12$ nm, with corresponding mesh steps of $0.75\\times 0.75\\times 0.5$ nm.",
"Numerical convergence for this meshing was verified by running calculations without the Coulomb term (a 3-dimensional problem, whose lower complexity facilitates convergence testing by comparison with finer meshes).",
"The computation domain includes several nm of GaN barrier material on either side of the QW; this is necessary due to the substantial penetration of electron wavefunctions in the barriers.",
"To avoid spurious reflections of wavefunctions at the edges of the computation domain, absorbing boundary conditions are included [59].",
"Multiple calculations are repeated with different random alloy distributions to obtain configuration-averaged values.",
"Computations are performed on a regular workstation; after optimization of the code for memory usage and computation speed, each calculation requires 32 Gb of memory and completes in about 2 to 10 hours, which is fast enough to perform sufficient configuration-averaging and explore the impact of the structure's parameters.",
"Figure: Numerical indium distribution profiles at three planes in a 3nm QW (with average composition [In]=0.12) (a) fine grid for initial generation; (b) final calculation grid.In the ABC scheme, the discretized radius operator is defined by: $\\begin{split}\\frac{1}{\\bar{r}} = & \\frac{1}{2} \\left( \\sqrt{\\frac{(i+1)^2}{dx^2}+\\frac{j^2}{dy^2}+\\frac{k^2}{dz^2}} + \\sqrt{\\frac{(i-1)^2}{dx^2}+\\frac{j^2}{dy^2}+\\frac{k^2}{dz^2}} + \\sqrt{\\frac{i^2}{dx^2}+\\frac{(j+1)^2}{dy^2}+\\frac{k^2}{dz^2}} \\right.",
"\\\\& \\left.",
"+ \\sqrt{\\frac{i^2}{dx^2}+\\frac{(j-1)^2}{dy^2}+\\frac{k^2}{dz^2}} + \\sqrt{\\frac{i^2}{dx^2}+\\frac{j^2}{dy^2}+\\frac{(k+1)^2}{dz^2}} + \\sqrt{\\frac{i^2}{dx^2}+\\frac{j^2}{dy^2}+\\frac{(k-1)^2}{dz^2}} \\right) \\\\& -3 \\sqrt{\\frac{i^2}{dx^2}+\\frac{j^2}{dy^2}+\\frac{k^2}{dz^2}},\\end{split}$ where $(dx, dy, dz)$ are the mesh steps, $i$ is defined as $i = \\frac{X_e-X_h}{dx},$ and similar equations define $j$ and $k$ .",
"Here, we discuss luminescence for free carriers, i.e.",
"in the absence of Coulomb interaction.",
"In this case, luminescence can simply be calculated by computing enough eigenstates and performing a sum over all transitions, weighed by their wavefunction overlap [60].",
"This approach is valid whether alloy disorder is considered or not, and enables us to evaluate the variation in the radiative rate caused by disorder.",
"For these calculations, we use a fine spatial mesh of 0.25 nm and a lateral domain size of 20 nm.",
"We find that, both without and with disorder, the radiative rate is bimolecular at low carrier density (with an additinoal field-screening effect at high density).",
"Hereafter, we consider the value of the radiative coefficient $B$ at low density.",
"$B$ strongly depends on the thickness of the QW, which modulates the electron-hole overlap.",
"Therefore, when comparing QWs without and with alloy disorder, the same QW thickness should be considered.",
"However, we are faced with an ambiguity in defining the thickness of a disordered QW, since its interfaces are not well-defined.",
"We consider two possible definitions: (i) the thickness is the full-width at half maximum of the average indium distribution across the epitaxial direction; (ii) the thickness is defined such the same total number of In atoms is present in QWs without and with disorder.",
"The results of these calculations are shown in Fig.",
"REF .",
"Depending on the definition for the disordered QW thickness, $B$ is either slightly smaller or larger than in a disorder-free QW.",
"Therefore, due to this ambiguity, one cannot state whether disorder increases or decreases $B$ .",
"This ambiguity is not addressed in other modeling work, which alternately predicted a decrease [53] or an increase [61] of $B$ .",
"Regardless, the modeled effect of disorder alone is small: a few tens of %.",
"This stands in contrast to the model including disorder and many-body interaction (also shown on Fig.",
"REF ), which predicts a much stronger variation with thickness and is in good agreement with our experimental data.",
"Figure: Modeled radiative coefficient BB against QW thickness.",
"Solid line: QW without disorder.",
"Dashed/dotted lines: QW with disorder, using definitions (i) and (ii) for the QW thickness respectively.",
"Line with symbols: QW with disorder and Coulomb interaction (using definition (i) for the QW thickness, and n=10 17 n=10^{17} cm -3 ^{-3}).In general, the electron-hole plasma can influence many-body interaction in various ways.",
"First, the magnitude of the Coulomb interaction is decreased by free-carrier screening.",
"This effect is included in our screened Coulomb potential model (see below).",
"Second, state-filling by carriers further influences the Coulomb interaction.",
"The interband (electron-hole) state-filling effect is accounted-for in the KMS relationship (Eq.",
"2).",
"The intraband terms (electron-electron and hole-hole repulsion) are ignored; this is suitable because their two main consequences are bandgap renormalization (whose consideration is not essential to compute the luminescence rate) and an influence on carrier populations (which we assume to be in quasi-equilibrium anyway).",
"The level of our approximations is comparable to that of [62].",
"For the screened Coulomb potential, we use the simple RPA approximation where $\\kappa ^2=e^2 (dn/d\\mu _{e}+dp/d\\mu _{h})/\\epsilon $ , with $n,p$ the electron and hole densities and $\\mu _{e,h}$ the respective quasi-Fermi levels.",
"More sophisticated models of the dielectric function could be considered, but would only bring second-order corrections to the present results.",
"The derivative terms in $\\kappa $ are evaluated numerically–incidentally, we find that the numerical derivatives closely match the analytical derivative formula for an ideal three-dimensional density of states (namely, $dn/d\\mu _{e} = n/k_BT $ ) [42].",
"We now discuss the KMS relationship (also know as van Roosbroeck-Shockley relation).",
"This relationship relates luminescence and absorption, and was originally expressed as [63], [64], [65]: $L=C \\frac{\\alpha }{e^{(E-E_f)/kT}-1}$ With $C=E^2 n^2/\\pi ^2 c^2 \\hbar ^3$ and $E_f$ the Fermi level splitting.",
"This form is only valid at low density, before the Fermi levels cross into the bands.",
"A more general form, also valid at high density, is [66]: $L=C \\frac{\\alpha (f_v-f_c)}{e^{(E-E_f)/kT}-1}=C \\alpha f_c (1-f_v)$ Here $\\alpha $ is the `bare' absorption coefficient in the absence of electrons and hole populations.",
"The `loaded' coefficient, taking occupation into account, is $\\alpha ^*=\\alpha (f_v-f_c)$ [66].",
"These expressions pertain to systems with in-plane symmetry, where each electron-hole pair state (with a given in-plane wavevector) corresponds to a specific transition energy.",
"In disordered systems, this is no longer the case: several electron-hole pairs can provide transition at the same energy, so that the band occupation $f_c (1-f_v)$ can no longer be assigned to a specific energy.",
"To deal with this, we return to the definition of luminescence and absorption in an energy band d$E$ , expressed as a sum over states (ignoring prefactors for simplicity): $L(E)\\text{d}E & \\propto & \\sum I^2 f_e(1-f_h),\\\\\\alpha (E)\\text{d}E & \\propto & \\sum I^2$ Here the sum runs over all states within the energy band d$E$ , and $I$ is the electron-hole overlap integral modulating the intensity of each transition.",
"Therefore, absorption and emission remain related by a detailed-balance relationship: $L = C \\alpha \\frac{\\sum I^2 f_e(1-f_h)}{\\sum I^2} = C \\alpha \\left< f_c (1-f_v) \\right>$ At low carrier density, $\\left< f_c (1-f_v) \\right>$ reduces to the Boltzmann limit for all electron-hole pairs and expression (REF ) is recovered.",
"In the absence of disorder, all pairs with allowed transitions have the same occupation factor and expression (REF ) is recovered.",
"For the many-body case, we replace the bare absorption coefficient $\\alpha $ with its many-body value, obtained from Eq. 1.",
"Note that $\\left< f_c (1-f_v) \\right>$ can only be calculated explicitly down to the lowest energy transition of the free electron-hole plasma, whereas the many-body absorption begins at a lower energy due to excitonic resonances.",
"However, this occupation function displays a smooth behavior on a log scale, as shown on Fig.",
"REF , so that it can be extrapolated to slightly lower energy (a few tens of meV) to apply the KMS relationship.",
"Figure: Carrier occupation function f e (1-f h )\\left< f_e(1-f_h) \\right> for a 3nm QW ([In]=12%), at various carrier densities (as labeled).",
"At low density, the usual Boltzmann limit is recovered."
]
] | 1906.04315 |
[
[
"An optimized twist angle to find the twist-averaged correlation energy\n applied to the uniform electron gas"
],
[
"Abstract We explore an alternative to twist averaging in order to obtain more cost-effective and accurate extrapolations to the thermodynamic limit (TDL) for coupled cluster doubles (CCD) calculations.",
"We seek a single twist angle to perform calculations at, instead of integrating over many random points or a grid.",
"We introduce the concept of connectivity, a quantity derived from the non-zero four-index integrals in an MP2 calculation.",
"This allows us to find a special twist angle that provides appropriate connectivity in the energy equation, and which yields results comparable to full twist averaging.",
"This special twist angle effectively makes the finite electron number CCD calculation represent the TDL more accurately, reducing the cost of twist-averaged CCD over $N_\\mathrm{s}$ twist angles from $N_s$ CCD calculations to $N_s$ MP2 calculations plus one CCD calculation."
],
[
"Introduction",
"In recent years, the use of wavefunction-based post-Kohn–Sham or post-Hartree–Fock methods to solve problems in materials science has proliferated.",
"[1] This is in part driven by an interest in obtaining precise energies (accurate to within 1mHa) for complex systems using hierarchies of methods found in quantum chemistry such as coupled cluster theory.",
"While growing in popularity, wavefunction methods have yet to see widespread adoption, in large part due to their significant computational cost scaling with system size.",
"This is especially of note in coupled cluster theory using a plane wave basis, and as a result, some authors are seeking methods to control finite size errors in order to run calculations using smaller system sizes.",
"[2] Finite size errors arise when attempts are made to simulate an infinite system Hamiltonian with a periodic supercell containing a necessarily finite particle number.",
"[3], [4] The finite size of a supercell places a limitation on the minimum momenta in Fourier sums (e.g., with a cubic box of length $L$ , the smallest momentum transfer is $2\\pi /L$ ).",
"These limitations ultimately lead to errors in the correlation energy; [2], [5] this has been attributed to long range van der Waals forces.",
"[2], [6] Since these finite size errors are large and slowly converging with increasing supercell size, which has been analyzed in detail for coupled cluster theory, [7] there has been significant interest in developing wavefunction methods with reduced computational cost to circumvent finite size error and allow the treatment of larger supercells.",
"These include embedding methods,[8] such as density matrix embedding,[9], [10], [11], [12], [13], [14], [15] wavefunction-in-DFT embedding,[16], [17], [18], [19], [20], [21], [22], [23], [24] electrostatic embedding,[25], [26], [27], [28], [29] QM/MM-inspired schemes,[30], [31], [32], [33], [34] and others.",
"[35], [36], [37], [38], [39] Local correlation methods[40], [41] such as fragment-based schemes,[42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53] incremental methods,[54], [55], [56], [57], [58], [59], [60], [61] and heirarchical methods,[62], [63], [64], [65] break the system into smaller subsystems, then extrapolate or stitch together the energies.",
"Some methods take advantage of range separation[66], [67], [68] or other distance-based schemes[69], [70], [71], [72], [73] to reduce computational cost.",
"In addition to work on developing or modifying electronic structure methods, much work on reducing the cost of wavefunction methods has been focused on modifying basis sets in order to accelerate convergence and decrease computation time.",
"Local orbital methods have been popular,[74], [75], [76], [77], [78], [79], [80], [81], [82], [83], [84], [85], [86], [87] often based on the local ansatz of Pulay and Saebo[88] or Stollhoff and Fulde.",
"[89] Other common methods include progressive downsampling,[90], [91], [92] downfolding,[93] use of explicitly-correlated basis sets[94], [95], [96], [97], [98] or natural orbitals,[99] and tensor manipulations.",
"[100], [101], [102], [103], [104] Discussion of the details and relative merits of these methods is beyond the scope of this paper; for a review, we direct the interested reader to Refs.",
"huangadvances2008,mullerwavefunction-based2012,beranmodeling2016,andreonicoupled2018.",
"However, there has been some work on developing corrections for finite size errors.",
"[3], [108], [109], [4], [110], [111] Many-body methods can sometimes be integrated to the thermodynamic limit (TDL),[112], [113], [114], [115], [116], [117], [118] allowing for the derivation of analytic finite-size correction expressions.",
"[119] Several studies from the last year have particular relevance to our work here.",
"Grüeneis et al.",
"[2], [6] employed a grid integration within periodic coupled cluster for ab initio Hamiltonians with applications to various solids.",
"In another study, Alavi et al.",
"[5] devised a novel extrapolation relationship that links different electron gas calculations through the density parameter.",
"Both of these papers use a technique known as twist averaging to try to remove finite size error.",
"Twist averaging is a method that attempts to control finite size errors by first offsetting the $k$ -point grid by a small amount, ${\\bf k}_s$ , and then averaging over all possible offsets.",
"[120] We refer to ${\\bf k}_s$ here as a twist angle.",
"One of the main purposes of twist averaging is to provide for a smoother extrapolation to the thermodynamic limit by reducing severe energy fluctuations as the particle number varies.",
"When performed with a fixed particle number and box length, this process is referred to as twist averaging in the canonical ensemble, which is what we study here.",
"When employed in stochastic methods, such as variational Monte Carlo,[120] diffusion Monte Carlo[4] or full configuration interaction quantum Monte Carlo,[5], [121] the grid can be stochastically sampled at the same time as the main stochastic algorithm, and both stochastic error and error in twist-averaging related to approximate integration can be removed at the same time.",
"As a result, the scaling with the number of twist angles sampled is extremely modest.",
"Unfortunately, the same cost savings cannot be realized for deterministic methods.",
"In this case, in order to achieve a reasonable estimate for the average, one must use a large number of individual energy calculations.",
"This results in the cost scaling linearly with the number of twist angles used, although the lessening of finite size effects with rising electron number would alleviate this scaling to some extent.",
"[7] Here, we seek to remedy the linear scaling of twist averaging for deterministic methods by devising a way to provide an energy that is as accurate as twist-averaging, but with single-calculation cost.",
"In principle, it is possible to find a single twist angle which exactly reproduces the total twist-averaged energy by recognizing that it is an integral of the energy over the twist angles for a system.",
"This was the same logic used in analysis by Baldereschi to find a special $k$ -point[122] and has been used by others in the QMC community to find a special twist angle.",
"[123], [124], [125] We are motivated similarly and wish to find a single twist angle that yields an energy approximately equal to the full twist-averaged energy for CCD and related wavefunction methods.",
"We take advantage of the similarity between the MP2 and CCD correlation energy expressions, using the much cheaper MP2 method to find a single twist angle that produces a system with the most similar number of allowed excitations to the twist-averaged system.",
"We refer to this set of allowed excitations as the `connectivity'.",
"We then use this twist angle to calculate the CCD energy, which is in good agreement with the fully twist-averaged CCD energy.",
"Finally, we compare our energies to those obtained using one twist angle at the Baldereschi point.",
"[122] We do not seek to completely remedy the whole of the finite size error, instead noting that other authors have come up with corrections or extrapolations that can be used after twist-averaging is applied.",
"[119], [126], [6]"
],
[
"Twist averaging & Connectivity",
"Both continuum/real-space and basis-set twist averaging have been used effectively in quantum Monte Carlo calculations;[120], [4], [5], [121] however, twist averaging remains relatively rare in coupled cluster calculations.",
"In Fig.",
"REF , the total $\\Gamma $ -point CCD energy ($N=38$ to $N=922$ ) and twist-averaged CCD energy ($N=38$ to $N=294$ ) are plotted alongside the extrapolation to the TDL for the uniform electron gas ($0.609(3)$ Ha/electron, where the error in the last digit is in parentheses).",
"The CCD calculation is performed in a finite basis that is analogous to a minimal basis.",
"[127] The $\\Gamma $ -point energy is highly non-monotonic; it does not fit well with the extrapolation.",
"The twist-averaged data shows a much better fit with the extrapolation, resulting in a better estimate of the TDL.",
"The drawback of twist averaging, however, is that it costs $N_\\mathrm {s}\\,\\mathcal {O}\\mathrm {[CCD]}$ for $N_\\mathrm {s}$ twist angles (here, 100).",
"The twist-averaged energy becomes too costly to calculate with CCD for system sizes above 294 electrons.",
"Figure: Comparison between the twist-averaged (TA) CCD energy and the Γ\\Gamma -point CCD energy for a uniform electron gas with r s =1.0r_s=1.0 as the system size changes (up to N=294N=294 and N=922N=922, respectively).",
"In general, an extrapolation (here, red line) is performed to calculate the TDL energy.",
"Twist averaging makes this extrapolation easier, because the noise around the extrapolation is smaller, leading to a smaller extrapolation error.",
"Twist averaging is performed over 100 twist angles.",
"Standard errors are calculated in the normal fashion for twist averaging, σ≈ Var (E CCD (𝐤 s ))/N s \\sigma \\approx \\sqrt{\\mathrm {Var}(E_{\\mathrm {CCD}}({{\\bf k}_s})) / N_s} (are too small to be shown on the graph, on average 0.2 mHa/el).Figure REF is a clear statement of the problem we wish to resolve here.",
"Twist averaging resolves some finite size errors that are present at an individual particle number $N$ , and allows for improved extrapolation to the thermodynamic limit.",
"That said, the scaling with the number of twist angles is cost-prohibitive.",
"We aim to develop an approximation to twist averaging that gives comparable accuracy at a fraction of the cost.",
"We begin by analyzing how the Hartree-Fock energy and the MP2 correlation energy are modified by twist averaging.",
"This analysis then allows us to build an algorithm that produces CCD twist-averaged accuracy/results for only MP2 cost."
],
[
"Hartree-Fock and single-particle eigenvalues",
"A finite-sized electron gas at the $\\Gamma $ -point is only closed-shell at certain so-called magic numbers, which are determined by the symmetry of the lattice (for example $N=2$ , 14, 38, and 54).",
"One of the reasons that the $\\Gamma $ -point calculations are so noisy (Fig.",
"REF ) is that there are degeneracies in the HF eigenvalues, which can be seen in Fig.",
"REF and has long been recognized.",
"[126] This can be partially remedied by modifying the Hartree–Fock eigenvalues.",
"The starting-point for this is writing the HF energy as follows: $E_\\mathrm {HF}({\\bf k}_s)= \\sum _{i} T_i ({\\bf k}_s)- \\frac{1}{2} \\sum _{ij} v_{ijji}({\\bf k}_s)$ where $T_i$ is the kinetic energy of orbital $i$ and $v_{ijji}$ is the exchange integral between electrons in orbitals $i$ and $j$ .",
"Here, we have included the explicit form of the dependence on the twist angle, ${\\bf k}_s$ .",
"The twist-averaged energy is found by summing Eq.",
"(REF ) over all possible ${\\bf k}_s$ : $\\langle E_\\mathrm {HF} \\rangle _{\\bf {k}_s} =\\frac{1}{N_\\mathrm {s}}\\sum _{{\\bf k}_s}^{N_\\mathrm {s}} \\sum _{i} T_i ({\\bf k}_s) - \\frac{1}{N_\\mathrm {s}}\\sum _{{\\bf k}_s}^{N_\\mathrm {s}}\\frac{1}{2} \\sum _{ij} v_{ijji}({\\bf k}_s)$ where $N_s$ indicates the number of twist angles used.",
"Swapping the sums yields: $\\langle E_\\mathrm {HF} \\rangle _{\\bf {k}_s} = \\sum _{i} \\left[\\frac{1}{N_\\mathrm {s}} \\sum _{{\\bf k}_s}^{N_\\mathrm {s}} T_i ({\\bf k}_s) \\right]- \\frac{1}{2} \\sum _{ij} \\left[ \\frac{1}{N_\\mathrm {s}}\\sum _{{\\bf k}_s}^{N_\\mathrm {s}} v_{ijji}({\\bf k}_s) \\right] .$ Therefore, twist averaging the HF energy is numerically identical to twist averaging the individual matrix elements: $\\langle E_\\mathrm {HF} \\rangle _{\\bf {k}_s} = \\sum _{i} \\langle T_i \\rangle _{\\bf {k}_s}- \\frac{1}{2} \\sum _{ij} \\langle v_{ijji} \\rangle _{\\bf {k}_s}$ Overall, then, we can use twist-averaged HF eigenvalues in place of twist-averaging the HF energy, obtaining a more reasonable density of states Fig.",
"REF .",
"We will use this in our subsequent scheme."
],
[
"Beyond Hartree–Fock",
"The above approach does not generalize to correlated theories because they have more complex energy expressions.",
"For example, averaging the second-order Møller-Plesset theory (MP2) correlation energy over all possible twist angles can be written: $\\langle E_\\mathrm {corr} \\rangle _{\\bf {k}_s}=\\frac{1}{N_\\mathrm {s}}\\sum _{{\\bf k}_s}^{N_\\mathrm {s}} \\frac{1}{4}\\sum _{ijab} \\bar{t}_{ijab}({\\bf k}_s) \\bar{v}_{ijab} ({\\bf k}_s),$ where $i$ and $j$ refer to occupied orbitals and $a$ and $b$ refer to unoccupied orbitals.",
"The symbols $\\bar{v}$ and $\\bar{t}$ refer to the antisymmetrized electron-repulsion integral and amplitude respectively.",
"For MP2: $\\bar{t}_{ijab} ({\\bf k}_s) \\bar{v}_{ijab} ({\\bf k}_s)=\\frac{ |\\bar{v}_{ijab}({\\bf k}_s)|^2 }{\\epsilon _i({\\bf k}_s)+\\epsilon _j({\\bf k}_s)-\\epsilon _a({\\bf k}_s)-\\epsilon _b({\\bf k}_s)}$ Even though MP2 diverges in the thermodynamic limit, the energy expression (Eq.",
"(REF )) has a similar structure to coupled cluster theory, the random phase approximation, and even full configuration interaction quantum Monte Carlo.",
"As such, we can make generalized observations using the MP2 energy expression, and then use these observations to derive a scheme to find an optimal $k_s$ twist angle that works for all of these methods."
],
[
"The connectivity approach",
"The MP2 correlation energy can vary substantially as the twist angle is changed.",
"For example, in the $N=14$ electron system with a basis set of $M=38$ orbitals, the MP2 energy can vary between $-0.0171$ Ha/electron to $-0.0001$ Ha/electron.",
"This arises, in particular, because the number of low-momentum excitations (minimum $|{\\bf k}_i-{\\bf k}_a|$ ) will vary significantly.",
"Since the contribution of each excitation to the MP2 sum is $|{\\bf k}_i-{\\bf k}_a|^{-4}$ , there is a rapid decay of an excitation's contribution to the correlation energy beyond the minimum vector.",
"This effect arises because, when the twist angle is changed, different orbitals now fall into the occupied ($ij$ ) space, and different orbitals fall into the virtual ($ab$ ) space.",
"This changes the value of the sum over both occupied and virtual orbitals, since many individual terms in the sum are now substantively different.",
"We illustrate this using a diagram in the Supplementary Information.",
"By contrast, the integrals themselves do not change; to show this, the integral can be written: $v_{ijab}=\\frac{4\\pi }{L^3} \\frac{1}{({\\bf k}_i-{\\bf k}_a)^2} \\delta _{{\\bf k}_i-{\\bf k}_a , {\\bf k}_b-{\\bf k}_j} \\delta _{\\sigma _i \\sigma _a}\\delta _{\\sigma _j \\sigma _b} .$ The Kronecker deltas, $\\delta $ , ensure that momentum and spin symmetry (denoted $\\sigma $ ) are conserved.",
"On changing ${\\bf k}_p \\rightarrow {\\bf k}_p+{\\bf k}_s$ for all ${\\bf k}$ 's, the difference in the denominator here does not change, since $({\\bf k}_i+{\\bf k}_s-{\\bf k}_a-{\\bf k}_s)^2=({\\bf k}_i-{\\bf k}_a)^2$ .",
"In general, our calculations were set up using details which can be found in our prior work e.g.",
"Ref.",
"shepherdconvergence2012.",
"At this stage, we conjecture that if one of the mechanisms by which twist averaging is affecting the MP2 energy (and other correlation energies) is to smooth out the inconsistent contributions between different momenta, then it might be possible for us to find a `special twist angle' where the number of low-momentum states for that single twist angle is a good match to the average number of momentum states across all twist angles.",
"Further, we will show this special twist angle is transferable to other, more sophisticated methods such as coupled cluster doubles theory.",
"To find this special twist angle, we proceed as follows: For a given twist angle ${\\bf k}_s$ , loop over the same $ijab$ as the MP2 sum $\\sum _{ijab}$ .",
"For each $ijab$ set: Determine the momentum transfer $x=|{\\bf n}_i-{\\bf n}_a|^2$ where ${\\bf n}_a$ is the integer equivalent of the quantum number: ${\\bf k}_a=\\frac{2\\pi }{L}{\\bf n}_a$ .",
"Increment a histogram element $h_x$ by one.",
"Create a vector ${\\bf h}$ , whose elements are $h_x$ , which correspond to the number of of $v_{ijab}$ matrix elements with magnitude $\\frac{1}{\\pi L}\\frac{1}{x}$ that are encountered during the MP2 sum.",
"Average ${\\bf h}$ over all twist angles, yielding $\\langle {\\bf h} \\rangle _{\\bf {k}_s}$ Loop over the twist angles again, and find the single ${\\bf h}$ (and corresponding twist angle) that best matches $\\langle {\\bf h} \\rangle _{\\bf {k}_s}$ using: $\\min _{\\bf {k}_s} \\sum _x \\frac{1}{x^2} \\left( h_x - \\langle h_x \\rangle _{\\bf {k}_s} \\right)^2$ The weight term $1/x^2$ was chosen empirically to diminish the contributions of large numbers of high-momentum weights that contribute relatively little to the energy.",
"Looking at Eq.",
"(REF ), there are two ways to proceed.",
"We could either use this special ${\\bf k}_s$ for all aspects of the calculation (e.g.",
"for both the integral evaluation and the eigenvalue difference), or we could use the special ${\\bf k}_s$ for the integral only, and twist-average the eigenvalues before performing the CCD calculation.",
"We found that the latter was more numerically effective for $N=14$ and decided to use this approach to generate the results presented here.",
"In general, though, for larger systems it does not make a large difference.",
"In practice, we implemented this algorithm within an MP2 and CCD code; we call the MP2 calculation at each twist angle and then the CCD calculation once at the end.",
"For the remainder of this work, we will call this application of the above algorithm the “connectivity scheme,\" referencing the idea that the pattern of non-zero matrix elements $v_{ijab}$ resembles a connected network.",
"We demonstrate the effectiveness of this algorithm for coupled cluster calculations on the uniform electron gas in Fig.",
"REF .",
"In general, our results as show that the connectivity scheme works for different electron numbers, basis sets, and $r_s$ values.",
"Furthermore, evaluation of the connectivity scheme is approximately 100x cheaper than twist averaging.",
"Figure: All energies shown represent the difference in correlation energy between the Γ\\Gamma -point and the relevant calculation, since, by design, the Hartree-Fock energy is identical between the connectivity scheme and standard twist averaging (TA).",
"The connectivity scheme delivers comparable corrections to the correlation energy (relative to the Γ\\Gamma -point) when compared with twist averaging across a wide range of (a) electron numbers (using a minimal basis set, where M≈2NM \\approx 2N, as mentioned in Ref.",
"shepherdmany-body2013 and tabulated in the Supplementary Information), (b) different basis sets (M=36-2838M=36-2838 orbitals, with N=54N=54 electrons), and (c) r s r_s values (0.01 – 50.0 a.u., with N=54N=54 electrons).",
"Twist averaging is performed over 100 twist angles.",
"Standard errors are calculated in the normal fashion for twist averaging, σ≈ Var (E CCD (𝐤 s ))/N s \\sigma \\approx \\sqrt{\\mathrm {Var}(E_{\\mathrm {CCD}}({{\\bf k}_s})) / N_s}.In Fig.",
"REF , we compare the connectivity scheme to full twist-averaging for CCD calculations on the uniform electron gas.",
"Energy differences from the $\\Gamma $ -point energy are plotted for each electron number.",
"Our results show that the connectivity scheme delivers comparable accuracy (mean absolute deviation = 0.3 mHa/electron) to twist averaging, with the benefit of being much faster to compute.",
"The connectivity scheme is substantially cheaper than the twist-averaging scheme: the $N=294$ twist-averaged calculation, for example, costs 58 hours, which is about the same time it takes to run the $N=922$ connectivity scheme calculation.",
"A complete set of timings is provided in the Supplementary Information.",
"In Fig.",
"REF , we compare our connectivity scheme to full twist-averaging over a range of basis set sizes ($M= 36 - 2838$ orbitals) for 54 electrons.",
"In Fig.",
"REF , we compare the connectivity scheme to full twist-averaging over a range of $r_s$ values ($0.01 -50.0$ a.u.)",
"for 54 electrons.",
"In both cases there is good agreement between the two methods for all system sizes, proving that the connectivity scheme delivers good accuracy when compared with twist averaging for a range of both basis set sizes (mean absolute deviation $<$ 0.35 mHa/electron) and $r_s$ values (mean absolute deviation $<$ 0.25 mHa/electron) at a decreased cost.",
"Figure: Connectivity scheme CCD correlation energies for electron numbers up to N=922N=922 for r s =1.0r_s=1.0 in the uniform electron gas (yellow triangles).",
"We fit 10 points (dotted red line) to the function E=a+bN -1 E=a+bN^{-1}, as proposed by other authors; we then use this fit to extrapolate to the thermodynamic limit.In Fig.",
"REF , we show the extrapolation of our connectivity scheme CCD correlation energy to the thermodynamic limit for the $r_s=1.0$ uniform electron gas.",
"We perform calculations up to $N=922$ electrons, and fit these results to the equation $E=a+bN^{-1}$ , as proposed by other authors.",
"[4] We then use this fit to extrapolate the correlation energy to the thermodynamic limit.",
"We also performed the same extrapolation for the twist-averaged data set up to $N=294$ electrons (not shown).",
"The extrapolations predict the TDL energy to be $-0.0340(8)$ Ha/electron for the connectivity scheme and $-0.033(4)$ Ha/electron for the twist-averaged scheme, a difference of $0.001(4)$ Ha/electron.",
"The numbers in parentheses are errors in the final digit.",
"These agree within error, and the connectivity scheme has an improved error due to having more data points.",
"Next, we demonstrate how to use this method to obtain a complete basis set and thermodynamic limit estimate for the uniform electron gas.",
"Connectivity scheme CCD energies were collected for the $N=54$ electron system with basis sets varying from $M=922$ to $M=2838$ orbitals, and for systems with electron numbers varying between $N=162$ to 610, with $M\\approx 4N$ .",
"These data allow us to extrapolate to both the complete basis set limit and the thermodynamic limit by using the numerical approach set out in our previous work.",
"[129] This yields an energy that is 0.0566(6), with the error in parentheses resulting from the extrapolations; this is in good agreement with our prior estimate with significantly less error.",
"[129] For more details the reader is referred to the Supplementary Information.",
"Finally, in Fig.",
"REF , we compare the CCD energies from full twist-averaging, our connectivity scheme, and performing single calculation using the Baldereschi point as a twist angle.",
"This point, first developed for insulators, is well known for the role it played in developing efficient thermodynamic integrations[122], [130], [131], [132] and was subsequently used for twist-averaging as the center-point of uniform grid twist-averaging by Drummond et al.[126].",
"At higher electron numbers ($N\\ge 162$ ) the difference between BP and the TA energies falls below 1mHa/electron as all of the approaches converge to the same energy.",
"At small electron numbers, however, the Baldereschi point significantly deviates from the twist-averaged energy, while the connectivity scheme is a much better approximation.",
"Figure: All energies shown reflect the difference in correlation energy between the Γ\\Gamma -point and the relevant calculation.",
"The connectivity algorithm delivers comparable corrections to the correlation energy (relative to the Γ\\Gamma -point) when compared with twist averaging across a wide range of electron numbers.",
"The Baldereschi point only delivers comparable corrections to the correlation energy (relative to the Γ\\Gamma -point) at higher electron numbers (N≥162N \\ge 162) when compared with twist averaging.",
"Twist averaging is performed over 100 twist angles.",
"Standard errors are calculated in the normal fashion for twist averaging, σ≈ Var (E CCD (𝐤 s ))/N s \\sigma \\approx \\sqrt{\\mathrm {Var}(E_{\\mathrm {CCD}}({{\\bf k}_s})) / N_s}."
],
[
"Discussion & concluding remarks",
"Our results show that a finite electron gas is best able to reproduce the twist-averaged total and correlation energies when a special $\\bf {k}_s$ -point is chosen to minimize the differences between the momentum connectivity of the finite system and a reference (here, a twist-averaged finite system).",
"Our interpretation of the connectivity-derived special $\\bf {k}_s$ -point's utility is that the low-momentum two-particle excitations from HF often suffer from finite size errors due to the shape of the Fermi surface in $k$ -space.",
"By finding a particularly representative $k_s$ -point, we aim to take the `best case' of a representative shape–or, at least, as best as can be managed by a truly finite system.",
"When we examine the occupied orbitals in $k$ -space at the special $\\bf {k}_s$ -point, they adopt low-symmetry patterns that tend more toward the shape of a sphere than the $\\Gamma $ -point distribution.",
"Though we have made significant progress here towards ameliorating finite size error, there are still two open questions.",
"First, could our method be modified in order to minimize the energy difference to the thermodynamic limit rather than just to the twist-averaged energy?",
"The second open question surrounds the extrapolation – in particular, what is the actual form of the energy as the system size tends to infinity?",
"We could investigate this source of error by comparing with the known high-density limit of RPA, which CCD is expected to be able to capture.",
"We leave both of these investigations for future work.",
"Overall, the results here should improve our ability to understand infinite-sized model systems that are necessarily represented as finite systems, such as the electron gas with varying dimensions, the Hubbard model, and the models of nuclear matter we previously studied.",
"[129], [133], [134] This communication is timely due to a resurgence of interest in the uniform electron gas [135], [136], [137], [138], [139], [140], [127], [68], [96] and of twist-averaged coupled cluster calculations.",
"[2], [141] We expect this work can immediately be applied to improve calculations.",
"Our long-term goals are to use this approach to study realistic systems.",
"Though calculations are left for future manuscripts, we expect to follow a similar approach to our prior work in this area.",
"In particular, we start by observing the similarity between how twist-averaging works in plane wave ab initio calculations where the energy is still obtained as a sum over matrix elements $v_{ijab}$ (as in Eq.",
"(REF )) which are offset by a twist angle.",
"Specifically, then, it should be possible to choose the twist angle in the same way as we propose here, so for a cubic system with $N$ electrons and a box length of $L$ , the same twist angle as used here should work.",
"As such, we will soon be applying this to real solids and leave this for a future study.",
"Supplementary Material.– The reader is directed to the supplementary material for raw data tables and illustrations mentioned in the text.",
"Acknowledgements.– JJS and TM acknowledge the University of Iowa for funding.",
"JJS thanks the University of Iowa for an Old Gold Award.",
"ARM was supported by the National Science Foundation Graduate Research Fellowship under Grant No.",
"1122374.",
"The code used throughout this work is a locally modified version of a github repository used in previous work [68], [142]: https://github.com/jamesjshepherd/uegccd."
]
] | 1906.04372 |
[
[
"Pure entropic regularization for metrical task systems"
],
[
"Abstract We show that on every $n$-point HST metric, there is a randomized online algorithm for metrical task systems (MTS) that is $1$-competitive for service costs and $O(\\log n)$-competitive for movement costs.",
"In general, these refined guarantees are optimal up to the implicit constant.",
"While an $O(\\log n)$-competitive algorithm for MTS on HST metrics was developed by Bubeck et al.",
"(SODA 2019), that approach could only establish an $O((\\log n)^2)$-competitive ratio when the service costs are required to be $O(1)$-competitive.",
"Our algorithm can be viewed as an instantiation of online mirror descent with the regularizer derived from a multiscale conditional entropy.",
"In fact, our algorithm satisfies a set of even more refined guarantees; we are able to exploit this property to combine it with known random embedding theorems and obtain, for any $n$-point metric space, a randomized algorithm that is $1$-competitive for service costs and $O((\\log n)^2)$-competitive for movement costs."
],
[
"Introduction",
"Let $(X,d_X)$ be a finite metric space with $|X|=n > 1$ .",
"The Metrical Task Systems (MTS) problem, introduced in [15] is described as follows.",
"The input is a sequence $\\langle c_t : X \\rightarrow R_+ : t \\geqslant 1\\rangle $ of nonnegative cost functions on the state space $X$ .",
"At every time $t$ , an online algorithm maintains a state $\\rho _t \\in X$ .",
"The corresponding cost is the sum of a service cost $c_t(\\rho _t)$ and a movement cost $d_X(\\rho _{t-1}, \\rho _t)$ .",
"Formally, an online algorithm is a sequence of mappings ${\\rho } = \\langle \\rho _1, \\rho _2, \\ldots , \\rangle $ where, for every $t \\geqslant 1$ , $\\rho _t : (R_+^X)^t \\rightarrow X$ maps a sequence of cost functions $\\langle c_1, \\ldots , c_t\\rangle $ to a state.",
"The initial state $\\rho _0 \\in X$ is fixed.",
"The total cost of the algorithm ${\\rho }$ in servicing ${c} = \\langle c_t : t \\geqslant 1\\rangle $ is defined as: $\\mathrm {cost}_{{\\rho }}({c}) \\mathrel {\\mathop :}=\\sum _{t \\geqslant 1} \\left[c_t\\!\\left(\\rho _t(c_1,\\ldots , c_t)\\right) + d_X\\!\\left(\\rho _{t-1}(c_1,\\ldots , c_{t-1}), \\rho _t(c_1,\\ldots , c_t)\\right)\\right].$ The cost of the offline optimum, denoted $\\mathrm {cost}^*({c})$ , is the infimum of $\\sum _{t \\geqslant 1} [c_t(\\rho _t)+d_X(\\rho _{t-1},\\rho _t)]$ over any sequence $\\left\\langle \\rho _t : t \\geqslant 1\\right\\rangle $ of states.",
"A randomized online algorithm ${\\rho }$ is said to be $\\alpha $ -competitive if for every $\\rho _0 \\in X$ , there is a constant $\\beta > 0$ such that for all cost sequences ${c}$ : $\\operatornamewithlimits{{E}}\\left[\\mathrm {cost}_{{\\rho }}({c})\\right] \\leqslant \\alpha \\cdot \\mathrm {cost}^*({c}) + \\beta \\,.$ For the $n$ -point uniform metric, a simple coupon-collector argument shows that the competitive ratio is $\\Omega (\\log n)$ , and this is tight [15].",
"A long-standing conjecture is that this $\\Theta (\\log n)$ competitive ratio holds for an arbitrary $n$ -point metric space.",
"The lower bound has almost been established [5], [14]; for any $n$ -point metric space, the competitive ratio is $\\Omega (\\log n / \\log \\log n)$ .",
"Following a long sequence of works (see, e.g., [19], [13], [4], [3], [17], [18]), an upper bound of $O((\\log n)^2)$ was shown in [11]."
],
[
"Relation to adversarial multi-arm bandits.",
"MTS is naturally related to the adversarial setting of the classical multi-arm bandits model in sequential decision making, and provides a very general framework for “bandits with switching costs.” Unlike in the setting of regret minimization, where one competes against the best static strategy in hindsight (see, e.g., [9]), competitive analysis compares the performance of an online algorithm to the best dynamical offline algorithm.",
"Thus this model emphasizes the importance of an adaptivity in the face of changing environments.",
"For MTS, the online algorithm has full information: access to the complete cost function $c_t$ is available when deciding on a point $\\rho _t(c_1,\\ldots ,c_t) \\in X$ at which to play.",
"And yet one of the fascinating relationships between MTS and adversarial bandits is the parallel between adaptivity—being willing to “try out” new strategies—and the classical exploration/exploitation tradeoff that occurs in models where one only has access to partial information about the loss functions."
],
[
"HST metrics.",
"The methods of [8] show that the competitive ratio for MTS is $O(\\log n)$ on weighted star metrics.",
"Recently, the authors of [11] generalized this result by designing an algorithm with competitive ratio $O(\\mathfrak {D}_T \\log n)$ on any weighted $n$ -point tree metric with combinatorial depth $\\mathfrak {D}_T$ .",
"We now discuss a special class of metrics.",
"Let $T=(V,E)$ be a finite tree with root ${r}$ and vertex weights $\\lbrace w_u > 0 : u \\in V\\rbrace $ , let $\\mathcal {L}\\subseteq V$ denote the leaves of $T$ , and suppose that the vertex weights on $T$ are non-increasing along root-leaf paths.",
"Consider the metric space $(\\mathcal {L},d_T)$ , where $d_T(\\ell ,\\ell ^{\\prime })$ is the weighted length of the path connecting $\\ell $ and $\\ell ^{\\prime }$ when the edge from a node $u$ to its parent is $w_u$ .",
"We will use $\\mathfrak {D}_T$ for the combinatorial (i.e., unweighted) depth of $T$ .",
"$(\\mathcal {L},d_T)$ is called an HST metric (or, equivalently for finite metric spaces, an ultrametric).",
"If, for some $\\tau > 1$ , the weights on $T$ satisfy the stronger inequality $w_v \\leqslant w_u/\\tau $ whenever $v$ is a child of $u$ , the space $(\\mathcal {L},d_T)$ is said to be a $\\tau $ -HST metric.",
"Such metric spaces play a special role in MTS since every $n$ -point metric space can be probabilistically approximated by a distribution over such spaces [3], [18].",
"Indeed, the $O((\\log n)^2)$ -competitive ratio for general metric spaces established in [11] is a consequence of their $O(\\log n)$ -competitive algorithm for HSTs."
],
[
"Refined guarantees",
"The authors of [7] observe that there is a more refined way to analyze competive algorithms for MTS.",
"For a randomized online algorithm ${\\rho }$ and a cost sequence ${c}$ , we denote, respectively, $\\mathsf {S}_{{\\rho }}({c})$ and $\\mathsf {M}_{{\\rho }}({c})$ for the (expected) service cost and movement cost, that is $\\mathsf {S}_{{\\rho }}({c}) \\mathrel {\\mathop :}=\\operatornamewithlimits{{E}}\\sum _{t \\geqslant 1} c_t(\\rho _t) \\quad \\text{and}\\quad \\mathsf {M}_{{\\rho }}({c}) \\mathrel {\\mathop :}=\\operatornamewithlimits{{E}}\\sum _{t\\geqslant 1}d_X(\\rho _{t-1}, \\rho _t) \\,.$ If there are numbers $\\alpha ,\\alpha ^{\\prime },\\beta ,\\beta ^{\\prime } > 0$ such that for every cost ${c}$ , it holds that S(c) cost*(c)+ M(c) ' cost*(c)+', one says that ${\\rho }$ is $\\alpha $ -competitive for service costs and $\\alpha ^{\\prime }$ -competitive for movement costs.",
"In [7], it is shown that on every $n$ -point HST metric, and for every $\\varepsilon > 0$ , there is an online algorithm that is simultaneously $(1+\\varepsilon )$ -competitive for service costs and $O((\\log (n/\\varepsilon ))^2)$ -competitive for movement costs.",
"The authors of [11] improve this slightly to show that actually there is an online algorithm that is simultaneously 1-competitive for service costs and $O((\\log n)^2)$ -competitive for movement costs.",
"We obtain the optimal refined guarantees.",
"Theorem 1.1 On any $n$ -point HST metric $X$ , there is a randomized online algorithm that is 1-competitive for service costs and $O(\\log n)$ -competitive for movement costs.",
"Remark 1.2 (Optimality of the refined guarantees) Any finitely competitive algorithm for MTS on an $n$ -point uniform metric cannot be better than $\\Omega (\\log n)$ -competitive for movement costs, regardless of its competitive ratio for service costs.",
"This is because this lower bound holds even if the cost functions only take values 0 and $\\infty $ .",
"Moreover, it cannot be better than 1-competitive for service costs, regardless of its competitive ratio for movement costs.",
"To see this, consider the case where each cost function is the constant function 1."
],
[
"Finely competitive guarantees.",
"Suppose that for some numbers $\\alpha _0,\\alpha _1,\\gamma ,\\beta ,\\beta ^{\\prime } > 0$ , a randomized online algorithm $\\rho $ satisfies, for every cost ${c}$ and every offline algorithm ${\\rho }^*$ : S(c) 0 S*(c) + 1 M*(c) + M(c) S(c) + ' .",
"In this case, we say that ${\\rho }$ is $(\\alpha _0,\\alpha _1,\\gamma )$ -finely competitive.",
"We establish the following.",
"Theorem 1.3 On any $n$ -point HST metric $X$ , for every $\\kappa \\geqslant 1$ , there is an online randomized algorithm ${\\rho }$ that is $\\left(1,1/\\kappa ,O(\\kappa \\log n)\\right)$ -finely competitive.",
"In fact, one can take $\\beta =0$ and $\\beta ^{\\prime } \\leqslant O(\\kappa \\mathrm {diam}(X))$ .",
"Combined with the random embedding from [18], this yields the following consequence for general $n$ -point metric spaces.",
"Corollary 1.4 On any $n$ -point metric space, there is an online randomized algorithm that is 1-competitive for service costs and $O((\\log n)^2)$ -competitive for movement costs.",
"Consider an $n$ -point metric space $(X,d_X)$ .",
"It is known [18] that there exists a random HST metric $(T,d_T)$ so that $\\mathcal {L}(T)=X$ and for all $x,y \\in X$ : $\\operatornamewithlimits{{P}}[d_T(x,y) \\geqslant d_X(x,y)]=1$ , $\\operatornamewithlimits{{E}}[d_T(x,y)] \\leqslant D\\cdot d_X(x,y)$ , and $D \\leqslant O(\\log n)$ .",
"Let $\\rho _T$ be the randomized algorithm for $(T,d_T)$ guaranteed by thm:super with $\\kappa =D$ .",
"Let $\\rho $ denote the algorithm that results from sampling $(T,d_T)$ and then using $\\rho _T$ .",
"We use $\\mathsf {M}^T$ to denote movement cost measured in $d_T$ and $\\mathsf {M}^X$ for movement cost measured in $d_X$ .",
"Then for any cost ${c}$ and any offline algorithm $\\rho ^*$ , we have S(c) = *E[ST(c)] S*(c) + -1 *E[MT*(c)] + O(1) S*(c) + -1 D MX*(c) + O(1) = S*(c) + MX*(c) + O(1) , and MX(c) = *E[MXT(c)] *E[MTT(c)] O(n) *E[ST(c)] + O(1), completing the proof."
],
[
"The fractional model on trees",
"We will work in the following deterministic fractional setting, which is equivalent to the randomized integral setting described earlier (see [11]).",
"The state of a fractional algorithm is given by a point in the polytope KT :={ x R+V : xr = 1, xu = v (u) xv u V L}, where we use $\\chi (u)$ for the set of children of $u$ in $T$ .",
"For $u \\ne {r}$ , we will also write $\\mathsf {p}(u)$ for the parent of $u$ in $T$ .",
"A state $x\\in \\mathsf {K}_T$ corresponds to the situation that the state of a randomized integral algorithm is a leaf descendant of $u$ with probability $x_u$ .",
"Note that $\\mathsf {K}_T$ is simply an affine encoding of the probability simplex on $\\mathcal {L}$ .",
"In the fractional setting, changing from state $x$ to $x^{\\prime }$ incurs movement cost $\\left\\Vert x-x^{\\prime }\\right\\Vert _{\\ell _1(w)}$ , where $\\left\\Vert z\\right\\Vert _{\\ell _1(w)} \\mathrel {\\mathop :}=\\sum _{u \\in V} w_u |z_u|$ denotes the weighted $\\ell _1$ -norm on $R^{V}$ ."
],
[
"Mirror descent, metric filtrations, and regularization",
"Following [11], our algorithm is based on the mirror descent framework as established in [10].",
"This is a method for regularized online convex optimization, an approach that was previously explored for competitive analysis in [1], [12].",
"A central component of mirror descent is choosing the appropriate mirror map (which we will often refer to as the “regularizer”).",
"This is a strictly convex function $\\Phi : \\mathsf {K}_T \\rightarrow R$ that endows $\\mathsf {K}_T$ with a geometric (Riemannian) structure, specifying how to perform constrained vector flow.",
"In other words, it specifies how one can move in a preferred direction while remaining inside $\\mathsf {K}_T$ .",
"The paper [11] employs the following regularizer: $\\Phi _0(x) \\mathrel {\\mathop :}=\\frac{1}{\\eta } \\sum _{u \\in V \\setminus \\lbrace {r}\\rbrace } w_u \\left(x_u+\\delta _u\\right) \\log \\left(x_u+\\delta _u\\right)\\,,$ with $\\eta =\\Theta (\\log |\\mathcal {L}|)$ and $\\delta _u = |\\mathcal {L}_u|/|\\mathcal {L}|$ , where $\\mathcal {L}_u$ is the set of leaves in the subtree rooted at $u$ ."
],
[
"Metric filtrations",
"It is straightforward that one can think of $\\Phi _0$ as a type of multiscale entropy (this is the negative of the associated Shannon entropy, since we use the analyst's convention that the entropy is convex).",
"To understand this notion, let us forget momentarily the weights on $T$ .",
"Then the structure of $T$ gives a natural filtration over probability measures on the leaves $\\mathcal {L}$ .",
"Suppose that ${X}$ is a random variable taking values in $\\mathcal {L}$ and, for $u \\in V$ , denote by $\\mathcal {E}_u$ the event $\\lbrace {X} \\in \\mathcal {L}_u\\rbrace $ .",
"Then the chain rule for Shannon entropy yields $\\sum _{\\ell \\in \\mathcal {L}} \\operatornamewithlimits{{P}}[\\mathcal {E}_\\ell ]\\log \\frac{1}{\\operatornamewithlimits{{P}}[\\mathcal {E}_{\\ell }]} =\\sum _{u\\in V\\setminus \\lbrace {r}\\rbrace }\\operatornamewithlimits{{P}}[\\mathcal {E}_u] \\log \\frac{\\operatornamewithlimits{{P}}[\\mathcal {E}_{\\mathsf {p}(u)}]}{\\operatornamewithlimits{{P}}[\\mathcal {E}_u]}.$ If we now imagine that uncertainty at higher scales is more costly than uncertainty at lower scales, then we might define an analogous weighted entropy by $\\sum _{u\\in V\\setminus \\lbrace {r}\\rbrace }w_u \\operatornamewithlimits{{P}}[\\mathcal {E}_u] \\log \\frac{\\operatornamewithlimits{{P}}[\\mathcal {E}_{\\mathsf {p}(u)}]}{\\operatornamewithlimits{{P}}[\\mathcal {E}_{u}]}.$ Such a notion is natural in the context of “metric learning” problems.",
"Ignoring the $\\lbrace \\delta _u\\rbrace $ values for a moment, consider that (REF ) is not analogous to (REF ).",
"Indeed, it corresponds to the quantity $\\sum _{u\\in V\\setminus \\lbrace {r}\\rbrace }w_u\\operatornamewithlimits{{P}}[\\mathcal {E}_u] \\log \\frac{1}{\\operatornamewithlimits{{P}}[\\mathcal {E}_u]},$ and now one can see a fundamental reason why the algorithm associated to (REF ) only achieves an $O(\\mathfrak {D}_T \\log n)$ competitive ratio, where $\\mathfrak {D}_T$ is the combinatorial depth of $T$ : The quantity (REF ) overmeasures the metric uncertainty.",
"Suppose that ${X}$ is a uniformly random leaf.",
"Then $\\sum _{\\ell \\in \\mathcal {L}} \\operatornamewithlimits{{P}}[\\mathcal {E}_\\ell ] \\log \\frac{1}{\\operatornamewithlimits{{P}}[\\mathcal {E}_{\\ell }]} = \\log n$ , where $n=|\\mathcal {L}|$ .",
"But, in general, one could have $\\sum _{u \\in V} \\operatornamewithlimits{{P}}[\\mathcal {E}_u] \\log \\frac{1}{\\operatornamewithlimits{{P}}[\\mathcal {E}_u]} \\geqslant \\Omega (\\mathfrak {D}_T \\log n)$ .",
"This fact was not lost on the authors of [11], but they bypass the problem by combining mirror descent on stars with a recursive composition method called “unfair gluing.”"
],
[
"Multiscale conditional entropy",
"We employ a regularizer that is a more faithful analog of (REF ): $\\Phi (x) \\mathrel {\\mathop :}=\\sum _{u \\in V \\setminus \\lbrace {r}\\rbrace } \\frac{w_u}{\\eta _u} \\left(x_u + \\delta _u x_{\\mathsf {p}(u)}\\right) \\log \\left(\\frac{x_u}{x_{\\mathsf {p}(u)}}+\\delta _u\\right),$ where $\\mathsf {p}(u)$ denotes the parent of $u$ .",
"If one ignores the additional parameters $\\lbrace \\eta _u \\geqslant 1,\\delta _u > 0\\rbrace $ , this is precisely the negative weighted Shannon entropy written according to the chain rule.",
"Here, we set u :=|Lu||Lp(u)| u :=1+(1/u) u :=u/u .",
"The numbers $\\lbrace \\theta _u\\rbrace $ are the conditional probabilites of the uniform distribution on leaves.",
"The $\\lbrace \\delta _u\\rbrace $ values are employed as “noise” added to the entropy calculation.",
"Such noise is a fundamental aspect for competitive analysis, and distinguishes it from the application of mirror descent to regret minimization problems (see, e.g., [9]).One finds aspects of this “mixing with the uniform distribution” in the bandits setting as well, but used for variance reduction, a seemingly very different purpose.",
"The effect of these noise parameters appears ubiquitously in applications of the primal-dual method to competitive analysis (see [16]), and manifests itself as an additive term in the update rules (see eq:evo below).",
"Intuitively, it ensures that the conditional probability $\\frac{x_u}{x_{\\mathsf {p}(u)}}$ is updated fast enough even when it is close to 0.",
"Finally, the numbers $\\lbrace \\eta _u : u \\in V\\rbrace $ are commonly referred to as “learning rates” in the study of online learning.",
"They represent the rate at which information is discounted in the resulting algorithm; for MTS, this corresponds to the relative importance of costs arriving now vs. costs that arrived in the past."
],
[
"The dynamics",
"We will derive in sec:derivation the following continuous time evolution of the resulting mirror descent algorithm $\\left(x(t) \\in \\mathsf {K}_T : t \\in [0,\\infty )\\right)$ for a cost path $c\\colon [0,\\infty )\\rightarrow R_+^{\\mathcal {L}}$ : $\\partial _t \\left(\\frac{x_u(t)}{x_{\\mathsf {p}(u)}(t)}\\right) = \\frac{\\eta _u}{w_u} \\left(\\frac{x_u(t)}{x_{\\mathsf {p}(u)}(t)}+\\delta _u\\right)\\left(\\beta _{\\mathsf {p}(u)}(t) - \\sum _{\\ell \\in \\mathcal {L}_u} \\frac{x_{\\ell }(t)}{x_u(t)} c_{\\ell }(t)\\right)$ Here, $\\beta _{\\mathsf {p}(u)}(t)$ is a Lagrangian multiplier that ensures conservation of conditional probability: $\\sum _{v \\in \\chi (\\mathsf {p}(u))} \\partial _t \\left(\\frac{x_v(t)}{x_{\\mathsf {p}(u)}(t)}\\right) = 0\\,.$ One can see that the evolution is being driven by the expected instantaneous cost incurred conditioned on the current state being in the subtree rooted at $u$ .",
"One should interpret (REF ) only when $x(t)$ lies in the relative interior of $\\mathsf {K}_T$ .",
"Otherwise, the conditional probabilities are ill-defined.",
"One way to rectify this is to prevent $x(t)$ from hitting the relative boundary of $\\mathsf {K}_T$ at all.",
"It is possible to adaptively modify the cost functions by a suitably small perturbation so as to guarantee this property and, at the same time, ensure that the total discrepancy between the modified and true service cost is a small additive constant.",
"Instead, we will follow a different approach, by extending the dynamics to an analogous system of conditional probabilities $\\lbrace q_u(t) : u \\in V \\setminus \\lbrace {r}\\rbrace \\rbrace $ : $\\partial _t q_u(t) = \\frac{\\eta _u}{w_u} \\left(q_u(t)+\\delta _u\\right) \\left(\\beta _{\\mathsf {p}(u)}(t) - \\hat{c}_u(t) + \\alpha _u(t)\\right),$ where $q_u(t) = \\frac{x_u(t)}{x_{\\mathsf {p}(u)}(t)}$ whenever $x_{\\mathsf {p}(u)}(t) > 0$ , $\\alpha _u(t)$ is a Lagrangian multiplier for the constraint $q_u(t)\\geqslant 0$ , and $\\hat{c}_u(t)$ is the “derived” cost in the subtree rooted at $u$ : cu(t) :=Lu qu(t) c(t) qu(t) :=v u, {u} qv(t) , where $\\gamma _{u,\\ell }$ is the unique simple $u$ -$\\ell $ path in $T$ .",
"Stated this way, the mirror descent algorithm can be envisioned as running a “weighted star” algorithm on the conditional probabilities at every internal node of $T$ , with the derived costs at an internal node $u$ given by the average cost of the current strategy for playing one unit of mass in the subtree rooted at $u$ .",
"In the next section, we will implement and analyze a discretization of (REF ) using Bregman projections.",
"Since our regularizer $\\Phi $ and convex body $\\mathsf {K}_T$ do not satisfy the assumptions underlying the existence and uniqueness theorem of [10], we need to construct a solution to (REF ) and, indeed, taking the discretization parameter in our algorithm to zero, one establishes a solution of bounded variation; see sec:continuous.",
"The major benefit of the formulations (REF ) and (REF ) is in motivating such an algorithm and prescribing the derived costs.",
"In sec:derivation, we describe how these dynamics can be predicted from the definition (REF )."
],
[
"The MTS algorithm",
"We will first establish some generic machinery which, at this point, is not specific to MTS yet.",
"Consider a convex polytope $\\mathsf {K}_0\\subseteq R^n$ , define $\\mathsf {K}\\mathrel {\\mathop :}=\\mathsf {K}_0 \\cap R_+^n$ , and assume that $\\mathsf {K}$ is compact.",
"Suppose additionally that $\\Phi : \\mathcal {D}\\rightarrow R$ is differentiable and strictly convex in an open neighborhood $\\mathcal {D}\\supseteq \\mathsf {K}$ .",
"Let us write $\\mathsf {D}_{\\Phi }$ for the corresponding Bregman divergence $\\mathsf {D}_{\\Phi }(y \\,\\Vert \\,x) \\mathrel {\\mathop :}=\\Phi (y) - \\Phi (x) - \\left\\langle \\nabla \\Phi (x), y-x\\right\\rangle ,$ which is non-negative due to convexity of $\\Phi $ .",
"Then for $x,y,z \\in \\mathsf {K}$ , we have: $\\mathsf {D}_{\\Phi }(z \\,\\Vert \\,y) - \\mathsf {D}_{\\Phi }(z \\,\\Vert \\,x) = -\\Phi (y)+\\Phi (x)-\\langle \\nabla \\Phi (y), z-y\\rangle +\\langle \\nabla \\Phi (x),z-x\\rangle .$ For a vector $c \\in R^n$ and $x \\in \\mathsf {K}$ , define the projection $\\Pi _{\\mathsf {K}}^c (x) \\mathrel {\\mathop :}=\\operatorname{\\mathrm {argmin}}\\left\\lbrace \\mathsf {D}_{\\Phi }(y \\,\\Vert \\,x) + \\langle c,y\\rangle : y \\in \\mathsf {K}\\right\\rbrace .$ Since $\\mathsf {K}$ is compact and $\\Phi $ is strictly convex, there is a unique minimizer $y^* \\in \\mathsf {K}$ .",
"For $x \\in \\mathsf {K}$ , recall the definition of the normal cone at $x$ : $\\mathsf {N}_{\\mathsf {K}}(x) = \\left\\lbrace p \\in R^n : \\langle p, y-x\\rangle \\leqslant 0 \\textrm { for all } y \\in \\mathsf {K}\\right\\rbrace .$ Given a representation of $\\mathsf {K}$ by inequality constraints, $\\mathsf {K}=\\lbrace x\\in R^n\\colon Ax\\leqslant b\\rbrace $ for $A\\in R^{m\\times n}$ and $b\\in R^n$ , it holds NK(x) = {ATyy0 and yT(Ax-b)=0}.",
"The KKT conditions yield $\\nabla \\Phi (y^*) = \\nabla \\Phi (x) - c - \\lambda ^*\\,,$ where $\\lambda ^* \\in \\mathsf {N}_{\\mathsf {K}}(y^*)$ .",
"Since $\\mathsf {N}_{\\mathsf {K}}(y^*) = \\mathsf {N}_{\\mathsf {K}_0}(y^*) + \\mathsf {N}_{R_+^n}(y^*)$ , we can can decompose $\\lambda ^* = \\beta - \\alpha $ with $\\beta \\in \\mathsf {N}_{\\mathsf {K}_0}(y^*)$ and $- \\alpha \\in \\mathsf {N}_{R_+^n}(y^*)$ .",
"In particular, we have $\\alpha \\geqslant 0$ and $\\alpha _i > 0 \\Rightarrow y^*_i = 0$ for every $i=1,\\ldots ,n$ .",
"Substituting this into (REF ) gives D(z y*) - D(z x) = -(y*)+(x)+(x),y*-x+ c-+ , z-y* -D(y* x) + c-, z-y*, where the inequality comes from $\\langle \\beta , z-y^*\\rangle \\leqslant 0$ since $z \\in \\mathsf {K}$ and $\\beta \\in \\mathsf {N}_{\\mathsf {K}}(y^*)$ .",
"We have proved the following.",
"Lemma 2.1 For any $x, z \\in \\mathsf {K}$ , and $c \\in R^n$ , let $y^* = \\Pi _{\\mathsf {K}}^c(x)$ and $\\lambda ^*$ be as in (REF ).",
"Then for any $\\alpha \\in -\\mathsf {N}_{R^+_n}(y^*)$ such that $\\lambda ^* + \\alpha \\in \\mathsf {N}_{\\mathsf {K}_0}(y^*)$ , it holds that $\\mathsf {D}_{\\Phi }(z \\,\\Vert \\,y^*) - \\mathsf {D}_{\\Phi }(z \\,\\Vert \\,x) \\leqslant \\langle c-\\alpha , z-y^*\\rangle .$"
],
[
"Iterative Bregman projections",
"We describe now a discretization of the algorithm from the introduction.",
"This discretization will mimic the continuous dynamics if the entries of each individual cost vector are small.",
"We can achieve this by splitting each cost vector into several copies of scaled down versions of itself, as discussed in sec:alg.",
"In sec:continuous, we will give a formal argument that this indeed yields a discretization of the continuous dynamics from the introduction.",
"Fix a tree $T$ and recall the definition of $\\mathsf {K}_T$ from (REF ).",
"Let $Q_{T}$ denote the collection of vectors $q \\in R_+^{V \\setminus \\lbrace {r}\\rbrace }$ such that for all $u \\in V \\setminus \\mathcal {L}$ , $\\sum _{v \\in \\chi (u) } q_v = 1.$ For $q \\in Q_T$ and $u \\in V \\setminus \\mathcal {L}$ , we use $q^{(u)} \\in R_+^{\\chi (u)}$ to denote the vector defined by $q^{(u)}_v \\mathrel {\\mathop :}=q_v$ for $v \\in \\chi (u)$ , and define the corresponding probability simplex $Q^{(u)}_T \\mathrel {\\mathop :}=\\lbrace q^{(u)} : q \\in Q_T \\rbrace $ .",
"We will use $\\Delta : Q_T \\rightarrow \\mathsf {K}_T$ for the map which sends $q \\in Q_T$ to the (unique) $x = \\Delta (q) \\in \\mathsf {K}_T$ such that $x_v = x_u q_v \\qquad \\forall u \\in V \\setminus \\mathcal {L}, v \\in \\chi (u).$ Note that $q$ contains more information than $x$ ; the map $\\Delta $ fails to be invertible whenever there is some $u \\in V \\setminus \\mathcal {L}$ with $x_u = 0$ .",
"Fix $\\kappa \\geqslant 1$ .",
"On the open domain $\\mathcal {D}^{(u)}=(-\\min _{v\\in \\chi (u)}\\delta _v,\\infty )^{\\chi (u)}$ , for $\\delta _v$ as given in (REF ), define the strictly convex function $\\Phi ^{(u)} : \\mathcal {D}^{(u)} \\rightarrow R$ by $\\Phi ^{(u)}(p) \\mathrel {\\mathop :}=\\frac{1}{\\kappa } \\sum _{v \\in \\chi (u)} \\frac{w_v}{\\eta _v} \\left(p_v + \\delta _v\\right) \\log \\left(p_v + \\delta _v\\right).$ Denote the corresponding Bregman divergence on $Q^{(u)}_T$ by $\\mathsf {D}^{(u)}\\!\\left(p \\,\\Vert \\,p^{\\prime } \\right) = \\frac{1}{\\kappa } \\sum _{v \\in \\chi (u)} \\frac{w_v}{\\eta _v} \\left[\\left(p_v + \\delta _v\\right) \\log \\frac{p_v+\\delta _v}{p^{\\prime }_v+\\delta _v}+ p^{\\prime }_v-p_v\\right].$ We now define an algorithm that takes a point $q \\in Q_T$ and a cost vector $c \\in R_+^{\\mathcal {L}}$ and outputs a point $p = \\mathcal {A}(q,c)\\in Q_T$ .",
"Fix $\\langle u_1,u_2,\\ldots , u_N\\rangle $ a topological ordering of $V \\setminus \\mathcal {L}$ such that every child in $T$ occurs before its parent.",
"We define $p$ inductively as follows.",
"Let $\\hat{c}_{\\ell } \\mathrel {\\mathop :}=c_{\\ell }$ for $\\ell \\in \\mathcal {L}$ .",
"For every $j=1,2,\\ldots ,N$ : c(uj)v :=cv v (uj) p(uj) :=argmin{ D(uj)(p q(uj)) + p, c(uj) | p Q(uj)T} cuj :=v (uj) p(uj)v cv Let $\\alpha ^{(u_j)}$ be the vector of Lagrange multipliers corresponding to the nonnegativity constraints in (REF ) (recall lem:bp).",
"One should note that in this setting (a probability simplex), the nonnegativity multipliers are unique and thus well-defined.",
"We denote $\\alpha = \\alpha ^{q,c} \\in R_+^{V}$ as the vector given by $\\alpha _v \\mathrel {\\mathop :}=\\alpha ^{(\\mathsf {p}(v))}_v$ for $v \\ne {r}$ and $\\alpha _{{r}} \\mathrel {\\mathop :}=0$ .",
"Recall the complementary slackness conditions: $\\alpha _v > 0 \\Rightarrow p_v = 0.$ For $v \\in \\chi (u)$ , calculate $\\left(\\nabla \\Phi ^{(u)}(p)\\right)_v = \\frac{1}{\\kappa } \\frac{w_v}{\\eta _v} \\left(1+\\log (p_v+\\delta _v)\\right).$ Then using (REF ), we can write the algorithm as follows: For j=1,2,...,N$: \\\\[0.1cm]\\:For v \\in \\chi (u_j)$ : $p^{(u_j)}_v \\mathrel {\\mathop :}=(q^{(u_j)}_v+\\delta _v) \\exp \\left(\\kappa \\frac{\\eta _v}{w_v} \\left(\\beta _{u_j} - (\\hat{c}_v - \\alpha _v)\\right)\\right)- \\delta _v$ , $\\hat{c}_{u_j} \\mathrel {\\mathop :}=\\sum _{v \\in \\chi (u_j)} p_v^{(u_j)} \\hat{c}_v$ .",
"where $\\beta _{u_j}\\geqslant 0$ is the multiplier for the constraint $\\sum _{v \\in \\chi (u_j)} q^{(u_j)}_v \\geqslant 1$ .",
"There is no multiplier for the constraint $\\sum _{v \\in \\chi (u_j)} q_v^{(u_j)} \\leqslant 1$ because this constraint will be satisfied automatically and is therefore not needed in (REF ): If it were violated, decreasing some $p_v$ with $p_v>q_v^{(u_j)}$ would yield a strictly better solution to the minimization problem (REF )."
],
[
"The global divergence",
"For $z \\in \\mathsf {K}_T$ and $q \\in Q_T$ , define the global divergence function $\\tilde{\\mathsf {D}}(z \\,\\Vert \\,q) \\mathrel {\\mathop :}=\\frac{1}{\\kappa } \\sum _{u \\notin \\mathcal {L}} \\sum _{v \\in \\chi (u)} \\frac{w_v}{\\eta _v} \\left[ \\left(z_v + \\delta _v z_u\\right) \\log \\left(\\frac{\\frac{z_v}{z_u}+\\delta _v}{q_v+\\delta _v}\\right)+ z_u q_v - z_v\\right],$ with the convention that $0 \\log \\left(\\frac{0}{0}+\\delta _v\\right) = \\lim _{\\varepsilon \\rightarrow 0} \\varepsilon \\log \\left(\\frac{0}{\\varepsilon }+\\delta _v\\right) = 0$ .",
"We remark that $\\tilde{\\mathsf {D}}$ is the Bregman divergence associated to the regularizer eq:reg1 (divided by $\\kappa $ ) when $\\frac{x_v}{x_{u}}$ is replaced by $q_v$ .",
"We will use $\\tilde{\\mathsf {D}}$ as a potential function to prove inequality eq:finesc.",
"Here, $z$ denotes the configuration of the offline algorithm.",
"Note that while the online configuration is encoded by $q\\in Q_T$ , we still use the polytope $\\mathsf {K}_T$ to encode the offline configuration.",
"This will be more convenient when expressing the offline movement cost.",
"The next lemma shows that when the offline algorithm moves, the change in potential is bounded by $O(1/\\kappa )$ times the offline movement cost.",
"Lemma 2.2 It holds that for any $q \\in Q_T$ and $z,z^{\\prime } \\in \\mathsf {K}_T$ , $\\left|\\tilde{\\mathsf {D}}(z \\,\\Vert \\,q) - \\tilde{\\mathsf {D}}(z^{\\prime } \\,\\Vert \\,q)\\right| \\leqslant \\frac{1}{\\kappa } \\left(2+\\frac{4}{\\tau }\\right) \\left\\Vert z-z^{\\prime }\\right\\Vert _{\\ell _1(w)}.$ Consider a differentiable map $z\\colon [0,1]\\rightarrow R_{++}^V$ such that $\\sum _{v\\in \\chi (u)}z_v(t)\\leqslant z_u(t)$ for each $t$ and $u\\notin \\mathcal {L}$ .",
"It suffices to show that for each $t$ and every fixed $q \\in Q_T$ , $\\kappa \\left|\\partial _t \\tilde{\\mathsf {D}}(z(t) \\,\\Vert \\,q)\\right| \\leqslant \\left(2+\\frac{4}{\\tau }\\right) \\left\\Vert z^{\\prime }(t)\\right\\Vert _{\\ell _1(w)}.$ Moreover, it suffices to address the case when there is at most one $u \\in V$ with $z^{\\prime }_u(t) \\ne 0$ .",
"A direct calculation gives t D(z(t) q) = wuu z'u(t) (zu(t)/zp(u)(t)+uqu+u) + v (u) wvv [v z'u(t) (zv(t)/zu(t)+vqv+v) + z'u(t) (qv - zv(t)zu(t)) ].",
"Let us now use definitions (REF ) and (REF ) to observe that $\\frac{1}{\\eta _v} \\left|\\log \\frac{p_v+\\delta _v}{q_v+\\delta _v}\\right| \\leqslant \\frac{1}{\\eta _v} \\log \\frac{1+\\delta _v}{\\delta _v} \\leqslant 2.$ Using this in (REF ) yields $\\kappa \\left|\\partial _t \\tilde{\\mathsf {D}}(z(t) \\,\\Vert \\,q)\\right| \\leqslant w_u |z_u^{\\prime }(t)| \\left(2 + \\frac{1}{\\tau } \\sum _{v \\in \\chi (u)}\\left(2\\delta _v + \\left|q_v-\\frac{z_v(t)}{z_u(t)}\\right|\\right)\\right)\\leqslant w_u |z_u^{\\prime }(t)| \\left(2+\\frac{4}{\\tau }\\right),$ where the last inequality uses $\\sum _{v \\in \\chi (u)} \\delta _v \\leqslant \\sum _{v \\in \\chi (u)} \\theta _v \\leqslant 1$ and $\\sum _{v\\in \\chi (u)}z_v(t)\\leqslant z_u(t)$ .",
"We will sometimes implicitly restrict vectors $x \\in R^{V}$ to the subspace spanned by $\\lbrace e_{\\ell } : \\ell \\in \\mathcal {L}\\rbrace $ .",
"In this case, we employ the notation $\\langle x,y \\rangle _{\\mathcal {L}} \\mathrel {\\mathop :}=\\sum _{\\ell \\in \\mathcal {L}} x_{\\ell } y_{\\ell },$ when either vector lies in $R^V$ or $R^{\\mathcal {L}}$ .",
"According to the following lemma, the change in potential due to movement of the online algorithm is bounded by the difference in service cost between the offline and online algorithm.",
"Lemma 2.3 For any cost vector $c \\in R_+^{\\mathcal {L}}$ , $z \\in \\mathsf {K}_T$ , and $q \\in Q_T$ , it holds that if $p = \\mathcal {A}(q,c)$ , then $\\tilde{\\mathsf {D}}(z \\,\\Vert \\,p) - \\tilde{\\mathsf {D}}(z \\,\\Vert \\,q) \\leqslant \\left\\langle c, z - \\Delta (p)\\right\\rangle _{\\mathcal {L}}.$ Fix $q \\in Q_T$ and $c \\in R_+^{\\mathcal {L}}$ .",
"Let $\\alpha = \\alpha ^{q,c}$ denote the vector of multipliers defined in sec:iter.",
"For $u \\in V \\setminus \\mathcal {L}$ with $z_u > 0$ , define $z^{(u)} \\in Q^{(u)}_T$ by $z^{(u)}_v \\mathrel {\\mathop :}=\\frac{z_v}{z_u}.$ Then lem:bp gives $\\mathsf {D}^{(u)}\\left(z^{(u)} \\,\\Vert \\,p^{(u)}\\right) - \\mathsf {D}^{(u)}\\left(z^{(u)} \\,\\Vert \\,q^{(u)}\\right) \\leqslant \\left\\langle \\hat{c}^{(u)} - \\alpha ^{(u)},z^{(u)} - p^{(u)}\\right\\rangle _{\\chi (u)},$ where we use $\\langle \\cdot ,\\cdot \\rangle _{\\chi (u)}$ for the standard inner product on $R^{\\chi (u)}$ .",
"Multiplying by $z_u$ and summing yields D(z p) - D(z q) u L zu c(u) - (u), z(u) - p(u) (u) = u L v (u) (c(u)v -(u)v) zv - u L zu v (u) (c(u)v-(u)v) pv .",
"Note that from (REF ), the latter expression is $\\sum _{u \\notin \\mathcal {L}} z_u \\sum _{v \\in \\chi (u)} \\hat{c}_v^{(u)} p_v\\stackrel{(\\ref {eq:hatc2})}{=}\\sum _{u \\notin \\mathcal {L}} z_u \\hat{c}_u.$ Noting that $\\hat{c}_{{r}} = \\sum _{\\ell \\in \\mathcal {L}} \\Delta (p)_{\\ell } c_{\\ell }$ , this gives $\\tilde{\\mathsf {D}}(z \\,\\Vert \\,p) - \\tilde{\\mathsf {D}}(z \\,\\Vert \\,q) \\leqslant \\sum _{u \\ne {r}} (\\hat{c}_u - \\alpha _u) z_u - \\sum _{u \\notin \\mathcal {L}} z_u \\hat{c}_u\\leqslant \\left\\langle c, z-\\Delta (p)\\right\\rangle _{\\mathcal {L}}.",
"$"
],
[
"Algorithm and competitive analysis",
"Let us now outline the proof of inequality eq:finemc.",
"First, we perform a standard reduction that allows us to bound only the “positive” movement costs when the algorithm moves from $x$ to $y$ .",
"Its proof is straightforward.",
"Lemma 2.4 For $x,y\\in \\mathsf {K}_T$ it holds that $\\Vert x-y\\Vert _{\\ell _1(w)} = 2 \\left\\Vert (x-y)_{+}\\right\\Vert _{\\ell _1(w)} + [\\psi (y)-\\psi (x)],$ where $\\psi (x) \\mathrel {\\mathop :}=\\sum _{u \\ne {r}} w_u x_u$ for $x \\in \\mathsf {K}_T$ .",
"We now state the key technical lemma which controls the positive movement cost by the service cost.",
"To this end, we employ an auxiliary potential function $\\Psi : Q_T \\rightarrow R$ defined by u(q) :=- (q)u D(u)((u) q(u)) (q) :=u L u(q).",
"Intuitively, $\\Psi (q)$ is a measure of difference between the online configuration $q$ and the uniform distribution over leaves (whose conditional probabilities are given by $\\theta $ ).",
"Let us give a brief explanation of the need for $\\Psi $ .",
"Our addition of “noise” to the multiscale conditional entropy is to achieve the smoothness property established in lem:lipschitz.",
"But this has the adverse effect of increasing the movement cost of the algorithm, as one can see from the $\\delta _u$ term in eq:evo.",
"This additional movement cannot be easily charged against the service cost in the regime where the noise term is dominant: $\\frac{x_u(t)}{x_{\\mathsf {p}(u)}(t)} \\ll \\delta _u$ .",
"On the other hand, this additional movement has the effect of further decreasing $\\frac{x_u(t)}{x_{\\mathsf {p}(u)}(t)}$ , which drives the conditional probabilities at $\\mathsf {p}(u)$ away from the uniform distribution, decreasing $\\Psi $ .",
"A formal statement appears later in lem:crucial.",
"For the next two results, take any $q \\in Q_T$ and cost $c \\in R_+^{\\mathcal {L}}$ , and denote $p=\\mathcal {A}(q,c), x =\\Delta (q),y=\\Delta (p)$ .",
"Lemma 2.5 (Movement analysis) It holds that $\\frac{\\tau -3}{\\kappa \\tau } \\left\\Vert \\left(x-y\\right)_+\\right\\Vert _{\\ell _1(w)} \\leqslant (2 \\mathfrak {D}_T + \\log n) \\langle c, x\\rangle _{\\mathcal {L}} +\\left[\\Psi (q)-\\Psi (p)\\right].$ This lemma will be proved in sec:movement-analysis.",
"Let us first see that it can be used to establish bounds on the competitive ratio.",
"Define $w_{\\min } \\mathrel {\\mathop :}=\\min \\lbrace w_{\\ell } : \\ell \\in \\mathcal {L}\\rbrace $ and $\\varepsilon _T \\mathrel {\\mathop :}=\\frac{w_{\\min }}{2 (2\\mathfrak {D}_T+\\log n)} \\frac{\\tau -3}{\\tau \\kappa }.$ Theorem 2.6 For any $z \\in \\mathsf {K}_T$ : c, yL c,zL + [D(z q) - D(z p)] -1 x-y1(w) [(y)-(x)] + 2 -3 ([(q)-(p)] + (2DT + n) c, xL) Moreover, if $\\Vert c\\Vert _{\\infty } \\leqslant \\varepsilon _T$ , then $\\kappa ^{-1} \\left\\Vert x-y\\right\\Vert _{\\ell _1(w)} \\leqslant [\\psi (y)-\\psi (x)] + \\frac{4 \\tau }{\\tau -3} \\left( [\\Psi (q)-\\Psi (p)] + (2\\mathfrak {D}_T + \\log n) \\langle c, y\\rangle _{\\mathcal {L}}\\right).$ The bound (REF ) follows from lem:sc, and (REF ) follows from lem:movement and lem:height.",
"To see that (REF ) follows from (REF ) and lem:movement, use the fact that $\\langle c,x \\rangle _{\\mathcal {L}} \\leqslant \\langle c,y\\rangle _{\\mathcal {L}} + \\frac{\\Vert c\\Vert _{\\infty }}{w_{\\min }} \\left\\Vert \\left(x-y\\right)_+\\right\\Vert _{\\ell _1(w)}.$ In light of thm:analysis, we can respond to a cost function $c \\in R_+^{\\mathcal {L}}$ by splitting it into $M$ pieces $c_1,c_2,\\ldots ,c_M$ where $M = \\lceil \\Vert c\\Vert _{\\infty }/\\varepsilon _T\\rceil $ .",
"Now define $q_i \\mathrel {\\mathop :}=\\mathcal {A}(q_{i-1}, c/M)$ , $q_0:=q$ and $\\bar{\\mathcal {A}}(q,c) \\mathrel {\\mathop :}=q_M$ .",
"Theorem 2.7 Fix $\\tau \\geqslant 4$ .",
"Consider the algorithm that begins in some configuration $q_0 \\in Q_T$ .",
"If $c_t \\in R_+^{\\mathcal {L}}$ is the cost function that arrives at time $t$ , denote $q_t \\mathrel {\\mathop :}=\\bar{\\mathcal {A}}(q_{t-1},c_t)$ .",
"Then the sequence $\\left\\langle \\Delta (q_0), \\Delta (q_1), \\ldots \\right\\rangle $ is an online algorithm that is $(1,O(1/\\kappa ), O(\\kappa (\\mathfrak {D}_T + \\log n)))$ -finely competitive.",
"We prove this momentarily.",
"The following fact is well-known and, in conjunction with the preceding theorem, yields the validity of thm:refined and thm:super.",
"Lemma 2.8 (cf.",
"[6]) If $(\\mathcal {L},d_T)$ is an HST metric, then there is another weighted tree $T^{\\prime }$ with leaf set $\\mathcal {L}$ such that $(\\mathcal {L},d_{T^{\\prime }})$ is a 7-HST metric.",
"$\\mathfrak {D}_{T^{\\prime }} \\leqslant \\log _2 |\\mathcal {L}|$ All the leaves of $T^{\\prime }$ have depth $\\mathfrak {D}_{T^{\\prime }}$ .",
"$d_T(\\ell ,\\ell ^{\\prime }) \\leqslant d_{T^{\\prime }}(\\ell ,\\ell ^{\\prime }) \\leqslant O(d_T(\\ell ,\\ell ^{\\prime }))$ for all $\\ell ,\\ell ^{\\prime } \\in \\mathcal {L}$ .",
"[Proof sketch] Replace every weight $w_v$ in $T$ with $\\hat{w}_v \\mathrel {\\mathop :}=7^{\\lceil \\log _7 w_v\\rceil }$ and iteratively contract every edge $(p(u),u)$ with $\\hat{w}_{p(u)}=\\hat{w}_u$ and $u \\notin \\mathcal {L}$ .",
"The resulting weighted tree $T_1$ is a 7-HST by construction.",
"Now iteratively contract every edge $(p(u),u)$ in $T_1$ for which $|\\mathcal {L}^{T_1}_{u}| > \\frac{1}{2} |\\mathcal {L}^{T_1}_{p(u)}|$ .",
"The resulting tree $T^{\\prime }$ has depth $\\mathfrak {D}_{T^{\\prime }} \\leqslant \\log _2 |\\mathcal {L}|$ .",
"Finally, one can achieve property (3) by increasing the depth of every root-leaf path to $\\mathfrak {D}_{T^{\\prime }}$ using vertex weights that decrease by a factor of 7 along the path.",
"[Proof of thm:main-technical] Consider a sequence $\\left\\langle c_t : t \\geqslant 1\\right\\rangle $ of cost functions.",
"By splitting the costs into smaller pieces, we may assume that $\\Vert c_t\\Vert _{\\infty } \\leqslant \\varepsilon _T$ for all $t \\geqslant 1$ .",
"Let $\\lbrace z_t^*\\rbrace $ denote some offline algorithm with $z_0^* = \\Delta (q_0)$ , and let $\\lbrace x_t = \\Delta (q_t)\\rbrace $ denote our online algorithm.",
"Then using $\\tilde{\\mathsf {D}}(z_{0}^* \\,\\Vert \\,x_0) = 0$ along with (REF ) and lem:lipschitz yields, for any time $t_1 \\geqslant 1$ , t=1t1 ct, xtL t=1t1 ct,zt*L - D(zt1* qt1) + O(1/) t=1t1 zt*-zt-1*1(w) t=1t1 ct,zt*L + O(1/) t=1t1 zt*-zt-1*1(w), where we have used $\\tilde{\\mathsf {D}}(z \\,\\Vert \\,q) \\geqslant 0$ for all $z \\in \\mathsf {K}_T$ and $q \\in Q_T$ .",
"This verifies (REF ) with $\\alpha _0=1$ , $\\alpha _1 = O(1/\\kappa )$ , and $\\beta = 0$ .",
"Moreover, (REF ) gives $\\frac{1}{\\kappa } \\sum _{t=1}^{t_1} \\Vert x_t-x_{t-1}\\Vert _{\\ell _1(w)} \\leqslant \\left[\\psi (x_{t_1})-\\psi (x_0)\\right]+\\frac{4\\tau }{\\tau -3} \\left[\\Psi (q_0)-\\Psi (q_{t_1})\\right] + \\left(2\\mathfrak {D}_T + \\log n\\right)\\sum _{t=1}^{t_1} \\langle c_t,x_t\\rangle _{\\mathcal {L}},$ verifying (REF ) with $\\alpha _1 \\leqslant O(\\kappa (\\mathfrak {D}_T + \\log n))$ and $\\beta ^{\\prime } \\leqslant O(\\kappa \\max _{v \\ne {r}} w_v)$ (see lem:dmax below)."
],
[
"Movement analysis",
"It remains to prove lem:movement.",
"The KKT conditions (cf.",
"(REF )) give: For every $v \\in \\chi (u)$ , $\\frac{1}{\\kappa } \\frac{w_v}{\\eta _v} \\log \\left(\\frac{p_v+\\delta _v}{q_v+\\delta _v}\\right) = \\beta _u - \\hat{c}_v + \\alpha _v\\,,$ where $\\beta _u \\geqslant 0$ is the multiplier corresponding to the constraint $\\sum _{v \\in \\chi (u)} q_v \\geqslant 1$ .",
"Lemma 2.9 It holds that $\\alpha _v \\leqslant \\hat{c}_v$ for all $v \\in V \\setminus \\lbrace {r}\\rbrace $ .",
"Note that $\\hat{c}_v \\geqslant 0$ by construction.",
"Thus if $\\alpha _v = 0$ , we are done.",
"Otherwise, by complementary slackness, it must be that $p_v = 0$ , and therefore $\\log (\\frac{p_v+\\delta _v}{q_v+\\delta _v}) \\leqslant 0$ .",
"Since $\\beta _{\\mathsf {p}(v)} \\geqslant 0$ , (REF ) implies that $\\alpha _v \\leqslant \\hat{c}_v$ .",
"Define $\\sigma _v \\mathrel {\\mathop :}=\\log \\left(\\frac{p_v+\\delta _v}{q_v+\\delta _v}\\right)$ so that qv - pv = (qv + v) (1-ev).",
"Recall that for $v \\in \\chi (u)$ , we have $x_v = q_v x_u$ and $y_v = p_v y_u$ , thus $x_v - y_v = x_u (q_v - p_v) + p_v (x_u - y_u) = (x_v+\\delta _v x_u) (1-e^{\\sigma _v}) + p_v (x_u - y_u).$ In particular, wv (xv-yv)+ wv (xv+v xu) (1-ev)+ + wv pv (xu-yu)+ wv (xv+v xu) (1-ev)+ + wu pv (xu-yu)+.",
"Using $\\sum _{v \\in \\chi (u)} p_v = 1$ and summing over all vertices yields $\\sum _{v \\ne {r}} w_v \\left(x_v-y_v\\right)_+ \\leqslant \\sum _{v \\ne {r}} w_v (x_v+\\delta _v x_{\\mathsf {p}(v)}) (1-e^{\\sigma _v})_{+} + \\frac{1}{\\tau } \\sum _{v \\ne {r}} w_v \\left(x_v-y_v\\right)_+\\,,$ hence v r wv (xv-yv)+ -1 v r wv (xv+v xp(v)) (1-ev)+ -1 v r wv (xv+v xp(v)) (v)- -1 (v r v xv cv + u L xu v (u) v (cv-v)), where the last line uses lem:alphas and (REF ), to bound $w_v (\\sigma _v)_{-} \\leqslant \\kappa \\eta _v\\left(\\hat{c}_v - \\alpha _v\\right)$ .",
"Note that v r v xv cv L c x v r, {r} v (DT + n) c,x, since for any $\\ell \\in \\mathcal {L}$ , it holds that v r, {r} v = DT() + v r, {r} |Lp(v)||Lv| = DT() + n, where $\\mathfrak {D}_T(\\ell )$ is the combinatorial depth of $\\ell $ .",
"The second sum in (REF ) can be interpreted as the service cost of hybrid configurations of $q$ and $\\theta $ : While $\\sum _{v\\in \\chi (u)}x_v\\hat{c}_v$ is the service cost of $x$ in $\\mathcal {L}_u$ , the term $x_u\\sum _{v\\in \\chi (u)}\\theta _v\\hat{c}_v$ is the service cost in $\\mathcal {L}_u$ of the modification of $x$ whose conditional probabilities at the children of $u$ are given by $\\theta ^{(u)}$ rather than $q^{(u)}$ .",
"To bound this hybrid service cost, we will employ the auxiliary potential $\\Psi $ ."
],
[
"The hybrid cost",
"We require the following elementary estimate.",
"Lemma 2.10 For $u \\notin \\mathcal {L}$ it holds that $\\max \\left\\lbrace \\mathsf {D}^{(u)}(r \\,\\Vert \\,p) : r,p \\in Q^{(u)}_T \\right\\rbrace \\leqslant \\frac{2}{\\kappa } \\frac{w_u}{\\tau }.$ Define $\\phi _v : (-\\delta _v,\\infty ) \\rightarrow R$ by $\\phi _v(p) \\mathrel {\\mathop :}=\\frac{1}{\\eta _v} (p_v+\\delta _v) \\log (p_v+\\delta _v),$ and let $\\mathsf {D}_{\\phi _v}(q_v \\,\\Vert \\,p_v) = \\frac{1}{\\eta _v} \\left[(q_v+\\delta _v) \\log \\frac{q_v+\\delta _v}{p_v+\\delta _v} + (p_v-q_v)\\right]$ denote the corresponding Bregman divergence.",
"Then for $q_v,p_v \\geqslant 0$ , it holds that $\\mathsf {D}_{\\phi _v}(q_v \\,\\Vert \\,p_v) \\geqslant 0$ since $\\phi _v$ is convex on $R_+$ .",
"Employing the $\\tau $ -HST property of $T$ , this implies that $\\mathsf {D}^{(u)}(r \\,\\Vert \\,p) = \\frac{1}{\\kappa } \\sum _{v \\in \\chi (u)} w_v \\mathsf {D}_{\\phi _v}(r_v \\,\\Vert \\,p_v) \\leqslant \\frac{w_u}{\\kappa \\tau } \\sum _{v \\in \\chi (u)} \\mathsf {D}_{\\phi _v}(r_v \\,\\Vert \\,p_v).$ Define $F : Q^{(u)}_T \\times Q^{(u)}_T \\rightarrow R_+$ by $F(r,p) \\mathrel {\\mathop :}=\\sum _{v \\in \\chi (u)} \\mathsf {D}_{\\phi _v}(r_v \\,\\Vert \\,p_v)$ .",
"The map $r \\mapsto F(r,p)$ is convex in general (for any Bregman divergence).",
"The map $p \\mapsto F(r,p)$ is convex as well, as this holds for each map $p_v \\mapsto \\mathsf {D}_{\\phi _v}(q_v \\,\\Vert \\,p_v)$ since $-\\log (x)$ is convex on $R_{++}$ .",
"Since the maximum of a convex function on the a polytope is achieved at an extreme point, we have { F(r,p) :r,p Q(u)T } v,v' (u) v v' [ 1v ((1+v) 1+vv -1) + 1v' (v' v'1+v' +1)] 2.",
"The next lemma is crucial: It relates the service cost (with respect to the reduced cost $\\hat{c}-\\alpha $ ) of the hybrid configurations to the service cost of the actual configuration and the movement cost.",
"Lemma 2.11 For any $u \\notin \\mathcal {L}$ , it holds that $\\Psi _u(p) - \\Psi _u(q) \\leqslant \\frac{2}{\\kappa } \\frac{w_u}{\\tau } \\left(x_u-y_u\\right)_+ +\\sum _{v \\in \\chi (u)} (\\hat{c}_v-\\alpha _v) \\left[x_v - \\theta _v x_u\\right].$ Write u(p) - u(q) = xu D(u)((u) q(u)) -yu D(u)((u) p(u)) = (xu-yu) D(u)((u) p(u)) + xu [D(u)((u) q(u))-D(u)((u) p(u))].",
"Using lem:dmax, the first term is bounded by $\\frac{2}{\\kappa } \\frac{w_u}{\\tau } (x_u-y_u)_+ $ .",
"Let us now bound the second term.",
"Using $1+t \\leqslant e^t$ , we have xu [D(u)((u) q(u))-D(u)((u) p(u))] = xu v (u) wvv [(v+v) pv+vqv+v+qv-pv] (REF )= xu v (u) wvv [(v+v) v + (qv+v) (1-ev)] xu v (u) wvv v (v-qv) = v (u) wvv v [v xu - xv].",
"To finish the proof, observe that from (REF ), $\\sum _{v \\in \\chi (u)} \\frac{w_v}{\\eta _v} \\sigma _v \\left[\\theta _v x_u - x_v\\right] =\\kappa \\sum _{v \\in \\chi (u)} (\\beta _u - \\hat{c}_v + \\alpha _v) \\left[\\theta _v x_u - x_v\\right] =\\kappa \\sum _{v \\in \\chi (u)} (\\alpha _v - \\hat{c}_v) \\left[\\theta _v x_u - x_v\\right],$ where the last equality uses $\\sum _{v \\in \\chi (u)} x_v = x_u$ and $\\sum _{v \\in \\chi (u)} \\theta _v = 1$ (from (REF )).",
"Using the lemma gives u L xu v (u) v (cv-v) (REF ) [(q)-(p)] + 2 ((q)-(p))+1(w) + v r cv xv [(q)-(p)] + 2 ((q)-(p))+1(w) + DT c,xL.",
"Combining this inequality with (REF ) and (REF ) gives $\\kappa ^{-1} \\left\\Vert \\left(x-y\\right)_+\\right\\Vert _{\\ell _1(w)}\\leqslant \\frac{\\tau }{\\tau -1}\\left[\\left(2\\mathfrak {D}_T + \\log n\\right) \\langle c,x\\rangle _{\\mathcal {L}} + \\left(\\Psi (q)-\\Psi (p)\\right)+ \\frac{2}{\\kappa \\tau } \\left\\Vert \\left(x-y\\right)_+\\right\\Vert _{\\ell _1(w)}\\right],$ completing the verification of lem:movement."
],
[
"Derivation of the dynamics and derived costs",
"For the sake of motivating the dynamics (REF ), we review the continuous-time mirror descent framework of [10].",
"Suppose that $\\mathsf {K}\\subseteq R^N$ is a convex set.",
"We recall again the definition of the normal cone to $\\mathsf {K}$ at $x \\in \\mathsf {K}$ which is given by 3 NK(x) :=(K-x) = { p RN : p,y-x0 for all y K}.",
"Suppose additionally that $\\Phi : \\mathcal {D}\\rightarrow R$ is $\\mathcal {C}^2$ and strictly convex on an open neighborhood $\\mathcal {D}\\supseteq \\mathsf {K}$ so that the Hessian $\\nabla ^2 \\Phi (x)$ is well-defined and positive definite on $\\mathcal {D}$ .",
"Given a control function $F : [0,\\infty ) \\times \\mathsf {K}\\rightarrow R^N$ and an initial point $x_0 \\in \\mathsf {K}$ , we will be concerned with absolutely continuous solutions $x : [0,\\infty ) \\rightarrow \\mathsf {K}$ to the differential inclusion x(0) = x0, 2 (x(t)) x'(t) F(t,x(t)) - NK(x(t)) .",
"In other words, a trajectory that satisfies $x(0) = x_0$ and for almost every $t \\geqslant 0$ : $x^{\\prime }(t) = \\nabla ^2 \\Phi (x(t))^{-1} \\left(F(t,x(t)) - \\gamma (t)\\right),$ with $\\gamma (t) \\in N_{\\mathsf {K}}(x(t))$ .",
"Under suitably strong conditions on $\\Phi $ and $F$ , there is a unique absolutely continuous solution to (REF ) [10].",
"In our setup, these conditions are actually not satisfied unless we prevent the path $x$ from hitting the relative boundary of $\\mathsf {K}$ .",
"Nevertheless, the formal calculation is elucidating and motivates the algorithm of sec:main.",
"For simplicity, we assume $\\kappa :=1$ in this section."
],
[
"Hessian computation",
"Let us take $\\Phi $ as in (REF ) and calculate $\\nabla ^2 \\Phi (x)$ for $x \\in R_{++}^{V}$ .",
"Fix $u \\ne {r}$ .",
"Then we have u(x) = wuu ((xuxp(u)+u) +1) + v (u)wvv(v(xvxu+v) - xvxu).",
"Moreover, $\\partial _{uv} \\Phi (x) = 0$ unless $u=v$ , $u \\in \\chi (v)$ , or $v \\in \\chi (u)$ , and in this case, uu(x) = wuu(xu+u xp(u)) + v(u) (xvxu)2wvv(xv+v xu) u,p(u)(x) = p(u),u(x) = -xuxp(u)wuu(xu+u xp(u))."
],
[
"Explicit dynamics",
"We are now in a position to calculate the formal dynamics.",
"Let us define the control by $F(\\cdot ,t) \\mathrel {\\mathop :}=-c(t)$ .",
"We claim that for $u \\ne {r}$ , $\\partial _t \\left(\\frac{x_u(t)}{x_{\\mathsf {p}(u)}(t)}\\right) = \\frac{\\eta _u}{w_u} \\left(\\frac{x_u(t)}{x_{\\mathsf {p}(u)}(t)}+\\delta _u\\right)\\left(\\beta _{\\mathsf {p}(u)}(t)- \\sum _{\\ell \\in \\mathcal {L}_u} \\frac{x_{\\ell }(t)}{x_{u}(t)} c_{\\ell }\\right),$ where $\\beta _u(t)\\geqslant 0$ denotes the Lagrange multiplier corresponding to the constraint $x_u = \\sum _{v\\in \\chi (u)} x_v$ .",
"To verify (REF ), let us define, for $u \\ne {r}$ , $\\mathcal {E}(u) \\mathrel {\\mathop :}=\\frac{w_u}{\\eta _u} \\frac{x_{\\mathsf {p}(u)}(t)}{x_u(t)+\\delta _u x_{\\mathsf {p}(u)}(t)} \\partial _t \\left(\\frac{x_u(t)}{x_{\\mathsf {p}(u)}(t)}\\right).$ Then (REF ) is equivalent to the assertion that $\\mathcal {E}(u) = \\beta _{\\mathsf {p}(u)}(t) - \\sum _{\\ell \\in \\mathcal {L}_u} \\frac{x_{\\ell }(t)}{x_u(t)} c_{\\ell }(t).$ Recalling (REF ), the equality $\\left(\\nabla ^2 \\Phi (x(t)) x^{\\prime }(t)\\right)_u = \\left(F(t,x(t))-\\gamma (t)\\right)_u$ is equivalent to 3 E() = p()(t) - c(t), L, E(u) - v (u) xv(t)xu(t) E(v) = p(u)(t) - u(t) , u V (L{r}).",
"Clearly (REF ) already confirms (REF ) for $\\ell \\in \\mathcal {L}$ .",
"Let us conclude by verifying (REF ) for all $u \\notin {r}$ by (reverse) induction on the depth.",
"Employing (REF ) along with the validity of (REF ) for $\\lbrace \\mathcal {E}(v) : v \\in \\chi (u)\\rbrace $ yields E(u) = p(u)(t) - u(t) + v (u) xv(t)xu(t) (u(t) - Lu x(t)xu(t) c(t)) = p(u)(t) - Lu x(t)xu(t) c(t), where we used the fact that $x_u = \\sum _{v \\in \\chi (u)} x_v$ for $x \\in \\mathsf {K}_{T}$ ."
],
[
"Relationship between discrete and continuous dynamics",
"Recall the setup from sec:dyn.",
"We consider a system of variables $\\lbrace q_u(t) : u \\in V \\setminus \\lbrace {r}\\rbrace \\rbrace $ satisfying the differential equations $\\partial _t q_u(t) = \\frac{\\eta _u}{w_u} \\left(q_u(t)+\\delta _u\\right) \\left(\\beta _{\\mathsf {p}(u)}(t) - \\hat{c}_u(t) + \\alpha _u(t)\\right),$ where $\\alpha _u(t)$ is a Lagrangian multiplier for the constraint $q_u(t)\\geqslant 0$ , and $\\hat{c}_u(t)$ is the “derived” cost in the subtree rooted at $u$ : cu(t) :=Lu qu(t) c(t) qu(t) :=v u, {u} qv(t) , where $\\gamma _{u,\\ell }$ is the unique simple $u$ -$\\ell $ path in $T$ .",
"Now the values $q_{\\ell \\mid {r}}$ give a probability distribution on the leaves.",
"Let us argue that when the discretization parameter of the algorithm presented in sec:main goes to zero, one arrives at a solution to (REF ).",
"Recall that in sec:alg, we split each cost function $c\\in R_+^{\\mathcal {L}}$ into $M$ pieces $M^{-1} c$ and computed a sequence of configurations $q_0,\\dots ,q_M \\in Q_T$ .",
"Define the piecewise-linear function $q_{(M)} : [0,1] \\rightarrow Q_T$ by $q_{(M)}\\left(\\frac{j+\\delta }{M}\\right) \\mathrel {\\mathop :}=(1-\\delta ) q_{j} + \\delta q_{j+1}, \\qquad \\delta \\in [0,1], j \\in \\lbrace 0,\\ldots ,M-1\\rbrace .$ Recalling sec:iter, we have qj(u) :=argmin{ D(u)(p qj-1(u)) + p, M-1 cj(u) | p Q(u)T}, where $\\hat{c}_j^{(u)} = \\sum _{\\ell \\in \\mathcal {L}_v} (q_j)_{\\ell \\mid u} c_{\\ell }.$ Thus for $v \\in \\chi (u)$ and $j \\geqslant 1$ , $\\left(q_j^{(u)}\\right)_v = \\left[\\left(q_{j-1}^{(u)}\\right)_v + \\delta _v\\right] \\exp \\left(\\frac{\\eta _v}{w_v} \\left(\\beta _u - (M^{-1}(\\hat{c}^{(u)}_j)_v - \\alpha _v)\\right)\\right) - \\delta _v.$ One can now verify that there is a constant $L=L(c,T)$ such that $\\left|\\left(q_j\\right)_v - \\left(q_{j-1}\\right)_v\\right| \\leqslant \\frac{L}{M}, \\quad j \\in \\lbrace 1,\\ldots ,M\\rbrace , v \\in V \\setminus \\lbrace {r}\\rbrace .$ In particular, we see that $q^{\\prime }_{(M)} \\in L^{\\infty }([0,1],R^{V \\setminus \\lbrace {r}\\rbrace })$ for every $M \\geqslant 1$ and, moreover, $\\sup _{M \\geqslant 1} \\left\\Vert q^{\\prime }_{(M)}\\right\\Vert _{L^{\\infty }} < \\infty .$ Therefore by Arzelà-Ascoli, there is a subsequence $\\lbrace M_k\\rbrace $ such that $q_{(M_k)}$ converges uniformly to a function $q : [0,1] \\rightarrow Q_T$ .",
"Since the unit ball of $L^{\\infty }([0,1], R^{V \\setminus \\lbrace {r}\\rbrace })$ is weakly compact (by the sequential Banach-Alaoglu Theorem), we can pass to a further subsequence $\\lbrace M^{\\prime }_{k}\\rbrace $ along which $q^{\\prime }_{(M^{\\prime }_k)}$ converges weakly to some $h \\in L^{\\infty }([0,1], R^{V \\setminus \\lbrace {r}\\rbrace })$ .",
"Moreover, since $q_{(M)}(b)-q_{(M)}(a) = \\int _a^b q^{\\prime }_{(M)}(t)\\,dt$ for all $0 \\leqslant a < b \\leqslant 1$ , it follows that $q(b)-q(a) = \\int _a^b h(t)\\,dt$ as well, and therefore for almost all $t \\in [0,1]$ , we have $q^{\\prime }(t) = h(t)$ .",
"If we similarly linearly interpolate the cost function to $\\hat{c}_{(M)} : [0,1] \\rightarrow R_+^{V \\setminus \\lbrace {r}\\rbrace }$ , then $\\hat{c}_{(M_k)} \\rightarrow \\hat{c}$ along this sequence as well, and $\\hat{c}^{(u)}(t) = \\sum _{\\ell \\in \\mathcal {L}_v} q_{\\ell \\mid u}(t) c_{\\ell }.$ Now the KKT conditions for optimality in (REF ) give $\\nabla \\Phi ^{(u)}\\left(q_j^{(u)}\\right) - \\nabla \\Phi ^{(u)}\\left(q_{j-1}^{(u)}\\right) + M^{-1} \\hat{c}^{(u)}_j \\in - \\mathsf {N}_{Q_T^{(u)}}\\left(q_j^{(u)}\\right),$ or equivalently, $\\frac{\\nabla \\Phi ^{(u)}\\left(q_j^{(u)}\\right) - \\nabla \\Phi ^{(u)}\\left(q_{j-1}^{(u)}\\right)}{M^{-1}} \\in -\\hat{c}^{(u)}_j - \\mathsf {N}_{Q_T^{(u)}}\\left(q_j^{(u)}\\right).$ By standard results in differential inclusion theory (e.g., the Convergence Theorem [2]), we conclude that $q : [0,1] \\rightarrow Q_T$ solves the differential inclusion $\\nabla ^2 \\Phi ^{(u)}\\!\\left(q^{(u)}(t)\\right) \\partial _t q^{(u)}(t) \\in -\\hat{c}^{(u)}(t) - \\mathsf {N}_{Q_T^{(u)}}(q^{(u)}(t)).$ Calculating the Hessian $\\nabla ^2 \\Phi ^{(u)}$ reveals that $q(t)$ is a solution to (REF )."
],
[
"Acknowledgments",
"Part of this work was carried out while C. Coester was visiting University of Washington, hosted by J. R. Lee.",
"C. Coester was partially supported by EPSRC Award 1652110.",
"J. R. Lee was partially supported by NSF grants CCF-1616297 and CCF-1407779 and a Simons Investigator Award."
]
] | 1906.04270 |
[
[
"Extensions of immersions of surfaces into $\\mathbb{R}^{3}$"
],
[
"Abstract This paper is to study the $\\mathbb{R}^{3}$ case of \\hyperref[Zhao]{[9]}.",
"We determine all equivalence classes of immersed $3$-manifolds bounded by an arbitrary immersed surface in $\\mathbb{R}^{3}$."
],
[
"Introduction",
"In this paper, we assume all 3-manifolds are oriented, and the 3-manifolds will be connected if not otherwise mentioned.",
"We assume all immersions are transverse immersions, and all graphs have no isolated point.",
"We work in PL category: all 3-manifolds are assumed to have a PL structure, and all maps (between 3 manifolds) are assumed to be PL maps.",
"Fix a closed oriented surface $\\Sigma $ and an immersion $f: \\Sigma \\rightarrow \\mathbb {R}^{3}$ .",
"We say an immersion $F: M \\rightarrow \\mathbb {R}^{3}$ ($M$ a compact, connected 3-manifold with boundary $\\Sigma $ ) extends $f$ if $F \\mid _{\\partial M} = f$ (toward the side that inward normal vectors point to).",
"Definition 1.1 Let $\\Sigma $ be a closed oriented surface and $f: \\Sigma \\rightarrow \\mathbb {R}^{3}$ an immersion.",
"Assume $g_1: M_1 \\rightarrow \\mathbb {R}^{3},g_2: M_2 \\rightarrow \\mathbb {R}^{3}$ are 2 extensions of $f$ .",
"$g_1,g_2$ are equivalent if there exists a (PL) homeomorphism $h: M_1 \\rightarrow M_2$ such that $g_1 = g_2 \\circ h$ .",
"(see [Pappas][7, Section 2], while it states this definition in smooth category) Question 1 Which immersed closed oriented surfaces in $\\mathbb {R}^{3}$ bound immersed 3-manifolds, and in how many inequivalent ways?",
"The 2-dimensional problems were solved by S. Blank ([Blank][3], for immersed disks bounded by the immersed planar circle), K. Bailey ([Bailey][2], for immersed surfaces bounded by the immersed planar circle).",
"But their algebraic approach does not readily generalize (see [Pappas][7, Section 1]).",
"In [Zhao][9] we presented the technique in 2-dimensional case.",
"We answer Question REF in this paper: Given an immersed surface in $\\mathbb {R}^{3}$ , we determine all equivalence classes of immersed 3-manifolds bounded by it (Theorem REF ).",
"The following question provides the author the basic motivation to accomplish this paper.",
"It includes the request to determine the equivalence classes of immersed 3-balls bounded by the immersed 2-spheres.",
"Question 2 [Kirby][5, Problem 3.19] Which immersed 2-spheres in $\\mathbb {R}^{3}$ bound immersed 3-balls?",
"By applying the algorithm [Preaux][8] after determining all inequivalent immersed 3-manifolds bounded by an immersed 2-sphere, we can determine all inequivalent immersed 3-balls bounded by the immersed 2-sphere (Corollary REF )."
],
[
"Main results",
"Fix a closed oriented surface $\\Sigma $ and an immersion $f: \\Sigma \\rightarrow \\mathbb {R}^{3}$ .",
"$f$ can't extend if there exists $x \\in \\mathbb {R}^{3} \\setminus f(\\Sigma )$ such that $\\omega (f,x) < 0$ (where $\\omega (f,x)$ denotes the winding number of $f$ around $x$ , see Deginition REF ).",
"If $\\omega (f,x) \\geqslant 0$ for every $x \\in \\mathbb {R}^{3} \\setminus f(\\Sigma )$ , an inscribed set $\\zeta $ of $f$ (Definition REF ) is a finite set, and $I(\\zeta )$ (Definition REF ) is a subset of $\\zeta $ ($\\zeta ,I(\\zeta )$ exist, and they can be obtained in finite steps).",
"Theorem 1 For a closed oriented surface $\\Sigma $ , let $f: \\Sigma \\rightarrow \\mathbb {R}^{3}$ be an immersion such that $\\omega (f,x) \\geqslant 0, \\forall x \\in \\mathbb {R}^{3} \\setminus f(\\Sigma )$ .",
"Assume $\\zeta $ is an inscribed set of $f$ .",
"Then there exists a bijection between $I(\\zeta )$ and all equivalence classes of immersions of 3-manifolds to extend $f$ .",
"[Preaux][8] (or, see [AFW][1, Section 4, C.29, C. 30]) provides an algorithm to detect if a 3-manifold with boundary $S^{2}$ is a 3-ball.",
"We apply this to determine the immersed 3-balls in an immersed 2-sphere: Assume $\\Sigma = S^{2}$ .",
"Assume $E(f)$ is the set of equivalence classes of immersions of 3-manifolds to extend $f$ .",
"Then Theorem REF gives a bijection $q: I(\\zeta ) \\rightarrow E(f)$ .",
"For each $A \\in I(\\zeta )$ , choose $g_A: M_A \\rightarrow \\mathbb {R}^{3}$ an extension to represent the equivalence class $q(A) \\in E(f)$ .",
"Definition REF provides $M_A$ a trangulation (determined by $A$ ).",
"By applying [Preaux][8], we can detect if $M_A$ is a 3-ball.",
"Hence we can detect $I_0(\\zeta ) = \\lbrace A \\in I(\\zeta ) \\mid M_A \\cong B^{3}\\rbrace $ .",
"Corollary 1.2 Let $f: S^{2} \\rightarrow \\mathbb {R}^{3}$ be an immersion such that $\\omega (f,x) \\geqslant 0, \\forall x \\in \\mathbb {R}^{3} \\setminus f(\\Sigma )$ .",
"Assume $\\zeta $ is an inscribed set of $f$ .",
"Then there is a bijection between $I_0(\\zeta )$ and all equivalence classes of immersions of 3-balls to extend $f$ ."
],
[
"Organization",
"We will give some basic definitions in Subsection REF , and we will introduce the branched immersion, good 2-complexes, cancellation operation in Subsection REF , Subsection REF , Subsection REF .",
"In Section , we will define the $(M,G)$ -simple 2-complex in a compact 3-manifold $M$ (with nonempty boundary) with a (trivalent) embedded graph $G \\subseteq \\partial M$ , and we will give the way to construct it.",
"In Section , we will define the inscribed set.",
"In Section , we will prove Theorem REF ."
],
[
"Preliminaries",
"In this section, we introduce some basic ingredients."
],
[
"The immersed surfaces in $\\mathbb {R}^{3}$",
"Definition 2.1 (Winding number) Let $\\Sigma $ be a closed oriented surface and $f: \\Sigma \\rightarrow \\mathbb {R}^{3}$ an immersion.",
"Chosen $x \\in \\mathbb {R}^{3} \\setminus f(\\Sigma )$ , assume $u: \\Sigma \\rightarrow S^{2}$ is the map such that $u(t) = \\dfrac{f(t) - x}{|f(t) - x|}$ ($\\forall t \\in \\Sigma $ ).",
"Let $\\omega (f,x) = deg_{u}(x)$ (see [GP][4, Page 144]).",
"Remark 2.2 If $F: M \\rightarrow \\mathbb {R}^{3}$ is an immersion to extend $f$ , then $\\omega (f,x)$ is the number of preimages under $F$ at each $x \\in \\mathbb {R}^{3} \\setminus f(\\Sigma )$ .",
"In the rest of this paper, if $f: \\Sigma \\rightarrow \\mathbb {R}^{3}$ is an immersion, we will always assume that $\\omega (f,x) \\geqslant 0, \\forall x \\in \\mathbb {R}^{3} \\setminus f(\\Sigma )$ (if not, then there is no immersed 3-manifold to extend $f$ ).",
"Definition 2.3 Let $\\Sigma $ be a closed oriented surface and $f: \\Sigma \\rightarrow \\mathbb {R}^{3}$ an immersion.",
"For each $1 \\leqslant k \\leqslant \\max _{x \\in \\mathbb {R}^{3} \\setminus f(\\Sigma )} \\omega (f,x)$ , let $D_k(f) = \\overline{\\lbrace x \\in \\mathbb {R}^{3} \\setminus f(\\Sigma ) \\mid \\omega (f,x) \\geqslant k\\rbrace }$ .",
"And we let $G_k(f) = \\partial D_k(f) \\cap \\partial D_{k-1}(f)$ ($2 \\leqslant k \\leqslant \\max _{x \\in \\mathbb {R}^{3} \\setminus f(\\Sigma )} \\omega (f,x)$ ) and $G_1(f) = \\emptyset $ .",
"Figure: ∂D k-1 (f),∂D k (f),∂D k+1 (f)\\partial D_{k-1}(f),\\partial D_k(f),\\partial D_{k+1}(f) intersect at one point (a triple point).Both Definition REF and Definition REF can be generalized to the case of an immersion of a disconnected surface.",
"We will apply this in Subsection REF .",
"In [LV][6], if $f: \\Sigma \\rightarrow \\mathbb {R}^{3}$ is a (transverse) immersion of a closed oriented surface $\\Sigma $ , the points in $f(\\Sigma )$ with $1,2,3$ preimages are called simple points, double points, triple points.",
"The non-simple points, triple points of $f(\\Sigma )$ are denoted by $S(f(\\Sigma ))$ , $T(f(\\Sigma ))$ .",
"Obviously, $G_k(f) \\subseteq S(f(\\Sigma )) \\cap \\partial D_{k}(f) = G_k(f) \\cup G_{k+1}(f)$ .",
"Actually, $G_k(f)$ is an embedded graph such that all vertices have degree 2 or 3 (in this paper, we assume that all embedded graphs have no isolated point), and $\\lbrace v \\in V(G_k(f)) \\mid deg_{G_k(f)}(v) = 3\\rbrace = G_k(f) \\cap T(f(\\Sigma ))$ .",
"We will not emphasize this in the rest of this paper.",
"Figure REF shows how $\\partial D_{k-1}(f),\\partial D_k(f),\\partial D_{k+1}(f)$ intersect at a tripe point.",
"To describe the relation between $G_k(f)$ and $G_{k+1}(f)$ in $\\partial D_k(f)$ (see the third picture in Figure REF ), we give the following statement: Definition 2.4 Let $\\Sigma $ be a closed oriented surface and $G,G^{^{\\prime }} \\subseteq \\Sigma $ an embedded graphs such that all vertices have degree 2 or 3.",
"$G^{^{\\prime }}$ is a thin trivalent graph of $G$ in $\\Sigma $ if: $\\bullet $ For each $x \\in G^{^{\\prime }} \\cap G$ , $x \\in \\lbrace v \\mid v \\in V(G), deg_G(v) = 3\\rbrace $ and $x \\in \\lbrace v \\mid v \\in V(G^{^{\\prime }}), deg_{G^{^{\\prime }}}(v) = 3\\rbrace $ .",
"Assume $a,b,c,d,e,f$ are the 6 edges of $G$ and $G^{^{\\prime }}$ at $x$ clockwise, and $a \\in E(G)$ .",
"Then $c,e \\in E(G)$ , $b,d,f \\in E(G^{^{\\prime }})$ ."
],
[
"Branched immersion",
"A (compact) topological space is a polyhedron if it is the underlying space of a simplicial complex.",
"In this paper, we say a polyhedron $K$ is a branched 3-manifold if there exists $M$ a compact oriented 3-manifold and $S_1,\\ldots ,S_n$ some components of $\\partial M$ , $S_1, \\ldots , S_n \\ncong S^{2}$ (and we allow $\\lbrace S_1,\\ldots ,S_n\\rbrace = \\emptyset $ ), $K = M \\cup _{i_1} C(S_1) \\cup _{i_2} \\ldots \\cup _{i_n} C(S_n)$ (where $i_1,\\ldots ,i_n$ are the identity maps of $S_1,\\ldots ,S_n$ , and $C(S) = S \\times I / S \\times \\lbrace 1\\rbrace $ for an arbitrary topological space $S$ ).",
"Moreover, we denote $\\partial M \\setminus (S_1 \\cup \\ldots \\cup S_n)$ by $\\partial K$ and say it is boundary of $K$ .",
"And we denote $\\lbrace the$ $vertices$ $of$ $the$ $cones$ $C(S_1),\\ldots ,C(S_n)\\rbrace $ by $B(K)$ (i.e.",
"$B(K) = \\lbrace the$ $points$ $in$ $K$ $that$ $have$ $no$ $open$ $neighborhood$ $homeomorphic$ $to$ $\\mathbb {R}^{3}$ $or$ $\\mathbb {R}^{3}_{+}\\rbrace $ ).",
"The following statement generalize the branched covers to the map of a branched 3-manifold $K$ into $\\mathbb {R}^{3}$ (we request $B(K)$ to lie in the singular set, then $K \\setminus B(K)$ is still a noncompact 3-manifold).",
"Definition 2.5 Let $K$ be a branched 3-manifold and $g: K \\rightarrow \\mathbb {R}^{3}$ a PL continuous map.",
"$g$ is called a branched immersion if there exists $F \\subseteq \\mathbb {R}^{3}$ an embedded graph such that $g^{-1}(F)$ is an embedded graph, $B(K) \\subseteq g^{-1}(F)$ , and $g \\mid _{K \\setminus g^{-1}(F)}$ is an immersion.",
"The singular set of $g$ is the set consisting of all $x \\in K$ such that $g$ is not a locally homeomorphism at $x$ , and the branch set of $g$ is the image of singular set under $g$ .",
"Remark 2.6 In this paper, if $g: K \\rightarrow \\mathbb {R}^{3}$ is a branched immersion of a branched 3-manifold $K$ and $S$ , $B$ are the singular set, branch set of $g$ , we will always assume that $g$ maps $S$ homeomorphically to $B$ .",
"For each branch point $y \\in B$ , assume $\\lbrace x\\rbrace = g^{-1}(y) \\cap S$ , we say $y$ has index $k$ if $g$ is $(k+1)$ -to-one near $x$ .",
"Remark 2.7 We explain the difference between the branched covers and our definition (branched immersion): we do not request it to be proper; we allow $x \\in K$ a singular point whose link with respect to $K$ is a not a 2-sphere, then $g \\mid _{lk(x,K)}$ is a branched cover of a surface to a 2-sphere (the number of such points is finite in total).",
"Actually, different from constructing 3-manifolds from branched covers that branched over links, the maps that branched over embedded graphs may construct branched 3-manifolds.",
"That's why we define the branched immersions in branched 3-manifolds.",
"We will introduce the cancellation operation in Subsection REF , which is defined in the branched immersions of branched 3-manifolds.",
"For a branched immersion $g: K \\rightarrow \\mathbb {R}^{3}$ (where $K$ is a branched 3-manifold, and $K$ is not a 3-manifold), $g$ can be transformed to a branched immersion of a 3-manifold into $\\mathbb {R}^{3}$ by deleting an open neighborhood at each $x \\in B(K)$ and filling a handlebody.",
"But this branched immersion does not send the singular set homeomorphically to the branch set (also, this branched immersion is not an open map).",
"So we do not do such transformation.",
"Example 2.8 Assume $C(T^{2}) = T^{2} \\times I / T^{2} \\times \\lbrace 1\\rbrace $ is a cone of a torus, and $B^{3} = S^{2} \\times I / S^{2} \\times \\lbrace 1\\rbrace $ is a 3-ball in $\\mathbb {R}^{3}$ .",
"Let $p: T^{2} \\rightarrow S^{2}$ be an arbitrary branched cover.",
"Let $g:C(T^{2}) \\rightarrow B^{3}$ ($B^{3} \\subseteq \\mathbb {R}^{3}$ ) be the map such that $g(x,t) = (p(x),t)$ ($\\forall x \\in T^{2}, t \\in [0,1)$ ), $g(T^{2},1) = (S^{2},1)$ .",
"Then $g$ is a branched immersion."
],
[
"Good 2-complexes",
"Definition 2.9 Let $M$ be a compact 3-manifold with nonempty boundary and $G \\subseteq \\partial M$ an embedded graph such that all vertices have degree 2 or 3.",
"Let $X \\subseteq M$ be an embedded 2-complex.",
"We say $X$ is a good 2-complex in $M$ with respect to $G$ if: $\\bullet $ Let $\\dot{\\varphi }^{2}_{X}: \\coprod _\\alpha \\partial D_{\\alpha }^{2} \\rightarrow X^{1}$ be the attaching map of all 2-cells of $X$ .",
"Then $\\dot{\\varphi }^{2}_{X}$ is surjective.",
"(i.e.",
"all points in $X$ have local dimension 2) $\\bullet $ For each 2-cell $e_{\\alpha }$ of $X$ , the characteristic map $\\varphi ^{2}_{\\alpha }: D_{\\alpha }^{2} \\rightarrow X$ is an embedding.",
"$\\bullet $ $X \\cap \\partial M = G$ , and $G \\setminus \\lbrace v \\in V(G) \\mid deg_G(v) = 3\\rbrace $ is the set consisting of all $t \\in X$ such that $\\exists N(t)$ an open neighborhood of $t$ in $M$ , $(N(t) \\cap X,t) \\cong (\\mathbb {R}^{2}_{+},0)$ .",
"$\\bullet $ For each $t \\in \\lbrace v \\in V(G) \\mid deg_G(v) = 3\\rbrace $ , $t$ has an open neighborhood $N(t)$ in $M$ such that $(N(t) \\cap X,t)$ is homeomorphic to $(\\lbrace x = 0, y \\geqslant 0, z \\geqslant 0\\rbrace \\cup \\lbrace y = 0, z \\geqslant 0\\rbrace ,0)$ (see Figure REF (a), where $\\lbrace x = 0, y \\geqslant 0, z \\geqslant 0\\rbrace \\cup \\lbrace y = 0, z \\geqslant 0\\rbrace $ denotes a subset of $\\mathbb {R}^{3}$ , $(x,y,z)$ is the coordinates of $\\mathbb {R}^{3}$ ).",
"Since the set of non-simple points in an immersed surface $f(\\Sigma )$ ($f: \\Sigma \\rightarrow \\mathbb {R}^{3}$ is an immersion of a surface $\\Sigma $ ) is denoted by $S(f(\\Sigma ))$ , we generalize this notation to an arbitrary 2-complex: Definition 2.10 For an arbitrary 2-complex $X$ , we denote by $S(X)$ the set consisting of all points in $X$ that have no open neighborhood in $X$ homeomorphic to $\\mathbb {R}^{2}$ or $\\mathbb {R}^{2}_{+}$ ."
],
[
"Cancellation operation",
"[Zhao][9] states the cancellation operation for a polymersion of a surface (with nonempty boundary) into a surface.",
"In this subsection, we define the cancellation operation for a branched immersion of a branched 3-manifold (with nonempty boundary) into $\\mathbb {R}^{3}$ .",
"Recall that Definition REF and Definition REF can be generalized to the case of an immersion of a disconnected surface.",
"Assume $K$ is a branched 3-manifold with nonempty boundary ($K$ may be disconnected), and $g: K \\rightarrow \\mathbb {R}^{3}$ is a branched immersion.",
"Assume $n = \\max _{x \\in \\mathbb {R}^{3} \\setminus g(\\partial K)} \\omega (g \\mid _{\\partial K},x)$ .",
"We denote $D_n(g \\mid _{\\partial K}), G_n(g \\mid _{\\partial K})$ by $R(g), G(g)$ .",
"Definition 2.11 (Cancellable domains) Let $K$ be a branched 3-manifold with nonempty boundary ($K$ may be disconnected) and $g: K \\rightarrow \\mathbb {R}^{3}$ a branched immersion.",
"(i) Assume $A_1, A_2, \\ldots , A_n \\subseteq K$ are closed domains (in this paper, the “domains” in the space are compact connected co-dimension 0 submanifolds).",
"$A_1, A_2, \\ldots , A_n$ are called cancellable if: $\\bullet $ $Int(A_1),$ $Int(A_2), \\ldots , Int(A_n)$ are homeomorphically embedded into $R(g)$ by $g$ .",
"$\\bullet $ There exists $X$ a good 2-complex in $R(g)$ with respect to $G(g)$ such that $\\lbrace g(Int(A_1)),$ $g(Int(A_2)),$ $\\ldots , g(Int(A_n))\\rbrace = \\lbrace the$ $components$ $of$ $Int(R(g)) \\setminus X\\rbrace $ (i.e.",
"$g$ maps $A_1, A_2, \\ldots , A_n$ homemomorphically to the closed components obtained by cutting off $X$ from $R(g)$ ).",
"We call $X$ the 2-complex associated to $A_1, A_2, \\ldots , A_n$ .",
"$\\bullet $ $(g \\mid _{A_i})^{-1}(g(A_i)\\cap \\partial R(g)) \\subseteq \\partial K$ if $g(A_i) \\cap \\partial R(g) \\ne \\emptyset $ ($\\forall i \\in \\lbrace 1,2,\\ldots ,n\\rbrace $ ).",
"(ii) We denote $g(\\overline{\\partial (A_1 \\cup A_2 \\cup \\ldots \\cup A_n) \\setminus \\partial K})$ by $X(A_1,A_2,\\ldots ,A_n)$ .",
"Obviously, $X(A_1,A_2,\\ldots ,A_n)$ is a subcomplex of $X$ , and it is a good 2-complex in $R(g)$ with respect to $G(g)$ .",
"Remark 2.12 The cancellable domains $A_1,\\ldots ,A_n$ can be determined uniquely in following 2 cases: (a) If each component of $R(g) \\setminus X$ contains a component of $\\partial R(g) \\setminus G(g)$ , then $A_1,\\ldots ,A_n$ are determined uniquely by $X$ (b) Fix the associated 2-complex $X$ .",
"Given a set $P \\subseteq K$ such that $A_i \\cap (\\partial K \\cup P) \\ne \\emptyset $ ($\\forall i \\in \\lbrace 1,2,\\ldots ,n\\rbrace $ ), then $A_1,\\ldots ,A_n$ are determined uniquely.",
"Definition 2.13 (Cancellation operation) Let $K$ be a branched 3-manifold with nonempty boundary ($K$ may be disconnected) and $g: K \\rightarrow \\mathbb {R}^{3}$ a branched immersion.",
"Assume that the closed domains $A_1, \\ldots , A_n \\subseteq K$ are cancellable, and $X$ is the 2-complex associated to $A_1,\\ldots ,A_n$ .",
"(i) A cancellation of $\\lbrace A_1,\\ldots ,A_n\\rbrace $ (canceling $\\lbrace A_1,\\ldots ,A_n\\rbrace $) $(g,K) \\stackrel{\\lbrace A_1,\\ldots ,A_n\\rbrace }{\\leadsto } (g_1,K_1)$ is the following procedure: $\\bullet $ Let $K_0$ be the space obtained by cutting out $A_1,\\ldots ,A_n$ from $K$ (i.e.",
"assume $\\lbrace P_1,\\ldots ,P_k\\rbrace = \\lbrace the$ $components$ $of$ $K \\setminus (A_1 \\cup A_2 \\cup \\ldots \\cup A_n)\\rbrace $ , and let $K_0 = \\coprod _{i=1}^{k} \\overline{P_i}$ ).",
"Let $g_0: K_0 \\rightarrow \\mathbb {R}^{3}$ be the map induced by $g$ .",
"For each $\\alpha $ a 2-cell of $X(A_1,A_2,\\ldots ,A_n)$ , there exists exactly two components of $g_{0}^{-1}(\\overline{\\alpha })$ lying in the boundary of $K_0$ .",
"We denote them by $D_{\\alpha }^{+},D_{\\alpha }^{-}$ .",
"Let $h$ be the equivalence relation such that $x \\stackrel{h}{\\sim } y$ if there exists $\\alpha $ a 2-cell of $X(A_1,A_2,\\ldots ,A_n)$ , $x \\in D_{\\alpha }^{+}$ , $y \\in D_{\\alpha }^{-}$ , and $g_0(x) = g_0(y)$ .",
"Let $K_1$ be the identification space $K_0 / \\sim _h$ .",
"Assume $h_*: K_0 \\rightarrow K_1$ is the identification map induced by $h$ .",
"Let $g_1: K_1 \\rightarrow \\mathbb {R}^{3}$ be the map given by following commutative diagram.",
"${& K_0 [d]^{h_*} [r]_{g_0}& \\mathbb {R}^{3} [d]_{id} \\\\& K_1 [r]^{g_1} & \\mathbb {R}^{3} }$ Hence the cancellation operation $(g,K) \\stackrel{\\lbrace A_1,\\ldots ,A_n\\rbrace }{\\leadsto } (g_1,K_1)$ has been defined ($K_1$ is a branched 3-manifold and $g_1: K_1 \\rightarrow \\mathbb {R}^{3}$ is a branched immersion).",
"(ii) For each $x \\in X(A_1,A_2,\\ldots ,A_n)$ , we say the cancellation $(g,K) \\stackrel{\\lbrace A_1,\\ldots ,A_n\\rbrace }{\\leadsto } (g_1,K_1)$ is regular at $x$ if $\\#(h_*(\\partial K_0 \\cap g^{-1}(x))) = 1$ (where $h_*: K_0 \\rightarrow K_1$ is the identification map of cancellation).",
"The cancellation $(g,K) \\stackrel{\\lbrace A_1,\\ldots ,A_n\\rbrace }{\\leadsto } (g_1,K_1)$ is called regular if it is regular at every $x \\in X(A_1,A_2,\\ldots ,A_n)$ .",
"(iii) Assume that the cancellation $(g,K) \\stackrel{\\lbrace A_1,\\ldots ,A_n\\rbrace }{\\leadsto } (g_1,K_1)$ is regular.",
"Let $T: X(A_1,$ $\\ldots ,A_n) \\rightarrow K_1$ be the map sending each $x \\in X(A_1,A_2,\\ldots ,A_n)$ to $h_*(\\partial K_0 \\cap g^{-1}(x))$ (then $X(A_1,A_2,\\ldots ,A_n)$ is homeomorphically embedded into $K_1$ by $T$ ).",
"We call $T$ the associated map of the cancellation of $\\lbrace A_1,\\ldots ,A_n\\rbrace $ .",
"Remark 2.14 We give some remarks on cancellations.",
"For $(g,K) \\stackrel{\\lbrace A_1,\\ldots ,A_n\\rbrace }{\\leadsto } (g_1,K_1)$ the cancellation of $A_1,\\ldots ,A_n$ , the following hold: (a) $g_1(\\partial K_1) = \\overline{g(\\partial K) \\setminus \\partial R(g)} = \\bigcup _{i=1}^{n-1}D_i(g \\mid _{\\partial K})$ (where $n = \\max _{x \\in \\mathbb {R}^{3} \\setminus g(\\partial K)} \\omega (g \\mid _{\\partial K},x)$ ).",
"(b) $G(g) = X(A_1,\\ldots ,A_n) \\cap \\partial R(g_1)$ .",
"$X(A_1,\\ldots ,A_n)$ is a good 2-complex in $R(g_1)$ with respect to $G(g)$ .",
"(c) The cancellation $(g,K) \\stackrel{\\lbrace A_1,\\ldots ,A_n\\rbrace }{\\leadsto } (g_1,K_1)$ is regular at every $x \\in X(A_1,A_2,\\ldots ,A_n) \\setminus S(X(A_1,A_2,\\ldots ,A_n))$ .",
"So $(g,K) \\stackrel{\\lbrace A_1,\\ldots ,A_n\\rbrace }{\\leadsto } (g_1,K_1)$ is regular if and only if it is regular at every $x \\in S(X(A_1,A_2,\\ldots ,A_n))$ .",
"(d) If the cancellation $(g,K) \\stackrel{\\lbrace A_1,\\ldots ,A_n\\rbrace }{\\leadsto } (g_1,K_1)$ is regular, then $T(G(g)) \\subseteq \\partial K_1$ (where $T$ is the associated map of the cancellation)."
],
[
"Embedded 2-complexes in 3-manifolds",
"In this section, we introduce some embedded 2-complexes in 3-manifolds, and give the steps to construct them.",
"We will let them be the associated 2-complexes of cancellable domains to yield cancellable domains in Section ."
],
[
"$(M,G)$ -simple 2-complex",
"Definition 3.1 Let $M$ be a compact 3-manifold with nonempty boundary and $G \\subseteq \\partial M$ an embedded graph such that all vertices have degree 2 or 3.",
"Let $X \\subseteq M$ be 2-complex in $M$ .",
"(i) We say $X$ is a $(M,G)$ -simple 2-complex if: $\\bullet $ $X \\cap \\partial M = G$ .",
"$\\bullet $ For each $t \\in G \\setminus \\lbrace v \\in V(G) \\mid deg_G(v) = 3\\rbrace $ , there exists an open neighborhood $N(t)$ of $t$ in $M$ such that $(N(t) \\cap X,t)$ is homeomorphic to $(\\mathbb {R}^{2}_{+},0)$ .",
"$\\bullet $ For each $t \\in \\lbrace v \\in V(G) \\mid deg_G(v) = 3\\rbrace $ , there exists an open neighborhood $N(t)$ of $t$ in $M$ such that $(N(t) \\cap X,t)$ is homeomorphic to $(\\lbrace x = 0, y \\geqslant 0, z \\geqslant 0\\rbrace \\cup \\lbrace y = 0, z \\geqslant 0\\rbrace ,0)$ (see Figure REF (a)).",
"(where $\\lbrace x = 0, y \\geqslant 0, z \\geqslant 0\\rbrace \\cup \\lbrace y = 0, z \\geqslant 0\\rbrace $ denotes a subset of $\\mathbb {R}^{3}$ , $(x,y,z)$ is the coordinates of $\\mathbb {R}^{3}$ , and the following is same) $\\bullet $ For each $t \\in X \\setminus G$ , there exists an open neighborhood $N(t)$ of $t$ in $M$ such that $(N(t) \\cap X,t)$ is homeomorphic to one of (a) $\\sim $ (c): (a) $(\\mathbb {R}^{2},0)$ .",
"(b) $(\\lbrace z = 0\\rbrace \\cup \\lbrace x = 0, z \\geqslant 0\\rbrace ,0)$ (see Figure REF (b)).",
"(c) $(\\lbrace z = 0\\rbrace \\cup \\lbrace x = 0, z \\geqslant 0\\rbrace \\cup \\lbrace y = 0, x \\geqslant 0, z \\geqslant 0\\rbrace ,0)$ (see Figure REF (c)).",
"$\\bullet $ $\\#(\\lbrace the$ $components$ $of$ $\\partial M \\setminus G\\rbrace ) = \\#(\\lbrace the$ $components$ $of$ $M \\setminus X\\rbrace )$ , and each component of $M \\setminus X$ contains exactly one component of $\\partial M \\setminus G$ .",
"$\\bullet $ Assume $\\lbrace A_1,A_2,\\ldots ,A_n\\rbrace = \\lbrace the$ $components$ $of$ $\\partial M \\setminus G\\rbrace $ , $\\lbrace B_1,B_2,\\ldots ,B_n\\rbrace = \\lbrace the$ $components$ $of$ $M \\setminus X\\rbrace $ , and $A_k \\subseteq B_k$ ($\\forall k \\in \\lbrace 1,2,\\ldots ,n\\rbrace $ ).",
"Choose $x_k \\in A_k$ , assume $i_*: \\pi _1(A_k,x_k) \\rightarrow \\pi _1(M,x_k),j_*: \\pi _1(B_k,x_k) \\rightarrow \\pi _1(M,x_k)$ are the maps induced by the inclusion maps of $A_k,B_k$ into $M$ .",
"Then $i_*(\\pi _1(A_k,x_k)) = j_*(\\pi _1(B_k,x_k))$ ($\\forall k \\in \\lbrace 1,2,\\ldots ,n\\rbrace $ ).",
"(ii) If $X$ is a $(M,G)$ -simple 2-complex, we say $Y$ is a good subcomplex of $X$ if: $Y$ is a subcomplex of $X$ such that $Y$ is a good complex in $M$ with respect to $G$ .",
"We denote by $sub(X)$ the set consisting of all good subcomplex of $X$ .",
"Obviously, a $(M,G)$ -simple 2-complex is a good 2-complex in $M$ with respect to $G$ .",
"Moreover, given a covering space $p: (\\tilde{M},\\tilde{x}_k) \\rightarrow (M,x_k)$ ($x_k \\in A_k$ ), then the inclusion map $i_B: (B_k,x_k) \\rightarrow (M,x_k)$ has a lift $\\tilde{i}_B: (B_k,x_k) \\rightarrow (\\tilde{M},\\tilde{x}_k)$ if and only if the inclusion map $i_A: (A_k,x_k) \\rightarrow (M,x_k)$ has a lift $\\tilde{i}_A: (A_k,x_k) \\rightarrow (\\tilde{M},\\tilde{x}_k)$ .",
"Definition 3.2 Let $M$ be a compact 3-manifold with nonempty boundary and $G \\subseteq \\partial M$ an embedded graph such that all vertices have degree 2 or 3.",
"$(M,G)$ is called appropriate if: for each $e \\in E(G)$ such that both 2 sides of $e$ lie in the same component $A$ of $\\partial M \\setminus G$ (i.e.",
"$Int(e) \\subseteq Int(\\overline{A})$ ), all immersed loops in $\\overline{A}$ that intersect with $e$ one time transversely are not null-homotopic in $M$ .",
"Figure: M=S 1 ×D 2 M = S^{1} \\times D^{2}, AA is a component of ∂M∖G\\partial M \\setminus G. Then (M,G)(M,G) is not appropriate.Figure REF gives an example of $(M,G)$ to be not appropriate.",
"And we can state Definition REF in a different way.",
"$(M,G)$ is appropriate if: for each $e \\in E(G)$ such that both 2 sides of $e$ lie in the same component $A$ of $\\partial M \\setminus G$ .",
"Choose $x \\in A$ , assume $i_*: \\pi _1(A,x) \\rightarrow \\pi _1(M,x),i_{*}^{^{\\prime }}: \\pi _1(A \\cup Int(e),x) \\rightarrow \\pi _1(M,x)$ are the maps induced by the inclusion maps of $A,A \\cup Int(e)$ into $M$ , then $i_*(\\pi _1(A,x)) \\ne i^{^{\\prime }}_{*}(\\pi _1(A \\cup Int(e),x))$ .",
"Lemma 3.3 Let $M$ be a compact 3-manifold with nonempty boundary and $G \\subseteq \\partial M$ an embedded graph such that all vertices have degree 2 or 3.",
"$(M,G)$ is appropriate if and only if: for each component $A$ of $\\partial M \\setminus G$ and $x \\in A$ , there exists a covering space $p: (\\tilde{M},\\tilde{x}) \\rightarrow (M,x)$ and $A_0 \\subseteq \\tilde{M}$ a closed domain such that $\\tilde{x} \\in A_0$ , and $p \\mid _{Int(A_0)}$ is a homeomorphism between $Int(A_0)$ and $A$ .",
"(i) We assume that for each $A$ a component of $\\partial M \\setminus G$ and $x \\in A$ , there exists $p: (\\tilde{M},\\tilde{x}) \\rightarrow (M,x)$ a covering space and $A_0 \\subseteq \\tilde{M}$ a closed domain such that $\\tilde{x} \\in A_0$ , and $p \\mid _{Int(A_0)}$ is a homeomorphism between $Int(A_0)$ and $A$ .",
"We will prove that $(M,G)$ is appropriate.",
"The inclusion map $i: (A,x) \\rightarrow (M,x)$ has a lift $\\tilde{i}: (A,x) \\rightarrow (\\tilde{M},\\tilde{x})$ such that $\\tilde{i}(A) = Int(A_0)$ .",
"But $p^{-1}(e) \\cap A_0$ has 2 different components for every $e \\in E(G)$ such that both 2 sides of $e$ lie in $A$ .",
"Hence $i_*(\\pi _1(A,x)) \\subseteq p_*(\\pi _1(\\tilde{M},\\tilde{x}))$ , and $i^{^{\\prime }}_{*}(\\pi _1(A \\cup Int(e),x)) \\nsubseteq p_*(\\pi _1(\\tilde{M},\\tilde{x}))$ (where $p_*: \\pi _1(\\tilde{M},\\tilde{x}) \\rightarrow \\pi _1(M,x),i_*: \\pi _1(A,x) \\rightarrow \\pi _1(M,x),i_{*}^{^{\\prime }}: \\pi _1(A \\cup Int(e),x) \\rightarrow \\pi _1(M,x)$ are the maps induced by $p$ , $i$ , and the inclusion map of $A \\cup Int(e)$ into $M$ ).",
"So $i_*(\\pi _1(A,x)) \\ne i^{^{\\prime }}_{*}(\\pi _1(A \\cup Int(e),x))$ .",
"Hence $(M,G)$ is appropriate.",
"(ii) Assume $(M,G)$ is appropriate.",
"For each component $A$ of $\\partial M \\setminus G$ and $x \\in A$ , let $p: (\\tilde{M},\\tilde{x}) \\rightarrow (M,x)$ be the covering space such that $p_*(\\pi _1(\\tilde{M},\\tilde{x})) = i_*(\\pi _1(A,x))$ ($p_*,i_*$ are same as (i)).",
"Let $\\tilde{i}: (A,x) \\rightarrow (\\tilde{M},\\tilde{x})$ be a lift of the inclusion map $i: (A,x) \\rightarrow (M,x)$ .",
"Let $A_0$ be the closure of $\\tilde{i}(A)$ .",
"For each edge $e \\in E(G)$ such that both 2 sides of $e$ lie in $A$ , $i^{^{\\prime }}_{*}(\\pi _1(A \\cup Int(e),x)) \\nsubseteq p_*(\\pi _1(\\tilde{M},\\tilde{x}))$ (where $i_{*}^{^{\\prime }}: \\pi _1(A \\cup Int(e),x) \\rightarrow \\pi _1(M,x)$ is the map induced by the inclusion map of $A \\cup Int(e)$ into $M$ ).",
"Then $p^{-1}(e) \\cap A_0$ has 2 different components.",
"Hence $Int(A_0) = \\tilde{i}(A)$ .",
"Lemma 3.4 (i) If $M$ is a compact 3-manifold with nonempty boundary and $g: M \\rightarrow \\mathbb {R}^{3}$ is an immersion, then $(R(g),G(g))$ is appropriate.",
"(ii) If $X$ is a $(R(g),G(g))$ -simple 2-complex, then there exists $A_1,\\ldots ,A_n$ cancellable domains such that $X$ is their associated 2-complex.",
"And the cancellation of $\\lbrace A_1,\\ldots ,A_n\\rbrace $ is regular.",
"(i) For each component $A$ of $\\partial R(g) \\setminus G(g)$ , let $S$ be $\\overline{(g \\mid _{\\partial M})^{-1}(A)}$ .",
"Then $g \\mid _{Int(S)}$ is a homeomorphism between $Int(S)$ and $A$ .",
"So $(R(g),G(g))$ is appropriate (by Lemma REF ).",
"(ii) For each component $A$ of $\\partial R(g) \\setminus G(g)$ , assume $B$ is the component of $R(g) \\setminus X$ containing $A$ , and $x \\in A$ .",
"Then $i_{A*}(\\pi _1(A,x)) = i_{B*}(\\pi _1(B,x))$ , where $i_{A*}: \\pi _1(A,x) \\rightarrow \\pi _1(R(g),x), i_{B*}: \\pi _1(B,x) \\rightarrow \\pi _1(R(g),x)$ are the maps induced by the inclusion maps of $A,B$ into $R(g)$ .",
"So there exists $\\tilde{i}_{B}: (B,x) \\rightarrow (M,g^{-1}(x) \\cap \\partial M)$ a lift of $i_B: (B,x) \\rightarrow (R(g),x)$ (the inclusion map of $B$ into $R(g)$ ), and $\\tilde{i}_B(B)$ contains $g^{-1}(A) \\cap \\partial M$ .",
"So there exist $A_1,\\ldots ,A_n$ cancellable domains such that $X$ is their associated 2-complex.",
"And we can verify that the cancellation of $\\lbrace A_1,\\ldots ,A_n\\rbrace $ is regular, since every point $t \\in S(X)$ has a neighborhood $N(t)$ such that $(N(t) \\cap X,t)$ is homeomorphic to one of Figure REF (a), (b), (c).",
"Definition 3.5 Let $M$ be a compact 3-manifold with nonempty boundary and $G_0 \\subseteq \\partial M$ an embedded graph such that all vertices have degree 2 or 3.",
"Let $X_0 \\subseteq M$ be a good 2-complex in $M$ with respect to $G_0$ .",
"Assume $N$ is a subgraph of $S(X_0)$ and $G \\subseteq \\partial M$ is a thin trivalent graph (Definition REF ) of $G_0$ in $\\partial M$ .",
"(i) Let $M_1$ , $M_2$ , ..., $M_s$ be the components obtained by cutting off $X_0$ from $M$ (“cut off” means to delete the set from the space and do a path compactification), and $i_k: M_k \\rightarrow M$ ($\\forall k \\in \\lbrace 1,2,\\ldots ,m\\rbrace $ ) is continuous map induced by the “cutting off” ($i_k \\mid _{Int(M_k)}$ is an inclusion map).",
"Let $G_k = \\lbrace x \\in \\partial M_k \\mid i_k(x) \\in G \\cup N\\rbrace $ .",
"$(M,X_0,G \\cup N)$ is called appropriate if: for each $k \\in \\lbrace 1,2,\\ldots ,m\\rbrace $ , $G_k$ is an embedded graph such that all vertices have degree 2 or 3, and $(M_k,G_k)$ is appropriate.",
"(ii) Assume $(M,X_0,G \\cup N)$ is appropriate.",
"If $X_k \\subseteq M_k$ is a $(M_k,G_k)$ -simple 2-complex ($\\forall k \\in \\lbrace 1,2,\\ldots ,s\\rbrace $ ), we say the 2-complex $X = \\bigcup _{k=1}^{s} i_k(X_k)$ is a $(M,X_0,G \\cup N)$ -simple 2-complex.",
"In addition, for all $k \\in \\lbrace 1,2,\\ldots ,s\\rbrace $ and $X^{^{\\prime }}_{k} \\in sub(X_k)$ , we say $\\bigcup _{k=1}^{s} i_k(X^{^{\\prime }}_{k})$ is an $X_0$ -good subcomplex of $X$ , and we denote by $sub_{X_0}(X)$ the set consisting of all $X_0$ -good subcomplexes of $X$ ."
],
[
"The construction of the $(M,G)$ -simple 2-complex",
"Proposition 3.6 Let $M$ be a compact 3-manifold with nonempty boundary and $G \\subseteq \\partial M$ an embedded graph such that all vertices have degree 2 or 3.",
"Assume $(M,G)$ is appropriate, then there exists a $(M,G)$ -simple 2-complex.",
"Assume $A_1,\\ldots ,A_n$ are the components of $\\partial M \\setminus G$ , $x_k \\in A_k$ ($k \\in \\lbrace 1,2,\\ldots ,n\\rbrace $ ).",
"Let $p_k: (\\tilde{M}_k,\\tilde{x}_k) \\rightarrow (M,x_k)$ be a covering space such that $p_{k*}(\\pi _1(\\tilde{M}_k,\\tilde{x}_k)) = i_{k*}(\\pi _1(A_k,x_k))$ , where $p_{k*}: \\pi _1(\\tilde{M}_k,\\tilde{x}_k) \\rightarrow \\pi _1(M,x_k)$ , $i_{k*}: \\pi _1(A_k,x_k) \\rightarrow \\pi _1(M,x_k)$ are induced by $p_k$ and the inclusion map of $A_k$ into $M$ .",
"Then there exists a closed domain $S_k \\subseteq \\partial \\tilde{M}_k$ such that $p_k \\mid _{Int(S_k)}$ is a homeomorphism between $Int(S_k)$ and $A_k$ .",
"Assume $p: \\coprod _{k=1}^{n} \\tilde{M}_k \\rightarrow M$ is the map such that $p \\mid _{\\tilde{M}_k} = p_k$ ($\\forall k \\in \\lbrace 1,2,\\ldots ,n\\rbrace $ ).",
"There exists $M_0 \\subseteq \\coprod _{k=1}^{n} \\tilde{M}_k$ such that: ($\\forall k \\in \\lbrace 1,2,\\ldots ,n\\rbrace $ ) assume $L_k = M_0 \\cap \\tilde{M}_k \\subseteq \\tilde{M}_k$ , then $L_k$ is connected, $S_k \\subseteq L_k$ , $Int(L_1),\\ldots ,Int(L_n)$ are homeomorphically embedded into $M$ by $p$ , and there exists $X(M_0)$ a good 2-complex in $M$ with respect to $G$ such that $\\lbrace p(Int(L_1)),\\ldots ,p(Int(L_n))\\rbrace = \\lbrace the$ $components$ $of$ $Int(M) \\setminus X(M_0)\\rbrace $ (i.e.",
"$p$ maps $L_1,\\ldots ,L_k$ homeomorphically to the closed components obtained by cutting off $X(M_0)$ from $M$ ).",
"Then $M_0$ is closed, $p(M_0) = M$ , $p(M_0 \\setminus U) \\ne M$ for any open set $U \\subseteq M_0$ .",
"Moreover, $X(M_0) = \\overline{p(\\partial M_0) \\setminus \\partial M}$ .",
"Assume $p_0: M_0 \\rightarrow M$ is the map such that $p_0 = p \\mid _{M_0}$ .",
"Then $X(M_0) = \\lbrace x \\in M \\mid \\#(p_{0}^{-1}(x)) \\geqslant 2\\rbrace $ , and the embedded graph $S(X(M_0)) = \\lbrace x \\in M \\mid \\#(p_{0}^{-1}(x)) \\geqslant 3\\rbrace $ .",
"Figure: Thickening an edge.In the following, we adjust $M_0$ step by step (after each step, $X(M_0)$ is also a good 2-complex in $M$ with respect to $G$ ).",
"$X(M_0)$ will be a $(M,G)$ -simple 2-complex after all steps finished.",
"(a) (Thicken an edge) If there exists an edge $e \\in E(S(X(M_0)))$ , $\\#(p_{0}^{-1}(x)) \\geqslant 4$ for an $x \\in Int(e)$ , we adjust $M_0$ by the process of thicken $e$ (Picture REF ): $\\bullet $ Choose $N(e)$ an arbitrarily small open regular neighborhood of $e$ in $M$ , and choose $e_0$ a component of $p_{0}^{-1}(e)$ .",
"Assume $N(e_0)$ is the component of $p_{0}^{-1}(N(e))$ containing $e_0$ .",
"Let $M^{^{\\prime }}_{0} = \\overline{(M_0 \\setminus p^{-1}_{0}(N(e))) \\cup N(e_0)}$ and replace $M_0$ by $M^{^{\\prime }}_{0}$ .",
"Obviously, $\\#(\\lbrace e \\in E(S(X(M_0))) \\mid \\exists x \\in Int(e),\\#(p_{0}^{-1}(x)) \\geqslant 4\\rbrace )$ reduces after thickening an edge.",
"Figure: Thickening a vertex.",
"(b) (Thicken a vertex) After all above thickenings (of edges), $M$ satisfies that for all $e \\in E(S(X(M_0)))$ and $x \\in Int(e)$ , $\\#(p_{0}^{-1}(x)) = 3$ .",
"If there exists $v \\in V(S(X(M_0)))$ such that $\\#(p_{0}^{-1}(v)) \\geqslant 5$ , we adjust $M_0$ by the process of thicken $v$ (Picture REF ): $\\bullet $ Choose $N(v)$ an arbitrarily small open regular neighborhood of $v$ in $M$ , and choose $v_0 \\in p_{0}^{-1}(v)$ .",
"Assume $N(v_0)$ is the component of $p_{0}^{-1}(N(v))$ containing $v_0$ .",
"Let $M^{^{\\prime }}_{0} = \\overline{(M_0 \\setminus p_{0}^{-1}(N(v))) \\cup N(v_0)}$ and replace $M_0$ by $M^{^{\\prime }}_{0}$ .",
"Obviously, the number of $v \\in V(S(X(M_0)))$ such that $\\#(p_{0}^{-1}(v)) \\geqslant 5$ reduces after thickening a vertex.",
"The edges produced in thickening a vertex satisfy that for each $x$ in their interior, $\\#(p_{0}^{-1}(x)) = 3$ .",
"After all above thickenings (of vertices), $M$ satisfies that $\\#(p_{0}^{-1}(v)) = 4$ for each $v \\in \\lbrace v \\in V(S(X(M_0))) \\mid deg_{S(X(M_0))}(v) > 2\\rbrace $ .",
"We denote $\\lbrace v \\in V(S(X(M_0))) \\mid deg_{S(X(M_0))}(v) > 2\\rbrace = \\lbrace v \\in V(S(X(M_0))) \\mid deg_{S(X(M_0))}(v) = 4\\rbrace $ by $T(X(M_0))$ in the following.",
"For each $t \\in S(X(M_0)) \\setminus (T(X(M_0)) \\cup \\lbrace v \\in V(G) \\mid deg_G(v) = 3\\rbrace )$ , (then $t$ is either in the interior of an edge of $S(X(M_0))$ or a vertex of $S(X(M_0))$ with degree 2) there exists $N(t)$ an open neighborhood of $t$ in $M$ such that $(N(t) \\cap X(M_0),t)$ is homeomorphic to $(\\lbrace z = 0\\rbrace \\cup \\lbrace x = 0, z \\geqslant 0\\rbrace ,0)$ .",
"For each $t \\in T(X(M_0))$ , there exists $N(t)$ an open neighborhood of $t$ in $M$ such that $(N(t) \\cap X(M_0),t)$ is homeomorphic to $(\\lbrace z = 0\\rbrace \\cup \\lbrace x = 0, z \\geqslant 0\\rbrace \\cup \\lbrace y = 0, x \\geqslant 0, z \\geqslant 0\\rbrace ,0)$ .",
"For each $t \\in \\lbrace v \\in V(G) \\mid deg_G(v) = 3\\rbrace $ , there exists an open neighborhood $N(t)$ of $t$ in $M$ such that $(N(t) \\cap X,t)$ is homeomorphic to $(\\lbrace x = 0, y \\geqslant 0, z \\geqslant 0\\rbrace \\cup \\lbrace y = 0, z \\geqslant 0\\rbrace ,0)$ (since $X(M_0)$ is a good 2-complex in $M$ with respect to $G$ ).",
"Obviously, $X(M_0)$ is a $(M,G)$ -simple 2-complex.",
"Corollary 3.7 Let $M$ be a compact 3-manifold with nonempty boundary and $G_0 \\subseteq \\partial M$ an embedded graph such that all vertices have degree 2 or 3.",
"Let $X_0 \\subseteq M$ be a good 2-complex in $M$ with respect to $G_0$ .",
"Assume $N$ is a subgraph of $S(X_0)$ and $G \\subseteq \\partial M$ is a thin trivalent graph of $G_0$ in $\\partial M$ .",
"If $(M,X_0,G \\cup N)$ is appropriate, then there exists a $(M,X_0,G \\cup N)$ -simple 2-complex.",
"Remark 3.8 In Proposition REF , We prove that there exists a $(M,G)$ -simple 2-complex if $(M,G)$ is appropriate.",
"Auctually, a $(M,G)$ -simple 2-complex can be constructed through the proof of Proposition REF .",
"Similarly, we can construct a $(M,X_0,G \\cup N)$ -simple 2-complex in Corollary REF .",
"In the rest of this paper, we will always assume that a $(M,G)$ -simple 2-complex can be constructed immediately when we know $(M,G)$ is appropriate, and assume a $(M,X_0,G \\cup N)$ -simple 2-complex can be constructed immediately when we know $(M,X_0,G \\cup N)$ is appropriate."
],
[
"Inscribed set",
"Definition 4.1 (Inscribed set) For a closed oriented surface $\\Sigma $ , let $f: \\Sigma \\rightarrow \\mathbb {R}^{3}$ be an immersion.",
"Assume $n = \\max _{x \\in \\mathbb {R}^{3} \\setminus f(\\Sigma )}\\omega (f,x)$ .",
"The following process induces decreasingly on $k$ , until $k = 1$ .",
"For step 1: If $(D_n(f),G_n(f))$ is appropriate, then there exists $\\tilde{X}_n$ a $(D_n(f),G_n(f))$ -simple 2 complex.",
"Let $\\zeta _n = \\lbrace (\\tilde{X}_n,X_n) \\mid X_n \\in sub(\\tilde{X}_n)\\rbrace $ .",
"If $(D_n(f),G_n(f))$ is not appropriate, then $\\zeta _n = \\emptyset $ .",
"For step $n-k+1$ ($1 \\leqslant k \\leqslant n-1$ ): Assume $\\zeta _{k+1}$ is obtained in the step $n-k$ .",
"$\\zeta _k$ is obtained as follows: For each $A = \\lbrace (\\tilde{X}_{k+1},X_{k+1}), \\ldots , (\\tilde{X}_n,X_n)\\rbrace \\in \\zeta _{k+1}$ , assume $N = \\overline{S(X_{k+1}) \\setminus S(X_{k+2})}$ .",
"And we define $Q(A)$ by the following rules: $\\bullet $ If $(D_k(f),X_{k+1},N \\cup G_k(f))$ is appropriate, choose $\\tilde{X}_k$ a $(D_k(f),X_{k+1},N \\cup G_k(f))$ -simple 2-complex.",
"Let $Q(A) =\\lbrace (\\tilde{X}_k,X_k) \\mid X_k \\in sub_{X_{k+1}}(\\tilde{X}_k)\\rbrace $ .",
"$\\bullet $ If $(D_k(f),X_{k+1},N \\cup G_k(f))$ is not appropriate, then $Q(A) = \\emptyset $ .",
"Let $\\zeta _k = \\bigcup _{A \\in \\zeta _{k+1}, Q(A) \\ne \\emptyset } \\bigcup _{B \\in Q(A)} (A \\cup B)$ .",
"($\\zeta _k = \\emptyset $ if $\\zeta _{k+1} = \\emptyset $ ) In the end, we obtain an inscribed set $\\zeta = \\zeta _1$ , and we obtain the sets $\\zeta _2,\\ldots ,\\zeta _n$ ($\\zeta _k = \\lbrace (\\tilde{X}_k,X_k), \\ldots , (\\tilde{X}_n,X_n)\\rbrace $ ) through the process (call $\\zeta _k$ the $k$ th-inscribed set of $\\zeta $ ).",
"Definition 4.2 For a closed oriented surface $\\Sigma $ , let $f: \\Sigma \\rightarrow \\mathbb {R}^{3}$ be an immersion.",
"Assume $n = \\max _{x \\in \\mathbb {R}^{3} \\setminus f(\\Sigma )}\\omega (f,x)$ .",
"Let $\\zeta $ be an inscribed set of $f$ .",
"An element $\\lbrace (\\tilde{X}_1,X_1),\\ldots ,(\\tilde{X}_n,X_n)\\rbrace \\in \\zeta $ is good if $X_1 = \\emptyset $ .",
"We denote $\\lbrace A \\in \\zeta \\mid A$ $is$ $good\\rbrace $ by $I(\\zeta )$ .",
"Figure: For each p∈{v∈V(S(X k ))∣deg S(X k ) (v)>2}∖X k+1 p \\in \\lbrace v \\in V(S(X_k)) \\mid deg_{S(X_k)}(v) > 2\\rbrace \\setminus X_{k+1} (k∈{2,...,n}k \\in \\lbrace 2,\\ldots ,n\\rbrace ),(a), (b), (c) describes X k ,X k-1 ,X k-2 X_k,X_{k-1},X_{k-2} near pp.Definition 4.3 (Inscribed map) For a closed oriented surface $\\Sigma $ , let $f: \\Sigma \\rightarrow \\mathbb {R}^{3}$ be an immersion.",
"Assume $n = \\max _{x \\in \\mathbb {R}^{3} \\setminus f(\\Sigma )}\\omega (f,x)$ , and $\\lbrace (\\tilde{X}_1,X_1),\\ldots ,(\\tilde{X}_n,X_n)\\rbrace \\in I(\\zeta )$ .",
"For each $k \\in \\lbrace 1,2,\\ldots ,n\\rbrace $ , let $g_k: D_k \\rightarrow \\mathbb {R}^{3}$ be an embedding such that $g_k(D_k) = D_k(f)$ , and let $A_k = g_{k}^{-1}(X_k)$ , $B_k = g_{k}^{-1}(X_{k+1})$ ($X_{n+1} = \\emptyset $ ).",
"Let $g: \\coprod _{k=1}^{n} D_k \\rightarrow \\mathbb {R}^{3}$ be a map such that $g \\mid _{D_k} = g_k, \\forall k \\in \\lbrace 1,2,\\ldots ,n\\rbrace $ .",
"We obtain a map $g_1: M \\rightarrow \\mathbb {R}^{3}$ by following procedure: $\\bullet $ We cut off $A_k \\cup B_k$ from $D_k$ to obtain a space $D^{^{\\prime }}_{k}$ ($k \\in \\lbrace 1,\\ldots ,n\\rbrace $ ).",
"Assume $g_0: \\coprod _{k=1}^{n} D^{^{\\prime }}_{k} \\rightarrow \\mathbb {R}^{3}$ is the map induced by $g$ .",
"For all $k \\in \\lbrace 2,\\ldots ,n\\rbrace $ and $\\alpha $ a 2-cell of $X_k$ , assume $\\alpha ^{+}_{1},\\alpha ^{-}_{1}$ (respectively, $\\alpha ^{+}_{2},\\alpha ^{-}_{2}$ ) are the 2 components of $(g_{0} \\mid _{D_{k}^{^{\\prime }}})^{-1}(\\overline{\\alpha })$ (respectively, $(g_{0} \\mid _{D_{k-1}^{^{\\prime }}})^{-1}(\\overline{\\alpha })$ ) which lie in the left and right side respectively.",
"Let $h$ be the equivalence relation such that $x \\stackrel{h}{\\sim } y$ if there exists $k \\in \\lbrace 2,\\ldots ,n\\rbrace $ and $\\alpha $ a 2-cell of $X_k$ , $x \\in \\alpha _{1}^{+}, y \\in \\alpha _{2}^{-}, g_0(x) = g_0(y)$ or $x \\in \\alpha _{2}^{+}, y \\in \\alpha _{1}^{-}, g_0(x) = g_0(y)$ .",
"Let $M = \\coprod _{k=1}^{n} D^{^{\\prime }}_{k} /\\sim _h$ , and $g_1: M \\rightarrow \\mathbb {R}^{3}$ is induced by $g_0$ .",
"We say $g_1$ is an inscribed map of $f$ associated to $\\lbrace (\\tilde{X}_1,X_1),\\ldots ,(\\tilde{X}_n,X_n)\\rbrace $ .",
"Lemma 4.4 $M$ is a (compact, connected) 3-manifold, and $g_1$ is an immersion.",
"We can verify that for each $p \\in \\mathbb {R}^{3}$ , every $t \\in g_{1}^{-1}(p)$ has an open neighborhood homeomorphic to $\\mathbb {R}^{3}$ or $\\mathbb {R}^{3}_{+}$ , and $g_1$ is a locally homeomorphism at $t$ .",
"We only explain this for the point $p$ that is a vertex of $S(X_k)$ with degree greater than 2: If $p \\in \\lbrace v \\in V(S(X_k)) \\mid deg_{S(X_k)}(v) > 2\\rbrace $ and $p \\notin X_{k+1}$ ($k \\in \\lbrace 2,\\ldots ,n\\rbrace $ ), then there exists $N(p)$ an open neighborhood of $p$ in $\\mathbb {R}^{3}$ such that $N(p) \\cap X_k, N(p) \\cap X_{k-1}, X(p) \\cap X_{k-2}$ are homeomorphic to Figure REF (a), (b), (c), and $p \\cap X_{k-3} = \\emptyset $ .",
"So we can verify that every point in $g_{1}^{-1}(p)$ has an open neighborhood which is homeomorphic to $\\mathbb {R}^{3}$ and homeomorphically embedded into $\\mathbb {R}^{3}$ by $g_1$ .",
"Moreover, if $p \\in g_1(M) \\subseteq \\mathbb {R}^{3}$ , assume $l \\subseteq \\mathbb {R}^{3}$ is a ray starting from $p$ and parallel to $x$ -axis.",
"For each $x \\in l \\setminus D_1(f)$ , $g_{1}^{-1}(x) = \\emptyset $ .",
"So every component of $g_{1}^{-1}(l)$ contains a point in $\\partial M$ .",
"Then every point in $g_{1}^{-1}(p)$ is in the same connected component with $\\partial M$ .",
"Hence $M$ is connected.",
"In the following , we say an inscribed map of $f$ associated to $\\lbrace (\\tilde{X}_1,X_1),\\ldots ,(\\tilde{X}_n,X_n)\\rbrace $ is an extension of $f$ related to $\\lbrace (\\tilde{X}_1,X_1),\\ldots ,(\\tilde{X}_n,X_n)\\rbrace $ .",
"Example 4.5 Let $f: S^{2} \\rightarrow \\mathbb {R}^{3}$ be an immersion described by Figure REF (a).",
"Figure REF (b) describes $D_3(f)$ , $D_2(f)$ , $D_1(f)$ , $G_3(f)$ , $G_2(f)$ .",
"Figure REF gives an inscribed set $\\zeta = \\lbrace \\lbrace (\\tilde{X}_1,X_1),(\\tilde{X}_2,X_2),$ $(\\tilde{X}_3,X_3)\\rbrace \\rbrace $ of $f$ ($\\#(\\zeta ) = 1$ ).",
"Then $\\lbrace (\\tilde{X}_1,X_1),(\\tilde{X}_2,X_2),(\\tilde{X}_3,X_3)\\rbrace \\in I(\\zeta )$ .",
"Hence we can construct (exactly) one extension of $f$ .",
"And Figure REF shows the construction of this extension (the extension of $f$ related to $\\lbrace (\\tilde{X}_1,X_1),(\\tilde{X}_2,X_2),(\\tilde{X}_3,X_3)\\rbrace $ ).",
"Figure: f:S 2 →ℝ 3 f: S^{2} \\rightarrow \\mathbb {R}^{3}.Figure: The inscribed set ζ={{(X ˜ 1 ,X 1 ),(X ˜ 2 ,X 2 ),\\zeta = \\lbrace \\lbrace (\\tilde{X}_1,X_1),(\\tilde{X}_2,X_2), (X ˜ 3 ,X 3 )}}(\\tilde{X}_3,X_3)\\rbrace \\rbrace of ff.Figure: The construction of the extension of ff related to {(X ˜ 1 ,X 1 ),(X ˜ 2 ,X 2 ),(X ˜ 3 ,X 3 )}\\lbrace (\\tilde{X}_1,X_1),(\\tilde{X}_2,X_2),(\\tilde{X}_3,X_3)\\rbrace ."
],
[
"The proof of Theorem 1",
"In this section, we will prove Theorem REF .",
"If $f: \\Sigma \\rightarrow \\mathbb {R}^{3}$ is an immersion of the closed oriented surface $\\Sigma $ and $\\zeta $ is an inscribed set of $f$ , then there exists a map $q: I(\\zeta ) \\rightarrow E(f)$ (where $E(f)$ is the set of equivalence classes of immersions of 3-manifolds to extend $f$ ) sending each element of $I(\\zeta )$ to the extension of $f$ related to it.",
"We prove that $q$ is injective in Lemma REF , and we prove that $q$ is surjective in Proposition REF .",
"Lemma 5.1 For a closed oriented surface $\\Sigma $ , let $f: \\Sigma \\rightarrow \\mathbb {R}^{3}$ be an immersion and $\\zeta $ an inscribed set of $f$ .",
"Assume $n = \\max _{x \\in \\mathbb {R}^{3} \\setminus f(\\Sigma )} \\omega (f,x)$ .",
"If $\\lbrace (\\tilde{X}_1,X_1),\\ldots ,(\\tilde{X}_n,X_n)\\rbrace ,\\lbrace (\\tilde{Y}_1,Y_1),\\ldots ,(\\tilde{Y}_n,Y_n)\\rbrace $ are 2 different elements of $I(\\zeta )$ , then the 2 extensions of $f$ related to $\\lbrace (\\tilde{X}_1,X_1),\\ldots ,(\\tilde{X}_n,X_n)\\rbrace ,\\lbrace (\\tilde{Y}_1,Y_1),\\ldots ,$ $(\\tilde{Y}_n,Y_n)\\rbrace $ are inequivalent.",
"The proof is similar to the proof of [Zhao][9, Lemma 6.1].",
"Assume $g_1: M_1 \\rightarrow \\mathbb {R}^{3},g_2: M_2 \\rightarrow \\mathbb {R}^{3}$ are the extensions related to $\\lbrace (\\tilde{X}_1,X_1),\\ldots ,(\\tilde{X}_n,X_n)\\rbrace $ , $\\lbrace (\\tilde{Y}_1,Y_1),\\ldots ,(\\tilde{Y}_n,Y_n)\\rbrace $ .",
"Then there exists $k \\in \\lbrace 2,3,\\ldots ,n\\rbrace $ such that $X_k \\ne Y_k$ and $X_i = Y_i$ for each $k+1 \\leqslant i \\leqslant n$ .",
"Note that $\\tilde{X}_k = \\tilde{Y}_k$ (since $\\tilde{X}_k$ is yielded by $\\lbrace (\\tilde{X}_{k+1},X_{k+1}),\\ldots ,(\\tilde{X}_n,X_n)\\rbrace $ , and $\\tilde{Y}_k$ is yielded by $\\lbrace (\\tilde{Y}_{k+1},Y_{k+1}),\\ldots ,(\\tilde{Y}_n,Y_n)\\rbrace $ ).",
"So there exists $\\alpha $ a 2-cell of $\\tilde{X}_k$ such that $\\alpha $ is contained by exactly one of $X_k, Y_k$ .",
"Assume without loss of generality that $\\alpha \\subseteq X_k, \\alpha \\nsubseteq Y_k$ .",
"For each $\\gamma $ a 2-cell of $\\tilde{X}_i$ ($i \\in \\lbrace 2,\\ldots ,n\\rbrace $ ), we denote by $D_+(\\gamma )$ (respectively $D_-(\\gamma )$ ) the closure of the component of $D_i(f) \\setminus (\\tilde{X_i} \\cup X_{i+1})$ which lie in the left side (respectively the right side) of $\\gamma $ .",
"Then $\\partial D_+(\\gamma ) \\cap (X_{i+1} \\cup \\partial D_{i}(f)), \\partial D_-(\\gamma ) \\cap (X_{i+1} \\cup \\partial D_{i}(f)) \\ne \\emptyset $ .",
"There exist $m \\in \\lbrace k,k+1,\\ldots ,n\\rbrace $ and $p_1 \\in \\partial D_m(f) \\setminus G_m(f)$ such that: for each $t \\in \\lbrace 1,\\ldots ,m-k-1\\rbrace $ , $\\exists $ $\\alpha _t$ a 2-cell in $X_{k+t}$ such that $\\alpha _t \\subseteq D_+(\\alpha _{t-1})$ , $\\alpha _0 = \\alpha $ , and $p_1 \\in D_+(\\alpha _{m-k-1})$ .",
"Choose $s \\in \\alpha $ .",
"There exists an immersion $h_1: [0,1] \\rightarrow \\mathbb {R}^{3}$ such that $h_1(\\frac{i}{m-k}) \\in \\alpha _i$ , $[\\frac{i}{m-k},\\frac{i+1}{m-k}]$ is homeomorphically embedded into $D_+(\\alpha _{i})$ by $h_1$ ($0 \\leqslant i \\leqslant m-k-1$ ), $h_1(0) = s$ , $h_1(1) = p_1$ .",
"Similarly, there exists $q \\in \\lbrace k,k+1,\\ldots ,n\\rbrace $ and $p_2 \\in \\partial D_q(f) \\setminus G_q(f)$ , such that: for each $t \\in \\lbrace 1,\\ldots ,q-k-1\\rbrace $ , $\\exists $ $\\beta _t$ a 2-cell in $X_{k+t}$ such that $\\beta _t \\subseteq D_-(\\beta _{t-1})$ , $\\beta _0 = \\alpha $ , and $p_2 \\in D_-(\\alpha _{q-k-1})$ .",
"Let $h_2: [0,1] \\rightarrow \\mathbb {R}^{3}$ be an immersion such that $h_2(\\frac{i}{q-k}) \\in \\beta _i$ , $[\\frac{i}{q-k},\\frac{i+1}{q-k}]$ is homeomorphically embedded into $D_-(\\alpha _{i})$ by $h_2$ ($0 \\leqslant i \\leqslant q-k-1$ ), $h_2(0) = s$ , $h_2(1) = p_2$ .",
"There exist embeddings $\\tilde{h}_{1}: [0,1] \\rightarrow M_1,\\tilde{h}_{2}: [0,1] \\rightarrow M_2$ such that $g_1 \\circ \\tilde{h}_1 = g_2 \\circ \\tilde{h}_2 = h_1$ , $\\tilde{h}_{1}(1) \\in \\partial M_1,\\tilde{h}_{2}(1) \\in \\partial M_2$ .",
"And there exists embeddings $\\tilde{h}_{3}: [0,1] \\rightarrow M_1,\\tilde{h}_{4}: [0,1] \\rightarrow M_2$ such that $g_1 \\circ \\tilde{h}_3 = g_2 \\circ \\tilde{h}_4 = h_2$ , $\\tilde{h}_{3}(1) \\in \\partial M_1,\\tilde{h}_{4}(1) \\in \\partial M_2$ .",
"Then $\\tilde{h}_1(0) \\ne \\tilde{h}_3(0)$ , $\\tilde{h}_2(0) = \\tilde{h}_4(0)$ (since $\\alpha \\subseteq X_k, \\alpha \\nsubseteq Y_k$ ).",
"So there exists $\\tilde{h}_2([0,1]) \\cup \\tilde{h}_4([0,1])$ a properly embedded arc of $M_2$ mapped to $h_1([0,1]) \\cup h_2([0,1])$ by $g_2$ , but there is no properly embedded arc of $M_1$ mapped to $h_1([0,1]) \\cup h_2([0,1])$ by $g_1$ .",
"Hence $g_1,g_2$ are inequivalent.",
"Proposition 5.2 For a closed oriented surface $\\Sigma $ , let $f: \\Sigma \\rightarrow \\mathbb {R}^{3}$ be an immersion and $\\zeta $ an inscribed set of $f$ .",
"Assume $n = \\max _{x \\in \\mathbb {R}^{3} \\setminus f(\\Sigma )} \\omega (f,x)$ .",
"If $g: M \\rightarrow \\mathbb {R}^{3}$ is an immersion of a compact 3-manifold $M$ such that $g \\mid _{\\partial M} = f$ , then there exists $\\lbrace (\\tilde{X}_1,X_1),\\ldots ,(\\tilde{X}_n,X_n)\\rbrace \\in I(\\zeta )$ , such that $g$ is the extension of $f$ related to $\\lbrace (\\tilde{X}_1,X_1),\\ldots ,(\\tilde{X}_n,X_n)\\rbrace $ .",
"First, we will construct a sequence of cancellation operations $(g,M) \\leadsto (g_{n-1},K_{n-1})\\leadsto (g_{n-2},K_{n-2}) \\leadsto \\ldots \\leadsto (g_1,K_1)$ in the following (where $K_j$ is a branched 3-manifold, $g_j$ is a branched immersion, $g_j(\\partial K_j) = \\bigcup _{i=1}^{j}\\partial D_i(f)$ , $K_1$ is a 3-manifold and $g_1$ is an embedding): Step 1.",
"By Lemma REF (i), $(R(g),G(g)) = (D_n(f),G_n(f))$ is appropriate.",
"So $\\zeta $ yields $\\tilde{X}_n$ a $(D_n(f),G_n(f))$ -simple 2-complex.",
"Assume $A_{(n,1)},\\ldots ,A_{(n,t_n)}$ are cancellable domains such that $\\tilde{X}_n$ is the 2-complex associated to them ($A_{(n,1)},\\ldots ,A_{(n,t_n)}$ exist and they are determined uniquely by $\\tilde{X}_n$ , see Lemma REF (ii) and Remark REF (a)).",
"Let $X_n = X(A_{(n,1)},\\ldots ,A_{(n,t_n)}) \\in sub(X_n)$ .",
"Then $\\lbrace (X_n,\\tilde{X}_n)\\rbrace \\in \\zeta _n$ .",
"We cancel $\\lbrace A_{(n,1)},\\ldots ,A_{(n,t_n)}\\rbrace $ .",
"The cancellation $(g,M) \\stackrel{\\lbrace A_{(n,1)},\\ldots ,A_{(n,t_n)}\\rbrace }{\\leadsto } (g_{n-1},K_{n-1})$ produces a branched immersion $g_{n-1}: K_{n-1} \\rightarrow \\mathbb {R}^{3}$ such that: Property (a): $g_{n-1}(\\partial K_{n-1}) = \\bigcup _{i=1}^{n-1}\\partial D_{i}(f)$ .",
"Property (b): The embedded graph $S(X_n)$ is the branch set of $g_{n-1}$ .",
"For each $x \\in S(X_n)$ , $x$ has index 1 if $x \\in S(X_n) \\setminus \\lbrace v \\in V(S(X_n)) \\mid deg_{S(X_n)}(v) > 2\\rbrace $ , and $x$ has index 2 if $x \\in \\lbrace v \\in V(S(X_n)) \\mid deg_{S(X_n)}(v) > 2\\rbrace $ .",
"Property (c): The cancellation is regular (Lemma REF (ii)).",
"Let $h_{n-1}: X_n \\rightarrow K_{n-1}$ denote the associated map (Definition REF (iii)) of the cancellation.",
"Then $h_{n-1}(X_n \\cap D_{n-1}(f)) = h_{n-1}(X_n) \\cap \\partial K_{n-1}$ (Remark REF (d)).",
"And $h_{n-1}(S(X_n))$ is the singular set of $g_{n-1}$ .",
"Property (d): For each $e \\in E(S(X_n))$ , there exists three 2-cells $\\alpha _1,\\alpha _2,\\alpha _3$ of $X_n$ such that $e \\subseteq \\overline{\\alpha _1}, \\overline{\\alpha _2}, \\overline{\\alpha _3}$ , and assume that $\\alpha _1,\\alpha _2,\\alpha _3$ are in clockwise.",
"Assume $e_0 = h_{n-1}(e)$ .",
"Assume $\\beta _1,\\ldots ,\\beta _6$ are the components of $g_{n-1}^{-1}(\\alpha _1),g_{n-1}^{-1}(\\alpha _2),g_{n-1}^{-1}(\\alpha _3)$ such that $e_0 \\subseteq \\overline{\\beta _i}$ ($i = 1,\\ldots ,6$ ), and $\\beta _1,\\ldots ,\\beta _6$ are in clockwise.",
"Assume without loss of generality $h_{n-1}(\\alpha _1) = \\beta _1$ (then $\\beta _1, \\beta _4 \\subseteq g_{n-1}^{-1}(\\alpha _1)$ , $\\beta _2, \\beta _5 \\subseteq g_{n-1}^{-1}(\\alpha _2)$ , $\\beta _3, \\beta _6 \\subseteq g_{n-1}^{-1}(\\alpha _3)$ ).",
"Then $h_{n-1}(\\alpha _2) = \\beta _5$ , $h_{n-1}(\\alpha _3) = \\beta _3$ .",
"Property (e): $(D_{n-1}(f),X_n,G_{n-1}(f) \\cup S(X_n))$ is appropriate.",
"We explain this as follows: If $L$ is one of the components obtained by cutting off $X_n$ from $D_{n-1}(f)$ , assume $i: L \\rightarrow D_{n-1}(f)$ is the continuous map induced by the “cutting off” ($i \\mid _{Int(L)}$ is an inclusion map, and $i(Int(L))$ is one of the components of $D_{n-1}(f) \\setminus X_n$ ).",
"Assume $S = \\overline{\\lbrace x \\in \\partial L \\mid i(x) \\in Int(i(L))\\rbrace }$ .",
"Let $L_0$ be the space obtained by cutting off $g_{n-1}^{-1}(i(L)) \\cap (h_{n-1}(S(X_n)) \\cup g_{n-1}^{-1}(i(S)))$ from $g_{n-1}^{-1}(i(L))$ , and assume $j: L_0 \\rightarrow g_{n-1}^{-1}(i(L))$ is the continuous map induced by the “cutting off” ($j \\mid _{Int(L_0)}$ is an inclusion map).",
"Then $L_0$ is a 3-manifold that may be disconnected, and there exists a covering map $g^{^{\\prime }}: L_0 \\rightarrow L$ such that $g^{^{\\prime }} \\mid _{g_{n-1}^{-1}(Int(i(L)))} = i^{-1} \\circ g_{n-1} \\circ j \\mid _{g_{n-1}^{-1}(Int(i(L)))}$ .",
"For each component $A$ of $\\partial L \\setminus i^{-1}(G_{n-1}(f) \\cup S(X_n))$ , assume $A_0 = \\overline{g^{^{\\prime }-1}(A) \\cap j^{-1}(h_{n-1}(X_n) \\cup \\partial K_{n-1})}$ , then $g^{^{\\prime }}$ maps $Int(A_0)$ homeomorphically to $A$ .",
"By Lemma REF , $(L, i^{-1}(G_{n-1}(f) \\cup S(X_n)))$ is appropriate.",
"So $(D_{n-1}(f),X_n,G_{n-1}(f) \\cup S(X_n))$ is appropriate.",
"Step $k+1$ .",
"After $k$ steps, assume we obtain a branched immersion $g_{n-k}: K_{n-k} \\rightarrow \\mathbb {R}^{3}$ from a sequence of cancellation operation $(g,M) \\stackrel{\\lbrace A_{(n,1)},\\ldots ,A_{(n,t_n)}\\rbrace }{\\leadsto } (g_{n-1},K_{n-1})\\stackrel{\\lbrace A_{(n-1,1)},\\ldots ,A_{(n-1,t_{n-1})}\\rbrace }{\\leadsto } \\ldots $ $\\stackrel{\\lbrace A_{(n-k+1,1)},\\ldots ,A_{(n-k+1,t_{n-k+1})}\\rbrace }{\\leadsto } (g_{n-k},K_{n-k})$ and the following hold: Induction hypothesis (i): $g(\\partial K_{n-k}) = \\bigcup _{i=1}^{n-k} \\partial D_{i}(f)$ (then $R(g_{n-k}) = D_{n-k}(f)$ , $G(g_{n-k}) = G_{n-k}(f)$ ).",
"Induction hypothesis (ii): $B_{n-k} = \\overline{S(X_{n-k+1}) \\setminus S(X_{n-k+2})}$ is the branched set of $g_{n-k}$ , and $B_{n-k}$ is an embedded graph.",
"For each $x \\in B_{n-k}$ , $x$ has index 2 if $x$ is a vertex of $S(X_{n-k+1})$ with degree greater than 2 and $x \\notin S(X_{n-k+2})$ , otherwise $x$ has degree 1.",
"Induction hypothesis (iii): There exists $h_{n-k}: X_{n-k+1} \\rightarrow K_{n-k}$ an embedding such that $g_{n-k} \\circ h_{n-k} = id$ , $h_{n-k}(X_{n-k+1} \\cap \\partial D_{n-k}(f)) = h_{n-k}(X_{n-k+1}) \\cap \\partial K_{n-k}$ , and $h_{n-k}(B_{n-k})$ is the singular set of $g_{n-k}$ .",
"Induction hypothesis (iv): For each $e \\in E(B_{n-k})$ , assume $\\alpha _1,\\alpha _2,\\alpha _3$ are the three 2-cells of $X_{n-k+1}$ such that $e \\subseteq \\overline{\\alpha _1}, \\overline{\\alpha _2}, \\overline{\\alpha _3}$ , and $\\alpha _1,\\alpha _2,\\alpha _3$ are in clockwise.",
"Assume $e_0 = h_{n-k}(e)$ .",
"Assume $\\beta _1,\\ldots ,\\beta _6$ are the components of $g_{n-k}^{-1}(\\alpha _1),g_{n-k}^{-1}(\\alpha _2),g_{n-k}^{-1}(\\alpha _3)$ such that $e_0 \\subseteq \\overline{\\beta _i}$ ($i = 1,\\ldots ,6$ ), and $\\beta _1,\\ldots ,\\beta _6$ are in clockwise.",
"Assume without loss of generality $h_{n-k}(\\alpha _1) = \\beta _1$ (then $\\beta _1, \\beta _4 \\subseteq g_{n-k}^{-1}(\\alpha _1)$ , $\\beta _2, \\beta _5 \\subseteq g_{n-k}^{-1}(\\alpha _2)$ , $\\beta _3, \\beta _6 \\subseteq g_{n-k}^{-1}(\\alpha _3)$ ).",
"Then $h_{n-k}(\\alpha _2) = \\beta _5$ , $h_{n-k}(\\alpha _3) = \\beta _3$ .",
"Induction hypothesis (v): $(D_{n-k}(f),X_{n-k+1},G_{n-k}(f) \\cup B_{n-k})$ is appropriate.",
"By Corollary REF , $\\lbrace (\\tilde{X}_{n-k+1},X_{n-k+1}),\\ldots (\\tilde{X}_n,X_n)\\rbrace $ yields a $(D_{n-k}(f),X_{n-k+1},G_{n-k}(f) \\cup B_{n-k})$ -simple 2-complex $\\tilde{X}_{n-k}$ .",
"Let $A_{(n-k,1)},\\ldots ,A_{(n-k,t_{n-k})}$ be the cancellable domains such that: $\\tilde{X}_{n-k}$ is the 2-complex associated to them, and for each $j \\in \\lbrace 1,2,\\ldots ,t_{n-k}\\rbrace $ , $A_{(n-k,j)} \\cap (\\partial M_{n-k} \\cup h_{n-k}(X_{n-k+1})) \\ne \\emptyset $ ($A_{(n-k,1)},\\ldots ,A_{(n-k,t_{n-k})}$ exist, similar to Lemma REF (ii); and $A_{(n-k,1)},\\ldots ,$ $A_{(n-k,t_{n-k})}$ are uniquely determined, see Remark REF (b)).",
"Let $X_{n-k} = X(A_{(n-k,1)},\\ldots ,A_{(n-k,t_{n-k})})$ .",
"Because of induction hypothesis (iv), $X_{n-k} \\in sub_{X_{n-k+1}}(\\tilde{X}_{n-k})$ .",
"So $\\lbrace (\\tilde{X}_{n-k},X_{n-k}),(\\tilde{X}_{n-k+1},X_{n-k+1}),$ $\\ldots ,(\\tilde{X}_n,X_n)\\rbrace \\in \\zeta _{n-k}$ .",
"We cancel $\\lbrace A_{(n-k,1)},\\ldots ,A_{(n-k,t_{n-k})}\\rbrace $ .",
"The cancellation $(g_{n-k},K_{n-k})\\stackrel{\\lbrace A_{(n-k,1)},\\ldots ,A_{(n-k,t_{n-k})}\\rbrace }{\\leadsto } (g_{n-k-1},$ $K_{n-k-1})$ gives a branched immersion $g_{n-k-1}: K_{n-k-1} \\rightarrow \\mathbb {R}^{3}$ .",
"First, we verify that the cancellation is regular: $\\bullet $ If $x \\in S(X_{n-k}) \\cap S(X_{n-k+1})$ , then $x \\in B_{n-k}$ and we denote by $y$ the singular point of $g_{n-k}$ mapped to $x$ .",
"Then $y \\in A_{(n-k,j)}$ if $x \\in g_{n-k}(A_{(n-k,j)})$ ($\\forall j \\in \\lbrace 1,2,\\ldots ,t_{n-k}\\rbrace $ ).",
"Hence the cancellation of $\\lbrace A_{(n-k,1)},\\ldots ,A_{(n-k,t_{n-k})}\\rbrace $ is regular at $x$ .",
"$\\bullet $ Similar to Lemma REF (ii), if $x \\in S(X_{n-k}) \\setminus S(X_{n-k+1})$ , then the cancellation of $\\lbrace A_{(n-k,1)},\\ldots ,$ $A_{(n-k,t_{n-k})}\\rbrace $ is regular at $x$ .",
"So the cancellation $(g_{n-k},K_{n-k})\\stackrel{\\lbrace A_{(n-k,1)},\\ldots ,A_{(n-k,t_{n-k})}\\rbrace }{\\leadsto } (g_{n-k-1},$ $K_{n-k-1})$ is regular.",
"Moreover, the following hold: Property (a): $g(\\partial K_{n-k-1}) = \\bigcup _{i=1}^{n-k-1} \\partial D_{i}(f)$ .",
"Property (b): The branch points of $g_{n-k}$ with index 1 are not branch points of $g_{n-k-1}$ , and the branch points of $g_{n-k}$ with index 2 are branch points of $g_{n-k-1}$ with index 1.",
"So the branch set $B_{n-k-1}$ of $g_{n-k-1}$ is $\\overline{S(X_{n-k}) \\setminus S(X_{n-k+1})}$ , and for each $x \\in B_{n-k-1}$ , $x$ has index 2 if $x \\in \\lbrace v \\in V(S(X_{n-k})) \\mid deg_{S(X_{n-k})}(v) > 2\\rbrace \\setminus S(X_{n-k+1})$ , and $x$ has index 1 otherwise.",
"Property (c): Since the cancellation $(g_{n-k},K_{n-k})$ $\\stackrel{\\lbrace A_{(n-k,1)},\\ldots ,A_{(n-k,t_{n-k})}\\rbrace }{\\leadsto } (g_{n-k-1},$ $K_{n-k-1})$ is regular, there exists $h_{n-k-1}: X_{n-k} \\rightarrow K_{n-k-1}$ the associated map of the cancellation.",
"$h_{n-k-1}(X_{n-k} \\cap \\partial D_{n-k-1}(f)) = h_{n-k-1}(X_{n-k}) \\cap \\partial K_{n-k-1}$ (Remark REF (d)).",
"And $h_{n-k-1}$ maps $B_{n-k-1}$ homeomorphically to the singular set of $g_{n-k-1}$ .",
"Property (d): Induction hypothesis (iv) is developed for $X_{n-k}$ and $B_{n-k-1}$ .",
"We state this again as follows: For each $e \\in E(B_{n-k-1})$ , there exists $\\alpha _1,\\alpha _2,\\alpha _3$ the three 2-cells of $X_{n-k}$ such that $e \\subseteq \\overline{\\alpha _1}, \\overline{\\alpha _2}, \\overline{\\alpha _3}$ , and $\\alpha _1,\\alpha _2,\\alpha _3$ are in clockwise.",
"Assume $e_0 = h_{n-k-1}(e)$ .",
"Assume $\\beta _1,\\ldots ,\\beta _6$ are the components of $g_{n-k-1}^{-1}(\\alpha _1),g_{n-k-1}^{-1}(\\alpha _2),g_{n-k-1}^{-1}(\\alpha _3)$ such that $e_0 \\subseteq \\overline{\\beta _i}$ ($i = 1,\\ldots ,6$ ), and $\\beta _1,\\ldots ,\\beta _6$ are in clockwise.",
"Assume without loss of generality $h_{n-k-1}(\\alpha _1) = \\beta _1$ (then $\\beta _1, \\beta _4 \\subseteq g_{n-k-1}^{-1}(\\alpha _1)$ , $\\beta _2, \\beta _5 \\subseteq g_{n-k-1}^{-1}(\\alpha _2)$ , $\\beta _3, \\beta _6 \\subseteq g_{n-k-1}^{-1}(\\alpha _3)$ ).",
"Then $h_{n-k-1}(\\alpha _2) = \\beta _5$ , $h_{n-k-1}(\\alpha _3) = \\beta _3$ .",
"Property (e): Similar to the Property (e) of Step 1, $(D_{n-k-1}(f),X_{n-k},G_{n-k-1}(f) \\cup B_{n-k-1})$ is appropriate.",
"Hence we can verify that all induction hypothesises will be developed in the next step (Step $k+2$).",
"In the end, we have constructed $\\lbrace (\\tilde{X}_1,X_1),\\ldots ,(\\tilde{X}_n,X_n)\\rbrace \\in \\zeta $ with a sequence of cancellation operations $(g,M) \\stackrel{\\lbrace A_{(n,1)},\\ldots ,A_{(n,t_n)}\\rbrace }{\\leadsto } (g_{n-1},K_{n-1})\\stackrel{\\lbrace A_{(n-1,1)},\\ldots ,A_{(n-1,t_{n-1})}\\rbrace }{\\leadsto } \\ldots \\stackrel{\\lbrace A_{(2,1)},\\ldots ,A_{(2,t_2)}\\rbrace }{\\leadsto } (g_1,K_1)$ .",
"Note that $\\#(g_{k-1}^{-1}(x)) + index_{g_{k-1}}(x) = \\#(g_{k}^{-1}(x)) + index_{g_k}(x) -1$ for each $x \\in D_k(x)$ (where $index_{g_j}(x)$ is the index of $x$ if $x$ is a branch point of the map $g_j$ , and $index_{g_j}(x) = 0$ if $x$ is not a branch point of the map $g_j$ ; and $g_n = g$ ).",
"So $g_1: K_1 \\rightarrow \\mathbb {R}^{3}$ is an embedding.",
"Then $X_1 = \\emptyset $ .",
"Hence $\\lbrace (\\tilde{X}_1,X_1),\\ldots ,(\\tilde{X}_n,X_n)\\rbrace \\in I(\\zeta )$ .",
"From this sequence of cancellation operations, we can verify that $g: M \\rightarrow \\mathbb {R}^{3}$ is the inscribed map of $f$ associated to $\\lbrace (\\tilde{X}_1,X_1),\\ldots ,(\\tilde{X}_n,X_n)\\rbrace $ , i.e.",
"$g$ is the extension of $f$ related to $\\lbrace (\\tilde{X}_1,X_1),\\ldots ,(\\tilde{X}_n,X_n)\\rbrace $ ."
],
[
"Acknowledgments",
" This paper is the motivation for the author to write [Zhao][9].",
"The author is grateful for Professor Shicheng Wang, Professor Yi Liu, Professor Jiajun Wang in these works, and in my study.",
"Especially, the author thanks Professor Jiajun Wang for his comment that help me a lot."
]
] | 1906.04294 |
[
[
"Causal Discovery with Reinforcement Learning"
],
[
"Abstract Discovering causal structure among a set of variables is a fundamental problem in many empirical sciences.",
"Traditional score-based casual discovery methods rely on various local heuristics to search for a Directed Acyclic Graph (DAG) according to a predefined score function.",
"While these methods, e.g., greedy equivalence search, may have attractive results with infinite samples and certain model assumptions, they are usually less satisfactory in practice due to finite data and possible violation of assumptions.",
"Motivated by recent advances in neural combinatorial optimization, we propose to use Reinforcement Learning (RL) to search for the DAG with the best scoring.",
"Our encoder-decoder model takes observable data as input and generates graph adjacency matrices that are used to compute rewards.",
"The reward incorporates both the predefined score function and two penalty terms for enforcing acyclicity.",
"In contrast with typical RL applications where the goal is to learn a policy, we use RL as a search strategy and our final output would be the graph, among all graphs generated during training, that achieves the best reward.",
"We conduct experiments on both synthetic and real datasets, and show that the proposed approach not only has an improved search ability but also allows a flexible score function under the acyclicity constraint."
],
[
"Introduction",
"Discovering and understanding causal mechanisms underlying natural phenomena are important to many disciplines of sciences.",
"An effective approach is to conduct controlled randomized experiments, which however is expensive or even impossible in certain fields such as social sciences [3] and bioinformatics [24].",
"Causal discovery methods that infer causal relationships from passively observable data are hence attractive and have been an important research topic in the past decades [25], [36], [28].",
"A major class of such causal discovery methods are score-based, which assign a score $\\mathcal {S}(\\mathcal {G})$ , typically computed with the observed data, to each directed graph $\\mathcal {G}$ and then search over the space of all Directed Acyclic Graphs (DAGs) for the best scoring: $\\min _{\\mathcal {G}}~\\mathcal {S}(\\mathcal {G}),~\\text{subject to}~\\mathcal {G} \\in \\mathsf {DAGs}.$ While there have been well-defined score functions such as the Bayesian Information Criterion (BIC) or Minimum Description Length (MDL) score [31], [7] and the Bayesian Gaussian equivalent (BGe) score [10], Problem (REF ) is generally NP-hard to solve [6], [8], largely due to the combinatorial nature of its acyclicity constraint with the number of DAGs increasing super-exponentially in the number of graph nodes.",
"To tackle this problem, most existing approaches rely on local heuristics to enforce the acyclicity.",
"For example, Greedy Equivalence Search (GES) enforces acyclicity one edge at a time, explicitly checking for the acyclicity constraint when an edge is added.",
"GES is known to find global minimizer with infinite samples under suitable assumptions [7], [23], but this is not guaranteed in the finite sample regime.",
"There are hybrid methods, e.g., the max-min hill climbing method [40], which use constraint-based approaches to reduce the search space before applying score-based methods.",
"However, this methodology generally lacks a principled way of choosing a problem-specific combination of score functions and search strategies.",
"Recently, [49] introduced a smooth characterization for the acyclicity.",
"With linear models, Problem (REF ) was then formulated as a continuous optimization problem w.r.t.",
"the weighted graph adjacency matrix by picking a proper loss function, e.g., the least squares loss.",
"Subsequent works [46] and [20] have also adopted the evidence lower bound and the negative log-likelihood as loss functions, respectively, and used Neural Networks (NNs) to model the causal relationships.",
"Note that the loss functions in these methods must be carefully chosen in order to apply continuous optimization methods.",
"Unfortunately, many effective score functions, e.g., the generalized score function proposed by [15] and the independence based score function from [27], either cannot be represented in closed forms or have very complicated equivalent loss functions, and thus cannot be easily combined with this approach.",
"We propose to use Reinforcement Learning (RL) to search for the DAG with the best score according to a predefined score function, as outlined in Figure REF .",
"The insight is that an RL agent with stochastic policy can determine automatically where to search given the uncertainty information of the learned policy, which can be updated promptly by the stream of reward signals.",
"To apply RL to causal discovery, we use an encoder-decoder NN model to generate directed graphs from the observed data, which are then used to compute rewards consisting of the predefined score function as well as two penalty terms to enforce acyclicity.",
"We resort to policy gradient and stochastic optimization methods to train the weights of the NNs, and our output is the graph that achieves the best reward, among all graphs generated in the training process.",
"Experiments on both synthetic and real datasets show that our approach has a much improved search ability without sacrificing any flexibility in choosing score functions.",
"In particular, the proposed approach with BIC score outperforms GES with the same score function on Linear Non-Gaussian Acyclic Model (LiNGAM) and linear-Gaussian datasets, and also outperforms recent gradient based methods when the causal relationships are nonlinear.",
"Figure: Reinforcement learning for score-based causal discovery."
],
[
"Related Work",
"Constraint-based causal discovery methods first use conditional independence tests to find causal skeleton and then determine the orientations of the edges up to the Markov equivalence class, which usually contains DAGs that can be structurally diverse and may still have many unoriented edges.",
"Examples include [37], [48] that use kernel-based conditional independence criteria and the well-known PC algorithm [36].",
"This class of methods involve a multiple testing problem where the tests are usually conducted independently.",
"The testing results may have conflicts and handling them is not easy, though there are certain works, e.g., [16], attempting to tackle this problem.",
"These methods are also not robust as small errors in building the graph skeleton can result in large errors in the inferred Markov equivalence class.",
"Another class of causal discovery methods are based on properly defined functional causal models.",
"Unlike constraint-based methods that assume faithfulness and identify only the Markov equivalence class, these methods are able to distinguish between different DAGs in the same equivalence class, thanks to the additional assumptions on data distribution and/or functional classes.",
"Examples include LiNGAM [32], [33], the nonlinear additive noise model [14], [27], [28], and the post-nonlinear causal model [47].",
"Besides [46], [20], other recent NN based approaches to causal discovery include [11] that proposes causal generative NNs to functional causal modeling with a prior knowledge of initial skeleton of the causal graph and [17] that learns causal generative models in an adversarial way but does not guarantee acyclicity.",
"Recent advances in sequence-to-sequence learning [38] have motivated the use of NNs for optimization in various domains [43], [50], [5].",
"A particular example is the traveling salesman problem that was revisited in the work of pointer networks [43].",
"Authors proposed a recurrent NN with nonparametric softmaxes trained in a supervised manner to predict the sequence of visited cities.",
"[2] further proposed to use the RL paradigm to tackle the combinatorial problems due to their relatively simple reward mechanisms.",
"It was shown that an RL agent can have a better generalization even when the optimal solutions are used as labeled data in the previous supervised approach.",
"Alternatively, the RL based approach in [9] considered combinatorial optimization problems on (undirected) graphs and achieved a promising performance by exploiting graph structures, in contrast with the general sequence-to-sequence modeling.",
"There are many other successful RL applications in recent years, e.g., AlphaGo [34], where the goal is to learn a policy for a given task.",
"As an exception, [50] applied RL to neural architecture search.",
"While we use a similar idea as the RL paradigm can naturally include the search task, our work is different in the actor and reward designs: our actor is an encoder-decoder model that generates graph adjacency matrices (cf.",
"Section ) and the reward is tailored for causal discovery by incorporating a score function and the acyclicity constraint (cf.",
"Section REF )."
],
[
"Model Definition",
"We assume the following model for data generating procedure, as in [14], [27].",
"Each variable $x_i$ is associated with a node $i$ in a $d$ -node DAG $\\mathcal {G}$ , and the observed value of $x_i$ is obtained as a function of its parents in the graph plus an independent additive noise $n_i$ , i.e., $x_i f_i(\\mathbf {x}_{\\mathrm {pa}(i)})+n_i,i=1,2,\\dots , d,$ where $\\mathbf {x}_{\\mathrm {pa}(i)}$ denotes the set of variables $x_j$ so that there is an edge from $x_j$ to $x_i$ in the graph, and the noises $n_i$ are assumed to be jointly independent.",
"We also assume causal minimality, which in this case reduces to that each function $f_i$ is not a constant in any of its arguments [27].",
"Without further assumption on the forms of functions and/or noises, the above model can be identified only up to Markov equivalence class under the usual Markov and faithful assumptions [36], [27]; in our experiments we will consider synthetic datasets that are generated from fully identifiable models so that it is practically meaningful to evaluate the estimated graph w.r.t.",
"the true DAG.",
"If all the functions $f_i$ are linear and the noises $n_i$ are Gaussian distributed, the above model yields the class of standard linear-Gaussian model that has been studied in [3], [10], [36], [28].",
"When the functions are linear but the noises are non-Gaussian, one can obtain the LiNGAM described in [32], [33] and the true DAG can be uniquely identified under favorable conditions.",
"In this paper, we consider that all the variables $x_i$ are scalars; extending to more complex cases is straightforward, provided with a properly defined score function.",
"The observed data $\\mathbf {X}$ , consisting of a number of vectors $\\mathbf {x}[x_1,x_2,\\ldots , x_d]^T\\in \\mathbb {R}^d$ , are then sampled independently according to the above model on an unknown DAG, with fixed functions $f_i$ and fixed distributions for $n_i$ .",
"The objective of causal discovery is to use the observed data $\\mathbf {X}$ , which gives the empirical version of the joint distribution of $\\mathbf {x}$ , to infer the underlying causal DAG $\\mathcal {G}$ ."
],
[
"Neural Network Architecture for Graph Generation",
"Given a dataset $\\mathbf {X}=\\lbrace \\mathbf {x}^k\\rbrace _{k=1}^m$ where $\\mathbf {x}^k$ denotes the $k$ -th observed sample, we want to infer the causal graph that best describes the data generating procedure.",
"We would like to use NNs to infer the causal graph from the observed data; specifically, we aim to design an NN based graph generator whose input is the observed data and the output is a graph adjacency matrix.",
"A naive choice would be using feed-forward NNs to output $d^2$ scalars and then reshape them to an adjacency matrix in $\\mathbb {R}^{d\\times d}$ .",
"However, this NN structure failed to produce promising results, possibly because the feed-forward NNs could not provide sufficient interactions amongst variables to capture the causal relations.",
"Motivated by recent advances in neural combinatorial optimization, particularly the pointer networks [2], [43], we draw $n$ random samples (with replacement) $\\lbrace \\mathbf {x}^{l}\\rbrace _{l=1}^n$ from $\\mathbf {X}$ and reshape them as $\\mathbf {s} \\lbrace \\tilde{\\mathbf {x}}_{i}\\rbrace _{i=1}^d$ where $\\tilde{\\mathbf {x}}_i\\in \\mathbb {R}^n$ is the vector concatenating all the $i$ -th entries of the vectors in $\\lbrace {\\mathbf {x}}^{l}\\rbrace _{l=1}^n$ .",
"In an analogy to the traveling salesman problem, this represents a sequence of $d$ cities lying in an $n$ -dim space.",
"We are concerned with generating a binary adjacency matrix $A \\in \\lbrace 0,1\\rbrace ^{d\\times d}$ so that the corresponding graph is acyclic and achieves the best score.",
"In this work we consider encoder-decoder models for graph generation: Encoder We use the attention based encoder in the Transformer structure proposed by [41].",
"We believe that the self-attention scheme, together with structural DAG constraint, is capable of finding the causal relations amongst variables.",
"Other attention based models such as graph attention network [42] may also be used, which will be considered in a future work.",
"Denote the outputs of the encoder by $enc_i,i=1,2,\\ldots ,d$ , with dimension $d_e$ .",
"Decoder Our decoder generates the graph adjacency matrix in an element-wise manner, by building relationships between two encoder outputs $enc_i$ and $enc_j$ .",
"We consider the single layer decoder $g_{ij}(W_1,W_2,u) = u^{T} \\tanh (W_1\\,enc_i+W_2\\,enc_j),\\nonumber $ where $W_1,W_2\\in \\mathbb {R}^{d_h\\times d_{e}}$ , $u\\in \\mathbb {R}^{d_h\\times 1}$ are trainable parameters and $d_h$ is the hidden dimension associated with the decoder.",
"To generate a binary adjacency matrix $A$ , we pass each entry $g_{ij}$ into a logistic sigmoid function $\\sigma (\\cdot )$ and then sample according to a Bernoulli distribution with probability $\\sigma (g_{ij})$ that indicates the probability of existing an edge from $x_i$ to $x_j$ .",
"To avoid self-loops, we simply mask the $(i,i)$ -th entry in the adjacency matrix.",
"Other decoder choices include the neural tensor network model [35] and the bilinear model that build the pairwise relationships between encoder outputs.",
"Another choice is the Transformer decoder which can generate an adjacency matrix in a row-wise manner.",
"Empirically, we find that the single layer decoder performs the best, possibly because it contains less parameters and is easier to train to find better DAGs while the self-attention based encoder has provided sufficient interactions amongst the variables for causal discovery.",
"Appendix provides more details regarding these decoders and their empirical results with linear-Gaussian data models."
],
[
"Reinforcement Learning for Search",
"In this section, we use RL as our search strategy to find the DAG with the best score, as outlined in Figure REF .",
"As one will see, the proposed method possesses an improved search ability over traditional score-based methods and also allows flexible score functions subject to the acyclicity constraint."
],
[
"Score Function, Acyclicity, and Reward",
"Score Function In this work, we consider only existing score functions to construct the reward that will be maximized by an RL agent.",
"Often score-based methods assume a parametric model for causal relationships (e.g., linear-Gaussian equations or multinomial distribution), which introduces a set of parameters $\\mathbf {\\theta }$ .",
"Among all score functions that can be directly included here, we focus on the BIC score that is not only consistent [13] but also locally consistent for its decomposability [6].",
"The BIC score for a given directed graph $\\mathcal {G}$ is $\\mathcal {S}_{\\textrm {BIC}}(\\mathcal {G})=-2\\log p(\\mathbf {X}; \\hat{\\theta },\\mathcal {G})+d_{\\theta }\\log m,$ where $\\hat{\\theta }$ is the maximum likelihood estimator and $d_{\\theta }$ denotes the dimensionality of the parameter $\\theta $ .",
"We assume i.i.d.",
"Gaussian additive noises throughout this paper.",
"If we apply linear models to each causal relationship and let $\\hat{x}^k_i$ be the corresponding estimate for $x_i^k$ , the $i$ -th entry in the $k$ -th observed sample, then we have the BIC score being (up to some additive constant) $\\mathcal {S}_{\\textrm {BIC}}(\\mathcal {G})=\\sum _{i=1}^d\\big (m\\log (\\mathrm {RSS}_i/m)\\big )+\\#(\\text{edges})\\log m,$ where $\\mathrm {RSS}_i=\\sum _{k=1}^m(x^k_i-\\hat{x}^k_i)^2$ denotes the residual sum of squares for the $i$ -th variable.",
"The first term in Eq.",
"(REF ) is equivalent to the log-likelihood objective used by GraN-DAG [20] and the second term adds penalty on the number of edges in the graph $\\mathcal {G}$ .",
"Further assuming that the noise variances are equal (despite the fact that they may be different), we have $\\mathcal {S}_{\\textrm {BIC}}(\\mathcal {G})=md\\log \\left(\\left(\\sum _{i=1}^d\\mathrm {RSS}_i\\right)/(md)\\right)+\\#(\\text{edges})\\log m.$ We notice that $\\sum _{i}\\mathrm {RSS}_i$ is the least squares loss used in NOTEARS [49].",
"Besides assuming linear models, other regression methods can also be used to estimate $x_i^k$ .",
"In Section , we will use quadratic regression and Gaussian Process Regression (GPR) to model causal relationships based on the observed data.",
"Acyclicity A remaining issue is the acyclicity constraint.",
"Other than GES that explicitly checks for acyclicity each time an edge is added, we add penalty terms w.r.t.",
"acyclicity to the score function to enforce acyclicity in an implicit manner and allow the generated graph to change more than one edges at each iteration.",
"In this work, we use a recent result from [49]: a directed graph $\\mathcal {G}$ with binary adjacency matrix $A$ is acyclic if and only if $h(A)\\operatorname{trace}\\left(e^A\\right)-d=0,$ where $e^A$ is the matrix exponential of $A$ .",
"We find that $h(A)$ , which is non-negative, can be small for cyclic graphs and the minimum over all non-DAGs is not easy to find.",
"We would require a very large penalty weight to guarantee acyclicity if only $h(A)$ is used.",
"We thus add another penalty term, the indicator function w.r.t.",
"acyclicity, to induce exact DAGs.",
"We remark that other functions (e.g., the total length of all cyclic paths in the graph), which compute some `distance' from a directed graph to $\\mathsf {DAGs}$ and need not be smooth, may also be used to construct the acyclicity penalty in our approach.",
"Reward Our reward incorporates both the score function and the acyclicity constraint: $\\mathrm {reward}-\\left[\\mathcal {S}(\\mathcal {G}) + \\lambda _1\\mathbf {I}(\\mathcal {G} \\notin \\mathsf {DAGs}) + \\lambda _2h(A)\\right],$ where $\\mathbf {I}(\\cdot )$ denotes the indicator function and $\\lambda _1,\\lambda _2\\ge 0$ are two penalty parameters.",
"It is not hard to see that the larger $\\lambda _1$ and $\\lambda _2$ are, the more likely a generated graph with a high reward is acyclic.",
"We then aim to maximize the reward over all possible directed graphs, or equivalently, we have $\\min _{\\mathcal {G}}~\\left[\\mathcal {S}(\\mathcal {G}) + \\lambda _1\\mathbf {I}(\\mathcal {G} \\notin \\mathsf {DAGs}) + \\lambda _2h(A)\\right].$ An interesting question is whether this new formulation is equivalent to the original problem with hard acyclicity constraint.",
"Fortunately, the following proposition guarantees that Problems (REF ) and (REF ) are equivalent with properly chosen $\\lambda _1$ and $\\lambda _2$ , which can be verified by showing that a minimizer of one problem is also a solution to the other.",
"A proof is provided in Appendix for completeness.",
"Proposition 1 Let $h_{\\min }>0$ be the minimum of $h(A)$ over all directed cyclic graphs, i.e., $h_{\\min } = \\min _{\\,\\mathcal {G} \\notin \\mathsf {DAGs}} h(A)$ .",
"Let $\\mathcal {S}^*$ denote the optimal score achieved by some DAG in Problem (REF ).",
"Assume that $\\mathcal {S}_L\\in \\mathbb {R}$ is a lower bound of the score function over all possible directed graphs, i.e., $S_L\\le \\min _{\\,\\mathcal {G}} \\mathcal {S}(\\mathcal {G})$ , and $S_U\\in \\mathbb {R}$ is an upper bound on the optimal score with $\\mathcal {S}^*\\le \\mathcal {S}_U$ .",
"Then Problems (REF ) and (REF ) are equivalent if $\\lambda _1 + \\lambda _2h_{\\min } \\ge \\mathcal {S}_U-\\mathcal {S}_L.$ For practical use, we need to find respective quantities in order to choose proper penalty parameters.",
"An upper bound $\\mathcal {S}_U$ can be easily found by drawing some random DAGs or using the results from other methods like NOTEARS.",
"A lower bound $\\mathcal {S}_L$ depends on the particular score function.",
"With BIC score, we can fit each variable $x_i$ against all the rest variables, and use only the $\\mathrm {RSS}_i$ terms but ignore the additive penalty on the number of edges.",
"With the independence based score function proposed by [27], we may simply set $\\mathcal {S}_L=0$ .",
"The minimum term $h_{\\min }$ , as previously mentioned, may not be easy to find.",
"Fortunately, with $\\lambda _1=\\mathcal {S}_U-\\mathcal {S}_L$ , Proposition REF guarantees the equivalence of Problems (REF ) and (REF ) for any $\\lambda _2\\ge 0$ .",
"However, simply setting $\\lambda _2=0$ could only get good performance with very small graphs (see a discussion in Appendix ).",
"We will pick a relatively small value for $\\lambda _2$ , which helps to generate directed graphs that become closer to $\\mathsf {DAGs}$ .",
"Empirically, we find that if the initial penalty weights are set too large, the score function would have little effect on the reward, which then limits the exploration of the RL agent and usually results in DAGs with high scores.",
"Similar to Lagrangian methods, we can start with small penalty weights and gradually increase them so that the condition in Proposition REF is satisfied.",
"Meanwhile, we notice that different score functions may have different ranges while the acyclicity penalty terms are independent of the particular range of the score function.",
"We hence also adjust the predefined scores to a certain range by using $\\mathcal {S}_{0}(\\mathcal {S}-\\mathcal {S}_L)/(\\mathcal {S}_U-\\mathcal {S}_L)$ for some $\\mathcal {S}_{0}>0$ and the optimal score will lie in $[0, \\mathcal {S}_0]$ .When $\\mathcal {S}_U-\\mathcal {S}_L=0$ , then we have obtained the solution if we know the graph that achieves $\\mathcal {S}_U$ or $\\mathcal {S}_L$ , or otherwise we may simply pick a new upper bound as $\\mathcal {S}_U+1$ .",
"Our algorithm is summarized in Algorithm REF , where $\\Delta _1$ and $\\Delta _2$ are the updating parameters associated with $\\lambda _1$ and $\\lambda _2$ , respectively, and $t_0$ denotes the updating frequency.",
"The weight $\\lambda _2$ is updated in a similar manner to the updating rule on the Lagrange multiplier used by NOTEARS and we set $\\Lambda _2$ as an upper bound on $\\lambda _2$ , as previously discussed.",
"In all our experiments that use BIC as score function, $\\mathcal {S}_L$ is obtained from a complete directed graph and $\\mathcal {S}_U$ is from an empty graph.",
"Since $\\mathcal {S}_U$ with the empty graph can be very high for large graphs, we also update it by keeping track of the lowest score achieved by DAGs generated during training.",
"Other parameter choices in this work are $\\mathcal {S}_0=5$ , $t_0=1,000$ , $\\lambda _1=0$ , $\\Delta _1=1$ , $\\lambda _2=10^{-\\lceil d/3\\rceil }$ , $\\Delta _2=10$ and $\\Lambda _2=0.01$ .",
"We comment that these parameter choices may be further tuned for specific applications, and the inferred causal graph would be the one that is acyclic and achieves the best score, among all the final outputs (cf.",
"Section REF ) of the RL approach with different parameter choices.",
"[t] The proposed RL approach to score-based causal discovery [1] score parameters: $\\mathcal {S}_L$ , $\\mathcal {S}_U$ , and $\\mathcal {S}_{0}$ ; penalty parameters: $\\lambda _1$ , $\\Delta _1$ , $\\lambda _2$ , $\\Delta _2$ , and $\\Lambda _2$ ; iteration number for parameter update: $t_0$ .",
"$t=1,2,\\ldots $ Run actor-critic algorithm, with score adjustment by $\\mathcal {S}\\leftarrow \\mathcal {S}_{0}(\\mathcal {S}-\\mathcal {S}_L)/(\\mathcal {S}_U-\\mathcal {S}_L)$ $t\\pmod {t_0}=0$ the maximum reward corresponds to a DAG with score $\\mathcal {S}_{\\min }$ update $\\mathcal {S}_U\\leftarrow \\min (\\mathcal {S}_U,\\mathcal {S}_{\\min })$ update $\\lambda _1\\leftarrow \\min (\\lambda _1+\\Delta _1,\\mathcal {S}_U)$ and $\\lambda _2\\leftarrow \\min (\\lambda _2\\Delta _2, \\Lambda _2)$ update recorded rewards according to new $\\lambda _1$ and $\\lambda _2$"
],
[
"Actor-Critic Algorithm",
"We believe that the exploitation and exploration scheme in the RL paradigm provides an appropriate way to guide the search.",
"Let $\\pi (\\cdot \\mid \\mathbf {s})$ and $\\psi $ denote the policy and NN parameters for graph generation, respectively.",
"Our training objective is the expected reward defined as $J(\\mathbf {\\psi }\\mid \\mathbf {s}) = \\mathbb {E}_{A\\sim \\pi (\\cdot \\mid \\mathbf {s})}\\left\\lbrace -\\left[\\mathcal {S}(\\mathcal {G}) + \\lambda _1\\mathbf {I}(\\mathcal {G} \\notin \\mathsf {DAGs}) + \\lambda _2h(A)\\right]\\right\\rbrace .$ During training, the input $\\mathbf {s}$ is constructed by randomly drawing samples from the observed dataset $\\mathbf {X}$ , as described in Section .",
"We resort to policy gradient methods and stochastic methods to optimize the parameters $\\psi $ .",
"The gradient $\\nabla _{\\psi } J(\\psi \\mid \\mathbf {s})$ can be obtained by the well-known REINFORCE algorithm [44], [39].",
"We draw $B$ samples $\\mathbf {s}_1, \\mathbf {s}_2, \\ldots ,\\mathbf {s}_B$ as a batch to estimate the gradient which is then used to train the NNs through stochastic optimization methods like Adam [18].",
"Using a parametric baseline to estimate the reward can also help training [19].",
"For the present work, our critic is a simple 2-layer feed-forward NN with ReLU units, with the input being the encoder outputs $\\lbrace enc_i\\rbrace _{i=1}^d$ .",
"The critic is trained with Adam on a mean squared error between its predictions and the true rewards.",
"An entropy regularization term [45], [22] is also added to encourage exploration of the RL agent.",
"Although policy gradient methods only guarantee local convergence under proper conditions [39], we remark that the inferred graphs from the actor-critic algorithm are all DAGs in our experiments.",
"Training an RL agent typically requires many iterations.",
"In the present work, we find that computing the rewards for generated graphs is much more time-consuming than training NNs.",
"Therefore, we record the computed rewards corresponding to different graph structures.",
"Moreover, the BIC score can be decomposed according to single causal relationships and we also record the corresponding $\\mathrm {RSS}_i$ to avoid repeated computations."
],
[
"Final Output",
"Since we are concerned with finding a DAG with the best score rather than a policy, we record all the graphs generated during the training process and output the one with the best reward.",
"In practice, the graph may contain spurious edges and further processing is needed.",
"To this end, we can prune the estimated edges in a greedy way, according to either the regression performance or the score function.",
"For an inferred causal relationship, we remove a parental variable and calculate the performance of the resulting graph, with all other causal relationships unchanged.",
"If the performance does not degrade or degrade within a predefined tolerance, we accept pruning and continue this process with the pruned causal relationship.",
"For linear models, pruning can be simply done by thresholding the estimated coefficients.",
"Related to the above pruning process is to add to the score function an increased penalty weight on the number of edges of a graph.",
"However, this weight is not easy to choose, as a large weight may incur missing edges.",
"In this work, we stick to the penalty weight $\\log m$ that is included in the BIC score and then apply pruning to the inferred graph in order to reduce false discoveries."
],
[
"Experimental Results",
"We report empirical results on synthetic and real datasets to compare our approach against both traditional and recent gradient based approaches, including GES (with BIC score) [7], [29], the PC algorithm (with Fisher-z test and $p$ -value $0.01$ ) [36], ICA-LiNGAM [32], the Causal Additive Model (CAM) based algorithm proposed by [4], NOTEARS [49], DAG-GNN [46], and GraN-DAG [20], among others.",
"All these algorithms have available implementations and we give a brief description on these algorithms and their implementations in Appendix .",
"Default hyper-parameters of these implementations are used unless otherwise stated.",
"For pruning, we use the same thresholding method for ICA-LiNGAM, NOTEARS, and DAG-GNN.",
"Since the authors of CAM and GraN-DAG propose to apply significance testing of covariates based on generalized additive models and then declare significance if the reported $p$ -values are lower than or equal to $0.001$ , we stick to the same pruning method for CAM and GraN-DAG.",
"The proposed RL based approach is implemented based on an existing Tensorflow [1] implementation of neural combinatorial optimizer (see Appendix for more details).",
"The decoder is modified as described in Section and the RL algorithm related hyper-parameters are left unchanged.",
"We pick $B=64$ as batch size at each iteration and $d_h=16$ as the hidden dimension with the single layer decoder.",
"Our approach is combined with the BIC scores under Gaussianity assumption given in Eqs.",
"(REF ) and (REF ), and are denoted as RL-BIC and RL-BIC2, respectively.",
"We evaluate the estimated graphs using three metrics: False Discovery Rate (FDR), True Positive Rate (TPR), and Structural Hamming Distance (SHD) which is the smallest number of edge additions, deletions, and reversals to convert the estimated graph into the true DAG.",
"The SHD takes into account both false positives and false negatives and a lower SHD indicates a better estimate of the causal graph.",
"Since GES and PC may output unoriented edges, we follow [49] to treat GES and PC favorably by regarding undirected edges as true positives as long as the true graph has a directed edge in place of the undirected edge."
],
[
"Linear Model with Gaussian and Non-Gaussian Noise",
"Given number of variables $d$ , we generate a $d\\times d$ upper triangular matrix as the graph binary adjacency matrix, in which the upper entries are sampled independently from $\\mathrm {Bern}(0.5)$ .",
"We assign edge weights independently from $\\mathrm {Unif}\\left([-2, -0.5]\\cup [0.5,2]\\right)$ to obtain a weight matrix $W\\in \\mathbb {R}^{d\\times d}$ , and then sample $\\mathbf {x} = W^T\\mathbf {x} + \\mathbf {n}\\in \\mathbb {R}^d$ from both Gaussian and non-Gaussian noise models.",
"The non-Gaussian noise is the same as the one used for ICA-LiNGAM [32], which generates samples from a Gaussian distribution and passes them through a power nonlinearity to make them non-Gaussian.",
"We pick unit noise variances in both models and generate $m=5,000$ samples as our datasets.",
"A random permutation of variables is then performed.",
"This data generating procedure is similar to that used by NOTEARS and DAG-GNN and the true causal graphs in both cases are known to be identifiable [32], [26].",
"Figure: negative rewardTable: Empirical results on LiNGAM and linear-Gaussian data models with 12-node graphs.We first consider graphs with $d=12$ nodes.",
"We use $n=64$ for constructing the input sample and set the maximum number of iterations to $20,000$ .",
"We use a threshold $0.3$ , same as NOTEARS and DAG-GNN with this data model, to prune the estimated edges.",
"Figure REF shows the learning process of the proposed method RL-BIC2 on a linear-Gaussian dataset.",
"In this example, RL-BIC2 generates $683,784$ different graphs during training, much lower than the total number (around $5.22\\times 10^{26}$ ) of DAGs.",
"The pruned DAG turns out to be exactly the same as the underlying causal graph.",
"We report the empirical results on LiNGAM and linear-Gaussian data models in Table REF .",
"Both PC and GES perform poorly, possibly because we consider relatively dense graphs for our data generating procedure.",
"CAM does not perform well either, as it assumes nonlinear causal relationships.",
"ICA-LiNGAM recovers all the true causal graphs for LiNGAM data but performs poorly on linear-Gaussian data.",
"This is not surprising because ICA-LiNGAM works for non-Gaussian noise and does not provide guarantee for linear-Gaussian datasets.",
"Both NOTEARS and DAG-GNN have good causal discovery results whereas GraN-DAG performs much worse.",
"We believe that it is because GraN-DAG uses 2-layer feed-forward NNs to model the causal relationships, which may not be able to learn a good linear relationship in this experiment.",
"Modifying the feed-forward NNs to linear functions reduces to NOTEARS with negative log-likelihood as loss function, which yields similar performance on these datasets (see Appendix REF for detailed results).",
"As to our proposed methods, we observe that RL-BIC2 recovers all the true causal graphs on both data models in this experiment while RL-BIC has a worse performance.",
"One may wonder whether this observation is due to the same noise variances that are used in our data models; we conduct additional experiments where the noise variances are randomly sampled and RL-BIC2 still outperforms RL-BIC by a large margin (see also Appendix REF ).",
"Nevertheless, with the same BIC score, RL-BIC performs much better than GES on both datasets, indicating that the RL approach brings in a greatly improved search ability.",
"Finally, we test the proposed method on larger graphs with $d=30$ nodes, where the upper entries are sampled independently from $\\mathrm {Bern}(0.2)$ .",
"This edge probability choice corresponds to the fact that large graphs usually have low edge degrees in practice; see, e.g., the experiment settings of [49], [46], [20].",
"To incorporate this prior information in our approach, we add to each $g_{ij}$ a common bias term initialized to $-10$ (see Appendix REF for details).",
"Considering the much increased search space, we also choose a larger number of observed samples, $n=128$ , to construct the input for graph generator and increase the training iterations to $40,000$ .",
"On LiNGAM datasets, RL-BIC2 has FDR, TPR, and SHD being $0.14\\,\\pm \\,0.15$ , $0.94\\pm 0.07$ , and $19.8\\pm 23.0$ , respectively, comparable to NOTEARS with $0.13\\pm 0.09$ , $0.94\\pm 0.04$ , and $17.2\\pm 13.12$ ."
],
[
"Nonlinear Model with Quadratic Functions",
"We now consider nonlinear causal relationships with quadratic functions.",
"We generate an upper triangular matrix in a similar way to the first experiment.",
"For a causal relationship with parents $\\mathbf {x}_{\\mathrm {pa}(i)}=[x_{i_1},x_{i_2},\\ldots ]^T$ at the $i$ -th node, we expand $\\mathbf {x}_{\\mathrm {pa}(i)}$ to contain both first- and second-order features.",
"The coefficient for each term is then either 0 or sampled from $\\mathrm {Unif}\\left([-1, -0.5]\\cup [0.5,1]\\right)$ , with equal probability.",
"If a parent variable does not appear in any feature term with a non-zero coefficient, then we remove the corresponding edge in the causal graph.",
"The rest follows the same as in first experiment and here we use the non-Gaussian noise model with 10-node graphs and $5,000$ samples.",
"The true causal graph is identifiable according to [27].",
"For this quadratic model, there may exist very large variable values which cause computation issues for quadratic regression.",
"We treat these samples as outliers and detailed processing is given in Appendix REF .",
"We use quadratic regression for a given causal relationship and calculate the BIC score (assuming equal noise variances) in Eq.",
"(REF ).",
"For pruning, we simply apply thresholding, with threshold as $0.3$ , to the estimated coefficients of both first- and second-order terms.",
"If the coefficient of a second-order term, e.g., $x_{i_1}x_{i_2}$ , is non-zero after thresholding, then we have two directed edges that are from $x_{i_1}$ to $x_i$ and from $x_{i_2}$ to $x_i$ , respectively.",
"We do not consider PC and GES in this experiment due to their poor performance in the first experiment.",
"Our results with 10-node graphs are reported in Table REF , which shows that RL-BIC2 achieves the best performance.",
"Table: Empirical results on nonlinear models with quadratic functions.For fair comparison, we apply the same quadratic regression based pruning method to the outputs of NOTEARS, denoted as NOTEARS-2.",
"We see that this pruning further reduces FDR, i.e., removes spurious edges, with little effect on TPR.",
"Since pruning does not help discover additional positive edges or increase TPR, we will not apply this pruning method to other methods as their TPRs are much lower than that of RL-BIC2.",
"Finally, with prior knowledge that the function form is quadratic, we can modify NOTEARS to apply quadratic functions to modeling the causal relationships, with an equivalent weighted adjacency matrix constructed using the coefficients of the first- and second-order terms, similar to the idea used by GraN-DAG (detailed derivations are given in Appendix REF ).",
"The problem then becomes a nonconvex optimization problem with $(d-1)d^2/2$ parameters (which are the coefficients of both first- and second-order features), compared to the original NOTEARS with $d^2$ parameters.",
"This method corresponds to NOTEARS-3 in Table REF .",
"Despite the fact that NOTEARS-3 did not achieve a better overall performance than RL-BIC2, we comment that it discovered almost correct causal graphs (with $\\text{SHD}\\le 2$ ) on more than half of the datasets, but performed poorly on the rest datasets.",
"We believe that it is due to the increased number of optimization parameters and the more complicated equivalent adjacency matrix which make the optimization problem harder to solve.",
"Meanwhile, we do not exclude that NOTEARS-3 can achieve a better causal discovery performance with other optimization algorithms."
],
[
"Nonlinear Model with Gaussian Processes",
"Given a randomly generated causal graph, we consider another nonlinear model where each causal relationship $f_i$ is a function sampled from a Gaussian process with RBF kernel of bandwidth one.",
"The additive noise $n_i$ is normally distributed with variance sampled uniformly.",
"This setting is known to be identifiable according to [27].",
"We use a setup that is also considered by GraN-DAG [20]: 10-node and 40-edge graphs with $1,000$ generated samples.",
"The empirical results are reported in Table REF .",
"One can see that ICA-LiNGAM, NOTEARS, and DAG-GNN perform poorly on this data model.",
"A possible reason is that they may not be able to model this type of causal relationship.",
"More importantly, these methods operate on a notion of weighted adjacency matrix, which is not obvious here.",
"For our method, we apply Gaussian Process Regression (GPR) with RBF kernel to model the causal relationships.",
"Notice that even though the observed data are from a function sampled from Gaussian process, it is not guaranteed that GPR with the same kernel can achieve a good performance.",
"Indeed, using a fixed kernel bandwidth would lead to severe overfitting that incurs many spurious edges and the graph with the highest reward is usually not a DAG.",
"To proceed, we normalize the observed data and apply median heuristics for kernel bandwidth.",
"Both our methods perform reasonably well, with RL-BIC outperforming all the other methods.",
"Table: Empirical results on nonlinear models with Gaussian processes."
],
[
"Real Data",
"We consider a real dataset to discover a protein signaling network based on expression levels of proteins and phospholipids [30].",
"This dataset is a common benchmark in graphical models, with experimental annotations well accepted by the biological community.",
"Both observational and interventional data are contained in this dataset.",
"Since we are interested in using observational data to infer causal mechanisms, we only consider the observational data with $m=853$ samples.",
"The ground truth causal graph given by [30] has 11 nodes and 17 edges.",
"Notice that the true graph is indeed sparse and an empty graph can have an SHD as low as 17.",
"Therefore, we report more detailed results regarding the estimated graph: number of total edges, number of correct edges, and the SHD.",
"Both PC and GES output too many unoriented edges, and we will not report their results here.",
"We apply GPR with RBF kernel to modeling the causal relationships, with the same data normalization and median heuristics for kernel bandwidth as in Section REF .",
"We also use CAM pruning on the inferred graph from the training process.",
"The empirical results are given in Table REF .",
"Both RL-BIC and RL-BIC2 achieve promising results, compared with other methods.",
"Table: Empirical results on Sachs dataset."
],
[
"Concluding Remarks and Future Works",
"We have proposed to use RL to search for the DAG with the optimal score.",
"Our reward is designed to incorporate a predefined score function and two penalty terms to enforce acyclicity.",
"We use the actor-critic algorithm as our RL algorithm, where the actor is constructed based on recently developed encoder-decoder models.",
"Experiments are conducted on both synthetic and real datasets to show the advantages of our method over other causal discovery methods.",
"We have also shown the effectiveness of the proposed method with 30-node graphs, yet dealing with large graphs (with more than 50 nodes) is still challenging.",
"Nevertheless, many real applications, like Sachs dataset [30], have a relatively small number of variables.",
"Furthermore, it is possible to decompose large causal discovery problems into smaller ones; see, e.g., [21].",
"Prior knowledge or constraint-based methods is also applicable to reduce the search space.",
"There are several future directions from the present work.",
"In our current implementation, computing scores is much more time consuming than training NNs.",
"We believe that developing a more efficient and effective score function will further improve the proposed approach.",
"Other powerful RL algorithms may also be used.",
"For example, the asynchronous advantage actor-critic algorithm has been shown to be effective in many applications [22], [50].",
"In addition, we observe that the total iteration numbers used in our experiments are usually more than needed (see, e.g., Figure REF ).",
"A proper early stopping criterion will be favored."
],
[
"Acknowledgments",
"The authors are grateful to the anonymous reviewers for valuable comments and suggestions.",
"The authors would also like to thank Prof. Jiji Zhang from Lingnan University, Dr. Yue Liu from Huawei Noah's Ark Lab, and Zhuangyan Fang from Peking University for many helpful discussions."
],
[
"More Details About Decoders",
"We briefly describe the NN based decoders for generating binary adjacency matrices: Single layer decoder: $g_{ij}(W_1,W_2,u) = u^{T} \\tanh (W_1\\,enc_i+W_2\\,enc_j),\\nonumber $ where $W_1,W_2\\in \\mathbb {R}^{d_h\\times d_{e}}$ , $u\\in \\mathbb {R}^{d_h\\times 1}$ are trainable parameters and $d_h$ denotes the hidden dimension associated with the decoder.",
"Bilinear decoder: $g_{ij}(W) = enc_i^TW enc_j,\\nonumber $ where $W\\in \\mathbb {R}^{d_{e}\\times d_{e}}$ is a trainable parameter.",
"Neural Tensor Network (NTN) decoder [35]: $g_{ij}(W^{[1:K]}, V, b) = u^T\\tanh \\left(enc_i^TW^{[1:K]}enc_j+V [enc_i^T, enc_j^T]^T+b\\right),\\nonumber $ where $W^{[1:k]}\\in \\mathbb {R}^{d_{e}\\times d_{e}\\times K}$ is a tensor and the bilinear tensor product $enc_i^TW^{[1:K]}enc_j$ results in a vector with each entry being $enc_i^TW^{[k]}enc_j$ for $k=1,2,\\ldots , K$ , $V\\in \\mathbb {R}^{K\\times 2d_{e}}$ , $u\\in \\mathbb {R}^{K\\times 1}$ , and $b\\in \\mathbb {R}^{K\\times 1}$ .",
"Transformer decoder uses the multi-head attention module to obtain the decoder outputs $\\lbrace dec_i\\rbrace $ , followed by a feed-forward NN whose weights are shared across all $dec_i$ [41].",
"The output consists of $d$ vectors in $\\mathbb {R}^d$ which are treated as the row vectors of a $d\\times d$ matrix.",
"We then pass each element of this matrix into sigmoid functions and sample a binary adjacency matrix accordingly.",
"Table REF provides the empirical results on linear-Gaussian data models with 12-node graphs and unit variances (see Section REF for more details on this data generating procedure).",
"In our implementation, we pick $d_{e}=64$ as the dimension of the encoder output, $d_h=16$ for the single layer decoder, and $K=2$ for the NTN decoder.",
"We find that single layer decoder performs the best, possibly because it has less parameters and is easier to train to find better DAGs, while the Transformer encoder has provided sufficient interactions amongst variables.",
"Table: Empirical results of different decoders on linear-Gaussian data models with 12-node graphs."
],
[
"Equivalence of Problems (",
"[Proof of Proposition REF ] Let $\\mathcal {G}$ be a solution to Problem (REF ).",
"Then we have $\\mathcal {S}^*=\\mathcal {S}(\\mathcal {G})$ and $\\mathcal {G}$ must be a DAG due to the hard acyclicity constraint.",
"Assume that $\\mathcal {G}$ is not a solution to Problem (REF ), which indicates that there exists a directed graph $\\mathcal {G}^{\\prime }$ (with binary adjacency matrix $A^{\\prime }$ ) so that $\\mathcal {S}^* > \\mathcal {S}(\\mathcal {G}^{\\prime }) + \\lambda _1\\mathbf {I}(\\mathcal {G}^{\\prime } \\notin \\mathsf {DAGs}) + \\lambda _2 h(A^{\\prime }).$ Clearly, $\\mathcal {G}^{\\prime }$ cannot be a DAG, for otherwise we would have a DAG that achieves a lower score than the minimum $\\mathcal {S}^*$ .",
"By our assumption, it follows that $\\text{r.h.s.~of Eq.~(\\ref {eqn:propo1})}\\ge \\mathcal {S}_L+\\lambda _1 +\\lambda _2 h_{\\min }\\ge \\mathcal {S}_U,$ which contradicts the fact that $\\mathcal {S}_U$ is an upper bound on $\\mathcal {S}^*$ .",
"For the other direction, let $\\mathcal {G}$ be a solution to Problem (REF ) but not a solution to Problem (REF ).",
"This indicates that either $\\mathcal {G}$ is not a DAG or $\\mathcal {G}$ is a DAG but has a higher score than the minimum score, i.e., $\\mathcal {S}(\\mathcal {G})>\\mathcal {S}^*$ .",
"The latter case clearly contradicts the definition of the minimum score.",
"For the former case, assume that some DAG $\\mathcal {G}^{\\prime }$ achieves the minimum score.",
"Then plugging $\\mathcal {G}^{\\prime }$ into the negative reward, we can get the same inequality in Eq.",
"(REF ) since both penalty terms are zeros for a DAG.",
"This then contradicts the assumption that $\\mathcal {G}$ minimizes the negative reward."
],
[
"Penalty Weight Choice",
"Although setting $\\lambda _2=0$ , or equivalently using only the indicator function w.r.t.",
"acyclicity, can still make Problem (REF ) equivalent to the original problem with hard acyclicity constraint, we remark that this choice usually does not result in good performance of the RL approach, largely due to that the reward with only the indicator term is likely to fail to guide the RL agent to generate DAGs.",
"To see why it is the case, consider two cyclic directed graphs, one with all the possible directed edges in place and the other with only two edges (i.e., $x_i\\rightarrow x_j$ and $x_j\\rightarrow x_i$ for some $i\\ne j$ ).",
"The latter is much `closer' to acyclicity in many senses, such as $h(A)$ given in Eq.",
"(REF ) and number of edge deletion, addition, and reversal to make a directed graph acyclic.",
"Assume a linear data model that has a relatively dense graph.",
"Then the former graph will have a lower BIC score when using linear regression for fitting causal relations, yet the penalty terms of acyclicity are the same with only the indicator function.",
"The former graph then has a higher reward, which does not help the agent to tend to generate DAGs.",
"Also notice that the graphs in our approach are generated according to Bernoulli distributions determined by NN parameters that are randomly initialized.",
"Without loss of generality, consider that each edge is drawn independently according to $\\mathrm {Bern}(0.5)$ .",
"For small graphs (with less than or equal to 6 nodes or so), a few hundreds of samples of directed graphs are very likely to contain a DAG.",
"Yet for large graphs, the probability of sampling a DAG is much lower.",
"If no DAG is generated during training, the RL agent can hardly learn to generate DAGs.",
"The above facts indeed motivate us to choose a small value of $\\lambda _2$ so that the agent can be trained to produce graphs closer to acyclicity and finally to generate exact DAGs.",
"A question is then what if the RL approach starts with a DAG, e.g., by initializing the probability of generating each edge to be nearly zero.",
"This setting did not lead to good performance, either.",
"The generated directed graphs at early iterations can be very different from the true graphs in that many true edges do not exist, and the resulting score is much higher than the minimum under the DAG constraint.",
"With small penalty weights of the acyclicity terms, the agent could be trained to produce cyclic graphs with better scores, similar to the case with randomly initialized NN parameters.",
"On the other hand, large initial penalty weights, as we have discussed in the paper, limit exploration of the RL agent and usually result in DAGs whose scores are far from optimum."
],
[
"Implementation Details",
"We use existing implementations of causal discovery algorithms in comparison, listed below: ICA-LiNGAM [32] assumes linear non-Gaussian additive model for data generating procedure and applies Independent Component Analysis (ICA) to recover the weighted adjacency matrix, followed by thresholding on the weights before outputting the inferred graph.",
"A Python implementation is available at the first author's website https://sites.google.com/site/sshimizu06/lingam.",
"GES and PC: we use the fast greedy search implementation of GES [29] which has been reported to outperform other techniques such as max-min hill climbing [12], [49].",
"Implementations of both methods are available through the py-causal package at https://github.com/bd2kccd/py-causal, written in highly optimized Java codes.",
"CAM [27] decouples the causal order search among the variables from feature or edge selection in a DAG.",
"CAM also assumes additive noise as in our work, with an additional condition that each function is nonlinear.",
"Codes are available through the CRAN R package repository at https://cran.r-project.org/web/packages/CAM.",
"NOTEARS [49] recovers the causal graph by estimating the weighted adjacency matrix with the least squares loss and the smooth characterization for acyclicity constraint, followed by thresholding on the estimated weights.",
"Codes are available at the first author's github repository https://github.com/xunzheng/notears.",
"We also re-implement the augmented Lagrangian method following the same updating rule on the Lagrange multiplier and the penalty parameter in Tensorflow, so that the augmented Lagrangian at each iteration can be readily minimized without the need of obtaining closed-form gradients.",
"We use this implementation in Sections REF and REF when the objective function and/or the acyclicity constraint are modified.",
"DAG-GNN [46] formulates causal discovery in the framework of variational autoencoder, where the encoder and decoder are two shallow graph NNs.",
"With a modified smooth characterization on acyclicity, DAG-GNN optimizes a weighted adjacency matrix with the evidence lower bound as loss function.",
"Python codes are available at the first author's github repository https://github.com/fishmoon1234/DAG-GNN.",
"GraN-DAG [20] uses feed-foward NNs to model each causal relationship and chooses the sum of all product paths between variables $x_i$ and $x_j$ as the $(i,j)$ -th element of an equivalent weighted adjacency matrix.",
"GraN-DAG uses the same smooth constraint from [49] to find a DAG that maximizes the log-likelihood of the observed samples.",
"Codes are available at the first author's github repository https://github.com/kurowasan/GraN-DAG.",
"Our implementation is based on an existing Tensorflow implementation of neural combinatorial optimizer available at https://github.com/MichelDeudon/neural-combinatorial-optimization-rl-tensorflow.",
"We add an entropy regularization term, and modify the reward and decoder as described in Sections and REF , respectively.",
"Our codes have been made available at https://github.com/huawei-noah/trustworthyAI/tree/master/Causal_Structure_Learning/Causal_Discovery_RL."
],
[
"Experiment 1 in Section ",
"We replace the feed-forward NNs with linear functions in GraN-DAG and obtain similar experimental results as NOTEARS (FDR, TPR, SHD): $0.05\\pm 0.04$ , $0.93\\pm 0.06$ , $3.2\\pm 2.93$ and $0.05\\pm 0.04$ , $0.95\\pm 0.03$ , $2.40\\pm 1.85$ for LiNGAM and linear-Gaussian data models, respectively.",
"We conduct additional experiments with linear models where the noise variances are uniformly sampled according to $\\mathrm {Unif}([0.5, 2])$ .",
"Results are given in Table REF .Notice that linear-Gaussian data models with different noise variances are generally not identifiable.",
"It turns out that the Markov equivalence class is small and has at most 5 DAGs for the datasets considered here.",
"Moreover, the SHD between the DAGs in the Markov equivalence class and the true DAG is bounded by 3, across all the datasets.",
"Consequently, we may still use the true causal graph to evaluate the estimation performance.",
"We thank Sebastien Lachapelle from Mila for pointing this out.",
"Table: Empirical results on LiNGAM and linear-Gaussian data models with 12-node graphs and different noise variances.Knowing a sparse true causal graph a priori is also helpful.",
"To incorporate this information in our experiment with 30-node graphs, we add an additional biased term $\\tilde{c}\\in \\mathbb {R}$ to each decoder output: for the single layer decoder, we have $g_{ij}(W_1,W_2,u) = u^{T} \\tanh (W_1\\,enc_i+W_2\\,enc_j) + \\tilde{c},\\nonumber $ where we let $\\tilde{c}$ be trainable and other parameters have been defined in Appendix .",
"In our experiments, $\\tilde{c}$ is initialized to be $-10$ ; this choice aims to set a good starting point for generating graph adjacency matrices, motivated by the fact that a good starting point is usually helpful to locally convergent algorithms."
],
[
"Experiment 2 in Section ",
"To remove `outlier' samples with large values that may cause computation issues for quadratic regression, we sort the samples in ascending order according to their $\\ell _2$ -norms and then pick the first $3,000$ samples for causal discovery.",
"We use a similar idea from GraN-DAG to build an equivalent weighted adjacency matrix for NOTEARS.",
"Take the first variable $x_1$ for example.",
"We first expand the rest variables $x_2,\\ldots , x_d$ to contain both first- and second-order features: $x_2,\\ldots , x_d, x_2x_3,\\ldots , x_{i}x_j, \\ldots , x_{d-1}x_d$ with $i,j=2,\\ldots ,d$ and $i\\le j$ .",
"There are in total $d(d-1)/2$ terms and we use $\\mathbf {x}_1$ to denote the vector that concatenates these feature terms.",
"Correspondingly, we use $c_i$ and $c_{ij}$ to denote the coefficients associated with these features and $\\mathbf {c}_1$ to denote the concatenating vector of the coefficients.",
"Notice that the variable $x_l$ , $l\\ne 1$ affects $x_1$ only through the terms $x_l$ , $x_ix_l$ with $i\\le l$ , and $x_lx_j$ with $j>l$ .",
"Therefore, an equivalent weighted adjacency matrix $W$ lying in $\\mathbb {R}^{d\\times d}$ can be constructed with the $(l, 1)$ -th entry $W_{l1}|c_l|+\\sum _{i=2}^l|c_{il}|+\\sum _{j=l+1}^d|c_{lj}|$ ; in this way, $W_{l1}=0$ implies that $x_l$ has no effect on $x_1$ .",
"The least squares term, corresponding to variable $x_1$ , in the loss function will become $\\sum _{k=1}^m\\left(x_1^k-\\mathbf {c}_1^T\\mathbf {x}^k_1\\right)^2$ where $m$ is the total number of samples.",
"In summary, we have the following optimization problem $\\min _{\\mathbf {c}_1,\\mathbf {c}_2,\\ldots ,\\mathbf {c}_d} \\quad & \\sum _{i=1}^d\\sum _{k=1}^m\\left(x_i^k-\\mathbf {c}_i^T\\mathbf {x}^k_i\\right)^2\\nonumber \\\\\\text{subject to} \\quad & \\mathrm {trace}\\left(e^{W\\circ W}\\right) - d = 0,\\nonumber $ where $\\circ $ denotes the element-wise product and the constraint enforces acyclicity w.r.t.",
"a weighted adjacency matrix [49].",
"The problem has $(d-1)d^2/2$ parameters to optimize, while the original NOTEARS optimizes $d^2$ parameters.",
"We solve this problem with augmented Lagrangian method where at each iteration the augmented Lagrangian is approximately minimized by Adam [18] with Tensorflow.",
"The Lagrange multiplier and the penalty parameter are updated in the same fashion as in the original NOTEARS."
]
] | 1906.04477 |
[
[
"Multi-level quantum Rabi model for anharmonic vibrational polaritons"
],
[
"Abstract We propose a cavity QED approach to describe light-matter interaction between an individual anharmonic molecular vibration and an infrared cavity field.",
"Starting from a generic Morse oscillator with quantized nuclear motion, we derive a multi-level quantum Rabi model to study vibrational polaritons beyond the rotating-wave approximation.",
"We analyze the spectrum of vibrational polaritons in detail and compare with available experiments.",
"For high excitation energies, the spectrum exhibits a dense manifold of true and avoided level crossings as the light-matter coupling strength and cavity frequency are tuned.",
"These crossings are governed by a pseudo parity selection rule imposed by the cavity field.",
"We also analyze polariton eigenstates in nuclear coordinate space.",
"We show that the bond length of a vibrational polariton at a given energy is never greater than the bond length of a bare Morse oscillator with the same energy.",
"This type of bond hardening of vibrational polaritons occurs at the expense of the creation of virtual infrared cavity photons, and may have implications in chemical reactivity."
],
[
"Morse oscillator",
"We model the nuclear motion of an anharmonic polar vibration (e.g.",
"carbonyl) with a Morse potential [41] $V(q) = D_e\\left(1-{\\rm exp}[-a(q-q_e)]\\right)^2,$ where $D_e$ is the classical dissociation energy (without zero point motion), $q_e$ is the equilibrium bond length, and $a$ is a parameter.",
"The vibrational Schrödinger equation with a Morse potential can be solved analytically in terms of associated Laguerre polynomials [41], [42], or numerically using grid-based methods [43].",
"We use DVR on a uniform grid with Fourier basis functions [44] to obtain the vibrational wavefunctions and eigenvalues of the Morse potential.",
"For the dimensionless parameters $D_e=12.0$ , $q_e = 4.0$ , and $a=0.2041$ , the corresponding potential is shown in Fig.",
"REF a.",
"For the dimensionless mass $\\mu =1$ , this potential has 24 bound states and is used throughout.",
"In the Supplemental Material, we show that our conclusions do not vary qualitatively for other Morse potential parameters.",
"Figure: (a) Morse potential V(q)V(q), in units of the classical dissociation energy D e D_e.",
"We use D e =12D_e = 12, q e =4q_e = 4, a=0.204a=0.204 (dimensionless) to generate 24 bound states.",
"(b) Anharmonicity of the energy spacing between adjacent Morse eigenstates ΔE≡ω ν,ν-1 -ω 10 \\Delta E\\equiv \\omega _{\\nu ,\\nu -1}-\\omega _{10}.",
"We use D e =12.0D_e=12.0, q e =4.0q_e = 4.0 for all points, with μ=3\\mu =3 and a=0.204a=0.204 (circles), μ=1\\mu =1 and a=0.175a=0.175 (diamonds), and μ=1\\mu =1 and a=0.233a=0.233 (squares).",
"The dashed line is the harmonic oscillator limit (ΔE=0)(\\Delta E = 0).As we explain below, the degree of anharmonicity in the potential has a profound effect in the behaviour of vibrational polaritons.",
"For the Morse potential, the anharmonicity can be easily tuned by changing the parameters $a$ and $\\mu $ , for fixed binding energy $D_e$ .",
"The relation between these parameters and the degree of anharmonicity can be understood from the exact eigenvalues of the Morse Hamiltonian [41] $E_{\\nu } =-D_e+a \\hbar \\sqrt{\\frac{2D_e}{\\mu }}\\,(\\nu +1/2)-\\frac{a^2\\hbar ^2}{2D_e \\mu } (\\nu +1/2)^2$ where $\\nu $ is the vibrational quantum number.",
"By comparing this expression with the Dunham expansion [45] $E_{\\nu } &=&Y_{00} +\\omega _{0}(\\nu +1/2)-\\omega _0\\chi _{e} (\\nu +1/2)^{2}+\\ldots $ where $\\omega _{0}$ is the vibrational frequency, the anharmonic coefficient $\\chi _e$ can be written as $\\chi _{e} = \\frac{\\pi \\hbar ^2 a}{(2\\mu )^{1/2} D_e ^{3/2} }.$ Vibrations with lower $\\mu $ and higher $a$ therefore have stronger spectral anharmonicity, for fixed dissociation energy.",
"We illustrate this dependence in Fig.",
"REF b, where the change in the vibrational energy level spacing relative to the fundamental frequency $\\omega _{10}$ for $\\nu =1\\leftarrow \\nu =0$ is shown for different values of $\\mu $ and $a$ .",
"The level spacing between adjacent vibrational states can be significantly smaller than the harmonic oscillator value $\\omega _{10}$ , even for relatively low values of $\\nu $ .",
"Figure: (a) Rayleigh distribution model for the electric dipole function of a polar bond d(q)d(q) normalized to its maximum value.",
"(b) Permanent dipole moment matrix elements 〈ν|d e (q)|ν〉\\mathinner {\\langle {\\nu }|}d_e(q)\\mathinner {|{\\nu }\\rangle } as a function of the vibrational quantum number ν\\nu .",
"(c) Dipole matrix elements 〈ν|d e (q)|ν ' 〉\\mathinner {\\langle {\\nu }|}d_e(q)\\mathinner {|{\\nu ^{\\prime }}\\rangle } as a function of ν\\nu , for ν ' =0\\nu ^{\\prime }=0 (circles) and ν ' =5\\nu ^{\\prime } = 5 (squares).",
"Dipole matrix elements are normalized to the maximum of d(q)d(q).",
"The Morse parameters used are D e =12.0D_e=12.0, q e =4.0q_e = 4.0, a=0.204a=0.204 and μ=1\\mu =1.The electronic wavefunction determines the contribution to the molecular dipole moment of the electron charge distribution, which in the Born-Oppenheimer approximation is a parametric function $d(\\lbrace \\mathbf {q}\\rbrace )$ of all nuclear coordinates $\\lbrace \\mathbf {q}\\rbrace $ .",
"In general, the dipole function $d(\\lbrace \\mathbf {q}\\rbrace )$ can be obtained using ab-initio quantum chemistry for simple molecular species.",
"Since we are interested in understanding universal features of anharmonic vibrational polaritons, we adopt a model functional form $d(q)$ that captures the correct physical behaviour of a one-dimensional polar bond.",
"The function must be: (i) continuous over the entire range of $q$ ; (ii) have a maximum at some value of $q$ , not necessarily the equilibrium distance; (iii) asymptotically vanish as the neutral bond dissociates into neutral species.",
"These requirements are satisfied by a Rayleigh distribution function of the form $d(q) = (q+c_0)\\exp {[-q^2 / 2\\sigma ^2]},$ which we show in Fig.",
"REF a for $c_0=0$ and $\\sigma /q_e=1.25$ .",
"These parameters are used throughout.",
"$\\sigma $ is the coordinate at which the electronic dipole moment is greater.",
"The parameter $c_0$ is the dipole moment for $q/q_e\\ll 1$ .",
"In the Supplemental Material, we show that our results and conclusions do not qualitatively vary for different choices of $\\sigma $ and $c_{0}$ .",
"In order to describe light-matter coupling properly, we not only need a reasonable description of the electric dipole moment near the equilibrium distance $q_e$ , but also in the long range up to the dissociation threshold.",
"This is because strong light-matter coupling in a cavity can strongly admix several vibrational eigenstates with high $\\nu $ .",
"Furthermore, we are interested in studying how highly exited polaritons behave near the energy dissociation threshold of the free-space molecular sub-system.",
"Therefore, dipole matrix elements between all bound and unbound states of the Morse potential must be accurately estimated.",
"In Figs.",
"REF b and REF c we show the scaling with $\\nu $ of the diagonal and off-diagonal vibrational dipole matrix elements.",
"The permanent dipole moments $\\mathinner {\\langle {\\nu }|}\\hat{d}(q)\\mathinner {|{\\nu }\\rangle }$ decrease with $\\nu $ (panel REF b), as expected from the behaviour of $d(q)$ on a neutral molecular system.",
"This trend also holds for Morse oscillators with different $a$ and $\\mu $ parameters.",
"The higher the oscillator's mass, the lower the rate of decrease.",
"Fig.",
"REF c shows that for a fixed vibrational eigenstate $\\mathinner {|{\\nu ^{\\prime }}\\rangle }$ , the transition dipole moments with neighbouring states $\\mathinner {|{\\nu }\\rangle }$ ($\\nu \\ne \\nu ^{\\prime }$ ) are not negligible, and must be taken into account in the light-matter coupling.",
"In linear infrared absorption of high-frequency modes (e.g., $\\omega _{10}\\approx 200$ meV for carbonyl), only the ground vibrational level $\\nu =0$ is populated at room temperature ($kT/\\hbar \\omega _{10}\\ll 1$ ).",
"The oscillator strength of the fundamental absorption peak ($\\nu =0\\rightarrow 1$ ) and its overtones ($|\\Delta \\nu | \\ge 2$ ) are thus proportional to $|\\mathinner {\\langle {\\nu }|}\\hat{d}(q)\\mathinner {|{0}\\rangle }|^2$ with $\\nu \\ge 1$ .",
"Figure REF c (circles) captures the typical IR absorption pattern of decreasing overtone strength for higher $\\Delta \\nu $ [46].",
"This qualitatively correct behaviour validates the dipole model function $d(q)$ in Eq.",
"(REF ).",
"Using strong infrared laser pulses, it is possible to prepare vibrational modes with high quantum numbers $\\nu \\gg 1$ , even when $kT/\\hbar \\omega _{10}\\ll 1$ .",
"This off-resonant driving is known as vibrational ladder climbing, and has been used in nonlinear spectroscopic measurements [47].",
"Vibrational ladder climbing is determined by the matrix elements $\\mathinner {\\langle {\\nu }|}\\hat{d}\\mathinner {|{\\nu ^{\\prime }}\\rangle }$ with $\\nu ^{\\prime }\\ne \\nu \\ge 1$ , corresponding to dipole transitions between overtones.",
"Fig.",
"REF c (squares) shows that these high-$\\nu $ matrix elements can be as strong as the first overtones of the fundamental transition (circles), over a range of neighbouring levels with $|\\nu -\\nu ^{\\prime }| \\le 4$ , for our choice of $d(q)$ .",
"We show below that ignoring dipole couplings between high-$\\nu $ overtones fails to describe the rich and complex physics of the excited polariton manifold up to the dissociation threshold.",
"Excited polariton levels can be expected to be relevant in the description of nonlinear cavity transmission signals, chemical reactions, and heat transport."
],
[
"Multi-level quantum Rabi model",
"We derive the total Hamiltonian for the molecule-cavity system starting using the Power-Zineau-Wolley (PZW) multipolar formulation of light-matter interaction [48].",
"The PZW frame is equivalent to minimal-coupling by a unitary transformation that eliminates the vector potential $\\mathbf {A}(\\mathbf {x})$ from the Hamiltonian [48].",
"We divide the total Hamiltonian $\\hat{\\mathcal {H}}$ in the three terms of the form $\\mathcal {\\hat{H}} = \\hat{H}_{\\mathrm {M}} + \\hat{H}_{\\mathrm {C}} + \\hat{H}_{\\mathrm {LM}}$ .",
"The molecular part is given by $\\hat{H}_{\\rm M} = \\hat{H}_{\\rm el} + \\hat{H}_{\\rm vib} + \\hat{H}_{\\rm rot} + \\int d\\mathbf {x}\\; |\\mathbf {P}(\\mathbf {x})|^{2},$ where the first three terms represent the electronic, vibrational and rotational contributions, respectively.",
"The last term corresponds to the dipole self-energy, with $\\mathbf {P}(\\mathbf {x})$ being the macroscopic polarization density.",
"The free cavity Hamiltonian $\\hat{H}_{\\rm C}$ is given by $\\hat{H}_{\\rm C} &=& \\frac{1}{2} \\int d\\mathbf {x}\\left( |\\mathbf {D}(\\mathbf {x})|^{2} + \\frac{1}{\\mu _{0}}|\\mathbf {H}(\\mathbf {x})|^{2}\\right) \\nonumber \\\\&= & \\sum _{\\xi } \\hbar \\omega _{\\xi } \\left(\\hat{a}_{\\xi }^{\\dag } \\hat{a}_{\\xi } + 1/2\\right),$ where $\\mathbf {D}(\\mathbf {x})$ and $\\mathbf {H}(\\mathbf {x})$ are the macroscopic displacement and magnetic fields, respectively.",
"$\\mu _{0}$ is the magnetic permeability.",
"In the second line, we imposed canonical field quantization into a set of normal modes with continuum label $\\xi $ , frequencies $\\omega _\\xi $ , and annihilation operators $\\hat{a}_\\xi $ .",
"Light-matter interaction in the PZW frame, ignoring magnetic moments, is given by [48] $\\hat{H}_{\\rm LM} = \\int d\\mathbf {x}\\; \\mathbf {P}(\\mathbf {x}) \\cdot \\mathbf {D}(\\mathbf {x}).$ We consider a non-rotating polar bond and therefore set $\\hat{H}_{\\rm rot}=0$ .",
"The solutions of the electronic Hamiltonian $\\hat{H}_{\\rm el}$ are assumed to be known within the Born-Oppenheimer approximation, such that they give the dipole function $d(q)$ .",
"The self-energy term in Eq.",
"(REF ) will be shown to produce a state-dependent vibrational shift that does not qualitatively affect the polariton spectrum and eigenstates, and can be ignored to simplify the analysis.",
"In Section we put the dipole self-energy back into the Hamiltonian and discuss its effect on the polariton spectrum.",
"We adopt a point-dipole approximation for the polarization density, i.e.",
"$\\mathbf {P}(\\mathbf {x})=\\mathbf {d}\\,\\delta (\\mathbf {x}-\\mathbf {x}_0)$ , where $\\mathbf {d}$ is the electric dipole vector and $\\mathbf {x}_0$ is the location of the molecule.",
"We use a single-mode approximation for the cavity Hamiltonian in Eq.",
"(REF ) by setting $\\omega _\\xi \\equiv \\omega _c$ for all $\\xi $ and defining the effective field operators $\\hat{a} =\\sum _\\xi \\hat{a}_\\xi $ (up to a normalization constant).",
"This simplification is justified in Fabry-Pérot cavities with a large free-spectral range (FSR $\\sim 300-500$ cm$^{-1}$ [9]), and low transmission linewidths (FWHM $\\sim 10-40$ cm$^{-1}$ [9]).",
"In this approximation, the intracavity displacement field operator can be approximated by $\\hat{\\mathbf {D}} \\approx \\mathcal {E}_{0}(\\hat{a}+\\hat{a}^{\\dag })$ , where $\\mathcal {E}_0$ can be considered as the amplitude of the vacuum field fluctuations, or the electric field per photon (ignoring vectorial character).",
"$\\mathcal {E}_0$ scales as $1/\\sqrt{V_m}$ with the effective cavity mode volume $V_m$ [49].",
"We thus write the light-matter interaction term as $ \\hat{H}_{\\rm LM} = \\mathcal {E}_{0}\\,(\\hat{d}_{+}+\\hat{d}_{-})\\otimes (\\hat{a}+\\hat{a}^{\\dag }),$ where the up-transition operator $\\hat{d}_+$ projected into the vibrational energy basis $\\mathinner {|{\\nu }\\rangle }$ is given by $\\hat{d}_{+} = \\sum _{\\nu ,\\nu ^{\\prime }>\\nu }\\mathinner {\\langle {\\nu ^{\\prime }}|}{d}(q)\\mathinner {|{\\nu }\\rangle }\\mathinner {|{\\nu ^{\\prime }}\\rangle }\\mathinner {\\langle {\\nu }|},$ with $\\hat{d}_{-}=(\\hat{d}_{+})^{\\dag }$ .",
"By combining Eqs.",
"(REF ) without self-energy, Eq.",
"(REF ) in the single-mode approximation, and Eq.",
"(REF ), we can arrive at the total system Hamiltonian $\\mathcal {\\hat{H}} &=& \\omega _{c} \\,\\hat{a}^{\\dag } \\hat{a} +\\sum _{\\nu }\\omega _{\\nu }\\mathinner {|{\\nu }\\rangle }\\mathinner {\\langle {\\nu }|} \\\\&&+\\sum _{\\nu }\\sum _{\\nu ^{\\prime }>\\nu } g_{\\nu ^{\\prime }\\nu }(\\mathinner {|{\\nu ^{\\prime }}\\rangle }\\mathinner {\\langle {\\nu }|}+\\mathinner {|{\\nu }\\rangle }\\mathinner {\\langle {\\nu ^{\\prime }}|})(\\hat{a}+\\hat{a}^{\\dag })\\nonumber $ where $\\omega _\\nu $ is the energy of the vibrational eigenstate $\\mathinner {|{\\nu }\\rangle }$ , and $g_{\\nu ^{\\prime }\\nu }=\\mathcal {E}_{0}\\mathinner {\\langle {\\nu ^{\\prime }}|}d(q)\\mathinner {|{\\nu }\\rangle }$ for $\\nu ^{\\prime }>\\nu $ is a state-dependent Rabi frequency.",
"The zero of energy is defined by the energy of the vibrational ground state ($\\nu =0$ ) in the cavity vacuum.",
"Equation (REF ) corresponds to a multi-level quantum Rabi (MLQR) model, which reduces to the quantum Rabi model for a two-level system [50], [51], [52], when the vibrational space is truncated to $\\nu =0,1$ , and the energy reference rescaled.",
"The vacuum field amplitude $\\mathcal {E}_0$ is considered here as a tunable parameter that determines the light-matter coupling strength.",
"In a cavity with small mode volume, the mode amplitude $\\mathcal {E}_0$ can be large and tunable by fabrication [12].",
"Moreover, the cavity detuning $\\Delta \\equiv \\omega _c-\\omega _{10}$ is another energy scale that can be tuned by fabrication.",
"For convenience, we define the state-independent Rabi frequency $g\\equiv g_{10}=E_0\\,\\langle 1| d(q) |0 \\rangle .$ Although we use the single parameter $g$ to quantify light-matter coupling strength throughout, we emphasize that dipole transitions $\\nu \\leftrightarrow \\nu ^{\\prime }$ in Eq.",
"(REF ) have in general different coupling strengths."
],
[
"Spectrum of vibrational polaritons",
"In order to gain some physical intuition about the structure of vibrational polaritons, in Fig.",
"REF we illustrate the light-matter coupling scheme implied by the uncoupled basis $\\mathinner {|{\\nu }\\rangle }\\mathinner {|{n}\\rangle }$ , where $\\mathinner {|{n}\\rangle }$ is a cavity Fock state.",
"We can associate a complete vibrational manifold $\\lbrace \\mathinner {|{\\nu }\\rangle }\\,; \\nu =0,1,2,\\ldots \\rbrace $ to every Fock state of the cavity $\\mathinner {|{n}\\rangle }$ .",
"The ground level in each vibrational manifold ($\\nu =0$ ) has energy $n\\omega _c$ in the Fock state $\\mathinner {|{n}\\rangle }$ , and the dissociation energy $E_\\infty $ becomes $E_\\infty =D_e+n\\omega _c,$ Only in the cavity vacuum ($n=0$ ), the bond dissociation energy coincides with the value expected for a Morse oscillator in free space.",
"In general, the energy required to break a chemical bond depends on quantum state of the cavity field.",
"Figure: (a) Illustration of resonant light-matter coupling between a Morse oscillator with dissociation energy D e D_e and a quantized cavity field with photon number nn (unbound).",
"Each Morse potential corresponds to the uncoupled subspace |ν〉|n〉\\mathinner {|{\\nu }\\rangle }\\mathinner {|{n}\\rangle }.",
"(b) Low energy couplings involving the subspace 𝒮 1 ={|1〉|0〉,|0〉|1〉}\\mathcal {S}_1=\\lbrace \\mathinner {|{1}\\rangle }\\mathinner {|{0}\\rangle }, \\mathinner {|{0}\\rangle }\\mathinner {|{1}\\rangle }\\rbrace at E≈ω 10 E\\approx \\omega _{10}, and 𝒮 2 ={|2〉|0〉,|1〉|1〉,|0〉|2〉}\\mathcal {S}_2=\\lbrace \\mathinner {|{2}\\rangle }\\mathinner {|{0}\\rangle },\\mathinner {|{1}\\rangle }\\mathinner {|{1}\\rangle }, \\mathinner {|{0}\\rangle }\\mathinner {|{2}\\rangle }\\rbrace at E≈2ω 10 E\\approx 2\\omega _{10}.",
"Dipole coupling within 𝒮 1 \\mathcal {S}_1 leads to the formation of the lower and upper polaritons, and coupling within 𝒮 2 \\mathcal {S}_2 to the formation of a polariton triplet.",
"(c) High energy couplings involving 𝒮 6 \\mathcal {S}_6 at E≈6ω 10 E\\approx 6\\omega _{10}.",
"State |6〉|0〉\\mathinner {|{6}\\rangle }\\mathinner {|{0}\\rangle } is red shifted with respect to |0〉|6〉\\mathinner {|{0}\\rangle }\\mathinner {|{6}\\rangle } by δ 6 \\delta _6, for ω 10 =ω c \\omega _{10}=\\omega _c.",
"The highlighted levels can strongly admix.Vibrational manifolds with different Fock states can couple each other via the light-matter term in Eq.",
"(REF ).",
"Since parity is broken for vibrational states due to anharmonicity, the only quasi selection rule that holds is $\\Delta n=\\pm 1$ , because the free cavity Hamiltonian $\\hat{H}_{\\rm C}$ commutes with parity.",
"Therefore, vibrational states $\\mathinner {|{\\nu }\\rangle }$ and $\\mathinner {|{\\nu ^{\\prime }}\\rangle }$ that differ by one photon number can admix due to light-matter coupling.",
"Because of anharmonicity, admixing of vibrational states with $|\\nu -\\nu ^{\\prime }| \\ge 1$ is allowed.",
"The amount of admixing that can occur between vibrational eigenstates in different manifolds is ultimately determined by the electric dipole function $d(q)$ .",
"The number bare states $\\mathinner {|{\\nu }\\rangle }\\mathinner {|{n}\\rangle }$ that can potentially admix to form vibrational polariton eigenstates grows as the total energy increases.",
"Figure REF b shows that for the lowest Fock states, resonant coupling at energy $E\\approx \\omega _{10}$ only involves the subspace $\\mathcal {S}_1=\\lbrace \\mathinner {|{1}\\rangle }\\mathinner {|{0}\\rangle }, \\mathinner {|{0}\\rangle }\\mathinner {|{1}\\rangle }\\rbrace $ for $g/\\omega _{10}\\ll 1$ .",
"This coupling results in the formation of the so-called lower polariton (LP) and upper polariton (UP), which are observable in linear spectroscopy [6], [9].",
"They can be written as $\\mathinner {|{\\Psi _{1}}\\rangle } = \\alpha \\mathinner {|{0}\\rangle }\\mathinner {|{1}\\rangle }- \\beta \\mathinner {|{1}\\rangle }\\mathinner {|{0}\\rangle } \\\\\\mathinner {|{\\Psi _{2}}\\rangle } = \\beta \\mathinner {|{0}\\rangle }\\mathinner {|{1}\\rangle }+ \\alpha \\mathinner {|{1}\\rangle }\\mathinner {|{0}\\rangle }$ where $\\mathinner {|{\\Psi _{1}}\\rangle }$ and $\\mathinner {|{\\Psi _{2}}\\rangle }$ correspond to LP and UP, respectively.",
"The orthonormal coefficients $\\alpha $ and $\\beta $ depend on $g$ and $\\Delta $ .",
"$\\mathinner {|{\\Psi _{1}}\\rangle }$ and $\\mathinner {|{\\Psi _{2}}\\rangle }$ in Eq.",
"() coincide with the first excitation manifold of the Jaynes-Cummings model [53].",
"Figure REF b also shows that for $g/\\omega _{10}\\ll 1$ , resonant coupling at energy $E\\approx 2\\omega _{10}$ only involves the subspace $\\mathcal {S}_2=\\lbrace \\mathinner {|{2}\\rangle }\\mathinner {|{0}\\rangle }, \\mathinner {|{1}\\rangle }\\mathinner {|{1}\\rangle }, \\mathinner {|{0}\\rangle }\\mathinner {|{2}\\rangle }\\rbrace $ , leading to the formation of three polariton branches, as discussed below.",
"For $g/\\omega _{10}\\sim 0.1$ coupling of bare states $\\mathinner {|{\\nu }\\rangle }\\mathinner {|{n}\\rangle }$ beyond $\\mathcal {S}_1$ and $\\mathcal {S}_2$ is allowed by counter-rotating terms in Eq.",
"(REF ).",
"In Fig.",
"REF c, we consider the coupling between vibrational manifolds around energy $E\\approx 6\\,\\omega _{10}$ .",
"If the molecular vibrations were harmonic, vibrational states $\\mathinner {|{\\nu }\\rangle }$ would have energy $\\nu \\omega _{10}$ .",
"Due to anharmonicity, vibrational levels in free space have energy $\\omega _{\\nu }=\\nu \\,\\omega _{10}-\\delta _\\nu ,$ where $\\delta _\\nu >0$ is the shift from a harmonic oscillator level, shown in Fig.",
"REF for a Morse oscillator.",
"For $\\nu =6$ , the anharmonic shift $\\delta _6$ is not negligible in comparison with $\\omega _{10}$ , which means that for the smaller couplings $g/\\omega _{10}\\ll 1$ , the number of bare states $\\mathinner {|{\\nu }\\rangle }\\mathinner {|{n}\\rangle }$ that can resonantly admix is relatively limited.",
"This resembles the role of anharmonicity in limiting the efficiency of vibrational ladder climbing using laser pulses [54], [47].",
"Figure: (a) Spectrum of anharmonic vibrational polaritons as a function of the coupling strength g/ω 10 g/\\omega _{10}, for resonant coupling ω c =ω 10 \\omega _{c} = \\omega _{10}.",
"The ground state (GS), lower (LP) and upper (UP) polaritons are highlighted.",
"(b) Spectrum of the lowest five excited polaritons obtained by three levels of theory: the multi-level quantum Rabi model of Eq.",
"() (solid line), the quantum Rabi model for a two-level vibration ν={0,1}\\nu =\\lbrace 0,1\\rbrace (open circles) and an anharmonic three-level quantum Rabi model with ν={0,1,2}\\nu = \\lbrace 0,1,2\\rbrace (dashed line).",
"(c) Vibrational polaritons spectrum as a function of the cavity detuning from the fundamental frequency Δ=ω c -ω 10 \\Delta = \\omega _c - \\omega _{10}.",
"The GS and LP are highlighted.",
"We set g=0.2ω 0 g=0.2 \\omega _0.",
"Energy is in units of ω 10 \\omega _{10}.On the other hand, Fig.",
"REF c suggests that for larger coupling ratios $g/\\omega _{10}$ there is a greater number of quasi-degenerate bare states $\\mathinner {|{\\nu }\\rangle }\\mathinner {|{n}\\rangle }$ that are energetically available to admix within an energy range $2\\delta $ .",
"As the total energy increases, the density of quasi-degenerate bare states that can strongly admix within a bandwidth $\\delta E$ grows.",
"We show below that this complex coupling structure leads to a large density of true and avoided crossings in the excited polariton manifold, even for relatively low values of the coupling ratio $g/\\omega _{10}$ .",
"In Fig.",
"REF , we show the spectrum of anharmonic vibrational polaritons as a function of $g$ and $\\Delta $ .",
"Figure REF a shows that the system has a unique non-degenerate ground state $\\mathinner {|{\\Psi _0}\\rangle }$ (GS).",
"The first excited manifold features a LP-UP doublet that scales linearly with $g$ over the range of couplings considered ($R^2=1.000$ for log-log fit).",
"However, the LP-UP splitting is not symmetric around $E=\\omega _{10}$ , which is the energy of the degenerate bare states $\\mathinner {|{1}\\rangle }\\mathinner {|{0}\\rangle }$ and $\\mathinner {|{0}\\rangle }\\mathinner {|{1}\\rangle }$ for $\\omega _c=\\omega _{10}$ .",
"Fig.",
"REF a shows the polariton triplet around $E=2\\omega _{10}$ , associated with light-matter coupling within the subspace $\\mathcal {S}_2$ discussed above (see Fig.",
"REF b).",
"Multiple true and avoided crossings occur at energies $E\\ge 2\\omega _{10}$ over the entire range of couplings considered.",
"The density of energy crossings grows with increasing energy.",
"In Fig.",
"REF b we compare the energies of the lowest five excited states obtained by three levels of theory: (i) the MLQR model in Eq.",
"(REF ); (ii) the quantum Rabi model for a two-level vibration involving states $\\nu =\\lbrace 0,1\\rbrace $ ; (iii) a three-level quantum Rabi model with $\\nu =\\lbrace 0,1,2\\rbrace $ which takes into account the anharmonicy shift of the transition $\\nu =1\\rightarrow \\nu =2$ , i.e., $\\omega _{21}=\\omega _{10}-\\delta _2$ .",
"The latter was used in Ref.",
"[38] to interpret the intracavity differential absorption spectrum of W(CO$_2)_6$ [55].",
"By construction, the qubit model can qualitatively match the asymmetric LP-UP splitting around $E=\\omega _{10}$ over the range $g/\\omega _{10}\\le 0.1$ , but deviations occur for larger coupling strengths.",
"Since $g/\\omega _{10}= 0.1$ is conventionally regarded as the onset of the ultrastrong coupling regime [31], the deviations of the two-level model from MLQR for $g/\\omega _{10}> 0.1$ , can be attributed to the inability of the truncated two-level model to capture counter-rotating overtone couplings properly.",
"By increasing the dimensionality of the vibrational basis by one additional state ($\\nu =2$ ), the three-level quantum Rabi model matches better the LP-UP spectrum predicted by the MLQR model, but is unable to correctly capture the splitting of the polariton triplet around energy $E=2\\,\\omega _{10}$ , except for the smallest coupling ratios ($g/\\omega _{10}\\ll 1$ ).",
"The comparison between models in Fig.",
"REF b suggests that for the excited vibrational polaritons considered, the onset of ultrastrong coupling–where counter-rotating terms in the light-matter interaction becomes important–occurs at much smaller values of $g$ than those expected for a qubit, and can involve off-resonant coupling to higher vibrational levels with $\\nu \\ge 3$ .",
"For excited polaritons with energies $E\\ge 3\\,\\omega _{10}$ , few-level truncations of the material Hamiltonian (e.g.",
"Ref.",
"[38]) fail to capture the multiple true and avoided crossings that the Hamiltonian allows.",
"We further discuss these excited state crossings in Section .",
"In Fig.",
"REF c, we show the polariton spectrum as a function of detuning $\\Delta \\equiv \\omega _{c}-\\omega _{10}$ , for $g/\\omega _{10}=0.2$ .",
"Several true and avoided crossings develop in the excited manifold.",
"When $\\Delta \\sim g$ , the energetic ordering of the excited polaritons can change in comparison with the resonant regime ($\\Delta /g\\ll 1$ ).",
"For example, there is an avoided crossing at $E\\approx 2.1\\,\\omega _{10}$ near $\\Delta \\approx -0.1\\omega _{10}$ .",
"The upper polariton (UP) also crosses with the next excited polariton level at $\\Delta \\approx -0.28\\,\\omega _{01}$ .",
"This raises concerns regarding the assignment of spectral lines in linear and nonlinear cavity transmission spectroscopy for light-matter coupling in the dispersive regime $|\\Delta |/g \\gtrsim 1$ ."
],
[
"Vibrational polaritons in nuclear coordinate space",
"In molecules and materials, the strength of a chemical bond is commonly associated with its vibration frequency $\\omega _0$ via the relation $\\omega _0 = \\sqrt{k/\\mu },$ where $k$ is the bond spring constant and $\\mu $ is the reduced mass of the vibrating nuclei.",
"Stronger bonds (higher $k$ ) thus lead to higher vibrational frequencies.",
"This simple argument has also been used to discuss the bonding character of vibrational polaritons under strong coupling [6].",
"In this Section, we show that the description of the bonding strength of vibrational polaritons is far more complex than the commonly used spring model suggests.",
"Figure: Conditional probability densities |Φ 6n (q)| 2 |\\Phi _{6n}(q)|^2 for the excited polariton eigenstate |Ψ 6 〉\\mathinner {|{\\Psi _6}\\rangle }, for coupling strengths g/ω 10 =0.002g/\\omega _{10}=0.002 (a) and g/ω 10 =0.2g/\\omega _{10}=0.2 (b).",
"Coordinates are in units of the bare equilibrium bond length q e q_e.",
"All densities are normalized.In order to analyze vibrational polaritons in nuclear coordinate space, keeping photons in Hilbert space, let us expand the eigenstates of Eq.",
"(REF ) in the uncoupled basis $\\lbrace \\mathinner {|{\\nu }\\rangle }\\mathinner {|{n}\\rangle }\\rbrace $ as $\\mathinner {|{\\Psi _j}\\rangle } = \\sum _{\\nu ,n}c_{\\nu n}^{j}\\mathinner {|{\\nu }\\rangle }\\mathinner {|{n}\\rangle },$ where $c_{\\nu n}^{j}$ are orthonormal coefficients associated with the $j$ -th eigenstate.",
"We can rewrite Eq.",
"(REF ) by combining vibrational components associated with a given photon number $n$ as $\\mathinner {|{\\Psi _j}\\rangle } = \\sum _{n}\\mathinner {|{\\Phi _n^j}\\rangle }\\mathinner {|{n}\\rangle },$ where $\\mathinner {|{\\Phi _n^j}\\rangle } = \\sum _\\nu c_{\\nu n}^j\\mathinner {|{\\nu }\\rangle }$ .",
"The state $\\mathinner {|{\\Phi _n^j}\\rangle }$ can be interpreted as a vibrational wavepacket conditional on the cavity photon number.",
"Its nuclear coordinate representation is simply given by the projection $\\Phi _{jn}(q)=\\langle q|\\Phi _{n}^j\\rangle .$ For concreteness, we show in Fig.",
"REF a set of normalized conditional probability distributions $|\\Phi _{jn}(q)|^2$ with $n\\le 4$ , for the excited polariton eigenstate $\\mathinner {|{\\Psi _6}\\rangle }$ under resonant light-matter coupling.",
"Since the energy of excited polariton $\\mathinner {|{\\Psi _6}\\rangle }$ tends asymptotically to $E_6\\approx 3\\omega _{10}$ as $g/\\omega _{10}\\rightarrow 0$ , one could expect the normalized probability distribution $|\\Phi _{6n}(q)|^2$ to resemble the behaviour of the Morse oscillator eigenfunction with $\\nu = 3$ for $g/\\omega _{10}\\ll 1$ .",
"Figure REF (lower panel) shows that indeed the vacuum component ($n=0$ ) of $\\mathinner {|{\\Psi _6}\\rangle }$ qualitatively matches the node structure of the bare Morse oscillator state $\\mathinner {|{\\nu =3}\\rangle }$ for $g/\\omega _{10}=0.002$ .",
"However, for the coupling ratio $g/\\omega _{10}=0.2$ , the nuclear density of the cavity vacuum $|\\Phi _{60}(q)|^2$ behaves qualitatively different from a Morse eigenfunction.",
"Similar deviations from the bare Morse behavior occurs also for nuclear components with higher photon numbers ($n\\ge 1$ ).",
"Figure: Probability amplitudes |c νn | 2 |c_{\\nu n}|^2 in the uncoupled basis |ν〉|n〉\\mathinner {|{\\nu }\\rangle }\\mathinner {|{n}\\rangle } for excited polariton eigenstates |Ψ 6 〉\\mathinner {|{\\Psi _6}\\rangle } (a) and |Ψ 8 〉\\mathinner {|{\\Psi _8}\\rangle } (b), as a function the coupling strength g/ω 10 g/\\omega _{10}.",
"Curves are labelled by the quantum numbers (ν,n)(\\nu ,n).",
"We set ω c =ω 10 \\omega _c=\\omega _{10}.Figure REF also shows that the nuclear densities $|\\Phi _{6n}(q)|^2$ associated with $n\\ge 1$ can also approximately resemble the node pattern of a bare Morse oscillator with the appropriate number of excitations, for small values of $g/\\omega _{10}$ .",
"For example, since the energy of $\\mathinner {|{\\Psi _6}\\rangle }$ tends to $E\\approx 3\\omega _{10}$ as $g\\rightarrow 0$ , its wave function should have components in the uncoupled basis $\\mathinner {|{\\nu }\\rangle }\\mathinner {|{n}\\rangle }$ such that $\\nu + n = 3$ at zero detuning ($\\omega _c=\\omega _{10}$ ).",
"For $g/\\omega _{10}=0.002$ , Fig.",
"REF shows that indeed for $n=1$ the nuclear density $|\\Phi _{61}(q)|^2$ of state $\\mathinner {|{\\Psi _6}\\rangle }$ has a node structure similar to the bare Morse eigenstate $\\mathinner {|{\\nu =2}\\rangle }$ , i.e., it has two nodes.",
"The nuclear densities associated with $n=2$ and $n=3$ also seem to satisfy a conservation rule for the total number of excitations $(\\nu +n)$ .",
"This rule however is broken for the $n=4$ nuclear wave packet $ \\Phi _{64}(q) $ (Fig.",
"REF , upper panel), which has a node structure similar to the bare Morse eigenstate $\\mathinner {|{\\nu =1}\\rangle }$ , corresponding to a total number of excitations $\\nu +n=5$ for all values of $g/\\omega _{10}$ considered.",
"Figure: Mean bond length 〈q ^〉\\langle \\hat{q}\\rangle and mean cavity photon number 〈a ^ † a ^〉\\langle \\hat{a}^\\dagger \\hat{a} \\rangle as a function of coupling strength g/ω 10 g/\\omega _{10} (a,c) and cavity detuning Δ\\Delta (b,d), for the system ground state (dashed line), lower polariton (solid line) and upper polariton (dot-dashed line).",
"We set Δ=0\\Delta =0 in panels a,c and g/ω 10 =0.1g/\\omega _{10}=0.1 in panels b,d.",
"Energy is in units of ω 10 \\omega _{10}.Figure: Mean bond length 〈q ^〉\\langle \\hat{q}\\rangle and mean cavity photon number 〈a ^ † a ^〉\\langle \\hat{a}^\\dagger \\hat{a} \\rangle as a function of coupling strength g/ω 10 g/\\omega _{10} (a,c) and cavity detuning Δ\\Delta (b,d), for excited polaritons |Ψ 6 〉\\mathinner {|{\\Psi _6}\\rangle } (solid line) and |Ψ 8 〉\\mathinner {|{\\Psi _8}\\rangle } (dashed line).",
"We set Δ=0\\Delta =0 in panels a,c and g/ω 10 =0.1g/\\omega _{10}=0.1 in panels b,d.",
"Energy is in units of ω 10 \\omega _{10}.In order to assess the contribution of each photon-number-dependent nuclear wave packet $\\Phi _{j n}(q)$ on the $j$ -th polariton eigenstate $\\mathinner {|{\\Psi _j}\\rangle }$ , we show in Fig.",
"REF the probability amplitudes $|c_{\\nu n}|^2$ [see Eq.",
"(REF )] as a function of the coupling ratio $g/\\omega _{10}$ , for the excited polariton eigenstates $\\mathinner {|{\\Psi _6}\\rangle }$ and $\\mathinner {|{\\Psi _8}\\rangle }$ .",
"These two excited states tend asymptotically to the energy $E\\approx 3\\,\\omega _{10}$ as $g\\rightarrow 0$ , and therefore can be expected to be mainly composed of uncoupled states $\\mathinner {|{\\nu }\\rangle }\\mathinner {|{n}\\rangle }$ such that $\\nu +n=3$ , for resonant coupling.",
"Figure REF shows that indeed occurs for $g/\\omega _{10}\\ll 1$ .",
"In this small coupling regime, the selected polariton eigenstates can be approximately written in the basis $\\mathinner {|{\\nu }\\rangle }\\mathinner {|{n}\\rangle }$ as $\\mathinner {|{\\Psi _6}\\rangle }\\approx \\mathinner {|{3}\\rangle }\\mathinner {|{0}\\rangle }$ and $\\mathinner {|{\\Psi _8}\\rangle }\\approx a\\mathinner {|{0}\\rangle }\\mathinner {|{3}\\rangle } +b\\mathinner {|{1}\\rangle }\\mathinner {|{2}\\rangle },$ where $|a|^2\\approx |b|^2=0.5$ .",
"As the coupling strength reaches the regime $g/\\omega _{10}\\sim 0.1$ , the near resonant coupling between vibrational manifolds with higher photon numbers in Fig.",
"REF leads to the emergence of wave function components with lower vibrational quantum numbers.",
"For instance, for $g/\\omega _{10}= 0.2$ the excited state $\\mathinner {|{\\Psi _6}\\rangle }$ is approximately given by $\\mathinner {|{\\Psi _6}\\rangle }\\approx a\\mathinner {|{1}\\rangle }\\mathinner {|{2}\\rangle } +b\\mathinner {|{0}\\rangle }\\mathinner {|{3}\\rangle } + c\\mathinner {|{2}\\rangle }\\mathinner {|{0}\\rangle } + d\\mathinner {|{1}\\rangle }\\mathinner {|{4}\\rangle }$ where $|a|^2\\sim |b|^2>|c|^2 \\gg |d|^2$ .",
"In other words, the state evolves from a bare Morse oscillator $\\mathinner {|{\\nu =3}\\rangle }$ in vacuum [see Eq.",
"(REF )], into a state with a lower mean vibrational excitation and higher mean photon number as $g/\\omega _{10}$ grows.",
"On the other hand, the state $\\mathinner {|{\\Psi _8}\\rangle }$ at $g/\\omega _{10}= 0.2$ can be written as $\\mathinner {|{\\Psi _8}\\rangle }\\approx a\\mathinner {|{3}\\rangle }\\mathinner {|{0}\\rangle }+ b \\mathinner {|{2}\\rangle }\\mathinner {|{1}\\rangle }+ c\\mathinner {|{0}\\rangle }\\mathinner {|{3}\\rangle }+d \\mathinner {|{1}\\rangle }\\mathinner {|{4}\\rangle },$ where $|a|^2\\approx 1/2> |b|^2>|c|^2 \\gg |d|^2$ , which also develops components with lower vibrational quanta and higher photon numbers in comparison with Eq.",
"REF .",
"The emergence of uncoupled components with $\\nu +n\\ne 3$ in Eqs.",
"(REF ) and (REF ) is a consequence of the counter-rotating terms in Eq.",
"(REF ).",
"Although the results in Figs.",
"REF and REF were obtained for specific polariton eigenstates, we find that they qualitatively describe the behavior of most excited polaritons $\\mathinner {|{\\Psi _j}\\rangle }$ with energies $E_j\\gg \\omega _{10}$ , i.e., above the LP and UP frequency region.",
"We can also understand the structure of vibrational polaritons in coordinate space and Fock space by analyzing the dependence of the mean bond distance $\\langle \\hat{q}\\rangle $ and the mean photon number $\\langle \\hat{a}^\\dagger \\hat{a} \\rangle $ with the coupling parameter $g$ and cavity detuning $\\Delta $ , for selected polariton eigenstates.",
"In Fig.",
"REF , we compare the evolution of these observables with $g$ and $\\Delta $ for the system ground state $\\mathinner {|{\\Psi _0}\\rangle }$ (GS), the lower polariton state $\\mathinner {|{\\Psi _1}\\rangle }$ and the upper polariton state $\\mathinner {|{\\Psi _2}\\rangle }$ .",
"In the regime $g/\\omega _{10}\\ll 1$ , both LP and UP have the approximately the same bond length, given by $\\langle \\hat{q}\\rangle \\approx \\frac{1}{2}\\left(\\langle 0|\\hat{q}|0\\rangle +\\langle 1| \\hat{q}| 1\\rangle \\right),$ with expectation value taken with respect to Morse eigenstates $\\mathinner {|{\\nu }\\rangle }$ .",
"Figure: (a) Spectral region with an avoided crossing (circled in grey) involving the excited polaritons |Ψ 9 〉\\mathinner {|{\\Psi _9}\\rangle } (blue) and |Ψ 10 〉\\mathinner {|{\\Psi _{10}}\\rangle } (red).",
"(b) and (c) Main components of |Ψ 9 〉\\mathinner {|{\\Psi _9}\\rangle } and |Ψ 10 〉\\mathinner {|{\\Psi _{10}}\\rangle }, respectively, in the uncoupled basis |ν〉|n〉\\mathinner {|{\\nu }\\rangle }\\mathinner {|{n}\\rangle }.",
"Curves are labelled by the quantum numbers (ν,n)(\\nu ,n).",
"We set ω c =ω 10 \\omega _c = \\omega _{10}.Figure REF a shows that as the coupling strength increases, the bond length of the LP decreases, reaching values even lower than the bond length of the Morse ground state $\\mathinner {|{\\nu =0}\\rangle }$ .",
"On the other hand, the bond length of the UP grows with increasing coupling strength.",
"The value of $\\langle \\hat{q}\\rangle $ for the UP is upper bounded by the bond length of the first excited Morse state $\\mathinner {|{\\nu =1}\\rangle }$ .",
"In other words, for resonant coupling the molecular bond in the LP state becomes stronger relative to the UP with increasing coupling strength, although both polariton states experience bond hardening in comparison with the Morse eigenstate $\\mathinner {|{\\nu =1}\\rangle }$ , which is in the same energy region as LP and UP ($E/\\omega _{10}\\approx 1$ ).",
"Bond hardening should be accompanied by the creation of virtual cavity photons, and bond softening by the decrease in the mean photon number.",
"Figure REF b shows that the GS, LP and UP states follow this behaviour as a function of $g/\\omega _{10}$ , for resonant coupling.",
"We show in panels REF b,d that for detuned cavities, the compromise between bond strength and cavity photon occupation also holds.",
"Within the range of system parameters considered, we find that this compromise also holds for higher excited vibrational polaritons, as Fig.",
"REF shows for states $\\mathinner {|{\\Psi _6}\\rangle }$ and $\\mathinner {|{\\Psi _8}\\rangle }$ .",
"Bond hardening of vibrational polaritons can be understood by recalling that an eigenstate $\\mathinner {|{\\Psi _j}\\rangle }$ in the vicinity of a bare Morse energy level $E_{\\nu ^{\\prime }}$ in general has non-vanishing components in the uncoupled basis $\\mathinner {|{\\nu }\\rangle }\\mathinner {|{n}\\rangle }$ with $\\nu <\\nu ^{\\prime }$ [see Eq.",
"(REF )].",
"These low-$\\nu $ components contribute to the stabilization of the molecular bond even at high excitation energies."
],
[
"Energy crossings in the excited polariton manifold",
"We discussed in Section how the density of polariton levels increases with energy, ultimately due to the large number of near-degenerate uncoupled subspaces $\\mathinner {|{\\nu }\\rangle }\\left|n\\right\\rangle $ (see Fig.",
"REF ).",
"Light-matter coupling leads to the formation of closely-spaced polariton levels that can become quasi-degenerate at specific values of $g$ and $\\Delta $ .",
"As the Hamiltonian parameters ($g,\\Delta $ ) are tuned across the degeneracy point, the polariton levels may undergo true or avoided crossing.",
"For a Hamiltonian like the quantum Rabi model for the qubit [51], [52], [56] and its multi-level generalizations [57], parity is a conserved quantity.",
"Therefore polaritons in the quantum Rabi model have well-defined parity and level crossings are analyzed in the usual way: states with opposite parity undergo true crossing under variation of a Hamiltonian parameter.",
"In particular, the crossing of the ground state with the lower polariton state at $g/\\omega _c=1$ marks the onset of the deep strong coupling regime [58], [59], [31].",
"Parity conservation in the quantum Rabi model ultimately emerges from the even character of the underlying microscopic Hamiltonians that describe the material system and the cavity field.",
"The harmonic oscillator Hamiltonian that describes the cavity field is invariant under the transformation $\\hat{a}\\rightarrow -\\hat{a}$ , and therefore commutes with parity (as any harmonic oscillator Hamiltonian).",
"For the material system, let $\\hat{q}$ and $\\hat{p}$ represent position and momentum operators in the material Hamiltonian $\\hat{H}_{\\rm M}$ .",
"Then polariton eigenstates of the coupled light-matter system would only have well-defined parity if $\\hat{H}_{\\rm M}$ is invariant under the parity transformation $\\hat{q}\\rightarrow -\\hat{q}$ and $\\hat{p}\\rightarrow -\\hat{p}$ .",
"The Morse potential in Eq.",
"(REF ) is not invariant under the transformation $ q\\rightarrow - q$ and $q_e\\rightarrow -q_e$ , and therefore breaks parity, which is the origin of vibrational overtones.",
"Polariton eigenstates of the MLQR model therefore do not have well-defined parity.",
"Figure: (a) Spectral region with true crossings involving the excited polaritons |Ψ 22 〉\\mathinner {|{\\Psi _{22}}\\rangle } (blue) |Ψ 23 〉\\mathinner {|{\\Psi _{23}}\\rangle } (red), and |Ψ 21 〉\\mathinner {|{\\Psi _{21}}\\rangle } and |Ψ 23 〉\\mathinner {|{\\Psi _{23}}\\rangle } (both crossing regions circled in grey).",
"(b) and (c) Main components of |Ψ 22 〉\\mathinner {|{\\Psi _{22}}\\rangle } and |Ψ 23 〉\\mathinner {|{\\Psi _{23}}\\rangle }, respectively, in the uncoupled basis |ν〉|n〉\\mathinner {|{\\nu }\\rangle }\\mathinner {|{n}\\rangle }.",
"Curves are labelled by the quantum numbers (ν,n)(\\nu ,n).",
"We set ω c =ω 10 \\omega _c=\\omega _{10}.Even parity is not a good quantum number, the vibrational polariton spectrum still exhibits true and avoided level crossings as the Hamiltonian parameters $g$ and $\\Delta $ vary.",
"We can track the origin of these crossings into an effective photonic parity selection rule imposed by the light-matter interaction term in the total Hamiltonian [Eq.",
"(REF )], which reads $\\Delta n=\\pm 1$ .",
"For two near-degenerate polariton levels $E_j$ and $E_k$ ($k\\ne j$ ), there will be a strong avoided crossing between them only if the largest probability amplitudes $c_{\\nu n}$ of their wavefunctions $\\mathinner {|{\\Psi _j}\\rangle }$ and $\\mathinner {|{\\Psi _k}\\rangle }$ in the uncoupled basis $\\mathinner {|{\\nu }\\rangle }\\mathinner {|{n}\\rangle }$ , differ by one photon number [see Eq.",
"REF ].",
"Otherwise, the levels will cross as the Hamiltonian is varied through the degeneracy.",
"We show this explicitly in Figs.",
"REF and REF , with examples of avoided and true crossings, respectively, in the excited polariton manifold.",
"In Figure REF a we highlight an avoided crossing between excited polaritons $\\mathinner {|{\\Psi _{9}}\\rangle }$ and $\\mathinner {|{\\Psi _{10}}\\rangle }$ , as the coupling strength is $g/\\omega _{10}\\approx 0.19$ .",
"Panels REF b and c show that the largest uncoupled components to the left of the avoided crossing are $\\lbrace \\mathinner {|{1}\\rangle }\\mathinner {|{2}\\rangle },\\mathinner {|{0}\\rangle }\\mathinner {|{3}\\rangle }\\rbrace $ for $\\mathinner {|{\\Psi _{9}}\\rangle }$ and $\\lbrace \\mathinner {|{1}\\rangle }\\mathinner {|{3}\\rangle },\\mathinner {|{0}\\rangle }\\mathinner {|{4}\\rangle }\\rbrace $ for $\\mathinner {|{\\Psi _{10}}\\rangle }$ , which indeed differ by one photon number.",
"Past the avoided crossing, the state $\\mathinner {|{\\Psi _{10}}\\rangle }$ dominantly acquires $\\mathinner {|{0}\\rangle }\\mathinner {|{3}\\rangle }$ character.",
"In Fig.",
"REF a we highlight a pair of level crossings as $g/\\omega _{10}$ increases.",
"For $g/\\omega _{10}\\simeq 0.1$ , the excited polariton states $\\mathinner {|{\\Psi _{22}}\\rangle }$ and $\\mathinner {|{\\Psi _{23}}\\rangle }$ undergo the first crossing.",
"Fig.",
"REF b shows that to the left of the crossing point, the largest uncoupled components of $\\mathinner {|{\\Psi _{22}}\\rangle }$ is $\\mathinner {|{5}\\rangle }\\mathinner {|{1}\\rangle }$ , while for $\\mathinner {|{\\Psi _{23}}\\rangle }$ the largest components are $\\lbrace \\mathinner {|{1}\\rangle }\\mathinner {|{5}\\rangle },\\mathinner {|{2}\\rangle }\\mathinner {|{4}\\rangle }\\rbrace $ .",
"Since $\\mathinner {|{\\Psi _{22}}\\rangle }$ and $\\mathinner {|{\\Psi _{23}}\\rangle }$ thus predominantly satisfy $\\Delta n > 1$ , they do not interact via by the light-matter term as $g$ is varied across the degeneracy.",
"Note that Fig.",
"REF b,c the state does undergo a small change of character at the two crossing points in Fig.",
"REF a.",
"This occurs because states $\\mathinner {|{\\Psi _{22}}\\rangle }$ and $\\mathinner {|{\\Psi _{23}}\\rangle }$ do have uncoupled components that interact via the photonic selection rule $\\Delta n\\pm 1$ , but their weight in the eigenfunction is comparatively small."
],
[
"Dipole self-energy",
"For simplicity, we have neglected the dipole self-energy term throughout.",
"This term which arises via the transformation from minimal-coupling light-matter interaction to the multipolar interaction through the Power-Zineau-Wolley (PZW) transformation $U_{\\rm PZW}= {\\rm exp}[i\\int d\\mathbf {x} \\mathbf {P}(\\mathbf {x}) \\cdot \\mathbf {A}(\\mathbf {x})]$ which eliminates the vector potential $\\mathbf {A}(\\mathbf {x})$ from the theory.",
"Even though the multipolar formalism is non-covariant, it has been widely used to describe light-molecule interaction in the non-relativistic regime [48].",
"For light-matter coupling in which counter-rotating terms become important, it has been argued that the dipole self-energy contribution should be taken into account in order to describe polaritons correctly [31].",
"In molecular polariton problems, self-energy terms have been given ad-hoc model treatments in previous work [14], [39].",
"There, the dipole self-energy is considered to be proportional to the Rabi frequency $g$ , which is in remarkable contrast with the PZW frame, in which a material Hamiltonian [see Eq.",
"(REF )] contains a dipole self-energy contribution even when light-matter coupling is perturbative.",
"In principle, relating the macroscopic polarization density $\\mathbf {P}(\\mathbf {x})$ with the molecular electric dipole operator $\\mathbf {d}(\\mathbf {x}_0)$ at position $\\mathbf {x}_0$ in the medium would require an ab-initio quantum electrodynamics formulation of field quantization in dispersive and absorptive media [60], which is beyond the scope of our work.",
"We ignore the contribution of the dielectric background and assume that for a single polar vibration the following ansatz holds $ \\hat{H}_{\\rm self}=\\int |\\mathbf {P}(\\mathbf {x})|^{2}dx \\equiv \\gamma \\sum _{\\nu }\\mathinner {\\langle {\\nu }|}\\hat{d}(q)\\mathinner {|{\\nu }\\rangle }^2 \\mathinner {|{\\nu }\\rangle }\\mathinner {\\langle {\\nu }|},$ where $\\mathinner {|{\\nu }\\rangle }$ are the anharmonic vibrational eigenstates.",
"In other words, we assume that self-energy leads to a state-dependent blue shift of every vibrational level.",
"We can thus build a new total Hamiltonian $\\hat{\\mathcal {H}}^{\\prime }=\\hat{\\mathcal {H}}+\\hat{H}_{\\rm self}$ , where $\\mathcal {H}$ is the MLQR model from Eq.",
"(REF ).",
"The parameter $\\gamma $ is introduced to control the numerical magnitude of $\\hat{H}_{\\rm self}$ relative to $\\hat{\\mathcal {H}}$ .",
"In Fig.",
"REF a, we show the polariton spectrum with increasing $\\gamma $ , for $g$ and $\\Delta $ fixed.",
"As expected [14], [39], $\\hat{H}_{\\rm self}$ only results in a state-dependent positive energy shift in the polariton levels.",
"Once the dipole-shifted ground state energy $E_{\\rm GS}$ is subtracted from the energies of $\\mathcal {H}^{\\prime }$ , Fig.",
"REF b shows that the energy spectrum of $\\mathcal {H}^{\\prime }$ has the same qualitative behavior with increasing $g$ as the spectrum of $\\hat{\\mathcal {H}}$ .",
"Quantitative deviations from the MLQR spectrum due to self-energy become important for relative energies $(E-E_{\\rm GS})\\gtrsim 2\\,\\omega _{10}$ , when the magnitude of the parameter $\\gamma $ is comparable with the coupling ratio $g/\\omega _{10}$ .",
"Figure: (a) Vibrational polariton spectrum as a function of the dipole self-energy parameter γ\\gamma , for g=0.2ω 10 g=0.2\\omega _{10} and ω c =ω 10 \\omega _c=\\omega _{10}.",
"(b) Comparison of the polariton spectrum as a function of the coupling strength g/ω 10 g/\\omega _{10} using the MLQR model without (solid line) and with (dashed line) the dipole self-energy term with γ=0.15\\gamma = 0.15.",
"Energy is given relative to the ground state (GS), in units of ω 10 \\omega _{10}."
],
[
"Conclusion and Outlook",
"In order to understand the microscopic behaviour of an individual anharmonic molecular vibration coupled to a single infrared cavity mode, we introduce and analyze the multi-level quantum Rabi (MLQR) model of vibrational polaritons [Eq.",
"(REF )].",
"We derive the model Hamiltonian starting from the exact anharmonic solutions of a free-space Morse oscillator, and treat light-matter interaction within the Power-Zineau-Wolley multipolar framework [48], which includes the dipole self-energy.",
"The model takes into account counter-rotating terms in the light-matter coupling and allows the analysis of vibrational polaritons both in Hilbert space and nuclear coordinate space.",
"Phase-space representations of the photon state follow directly from the QED formulation of the model [61].",
"Such phase-space analysis would be closely related to previous coordinate-only treatments of photon-nuclei coupling [39], [28], [62], although a systematic comparison has yet to be done.",
"The model is consistent with previous work based on few-level vibrational systems [38], and therefore is also able to describe the spectral features observed in linear and nonlinear transmission spectroscopy [63], [55], [26], which due to the relatively weak intensities involved, can only probe up to the second excited polariton triplet around $E\\approx 2\\,\\omega _{10}$ , where $\\omega _{10}$ is the fundamental vibration frequency.",
"Few-level vibrational truncations are however unable to capture the dense and complex polariton level structure predicted by the MLQR model at energies $E\\gtrsim 3\\,\\omega _{10}$ .",
"The system Hamiltonian allows the emergence of an ensemble of avoided and true crossings as the Rabi frequency $g$ and cavity detuning $\\Delta $ are tuned.",
"The density of these level crossings increases with energy.",
"These crossings are governed by a pseudo-parity selection rule in the photonic degree of freedom (details in Sec.",
").",
"The nuclear coordinate analysis of vibrational polaritons within the MLQR model unveils a few general trends accross the entire energy spectrum.",
"First, it is no longer possible to define a unique bond dissociation energy in an infrared cavity as is commonly done in free space.",
"The dissociation energy depends on the quantum state of the cavity field.",
"Second, within any given energy range $E_j+\\Delta E$ , it is always possible to find a vibrational polariton eigenstate with small mean photon number $\\langle \\hat{a}^\\dagger \\hat{a}\\rangle $ and large mean bond distance $\\langle \\hat{q}\\rangle $ , and vice-versa.",
"Third, the bond distance $\\langle \\hat{q}\\rangle $ of an arbitrary vibrational polariton state with energy $E_j$ , never exceeds the bond length of a free-space Morse eigenstate $\\mathinner {|{\\nu }\\rangle }$ with similar energy ($E_\\nu \\approx E_j$ ).",
"In other words, the formation of vibrational polaritons inside the cavity leads to a type of bond-hardening effect that may have consequences in the reactivity of chemical bonds.",
"The generalization of the multi-level quantum Rabi model developed here to the many-molecule and multi-mode scenarios is straightforward.",
"Since it is formulated in the energy eigenbasis, treating the dissipative dynamics of vibrational polaritons due to cavity photon decay and vibrational relaxation is also straightforward to formulate within a Markovian approach [64].",
"The dynamics of vibrational polaritons in the many-body regime has been previously discussed in Refs.",
"[33], [65], using truncated vibrational subspaces.",
"The main qualitatively new effect that the many-body system introduces to the problem, is the formation of collective molecular states that are not symmetric with respect to particle permutations.",
"These so-called “dark exciton states\" [66] arise naturally from state classification by permutation symmetry in the Hilbert space of the Dicke model [67], [68].",
"It has been shown originally within a quasi-particle approach for systems with macroscopic translational invariance [69], and later using a cavity QED approach [32], [70], [71], [72], that totally-symmetric and non-symmetric collective molecular states can strongly admix due to ever-present inhomogeneous broadening of molecular energy levels, inhomogeneities in the light-matter interaction energy across the medium, or any local coherent term such as intramolecular electron-vibration coupling (in the case of electronic strong coupling [72]).",
"In general, the role of quasi-dark collective states in determining the rate of chemical reactions and also spectroscopic signals of vibrational polaritons is yet to be fully understood.",
"We thank Guillermo Romero, Blake Simpkins and Jeffrey Owrutsky for discussions.",
"This work is supported by CONICYT through the Proyecto REDES ETAPA INICIAL, Convocatoria 2017 no.",
"REDI 170423, FONDECYT Regular No.",
"1181743, and also thank support by Iniciativa Científica Milenio (ICM) through the Millennium Institute for Research in Optics (MIRO)."
]
] | 1906.04374 |
[
[
"Federated Learning for Emoji Prediction in a Mobile Keyboard"
],
[
"Abstract We show that a word-level recurrent neural network can predict emoji from text typed on a mobile keyboard.",
"We demonstrate the usefulness of transfer learning for predicting emoji by pretraining the model using a language modeling task.",
"We also propose mechanisms to trigger emoji and tune the diversity of candidates.",
"The model is trained using a distributed on-device learning framework called federated learning.",
"The federated model is shown to achieve better performance than a server-trained model.",
"This work demonstrates the feasibility of using federated learning to train production-quality models for natural language understanding tasks while keeping users' data on their devices."
],
[
"Introduction",
"Emoji have become an important mode of expression on smartphones as users increasingly use them to communicate on social media and chat applications.",
"Easily accessible emoji suggestions have therefore become an important feature for mobile keyboards.",
"Gboard is a mobile keyboard with more than 1 billion installs and support for over 600 language varieties.",
"With this work, we provide a mechanism by which Gboard offers emoji as predictions based on the text previously typed, as shown in Figure REF .",
"Figure: Emoji predictions in Gboard.",
"Based on the context “This party is lit”,Gboard predicts both emoji and words.Mobile devices are constrained by both memory and CPU.",
"Low-latency is also required, since users typically expect a keyboard response within 20 ms of an input event [10].",
"A unidirectional recurrent neural network architecture (RNN) is used in this work.",
"Since forward RNNs only include dependencies backwards in time, the model state can be cached at each timestep during inference to reduce latency."
],
[
"Federated Learning",
"Federated Learning (FL) [3] is a new computation paradigm in which data is kept on users' devices and never collected centrally.",
"Instead, minimal and focused model updates are transmitted to the server.",
"This allows us to train models while keeping users' data on their devices.",
"FL can be combined with other privacy-preserving techniques like secure multi-party computation [4] and differential privacy [14], [2], [1].",
"FL has been shown to be robust to unbalanced and non-IID data.",
"We use the FederatedAveraging algorithm presented in [13] to aggregate client updates after each round of local, on-device training to produce a new global model.",
"At training round $t$ , a global model with parameters $w_t$ , is sent to $K$ devices selected from the device population.",
"Each device has a local dataset $P_k$ which is split into batches of size $B$ .",
"Stochastic gradient descent (SGD) is used on the clients to compute new model parameters $w_{k}^{t+1}$ .",
"These client weights are then averaged across devices, on the server, to compute the new model parameters $w_{t+1}$ ."
],
[
"Network architecture",
"The Long-Short-Term Memory (LSTM) [11] architecture has been shown to achieve state-of-the art performance of a number of sentiment prediction and language modeling tasks [16].",
"We use an LSTM variant called the Coupled Input and Forget Gate (CIFG) [8].",
"As with Gated Recurrent Units [6], the CIFG uses a single gate to control both the input and recurrent cell self-connections.",
"The input gate ($i$ ) and the forget gate ($f$ ) are related by $f = 1 - i$ .",
"This coupling reduces the number of parameters per cell by 25%, compared to an LSTM.",
"We use an input word vocabulary size of 10,000, an input embedding size of 96, and a two-layer CIFG with 256 units per layer.",
"The logits are passed through a softmax layer to predict probabilities over 100 emoji."
],
[
"Pretraining",
" [12] demonstrated that pretraining parameters on a language modeling task can improve performance on other tasks.",
"We pretrain all layers except the output projection layer, using a language model trained to predict the next word in a sentence.",
"For the output projection, we reuse the input embeddings.",
"This type of sharing of input and output embeddings has been shown to improve performance of language models [15].",
"Pretraining is done with federated learning using techniques similar to those described by [9].",
"The language model achieves an Accuracy@1 of 13.7%, on the same vocabulary.",
"Pretraining with a language model task leads to much faster convergence for the emoji model, as seen in Figure REF .",
"Figure: Accuracy@1 vs. training step with and withoutpretraining, using server-based evaluations."
],
[
"Triggering",
"In addition to predicting the correct emoji, a triggering mechanism must determine when to show emoji predictions to users.",
"For instance, a user is likely to type Figure: NO_CAPTION One way to handle this would be to use a single language model that can predict both words and emojis.",
"However, we want to separate the task of predicting relevant emoji from that of deciding how much we wanted emoji to trigger, since the latter is more of a product decision, rather than a technical challenge.",
"For instance, if we want to allow users to control how often emoji predictions are offered, it's easier to do with a separate model.",
"Another way to handle triggering is to use a separate binary classification model that predicts the likelihood of the user typing any emoji after a given phrase.",
"However, using a separate model for triggering leads to additional overhead in terms of memory and latency.",
"Instead, we adjust the softmax layer of the model to predict over $N$ emoji and an additional unknown token <UNK> class.",
"The <UNK> class is set as the target output for inputs without emoji labels.",
"At inference, we show the predictions from the model only if the probability of the <UNK> class is less than a certain threshold.",
"During training, sentences without emoji are truncated to a random length in the range [1, length of sentence].",
"Truncation allows the model to learn to not predict emoji after conjunctions, prepositions etc.",
"which typically occur in the middle of sentences."
],
[
"Diversification",
"The distribution of emoji usage frequency is very light-tailed as seen in Figure REF .",
"As a result, the top predictions from the model are almost always the most frequent emoji regardless of the input context.",
"To overcome this, the probability of each emoji($\\widehat{P}$ ) is scaled by the empirical probability of that emoji($P$ ) in the training data as follows.",
"$S_i = \\frac{\\widehat{P} \\left(\\textrm {emoji}=i \\vert \\textrm {text} \\right)}{{P\\left(\\textrm {emoji}=i\\right)}^\\alpha }$ where $\\alpha $ is a scaling factor, determined empirically through experiments on live traffic.",
"Setting $\\alpha $ to 0 removes diversification.",
"Table REF provides examples with and without diversification.",
"Table: Examples of emoji predictions with and without diversification"
],
[
"Server-based Training",
"Server-based training of models is done on data logged from Gboard users who have opted to periodically share anonymized snippets of text typed in selected apps.",
"All personally identifiable information is stripped from these logs.",
"The logs are filtered further to only include sentences that are labeled as English with high confidence by a language detection model [5], [21].",
"The subset of logs used for training contain approximately 370 million snippets, approximately 11 million of which contain emoji.",
"Hyperparameters for server-based training are optimized using a black-box optimization technique [7]."
],
[
"Federated Training",
"The data used for federated training is stored in local caches on client devices.",
"For a device to participate in training, it must have at least 2 GB of RAM, must be located in United States or Canada, and must be using English (US) as the primary language.",
"In addition, only devices that are connected to un-metered networks, idle, and charging are eligible for participation at any given time.",
"On average, each client has approximately 400 sentences.",
"The model is trained for one epoch on each client, in each round.",
"The model typically converges after 2000 training rounds.",
"In federated training, there is no explicit split of data into train and eval samples.",
"Instead, a separate evaluation task runs on a different subset of client devices in parallel to the training task.",
"The eval task uses model checkpoints generated by the federated training task during a 24-hour period and aggregates the metrics across evaluation rounds."
],
[
"Evaluation",
"Model quality is evaluated using Accuracy@1, defined as the ratio of accurate top-1 emoji predictions to the total number of examples containing emoji.",
"Area Under ROC Curve (AUC) is used to evaluate the quality of the triggering mechanism.",
"Computing the AUC involves numerical integration and is not straightforward to do in the FL setting.",
"Therefore, we report AUC only on logs data that is collected on the server.",
"All evaluation metrics are computed prior to diversification."
],
[
"Federated Experiments",
"In FL, the contents of the client caches are constantly changing as old entries are cleared and replaced by new activity.",
"Since these experiments were conducted non-concurrently, the client cache contents are different and therefore numbers cannot be compared across experiments.",
"We conduct experiments to study the effect of client batch size ($B$ ), devices per round ($K$ ) and server optimizer configuration on model quality.",
"We then take the best model and compare it with a server trained model.",
"The results are summarized in Table REF .",
"Table: The results from federated experiments.",
"All numbers reported areafter 2000 training rounds.",
"η s \\eta _s refers to the learning rate usedon the server for applying the update aggregated across users in each round.Because of the sparsity of sentences containing emoji in the client caches, the model quality is improved to a large degree by using large client batch sizes.",
"This is not entirely surprising, since gradient updates are more accurate with larger batch sizes [17].",
"This is particularly true when the target classes are heavily imbalanced.",
"The accuracy of the model also increases with the number of devices per round but there are diminishing returns beyond $K=500$ .",
"We experimented with various optimizers for the server update after each round of federated training and found that using momentum of 0.9 with Nesterov accelerated gradients [18] gives significant benefits over using SGD, both in terms of speed of convergence and model performance.",
"The best federated model, which runs in production, uses $B = 1000, K=1000$ , and is trained with momentum.",
"We assign a weight of 0 to 99% of the <UNK> examples at training time so as to balance the triggering and emoji prediction losses.",
"We ran federated evaluation tasks of the best server-trained model on the client caches in order to fairly compare the two training approaches.",
"The federated model achieved better Accuracy@1 in the federated evaluation, as shown in Figure REF .",
"However, the AUC achieved by the federated model is lower than that of the server trained model.",
"AUC is only computed on the logs collected on the server.",
"These logs are restricted to short snippets of text typed in selected apps, therefore the data is not believed to be as representative of the text typed by users as data that resides on the client caches.",
"The lower AUC of the federated model is likely because of this bias.",
"Figure: Evaluation Accuracy@1 vs.",
"Round for federated and server trained models."
],
[
"Live experiment",
"At inference time, we use a quantized TensorFlow Lite [19] model format.",
"The average inference latency is around 1 ms. We ran a live-traffic experiment for users in USA and Canada typing in English (US).",
"We observed that both the federated and the server trained model lead to significant increases in the overall click-through rate (CTR) of predictions, total emoji shares, and daily active users (DAU) of emoji (see Table REF ).",
"We also observed that the federated model did better than the server trained model on all of the metrics.",
"Given that emoji are triggered rarely, the increase in CTR is quite large, for both the models.",
"Table: Relative changes to metrics as a result of the server trained andfederated emoji prediction models, measured in experiments on live user traffic.The baseline does not have any emoji predictions.Quoted 95% confidence interval errors for allresults are derived using the jackknife method with user buckets."
],
[
"Conclusions",
"In this paper, we train an emoji prediction model using a CIFG-LSTM network.",
"We demonstrate that this model can be trained using FL to achieve better performance than a server trained model.",
"This work builds on previous practical applications of federated learning in [20], [9], [3].",
"We show that FL works even with sparse data and poorly balanced classes."
]
] | 1906.04329 |
[
[
"Counterfactual Data Augmentation for Mitigating Gender Stereotypes in\n Languages with Rich Morphology"
],
[
"Abstract Gender stereotypes are manifest in most of the world's languages and are consequently propagated or amplified by NLP systems.",
"Although research has focused on mitigating gender stereotypes in English, the approaches that are commonly employed produce ungrammatical sentences in morphologically rich languages.",
"We present a novel approach for converting between masculine-inflected and feminine-inflected sentences in such languages.",
"For Spanish and Hebrew, our approach achieves F1 scores of 82% and 73% at the level of tags and accuracies of 90% and 87% at the level of forms.",
"By evaluating our approach using four different languages, we show that, on average, it reduces gender stereotyping by a factor of 2.5 without any sacrifice to grammaticality."
],
[
"Introduction",
"One of the biggest challenges faced by modern natural language processing (NLP) systems is the inadvertent replication or amplification of societal biases.",
"This is because NLP systems depend on language corpora, which are inherently “not objective; they are creations of human design” [8].",
"One type of societal bias that has received considerable attention from the NLP community is gender stereotyping [13], [25], [27].",
"Gender stereotypes can manifest in language in overt ways.",
"For example, the sentence he is an engineer is more likely to appear in a corpus than she is an engineer due to the current gender disparity in engineering.",
"Consequently, any NLP system that is trained such a corpus will likely learn to associate engineer with men, but not with women [9].",
"To date, the NLP community has focused primarily on approaches for detecting and mitigating gender stereotypes in English [1], [10], [30].",
"Yet, gender stereotypes also exist in other languages because they are a function of society, not of grammar.",
"Moreover, because English does not mark grammatical gender, approaches developed for English are not transferable to morphologically rich languages that exhibit gender agreement [4].",
"In these languages, the words in a sentence are marked with morphological endings that reflect the grammatical gender of the surrounding nouns.",
"This means that if the gender of one word changes, the others have to be updated to match.",
"As a result, simple heuristics, such as augmenting a corpus with additional sentences in which he and she have been swapped [31], will yield ungrammatical sentences.",
"Consider the Spanish phrase el ingeniero experto (the skilled engineer).",
"Replacing ingeniero with ingeniera is insufficient—el must also be replaced with la and experto with experta.",
"Figure: Dependency tree for the sentence El ingeniero alemán es muy experto.In this paper, we present a new approach to counterfactual data augmentation [18] for mitigating gender stereotypes associated with animateSpecifically, we consider a noun to be animate if WordNet considers person to be a hypernym of that noun.",
"nouns (i.e., nouns that represent people) for morphologically rich languages.",
"We introduce a Markov random field with an optional neural parameterization that infers the manner in which a sentence must change when altering the grammatical gender of particular nouns.",
"We use this model as part of a four-step process, depicted in fig:pipeline, to reinflect entire sentences following an intervention on the grammatical gender of one word.",
"We intrinsically evaluate our approach using Spanish and Hebrew, achieving tag-level $F_1$ scores of 83% and 72% and form-level accuracies of 90% and 87%, respectively.",
"We also conduct an extrinsic evaluation using four languages.",
"Following DBLP:journals/corr/abs-1807-11714, we show that, on average, our approach reduces gender stereotyping in neural language models by a factor of 2.5 without sacrificing grammaticality."
],
[
"Gender Stereotypes in Text",
"Men and women are mentioned at different rates in text [3].",
"This problem is exacerbated in certain contexts.",
"For example, the sentence he is an engineer is more likely to appear in a corpus than she is an engineer due to the current gender disparity in engineering.",
"This imbalance in representation can have a dramatic downstream effect on NLP systems trained on such a corpus, such as giving preference to male engineers over female engineers in an automated resumé filtering system.",
"Gender stereotypes of this sort have been observed in word embeddings [1], [27], contextual word embeddings [29], and co-reference resolution systems [26], [31] inter alia."
],
[
"A quick fix: swapping gendered words.",
"One approach to mitigating such gender stereotypes is counterfactual data augmentation [18].",
"In English, this involves augmenting a corpus with additional sentences in which gendered words, such as he and she, have been swapped to yield a balanced representation.",
"Indeed, zhao2018gender showed that this simple heuristic significantly reduces gender stereotyping in neural co-reference resolution systems, without harming system performance.",
"Unfortunately, this approach is only applicable to English and other languages with little morphological inflection.",
"When applied to morphologically rich languages that exhibit gender agreement, it yields ungrammatical sentences."
],
[
"The problem: inflected languages.",
"Many languages, including Spanish and Hebrew, have gender inflections for nouns, verbs, and adjectives—i.e., the words in a sentence are marked with morphological endings that reflect the grammatical gender of the surrounding nouns.The number of grammatical genders varies for different languages, with two being the most common non-zero number [12].",
"The languages that we use in our evaluation have two grammatical genders (male, female).",
"This means that if the gender of one word changes, the others have to be updated to preserve morpho-syntactic agreement [5].",
"Consider the following example from Spanish, where we wish to transform sent:msc to sent:fem.",
"(Parts of words that mark gender are depicted in bold.)",
"This task is not as simple as replacing el with la—ingeniero and experto must also be reinflected.",
"Moreover, the changes required for one language are not the same as those required for another (e.g., verbs are marked with gender in Hebrew, but not in Spanish).",
".",
"El ingeniero alemán es muy experto.",
"The.msc.sg engineer.msc.sg German.msc.sg is.in.pr.sg very skilled.msc.sg gray (The German engineer is very skilled.)",
".",
"La ingeniera alemana es muy experta.",
"The.fem.sg engineer.fem.sg German.fem.sg is.in.pr.sg very skilled.fem.sg gray (The German engineer is very skilled.)"
],
[
"Our approach.",
"Our goal is to transform sentences like sent:msc to sent:fem and vice versa.",
"To the best of our knowledge, this task has not been studied previously.",
"Indeed, there is no existing annotated corpus of paired sentences that could be used to train a supervised model.",
"As a result, we take an unsupervisedBecause we do not have any direct supervision for the task of interest, we refer to our approach as being unsupervised even though it does rely on annotated linguistic resources.",
"approach using dependency trees, lemmata, part-of-speech (POS) tags, and morpho-syntactic tags from Universal Dependencies corpora [21].",
"Specifically, we propose the following process: Analyze the sentence (including parsing, morphological analysis, and lemmatization).",
"Intervene on a gendered word.",
"Infer the new morpho-syntactic tags.",
"Reinflect the lemmata to their new forms.",
"This process is depicted in fig:pipeline.",
"The primary technical contribution is a novel Markov random field for performing step 3, described in the next section."
],
[
"A Markov Random Field for Morpho-Syntactic Agreement",
"In this section, we present a Markov random field [17] for morpho-syntactic agreement.",
"This model defines a joint distribution over sequences of morpho-syntactic tags, conditioned on a labeled dependency tree with associated part-of-speech tags.",
"Given an intervention on a gendered word, we can use this model to infer the manner in which the remaining tags must be updated to preserve morpho-syntactic agreement.",
"A dependency tree for a sentence (see fig:tree for an example) is a set of ordered triples $(i, j, \\ell )$ , where $i$ and $j$ are positions in the sentence (or a distinguished root symbol) and $\\ell \\in L$ is the label of the edge $i \\rightarrow j$ in the tree; each position occurs exactly once as the first element in a triple.",
"Each dependency tree $T$ is associated with a sequence of morpho-syntactic tags $\\mathbf {m}= m_1, \\ldots , m_{|T|}$ and a sequence of part-of-speech (POS) tags $\\mathbf {p}= p_1, \\ldots , p_{|T|}$ .",
"For example, the tags $m \\in M$ and $p \\in P$ for ingeniero are $[\\textsc {msc};\\textsc {sg}]$ and $\\textsc {noun}$ , respectively, because ingeniero is a masculine, singular noun.",
"For notational simplicity, we define ${\\cal M} = M^{|T|}$ to be the set of all length-$|T|$ sequences of morpho-syntactic tags.",
"We define the probability of $\\mathbf {m}$ given $T$ and $\\mathbf {p}$ as $&\\text{Pr}(\\mathbf {m}\\,|\\, T, \\mathbf {p}) \\propto {}\\\\&\\prod _{(i, j, \\ell ) \\in T} {black!60!green}{\\phi _i}(m_i)\\cdot {black!40!red!30!blue}{\\psi }(m_i, m_j \\,|\\, p_i, p_j, \\ell ),$ where the binary factor ${black!40!red!30!blue}{\\psi }(\\cdot , \\cdot \\,|\\, \\cdot , \\cdot , \\cdot )\\ge 0$ scores how well the morpho-syntactic tags $m_i$ and $m_j$ agree given the POS tags $p_i$ and $p_j$ and the label $\\ell $ .",
"For example, consider the $\\mathrm {amod}$ (adjectival modifier) edge from experto to ingeniero in fig:tree.",
"The factor ${black!40!red!30!blue}{\\psi }(m_i, m_j\\,|\\, \\textsc {a}, \\textsc {n}, \\mathrm {amod})$ returns a high score if the corresponding morpho-syntactic tags agree in gender and number (e.g., $m_i=[\\textsc {msc};\\textsc {sg}]$ and $m_j=[\\textsc {msc};\\textsc {sg}]$ ) and a low score if they do not (e.g., $m_i=[\\textsc {msc};\\textsc {sg}]$ and $m_j=[\\textsc {fem};\\textsc {pl}]$ ).",
"The unary factor ${black!60!green}{\\phi _i}(\\cdot ) \\ge 0$ scores a morpho-syntactic tag $m_i$ outside the context of the dependency tree.",
"As we explain in sec:constraint, we use these unary factors to force or disallow particular tags when performing an intervention; we do not learn them.",
"eq:dist is normalized by the following partition function: $&Z(T, \\mathbf {p}) = {} \\\\&\\sum _{\\mathbf {m}^{\\prime } \\in {\\cal M}}\\prod _{(i, j, \\ell ) \\in T} {black!60!green}{\\phi _i}(m^{\\prime }_i)\\cdot {black!40!red!30!blue}{\\psi }(m^{\\prime }_i, m^{\\prime }_j \\mid p_i, p_j, \\ell ).$ $Z(T, \\mathbf {p})$ can be calculated using belief propagation; we provide the update equations that we use in sec:bp.",
"Our model is depicted in fig:fg.",
"It is noteworthy that this model is delexicalized—i.e., it considers only the labeled dependency tree and the POS tags, not the actual words themselves.",
"Figure: Factor graph for the sentence El ingeniero alemán es muy experto."
],
[
"Parameterization",
"We consider a linear parameterization and a neural parameterization of the binary factor ${black!40!red!30!blue}{\\psi }(\\cdot , \\cdot \\,|\\, \\cdot , \\cdot , \\cdot )$ ."
],
[
"Linear parameterization.",
"We define a matrix $W(p_i, p_j, \\ell )\\in \\mathbb {R}^{c\\times c}$ for each triple $(p_i, p_j, \\ell )$ , where $c$ is the number of morpho-syntactic subtags.",
"For example, $[\\textsc {msc};\\textsc {sg}]$ has two subtags $\\textsc {msc}$ and $\\textsc {sg}$ .",
"We then define ${black!40!red!30!blue}{\\psi }(\\cdot , \\cdot \\,|\\, \\cdot , \\cdot , \\cdot )$ as follows: ${black!40!red!30!blue}{\\psi }(m_i, m_j \\mid p_i, p_j, \\ell ) = \\exp {(\\hspace{0.83328pt}\\overline{\\hspace{-0.83328pt}m\\hspace{-0.83328pt}}\\hspace{0.83328pt}_i^{\\top } W(p_i, p_j, \\ell ) \\hspace{0.83328pt}\\overline{\\hspace{-0.83328pt}m\\hspace{-0.83328pt}}\\hspace{0.83328pt}_j)}, \\nonumber $ where $\\hspace{0.83328pt}\\overline{\\hspace{-0.83328pt}m\\hspace{-0.83328pt}}\\hspace{0.83328pt}_i \\in \\lbrace 0, 1\\rbrace ^c$ is a multi-hot encoding of $m_i$ ."
],
[
"Neural parameterization.",
"As an alternative, we also define a neural parameterization of $W(p_i,p_j, \\ell )$ to allow parameter sharing among edges with different parts of speech and labels: $&W(p_i, p_j, \\ell ) = \\\\&\\quad \\exp {(U \\tanh (V\\,[\\mathbf {e}(p_i); \\mathbf {e}(p_j); \\mathbf {e}(\\ell )]))} \\nonumber $ where $U \\in \\mathbb {R}^{c\\times c\\times n_1}$ , $V \\in \\mathbb {R}^{n_1\\times 3n_2}$ , and $n_1$ and $n_2$ define the structure of the neural parameterization and each $\\mathbf {e}(\\cdot ) \\in \\mathbb {R}^{n_2}$ is an embedding function."
],
[
"Parameterization of ${black!60!green}{\\phi _i}$ .",
"We use the unary factors only to force or disallow particular tags when performing an intervention.",
"Specifically, we define ${black!60!green}{\\phi _i}(m) = {\\left\\lbrace \\begin{array}{ll}\\alpha & \\text{if } m = m_i \\\\1 & \\text{otherwise},\\end{array}\\right.",
"}$ where $\\alpha >1$ is a strength parameter that determines the extent to which $m_i$ should remain unchanged following an intervention.",
"In the limit as $\\alpha \\rightarrow \\infty $ , all tags will remain unchanged except for the tag directly involved in the intervention.In practice, $\\alpha $ is set using development data."
],
[
"Inference",
"Because our MRF is acyclic and tree-shaped, we can use belief propagation [22] to perform exact inference.",
"The algorithm is a generalization of the forward-backward algorithm for hidden Markov models [23].",
"Specifically, we pass messages from the leaves to the root and vice versa.",
"The marginal distribution of a node is the point-wise product of all its incoming messages; the partition function $Z(T, \\mathbf {p})$ is the sum of any node's marginal distribution.",
"Computing $Z(T, \\mathbf {p})$ takes polynomial time [22]—specifically, ${\\cal O}(n\\cdot |M|^2)$ where $M$ is the number of morpho-syntactic tags.",
"Finally, inferring the highest-probability morpho-syntactic tag sequence $\\mathbf {m}^{\\star }$ given $T$ and $\\mathbf {p}$ can be performed using the max-product modification to belief propagation."
],
[
"Parameter Estimation",
"We use gradient-based optimization.",
"We treat the negative log-likelihood ${}-\\log {(\\text{Pr}(\\mathbf {m}\\,|\\, T, \\mathbf {p}))}$ as the loss function for tree $T$ and compute its gradient using automatic differentiation [24].",
"We learn the parameters of sec:param by optimizing the negative log-likelihood using gradient descent."
],
[
"Intervention",
"As explained in sec:gender, our goal is to transform sentences like sent:msc to sent:fem by intervening on a gendered word and then using our model to infer the manner in which the remaining tags must be updated to preserve morpho-syntactic agreement.",
"For example, if we change the morpho-syntactic tag for ingeniero from [msc;sg] to [fem;sg], then we must also update the tags for el and experto, but do not need to update the tag for es, which should remain unchanged as [in; pr; sg].",
"If we intervene on the $i^\\text{th}$ word in a sentence, changing its tag from $m_i$ to $m_i^{\\prime }$ , then using our model to infer the manner in which the remaining tags must be updated means using $\\text{Pr}(\\mathbf {m}_{-i} \\,|\\,m^{\\prime }_i, T, \\mathbf {p})$ to identify high-probability tags for the remaining words.",
"Crucially, we wish to change as little as possible when intervening on a gendered word.",
"The unary factors ${black!60!green}{\\phi _i}$ enable us to do exactly this.",
"As described in the previous section, the strength parameter $\\alpha $ determines the extent to which $m_i$ should remain unchanged following an intervention—the larger the value, the less likely it is that $m_i$ will be changed.",
"Table: Morphological reinflection accuracies.Once the new tags have been inferred, the final step is to reinflect the lemmata to their new forms.",
"This task has received considerable attention from the NLP community [7], [6].",
"We use the inflection model of D18-1473.",
"This model conditions on the lemma $\\mathbf {x}$ and morpho-syntactic tag $m$ to form a distribution over possible inflections.",
"For example, given experto and $[\\textsc {a};\\textsc {fem};\\textsc {pl}]$ , the trained inflection model will assign a high probability to expertas.",
"We provide accuracies for the trained inflection model in tab:reinflect."
],
[
"Experiments",
"We used the Adam optimizer [16] to train both parameterizations of our model until the change in dev-loss was less than $10^{-5}$ bits.",
"We set $\\beta =(0.9, 0.999)$ without tuning, and chose a learning rate of $0.005$ and weight decay factor of $0.0001$ after tuning.",
"We tuned $\\log \\alpha $ in the set $\\lbrace 0.5, 0.75, 1, 2, 5, 10\\rbrace $ and chose $\\log \\alpha =1$ .",
"For the neural parameterization, we set $n_1 = 9$ and $n_2 = 3$ without any tuning.",
"Finally, we trained the inflection model using only gendered words.",
"We evaluate our approach both intrinsically and extrinsically.",
"For the intrinsic evaluation, we focus on whether our approach yields the correct morpho-syntactic tags and the correct reinflections.",
"For the extrinsic evaluation, we assess the extent to which using the resulting transformed sentences reduces gender stereotyping in neural language models."
],
[
"Intrinsic Evaluation",
"To the best of our knowledge, this task has not been studied previously.",
"As a result, there is no existing annotated corpus of paired sentences that can be used as “ground truth.” We therefore annotated Spanish and Hebrew sentences ourselves, with annotations made by native speakers of each language.",
"Specifically, for each language, we extracted sentences containing animate nouns from that language's UD treebank.",
"The average length of these extracted sentences was 37 words.",
"We then manually inspected each sentence, intervening on the gender of the animate noun and reinflecting the sentence accordingly.",
"We chose Spanish and Hebrew because gender agreement operates differently in each language.",
"We provide corpus statistics for both languages in the top two rows of tab:data.",
"We created a hard-coded ${black!40!red!30!blue}{\\psi }(\\cdot , \\cdot \\,|\\, \\cdot , \\cdot ,\\cdot )$ to serve as a baseline for each language.",
"For Spanish, we only activated values (i.e.",
"set to numbers greater than zero) that relate adjectives and determiners to nouns; for Hebrew, we only activated values that relate adjectives and verbs to nouns.",
"We created these language-specific baselines because gender agreement operates differently in each language.",
"To evaluate our approach, we held all morpho-syntactic subtags fixed except for gender.",
"For each annotated sentence, we intervened on the gender of the animate noun.",
"We then used our model to infer which of the remaining tags should be updated (updating a tag means swapping the gender subtag because all morpho-syntactic subtags were held fixed except for gender) and reinflected the lemmata.",
"Finally, we used the annotations to compute the tag-level $F_1$ score and the form-level accuracy, excluding the animate nouns on which we intervened.",
"Table: Tag-level precision, recall, F 1 F_1 score, and accuracyand form-level accuracy for the baselines (“–BASE”) and forour approach (“–LIN” is the linear parameterization,“–NN” is the neural parameterization).Figure: Gender stereotyping (left) and grammaticality (right) using the original corpus, thecorpus following CDA using naïve swapping of gendered words(“Swap”), and the corpus following CDA using our approach(“MRF”)."
],
[
"Results.",
"We present the results in tab:intrinsic.",
"Recall is consistently significantly lower than precision.",
"As expected, the baselines have the highest precision (though not by much).",
"This is because they reflect well-known rules for each language.",
"That said, they have lower recall than our approach because they fail to capture more subtle relationships.",
"For both languages, our approach struggles with conjunctions.",
"For example, consider the phrase él es un ingeniero y escritor (he is an engineer and a writer).",
"Replacing ingeniero with ingeniera does not necessarily result in escritor being changed to escritora.",
"This is because two nouns do not normally need to have the same gender when they are conjoined.",
"Moreover, our MRF does not include co-reference information, so it cannot tell that, in this case, both nouns refer to the same person.",
"We do not include co-reference information in our MRF because this would create cycles and inference would no longer be exact.",
"Additionally, the lack of co-reference information means that, for Spanish, our approach fails to convert nouns that are noun-modifiers or indirect objects of verbs.",
"Somewhat surprisingly, the neural parameterization does not outperform the linear parameterization.",
"We proposed the neural parameterization to allow parameter sharing among edges with different parts of speech and labels; however, this parameter sharing does not seem to make a difference in practice, so the linear parameterization is sufficient."
],
[
"Extrinsic Evaluation",
"We extrinsically evaluate our approach by assessing the extent to which it reduces gender stereotyping.",
"Following DBLP:journals/corr/abs-1807-11714, focus on neural language models.",
"We choose language models over word embeddings because standard measures of gender stereotyping for word embeddings cannot be applied to morphologically rich languages.",
"As our measure of gender stereotyping, we compare the log ratio of the prefix probabilities under a language model $P_{\\textrm {lm}}$ for gendered, animate nouns, such as ingeniero, combined with four adjectives: good, bad, smart, and beautiful.",
"The translations we use for these adjectives are given in sec:translation.",
"We chose the first two adjectives because they should be used equally to describe men and women, and the latter two because we expect that they will reveal gender stereotypes.",
"For example, consider $\\log \\frac{\\sum _{\\mathbf {x} \\in \\Sigma ^*} P_{\\textit {lm}}(\\textit {\\textsc {bos} } \\textit {El } \\textit {ingeniero } \\textit {bueno } \\mathbf {x})}{\\sum _{\\mathbf {x} \\in \\Sigma ^*}P_{\\textit {lm}}(\\textit {\\textsc {bos} } \\textit {La } \\textit {ingeniera } \\textit {buena }\\mathbf {x})}.$ If this log ratio is close to 0, then the language model is as likely to generate sentences that start with el ingeniero bueno (the good male engineer) as it is to generate sentences that start with la ingeniera bueno (the good female engineer).",
"If the log ratio is negative, then the language model is more likely to generate the feminine form than the masculine form, while the opposite is true if the log ratio is positive.",
"In practice, given the current gender disparity in engineering, we would expect the log ratio to be positive.",
"If, however, the language model were trained on a corpus to which our CDA approach had been applied, we would then expect the log ratio to be much closer to zero.",
"Because our approach is specifically intended to yield sentences that are grammatical, we additionally consider the following log ratio (i.e., the grammatical phrase over the ungrammatical phrase): $\\log \\frac{\\sum _{\\mathbf {x} \\in \\Sigma ^*} P_{\\textit {lm}}(\\textit {\\textsc {bos} } \\textit {El } \\textit {ingeniero } \\textit {bueno } \\mathbf {x})}{\\sum _{\\mathbf {x} \\in \\Sigma ^*}P_{\\textit {lm}}(\\textit {\\textsc {bos} } \\textit {El } \\textit {ingeniera } \\textit {bueno }\\mathbf {x})}.$ Table: Animate noun statistics.We trained the linear parameterization using UD treebanks for Spanish, Hebrew, French, and Italian (see tab:data).",
"For each of the four languages, we parsed one million sentences from Wikipedia (May 2018 dump) using [11]'s parser and extracted taggings and lemmata using the method of [20].",
"We automatically extracted an animacy gazetteer from WordNet [2] and then manually filtered the output for correctness.",
"We provide the size of the languages' animacy gazetteers and the percentage of automatically parsed sentences that contain an animate noun in tab:anim.",
"For each sentence containing a noun in our animacy gazetteer, we created a copy of the sentence, intervened on the noun, and then used our approach to transform the sentence.",
"For sentences containing more than one animate noun, we generated a separate sentence for each possible combination of genders.",
"Choosing which sentences to duplicate is a difficult task.",
"For example, alemán in Spanish can refer to either a German man or the German language; however, we have no way of distinguishing between these two meanings without additional annotations.",
"Multilingual animacy detection [15] might help with this challenge; co-reference information might additionally help.",
"For each language, we trained the BPE-RNNLM baseline open-vocabulary language model of [19] using the original corpus, the corpus following CDA using naïve swapping of gendered words, and the corpus following CDA using our approach.",
"We then computed gender stereotyping and grammaticality as described above.",
"We provide example phrases in tab:lm; we provide a more extensive list of phrases in app:queries."
],
[
"Results",
"fig:bias demonstrates depicts gender stereotyping and grammaticality for each language using the original corpus, the corpus following CDA using naïve swapping of gendered words, and the corpus following CDA using our approach.",
"It is immediately apparent that our approch reduces gender stereotyping.",
"On average, our approach reduces gender stereotyping by a factor of 2.5 (the lowest and highest factors are 1.2 (Ita) and 5.0 (Esp), respectively).",
"We expected that naïve swapping of gendered words would also reduce gender stereotyping.",
"Indeed, we see that this simple heuristic reduces gender stereotyping for some but not all of the languages.",
"For Spanish, we also examine specific words that are stereotyped toward men or women.",
"We define a word to be stereotyped toward one gender if 75% of its occurrences are of that gender.",
"fig:espbias suggests a clear reduction in gender stereotyping for specific words that are stereotyped toward men or women.",
"The grammaticality of the corpora following CDA differs between languages.",
"That said, with the exception of Hebrew, our approach either sacrifices less grammaticality than naïve swapping of gendered words and sometimes increases grammaticality over the original corpus.",
"Given that we know the model did not perform as accurately for Hebrew (see tab:intrinsic), this finding is not surprising.",
"Table: Prefix log-likelihoods of Spanish phrases using the original corpus, the corpus following CDA using naïve swapping of gendered words (“Swap”), and the corpus following CDA using our approach (“MRF”).",
"Phrases 1 and 2 are grammatical, while phrases 3 and 4 are not (dentoted by “*”).Gender stereotyping is measured using phrases 1 and 2.",
"Grammaticality is measured using phrases 1 and 3 and using phrases 2 and 4; these scores are then averaged."
],
[
"Related Work",
"In contrast to previous work, we focus on mitigating gender stereotypes in languages with rich morphology—specifically languages that exhibit gender agreement.",
"To date, the NLP community has focused on approaches for detecting and mitigating gender stereotypes in English.",
"For example, [1] proposed a way of mitigating gender stereotypes in word embeddings while preserving meanings; [18] studied gender stereotypes in language models; and [26] introduced a novel Winograd schema for evaluating gender stereotypes in co-reference resolution.",
"The most closely related work is that of [31], who used CDA to reduce gender stereotypes in co-reference resolution; however, their approach yields ungrammatical sentences in morphologically rich languages.",
"Our approach is specifically intended to yield grammatical sentences when applied to such languages.",
"[14] also focused on morphologically rich languages, specifically Arabic, but in the context of gender identification in machine translation."
],
[
"Conclusion",
"We presented a new approach for converting between masculine-inflected and feminine-inflected noun phrases in morphologically rich languages.",
"To do this, we introduced a Markov random field with an optional neural parameterization that infers the manner in which a sentence must change to preserve morpho-syntactic agreement when altering the grammatical gender of particular nouns.",
"To the best of our knowledge, this task has not been studied previously.",
"As a result, there is no existing annotated corpus of paired sentences that can be used as “ground truth.” Despite this limitation, we evaluated our approach both intrinsically and extrinsically, achieving promising results.",
"For example, we demonstrated that our approach reduces gender stereotyping in neural language models.",
"Finally, we also identified avenues for future work, such as the inclusion of co-reference information."
],
[
"Acknowledgments",
"R. Cotterell acknowledges a Facebook Fellowship."
],
[
"Belief Propagation Update Equations",
"Our belief propagation update equations are $\\mu _{i\\rightarrow f}(m) &=\\prod _{f^{\\prime } \\in N(i)\\setminus \\lbrace f\\rbrace }\\mu _{f^{\\prime }\\rightarrow i}(m) \\\\\\mu _{f_i\\rightarrow i}(m) &= {black!60!green}{\\phi _i}(m)\\,\\mu _{i\\rightarrow f_i}(m)$ $&\\mu _{f_{ij}\\rightarrow i}(m) = {} \\\\&\\sum _{m^{\\prime }\\in M} {black!40!red!30!blue}{\\psi }(m^{\\prime }, m \\mid p_i, p_j, \\ell )\\,\\mu _{j\\rightarrow f_{ij}}(m^{\\prime })\\\\&\\mu _{f_{ij}\\rightarrow j}(m) ={} \\\\&\\sum _{m^{\\prime }\\in M} {black!40!red!30!blue}{\\psi }(m, m^{\\prime } \\mid p_i, p_j, \\ell )\\,\\mu _{i\\rightarrow f_{ij}}(m^{\\prime })$ where $N(i)$ returns the set of neighbouring nodes of node $i$ .",
"The belief at any node is given by $\\beta (v)= \\prod _{f\\in N(v)}\\mu _{f\\rightarrow v}(m).$"
],
[
"Adjective Translations",
"tab:fem and tab:masc contain the feminine and masculine translations of the four adjectives that we used.",
"Table: Feminine translations of good, bad, smart, beautiful in French, Hebrew, Italian, and SpanishTable: Masculine translations of good, bad, smart, beautiful in French, Hebrew, Italian, and Spanish"
],
[
"Extrinsic Evaluation Example Phrases",
"For each noun in our animacy gazetteer, we generated sixteen phrases.",
"Consider the noun engineer as an example.",
"We created four phrases—one for each translation of The good engineer, The bad engineer, The smart engineer, and The beautiful engineer.",
"These phrases, as well as their prefix log-likelihoods are provided below in tab:query.",
"Table: Prefix log-likelihoods of Spanish phrases using the original corpus, the corpus following CDA using naïve swapping of gendered words (“Swap”), and the corpus following CDA using our approach (“MRF”).",
"Ungrammatical phrases are denoted by “*”."
]
] | 1906.04571 |
[
[
"Two-dimensional partial cubes"
],
[
"Abstract We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hypercubes whose vertex set defines a set family of VC-dimension at most 2.",
"Equivalently, those are the partial cubes which are not contractible to the 3-cube $Q_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube).",
"We show that our graphs can be obtained from two types of combinatorial cells (gated cycles and gated full subdivisions of complete graphs) via amalgams.",
"The cell structure of two-dimensional partial cubes enables us to establish a variety of results.",
"In particular, we prove that all partial cubes of VC-dimension 2 can be extended to ample aka lopsided partial cubes of VC-dimension 2, yielding that the set families defined by such graphs satisfy the sample compression conjecture by Littlestone and Warmuth (1986).",
"Furthermore we point out relations to tope graphs of COMs of low rank and region graphs of pseudoline arrangements."
],
[
"Introduction",
"Set families are fundamental objects in combinatorics, algorithmics, machine learning, discrete geometry, and combinatorial optimization.",
"The Vapnik-Chervonenkis dimension (the VC-dimension for short) $\\operatorname{VC-dim}(\\mathcal {S})$ of a set family $\\mathcal {S}\\subseteq 2^U$ is the size of a largest subset of $X\\subseteq U$ which can be shattered by $\\mathcal {S}$ [51], i.e., $2^{X}=\\lbrace X\\cap S: S\\in \\mathcal {S}\\rbrace $ .",
"Introduced in statistical learning by Vapnik and Chervonenkis [51], the VC-dimension was adopted in the above areas as complexity measure and as a combinatorial dimension of $\\mathcal {S}$ .",
"Two important inequalities relate a set family $\\mathcal {S}\\subseteq 2^{U}$ with its VC-dimension.",
"The first one, the Sauer-Shelah lemma [49], [50] establishes that if $|U|=m$ , then the number of sets in a set family $\\mathcal {S}\\subseteq 2^{U}$ with VC-dimension $d$ is upper bounded by $\\binom{m}{\\le d}$ .",
"The second stronger inequality, called the sandwich lemma, proves that $|\\mathcal {S}|$ is sandwiched between the number of strongly shattered sets (i.e., sets $X$ such that $\\mathcal {S}$ contains an $X$ -cube, see Section REF ) and the number of shattered sets [2], [11], [21], [44].",
"The set families for which the Sauer-Shelah bounds are tight are called maximum families [27], [25] and the set families for which the upper bounds in the sandwich lemma are tight are called ample, lopsided, and extremal families [5], [11], [35].",
"Every maximum family is ample, but not vice versa.",
"To take a graph-theoretical point of view on set families, one considers the subgraph $G(\\mathcal {S})$ of the hypercube $Q_m$ induced by the subsets of $\\mathcal {S}\\subseteq 2^{U}$ .",
"(Sometimes $G(\\mathcal {S})$ is called the 1-inclusion graph of $\\mathcal {S}$ [30], [31].)",
"Each edge of $G(\\mathcal {S})$ corresponds to an element of $U$ .",
"Then analogously to edge-contraction and minors in graph theory, one can consider the operation of simultaneous contraction of all edges of $G(\\mathcal {S})$ defined by the same element $e\\in U$ .",
"The resulting graph is the 1-inclusion graph $G(\\mathcal {S}_e)$ of the set family $\\mathcal {S}_e\\subseteq 2^{U\\setminus \\lbrace e\\rbrace }$ obtained by identifying all pairs of sets of $\\mathcal {S}$ differing only in $e$ .",
"Given $Y\\subseteq U$ , we call the set family $\\mathcal {S}_Y$ and its 1-inclusion graph $G(\\mathcal {S}_Y)$ obtained from $\\mathcal {S}$ and $G(\\mathcal {S})$ by successively contracting the edges labeled by the elements of $Y$ the Q-minors of $\\mathcal {S}$ and $G(\\mathcal {S})$ .",
"Then $X\\subseteq U$ is shattered by $\\mathcal {S}$ if and only if the Q-minor $G(\\mathcal {S}_{U\\setminus X})$ is a full cube.",
"Thus, the cubes play the same role for Q-minors as the complete graphs for classical graph minors.",
"To take a metric point of view on set families, one restricts to set families whose 1-inclusion graph satisfies further properties.",
"The typical property here is that the 1-inclusion graph $G(\\mathcal {S})$ of $\\mathcal {S}$ is an isometric (distance-preserving) subgraph of the hypercube $Q_m$ .",
"Such graphs are called partial cubes.",
"Partial cubes can be characterized in a pretty and efficient way [20] and can be recognized in quadratic time [24].",
"Partial cubes comprise many important and complex graph classes occurring in metric graph theory and initially arising in completely different areas of research such as geometric group theory, combinatorics, discrete geometry, and media theory (for a comprehensive presentation of partial cubes and their classes, see the survey [4] and the books [19], [29], [43]).",
"For example, 1-inclusion graphs of ample families (and thus of maximum families) are partial cubes [5], [35] (in view of this, we will call such graphs ample partial cubes and maximum partial cubes, respectively).",
"Other important examples comprise median graphs (aka 1-skeletons of CAT(0) cube complexes [17], [48]) and, more generally, 1-skeletons of CAT(0) Coxeter zonotopal complexes [28], the tope graphs of oriented matroids (OMs) [8], of affine oriented matroids (AOMs) [34], and of lopsided sets (LOPs) [34], [35], where the latter coincide with ample partial cubes (AMPs).",
"More generally, tope graphs of complexes of oriented matroids (COMs) [6], [34] capture all of the above.",
"Other classes of graphs defined by distance or convexity properties turn out to be partial cubes: bipartite cellular graphs (aka bipartite graphs with totally decomposable metrics) [3], bipartite Pasch [14], [16] and bipartite Peano [46] graphs, netlike graphs [45], and hypercellular graphs [18].",
"Many mentioned classes of partial cubes can be characterized via forbidden $Q$ -minors; in case of partial cubes, $Q$ -minors are endowed with a second operation called restriction and are called partial cube minors, or pc-minors [18].",
"The class of partial cubes is closed under pc-minors.",
"Thus, given a set $G_1,G_2,\\ldots ,G_n$ of partial cubes, one considers the set ${\\mathcal {F}}(G_1,\\ldots ,G_n)$ of all partial cubes not having any of $G_1,G_2,\\ldots ,G_n$ as a pc-minor.",
"Then $\\mathcal {F}(Q_2)$ is the class of trees, $\\mathcal {F}(P_3)$ is the class of hypercubes, and $\\mathcal {F}(K_2\\square P_3)$ consists of bipartite cacti [37].",
"Other obstructions lead to more interesting classes, e.g., almost-median graphs ($\\mathcal {F}(C_6)$ [37]), hypercellular graphs ($\\mathcal {F}(Q_3^-)$ [18]), median graphs ($\\mathcal {F}(Q_3^-, C_6)$ [18]), bipartite cellular graphs ($\\mathcal {F}(Q_3^-, Q_3)$ [18]), rank two COMs ($\\mathcal {F}(SK_4, Q_3)$ [34]), and two-dimensional ample graphs ($\\mathcal {F}(C_6, Q_3)$ [34]).",
"Here $Q_3^-$ denotes the 3-cube $Q_3$ with one vertex removed and $SK_4$ the full subdivision of $K_4$ , see Figure REF .",
"Bipartite Pasch graphs have been characterized in [14], [16] as partial cubes excluding 7 isometric subgraphs of $Q_4$ as pc-minors.",
"Littlestone and Warmuth [36] introduced the sample compression technique for deriving generalization bounds in machine learning.",
"Floyd and Warmuth [25] asked whether any set family $\\mathcal {S}$ of VC-dimension $d$ has a sample compression scheme of size $O(d)$ .",
"This question remains one of the oldest open problems in computational machine learning.",
"It was recently shown in [40] that labeled compression schemes of size $O(2^d)$ exist.",
"Moran and Warmuth [39] designed labeled compression schemes of size $d$ for ample families.",
"Chalopin et al.",
"[13] designed (stronger) unlabeled compression schemes of size $d$ for maximum families and characterized such schemes for ample families via unique sink orientations of their 1-inclusion graphs.",
"For ample families of VC-dimension 2 such unlabeled compression schemes exist because they admit corner peelings [13], [41].",
"In view of this, it was noticed in [47] and [39] that the original sample compression conjecture of [25] would be solved if one can show that any set family $\\mathcal {S}$ of VC-dimension $d$ can be extended to an ample (or maximum) partial cube of VC-dimension $O(d)$ or can be covered by $exp(d)$ ample partial cubes of VC-dimension $O(d)$ .",
"These questions are already nontrivial for set families of VC-dimension 2.",
"In this paper, we investigate the first question for partial cubes of VC-dimension 2, i.e., the class $\\mathcal {F}(Q_3)$ , that we will simply call two-dimensional partial cubes.",
"We show that two-dimensional partial cubes can be extended to ample partial cubes of VC-dimension 2 – a property that is not shared by general set families of VC-dimension 2.",
"In relation to this result, we establish that all two-dimensional partial cubes can be obtained via amalgams from two types of combinatorial cells: maximal full subdivisions of complete graphs and convex cycles not included in such subdivisions.",
"We show that all such cells are gated subgraphs.",
"On the way, we detect a variety of other structural properties of two-dimensional partial cubes.",
"Since two-dimensional partial cubes are very natural from the point of view of pc-minors and generalize previously studied classes such as bipartite cellular graphs [18], we consider these results of independent interest also from this point of view.",
"In particular, we point out relations to tope graphs of COMs of low rank and region graphs of pseudoline arrangements.",
"See Theorem REF for a full statement of our results on two-dimensional partial cubes.",
"Figure REF presents an example of a two-dimensional partial cube which we further use as a running example.",
"We also provide two characterizations of partial cubes of VC-dimension $\\le d$ for any $d$ (i.e., of the class ${\\mathcal {F}}(Q_{d+1})$ ) via hyperplanes and isometric expansions.",
"However, understanding the structure of graphs from ${\\mathcal {F}}(Q_{d+1})$ with $d\\ge 3$ remains a challenging open question.",
"Figure: A two-dimensional partial cube MM"
],
[
"Metric subgraphs and partial cubes",
"All graphs $G=(V,E)$ in this paper are finite, connected, and simple.",
"The distance $d(u,v):=d_G(u,v)$ between two vertices $u$ and $v$ is the length of a shortest $(u,v)$ -path, and the interval $I(u,v)$ between $u$ and $v$ consists of all vertices on shortest $(u,v)$ -paths: $I(u,v):=\\lbrace x\\in V: d(u,x)+d(x,v)=d(u,v)\\rbrace .$ An induced subgraph $H$ of $G$ is isometric if the distance between any pair of vertices in $H$ is the same as that in $G.$ An induced subgraph of $G$ (or the corresponding vertex set $A$ ) is called convex if it includes the interval of $G$ between any two of its vertices.",
"Since the intersection of convex subgraphs is convex, for every subset $S\\subseteq V$ there exists the smallest convex set ${\\rm conv}(S)$ containing $S$ , referred to as the convex hull of $S$ .",
"A subset $S\\subseteq V$ or the subgraph $H$ of $G$ induced by $S$ is called gated (in $G$ ) [23] if for every vertex $x$ outside $H$ there exists a vertex $x^{\\prime }$ (the gate of $x$ ) in $H$ such that each vertex $y$ of $H$ is connected with $x$ by a shortest path passing through the gate $x^{\\prime }$ .",
"It is easy to see that if $x$ has a gate in $H$ , then it is unique and that gated sets are convex.",
"Since the intersection of gated subgraphs is gated, for every subset $S\\subseteq V$ there exists the smallest gated set ${\\rm gate}( S)$ containing $S,$ referred to as the gated hull of $S$ .",
"A graph $G=(V,E)$ is isometrically embeddable into a graph $H=(W,F)$ if there exists a mapping $\\varphi : V\\rightarrow W$ such that $d_H(\\varphi (u),\\varphi (v))=d_G(u,v)$ for all vertices $u,v\\in V$ , i.e., $\\varphi (G)$ is an isometric subgraph of $H$ .",
"A graph $G$ is called a partial cube if it admits an isometric embedding into some hypercube $Q_m$ .",
"For an edge $e=uv$ of $G$ , let $W(u,v)=\\lbrace x\\in V: d(x,u)<d(x,v)\\rbrace $ .",
"For an edge $uv$ , the sets $W(u,v)$ and $W(v,u)$ are called complementary halfspaces of $G$ .",
"Theorem 1 [20] A graph $G$ is a partial cube if and only if $G$ is bipartite and for any edge $e=uv$ the sets $W(u,v)$ and $W(v,u)$ are convex.",
"To establish an isometric embedding of $G$ into a hypercube, Djoković [20] introduced the following binary relation $\\Theta $ (called Djoković-Winkler relation) on the edges of $G$ : for two edges $e=uv$ and $e^{\\prime }=u^{\\prime }v^{\\prime }$ we set $e\\Theta e^{\\prime }$ if and only if $u^{\\prime }\\in W(u,v)$ and $v^{\\prime }\\in W(v,u)$ .",
"Under the conditions of the theorem, $e\\Theta e^{\\prime }$ if and only if $W(u,v)=W(u^{\\prime },v^{\\prime })$ and $W(v,u)=W(v^{\\prime },u^{\\prime })$ , i.e.",
"$\\Theta $ is an equivalence relation.",
"Let $E_1,\\ldots ,E_m$ be the equivalence classes of $\\Theta $ and let $b$ be an arbitrary vertex taken as the basepoint of $G$ .",
"For a $\\Theta $ -class $E_i$ , let $\\lbrace G^-_i,G^+_i\\rbrace $ be the pair of complementary convex halfspaces of $G$ defined by setting $G^-_i:=G(W(u,v))$ and $G^+_i:=G(W(v,u))$ for an arbitrary edge $uv\\in E_i$ such that $b\\in G^-_i$ .",
"Then the isometric embedding $\\varphi $ of $G$ into the $m$ -dimensional hypercube $Q_m$ is obtained by setting $\\varphi (v):=\\lbrace i: v\\in G^+_i\\rbrace $ for any vertex $v\\in V$ .",
"Then $\\varphi (b)=\\varnothing $ and for any two vertices $u,v$ of $G$ , $d_G(u,v)=|\\varphi (u)\\Delta \\varphi (v)|.$ The bipartitions $\\lbrace G^-_i,G^+_i\\rbrace , i=1,\\ldots ,m,$ can be canonically defined for all subgraphs $G$ of the hypercube $Q_m$ , not only for partial cubes.",
"Namely, if $E_i$ is a class of parallel edges of $Q_m$ , then removing the edges of $E_i$ from $Q_m$ but leaving their end-vertices, $Q_m$ will be divided into two $(m-1)$ -cubes $Q^{\\prime }$ and $Q^{\\prime \\prime }$ .",
"Then $G^-_i$ and $G^+_i$ are the intersections of $G$ with $Q^{\\prime }$ and $Q^{\\prime \\prime }$ .",
"For a $\\Theta $ -class $E_i$ , the boundary $\\partial G^-_i$ of the halfspace $G^-_i$ consists of all vertices of $G^-_i$ having a neighbor in $G^+_i$ ($\\partial G^+_i$ is defined analogously).",
"Note that $\\partial G^-_i$ and $\\partial G^+_i$ induce isomorphic subgraphs (but not necessarily isometric) of $G$ .",
"Figure REF (a) illustrates a $\\Theta $ -class $E_i$ of the two-dimensional partial cube $M$ , the halfspaces $M_i^-, M_i^+$ and their boundaries $\\partial M_i^-,\\partial M_i^+$ .",
"Figure: (a) The halfspaces and their boundaries defined by a Θ\\Theta -class E i E_i of MM.",
"(b) The two-dimensional partial cube M * =π i (M)M_*=\\pi _i(M) obtained from MM by contracting E i E_i.An antipode of a vertex $v$ in a partial cube $G$ is a vertex $-v$ such that $G={\\rm conv}(v,-v)$ .",
"Note that in partial cubes the antipode is unique and ${\\rm conv}(v,-v)$ coincides with the interval $I(v,-v)$ .",
"A partial cube $G$ is antipodal if all its vertices have antipodes.",
"A partial cube $G$ is said to be affine if there is an antipodal partial cube $G^{\\prime }$ , such that $G$ is a halfspace of $G^{\\prime }$ ."
],
[
"Partial cube minors",
"Let $G$ be a partial cube, isometrically embedded in the hypercube $Q_m$ .",
"For a $\\Theta $ -class $E_i$ of $G$ , an elementary restriction consists of taking one of the complementary halfspaces $G^-_i$ and $G^+_i$ .",
"More generally, a restriction is a subgraph of $G$ induced by the intersection of a set of (non-complementary) halfspaces of $G$ .",
"Such an intersection is a convex subgraph of $G$ , thus a partial cube.",
"Since any convex subgraph of a partial cube $G$ is the intersection of halfspaces [1], [14], the restrictions of $G$ coincide with the convex subgraphs of $G$ .",
"For a $\\Theta $ -class $E_i$ , we say that the graph $\\pi _i(G)$ obtained from $G$ by contracting the edges of $E_i$ is an ($i$ -)contraction of $G$ ; for an illustration, see Figure REF (b).",
"For a vertex $v$ of $G$ , we will denote by $\\pi _i(v)$ the image of $v$ under the $i$ -contraction, i.e., if $uv$ is an edge of $E_i$ , then $\\pi _i(u)=\\pi _i(v)$ , otherwise $\\pi _i(u)\\ne \\pi _i(v)$ .",
"We will apply $\\pi _i$ to subsets $S\\subset V$ , by setting $\\pi _i(S):=\\lbrace \\pi _i(v): v\\in S\\rbrace $ .",
"In particular we denote the $i$ -contraction of $G$ by $\\pi _i(G)$ .",
"From the proof of the first part of [15] it easily follows that $\\pi _i(G)$ is an isometric subgraph of $Q_{m-1}$ , thus the class of partial cubes is closed under contractions.",
"Since edge contractions in graphs commute, if $E_i,E_j$ are two distinct $\\Theta $ -classes, then $\\pi _j(\\pi _i(G))=\\pi _i(\\pi _j(G))$ .",
"Consequently, for a set $A$ of $k$ $\\Theta $ -classes, we can denote by $\\pi _A(G)$ the isometric subgraph of $Q_{m-k}$ obtained from $G$ by contracting the equivalence classes of edges from $A$ .",
"Contractions and restrictions commute in partial cubes [18].",
"Consequently, any set of restrictions and any set of contractions of a partial cube $G$ provide the same result, independently of the order in which we perform them.",
"The resulting graph $G^{\\prime }$ is a partial cube, and $G^{\\prime }$ is called a partial cube minor (or pc-minor) of $G$ .",
"For a partial cube $H$ we denote by ${\\mathcal {F}}(H)$ the class of all partial cubes not having $H$ as a pc-minor.",
"In this paper we investigate the class ${\\mathcal {F}}(Q_3)$ .",
"Figure: The excluded pc-minors of isometric dimension ≤4\\le 4 for COMs.With the observation that a convex subcube of a partial cube can be obtained by contractions as well, the proof of the following lemma is straightforward.",
"Lemma 2 A partial cube $G$ belongs to ${\\mathcal {F}}(Q_{d+1})$ if and only if $G$ has VC-dimension $\\le d$ .",
"Let $G$ be a partial cube and $E_i$ be a $\\Theta $ -class of $G$ .",
"Then $E_i$ crosses a convex subgraph $H$ of $G$ if $H$ contains an edge $uv$ of $E_i$ and $E_i$ osculates $H$ if $E_i$ does not cross $H$ and there exists an edge $uv$ of $E_i$ with $u\\in H$ and $v\\notin H$ .",
"Otherwise, $E_i$ is disjoint from $H$ .",
"The following results summarize the properties of contractions of partial cubes established in [18] and [34]: Lemma 3 Let $G$ be a partial cube and $E_i$ be a $\\Theta $ -class of $G$ .",
"(i) [18] If $H$ is a convex subgraph of $G$ and $E_i$ crosses or is disjoint from $H$ , then $\\pi _i(H)$ is also a convex subgraph of $\\pi _i(G)$ ; (ii) [18] If $S$ is a subset of vertices of $G$ , then $\\pi _i({\\rm conv}(S))\\subseteq {\\rm conv}(\\pi _i(S))$ .",
"If $E_i$ crosses $S$ , then $\\pi _i({\\rm conv}(S))= {\\rm conv}(\\pi _i(S))$ ; (iii) [18] If $S$ is a gated subgraph of $G$ , then $\\pi _i(S)$ is a gated subgraph of $\\pi _i(G)$ .",
"Lemma 4 [34] Affine and antipodal partial cubes are closed under contractions."
],
[
"OMs, COMs, and AMPs",
"In this subsection, we recall the definitions of oriented matroids, complexes of oriented matroids, and ample set families."
],
[
"OMs: oriented matroids",
"Co-invented by Bland $\\&$ Las Vergnas [10] and Folkman $\\&$ Lawrence [26], and further investigated by many other authors, oriented matroids represent a unified combinatorial theory of orientations of (orientable) ordinary matroids.",
"OMs capture the basic properties of sign vectors representing the circuits in a directed graph or more generally the regions in a central hyperplane arrangement in ${\\mathbb {R}}^d$ .",
"OMs obtained from a hyperplane arrangement are called realizable.",
"Just as ordinary matroids, oriented matroids may be defined in a multitude of distinct but equivalent ways, see the book by Björner et al. [8].",
"Let $U$ be a finite set and let $\\mathcal {L}$ be a system of sign vectors, i.e., maps from $U$ to $\\lbrace \\pm 1,0\\rbrace = \\lbrace -1,0,+1\\rbrace $ .",
"The elements of $\\mathcal {L}$ are also referred to as covectors and denoted by capital letters $X, Y, Z$ , etc.",
"We denote by $\\le $ the product ordering on $\\lbrace \\pm 1,0\\rbrace ^{U} $ relative to the standard ordering of signs with $0 \\le -1$ and $0 \\le +1$ .",
"The composition of $X$ and $Y$ is the sign vector $X\\circ Y$ , where for all $e\\in U$ one defines $(X\\circ Y)_e = X_e$ if $X_e\\ne 0$ and $(X\\circ Y)_e=Y_e$ if $X_e=0$ .",
"The topes of $\\mathcal {L}$ are the maximal elements of $\\mathcal {L}$ with respect to $\\le $ .",
"A system of sign vectors $(U,\\mathcal {L})$ is called an oriented matroid (OM) if $\\mathcal {L}$ satisfies the following three axioms: (C) (Composition) $X\\circ Y \\in \\mathcal {L}$ for all $X,Y \\in \\mathcal {L}$ .",
"(SE) (Strong elimination) for each pair $X,Y\\in \\mathcal {L}$ and for each $e\\in U$ such that $X_eY_e=-1$ , there exists $Z \\in \\mathcal {L}$ such that $Z_e=0$ and $Z_f=(X\\circ Y)_f$ for all $f\\in U$ with $X_fY_f\\ne -1$ .",
"(Sym) (Symmetry) $-\\mathcal {L}=\\lbrace -X: X\\in \\mathcal {L}\\rbrace =\\mathcal {L},$ that is, $\\mathcal {L}$ is closed under sign reversal.",
"Furthermore, a system of sign-vectors $(U,\\mathcal {L})$ is simple if it has no “redundant” elements, i.e., for each $e \\in U$ , $\\lbrace X_e: X\\in \\mathcal {L}\\rbrace =\\lbrace +, -,0 \\rbrace $ and for each pair $e\\ne f$ in $U$ , there exist $X,Y \\in \\mathcal {L}$ with $\\lbrace X_eX_f,Y_eY_f\\rbrace =\\lbrace +, -\\rbrace $ .",
"From (C), (Sym), and (SE) it easily follows that the set ${\\mathcal {T}}$ of topes of any simple OM $\\mathcal {L}$ are $\\lbrace -1,+1\\rbrace $ -vectors.",
"Therefore ${\\mathcal {T}}$ can be viewed as a set family (where $-1$ means that the corresponding element does not belong to the set and $+1$ that it belongs).",
"We will only consider simple OMs, without explicitly stating it every time.",
"The tope graph of an OM $\\mathcal {L}$ is the 1-inclusion graph $G({\\mathcal {T}})$ of $\\mathcal {T}$ viewed as a set family.",
"The Topological Representation Theorem of Oriented Matroids of [26] characterizes tope graphs of OMs as region graphs of pseudo-sphere arrangements in a sphere $S^d$ [8].",
"See the bottom part of Figure REF for an arrangement of pseudo-circles in $S^2$ .",
"It is also well-known (see for example [8]) that tope graphs of OMs are partial cubes and that $\\mathcal {L}$ can be recovered from its tope graph $G({\\mathcal {T}})$ (up to isomorphism).",
"Therefore, we can define all terms in the language of tope graphs.",
"In particular, the isometric dimension of $G({\\mathcal {T}})$ is $|U|$ and its VC-dimension coincides with the dimension $d$ of the sphere $S^d$ hosting a representing pseudo-sphere arrangement.",
"Moreover a graph $G$ is the tope graph of an affine oriented matroid (AOM) if $G$ is a halfspace of a tope graph of an OM.",
"In particular, tope graphs of AOMs are partial cubes as well."
],
[
"COMs: complexes of oriented matroids",
"Complexes of oriented matroids (COMs) have been introduced and investigated in [6] as a far-reaching natural common generalization of oriented matroids, affine oriented matroids, and ample systems of sign-vectors (to be defined below).",
"Some research has been connected to COMs quite quickly, see e.g.",
"[7], [32], [38] and the tope graphs of COMs have been investigated in depth in [34], see Subsection REF .",
"COMs are defined in a similar way as OMs, simply replacing the global axiom (Sym) by a weaker local axiom (FS) of face symmetry: a complex of oriented matroids (COMs) is a system of sign vectors $(U,\\mathcal {L})$ satisfying (SE), and the following axiom: (FS) (Face symmetry) $X\\circ -Y \\in \\mathcal {L}$ for all $X,Y \\in \\mathcal {L}$ .",
"As for OMs we generally restrict ourselves to simple COMs, i.e., COMs defining simple systems of sign-vectors.",
"It is easy to see that (FS) implies (C), yielding that OMs are exactly the COMs containing the zero sign vector ${\\bf 0}$ , see [6].",
"Also, AOMs are COMs, see [6] or [7].",
"In analogy with realizable OMs, a COM is realizable if it is the systems of sign vectors of the regions in an arrangement $U$ of (oriented) hyperplanes restricted to a convex set of ${\\mathbb {R}}^d$ .",
"See Figure REF for an example in ${\\mathbb {R}}^2$ .",
"For other examples of COMs, see [6].",
"Figure: The system of sign-vectors associated to an arrangement of hyperplanes restricted to a convex set and the tope graph of the resulting realizable COM.The simple twist between (Sym) and (FS) leads to a rich combinatorial and geometric structure that is build from OM cells but is quite different from OMs.",
"Let $(U,\\mathcal {L})$ be a COM and $X$ be a covector of $\\mathcal {L}$ .",
"The face of $X$ is $F(X):=\\lbrace X\\circ Y: Y\\in \\mathcal {L}\\rbrace $ .",
"By [6], each face $F(X)$ of $\\mathcal {L}$ is an OM.",
"Moreover, it is shown in [6] that replacing each combinatorial face $F(X)$ of $\\mathcal {L}$ by a PL-ball, we obtain a contractible cell complex associated to each COM.",
"The topes and the tope graphs of COMs are defined in the same way as for OMs.",
"Again, the topes ${\\mathcal {T}}$ are $\\lbrace -1,+1\\rbrace $ -vectors, the tope graph $G({\\mathcal {T}})$ is a partial cubes, and the COM $\\mathcal {L}$ can be recovered from its tope graph, see [6] or [34].",
"As for OMs, the isometric dimension of $G({\\mathcal {T}})$ is $|U|$ .",
"If a COM is realizable in $\\mathbb {R}^d$ , then the VC-dimension of $G({\\mathcal {T}})$ is at most $d$ .",
"For each covector $X\\in \\mathcal {L}$ , the tope graph of its face $F(X)$ is a gated subgraph of the tope graph of $\\mathcal {L}$ [34]: the gate of any tope $Y$ in $F(X)$ is the covector $X\\circ Y$ (which is obviously a tope).",
"All this implies that the tope graph of any COM $\\mathcal {L}$ is obtained by amalgamating gated tope subgraphs of its faces, which are all OMs.",
"Let $\\hspace{3.0pt}\\uparrow \\hspace{-2.0pt}\\mathcal {L}:=\\lbrace Y \\in \\lbrace \\pm 1,0\\rbrace ^{U}: X \\le Y \\text{ for some } X \\in \\mathcal {L}\\rbrace $ .",
"Then the ample systems (AMPs)In the papers on COMs, these systems of sign-vectors are called lopsided (LOPs).",
"of sign vectors are those COMs such that $\\hspace{3.0pt}\\uparrow \\hspace{-2.0pt}\\mathcal {L}=\\mathcal {L}$ [6].",
"From the definition it follows that any face $F(X)$ consists of the sign vectors of all faces of the subcube of $[-1,+1]^{U}$ with barycenter $X$ ."
],
[
"AMPs: ample set families",
"Just above we defined ample systems as COMs satisfying $\\hspace{3.0pt}\\uparrow \\hspace{-2.0pt}\\mathcal {L}=\\mathcal {L}$ .",
"This is not the first definition of ample systems; all previous definitions define them as families of sets and not as systems of sign vectors.",
"Ample sets have been introduced by Lawrence [35] as asymmetric counterparts of oriented matroids and have been re-discovered independently by several works in different contexts [5], [11], [52].",
"Consequently, they received different names: lopsided [35], simple [52], extremal [11], and ample [5], [21].",
"Lawrence [35] defined ample sets for the investigation of the possible sign patterns realized by points of a convex set of $\\mathbb {R}^d$ .",
"Ample set families admit a multitude of combinatorial and geometric characterizations [5], [11], [35] and comprise many natural examples arising from discrete geometry, combinatorics, graph theory, and geometry of groups [5], [35] (for applications in machine learning, see [13], [39]).",
"Let $X$ be a subset of a set $U$ with $m$ elements and let $Q_m=Q(U)$ .",
"A $X$ -cube of $Q_m$ is the 1-inclusion graph of the set family $\\lbrace Y\\cup X^{\\prime }: X^{\\prime }\\subseteq X\\rbrace $ , where $Y$ is a subset of $U\\setminus X$ .",
"If $|X|=m^{\\prime }$ , then any $X$ -cube is a $m^{\\prime }$ -dimensional subcube of $Q_m$ and $Q_m$ contains $2^{m-m^{\\prime }}$ $X$ -cubes.",
"We call any two $X$ -cubes parallel cubes.",
"Recall that $X\\subseteq U$ is shattered by a set family ${\\mathcal {S}}\\subseteq 2^{U}$ if $\\lbrace X\\cap S: S\\in {\\mathcal {S}}\\rbrace =2^X$ .",
"Furthermore, $X$ is strongly shattered by $\\mathcal {S}$ if the 1-inclusion graph $G({\\mathcal {S}})$ of $\\mathcal {S}$ contains a $X$ -cube.",
"Denote by $\\overline{\\mathcal {X}} (\\mathcal {S})$ and $\\underline{\\mathcal {X}} (\\mathcal {S})$ the families consisting of all shattered and of all strongly shattered sets of $\\mathcal {S}$ , respectively.",
"Clearly, $\\underline{\\mathcal {X}} (\\mathcal {S})\\subseteq \\overline{\\mathcal {X}} (\\mathcal {S})$ and both $\\overline{\\mathcal {X}} (\\mathcal {S})$ and $\\underline{\\mathcal {X}} (\\mathcal {S})$ are closed by taking subsets, i.e., $\\overline{\\mathcal {X}} (\\mathcal {S})$ and $\\underline{\\mathcal {X}} (\\mathcal {S})$ are abstract simplicial complexes.",
"The VC-dimension [51] $\\operatorname{VC-dim}(\\mathcal {S})$ of $\\mathcal {S}$ is the size of a largest set shattered by $\\mathcal {S}$ , i.e., the dimension of the simplicial complex $\\overline{\\mathcal {X}} (\\mathcal {S})$ .",
"The fundamental sandwich lemma (rediscovered independently in [2], [11], [21], [44]) asserts that $\\vert \\underline{\\mathcal {X}} (\\mathcal {S})\\vert \\le \\vert \\mathcal {S}\\vert \\le \\vert \\overline{\\mathcal {X}} (\\mathcal {S})\\vert $ .",
"If $d=\\operatorname{VC-dim}(\\mathcal {S})$ and $m=\\vert U\\vert $ , then $\\overline{\\mathcal {X}} (\\mathcal {S})$ cannot contain more than $\\Phi _d(m):=\\sum _{i=0}^d \\binom{m}{i}$ simplices.",
"Thus, the sandwich lemma yields the well-known Sauer-Shelah lemma [49], [50], [51] that $\\vert \\mathcal {S}\\vert \\le \\Phi _d(m)$ .",
"A set family $\\mathcal {S}$ is called ample if $\\vert \\mathcal {S}\\vert =\\vert \\overline{\\mathcal {X}} (\\mathcal {S})\\vert $ [11], [5].",
"As shown in those papers this is equivalent to the equality $\\overline{\\mathcal {X}} (\\mathcal {S})=\\underline{\\mathcal {X}} (\\mathcal {S})$ , i.e., $\\mathcal {S}$ is ample if and only if any set shattered by $\\mathcal {S}$ is strongly shattered.",
"Consequently, the VC-dimension of an ample family is the dimension of the largest cube in its 1-inclusion graph.",
"A nice characterization of ample set families was provided in [35]: $\\mathcal {S}$ is ample if and only if for any cube $Q$ of $Q_m$ if $Q\\cap \\mathcal {S}$ is closed by taking antipodes, then either $Q\\cap \\mathcal {S}=\\varnothing $ or $Q$ is included in $G(\\mathcal {S})$ .",
"The paper [5] provides metric and recursive characterizations of ample families.",
"For example, it is shown in [5] that $\\mathcal {S}$ is ample if and only if any two parallel $X$ -cubes of the 1-inclusion graph $G(\\mathcal {S})$ of $\\mathcal {S}$ can be connected in $G(\\mathcal {S})$ by a shortest path of $X$ -cubes.",
"This implies that 1-inclusion graphs of ample set families are partial cubes; therefore further we will speak about ample partial cubes.",
"Note that maximum set families (i.e., those which the Sauer-Shelah lemma is tight) are ample."
],
[
"Characterizing tope graphs of OMs, COMS, and AMPs",
"In this subsection we recall the characterizations of [34] of tope graphs of COMs, OMs, and AMPs.",
"We say that a partial cube $G$ is a COM, an OM, an AOM, or an AMP if $G$ is the tope graph of a COM, OM, AOM, or AMP, respectively.",
"Tope graphs of COMs and AMPs are closed under pc-minors and tope graphs of OMs and AOMs are closed under contractions but not under restrictions.",
"Convex subgraphs of OMs are COMs and convex subgraphs of tope graphs of uniform OMs are ample.",
"The reverse implications are conjectured in [6] and [35], respectively.",
"As shown in [34], a partial cube is the tope graph of a COM if and only if all its antipodal subgraphs are gated.",
"Another characterization from the same paper is by an infinite family of excluded pc-minors.",
"This family is denoted by $\\mathcal {Q}^-$ and defined as follows.",
"For every $m \\ge 4$ there are partial cubes $X_m^1, \\ldots , X_m^{m+1}\\in \\mathcal {Q}^-$ .",
"Here, $X_m^{m+1}:=Q_m\\setminus \\lbrace (0,\\ldots ,0),(0,\\ldots ,1,0)\\rbrace $ , $X_m^{m}=X_m^{m+1}\\setminus \\lbrace (0,\\ldots ,0,1)\\rbrace $ , and $X_m^{m-i}=X_m^{m-i+1}\\setminus \\lbrace e_{im}\\rbrace $ .",
"Here $e_{im}$ denotes the vector with all zeroes except positions, $i$ and $m$ , where it is one.",
"See Figure REF for the members of $\\mathcal {Q}^-$ of isometric dimension at most 4.",
"Note in particular that $X_4^1=SK_4$ .",
"Ample partial cubes can be characterized by the excluding set $\\mathcal {Q}^{- -}=\\lbrace Q_m^{- -}: m\\ge 4\\rbrace $ , where $Q_m^{- -}=Q_m\\setminus \\lbrace (0,\\ldots ,0), (1,\\ldots ,1)\\rbrace $ [34].",
"Further characterizations from [34] yield that OMs are exactly the antipodal COMs, and (as mentioned at the end of Subsection REF ) AOMs are exactly the halfspaces of OMs.",
"On the other hand, ample partial cubes are exactly the partial cubes in which all antipodal subgraphs are hypercubes.",
"A central notion in COMs and OMs is the one of the rank of $G$ , which is the largest $d$ such that $G\\in \\mathcal {F}(Q_{d+1})$ .",
"Hence this notion fits well with the topic of the present paper and combining the families of excluded pc-minors $\\mathcal {Q}^{- -}$ and $\\mathcal {Q}^{-}$ , respectively, with $Q_3$ one gets: Proposition 5 [34] The class of two-dimensional ample partial cubes coincides with $\\mathcal {F}(Q_3, C_6)$ .",
"The class of two-dimensional COMs coincides with $\\mathcal {F}(Q_3, SK_4)$ .",
"Figure: From upper left to bottom right: a disk GG, a pseudoline arrangement UU whose region graph is GG, adding a line ℓ ∞ \\ell _{\\infty } to UU, the pseudocircle arrangement U ' U^{\\prime } obtained from U∪{ℓ ∞ }U\\cup \\lbrace \\ell _{\\infty }\\rbrace with a centrally mirrored copy, the pseudocircle arrangement U ' U^{\\prime } with region graph G ' G^{\\prime }, the OM G ' G^{\\prime } with halfspace GG."
],
[
"Disks",
"A pseudoline arrangement $U$ is a family of simple non-closed curves where every pair of curves intersects exactly once and crosses in that point.",
"Moreover, the curves must be extendable to infinity without introducing further crossings.",
"Note that several curves are allowed to cross in the same point.",
"Figure REF for an example.",
"We say that a partial cube $G$ is a disk if it is the region graph of a pseudoline arrangement $U$ .",
"The $\\Theta $ -classes of $G$ correspond to the elements of $U$ .",
"Contrary to a convention sometimes made in the literature, we allow a pseudoline arrangement $U$ to be empty, consisting of only one element, or all pseudolines to cross in a single point.",
"These situations yield the simplest examples of disks, namely: $K_1$ , $K_2$ , and the even cycles.",
"Disks are closed under contraction, since contracting a $\\Theta $ -class correspond to removing a line from the pseudoline arrangement.",
"It is well-known that disks are tope graphs of AOMs of rank at most 2.",
"A quick explanation can be found around [8].",
"The idea is to first add a line $\\ell _{\\infty }$ at infinity to the pseudoline arrangement $U$ representing $G$ .",
"Then embed the disk enclosed by $\\ell _{\\infty }$ on a hemisphere of $S^2$ , such that $\\ell _{\\infty }$ maps on the equator.",
"Now, mirror the arrangement through the origin of $S^2$ in order to obtain a pseudocircle arrangement $U^{\\prime }$ .",
"The region graph of $U^{\\prime }$ is an OM $G^{\\prime }$ , and the regions on one side of $\\ell _{\\infty }$ correspond to a halfspace of $G^{\\prime }$ isomorphic to $G$ .",
"See Figure REF for an illustration."
],
[
"Hyperplanes and isometric expansions",
"In this section we characterize the graphs from ${\\mathcal {F}}(Q_{d+1})$ (i.e., partial cubes of VC-dimension $\\le d$ ) via the hyperplanes of their $\\Theta $ -classes and via the operation of isometric expansion."
],
[
"Hyperplanes",
"Let $G$ be isometrically embedded in the hypercube $Q_m$ .",
"For a $\\Theta $ -class $E_i$ of $G$ , recall that $G^-_i,G^+_i$ denote the complementary halfspaces defined by $E_i$ and $\\partial G^-_i, G^+_i$ denote their boundaries.",
"The hyperplane $H_i$ of $E_i$ has the middles of edges of $E_i$ as the vertex-set and two such middles are adjacent in $H_i$ if and only if the corresponding edges belong to a common square of $G$ , i.e., $H_i$ is isomorphic to $\\partial G^-_i$ and $\\partial G^+_i$ .",
"Combinatorially, $H_i$ is the 1-inclusion graph of the set family defined by $\\partial H^-_i\\cup \\partial H^+_i$ by removing from each set the element $i$ .",
"Proposition 6 A partial cube $G$ has VC-dimension $\\le d$ (i.e., $G$ belongs to ${\\mathcal {F}}(Q_{d+1})$ ) if and only if each hyperplane $H_i$ of $G$ has VC-dimension $\\le d-1$ .",
"If some hyperplane $H_i$ of $G\\in {\\mathcal {F}}(Q_{d+1})$ has VC-dimension $d$ , then $\\partial G^-_i$ and $\\partial G^+_i$ also have VC-dimension $d$ and their union $\\partial H^-_i\\cup \\partial H^+_i$ has VC-dimension $d+1$ .",
"Consequently, $G$ has VC-dimension $\\ge d+1$ , contrary to Lemma REF .",
"To prove the converse implication, denote by ${\\mathcal {H}}_{d-1}$ the set of all partial cubes of $G$ in which the hyperplanes have VC-dimension $\\le d-1$ .",
"We assert that ${\\mathcal {H}}_{d-1}$ is closed under taking pc-minors.",
"First, ${\\mathcal {H}}_{d-1}$ is closed under taking restrictions because the hyperplanes $H^{\\prime }_i$ of any convex subgraph $G^{\\prime }$ of a graph $G\\in {\\mathcal {H}}_{d-1}$ are subgraphs of the respective hyperplanes $H_i$ of $G$ .",
"Next we show that ${\\mathcal {H}}_{d-1}$ is closed under taking contractions.",
"Let $G\\in {\\mathcal {H}}_{d-1}$ and let $E_i$ and $E_j$ be two different $\\Theta $ -classes of $G$ .",
"Since $\\pi _{j}(G)$ is a partial cube, to show that $\\pi _{j}(G)$ belongs to ${\\mathcal {H}}_{d-1}$ it suffices to show that $\\partial \\pi _{j}(G)^-_i=\\pi _{j}(\\partial G^-_i)$ .",
"Indeed, this would imply that the $i$ th hyperplane of $\\pi _{j}(G)$ coincides with the $j$ th contraction of the $i$ th hyperplane of $G$ .",
"Consequently, this would imply that the VC-dimension of all hyperplanes of $\\pi _{j}(G)$ is at most $d-1$ .",
"Pick $v\\in \\pi _{j}(\\partial G^-_i)$ .",
"Then $v$ is the image of the edge $v^{\\prime }v^{\\prime \\prime }$ of the hypercube $Q_m$ such that at least one of the vertices $v^{\\prime },v^{\\prime \\prime }$ , say $v^{\\prime }$ , belongs to $\\partial G^-_i$ .",
"This implies that the $i$ th neighbor $u^{\\prime }$ of $v^{\\prime }$ in $Q_m$ belongs to $\\partial G^+_i$ .",
"Let $u^{\\prime \\prime }$ be the common neighbor of $u^{\\prime }$ and $v^{\\prime \\prime }$ in $Q_m$ and $u$ be the image of the edge $u^{\\prime }u^{\\prime \\prime }$ by the $j$ -contraction.",
"Since $u^{\\prime }\\in \\partial G^+_i$ , the $i$ th edge $vu$ belongs to $\\pi _j(G)$ , whence $v\\in \\partial \\pi _{j}(G)^-_i$ and $u\\in \\partial \\pi _{j}(G)^+_i$ .",
"This shows $\\pi _{j}(\\partial G^-_i)\\subseteq \\partial \\pi _{j}(G)^-_i$ .",
"To prove the converse inclusion, pick a vertex $v\\in \\partial \\pi _{j}(G)^-_i$ .",
"This implies that the $i$ -neighbor $u$ of $v$ in $Q_m$ belongs to $\\partial \\pi _{j}(G)^+_i$ .",
"As in the previous case, let $v$ be the image of the $j$ -edge $v^{\\prime }v^{\\prime \\prime }$ of the hypercube $Q_m$ and let $u^{\\prime }$ and $u^{\\prime \\prime }$ be the $i$ -neighbors of $v^{\\prime }$ and $v^{\\prime \\prime }$ in $Q_m$ .",
"Then $u$ is the image of the $j$ -edge $u^{\\prime }u^{\\prime \\prime }$ .",
"Since the vertices $u$ and $v$ belong to $\\pi _{j}(G)$ , at least one vertex from each of the pairs $\\lbrace u^{\\prime },u^{\\prime \\prime }\\rbrace $ and $\\lbrace v^{\\prime },v^{\\prime \\prime }\\rbrace $ belongs to $G$ .",
"If one of the edges $u^{\\prime }v^{\\prime }$ or $u^{\\prime \\prime }v^{\\prime \\prime }$ of $Q_m$ is an edge of $G$ , then $u\\in \\pi _{j}(\\partial G^+_i)$ and $v\\in \\pi _{j}(\\partial G^-_i)$ and we are done.",
"Finally, suppose that $u^{\\prime }$ and $v^{\\prime \\prime }$ are vertices of $G$ .",
"Since $G$ is an isometric subgraph of $Q_m$ and $d(u^{\\prime },v^{\\prime \\prime })=2$ , a common neighbor $v^{\\prime },u^{\\prime \\prime }$ of $u^{\\prime }$ and $v^{\\prime \\prime }$ also belongs to $G$ and we fall in the previous case.",
"This shows that $\\partial \\pi _{j}(G)^-_i\\subseteq \\pi _{j}(\\partial G^-_i)$ .",
"Consequently, ${\\mathcal {H}}_{d-1}$ is closed under taking pc-minors.",
"Since $Q_{d+1}$ does not belong to ${\\mathcal {H}}_{d-1}$ , if $G$ belongs to ${\\mathcal {H}}_{d-1}$ , then $G$ does not have $Q_{d+1}$ as a pc-minor, i.e., $G\\in {\\mathcal {F}}(Q_{d+1})$ .",
"Corollary 7 A partial cube $G$ belongs to ${\\mathcal {F}}(Q_{3})$ if and only if each hyperplane $H_i$ of $G$ has VC-dimension $\\le 1$ .",
"Remark 8 In Proposition REF it is essential for $G$ to be a partial cube.",
"For example, let ${\\mathcal {S}}$ consist of all subsets of even size of an $m$ -element set.",
"Then the 1-inclusion graph $G({\\mathcal {S}})$ of ${\\mathcal {S}}$ consists of isolated vertices (i.e., $G({\\mathcal {S}})$ does not contain any edge).",
"Therefore, any hyperplane of $G({\\mathcal {S}})$ is empty, however the VC-dimension of $G({\\mathcal {S}})$ depends on $m$ and can be arbitrarily large.",
"By Corollary REF , the hyperplanes of graphs from ${\\mathcal {F}}(Q_{3})$ have VC-dimension 1.",
"However they are not necessarily partial cubes: any 1-inclusion graph of VC-dimension 1 may occur as a hyperplane of a graph from ${\\mathcal {F}}(Q_{3})$ .",
"Thus, it will be useful to establish the metric structure of 1-inclusion graphs of VC-dimension 1.",
"We say that a 1-inclusion graph $G$ is a virtual isometric tree of $Q_m$ if there exists an isometric tree $T$ of $Q_m$ containing $G$ as an induced subgraph.",
"Clearly, each virtually isometric tree is a forest in which each connected component is an isometric subtree of $Q_m$ .",
"Proposition 9 An induced subgraph $G$ of $Q_m$ has VC-dimension 1 if and only if $G$ is a virtual isometric tree of $Q_m$ .",
"Each isometric tree of $Q_m$ has VC-dimension 1, thus any virtual isometric tree has VC-dimension $\\le 1$ .",
"Conversely, let $G$ be an induced subgraph of $Q_m$ of VC-dimension $\\le 1$ .",
"We will say that two parallelism classes $E_i$ and $E_j$ of $Q_m$ are compatible on $G$ if one of the four intersections $G^-_i\\cap G^-_j, G^-_i\\cap G^+_j,G^+_i\\cap G^-_j, G^+_i\\cap G^+_j$ is empty and incompatible if the four intersections are nonempty.",
"From the definition of VC-dimension immediately follows that $G$ has VC-dimension 1 if and only if any two parallelism classes of $Q_m$ are compatible on $G$ .",
"By a celebrated result by Buneman [12] (see also [22]), on the vertex set of $G$ one can define a weighted tree $T_0$ with the same vertex-set as $G$ and such that the bipartitions $\\lbrace G^-_i,G^+_i\\rbrace $ are in bijection with the splits of $T_0$ , i.e., bipartitions obtained by removing edges of $T_0$ .",
"The length of each edge of $T_0$ is the number of $\\Theta $ -classes of $Q_m$ defining the same bipartition of $G$ .",
"The distance $d_{T_0}(u,v)$ between two vertices of $T_0$ is equal to the number of parallelism classes of $Q_m$ separating the vertices of $T_0$ .",
"We can transform $T_0$ into an isometrically embedded tree $T$ of $Q_m$ in the following way: if the edge $uv$ of $T_0$ has length $k>1$ , then replace this edge by any shortest path $P(u,v)$ of $Q_m$ between $u$ and $v$ .",
"Then it can be easily seen that $T$ is an isometric tree of $Q_m$ , thus $G$ is a virtual isometric tree."
],
[
"Isometric expansions",
"In order to characterize median graphs Mulder [42] introduced the notion of a convex expansion of a graph.",
"A similar construction of isometric expansion was introduced in [14], [15], with the purpose to characterize isometric subgraphs of hypercubes.",
"A triplet $(G^1,G^0,G^2)$ is called an isometric cover of a connected graph $G$ , if the following conditions are satisfied: $G^1$ and $G^2$ are two isometric subgraphs of $G$ ; $V(G)=V(G^1)\\cup V(G^2)$ and $E(G)=E(G^1)\\cup E(G^2)$ ; $V(G^1)\\cap V(G^2)\\ne \\varnothing $ and $G^0$ is the subgraph of $G$ induced by $V(G^1)\\cap V(G^2)$ .",
"A graph $G^{\\prime }$ is an isometric expansion of $G$ with respect to an isometric cover $(G^1,G^0,G^2)$ of $G$ (notation $G^{\\prime }=\\psi (G)$ ) if $G^{\\prime }$ is obtained from $G$ in the following way: replace each vertex $x$ of $V(G^1)\\setminus V(G^2)$ by a vertex $x_1$ and replace each vertex $x$ of $V(G^2)\\setminus V(G^1)$ by a vertex $x_2$ ; replace each vertex $x$ of $V(G^1)\\cap V(G^2)$ by two vertices $x_1$ and $x_2$ ; add an edge between two vertices $x_i$ and $y_i,$ $i=1,2$ if and only if $x$ and $y$ are adjacent vertices of $G^i$ , $i=1,2$ ; add an edge between any two vertices $x_1$ and $x_2$ such that $x$ is a vertex of $V(G^1)\\cap V(G^2)$ .",
"Figure: (a) The graph M * M_*.",
"(b) The isometric expansion M * ' M^{\\prime }_* of M * M_*.In other words, $G^{\\prime }$ is obtained by taking a copy of $G^1$ , a copy of $G^2$ , supposing them disjoint, and adding an edge between any two twins, i.e., two vertices arising from the same vertex of $G^0$ .",
"The following result characterizes all partial cubes by isometric expansions: Proposition 10 [14], [15] A graph is a partial cube if and only if it can be obtained by a sequence of isometric expansions from a single vertex.",
"We also need the following property of isometric expansions: Lemma 11 [18] If $S$ is a convex subgraph of a partial cube $G$ and $G^{\\prime }$ is obtained from $G$ by an isometric expansion $\\psi $ , then $S^{\\prime }:=\\psi (S)$ is a convex subgraph of $G^{\\prime }$ .",
"Example 12 For partial cubes, the operation of isometric expansion can be viewed as the inverse to the operation of contraction of a $\\Theta $ -class.",
"For example, the two-dimensional partial cube $M$ can be obtained from the two-dimensional partial cube $M_*$ (see Figure REF (b)) via an isometric expansion.",
"In Figure REF we present another isometric expansion $M^{\\prime }_*$ of $M_*$ .",
"By Proposition REF , $M^{\\prime }_*$ is a partial cube but one can check that it is no longer two-dimensional.",
"Therefore, contrary to all partial cubes, the classes ${\\mathcal {F}}(Q_{d+1})$ are not closed under arbitrary isometric expansions.",
"In this subsection, we characterize the isometric expansions which preserve the class ${\\mathcal {F}}(Q_{d+1})$ .",
"Let $G$ be isometrically embedded in the hypercube $Q_m=Q(X)$ .",
"Suppose that $G$ shatters the subset $Y$ of $X$ .",
"For a vertex $v_A$ of $Q(Y)$ (corresponding to a subset $A$ of $Y$ ), denote by $F(v_A)$ the set of vertices of the hypercube $Q_m$ which projects to $v_A$ .",
"In set-theoretical language, $F(v_A)$ consists of all vertices $v_B$ of $Q(X)$ corresponding to subsets $B$ of $X$ such that $B\\cap Y=A$ .",
"Therefore, $F(v_A)$ is a subcube of dimension $m-|Y|$ of $Q_m$ .",
"Let $G(v_A)=G\\cap F(v_A)$ .",
"Since $F(v_A)$ is a convex subgraph of $Q_m$ and $G$ is an isometric subgraph of $Q_m$ , $G(v_A)$ is also an isometric subgraph of $Q_m$ .",
"Summarizing, we obtain the following property: Lemma 13 If $G$ is an isometric subgraph of $Q_m=Q(X)$ which shatters $Y\\subseteq X$ , then for any vertex $v_A$ of $Q(Y)$ , $G(v_A)$ is a nonempty isometric subgraph of $G$ .",
"The following lemma establishes an interesting separation property in partial cubes: Lemma 14 If $(G^1,G^0,G^2)$ is an isometric cover of an isometric subgraph $G$ of $Q_m=Q(X)$ and $G^1$ and $G^2$ shatter the same subset $Y$ of $X$ , then $G^0$ also shatters $Y$ .",
"To prove that $G^0$ shatters $Y$ it suffices to show that for any vertex $v_A$ of $Q(Y)$ , $G^0\\cap F(v_A)$ is nonempty.",
"Since $G^1$ and $G^2$ both shatter $Q(Y)$ , $G^1\\cap F(v_A)$ and $G^2\\cap F(v_A)$ are nonempty subgraphs of $G$ .",
"Pick any vertices $x\\in V(G^1\\cap F(v_A))$ and $y\\in V(G^2\\cap F(v_A))$ .",
"Then $x$ and $y$ are vertices of $G(v_A)$ .",
"Since by Lemma REF , $G(v_A)$ is an isometric subgraph of $Q_m$ , there exists a shortest path $P(x,y)$ of $Q_m$ belonging to $G(v_A)$ .",
"Since $(G^1,G^0,G^2)$ is an isometric cover of $G$ , $P(x,y)$ contains a vertex $z$ of $G^0$ .",
"Consequently, $z\\in V(G^0\\cap F(v_A))$ , and we are done.",
"Proposition 15 Let $G^{\\prime }$ be obtained from $G\\in {\\mathcal {F}}(Q_{d+1})$ by an isometric expansion with respect to $(G^1,G^0,G^2)$ .",
"Then $G^{\\prime }$ belongs to ${\\mathcal {F}}(Q_{d+1})$ if and only if $G^0$ has VC-dimension $\\le d-1$ .",
"The fact that $G^{\\prime }$ is a partial cube follows from Proposition REF .",
"Let $E_{m+1}$ be the unique $\\Theta $ -class of $G^{\\prime }$ which does not exist in $G$ .",
"Then the halfspaces $(G^{\\prime })^-_{m+1}$ and $(G^{\\prime })^+_{m+1}$ of $G^{\\prime }$ are isomorphic to $G^1$ and $G^2$ and their boundaries $\\partial (G^{\\prime })^-_{m+1}$ and $\\partial (G^{\\prime })^+_{m+1}$ are isomorphic to $G^0$ .",
"If $G^{\\prime }$ belongs to ${\\mathcal {F}}(Q_{d+1})$ , by Proposition REF necessarily $G^0$ has VC-dimension $\\le d-1$ .",
"Conversely, let $G^0$ be of VC-dimension $\\le d-1$ .",
"Suppose that $G^{\\prime }$ has VC-dimension $d+1$ .",
"Since $G$ has VC-dimension $d$ , this implies that any set $Y^{\\prime }$ of size $d+1$ shattered by $G^{\\prime }$ contains the element $m+1$ .",
"Let $Y=Y^{\\prime }\\setminus \\lbrace m+1\\rbrace $ .",
"The $(m+1)$ th halfspaces $(G^{\\prime })^-_{m+1}$ and $(G^{\\prime })^+_{m+1}$ of $G^{\\prime }$ shatter the set $Y$ .",
"Since $(G^{\\prime })^-_{m+1}$ and $(G^{\\prime })^+_{m+1}$ are isomorphic to $G^1$ and $G^2$ , the subgraphs $G^1$ and $G^2$ of $G$ both shatter $Y$ .",
"By Lemma REF , the subgraph $G^0$ of $G$ also shatters $Y$ .",
"Since $|Y|=d$ , this contradicts our assumption that $G^0$ has VC-dimension $\\le d-1$ .",
"Let us end this section with a useful lemma with respect to antipodal partial cubes: Lemma 16 If $G$ is a proper convex subgraph of an antipodal partial cube $H\\in \\mathcal {F}(Q_{d+1})$ , then $G\\in \\mathcal {F}(Q_{d})$ .",
"Suppose by way of contradiction that $G$ has $Q_d$ as a pc-minor.",
"Since convex subgraphs of $H$ are intersections of halfspaces, there exists a $\\Theta $ -class $E_i$ of $H$ such that $G$ is included in the halfspace $H_i^+$ .",
"Since $H$ is antipodal, the subgraph $-G \\subseteq H_i^-$ consisting of antipodes of vertices of $G$ is isomorphic to $G$ .",
"As $G\\subseteq H^+_i$ , $-G$ and $G$ are disjoint.",
"Since $G$ has $Q_d$ as a pc-minor, $-G$ also has $Q_{d}$ as a pc-minor: both those minors are obtained by contracting the same set $I$ of $\\Theta $ -classes of $H$ ; note that $E_i\\notin I$ .",
"Thus, contracting the $\\Theta $ -classes from $I$ and all other $\\Theta $ -classes not crossing the $Q_d$ except $E_i$ , we will get an antipodal graph $H^{\\prime }$ , since antipodality is preserved by contractions.",
"Now, $H^{\\prime }$ consists of two copies of $Q_d$ separated by $E_i$ .",
"Take any vertex $v$ in $H^{\\prime }$ .",
"Then there is a path from $v$ to $-v$ first crossing all $\\Theta $ -classes of the cube containing $v$ and then $E_i$ , to finally reach $-v$ .",
"Thus, $-v$ is adjacent to $E_i$ and hence every vertex of $H^{\\prime }$ is adjacent to $E_i$ .",
"Thus $H^{\\prime }=Q_{d+1}$ , contrary to the assumption that $H\\in \\mathcal {F}(Q_{d+1})$ ."
],
[
"Gated hulls of 6-cycles",
"In this section, we prove that in two-dimensional partial cubes the gated hull of any 6-cycle $C$ is either $C$ , or $Q^-_3$ , or a maximal full subdivision of $K_n$ ."
],
[
"Full subdivisions of $K_n$",
"A full subdivision of $K_n$ (or full subdivision for short) is the graph $SK_n$ obtained from the complete graph $K_n$ on $n$ vertices by subdividing each edge of $K_n$ once; $SK_n$ has $n+\\binom{n}{2}$ vertices and $n(n-1)$ edges.",
"The $n$ vertices of $K_n$ are called original vertices of $SK_n$ and the new vertices are called subdivision vertices.",
"Note that $SK_3$ is the 6-cycle $C_6$ .",
"Each $SK_n$ can be isometrically embedded into the $n$ -cube $Q_n$ in such a way that each original vertex $u_i$ is encoded by the one-element set $\\lbrace i\\rbrace $ and each vertex $u_{i,j}$ subdividing the edge $ij$ of $K_n$ is encoded by the 2-element set $\\lbrace i,j\\rbrace $ (we call this embedding of $SK_n$ a standard embedding).",
"If we add to $SK_n$ the vertex $v_{\\varnothing }$ of $Q_n$ which corresponds to the empty set $\\varnothing $ , we will obtain the partial cube $SK^*_n$ .",
"Since both $SK_n$ and $SK^*_n$ are encoded by subsets of size $\\le 2$ , those graphs have VC-dimension 2.",
"Consequently, we obtain: Lemma 17 For any $n$ , $SK_n$ and $SK^*_n$ are two-dimensional partial cubes.",
"Figure: (a) An isometric embedding of SK 4 SK_4 into Q 4 Q_4.",
"(b) A standard embedding of SK 4 SK_4.",
"(c) A completion of SK 4 SK_4 to SK 4 * SK^*_4.Example 18 Our running example $M$ contains two isometrically embedded copies of $SK_4$ .",
"In Figure REF (a)&(b) we present two isometric embeddings of $SK_4$ into the 4-cube $Q_4$ , the second one is the standard embedding of $SK_4$ .",
"The original and subdivision vertices are illustrated by squares and circles, respectively.",
"Figure REF (c) describes the completion of $SK_4$ to $SK^*_4$ .",
"Lemma 19 If $H=SK_n$ with $n \\ge 4$ is an isometric subgraph of a partial cube $G$ , then $G$ admits an isometric embedding into a hypercube such that the embedding of $H$ is standard.",
"Pick any original vertex of $H$ as the base point $b$ of $G$ and consider the standard isometric embedding $\\varphi $ of $G$ into $Q_m$ .",
"Then $\\varphi (b)=\\varnothing $ .",
"In $H$ the vertex $b$ is adjacent to $n-1\\ge 3$ subdivision vertices of $H$ .",
"Then for each of those vertices $v_i, i=1,\\ldots ,n-1,$ we can suppose that $\\varphi (v_i)=\\lbrace i\\rbrace $ .",
"Each $v_i$ is adjacent in $H$ to an original vertex $u_i\\ne b$ .",
"Since $H$ contains at least three such original vertices and they have pairwise distance 2, one can easily check that the label $\\varphi (u_i)$ consists of $i$ and an element common to all such vertices, denote it by $n$ .",
"Finally, the label of any subdivision vertex $u_{i,j}$ adjacent to the original vertices $u_i$ and $u_j$ is $\\lbrace i,j\\rbrace $ .",
"Now consider an isometric embedding $\\varphi ^{\\prime }$ of $G$ defined by setting $\\varphi ^{\\prime }(v)=\\varphi (v)\\Delta \\lbrace n\\rbrace $ for any vertex $v$ of $G$ .",
"Then $\\varphi ^{\\prime }$ provides a standard embedding of $H$ : $\\varphi ^{\\prime }(b)=\\lbrace n\\rbrace ,$ $\\varphi ^{\\prime }(u_i)=\\lbrace i\\rbrace $ for any original vertex $u_i$ , and $\\varphi ^{\\prime }(v_i)=\\lbrace i,n\\rbrace $ for any subdivision vertex $v_i$ adjacent to $b$ and $\\varphi ^{\\prime }(u_{i,j})=\\lbrace i,j\\rbrace $ for any other subdivision vertex $u_{i,j}$ .",
"By Lemma REF , when a full subdivision $H=SK_n$ of a graph $G\\in {\\mathcal {F}}(Q_3)$ is fixed, we assume that $G$ is isometrically embedded in a hypercube so that $H$ is standardly embedded.",
"We describe next the isometric expansions of $SK_n$ which result in two-dimensional partial cubes.",
"An isometric expansion of a partial cube $G$ with respect to $(G^1,G^0,G^2)$ is called peripheral if at least one of the subgraphs $G^1,G^2$ coincides with $G^0$ , i.e., $G^1\\subseteq G^2$ or $G^2\\subseteq G^1$ .",
"Lemma 20 If $G^{\\prime }$ is obtained from $G:=SK_n$ with $n \\ge 4$ by an isometric expansion with respect to $(G^1,G^0,G^2)$ , then $G^{\\prime }\\in {\\mathcal {F}}(Q_3)$ if and only if this is a peripheral expansion and $G^0$ is an isometric tree of $SK_n$ .",
"The fact that an isometric expansion of $SK_n$ , such that $G^0$ is an isometric tree, belongs to ${\\mathcal {F}}(Q_3)$ follows from Proposition REF and Lemma REF .",
"Conversely, suppose that $G^{\\prime }$ belongs to ${\\mathcal {F}}(Q_3)$ .",
"By Proposition REF , $G^0$ has VC-dimension $\\le 1$ and by Proposition REF $G^0$ is a virtual tree.",
"It suffices to prove that $G^1$ or $G^2$ coincides with $G^0$ .",
"Indeed, since $G^1$ and $G^2$ are isometric subgraphs of $SK_n$ , this will also imply that $G^0$ is an isometric tree.",
"We distinguish two cases.",
"Case 1.",
"First, let $G^0$ contain two original vertices $u_{i}$ and $u_{j}$ .",
"Since $u_{i}$ and $u_{j}$ belong to $G^1$ and $G^2$ and those two subgraphs are isometric subgraphs of $G$ , the unique common neighbor $u_{i,j}$ of $u_{i}$ and $u_{j}$ must belong to $G^1$ and $G^2$ , and thus to $G^0$ .",
"If another original vertex $u_{k}$ belongs to $G^0$ , then the four vertices $u_{i,j}, u_{i}, u_{j},u_{k}$ of $G^0$ shatter the set $\\lbrace i,j\\rbrace $ , contrary to the assumption that $G^0$ has VC-dimension $\\le 1$ (Proposition REF ).",
"This implies that each other original vertex $u_{k}$ either belongs to $G^1\\setminus G^2$ or to $G^2\\setminus G^1$ .",
"If there exist original vertices $u_{k}$ and $u_{\\ell }$ such that $u_{k}$ belongs to $G^1\\setminus G^2$ and $u_{\\ell }$ belongs to $G^2\\setminus G^1$ , then their unique common neighbor $u_{k,\\ell }$ necessarily belongs to $G^0$ .",
"But in this case the four vertices $u_{i,j}, u_{i}, u_{j},u_{k,\\ell }$ of $G^0$ shatter the set $\\lbrace i,j\\rbrace $ .",
"Thus we can suppose that all other original vertices $u_{k}$ belong to $G^1\\setminus G^2$ .",
"Moreover, for the same reason and since $G^1$ is an isometric subgraph of $G$ , any vertex $u_{k,\\ell }$ with $\\lbrace k,\\ell \\rbrace \\ne \\lbrace i,j\\rbrace $ also belongs to $G^1\\setminus G^2$ .",
"Since $G^1$ is an isometric subgraph of $G$ , for any $k\\ne i,j$ , the vertices $u_{i,k}, u_{j,k}$ belong to $G^1$ .",
"Therefore $G^1=G$ and $G^0=G^2$ .",
"Since $G^2$ is an isometric subgraph of $G$ and $G^0$ has VC-dimension $\\le 1$ , $G^0$ is an isometric subtree of $G$ .",
"Case 2.",
"Now, suppose that $G^0$ contains at most one original vertex.",
"Let $A^1$ be the set of original vertices belonging to $G^1\\setminus G^2$ and $A^2$ be the set of original vertices belonging to $G^2\\setminus G^1$ .",
"First suppose that $|A^1|\\ge 2$ and $|A^2|\\ge 2$ , say $u_{1},u_{2}\\in A^1$ and $u_{3},u_{4}\\in A^2$ .",
"But then the vertices $u_{1,3},u_{1,4},u_{2,3},u_{2,4}$ must belong to $G^0$ .",
"Since those four vertices shatter the set $\\lbrace 1,3\\rbrace $ , we obtain a contradiction that $G^0$ has VC-dimension $\\le 1$ .",
"Hence, one of the sets $A^1$ or $A^2$ contains at most one vertex.",
"Suppose without loss of generality that $A^1$ contains at least $n-2$ original vertices $u_{1},u_{2},\\ldots , u_{n-2}$ .",
"First suppose that $G^1$ contains all original vertices.",
"Then since $G^1$ is an isometric subgraph of $G$ , each subdivision vertex $u_{i,j}$ also belongs to $G^1$ .",
"This implies that $G^1=G$ and we are done.",
"Thus suppose that the vertex $u_{n}$ does not belong to $A^1$ .",
"Since $G^0$ contains at most one original vertex, one of the vertices $u_{n-1},u_{n}$ , say $u_{n}$ , must belong to $A^2$ (i.e., to $G^2\\setminus G^1$ ).",
"This implies that all vertices $u_{i,n}, i=1,\\ldots ,n-2$ belong to $G^0$ .",
"Since $n\\ge 4$ and $u_{n}$ is the unique common neighbor of the vertices $u_{i,n}$ and $u_{j,n}$ with $i\\ne j$ and $1\\le i,j\\le n-2$ and $G^1$ is an isometric subgraph of $G$ , necessarily $u_{n}$ must be a vertex of $G^1$ , contrary to our assumption that $u_{n}\\in A^2$ .",
"This contradiction concludes the proof of the lemma.",
"Corollary 21 If $G \\in \\mathcal {F}(Q_3)$ and $G$ contains $SK_n$ with $n \\ge 4$ as a pc-minor, then $G$ contains $SK_n$ as a convex subgraph.",
"Suppose by way of contradiction that $G^{\\prime }$ is a smallest graph in $\\mathcal {F}(Q_3)$ which contains $SK_n$ as a pc-minor but does not contain $SK_n$ as a convex subgraph.",
"This means that any contraction of $G^{\\prime }$ along a $\\Theta $ -class of $G^{\\prime }$ that do not cross the $SK_n$ pc-minor, also contains this $SK_n$ as a pc-minor.",
"We denote the resulting graph by $G$ .",
"Since $G\\in \\mathcal {F}(Q_3)$ , by minimality choice of $G^{\\prime }$ , $G$ contains $SK_n$ as a convex subgraph, denote this subgraph by $H$ .",
"Now, $G^{\\prime }$ is obtained from $G$ by an isometric expansion.",
"By Lemma REF , $H^{\\prime }=\\psi (H)$ is a convex subgraph of $G^{\\prime }$ .",
"Since $G^{\\prime }\\in \\mathcal {F}(Q_3)$ , by Lemma REF this isometric expansion restricted to $H=SK_n$ is a peripheral expansion.",
"This implies that the image of $H$ under this expansion is a convex subgraph $H^{\\prime }$ of $G^{\\prime }$ which contains a copy of $SK_n$ as a convex subgraph, and thus $G^{\\prime }$ contains a convex copy of $SK_n$ .",
"Lemma 22 If $C=SK_3$ is an isometric 6-cycle of $G\\in {\\mathcal {F}}(Q_3)$ , then $C$ is convex or its convex hull is $Q^-_3$ .",
"The convex hull of $C$ in $Q_m$ is a 3-cube $Q$ and ${\\rm conv}(C)=Q\\cap V(G)$ .",
"Since $G$ belongs to ${\\mathcal {F}}(Q_3)$ , $Q$ cannot be included in $G$ .",
"Hence either ${\\rm conv}(C)=C$ or ${\\rm conv}(C)=Q^-_3$ ."
],
[
"Gatedness of full subdivisions of $K_n$",
"The goal of this subsection is to prove the following result: Proposition 23 If $H=SK_n$ with $n \\ge 4$ is a convex subgraph of $G\\in {\\mathcal {F}}(Q_3)$ and $H$ is not included in a larger full subdivision of $G$ , then $H$ is a gated subgraph of $G$ .",
"The proof of Proposition REF uses the results of previous subsection and two claims.",
"Claim 24 If $H=SK_n$ with $n \\ge 4$ is an isometric subgraph of $G\\in {\\mathcal {F}}(Q_3)$ , then either $H$ extends in $G$ to $SK^*_n$ or $H$ is a convex subgraph of $G$ .",
"Suppose by way of contradiction that $H=SK_n$ does not extend in $G$ to $SK^*_n$ however $H$ is not convex.",
"Then there exists a vertex $v\\in V(G)\\setminus V(H)$ such that $v\\in I(x,y)$ for two vertices $x,y\\in V(H)$ .",
"First note that $x$ and $y$ cannot be both original vertices.",
"Indeed, if $x=u_i$ and $y=u_j$ , then in $Q_m$ the vertices $x$ and $y$ have two common neighbors: the subdivision vertex $u_{i,j}$ and $v_{\\varnothing }$ .",
"But $v_{\\varnothing }$ is adjacent in $Q_m$ to all original vertices of $H$ , thus it cannot belong to $G$ because $H=SK_n$ does not extend to $SK^*_n$ .",
"Thus, further we can suppose that the vertex $x$ is a subdivision vertex, say $x=u_{i,j}$ .",
"We distinguish several cases depending of the value of $d(x,y)$ .",
"Case 1.",
"$d(x,y)=2$ .",
"This implies that $y=u_{i,k}$ is also a subdivision vertex and $x$ and $y$ belong in $H$ to a common isometric 6-cycle $C$ .",
"Since $v$ belongs to ${\\rm conv}(C)$ , Lemma REF implies that $v$ is adjacent to the third subdivision vertex $z=u_{j,k}$ of $C$ .",
"Hence $v=\\lbrace i,j,k\\rbrace $ .",
"Since $n\\ge 4$ , there exists $\\ell \\ne i,j,k$ such that $\\lbrace \\ell \\rbrace $ is an original vertex of $H$ and $\\lbrace i,\\ell \\rbrace ,\\lbrace j,\\ell \\rbrace ,$ and $\\lbrace k,\\ell \\rbrace $ are subdivision vertices of $H$ .",
"Contracting $\\ell $ , we will obtain a forbidden $Q_3$ .",
"Case 2.",
"$d(x,y)=3$ .",
"This implies that $y=u_k$ is an original vertex with $k\\ne i,j$ .",
"Then again the vertices $x$ and $y$ belong in $H$ to a common isometric 6-cycle $C$ .",
"Since $v$ belongs to ${\\rm conv}(C)$ , Lemma REF implies that either $v$ is adjacent to $u_i,u_j$ , and $u_k$ or to $u_{i,j},u_{i,k}$ , and $u_{j,k}$ , which was covered by the Case 1.",
"Case 3.",
"$d(x,y)=4$ .",
"This implies that $y=u_{ k,\\ell }$ is a subdivision vertex with $k,\\ell \\ne i,j$ .",
"In view of the previous cases, we can suppose that $v$ is adjacent to $x$ or to $y$ , say $v$ is adjacent to $x$ .",
"Let $Q$ be the convex hull of $\\lbrace x,y\\rbrace $ in $Q_m$ .",
"Then $Q$ is a 4-cube and $x=\\lbrace i,j\\rbrace $ has 4 neighbors in $Q$ : $\\lbrace i\\rbrace , \\lbrace j\\rbrace , \\lbrace i,j,k\\rbrace $ and $\\lbrace i,j,\\ell \\rbrace $ .",
"The vertices $\\lbrace i\\rbrace , \\lbrace j\\rbrace $ are original vertices of $H$ .",
"Thus suppose that $v$ is one of the vertices $\\lbrace i,j,k\\rbrace ,\\lbrace i,j,\\ell \\rbrace $ , say $v=\\lbrace i,j,k\\rbrace $ .",
"But then $v$ is adjacent to $\\lbrace j,k\\rbrace $ , which is a subdivision vertex of $H$ and we are in the conditions of Case 1.",
"Hence $H$ is a convex subgraph of $G$ .",
"Claim 25 If $H=SK_n$ with $n \\ge 4$ is a convex subgraph of $G\\in {\\mathcal {F}}(Q_3)$ and $H$ is not included in a larger full subdivision in $G$ , then the vertex $v_{\\varnothing }$ of $Q_m$ is adjacent only to the original vertices $u_1,\\ldots ,u_n$ of $H$ .",
"Since $H$ is convex, the vertex $v_{\\varnothing }$ of $Q_m$ is not a vertex of $G$ .",
"Let $u_i=\\lbrace i\\rbrace , i=1,\\ldots ,n$ be the original vertices of $H$ .",
"Suppose that in $Q_m$ the vertex $v_{\\varnothing }$ is adjacent to a vertex $u$ of $G$ , which is not included in $H$ , say $u=\\lbrace n+1\\rbrace $ .",
"Since $u$ and each $u_i$ has in $Q_m$ two common neighbors $v_{\\varnothing }$ and $u_{i,n+1}=\\lbrace i,n+1\\rbrace $ and since $G$ is an isometric subgraph of $Q_m$ , necessarily each vertex $u_{i,n+1}$ is a vertex of $G$ .",
"Consequently, the vertices of $H$ together with the vertices $u,u_{1,n+1},\\ldots ,u_{n,n+1}$ define an isometric subgraph $H^{\\prime }=SK_{n+1}$ of $Q_m$ .",
"Since $v_{\\varnothing }$ does not belong to $G$ , by Claim REF $H^{\\prime }$ is convex, contrary to the assumption that $H$ is not included in a larger convex full subdivision of $G$ .",
"Consequently, the neighbors in $G$ of $v_{\\varnothing }$ are only the original vertices $u_1,\\ldots ,u_n$ of $H$ .",
"Now, we prove Proposition REF .",
"Let $G\\in {\\mathcal {F}}(Q_3)$ be an isometric subgraph of the cube $Q_m$ in such that the embedding of $H$ is standard.",
"Let $Q$ be the convex hull of $H$ in $Q_m$ ; $Q$ is a cube of dimension $n$ and a gated subgraph of $Q_m$ .",
"Let $v$ be a vertex of $G$ and $v_0$ be the gate of $v$ in $Q$ .",
"To prove that $H$ is gated it suffices to show that $v_0$ is a vertex of $H$ .",
"Suppose by way of contradiction that $H$ is not gated in $G$ and among the vertices of $G$ without a gate in $H$ pick a vertex $v$ minimizing the distance $d(v,v_0)$ .",
"Suppose that $v$ is encoded by the set $A$ .",
"Then its gate $v_0$ in $Q_m$ is encoded by the set $A_0:=A\\cap \\lbrace 1,\\ldots ,n\\rbrace $ .",
"If $|A_0|=1,2$ , then $A_0$ encodes an original or subdivided vertex of $H$ , therefore $v_0$ would belong to $H$ , contrary to the choice of $v$ .",
"So, $A_0=\\varnothing $ or $|A_0|>2$ .",
"First suppose that $A_0=\\varnothing $ , i.e., $v_0=v_{\\varnothing }$ .",
"Since $v_\\varnothing $ is adjacent only to the original vertices of $H$ , by Claim REF all original vertices of $H$ have distance $k=d(v,v_{\\varnothing })+1\\ge 3$ to $v$ .",
"From the choice of $v$ it follows that $I(v,u_i)\\cap I(v,u_j)=\\lbrace v\\rbrace $ for any two original vertices $u_i$ and $u_j$ , $i\\ne j$ .",
"Indeed, if $I(v,u_i)\\cap I(v,u_j)\\ne \\lbrace v\\rbrace $ and $w$ is a neighbor of $v$ in $I(v,u_i)\\cap I(v,u_j)$ , then $d(w,u_i)=d(w,u_j)=k-1$ .",
"Therefore the gate $w_0$ of $w$ in $Q$ has distance at most $k-2$ from $w$ , yielding that $d(v,w_0)=k-1$ .",
"This is possible only if $w_0=v_0$ .",
"Therefore, replacing $v$ by $w$ we will get a vertex of $G$ whose gate $w_0=v_0$ in $Q$ does not belong to $H$ and for which $d(w,w_0)<d(v,v_0)$ , contrary to the minimality in the choice of $v$ .",
"Thus $I(v,u_i)\\cap I(v,u_j)=\\lbrace v\\rbrace $ .",
"Let $A=\\lbrace n+1,\\ldots , n+k-1\\rbrace $ .",
"If $k=3$ , then $v$ is encoded by $A=\\lbrace n+1,n+2\\rbrace $ .",
"By Claim REF , any shortest path of $G$ from $u_i=\\lbrace i\\rbrace $ to $v$ must be of the form $(\\lbrace i\\rbrace , \\lbrace i,\\ell \\rbrace , \\lbrace \\ell \\rbrace , \\lbrace n+1,n+2\\rbrace )$ , where $\\ell \\in \\lbrace n+1,n+2\\rbrace $ .",
"Since we have at least four original vertices, at least two of such shortest paths of $G$ will pass via the same neighbor $\\lbrace n+1\\rbrace $ or $\\lbrace n+2\\rbrace $ of $v$ , contrary to the assumption that $I(v,u_i)\\cap I(v,u_j)=\\lbrace v\\rbrace $ for any $u_i$ and $u_j$ , $i\\ne j$ .",
"If $k\\ge 4$ , let $G^{\\prime }=\\pi _{n+1}(G)$ and $H^{\\prime }=\\pi _{n+1}(H)$ be the images of $G$ and $H$ by contracting the edges of $Q_m$ corresponding to the coordinate $n+1$ .",
"Then $G^{\\prime }$ is an isometric subgraph of the hypercube $Q_{m-1}$ and $H^{\\prime }$ is a full subdivision isomorphic to $SK_n$ isometrically embedded in $G^{\\prime }$ .",
"Let also $v^{\\prime },v^{\\prime }_{\\varnothing ,}$ and $u^{\\prime }_i, i=1,\\ldots ,n,$ denote the images of the vertices $v, v_{\\varnothing },$ and $u_i$ of $G$ .",
"Then $u^{\\prime }_1,\\ldots , u^{\\prime }_n$ are the original vertices of $H^{\\prime }$ .",
"Notice also that $v^{\\prime }$ has distance $k-1$ to all original vertices of $H^{\\prime }$ and distance $k-2$ to $v^{\\prime }_{\\varnothing }$ .",
"Thus in $G^{\\prime }$ the vertex $v^{\\prime }$ does not have a gate in $H^{\\prime }$ .",
"By the minimality in the choice of $v$ and $H$ , either $H^{\\prime }$ is not convex in $G^{\\prime }$ or $H^{\\prime }$ is included in a larger full subdivision of $G^{\\prime }$ .",
"If $H^{\\prime }$ is not convex in $G^{\\prime }$ , by Claim REF $v^{\\prime }_{\\varnothing }$ must be a vertex of $G^{\\prime }$ .",
"Since $v_{\\varnothing }$ is not a vertex of $G$ , this is possible only if the set $\\lbrace n+1\\rbrace $ corresponds to a vertex of $G$ .",
"But we showed in Claim REF that the only neighbors of $v_{\\varnothing }$ in $G$ are the original vertices of $H$ .",
"This contradiction shows that $H^{\\prime }$ is a convex.",
"Therefore, suppose that $H^{\\prime }$ is included in a larger full subdivision $H^{\\prime \\prime }=SK_{n+1}$ of $G^{\\prime }$ .",
"Denote by $u^{\\prime }_{\\ell }=\\lbrace \\ell \\rbrace $ the original vertex of $H^{\\prime \\prime }$ different from the vertices $u^{\\prime }_i, i=1,\\ldots ,n$ ; hence $\\ell \\notin \\lbrace 1,\\ldots ,n\\rbrace $ .",
"Since $u^{\\prime }_{\\ell }$ is a vertex of $G^{\\prime }$ and in $Q_m$ the set $\\lbrace \\ell \\rbrace $ does not correspond to a vertex of $G$ , necessarily the set $\\lbrace n+1,\\ell \\rbrace $ is a vertex of $G$ in $Q_m$ .",
"Therefore, we are in the conditions of the previous subcase, which was shown to be impossible.",
"This concludes the analysis of case $A_0=\\varnothing $ .",
"Now, suppose that $|A_0|\\ge 3$ and let $A_0=\\lbrace 1,2,3,\\ldots ,k\\rbrace $ .",
"This implies that the vertices $u_1,u_2,u_3$ are original vertices and $u_{1,2},u_{1,3},u_{2,3}$ are subdivision vertices of $H$ .",
"Since $H=SK_n$ with $n\\ge 4$ , $H$ contains an original vertex $u_{\\ell }$ with $\\ell \\ge 4$ , say $\\ell =4$ .",
"But then the sets corresponding to the vertices $u_1,u_2,u_3,u_4,u_{1,2},u_{1,3},u_{2,3},$ and $v$ of $G$ shatter the set $\\lbrace 1,2,3\\rbrace $ , contrary to the assumption that $G\\in {\\mathcal {F}}(Q_3)$ .",
"This concludes the case $|A_0|\\ge 3$ .",
"Consequently, for any vertex $v$ of $G$ the gate $v_0$ of $v$ in $Q$ belongs to $H$ .",
"This shows that $H$ is a gated subgraph of $G$ and concludes the proof of the proposition."
],
[
"Gated hulls of 6-cycles",
"The goal of this subsection is to prove the following result: Proposition 26 If $C$ is an induced (and thus isometric) 6-cycle of $G\\in {\\mathcal {F}}(Q_3)$ , then the gated hull ${\\rm gate}(C)$ of $C$ is either $C$ , or $Q_3^-$ , or a full subdivision.",
"If $C$ is included in a maximal full subdivision $H=SK_n$ with $n\\ge 4$ , by Proposition REF $H$ is gated.",
"Moreover, one can directly check that any vertex of $H\\setminus C$ must be included in the gated hull of $C$ , whence ${\\rm gate}( C )=H$ .",
"Now suppose that $C$ is not included in any full subdivision $SK_n$ with $n\\ge 4$ .",
"By Lemma REF , $S:={\\rm conv}(C)$ is either $C$ or $Q^-_3$ .",
"In this case we assert that $S$ is gated and thus ${\\rm gate}(C)={\\rm conv}(C)$ .",
"Suppose that $G$ is a two-dimensional partial cube of smallest size for which this is not true.",
"Let $v$ be a vertex of $G$ that has no gate in $S$ and is as close as possible to $S$ , where $d_G(v,S)=\\min \\lbrace d_G(v,z): z\\in S\\rbrace $ is the distance from $v$ to $S$ .",
"Given a $\\Theta $ -class $E_i$ of $G$ , let $G^{\\prime }:=\\pi _i(G)$ , $S^{\\prime }:=\\pi _i(S)$ , and $C^{\\prime }=\\pi _i(C)$ .",
"For a vertex $u$ of $G$ , let $u^{\\prime }:=\\pi _i(u)$ .",
"Since any convex subgraph of $G$ is the intersection of halfspaces, if all $\\Theta $ -classes of $G$ cross $S$ , then $S$ coincides with $G$ , contrary to the choice of $G$ .",
"Thus $G$ contains $\\Theta $ -classes not crossing $S$ .",
"First suppose that there exists a $\\Theta $ -class $E_i$ of $G$ not crossing $S$ such that $S^{\\prime }$ is convex in $G^{\\prime }$ .",
"Since $G^{\\prime }\\in {\\mathcal {F}}(Q_3)$ , by Lemma REF either the 6-cycle $C^{\\prime }$ is convex or its convex hull in $G^{\\prime }$ is $Q^-_3$ .",
"Since the distance in $G^{\\prime }$ between $v^{\\prime }$ and any vertex of $S^{\\prime }$ is either the same as the distance in $G$ between $v$ and the corresponding vertex of $S$ (if $E_i$ does not separate $v$ from $S$ ) or is one less than the corresponding distance in $G$ (if $v$ and $S$ belong to complementary halfspaces defined by $E_i$ ), $S^{\\prime }$ is not gated in $G^{\\prime }$ , namely the vertex $v^{\\prime }$ has no gate in $S^{\\prime }$ .",
"Therefore, if $S^{\\prime }=Q^-_3$ , then contracting all $\\Theta $ -classes of $G^{\\prime }$ separating $S^{\\prime }$ from $v^{\\prime }$ , we will get $Q_3$ as a pc-minor, contrary to the assumption that $G$ and $G^{\\prime }$ belong to $\\mathcal {F}(Q_3)$ .",
"This implies that $S^{\\prime }=C^{\\prime }$ and thus that $S=C$ .",
"Moreover, by minimality of $G$ , the 6-cycle $C^{\\prime }$ is included in a maximal full subdivision $H^{\\prime }=SK_n$ of $G^{\\prime }$ .",
"By Proposition REF , $H^{\\prime }$ is a gated subgraph of $G^{\\prime }$ .",
"Let $w^{\\prime }$ be the gate of $v^{\\prime }$ in $H^{\\prime }$ (it may happen that $w^{\\prime }=v^{\\prime }$ ).",
"Since $C^{\\prime }$ is not gated, necessarily $w^{\\prime }$ is not a vertex of $C^{\\prime }$ .",
"For the same reason, $w^{\\prime }$ is not adjacent to a vertex of $C^{\\prime }$ .",
"The graph $G$ is obtained from $G^{\\prime }$ by an isometric expansion $\\psi _i$ (inverse to $\\pi _i$ ).",
"By Lemma REF , $\\psi _i$ , restricted to $H^{\\prime }$ , is a peripheral expansion along an isometric tree of $H^{\\prime }$ .",
"By Corollary REF , $G$ contains an isometric subgraph isomorphic to $H^{\\prime }$ .",
"By the choice of $E_i$ , $C$ does not cross $E_i$ , and this implies that in $G$ the convex cycle $C$ is contained in a full subdivision of $K_n$ , contrary to the choice of $C$ .",
"Now, suppose that for any $\\Theta $ -class $E_i$ of $G$ not crossing $S$ , $S^{\\prime }$ is not convex in $G^{\\prime }$ .",
"Since $C^{\\prime }$ is an isometric 6-cycle of $G^{\\prime }$ , $G^{\\prime }\\in {\\mathcal {F}} (Q_3)$ , and the 6-cycle $C^{\\prime }$ is not convex in $G^{\\prime }$ , by Lemma REF we conclude that the convex hull of $C^{\\prime }$ in $G^{\\prime }$ is $Q^-_3$ and this $Q^-_3$ is different from $S^{\\prime }$ .",
"Hence $S^{\\prime }=C^{\\prime }$ and $S=C$ .",
"This implies that there exists a vertex $z^{\\prime }$ of $G^{\\prime }$ adjacent to three vertices $z^{\\prime }_1,z^{\\prime }_2,$ and $z^{\\prime }_3$ of $C^{\\prime }$ .",
"Let $z_1,z_2,z_3$ be the three preimages in $C$ of the vertices $z^{\\prime }_1,z^{\\prime }_2,z^{\\prime }_3$ .",
"Let also $y,z$ be the preimages in the hypercube $Q_m$ of the vertex $z^{\\prime }$ .",
"Suppose that $y$ is adjacent to $z_1,z_2,z_3$ in $Q_m$ .",
"Since $C^{\\prime }$ is the image of the convex 6-cycle of $G$ , this implies that $y$ is not a vertex of $G$ while $z$ is a vertex of $G$ .",
"Since $G$ is an isometric subgraph of $Q_m$ , $G$ contains a vertex $w_1$ adjacent to $z$ and $z_1$ , a vertex $w_2$ adjacent to $z$ and $z_2$ , and a vertex $w_3$ adjacent to $z$ and $z_3$ .",
"Consequently, the vertices of $C$ together with the vertices $z,w_1,w_2,w_3$ define a full subdivision $SK_4$ , contrary to our assumption that $C$ is not included in such a subdivision.",
"This shows that the convex hull of the 6-cycle $C$ is gated."
],
[
"Convex and gated hulls of long isometric cycles",
"In the previous section we described the structure of gated hulls of 6-cycles in two-dimensional partial cubes.",
"In this section, we provide a description of convex and gated hulls of long isometric cycles, i.e., of isometric cycles of length $\\ge 8$ .",
"We prove that convex hulls of long isometric cycles are disks, i.e., the region graphs of pseudoline arrangements.",
"Then we show that all such disks are gated.",
"In particular, this implies that convex long cycles in two-dimensional partial cubes are gated."
],
[
"Convex hulls of long isometric cycles",
"A two-dimensional partial cube $D$ is called a pseudo-disk if $D$ contains an isometric cycle $C$ such that ${\\rm conv}(C)=D$ ; $C$ is called the boundary of $D$ and is denoted by $\\partial D$ .",
"If $D$ is the convex hull of an isometric cycle $C$ of $G$ , then we say that $D$ is a pseudo-disk of $G$ .",
"Admitting that $K_1$ and $K_2$ are pseudo-disks, the class of all pseudo-disks is closed under contractions.",
"The main goal of this subsection is to prove the following result: Proposition 27 A graph $D\\in {\\mathcal {F}}(Q_3)$ is a pseudo-disk if and only if $D$ is a disk.",
"In particular, the convex hull ${\\rm conv}(C)$ of an isometric cycle $C$ of any graph $G\\in {\\mathcal {F}}(Q_3)$ is an $\\mathrm {AOM}$ of rank 2.",
"The fact that disks are pseudo-disks follows from the next claim: Claim 28 If $D\\in \\mathcal {F}(Q_3)$ is a disk, then $D$ is the convex hull of an isometric cycle $C$ of $D$ .",
"By definition, $D$ is the region graph of an arrangement $\\mathcal {A}$ of pseudolines.",
"The cycle $C$ is obtained by traversing the unbounded cells of the arrangement in circular order, i.e., $C=\\partial D$ .",
"This cycle $C$ is isometric in $D$ because the regions corresponding to any two opposite vertices $v$ and $-v$ of $C$ are separated by all pseudolines of $\\mathcal {A}$ , thus $d_D(v,-v)=|\\mathcal {A}|$ .",
"Moreover, ${\\rm conv}(C)=D$ because for any other vertex $u$ of $D$ , any pseudoline $\\ell \\in \\mathcal {A}$ separates exactly one of the regions corresponding to $v$ and $-v$ from the region corresponding to $u$ , whence $d_D(v,u)+d_D(u,-v)=d_D(v,-v)$ .",
"The remaining part of the proof is devoted to prove that any pseudo-disk is a disk.",
"Let $D$ be a pseudo-disk with boundary $C$ .",
"Let $A_D:=\\lbrace v\\in D: v \\text{ has an antipode}\\rbrace $ .",
"As before, for a $\\Theta $ -class $E_i$ of $D$ , by $D^+_i$ and $D^-_i$ we denote the complementary halfspaces of $D$ defined by $E_i$ .",
"Claim 29 If $D$ is a pseudo-disk with boundary $C$ , then $A_D=C$ .",
"Clearly, $C\\subseteq A_D$ .",
"To prove $A_D\\subseteq C$ , suppose by way of contradiction that $v,-v$ are antipodal vertices of $D$ not belonging to $C$ .",
"Contract the $\\Theta $ -classes until $v$ is adjacent to a vertex $u\\in C$ , say via an edge in class $E_i$ (we can do this because all such classes crosses $C$ and by Lemma REF (ii) their contraction will lead to a disk).",
"Let $u\\in D^+_i$ and $v\\in D^-_i$ .",
"Since $D={\\rm conv}(C)$ , the $\\Theta $ -class $E_i$ crosses $C$ .",
"Let $xy$ and $zw$ be the two opposite edges of $C$ belonging to $E_i$ and let $x,z\\in D^+_i, y,w\\in D^-_i$ .",
"Let $P,Q$ be two shortest paths in $D^-_i$ connecting $v$ with $y$ and $w$ , respectively.",
"Since the total length of $P$ and $Q$ is equal to the shortest path of $C$ from $x$ to $z$ passing through $u$ , the paths $P$ and $Q$ intersect only in $v$ .",
"Extending $P$ and $Q$ , respectively within $D^-_i\\cap C$ until $-u$ , yields shortest paths $P^{\\prime }, Q^{\\prime }$ that are crossed by all $\\Theta $ -classes except $E_i$ .",
"Therefore, both such paths can be extended to shortest $(v,-v)$ -paths by adding the edge $-u-v$ of $E_i$ .",
"Similarly to the case of $v$ , there are shortest paths $P^{\\prime \\prime }, Q^{\\prime \\prime }$ from the vertex $-v\\in D^+_i$ to the vertices $x,z\\in C\\cap D^+_i$ .",
"Again, $P^{\\prime \\prime }$ and $Q^{\\prime \\prime }$ intersect only in $-v$ .",
"Let $E_j$ be any $\\Theta $ -class crossing $P$ and $E_k$ be any $\\Theta $ -class crossing $Q$ .",
"We assert that the set $S:=\\lbrace u,v,x,y,z,w,-u,-v\\rbrace $ of vertices of $D$ shatter $\\lbrace i,j,k\\rbrace $ , i.e., that contracting all $\\Theta $ -classes except $E_i,E_j$ , and $E_k$ yields a forbidden $Q_3$ .",
"Indeed, $E_i$ separates $S$ into the sets $\\lbrace u,x,-v,z\\rbrace $ and $\\lbrace v,y,-u,w\\rbrace $ , $E_j$ separates $S$ into the sets $\\lbrace x,y,-v,-u\\rbrace $ and $\\lbrace u,v,z,w\\rbrace $ , and $E_k$ separates $S$ into the sets $\\lbrace u,v,x,y\\rbrace $ and $\\lbrace -v,-u,z,w\\rbrace $ .",
"This contradiction shows that $A_D\\subseteq C$ , whence $A_D=C$ .",
"Claim 30 If $D$ is a pseudo-disk with boundary $C$ , then $D$ is an affine partial cube.",
"Moreover, there exists an antipodal partial cube $D^{\\prime }\\in \\mathcal {F}(Q_{4})$ containing $D$ as a halfspace.",
"First we show that $D$ is affine.",
"Let $u,v\\in D$ .",
"Using the characterization of affine partial cubes provided by [34] we have to show that for all vertices $u,v$ of $D$ one can find $w,-w\\in A_D$ such that the intervals $I(w,u)$ and $I(v,-w)$ are not crossed by the same $\\Theta $ -class of $D$ .",
"By Claim REF this is equivalent to finding such $w,-w$ in $C$ .",
"Let $I$ be the index set of all $\\Theta $ -classes crossing $I(u,v)$ .",
"Without loss of generality assume that $u\\in D_i^+$ (and therefore $v\\in D^-_i$ ) for all $i\\in I$ .",
"We assert that $(\\bigcap _{i\\in I}D_i^+)\\cap C\\ne \\emptyset $ .",
"Then any vertex from this intersection can play the role of $w$ .",
"For $i\\in I$ , let $C^+_i=C\\cap D^+_i$ and $C^-_i=C\\cap D^-_i$ ; $C^+_i$ and $C^-_i$ are two disjoint shortest paths of $C$ covering all vertices of $C$ .",
"Viewing $C$ as a circle, $C^+_i$ and $C^-_i$ are disjoint arcs of this circle.",
"Suppose by way of contradiction that $\\bigcap _{i\\in I}C_i^+=\\bigcap _{i\\in I}D_i^+\\cap C=\\emptyset $ .",
"By the Helly property for arcs of a circle, there exist three classes $i,j,k\\in I$ such that the paths $C^+_i,C^+_j,$ and $C^+_k$ pairwise intersect, together cover all the vertices and edges of the cycle $C$ , and all three have empty intersection.",
"This implies that $C$ is cut into 6 nonempty paths: $C_i^+\\cap C_j^+\\cap C_k^-$ , $C_i^+\\cap C_j^-\\cap C_k^-$ , $C_i^+\\cap C_j^-\\cap C_k^+$ , $C_i^-\\cap C_j^-\\cap C_k^+$ , $C_i^-\\cap C_j^+\\cap C_k^+$ , and $C_i^-\\cap C_j^+\\cap C_k^-$ .",
"Recall also that $u\\in D_i^+\\cap D_j^+\\cap D_k^+$ and $v\\in D_i^-\\cap D_j^-\\cap D_k^-$ .",
"But then the six paths partitioning $C$ together with $u,v$ will shatter the set $\\lbrace i,j,k\\rbrace $ , i.e., contracting all $\\Theta $ -classes except $i,j,k$ yields a forbidden $Q_3$ .",
"Consequently, $D$ is an affine partial cube, i.e., $D$ is a halfspace of an antipodal partial cube $G$ , say $D=G^+_i$ for a $\\Theta $ -class $E_i$ .",
"Suppose that $G$ can be contracted to the 4-cube $Q_4$ .",
"If $E_i$ is a coordinate of $Q_4$ (i.e., the class $E_i$ is not contracted), since $D=G^+_i$ , we obtain that $D$ can be contracted to $Q_3$ , which is impossible because $D\\in {\\mathcal {F}}(Q_3)$ .",
"Therefore $E_i$ is contracted.",
"Since the contractions of $\\Theta $ -classes commute, suppose without loss of generality that $E_i$ was contracted last.",
"Let $G^{\\prime }$ be the partial cube obtained at the step before contracting $E_i$ .",
"Let $D^{\\prime }$ be the isometric subgraph of $G^{\\prime }$ which is the image of $D$ under the performed contractions.",
"Since the property of being a pseudo-disk is preserved by contractions, $D^{\\prime }$ is a pseudo-disk, moreover $D^{\\prime }$ is one of the two halfspaces of $G^{\\prime }$ defined by the class $E_i$ restricted to $G^{\\prime }$ .",
"Analogously, by Lemma REF antipodality is preserved by contractions, whence $G^{\\prime }$ is an antipodal partial cube such that $\\pi _i(G^{\\prime })=Q_4$ .",
"This implies that $G^{\\prime }$ was obtained from $H:=Q_4$ by an isometric antipodal expansion $(H^1,H^0,H^2)$ .",
"Notice that one of the isometric subgraphs $H^1$ or $H^2$ of the 4-cube $H$ , say $H_1$ coincides with the disk $D^{\\prime \\prime }:=\\pi _i(D^{\\prime })$ .",
"Since $H$ is antipodal, by [34], $H_0$ is closed under antipodes in $Q_4$ and $-(H_1\\setminus H_0)=H_2\\setminus H_0$ .",
"Since $H_0$ is included in the isometric subgraph $H_1=D^{\\prime \\prime }$ of $H$ , $H_0$ is closed under antipodes also in $D^{\\prime \\prime }$ .",
"By Claim REF we obtain $H_0=A_{D^{\\prime \\prime }}=\\partial D^{\\prime \\prime }$ .",
"Consequently, $H_0$ is an isometric cycle of $H=Q_{4}$ that separates $Q_{4}$ in two sets of vertices.",
"However, no isometric cycle of $Q_4$ separates the graph.",
"Figure: An OM containing Q 3 - Q_3^- as a halfspace.If $D\\notin \\mathcal {F}(Q_3)$ is the convex hull of an isometric cycle, then $D$ is not necessarily affine, see $X_4^5$ in Figure REF .",
"On the other hand, $SK_4\\in \\mathcal {F}(Q_3)$ is affine but is not a pseudo-disk.",
"Let us introduce the distinguishing feature.",
"Claim 31 If $D$ is a pseudo-disk with boundary $C$ , then $D$ is a disk, i.e., the region graph of a pseudoline arrangement.",
"By Claim REF we know that $D$ is the halfspace of an antipodal partial cube $G$ .",
"Suppose by contradiction that $G$ is not an $\\mathrm {OM}$ .",
"By [34] $G$ has a minor $X$ from the family $\\mathcal {Q}^{-}$ .",
"Since the members of this class are non-antipodal, to obtain $X$ from $G$ not only contractions but also restrictions are necessary.",
"We perform first all contractions $I$ to obtain a pseudo-disk $D^{\\prime }:=\\pi _I(D)\\in \\mathcal {F}(Q_3)$ that is a halfspace of the antipodal graph $G^{\\prime }:=\\pi _I(G)$ .",
"By the second part of Claim REF we know that $G^{\\prime }\\in \\mathcal {F}(Q_4)$ .",
"Now, since $G^{\\prime }$ contains $X$ as a proper convex subgraph, by Lemma REF we get $X\\in \\mathcal {F}(Q_3)$ .",
"Since $SK_4$ is the only member of the class $\\mathcal {Q}^-$ containing $SK_4$ as a convex subgraph, by Proposition REF , we obtain $X=SK_4$ .",
"Assume minimality in this setting, in the sense that any further contraction destroys all copies of $X$ present in $D^{\\prime }$ .",
"We distinguish two cases.",
"First, suppose that there exists a copy of $X$ which is a convex subgraph of $D^{\\prime }$ .",
"Let $n\\ge 4$ be maximal such that there is a convex $H=SK_n$ in $D^{\\prime }$ extending a convex copy of $X$ .",
"By Proposition REF , $H$ is gated.",
"If $H\\ne D^{\\prime }$ , there exists a $\\Theta $ -class $E_i$ of $D^{\\prime }$ not crossing $H$ .",
"Contracting $E_i$ , by Lemma REF (iii) we will obtain a gated full subdivision $\\pi _i(H)=SK_n$ contrary to the minimality in the choice of $D^{\\prime }$ .",
"Therefore $D^{\\prime }=H=SK_n$ , but it is easy to see that all $SK_n, n\\ge 4,$ are not pseudo-disks, a contradiction.",
"Now, suppose that no copy of $X$ is a convex subgraph of $D^{\\prime }$ .",
"Since $G^{\\prime }$ contains $X$ as a convex subgraph, $D^{\\prime }$ is a halfspace of $G^{\\prime }$ (say $D^{\\prime }=(G^{\\prime })^+_i$ ) defined by a $\\Theta $ -class $E_i$ , and $G^{\\prime }$ is an antipodal partial cube, we conclude that $E_i$ crosses all convex copies $H$ of $X=SK_4$ .",
"Then $E_i$ partitions $H$ into a 6-cycle $C$ and a $K_{1,3}$ such that all edges between them belong to $E_i$ .",
"The antipodality map of $G^{\\prime }$ maps the vertices of $(G^{\\prime })^+_i$ to vertices of $(G^{\\prime })^-_i$ and vice-versa.",
"Therefore in $D^{\\prime }$ there must be a copy of $K_{1,3}$ and a copy of $C=C_6$ , and both such copies belong to the boundary $\\partial (G^{\\prime })^+_i$ .",
"The antipodality map is also edge-preserving.",
"Therefore, it maps edges of $E_i$ to edges of $E_i$ and vertices of $(G^{\\prime })^+_i\\setminus \\partial (G^{\\prime })^+_i$ to vertices of $(G^{\\prime })^-_i\\setminus \\partial (G^{\\prime })^-_i$ .",
"Consequently, all vertices of $\\partial (G^{\\prime })^-_i$ have antipodes in the pseudo-disk $D^{\\prime }=(G^{\\prime })^+_i$ and their antipodes also belong to $\\partial (G^{\\prime })^+_i$ .",
"This and Claim REF imply that $\\partial (G^{\\prime })^+_i\\subset A_{D^{\\prime }}=\\partial D^{\\prime }$ .",
"Therefore the isometric cycle $\\partial D^{\\prime }$ contains an isometric copy of $C_6$ , whence $\\partial D^{\\prime }=C_6$ .",
"Since $\\partial D^{\\prime }$ also contains the leafs of a $K_{1,3}$ we conclude that the pseudo-disk $D^{\\prime }$ coincides with $Q^-_3$ .",
"However, the only antipodal partial cube containing $Q_3^-$ as a halfspace is depicted in Figure REF and it is an $\\mathrm {OM}$ , leading to a contradiction.",
"Note that Claim REF generalizes Lemma REF .",
"Together with Claim REF it yields that pseudo-disks are disks, i.e., tope graphs of $\\mathrm {AOM}$ s of rank two, concluding the proof of Proposition REF ."
],
[
"Gated hulls of long isometric cycles",
"By Proposition REF disks and pseudo-disks are the same, therefore, from now on we use the name “disk” for both.",
"We continue by showing that in two-dimensional partial cubes all disks with boundary of length $>6$ are gated.",
"Proposition 32 If $D$ is a disk of $G\\in {\\mathcal {F}}(Q_3)$ and $|\\partial D|>6$ , then $D$ is a gated subgraph of $G$ .",
"In particular, convex long cycles of $G$ are gated.",
"Let $G$ be a minimal two-dimensional partial cube in which the assertion does not hold.",
"Let $D$ be a non-gated disk of $G$ whose boundary $C:=\\partial D$ is a long isometric cycle.",
"Let $v$ be a vertex of $G$ that has no gate in $D$ and is as close as possible to $D$ , where $d_G(v,D)=\\min \\lbrace d_G(v,z): z\\in D\\rbrace $ .",
"We use some notations from the proof of [18].",
"Let $P_v:=\\lbrace x\\in D: d_G(v,x)=d_G(v,D)\\rbrace $ be the metric projection of $v$ to $D$ .",
"Let also $R_v:=\\lbrace x\\in D: I(v,x)\\cap D=\\lbrace x\\rbrace \\rbrace .$ Since $D$ is not gated, $R_v$ contains at least two vertices.",
"Obviously, $P_v\\subseteq R_v$ and the vertices of $R_v$ are pairwise nonadjacent.",
"We denote the vertices of $P_v$ by $x_1,\\ldots ,x_k$ .",
"For any $x_i\\in P_v$ , let $v_i$ be a neighbor of $v$ on a shortest $(v,x_i)$ -path.",
"By the choice of $v$ , each $v_i$ has a gate in $D$ .",
"By the definition of $P_v$ , $x_i$ is the gate of $v_i$ in $D$ .",
"This implies that the vertices $v_1,\\ldots ,v_k$ are pairwise distinct.",
"Moreover, since $x_i$ is the gate of $v_i$ in $D$ , for any two distinct vertices $x_i,x_j\\in P_v$ , we have $d_G(v_i,x_i)+d_G(x_i,x_j)=d_G(v_i,x_j)\\le 2+d_G(v_j,x_j)$ .",
"Since $d_G(x_i,v_i)=d_G(x_j,v_j)$ , necessarily $d_G(x_i,x_j)=2$ .",
"We assert that any three distinct vertices $x_j,x_k,x_\\ell \\in P_v$ do not have a common neighbor.",
"Suppose by way of contradiction that there exists a vertex $x$ adjacent to $x_j,x_k,x_\\ell $ .",
"Then $x$ belongs to $D$ by convexity of $D$ and $x_j,x_k,x_\\ell \\in I(x,v)$ since $x_j,x_k,x_\\ell \\in P_v$ .",
"Let $E_j$ be the $\\Theta $ -class of the edge $v_jv$ and let $C_k$ be the cycle of $G$ defined by a $(v,x_j)$ -shortest path $P$ passing via $v_j$ , the 2-path $(x_j,x,x_k)$ , and a shortest $(x_k,v)$ -path $Q$ passing via $v_k$ .",
"Then $E_j$ must contain another edge of $C_k$ .",
"Necessarily this cannot be an edge of $P$ .",
"Since $v$ is a closest vertex to $D$ without a gate, this edge cannot be an edge of $Q$ .",
"Since $x_j\\in I(x,v)$ , this edge is not $xx_j$ .",
"Therefore the second edge of $E_j$ in $C_k$ is the edge $xx_k$ .",
"This implies that $v$ and $x_k$ belong to the same halfspace defined by $E_j$ , say $G^+_j$ , and $v_j$ and $x$ belong to its complement $G^-_j$ .",
"Using an analogously defined cycle $C_{\\ell }$ , one can show that the edge $xx_{\\ell }$ also belong to $E_j$ , whence the vertices $x_k$ and $x_{\\ell }$ belong to the same halfspace $G^+_j$ .",
"Since $x\\in I(x_k,x_{\\ell })$ and $x\\in G_j^-$ , we obtain a contradiction with convexity of $G^+_j$ .",
"Therefore, if $x_j,x_k,x_\\ell \\in P_v$ , then ${\\rm conv}(x_j,x_k,x_\\ell )$ is an isometric 6-cycle of $D$ .",
"In particular, this implies that each of the intervals $I(x_j,x_k),I(x_k,x_{\\ell }), I(x_j,x_{\\ell })$ consists of a single shortest path.",
"Next we show that $|P_v|\\le 3$ .",
"Suppose by way of contradiction that $|P_v|\\ge 4$ and pick the vertices $x_1,x_2,x_3,x_4\\in P_v$ .",
"Let $H$ be the subgraph of $D$ induced by the union of the intervals $I(x_j,x_k)$ , with $j,k\\in \\lbrace 1,2,3,4\\rbrace $ .",
"Since these intervals are 2-paths intersecting only in common end-vertices, $H$ is isomorphic to $SK_4$ with $x_1,x_2,x_3,x_4$ as original vertices.",
"Since $D$ is a two-dimensional partial cube, one can directly check that $H$ is an isometric subgraph of $D$ .",
"Since the intervals $I(x_j,x_k)$ are interiorly disjoint paths, $H=SK_4$ cannot be extended to $SK_4^*$ .",
"By Claim REF , $H=SK_4$ is a convex subgraph of $D$ .",
"Since $D$ is an AOM of rank 2 and thus a COM of rank 2, by Proposition REF , $D$ cannot contain $SK_4$ as a pc-minor.",
"This contradiction shows that $|P_v|\\le 3$ .",
"Let $S:= {\\rm conv}(P_v)$ .",
"Since $|P_v|\\le 3$ and $d_G(x_j,x_k)=2$ for any two vertices $x_j,x_k$ of $P_v$ , there exists at most three $\\Theta $ -classes crossing $S$ .",
"Since the length of the isometric cycle $C$ is at least 8, there exists a $\\Theta $ -class $E_i$ crossing $C$ (and $D$ ) and not crossing $S$ .",
"We assert that $v$ and the vertices of $P_v$ belong to the same halfspace defined by $E_i$ .",
"Indeed, if $E_i$ separates $v$ from $S$ , then for any $j$ , $E_i$ has an edge on any shortest $(v_j,x_j)$ -path.",
"This contradicts the fact that $x_j$ is the gate of $v_j$ in $D$ .",
"Consequently, $v$ and the set $S$ belong to the same halfspace defined by $E_i$ .",
"Consider the graphs $G^{\\prime }:=\\pi _i(G)$ , $D^{\\prime }:=\\pi _i(D)$ and the cycle $C^{\\prime }:=\\pi _i(C)$ .",
"By Lemma REF (i), $D^{\\prime }$ is a disk with boundary $C^{\\prime }$ (and thus an $\\mathrm {AOM}$ ) of the two-dimensional partial cube $G^{\\prime }$ .",
"Notice that the distance in $G^{\\prime }$ between $v^{\\prime }$ and the vertices $x^{\\prime }_j$ of $P_v$ is the same as the distance between $v$ and $x_j$ in $G$ and that the distance between $v^{\\prime }$ and the images of vertices of $R_v\\setminus P_v$ may eventually decrease by 1.",
"This implies that $D^{\\prime }$ is not gated.",
"By minimality of $G$ , this is possible only if $C^{\\prime }$ is a 6-cycle.",
"In this case, by Proposition REF , we conclude that $D^{\\prime }$ is included in a maximal full subdivision $H^{\\prime }=SK_n$ , which is a gated subgraph of $G^{\\prime }$ .",
"The graph $G$ is obtained from $G^{\\prime }$ by an isometric expansion $\\psi _i$ (inverse to $\\pi _i$ ).",
"By Lemma REF , $\\psi _i$ , restricted to $H^{\\prime }$ , is a peripheral expansion along an isometric tree of $H^{\\prime }$ .",
"This implies that in $G$ the convex $\\mathrm {AOM}$ $D$ is contained in a full subdivision of $K_n$ , contrary to the assumption that $D$ is the convex hull of the isometric cycle $C$ of length at least 8.",
"Summarizing Propositions REF , REF , and REF , we obtain the following results: Theorem 33 Let $G$ be a two-dimensional partial cube and $C$ be an isometric cycle of $G$ .",
"If $C=C_6$ , then the gated hull of $C$ is either $C$ , $Q^-_3$ , or a maximal full subdivision.",
"If otherwise $C$ is long, then ${\\rm conv}(C)$ is a gated disk.",
"Corollary 34 Maximal full subdivisions, convex disks with long cycles as boundaries (in particular, long convex cycles) are gated subgraphs in two-dimensional partial cubes."
],
[
"Completion to ample partial cubes",
"In this section, we prove that any partial cube $G$ of VC-dimension 2 can be completed to an ample partial cube $G^{\\top }$ of VC-dimension 2.",
"We perform this completion in two steps.",
"First, we canonically extend $G$ to a partial cube $G\\in {\\mathcal {F}}(Q_3)$ not containing convex full subdivisions.",
"The resulting graph $G$ is a COM of rank 2: its cells are the gated cycles of $G$ and the 4-cycles created by extensions of full subdivisions.",
"Second, we transform $G$ into an ample partial cube $(G)\\in {\\mathcal {F}}(Q_3)$ by filling each gated cycle $C$ of length $\\ge 6$ of $G$ (and of $G$ ) by a planar tiling with squares.",
"Here is the main result of this section and one of the main results of the paper: Theorem 35 Any $G\\in \\mathcal {F}(Q_3)$ can be completed to an ample partial cube $G^{\\top }:=(G)\\in \\mathcal {F}(Q_3)$ ."
],
[
"Canonical completion to two-dimensional COMs",
"The 1-extension graph of a partial cube $G\\in {\\mathcal {F}}(Q_3)$ of $Q_m$ is a subgraph $G^{\\prime }$ of $Q_m$ obtained by taking a maximal by inclusion convex full subdivision $H=SK_n$ of $G$ such that $H$ is standardly embedded in $Q_m$ and adding to $G$ the vertex $v_{\\varnothing }$ .",
"Lemma 36 If $G^{\\prime }$ is the 1-extension of $G\\in {\\mathcal {F}}(Q_3)$ and $G^{\\prime }$ is obtained with respect to the maximal by inclusion convex full subdivision $H=SK_n$ of $G$ , then $G^{\\prime }\\in {\\mathcal {F}}(Q_3)$ and $G$ is an isometric subgraph of $G^{\\prime }$ .",
"Moreover, any convex full subdivision $SK_{r}$ with $r\\ge 3$ of $G^{\\prime }$ is a convex full subdivision of $G$ and any convex cycle of length $\\ge 6$ of $G^{\\prime }$ is a convex cycle of $G$ .",
"Let $G$ be an isometric subgraph of $Q_m$ .",
"To show that $G^{\\prime }$ is an isometric subgraph of $Q_m$ it suffices to show that any vertex $v$ of $G$ can be connected in $G^{\\prime }$ with $v_{\\varnothing }$ by a shortest path.",
"By Proposition REF $H$ is a gated subgraph of $G$ and the gate $v_0$ of $v$ in $Q={\\rm conv}(H)$ belongs to $H$ .",
"This means that if $v$ is encoded by the set $A$ and $v_0$ is encoded by the set $A_0=A\\cap \\lbrace 1,\\ldots ,n\\rbrace $ , then either $A_0=\\lbrace i\\rbrace $ or $A_0=\\lbrace i,j\\rbrace $ for an original vertex $u_i$ or a subdivision vertex $u_{i,j}$ .",
"This means that $d(v,v_0)=d(v,u_i)=|A|-1$ in the first case and $d(v,v_0)=d(v,u_{i,j})=|A|-2$ in the second case.",
"Since $d(v,v_{\\varnothing })=|A|$ , we obtain a shortest $(v,v_{\\varnothing })$ -path in $G^{\\prime }$ first going from $v$ to $v_0$ and then from $v_0$ to $v_{\\varnothing }$ via an edge or a path of length 2 of $H$ .",
"This establishes that $G^{\\prime }$ is an isometric subgraph of $Q_m$ .",
"Since any two neighbors of $v_{\\varnothing }$ in $H$ have distance 2 in $G$ and $v_{\\varnothing }$ is adjacent in $G$ only to the original vertices of $H$ , we also conclude that $G$ is an isometric subgraph of $G^{\\prime }$ .",
"Now we will show that $G^{\\prime }$ belongs to ${\\mathcal {F}}(Q_3)$ .",
"Suppose by way of contradiction that the sets corresponding to some set $S$ of 8 vertices of $G^{\\prime }$ shatter the set $\\lbrace i,j,k\\rbrace $ .",
"Since $G\\in {\\mathcal {F}}(Q_3)$ , one of the vertices of $S$ is the vertex $v_{\\varnothing }$ : namely, $v_{\\varnothing }$ is the vertex whose trace on $\\lbrace i,j,k\\rbrace $ is $\\varnothing $ .",
"Thus the sets corresponding to the remaining 7 vertices of $S$ contain at least one of the elements $i,j,k$ .",
"Now, since $H=SK_n$ with $n\\ge 4$ , necessarily there exists an original vertex $u_{\\ell }$ of $H$ with $\\ell \\notin \\lbrace i,j,k\\rbrace $ .",
"Clearly, $u_{\\ell }$ is not a vertex of $S$ .",
"Since the trace of $\\lbrace \\ell \\rbrace $ on $\\lbrace i,j,k\\rbrace $ is $\\varnothing $ , replacing in $S$ the vertex $v_{\\varnothing }$ by $u_{\\ell }$ we will obtain a set of 8 vertices of $G$ still shattering the set $\\lbrace i,j,k\\rbrace $ , contrary to $G\\in {\\mathcal {F}}(Q_3)$ .",
"It remains to show that any convex full subdivision of $G^{\\prime }$ is a convex full subdivision of $G$ .",
"Suppose by way of contradiction that $H^{\\prime }=SK_r, r\\ge 3,$ is a convex full subdivision of $G^{\\prime }$ containing the vertex $v_{\\varnothing }$ .",
"By Claim REF , in $G^{\\prime }$ $v_{\\varnothing }$ is adjacent only to the original vertices of $H$ .",
"Hence, if $v_{\\varnothing }$ is an original vertex of $H^{\\prime }$ then at least two original vertices of $H$ are subdivision vertices of $H^{\\prime }$ and if $v_{\\varnothing }$ is a subdivision vertex of $H^{\\prime }$ then the two original vertices of $H^{\\prime }$ adjacent to $v_{\\varnothing }$ are original vertices of $H$ .",
"In both cases, denote those two original vertices of $H$ by $x=u_i$ and $y=u_j$ .",
"Since $H^{\\prime }$ is convex and $u_{i,j}$ is adjacent to $u_i$ and $u_j$ , $u_{i,j}$ must belong to $H^{\\prime }$ .",
"But this implies that $H^{\\prime }$ contains the 4-cycle $(x=u_i,v_{\\varnothing },y=u_j,u_{i,j})$ , which is impossible in a convex full subdivision.",
"In a similar way, using Claim REF , one can show that any convex cycle of length $\\ge 6$ of $G^{\\prime }$ is a convex cycle of $G$ .",
"Now, suppose that we consequently perform the operation of 1-extension to all gated full subdivisions and to the occurring intermediate partial cubes.",
"By Lemma REF all such isometric subgraphs of $Q_m$ have VC-dimension 2 and all occurring convex full subdivisions are already convex full subdivisions of $G$ .",
"After a finite number of 1-extension steps (by the Sauer-Shelah-Perles lemma, after at most $\\binom{m}{\\le 2}$ 1-extensions), we will get an isometric subgraph $G$ of $Q_m$ such that $G\\in {\\mathcal {F}}(Q_3)$ , $G$ is an isometric subgraph of $G$ , and all maximal full subdivisions $SK_n$ of $G$ are included in $SK^*_n$ .",
"We call $G$ the canonical 1-completion of $G$ .",
"We summarize this result in the following proposition: Proposition 37 If $G\\in {\\mathcal {F}}(Q_3)$ is an isometric subgraph of the hypercube $Q_m$ , then after at most $\\binom{m}{\\le 2}$ 1-extension steps, $G$ can be canonically completed to a two-dimensional COM $G$ and $G$ is an isometric subgraph of $G$ .",
"To prove that $G$ is a two-dimensional COM, by second assertion of Proposition REF we have to prove that $G$ belongs to $\\mathcal {F}(Q_3, SK_4)=\\mathcal {F}(Q_3)\\cap \\mathcal {F}(SK_4)$ .",
"The fact that $G$ belongs to $\\mathcal {F}(Q_3)$ follows from Lemma REF .",
"Suppose now that $G$ contains $SK_4$ as a pc-minor.",
"By Corollary REF , $G$ contains a convex subgraph $H$ isomorphic to $SK_4$ .",
"Then $H$ extends in $G$ to a maximal by inclusion $SK_n$ , which we denote by $H^{\\prime }$ .",
"Since $G\\in \\mathcal {F}(Q_3)$ and $H$ does not extend to $SK^*_4$ , $H^{\\prime }$ does not extend to $SK^*_n$ either.",
"By Claim REF and Proposition REF applied to $G$ , we conclude that $H^{\\prime }$ is a convex and thus gated subgraph of $G$ .",
"Applying the second assertion of Lemma REF (in the reverse order) to all pairs of graphs occurring in the construction transforming $G$ to $G$ , we conclude that $H^{\\prime }$ is a convex and thus gated full subdivision of $G$ .",
"But this is impossible because all maximal full subdivisions $SK_n$ of $G$ are included in $SK^*_n$ .",
"This shows that $G$ belongs to $\\mathcal {F}(SK_4)$ , thus $G$ is a two-dimensional COM.",
"That $G$ is isometrically embedded in $G$ follows from Lemma REF and the fact that if $G$ is an isometric subgraph of $G^{\\prime }$ and $G^{\\prime }$ is an isometric subgraph of $G^{\\prime \\prime }$ , then $G$ is an isometric subgraph of $G^{\\prime \\prime }$ ."
],
[
"Completion to ample two-dimensional partial cubes",
"Let $G\\in {\\mathcal {F}}(Q_3)$ , $C$ a gated cycle of $G$ , and $E_j$ a $\\Theta $ -class crossing $C$ .",
"Set $C:=(v_1,v_2,\\ldots , v_{2k})$ , where the edges $v_{2k}v_1$ and $v_kv_{k+1}$ are in $E_j$ .",
"The graph $G_{C,E_j}$ is defined by adding a path on vertices $v_{2k}=v^{\\prime }_1, \\ldots , v^{\\prime }_{k}=v_{k+1}$ and edges $v_iv^{\\prime }_i$ for all $2\\le i\\le k-1$ .",
"Let $C^{\\prime }=(v^{\\prime }_1,\\ldots ,v^{\\prime }_k,v_{k+2},\\ldots , v_{2k-1})$ .",
"Then we recursively apply the same construction to the cycle $C^{\\prime }$ and we call the resulting graph a cycle completion of $G$ along a gated cycle $C$; see Figure REF for an illustration.",
"Proposition REF establishes the basic properties of this construction, in particular it shows that the cycle completion along a gated cycle is well defined.",
"Figure: (a) G C,E j G_{C,E_j} is obtained by adding the white vertices to a graph GG with a gated cycle C=(v 1 ,v 2 ,...,v 8 )C=(v_1,v_2,\\ldots , v_{8}).",
"(b) A cycle completion of GG along the cycle C=(v 1 ,v 2 ,...,v 8 )C=(v_1,v_2,\\ldots , v_{8}).Proposition 38 Let $G$ be a partial cube, $C$ a gated cycle of $G$ , and $E_j$ a $\\Theta $ -class crossing $C$ .",
"$G_{C,E_j}$ is a partial cube and $G$ is an isometric subgraph of $G_{C,E_j}$ , $C^{\\prime }=(v^{\\prime }_1,\\ldots ,v^{\\prime }_k,v_{k+2},\\ldots , v_{2k-1})$ is a gated cycle, If $G\\in {\\mathcal {F}}(Q_3)$ , then so is $G_{C,E_j}$ , If $G$ contains no convex $SK_n$ , then neither does $G_{C,E_j}$ .",
"To prove (REF ), notice that the $\\Theta $ -classes of $G$ extend to $G_{C,E_j}$ is a natural way, i.e., edges of the form $v_iv^{\\prime }_i$ for all $2\\le i\\le k-1$ belong to $E_j$ , while an edge $v^{\\prime }_iv^{\\prime }_{i+1}$ belongs to the $\\Theta $ -class of the edge $v_iv_{i+1}$ for all $1\\le i\\le k-1$ .",
"Clearly, among the old vertices distances have not changed and the new vertices are embedded as an isometric path.",
"If $w\\in C$ and $u\\in C^{\\prime }$ is a new vertex, then it is easy to see that there is a shortest path using each $\\Theta $ -class at most once.",
"In fact, since $w$ is at distance at most one from $C^{\\prime }$ it has a gate in $C^{\\prime }$ , i.e., the path only uses $E_j$ .",
"Finally, let $v$ be an old vertex of $G \\setminus C$ , $w$ be its gate in $C$ , and $u$ be a new vertex, i.e., $u\\in G_{C, E_j} \\setminus G$ .",
"Let $P$ be a path from $v$ to $u$ that is a concatenation of a shortest $(v,w)$ -path $P_1$ and a shortest $(w,u)$ -path $P_2$ .",
"Since $C$ is gated and all $\\Theta $ -classes crossing $P_2$ also cross $C$ , the $\\Theta $ -classes of $G$ crossing $P_1$ and the $\\Theta $ -classes crossing $P_2$ are distinct.",
"Since $P_1$ and $P_2$ are shortest paths, the $\\Theta $ -classes in each of two groups are also pairwise different.",
"Consequently, $P$ is a shortest $(v,u)$ -path and thus $G_{C,E_j}$ is a partial cube.",
"Finally, $G$ is an isometric subgraph of $G_{C,E_j}$ by construction.",
"To prove (REF ), let $v \\in G \\setminus C^{\\prime }$ .",
"If $v \\in G \\setminus C$ , let $w$ be its gate in $C$ .",
"Thus there is a shortest $(v,w)$ -path which does not cross the $\\Theta $ -classes crossing $C$ .",
"Suppose that $w \\notin C^{\\prime }$ , otherwise we are done.",
"Then there exists a vertex $w^{\\prime }$ such that the edge $ww^{\\prime }$ belongs to $E_j$ .",
"Since $E_j$ crosses $C$ and not $C^{\\prime }$ , $w^{\\prime }$ is the gate of $v$ in $C^{\\prime }$ .",
"If $v \\in C \\setminus C^{\\prime }$ , using the previous argument, there exists an edge $vv^{\\prime }$ belonging to $E_j$ and we conclude that $v^{\\prime }$ is the gate of $v$ in $C^{\\prime }$ .",
"To prove (REF ), suppose by way of contradiction that $G_{C,E_j}$ has a $Q_3$ as a pc-minor.",
"Then there exists a sequence $s$ of restrictions $\\rho _s$ and contractions $\\pi _s$ such that $s(G) = Q_3$ .",
"Recall that restrictions and contractions commute in partial cube [18].",
"Hence, we get a graph $G^{\\prime }=\\pi _s(G)$ which contains a convex $Q_3$ .",
"Thus, this pc-minor $Q_3$ can be obtained by contractions.",
"Clearly, $E_j$ must be among the uncontracted classes, because $\\pi _j(G_{C,E_j})=\\pi _j(G)$ .",
"Furthermore, if only one other $\\Theta $ -class of $C$ is not contracted in $G_{C,E_j}$ , then contraction will identify all new vertices with (contraction) images of old vertices and again by the assumption $G\\in {\\mathcal {F}}(Q_3)$ we get a contradiction.",
"Thus, the three classes that constitute the copy of $Q_3$ are $E_j$ and two other classes say $E_j^{\\prime }, E_j^{\\prime \\prime }$ of $C$ .",
"Thus, the augmented $C$ yields a $Q_3^-$ in the contraction of $G_{C,E_j}$ , but the last vertex of the $Q_3$ comes from a part of $G$ .",
"In other words, there is a vertex $v\\in G$ , such that all shortest paths from $v$ to $C$ cross $E_j$ , $E_j^{\\prime }$ , or $E_j^{\\prime \\prime }$ .",
"This contradicts that $C$ was gated, establishing that $G_{C,E_j}\\in {\\mathcal {F}}(Q_3)$ .",
"To prove (REF ), suppose by way of contradiction that $G_{C,E_j}$ contains a convex $SK_n$ .",
"Since $SK_n$ has no 4-cycles nor vertices of degree one, the restrictions leading to $SK_n$ must either include $E_j$ or the class of the edge $v_1v_2$ or $v_{2k-1}v_{2k}$ .",
"The only way to restrict here in order to obtain a graph that is not a convex subgraph of $G$ is restricting to the side of $E_j$ , that contains the new vertices.",
"But the obtained graph cannot use new vertices in a convex copy of $SK_n$ because they form a path of vertices of degree two, which does not exist in a $SK_n$ .",
"Thus $G_{C,E_j}$ does not contain a convex $SK_n$ .",
"Propositions REF and REF allow us to prove Theorem REF .",
"Namely, applying Proposition REF to a graph $G \\in \\mathcal {F}(Q_3)$ , we obtain a two-dimensional COM $G$ , i.e.",
"a graph $G\\in \\mathcal {F}(Q_3,SK_4)$ .",
"Then, we recursively apply the cycle completion along gated cycles to the graph $G$ and to the graphs resulting from $G$ .",
"By Proposition REF (REF ), (REF ), all intermediate graphs belong to $\\mathcal {F}(Q_3,SK_4)$ , i.e.",
"they are two-dimensional COMs.",
"This explain why we can recursively apply the cycle completion construction cycle-by-cycle.",
"Since this construction does not increase the VC-dimension, by Sauer-Shelah lemma after a finite number of steps, we will get a graph $(G)\\in \\mathcal {F}(Q_3,SK_4)$ in which all convex cycles must be gated (by Propositions REF and REF ) and must have length 4.",
"This implies that $(G)\\in \\mathcal {F}(C_6)$ .",
"Consequently, $(G)\\in \\mathcal {F}(Q_3,C_6)$ and by Proposition REF the final graph $G^{\\top }=(G)$ is a two-dimensional ample partial cube.",
"This completes the proof of Theorem REF .",
"For an illustration, see Figure REF .",
"Figure: An ample completion M ⊤ M^{\\top } of the running example MM.Remark 39 One can generalize the construction in Proposition REF by replacing a gated cycle $C$ by a gated $\\mathrm {AOM}$ that is the convex hull of $C$ , such that all its convex cycles are gated.",
"In a sense, this construction captures the set of all possible extensions of the graph $G$ ."
],
[
"Cells and carriers",
"This section uses concepts and techniques developed for COMs [6] and for hypercellular graphs [18].",
"Let ${\\mathcal {C}}(G)$ denote the set of all convex cycles of a partial cube $G$ and let ${\\mathbf {C}}(G)$ be the 2-dimensional cell complex whose 2-cells are obtained by replacing each convex cycle $C$ of length $2j$ of $G$ by a regular Euclidean polygon $[C]$ with $2j$ sides.",
"It was shown in [33] that the set ${\\mathcal {C}}(G)$ of convex cycles of any partial cube $G$ constitute a basis of cycles.",
"This result was extended in [18] where it has been shown that the 2-dimensional cell complex ${\\mathbf {C}}(G)$ of any partial cube $G$ is simply connected.",
"Recall that a cell complex $\\bf X$ is simply connected if it is connected and if every continuous map of the 1-dimensional sphere $S^1$ into $\\bf X$ can be extended to a continuous mapping of the (topological) disk $D^2$ with boundary $S^1$ into $\\bf X$ .",
"Let $G$ be a partial cube.",
"For a $\\Theta $ -class $E_i$ of $G$ , we denote by $N(E_i)$ the carrier of $E_i$ in ${\\mathbf {C}}(G)$ , i.e., the subgraph of $G$ induced by the union of all cells of ${\\mathbf {C}}(G)$ crossed by $E_i$ .",
"The carrier $N(E_i)$ of $G$ splits into its positive and negative parts $N^+(E_i):=N(E_i)\\cap G^+_i$ and $N^-(E_i):=N(E_i)\\cap G^-_i$ , which we call half-carriers.",
"Finally, call $G^+_i\\cup N^-(E_i)$ and $G^-_i\\cup N^+(E_i)$ the extended halfspaces of $E_i$ .",
"By Djoković's Theorem REF , halfspaces of partial cubes $G$ are convex subgraphs and therefore are isometrically embedded in $G$ .",
"However, this is no longer true for carriers, half-carriers, and extended halfspaces of all partial cubes.",
"However this is the case for two-dimensional partial cubes: Proposition 40 If $G\\in {\\mathcal {F}}(Q_3)$ and $E_i$ is a $\\Theta $ -class of $G$ , then the carrier $N(E_i)$ , its halves $N^+(E_i), N^-(E_i)$ , and the extended halfspaces $G^+_i\\cup N^-(E_i), G^-_i\\cup N^+(E_i)$ are isometric subgraphs of $G$ , and thus belong to ${\\mathcal {F}}(Q_3)$ .",
"Since the class ${\\mathcal {F}}(Q_3)$ is closed under taking isometric subgraphs, it suffices to show that each of the mentioned subgraphs is an isometric subgraph of $G$ .",
"The following claim reduces the isometricity of carriers and extended halfspaces to isometricity of half-carriers: Claim 41 Carriers and extended halfspaces of a partial cube $G$ are isometric subgraphs of $G$ if and only if half-carriers are isometric subgraphs of $G$ .",
"One direction is implied by the equality $N^+(E_i):=N(E_i)\\cap G^+_i$ and the general fact that the intersection of a convex subgraph and an isometric subgraph of $G$ is an isometric subgraph of $G$ .",
"Conversely, suppose that $N^+(E_i)$ and $N^-(E_i)$ are isometric subgraphs of $G$ and we want to prove that the carrier $N(E_i)$ is isometric (the proof for $G^+_i\\cup N^-(E_i)$ and $G^-_i\\cup N^+(E_i)$ is similar).",
"Pick any two vertices $u,v\\in N(E_i)$ .",
"If $u$ and $v$ belong to the same half-carrier, say $N^+(E_i)$ , then they are connected in $N^+(E_i)$ by a shortest path and we are done.",
"Now, let $u\\in N^+(E_i)$ and $v\\in N^-(E_i)$ .",
"Let $P$ be any shortest $(u,v)$ -path of $G$ .",
"Then necessarily $P$ contains an edge $u^{\\prime },v^{\\prime }$ with $u^{\\prime }\\in \\partial G^+_i\\subseteq N^+(E_i)$ and $v^{\\prime }\\in \\partial G^-_i\\subseteq N^-(E_i)$ .",
"Then $u,u^{\\prime }$ can be connected in $N^+(E_i)$ by a shortest path $P^{\\prime }$ and $v,v^{\\prime }$ can be connected in $N^-(E_i)$ by a shortest path $P^{\\prime \\prime }$ .",
"The path $P^{\\prime }$ , followed by the edge $u^{\\prime }v^{\\prime }$ , and by the path $P^{\\prime \\prime }$ is a shortest $(u,v)$ -path included in $N(E_i)$ .",
"By Claim REF it suffices to show that the half-carriers $N^+(E_i)$ and $N^-(E_i)$ of a two-dimensional partial cube $G$ are isometric subgraphs of $G$ .",
"By Proposition REF , $G$ is an isometric subgraph of its canonical COM-extension $G$ .",
"Moreover from the construction of $G$ it follows that the carrier $N(E_i)$ and its half-carriers $N^+(E_i)$ and $N^-(E_i)$ are subgraphs of the carrier $N(E_i)$ and its half-carriers $N^+(E_i), N^-(E_i)$ in the graph $G$ .",
"By [6], carriers and their halves of COMs are also COMs.",
"Consequently, $N^+(E_i)$ and $N^-(E_i)$ are isometric subgraphs of $G$ .",
"Since the graph $G$ is obtained from $G$ via a sequence of 1-extensions, it easily follows that any shortest path $P\\subset N^+(E_i)$ between two vertices of $N^+(E_i)$ can be replaced by a path $P^{\\prime }$ of the same length lying entirely in $N^+(E_i)$ .",
"Therefore $N^+(E_i)$ is an isometric subgraph of the partial cube $N^+(E_i)$ , thus the half-carrier $N^+(E_i)$ is also an isometric subgraph of $G$ .",
"A partial cube $G=(V,E)$ is a 2d-amalgam of two-dimensional partial cubes $G_1=(V_1,E_1), G_2=(V_2,E_2)$ both isometrically embedded in the cube $Q_m$ if the following conditions are satisfied: (1) $V=V_1\\cup V_2, E=E_1\\cup E_2$ and $V_2\\setminus V_1,V_1\\setminus V_2,V_1\\cap V_2\\ne \\varnothing ;$ (2) the subgraph $G_{12}$ of $Q_m$ induced by $V_1\\cap V_2$ is a two-dimensional partial cube and each maximal full subdivision $SK_n$ of $G_{12}$ is maximal in $G$ ; (3) $G$ is a partial cube.",
"As a last ingredient for the next proposition we need a general statement about COMs.",
"Lemma 42 If $G$ is a COM and the cube $Q_d$ is a pc-minor of $G$ , then there is an antipodal subgraph $H$ of $G$ that has $Q_d$ as a pc-minor.",
"By [34], if $H$ is an antipodal subgraph of a COM $G$ and $G^{\\prime }$ is an expansion of $G$ , then the expansion $H^{\\prime }$ of $H$ in $G^{\\prime }$ is either antipodal as well or is peripheral, where the latter implies that $H^{\\prime }$ contains $H$ as a convex subgraph.",
"In either case $G^{\\prime }$ contains an antipodal subgraph, that has $H$ as minor.",
"Since $Q_d$ is antipodal, considering a sequence of expansions from $Q_d=G_0, \\ldots G_k=G$ every graph at an intermediate step contains an antipodal subgraph having $Q_d$ as a minor.",
"Proposition 43 Two-dimensional partial cubes are obtained via successive 2d-amalgamations from their gated cycles and gated full subdivisions.",
"Conversely, the 2d-amalgam of two-dimensional partial cubes $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ of $Q_m$ is a two-dimensional partial cube of $Q_m$ in which every gated cycle or gated full subdivision belongs to at least one of the two constituents.",
"Let $G=(V,E)$ be a two-dimensional partial cube which is not a single cell.",
"We can suppose that $G$ is 2-connected, otherwise we can do an amalgam along an articulation vertex.",
"We assert that $G$ contains two gated cells intersecting in an edge.",
"Since the intersection of two gated sets is gated and any cell does not contain any proper gated subgraph except vertices and edges, the intersection of any two cells of $G$ is either empty, a vertex, or an edge.",
"If the last case never occur, since any convex cycle of $G$ is included in a single cell, any cycle of $G$ containing edges of several cells (such a cycle exists because $G$ is 2-connected) cannot be written as a modulo 2 sum of convex cycles.",
"This contradicts the result of [33] that the set of convex cycles of any partial cube $G$ constitute a basis of cycles.",
"Pick two gated cells $C_1$ and $C_2$ intersecting in an edge $e$ .",
"Let $E_i$ be a $\\Theta $ -class crossing $C_1$ and not containing $e$ .",
"Since $C_2$ is gated, $C_2$ is contained in one of the halfspaces $G^+_i$ or $G^-_i$ , say $C_2\\subseteq G^+_i$ .",
"Notice also that $C_2$ is not included in the carrier $N(E_i)$ .",
"Set $G_1:=G^-_i\\cup N^+(E_i)$ and $G_2:=G^+_i$ .",
"By Proposition REF , $G_1,G_2,$ and $G_1\\cap G_2=N^+(E_i)$ are two-dimensional partial cubes, thus $G$ is a 2d-amalgam of $G_1$ and $G_2$ .",
"Conversely, suppose that a partial cube $G$ is a 2d-amalgam of two-dimensional partial cubes $G_1$ and $G_2$ .",
"Consider the canonical COM completions $G_1$ and $G_2$ of $G_1$ and $G_2$ , which are in $\\mathcal {F}(Q_3)$ by the Lemma REF .",
"Then $G_1\\cap G_2$ coincides with $G_{12}$ .",
"Therefore, by [6] this provides a COM $G^{\\prime }$ , which is a COM amalgam of $G_1$ and $G_2$ along $G_{12}$ without creating new antipodal subgraphs.",
"Using the Lemma REF , we deduce that $G^{\\prime }\\in {\\mathcal {F}}(Q_3)$ .",
"Since the graph $G$ is isometrically embedded in $G^{\\prime }$ , $G\\in {\\mathcal {F}}(Q_3)$ , which concludes the proof.",
"The 2-dimensional cell complex ${\\mathbf {C}}(G)$ of a partial cube $G$ is simply connected but not contractible even if $G$ is two-dimensional.",
"However, for a two-dimensional partial cube $G$ there is a simple remedy: one can consider the (combinatorial) cell complex having gated cycles and gated full subdivisions of $G$ as cells.",
"However, since full subdivisions cannot be directly represented by Euclidean cells, this complex does not have a direct geometric meaning.",
"One possibility is to replace each gated full subdivision $SK_n$ by a regular Euclidean simplex with sides of length 2 and each gated cycle by a regular Euclidean polygon.",
"Denote the resulting polyhedral complex by ${\\mathbf {X}}(G)$ .",
"Notice that two cells of ${\\mathbf {X}}(G)$ can intersect in an edge of a polygonal cell or in a half-edge of a simplex.",
"This way, with each two-dimensional partial cube $G$ we associate a polyhedral complex ${\\mathbf {X}}(G)$ which may have cells of arbitrary dimensions.",
"Alternatively, one can associate to $G$ the cell complex ${\\mathbf {C}}(G)$ of the canonical COM completion $G$ of $G$ .",
"Recall that in ${\\mathbf {C}}(G)$ , each gated cycle of $G$ is replaced by a regular Euclidean polygon and each gated full subdivision $SK_n$ of $G$ is extended in $G$ to $SK^*_n$ and this correspond to a bouquet of squares in ${\\mathbf {C}}(G)$ .",
"Thus ${\\mathbf {C}}(G)$ is a 2-dimensional cell complex.",
"Corollary 44 If $G\\in {\\mathcal {F}}(Q_3)$ , then the complexes ${\\mathbf {X}}(G)$ and ${\\mathbf {C}}(G)$ are contractible.",
"That ${\\mathbf {C}}(G)$ is contractible follows from the fact that $G$ is a two-dimensional COM (Proposition REF ) and that the cell complexes of COMs are contractible (Proposition 15 of [6]).",
"The proof that ${\\mathbf {X}}(G)$ is contractible uses the same arguments as the proof of [6].",
"We prove the contractibility of ${\\mathbf {X}}(G)$ by induction on the number of maximal cells of ${\\mathbf {X}}(G)$ by using the gluing lemma [9] and Proposition REF .",
"By the gluing lemma, if $\\bf X$ is a cell complex which is the union of two contractible cell complexes ${\\bf X}_1$ and ${\\bf X}_2$ such that their intersection ${\\bf X}_1\\cap {\\bf X}_2$ is contractible, then $\\bf X$ is contractible.",
"If ${\\mathbf {X}}(G)$ consists of a single maximal cell, then this cell is either a polygon or a simplex, thus is contractible.",
"If ${\\mathbf {X}}(G)$ contains at least two cells, then by the first assertion of Proposition REF $G$ is a 2d-amalgam of two-dimensional partial cubes $G_1$ and $G_2$ along a two-dimensional partial cube $G_{12}$ .",
"By induction assumption, the complexes ${\\mathbf {X}}(G_1)$ , ${\\mathbf {X}}(G_1)$ , and ${\\mathbf {X}}(G_{12})={\\mathbf {X}}(G_1)\\cap {\\mathbf {X}}(G_2)$ are contractible, thus ${\\mathbf {X}}(G)$ is contractible by gluing lemma."
],
[
"Characterizations of two-dimensional partial cubes",
"The goal of this section is to give a characterization of two-dimensional partial cubes, summarizing all the properties established in the previous sections: Theorem 45 For a partial cube $G=(V,E)$ the following conditions are equivalent: (i) $G$ is a two-dimensional partial cube; (ii) the carriers $N(E_i)$ of all $\\Theta $ -classes of $G$ , defined with respect to the cell complex ${\\mathbf {C}}(G)$ , are two-dimensional partial cubes; (iii) the hyperplanes of $G$ are virtual isometric trees; (iv) $G$ can be obtained from the one-vertex graph via a sequence $\\lbrace (G_i^1,G^0_i,G^2_i): i=1,\\ldots ,m\\rbrace $ of isometric expansions, where each $G^0_i, i=1,\\ldots ,m$ has VC-dimension $\\le 1$ ; (v) $G$ can be obtained via 2d-amalgams from even cycles and full subdivisions; (vi) $G$ has an extension to a two-dimensional ample partial cube.",
"Moreover, any two-dimensional partial cube $G$ satisfies the following condition: (vii) the gated hull of each isometric cycle of $G$ is a disk or a full subdivision.",
"The implication (i)$\\Rightarrow $ (ii) is the content of Proposition REF .",
"To prove that (ii)$\\Rightarrow $ (iii) notice that since $N(E_i)$ is a two-dimensional partial cube, by Propositions REF and REF it follows that the hyperplane of the $\\Theta $ -class $E_i$ of $N(E_i)$ is a virtual isometric tree.",
"Since this hyperplane of $N(E_i)$ coincides with the hyperplane $H_i$ of $G$ , we deduce that all hyperplanes of $G$ are virtual isometric trees, establishing (ii)$\\Rightarrow $ (iii).",
"The implication (iii)$\\Rightarrow $ (i) follows from Propositions REF and REF .",
"The equivalence (i)$\\Leftrightarrow $ (iv) follows from Proposition REF .",
"The equivalence (i)$\\Leftrightarrow $ (v) follows from Proposition REF .",
"The implication (i)$\\Rightarrow $ (vi) follows from Theorem REF and the implication (vi)$\\Rightarrow $ (i) is evident.",
"Finally, the implication (i)$\\Rightarrow $ (vii) is the content of Theorem REF .",
"Note that it is not true that if in a partial cube $G$ the convex hull of every isometric cycle is in $\\mathcal {F}(Q_3)$ , then $G\\in \\mathcal {F}(Q_3)$ ; see $X_2^4$ in Figure REF .",
"However, we conjecture that the condition (vii) of Theorem REF is equivalent to conditions (i)-(vi): Conjecture 46 Any partial cube $G$ in which the gated hull of each isometric cycle is a disk or a full subdivision is two-dimensional."
],
[
"Final remarks",
"In this paper, we provided several characterizations of two-dimensional partial cubes via hyperplanes, isometric expansions and amalgamations, cells and carriers, and gated hulls of isometric cycles.",
"One important feature of such graphs is that gated hulls of isometric cycles have a precise structure: they are either full subdivisions of complete graphs or disks, which are plane graphs representable as graphs of regions of pseudoline arrangements.",
"Using those results, first we show that any two-dimensional partial cube $G$ can be completed in a canonical way to a COM $G$ of rank 2 and that $G$ can be further completed to an ample partial cube $G^{\\top }:=(G)$ of VC-dimension 2.",
"Notice that $G$ is isometrically embedded in $G$ and that $G$ is isometrically embedded in $G^{\\top }$ .",
"This answers in the positive (and in the strong way) the question of [39] for partial cubes of VC-dimension 2.",
"However, for Theorem REF it is essential that the input is a partial cube: Figure REF presents a (non-isometric) subgraph $Z$ of $Q_4$ of VC-dimension 2, such that any ample partial cube containing $Z$ has VC-dimension 3.",
"Therefore, it seems to us interesting and nontrivial to solve the question of [47] and [39] for all (non-isometric) subgraphs of hypercubes of VC-dimension 2 (alias, for arbitrary set families of VC-dimension 2).",
"Figure: A subgraph ZZ of Q 4 Q_4 of VC-dimension 2, such that any ample partial cube containing ZZ has VC-dimension 3.It is also important to investigate the completion questions of [39] and [47] for all partial cubes from ${\\mathcal {F}}(Q_{d+1})$ (i.e., for partial cubes of VC-dimension $\\le d$ ).",
"For this, it will be interesting to see which results for partial cubes from ${\\mathcal {F}}(Q_3)$ can be extended to graphs from ${\\mathcal {F}}(Q_{d+1})$ .",
"We have the impression, that some of the results on disks can be extended to balls; a partial cube is a $d$ -ball if $G\\in \\mathcal {F}(Q_{d+1})$ and $G$ contains an isometric antipodal subgraph $C\\in \\mathcal {F}(Q_{d+1})$ such that $G={\\rm conv}(C)$ .",
"With this is mind, one next step would be to study the class $\\mathcal {F}(Q_{4})$ ."
],
[
"Acknowledgements",
"We are grateful to the anonymous referees for a careful reading of the paper and numerous useful comments and improvements.",
"This work was supported by the ANR project DISTANCIA (ANR-17-CE40-0015).",
"The second author moreover was supported by the Spanish Ministerio de Economía, Industria y Competitividad through grant RYC-2017-22701."
]
] | 1906.04492 |
[
[
"Upper envelopes of families of Feller semigroups and viscosity solutions\n to a class of nonlinear Cauchy problems"
],
[
"Abstract In this paper, we consider the (upper) semigroup envelope, i.e.",
"the least upper bound, of a given family of linear Feller semigroups.",
"We explicitly construct the semigroup envelope and show that, under suitable assumptions, it yields viscosity solutions to abstract Hamilton-Jacobi-Bellman-type partial differential equations related to stochastic optimal control problems arising in the field of Robust Finance.",
"We further derive conditions for the existence of a Markov process under a nonlinear expectation related to the semigroup envelope for the case where the state space is locally compact.",
"The procedure is then applied to numerous examples, in particular, nonlinear PDEs that arise from control problems for infinite dimensional Ornstein-Uhlenbeck and L\\'evy processes."
],
[
"Introduction",
"Assume that we are given a “nice” Feller process and that there are some features, for example some parameters (drift, volatility, etc.",
"), of the process that cannot be determined precisely.",
"In this case, one typically speaks of model uncertainty or ambiguity.",
"This topic has been studied extensively in the context of Economics and Mathematical Finance in the last decades.",
"Prominent examples include a Brownian motion (Bachelier model) with drift uncertainty (cf.",
"Coquet et al.",
"[7]) or volatility uncertainty (cf.",
"Peng [30],[31]), a Black-Scholes model with volatility uncertainty (cf.",
"Epstein and Ji [13], Vorbrink [36]), and Lévy processes with uncertainty in the Lévy triplet (cf.",
"Hu and Peng [18], Neufeld and Nutz [25], Hollender [17], Kühn [22], Denk et al. [10]).",
"Under this type of uncertainty, worst case considerations together with dynamic consistency requirements lead to a stochastic optimal control problem, where, intuitively speaking, “nature” tries to control the system into the worst possible scenario, and to the consideration of so-called nonlinear expectations.",
"In the case of a Brownian Motion with uncertain volatility within an interval $[\\sigma _\\ell ,\\sigma _h]$ with $0<\\sigma _\\ell <\\sigma _h$ , this leads, for instance, to the control problem $V(t,x;u_0):=\\sup _{\\sigma \\in \\Sigma } \\mathbb {E}\\bigg [u_0\\bigg (x+\\int _0^t \\sigma _s\\, {\\rm d}B_s\\bigg )\\bigg ],$ where $B$ is a standard Brownian Motion on a suitable filtered probability space and $\\Sigma $ consists of all progressively measurable stochastic processes $\\sigma =(\\sigma _t)_{t\\ge 0}$ with values in $[\\sigma _\\ell ,\\sigma _h]$ .",
"Solving the optimal control problem (REF ) then results in the HJB equation $\\partial _t u(t,x)=\\sup _{\\sigma \\in [\\sigma _\\ell ,\\sigma _h]} \\frac{\\sigma ^2}{2}\\partial _{xx}u(t,x)\\quad \\text{for }t\\ge 0\\text{ and }x\\in \\mathbb {R}, \\quad u(0)=u_0,$ which is typically referred to as $G$ -heat equation.",
"We refer to Denis et al.",
"[8] for a detailed illustration of this relation.",
"Moreover, one can show that the value function (REF ) admits a representation of the form $V(t,x;u_0)=\\mathcal {E}\\big (u_0(x+X_t)\\big ),$ where $\\mathcal {E}$ is a sublinear expectation, more precisely a $G$ -expectation, and $X$ is a so-called $G$ -Brownian Motion (cf.",
"Denis et al.",
"[8] and Peng [30],[31]).",
"Motivated by this example, we choose a semigroup-theoretic approach, formally separating the space and time variable, in order to prove the existence of viscosity solutions to abstract Hamilton-Jacobi-Bellman-type equations of the form $\\partial _t u(t)=\\sup _{\\lambda \\in \\Lambda } A_\\lambda u(t)\\quad \\text{for }t\\ge 0, \\quad u(0)=u_0,$ where $(A_\\lambda )_{\\lambda \\in \\Lambda }$ is a family of generators of Feller processes indexed by a nonempty index set $\\Lambda $ .",
"We refer to Engel and Nagel [12] or Pazy [29] for more details on semigroup theory related to linear PDEs and the idea of formally separating space and time.",
"Our approach is based on an explicit construction and approximation of the solution due to Nisio [26], which adds a primal description to the dual representation in terms of a stochastic optimal control problem.",
"In a second step, we discuss how a stochastic process under a sublinear expectation can be obtained from the nonlinear semigroup which describes the transition of the process, using a nonlinear version of Kolmogorov's extension theorem by Denk et al.",
"[9].",
"Finally, we link semigroup envelopes to the value functions of abstract versions of Meyer-type control problems.",
"We thus provide a nonlinear analogue to the classical relation between Feller processes, partial differential equations and semigroups.",
"It is worth noting that stochastic optimal control problems and nonlinear PDEs of the form (REF ) are intimately related to BSDEs (cf.",
"Pardoux and Peng [27],[28], El Karoui et al.",
"[11], Coquet et al.",
"[7]), 2BSDEs (cf.",
"Cheridito et al.",
"[6], Soner et al.",
"[32],[33]) and BSDEs with jumps (cf.",
"Kazi-Tani et al.",
"[20],[21]) resulting in a stochastic representation of solutions to nonlinear Cauchy problems of the form (REF ).",
"The present paper can be seen as an analytic counter part to these approaches, which are based on mainly stochastic methods, and the techniques we use might pave the way for further applications in control theory.",
"For two (possibly nonlinear) semigroups $S=(S(t))_{t\\ge 0}$ and $T=(T(t))_{t\\ge 0}$ on a Banach lattice $X$ , we write $S\\le T$ if $S(t) x\\le T(t)x$ for all $t\\ge 0$ and $x\\in X$ .",
"For a nonempty index set $\\Lambda $ and a family $(S_\\lambda )_{\\lambda \\in \\Lambda }$ of semigroups on $X$ we call a semigroup $T$ an upper bound of $(S_\\lambda )_{\\lambda \\in \\Lambda }$ if $T\\ge S_\\lambda $ for all $\\lambda \\in \\Lambda $ .",
"We call ${S}$ the least upper bound of $(S_\\lambda )_{\\lambda \\in \\Lambda }$ if ${S}$ is an upper bound of $(S_\\lambda )_{\\lambda \\in \\Lambda }$ and ${S}\\le T$ for any other upper bound $T$ of $(S_\\lambda )_{\\lambda \\in \\Lambda }$ .",
"Then, the question arises under which conditions the family $(S_\\lambda )_{\\lambda \\in \\Lambda }$ has a least upper bound.",
"To the best of our knowledge this question has first been addressed by Nisio [26], in the case every $S_\\lambda $ is a strongly continuous semigroup on the space of all bounded measurable functions, which is why we call the least upper bound ${S}$ of $(S_\\lambda )_{\\lambda \\in \\Lambda }$ the Nisio semigroup or the (upper) semigroup envelope of $(S_\\lambda )_{\\lambda \\in \\Lambda }$ .",
"Due to a Theorem of Lotz [24] it is known that strongly continuous linear semigroups on the space of all bounded measurable functions always have a bounded generator, which is why the result of Nisio is not applicable for most semigroups related to partial differential equations.",
"However, using a similar approach to the one by Nisio on the space of bounded and uniformly continuous functions, Denk et al.",
"[10] proved the existence of a least upper bound for transition semigroups of Lévy processes.",
"In the present paper, we use the idea of Nisio in a more general framework than Denk et al.",
"[10] in order to go beyond Lévy processes.",
"Main examples will be transition semigroups of Ornstein-Uhlenbeck processes and Lévy processes on real separable Hilbert spaces, Geometric Brownian Motions, and Koopman semigroups with semiflows in real separable Banach spaces.",
"A fundamental result from semigroup theory is the fact that for a strongly continuous semigroup $S=(S(t))_{t\\ge 0}$ of linear operators with generator $A$ the function $u(t):=S(t)u_0$ , for sufficiently regular initial data $u_0$ , is a solution to the abstract Cauchy problem $\\partial _t u(t)=A u(t)\\quad \\text{for }t\\ge 0, \\quad u(0)=u_0.$ We refer to Engel and Nagel [12] or Pazy [29] for more details on this relation.",
"Similar as in the work by Denk et al.",
"[10], we show that the semigroup envelope yields a viscosity solution to the nonlinear Cauchy problem (REF ) if $A_\\lambda $ is the generator of $S_\\lambda $ for all $\\lambda \\in \\Lambda $ .",
"On one side, this is interesting from a structural point of view, since it establishes a relation between the least upper bound of a family of semigroups and the least upper bound of their generators.",
"On the other side, this shows that semigroup envelopes are closely related to solutions to possibly infinite-dimensional stochastic optimal control problems as well as local and non-local Hamilton-Jacobi-Bellman equations in Hilbert spaces, cf.",
"Barbu and Da Prato [1],[2],[3],[4], Fabbri et al.",
"[14], Federico and Gozzi [15], Świech and Zabczyk [34],[35].",
"We point out that, in comparison to the standard literature on control theory and viscosity theory, our approach covers a different spectrum of applications.",
"While in the standard theory on viscosity solutions very general types of HJB equations of the form $u_t=F\\big (t,x,u(t,x),D_x u(t,x), D_{xx}u(t,x)\\big )$ with a suitable function $F$ are considered, our approach uses very much the particular structure of the equation (REF ).",
"On the other hand, we allow for very general forms of generators, which are not covered by standard results.",
"However, as we discuss in Section , in most cases that are covered by, both, the standard approach and our approach, the solution concepts coincide.",
"We thus propose a different yet consistent solution concept, which allows to cover a different range of examples, in particular, completely non-standard control problems.",
"In order to come up with control problems that are somewhat closer to reality, in the past decades, an increasing interest has been paid to infinite-dimensional control problems with a particular focus on infinite-dimensional controlled Ornstein-Uhlenbeck processes.",
"We refer to Fabbri et al.",
"[14] and the references therein for a detailed discussion on this topic.",
"Considering a family $(A_\\lambda )_{\\lambda \\in \\Lambda }$ of generators of infinite-dimensional Ornstein-Uhlenbeck processes, we cover a certain range of examples for Ornstein-Uhlenbeck control problems.",
"In the standard theory on controlled Ornstein-Uhlenbeck processes (cf.",
"Fabbri et al.",
"[14]) the drift term consists of an expression of the form $\\big (BX_t+m \\big ){\\rm d}t$ with a fixed unbounded generator $B$ and a controlled vector $m$ .",
"Under certain conditions, the existence of mild solutions and $C^1$ -regularity of the related HJB equation can be obtained using smoothing properties of the linear semigroup related to $B$ and perturbation results from semigroup theory for semilinear equations.",
"Our approach allows to consider controlled Ornstein-Uhlenbeck processes with bounded generators in the drift term with controls in terms of $B$ , $m$ and the covariance operator in the diffusion part (see Example REF ).",
"In a forthcoming paper with Ben Goldys and the authors we show that our approach also extends to unbounded operators $B$ Throughout, we consider a nonempty index set $\\Lambda $ , a fixed separable metric space $(M,d)$ and a fixed weight function $\\kappa \\colon M\\rightarrow (0,\\infty )$ , which is assumed to be continuous and bounded.",
"Let ${\\rm {C}}={\\rm {C}}(M)$ be the space of all continuous functions $M\\rightarrow \\mathbb {R}$ .",
"We denote the space of all $u\\in {\\rm {C}}$ with norm $\\Vert u\\Vert _\\infty :=\\sup _{x\\in M}|u(x)|<\\infty $ by ${\\rm {C}}_{\\rm b}$ and the space of all $u\\in {\\rm {C}}$ with seminorm $\\Vert u\\Vert _{{\\rm {Lip}}}:=\\inf \\big \\lbrace L\\ge 0\\, |\\, \\forall x,y\\in M:|u(x)-u(y)|\\le Ld(x,y)\\big \\rbrace <\\infty $ by ${\\rm {Lip}}$ .",
"Finally, we denote the space of all $u\\in C$ with norm $\\Vert u\\Vert _\\kappa :=\\Vert \\kappa u\\Vert _\\infty <\\infty $ by ${\\rm {C}}_\\kappa $ and the closure of ${\\rm {Lip}}_{\\rm b}:={\\rm {Lip}}\\cap {\\rm {C}}_{\\rm b}$ in the space ${\\rm {C}}_\\kappa $ by $\\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ .",
"If $\\kappa $ is bounded below by some positive constant, then $C_\\kappa =C_{\\rm b}$ and $\\Vert \\cdot \\Vert _\\kappa $ is equivalent to $\\Vert \\cdot \\Vert _\\infty $ .",
"In this case, $\\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ is the closure of ${\\rm {Lip}}_{\\rm b}$ w.r.t.",
"$\\Vert \\cdot \\Vert _\\infty $ , which is the space $\\mathop {\\text{\\upshape {UC}}}\\nolimits _{\\rm b}$ of all bounded and uniformly continuous functions $M\\rightarrow \\mathbb {R}$ .",
"If $M$ has the Heine-Borel property, i.e if every closed bounded subset of $M$ is compact, and $\\kappa \\in {\\rm {C}}_0$ , then $\\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa =\\lbrace u\\in {\\rm {C}}\\, |\\, \\kappa u\\in {\\rm {C}}_0\\rbrace $ , where ${\\rm {C}}_0$ is the closure of the space ${\\rm {C}}_c$ of all continuous functions with compact support w.r.t.",
"$\\Vert \\cdot \\Vert _\\infty $ .",
"We refer to Example REF b) for more details.",
"For a sequence $(u_n)_{n\\in \\mathbb {N}}\\subset \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ and $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ , we write $u_n\\nearrow u$ as $n\\rightarrow \\infty $ if $u_n\\le u_{n+1}$ for all $n\\in \\mathbb {N}$ and $u_n(x)\\rightarrow u(x)$ as $n\\rightarrow \\infty $ for all $x\\in M$ .",
"Analogously, we write $u_n\\searrow u$ as $n\\rightarrow \\infty $ if $u_n\\ge u_{n+1}$ for all $n\\in \\mathbb {N}$ and $u_n(x)\\rightarrow u(x)$ as $n\\rightarrow \\infty $ for all $x\\in M$ .",
"We are now ready to introduce the central objects of our discussion.",
"Definition 1.1 We call a family ${S}=({S}(t))_{t\\ge 0}$ of possibly nonlinear operators a Feller semigroup if the following conditions are satisfied: ${S}(t)\\colon \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa \\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ is continuous for all $t\\ge 0$ , ${S}(0)u=u$ and ${S}(s+t)u={S}(s){S}(t)u$ for all $s,t\\ge 0$ and $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ , ${S}(t)$ is monotone and continuous from below for all $t\\ge 0$ , i.e.",
"for any sequence $(u_n)_{n\\in \\mathbb {N}}\\subset \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ and $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ with $u_n\\nearrow u$ as $n\\rightarrow \\infty $ it holds ${S}(t) u_n\\nearrow {S}(t) u$ as $n\\rightarrow \\infty $ .",
"Let $D\\subset \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ .",
"We then say that a Feller semigroup ${S}$ is strongly continuous on $D$ if the map $[0,\\infty )\\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa ,\\quad t\\mapsto {S}(t)u$ is continuous for all $u\\in D$ .",
"If $D=\\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ , we say that ${S}$ is strongly continuous.",
"Note that our definition of a Feller semigroup is somewhat different from the standard notion in the literature.",
"First of all, we do not require strong continuity or linearity of the semigroup a priori, as it is usually the case.",
"Moreover, Feller semigroups are oftentimes related to functions vanishing at infinity.",
"In order to treat situations, where the state space $M$ is infinite-dimensional, we do not require any condition related to compact sets but rather a certain growth condition in terms of the weight function $\\kappa $ .",
"Throughout this work, we assume the following setup: For all $\\lambda \\in \\Lambda $ let $S_\\lambda $ be a Feller semigroup of linear operators with $S_\\lambda (t)1=1$ , where 1 denotes the constant 1-function.",
"There exist constants $\\alpha ,\\beta \\in \\mathbb {R}$ such that $\\Vert S_\\lambda (t)u\\Vert _\\kappa \\le e^ {\\alpha t}\\Vert u\\Vert _\\kappa \\quad \\text{and}\\quad \\Vert S_\\lambda (t)u\\Vert _{{\\rm {Lip}}}\\le e^{\\beta t}\\Vert u\\Vert _{{\\rm {Lip}}}$ for all $u\\in {\\rm {Lip}}_{\\rm b}$ , $\\lambda \\in \\Lambda $ and $t\\ge 0$ .",
"At this point, we would like to briefly discuss the assumptions (A1) and (A2) and explain the key differences between the present paper and the paper by Denk et al. [10].",
"First, we would like to mention that the assumptions (A1) and (A2) are satisfied with $\\kappa =1$ , $\\alpha =0$ and $\\beta =0$ for Markovian convolution semigroups (semigroups arising from Lévy processes).",
"Different from [10], we do not make any assumption on strong continuity of the semigroups $(S_\\lambda )_{\\lambda \\in \\Lambda }$ or their generators at this point.",
"Strong continuity was a key ingredient in the proof of the dynamic programming principle (the semigroup property of the semigroup envelope) in [10] and also in the paper by Nisio [26].",
"In this paper, we provide an alternative proof for the dynamic programming principle, which does not require any strong continuity assumptions, and covers a more general setup.",
"In particular, we prove the existence of the semigroup envelope of the family $(S_\\lambda )_{\\lambda \\in \\Lambda }$ (Theorem REF ) solely under the assumptions (A1) and (A2).",
"In Section , we then provide three conditions that imply the strong continuity of the Nisio semigroup, which in turn implies that the Nisio semigroup is a viscosity solution to a nonlinear Cauchy problem (cf.",
"Section ).",
"The key assumption in order to obtain the strong continuity in [10] and [26] is a joint density assumption on the domains of the generators, which, in some infinite-dimensional applications, is not satisfied.",
"In particular, uncertainty in the covariance operator of infinite-dimensional Brownian Motions leads to major restrictions, see [10].",
"The conditions for strong continuity and the generalised setup, we present in this paper, allow us to treat, both, finite and infinite-dimensional applications (Koopman semigroups, geometric dynamics, Ornstein-Uhlenbeck processes and Lévy processes) in full generality concerning the uncertainty, and to improve [10] in such a way that no Lévy triplet is excluded a priori.",
"The assumption in order to obtain the strong continuity in [10] is a special case of Proposition REF in the present paper.",
"Finally, we would like to point out that the setup we choose is also more flexible regarding the tail behaviour of solutions.",
"More precisely, the choice of the weight function $\\kappa $ enables us to consider also unbounded initial data (contingent claims), which was not possible in the setup chosen by Denk et al.",
"The paper is structured as follows.",
"In Section , we show the existence of the semigroup envelope ${S}$ of the family $(S_\\lambda )_{\\lambda \\in \\Lambda }$ under the assumptions (A1) and (A2), and provide approximation results for the Nisio semigroup.",
"The main result of this section is Theorem REF .",
"In Section , we provide conditions that guarantee the strong continuity of the semigroup envelope (Propositions REF - REF ).",
"In Section , we discuss the connection between semigroup envelopes and viscosity solutions to a nonlinear abstract Cauchy problem.",
"The main result of this section is Theorem REF .",
"In Section , we give a stochastic representation of the semigroup envelope via a stochastic process under a sublinear expectation (cf.",
"Theorem REF ).",
"Section is devoted to the connection between the results obtained in the present paper and the field of control theory.",
"In particular, we explain the link between semigroup envelopes and value functions of abstract control problems.",
"In Section , we apply the results from Sections , and to several non-standard examples."
],
[
"Construction of the semigroup envelope",
"Let $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ , $\\lambda \\in \\Lambda $ and $h\\ge 0$ .",
"Then, $\\Vert S_\\lambda (h)u\\Vert _\\kappa \\le e^{\\alpha h}\\Vert u\\Vert _\\kappa $ since the map $S_\\lambda (h)\\colon \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa \\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ is continuous, which implies that $\\big (\\mathcal {E}_h u\\big )(x):=\\sup _{\\lambda \\in \\Lambda } \\big (S_\\lambda (h)u\\big )(x)$ is well-defined for all $x\\in M$ .",
"Lemma 2.1 Let $h\\ge 0$ .",
"$\\Vert \\mathcal {E}_h u-\\mathcal {E}_h v\\Vert _\\kappa \\le e^{\\alpha h} \\Vert u-v\\Vert _\\kappa $ for all $u,v\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ .",
"$\\Vert \\mathcal {E}_h u\\Vert _{{\\rm {Lip}}}\\le e^{\\beta h}\\Vert u\\Vert _{{\\rm {Lip}}}$ for all $u\\in {\\rm {Lip}}_{\\rm b}$ .",
"The map $\\mathcal {E}_h\\colon \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa \\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ is well-defined and Lipschitz continuous with Lipschitz constant $e^{\\alpha h}$ .",
"$\\mathcal {E}_h$ is sublinear, monotone, and continuous from below with $\\mathcal {E}_h 1=1$ .",
"Let $u,v\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ and $h\\ge 0$ .",
"Then, for all $\\lambda \\in \\Lambda $ , $\\kappa \\big (S_\\lambda (h)u-\\mathcal {E}_h v\\big )&\\le \\kappa \\big (S_\\lambda (h)u-S_\\lambda (h)v\\big )= \\kappa S_\\lambda (h)(u-v)\\\\&\\le \\Vert S_\\lambda (h)(u-v)\\Vert _\\kappa \\le e^{\\alpha h}\\Vert u-v\\Vert _\\kappa .$ Taking the supremum over $\\lambda \\in \\Lambda $ and a symmetry argument imply that $\\Vert \\mathcal {E}_h u-\\mathcal {E}_h v\\Vert _\\kappa \\le e^{\\alpha h}\\Vert u-v\\Vert _\\kappa .$ Let $u\\in {\\rm {Lip}}_{\\rm b}$ and $x,y\\in M$ .",
"Then, for all $\\lambda \\in \\Lambda $ , $(S_\\lambda (h)u\\big )(x)-\\big (\\mathcal {E}_h u\\big )(y)\\le (S_\\lambda (h)u\\big )(x)-(S_\\lambda (h)u\\big )(y)\\le e^{\\beta h}\\Vert u\\Vert _{{\\rm {Lip}}}d(x,y).$ Taking the supremum over $\\lambda \\in \\Lambda $ and a symmetry argument yield that $\\big |\\big (\\mathcal {E}_h u\\big )(x)-\\big (\\mathcal {E}_h u\\big )(y)\\big |\\le e^{\\beta h}\\Vert u\\Vert _{{\\rm {Lip}}}d(x,y).$ By part b) and Assumption (A1), we have that $\\mathcal {E}_h u\\in {\\rm {Lip}}_{\\rm b}$ for all $u\\in {\\rm {Lip}}_{\\rm b}$ .",
"Since ${\\rm {Lip}}_{\\rm b}$ is dense in $\\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ , part a) implies that $\\mathcal {E}_h\\colon \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa \\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ is well-defined and Lipschitz continuous with Lipschitz constant $e^{\\alpha h}$ .",
"All these properties directly carry over to the supremum.",
"In the sequel, we consider the set $P:= \\lbrace \\pi \\subset [0,\\infty )\\colon 0\\in \\pi , \\, |\\pi |<\\infty \\rbrace $ of finite partitions of the positive half line.",
"The set of partitions with end-point $t$ will be denoted by $P_t$ , i.e.",
"$P_t := \\lbrace \\pi \\in P: \\max \\pi = t\\rbrace $ .",
"Let $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ and $\\pi \\in P\\setminus \\big \\lbrace \\lbrace 0\\rbrace \\big \\rbrace $ .",
"Then, there exist $0=t_0<t_1<\\ldots <t_m$ such that $\\pi =\\lbrace t_0,t_1,\\dots ,t_m\\rbrace $ and we set $\\mathcal {E}_\\pi u := \\mathcal {E}_{t_1-t_0} \\ldots \\mathcal {E}_{t_m-t_{m-1}} u.$ Moreover, we set $\\mathcal {E}_{\\lbrace 0\\rbrace }u := u$ .",
"Note that, by definition, $\\mathcal {E}_h = \\mathcal {E}_{\\lbrace 0,h\\rbrace }$ for $h>0$ .",
"Since $\\mathcal {E}_h\\colon \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa \\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ is well-defined, the map $\\mathcal {E}_\\pi \\colon \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa \\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ is well-defined, too.",
"Lemma 2.2 For all $\\pi \\in P$ , the operator $\\mathcal {E}_\\pi $ is sublinear, monotone and continuous from below with $\\mathcal {E}_\\pi 1=1$ .",
"Moreover, $\\Vert \\mathcal {E}_\\pi u-\\mathcal {E}_\\pi v\\Vert _\\kappa \\le e^{\\alpha \\max \\pi } \\Vert u-v\\Vert _\\kappa $ for all $u,v\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ and $\\Vert \\mathcal {E}_\\pi u\\Vert _{{\\rm {Lip}}}\\le e^{\\beta \\max \\pi }\\Vert u\\Vert _{{\\rm {Lip}}}$ for all $u\\in {\\rm {Lip}}_{\\rm b}$ .",
"Since $\\mathcal {E}_h$ is a sublinear, monotone and continuous from below with $\\mathcal {E}_h1=1$ for all $h\\ge 0$ , the same holds for $\\mathcal {E}_\\pi $ as these properties are preserved under compositions.",
"The Lipschitz continuity follows from Lemma REF and the behaviour of Lipschitz constants under composition.",
"Let $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ .",
"In the following, we consider the limit of $\\mathcal {E}_\\pi u$ when the mesh size of the partition $\\pi \\in P$ tends to zero.",
"First note that, for $h_1,h_2\\ge 0$ and $x\\in M$ , $\\big (\\mathcal {E}_{h_1+h_2}u\\big )(x)&=\\sup _{\\lambda \\in \\Lambda } \\big (S_\\lambda (h_1+h_2)u\\big )(x)=\\sup _{\\lambda \\in \\Lambda } \\big (S_\\lambda (h_1)S_\\lambda (h_2)u\\big )(x)\\\\&\\le \\sup _{\\lambda \\in \\Lambda } \\big (S_\\lambda (h_1)\\mathcal {E}_{h_2}u\\big )(x)=\\big (\\mathcal {E}_{h_1}\\mathcal {E}_{h_2}u\\big )(x),$ which implies the pointwise inequality $\\mathcal {E}_{\\pi _1}u\\le \\mathcal {E}_{\\pi _2}u \\quad \\text{for }\\pi _1,\\pi _2\\in P\\text{ with }\\pi _1\\subset \\pi _2.$ In particular, for $\\pi _1,\\pi _2\\in P$ and $\\pi :=\\pi _1\\cup \\pi _2$ it follows that $\\pi \\in P$ with $\\big (\\mathcal {E}_{\\pi _1}u\\big )\\vee \\big (\\mathcal {E}_{\\pi _2}u\\big )\\le \\mathcal {E}_\\pi u.$ Recall that we denote the set of all finite partitions with end point $t\\ge 0$ by $P_t$ .",
"For $t\\ge 0$ , $x\\in M$ and $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ , we define $\\big ({S}(t)u\\big )(x):=\\sup _{\\pi \\in P_t} \\big (\\mathcal {E}_\\pi u\\big )(x).$ The family ${S}=({S}(t))_{t\\ge 0}$ is called the (upper) semigroup envelope or Nisio semigroup of the family $(S_\\lambda )_{\\lambda \\in \\Lambda }$ .",
"Note that, by definition, ${S}(0)u=u$ for all $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ .",
"We observe the following basic facts, which are a direct consequence of Lemma REF .",
"Lemma 2.3 Let $t\\ge 0$ .",
"Then, the map ${S}(t)\\colon \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa \\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ is well-defined and Lipschitz continuous with Lipschitz constant $e^{\\alpha t}$ .",
"Moreover, ${S}(t)$ is sublinear, monotone and continuous from below with ${S}(t)1=1$ .",
"By Lemma REF , $\\Vert {S}(t) u-{S}(t) v\\Vert _\\kappa \\le e^{\\alpha t} \\Vert u-v\\Vert _\\kappa \\quad \\text{for all }u,v\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ and $\\Vert {S}(t)u\\Vert _{{\\rm {Lip}}}\\le e^{\\beta t}\\Vert u\\Vert _{{\\rm {Lip}}}$ for all $u\\in {\\rm {Lip}}_{\\rm b}$ .",
"In particular, ${S}(t)u\\in {\\rm {Lip}}_{\\rm b}$ for all $u\\in {\\rm {Lip}}_{\\rm b}$ .",
"Now, the estimate (REF ) implies that ${S}(t)\\colon \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa \\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ is well-defined and Lipschitz continuous with Lipschitz constant $e^{\\alpha t}$ .",
"The remaining properties follow directly from the observation that, by Lemma REF they are satisfied by $\\mathcal {E}_\\pi $ , for $\\pi \\in P_t$ , and carry over to the supremum over all $\\pi \\in P_t$ .",
"In the following, we show that the Nisio semigroup ${S}$ is in fact a semigroup.",
"We start with the following lemma, which shows that ${S}(t)u$ can be approximated by a monotone sequence of partitions depending on $u$ .",
"We would like to point out that, under additional assumptions, the dependence of the sequence on $u$ can be dropped (see Proposition REF , below).",
"Lemma 2.4 Let $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ and $t>0$ .",
"Then, there exists a sequence $(\\pi _n)_{n\\in \\mathbb {N}}\\subset P_t$ (depending on $u$ ) with $\\mathcal {E}_{\\pi _n}u\\nearrow {S}(t)u$ as $n\\rightarrow \\infty $ .",
"Let $(x_k)_{k\\in \\mathbb {N}}\\subset M$ such that the set $\\lbrace x_k\\,|\\, k\\in \\mathbb {N}\\rbrace $ is dense in $M$ .",
"Then, for every $k\\in \\mathbb {N}$ , there exists a sequence $(\\pi _n^k)_{n\\in \\mathbb {N}}\\subset P_t$ with $\\pi _n^k\\subset \\pi _{n+1}^k$ for all $n\\in \\mathbb {N}$ and $\\big (\\mathcal {E}_{\\pi _n^k}u\\big )(x_k)\\nearrow \\big ({S}(t)u\\big )(x_k)\\quad \\text{as } n\\rightarrow \\infty .$ Now, let $\\pi _n:=\\bigcup _{k=1}^n\\pi _n^k$ for all $n\\in \\mathbb {N}$ .",
"Then, $\\pi _n^k\\subset \\pi _n\\subset \\pi _{n+1}$ for all $n\\in \\mathbb {N}$ and $k\\in \\lbrace 1,\\ldots ,n\\rbrace $ .",
"Hence, $\\mathcal {E}_{\\pi _n^k}u\\le \\mathcal {E}_{\\pi _n}u\\le \\mathcal {E}_{\\pi _{n+1}}u\\quad \\text{for all }n\\in \\mathbb {N}\\text{ and }k\\in \\lbrace 1,\\ldots ,n\\rbrace .$ Let $\\big (\\mathcal {E}_\\infty v\\big )(x):= \\sup _{n\\in \\mathbb {N}}\\big (\\mathcal {E}_{\\pi _n}v\\big )(x)$ for all $v\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ and $x\\in M$ .",
"Then, by Lemma REF , the map $\\mathcal {E}_\\infty \\colon \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa \\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ is well-defined.",
"In particular, $\\mathcal {E}_\\infty u\\colon M\\rightarrow \\mathbb {R}$ is continuous and, by (REF ), $\\mathcal {E}_{\\pi _n}u\\nearrow \\mathcal {E}_\\infty u$ as $n\\rightarrow \\infty $ .",
"Again, by (REF ), $\\big ({S}(t)u\\big )(x_k)=\\lim _{n\\rightarrow \\infty }\\big (\\mathcal {E}_{\\pi _n^k}u\\big )(x_k)\\le \\lim _{n\\rightarrow \\infty }\\big (\\mathcal {E}_{\\pi _n}u\\big )(x_k)=\\big (\\mathcal {E}_\\infty u\\big )(x_k)\\le \\big ({S}(t)u\\big )(x_k)$ for all $k\\in \\mathbb {N}$ .",
"Since, ${S}(t)u$ and $\\mathcal {E}_\\infty u$ are both continuous and the set $\\lbrace x_k\\,|\\, k\\in \\mathbb {N}\\rbrace $ is dense in $M$ , it follows that ${S}(t)u= \\mathcal {E}_\\infty u$ , which shows that $\\mathcal {E}_{\\pi _n}u\\nearrow {S}(t)u \\quad \\text{as } n\\rightarrow \\infty .$ We obtain the following main theorem.",
"Theorem 2.5 The family ${S}$ is a Feller semigroup of sublinear operators and the least upper bound of the family $(S_\\lambda )_{\\lambda \\in \\Lambda }$ .",
"We first show that, for all $s,t\\ge 0$ , ${S}(s+t)={S}(s){S}(t).$ If $s=0$ or $t=0$ the statement is trivial.",
"Therefore, let $s,t>0$ , $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ , $\\pi _0\\in P_{s+t}$ , and $\\pi :=\\pi _0\\cup \\lbrace s\\rbrace $ .",
"Then, $\\pi \\in P_{s+t}$ with $\\pi _0\\subset \\pi $ and, by (REF ), $\\mathcal {E}_{\\pi _0}u\\le \\mathcal {E}_{\\pi }u$ .",
"Let $m\\in \\mathbb {N}$ , $0=t_0<t_1<\\ldots t_m=s+t$ with $\\pi =\\lbrace t_0,\\ldots , t_m\\rbrace $ , and $i\\in \\lbrace 1,\\ldots , m\\rbrace $ with $t_i=s$ .",
"Then, $\\pi _1:=\\lbrace t_0,\\ldots , t_i\\rbrace \\in P_s$ and $\\pi _2:=\\lbrace t_i-s,\\ldots , t_n-s\\rbrace \\in P_t$ with $\\mathcal {E}_{\\pi _1}=\\mathcal {E}_{t_1-t_0}\\cdots \\mathcal {E}_{t_i-t_{i-1}}\\quad \\text{and}\\quad \\mathcal {E}_{\\pi _2}=\\mathcal {E}_{t_{i+1}-t_i}\\cdots \\mathcal {E}_{t_m-t_{m-1}}.$ We thus obtain that $\\mathcal {E}_{\\pi _0}u&\\le \\mathcal {E}_\\pi u=\\mathcal {E}_{t_1-t_0}\\cdots \\mathcal {E}_{t_m-t_{m-1}}u=\\big (\\mathcal {E}_{t_1-t_0}\\cdots \\mathcal {E}_{t_i-t_{i-1}}\\big )\\big (\\mathcal {E}_{t_{i+1}-t_i}\\cdots \\mathcal {E}_{t_m-t_{m-1}}u\\big )\\\\&=\\mathcal {E}_{\\pi _1}\\mathcal {E}_{\\pi _2}u\\le \\mathcal {E}_{\\pi _1}{S}(t)u\\le {S}(s){S}(t)u.$ Taking the supremum over all $\\pi _0\\in P_{s+t}$ yields that ${S}(s+t)u\\le {S}(s){S}(t)u$ .",
"Now, let $(\\pi _n)_{n\\in \\mathbb {N}}\\subset P_t$ with $\\mathcal {E}_{\\pi _n}u\\nearrow {S}(t)u$ as $n\\rightarrow \\infty $ (see Lemma REF ) and fix $\\pi _0\\in P_s$ .",
"Then, for all $n\\in \\mathbb {N}$ , $\\pi _n^{\\prime }:=\\pi _0\\cup \\lbrace s+\\tau \\colon \\tau \\in \\pi _n\\rbrace \\in P_{s+t}\\quad \\text{with}\\quad \\mathcal {E}_{\\pi _n^{\\prime }}=\\mathcal {E}_{\\pi _0}\\mathcal {E}_{\\pi _n}.$ As $\\mathcal {E}_{\\pi _0}$ is continuous from below, it follows that $\\mathcal {E}_{\\pi _0}\\big ({S}(t)u\\big )=\\lim _{n\\rightarrow \\infty }\\mathcal {E}_{\\pi _0}\\mathcal {E}_{\\pi _n}u=\\lim _{n\\rightarrow \\infty }\\mathcal {E}_{\\pi _n^{\\prime }}u\\le {S}(s+t)u.$ Taking the supremum over all $\\pi _0\\in P_s$ , we get that ${S}(s){S}(t)u\\le {S}(s+t)u$ , and therefore (REF ) follows.",
"From the definition of ${S}$ in Equation (REF ) and Lemma REF , we now may conclude that ${S}$ defines a Feller semigroup of sublinear operators.",
"It remains to show that ${S}$ is the least upper bound of the family $(S_\\lambda )_{\\lambda \\in \\Lambda }$ .",
"To this end, let $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ , $x\\in M$ , and $T$ be an upper bound of the family $(S_\\lambda )_{\\lambda \\in \\Lambda }$ , i.e $\\big (S_\\lambda (t)u\\big )(x)\\le \\big (T(t)u\\big )(x)$ for all $\\lambda \\in \\Lambda $ , $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ , $t\\ge 0$ and $x\\in M$ .",
"Then, $\\big (S_\\lambda (h)u\\big )(x)\\le \\big (\\mathcal {E}_h u\\big )(x) \\le \\big (T(h)u\\big )(x)\\quad \\text{for all }\\lambda \\in \\Lambda \\text{ and }h\\ge 0.$ Since $S_\\lambda $ and $T$ are semigroups, it follows that $\\big (S_\\lambda (t)u\\big )(x)\\le \\big (\\mathcal {E}_\\pi u\\big )(x) \\le \\big (T(t)u\\big )(x)\\quad \\text{for all }\\lambda \\in \\Lambda ,\\,t\\ge 0,\\text{ and }\\pi \\in P_t.$ Taking the supremum over all $\\pi \\in P_t$ , we obtain that $\\big (S_\\lambda (t)u\\big )(x)\\le \\big ({S}(t) u\\big )(x) \\le \\big (T(t)u\\big )(x)\\quad \\text{for all }\\lambda \\in \\Lambda \\text{ and }t\\ge 0.$ The remainder of this section is devoted to show that the approximation result of Lemma REF , where the approximating sequence was dependent on the function $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ , can be made stronger under the additional assumption that the map $[0,\\infty )\\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa ,\\quad h\\mapsto \\mathcal {E}_h u$ is continuous for all $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ .",
"More precisely, under this condition every sequence of partitions with mesh size tending to 0 can be used for the approximation of the semigroup envelope.",
"Note that (REF ) is, for example, implied by the condition that $\\sup _{\\lambda \\in \\Lambda }\\Vert S_\\lambda (h)u-u\\Vert _\\kappa \\rightarrow 0\\quad \\text{as }h\\rightarrow 0,$ for all $u\\in {\\rm {Lip}}_{\\rm b}$ , which, in most applications, is satisfied.",
"The following lemma shows that $\\mathcal {E}_\\pi $ depends continuously on the partition $\\pi \\in P$ .",
"Lemma 2.6 Assume that the map (REF ) is continuous for all $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ .",
"Let $m\\in \\mathbb {N}$ and $\\pi =\\lbrace t_0,t_1,\\ldots , t_m\\rbrace \\in P$ with $0=t_0<\\ldots <t_m$ .",
"For each $n\\in \\mathbb {N}$ let $\\pi _n=\\lbrace t_0^n,t_1^n,\\ldots , t_m^n\\rbrace \\in P$ with $0=t_0^n<t_1^n<\\ldots < t_m^n$ and $t_i^n\\rightarrow t_i$ as $n\\rightarrow \\infty $ for all $i\\in \\lbrace 1,\\ldots , m\\rbrace $ .",
"Then, for all $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ we have that $\\Vert \\mathcal {E}_{\\pi }u-\\mathcal {E}_{\\pi _n}u\\Vert _\\kappa \\rightarrow 0,\\quad n\\rightarrow \\infty .$ First note that the set of all partitions with cardinality $m+1$ can be identified with the set $S^m:=\\big \\lbrace (s_1,\\ldots , s_m)\\in \\mathbb {R}^m\\, \\big |\\, 0<s_1<\\ldots <s_m\\big \\rbrace \\subset \\mathbb {R}^m.$ Therefore, the assertion is equivalent to the continuity of the map $S^m\\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa ,\\quad (s_1,\\ldots , s_m)\\rightarrow \\mathcal {E}_{\\lbrace 0,s_1,\\ldots , s_m\\rbrace }u.$ Since the mapping $[0,\\infty )\\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa ,\\; h\\mapsto \\mathcal {E}_h u$ is continuous for all $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ , and $\\Vert \\mathcal {E}_h u-\\mathcal {E}_h v\\Vert _\\kappa \\le e^{\\alpha h}\\Vert u-v\\Vert _\\kappa $ for all $h\\ge 0$ and $u,v\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ , it follows that (REF ) is continuous.",
"Let $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ .",
"In the following, we consider the limit of $\\mathcal {E}_\\pi u$ when the mesh size $|\\pi |_\\infty := \\max _{j=1,\\dots ,m} (t_j-t_{j-1})$ of the partition $\\pi =\\lbrace t_0,t_1,\\dots ,t_m\\rbrace \\in P$ with $0=t_0< t_1< \\ldots < t_m$ tends to zero.",
"For the sake of completeness, we define $|\\lbrace 0\\rbrace |_\\infty :=0$ .",
"The following lemma shows that ${S}(t)u$ can be obtained by a pointwise monotone approximation with finite partitions letting the mesh size tend to zero.",
"Proposition 2.7 Assume that the map (REF ) is continuous for all $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ .",
"Let $t\\ge 0$ and $(\\pi _n)_{n\\in \\mathbb {N}}\\subset P_t$ with $\\pi _n\\subset \\pi _{n+1}$ for all $n\\in \\mathbb {N}$ and $|\\pi _n|_\\infty \\searrow 0$ as $n\\rightarrow \\infty $ .",
"Then, for all $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ , $\\mathcal {E}_{\\pi _n}u\\nearrow {S}(t)u \\quad \\text{as }n\\rightarrow \\infty .$ In particular, ${S}(t)u=\\sup _{n\\in \\mathbb {N}} \\mathcal {E}_{\\frac{t}{n}}^n u=\\lim _{n\\rightarrow \\infty }\\mathcal {E}_{2^{-n}t}^{2^n}u\\quad \\text{for all }u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa ,$ where the supremum and the limit are to be understood in a pointwise sense.",
"For $t=0$ the statement is trivial.",
"Therefore, assume that $t>0$ , and let $\\big (\\mathcal {E}_\\infty u\\big )(x):=\\sup _{n\\in \\mathbb {N}} \\big (\\mathcal {E}_{\\pi _n}u\\big )(x)\\quad \\text{for }u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa \\text{ and }x\\in M.$ As in the proof of Lemma REF , the map $\\mathcal {E}_\\infty \\colon \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa \\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ is well-defined.",
"Let $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ .",
"Since $\\pi _n\\subset \\pi _{n+1}$ for all $n\\in \\mathbb {N}$ , it follows that $\\mathcal {E}_{\\pi _n}u\\nearrow \\mathcal {E}_\\infty u$ as $n\\rightarrow \\infty $ .",
"Since $(\\pi _n)_{n\\in \\mathbb {N}}\\subset P_t$ , we obtain that $\\mathcal {E}_\\infty u \\le {S}(t)u.$ Let $\\pi =\\lbrace t_0,t_1,\\ldots , t_m\\rbrace \\in P_t$ with $m\\in \\mathbb {N}$ and $0=t_0<t_1<\\ldots <t_m=t$ .",
"Since $|\\pi _n|_\\infty \\searrow 0$ as $n\\rightarrow \\infty $ , we may w.l.o.g.",
"assume that $\\#\\pi _n\\ge m+1$ for all $n\\in \\mathbb {N}$ .",
"Let $0=t_0^n<t_1^n<\\ldots <t_m^n=t$ for all $n\\in \\mathbb {N}$ with $\\pi _n^{\\prime }:=\\lbrace t_0^n,t_1^n,\\ldots , t_m^n\\rbrace \\subset \\pi _n$ and $t_i^n\\rightarrow t_i$ as $n\\rightarrow \\infty $ for all $i\\in \\lbrace 1,\\ldots , m\\rbrace $ .",
"Then, by Lemma REF , $\\Vert \\mathcal {E}_\\pi u-\\mathcal {E}_{\\pi _n^{\\prime }}u\\Vert _\\kappa \\rightarrow 0 \\quad \\text{as }n\\rightarrow \\infty .$ Therefore, $\\mathcal {E}_\\infty u-\\mathcal {E}_\\pi u\\ge \\mathcal {E}_{\\pi _n}u-\\mathcal {E}_\\pi u\\ge \\mathcal {E}_{\\pi _n^{\\prime }}u-\\mathcal {E}_\\pi u\\rightarrow 0\\quad \\text{as }n\\rightarrow \\infty ,$ showing that $\\mathcal {E}_\\infty u\\ge \\mathcal {E}_\\pi u$ .",
"Taking the supremum over all $\\pi \\in P_t$ , we obtian that $\\mathcal {E}_\\infty u={S}(t) u$ .",
"Now, let $\\pi _n:=\\big \\lbrace \\tfrac{kt}{2^n}\\, \\big |\\, k\\in \\lbrace 0,\\ldots ,2^n\\rbrace \\big \\rbrace $ for all $n\\in \\mathbb {N}$ .",
"Then, ${S}(t)u=\\lim _{m\\rightarrow \\infty }\\mathcal {E}_{\\pi _m}u=\\lim _{m\\rightarrow \\infty }\\mathcal {E}_{2^{-m}t}^{2^m}u\\le \\sup _{n\\in \\mathbb {N}} \\mathcal {E}_{\\frac{t}{n}}^n u\\le {S}(t)u,$ where we used the basic fact that $n=2^m\\in \\mathbb {N}$ for all $m\\in \\mathbb {N}$ ."
],
[
"Strong continuity",
"Let ${S}$ be the Feller semigroup from the previous section, i.e.",
"the semigroup envelope of the family $(S_\\lambda )_{\\lambda \\in \\Lambda }$ .",
"The aim of this section is to give conditions that ensure the strong continuity of the semigroup envelope ${S}$ .",
"Remark 3.1 Let $D\\subset \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ be the set of all $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ , for which the map $[0,\\infty )\\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa ,\\quad t\\mapsto {S}(t)u$ is continuous.",
"Then, by the semigroup property (REF ), $[0,\\infty )\\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa ,\\quad s\\mapsto {S}(s){S}(t)u= {S}(s+t)u$ is continuous for all $u\\in D$ .",
"Therefore, the set $D$ is invariant under the semigroup ${S}$ , i.e.",
"${S}(t)u\\in D$ for all $u\\in D$ and all $t\\ge 0$ .",
"Lemma 3.2 Let $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ .",
"Then, the following statements are equivalent: $\\lim _{h\\rightarrow 0}\\Vert {S}(h)u-u\\Vert _\\kappa =0$ .",
"The map $[0,\\infty )\\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa ,\\; t\\mapsto {S}(t)u$ is continuous.",
"Clearly, (ii) implies (i).",
"Therefore, assume that $\\lim _{h\\rightarrow 0}\\Vert {S}(h)u-u\\Vert _\\kappa =0$ .",
"Let $t\\ge 0$ and $\\varepsilon >0$ .",
"W.l.o.g.",
"we may assume that in (A2) we have $\\alpha \\ge 0$ .",
"By assumption, there exists some $\\delta >0$ such that $\\Vert {S}(h)u-u\\Vert _\\kappa <e^{-\\alpha t}\\varepsilon $ for all $h\\in [0,\\delta )$ .",
"Now, let $s\\ge 0$ with $|t-s|<\\delta $ .",
"Then, for $\\tau :=s\\wedge t$ , $\\Vert {S}(t)u-{S}(s)u\\Vert _\\kappa &=\\big \\Vert {S}(\\tau )\\big ({S}(|t-s|)u\\big )-{S}(\\tau )u\\big \\Vert _\\kappa \\\\&\\le e^{\\alpha \\tau }\\big \\Vert {S}\\big (|t-s|\\big )u-u\\big \\Vert _\\kappa <\\varepsilon ,$ where we used the Lipschitz continuity of ${S}(\\tau )$ with Lipschitz constant $e^{\\alpha \\tau }$ .",
"Remark 3.3 Let $D\\subset \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ arbitrary, and assume that ${S}$ is strongly continuous on $D$ .",
"Then, ${S}$ is also strongly continuous on the closure $\\overline{D}$ of $D$ .",
"In order to see this, let $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ and $(u_n)_{n\\in \\mathbb {N}}\\subset D$ with $\\Vert u_n-u\\Vert _\\kappa \\rightarrow 0$ as $n\\rightarrow \\infty $ .",
"W.l.o.g.",
"we may assume that $\\alpha \\ge 0$ .",
"Let $\\varepsilon >0$ .",
"Then, there exists some $n_0\\in \\mathbb {N}$ such that $\\Vert u_{n_0}-u\\Vert _\\kappa \\le \\tfrac{\\varepsilon }{3}e^{-\\alpha }$ .",
"Since $u_{n_0}\\in D$ , there exists some $\\delta \\in (0,1]$ such that $\\Vert {S}(h)u_{n_0}-u_{n_0}\\Vert _\\kappa <\\tfrac{\\varepsilon }{3}$ for all $h\\in [0,\\delta )$ .",
"Hence, for $h\\in [0,\\delta )$ , it follows that $\\Vert {S}(h)u-u\\Vert _\\kappa \\le \\frac{2\\varepsilon }{3} + \\Vert {S}(h)u_{n_0}-u_{n_0}\\Vert _\\kappa <\\varepsilon .$ Now, the previous lemma implies that $[0,\\infty )\\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa ,\\; t\\mapsto {S}(t)u$ is continuous.",
"We start with the first result ensuring the strong continuity of the semigroup envelope ${S}$ .",
"Proposition 3.4 Assume that, for every $\\delta >0$ , there exists a family of functions $(\\varphi _x)_{x\\in M}\\subset \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ satisfying the following: $0\\le \\varphi _x(y)\\le 1$ for all $y\\in M$ , $\\varphi _x(x)=0$ , $\\varphi _x(y)=1$ for all $y\\in M$ with $d(x,y)\\ge \\delta $ , $\\sup _{x\\in M} \\kappa (x)\\big [\\big ({S}(h)\\varphi _x\\big )(x)\\big ] \\rightarrow 0$ as $h\\searrow 0$ .",
"Then, the semigroup ${S}$ is strongly continuous.",
"Let $u\\in {\\rm {Lip}}_{\\rm b}\\setminus \\lbrace 0\\rbrace $ and $\\varepsilon >0$ .",
"Then, since $\\kappa $ is bounded, there exists some $\\delta >0$ such that $\\kappa (y)|u(y)-u(x)|\\le \\tfrac{\\varepsilon }{2e^{\\alpha }}\\quad \\text{for all }x,y\\in M\\text{ with }d(x,y)<\\delta .$ By assumption, there exists a family $(\\varphi _x)_{x\\in M}\\subset \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ with $0\\le \\varphi _x(y)\\le 1$ for all $y\\in M$ , $\\varphi _x(x)=0$ , $\\varphi _x(y)=1$ for all $y\\in M$ with $d(x,y)\\ge \\delta $ , and some $h_0\\in (0,1]$ such that $\\sup _{x\\in M} \\kappa (x)\\big [\\big ({S}(h)\\varphi _x\\big )(x)\\big ]< \\frac{\\varepsilon }{4\\Vert u\\Vert _\\infty }\\quad \\text{for all }h\\in [0,h_0).$ For all $h\\in (0,1]$ and $x\\in M$ , $\\Big \\Vert {S}(h)\\big ((1-\\varphi _x)|u-u(x)|\\big )\\Big \\Vert _\\kappa &\\le e^\\alpha \\big \\Vert (1-\\varphi _x)|u-u(x)|\\big \\Vert _\\kappa \\\\&\\le e^\\alpha \\sup _{\\begin{array}{c}y\\in M\\\\d(x,y)\\le \\delta \\end{array}} \\kappa (y)|u(y)-u(x)|\\le \\frac{\\varepsilon }{2}.$ Hence, for all $h\\in [0,h_0)$ and $x\\in M$ , since ${S}(h)1=1$ , $\\kappa (x)\\big |\\big ({S}(h)u\\big )(x)-u(x)\\big |&=\\kappa (x)\\big |\\big ({S}(h)(u-u(x))\\big )(x)\\big |\\\\&\\le \\kappa (x)\\big ({S}(h)|u-u(x)|\\big )(x)\\\\&\\le \\Big \\Vert {S}(h)\\big ((1-\\varphi _x)|u-u(x)|\\big )\\Big \\Vert _\\kappa \\\\&\\quad +\\kappa (x)\\big ({S}(h)(\\varphi _x|u-u(x)|)\\big )(x)\\\\&\\le \\frac{\\varepsilon }{2}+2\\kappa (x)\\Vert u\\Vert _\\infty \\big ({S}(h)\\varphi _x\\big )(x)< \\varepsilon .$ This shows that $\\Vert {S}(h)u-u\\big \\Vert _\\kappa <\\varepsilon $ for all $h\\in [0,h_0)$ , and therefore ${S}$ is strongly continuous on ${\\rm {Lip}}_{\\rm b}$ .",
"Since ${\\rm {Lip}}_{\\rm b}$ is, by definition, dense in $\\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ , Remark REF implies that ${S}$ is strongly continuous.",
"The function $\\varphi _x$ , for $x\\in M$ , in the previous proposition plays the role of a cut-off function.",
"Proposition REF is a generalisation of the well-known fact that transition semigroups of Lévy processes are strongly continuous, where the strong continuity is intimately related to the convergence in law of the process.",
"Note that, for transition semigroups of Lévy processes, the translation invariance together with the convergence in law ensures that the assumptions of Proposition REF are satisfied.",
"We denote by $D_\\Lambda \\subset \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ the linear space of all $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ for which there exist $L_u\\ge 0$ and $h_u>0$ such that $\\sup _{\\lambda \\in \\Lambda }\\Vert S_\\lambda (h)u-u\\Vert _\\kappa \\le L_u h\\quad \\text{for all }h\\in [0,h_u).$ Proposition 3.5 The semigroup ${S}$ is strongly continuous on $\\overline{D_\\Lambda }$ .",
"In particular, ${S}$ is strongly continuous if $D_\\Lambda $ is dense in $\\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ .",
"Let $u\\in D_\\Lambda $ and $0\\le h_1<h_2$ with $h_2-h_1<h_u$ .",
"Then, $\\big (S_{\\lambda _0}(h_1)u\\big )(x)-\\big (\\mathcal {E}_{h_2}u\\big )(x)\\le \\big (S_{\\lambda _0}(h_1)u\\big )(x)-\\big (S_{\\lambda _0}(h_2)u\\big )(x)$ for all $x\\in M$ and $\\lambda _0\\in \\Lambda $ .",
"Taking the supremum over $\\lambda _0\\in \\Lambda $ , it follows that $\\big (\\mathcal {E}_{h_1}u\\big )(x)-\\big (\\mathcal {E}_{h_2}u\\big )(x)\\le \\sup _{\\lambda \\in \\Lambda }\\big |\\big (S_\\lambda (h_1)u\\big )(x)-\\big (S_\\lambda (h_2)u\\big )(x)\\big |$ for all $x\\in M$ .",
"By a symmetry argument, multiplying by $\\kappa (x)$ and taking the supremum over all $x\\in M$ , we obtain that $\\Vert \\mathcal {E}_{h_1}u-\\mathcal {E}_{h_2}u\\Vert _\\kappa \\le \\sup _{\\lambda \\in \\Lambda }\\Vert S_\\lambda (h_1)u-S_\\lambda (h_2)u\\Vert _\\kappa $ .",
"Moreover, $\\Vert S_\\lambda (h_1)u-S_\\lambda (h_2)u\\Vert _\\kappa \\le e^{\\alpha h_1}\\Vert S_\\lambda (h_2-h_1)u-u\\Vert _\\kappa \\le L_ue^{\\alpha h_1} (h_2-h_1).$ Taking the supremum over all $\\lambda \\in \\Lambda $ , we obtain that $\\Vert \\mathcal {E}_{h_1}u -\\mathcal {E}_{h_2}u\\Vert _\\kappa \\le L_u e^{\\alpha h_1} (h_2-h_1).$ Next, we show that $\\Vert \\mathcal {E}_\\pi u -u\\Vert _\\kappa \\le L_ue^{\\alpha \\max \\pi } \\max \\pi $ for all $\\pi \\in P$ with $\\max \\pi \\in [0,h_u)$ by an induction on $\\# \\pi \\in \\mathbb {N}$ .",
"First, let $\\pi \\in P$ with $\\#\\pi =1$ , i.e.",
"$\\pi =\\lbrace 0\\rbrace $ .",
"Then, $\\Vert \\mathcal {E}_\\pi u -u\\Vert _\\kappa =\\Vert \\mathcal {E}_{\\lbrace 0\\rbrace }u-u\\Vert _\\kappa =0=L_u e^{\\alpha \\max \\pi } \\max \\pi .$ Now, let $m\\in \\mathbb {N}$ , and assume that (REF ) holds for all $\\pi \\in P$ with $\\max \\pi \\in [0,h_u)$ and $\\#\\pi =m$ .",
"Let $\\pi \\in P$ with $\\#\\pi =m+1$ and $t_m:=\\max \\pi \\in [0,h_u)$ .",
"Then, $\\pi ^{\\prime }:=\\pi \\setminus \\lbrace t_m\\rbrace \\in P$ with $\\#\\pi ^{\\prime }=m$ and $t_{m-1}:=\\max \\pi ^{\\prime }\\in [0,t_m)$ .",
"Therefore, by induction hypothesis and (REF ), it follows that $\\Vert \\mathcal {E}_\\pi u-u\\Vert _\\kappa &\\le \\Vert \\mathcal {E}_\\pi u-\\mathcal {E}_{\\pi ^{\\prime }}u\\Vert _\\kappa +\\Vert \\mathcal {E}_{\\pi ^{\\prime }}u-u\\Vert _\\kappa \\\\&=\\Vert \\mathcal {E}_{\\pi ^{\\prime }}\\mathcal {E}_{t_m-t_{m-1}}u-\\mathcal {E}_{\\pi ^{\\prime }}u\\Vert _\\kappa +\\Vert \\mathcal {E}_{\\pi ^{\\prime }}u-u\\Vert _\\kappa \\\\&\\le e^{\\alpha t_{m-1}}\\Vert \\mathcal {E}_{t_m-t_{m-1}}u-u\\Vert _\\kappa +\\Vert \\mathcal {E}_{\\pi ^{\\prime }}u-u\\Vert _\\kappa \\\\&\\le L_u e^{\\alpha t_{m-1}}(t_m-t_{m-1})+L_ue^{\\alpha t_{m-1}}t_{m-1}\\\\&= L_u e^{\\alpha t_{m-1}}t_m\\le L_ue^{\\alpha \\max \\pi }\\max \\pi .$ By definition of the semigroup ${S}$ , we thus obtain that $\\Vert {S}(h)u-u\\Vert _\\kappa \\le L_u e^{\\alpha h}h \\rightarrow 0\\quad \\text{as }h\\rightarrow 0.$ The following proposition is somewhat similar to Proposition REF .",
"Note that (ii) in Proposition REF is a condition related to the semigroup envelope ${S}$ , and its verification is typically nontrivial.",
"The following proposition replaces condition (ii) in Proposition REF by a smoothness condition on the cut-off functions $(\\varphi _x)_{x\\in M}$ , where smoothness is given in terms of the family of generators $(A_\\lambda )_{\\lambda \\in \\Lambda }$ .",
"Proposition 3.6 Assume that for every $\\delta >0$ there exists a family of functions $(\\varphi _x)_{x\\in M}\\subset \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ satisfying the following: $0\\le \\varphi _x(y)\\le 1$ for all $y\\in M$ , $\\varphi _x(x)=0$ , and $\\varphi _x(y)=1$ for all $y\\in M$ with $d(x,y)\\ge \\delta $ , There exist $L\\ge 0$ and $h_0>0$ such that, for all $h\\in [0,h_0)$ and $x\\in M$ , $\\sup _{\\lambda \\in \\Lambda }\\Vert S_\\lambda (h)\\varphi _x-\\varphi _x\\Vert _\\kappa \\le L h.$ Then, ${S}$ is strongly continuous.",
"By assumption, the family $(\\varphi _x)_{x\\in M}$ satisfies condition (i) from Proposition REF .",
"We now verify that (ii') implies condition (ii) from Proposition REF .",
"Observe that $\\big (\\mathcal {E}_h \\varphi _x\\big )(y)\\le \\varphi _x(y)+ \\big | \\big (\\mathcal {E}_h \\varphi _x\\big )(y)-\\varphi _x(y)\\big |$ for all $h\\in [0,h_0)$ and $x,y\\in M$ .",
"W.l.o.g.",
"we assume that $\\alpha \\ge 0$ in (A2).",
"Then, by (REF ), we obtain that $\\big ( \\mathcal {E}_\\pi \\mathcal {E}_h \\varphi _x\\big )(x)&\\le \\big (\\mathcal {E}_\\pi \\varphi _x\\big )(x)+\\big (\\mathcal {E}_\\pi \\big | \\mathcal {E}_h \\varphi _x-\\varphi _x\\big |\\big )(x)\\\\&\\le \\big (\\mathcal {E}_\\pi \\varphi _x\\big )(x)+\\frac{e^{\\alpha h_0}}{\\kappa (x)}\\Vert \\mathcal {E}_h \\varphi _x-\\varphi _x\\Vert _\\kappa \\\\&\\le \\big (\\mathcal {E}_\\pi \\varphi _x\\big )(x)+\\frac{Le^{\\alpha h_0}h}{\\kappa (x)}$ for all $\\pi \\in P$ with $\\max \\pi \\in [0,h_0)$ and $h\\in [0,h_0)$ .",
"Inductively, it follows that $\\big (\\mathcal {E}_\\pi \\varphi _x\\big )(x)\\le \\varphi _x(x)+\\frac{Le^{\\alpha h_0}\\max \\pi }{\\kappa (x)}=\\frac{Le^{\\alpha h_0}\\max \\pi }{\\kappa (x)}$ for all $\\pi \\in P$ with $\\max \\pi \\in [0,h_0)$ .",
"Taking the supremum over all $\\pi \\in P_h$ for $h\\in [0,h_0)$ yields that $\\sup _{x\\in M}\\kappa (x)\\big [\\big ({S}(h)\\varphi _x\\big )(x)\\big ]\\le Le^{\\alpha h_0}h\\rightarrow 0\\quad \\text{as }h\\searrow 0.$ Therefore, condition (ii) from Proposition REF is satisfied and the strong continuity of ${S}$ follows."
],
[
"Related HJB equation and viscosity solutions",
"Let $\\lambda \\in \\Lambda $ .",
"Then, we denote by $D_\\lambda \\subset \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ the space of all $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ such that the map $[0,\\infty )\\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa ,\\; t\\mapsto S_\\lambda (t)u$ is continuous.",
"Further, let $D(A_\\lambda )$ denote the space of all $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ for which $A_\\lambda u:=\\lim _{h\\searrow 0}\\frac{S_\\lambda (h)u-u}{h}\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ exists w.r.t.",
"$\\Vert \\cdot \\Vert _\\kappa $ .",
"Note that, by definition, $D(A_\\lambda )\\subset D_\\lambda $ .",
"Let $u\\in \\bigcap _{\\lambda \\in \\Lambda }D(A_\\lambda )$ with $C_u:=\\sup _{\\lambda \\in \\Lambda }\\Vert A_\\lambda u\\Vert _\\kappa <\\infty $ .",
"Then, it follows that (see e.g.",
"[12]) $\\Vert S_\\lambda (h)u-u\\Vert _\\kappa \\le \\int _0^h\\Vert S_\\lambda (s) A_\\lambda u\\Vert _\\kappa \\, {\\rm d}s\\le C_ue^{\\alpha h}h\\quad \\text{for all }\\lambda \\in \\Lambda .$ This shows that $u\\in D_\\Lambda $ .",
"Moreover, since $\\sup _{\\lambda \\in \\Lambda } \\Vert A_\\lambda u\\Vert _\\kappa <\\infty $ , it follows that $\\big (\\mathcal {A}u\\big )(x):=\\sup _{\\lambda \\in \\Lambda } \\big (A_\\lambda u\\big )(x)$ is well-defined for all $x\\in M$ .",
"Lemma 4.1 Let $u\\in \\bigcap _{\\lambda \\in \\Lambda }D(A_\\lambda )$ with $\\sup _{\\lambda \\in \\Lambda }\\Vert A_\\lambda u\\Vert _\\kappa <\\infty \\quad \\text{and}\\quad \\sup _{\\lambda \\in \\Lambda }\\Vert S_\\lambda (h)A_\\lambda u-A_\\lambda u\\Vert _\\kappa \\rightarrow 0\\quad \\text{as }h\\rightarrow 0.$ Then, $\\lim _{h\\searrow 0}\\big \\Vert \\frac{\\mathcal {E}_h u-u}{h}- \\mathcal {A}u\\big \\Vert _\\kappa =0$ .",
"In particular, $\\mathcal {A}u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ .",
"Let $\\varepsilon >0$ .",
"Then, by assumption, there exists some $h_0>0$ such that $\\sup _{\\lambda \\in \\Lambda } \\Vert S_\\lambda (s) A_\\lambda u-A_\\lambda u\\Vert _\\kappa \\le \\varepsilon \\quad \\text{for all }s\\in [0,h_0].$ Hence, for all $h\\in (0,h_0]$ , it follows that $\\bigg \\Vert \\frac{\\mathcal {E}_hu-u}{h}- \\mathcal {A}u\\bigg \\Vert _\\kappa &\\le \\sup _{\\lambda \\in \\Lambda }\\bigg \\Vert \\frac{S_\\lambda (h)u-u}{h}- A_\\lambda u\\bigg \\Vert _\\kappa \\\\&=\\sup _{\\lambda \\in \\Lambda }\\frac{1}{h} \\bigg \\Vert \\int _0^hS_\\lambda (s)A_\\lambda u-A_\\lambda u\\, {\\rm d}s\\bigg \\Vert _\\kappa \\\\&\\le \\sup _{\\lambda \\in \\Lambda }\\frac{1}{h} \\int _0^h\\Vert S_\\lambda (s)A_\\lambda u-A_\\lambda u\\Vert _\\kappa \\, {\\rm d}s\\le \\varepsilon .$ Proposition 4.2 Let $u\\in \\bigcap _{\\lambda \\in \\Lambda }D(A_\\lambda )$ with $\\sup _{\\lambda \\in \\Lambda }\\Vert A_\\lambda u\\Vert _\\kappa <\\infty \\quad \\text{and}\\quad \\sup _{\\lambda \\in \\Lambda }\\Vert S_\\lambda (h)A_\\lambda u-A_\\lambda u\\Vert _\\kappa \\rightarrow 0\\quad \\text{as }h\\rightarrow 0.$ Then, $\\mathcal {A}u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ and the following statements are equivalent: The map $[0,\\infty )\\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa ,\\; t\\mapsto {S}(t)\\mathcal {A}u$ is continuous, $\\lim _{h\\searrow 0}\\big \\Vert \\tfrac{{S}(h)u-u}{h}- \\mathcal {A}u\\big \\Vert _\\kappa =0$ , i.e.",
"$\\mathcal {A}u=\\lim _{h\\searrow 0} \\frac{{S}(h)u-u}{h}$ , where the limit is w.r.t.",
"$\\Vert \\cdot \\Vert _\\kappa $ .",
"By Lemma REF , we already know that $\\mathcal {A}u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ .",
"Let $D$ denote the set of all $v\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ , for which the map $[0,\\infty )\\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa ,\\; t\\mapsto {S}(t)v$ is continuous.",
"Our assumptions imply that $u\\in D_\\Lambda $ .",
"Therefore, by Proposition REF , $u\\in D$ and, by Remark REF , ${S}(h)u\\in D$ for all $h\\ge 0$ .",
"Hence, by Remark REF , statement (ii) implies (i).",
"By Lemma REF , $\\mathcal {A}u-\\frac{{S}(h)u-u}{h}\\le \\mathcal {A}u -\\frac{\\mathcal {E}_hu-u}{h}\\rightarrow 0,\\quad \\text{as }h\\searrow 0.$ Assuming that the map $[0,\\infty )\\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa ,\\; t\\mapsto {S}(t)\\mathcal {A}u$ is continuous, it follows that $\\bigg \\Vert \\frac{1}{h}\\int _0^h {S}(s)\\mathcal {A}u \\, {\\rm d}s-\\mathcal {A}u\\bigg \\Vert _\\kappa \\rightarrow 0,\\quad \\text{as }h\\searrow 0.$ Hence, it is sufficient to show that ${S}(t)u-u\\le \\int _0^t {S}(s)\\mathcal {A}u \\, {\\rm d}s\\quad \\text{for all }t\\ge 0.$ Let $t\\ge 0$ and $h>0$ .",
"Then, $\\mathcal {E}_h u-u=\\sup _{\\lambda \\in \\Lambda } \\int _0^h S_\\lambda (s)A_\\lambda u\\, {\\rm d} s \\le \\int _0^h{S}(s)\\mathcal {A}u\\,{\\rm d} s=\\int _t^{t+h}{S}(s-t)\\mathcal {A}u\\, {\\rm d}s.$ Next, we prove that $\\mathcal {E}_\\pi u-u\\le \\int _0^{\\max \\pi } {S}(s)\\mathcal {A}u \\, {\\rm d}s \\quad \\text{for all }\\pi \\in P$ by an induction on $m=\\# \\pi $ .",
"If $m=1$ , i.e.",
"if $\\pi =\\lbrace 0\\rbrace $ , the statement is trivial.",
"Hence, assume that $\\mathcal {E}_{\\pi ^{\\prime }} u-u\\le \\int _0^{\\max \\pi ^{\\prime }} {S}(s)\\mathcal {A}u \\, {\\rm d}s$ for all $\\pi ^{\\prime }\\in P$ with $\\#\\pi ^{\\prime }=m$ for some $m\\in \\mathbb {N}$ .",
"Let $\\pi =\\lbrace t_0,t_1,\\ldots , t_m\\rbrace \\in P$ with $0=t_0< t_1< \\ldots < t_m$ and $\\pi ^{\\prime }:=\\pi \\setminus \\lbrace t_m\\rbrace $ .",
"Then, it follows from (REF ) that $\\mathcal {E}_\\pi u -\\mathcal {E}_{\\pi ^{\\prime }}u&\\le {S}(t_{m-1})\\big (\\mathcal {E}_{t_m-t_{m-1}}u-u\\big )\\le {S}(t_{m-1})\\bigg (\\int _{t_{m-1}}^{t_m} {S}(s-t_{m-1})\\mathcal {A}u\\, {\\rm d}s\\bigg )\\\\&\\le \\int _{t_{m-1}}^{t_m}{S}(s)\\mathcal {A}u\\, {\\rm d}s,$ where the last inequality follows from Jensen's inequality.",
"By induction hypothesis, we thus obtain that $\\mathcal {E}_\\pi u -u&= \\big (\\mathcal {E}_\\pi u -\\mathcal {E}_{\\pi ^{\\prime }}u\\big )+\\big (\\mathcal {E}_{\\pi ^{\\prime }} u-u\\big )\\le \\int _{t_{m-1}}^{t_m}{S}(s)\\mathcal {A}u\\, {\\rm d}s+\\int _{0}^{t_{m-1}}{S}(s)\\mathcal {A}u\\, {\\rm d}s\\\\&=\\int _0^{\\max \\pi }{S}(s)\\mathcal {A}u\\, {\\rm d}s.$ In particular, $\\mathcal {E}_\\pi u-u\\le \\int _0^t{S}(s)\\mathcal {A}u\\, {\\rm d}s$ for every $\\pi \\in P_t$ .",
"Taking the supremum over all $\\pi \\in P_t$ yields the assertion.",
"We now introduce the class of test functions, which will be used for the definition of a viscosity solution.",
"Let $\\mathcal {D}:=\\bigg \\lbrace u\\in \\bigcap _{\\lambda \\in \\Lambda }D(A_\\lambda )\\, \\bigg |\\, \\sup _{\\lambda \\in \\Lambda }\\Vert A_\\lambda u\\Vert _\\kappa <\\infty \\text{ and } \\lim _{h\\searrow 0}\\bigg \\Vert \\frac{{S}(h)u-u}{h}-\\mathcal {A}u\\bigg \\Vert _\\kappa = 0 \\bigg \\rbrace .$ In the sequel, we are interested in viscosity solutions to the differential equation $u^{\\prime }(t)= \\mathcal {A}u(t), \\quad \\text{for }t> 0,$ where we use the following notion of a viscosity solution.",
"Definition 4.3 We say that $u\\colon [0,\\infty )\\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ is a viscosity subsolution to (REF ) if $u$ is continuous, and for every $t>0$ , $x\\in M$ , and every differentiable function $\\psi \\colon (0,\\infty )\\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ with $\\psi (t)\\in \\mathcal {D}$ , $\\big (\\psi (t)\\big )(x)=\\big (u(t)\\big )(x)$ and $\\psi (s)\\ge u(s)$ for all $s>0$ , $\\big (\\psi ^{\\prime }(t)\\big )(x)\\le \\big (\\mathcal {A}\\psi (t)\\big )(x).$ Analogously, $u$ is called a viscosity supersolution to (REF ) if $u\\colon [0,\\infty )\\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ is continuous, and for every $t>0$ , $x\\in M$ , and every differentiable function $\\psi \\colon (0,\\infty )\\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ with $\\psi (t)\\in \\mathcal {D}$ , $\\big (\\psi (t)\\big )(x)=\\big (u(t)\\big )(x)$ and $\\psi (s)\\le u(s)$ for all $s>0$ , $\\big (\\psi ^{\\prime }(t)\\big )(x)\\ge \\big (\\mathcal {A}\\psi (t)\\big )(x).$ We say that $u$ is a viscosity solution to (REF ) if $u$ is a viscosity subsolution and a viscosity supersolution.",
"Remark 4.4 In general it is not clear how rich the class of test functions for a viscosity solution from the previous definition is.",
"However, in the examples in Section , we will see that, in most cases, where $M$ is a Banach space, ${\\rm {Lip}}_{\\rm b}^k\\subset \\mathcal {D}$ with $k\\in \\lbrace 0,1,2\\rbrace $ , where ${\\rm {Lip}}_{\\rm b}^k$ denotes the set of all $k$ -times (Fréchet) differentiable functions $M\\rightarrow \\mathbb {R}$ with bounded and Lipschitz continuous derivatives.",
"For a function $\\psi \\colon (0,\\infty )\\times M\\rightarrow \\mathbb {R}$ , which is differentiable w.r.t.",
"$t$ and $\\partial _t\\psi \\colon (0,\\infty )\\times M\\rightarrow \\mathbb {R}$ uniformly w.r.t.",
"$x$ Lipschitz continuous in $t$ with Lipschitz constant $L\\ge 0$ , it follows that $\\sup _{x\\in M}\\bigg |\\frac{\\psi (t+h,x)-\\psi (t,x)}{h}-\\partial _t\\psi (t,x)\\bigg | \\le Lh \\rightarrow 0 \\quad \\text{as }h\\searrow 0$ for all $t>0$ .",
"Hence, if ${\\rm {Lip}}_{\\rm b}^k\\subset \\mathcal {D}$ for some $k\\in \\mathbb {N}_0$ , every $\\psi \\in {\\rm {Lip}}_{\\rm b}^{1,k}\\big ((0,\\infty )\\times M\\big )$ is differentiable as a map $(0,\\infty )\\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ and satisfies $\\psi (t)\\in \\mathcal {D}$ for all $t>0$ .",
"In most applications, the class ${\\rm {Lip}}_{\\rm b}^{1,k}\\big ((0,\\infty )\\times M\\big )$ of test functions is sufficiently large in order to obtain uniqueness of a viscosity solution.",
"For more details concerning our notion of a viscosity solution and the uniqueness of solutions, we refer to Section REF .",
"We conclude this section with the following main theorem.",
"Theorem 4.5 Assume that the semigroup ${S}$ is strongly continuous.",
"Then, for every $u_0\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ , the function $u\\colon [0,\\infty )\\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa , \\;t\\mapsto {S}(t)u_0$ is a viscosity solution to the abstract initial value problem $u^{\\prime }(t)&=&\\mathcal {A}u(t), \\quad \\text{for }t> 0,\\\\u(0)&=&u_0.$ Fix $t>0$ and $x\\in M$ .",
"We first show that $u$ is a viscosity subsolution.",
"Let $\\psi \\colon (0,\\infty )\\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ differentiable with $\\psi (t)\\in \\mathcal {D}$ , $\\big (\\psi (t)\\big )(x)=\\big (u(t)\\big )(x)$ and $\\psi (s)\\ge u(s)$ for all $s>0$ .",
"Then, for every $h\\in (0,t)$ , it follows from Equation (REF ) that $0&=\\frac{{S}(h){S}(t-h)u_0-{S}(t)u_0}{h}=\\frac{{S}(h)u(t-h)-u(t)}{h}\\\\&\\le \\frac{{S}(h)\\psi (t-h)-u(t)}{h} \\le \\frac{{S}(h)\\big (\\psi (t-h)-\\psi (t)\\big )+{S}(h)\\psi (t)-u(t)}{h}\\\\&= {S}(h)\\bigg (\\frac{\\psi (t-h)-\\psi (t)}{h}\\bigg )+\\frac{{S}(h)\\psi (t)-\\psi (t)}{h}+\\frac{\\psi (t)-u(t)}{h}.$ Moreover, $&\\bigg \\Vert {S}(h)\\bigg (\\frac{\\psi (t-h)-\\psi (t)}{h}\\bigg )+\\psi ^{\\prime }(t)\\bigg \\Vert _\\kappa \\rightarrow 0\\quad \\text{and}\\\\&\\bigg \\Vert \\frac{{S}(h)\\psi (t)-\\psi (t)}{h}-\\mathcal {A}\\psi (t)\\bigg \\Vert _\\kappa \\rightarrow 0.$ as $h\\searrow 0$ .",
"Since $\\big (u(t)\\big )(x)=\\big (\\psi (t)\\big )(x)$ , it follows that $0\\le -\\big (\\psi ^{\\prime }(t)\\big )(x)+\\big (\\mathcal {A}\\psi (t)\\big )(x).$ In order to show that $u$ is a viscosity supersolution, let $\\psi \\colon (0,\\infty )\\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ differentiable with $\\psi (t)\\in \\mathcal {D}$ , $\\big (\\psi (t)\\big )(x)=\\big (u(t)\\big )(x)$ and $\\psi (s)\\le u(s)$ for all $s>0$ .",
"By Equation (REF ), for all $h>0$ with $0<h<t$ , we obtain that $0&=\\frac{{S}(t)u_0-{S}(h){S}(t-h)u_0}{h}\\\\&=\\frac{u(t)-{S}(h)u(t-h)}{h}\\le \\frac{u(t)-{S}(h)\\psi (t-h)}{h}\\\\&= \\frac{u(t)-\\psi (t)}{h}+\\frac{\\psi (t)-{S}(h)\\psi (t)}{h}+\\frac{{S}(h)\\psi (t)-{S}(h)\\psi (t-h)}{h}\\\\&\\le \\frac{u(t)-\\psi (t)}{h}+\\frac{\\psi (t)-{S}(h)\\psi (t)}{h}+{S}(h)\\bigg (\\frac{\\psi (t)-\\psi (t-h)}{h}\\bigg ).$ Furthermore, $&\\bigg \\Vert {S}(h)\\bigg (\\frac{\\psi (t)-\\psi (t-h)}{h}\\bigg )- \\psi ^{\\prime }(t)\\bigg \\Vert _\\kappa \\rightarrow 0\\quad \\text{and}\\\\&\\bigg \\Vert \\frac{\\psi (t)-{S}(h)\\psi (t)}{h} +\\mathcal {A}\\psi (t)\\bigg \\Vert _\\kappa \\rightarrow 0.$ Since $\\big (u(t)\\big )(x)=\\big (\\psi (t)\\big )(x)$ , we obtain that $0\\le -\\big (\\mathcal {A}\\psi (t)\\big )(x)+\\big (\\psi ^{\\prime }(t)\\big )(x)$ ."
],
[
"Stochastic representation",
"In this section, we derive a stochastic representation for the semigroup envelope ${S}$ using sublinear expectations.",
"Such stochastic representations are of fundamental interest in various fields and, in particular, in the field of robust finance.",
"The prime example for a sublinear expectation arising from a semigroup envelope for a particular family of semigroups is the $G$ -expectation, cf.",
"Denis et al.",
"[8] and Peng [30],[31], and the corresponding Markov process, the $G$ -Brownian Motion, is the analogue of a Brownian Motion in the presence of volatility uncertainty.",
"More general forms of stochastic processes arising from semigroups are given by the class of so-called $G$ -Lévy processes, cf.",
"Hu and Peng [18], Neufeld and Nutz [25], and Denk et al.",
"[10].",
"In this section, we provide a similar representation for ${S}$ under an additional continuity assumption.",
"We point out that our setup covers the aforementioned existing approaches.",
"We start with a short introduction to the theory of nonlinear expectations.",
"For a measurable space $(\\Omega ,\\mathcal {F})$ , we denote the space of all bounded $\\mathcal {F}$ -measurable functions (random variables) $\\Omega \\rightarrow \\mathbb {R}$ by $\\mathcal {L}^\\infty (\\Omega ,\\mathcal {F})$ .",
"For two bounded random variables $X,Y\\in \\mathcal {L}^\\infty (\\Omega ,\\mathcal {F})$ we write $X\\le Y$ if $X(\\omega )\\le Y(\\omega )$ for all $\\omega \\in \\Omega $ .",
"For a constant $\\alpha \\in \\mathbb {R}$ , we do not distinguish between $\\alpha $ and the constant function taking the value $\\alpha $ .",
"Definition 5.1 Let $(\\Omega ,\\mathcal {F})$ be a measurable space.",
"A functional $\\mathcal {E}\\colon \\mathcal {L}^\\infty (\\Omega ,\\mathcal {F})\\rightarrow \\mathbb {R}$ is called a sublinear expectation if for all $X,Y\\in \\mathcal {L}^\\infty (\\Omega ,\\mathcal {F})$ and $\\lambda >0$ $\\mathcal {E}(X)\\le \\mathcal {E}(Y)$ if $X\\le Y$ , $\\mathcal {E}(\\alpha )=\\alpha $ for all $\\alpha \\in \\mathbb {R}$ , $\\mathcal {E}(X+Y)\\le \\mathcal {E}(X)+\\mathcal {E}(Y)$ and $\\mathcal {E}(\\lambda X)=\\lambda \\mathcal {E}(X)$ .",
"We say that $(\\Omega ,\\mathcal {F},\\mathcal {E})$ is a sublinear expectation space if there exists a set of probability measures $\\mathcal {P}$ on $(\\Omega ,\\mathcal {F})$ such that $\\mathcal {E}(X)=\\sup _{\\mathbb {P}\\in \\mathcal {P}} \\mathbb {E}_\\mathbb {P}(X)\\quad \\text{for all }X\\in \\mathcal {L}^\\infty (\\Omega ,\\mathcal {F}),$ where $\\mathbb {E}_\\mathbb {P}(\\cdot )$ denotes the expectation w.r.t.",
"to the probability measure $\\mathbb {P}$ .",
"Definition 5.2 Let $L\\subset \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ be a linear space.",
"We say that ${S}$ is continuous from above on $L$ if ${S}(t)u_n\\searrow 0$ for all $t\\ge 0$ and all $(u_n)_{n\\in \\mathbb {N}}\\subset L$ with $u_n\\searrow 0$ as $n\\rightarrow \\infty $ .",
"Remark 5.3 Assume that $M$ is compact.",
"Then, by Dini's lemma, ${S}$ is continuous from above on $\\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa =\\mathop {\\text{\\upshape {UC}}}\\nolimits _{\\rm b}$ .",
"Assume that $M$ satisfies the Heine-Borel property, i.e.",
"every closed and bounded subset of $M$ is compact, and that $\\kappa \\in {\\rm {C}}_0$ .",
"Then, $\\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa =\\lbrace u\\in {\\rm {C}}\\, |\\, \\kappa u\\in {\\rm {C}}_0\\rbrace $ , where ${\\rm {C}}_0$ denotes the closure of the space ${\\rm {C}}_c$ of all continuous functions with compact support w.r.t.",
"$\\Vert \\cdot \\Vert _\\infty $ .",
"In fact, let $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ .",
"Then, there exists a sequence $(u_n)_{n\\in \\mathbb {N}}\\subset {\\rm {Lip}}_{\\rm b}$ with $\\Vert u-u_n\\Vert _\\kappa \\rightarrow 0$ as $n\\rightarrow \\infty $ .",
"Since $\\kappa \\in {\\rm {C}}_0$ , it follows that $v_n:=\\kappa u_n\\in {\\rm {C}}_0$ for all $n\\in \\mathbb {N}$ .",
"Since ${\\rm {C}}_0$ endowed with $\\Vert \\cdot \\Vert _\\infty $ is a Banach space and $\\Vert \\kappa u-v_n\\Vert _\\infty =\\Vert u-u_n\\Vert _\\kappa \\rightarrow 0\\quad \\text{as }n\\rightarrow \\infty ,$ we find that $\\kappa u\\in {\\rm {C}}_0$ .",
"Now, assume that $\\kappa u\\in {\\rm {C}}_0$ .",
"Then, there exists a sequence $(v_n)_{n\\in \\mathbb {N}}\\subset {\\rm {C}}_c$ with $\\Vert \\kappa u-v_n\\Vert _\\infty \\rightarrow 0$ .",
"Defining $u_n:=\\frac{v_n}{\\kappa }$ for $n\\in \\mathbb {N}$ , we see that $u_n\\in {\\rm {C}}_0\\subset \\mathop {\\text{\\upshape {UC}}}\\nolimits _{\\rm b}$ .",
"Since $\\mathop {\\text{\\upshape {UC}}}\\nolimits _{\\rm b}\\subset \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ and $\\Vert u-u_n\\Vert _\\kappa =\\Vert \\kappa u-v_n\\Vert _\\infty \\rightarrow 0,$ it follows that $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ .",
"We have therefore established the equality $\\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa =\\lbrace u\\in {\\rm {C}}\\, |\\, \\kappa u\\in {\\rm {C}}_0\\rbrace $ .",
"Let $(u_n)_{n\\in \\mathbb {N}}\\subset \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ with $u_n\\searrow 0$ as $n\\rightarrow \\infty $ .",
"Since $v_n:=\\kappa u_n\\in {\\rm {C}}_0$ for all $n\\in \\mathbb {N}$ with $v_n\\searrow 0$ as $n\\rightarrow \\infty $ , it follows that $\\Vert u_n\\Vert _\\kappa =\\Vert v_n\\Vert _\\infty \\rightarrow 0$ as $n\\rightarrow \\infty $ by Dini's lemma.",
"In particular, the semigroup ${S}$ and in fact every continuous map $\\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa \\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ is continuous from above on $\\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ .",
"Assume that ${S}$ is continuous from above on ${\\rm {Lip}}_{\\rm b}$ .",
"the space ${\\rm {Lip}}_{\\rm b}$ is invariant under ${S}(t)$ for all $t\\ge 0$ .",
"Note that ${S}(t)u\\in {\\rm {Lip}}_{\\rm b}$ for all $u\\in {\\rm {Lip}}_{\\rm b}$ and $t\\ge 0$ .",
"Therefore, by [9], ${S}(t)$ uniquely extends to an operator ${S}(t)\\colon {\\rm {C}}_{\\rm b}\\rightarrow {\\rm {C}}_{\\rm b}$ , which is again continuous from above.",
"Moreover, for every $n\\in \\mathbb {N}$ , $v\\in {\\rm {C}}_{\\rm b}(M^{n+1})$ the mapping $M^{n+1}\\rightarrow \\mathbb {R},\\quad (x_1,\\ldots , x_n, x_{n+1})\\mapsto \\big ({S}(t)v(x_1,\\ldots , x_n,\\, \\cdot \\,)\\big )(x_{n+1})$ is bounded and continuous.",
"Continuity from above on ${\\rm {Lip}}_{\\rm b}$ will be crucial for the existence of a stochastic representation.",
"In Remark REF b), we have seen that, if $M$ satisfies the Heine-Borel property and $\\kappa \\in {\\rm {C}}_0$ , then ${S}$ is continuous from above on $\\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ .",
"The following proposition, which is a generalisation of [10], gives a sufficient condition for the continuity from above on ${\\rm {Lip}}_{\\rm b}$ in the case that $\\kappa $ does not vanish at infinity and $M$ is (only) locally compact.",
"Recall that ${\\rm {C}}_0$ is the closure of the space ${\\rm {Lip}}_c$ of all Lipschitz continuous functions with compact support w.r.t.",
"the supremum norm $\\Vert \\cdot \\Vert _\\infty $ , and that ${\\rm {C}}_0\\subset \\mathop {\\text{\\upshape {UC}}}\\nolimits _{\\rm b}\\subset \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ .",
"Proposition 5.4 Suppose that for every $x\\in M$ and every $\\delta >0$ there exists a function $\\varphi _x\\in {\\rm {C}}_0$ satisfying the following: $\\varphi _x(x)=1$ and $0\\le \\varphi _x\\le 1$ , $\\varphi _x\\in \\bigcap _{\\lambda \\in \\Lambda } D(A_\\lambda )$ with $\\sup _{\\lambda \\in \\Lambda }\\Vert A_\\lambda \\varphi _x\\Vert _\\kappa \\le \\delta $ .",
"Then, ${S}$ is continuous from above on ${\\rm {Lip}}_{\\rm b}$ .",
"Fix $t> 0$ , $x\\in M$ and $\\delta >0$ .",
"Notice that $1\\in D(A_\\lambda )$ with $A_\\lambda 1=0$ since $S_\\lambda (t) 1 =1$ for all $\\lambda \\in \\Lambda $ .",
"Therefore, $(1-\\varphi _x)\\in \\bigcap _{\\lambda \\in \\Lambda }D(A_\\lambda )$ with $A_\\lambda (1-\\varphi _x)=-A_\\lambda \\varphi _x$ .",
"Since $\\varphi _x(x)=1$ , it follows that $\\kappa (x)\\big [\\big ({S}(t)(1-\\varphi _x)\\big )(x)\\big ]&\\le \\Vert {S}(t)(1-\\varphi _x)-(1-\\varphi _x)\\Vert _\\kappa \\le te^{\\alpha t}\\sup _{\\lambda \\in \\Lambda }\\Vert A_\\lambda \\varphi _x\\Vert _\\kappa \\\\&\\le \\delta te^{\\alpha t}.$ Let $(u_n)_{n\\in \\mathbb {N}}\\subset {\\rm {Lip}}_{\\rm b}$ with $u_n\\searrow 0$ as $n\\rightarrow \\infty $ and $\\varepsilon >0$ .",
"Then, there exists some $\\varphi _x\\in {\\rm {C}}_0$ satisfying (i) and (ii) with $\\delta =\\frac{\\varepsilon \\kappa (x)}{2te^{\\alpha t}c}$ , where $c:=\\max \\big \\lbrace 1,\\Vert u_1\\Vert _\\infty \\big \\rbrace $ .",
"Then, $\\Vert u_n\\Vert _\\infty \\big ({S}(t)(1-\\varphi _x)\\big )(x)\\le \\frac{\\varepsilon }{2}\\quad \\text{for all }n\\in \\mathbb {N}.$ Moreover, there exists some $n\\in \\mathbb {N}$ such that $\\Vert u_n\\varphi _x\\Vert _\\kappa <\\frac{\\varepsilon }{2}$ since $\\varphi _x\\in {\\rm {C}}_0$ .",
"Hence, $\\big ({S}(t)u_n\\big )(x)\\le \\Vert u_n\\Vert _\\infty \\big ({S}(t)(1-\\varphi _x)\\big )(x)+\\big ({S}(t)(u_n\\varphi _x)\\big )(x)<\\varepsilon .$ This shows that ${S}(t)u_n\\searrow 0$ as $n\\rightarrow \\infty $ .",
"Now, let $(u_n)_{n\\in \\mathbb {N}}\\subset {\\rm {Lip}}_{\\rm b}$ and $u\\in {\\rm {Lip}}_{\\rm b}$ with $u_n\\searrow u$ as $n\\rightarrow \\infty $ .",
"Then, $|{S}(t)u_n-{S}(t)u|\\le {S}(t)(u_n-u)\\searrow 0 \\quad \\text{as }n\\rightarrow \\infty .$ Note that, although not explicitly stated in Proposition REF , the existence of a function $\\varphi _x\\in {\\rm {C}}_0$ with $\\varphi _x(x)\\ne 0$ for all $x\\in M$ implies that $M$ is locally compact.",
"Thus, Proposition REF is thus only applicable for locally compact $M$ .",
"The following theorem is a direct consequence of [9].",
"Theorem 5.5 Assume that $M$ is a Polish space and that ${S}$ is continuous from above on ${\\rm {Lip}}_{\\rm b}$ .",
"Then, there exists a quadruple $(\\Omega ,\\mathcal {F},(\\mathcal {E}^x)_{x\\in M},(X_t)_{t\\ge 0})$ such that $X_t\\colon \\Omega \\rightarrow M$ is $\\mathcal {F}$ -$\\mathcal {B}$ -measurable for all $t\\ge 0$ , $(\\Omega ,\\mathcal {F},\\mathcal {E}^x)$ is a sublinear expectation space with $\\mathcal {E}^x(u(X_0))=u(x)$ for all $x\\in M$ and $u\\in {\\rm {C}}_{\\rm b}$ , For all $0\\le s<t$ , $n\\in \\mathbb {N}$ , $0\\le t_1<\\ldots <t_n\\le s$ and $v\\in {\\rm {C}}_{\\rm b}(M^{n+1})$ , $\\mathcal {E}^x\\big (v(X_{t_1},\\ldots ,X_{t_n},X_t)\\big )=\\mathcal {E}^x\\left(\\big ({S}(t-s)v(X_{t_1},\\ldots ,X_{t_n},\\, \\cdot \\,)\\big )(X_s)\\right).$ In particular, $\\big ({S}(t)u\\big )(x)=\\mathcal {E}^x(u(X_t)).$ for all $t\\ge 0$ , $x\\in M$ and $u\\in {\\rm {C}}_{\\rm b}$ .",
"Remark 5.6 The quadruple $(\\Omega ,\\mathcal {F},(\\mathcal {E}^x)_{x\\in M},(X_t)_{t\\ge 0})$ can be seen as a nonlinear version of a Markov process.",
"As an illustration, we consider the case, where the semigroup ${S}$ and thus $\\mathcal {E}^x$ is linear for all $x\\in M$ , and choose $v=u(X_t)1_B(Y)$ with $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _{\\rm b}$ and $B\\in \\mathcal {B}^n$ , where $\\mathcal {B}^n$ denotes the product $\\sigma $ -algebra of the Borel $\\sigma $ -algebra $\\mathcal {B}$ .",
"Then, $\\mathcal {E}^x=\\mathbb {E}_{\\mathbb {P}^x}$ is the expectation w.r.t.",
"a probability measure $\\mathbb {P}^x$ on $(\\Omega ,\\mathcal {F})$ for all $x\\in M$ .",
"Using the continuity from above and Dynkin's lemma, Equation (REF ) reads as $\\mathbb {E}_{\\mathbb {P}^x}\\big (u(X_t)1_B(X_{t_1},\\ldots , X_{t_n})\\big )=\\mathbb {E}_{\\mathbb {P}^x}\\big [\\big ({S}(t-s)u\\big )(X_s)1_B(X_{t_1},\\ldots , X_{t_n})\\big ],$ which is equivalent to the Markov property $\\mathbb {E}_{\\mathbb {P}^x}\\big (u(X_t)|\\mathcal {F}_s\\big )=\\big ({S}(t-s)u\\big )(X_s) \\quad \\mathbb {P}^x\\text{-a.s.},$ where $\\mathcal {F}_s:=\\sigma \\big (\\lbrace X_u\\, |\\, 0\\le u\\le s\\rbrace \\big )$ .",
"On the other hand, if $\\mathcal {E}^x=\\mathbb {E}_{\\mathbb {P}^x}$ , the Markov property (REF ) implies Property (iii) from Theorem REF .",
"A natural question, in particular in view of (REF ) is, if the nonlinear expectation $\\mathcal {E}^x$ can be extended to unbounded functions satisfying a certain growth condition.",
"We would like to point out that [9] a priori only applies to bounded functions.",
"Using the fact that $\\mathcal {E}^x$ admits a representation in terms of a nonempty set $\\mathcal {P}^x$ of probability measures on $(\\Omega ,\\mathcal {F})$ , i.e.",
"$\\mathcal {E}^x(Y)=\\sup _{\\mathbb {P}\\in \\mathcal {P}^x} \\mathbb {E}_\\mathbb {P}(Y)\\quad \\text{for all }Y\\in \\mathcal {L}^\\infty (\\Omega ,\\mathcal {F}),$ allows to define $\\mathcal {E}^x(Y):=\\sup _{\\mathbb {P}\\in \\mathcal {P}^x} \\mathbb {E}_\\mathbb {P}(Y)\\in \\mathbb {R}$ for $\\mathcal {F}$ -measurable functions $Y\\colon \\Omega \\rightarrow \\mathbb {R}$ with $\\sup _{\\mathbb {P}\\in \\mathcal {P}^x} \\mathbb {E}_\\mathbb {P}(|Y|)<\\infty $ .",
"On the other hand, (REF ) gives rise to a well-defined notion of $\\mathcal {E}^x$ for functions of the form $u(X_t)$ with $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ and $t\\ge 0$ .",
"Consider a weight function $w\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ with $w(x)\\ge 0$ for all $x\\in M$ and a measurable function $u\\colon M\\rightarrow \\mathbb {R}$ with $|u(x)|\\le w(x)$ for all $x\\in M$ .",
"Then, (REF ) implies that $\\mathcal {E}^x\\big (|u(X_t)|\\big )\\le \\mathcal {E}^x\\big (w(X_t)\\big )=\\big ({S}(t)w\\big )(x)<\\infty $ for all $t\\ge 0$ and $x\\in M$ ."
],
[
"Connection to control theory",
"In this section, we discuss our results in light of the standard literature and standard examples in control theory.",
"In particular, we discuss the relation between the semigroup envelope and the value function of Meyer-type control problems.",
"We further go into more detail on our notion of a viscosity solution in view of the standard one and uniqueness results for the latter."
],
[
"The notion of viscosity solution and uniqueness",
"A priori, our notion of a viscosity solution is somewhat different from the classical one related to (standard) parabolic HJB equations.",
"The key difference between both notions is the class of test functions.",
"While in a standard setting, the class of test functions typically consists of sufficiently smooth functions defined on the parabolic domain $[0,\\infty )\\times M$ , in our notion, we formally separate the space and time variable and consider differentiable functions $\\psi \\colon [0,\\infty )\\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ taking values in a function space.",
"Here, time regularity is given in terms of differentiability in $t$ w.r.t.",
"the norm $\\Vert \\cdot \\Vert _\\kappa $ , and the convergence of the difference quotient to the derivative is thus up to the weight $\\kappa $ uniform in the space variable.",
"Space regularity is given in terms of the abstract condition $\\psi (t)\\in \\mathcal {D}$ , where $\\mathcal {D}:=\\bigg \\lbrace u\\in \\bigcap _{\\lambda \\in \\Lambda }D(A_\\lambda )\\, \\bigg |\\, \\sup _{\\lambda \\in \\Lambda }\\Vert A_\\lambda u\\Vert _\\kappa <\\infty \\text{ and } \\lim _{h\\searrow 0}\\bigg \\Vert \\frac{{S}(h)u-u}{h}-\\mathcal {A}u\\bigg \\Vert _\\kappa = 0 \\bigg \\rbrace .$ Let us consider as an illustrative example, the case where $M=\\mathbb {R}$ , $\\kappa =1$ , and $A_\\lambda =\\frac{\\lambda ^2}{2}\\partial _{xx}$ for $\\lambda \\in \\Lambda :=[\\sigma _\\ell ,\\sigma _h]$ with $0<\\sigma _\\ell \\le \\sigma _h$ .",
"That is, our control parameter is the volatility of a Brownian Motion.",
"In this case, $\\mathcal {D}=\\mathop {\\text{\\upshape {UC}}}\\nolimits _b^2$ is the space of all twice differentiable functions with bounded and uniformly continuous derivatives.",
"We therefore see that, in the case of partial differential equations, the set $\\mathcal {D}$ typically encodes some sort of space regularity in terms of differentiability in the space variable.",
"This will become also clear in the examples in Section REF .",
"As we point out in Remark REF , it is, in general, unclear how rich the class of test functions for a viscosity solution from Definition REF is.",
"Therefore, uniqueness is not given a priori and has to be checked on a case by case basis.",
"However, it is worth noting that, if $M$ is, for example, an open subset of $\\mathbb {R}^d$ with $d\\in \\mathbb {N}$ , the standard notion of a viscosity solution is very robust in view of the considered class of test functions, cf.",
"Ishii [19].",
"Typically, one chooses functions that are twice differentiable on $M\\times [0,\\infty )$ with continuous derivatives up to order 2 as test functions.",
"However, the notion of a viscosity solution and, in particular, uniqueness is not affected by replacing ${\\rm {C}}^2(M\\times [0,\\infty ))$ , e.g., by ${\\rm {C}}^\\infty _c([0,\\infty )\\times M)$ , i.e.",
"functions that are compactly supported and infinitely smooth functions.",
"Roughly speaking this is due to the fact that the notion of a viscosity solution is a very local solution concept, and therefore only the local behaviour of test functions matters.",
"We point out that under very mild conditions, e.g., for all $\\delta >0$ and $x\\in M$ , the existence of a cut-off function $\\varphi \\in \\mathcal {D}$ with $0\\le \\varphi \\le 1$ , $\\varphi (x)=0$ , and $\\varphi (y)=1$ for $y\\in M$ with $d(x,y)\\ge \\delta $ , our notion of a viscosity solution can also be formulated in terms of local extrema instead of global extrema; thus leading to a local solution concept as well.",
"We build on Remark REF in the case that $M$ is an open subset of $\\mathbb {R}^d$ with $d\\in \\mathbb {N}$ .",
"Assume that ${\\rm {C}}_c^\\infty \\subset \\mathcal {D}$ , where ${\\rm {C}}^\\infty _c$ denotes the space of all infinitely differentiable functions $M\\rightarrow \\mathbb {R}$ with compact support, and let $\\psi \\in {\\rm {C}}^\\infty _c([0,\\infty )\\times M)$ .",
"Since $\\psi $ has a compact support and $\\kappa $ is continuous, it follows that $\\sup _{x\\in M}\\kappa (x)\\bigg |\\frac{\\psi (t+h,x)-\\psi (t,x)}{h}-\\partial _t\\psi (t,x)\\bigg | \\le Lh \\rightarrow 0 \\quad \\text{as }h\\searrow 0$ for all $t>0$ .",
"In particular, the function $\\psi \\colon [0,\\infty )\\rightarrow \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa , \\; t\\mapsto \\psi (t):=\\psi (t,\\, \\cdot \\,)$ is differentiable.",
"Moreover $\\psi (t)=\\psi (t,\\,\\cdot \\, )\\in {\\rm {C}}_c^\\infty \\subset \\mathcal {D}$ .",
"Therefore, assuming that (at least) ${\\rm {C}}_c^\\infty \\subset \\mathcal {D}$ , any $\\psi \\in {\\rm {C}}^\\infty _c([0,\\infty )\\times M)$ is a test function in the sense of Definition REF .",
"Thus, the notion of a viscosity solution from Definition REF coincides with the usual notion in most cases covered by the standard theory.",
"As a consequence, uniqueness of viscosity solutions can be obtained from Ishii's lemma."
],
[
"Semigroup envelopes as value functions to optimal control problems",
"In this section, we identify the semigroup envelope as the value function of a space-time discrete Meyer-type optimal control problem under the additional assumption that each semigroup $S_\\lambda $ is a family of transition kernels of a stochastic process.",
"In the following, we describe the broad idea behind the approach using semigroup envelopes.",
"Assume that, $S_\\lambda $ is a semigroup of transition kernels of a controlled stochastic process $(X_t^{x, \\lambda })_{t\\ge }$ (for the sake of a simplified notation defined on the same probability space) with control set $\\Lambda $ and control parameter $\\lambda \\in \\Lambda $ , i.e.",
"$\\big (S_\\lambda (t)u\\big )(x)=\\mathbb {E}\\big [u\\big (X_t^{\\lambda ,x}\\big )\\big ]$ for $x\\in M$ , $\\lambda \\in \\Lambda $ , $t\\ge 0$ , and $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ .",
"Then, for a fixed time-horizon $t\\ge 0$ , one typically considers a (suitably defined) set of admissible controls $\\Lambda _{\\rm ad}^t$ and the value function $V(u,t,x):=\\sup _{\\lambda \\in \\Lambda _{\\rm ad}^t}\\mathbb {E}\\big [u\\big (X_t^{\\lambda ,x}\\big )\\big ]$ of the related Meyer-type optimal control problem.",
"Note that this is usually only possible if the controlled dynamics satisfy a certain structure.",
"The idea behind the semigroup envelope is to transform the dynamic optimization problem given in terms of the value function (REF ) into a series of static optimization problems with value functions of the form $\\sup _{\\lambda \\in \\Lambda }\\mathbb {E}\\big [u\\big (X_t^{\\lambda ,x}\\big )\\big ]=\\sup _{\\lambda \\in \\Lambda }\\big (S_\\lambda (t)u\\big )(x)=:\\big (\\mathcal {E}_t u\\big )(x),$ Now, one considers a partition $\\pi =\\lbrace t_0,\\ldots , t_m\\rbrace \\in P_t$ with $0=t_0<\\ldots <t_m=t$ of the time-interval $[0,t]$ , and one optimizes after each time-step, leading to the expression $\\big (\\mathcal {E}_\\pi u\\big )(x)\\big (\\mathcal {E}_{t_1-t_0}\\cdots \\mathcal {E}_{t_m-t_{m-1}} u\\big )(x).$ Letting the mesh size $|\\pi |$ of the partition $\\pi $ tend to zero or taking the supremum over all partitions $\\pi \\in P_t$ leads to a formal approximation of the dynamic optimization problem (REF ) in terms of a series of static control problems on a grid that becomes finer and finer as the mesh size tends to zero.",
"In the sequel, we will make this approximation rigorous by choosing the set of admissible controls as space-time discrete controls.",
"To that end, we consider static controls of the form $\\Lambda _M:=\\bigg \\lbrace (\\lambda _i, B_i)_{i\\in \\mathbb {N}}\\in (\\Lambda \\times \\mathcal {B})^\\mathbb {N}\\, \\bigg |\\, B_i\\cap B_j=\\emptyset \\text{ for }i\\ne j\\text{ and }\\bigcup _{i\\in \\mathbb {N}}B_i=M\\bigg \\rbrace $ One can think of $\\lambda =(\\lambda _i, B_i)_{i\\in \\mathbb {N}}\\in (\\Lambda \\times \\mathcal {B})^\\mathbb {N}\\in \\Lambda _M$ as a function taking the value $\\lambda _i$ on $B_i$ for each $i\\in \\mathbb {N}$ .",
"For $\\lambda =(\\lambda _i,B_i)_{i\\in \\mathbb {N}}\\in (\\Lambda \\times \\mathcal {B})^\\mathbb {N}\\in \\Lambda _M$ , we define $\\big (S_\\lambda (t) u\\big )(x):=\\sum _{i\\in \\mathbb {N}}1_{B_i}(x)\\big (S_{\\lambda ^i}(t)u\\big )(x)$ for all $x\\in M$ and $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ .",
"We now add a dynamic component, and define $\\Lambda _{\\rm ad}^t:=\\bigg \\lbrace (\\lambda _k,h_k)_{k=1,\\ldots , m}\\in \\big (\\Lambda _M\\times [0,t]\\big )^m\\, \\bigg |\\, m\\in \\mathbb {N},\\; \\sum _{k=1}^m h_k=t\\bigg \\rbrace .$ Roughly speaking, the set $\\Lambda _{\\rm ad}^t$ corresponds to the set of all space-time discrete admissible controls for the control set $\\Lambda $ .",
"For $\\lambda =(\\lambda _k,h_k)_{k=1,\\ldots , m}\\in \\Lambda _{\\rm ad}^t$ with $m\\in \\mathbb {N}$ and $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ , we define $J_\\lambda u:=S_{\\lambda _1}(h_1)\\cdots S_{\\lambda _m}(h_m)u,$ where $S_{\\lambda _k}(h_k)$ is defined as in (REF ) for $k=1,\\ldots , m$ .",
"Then, for all $t\\ge 0$ , $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ , and $x\\in M$ , $\\big ({S}(t)u\\big )(x)=\\sup _{\\lambda \\in \\Lambda _{\\rm ad}^t}\\big (J_\\lambda u\\big )(x).$ That is, the semigroup envelope is the value function of an abstract analogue of the optimal control problem (REF ) with $\\Lambda _{\\rm ad}^t$ given as in (REF ).",
"In fact, by definition of $\\Lambda _{\\rm ad}^t$ , it follows that $\\sup _{\\lambda \\in \\Lambda _{\\rm ad}^t}J_\\lambda u\\le {S}(t)u$ for all $t\\ge 0$ and $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ .",
"On the other hand, let $\\varepsilon >0$ and $\\pi =\\lbrace t_0,\\ldots , t_m\\rbrace \\in P_t$ with $0=t_0<\\ldots < t_m=t$ , and define $h_k:=t_k-t_{k-1}$ for $k =1,\\ldots , m$ .",
"By a backward recursion, we may choose an $\\frac{\\varepsilon }{2m}$ -optimizer of $\\mathcal {E}_{h_k}\\cdots \\mathcal {E}_{h_m} u$ for each $x\\in M$ and $k=m,\\ldots , 1$ .",
"Since $M$ is separable, there exist $\\lambda _1,\\ldots , \\lambda _m\\in \\Lambda _M$ such that $\\mathcal {E}_\\pi u=\\mathcal {E}_{h_1}\\cdots \\mathcal {E}_{h_m} u\\le S_{\\lambda _1}(h_1)\\cdots S_{\\lambda _m}(h_m)u+\\varepsilon = J_\\lambda u+\\varepsilon $ where $\\lambda :=(\\lambda _k,t_k-t_{k-1})_{k=1,\\ldots , m}\\in \\Lambda _{\\rm ad}^t$ .",
"Letting $\\varepsilon \\rightarrow 0$ and taking the supremum over all $\\pi \\in P_t$ and $\\lambda \\in \\Lambda _{\\rm ad}^t$ , yields ${S}(t)u\\le \\sup _{\\lambda \\in \\Lambda _{\\rm ad}^t}J_\\lambda u$ .",
"Considering standard cases in optimal control, the connection between semigroup envelopes and the value function of a Meyer-type optimal control problem can also be established a posteriori, since both lead to a viscosity solution to the same HJB-equation.",
"In these cases, one thus sees that the optimizing over space-time discrete admissible controls, which we have discussed in this section, is equivalent to optimizing over usual admissible controls, which typically possess a nondiscrete structure."
],
[
"Some illustrative examples from control theory",
"In this section, we discuss two examples in the context of control theory.",
"For $k\\in \\mathbb {N}_0$ , let ${\\rm {Lip}}_{\\rm b}^k$ denote the space of all $k$ -times differentiable functions with bounded and Lipschitz continuous (Fréchet) derivatives up to order $k$ .",
"Example 6.1 (Geometric Brownian Motion) Let $M=\\mathbb {R}$ and $\\Lambda $ be a nonempty set of tuples $(\\mu ,\\sigma )\\subset \\mathbb {R}\\times [0,\\infty )$ with $\\beta :=\\sup _{(\\mu ,\\sigma )\\in \\Lambda }|\\mu |+\\frac{\\sigma ^2}{2}<\\infty $ Let $\\lambda =(\\mu ,\\sigma )\\in \\Lambda $ , $p\\ge 1$ , and $W$ be a Brownian Motion on a probability space $(\\Omega ,\\mathcal {F},\\mathbb {P})$ .",
"Define $X_t^\\lambda :=\\exp \\bigg (t\\big (\\mu -\\tfrac{\\sigma ^2}{2}\\big ) +\\sigma W_t\\bigg )$ for $t\\ge 0$ and $x\\in \\mathbb {R}$ .",
"Then, $\\mathbb {E}(|X_t^\\lambda |^p)^{\\tfrac{1}{p}}= e^{\\left(\\mu +\\tfrac{(p-1)\\sigma ^2}{2}\\right) t}\\le e^{p \\beta t}.$ Moreover, $\\mathbb {E}\\big (|X_t^\\lambda -1|^2\\big )&=1-2\\mathbb {E}\\big (X_t^\\lambda \\big )+\\mathbb {E}\\big (|X_t^\\lambda |^2\\big )=1-2e^{\\mu t}+e^{(2\\mu +\\sigma ^2)t}\\\\& \\le 1-2e^{-\\beta t}+e^{2\\beta t}.$ Let $\\kappa (x):=(1+|x|)^{-p}$ for $x\\in \\mathbb {R}$ and $S_\\lambda $ be given by $\\big (S_\\lambda (t)u\\big )(x):=\\mathbb {E}\\big (u(x X_t^\\lambda )\\big )$ for $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ , $t\\ge 0$ , and $x\\in \\mathbb {R}$ .",
"Then, it follows that $\\Vert S_\\lambda (t)u\\Vert _\\kappa \\le e^{p \\beta t} \\Vert u\\Vert _\\kappa $ for $t\\ge 0$ and $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ .",
"Moreover, for $u\\in {\\rm {Lip}}_{\\rm b}$ , $\\Vert u\\Vert _{{\\rm {Lip}}}\\le e^{\\beta t}\\Vert u\\Vert _{{\\rm {Lip}}}$ and $\\Vert S_\\lambda (t)u-u\\Vert _\\kappa \\le \\Vert u\\Vert _{{\\rm {Lip}}}\\mathbb {E}\\big (|X_t^1-1|\\big )\\le \\sqrt{1-2e^{-\\beta t}+e^{2\\beta t}}\\rightarrow 0\\quad \\text{as }t\\rightarrow 0.$ Therefore, by Theorem REF and Proposition REF , the semigroup envelope ${S}$ for the family $(S_\\lambda )_{\\lambda \\in \\Lambda }$ exists and is a strongly continuous Feller semigroup.",
"Let $u\\in {\\rm {Lip}}_{\\rm b}^2$ with compact support ${\\rm supp}(u)$ and $A_\\lambda u\\in {\\rm {Lip}}_{\\rm b}$ be given by $\\big (A_\\lambda u\\big )(x):= \\mu xu^{\\prime }(x) +\\frac{\\sigma ^2x^2}{2}u^{\\prime \\prime }(x)\\quad \\text{for }x\\in \\mathbb {R}.$ Since ${\\rm supp}(u)$ is compact, $\\sup _{\\lambda \\in \\Lambda }\\Vert A_\\lambda u\\Vert _\\infty <\\infty \\quad \\text{and}\\quad C_u:=\\sup _{\\lambda \\in \\Lambda }\\Vert A_\\lambda u\\Vert _{\\rm {Lip}}<\\infty .$ By Ito's formula, it follows that $\\frac{\\big (S_\\lambda (h)u\\big )(x)-u(x)}{h}=\\frac{1}{h}\\int _0^h \\big (S_\\lambda (s)A_\\lambda u\\big )(x)\\, {\\rm d}s$ for all $h>0$ and $x\\in \\mathbb {R}$ , which, together with (REF ), implies that $\\bigg \\Vert \\frac{S_\\lambda (h)u-u}{h}-A_\\lambda u\\bigg \\Vert _\\kappa \\le C_u\\sqrt{1-2e^{-\\beta h}+e^{2\\beta h}}\\rightarrow 0\\quad \\text{as }h\\searrow 0.$ It follows that the set of all $u\\in {\\rm {Lip}}_{\\rm b}^2$ with compact support ${\\rm supp}(u)$ is contained in $\\mathcal {D}$ .",
"By Theorem REF , we thus obtain that $u(t):={S}(t)u_0$ , for $t\\ge 0$ , defines a viscosity solution to the fully nonlinear Cauchy problem $\\partial _t u(t,x)&=&\\sup _{(\\mu ,\\sigma )\\in \\Lambda } \\left(\\mu x\\partial _xu(t,x)+\\frac{\\sigma ^2x^2}{2}\\partial _{xx}u(t,x)\\right), \\quad (t,x)\\in (0,\\infty )\\times \\mathbb {R},\\\\u(0,x)&=&u_0(x),\\quad x\\in \\mathbb {R}.$ Under the nondegeneracy condition $\\inf _{(\\mu ,\\sigma )\\in \\Lambda } |\\sigma |>0$ , the above HJB equation has a unique viscosity solution.",
"By Remark REF b), the semigroup ${S}$ is continuous from above.",
"The nonlinear Markov process related to ${S}$ can be seen as a geometric $G$ -Brownian Motion (cf.",
"Theorem REF ).",
"Example 6.2 (Ornstein-Uhlenbeck processes on separable Hilbert spaces) We consider the case where $M=H$ is a real separable Hilbert space.",
"Let $\\Lambda $ be a set of triplets $(B,m,C)$ , where $m\\in H$ , $B\\in L(H)$ , and $C\\in L(H)$ is a self-adjoint positive semidefinite trace class operator, with $\\beta :=\\sup _{(B,m,C)\\in \\Lambda } \\big (\\Vert B\\Vert + \\Vert m\\Vert +\\Vert C\\Vert _{\\mathop {\\text{\\upshape {tr}}}\\nolimits }\\big )<\\infty .$ Let $\\lambda =(B,m,C)\\in \\Lambda $ , $T_B(t):=e^{tB}\\in L(H)$ , for $t\\ge 0$ , and $W^C$ be an $H$ -valued Brownian Motion with covariance operator $C$ on a probability space $(\\Omega ,\\mathcal {F},\\mathbb {P})$ .",
"For $t\\ge 0$ , we define $X_t^\\lambda :=\\int _0^t T_B(s)m\\, {\\rm d}s+\\int _0^t T_B(t-s)\\, {\\rm d}W^C_s$ and $S_\\lambda $ by $\\big (S_\\lambda (t)u\\big )(x):=\\mathbb {E}\\big (u(T_B(t)x+X_t^\\lambda )\\big )$ for $x\\in H$ , $t\\ge 0$ , and $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ .",
"Moreover, let $\\kappa :=(1+\\Vert x\\Vert ^2)^{-1}$ for $x\\in H$ .",
"Using basic facts from (infinite-dimensional) stochastic calculus and (REF ), below, for $F(x)=B(x)+m$ , $1+\\mathbb {E}(\\Vert T_B(t)x+X_t^\\lambda \\Vert ^2)&\\le 1+\\bigg \\Vert T_B(t)x+\\int _0^t T_B(s)m\\, {\\rm d}s\\bigg \\Vert ^2+\\int _0^t e^{2\\Vert B\\Vert s}\\Vert C\\Vert _{\\mathop {\\text{\\upshape {tr}}}\\nolimits }\\, {\\rm d}s\\\\&\\le (1+\\Vert x\\Vert ^2)e^{2\\big (\\Vert B\\Vert +\\Vert m\\Vert \\big )t}+e^{2\\Vert B\\Vert t}\\Vert C\\Vert _{\\mathop {\\text{\\upshape {tr}}}\\nolimits }t\\\\&\\le (1+\\Vert x\\Vert ^2)e^{2\\beta t}$ for all $t\\ge 0$ and $x\\in H$ , which implies that $\\Vert S_\\lambda (t) u\\Vert _{\\kappa }\\le e^{2\\beta t }\\Vert u\\Vert _\\kappa $ for all $t\\ge 0$ and $u\\in {\\rm {Lip}}_{\\rm b}$ .",
"By (REF ), below, $\\Vert S_\\lambda (t) u\\Vert _{\\rm {Lip}}\\le e^{\\beta t}\\Vert u\\Vert _{\\rm {Lip}}$ for all $t\\ge 0$ and $u\\in {\\rm {Lip}}_{\\rm b}$ .",
"For $u\\in {\\rm {Lip}}_{\\rm b}^2$ , let $C_u:=\\max \\lbrace \\Vert D_x u\\Vert ,\\Vert D_x^2 u\\Vert _\\infty ,\\Vert D_x^2 u\\Vert _{{\\rm {Lip}}}\\rbrace ,$ where $D_x$ and $D_x^2$ denote the first and second Fréchet derivative in the space-variable, and $A_\\lambda u\\in C_\\kappa $ be given by $\\big (A_\\lambda u\\big )(x) =D_x u(x)(Bx+m)+\\frac{1}{2}\\mathop {\\text{\\upshape {tr}}}\\nolimits \\big (CD_x^2u(x)\\big )$ for $x\\in H$ .",
"Then, for all $h\\ge 0$ and $x\\in H$ , $\\big |\\big (S_\\lambda (h)A_\\lambda u\\big )(x)-\\big (A_\\lambda u\\big )(x)\\big |\\le C_u \\beta (1+\\Vert x\\Vert ) \\mathbb {E}\\big (\\Vert T_B(t)x+X_t^\\lambda -x\\Vert \\big ).$ We estimate the last term using (REF ), below, and obtain that $\\mathbb {E}\\big (\\Vert T_B(t)x+X_t^\\lambda -x\\Vert \\big )&\\le \\big (e^{(\\Vert B\\Vert +\\Vert m\\Vert )t}-1\\big )\\big (1+\\Vert x\\Vert \\big )+\\sqrt{\\Vert C\\Vert _{\\mathop {\\text{\\upshape {tr}}}\\nolimits }t}\\\\&\\le (1+\\Vert x\\Vert )\\big (e^{\\beta t}-1+\\sqrt{\\beta t}\\big ).$ Therefore, $\\Vert S_\\lambda (h)A_\\lambda u-A_\\lambda u\\Vert _\\kappa \\le C_u \\sqrt{2}\\beta \\big (e^{\\beta h}-1+\\sqrt{\\beta h}\\big ).$ By Ito's formula, it follows that $\\frac{\\big (S_\\lambda (h)u\\big )(x)-u(x)}{h}=\\frac{1}{h}\\int _0^h \\big (S_\\lambda (s)Au\\big )(x)\\, {\\rm d}s$ for all $h>0$ and $x\\in H$ , which implies that $\\bigg \\Vert \\frac{S_\\lambda (h)u-u}{h}-A_\\lambda u\\bigg \\Vert _\\kappa \\le C_u \\sqrt{2}\\beta \\big (e^{\\beta h}-1+\\sqrt{\\beta h}\\big )\\rightarrow 0 \\quad \\text{as }h\\searrow 0.$ In order to show that ${\\rm {Lip}}_{\\rm b}^2\\subset \\mathcal {D}$ , it remains to show that ${S}$ is strongly continuous.",
"For this we invoke Proposition REF .",
"Note that ${\\rm {Lip}}_{\\rm b}^2$ is not dense in ${\\rm {Lip}}_{\\rm b}$ if $H$ is infinite-dimensional.",
"Let $\\delta \\in (0,1]$ and $\\varphi \\colon [0,\\infty )\\rightarrow [0,1]$ infinitely smooth with $\\varphi (s)=1$ for $x\\in \\big [0,\\tfrac{\\delta }{2}\\big ]$ and $\\varphi (s)=0$ for $s\\in [\\delta ,\\infty )$ .",
"For $x,y\\in H$ , let $\\varphi _x(y):=\\varphi (\\Vert y-x\\Vert )$ .",
"Then, $\\varphi _x\\in {\\rm {Lip}}_{\\rm b}^2$ with $\\Vert D_x\\varphi _x \\Vert _\\infty \\le \\Vert \\varphi ^{\\prime }\\Vert _\\infty \\quad \\text{and}\\quad \\Vert D_x^2\\varphi _x\\Vert _\\infty \\le \\frac{3}{\\delta }\\Vert \\varphi ^{\\prime }\\Vert _\\infty +\\Vert \\varphi ^{\\prime \\prime }\\Vert _\\infty \\quad \\text{for all }x\\in M.$ Hence, $\\Vert A_\\lambda \\varphi _x\\Vert _\\kappa \\le \\frac{5\\beta }{2\\delta }\\max \\big (\\Vert \\varphi ^{\\prime }\\Vert _\\infty +\\Vert \\varphi ^{\\prime \\prime }\\Vert _\\infty \\big )=:L$ for all $x\\in M$ .",
"Therefore, by Proposition REF , the semigroup ${S}$ is strongly continuous.",
"Altogether, we have shown that the assumptions (A1) and (A2) are satisfied, the semigroup envelope ${S}$ is strongly continuous and ${\\rm {Lip}}_{\\rm b}^2\\subset \\mathcal {D}$ .",
"By Theorem REF , we thus obtain that $u(t):={S}(t)u_0$ , for $t\\ge 0$ , defines a viscosity solution to the fully nonlinear PDE $\\partial _t u(t,x)&=&\\sup _{(B,m,C)\\in \\Lambda }\\left( D_x u(t,x)(Bx+m)+\\frac{1}{2}\\mathop {\\text{\\upshape {tr}}}\\nolimits \\big (CD_x^2u(t,x)\\big )\\right),\\\\&&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\quad (t,x)\\in (0,\\infty )\\times H,\\\\u(0,x)&=&u_0(x),\\quad x\\in H.$ We point out that, by Remark REF , the class of test functions in the definition of a viscosity solution contains the set ${\\rm {Lip}}_{\\rm b}^{1,2}[0,\\infty )\\times H)$ .",
"If $H=\\mathbb {R}^d$ , the semigroup ${S}$ is continuous from above by Remark REF b), which implies the existence of an O-U-process under a nonlinear expectation which represents ${S}$ (cf.",
"Theorem REF )."
],
[
"Further examples",
"For $k\\in \\mathbb {N}_0$ , let ${\\rm {Lip}}_{\\rm b}^k$ denote the space of all $k$ -times differentiable functions with bounded and Lipschitz continuous derivatives up to order $k$ .",
"Example 7.1 (Koopman semigroups on real separable Banach spaces) We consider the case, where the state space $M=X$ is a real separable Banach space.",
"We denote topological dual space of $X$ by $X^{\\prime }$ and the operator norm on $X^{\\prime }$ by $\\Vert \\cdot \\Vert _{X^{\\prime }}$ .",
"We consider a nonempty set $\\Lambda $ of Lipschitz continuous functions $F\\colon X\\rightarrow X$ with $\\beta :=\\sup _{F\\in \\Lambda }\\bigg (\\sup _{\\begin{array}{c}x,y\\in M\\\\ x\\ne y\\end{array}}\\frac{\\Vert F(x)-F(y)\\Vert }{\\Vert x-y\\Vert }\\bigg )<\\infty \\quad \\text{and}\\quad \\alpha :=\\beta +\\sup _{F\\in \\Lambda }\\Vert F(0)\\Vert <\\infty .$ Let $F\\in \\Lambda $ , and denote by $\\Phi _F\\colon [0,\\infty )\\times X\\rightarrow X$ the continuous semiflow related to the ODE $x^{\\prime }=F(x)$ , i.e., for $x\\in X$ , $\\Phi (\\,\\cdot \\,,x)$ is the unique solution to the initial value problem $\\partial _t\\Phi _F(t,x)&=F\\big (\\Phi _F(t,x)\\big ),\\quad \\text{for }t\\ge 0,\\\\\\Phi _F(0,x)&=x.$ Then, by Gronwall's lemma, $\\Vert \\Phi _F(t,x)-\\Phi _F(t,y)\\Vert \\le e^{\\beta t}\\Vert x-y\\Vert $ for all $t\\ge 0$ and $x,y\\in X$ .",
"Moreover, $1+\\Vert x\\Vert +\\Vert \\Phi _F(t,x)-x\\Vert \\le 1+\\Vert x\\Vert +\\alpha \\int _0^t 1+\\Vert x\\Vert +\\Vert \\Phi _F(s,x)-x\\Vert \\, {\\rm d}s$ for all $t\\ge 0$ and $x\\in X$ .",
"Again, by Gronwall's lemma, it follows that $1+\\Vert \\Phi _F(t,x)\\Vert \\le 1+\\Vert x\\Vert +\\Vert \\Phi _F(t,x)-x\\Vert \\le \\big (1+\\Vert x\\Vert \\big )e^{\\alpha t}$ for all $t\\ge 0$ and $x\\in X$ .",
"Let $p\\in (0,\\infty )$ and $\\kappa (x):=\\big (1+\\Vert x\\Vert \\big )^{-p}$ for all $x\\in X$ .",
"For $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ , $t\\ge 0$ , and $x\\in X$ , we define $\\big (S_F(t)u\\big )(x):=u\\big (\\Phi _F(t,x)\\big ).$ Then, by (REF ), for $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ , $t\\ge 0$ , and $x\\in X$ , $\\big |\\big (S_F(t)u\\big )(x)\\big |\\le \\Vert u\\Vert _\\kappa \\big (1+\\Vert \\Phi _F(t,x)\\Vert \\big )^p\\le \\Vert u\\Vert _\\kappa \\big (1+\\Vert x\\Vert \\big )^pe^{\\alpha p t},$ which implies that $\\Vert S_F(t)u\\Vert _\\kappa \\le e^{\\alpha pt}\\Vert u\\Vert _\\kappa $ .",
"Moreover, Equation (REF ) yields that $\\Vert S_F(t)u\\Vert _{{\\rm {Lip}}}\\le e^{\\beta t}\\Vert u\\Vert _{{\\rm {Lip}}}$ for all $u\\in {\\rm {Lip}}_{\\rm b}$ .",
"We have therefore shown that the family of semigroups $(S_F)_{F\\in \\Lambda }$ satisfies the assumptions (A1) and (A2), so that the semigroup envelope of the family $(S_F)_{F\\in \\Lambda }$ exists.",
"We continue by showing that the semigroup envelope ${S}$ is strongly continuous.",
"Let $\\theta :=\\min \\lbrace 1,p\\rbrace $ and $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _b$ with $C_{u,\\theta }:=\\sup _{x,y\\in M}\\frac{|u(x)-u(y)|}{|x-y|^\\theta }<\\infty .$ Again, by (REF ), $\\big |\\big (S_F(t)u\\big )(x)-u(x)\\big |\\le C_{u,\\theta } \\Vert \\Phi _F(t,x)-x\\Vert ^\\theta \\le C_{u,\\theta }\\big (1+C+\\Vert x\\Vert \\big )^\\theta \\big (e^{\\alpha t}-1\\big )^\\theta .$ Therefore, $\\Vert S_F(t)u-u\\Vert _\\kappa \\le C_{u,\\theta } \\big (e^{\\alpha t}-1\\big )^\\theta .$ Since the set of all Hölder continuous functions of degree $\\theta $ is dense in $\\mathop {\\text{\\upshape {UC}}}\\nolimits _b$ w.r.t.",
"$\\Vert \\cdot \\Vert _\\infty $ (and consequently w.r.t.",
"$\\Vert \\cdot \\Vert _\\kappa $ ), and $\\mathop {\\text{\\upshape {UC}}}\\nolimits _b$ is dense in $\\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ w.r.t.",
"$\\Vert \\cdot \\Vert _\\kappa $ , Proposition REF implies that the semigroup envelope ${S}$ is strongly continuous.",
"Let $u\\in {\\rm {Lip}}_{\\rm b}^1$ with bounded support ${\\rm supp}( u):=\\overline{\\lbrace x\\in X\\, |\\, u(x)\\ne 0\\rbrace }\\subset X$ , i.e.",
"${\\rm supp}( u)\\subset B(0,R)$ for some $R>0$ , and let $A_Fu\\in {\\rm {Lip}}_{\\rm b}$ be given by $\\big (A_Fu\\big )(x):=u^{\\prime }(x)F(x)\\quad \\text{for }x\\in X,$ where $u^{\\prime }\\colon X\\rightarrow X^{\\prime }$ denotes the (first) Fréchet derivative of $u$ .",
"Since ${\\rm supp}( u)$ is bounded, $\\sup _{F\\in \\Lambda }\\Vert A_Fu\\Vert _\\infty <\\infty \\quad \\text{and}\\quad C_u:=\\sup _{F\\in \\Lambda } \\Vert A_Fu\\Vert _{\\rm {Lip}}<\\infty .$ By the chain rule and the fundamental theorem of infinitesimal calculus, it follows that $\\frac{\\big (S_F(h)u\\big )(x)-u(x)}{h}=\\frac{1}{h}\\int _0^h \\big (S_F(s)A_Fu\\big )(x)\\, {\\rm d}s$ for all $h>0$ and $x\\in X$ , which, together with (REF ), implies that $\\bigg \\Vert \\frac{S_F(h)u-u}{h}-A_Fu\\bigg \\Vert _\\kappa \\le C_u\\big (e^{\\alpha h}-1\\big )\\rightarrow 0\\quad \\text{as }h\\searrow 0.$ Hence, $\\mathcal {D}$ contains the set of all $u\\in {\\rm {Lip}}_{\\rm b}^1$ with bounded support ${\\rm supp}(u)$ .",
"By Theorem REF , we thus obtain that $u(t):={S}(t)u_0$ , for $t\\ge 0$ , defines a viscosity solution to the fully nonlinear PDE $\\partial _t u(t,x)&=&\\sup _{F\\in \\Lambda } D_x u(t,x)F(x), \\quad (t,x)\\in (0,\\infty )\\times X,\\\\u(0,x)&=&u_0(x),\\quad x\\in X,$ where $D_x$ denotes the (first) Fréchet derivative in the space-variable.",
"If $X=\\mathbb {R}^d$ , the semigroup envelope ${S}$ is continuous from above by Remark REF b).",
"In this case, Theorem REF implies the existence of a Markov process under a nonlinear expectation related to ${S}$ .",
"This Markov process can be viewed as a nonlinear drift process.",
"Example 7.2 (Lévy Processes on abelian groups) Let $M=G$ be an abelian group with a translation invariant metric $d$ and $\\kappa (x):=1$ for all $x\\in M$ .",
"Let $(S(t))_{t\\ge 0}$ be a Markovian convolution semigroup, i.e.",
"a semigroup arising from a Lévy process.",
"Then, $(S(t))_{t\\ge 0}$ is a strongly continuous Feller semigroup of linear contractions (cf.",
"[10]).",
"Moreover, due to the translation invariance, $\\Vert S(t)u\\Vert _{{\\rm {Lip}}}\\le \\Vert u\\Vert _{{\\rm {Lip}}}$ for all $t\\ge 0$ and $u\\in {\\rm {Lip}}_{\\rm b}$ .",
"Now, let $(S_\\lambda )_{\\lambda \\in \\Lambda }$ be a family of Markovian convolution semigroups with generators $(A_\\lambda )_{\\lambda \\in \\Lambda }$ .",
"Then, the assumptions (A1) - (A2) are satisfied.",
"We refer to [10] for examples, where the semigroup envelope is strongly continuous.",
"In particular, all examples from [10] fall into our theory.",
"In the case, where $G=H$ is a real separable Hilbert space, we can improve the result obtained in [10].",
"In this case, by the Lévy-Khintchine formula (see e.g.",
"[23]), every generator $A$ of a Markovian convolution semigroup is characterized by a Lévy triplet $(b,\\Sigma ,\\mu )$ , where $b\\in H$ , $\\Sigma \\in L(H)$ is a self-adjoint positive semidefinite trace-class operator and $\\mu $ is a Lévy measure on $H$ .",
"For $u \\in {\\rm {Lip}}_{\\rm b}^2(H)$ and a Lévy triplet $(b,\\Sigma ,\\mu )$ , the generator $A_{b,\\Sigma ,\\mu }$ is given by $\\big (A_{b,\\Sigma ,\\mu }u\\big )(x)= \\langle &b, D_x u(x)\\rangle + \\frac{1}{2}{\\rm tr} \\big (\\Sigma D_x^2u(x)\\big )\\\\&+\\int _{H} u(x+y)-u(x)-\\langle D_x u(x), h(y)\\rangle \\, {\\rm d}\\mu (y)$ for $x\\in H$ .",
"Here, $D_x$ and $D_x^2$ denote the first and second Fréchet derivative in the space-variable, respectively, and the function $h\\colon H\\rightarrow H$ is defined by $h(y)=y$ for $\\Vert y\\Vert \\le 1$ and $h(y)=0$ whenever $\\Vert y\\Vert > 1$ .",
"Let $\\Lambda $ be a nonempty set of Lévy triplets.",
"We assume that $C:=\\sup _{(b,\\Sigma ,\\mu )\\in \\Lambda } \\bigg (\\Vert b\\Vert +\\Vert \\Sigma \\Vert _{\\mathop {\\text{\\upshape {tr}}}\\nolimits }+\\int _{H} 1\\wedge \\Vert y\\Vert ^2\\, {\\rm d}\\mu (y)\\bigg )<\\infty .$ Note that (REF ) does not exclude any Lévy triplet a priori.",
"Under (REF ), the semigroup envelope ${S}$ is strongly continuous on ${\\rm {Lip}}_{\\rm b}^2$ .",
"In order to show that ${\\rm {Lip}}_{\\rm b}^2\\subset \\mathcal {D}$ , by the computations in [10], it suffices to show that ${S}$ is strongly continuous.",
"For this we invoke Proposition REF .",
"For $\\delta >0$ , we choose the family $\\big (\\varphi _x)_{x\\in H}$ as in the previous example.",
"Since $\\big ({S}(t)v\\big )(x)=\\big ({S}(t)v(x+\\cdot )\\big )(0)$ for all $v\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _\\kappa $ , $x\\in H$ and $t\\ge 0$ , it follows that $\\big ({S}(t)(1-\\varphi _x)\\big )(x)=\\big ({S}(t)(1-\\varphi _0) \\big )(0)$ for all $x\\in H$ and $t\\ge 0$ .",
"Defining $f(t):=\\big ({S}(t)(1-\\varphi _0) \\big )(0)$ for $t\\ge 0$ , it follows that $f$ is continuous with $f(0)=0$ .",
"Therefore, by Proposition REF , the semigroup ${S}$ is strongly continuous.",
"Altogether, we have shown that under the condition (REF ), the assumptions (A1) and (A2) are satisfied, the semigroup envelope ${S}$ is strongly continuous and ${\\rm {Lip}}_{\\rm b}^2\\subset \\mathcal {D}$ .",
"By Theorem REF , we thus obtain that $u(t):={S}(t)u_0$ , for $t\\ge 0$ , defines a viscosity solution to the fully nonlinear Cauchy problem $u_t(t,x)&=&\\sup _{(b,\\Sigma ,\\mu )\\in \\Lambda } \\big (A_{b,\\Sigma ,\\mu } u(t)\\big )(x), \\quad (t,x)\\in (0,\\infty )\\times H,\\\\u(0,x)&=&u_0(x),\\quad x\\in H.$ If $H=\\mathbb {R}^d$ and the set of Lévy measures within the set of Lévy triplets $\\Lambda $ is tight, Proposition REF implies that the semigroup envelope ${S}$ is continuous from above, leading to the existence of a nonlinear Lévy process related to ${S}$ .",
"However, due to the translation invariance of the semigroups, the continuity from above is actually not necessary in order to obtain the existence of a Lévy process under a nonlinear expectation.",
"The nonlinear Lévy process can be explicitly constructed via space-time discrete stochastic integrals w.r.t.",
"Lévy processes with Lévy triplet contained in $\\Lambda $ .",
"We refer to [10] for the details of the construction.",
"Example 7.3 ($\\alpha $ -stable Lévy processes) Consider the setup of the previous example, with $G=\\mathbb {R}^d$ for some $d\\in \\mathbb {N}$ and let $A_\\alpha :=-(-\\Delta )^\\alpha $ be fractional Laplacian for $0<\\alpha <1$ .",
"Then, for any compact subset $\\Lambda \\subset (0,1)$ , condition (REF ) is satisfied.",
"Hence, the assumptions (A1) and (A2) are satisfied and the semigroup envelope ${S}$ is strongly continuous with ${\\rm {Lip}}_{\\rm b}^2\\subset \\mathcal {D}$ .",
"By Theorem REF , we thus obtain that $u(t):={S}(t)u_0$ , for $t\\ge 0$ , defines a viscosity solution to the nonlinear Cauchy problem $u_t(t,x)&=&\\sup _{\\alpha \\in \\Lambda } -(-\\Delta )^\\alpha u(t,x), \\quad (t,x)\\in (0,\\infty )\\times \\mathbb {R}^d,\\\\u(0,x)&=&u_0(x),\\quad x\\in \\mathbb {R}^d.$ The related nonlinear Lévy process can be interpreted as a $\\Lambda $ -stable Lévy process.",
"Example 7.4 (Mehler semigroups) Consider the case, where the state space $M=H$ is a real separable Hilbert space and $\\kappa =1$ .",
"Let $(T,\\mu )$ be a tuple consisting of a $C_0$ -semigroup $T=(T(t))_{t\\ge 0}$ of linear operators on $H$ with $\\Vert T(t)\\Vert \\le e^{\\alpha t}$ for all $t\\ge 0$ and some $\\alpha \\in \\mathbb {R}$ and a family $\\mu =(\\mu _t)_{t\\ge 0}$ of probability measures on $H$ such that $\\mu _0=\\delta _0\\quad \\text{and}\\quad \\mu _{t+s}=\\mu _s \\ast \\mu _t\\circ T(s)^{-1}\\quad \\text{for all }s,t\\ge 0.$ We then define the generalized Mehler semigroup $S=S_{(T,\\mu )}$ by $\\big (S(t)u\\big )(x):=\\int _H u(T(t)x+y)\\, {\\rm d}\\mu _t(y)$ for $u\\in \\mathop {\\text{\\upshape {UC}}}\\nolimits _{\\rm b}$ , $t\\ge 0$ and $x\\in H$ , see e.g. [5],[16].",
"Then, $\\Vert S(t)u\\Vert _\\infty \\le \\Vert u\\Vert _\\infty $ for all $u\\in {\\rm {C}}_{\\rm b}$ and $\\Vert S(t)u\\Vert _{{\\rm {Lip}}}\\le e^{\\alpha t}\\Vert u\\Vert _{{\\rm {Lip}}}$ for $u\\in {\\rm {Lip}}_{\\rm b}$ .",
"Hence, for any nonempty family $\\Lambda $ of tuples $(T,\\mu )$ with $\\Vert T(t)\\Vert \\le e^{\\alpha t}$ for all $t\\ge 0$ the assumptions (A1) and (A2) are satisfied.",
"Example 7.5 (Bounded generators on $\\ell ^\\infty $ ) Let $M=\\mathbb {N}$ and $\\kappa (i)=1$ for all $i\\in \\mathbb {N}$ .",
"Let $(A_\\lambda )_{\\lambda \\in \\Lambda }\\subset L(\\ell ^\\infty )$ be a family of operators satisfying the positive maximum principle and $\\sup _{\\lambda \\in \\Lambda }\\Vert A_\\lambda \\Vert _{L(\\ell ^\\infty )}<\\infty .$ Here, we say that an operator $A\\in L(\\ell ^\\infty )$ satisfies the positive maximum principle if $A_{ii}<0$ for all $i\\in \\mathbb {N}$ and $A_{ij}\\ge 0$ for all $i,j\\in \\mathbb {N}$ with $i\\ne j$ .",
"Then, the family $(A_\\lambda )_{\\lambda \\in \\Lambda }$ satisfies the assumptions (A1) and (A2) with $\\mathcal {D}=\\ell ^\\infty $ .",
"In particular, the semigroup envelope is strongly continuous.",
"If $A_\\lambda 1=0$ for all $\\lambda \\in \\Lambda $ , then the semigroup envelope admits a stochastic representation.",
"This representation can be seen as a nonlinear Markov chain with state space $\\mathbb {N}$ .",
"Example 7.6 (Multiples of generators of Feller semigroups) Let $A$ be the generator of a strongly continuous Feller semigroup $(S(t))_{t\\ge 0}$ of linear operators.",
"Assume that there exist constants $\\alpha ,\\beta \\in \\mathbb {R}$ such that $\\Vert S(t)u\\Vert _\\kappa \\le e^ {\\alpha t}\\Vert u\\Vert _\\kappa \\quad \\text{and}\\quad \\Vert S(t)u\\Vert _{{\\rm {Lip}}}\\le e^{\\beta t}\\Vert u\\Vert _{{\\rm {Lip}}}$ for all $u\\in {\\rm {Lip}}_{\\rm b}$ and $t\\ge 0$ .",
"For $\\lambda \\ge 0$ let $A_\\lambda :=\\lambda A$ for all $\\lambda $ .",
"Then, $A_\\lambda $ generates the semigroup $S_\\lambda $ given by $S_\\lambda (t):=S(\\lambda t)$ for all $t\\ge 0$ and $\\lambda \\ge 0$ .",
"Then, for any compact set $\\Lambda \\subset [0,\\infty )$ the family $(S_\\lambda )_{\\lambda \\in \\Lambda }$ satisfies the assumptions (A1) and (A2) with $D(A)\\subset \\mathcal {D}$ and the semigroup envelope is strongly continuous.",
"Hence, by Theorem REF , we obtain that $u(t):={S}(t)u_0$ , for $t\\ge 0$ , defines a viscosity solution to the abstract Cauchy problem $u^{\\prime }(t)&=&\\sup _{\\lambda \\in \\Lambda }\\lambda A u(t), \\quad \\text{for }t> 0,\\\\u(0)&=&u_0.$"
]
] | 1906.04430 |
[
[
"Latent Channel Networks"
],
[
"Abstract Latent Euclidean embedding models a given network by representing each node in a Euclidean space, where the probability of two nodes sharing an edge is a function of the distances between the nodes.",
"This implies that for two nodes to share an edge with high probability, they must be relatively close in all dimensions.",
"This constraint may be overly restrictive for describing modern networks, in which having similarities in at least one area may be sufficient for having a high edge probability.",
"We introduce a new model, which we call Latent Channel Networks, which allows for such features of a network.",
"We present an EM algorithm for fitting the model, for which the computational complexity is linear in the number of edges and number of channels and apply the algorithm to both synthetic and classic network datasets."
],
[
"Abstract",
"In modern social networks, individuals often belong to several communities with widely varying number of connections made through each community.",
"To capture this social structure, we introduce the Latent Channel Networks model, in which nodes share observed edges if they make a connection through at least one latent channel.",
"We present the mathematical model, interpretations of various parameters and an EM algorithm whose computational complexity is linear in the number of edges and channels and can handle a graph in which some of the edge statuses may be unknown.",
"We compare the performance of our model to a similarly structured competing model on both synthetic and and a subsample of the Facebook100 graphs.",
"Although both models show strong predictive capabilities, we find that our model tends to suffer less from overfitting and consistently achieves a moderate improvement in out-of-sample AUC when estimating masked edge status on the Facebook data.",
"We discuss insights the model provides into the graph structure of the Facebook data."
],
[
"Relevant Work",
"In the analysis of graph data, a common goal is describing a network in a reduced-order space that provides insight into the underlying graph structure.",
"One of the simplest structures is the stochastic block model (SBM) [16].",
"In this model, each node belongs to an unobserved block, and nodes have a fixed probability of having an edge with nodes within their block and another fixed probability of having an edge with nodes outside their block.",
"Typically, the within-block edge probabilities are greater than the between-block edge probabilities, so nodes are more likely to share an edge with nodes within the same block, and each block may be considered a cluster or community.",
"Recent work covers efficient estimation of the parameters of SBMs [1], [9], time-evolving or dynamic SBMs [35], statistical characteristics of the estimators [21], [8] model selection [33] and hierarchical SBMs [24].",
"One disadvantage of SBMs is that the expected within-block degree of a node is constant, with a variance implied by a binomial distribution.",
"This fails to capture a commonly observed phenomenon in social networks; often, a small number of nodes express an extremely high degree relative to most other nodes.",
"Other models have been proposed to capture this property, such as the degree-corrected SBM [19], in which edge probabilities are based on block membership and a given node's degree.",
"Another limitation of SBMs is that it is hard clustering approach: each node deterministically belongs to a single block.",
"Several alternatives have been considered that allow for soft clustering, including the mixture SBM [2] in which each node belongs to a each block with a given probability.",
"Another approach is to maximize the modularity score [22], but with community membership described as a probability vector rather than as a categorical variable [12], [14], [18].",
"Other metrics such as the overlapping correlation coefficient [7] may also be used.",
"Ball, Karrer, and Newman presented a soft clustering-related model that uses overlapping clusters [5]; we refer to this model as the BKN model.",
"In this model, several latent communities exist, and each node has a community intensity parameter associated with each community.",
"The number of edges between two nodes is modeled as a Poisson distribution with a mean given by the sum over all communities of the product of community intensities between the two nodes.",
"Although this modeling assumption implies counts rather than binary values, the authors present the Poisson distribution as a reasonable approximation when edges are binary.",
"The Euclidean embedding model [15] is another popular probabilistic model.",
"In this model, each node is represented as a point in a latent Euclidean space, with edge probabilities being inversely proportional to distances.",
"Because these probabilities are directly modeled, one can naturally allow them to be a functions of both latent distance and linear predictors associated with each node.",
"This model also allows for both very high and low degree nodes; each node has its own intercept, and high degree nodes are simply nodes whose intercept is exceptionally high.",
"Traditional MCMC approaches were initially presented for inference, and methods to accelerate this have also been proposed, including using variational Bayes [27] and stratified case-control sampling [26].",
"A similar model is the random dot product graph [23], [36], in which nodes are represented in a latent space and edge probabilities between two nodes are given by the dot product of their latent positions.",
"These positions can be estimated via eigendecompositions of the adjacency matrix [28].",
"While clusters are not explicitly modeled in latent space embedding, clustering may still be performed on the lower-dimensional latent embedding.",
"One last class of models that is related to our work is the sender/receiver model [17], [30].",
"These models are applied to directed graphs, and each node has a parameter that controls how frequently it broadcasts and receives connections.",
"The probability of an edge from a source node to target node is then a function of the source's broadcast strength and the target's receiving strength."
],
[
"Latent Channel Network",
"One major disadvantage of an Euclidean embedding is that two nodes must be close in all dimensions in order to have a high edge probability.",
"The suitability of this assumption has been called into question for web-based applications [20] and legislative voting patterns [37].",
"In modern social networks, being similar in at least one social dimension may be sufficient for high edge probability.",
"For instance, [6] demonstrated that while political retweeting falls very tightly along political lines, it is not strongly predictive of retweeting of non-political topics.",
"While one could relax this requirement by embedding the network in a non-Euclidean space, say $L^p$ , a few issues arise; picking $p$ , unstable and computationally expensive likelihoods and interpretation of the parameters.",
"This shortcoming motivated us to work on a new model that more naturally allows two nodes to share a connection if they share some social overlap.",
"To capture this, we present the Latent Channel Network (LCN) model.",
"Here, we assume that two nodes will share an edge in the graph if they are connected through at least one of potentially several unobserved latent channels.",
"The probability of two nodes connecting through a given channel is the product of each node's frequency of use of the given channel.",
"Thus, two nodes may differ strongly in many social areas, i.e.",
"they do not need to frequently use all of the same channels, yet the model still allows them to have a high edge probability if they both frequently communicate through one or more shared channels."
],
[
"Relation to Previous Work",
"The LCN is closely related to the BKN model [5].",
"The fundamental difference between the two models is that rather than modeling edge counts as a Poisson variable with mean equal to the sum of means across all channels, the LCN instead models binary edges as a Bernoulli distribution, where the probability of having an edge is one minus the probability of having no edges through any of the channels.",
"In Section REF , we show that this produces a model that encourages overlap in channels between edge-pairs, while the BKN model discourages edge-pairs in the model from having strong connections through multiple channels, and we empirically show that our model leads to better edge predictions on social network data.",
"If one considers channels to represent communities, both our model and the BKN model can be viewed as similar to an overlapping communities model with one key difference.",
"In particular, in the formal definition overlapping communities provided in [34], each node has a probability vector containing probabilities of association to each community which sums to one for each node.",
"In contrast, both the LCN and BKN community membership parameters are attachment parameters for independent communities, and thus the vector of probabilities/intensities is not constrained to sum to one for each node.",
"Thus, being strongly attached to one community does not prevent a node from being strongly attached to another community.",
"These models naturally allow networks that contain a mix of high degree nodes (those that use multiple channels with high frequency) and low degree nodes (those that use all channels with low frequency).",
"One can also view the LCN model as a modified sender/receiver model, in which each node has multiple opportunities to make a connection and direction is ignored."
],
[
"Structure of Paper",
"In Section , we formally define our model and present various ways to interpret meaningful parameters from the model.",
"We also compare and contrast our model with the BKN model.",
"In Section , we present both a simple and more computationally efficient algorithm to compute the maximum likelihood estimate of the model parameters.",
"In Section , we apply the model to synthetic data and several graphs from the Facebook100 dataset [29].",
"In Section , we review our work and discuss potential further directions."
],
[
"Model Parameterization",
"Let $G$ be undirected graph with nodes $n_1$ ,...,$n_{N_n}$ and edges $e_{ij} = 1$ if $n_i$ and $n_j$ are connected and 0 otherwise.",
"For simplicity, we assume the graph has no self-loops.",
"Let $N_n$ to be the number of nodes and $N_e$ to be the number of edges of the graph.",
"We augment this observed graph with a set of latent channels $C_1$ ,...,$C_K$ that provide intermediate connections between nodes.",
"In particular, we introduce latent edges $\\tilde{e}_{ikj}$ , which is equal to 1 if node $n_i$ shares a latent edge to channel $C_k$ toward node $n_j$ and 0 otherwise.",
"Our model then dictates that a pair of nodes share an observed edge if they are both fully connected through one or more latent channels.",
"More formally, $e_{ij} ={\\left\\lbrace \\begin{array}{ll}1 & \\text{if there exists } k \\text{ such that } \\tilde{e}_{ikj} = \\tilde{e}_{jki} = 1 \\\\0 & \\text{otherwise}.\\\\\\end{array}\\right.",
"}$ Examples of this architecture are illustrated in Figures REF and REF .",
"For simplicity, we define $c_{ijk} = \\mathbb {I}( \\tilde{e}_{ikj} = \\tilde{e}_{jki} = 1 ).$ In other words, $c_{ijk} $ is an indicator that nodes $n_i$ and $n_j$ are connected through channel $C_k$ .",
"Figure: Nodes n i n_i and n j n_j share an edge as they are connected through at least one channel.In general, $\\tilde{e}_{ikj}$ and $c_{ijk} $ cannot be observed directly.",
"However, our model assumes that for all $n_j$ , $\\tilde{e}_{ikj}$ are independently distributed Bernoulli distributions with probability $p_{ik}$ .",
"Thus, the marginal probability that $n_i$ will share a edge with $n_j$ through channel $C_k$ is $p_{ik} p_{jk}$ .",
"To determine the probability that nodes $n_i$ and $n_j$ share an edge, we compute $ \\begin{split}P(e_{ij} = 1) & = 1 - P(e_{ij} = 0) \\\\& = 1 - \\prod _{k = 1}^K \\left(1 - P( c_{ijk} ) \\right) \\\\& = 1 - \\prod _{k = 1}^K (1 - p_{ik} p_{jk}).",
"\\\\\\end{split}$ The log-likelihood of a latent channel graph can then be written as $L(G | p) = \\sum _{i = 2}^n \\sum _{j = 1}^{i-1}e_{ij} \\log \\left( 1 - \\prod _{i = 1}^K (1 - p_{ik} p_{jk}) \\right) +(1 - e_{ij}) \\log \\left( \\prod _{i = 1}^K (1 - p_{ik} p_{jk}) \\right).$"
],
[
"Interpretation of Model",
"If channels correspond to latent communities, then $p_{ik}$ informally represents the strength of node $n_i$ 's attachment to community $C_k$ .",
"However, this parameter alone can be difficult to interpret, as it is unclear how large $p_{ik}$ should be to be considered a strong connection.",
"To aid in the interpretation of the model, we present several derived values.",
"We first consider the parameter $\\theta _{ijk}$ : $ \\theta _{ijk} = P( c_{ijk} = 1 | e_{ij} = 1) = \\frac{p_{ik} p_{jk} }{ 1 - \\sum _{k = 1}^K (1 - p_{ik} p_{jk}) }.$ $\\theta _{ijk}$ represents the probability that nodes $n_i$ and $n_j$ are connected through channel $C_k$ , given that the graph contains an edge between $n_i$ and $n_j$ .",
"This is especially relevant in the case that channel $C_k$ has a meaningful interpretation, such as the attachment strength parameter $p_{ik}$ being correlated with meta-data on the nodes.",
"For example, if the attachment strength to channel $C_k$ is strongly associated with nodes whose occupations are statisticians and $\\theta _{ijk}$ is high, this suggests that given that nodes $n_i$ and $n_j$ share a connection, they have a high probability of having an edge through the statistical community.",
"It is worth noting that $\\sum _{k = 1}^K \\theta _{ijk} \\ge 1,$ and typically with strict inequality.",
"This is because two nodes that share an edge must share at least one edge through a latent channel, but may share many.",
"For example, if channel $C_k$ represents the statistical community and $C_{k^{\\prime }}$ represents associations through a given research institution, statisticians at the same institution are likely to be connected through both channels $C_k$ and $C_{k^{\\prime }}$ .",
"Next, we consider $S_k = \\sum _{i = 1}^{N_n} p_{ik},$ which we refer to as the size of the channel.",
"To interpret this parameter, note that if a new node $n_{i^{\\prime }}$ were to be fully connected to channel $C_k$ , i.e.",
"$p_{i^{\\prime } k} = 1$ , it is expected to have $S_k$ connections through $C_k$ .",
"More generally, the expected number of connections for a new node is $p_{i^{\\prime } k} S_k$ .",
"Another particularly useful parameter is $C_{ik}$ : $C_{ik} = \\mathbb {E}\\left[ \\scriptstyle \\sum _{j \\ne i} c_{ijk} | G \\right] = \\displaystyle \\sum _{j \\ne i} e_{ij} \\theta _{ijk}.$ $C_{ik}$ represents the expected number of connections node $n_i$ has through channel $C_k$ , conditional on the edges observed in the graph.",
"While $p_{ik}$ tells us the strength of attachment node $n_i$ has to channel $C_k$ , it is not sufficient to determine how many connections node $n_i$ may have through channel $C_k$ .",
"For example, a strong attachment to a small channel may result in fewer edges than a weak attachment to a large channel.",
"As such, this statistic can provide insight into the number of connections a node has through a given community, which is a function of both that individual's strength of attachment to the community and the size of the community.",
"Similar to Equation REF , we note that $ \\mathbb {E}\\left[ \\scriptstyle \\sum _{j \\ne i} c_{ijk} | G \\right] \\ge \\sum _{j \\ne i} e_{ij},$ or that for node $n_i$ , the expected sum of connections through all channels is typically greater than the sum of all observed edges in the graph associated with that node.",
"Again, this is because a single edge can be the result of connections through multiple channels.",
"Caution should be taken in interpreting such parameters based on fitted data.",
"As is the case for many probabilistic network models, we currently estimate the parameters via maximum likelihood estimation.",
"Given the high dimensional parameter space, standard asymptotic normality results should not be considered a reliable method for estimating uncertainty.",
"Therefore, we suggest using these methods for exploratory data analysis rather than making strong inference statements about a given network.",
"Alternatively, Bayesian methods may be used to determine uncertainty.",
"However, to do so, one must first address the unidentifiability issue that arises due to label switching of the channels."
],
[
"Comparisons with BKN Model",
"In the LCN model, each edge in the graph is independently distributed as $ e_{ij} \\sim \\text{Bernoulli}\\left( p = 1 - \\prod _{k = 1}^K (1 - p_{ik} p_{jk} ) \\right),$ while the BKN model assumes that it is independently distributed as $ e_{ij} \\sim \\text{Poisson}\\left( \\lambda = \\sum _{k = 1}^K \\theta _{ik} \\theta _{jk} \\right).$ While both models have similar structures, they have important fundamental differences.",
"Most notably, LCNs can only be used with binary edges, whereas the BKN can handle edge counts.",
"Therefore, when the data contains edges counts, the BKN model should be preferred.",
"When the graph contains binary edges, the BKN model can be seen as a useful approximation.",
"Indeed, the authors present it as such: “However, allowing multiedges makes the model enormously simpler to treat [when edges should be binary] and in practice the number of multiedges tends to be small, so the error introduced is also small, typically vanishing as $1/n$ in the limit of large network size.",
"[5]\" In the light of this, we view the LCN as as a model for treating the edges exactly as Bernoulli data, rather than using a Poisson approximation.",
"We note that the EM algorithm we present has the same computational complexity as that of BKN's algorithm.",
"In addition to philosophical motivations, the two models also differ in how they treat multiple channel usage.",
"The BKN model discourages two nodes that share an edge from being too strongly attached through many channels.",
"In particular, because the mean number of edges is $\\sum \\theta _{ik} \\theta _{jk}$ , the log-likelihood contribution of a single edge is maximized when $\\sum \\theta _{ik} \\theta _{jk} = 1$ .",
"In contrast, for the LCN, the log-likelihood contribution of a single edge is non-decreasing with $p_{ik}p_{jk}$ , although as a single $p_{ik} p_{jk}$ approaches 1, the derivative of the other $p_{ik^{\\prime }} p_{jk^{\\prime }}$ approaches 0.",
"Therefore, the BKN model discourages node pairs from being too strongly attached to the same channels if this leads to $\\sum \\theta _{ik} \\theta _{jk} > 1$ , while the LCN does not penalize two nodes that share an edge from being overly similar.",
"Another difference between the two models is the BKN model allows one node to force another node to use a given channel.",
"If $\\theta _{ik}$ is very small, one might assume that this implies that node $i$ infrequently uses channel $C_k$ .",
"However, if $\\theta _{jk}$ is very large, then the expected number of edges between nodes $i$ and $j$ through channel $C_k$ may still be large as $\\theta _{ik}\\theta _{jk}$ is unbounded.",
"In contrast, with the LCN model, the expected number of edges through channel $C_k$ between nodes $i$ and $j$ is $p_{ik} p_{jk}$ , which is bounded above by $p_{ik}$ .",
"Finally, we note that edge probability estimates, both latent and observed, are more interpretable in our model with binary edges.",
"For example, we can directly compute the observed edge probabilities from our model if we wish to make inferences about the existence of an edge between two nodes.",
"With the BKN model, it is not quite clear how to interpret the edge predictions.",
"To illustrate, suppose the BKN model estimates that the expected edge count between two nodes is 1.",
"If we interpret this as the expected value of a binary distribution, this is equivalent to saying that there exists an edge with probability one.",
"However, if we interpret it as a parameter of a Poisson distribution, then the probability of one or more edges is $1 - e^{-1}$ .",
"Similarly, it is not clear how to interpret the difference between estimated edge counts greater than 1 in a BKN model.",
"Finally, if one were to use a BKN model to generate a synthetic binary network, one would need to truncate any edge counts greater than 1, which would induce bias when estimating the parameters with a BKN model that does not account for the truncation."
],
[
"Algorithm",
"We appeal to maximum likelihood estimation to estimate the values of $p_{ik}$ .",
"In general, the problem is non-identifiable and highly non-concave.",
"We will use an EM algorithm [10] to fit the parameters of the model."
],
[
"Fundamental EM Algorithm",
"To determine the steps of the EM algorithm, we make several observations.",
"First, we note that if the values of $\\tilde{e}_{ikj}$ were known, the log-likelihood would be simplified to $ L(G, \\tilde{e}| p) = \\sum _{i = 1}^{N_n} \\sum _{j \\ne i}^{N_n} \\sum _{k = 1}^{K} \\tilde{e}_{ikj} \\log ( p_{ik} ) + (1 - \\tilde{e}_{ikj}) \\log (1 - p_{ik}),$ which is maximized when $\\hat{p}_{ik} = \\sum _{j \\ne i}^{N_n} \\tilde{e}_{ikj} / (N_n - 1),$ providing our M-step in the EM algorithm.",
"For the E-step, we recognize that $\\displaystyle P(\\tilde{e}_{ikj} = 1 | e_{ij} = 1) =\\frac{ p_{ik} p_{jk} + p_{ik} (1 - p_{jk}) \\left( 1 - \\prod _{k^{\\prime } \\ne k} ( 1 - p_{ik^{\\prime }} p_{jk^{\\prime }} ) \\right) }{ 1 - \\prod _{k = 1}^K( 1 - p_{ik} p_{jk}) },$ $P( \\tilde{e}_{ikj} = 1 | e_{ij} = 0) =p_{ik} - p_{ik} p_{jk}.$ For clarity, we first present a simple but computationally inefficient implementation in Algorithm REF .",
"Noting that computing $P( \\tilde{e}_{ikj} | e_{ij} = 1)$ requires $O(K)$ operations and $P( \\tilde{e}_{ikj} | e_{ij} = 0)$ requires $O(1)$ operations, this implementation then requires $O(N_e K^2 + (N_n^2 - N_e)K)$ computations per iteration.",
"Fixed point estimate of $N \\times K$ matrix $p$ Adjacency Matrix $e$ ; K N = nrow(e) $p$ = RandomUniform(min = 0, max = 1, nrow = N, ncol = K) maxIters = 1,000; iter = 0; tol = $10^{-4}$ ; maxDiff = tol + 1 iter $<$ maxIters & tol $>$ maxDiff iter++ i in 1:N k in 1:K j in 1:N $\\tilde{e}_{ijk} ={\\left\\lbrace \\begin{array}{ll}0 & \\text{ if } i = j \\\\P( \\tilde{e}_{ikj} | e[i,j] = 1) & \\text{ else if } e[i,j] = 1\\\\P( \\tilde{e}_{ikj} | e[i,j] = 0) & \\text{ otherwise} \\\\\\end{array}\\right.",
"}$ pNew[i,k] = $\\frac{\\sum _{j = 1}^N \\tilde{e}_{ikj}}{N-1}$ maxDiff = max($|$ pNew - p $|$ ) p = pNew (p) Simple EM Algorithm"
],
[
"Unknown Edge Status",
"In many applications, the edge status between two nodes may be unknown, and one may wish to predict missing edges from the data.",
"This can also be used to assess model fit by evaluating edge predictions on masked edges.",
"With our EM algorithm, this can be easily accommodated.",
"If we define $M(i,j) ={\\left\\lbrace \\begin{array}{ll}0 & \\text{if edge status for node pair $i,j$ is known} \\\\1 & \\text{if edge status for node pair $i,j$ is missing}, \\\\\\end{array}\\right.",
"}$ then we can redefine the M-step update as $\\hat{p}_{ik} = \\frac{ \\sum _{j \\ne i}^{N_n} \\tilde{e}_{ikj} (1 - M(i,j) )}{ \\sum _{j \\ne i} (1 - M(i,j)) }.",
"$"
],
[
"Efficient Updates",
"While the algorithm described in Algorithm REF is straightfoward, many of the computations in this algorithm are redundant and the order of complexity of this algorithm can be reduced by caching various statistics.",
"Let $E_i$ to be the set of nodes that are known share an edge with node $n_i$ and $E_i^c $ to be the set of nodes that are known to lack an edge with node $i$ .",
"We explicitly store $E_1,...,E_{N_n}$ in a list but do not explicitly store $E_i^c$ .",
"Note that $n_i$ is neither in $E_i$ nor $E_i^c$ , and if the edge status between node $n_i$ and $n_j$ is unknown, $n_j$ appears in neither set.",
"We first note that the EM steps can be combined in the form $ p^{new}_{ik} =\\frac{ \\displaystyle \\sum _{j \\in E_i^c} P( \\tilde{e}_{ikj} | e_{ij} = 0) + \\sum _{j \\in E_i } P( \\tilde{e}_{ikj} | e_{ij} = 1) }{|E_i| + |E^c_i|} .$ Defining $M_i$ to be the set of nodes such that edge status with node $n_i$ is unknown, the first term of the numerator can then be rearranged as $ \\begin{split}\\displaystyle \\sum _{j \\in E_i^c} P( \\tilde{e}_{ikj} | e_{ij} = 0) & = \\sum _{j \\in E_i^c} p_{ik} - p_{ik} p_{jk} \\\\& = N_n p_{ik} (1 - \\bar{p}_{.k} ) - p_{ik} \\left( (1 - p_{ik}) + \\sum _{j \\in E_i \\cup M_i} (1 - p_{jk}) \\right),\\end{split}$ where $\\bar{p}_{.k}$ represents the column mean of the matrix $p$ .",
"Assuming $| E_i ^c | > | E_i \\cup M_i |$ , this reduces the computation required to compute the first term from $O( | E_i ^ c | )$ to $O( | E_i \\cup M_i |)$ as long as $\\bar{p}_{.k}$ is cached.",
"Next, if we define $ \\pi _{ij} \\equiv P(e_{ij} = 1) = 1 - \\prod _{k = 1}^K( 1 - p_{ik}p_{jk} )$ we can write $ \\begin{split}\\sum _{j \\in E_i } P( \\tilde{e}_{ikj} | e_{ij} = 1) &= \\sum _{j \\in E_i } \\left( \\frac{ p_{ik} p_{jk} + p_{ik} (1 - p_{jk}) \\left( 1 - \\prod _{k^{\\prime } \\ne k} ( 1 - p_{ik^{\\prime }} p_{jk^{\\prime }} ) \\right) }{\\pi _{ij} } \\right) \\\\& = \\sum _{j \\in E_i } \\left( \\frac{ p_{ik} p_{jk} + p_{ik} (1 - p_{jk}) \\left( 1 - \\frac{ 1 -\\pi _{ij} }{ 1 - p_{ik} p_{jk} } \\right) }{ \\pi _{ij} } \\right).",
"\\\\\\end{split}$ Using cached values of $\\pi _{ij}$ reduces the computations required for the second term of Equation REF from $O(K | E_i |)$ to $O( | E_i | )$ .",
"Precomputing all $\\pi _{ij}$ within the edge list $E$ requires $O(KN_e)$ time.",
"Similar to [5], it should be noted that if $p_{ik} = 0$ , then the EM algorithm will leave $p_{ik}$ unchanged.",
"This can be exploited for additional speedup by skipping the update for $p_{ik}$ if $p_{ik} < \\epsilon _p$ for a preset tolerance level $\\epsilon _p$ .",
"Finally, it should be noted that while updates of a given row of the matrix $p$ will share the same $\\pi _{ij}$ , the computations across rows are independent.",
"Thus, it is simple to parallelize the computation by splitting up the updates per thread by row of $p$ .",
"Pseudocode for our full EM algorithm is shown in Algorithm REF .",
"The initial computational complexity of each step of this algorithm is $O(K(N_n + N_e + N_m ) )$ , where $N_m$ the total number of unknown edges, but the complexity of later steps of the algorithm can be reduced by skipping updates where $p_{ik} < \\epsilon _p$ .",
"Fixed point estimate of $N \\times K$ matrix $p$ Edge list $E$ s.t.",
"$E[i][j] \\equiv j^{th}$ index of node sharing $j^{th}$ edge with node $i$ Missing list $M$ s.t.",
"$M[i][j] \\equiv j^{th}$ index of node sharing $j^{th}$ with unknown edge status with node $n_i$ $pNew$ = RandomUniform(min = 0, max = 1, nrow = N, ncol = K) maxIters = 10,000; iter = 0 tol = $10^{-4}$ ; pTol = $10^{-10}$ ; maxDiff = tol + 1 p = pNew iter $<$ maxIters & tol $>$ maxDiff pBar = ColumnMeans(p) iter++ i in 1:N (in parallel) nEdges = length(E[i]) edgeProbs = vector(nEdges) ii in 1:nEdges j = E[i][ii] edgeProbs[ii] = computeEdgeProb(i, j, p) k in 1:K pik = p[i,k] pik $<$ pTol skip edgeSum = 0.0 noEdgeSum = N * pik * (1 - pBar[k]) - pik *(1 - pik) nMissing = length(M[i]) ii in 1:nMissing pjk = p[M[i][ii], k] noEdgeSum -= pik * (1 - pjk) ii in 1:nEdges pjk = p[E[i][ii],k] noEdgeSum -= pik * (1 - pjk) ep = edgeProbs[ii] edgeSum += pik * (pjk+(1-pjk) * (1-$\\frac{\\text{1- ep}}{ \\text{1-pik * pjk} }$ ) ) / ep pNew[i,k] = (edgeSum + noEdgeSum) / (N - nMissing - 1) maxDiff = max($|$ pNew - p $|$ ) p = pNew; p Efficient Algorithm"
],
[
"Missing Edges with BKN Model",
"To quantitatively compare the LCN model to the BKN model, we compare AUCs on missing link predictions for which the edge status is masked from the model.",
"In [5], the EM algorithm presented assumes all edge statuses are known, and unlike the LCN model, we were not able to derive a simple closed form solution to the M-step when the edges statuses are unknown.",
"However, because the unknown edge statuses appear in the complete data likelihood function in a linear manner, we can simply add an imputation step to their algorithm that imputes the expected edge count of the unknown edges.",
"Notably, this does not increase the computational complexity of the algorithm when compared with our LCN algorithm, as imputing the missing edges is $O(KN_m)$ , so the standard update is $O(K(N_n + N_e + N_m))$ , similar to our algorithm."
],
[
"Evaluating Performance",
"We compared the LCN and BKN models on a variety of synthetic and real datasets.",
"We assessed performance in both a quantitative and qualitative manner.",
"For quantitative assessment, we randomly masked the edge status of 500 node pairs that shared an edge and 500 node pairs that lacked an edge during fitting and then predicted the edge status of node pairs in both sets.",
"We compared the area under the curve of the receiver operating characteristic curve (AUC), since this measure relates to the predictive power of the model without being overly sensitive to overfitting as metrics such as out-of-sample log-likelihood are.",
"Indeed, although it was very rare, both models occasionally estimated that a masked edge had zero probability, leading to an out-of-sample log-likelihood of $-\\infty $ , as one should expect for such a high dimensional problem fit with maximum likelihood.",
"For qualitative assessment, we visually examined the fit to see if the LCN model captured relations we would expect from metadata about the nodes.",
"This was done by creating a heatmap of $\\hat{p}$ .",
"To assist in visualizing whether categorical metadata was associated with given channels, we first reordered the rows of the parameter matrix by the node's metadata and then reordered the columns of the matrix by the ratio of between-group and within-group variance of the parameter values.",
"This somewhat crudely sorts the channels such that those near the top of the heatmap will be mostly strongly differentially expressed by category, while those near the bottom will be very weakly differentiated by category."
],
[
"Stochastic Block Model",
"To compare the LCN and BKN models on synthetic data, we simulated stochastic block models with $p_{in} = 0.5$ , $p_{out} = 0.02$ , 8 blocks with 32 nodes per block.",
"We fit both models with $1, 2, 4, \\ldots , 64$ number of channels.",
"Note that the stochastic block model can be perfectly reproduced with 8 channels.",
"For each set of channels used, 25 stochastic block models were simulated and the LCN and BKN models were fit with the edge status of 500 edges and 500 non-edges masked from the algorithm.",
"Out-of-sample errors were calculated on these withheld sets and in-sample errors were calculated on 500 randomly chosen edges and non-edges that were used to fit the model.",
"Mean squared errors were computed on the true edge probabilities rather than the actual edge statuses themselves.",
"Each setup was repeated 10 times, and the mean and standard errors of the AUC were recorded.",
"Figure: AUC for synthetic stochastic block model.The summarized results from the simulations can be seen on Figure REF .",
"We see that both models appear to achieve near the Bayes Error when the number of channels is equal to 8, the minimum necessary to fully parameterize the stochastic block model.",
"We note that the out-of-sample performance of the LCN model appears to degrade significantly slower than the BKN model as unnecessary channels are added.",
"Both models greatly overstate the AUC using the in-sample data.",
"The heavy overfitting of AUCs on the SBMs is partially due to the SBMs having only two edge probabilities in the model ($p_{in}$ and $p_{out}$ ).",
"This is described in more detail in the Appendix.",
"An oddity that deserves some explanation is that when a single channel is used, the AUC drops below 0.5 for both models.",
"Our explanation for this behavior is that with only one channel, each node has a single parameter dictating the frequency with which it makes connections with all other nodes.",
"When we mask a single connection, we induce a very slight downward bias in the node pair's parameters.",
"Likewise, when we mask the lack of an edge, we induce a slight upward bias in the node pair's parameters.",
"Since the probability of any node connecting with any other randomly sample is constant across nodes in our simple SBM example, this bias induced by masking makes the predictor's performance worse than that with random guessing when the number of channels is one."
],
[
"Facebook100",
"For an application to real data, we considered a sample of the the Facebook 100 datasets [29].",
"These graphs contain historic Facebook data from 100 different universities.",
"We picked the first ten graphs on the list for comparing optimal out-of-sample AUC and focussed on the top four for visualizations.",
"Basic summaries of each graph are shown in Table REF .",
"Several metadata variables are available on each node.",
"For the qualitative analysis, we consider enrollment year.",
"Table: Size of Facebook 100 GraphsTable: Optimal AUC results.We fit both models with $1, 2, 4, \\ldots , 256$ channels and masked 500 edges and non-edges to estimate an out-of-sample AUC.",
"Each setup was repeated 10 times, and the mean and standard error of the AUC, both in-sample and out-of-sample, was recorded.",
"The optimal out-of-sample AUC and number of channels for the different networks can be found in Table REF .",
"The AUC's by number of channels for the first four universities are shown on Figure REF .",
"We make several observations: Both models were able to achieve out-of-sample AUC's over 0.929 on all ten networks.",
"The best performance was usually seen with a large number of channels.",
"The performance of the BKN and LCN models was nearly identical on the Facebook networks when the number of channels was small to moderate ($\\le 32$ channels).",
"In larger models, ($ \\ge 64$ channels), the BKN models consistently showed more overfitting when compared with the LCN model.",
"This consistently led to the highest out-of-sample AUC for LCN by a moderate amount.",
"Note that the optimal number of channels for the LCN model was consistently more than for the optimal number for the BKN model.",
"In fact, in all ten networks, the optimal number of channels for the LCN model was the maximum number fit (256), although the improvement over 128 channels was minimal.",
"Both models had moderate predictive power (out-of-sample AUC $>$ 0.8) with a single channel, which is the opposite of what we saw with our synthetic SBM.",
"This can be explained by the skew of the degree distribution of the real networks, compared with a constant expected degree in the SBM graphs.",
"Figure: AUC for top four datasets from Facebook 100Next, we fit each network with the full edge list and plotted heat maps of the estimated $p$ matrix.",
"For visualization purposes, we used 32 rather 256 channels which can be seen on Figure REF .",
"Plots with 256 channels can be found in the Appendix on Figure REF .",
"Table REF shows the average usage of channels by year of enrollment, where we count a node as using a channel if the $\\hat{p}$ entry is greater than 0.01.",
"Across all four networks, we see some common trends: Many of the channels were strongly tied to year of enrollment.",
"Some channels were essentially exclusively used by a cohort.",
"Others were strongly used by multiple cohorts, with the upper cohorts using more frequently than the lower cohorts.",
"With rare exceptions, freshmen (2009) used the channels associated with the freshmen class only.",
"This differed with upperclassmen, who used both channels attached strongly to their cohort and channels strongly used by many cohorts.",
"Sophomores tended to use multi-cohort channels less than juniors and seniors did, but more than freshmen did.",
"Freshmen, and to a lesser extent sophomores, tended to make edges through fewer channels than juniors and seniors.",
"Overall, the $\\hat{p}$ matrix was fairly sparse.",
"Across the four networks, the proportion of exact zeros in $\\hat{p}$ ranged from 0.839 and 0.860.",
"When the model was fit with more channels, $\\hat{p}$ was more sparse, as expected.",
"Table: Average number of channels used per node by cohort.Figure: Heatmaps for estimated pp matrix for each Facebook 100 dataset.",
"Thirty-two channels were used for visualization purposes.We note that these results seem consistent with what one might expect from a university network; freshmen are very likely to make friends in their lower division courses, whereas by the time students become upperclassmen, many friendships and groups will haved formed that are less strictly determined by cohort."
],
[
"Synthetic Latent Channel Graphs",
"Interestingly, our real data examples demonstrated significantly higher out-of-sample AUC than the Bayes Error of the SBM example we evaluated earlier.",
"Initially, this was slightly perplexing, as the parameters that we specified for the SBM ($ p_{in} = 0.5$ , $p_{out} = 0.02$ ) implied quite strong individual communities.",
"In addition, the synthetic data appeared to show more issues with overfitting than the real data.",
"While it was encouraging to learn that our model worked better on real data than synthetic data, it was not immediately clear why both the LCN and BKN models performed better on the Facebook100 data than on stochastic block model data with strong communities.",
"After reviewing the fitted models, we pinpointed two factors that seemed to contribute to better performance of the LCN and BKN models: Sparse $p$ matrix.",
"Small number of large degree nodes.",
"To demonstrate this, we simulated data according an LCN generative model under different conditions; sparse $p$ vs dense $p$ and skewed degrees vs uniform degrees.",
"The conditions dictated how the parameters of the $p$ matrix were drawn, which was then used to a draw a graph according to the LCN model.",
"Associated with each node were two types of channel usages: main channels and background channels.",
"To simulate sparse $p$ 's, all the background channels were set to 0.",
"With dense $p$ 's, the background channels follow a Beta($a = 1, b = 20$ ).",
"To simulate skewed degrees, the number of main channels is simulated as 1 + Beta-binomial($a = 1, b = 10, n = 15$ ).",
"With the relatively uniform degrees, all nodes have 3 main channels.",
"The main channels followed a Uniform(0,1) distribution.",
"Each simulated graph had 1,000 nodes and 16 channels.",
"After the data was simulated, it was fit with an LCN model with $1, 2, 4, \\ldots , 128$ channels.",
"Each scenario was repeated 10 times and the mean and standard error of the in-sample and out-of-sample AUC was recorded.",
"The results were plotted on figure REF .",
"We make several notes about this results.",
"Figure: Simulated in-sample and out-of-sample AUC based on different generative Latent Channel Network pp structures.",
"The out-of-sample AUC for sparse, skewed degree scenario was quite similar to the results in the Facebook100 datasets.",
"With a dense $p$ matrix, the optimal out-of-sample AUC was poor, never doing much better than 0.7.",
"With a skew degree, fits with only a single channel still had some predictive power, despite being a heavily under-parameterized model.",
"We hypothesize this is the single channel being used to identify the high degree nodes.",
"With sparse $p$ and skewed degrees, the model was fairly resistant to overfitting.",
"The model was fully parameterized with 16 channels, yet little loss in out-of-sample AUC was seen when 32 and 64 channels were used to fit the data.",
"Only at 128 channels was a moderate drop in AUC observed.",
"These observations suggest some serendipity on our part.",
"Data may be exactly generated by a Latent Channel Network model and fit with the correct dimensional space, yet the fitted model can still lead to poor predictive behavior for dense $p$ matrices.",
"The fact that the Facebook100 datasets appear well approximated by a sparse $p$ matrix with a highly skewed degree distribution implies they are the type of graphs for which the unknown edge status can be predicted well."
],
[
"Discussion",
"We have presented the latent channel network, a model that allows nodes to share an edge if they connect through at least one unobserved channel, which we believe captures an important aspect of social networks.",
"We implemented an EM algorithm that scales linearly in the number of edges in the graph and number of channels in the model.",
"We applied this model to the top ten Facebook100 networks and consistently found a moderate improvement in predicting unknown edge status in comparison with the BKN model.",
"We found the channels uncovered by this model tended to correspond with meaningful features of the data and gave insight into the structure of the graph.",
"An R [25] implementation of the algorithm has been made available in a GitHub repo [4].",
"The code uses Rcpp [11] and RcppParallel [3] for efficient computations.",
"There are several ways that this work can be further expanded.",
"In regards to efficient computation, although each iteration of the EM algorithm is relatively computationally cheap, typically, several thousand iterations are required.",
"There are several ways in which the EM-algorithm can be accelerated.",
"One generic method is the SquareEM algorithm [32] for reducing the iterations required until convergence.",
"Unfortunately, this algorithm requires computating the observed log-likelihood.",
"Given that each iteration of our cached EM-algorithm requires $O(KN_e)$ observations and computation of the observed log-likelihood requires $O(K N_n^2)$ , this approach is likely to only reduce wall-time computations (rather than iterations) if we have a dense graph with $N_e \\approx N_n^2$ .",
"A more promising approach is using efficient data augmentation algorithms [31] [38], in which the missing data is less informative but still provides a closed form solution, reducing the number of iterations required without significantly increasing the computational cost per iteration.",
"This approach requires clever data augmentation schemes and while several have been proposed for mixture models, it is not obvious how our problem may be viewed as that of a mixture model, so novel data imputation methods would be required to specialize to our problem."
],
[
"Acknowledgements",
"We would like to thank Goran Konjevod for informative discussions on graph algorithms.",
"All heatmaps in this paper were generated by [13]."
],
[
"Bayes Error for Stochastic Block Model",
"One definition of the AUC measure is $P(\\hat{y} > \\hat{y}^{\\prime } | y = 1, y^{\\prime } = 0).$ In terms of edge prediction, we can write this as $P(\\hat{e}_{ij} > \\hat{e}_{i^{\\prime }j^{\\prime }} | e_{ij} = 1, e_{i^{\\prime }j^{\\prime }} = 0).$ To compute the Bayes error for Stochastic Block Models, we clarify how to calculate this metric with ties in the predictor as $P(\\hat{e}_{ij} > \\hat{e}_{i^{\\prime }j^{\\prime }} | e_{ij} = 1, e_{i^{\\prime }j^{\\prime }} = 0) + \\frac{1}{2} P(\\hat{e}_{ij} = \\hat{e}_{i^{\\prime }j^{\\prime }} | e_{ij} = 1, e_{i^{\\prime }j^{\\prime }} = 0)$ Note that this definition is equivalent to the previous, where ties in the the estimator are broken up at random.",
"In a stochastic block model, the Bayes estimator for an edge is an indicator function of whether the node pair belongs to the same block.",
"We define this as $B(i,j)$ , where $B(i,j) = 1$ if nodes $i$ and $j$ are from the same block and $B(i,j) = 0$ if they are from different blocks.",
"We consider the case in which $p_{in}$ , $p_{out}$ and the block size is the same across all blocks.",
"If we define $n_b$ as the number blocks and $n_s$ as the size of the blocks and $\\pi _B \\equiv P(B(i,j) = 1) = \\frac{n_s - 1}{n_b n_s - 1} $ , we can write the AUC as $\\frac{ P(B(i,j) > B(i^{\\prime },j^{\\prime }) , e_{ij} = 1, e_{i^{\\prime }j^{\\prime }} = 0) + \\frac{1}{2} P(B(i,j) = B(i^{\\prime },j^{\\prime }) , e_{ij} = 1, e_{i^{\\prime }j^{\\prime }} = 0) }{ P( e_{ij} = 1, e_{ij} = 0) }.$ To evaluate and simplify this equation, we define $q_{in} = 1 - p_{in}$ , $q_{out} = 1 - p_{out}$ and $q_{B} = 1 - \\pi _{B}$ .",
"Then the Bayes Error for the AUC can be written in closed form as $\\frac{p_{in} q_{out} \\pi _B q_B +\\frac{1}{2} (p_{in} q_{in} \\pi _B^2 +p_{out} q_{out} q_B^2 )}{p_{in} q_{out} \\pi _B q_B +p_{in} q_{in} \\pi _B^2 +p_{out} q_{out} q_B^2 +p_{out} q_{in} q_B \\pi _B}.$"
],
[
"Overfitting AUC with SBMs",
"In the applications section, it was observed that the in-sample AUC greatly estimated the predictive power of both the LCN and BKN models.",
"We note that the AUC is defined as $P(\\hat{e}_{ij} > \\hat{e}_{i^{\\prime }j^{\\prime }} | e_{ij} = 1, e_{i^{\\prime }j^{\\prime }} = 0) + \\frac{1}{2} P(\\hat{e}_{ij} = \\hat{e}_{i^{\\prime }j^{\\prime }} | e_{ij} = 1, e_{i^{\\prime }j^{\\prime }} = 0).$ If we know the predictor $\\hat{e}_{ij}$ only up to a very small amount of continuous noise, all the ties in the data would be broken, and the estimated AUC would be $P(\\hat{e}_{ij} > \\hat{e}_{i^{\\prime }j^{\\prime }} | e_{ij} = 1, e_{i^{\\prime }j^{\\prime }} = 0) + P(\\hat{e}_{ij} = \\hat{e}_{i^{\\prime }j^{\\prime }} | e_{ij} = 1, e_{i^{\\prime }j^{\\prime }} = 0).$ Because a simple SBM only has two unique edge probabilities, a very large number of ties in the true edges probabilities exist.",
"Since the LCN and BKN models do not constrain any of the edge probabilities to be exactly equal, any error in the estimated edge probabilities automatically upwardly bias the in-sample AUC by $ \\frac{1}{2} P(\\hat{e}_{ij} = \\hat{e}_{i^{\\prime }j^{\\prime }} | e_{ij} = 1, e_{i^{\\prime }j^{\\prime }} = 0)$ ."
],
[
"Extending BKN Algorithm for Unknown Edge Status",
"The algorithm presented by Ball, Karrer and Newman can be presented as an EM algorithm, in which the E-step consists of computing $q_{ijk} = \\frac{ \\theta _{ik}\\theta _{jk}}{ \\sum _k \\theta _{ik}\\theta _{jk} }$ and the M-step consists of computing $\\theta _{ik} = \\frac{ \\sum _j A_{ij} q_{ijk} }{ \\sqrt{\\sum _{ij} A_{ij} q_{ijk} } }.$ If the edge status of a set of node pairs is unknown, then the value of $ A_{ij} $ will be unknown for a set of $ij$ pairs.",
"To account this, we add a step to estimate the missing data with $\\tilde{A}_{ij} = \\sum _{k} \\theta _{ik} \\theta _{jk}.$ and use $\\tilde{A}_{ij}$ for the unknown edges in the M-step."
],
[
"Plots of $\\hat{p}$ with 256 Channels",
"In Figure REF , we show the $\\hat{p}$ matrix with 256 channels.",
"We chose not to focus on these plots as the minute cell size makes visualization difficult.",
"If one wished to visualize $\\hat{p}$ for such a large matrix, we suggest focusing on a subset of interesting channels.",
"Figure: Heatmaps for estimated pp matrix for each Facebook 100 dataset with 256 channels."
]
] | 1906.04563 |
[
[
"A short introduction to the Lindblad Master Equation"
],
[
"Abstract The theory of open quantum system is one of the most essential tools for the development of quantum technologies.",
"Furthermore, the Lindblad (or Gorini-Kossakowski-Sudarshan-Lindblad) Master Equation plays a key role as it is the most general generator of Markovian dynamics in quantum systems.",
"In this paper, we present this equation together with its derivation and methods of resolution.",
"The presentation tries to be as self-contained and straightforward as possible to be useful to readers with no previous knowledge of this field."
],
[
"Introduction",
"Open quantum system techniques are vital for many studies in quantum mechanics [1], [2], [3].",
"This happens because closed quantum systems are just an idealisation of real systemsThe same happens with closed classical systems., as in Nature nothing can be isolated.",
"In practical problems, the interaction of the system of interest with the environment cannot be avoided, and we require an approach in which the environment can be effectively removed from the equations of motion.",
"The general problem addressed by Open Quantum Theory is sketched in Figure REF .",
"In the most general picture, we have a total system that conforms a closed quantum system by itself.",
"We are mostly interested in a subsystem of the total one (we call it just “system” instead “total system”).",
"Therefore, the whole system is divided into our system of interest and an environment.",
"The goal of Open Quantum Theory is to infer the equations of motions of the reduced systems from the equation of motion of the total system.",
"For practical purposes, the reduced equations of motion should be easier to solve than the full dynamics of the system.",
"Because of his requirement, several approximations are usually made in the derivation of the reduced dynamics.",
"Figure: A total system divided into the system of interest, “System”, and the environment.One particular, and interesting, case of study is the dynamics of a system connected to several baths modelled by a Markovian interaction.",
"In this case the most general quantum dynamics is generated by the Lindblad equation (also called Gorini-Kossakowski-Sudarshan-Lindblad equation) [4], [5].",
"It is difficult to overemphasize the importance of this Master Equation.",
"It plays an important role in fields as quantum optics [1], [6], condensed matter [7], [8], [9], [10], atomic physics [11], [12], quantum information [13], [14], decoherence [15], [16], and quantum biology [17], [18], [19].",
"The purpose of this paper is to provide basic knowledge about the Lindblad Master Equation.",
"In Section , the mathematical requirements are introduced while in Section there is a brief review of quantum mechanical concepts that are required to understand the paper.",
"Section , includes a description of a mathematical framework, the Fock-Liouville space, that is especially useful to work in this problem.",
"In Section , we define the concept of CPT-Maps, derive the Lindblad Master Equation from two different approaches, and we discus several properties of the equation.",
"Finally, Section is devoted to the resolution of the master equation using different methods.",
"To deepen in the techniques of solving the Lindblad equation, an example consisting of a two-level system with decay is analysed, illustrating the content of every section.",
"The problems proposed are solved by the use of Mathematica notebooks that can be found at [20]."
],
[
"Mathematical basis",
"The primary mathematical tool in quantum mechanics is the theory of Hilbert spaces.",
"This mathematical framework allows extending many results from finite linear vector spaces to infinite ones.",
"In any case, this tutorial deals only with finite systems and, therefore, the expressions `Hilbert space' and `linear space' are equivalent.",
"We assume that the reader is skilled in operating in Hilbert spaces.",
"To deepen in the field of Hilbert spaces we recommend the book by Debnath and Mikusińki [21].",
"If the reader needs a brief review of the main concepts required for understanding this paper, we may recommend Nielsen and Chuang's Quantum Computing book [22].",
"It is also required some basic knowledge about infinitesimal calculus, like integration, derivation, and the resolution of simple differential equations, To help the readers, we have made a glossary of the most used mathematical terms.",
"It can be used also as a checklist of terms the reader should be familiar with.",
"Glossary: ${\\cal H}$ represents a Hilbert space, usually the space of pure states of a system.",
"$\\vert {\\psi }\\rangle \\in {\\cal H}$ represents a vector of the Hilbert space ${\\cal H}$ (a column vector).",
"$\\langle {\\psi }\\vert \\in {\\cal H}$ represents a vector of the dual Hilbert space of ${\\cal H}$ (a row vector).",
"$\\langle {\\psi }\\vert {\\phi }\\rangle \\in \\mathbb {C}$ is the scalar product of vectors $\\vert {\\psi }\\rangle $ and $\\vert {\\phi }\\rangle $ .",
"$\\left| \\left| \\vert {\\psi }\\rangle \\right|\\right|$ is the norm of vector $\\vert {\\psi }\\rangle $ .",
"$\\left| \\left| \\vert {\\psi }\\rangle \\right|\\right|\\equiv \\sqrt{\\langle {\\psi }\\vert {\\psi }\\rangle }$ .",
"$B({\\cal H})$ represents the space of bounded operators acting on the Hilbert space $B:{\\cal H}\\rightarrow {\\cal H}$ .",
"$\\mathbb {1}_{{\\cal H}}\\in B({\\cal H})$ is the Identity Operator of the Hilbert space ${\\cal H}$ s.t.",
"$\\mathbb {1}_{{\\cal H}}\\vert {\\psi }\\rangle =\\vert {\\psi }\\rangle ,\\; \\; \\forall \\vert {\\psi }\\rangle \\in {\\cal H}$ .",
"$\\vert {\\psi }\\rangle \\!\\langle {\\phi }\\vert \\in B({\\cal H})$ is the operator such that $\\left( \\vert {\\psi }\\rangle \\!\\langle {\\phi }\\vert \\right) \\vert {\\varphi }\\rangle =\\langle {\\phi }\\vert {\\varphi }\\rangle \\vert {\\psi }\\rangle ,\\; \\; \\forall \\vert {\\varphi }\\rangle \\in {\\cal H}$ .",
"$O^\\dagger \\in B({\\cal H})$ is the Hermitian conjugate of the operator $O\\in B({\\cal H})$ .",
"$U\\in B({\\cal H})$ is a unitary operator iff $U U^{\\dagger }=U^{\\dagger }U=\\mathbb {1}$ .",
"$H\\in B({\\cal H})$ is a Hermitian operator iff $H=H^{\\dagger }$ .",
"$A\\in B({\\cal H})$ is a positive operator $\\left( A> 0 \\right)$ iff $\\langle {\\phi }\\vert A \\vert {\\phi }\\rangle \\ge 0,\\;\\; \\forall \\vert {\\phi }\\rangle \\in {\\cal H}$ $P\\in B({\\cal H})$ is a proyector iff $P P=P$ .",
"$\\textrm {Tr}\\left[ B \\right]$ represents the trace of operator $B$ .",
"$\\rho \\left( {\\cal L} \\right)$ represents the space of density matrices, meaning the space of bounded operators acting on ${\\cal H}$ with trace 1 and positive.",
"$\\vert {\\rho }\\rangle \\rangle $ is a vector in the Fock-Liouville space.",
"$\\langle \\langle {A}\\vert {B}\\rangle \\rangle =\\textrm {Tr}\\left[ A^\\dagger B \\right]$ is the scalar product of operators $A,B\\in B({\\cal H})$ in the Fock-Liouville space.",
"$\\tilde{{\\cal L}}$ is the matrix representation of a superoperator in the Fock-Liouville space."
],
[
"(Very short) Introduction to quantum mechanics",
"The purpose of this chapter is to refresh the main concepts of quantum mechanics necessary to understand the Lindblad Master Equation.",
"Of course, this is NOT a full quantum mechanics course.",
"If a reader has no background in this field, just reading this chapter would be insufficient to understand the remaining of this tutorial.",
"Therefore, if the reader is unsure of his/her capacities, we recommend to go first through a quantum mechanics course or to read an introductory book carefully.",
"There are many great quantum mechanics books in the market.",
"For beginners, we recommend Sakurai's book [23] or Nielsen and Chuang's Quantum Computing book [22].",
"For more advanced students, looking for a solid mathematical description of quantum mechanics methods, we recommend Galindo and Pascual [24].",
"Finally, for a more philosophical discussion, you should go to Peres' book [25].",
"We start stating the quantum mechanics postulates that we need to understand the derivation and application of the Lindblad Master Equation.",
"The first postulate is related to the concept of a quantum state.",
"Postulate 1 Associated to any isolated physical system, there is a complex Hilbert space ${\\cal H}$ , known as the state space of the system.",
"The state of the system is entirely described by a state vector, which is a unit vector of the Hilbert space $(\\vert {\\psi }\\rangle \\in {\\cal H})$ .",
"As quantum mechanics is a general theory (or a set of theories), it does not tell us which is the proper Hilbert space for each system.",
"This is usually done system by system.",
"A natural question to ask is if there is a one-to-one correspondence between unit vectors and physical states, meaning that if every unit vector corresponds to a physical system.",
"This is resolved by the following corollary that is a primary ingredient for quantum computation theory (see Ref.",
"[22] Chapter 7).",
"Corollary 1 All unit vectors of a finite Hilbert space correspond to possible physical states of a system.",
"Unit vectors are also called pure states.",
"If we know the pure state of a system, we have all physical information about it, and we can calculate the probabilistic outcomes of any potential measurement (see the next postulate).",
"This is a very improbable situation as experimental settings are not perfect, and in most cases, we have only imperfect information about the state.",
"Most generally, we may know that a quantum system can be in one state of a set $\\left\\lbrace \\vert {\\psi _i}\\rangle \\right\\rbrace $ with probabilities $p_i$ .",
"Therefore, our knowledge of the system is given by an ensemble of pure states described by the set $\\left\\lbrace \\vert {\\psi _i}\\rangle , \\; p_i \\right\\rbrace $ .",
"If more than one $p_i$ is different from zero the state is not pure anymore, and it is called a mixed state.",
"The mathematical tool that describes our knowledge of the system, in this case, is the density operator (or density matrix).",
"$\\rho \\equiv \\sum _i p_i \\vert {\\psi _i}\\rangle \\!\\langle {\\psi _i}\\vert .$ Density matrices are bounded operators that fulfil two mathematical conditions A density matrix $\\rho $ has unit trace $\\left( \\textrm {Tr}[\\rho ]=1 \\right)$ .",
"A density matrix is a positive matrix $\\rho >0$ .",
"Any operator fulfilling these two properties is considered a density operator.",
"It can be proved trivially that density matrices are also Hermitian.",
"If we are given a density matrix, it is easy to verify if it belongs to a pure or a mixed state.",
"For pure states, and only for them, $\\textrm {Tr}[\\rho ^2]=\\textrm {Tr}[\\rho ]=1$ .",
"Therefore, if $\\textrm {Tr}[\\rho ^2]<1$ the system is mixed.",
"The quantity $\\textrm {Tr}[\\rho ^2]$ is called the purity of the states, and it fulfils the bounds $\\frac{1}{d} \\le \\textrm {Tr}[\\rho ^2] \\le 1$ , being $d$ the dimension of the Hilbert space.",
"If we fix an arbitrary basis $\\left\\lbrace \\vert {i}\\rangle \\right\\rbrace _{i=1}^N$ of the Hilbert space the density matrix in this basis is written as $\\rho =\\sum _{i,j=1}^N \\rho _{i,j} \\vert {i}\\rangle \\!\\langle {j}\\vert $ , or $\\rho =\\begin{pmatrix}\\rho _{00} & \\rho _{01} & \\cdots & \\rho _{0N} \\\\\\rho _{10} & \\rho _{11} & \\cdots & \\rho _{1N} \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\\\rho _{N0} & \\rho _{N1} & \\cdots & \\rho _{NN}\\end{pmatrix},$ where the diagonal elements are called populations $\\left( \\rho _{ii}\\in \\mathbb {R}_0^+\\text{ and } \\sum _{i} \\rho _{i,i}=1 \\right)$ , while the off-diagonal elements are called coherences $\\left( \\rho ^{\\phantom{*}}_{i,j} \\in \\mathbb {C} \\text{ and } \\rho ^{\\phantom{*}}_{i,j}=\\rho _{j,i}^* \\right)$ .",
"Note that this notation is base-dependent.",
"Box 1.",
"State of a two-level system (qubit) The Hilbert space of a two-level system is just the two-dimension lineal space ${\\cal H}_2$ .",
"Examples of this kind of system are $\\frac{1}{2}$ -spins and two-level atoms.",
"We can define a basis of it by the orthonormal vectors: $\\left\\lbrace \\vert {0}\\rangle ,\\;\\vert {1}\\rangle \\right\\rbrace $ .",
"A pure state of the system would be any unit vector of ${\\cal H}_2$ .",
"It can always be expressed as a $\\vert {\\psi }\\rangle =a\\vert {0}\\rangle + b\\vert {1}\\rangle $ with $a,b \\in \\mathbb {C}$ s. t. $\\left|a\\right|^2 + \\left|b\\right|^2=1$ .",
"A mixed state is therefore represented by a positive unit trace operator $ \\rho \\in O({\\cal H}_2)$ .",
"$\\rho =\\begin{pmatrix}\\rho _{00} & \\rho _{01} \\\\\\rho _{10} & \\rho _{11}\\end{pmatrix}= \\rho _{00} \\vert {0}\\rangle \\!\\langle {0}\\vert + \\rho _{01} \\vert {0}\\rangle \\!\\langle {1}\\vert + \\rho _{10} \\vert {1}\\rangle \\!\\langle {0}\\vert + \\rho _{11} \\vert {1}\\rangle \\!\\langle {1}\\vert ,$ ant it should fulfil $\\rho _{00}+\\rho _{11}=1$ and $\\rho _{01}^{\\phantom{*}}=\\rho _{10}^*$ .",
"Once we know the state of a system, it is natural to ask about the possible outcomes of experiments (see Ref.",
"[23], Section 1.4).",
"Postulate 2 All possible measurements in a quantum system are described by a Hermitian operator or observable.",
"Due to the Spectral Theorem we know that any observable $O$ has a spectral decomposition in the formFor simplicity, we assume a non-degenerated spectrum.",
"$O=\\sum _i a_i \\vert {a_i}\\rangle \\!\\langle {a_i}\\vert ,$ being $a_i\\in \\mathbb {R}$ the eigenvalues of the observable and $\\vert {a_i}\\rangle $ their corresponding eigenvectors.",
"The probability of obtaining the result $a_i$ when measuring the property described by observable $O$ in a state $\\vert {\\psi }\\rangle $ is given by $P(a_i)= \\left| \\langle {\\psi }\\vert {a_i}\\rangle \\right|^2.$ After the measurement we obtain the state $\\vert {a_i}\\rangle $ if the outcome $a_i$ was measured.",
"This is called the post-measurement state.",
"This postulate allow us to calculate the possible outputs of a system, the probability of these outcomes, as well as the after-measurement state.",
"A measurement usually changes the state, as it can only remain unchanged if it was already in an eigenstate of the observable.",
"It is possible to calculate the expectation value of the outcome of a measurement defined by operator $O$ in a state $\\vert {\\psi }\\rangle $ by just applying the simple formula $\\langle O \\rangle = \\langle {\\psi }\\vert O \\vert {\\psi }\\rangle .$ With a little algebra we can translate this postulate to mixed states.",
"In this case, the probability of obtaining an output $a_i$ that corresponds to an eigenvector $\\vert {a_i}\\rangle $ is $P(a_i)=\\textrm {Tr}\\left[ \\vert {a_i}\\rangle \\!\\langle {a_i}\\vert \\rho \\right],$ and the expectation value of operator $O$ is $\\langle O \\rangle =\\textrm {Tr}\\left[ O\\rho \\right].$ Box 2.",
"Measurement in a two-level system.",
"A possible test to perform in our minimal model is to measure the energetic state of a system, assuming that both states have a different energy.",
"The observable corresponding to this measurement would be $H=E_0 \\vert {0}\\rangle \\!\\langle {0}\\vert + E_1 \\vert {1}\\rangle \\!\\langle {1}\\vert .$ This operator has two eigenvalues $\\left\\lbrace E_0,\\;E_1 \\right\\rbrace $ with two corresponding eigenvectors $\\left\\lbrace \\vert {0}\\rangle ,\\; \\vert {1}\\rangle \\right\\rbrace $ .",
"If we have a pure state $\\psi =a\\vert {0}\\rangle + b \\vert {1}\\rangle $ the probability of measuring the energy $E_0$ would be $P(E_0)=\\left|\\langle {0}\\vert {\\psi }\\rangle \\right|^2=\\left|a\\right|^2$ .",
"The probability of finding $E_1$ would be $P(E_1)=\\left|\\langle {1}\\vert {\\psi }\\rangle \\right|^2=\\left|b\\right|^2$ .",
"The expected value of the measurement is $\\langle H \\rangle = E_0\\left|a\\right|^2+ E_1\\left|b\\right|^2$ .",
"In the more general case of having a mixed state $\\rho =\\rho _{00} \\vert {0}\\rangle \\!\\langle {0}\\vert + \\rho _{01} \\vert { 0}\\rangle \\!\\langle {1}\\vert + \\rho _{10} \\vert {1}\\rangle \\!\\langle {0}\\vert + \\rho _{11} \\vert {1}\\rangle \\!\\langle {1}\\vert $ the probability of finding the ground state energy is $P(0)=\\textrm {Tr}\\left[ \\vert {0}\\rangle \\!\\langle {0}\\vert \\rho \\right]= \\rho _{00}$ , and the expected value of the energy would be $\\langle H \\rangle =\\textrm {Tr}\\left[ H\\rho \\right]= E_0 \\rho _{00} + E_1 \\rho _{11}$ .",
"Another natural question to ask is how quantum systems evolve.",
"The time-evolution of a pure state of a closed quantum system is given by the Schrödinger equation (see [24], Section 2.9).",
"Postulate 3 Time evolution of a pure state of a closed quantum system is given by the Schrödinger equation $\\frac{d}{dt} \\vert {\\psi (t)}\\rangle = -i\\hbar H\\vert {\\psi (t)}\\rangle ,$ where $H$ is the Hamiltonian of the system and it is a Hermitian operator of the Hilbert space of the system state (from now on we avoid including Planck's constant by selecting the units such that $\\hbar =1)$ .",
"The Hamiltonian of a system is the operator corresponding to its energy, and it can be non-trivial to realise.",
"Schrödinger equation can be formally solved in the following way.",
"If at $t=0$ the state of a system is given by $\\vert {\\psi (0)}\\rangle $ at time $t$ it will be $\\vert {\\psi (t)}\\rangle =e^{-i Ht } \\vert {\\psi (0)}\\rangle .$ As $H$ is a Hermitian operator, the operator $U=e^{-i Ht }$ is unitary.",
"This gives us another way of phrasing Postulate REF .",
"Postulate 3' The evolution of a closed system is given by a unitary operator of the Hilbert space of the system $\\vert {\\psi (t)}\\rangle =U \\vert {\\psi (0)}\\rangle ,$ with $U\\in {\\cal B}\\left( {\\cal H} \\right)$ s.t.",
"$U U^{\\dagger }=U^{\\dagger }U=\\mathbb {1}$ .",
"It is easy to prove that unitary operators preserve the norm of vectors and, therefore, transform pure states into pure states.",
"As we did with the state of a system, it is reasonable to wonder if any unitary operator corresponds to the evolution of a real physical system.",
"The answer is yes.",
"Lemma 1 All unitary evolutions of a state belonging to a finite Hilbert space can be constructed in several physical realisations like photons and cold atoms.",
"The proof of this lemma can be found at [22].",
"The time evolution of a mixed state can be calculated just by combining Eqs.",
"(REF ) and (REF ), giving the von-Neumann equation.",
"$\\dot{\\rho } = - i \\left[ H,\\rho \\right]\\equiv {\\cal L}\\rho ,$ where we have used the commutator $\\left[ A,B \\right]=AB-BA$ , and ${\\cal L}$ is the so-called Liouvillian superoperator.",
"It is easy to prove that the Hamiltonian dynamics does not change the purity of a system $\\frac{d}{dt} \\textrm {Tr}\\left[ \\rho ^2 \\right] = \\textrm {Tr}\\left[ \\frac{d \\rho ^2}{dt} \\right] = \\textrm {Tr}\\left[ 2\\rho \\dot{\\rho } \\right] = -2 i \\textrm {Tr}\\left[ \\rho \\left( H\\rho -\\rho H \\right) \\right]=0,$ where we have used the cyclic property of the trace.",
"This result illustrates that the mixing rate of a state does not change due to the quantum evolution.",
"Box 3.",
"Time evolution of a two-level system.",
"The evolution of our isolated two-level system is described by its Hamiltonian $H_{\\text{free}}=E_0 \\vert {0}\\rangle \\!\\langle {0}\\vert + E_1 \\vert {1}\\rangle \\!\\langle {1}\\vert ,$ As the states $\\vert {0}\\rangle $ and $\\vert {1}\\rangle $ are Hamiltonian eigenstates if at $t=0$ the atom is at the excited state $\\vert {\\psi (0)}\\rangle =\\vert {1}\\rangle $ after a time $t$ the state would be $\\vert {\\psi (t)}\\rangle =e^{-iHt} \\vert {1}\\rangle =e^{-i E_1 t} \\vert {1}\\rangle $ .",
"As the system was already in an eigenvector of the Hamiltonian, its time-evolution consists only in adding a phase to the state, without changing its physical properties.",
"(If an excited state does not change, why do atoms decay?)",
"Without losing any generality we can fix the energy of the ground state as zero, obtaining $H_{\\text{free}}= E \\vert {1}\\rangle \\!\\langle {1}\\vert ,$ with $E\\equiv E_1$ .",
"To make the model more interesting we can include a driving that coherently switches between both states.",
"The total Hamiltonian would be then $H=E \\vert {1}\\rangle \\!\\langle {1}\\vert + \\Omega \\left( \\vert {0}\\rangle \\!\\langle {1}\\vert +\\vert {1}\\rangle \\!\\langle {0}\\vert \\right),$ where $\\Omega $ is the frequency of driving.",
"By using the von-Neumann equation (REF ) we can calculate the populations $\\left( \\rho _{00},\\rho _{11} \\right)$ as a function of time.",
"The system is then driven between the states, and the populations present Rabi oscillations, as it is shown in Fig.",
".",
"Figure: NO_CAPTION$\\Omega =1,\\; E=1$ ).",
"The blue line represents $\\rho _{11}$ and the orange one $\\rho _{00}$ .",
"Finally, as we are interested in composite quantum systems, we need to postulate how to work with them.",
"Postulate 4 The state-space of a composite physical system, composed by $N$ subsystems, is the tensor product of the state space of each component ${\\cal H}={\\cal H}_1 \\otimes {\\cal H}_2 \\otimes \\cdots \\otimes {\\cal H}_N$ .",
"The state of the composite physical system is given by a unit vector of ${\\cal H}$ .",
"Moreover, if each subsystem belonging to ${\\cal H}_i$ is prepared in the state $\\vert {\\psi _i}\\rangle $ the total state is given by $\\vert {\\psi }\\rangle =\\vert {\\psi _1}\\rangle \\otimes \\vert {\\psi _2}\\rangle \\otimes \\cdots \\otimes \\vert {\\psi _N}\\rangle $ .",
"The symbol $\\otimes $ represents the tensor product of Hilbert spaces, vectors, and operators.",
"If we have a composited mixed state where each component is prepared in the state $\\rho _i$ the total state is given by $\\rho =\\rho _1 \\otimes \\rho _2 \\otimes \\cdots \\otimes \\rho _N$ .",
"States that can be expressed in the simple form $\\vert {\\psi }\\rangle =\\vert {\\psi _1}\\rangle \\otimes \\vert {\\psi _2}\\rangle $ , in any specific basis, are very particular and they are called separable states (For this discussion, we use a bipartite system as an example.",
"The extension to a general multipartite system is straightforward.)",
".",
"In general, any arbitrary state should be described as $\\vert {\\psi }\\rangle =\\sum _{i,j} \\vert {\\psi _i}\\rangle \\otimes \\vert {\\psi _j}\\rangle $ (or $\\rho =\\sum _{i,j} \\rho _i \\otimes \\rho _j$ for mixed states).",
"Non-separable states are called entangled states.",
"Now that we know how to compose systems, but we can be interested in going the other way around.",
"If we have a system belonging to a bipartite Hilbert space in the form ${\\cal H}={\\cal H}_a \\otimes {\\cal H}_b$ we can be interested in studying some properties of the subsystem corresponding to one of the subspaces.",
"To do so, we define the reduced density matrix.",
"If the state of our system is described by a density matrix $\\rho $ the reduced density operator of the subsystem $a$ is defined by the operator $\\rho _{a} \\equiv \\textrm {Tr}_{b} \\left[ \\rho \\right],$ were $\\textrm {Tr}_b$ is the partial trace over subspace $b$ and it is defined as [22] $\\textrm {Tr}_b \\left[ \\sum _{i,j,k,l} \\vert {a_i}\\rangle \\!\\langle {a_j}\\vert \\otimes \\vert {b_k}\\rangle \\!\\langle {b_l}\\vert \\right] \\equiv \\sum _{i,j} \\vert {a_i}\\rangle \\!\\langle {a_j}\\vert \\textrm {Tr}\\left[ \\sum _{k,l} \\vert {b_k}\\rangle \\!\\langle {b_l}\\vert \\right].$ The concepts of reduced density matrix and partial trace are essential in the study of open quantum systems.",
"If we want to calculate the equation of motions of a system affected by an environment, we should trace out this environment and deal only with the reduced density matrix of the system.",
"This is the main idea of the theory of open quantum systems.",
"Box 4.",
"Two two-level atoms If we have two two-level systems, the total Hilbert space is given by ${\\cal H}={\\cal H}_2\\otimes {\\cal H}_2$ .",
"A basis of this Hilbert space would be given by the set $\\left\\lbrace \\vert {00}\\rangle \\equiv \\vert {0}\\rangle _1 \\otimes \\vert {0}\\rangle _2,\\; \\vert {01}\\rangle \\equiv \\vert {0}\\rangle _1 \\otimes \\vert {1}\\rangle _2,\\; \\vert {10}\\rangle \\equiv \\vert {1}\\rangle _1 \\otimes \\vert {0}\\rangle _2,\\;\\vert {11}\\rangle \\equiv \\vert {1}\\rangle _1 \\otimes \\vert {1}\\rangle _2 \\right\\rbrace $ .",
"If both systems are in their ground state, we can describe the total state by the separable vector $\\vert {\\psi }\\rangle _G=\\vert {00}\\rangle .$ A more complex, but still separable, state can be formed if both systems are in superposition.",
"$\\vert {\\psi }\\rangle _S&=&\\frac{1}{\\sqrt{2}} \\left( \\vert {0}\\rangle _1 +\\vert {1}\\rangle _1 \\right) \\otimes \\frac{1}{ \\sqrt{2}} \\left( \\vert {0}\\rangle _2 +\\vert {1}\\rangle _2 \\right) \\nonumber \\\\&=& \\frac{1}{2} \\left( \\vert {00}\\rangle + \\vert {10}\\rangle + \\vert {01}\\rangle + \\vert {11}\\rangle \\right)$ An entangled state would be $\\vert {\\psi }\\rangle _E=\\frac{1}{ \\sqrt{2}} \\left( \\vert {00}\\rangle +\\vert {11}\\rangle \\right).$ This state cannot be separated into a direct product of each subsystem.",
"If we want to obtain a reduced description of subsystem 1 (or 2) we have to use the partial trace.",
"To do so, we need first to calculate the density matrix corresponding to the pure state $\\vert {\\psi }\\rangle _E$ .",
"$\\rho _E=\\vert {\\psi }\\rangle \\langle {\\psi }\\vert _E = \\frac{1}{2} \\left( \\vert {00}\\rangle \\!\\langle {00}\\vert + \\vert {00}\\rangle \\!\\langle {11}\\vert + \\vert {11}\\rangle \\!\\langle {00}\\vert + \\vert {11}\\rangle \\!\\langle {11}\\vert \\right).$ We can now calculate the reduced density matrix of the subsystem 1 by using the partial trace.",
"$\\rho _E^{(1)}= \\langle {0}\\vert _2 \\rho _E \\vert {0}\\rangle _2 + \\langle {1}\\vert _2 \\rho _E \\vert {1}\\rangle _2 = \\frac{1}{2} \\left( \\vert {00}\\rangle \\!\\langle {00}\\vert _1 + \\vert {11}\\rangle \\!\\langle {11}\\vert _2 \\right).$ From this reduced density matrix, we can calculate all the measurement statistics of subsystem 1."
],
[
"The Fock-Liouville Hilbert space. The Liouville superoperator",
"In this section, we revise a useful framework for both analytical and numerical calculations.",
"It is clear that some linear combinations of density matrices are valid density matrices (as long as they preserve positivity and trace 1).",
"Because of that, we can create a Hilbert space of density matrices just by defining a scalar product.",
"This is clear for finite systems because in this case scalar space and Hilbert space are the same things.",
"It also happens to be true for infinite spaces.",
"This allows us to define a linear space of matrices, converting the matrices effectively into vectors ($\\rho \\rightarrow \\vert {\\rho }\\rangle \\rangle $ ).",
"This is called Fock-Liouville space (FLS).",
"The usual definition of the scalar product of matrices $\\phi $ and $\\rho $ is defined as $\\langle \\langle {\\phi }\\vert {\\rho }\\rangle \\rangle \\equiv \\textrm {Tr}\\left[ \\phi ^\\dagger \\rho \\right]$ .",
"The Liouville super-operator from Eq.",
"(REF ) is now an operator acting on the Hilbert space of density matrices.",
"The main utility of the FLS is to allow the matrix representation of the evolution operator.",
"Box 5.",
"Time evolution of a two-level system.",
"The density matrix of our system (REF ) can be expressed in the FLS as $\\vert {\\rho }\\rangle \\rangle =\\begin{pmatrix}\\rho _{00} \\\\\\rho _{01} \\\\\\rho _{10} \\\\\\rho _{11}\\end{pmatrix}.$ The time evolution of a mixed state is given by the von-Neumann equation (REF ).",
"The Liouvillian superoperator can now be expressed as a matrix $\\tilde{{\\cal L}}=\\left(\\begin{array}{cccc}0 & i \\Omega & -i\\Omega & 0 \\\\i\\Omega & i E &0 & -i\\Omega \\\\-i\\Omega & 0 & -iE & i\\Omega \\\\0 & -i\\Omega & i\\Omega & 0\\end{array}\\right),$ where each row is calculated just by observing the output of the operation $-i \\left[ H,\\rho \\right]$ in the computational basis of the density matrices space.",
"The time evolution of the system now corresponds to the matrix equation $\\frac{d \\vert {\\rho }\\rangle \\rangle }{dt}=\\tilde{{\\cal L}} \\vert {\\rho }\\rangle \\rangle $ , that in matrix notation would be $\\begin{pmatrix}\\dot{\\rho }_{00} \\\\\\dot{\\rho }_{01} \\\\\\dot{\\rho }_{10} \\\\\\dot{\\rho }_{11}\\end{pmatrix}=\\left(\\begin{array}{cccc}0 & i \\Omega & -i\\Omega & 0 \\\\i\\Omega & i E &0 & -i\\Omega \\\\-i\\Omega & 0 & -i E & i\\Omega \\\\0 & -i\\Omega & i\\Omega & 0\\end{array}\\right)\\begin{pmatrix}\\rho _{00} \\\\\\rho _{01} \\\\\\rho _{10} \\\\\\rho _{11}\\end{pmatrix}$"
],
[
"Completely positive maps",
"The problem we want to study is to find the most general Markovian transformation set between density matrices.",
"Until now, we have seen that quantum systems can evolve in two way, by a coherent evolution given (Postulate REF ) and by collapsing after a measurement (Postulate REF ).",
"Many efforts have been made to unify these two ways of evolving [16], without giving a definite answer so far.",
"It is reasonable to ask what is the most general transformation that can be performed in a quantum system, and what is the dynamical equation that describes this transformation.",
"We are looking for maps that transform density matrices into density matrices.",
"We define $\\rho ({\\cal H})$ as the space of all density matrices in the Hilbert space ${\\cal H}$ .",
"Therefore, we are looking for a map of this space onto itself, ${\\cal V}:\\rho ({\\cal H})\\rightarrow \\rho ({\\cal H})$ .",
"To ensure that the output of the map is a density matrix this should fulfil the following properties Trace preserving.",
"$\\textrm {Tr}\\left[ {\\cal V}A \\right]=\\textrm {Tr}\\left[ A \\right],$ $\\forall A\\in O({\\cal H})$ .",
"Completely positive (see below).",
"Any map that fulfils these two properties is called a completely positive and trace-preserving map (CPT-maps).",
"The first property is quite apparent, and it does not require more thinking.",
"The second one is a little more complicated, and it requires an intermediate definition.",
"Definition 1 A map ${\\cal V}$ is positive iff $\\forall A\\in B({\\cal H})$ s.t.",
"$A \\ge 0 \\Rightarrow {\\cal V}A \\ge 0$ .",
"This definition is based in the idea that, as density matrices are positive, any physical map should transform positive matrices into positive matrices.",
"One could naively think that this condition must be sufficient to guarantee the physical validity of a map.",
"It is not.",
"As we know, there exist composite systems, and our density matrix could be the partial trace of a more complicated state.",
"Because of that, we need to impose a more general condition.",
"Definition 2 A map ${\\cal V}$ is completely positive iff $\\forall n\\in \\mathbb {N}$ , ${\\cal V}\\otimes \\mathbb {1}_n$ is positive.",
"To prove that not all positive maps are completely positive, we need a counterexample.",
"A canonical example of an operation that is positive but fails to be completely positive is the matrix transposition.",
"If we have a Bell state in the form $\\vert {\\psi _B}\\rangle =\\frac{1}{\\sqrt{2}} \\left( \\vert {01}\\rangle +\\vert {10}\\rangle \\right)$ its density matrix can be expressed as $\\rho _B=\\frac{1}{2} \\left( \\vert {0}\\rangle \\!\\langle {0}\\vert \\otimes \\vert {1}\\rangle \\!\\langle {1}\\vert + \\vert {1}\\rangle \\!\\langle {1}\\vert \\otimes \\vert {0}\\rangle \\!\\langle {0}\\vert + \\vert {0}\\rangle \\!\\langle {1}\\vert \\otimes \\vert {1}\\rangle \\!\\langle {0}\\vert + \\vert {1}\\rangle \\!\\langle {0}\\vert \\otimes \\vert {0}\\rangle \\!\\langle {1}\\vert \\right),$ with a matrix representation $\\rho _B= \\frac{1}{2} \\left\\lbrace \\left(\\begin{array}{cc}1 & 0 \\\\0 & 0 \\\\\\end{array}\\right)\\otimes \\left(\\begin{array}{cc}0 & 0 \\\\0 & 1 \\\\\\end{array}\\right)+\\left(\\begin{array}{cc}0 & 0 \\\\0 & 1 \\\\\\end{array}\\right)\\otimes \\left(\\begin{array}{cc}1 & 0 \\\\0 & 0 \\\\\\end{array}\\right)\\right.\\nonumber \\\\\\left.\\otimes \\left(\\begin{array}{cc}0 & 0 \\\\1 & 0 \\\\\\end{array}\\right)\\otimes \\left(\\begin{array}{cc}0 & 1 \\\\0 & 0 \\\\\\end{array}\\right)+\\left(\\begin{array}{cc}0 & 1 \\\\0 & 0\\\\\\end{array}\\right)\\otimes \\left(\\begin{array}{cc}0 & 0 \\\\1 & 0 \\\\\\end{array}\\right)\\right\\rbrace .$ A little algebra shows that the full form of this matrix is $\\rho _B=\\left(\\begin{array}{cccc}0 & 0 & 0 & 0\\\\0 & 1 & 1 & 0\\\\0 & 1 & 1 & 0\\\\0 & 0 & 0 & 0\\end{array}\\right),$ and it is positive.",
"It is easy to check that the transformation $\\mathbb {1}\\otimes T_2 $ , meaning that we transpose the matrix of the second subsystem leads to a non-positive matrix $\\left( \\mathbb {1}\\otimes T_2 \\right) \\rho _B= \\frac{1}{2} \\left\\lbrace \\left(\\begin{array}{cc}1 & 0 \\\\0 & 0 \\\\\\end{array}\\right)\\otimes \\left(\\begin{array}{cc}0 & 1 \\\\0 & 0 \\\\\\end{array}\\right)+\\left(\\begin{array}{cc}0 & 0 \\\\0 & 1 \\\\\\end{array}\\right)\\otimes \\left(\\begin{array}{cc}0 & 0 \\\\1 & 0 \\\\\\end{array}\\right)\\right.\\nonumber \\\\\\left.\\otimes \\left(\\begin{array}{cc}0 & 0 \\\\1 & 0 \\\\\\end{array}\\right)\\otimes \\left(\\begin{array}{cc}0 & 0 \\\\1 & 0 \\\\\\end{array}\\right)+\\left(\\begin{array}{cc}0 & 0 \\\\0 & 1 \\\\\\end{array}\\right)\\otimes \\left(\\begin{array}{cc}0 & 1 \\\\0 & 0 \\\\\\end{array}\\right)\\right\\rbrace .$ The total matrix is $\\left( \\mathbb {1}\\otimes T_2 \\right) \\rho _B=\\left(\\begin{array}{cccc}0 & 0 & 0 & 1\\\\0 & 1 & 0 & 0\\\\0 & 0 & 1 & 0\\\\1 & 0 & 0 & 0\\end{array}\\right),$ with $-1$ as an eigenvalue.",
"This example illustrates how the non-separability of quantum mechanics restrict the operations we can perform in a subsystem.",
"By imposing this two conditions, we can derive a unique master equation as the generator of any possible Markovian CPT-map."
],
[
"Derivation of the Lindblad Equation from microscopic dynamics",
"The most common derivation of the Lindblad master equation is based on Open Quantum Theory.",
"The Lindblad equation is then an effective motion equation for a subsystem that belongs to a more complicated system.",
"This derivation can be found in several textbooks like Breuer and Petruccione's [2] as well as Gardiner and Zoller's [1].",
"Here, we follow the derivation presented in Ref.",
"[26].",
"Our initial point is displayed in Figure REF .",
"A total system belonging to a Hilbert space ${\\cal H}_T$ is divided into our system of interest, belonging to a Hilbert space ${\\cal H}$ , and the environment living in ${\\cal H}_E$ .",
"Figure: A total system (belonging to a Hilbert space ℋ T {\\cal H}_T, with states described by density matrices ρ T \\rho _T, and with dynamics determined by a Hamiltonian H T H_T) divided into the system of interest, `System', and the environment.The evolution of the total system is given by the von Neumann equation (REF ).",
"$\\dot{\\rho _T}(t)=-i \\left[ H_T,\\rho _T(t) \\right].$ As we are interested in the dynamics of the system, without the environment, we trace over the environment degrees of freedom to obtain the reduced density matrix of the system $\\rho (t)=\\textrm {Tr}_E[\\rho _T]$ .",
"To separate the effect of the total hamiltonian in the system and the environment we divide it in the form $H_T=H_S \\otimes \\mathbb {1}_E + \\mathbb {1}_S \\otimes H_E + \\alpha H_I $ , with $H\\in {\\cal H}$ , $H_E\\in {\\cal H}_E$ , and $H_I\\in {\\cal H}_T$ , and being $\\alpha $ a measure of the strength of the system-environment interaction.",
"Therefore, we have a part acting on the system, a part acting on the environment, and the interaction term.",
"Without losing any generality, the interaction term can be decomposed in the following way $H_I = \\sum _i S_i \\otimes E_i,$ with $S_i\\in B({\\cal H}$ ) and $E_i\\in B({\\cal H}_E)$From now on we will not writethe identity operators of the Hamiltonian parts explicitly when they can be inferred from the context.. To better describe the dynamics of the system, it is useful to work in the interaction picture (see Ref.",
"[24] for a detailed explanation about Schrödinger, Heisenberg, and interaction pictures).",
"In the interaction picture, density matrices evolve with time due to the interaction Hamiltonian, while operators evolve with the system and environment Hamiltonian.",
"An arbitrary operator $O\\in {\\cal B}({\\cal H}_T)$ is represented in this picture by the time-dependent operator $\\hat{O}(t)$ , and its time evolution is $\\hat{O}(t) = e^{i(H+H_E)t} \\,O\\, e^{-i(H+H_E)t}.$ The time evolution of the total density matrix is given in this picture by $\\frac{d \\hat{\\rho }_T(t)}{dt} = -i \\alpha \\left[ \\hat{H}_I(t),\\hat{\\rho }_T(t) \\right].$ This equation can be easily integrated to give $\\hat{\\rho }_T(t) = \\hat{\\rho }_T(0) -i \\alpha \\int _0^t ds \\left[ \\hat{H}_I(s),\\hat{\\rho }_T(s) \\right].$ By this formula, we can obtain the exact solution, but it still has the complication of calculating an integral in the total Hilbert space.",
"It is also troublesome the fact that the state $\\tilde{\\rho }(t)$ depends on the integration of the density matrix in all previous time.",
"To avoid that we can introduce Eq.",
"(REF ) into Eq.",
"(REF ) giving $\\frac{d \\hat{\\rho }_T(t)}{dt} = -i \\alpha \\left[ \\hat{H}_I(t),\\hat{\\rho }_T(0) \\right] -\\alpha ^2 \\int _0^t ds \\left[ \\hat{H}_I(t),\\left[ \\hat{H}_I(s),\\hat{\\rho }_T(s) \\right] \\right].$ By applying this method one more time we obtain $\\frac{d \\hat{\\rho }_T(t)}{dt} = -i \\alpha \\left[ \\hat{H}_I(t),\\hat{\\rho }_T(0) \\right] -\\alpha ^2 \\int _0^t ds \\left[ \\hat{H}_I(t),\\left[ \\hat{H}_I(s),\\hat{\\rho }_T(t) \\right] \\right] + O(\\alpha ^3).$ After this substitution, the integration of the previous states of the system is included only in the terms that are $O(\\alpha ^3)$ or higher.",
"At this moment, we perform our first approximation by considering that the strength of the interaction between the system and the environment is small.",
"Therefore, we can avoid high-orders in Eq.",
"(REF ).",
"Under this approximation we have $\\frac{d \\hat{\\rho }_T(t)}{dt} = -i \\alpha \\left[ \\hat{H}_I(t),\\hat{\\rho }_T(0) \\right] -\\alpha ^2 \\int _0^t ds \\left[ \\hat{H}_I(t),\\left[ \\hat{H}_I(s),\\hat{\\rho }_T(t) \\right] \\right].$ We are interested in finding an equation of motion for $\\rho $ , so we trace over the environment degrees of freedom $\\frac{d \\hat{\\rho }(t)}{dt}= \\textrm {Tr}_E \\left[\\frac{d \\hat{\\rho }_T(t)}{dt} \\right] = -i \\alpha \\textrm {Tr}_E\\left[ \\hat{H}_I(t),\\hat{\\rho }_T(0) \\right] -\\alpha ^2 \\int _0^t ds \\textrm {Tr}_E\\left[ \\hat{H}_I(t),\\left[ \\hat{H}_I(s),\\hat{\\rho }_T(t) \\right] \\right].$ This is not a closed time-evolution equation for $\\hat{\\rho }(t)$ , because the time derivative still depends on the full density matrix $\\hat{\\rho }_T(t)$ .",
"To proceed, we need to make two more assumptions.",
"First, we assume that $t=0$ the system and the environment have a separable state in the form $\\rho _T(0)=\\rho (0) \\otimes \\rho _E(0)$ .",
"This means that there are not correlations between the system and the environment.",
"This may be the case if the system and the environment have not interacted at previous times or if the correlations between them are short-lived.",
"Second, we assume that the initial state of the environment is thermal, meaning that it is described by a density matrix in the form $\\rho _E(0)=\\exp \\left( -H_E/T \\right)/\\textrm {Tr}[\\exp \\left( -H_E/T \\right)]$ , being $T$ the temperature and taking the Boltzmann constant as $k_B=1$ .",
"By using these assumptions, and the expansion of $H_I$ (REF ), we can calculate an expression for the first element of the r.h.s of Eq.",
"(REF ).",
"$\\textrm {Tr}_E\\left[ \\hat{H}_I(t),\\hat{\\rho }_T(0) \\right] = \\sum _i \\left( \\hat{S}_i(t) \\hat{\\rho }(0) \\textrm {Tr}_E \\left[ \\hat{E}_i(t) \\hat{\\rho }_E(0) \\right] -\\hat{\\rho }(0) \\hat{S}_i(t) \\textrm {Tr}_E \\left[ \\hat{\\rho }_E(0) \\hat{E}_i(t) \\right] \\right).$ To calculate the explicit value of this term, we may use that $\\left< E_i\\right>=\\textrm {Tr}[E_i \\rho _E(0)]=0$ for all values of $i$ .",
"This looks like a strong assumption, but it is not.",
"If our total Hamiltonian does not fulfil it, we can always rewrite it as $H_T=\\left( H+ \\alpha \\sum _i \\left< E_i\\right> S_i\\right) + H_E + \\alpha H_i^{\\prime }$ , with $H^{\\prime }_i= \\sum _i S_i \\otimes (E_i- \\left< E_i\\right>)$ .",
"It is clear that now $\\left< E^{\\prime }_i \\right>=0$ , with $E^{\\prime }_i=E_i- \\left< E_i\\right>$ , and the system Hamiltonian is changed just by the addition of an energy shift that does no affect the system dynamics.",
"Because of that, we can assume that $\\left< E_i\\right>=0$ for all $i$ .",
"Using the cyclic property of the trace, it is easy to prove that the term of Eq.",
"(REF ) is equal to zero, and the equation of motion (REF ) reduces to $\\dot{\\hat{\\rho }}(t)= -\\alpha ^2 \\int _0^t ds \\textrm {Tr}_E\\left[ \\hat{H}_I(t),\\left[ \\hat{H}_I(s),\\hat{\\rho }_T(t) \\right] \\right].$ This equation still includes the entire state of the system and environment.",
"To unravel the system from the environment, we have to make a more restrictive assumption.",
"As we are working in the weak coupling regime, we may suppose that the system and the environment are non-correlated during all the time evolution.",
"Of course, this is only an approximation.",
"Due to the interaction Hamiltonian, some correlations between system and environment are expected to appear.",
"On the other hand, we may assume that the timescales of correlation ($\\tau _\\text{corr}$ ) and relaxation of the environment ($\\tau _\\text{rel}$ ) are much smaller than the typical system timescale ($\\tau _\\text{sys}$ ), as the coupling strength is very small ($\\alpha <<$ ).",
"Therefore, under this strong assumption, we can assume that the environment state is always thermal and is decoupled from the system state, $\\hat{\\rho }_T(t)=\\hat{\\rho }(t) \\otimes \\hat{\\rho }_E(0)$ .",
"Eq.",
"(REF ) then transforms into $\\dot{\\hat{\\rho }}(t)= -\\alpha ^2 \\int _0^t ds \\textrm {Tr}_E\\left[ \\hat{H}_I(t),\\left[ \\hat{H}_I(s),\\hat{\\rho }(t) \\otimes \\hat{\\rho }_E(0) \\right] \\right].$ The equation of motion is now independent for the system and local in time.",
"It is still non-Markovian, as it depends on the initial state preparation of the system.",
"We can obtain a Markovian equation by realising that the kernel in the integration and that we can extend the upper limit of the integration to infinity with no real change in the outcome.",
"By doing so, and by changing the integral variable to $s\\rightarrow t-s$ , we obtain the famous Redfield equation .",
"$\\dot{\\hat{\\rho }}(t)= -\\alpha ^2 \\int _0^{\\infty } ds \\textrm {Tr}_E\\left[ \\hat{H}_I(t),\\left[ \\hat{H}_I(s-t),\\hat{\\rho }(t) \\otimes \\hat{\\rho }_E(0) \\right] \\right].$ It is known that this equation does not warrant the positivity of the map, and it sometimes gives rise to density matrices that are non-positive.",
"To ensure complete positivity, we need to perform one further approximation, the rotating wave approximation.",
"To do so, we need to use the spectrum of the superoperator $\\tilde{H}A\\equiv \\left[ H,A \\right]$ , $\\forall A\\in {\\cal B}({\\cal H})$ .",
"The eigenvectors of this superoperator form a complete basis of space ${\\cal B}({\\cal H})$ and, therefore, we can expand the system-environment operators from Eq.",
"(REF ) in this basis $S_i = \\sum _{\\omega } S_i(\\omega ),$ where the operators $S_i(\\omega )$ fulfils $\\left[ H,S_i(\\omega ) \\right]= -\\omega S_i(\\omega ),$ being $\\omega $ the eigenvalues of $\\tilde{H}$ .",
"It is easy to take also the Hermitian conjugated $\\left[ H,S_i^{\\dagger }(\\omega ) \\right]= \\omega S_i^{\\dagger }(\\omega ).$ To apply this decomposition, we need to change back to the Schrödinger picture for the term of the interaction Hamiltonian acting on the system's Hilbert space.",
"This is done by the expression $\\hat{S}_k= e^{it H} S_k e^{-it H}$ .",
"By using the eigen-expansion (REF ) we arrive to $\\tilde{H}_i(t) = \\sum _{k,\\omega } e^{-i\\omega t} S_k(\\omega ) \\otimes \\tilde{E}_k (t) = \\sum _{k,\\omega } e^{i\\omega t} S_k^{\\dagger }(\\omega ) \\otimes \\tilde{E}_k^{\\dagger } (t).$ To combine this decomposition with Redfield equation (REF ), we first may expand the commutators.",
"$\\hspace{-56.9055pt}\\dot{\\hat{\\rho }}(t)= -\\alpha ^2 \\textrm {Tr}\\left[ \\int _0^\\infty ds\\, \\hat{H}_I (t) \\hat{H}_I (t-s) \\hat{\\rho } (t) \\otimes \\hat{\\rho }_E(0)- \\int _0^\\infty ds\\, \\hat{H}_I (t) \\hat{\\rho } (t) \\otimes \\hat{\\rho }_E(0) \\hat{H}_I (t-s) \\right.",
"\\nonumber \\\\\\hspace{-56.9055pt}\\left.",
"- \\int _0^\\infty ds\\, \\hat{H}_I (t-s) \\hat{\\rho } (t) \\otimes \\hat{\\rho }_E(0) \\hat{H}_I (t)+ \\int _0^\\infty ds\\, \\hat{\\rho } (t) \\otimes \\hat{\\rho }_E(0) \\hat{H}_I (t-s) \\hat{H}_I (t)\\right].$ We now apply the eigenvalue decomposition in terms of $S_k(\\omega )$ for $\\hat{H}_I(t-s)$ and in terms of $S_k^{\\dagger }(\\omega ^{\\prime })$ for $\\hat{H}_I(t)$ .",
"By using the permutation property of the trace and the fact that $\\left[ H_E,\\rho _E(0) \\right]=0$ , and after some non-trivial algebra we obtain $\\dot{\\hat{\\rho }}(t) =\\sum _{\\begin{array}{c}\\omega ,\\omega ^{\\prime }\\\\ k,l \\end{array}} \\left( e^{i (\\omega ^{\\prime }-\\omega )t }\\, \\Gamma _{kl} (\\omega ) \\left[ S_l(\\omega )\\hat{\\rho } (t), S_k^\\dagger (\\omega ^{\\prime }) \\right] +e^{i (\\omega -\\omega ^{\\prime })t } \\, \\Gamma _{lk}^* (\\omega ^{\\prime }) \\left[ S_l(\\omega ), \\hat{\\rho } (t) S_k^\\dagger (\\omega ^{\\prime }) \\right] \\right),$ where the effect of the environment has been absorbed into the factors $\\Gamma _{kl} (\\omega ) \\equiv \\int _0^{\\infty } ds\\, e^{i\\omega s} \\textrm {Tr}_E \\left[ \\tilde{E}_k^\\dagger (t) \\tilde{E}_l (t-s) \\rho _E(0) \\right],$ where we are writing the environment operators of the interaction Hamiltonian in the interaction picture ($\\hat{E}_l(t)=e^{iH_Et} E_l e^{-iH_Et} $ ).",
"At this point, we can already perform the rotating wave approximation.",
"By considering the time-dependency on Eq.",
"(REF ), we conclude that the terms with $\\left| \\omega -\\omega ^{\\prime } \\right|>>\\alpha ^2$ will oscillate much faster than the typical timescale of the system evolution.",
"Therefore, they do not contribute to the evolution of the system.",
"In the low-coupling regime $(\\alpha \\rightarrow 0)$ we can consider that only the resonant terms, $\\omega =\\omega ^{\\prime }$ , contribute to the dynamics and remove all the others.",
"By applying this approximation to Eq.",
"(REF ) reduces to $\\dot{\\hat{\\rho }}(t) =\\sum _{\\begin{array}{c}\\omega \\\\ k,l \\end{array}} \\left( \\Gamma _{kl} (\\omega ) \\left[ S_l(\\omega )\\hat{\\rho } (t), S_k^\\dagger (\\omega ) \\right] +\\Gamma _{lk}^* (\\omega ) \\left[ S_l(\\omega ), \\hat{\\rho } (t) S_k^\\dagger (\\omega ) \\right] \\right).$ To divide the dynamics into Hamiltonian and non-Hamiltonian we now decompose the operators $\\Gamma _{kl}$ into Hermitian and non-Hermitian parts, $\\Gamma _{kl}(\\omega ) = \\frac{1}{2} \\gamma _{kl}(\\omega ) + i\\pi _{kl} $ , with $\\pi _{kl}(\\omega ) \\equiv \\frac{-i}{2} \\left(\\Gamma _{kl} (\\omega ) - \\Gamma _{kl}^*(\\omega ) \\right) \\nonumber \\\\\\gamma _{kl}(\\omega ) \\equiv \\Gamma _{kl} (\\omega ) + \\Gamma _{kl}^*(\\omega ) = \\int _{-\\infty }^{\\infty } ds e^{i\\omega s}\\textrm {Tr}\\left[ \\hat{E}_k^\\dagger (s) E_l \\hat{\\rho }_E(0)\\right].$ By these definitions we can separate the Hermitian and non-Hermitian parts of the dynamics and we can transform back to the Schrödinger picture $\\dot{\\rho }(t) = -i \\left[ H+H_{Ls} ,\\rho (t) \\right] +\\sum _{\\begin{array}{c}\\omega \\\\ k,l \\end{array}} \\gamma _{kl} (\\omega ) \\left( S_l (\\omega ) \\rho (t) S_k^\\dagger (\\omega ) -\\frac{1}{2} \\left\\lbrace S_k^\\dagger S_l (\\omega ) , \\rho (t) \\right\\rbrace \\right).$ The Hamiltonian dynamics now is influenced by a term $H_{Ls} = \\sum _{\\omega ,k,l} \\pi _{kl} (\\omega ) S_k^\\dagger (\\omega )S_l (\\omega )$ .",
"This is usually called a Lamb shift Hamiltonian and its role is to renormalize the system energy levels due to the interaction with the environment.",
"Eq.",
"(REF ) is the first version of the Markovian Master Equation, but it is not in the Lindblad form yet.",
"It can be easily proved that the matrix formed by the coefficients $\\gamma _{kl}(\\omega )$ is positive as they are the Fourier's transform of a positive function $\\left(\\textrm {Tr}\\left[ \\hat{E}_k^\\dagger (s) E_l \\hat{\\rho }_E(0)\\right]\\right)$ .",
"Therefore, this matrix can be diagonalised.",
"This means that we can find a unitary operator, $O$ , s.t.",
"$O\\gamma (\\omega ) O^\\dagger =\\left(\\begin{array}{cccc}d_1(\\omega ) & 0 & \\cdots & 0\\\\0 & d_2(\\omega ) & \\cdots & 0 \\\\\\vdots & \\vdots & \\ddots & 0\\\\0 & 0 & \\cdots & d_N(\\omega )\\end{array}\\right).$ We can now write the master equation in a diagonal form $\\dot{\\rho }(t) = -i \\left[ H+H_{Ls} ,\\rho (t) \\right] +\\sum _{i,\\omega } \\left( L_i (\\omega ) \\rho (t) L_i^\\dagger (\\omega ) -\\frac{1}{2} \\left\\lbrace L_i^\\dagger L_i (\\omega ) , \\rho (t) \\right\\rbrace \\right)\\equiv {\\cal L}\\rho (t).$ This is the celebrated Lindblad (or Lindblad-Gorini-Kossakowski-Sudarshan) Master Equation.",
"In the simplest case, there will be only one relevant frequency $\\omega $ , and the equation can be further simplified to $\\dot{\\rho }(t) = -i \\left[ H+H_{Ls} ,\\rho (t) \\right] +\\sum _{i} \\left( L_i \\rho (t) L_i^\\dagger -\\frac{1}{2} \\left\\lbrace L_i^\\dagger L_i, \\rho (t) \\right\\rbrace \\right)\\equiv {\\cal L}\\rho (t).$ The operators $L_i$ are usually referred to as jump operators."
],
[
"Derivation of the Lindblad Equation as a CPT generator",
"The second way of deriving Lindblad equation comes from the following question: What is the most general (Markovian) way of mapping density matrix onto density matrices?",
"This is usually the approach from quantum information researchers that look for general transformations of quantum systems.",
"We analyse this problem following mainly Ref.",
"[28].",
"To start, we need to know what is the form of a general CPT-map.",
"Lemma 2 Any map ${\\cal V}:B\\left( {\\cal H} \\right)\\rightarrow B\\left( {\\cal H} \\right)$ that can be written in the form ${\\cal V}\\rho =V^\\dagger \\rho V^{\\phantom{\\dagger }}$ with $V\\in B\\left( {\\cal H} \\right)$ is positive.",
"The proof of the lemma requires a little algebra and a known property of normal matrices Proof.",
"If $\\rho \\ge 0 \\Rightarrow \\rho =A^\\dagger A^{\\phantom{\\dagger }}$ , with $A\\in B({\\cal H})$ .",
"Therefore, ${\\cal V}\\rho =V^\\dagger \\rho V^{\\phantom{\\dagger }}\\Rightarrow \\langle {\\psi }\\vert V^\\dagger \\rho V\\vert {\\psi }\\rangle = \\langle {\\psi }\\vert V^\\dagger A^{\\dagger }A V\\vert {\\psi }\\rangle =\\left| \\left| AV\\vert {\\psi }\\rangle \\right|\\right|\\ge 0$ .",
"Therefore, if $\\rho $ is positive, the output of the map is also positive.",
"End of the proof.",
"This is a sufficient condition for the positivity of a map, but it is not necessary.",
"It could happen that there are maps that cannot be written in this form, but they are still positive.",
"To go further, we need a more general condition, and this comes in the form of the next theorem.",
"Theorem 1 Choi's Theorem.",
"A linear map ${\\cal V}:B({\\cal H})\\rightarrow B({\\cal H})$ is completely positive iff it can be expressed as ${\\cal V}\\rho =\\sum _i V_i^\\dagger \\rho V^{\\phantom{\\dagger }}_i$ with $V_i\\in B({\\cal H})$ .",
"The proof of this theorem requires some algebra.",
"Proof The `if' implication is a trivial consequence of the previous lemma.",
"To prove the converse, we need to extend the dimension of our system by the use of an auxiliary system.",
"If $d$ is the dimension of the Hilbert space of pure states, ${\\cal H}$ , we define a new Hilbert space of the same dimension ${\\cal H}_A$ .",
"We define a maximally entangled pure state in the bipartition ${\\cal H}_A \\otimes {\\cal H}$ in the way $\\vert {\\Gamma }\\rangle \\equiv \\sum _{i=0}^{d-1} \\vert {i}\\rangle _A \\otimes \\vert {i}\\rangle ,$ being $\\left\\lbrace \\vert {i}\\rangle \\right\\rbrace $ and ${\\left\\lbrace \\vert {i}\\rangle _A \\right\\rbrace }$ arbitrary orthonormal bases for ${\\cal H}$ and ${\\cal H}_A$ .",
"We can extend the action of our original map ${\\cal V}$ , that acts on ${\\cal B}({\\cal H})$ to our extended Hilbert space by defining the map ${\\cal V}_2:{\\cal B}( {\\cal H}_A) \\otimes {\\cal B}({\\cal H}) \\rightarrow {\\cal B}( {\\cal H}_A) \\otimes {\\cal B}({\\cal H})$ as ${\\cal V}_2\\equiv \\mathbb {1}_{{\\cal B}( {\\cal H}_A)} \\otimes {\\cal V}.$ Note that the idea behind this map is to leave the auxiliary subsystem invariant while applying the original map to the original system.",
"This map is positive because ${\\cal V}$ is completely positive.",
"This may appear trivial, but as it has been explained before complete positivity is a more restrictive property than positivity, and we are looking for a condition to ensure complete positivity.",
"We can now apply the extended map to the density matrix corresponding to the maximally entangled state (REF ), obtaining ${\\cal V}_2 \\vert {\\Gamma }\\rangle \\!\\langle {\\Gamma }\\vert = \\sum _{i,j=0}^{d-1} \\vert {i}\\rangle \\!\\langle {j}\\vert \\otimes {\\cal V}\\vert {i}\\rangle \\!\\langle {j}\\vert .$ Now we can use the maximal entanglement of the state $\\vert {\\Gamma }\\rangle $ to relate the original map ${\\cal V}$ and the action ${\\cal V}_2 \\vert {\\Gamma }\\rangle \\!\\langle {\\Gamma }\\vert $ by taking the matrix elements with respect to ${\\cal H}_A$ .",
"${\\cal V}\\vert {i}\\rangle \\!\\langle {j}\\vert = \\langle {i}\\vert _A \\left( {\\cal V}_2\\vert {\\Gamma }\\rangle \\!\\langle {\\Gamma }\\vert \\right)\\vert {j}\\rangle _A.$ To relate this operation to the action of the map to an arbitrary vector $\\vert {\\psi }\\rangle \\in {\\cal H}_A \\otimes {\\cal H}$ , we can expand it in this basis as $\\vert {\\psi }\\rangle = \\sum _{i=0}^{d-1} \\sum _{j=0}^{d-1} \\alpha _{ij} \\vert {i}\\rangle _A \\otimes \\vert {j}\\rangle .$ We can also define an operator $V_{\\vert {\\psi }\\rangle } \\in {\\cal B}\\left( {\\cal H} \\right)$ s.t.",
"it transforms $\\vert {\\Gamma }\\rangle $ into $\\vert {\\psi }\\rangle $ .",
"Its explicit action would be written as $\\hspace{-56.9055pt}\\left( \\mathbb {1}_A \\otimes V_{\\vert {\\psi }\\rangle } \\right) \\vert {\\Gamma }\\rangle =&\\sum _{i,j=0}^{d-1} \\alpha _{ij} \\left( \\mathbb {1}_A \\otimes \\vert {j}\\rangle \\!\\langle {i}\\vert \\right) \\left( \\sum _{k=0}^{d-1} \\vert {k}\\rangle \\otimes \\vert {k}\\rangle \\right)= \\sum _{i,j,k=0}^{d-1} \\alpha _{ij} \\left( \\vert {k}\\rangle \\otimes \\vert {j}\\rangle \\right) \\langle {i}\\vert {k}\\rangle \\nonumber \\\\&= \\sum _{i,j,k=0}^{d-q} \\alpha _{ij} \\left( \\vert {k}\\rangle \\otimes \\vert {j}\\rangle \\right) \\delta _{i,k} = \\sum _{i,j=0}^{d-1} \\alpha _{ij} \\vert {i}\\rangle \\otimes \\vert {j}\\rangle = \\vert {\\psi }\\rangle .$ At this point, we have related the vectors in the extended space ${\\cal H}_A \\otimes {\\cal H}$ to operators acting on ${\\cal H}$ .",
"This can only be done because the vector $\\vert {\\Gamma }\\rangle $ is maximally entangled.",
"We go now back to our extended map ${\\cal V}_2$ .",
"Its action on $\\vert {\\Gamma }\\rangle \\!\\langle {\\Gamma }\\vert $ is given by Eq.",
"(REF ) and as it is a positive map it can be expanded as ${\\cal V}_2\\left( \\vert {\\Gamma }\\rangle \\!\\langle {\\Gamma }\\vert \\right) =\\sum _{l=0}^{d^2-1}\\vert {v_l}\\rangle \\!\\langle {v_l}\\vert .$ with $\\vert {v_l}\\rangle \\in {\\cal H}_A\\otimes {\\cal H}$ .",
"The vectors $\\vert {v_l}\\rangle $ can be related to operators in ${\\cal H}$ as in Eq.",
"(REF ).",
"$\\vert {v_l}\\rangle =\\left( \\mathbb {1}_A\\otimes \\ V_l \\right)\\vert {\\Gamma }\\rangle .$ Based on this result we can calculate the product of an arbitrary vector $\\vert {i}\\rangle _A\\in {\\cal H}_A$ with $\\vert {v_l}\\rangle $ .",
"$\\langle {i}\\vert _A \\vert {v_l}\\rangle =\\langle {i}\\vert _A \\left( \\mathbb {1}_A \\otimes V_l \\right) \\vert {\\Gamma }\\rangle =V_l \\sum _{k=0}^{d-1} \\langle {i}\\vert {k}\\rangle _A \\otimes \\vert {k}\\rangle .$ This is the last ingredient we need for the proof.",
"We come back to the original question, we want to characterise the map ${\\cal V}$ .",
"We do so by applying it to an arbitrary basis element $\\vert {i}\\rangle \\!\\langle {j}\\vert $ of ${\\cal B}\\left( {\\cal H} \\right)$ .",
"$\\hspace{-56.9055pt}{\\cal V}\\left( \\vert {i}\\rangle \\!\\langle {j}\\vert \\right) = \\left( \\langle {i}\\vert _A \\otimes \\mathbb {1}_A \\right) {\\cal V}_2\\left( \\vert {\\Gamma }\\rangle \\!\\langle {\\Gamma }\\vert \\right) \\left( \\vert {j}\\rangle _A\\otimes \\mathbb {1}_A \\right)= \\left( \\langle {i}\\vert _A \\otimes \\mathbb {1}_A \\right) \\left[ \\sum _{l=0}^{d^2-1} \\vert {v_l}\\rangle \\!\\langle {v_l}\\vert \\right] \\left( \\vert {j}\\rangle _A\\otimes \\mathbb {1}_A \\right) \\nonumber \\\\= \\sum _{l=0}^{d^2-1} \\left[ \\left( \\langle {i}\\vert _A \\otimes \\mathbb {1}_A \\right)\\vert {v_l}\\rangle \\right] \\left[ \\langle {v_l}\\vert \\left( \\vert {j}\\rangle _A \\otimes \\mathbb {1}_A \\right) \\right]= \\sum _{l=0}^{d^2-1} V_l\\vert {i}\\rangle \\!\\langle {j}\\vert V_l.$ As $\\vert {i}\\rangle \\!\\langle {j}\\vert $ is an arbitrary element of a basis any operator can be expanded in this basis.",
"Therefore, it is straightforward to prove that ${\\cal V}\\rho =\\sum _l^{d^2-l} V^{\\dagger }_l\\rho V^{\\phantom{\\dagger }}_l.\\nonumber $ End of the proof.",
"Thanks to Choi's Theorem, we know the general form of CP-maps, but there is still an issue to address.",
"As density matrices should have trace one, we need to require any physical maps to be also trace-preserving.",
"This requirement gives as a new constraint that completely defines all CPT-maps.",
"This requirement comes from the following theorem.",
"Theorem 2 Choi-Kraus' Theorem.",
"A linear map ${\\cal V}:B({\\cal H})\\rightarrow B({\\cal H})$ is completely positive and trace-preserving iff it can be expressed as ${\\cal V}\\rho =\\sum _l V_l^\\dagger \\rho V^{\\phantom{\\dagger }}_l$ with $V_l\\in B({\\cal H})$ fulfilling $\\sum _l V^{\\phantom{\\dagger }}_l V_l^\\dagger =\\mathbb {1}_{{\\cal H}}.$ Proof.",
"We have already proved that this is a completely positive map, we only need to prove that it is also trace-preserving and that all trace preserving-maps fulfil Eq.",
"(REF ).",
"The `if' proof is quite simple by applying the cyclic permutations and linearity properties of the trace operator.",
"$\\textrm {Tr}\\left[ {\\cal V}\\rho \\right]=\\textrm {Tr}\\left[ \\sum _{l=1}^{d^2-1} V^{\\phantom{\\dagger }}_l\\rho V_l^\\dagger \\right] = \\textrm {Tr}\\left[ \\left( \\sum _{l=1}^{d^2-1} V_l^\\dagger V^{\\phantom{\\dagger }}_l \\right)\\rho \\right] =\\textrm {Tr}\\left[ \\rho \\right].$ We have to prove also that any map in the form (REF ) is trace-preserving only if the operators $V_l$ fulfil (REF ).",
"We start by stating that if the map is trace-preserving by applying it to an any arbitrary element of a basis of ${\\cal B}\\left( {\\cal H} \\right)$ we should obtain $\\textrm {Tr}\\left[ {\\cal V}\\left( \\vert {i}\\rangle \\!\\langle {j}\\vert \\right) \\right]=\\textrm {Tr}\\left[ \\vert {i}\\rangle \\!\\langle {j}\\vert \\right]=\\delta _{i,j}.$ As the map has a form given by (REF ) we can calculate this same trace in an alternative way.",
"$\\textrm {Tr}\\left[ {\\cal V}\\left( \\vert {i}\\rangle \\!\\langle {j}\\vert \\right) \\right] &=& \\textrm {Tr}\\left[ \\sum _{l=1}^{d^2-1} V^{\\phantom{\\dagger }}_l \\vert {i}\\rangle \\!\\langle {j}\\vert V_l^\\dagger \\right]= \\textrm {Tr}\\left[ \\sum _{l=1}^{d^2-1} V_l^\\dagger V^{\\phantom{\\dagger }}_l \\vert {i}\\rangle \\!\\langle {j}\\vert \\right] \\nonumber \\\\&= &\\sum _{k} \\langle {k}\\vert \\left( \\sum _{l=1}^{d^2-1} V_l^\\dagger V^{\\phantom{\\dagger }}_l \\vert {i}\\rangle \\!\\langle {j}\\vert \\right) \\vert {k}\\rangle = \\langle {j}\\vert \\left( \\sum _{l=1}^{d^2-1} V_l^\\dagger V^{\\phantom{\\dagger }}_l \\right)\\vert {i}\\rangle ,$ where $\\left\\lbrace \\vert {k}\\rangle \\right\\rbrace $ is an arbitrary basis of ${\\cal H}$ .",
"As both equalities should be right we obtain $\\langle {j}\\vert \\left( \\sum _{l=1}^{d^2-1} V^{\\phantom{\\dagger }}_l V^\\dagger _l \\right)\\vert {i}\\rangle = \\delta _{i,j},$ and therefore, the condition (REF ) should be fulfilled.",
"End of the proof.",
"Operators $V_i$ of a map fulfilling condition (REF ) are called Krauss operators.",
"Because of that, sometimes CPT-maps are also called Krauss maps, especially when they are presented as a collection of Krauss operators.",
"Both concepts are ubiquitous in quantum information science.",
"Krauss operators can also be time-dependent as long as they fulfil relation (REF ) for all times.",
"At this point, we already know the form of CPT-maps, but we do not have a master equation, that is a continuous set of differential equations.",
"This means that we know how to perform an arbitrary operation in a system, but we do not have an equation to describe its time evolution.",
"To do so, we need to find a time-independent generator ${\\cal L}$ such that $\\frac{d}{dt} \\rho \\left( t \\right)= {\\cal L}\\rho (t),$ and therefore our CPT-map could be expressed as ${\\cal V}(t)=e^{{\\cal L}t}$ .",
"The following calculation is about founding the explicit expression of ${\\cal L}$ .",
"We start by choosing an orthonormal basis of the bounded space of operators ${\\cal B}({\\cal H})$ , $\\left\\lbrace F_i \\right\\rbrace _{i=1}^{d^2}$ .",
"To be orthonormal it should satisfy the following condition $\\langle \\langle {F_i}\\vert {F_j}\\rangle \\rangle \\equiv \\textrm {Tr}\\left[ F_i^\\dagger F_j \\right]=\\delta _{i,j}.$ Without any loss of generality, we select one of the elements of the basis to be proportional to the identity, $F_{d^2}=\\frac{1}{\\sqrt{d}} \\mathbb {1}_{{\\cal H}}$ .",
"It is trivial to prove that the norm of this element is one, and it is easy to see from Eq.",
"(REF ) that all the other elements of the basis should have trace zero.",
"$\\textrm {Tr}\\left[ F_i \\right]=0 \\qquad \\forall i=1,\\dots ,d^2-1.$ The closure relation of this basis is $\\mathbb {1}_{{\\cal B}({\\cal H})}=\\sum _i \\vert {F_i}\\rangle \\rangle \\!\\langle \\langle {F_i}\\vert $ .",
"Therefore, the Krauss operators can be expanded in this basis by using the Fock-Liouville notation $V_l(t)= \\sum _{i=1}^{d^2} \\langle \\langle {F_i}\\vert {V_l(t)}\\rangle \\rangle \\vert {F_i}\\rangle \\rangle .$ As the map ${\\cal V}(t)$ is in the form (REF ) we can apply (REF ) to obtainFor simplicity, in this discussion we omit the explicit time-dependency of the density matrix.. $\\hspace{-56.9055pt}{\\cal V}(t) \\rho = \\sum _l\\left[ \\sum _{i=1}^{d^2} \\langle \\langle {F_i}\\vert {V_l(t)}\\rangle \\rangle F_i \\;\\rho \\sum _{j=1}^{d^2} F_j^\\dagger \\langle \\langle {V_l(t)}\\vert {F_j}\\rangle \\rangle \\right]= \\sum _{i,j=1}^{d^2} c_{i,j}(t) F_i^{\\phantom{\\dagger }}\\rho F_j^{\\dagger },$ where we have absorved the sumation over the Krauss operators in the terms $c_{i,j}(t)= \\sum _l \\langle \\langle {F_i}\\vert {V_l}\\rangle \\rangle \\langle \\langle {V_l}\\vert {F_j}\\rangle \\rangle $ .",
"We go back now to the original problem by applying this expansion into the time-derivative of Eq.",
"(REF ) $\\frac{d \\rho }{dt} &=&\\lim _{\\Delta t\\rightarrow 0} \\frac{1}{\\Delta t} \\left( {\\cal V}(\\Delta t)\\rho -\\rho \\right)= \\lim _{\\Delta t\\rightarrow 0} \\left( \\sum _{i,j=1}^{d^2} c_{i,j}(\\Delta t) F_i^{\\phantom{\\dagger }} \\rho F_j^\\dagger - \\rho \\right) \\nonumber \\\\&=& \\lim _{\\Delta t\\rightarrow 0} \\left( \\sum _{i,j=0}^{d^2-1} c_{i,j}(\\Delta t) F_i^{\\phantom{\\dagger }}\\rho F_j^\\dagger + \\sum _{i=1}^{d^2-1} c_{i,d^2} F_i^{\\phantom{\\dagger }}\\rho F_{d^2}^\\dagger \\right.",
"\\nonumber \\\\&&\\left.",
"+ \\sum _{j=1}^{d^2-1} c_{d^2,j} (\\Delta t) F_{d^2}^{\\phantom{\\dagger }} \\rho F_j^\\dagger + c_{d^2,d^2}(\\Delta t) F_{d^2}^{\\phantom{\\dagger }} \\rho F_{d^2}^\\dagger - \\rho \\right),$ where we have separated the summations to take into account that $F_{d^2}=\\frac{1}{\\sqrt{d}}\\mathbb {1}_{{\\cal H}}$ .",
"By using this property this equation simplifies to $\\frac{d \\rho }{dt} &=&\\lim _{\\Delta t\\rightarrow 0} \\frac{1}{\\Delta t} \\left( \\sum _{i,j=1}^{d^2-1} c_{i,j}(\\Delta t) F_i^{\\phantom{\\dagger }} \\rho F_j^\\dagger + \\frac{1}{\\sqrt{d}} \\sum _{i=1}^{d^2-1}c_{i,d^2}(\\Delta t) F_i^{\\phantom{\\dagger }}\\rho \\right.",
"\\nonumber \\\\&+&\\left.",
"\\frac{1}{\\sqrt{d}} \\sum _{j=1}^{d^2-1} c_{d^2,j} (\\Delta t) \\rho F_j^\\dagger + \\frac{1}{d} c_{d^2,d^2}(\\Delta t) \\rho -\\rho \\right).$ The next step is to eliminate the explicit dependence with time.",
"To do so, we define new constants to absorb all the time intervals.",
"$g_{i,j}&\\equiv & \\lim _{\\Delta t\\rightarrow 0} \\frac{c_{i,j} (\\Delta t) }{\\Delta t} \\qquad (i,j<d^2), \\nonumber \\\\g_{i,d^2}&\\equiv & \\lim _{\\Delta t \\rightarrow 0} \\frac{c_{i,d^2} (\\Delta t) }{\\Delta t} \\qquad (i<d^2), \\nonumber \\\\g_{d^2,j}&\\equiv & \\lim _{\\Delta t \\rightarrow 0} \\frac{c_{d^2,j} (\\Delta t) }{\\Delta t} \\qquad (j<d^2), \\\\g_{d^2,d^2}&\\equiv & \\lim _{\\Delta t\\rightarrow 0} \\frac{c_{d^2,d^2}(\\Delta t)-d}{\\Delta t}.",
"\\nonumber $ Introducing these coefficients in Eq (REF ) we obtain an equation with no explicit dependence in time.",
"$\\frac{d \\rho }{dt} = \\sum _{i,j=1}^{d^2-1} g_{i,j} F_i \\rho F_j^\\dagger + \\frac{1}{\\sqrt{d}} \\sum _{i=1}^{d^2-1} g_{i,d^2} F_i \\rho + \\frac{1}{\\sqrt{d}} \\sum _{j=1}^{d^2-1} g_{d^2,j} \\rho F_j^\\dagger + \\frac{g_{d^2,d^2}}{d} \\rho .\\nonumber \\\\$ As we are already summing up over all the Krauss operators it is useful to define a new operator $F\\equiv \\frac{1}{\\sqrt{d}} \\sum _{i=1}^{d^2-1} g_{i,d^2} F_i.$ Applying it to Eq.",
"(REF ).",
"$\\frac{d \\rho }{dt} = \\sum _{i,j=1}^{d^2-1} g_{i,j} F_i \\rho F_j^\\dagger + F \\rho + \\rho F^\\dagger + \\frac{g_{d^2,d^2}}{d} \\rho .$ At this point, we want to separate the dynamics of the density matrix into a Hermitian (equivalent to von Neunmann equation) and an incoherent part.",
"We split the operator $F$ in two to obtain a Hermitian and anti-Hermitian part.",
"$F=\\frac{F+F^\\dagger }{2} + i\\frac{F-F^\\dagger }{2i} \\equiv G-iH,$ where we have used the notation $H$ for the Hermitian part for obvious reasons.",
"If we take this definition to Eq.",
"(REF ) we obtain $\\frac{d \\rho }{dt} = g_{i,j} F_i \\rho F_j^\\dagger + \\left\\lbrace G,\\rho \\right\\rbrace - i \\left[ H,\\rho \\right] + \\frac{g_{d^2,d^2}}{d} \\rho .$ We define now the last operator for this proof, $G_2\\equiv G+\\frac{g_{d^2,d^2}}{2d}$ , and the expression of the time derivative leads to $\\frac{d \\rho }{dt} = \\sum _{i,j=1}^{d^2-1} g_{i,j} F_i \\rho F_j^\\dagger + \\left\\lbrace G_2,\\rho \\right\\rbrace -i\\left[ H,\\rho \\right].$ Until now we have imposed the complete positivity of the map, as we have required it to be written in terms of Krauss maps, but we have not used the trace-preserving property.",
"We impose now this property, and by using the cyclic property of the trace, we obtain a new condition $\\textrm {Tr}\\left[ \\frac{d \\rho }{dt} \\right]=\\textrm {Tr}\\left[ \\sum _{i,j=1}^{d^2-1} F_j^\\dagger F_i \\rho + 2 G_2 \\rho \\right]=0.$ Therefore, $G_2$ should fulfil $G_2=\\frac{1}{2} \\sum _{i,j=1}^{d^2-1} g_{i,j} F_j^\\dagger F_i\\rho .$ By applying this condition, we arrive at the Lindblad master equation $\\frac{d \\rho }{dt}= -i\\left[ H,\\rho \\right] + \\sum _{i,j=1}^{d^2-1} g_{i,j} \\left( F_i^{\\phantom{\\dagger }}\\rho F_j^\\dagger - \\frac{1}{2} \\left\\lbrace F_j^\\dagger F_i^{\\phantom{\\dagger }},\\rho \\right\\rbrace \\right).$ Finally, by definition the coefficients $g_{i,j}$ can be arranged to form a Hermitian, and therefore diagonalisable, matrix.",
"By diagonalising it, we obtain the diagonal form of the Lindblad master equation.",
"$\\frac{d}{dt} \\rho = -i \\left[ H,\\rho \\right] + \\sum _{k} \\Gamma _k \\left( L_k^{\\phantom{\\dagger }} \\rho L_k^\\dagger - \\frac{1}{2} \\left\\lbrace L_k^{\\phantom{\\dagger }} L_k^\\dagger ,\\rho \\right\\rbrace \\right) \\equiv {\\cal L}\\rho .$"
],
[
"Properties of the Lindblad Master Equation",
"Some interesting properties of the Lindblad equation are: Under a Lindblad dynamics, if all the jump operators are Hermitian, the purity of a system fulfils $\\frac{d}{dt}\\left( \\textrm {Tr}\\left[ \\rho ^2 \\right] \\right) \\le 0$ .",
"The proof is given in .",
"The Lindblad Master Equation is invariant under unitary transformations of the jump operators $\\sqrt{\\Gamma _i} L_i\\rightarrow \\sqrt{\\Gamma ^{\\prime }_i} L_i^{\\prime }= \\sum _j v_{ij} \\sqrt{\\Gamma _j} L_j,$ with $v$ representing a unitary matrix.",
"It is also invariant under inhomogeneous transformations in the form $L_i &\\rightarrow & L^{\\prime }_i= L_i + a_i \\nonumber \\\\H&\\rightarrow & H^{\\prime }=H+\\frac{1}{2i} \\sum _j \\Gamma _j \\left( a_j^* A_j - a_j A_j^\\dagger \\right)+ b,$ where $a_i \\in \\mathbb {C}$ and $b \\in \\mathbb {R}$ .",
"The proof of this can be found in Ref.",
"[2] (Section 3).",
"Thanks to the previous properties it is possible to find traceless jump operators without loss of generality.",
"Box 6.",
"A master equation for a two-level system with decay.",
"Continuing our example of a two-level atom, we can make it more realistic by including the possibility of atom decay by the emission of a photon.",
"This emission happens due to the interaction of the atom with the surrounding vacuum stateThis is why atoms decay..",
"The complete quantum system would be in this case the `atom+vacuum' system and its time evolution should be given by the von Neumann equation (REF ), where $H$ represents the total `atom+vacuum' Hamiltonian.",
"This system belongs to an infinite-dimension Hilbert space, as the radiation field has infinite modes.",
"If we are interested only in the time dependence state of the atom, we can derive a Markovian master equation for the reduced density matrix of the atom (see for instance Refs.",
"[2], [1]).",
"The master equation we will study is $\\frac{d}{dt}\\rho (t) = -i\\left[ H,\\rho \\right] + \\Gamma \\left( \\sigma ^- \\rho \\sigma ^+ -\\frac{1}{2} \\left\\lbrace \\sigma ^+ \\sigma ^-,\\rho \\right\\rbrace \\right),$ where $\\Gamma $ is the coupling between the atom and the vacuum.",
"In the Fock-Liouvillian space (following the same ordering as in Eq.",
"(REF )) the Liouvillian corresponding to evolution (REF ) is ${\\cal L}=\\left(\\begin{array}{cccc}0 & i \\Omega & -i\\Omega & \\Gamma \\\\i\\Omega & -i E - \\frac{\\Gamma }{2} & 0 & -i\\Omega \\\\-i\\Omega & 0 & -i E -\\frac{\\Gamma }{2} & i\\Omega \\\\0 & -i\\Omega & i\\Omega & -\\Gamma \\\\\\end{array}\\right).$ Expressing explicitly the set of differential equations we obtain $\\dot{\\rho }_{00} &= & i \\Omega \\rho _{01} -i\\Omega \\rho _{10} + \\Gamma \\rho _{11} \\nonumber \\\\\\dot{\\rho }_{01} &=& i\\Omega \\rho _{00} - \\left( iE - \\frac{\\Gamma }{2} \\right) \\rho _{01} -i\\Omega \\rho _{11} \\nonumber \\\\\\dot{\\rho }_{10} &=& -i\\Omega \\rho _{00} \\left( -i E - \\frac{\\Gamma }{2} \\right) \\rho _{10} + i\\Omega \\rho _{11} \\\\\\dot{\\rho }_{10} &=& -i\\Omega \\rho _{01} + i\\Omega \\rho _{10} -\\Gamma \\rho _{11} \\nonumber $"
],
[
"Integration",
"To calculate the time evolution of a system determined by a Master Equation in the form (REF ) we need to solve a set of equations with as many equations as the dimension of the density matrix.",
"In our example, this means to solve a 4 variable set of equations, but the dimension of the problem increases exponentially with the system size.",
"Because of this, for bigger systems techniques for dimension reduction are required.",
"To solve systems of partial differential equations there are several canonical algorithms.",
"This can be done analytically only for a few simple systems and by using sophisticated techniques as damping bases [29].",
"In most cases, we have to rely on numerical approximated methods.",
"One of the most popular approaches is the 4th-order Runge-Kutta algorithm (see, for instance, [30] for an explanation of the algorithm).",
"By integrating the equations of motion, we can calculate the density matrix at any time $t$ .",
"The steady-state of a system can be obtained by evolving it for a long time $\\left( t \\rightarrow \\infty \\right)$ .",
"Unfortunately, this method presents two difficulties.",
"First, if the dimension of the system is big, the computing time would be huge.",
"This means that for systems beyond a few qubits, it will take too long to reach the steady-state.",
"Even worse is the problem of stability of the algorithms for integrating differential equations.",
"Due to small errors in the calculation of derivatives by the use of finite differences, the trace of the density matrix may not be constantly equal to one.",
"This error accumulates during the propagation of the state, giving non-physical results after a finite time.",
"One solution to this problem is the use of algorithms specifically designed to preserve the trace, as Crank-Nicholson algorithm [31].",
"The problem with this kind of algorithms is that they consume more computational power than Runge-Kutta, and therefore they are not useful to calculate the long-time behaviour of big systems.",
"An analysis of different methods and their advantages and disadvantages can be found at Ref.",
"[32].",
"Box 7.",
"Time dependency of the two-level system with decay.",
"In this box we show some results of solving Eq (REF ) and calculating the density matrix as a function of time.",
"A Mathematica notebook solving this problem can be found at [20].",
"To illustrate the time behaviour of this system, we calculate the evolution for different state parameters.",
"In all cases, we start with an initial state that represents the state being excited $\\rho _{11}=1$ , with no coherence between different states, meaning $\\rho _{01}=\\rho _{10}=0$ .",
"If the decay parameter $\\Gamma $ is equal to zero, the problem reduces to solve von Neumann equation, and the result is displayed in Figure .",
"The other extreme case would be a system with no coherent dynamics ($\\Omega =0$ ) but with decay.",
"In this case, we observe an exponential decay of the population of the excited state.",
"Finally, we can calculate the dynamics of a system with both coherent driving and decay.",
"In this case, both behaviours coexist, and there are oscillations and decay.",
"Figure: NO_CAPTIONFigure: NO_CAPTION$\\Gamma =0.1,\\; n=1,\\; \\Omega =0,\\; E=1$ ).",
"Right: Population dynamics under both coherent and incoherent dynamics ($\\Gamma =0.1,\\; n=1,\\; \\Omega =1,\\; E=1)$ .",
"In both the blue lines represent $\\rho _{11}$ and the orange one $\\rho _{00}$ ."
],
[
"Diagonalisation",
"As we have discussed before, in the Fock-Liouville space the Liouvillian corresponds to a complex matrix (in general complex, non-hermitian, and non-symmetric).",
"By diagonalising it we can calculate both the time-dependent and the steady-state of the density matrices.",
"For most purposes, in the short time regime integrating the differential equations may be more efficient than diagonalising.",
"This is due to the high dimensionality of the Liouvillian that makes the diagonalisation process very costly in computing power.",
"On the other hand, in order to calculate the steady-state, the diagonalisation is the most used method due to the problems of integrating the equation of motions discussed in the previous section.",
"Let see first how we use diagonalisation to calculate the time evolution of a system.",
"As the Liouvillian matrix is non-Hermitian, we cannot apply the spectral theorem to it, and it may have different left and right eigenvectors.",
"For a specific eigenvalue $\\Lambda _i$ we can obtain the eigenvectors $\\vert {\\Lambda _i^R}\\rangle \\rangle $ and $\\vert {\\Lambda _i^L}\\rangle \\rangle $ s. t. $\\hspace{56.9055pt}\\tilde{{\\cal L}} \\; \\vert {\\Lambda _i^R}\\rangle \\rangle = \\Lambda _i \\vert {\\Lambda _i^R}\\rangle \\rangle \\nonumber \\\\\\hspace{56.9055pt}\\langle \\langle {\\Lambda _i^L}\\vert \\; \\tilde{{\\cal L}} = \\Lambda _i \\langle \\langle {\\Lambda _i^L}\\vert $ An arbitrary system can be expanded in the eigenbasis of $\\tilde{{\\cal L}}$ as [33], [1] $\\vert {\\rho (0)}\\rangle \\rangle = \\sum _i \\vert {\\Lambda _i^R}\\rangle \\rangle \\langle \\langle {\\Lambda _i^L}\\vert {\\rho (0)}\\rangle \\rangle .$ Therefore, the state of the system at a time $t$ can be calculated in the form $\\vert {\\rho (t)}\\rangle \\rangle = \\sum _i e^{\\Lambda _i t} \\vert {\\Lambda _i^R}\\rangle \\rangle \\langle \\langle {\\Lambda _i^L}\\vert {\\rho (0)}\\rangle \\rangle .$ Note that in this case to calculate the state a time $t$ we do not need to integrate into the interval $\\left[ 0,t \\right]$ , as we have to do if we use a numerical solution of the differential set of equations.",
"This is an advantage when we want to calculate long-time behaviour.",
"Furthermore, to calculate the steady-state of a system, we can look to the eigenvector that has zero eigenvalue, as this is the only one that survives when $t\\rightarrow \\infty $ .",
"For any finite system, Evans' Theorem ensures the existence of at least one zero eigenvalue of the Liouvillian matrix [34], [35].",
"The eigenvector corresponding to this zero eigenvalue would be the steady-state of the system.",
"In exceptional cases, a Liouvillian can present more than one zero eigenvalues due to the presence of symmetry in the system [36], [27], [26].",
"This is a non-generic case, and for most purposes, we can assume the existence of a unique fixed point in the dynamics of the system.",
"Therefore, diagonalising can be used to calculate the steady-state without calculating the full evolution of the system.",
"This can be done even analytically for small systems, and when numerical approaches are required this technique gives better precision than integrating the equations of motion.",
"The spectrum of Liouvillian superoperators has been analysed in several recent papers [37], [33].",
"Box 8.",
"Spectrum-analysis of the Liouvillian for the two-level system with decay.",
"Here we diagonalise (REF ) and obtain its steady state.",
"A Mathematica notebook solving this problem can be downloaded from [20].",
"This specific case is straightforward to diagonalize as the dimension of the system is very low.",
"We obtain 4 different eigenvalues, two of them are real while the other two form a conjugated pair.",
"Figure REF sisplays the spectrum of the superoperator ${\\cal L}$ given in (REF ).",
"Figure: NO_CAPTIONREF ) for the general case of both coherent and incoherent dynamics ($\\Gamma =0.2,\\; n=1,\\; \\Omega =0,\\; E=1$ ).",
"As there only one zero eigenvalue we can conclude that there is only one steady-state, and any initial density matrix will evolve to it after an infinite-time evolution.",
"By selecting the right eigenvector corresponding to the zero-eigenvalue and normalizing it we obtain the density matrix.",
"This can be done even analytically.",
"The result is the matrix: $\\rho _{SS}=\\left(\\begin{array}{cc}\\frac{(1+n) \\left(4\\, E^2+(\\Gamma +2 n\\, \\Gamma )^2\\right)+4 (1+2 n) \\Omega ^2}{(1+2 n) \\left(4 \\,E^2+(\\Gamma +2 n \\,\\Gamma )^2+8 \\Omega ^2\\right)} & \\frac{2 (-2 \\,E-i (\\Gamma +2 n \\Gamma )) \\Omega }{(1+2 n) \\left(4 \\,E^2+(\\Gamma +2 n \\,\\Gamma )^2+8 \\,\\Omega ^2\\right)} \\\\\\frac{2 (-2 \\,E+i (\\Gamma +2 n\\, \\Gamma )) \\Omega }{(1+2 n) \\left(4\\, E^2+(\\Gamma +2 n\\, \\Gamma )^2+8 \\Omega ^2\\right)} & \\frac{n \\left(4E^2+(\\Gamma +2 n \\Gamma )^2\\right)+4 (1+2 n) \\Omega ^2}{(1+2 n) \\left(4\\, E^2+(\\Gamma +2 n \\Gamma )^2+8\\, \\Omega ^2\\right)} \\\\\\end{array}\\right)$"
],
[
"Acknowledgements",
"The author wants to acknowledge the Spanish Ministry and the Agencia Española de Investigación (AEI) for financial support under grant FIS2017-84256-P (FEDER funds)."
],
[
" Proof of $\\frac{d}{dt} \\textrm {Tr}\\left[ \\rho ^2 \\right] \\le 0$",
"In this appendix we proof that under the Lindblad dynamics given by Eq.",
"(REF ) the purity of a density matrix fulfils that $\\frac{d}{dt} \\textrm {Tr}\\left[ \\rho ^2 \\right] \\le 0$ if all the jump operators of the Lindblad dynamics are Hermitian.",
"We start just by interchanging the trace and the derivative.",
"As the trace is a linear operation it commutes with the derivation, and we have $\\frac{d}{dt}\\left( \\textrm {Tr}\\left[ \\rho ^2 \\right] \\right) = \\textrm {Tr}\\left[ \\frac{ d \\rho ^2}{dt} \\right] = \\textrm {Tr}\\left[ 2 \\rho \\dot{\\rho } \\right],$ where we have used the cyclic property of the trace operatorThis property is used along all the demonstration without explicitly mentioning it.. By inserting the Lindblad Eq.",
"(REF ) into the r.h.s of (REF ) we obtain $\\frac{d}{dt}\\left( \\textrm {Tr}\\left[ \\rho ^2 \\right] \\right) &=& - \\frac{i}{\\hbar } \\textrm {Tr}\\left[ \\left( 2\\rho \\left( H\\rho -\\rho H \\right) \\right) \\right] \\nonumber \\\\&+& 2 \\sum _k \\Gamma _k \\textrm {Tr}\\left[ \\rho \\, L_k^{\\phantom{\\dagger }} \\, \\rho \\, L_k^\\dagger \\right] - 2\\sum _k \\Gamma _k \\textrm {Tr}\\left[ \\rho ^2 L_k^\\dagger L_k^{\\phantom{\\dagger }} \\right].$ The first term is zero.",
"Therefore, the inequality we want to prove becomes equivalent to $\\sum _k \\Gamma _k \\textrm {Tr}\\left[ \\rho \\, L_k^{\\phantom{\\dagger }}\\, \\rho \\, L_k^\\dagger \\right] \\le \\sum _k \\Gamma _k \\textrm {Tr}\\left[ \\rho ^2 L_k^\\dagger L_k^{\\phantom{\\dagger }} \\right]$ As the density matrix is Hermitian we can diagonalize it to obtain its eigenvalues ($\\Lambda _i \\in \\mathbb {R}$ ) and its corresponding eigenvectors ($\\vert {\\Lambda _i}\\rangle $ ).",
"The density matrix is diagonal in its own eigenbasis and can be expressed asThis eigenbasis changes with time, of course, but the proof is valid as the inequality should be fulfilled at any time.",
"$\\rho \\rightarrow \\tilde{\\rho }=\\sum _i \\Lambda _i \\vert {\\Lambda _i}\\rangle \\!\\langle {\\Lambda _i}\\vert ,$ where we assume an ordering of the eigenvalues in the form $\\Lambda _0\\ge \\Lambda _1 \\ge \\cdots \\ge \\Lambda _d$ .",
"We rename the jump operators in this basis as $\\tilde{L}_i$ a.",
"Expanding each term of the inequality (REF ) in this basis we obtain $\\sum _k \\Gamma _k \\textrm {Tr}\\left[ \\rho \\, L_k\\, \\rho \\, L_k^\\dagger \\right] = \\sum _k \\Gamma _k \\textrm {Tr}\\left[ \\left( \\sum _i \\Lambda _i \\vert {\\Lambda _i}\\rangle \\!\\langle {\\Lambda _i}\\vert \\right) \\tilde{L}_k \\left( \\sum _j \\Lambda _j \\vert {\\Lambda _j}\\rangle \\!\\langle {\\Lambda _j}\\vert \\right) \\tilde{L}_k \\right] \\nonumber \\\\=\\sum _k \\Gamma _k \\sum _{i,j} \\Lambda _i \\Lambda _j \\textrm {Tr}\\left[ \\tilde{L}_k^\\dagger \\vert {\\Lambda _i}\\rangle \\!\\langle {\\Lambda _i}\\vert \\tilde{L}_k \\vert {\\Lambda _j}\\rangle \\!\\langle {\\Lambda _j}\\vert \\right]=\\sum _k \\Gamma _k \\sum _{i,j} \\Lambda _i \\Lambda _j \\textrm {Tr}\\left[ \\left| \\langle {\\Lambda _i}\\vert \\tilde{L}_k \\vert {\\Lambda _j}\\rangle \\right|^2 \\right]\\nonumber \\\\= \\sum _k \\Gamma _k \\sum _{i,j} \\Lambda _i \\Lambda _j x_{ij}^{(k)},$ where we have introduced the oefficients $x_{ij}^{(k)} \\equiv \\left| \\langle {\\Lambda _i}\\vert \\tilde{L}_k \\vert {\\Lambda _j}\\rangle \\right|^2 $ .",
"As the operators $L_k$ are Hermitian these coefficients fulfil $x_{ij}^{(k)}=x_{ji}^{(k)}$ The second term is expanded as $\\sum _k \\Gamma _k \\textrm {Tr}\\left[ \\rho ^2 L_k^\\dagger L_k \\right] = \\sum _k \\Gamma _k \\textrm {Tr}\\left[ \\left( \\sum _i \\Lambda _i \\vert {\\Lambda _i}\\rangle \\!\\langle {\\Lambda _i}\\vert \\right) \\left( \\sum _j \\Lambda _j \\vert {\\Lambda _j}\\rangle \\!\\langle {\\Lambda _j}\\vert \\right) \\tilde{L}^\\dagger _k \\tilde{L}_k \\right] \\nonumber \\\\= \\sum _k \\Gamma _k \\sum _{ij} \\Lambda _i \\Lambda _j \\textrm {Tr}\\left[ \\tilde{L}_k \\vert {\\Lambda _i}\\rangle \\!\\langle {\\Lambda _j}\\vert \\tilde{L}_k^\\dagger \\langle {\\Lambda _i}\\vert {\\Lambda _j}\\rangle \\right]= \\sum _k \\Gamma _k \\sum _i \\Lambda _i^2 \\textrm {Tr}\\left[ \\tilde{L}_k \\vert {\\Lambda _i}\\rangle \\!\\langle {\\Lambda _i}\\vert \\tilde{L}_k^\\dagger \\; \\right] \\nonumber \\\\= \\sum _k \\Gamma _k \\sum _i \\Lambda _i^2 \\textrm {Tr}\\left[ \\tilde{L}_k \\vert {\\Lambda _i}\\rangle \\!\\langle {\\Lambda _i}\\vert \\tilde{L}_k^\\dagger \\left( \\sum _j \\vert {\\Lambda _j}\\rangle \\!\\langle {\\Lambda _j}\\vert \\right) \\right] \\nonumber \\\\= \\sum _k \\Gamma _k \\sum _{ij} \\Lambda _i^2 \\textrm {Tr}\\left[ \\langle {\\Lambda _j}\\vert \\tilde{L}_k \\vert {\\Lambda _i}\\rangle + \\langle {\\Lambda _i}\\vert \\tilde{L}_k \\vert {\\Lambda _j}\\rangle \\right]= \\sum _k \\Gamma _k \\sum _{ij} \\Lambda _i^2 x_{ij},$ where we have used the closure relation in the density matrix eigenbasis, $\\mathbb {1}_{{\\cal H}}=\\sum _j \\vert {\\Lambda _j}\\rangle \\!\\langle {\\Lambda _j}\\vert $ .",
"The inequality can be written now as $\\sum _k \\Gamma _k \\sum _{ij} \\Lambda _i \\Lambda _j x_{ij} \\le \\sum _k \\Gamma _k \\sum _{ij} \\Lambda _i^2 x_{ij}.$ As $x_{ij}=x_{ji}$ we can re-order the $ij$ sum in the following way $\\sum _k \\Gamma _k \\sum _i \\left( \\sum _{j\\le i} 2 \\Lambda _i \\Lambda _j x_{ij}^{(k)} + \\Lambda _i^2 x_{ii}^{(k)} \\right) \\le \\sum _k \\Gamma _k\\sum _i \\left( \\sum _{j<i} \\left( \\Lambda _i^2 + \\Lambda _j^2 \\right) x_{ij}^{(k)} + \\Lambda _i^2 x_{ii}^{(k)} \\right).$ Therefore, we can reduce the proof of this inequality to the proof of a set of inequalities $2\\Lambda _i\\Lambda _j x_{ij}^{(k)} \\le \\left( \\Lambda _i^2 + \\Lambda _j^2 \\right) x_{ij}^{(k)} \\qquad \\forall \\left( k,i,j \\right).$ It is obvious that (REF ) $\\Rightarrow $ (REF ) (but not the other way around).",
"The inequalities (REF ) are easily proved just by taking into account that $x_{ij}^{(k)} \\ge 0$ and applying the Triangular Inequality."
]
] | 1906.04478 |
[
[
"Normality of the dual nilcone in positive characteristic"
],
[
"Abstract Let $G$ be a connected semisimple algebraic group of adjoint type defined over an algebraically closed field $K$ of positive characteristic.",
"The characteristic $p$ is very good for $G$ when $p$ is suitably large and, if $G$ is of type $A_n$, $p$ does not divide $n+1$.",
"The majority of results concerning the geometric structure of algebraic groups in positive characteristic are valid only in very good characteristic.",
"We demonstrate that the dual nilcone $\\mathcal{N}^* \\subseteq \\mathfrak{g}^*$ is a normal variety in certain characteristics which are not very good for $G$.",
"As an application, we extend the results of Ardakov and Wadsley on representations of $p$-adic Lie groups.",
"Under further restrictions on the characteristic, we show that the canonical dimension of a coadmissible representation of a semisimple $p$-adic Lie group in a $p$-adic Banach space is either zero or at least half the dimension of a nonzero coadjoint orbit."
],
[
"Introduction",
"Let $G$ be a connected semisimple algebraic group of adjoint type defined over an algebraically closed field $K$ of positive characteristic $p > 0$ .",
"In case the characteristic of $K$ is very good for $G$ , which, broadly speaking, implies that $G$ is not of type $A$ and $p > 5$ , it is known that the dual nilpotent cone $\\mathcal {N}^* \\subseteq \\mathfrak {g}^*$ is a normal variety, and it admits a desingularisation $\\mu : T^*\\mathcal {B} \\rightarrow \\mathcal {N}$ from the cotangent bundle of the flag variety $\\mathcal {B}$ of $G$ ; the so-called $\\emph {Springer resolution}$ of $\\mathcal {N}$ .",
"When $p$ is small, the classical proofs of these results break down.",
"The goal of this paper is to investigate in which bad characteristics the dual nilcone $\\mathcal {N}^*$ remains a normal variety and the Springer map is a resolution of singularities.",
"In case $G$ is of type $A_n$ , the picture is a little different.",
"Here, the classical proofs are valid when $p$ does not divide $n+1$ .",
"We have the following main theorems: Theorem A Let $G = PGL_n$ and suppose $p|n$ .",
"Then the dual nilpotent cone $\\mathcal {N}^* \\subseteq \\mathfrak {g}^*$ is a normal variety.",
"Theorem B Suppose $G$ is of type $G_2$ and $p = 2$ .",
"Then the dual nilpotent cone $\\mathcal {N}^* \\subseteq \\mathfrak {g}^*$ is a normal variety.",
"As an application, let $p$ be a prime, $G$ be a semisimple compact $p$ -adic Lie group and let $K$ be a finite extension of $\\mathbb {Q}_p$ .",
"Ardakov and Wadsley studied the coadmissible representations of $G$ , which are finitely generated modules over the completed group ring $KG$ with coefficients in $K$ , in [2].",
"These completed group rings may be realised as Iwasawa algebras, which are important objects in noncommutative Iwasawa theory.",
"One of the central results in [2] is an estimate for the canonical dimension of a coadmissible representation of a semisimple $p$ -adic Lie group in a $p$ -adic Banach space.",
"When $p$ is very good for $G$ , Ardakov and Wadsley showed that this canonical dimension is either zero or at least half the dimension of a nonzero coadjoint orbit.",
"We extend their results to the case where $G = PGL_n$ , $p|n$ , and $n > 2$ .",
"The main result of this section is as follows: Theorem C Let $G$ be a compact $p$ -adic analytic group whose Lie algebra is semisimple.",
"Suppose that $G = PGL_n$ , $p|n$ , and $n > 2$ .",
"Let $G_{\\mathbb {C}}$ be a complex semisimple algebraic group with the same root system as $G$ , and let $r$ be half the smallest possible dimension of a nonzero coadjoint $G_{\\mathbb {C}}$ -orbit.",
"Then any coadmissible $KG$ -module $M$ that is infinite-dimensional over $K$ satisfies $d(M) \\ge r$ .",
"$\\textbf {Acknowledgments.",
"}$ I would like to thank Konstantin Ardakov for suggesting this research project, and Kevin McGerty for his interest in my work and his helpful contributions."
],
[
"Characteristic",
"In this section, we study the geometric structure of the nilpotent cone $\\mathcal {N}$ of the Lie algebra $\\mathfrak {g}$ of a reductive algebraic group $G$ in arbitrary characteristic.",
"We begin with a discussion of the ordinary nilpotent cone, defined as a subvariety of $\\mathfrak {g}$ , and then give a characterisation of the dual nilpotent cone $\\mathcal {N}^*$ .",
"Our treatment of the material on $\\mathcal {N}$ is based on that of Jantzen in [18].",
"We generalise some of his arguments which are dependent on certain restrictions on the characteristic.",
"Later, we will specialise further to the case $G = PGL_n$ and $p|n$ at certain points of the argument.",
"The last subsection of the section discusses analogues of the results presented here when we consider a more general algebraic group $G$ .",
"Let $\\textbf {G}$ be a split reductive algebraic group scheme, defined over $\\mathbb {Z}$ , and $K$ an algebraically closed field of characteristic $p > 0$ .",
"Let $G := \\textbf {G}(K)$ .",
"Let $\\mathfrak {g}$ denote the Lie algebra of $G$ and $W(G)$ the Weyl group of $G$ .",
"When $G$ is clear from context, we will abbreviate $W(G)$ to $W$ .",
"Since $G$ is a linear algebraic group, we fix an embedding $G \\subseteq GL(V)$ for some $n$ -dimensional $K$ -vector space $V$ .",
"Definition 2.1.1 Let $\\alpha _i$ be the simple roots of the root system $R$ of $G$ , and let $\\beta $ be the highest-weight root.",
"Writing $\\beta = \\sum _i m_i \\alpha _i$ , $p$ is $\\textit {bad}$ for $G$ if $p = m_i$ for some $i$ .",
"$p$ is $\\emph {good}$ if $p$ is not bad.",
"The prime $p$ is $\\emph {very good}$ if one of the following conditions hold: (a) $G$ is not of type $A$ and $p$ is good, (b) $G$ is of type $A_n$ and $p$ does not divide $n+1$ .",
"In practice, we have the following classification.",
"In types $B, C$ and $D$ , the only bad prime is 2.",
"For the exceptional Lie algebras, the bad primes are 2 and 3 for types $E_6, E_7, F_4$ and $G_2$ , and 2,3 and 5 for type $E_8$ .",
"In type $A$ , there are no bad primes.",
"For more details of this classification, see [26].",
"Definition 2.1.2 A prime $p$ is $\\emph {special}$ for $G$ if the pair (Dynkin diagram of $G$ , $p$ ) lies in the following list: (a) ($B$ , 2), (b) ($C$ , 2), (c) ($F_4$ , 2), (d) ($G_2$ , 3).",
"A prime $p$ is $\\emph {nonspecial}$ for $G$ if it is not special.",
"This definition, and material on the importance of nonspecial primes, can be found in [23]."
],
[
"The $W$ -invariants of {{formula:1c624ae5-e4e2-4fa2-8afd-0fd2d1cb7d8d}}",
"Let $G = PGL_n$ and suppose $p|n$ .",
"This short section investigates the structure of the invariants of the Weyl group action on the symmetric algebra $S(\\mathfrak {h})$ .",
"Let $\\mathfrak {g}^*$ be the dual vector space of $\\mathfrak {g}$ .",
"Since $G$ is of type $A$ and the prime $p$ is always good for $G$ , there is a $G$ -equivariant isomorphism $\\kappa : \\mathfrak {g} \\rightarrow \\mathfrak {g}^*$ by the argument in [18].",
"Since $\\mathfrak {g}$ is a finite-dimensional vector space, we naturally identify the symmetric algebra $S(\\mathfrak {g})$ and the algebra of polynomial functions $K[\\mathfrak {g}^*]$ .",
"Let $\\mathfrak {h}$ be a fixed Cartan subalgebra of $\\mathfrak {g}$ .",
"The Weyl group $W$ has a natural action on $\\mathfrak {h}$ , which can be extended linearly to an action of $W$ on the symmetric algebra $S(\\mathfrak {h})$ .",
"The identification $S(\\mathfrak {h}) \\cong K[\\mathfrak {h}^*]$ is compatible with the $W$ -action.",
"We begin this section by studying the $W$ -invariants under this action.",
"Theorem 2.2.1 Suppose $G = PGL_n$ and $p|n$ .",
"Then $S(\\mathfrak {h})^W$ is a polynomial ring.",
"Recall the Weyl group $W$ is isomorphic to $S_n$ , and let $\\mathfrak {t}$ be the image of the diagonal matrices in $\\mathfrak {g} = \\mathfrak {pgl}_n$ .",
"Then $\\mathfrak {t}$ is the quotient of the natural $S_n$ -module $V$ with basis $\\lbrace e_1, \\cdots , e_n \\rbrace $ , permuted by $S_n$ , by the trivial submodule $U := K(\\sum _{i=1}^n e_i)$ .",
"Let $X = V/U$ .",
"The quotient map $V \\rightarrow X$ induces a surjective map $S(V) \\rightarrow S(X)$ .",
"Suppose $p = n = 2$ and let $\\lbrace \\overline{e_1}, \\overline{e_2} \\rbrace $ be the images of the vector space basis $\\lbrace e_1, e_2 \\rbrace $ of $V$ inside $X$ .",
"Let $\\sigma $ denote the non-identity element of $S_2$ .",
"Then $\\sigma \\cdot \\overline{e_1} = \\overline{e_2}$ and $\\sigma \\cdot \\overline{e_2} = \\overline{e_1}$ .",
"Since $\\overline{e_1} + \\overline{e_2} = 0$ , it follows that $\\overline{e_1} = \\overline{e_2}$ .",
"Hence $S(X)^{S_2} = S(X)$ , which is a polynomial ring.",
"Now suppose $n > 2$ and $p|n$ .",
"We claim that the $S_n$ -action on $V$ and on $X$ is faithful.",
"The $S_n$ -action on $V$ is by permutation and therefore is faithful.",
"To see the claim for the $S_n$ -action on $X$ , let $N := \\lbrace g \\in S_n \\mid g \\cdot x = x \\text{ } \\forall x \\in S(X) \\rbrace $ denote the kernel of the natural map $S_n \\rightarrow S(X)$ .",
"Suppose $g$ is some non-identity element of $N$ .",
"Then, relabelling the elements $\\overline{e_i}$ if necessary, $g \\cdot \\overline{e_1} = \\overline{e_2}$ .",
"Hence it suffices to show that $\\overline{e_1} \\ne \\overline{e_2}$ .",
"If $\\overline{e_1} = \\overline{e_2}$ , then since $\\sum _{i=1}^n \\overline{e_i} = 0$ , $\\sum _{i=2}^n \\overline{e_i} = \\overline{e_1}$ and $e_1 + \\sum _{i=3}^n \\overline{e_i} = \\overline{e_2}$ .",
"Rearranging, $\\sum _{i=3}^n \\overline{e_i} = (p-2)\\overline{e_1}$ .",
"Hence the set $\\lbrace \\overline{e_1}, \\overline{e_3}, \\cdots , \\overline{e_{n-1}} \\rbrace $ spans $X$ , but $X$ is an $(n-1)$ -dimensional vector space, a contradiction.",
"It follows that the $S_n$ -action on $X$ is faithful.",
"The ring of invariants $S(V)^{S_n}$ is generated by the elementary symmetric polynomials $s_1(e_1, \\cdots , e_n), \\cdots , s_n(e_1, \\cdots , e_n)$ , which are algebraically independent by [5].",
"Applying [22], we see that $S(X)^{S_n}$ is also a polynomial ring.",
"The proof of [20] also demonstrates that $S(X)^{S_n}$ is generated by the images of $s_2(e_1, \\cdots , e_n), \\cdots , s_n(e_1, \\cdots , e_n)$ under the map $S(V) \\rightarrow S(X)$ .",
"To finish the proof, it suffices to note that we may identify $\\mathfrak {t} \\cong \\mathfrak {h}$ and that $\\mathfrak {h} \\cong V/U = X$ .",
"We state a version of Kostant's freeness theorem that will be useful for our applications.",
"Theorem 2.2.2 $S(\\mathfrak {h})$ is a free $S(\\mathfrak {h})^W$ -module if and only if $S(\\mathfrak {h})^W$ is a polynomial ring.",
"See [24]."
],
[
"Properties of the nilpotent cone",
"We now outline some general preliminaries on the structure theory of groups acting on varieties.",
"At first, we do not impose any restriction on the characteristic.",
"Let $M$ be a variety which admits an algebraic group action by $G$ , and let $x \\in M$ .",
"The closure $\\overline{Gx}$ of the orbit $Gx$ of $x$ is a closed subvariety of $M$ .",
"By [16], $Gx$ is open in $\\overline{Gx}$ and so $Gx$ has the structure of an algebraic variety.",
"The orbit map $\\pi _x: G \\rightarrow Gx$ , $\\pi _x(g) = gx$ , is a surjective morphism of varieties.",
"The stabiliser $G_x := \\lbrace g \\in G \\mid gx = x \\rbrace $ is a closed subgroup of $G$ , and $\\pi _x$ induces a bijective morphism: $\\overline{\\pi _x}: G/G_x \\rightarrow Gx$ by [16].",
"We now specialise to the case where $M = \\mathfrak {g}$ and $G$ acts on $\\mathfrak {g}$ via the adjoint action.",
"Let $X \\in \\mathfrak {g}$ and let $GX$ denote the $G$ -orbit of $X$ under the adjoint action $\\text{Ad}: G \\rightarrow \\text{Ad}(\\mathfrak {g})$ .",
"Recall that an element $g \\in \\mathfrak {g}$ is $\\emph {nilpotent}$ if the operator $\\text{ad}_g(y)$ is nilpotent for each $y \\in \\mathfrak {g}$ .",
"The set of nilpotent elements is denoted $\\mathcal {N}$ .",
"Since $G$ is a linear algebraic group, fix an embedding $G \\subseteq GL(V)$ for some $n$ -dimensional $K$ -vector space $V$ .",
"Then $\\mathcal {N} = \\mathfrak {g} \\cap \\mathcal {N}(\\mathfrak {gl}(V))$ , where $\\mathcal {N}(\\mathfrak {gl}(V))$ denotes the set of nilpotent elements of the Lie algebra of $GL(V)$ .",
"It follows that $\\mathcal {N}$ is closed in $\\mathfrak {g}$ , and hence $\\mathcal {N}$ has the structure of a subvariety of the algebraic variety $\\mathfrak {g}$ .",
"Let: $P_X(t) := \\text{det}(tI - X)$ denote the characteristic polynomial of $X$ in the variable $t$ .",
"Then: $P_X(t) := t^n + \\sum _{i=1}^n (-1)^i s_i(X) t^{n-i}$ where each $s_i$ is a homogeneous polynomial of degree $i$ in the entries of $X$ .",
"If $a_1, \\cdots , a_n$ are the eigenvalues of $X$ , counted with algebraic multiplicity, then, since $K$ is algebraically closed, $P_X(t) = \\prod _{i=1}^n (t-a_i)$ , and so $s_i(X)$ can be identified with the $i$ th elementary symmetric function in the $a_j$ .",
"It follows that $X$ is nilpotent if and only if $P_X(t) = t^n$ if and only if $s_i(X) = 0$ for each $i$ : $\\mathcal {N}(\\mathfrak {gl}(V)) = \\lbrace X \\in \\mathfrak {gl}(V) \\mid s_i(X) = 0 \\text{ for all } i \\rbrace .$ Let $S(V)$ denote the algebra of polynomial functions on $V$ .",
"This has a natural grading by degree, with $S(V) = \\bigoplus _{i \\ge 0} S^i(V)$ .",
"Set $S^+(V) := \\bigoplus _{i \\ge 1} S^i(V)$ .",
"Now the restrictions of the $s_i$ to $\\mathfrak {g}$ are $G$ -invariant polynomial functions on $\\mathfrak {g}$ , and so $s_{i|\\mathfrak {g}} \\in S^i(\\mathfrak {g}^*)^G$ .",
"It follows that there exist $f_1, \\cdots , f_n \\in S^+(\\mathfrak {g}^*)^G$ such that: $\\mathcal {N} = \\lbrace X \\in \\mathfrak {g} \\mid f_i(X) = 0 \\text{ for all } i \\rbrace .$ Proposition 2.3.1 The nilpotent cone $\\mathcal {N}$ may be realised as: $\\mathcal {N} = \\lbrace X \\in \\mathfrak {g} \\mid f(X) = 0 \\text{ for all } f \\in S^+(\\mathfrak {g}^*)^G \\rbrace .$ Hence $\\mathcal {N} = V(S^+(\\mathfrak {g}^*)^G)$ is an affine variety.",
"It is clear that $\\lbrace X \\in \\mathfrak {g} \\mid f(X) = 0 \\text{ for all } f \\in S^+(\\mathfrak {g}^*)^G \\rbrace \\subseteq \\mathcal {N}$ by the above discussion.",
"Conversely, given $f \\in S^+(\\mathfrak {g}^*)^G$ , $f(0) = 0$ and $f$ is constant on the closure of the orbits under the adjoint action.",
"Then $f$ is constant on $\\overline{GX}$ , the closure of the regular orbit under the adjoint action, and $0 \\in \\overline{GX}$ by [18].",
"Lemma 2.3.2 Let $\\mathcal {B}$ be the set of all Borel subalgebras of $\\mathfrak {g}$ .",
"Then there is a bijection $G/B \\leftrightarrow \\mathcal {B}$ .",
"$\\mathcal {B}$ is the closed subvariety of the Grassmannian of $\\text{dim } \\mathfrak {b}$ -dimensional subspaces in $\\mathfrak {g}$ formed by solvable Lie algebras.",
"Hence $\\mathcal {B}$ is a projective variety.",
"All Borel subalgebras are conjugate under the adjoint action of $G$ , and the stabiliser subgroup $G_{\\mathfrak {b}}$ of $\\mathfrak {b}$ in $G$ is equal to $B$ by [4].",
"Hence the claimed bijection follows via the assignment $g \\mapsto g \\cdot \\mathfrak {b} \\cdot g^{-1}$ .",
"Definition 2.3.3 Set $\\widetilde{\\mathfrak {g}} := \\lbrace (x, \\mathfrak {b}) \\in \\mathfrak {g} \\times \\mathcal {B} \\mid x \\in \\mathfrak {b} \\rbrace $ , and let $\\mu : \\widetilde{\\mathfrak {g}} \\rightarrow \\mathfrak {g}$ be the projection onto the first coordinate.",
"The $\\emph {enhanced nilpotent cone}$ is the preimage of $\\mathcal {N}$ under the map $\\mu $ : $\\widetilde{\\mathcal {N}} := \\mu ^{-1}(\\mathcal {N}) = \\lbrace (x, \\mathfrak {b}) \\in \\mathcal {N} \\times \\mathcal {B} \\mid x \\in \\mathfrak {b} \\rbrace .$ Lemma 2.3.4 $\\widetilde{\\mathcal {N}}$ is a smooth irreducible variety.",
"Let $\\mathfrak {b} \\in \\mathcal {B}$ be a fixed Borel subalgebra.",
"The fibre over $\\mathfrak {b}$ of the second projection $\\pi : \\widetilde{\\mathcal {N}} \\rightarrow \\mathcal {B}$ is the set of nilpotent elements of $\\mathfrak {b}$ .",
"Decomposing $\\mathfrak {b} = \\mathfrak {h} \\oplus \\mathfrak {n}$ , where $\\mathfrak {n}:= [\\mathfrak {b}, \\mathfrak {b}]$ is the nilradical of $\\mathfrak {b}$ , an element $x \\in \\mathfrak {b}$ is nilpotent if and only if it is has no component in the Cartan subalgebra $\\mathfrak {h}$ .",
"Hence $\\pi $ makes $\\widetilde{\\mathcal {N}}$ a vector bundle over $\\mathcal {B}$ with fibre $\\mathfrak {n}$ .",
"The canonical map $G \\rightarrow G/B$ is locally trivial by [17], so the set of $B$ -orbits on $G \\times \\mathfrak {n}$ has a natural structure of a variety, denoted $G \\times _B \\mathfrak {n}$ .",
"The above construction yields a $G$ -equivariant vector bundle isomorphism: $\\widetilde{\\mathcal {N}} \\cong G \\times _B \\mathfrak {n},$ where $B$ is the Borel subgroup of $G$ corresponding to $\\mathfrak {b}$ .",
"It follows that we may view $\\widetilde{\\mathcal {N}}$ as a vector bundle over the smooth variety $G/B$ , and so $\\widetilde{\\mathcal {N}}$ is smooth.",
"Using Lemma REF , identify $\\mathcal {B}$ with $G/B$ and consider the morphism $f: \\mathfrak {g} \\times G \\rightarrow \\mathfrak {g} \\times \\mathcal {B}$ defined by $f(x,g) = (x,gB)$ .",
"The inverse image: $f^{-1}(\\widetilde{\\mathcal {N}}) = \\lbrace (x,g) \\in \\mathcal {N} \\times G \\mid \\text{Ad}(g^{-1})(x) \\in \\mathfrak {n} \\rbrace $ is closed in $\\mathfrak {g} \\times G$ since it is the inverse image of $\\mathfrak {n}$ under the natural map $\\mathfrak {g} \\times G \\rightarrow \\mathfrak {g}$ , $(x,g) \\mapsto \\text{Ad}(g^{-1})(x)$ .",
"Since $f$ is an open map and $f^{-1}(\\widetilde{\\mathcal {N}})$ is closed, $\\widetilde{\\mathcal {N}}$ is a closed subvariety of $\\mathcal {N} \\times \\mathcal {B}$ .",
"The morphism $\\mathfrak {n} \\times G \\rightarrow \\widetilde{\\mathcal {N}}, (x,g) \\mapsto (\\text{ad}(g)(x), gB)$ is surjective by definition.",
"Hence $\\widetilde{\\mathcal {N}}$ is irreducible.",
"By [18], there are only finitely many orbits for the $G$ -action in the nilpotent cone $\\mathcal {N}$ .",
"Let $X_1, \\cdots , X_r$ be representatives for these orbits.",
"Then: $\\mathcal {N} = \\bigcup _{i=1}^r \\overline{\\mathcal {O}_{X_i}}$ Since $\\mathcal {N}$ is irreducible by Lemma REF , one of these closed subsets must be all of $\\mathcal {N}$ : let $\\overline{GZ} = \\mathcal {N}$ .",
"Then, by [27], this orbit is open in $\\mathcal {N}$ and $\\text{dim}(GZ) = \\text{dim}(\\mathcal {N})$ , while $\\text{dim}(GY) < \\text{dim}(GZ)$ for any $GY \\ne GZ$ .",
"Hence $GZ$ is unique with respect to this property.",
"Definition 2.3.5 An element $X \\in \\mathfrak {g}$ is $\\emph {regular}$ if it lies in $GZ$ , the unique open dense $G$ -orbit of $\\mathcal {N}$ .",
"We now specialise to the case where $G = PGL_n$ and $p|n$ .",
"Lemma 2.3.6 There is a natural $G$ -equivariant vector bundle isomorphism: $\\widetilde{\\mathcal {N}} \\cong T^*\\mathcal {B}.$ This follows from [18].",
"Definition 2.3.7 The map $\\mu : T^*\\mathcal {B} \\rightarrow \\mathcal {N}$ is the $\\emph {Springer resolution}$ for the nilpotent cone $\\mathcal {N}$ .",
"Lemma 2.3.8 Let $\\mathcal {N}_s$ denote the set of smooth points of $\\mathcal {N}$ .",
"Then $\\mu ^{-1}(\\mathcal {N}_s)$ is dense in $\\widetilde{\\mathcal {N}}$ .",
"$\\mathcal {N}_s$ is an open and non-empty subset of $\\mathcal {N}$ .",
"Hence it is dense, and its preimage is open and non-empty in $\\widetilde{\\mathcal {N}}$ .",
"By Lemma REF , $\\widetilde{\\mathcal {N}}$ is irreducible and so $\\mu ^{-1}(\\mathcal {N}_s)$ is dense.",
"Lemma 2.3.9 Let $GZ$ denote the orbit of all regular nilpotent elements.",
"The morphism $\\mu ^{-1}(GZ) \\rightarrow GZ$ is an isomorphism of varieties.",
"By [18], $GZ$ is an open subset of $\\mathcal {N}$ , and $|\\mu ^{-1}(X)| = 1$ for $X \\in GZ$ .",
"Hence $\\mu $ induces a bijection $\\mu ^{-1}(GZ) \\rightarrow GZ$ .",
"Since the morphism $\\mu : T^*\\mathcal {B} \\rightarrow \\mathcal {N}$ is given by projection onto the first coordinate, from Definition REF , it is a morphism of varieties and hence so is the restriction $\\mu \\mid _{\\mu ^{-1}(GZ)}: \\mu ^{-1}(GZ) \\rightarrow GZ$ .",
"The result follows.",
"Recall from Theorem REF that $S(\\mathfrak {h})^W$ is a polynomial ring, with algebraically independent generators $f_1, \\cdots , f_n$ .",
"Theorem 2.3.10 Let $G$ be a simple algebraic group, and suppose $(G,p) \\ne (B,2)$ .",
"There is a projection map $\\mathfrak {g} = \\mathfrak {n}^- \\oplus \\mathfrak {h} \\oplus \\mathfrak {n} \\rightarrow \\mathfrak {h}$ , which induces a map $S(\\mathfrak {g}) \\rightarrow S(\\mathfrak {h})$ .",
"This map induces a map $\\eta : S(\\mathfrak {g})^G \\rightarrow S(\\mathfrak {h})^W$ , which is an isomorphism.",
"This is [19].",
"When $G = PGL_n$ and $p|n$ , the hypotheses of Theorem REF are satisfied.",
"This allows us to make sense of the following definition.",
"Definition 2.3.11 The $\\emph {Steinberg quotient}$ is the map $\\chi : \\mathfrak {g} \\rightarrow K^n$ defined by $\\chi (Z) = (\\eta ^{-1}(f_1)(Z), \\cdots , \\eta ^{-1}(f_n)(Z))$ .",
"Note that the nilpotent cone $\\mathcal {N} = \\chi ^{-1}(0)$ .",
"Lemma 2.3.12 The smooth points of $\\mathcal {N}$ are precisely the regular nilpotent elements.",
"By the assumptions on the prime $p$ , applying Theorem REF and Theorem REF shows that $S(\\mathfrak {h})$ is a free $S(\\mathfrak {h})^W$ -module and $S(\\mathfrak {h})^W$ is a polynomial ring, with generators $f_1, \\cdots , f_n$ .",
"Hence the argument for [7] applies and the Steinberg quotient $\\chi $ satisfies, for $Z \\in \\mathfrak {g}$ , the condition that $(d\\chi )_Z$ is surjective if and only if $Z$ is regular.",
"By [18], for each $b = (b_1, \\cdots , b_n) \\in K^n$ , the ideal of $\\chi ^{-1}(b)$ is generated by all $\\eta ^{-1}(f_i) - b_i$ .",
"By [12], $Z \\in \\chi ^{-1}(b)$ is a smooth point if and only if the $d(\\eta (f_i) - b_i)$ are linearly independent at $Z$ if and only if the map $(d\\chi )_Z$ is surjective.",
"Let $b=0$ .",
"Then the smooth points in $\\chi ^{-1}(0)$ are the regular elements contained in $\\chi ^{-1}(0)$ , and so the smooth points of $\\mathcal {N}$ are precisely the regular nilpotent elements.",
"Theorem 2.3.13 $\\mu : T^*\\mathcal {B} \\rightarrow \\mathcal {N}$ is a resolution of singularities for $\\mathcal {N}$ .",
"By Lemma REF and Lemma REF , $\\widetilde{\\mathcal {N}}$ is a smooth irreducible variety.",
"Furthermore, $\\mu $ is proper by [18].",
"By Lemma REF , $\\mu ^{-1}(\\mathcal {N}_s)$ is dense in $\\widetilde{\\mathcal {N}}$ , and by Lemma REF , $\\mu $ is a birational morphism between $\\mu ^{-1}(\\mathcal {N}_s)$ and $\\mathcal {N}_s$ .",
"Hence $\\mu $ is a resolution of singularities."
],
[
"The dual nilpotent cone is a normal variety",
"In this section, we demonstrate that the dual nilcone $\\mathcal {N}^*$ is a normal variety in the case $G = PGL_n$ , $p|n$ .",
"Definition 2.4.1 Since we have a $G$ -equivariant isomorphism $\\kappa : \\mathfrak {g} \\rightarrow \\mathfrak {g}^*$ by [18], the $\\emph {dual nilcone}$ $\\mathcal {N}^*$ may be defined as: $\\mathcal {N}^* = \\lbrace X \\in \\mathfrak {g}^* \\mid f(X) = 0 \\text{ for all } f \\in S^+(\\mathfrak {g})^G \\rbrace .$ The same argument as in Proposition REF shows that $\\mathcal {N} = V(S^+(\\mathfrak {g})^G)$ is an affine variety.",
"We next review some basic properties of normal rings and varieties.",
"Definition 2.4.2 [7] A finitely generated commutative $K$ -algebra $A$ is $\\textit {Cohen-Macaulay}$ if it contains a subalgebra of the form $\\mathcal {O}(V)$ such that $A$ is a free $\\mathcal {O}(V)$ -module of finite rank, and $V$ is a smooth affine scheme.",
"A scheme $X$ defined over $K$ is $\\emph {Cohen-Macaulay}$ if, at each point $x \\in X$ , the local ring $\\mathcal {O}_{X,x}$ is a Cohen-Macaulay ring.",
"Definition 2.4.3 A commutative ring $A$ is $\\textit {normal}$ if the localization $A_{\\mathfrak {p}}$ for each prime ideal $\\mathfrak {p}$ is an integrally closed domain.",
"A variety $V$ is $\\textit {normal}$ if, for any $x \\in V$ , the local ring $\\mathcal {O}_{V,x}$ is a normal ring.",
"We now begin the proof of the normality of the dual nilpotent cone $\\mathcal {N}^*$ .",
"We adapt the arguments in [3] to our situation.",
"Theorem 2.4.4 Let $X$ be an irreducible affine Cohen-Macaulay scheme defined over $K$ and $U \\subseteq X$ an open subscheme.",
"Suppose $\\text{dim } (X/U) \\le \\text{dim } X - 2$ and that the scheme $U$ is normal.",
"Then the scheme $X$ is normal.",
"This is [3].",
"We aim to apply Theorem REF to our situation.",
"We begin with the following lemma, a variant on Hartogs' lemma.",
"Lemma 2.4.5 Let $Y$ be an affine normal variety and $Z \\subseteq Y$ be a subvariety of codimension at least 2.",
"Then any rational function on $Y$ which is regular on $Y \\setminus Z$ can be extended to a regular function on $Y$ .",
"Write $Y = \\text{Spec } B$ , where $B$ is a normal domain.",
"Set $Z := V(I)$ for some ideal $I$ , and write $U := Y \\setminus Z$ .",
"Then $U = \\bigcup _{f \\in I} D(f)$ , where $D(f)$ denotes the basic open sets in the Zariski topology.",
"Let $\\mathfrak {p}$ be a prime ideal of height 1.",
"By assumption, $\\text{ht } I \\ge 2$ , and so there exists some $f \\in I$ with $f \\notin \\mathfrak {p}$ .",
"It follows that $B_f \\subseteq B_{\\mathfrak {p}}$ .",
"Let $a/b$ be a regular function on $U$ , with $a/b \\in \\text{Frac} B$ , the field of fractions of $B$ .",
"Since $\\mathfrak {p}$ has height 1, we can find $f \\in I \\setminus \\mathfrak {p}$ .",
"Then $a/b$ is regular on $D(f)$ , and so $a/b \\in \\mathcal {O}(D(f)) = B_f \\subseteq B_{\\mathfrak {p}}$ .",
"As $\\mathfrak {p}$ was arbitrary, $a/b \\in \\bigcap _{\\text{ht}\\mathfrak {p} = 1} B_{\\mathfrak {p}} = B$ .",
"Hence $a/b$ can be extended to a regular function on $Y$ .",
"Lemma 2.4.6 Let $X$ be an affine Cohen-Macaulay scheme with an open subscheme $U$ .",
"Let $r: \\mathcal {O}(X) \\rightarrow \\mathcal {O}(U)$ be the restriction morphism.",
"Then: (a) if $\\text{dim }(X \\setminus U) < \\text{dim } X$ , then $r$ is injective, (b) if $\\text{dim } (X \\setminus U) \\le \\text{dim } X - 2$ , then $r$ is an isomorphism.",
"We expand on the proof given in [3].",
"For ease of notation, we suppose $\\mathcal {O}(X)$ is a finitely generated $\\mathcal {O}(Y)$ -module for some smooth affine scheme $Y$ .",
"Now the projection map $p: X \\rightarrow Y$ is a finite morphism and hence is closed.",
"Without loss of generality, we can shrink $U$ , replacing it by a smaller open subset $p^{-1}(W)$ , where $W = Y \\setminus p(X \\setminus U)$ is an open subset of $Y$ .",
"Let $F := p_*(\\mathcal {O}_X)$ .",
"This is a free $\\mathcal {O}_Y$ -module and we clearly have $\\Gamma (Y,F) = p_*(\\mathcal {O}_X)(Y) = \\mathcal {O}_X(p^{-1}(Y)) = \\mathcal {O}_X(X)$ , and similarly $\\Gamma (W,F) = p_*(\\mathcal {O}_X)(W) = \\mathcal {O}_X(p^{-1}(W)) = \\mathcal {O}_X(U)$ .",
"Hence the restriction morphism $r$ agrees with the natural restriction map $r: \\Gamma (Y,F) \\rightarrow \\Gamma (W,F)$ .",
"If $\\text{dim }(X \\setminus U) < \\text{dim } X$ , then $\\text{dim }(p(U \\setminus X)) < \\text{dim } (p(X))$ , so $\\text{dim }(Y \\setminus W) < \\text{dim } Y$ , and so $r$ is injective.",
"Similarly, if $\\text{dim } (X \\setminus U) \\le \\text{dim } X - 2$ , then $\\text{dim } (Y \\setminus W) \\le \\text{dim } Y - 2$ .",
"Hence, by Lemma REF , any regular function on $W$ can be extended to a regular function on $Y$ .",
"Furthermore, $F$ is a free $\\mathcal {O}_Y$ -module; it follows that $r$ is surjective.",
"As an immediate consequence, we see that if the scheme $U$ is reduced and normal, then so is $X$ .",
"We now demonstrate that the hypotheses in Theorem REF are satisfied in our situation.",
"Recall that $\\mathcal {N}^*$ is an affine variety.",
"It suffices to show that $\\mathcal {N}^*$ is irreducible and Cohen-Macaulay.",
"Definition 2.4.7 $\\lambda \\in \\mathfrak {h}^*$ is $\\emph {regular}$ if its centraliser in $\\mathfrak {g}$ under the natural $\\mathfrak {g}$ -action on $\\mathfrak {g}^*$ coincides with the Cartan subalgebra $\\mathfrak {h}$ .",
"A general $\\lambda \\in \\mathfrak {g}^*$ is $\\emph {regular}$ if its coadjoint orbit contains a regular element of $\\mathfrak {h}^*$ .",
"The subvariety $U$ in Lemma REF will be taken to be the subset of regular nilpotent elements.",
"Proposition 2.4.8 Suppose $p$ is nonspecial for $G$ .",
"Then: (a) the dual nilcone $\\mathcal {N}^* \\subseteq \\mathfrak {g}^*$ is a closed irreducible subvariety of $\\mathfrak {g}^*$ , and it has codimension $r$ in $G$ , where $r$ is the rank of $G$ .",
"(b) Let $U$ denote the set of regular elements of $\\mathcal {N}^*$ .",
"Then $U$ is a single coadjoint orbit, which is open in $\\mathcal {N}^*$ , and its complement has codimension $\\ge 2$ .",
"(a) We define an auxiliary variety $S$ via: $S := \\lbrace (gB, \\zeta ) \\in G/B \\times \\mathfrak {g}^* \\mid g \\cdot \\zeta \\in \\mathfrak {b}^{\\perp } \\rbrace .$ This subset of $G/B \\times \\mathfrak {g}^*$ is closed.",
"Define a map $\\phi : G \\times \\mathfrak {b}^{\\perp } \\rightarrow G/B \\times \\mathfrak {g}^*$ by $\\phi (g, \\zeta ) = (gB, g^{-1} \\cdot \\zeta )$ .",
"Now the image of $\\phi $ is contained in $S$ , and we can also see that $\\text{im}(\\phi ) \\cong S$ since we have a linear isomorphism $\\mathfrak {b}^{\\perp } \\rightarrow g^{-1} \\cdot \\mathfrak {b}^{\\perp }$ .",
"Hence the image of $\\phi $ coincides with $S$ .",
"It follows that $S$ is a morphic image of an irreducible variety, and hence $S$ is itself an irreducible subvariety.",
"Let $p_1: G/B \\times \\mathfrak {g}^* \\rightarrow G/B$ and $p_2: G/B \\times \\mathfrak {g}^* \\rightarrow \\mathfrak {g}^*$ be the obvious projection maps.",
"Clearly $p_1(S) = G/B$ .",
"The fiber of $gB$ under the map $p_1$ is $g^{-1} \\cdot \\mathfrak {n}$ , which is isomorphic to $\\mathfrak {n}$ .",
"Hence the fibers are equidimensional, and we have: $\\text{dim } S = \\text{dim } G/B + \\text{dim } \\mathfrak {n}, \\\\\\text{dim } S = \\text{dim } G/B + \\text{dim } U \\\\= \\text{dim } G - r.$ Using the second projection, $\\text{dim } p_2(S) \\le \\text{dim } G - r$ , with equality if some fibre is finite (as a set).",
"First notice that: $p_2(S) = \\lbrace \\zeta \\in \\mathfrak {g}^* \\mid \\exists g \\in G \\text{ s.t. }",
"g \\cdot \\zeta \\in \\mathfrak {b}^{\\perp } \\rbrace = \\mathcal {N}^*.$ Hence $\\mathcal {N}^*$ is irreducible, and, since the flag variety $G/B$ is complete by [4], $\\mathcal {N}^*$ is closed.",
"We show that there exists some $\\zeta \\in \\mathfrak {g}^*$ with: $| \\lbrace gB \\mid g \\cdot \\zeta \\in \\mathfrak {b}^{\\perp } \\rbrace | < \\infty , \\\\| \\lbrace gB \\mid \\zeta (\\text{Ad}_g^{-1}(\\mathfrak {b})) = 0 \\rbrace | < \\infty .$ By [14], we have the following dimension formula: $\\text{dim } p_1(p_2^{-1}(\\zeta )) = \\frac{\\text{dim } Z_G(\\zeta ) - r}{2}.$ Since $p$ is nonspecial for $G$ , the set of regular nilpotent elements $U$ in $\\mathcal {N}^*$ is non-empty, by [11], and thus we can always pick some $\\zeta \\in \\mathfrak {g}^*$ such that $\\text{dim } Z_G(\\zeta ) - r = 0$ .",
"Thus there exists $\\zeta $ with $| \\lbrace p_1(p_2^{-1}(\\zeta )) \\rbrace | < \\infty $ .",
"Now consider two points $(gB, \\zeta ), (hB, \\zeta ) \\in S$ .",
"By definition, $g \\cdot \\zeta \\in \\mathfrak {b}^{\\perp }$ and $h \\cdot \\zeta \\in \\mathfrak {b}^{\\perp }$ .",
"The coadjoint action then gives $\\zeta (\\text{ad}_g^{-1}(\\mathfrak {b})) = \\zeta (\\text{ad}_h^{-1}(\\mathfrak {b})) = 0$ .",
"It follows that $gB = hB$ , and so $p_1$ is injective when restricted to the fibre $p_2^{-1}(\\zeta )$ .",
"It follows that there is a fibre of $p_2$ which is finite as a set.",
"Given the existence of a finite fibre of $p_2$ , we have $\\text{dim } S = \\text{dim } p_2(S) = \\text{dim } \\mathcal {N}^* = \\text{dim } G - r$ .",
"(b) Now $\\mathcal {N}^*$ has only finitely many $G$ -orbits by [29] and [28], so the dimension of $\\mathcal {N^*}$ is equal to the dimension of at least one of these orbits.",
"Since $\\text{dim } \\mathcal {N^*} = \\text{dim } G - r$ , some orbit in $\\mathcal {N^*}$ also has dimension equal to $\\text{dim } G - r$ .",
"This orbit is regular and its closure is all of $\\mathcal {N^*}$ , since the dimensions are equal and $\\mathcal {N^*}$ is irreducible.",
"Since any $G$ -orbit is open in its closure, by [27], this class is open in $\\mathcal {N^*}$ and thus is dense.",
"Let $R$ be the root system of $G$ and fix a subset of positive roots $R^+ \\subseteq R$ .",
"Let $\\alpha _i$ be a simple root, $X_{\\alpha }$ the corresponding root subgroup, and set $U_i := \\prod _{\\alpha \\in R^+, \\alpha \\ne \\alpha _i} X_{\\alpha }$ .",
"Let $T$ be the maximal torus of $G$ defined by this root system and let $P_i := T \\cdot \\langle X_{\\alpha _i}, X_{-\\alpha _i} \\rangle \\cdot U_i$ .",
"Since both $T$ and $\\langle X_{\\alpha _i}, X_{-\\alpha _i} \\rangle $ normalise $U_i$ by the commutation formulae in [27], we see that $P_i$ is a rank 1 parabolic subgroup of $G$ , $U_i$ is its unipotent radical and $T \\cdot \\langle X_{\\alpha _i}, X_{-\\alpha _i} \\rangle $ is a Levi subgroup of $P_i$ .",
"Note that $\\text{dim } T \\cdot \\langle X_{\\alpha _i}, X_{-\\alpha _i} \\rangle = r+2$ and so $\\text{dim } P_i - \\text{dim } U_i = r+2$ .",
"Parallel to the definition of the variety $S$ , we set: $S_i := \\lbrace (gP_i, \\zeta ) \\in G/P_i \\times \\mathfrak {g}^* \\mid g \\cdot \\zeta \\in \\mathfrak {b}^{\\perp }_i \\rbrace $ where $\\mathfrak {b}^{\\perp }_i = \\lbrace \\zeta \\in \\mathfrak {g}^* \\mid \\zeta (\\text{Lie}(U_iT)) = 0 \\rbrace $ .",
"Then $S_i$ is a closed and irreducible variety and, by the same argument as in part (a) of the proposition: $\\text{dim } S_i = \\text{dim } G/P_i + \\text{dim } U_i \\\\= \\text{dim } G - (r+2).$ Projecting onto the second factor, we see that: $\\text{dim } p_2(S_i) \\le \\text{dim } S_i = \\text{dim } G - (r+2).$ But an element $\\zeta \\in \\mathcal {N}^*$ fails to be regular if and only if $G \\cdot \\zeta \\cap \\mathfrak {h}^*_{\\text{reg}} = \\emptyset $ .",
"By the decomposition in [11], this occurs precisely when the centraliser of each $\\xi \\in G \\cdot \\zeta \\cap \\mathfrak {h}^*$ contains some non-zero root $\\alpha $ such that $\\xi (\\alpha ^{\\vee }(1)) = 0$ , where $\\alpha ^{\\vee }$ is the coroot corresponding to $\\alpha $ .",
"It follows that $\\zeta \\in \\mathcal {N}^*$ fails to be regular if and only if it lies in $p_2(S_i)$ for some $i$ .",
"Then: $\\text{dim }(\\mathcal {N}^* \\setminus U) = \\text{sup}_i \\text{ dim } p_2(S_i) \\le \\text{dim } G - (r+2).$ Lemma 2.4.9 Let $r: S(\\mathfrak {g}) \\rightarrow S(\\mathfrak {h})$ be the natural map, and $r^{\\prime }$ its restriction to the graded subalgebra $S(\\mathfrak {g})^G$ .",
"Suppose that $r^{\\prime }$ is an isomorphism onto its image $S(\\mathfrak {h})^W$ and $S(\\mathfrak {h})$ is a free $S(\\mathfrak {h})^W$ -module.",
"Then $S(\\mathfrak {g})$ is a free $R$ -module, where $R := S(\\mathfrak {g}/\\mathfrak {h}) \\otimes S(\\mathfrak {g})^G$ , and hence is a free $S(\\mathfrak {g})^G$ -module.",
"The argument is similar to that which is set out in [7] and the following discussion.",
"Consider the projection map $\\mathfrak {g} \\rightarrow \\mathfrak {g}/\\mathfrak {h}$ .",
"This makes $\\mathfrak {g}$ a vector bundle over $\\mathfrak {g}/\\mathfrak {h}$ , and defines a natural increasing filtration on $S(\\mathfrak {g})$ via: $F_pS(\\mathfrak {g}) = \\lbrace P \\in S(\\mathfrak {g}) \\mid P \\text{ has degree } \\le p \\text{ along the fibers} \\rbrace .$ Let $\\text{gr}_F(S(\\mathfrak {g}))$ denote the associated graded ring corresponding to this filtration, and set $S(\\mathfrak {g})(p)$ to denote the $p$ -th graded component.",
"Clearly $S(\\mathfrak {g})(0) = S(\\mathfrak {g}/\\mathfrak {h})$ , and each graded component is an infinite-dimensional free $S(\\mathfrak {g}/\\mathfrak {h})$ -module.",
"There is a $K$ -algebra isomorphism: $S(\\mathfrak {g})(p) \\cong S(\\mathfrak {g}/\\mathfrak {h}) \\otimes _K S^p(\\mathfrak {h}),$ where $S^p(\\mathfrak {h})$ denotes the space of degree $p$ homogeneous polynomials on $\\mathfrak {h}$ .",
"Let $\\sigma _p: F_pS(\\mathfrak {g}) \\rightarrow S(\\mathfrak {g})(p)$ be the principal symbol map.",
"Suppose $f \\in F_pS(\\mathfrak {g})$ is a homogeneous degree $p$ polynomial whose restriction $r(f)$ to $\\mathfrak {h}$ is non-zero.",
"Then $\\sigma _p(f)$ equals the image of the element $1 \\otimes _k r(f)$ under the above isomorphism, and so is non-zero in $S(\\mathfrak {g})(p)$ .",
"To see this, choose a vector subspace $\\mathfrak {j}$ of $\\mathfrak {g}$ such that $\\mathfrak {g} = \\mathfrak {h} \\oplus \\mathfrak {j}$ .",
"This yields a graded algebra isomorphism $S(\\mathfrak {g}) = S(\\mathfrak {h}) \\otimes S(\\mathfrak {j})$ , and so one writes $F_pS(\\mathfrak {g}) = \\sum _{i \\le p} S^i(\\mathfrak {h}) \\otimes S^{p-i}(\\mathfrak {j})$ .",
"Hence $f \\in F_pS(\\mathfrak {g})$ has the form: $f = e_p \\otimes 1 + \\sum _{i \\le p} e_i \\otimes w_{p-i},$ where $e_i \\in S^i(\\mathfrak {h})$ and $w_{p-i} \\in S^{p-i}(\\mathfrak {j})$ .",
"Hence $r(f) = e_p$ and $\\sigma _p(f) = e_p \\otimes 1$ , as required.",
"Given this claim, consider the filtration $F^pS(\\mathfrak {g})^G$ in $S(\\mathfrak {g})$ .",
"For any homogeneous element $f \\in S(\\mathfrak {g})^G$ , its symbol $\\sigma _p(f)$ coincides with $r(f) \\in S(\\mathfrak {h}) \\subseteq \\text{gr}_{\\Phi } (S(\\mathfrak {g}))$ .",
"Hence the subalgebra $\\sigma _{\\Phi }(f) \\subseteq \\text{gr}_{\\Phi } (S(\\mathfrak {g}))$ coincides with $r(S(\\mathfrak {g})^G) = S(\\mathfrak {h})^W$ .",
"Let $\\lbrace a_k \\rbrace $ be a free basis for the $S(\\mathfrak {h})^W$ -module $S(\\mathfrak {h})$ , and fix $b_k \\in S(\\mathfrak {g})$ with $r(b_k) = a_k$ .",
"Then $\\sigma _p(b_k) = a_k$ .",
"The $a_k$ form a free basis of the $\\text{gr}_pR$ -module $\\text{gr}_p (S(\\mathfrak {g})) = S(\\mathfrak {h}) \\otimes S(\\mathfrak {g}/\\mathfrak {h})$ , via tensoring on the right and applying the second part of the claim.",
"It follows that the $\\lbrace b_k \\rbrace $ form a free basis of the $R$ -module $S(\\mathfrak {g})$ .",
"Theorem 2.4.10 Let $G = PGL_n$ and suppose $p|n$ .",
"Then the dual nilpotent cone $\\mathcal {N}^* \\subseteq \\mathfrak {g}^*$ is a normal variety.",
"Recall that $\\mathcal {N}^*$ is an affine variety with defining ideal $J := V(S^+(\\mathfrak {g})^G))$ .",
"It follows that its algebra of global functions $\\mathcal {O}(\\mathcal {N}^*) = S(\\mathfrak {g})/J$ .",
"Consider $Y := \\mathfrak {g}/\\mathfrak {h}$ as an affine variety.",
"Then Lemma $\\ref {Kostant}$ implies that $\\mathcal {O}(\\mathcal {N}^*)$ is a free finitely generated module over the polynomial algebra $S(Y)$ .",
"Hence $\\mathcal {N}^*$ is a Cohen-Macaulay variety.",
"By Proposition REF , $\\mathcal {N}^*$ is a closed irreducible subvariety of $\\mathfrak {g}^*$ , and the complement of the set of regular elements $U$ in $\\mathcal {N}^*$ has codimension $\\ge 2$ .",
"Hence all conditions in the statement of Theorem REF are satisfied, and so $\\mathcal {N}^*$ is normal.",
"$\\textbf {Proof of Theorem A:}$ This is immediate from Theorem REF .",
"We conclude this section with an application of this result, which will be used in later sections.",
"Corollary 2.4.11 We have an isomorphism $\\mu ^*: \\mathcal {O}(\\mathcal {N}^*) \\rightarrow \\mathcal {O}(T^*\\mathcal {B})$ .",
"The map $\\mu : T^*\\mathcal {B} \\rightarrow \\mathcal {N}$ is a resolution of singularities by Theorem REF .",
"Let $\\tau : T^*\\mathcal {B} \\rightarrow \\mathcal {N}^*$ be the composition of $\\mu $ with the $G$ -equivariant isomorphism $\\kappa : \\mathcal {N} \\rightarrow \\mathcal {N}^*$ from [18].",
"This induces an isomorphism $\\tau ^s: \\tau ^{-1}((\\mathcal {N^*})^s) \\rightarrow (\\mathcal {N^*})^s$ on the smooth points.",
"These are non-empty open subsets of $T^*\\mathcal {B}$ and $\\mathcal {N^*}$ respectively, and so $T^*\\mathcal {B}$ and $\\mathcal {N^*}$ are birationally equivalent.",
"Let $\\mathcal {Q}(A)$ denote the field of fractions of an integral domain $A$ .",
"By [12], $\\mu $ induces an isomorphism $\\mathcal {Q}(\\mathcal {O}(\\mathcal {N^*})) \\rightarrow \\mathcal {Q}(\\mathcal {O}(T^*\\mathcal {B}))$ , and so $\\mathcal {O}(T^*\\mathcal {B})$ can be considered as a subring of $\\mathcal {Q}(\\mathcal {O}(\\mathcal {N^*}))$ .",
"Since the map $T^*\\mathcal {B} \\rightarrow \\mathcal {N^*}$ is surjective, and $\\mathcal {O}(T^*\\mathcal {B})$ , $\\mathcal {O}(\\mathcal {N^*})$ are integral domains, there is an inclusion $\\mathcal {O}(\\mathcal {N^*}) \\rightarrow \\mathcal {O}(T^*\\mathcal {B})$ .",
"The map $\\tau $ is proper, and so the direct image sheaf $\\tau _*\\mathcal {O}_{T^*\\mathcal {B}}$ is a coherent $\\mathcal {O}_{\\mathcal {N^*}}$ -module.",
"In particular, taking global sections, we have that $\\Gamma (\\mathcal {N^*}, \\tau _*\\mathcal {O}_{T^*\\mathcal {B}})$ is a finitely generated $\\mathcal {O}(\\mathcal {N^*})$ -module.",
"By definition, $\\Gamma (\\mathcal {N^*}, \\tau _*\\mathcal {O}_{T^*\\mathcal {B}}) = \\mathcal {O}(T^*\\mathcal {B})$ , so $\\mathcal {O}(T^*\\mathcal {B})$ is a finitely generated $\\mathcal {O}(\\mathcal {N^*})$ -module.",
"The variety $\\mathcal {N}^*$ is normal, and so $\\mathcal {O}(\\mathcal {N}^*)$ is an integrally closed domain.",
"Let $b \\in \\mathcal {O}(T^*\\mathcal {B})$ .",
"Then clearly $\\mathcal {O}(T^*\\mathcal {B})b \\subseteq \\mathcal {O}(T^*\\mathcal {B})$ , and hence $b$ is integral over $\\mathcal {O}(\\mathcal {N}^*)$ .",
"Hence, by integral closure, $b \\in \\mathcal {N}^*$ and there is an isomorphism $\\mathcal {O}(\\mathcal {N}^*) \\rightarrow \\mathcal {O}(T^*\\mathcal {B})$ ."
],
[
"Analogous results when $G$ is not of type A",
"The restriction that $G = PGL_n$ , $p|n$ plays a role in only a few places in the argument that $\\mathcal {N}^*$ is a normal variety.",
"In this section, we indicate some of the issues that arise when we replace $PGL_n$ by a more general simple algebraic group of adjoint type.",
"Theorem REF demonstrated that, in case $G = PGL_n$ , $p|n$ , the Weyl group invariants $S(\\mathfrak {h})^W$ is a polynomial ring.",
"This result is usually false in bad characteristic.",
"In case the $W$ -action on $\\mathfrak {h}$ is irreducible,[6] gives a full classification of the types in which this result holds, drawing on [20].",
"Proposition 2.5.1 Suppose the pair (Dynkin diagram of $G$ , $p$ ) lies in the following list: (a) ($E_7$ , 3), (b) ($E_8$ , 2), (c) ($E_8$ , 3), (d) ($E_8$ , 5), (e) ($F_4$ , 3), (f) ($G_2$ , 2).",
"Then the $W$ -action on $\\mathfrak {h}$ is irreducible.",
"In all of these cases, the argument in [18] demonstrates that there is a $G$ -equivariant bijection $\\kappa : \\mathfrak {g} \\rightarrow \\mathfrak {g}^*$ , which restricts to a $G$ -equivariant bijection $\\mathfrak {h} \\rightarrow \\mathfrak {h}^*$ .",
"Furthermore, the classification in [16] demonstrates that $\\mathfrak {g}$ is simple.",
"Given these two statements, we may apply the same proof as that given in [10] to obtain the result.",
"Theorem 2.5.2 Suppose $G$ is of type $G_2$ and $p = 2$ .",
"Then the invariant ring $S(\\mathfrak {h})^W$ is polynomial.",
"This follows from the calculations in [20].",
"In case $G$ is of type $G_2$ and $p = 2$ , we may apply the same argument as for $G = PGL_n$ to obtain the following result.",
"Theorem 2.5.3 Let $(G,p) = (G_2, 2)$ .",
"Then the dual nilpotent cone $\\mathcal {N}^* \\subseteq \\mathfrak {g}^*$ is a normal variety.",
"$\\textbf {Proof of Theorem B:}$ This is immediate from Theorem REF .",
"If $S(\\mathfrak {h})^W$ is not polynomial, there are significant obstacles to generalising the result that $\\mathcal {N}^*$ is a normal variety.",
"In particular, the following behaviour may be observed.",
"- Kostant's freeness theorem, stated as Theorem REF , fails.",
"This means that $S(\\mathfrak {h})$ is not free as an $S(\\mathfrak {h})^W$ -module, meaning that we cannot apply the argument in Lemma REF to show that $\\mathcal {N^*}$ is a Cohen-Macaulay variety.",
"- The Steinberg quotient $\\chi : \\mathfrak {g} \\rightarrow K^n$ , defined in Definition REF , makes sense as an abstract function, but since the generators $\\lbrace f_1, \\cdots , f_n \\rbrace $ of $S(\\mathfrak {h})^W$ are not algebraically independent, we cannot apply the argument in Lemma REF to show that the smooth elements of $\\mathcal {N}$ coincide with the regular elements, which is a key step in the proof that the Springer resolution $\\mu : T^*\\mathcal {B} \\rightarrow \\mathcal {N}$ is a resolution of singularities for $\\mathcal {N}$ .",
"Calculations in [6] show that, in the following cases (Dynkin diagram of $G$ , $p$ ), the invariant ring $S(\\mathfrak {h})^W$ is not even Cohen-Macaulay.",
"(a) $(E_7, 3)$ , (b) $(E_8, 3)$ , (c) $(E_8, 5)$ .",
"Conjecture 2.5.4 In case the invariant ring $S(\\mathfrak {h})^W$ is not Cohen-Macaulay, is it true that the dual nilpotent cone $\\mathcal {N}^*$ is not a normal variety?"
],
[
"Generalising the Beilinson-Bernstein theorem for $\\widehat{\\mathcal {D}^{\\lambda }_{n,K}}$",
"In this section, we apply the results of Section to the constructions given in [2].",
"This allows us to weaken the restrictions on the characteristic of the base field given in [2], thereby providing us with generalisations of their results.",
"Throughout Section , we suppose $R$ is a fixed complete discrete valuation ring with uniformiser $\\pi $ , residue field $k$ and field of fractions $K$ .",
"Assume throughout this section that $K$ has characteristic 0 and $k$ is algebraically closed.",
"We recall some of the arguments from [2], to define the sheaf of enhanced vector fields $\\widetilde{\\mathcal {T}}$ on a smooth scheme $X$ , and the relative enveloping algebra $\\widetilde{\\mathcal {D}}$ of an $\\textbf {H}$ -torsor $\\xi : \\widetilde{X} \\rightarrow X$ .",
"Let $X$ be a smooth separated $R$ -scheme that is locally of finite type.",
"Let $\\textbf {H}$ be a flat affine algebraic group defined over $R$ of finite type, and let $\\widetilde{X}$ be a scheme equipped with an $\\textbf {H}$ -action.",
"Definition 3.1.1 A morphism $\\xi : \\widetilde{X} \\rightarrow X$ is an $\\textbf {H}-\\emph {torsor}$ if: (i) $\\xi $ is faithfully flat and locally of finite type, (ii) the action of $\\textbf {H}$ respects $\\xi $ , (iii) the map $\\widetilde{X} \\times \\textbf {H} \\rightarrow \\widetilde{X} \\times _X \\widetilde{X}$ , $(x, h) \\rightarrow (x, hx)$ is an isomorphism.",
"An open subscheme $U$ of $X$ $\\emph {trivialises the torsor}$ $\\xi $ if there is an $\\textbf {H}$ -invariant isomorphism: $U \\times \\textbf {H} \\rightarrow \\xi ^{-1}(U)$ where $\\textbf {H}$ acts on $U \\times \\textbf {H}$ by left translation on the second factor.",
"Definition 3.1.2 Let $\\mathcal {S}_X$ denote the set of open subschemes $U$ of $X$ such that: (i) $U$ is affine, (ii) $U$ trivialises $\\xi $ , (iii) $\\mathcal {O}(U)$ is a finitely generated $R$ -algebra.",
"$\\xi $ is $\\emph {locally trivial}$ for the Zariski topology if $X$ can be covered by open sets in $\\mathcal {S}_X$ .",
"Lemma 3.1.3 If $\\xi $ is locally trivial, then $\\mathcal {S}_X$ is a base for $X$ .",
"Since $X$ is separated, $\\mathcal {S}_X$ is stable under intersections.",
"If $U \\in \\mathcal {S}_X$ and $W$ is an open affine subscheme of $U$ , then $W \\in \\mathcal {S}_X$ .",
"Hence $\\mathcal {S}_X$ is a base for $X$ .",
"The action of $\\textbf {H}$ on $\\widetilde{X}$ induces a rational action of $\\textbf {H}$ on $\\mathcal {O}(V)$ for any $\\textbf {H}$ -stable open subscheme $V \\subseteq \\widetilde{X}$ , and therefore induces an action of $\\textbf {H}$ on $\\mathcal {T}_{\\widetilde{X}}$ via: $(h \\cdot \\partial )(f) = h \\cdot \\partial (h^{-1} \\cdot f)$ for $\\partial \\in \\mathcal {T}_{\\widetilde{X}}, f \\in \\mathcal {O}(\\widetilde{X})$ and $h \\in \\textbf {H}$ .",
"The $\\emph {sheaf of enhanced vector fields}$ on $X$ is: $\\widetilde{\\mathcal {T}} := (\\xi _*\\mathcal {T}_{\\widetilde{X}})^{\\textbf {H}}.$ Differentiating the $\\textbf {H}$ -action on $\\widetilde{X}$ gives an $R$ -linear Lie algebra homomorphism: $j: \\mathfrak {h} \\rightarrow \\mathcal {T}_{\\widetilde{X}}$ where $\\mathfrak {h}$ is the Lie algebra of $\\textbf {H}$ .",
"Definition 3.1.4 Let $\\xi : \\widetilde{X} \\rightarrow X$ be an $\\textbf {H}$ -torsor.",
"Then $\\xi _*\\mathcal {D}_{\\widetilde{X}}$ is a sheaf of $R$ -algebras with an $\\textbf {H}$ -action.",
"The $\\emph {relative enveloping algebra}$ of the torsor is the sheaf of $\\textbf {H}$ -invariants of $\\xi _*\\mathcal {D}_{\\widetilde{X}}$ : $\\widetilde{\\mathcal {D}} := (\\xi _*\\mathcal {D}_{\\widetilde{X}})^{\\textbf {H}}.$ This sheaf has a natural filtration: $F_m\\widetilde{\\mathcal {D}} := (\\xi _*F_m\\mathcal {D}_{\\widetilde{X}})^{\\textbf {H}}$ induced by the filtration on $\\mathcal {D}_{\\widetilde{X}}$ by order of differential operator.",
"Let $\\textbf {B}$ be a Borel subgroup of $\\textbf {G}$ .",
"Let $\\textbf {N}$ be the unipotent radical of $\\textbf {B}$ , and $\\textbf {H} := \\textbf {B}/\\textbf {N}$ the abstract Cartan group.",
"Let $\\widetilde{\\mathcal {B}}$ denote the homogeneous space $\\textbf {G}/\\textbf {N}$ .",
"There is an $\\textbf {H}$ -action on $\\widetilde{\\mathcal {B}}$ defined by: $b\\textbf {N} \\cdot g \\textbf {N} := gb\\textbf {N}$ which is well-defined since $[\\textbf {B}, \\textbf {B}]$ is contained in $\\textbf {N}$ .",
"$\\mathcal {B} := \\textbf {G}/\\textbf {B}$ is the $\\emph {flag variety}$ of $\\textbf {G}$ .",
"$\\widetilde{\\mathcal {B}}$ is the $\\emph {basic affine space}$ of $\\textbf {G}$ .",
"By the splitting assumption of $\\textbf {G}$ , we can find a Cartan subgroup $\\textbf {T}$ of $\\textbf {G}$ complementary to $\\textbf {N}$ in $\\textbf {B}$ .",
"This is naturally isomorphic to $\\textbf {H}$ , and induces an isomorphism of the corresponding Lie algebras $\\mathfrak {t} \\rightarrow \\mathfrak {h}$ .",
"We let $\\textbf {W}$ denote the Weyl group of $\\textbf {G}$ , and let $\\textbf {W}_k$ denote the Weyl group of $\\textbf {G}_k$ , the $k$ -points of the algebraic group $\\textbf {G}$ .",
"We may differentiate the natural $\\textbf {G}$ -action on $\\widetilde{\\mathcal {B}}$ to obtain an $R$ -linear Lie homomorphism: $\\varphi : \\mathfrak {g} \\rightarrow \\mathcal {T}_{\\widetilde{\\mathcal {B}}}.$ Since the $\\textbf {G}$ -action commutes with the $\\textbf {H}$ -action on $\\widetilde{\\mathcal {B}}$ , this map descends to an $R$ -linear Lie homomorphism $\\varphi : \\mathfrak {g} \\rightarrow \\widetilde{\\mathcal {T}}_{\\mathcal {B}}$ and an $\\mathcal {O}_{\\mathcal {B}}$ -linear morphism: $\\varphi : \\mathcal {O}_{\\mathcal {B}} \\otimes \\mathfrak {g} \\rightarrow \\widetilde{\\mathcal {T}}_{\\mathcal {B}}$ of locally free sheaves on $\\mathcal {B}$ .",
"Dualising, we obtain a morphism of vector bundles over $\\mathcal {B}$ : $\\varphi ^*: \\widetilde{T^*\\mathcal {B}} \\rightarrow \\mathcal {B} \\times \\mathfrak {g}^*$ from the enhanced cotangent bundle to the trivial vector bundle of rank dim $\\mathfrak {g}$ .",
"Definition 3.1.5 The $\\emph {enhanced moment map}$ is the composition of $\\varphi ^*$ with the projection onto the second coordinate: $\\beta : \\widetilde{T^*\\mathcal {B}} \\rightarrow \\mathfrak {g}^*.$ We may apply the deformation functor ([2]) to the map $j: U(\\mathfrak {h}) \\rightarrow \\widetilde{\\mathcal {D}}$ , defined above Definition REF , to obtain a central embedding of the constant sheaf $U(\\mathfrak {h})_n$ into $\\widetilde{\\mathcal {D}}_n$ .",
"This gives $\\widetilde{\\mathcal {D}}_n$ the structure of a $U(\\mathfrak {h})_n$ -module.",
"Let $\\lambda \\in \\text{Hom}_R(\\pi ^n\\mathfrak {h}, R)$ be a linear functional.",
"This extends to an $R$ -algebra homomorphism $U(\\mathfrak {h})_n \\rightarrow R$ , which gives $R$ the structure of a $U(\\mathfrak {h})_n$ -module, denoted $R_{\\lambda }$ .",
"Definition 3.1.6 The $\\emph {sheaf of deformed twisted differential operators}$ $\\mathcal {D}^{\\lambda }_n$ on $\\mathcal {B}$ is the sheaf: $\\mathcal {D}^{\\lambda }_n := \\widetilde{\\mathcal {D}_n} \\otimes _{U(\\mathfrak {h})_n} R_{\\lambda }$ By [2], this is a sheaf of deformable $R$ -algebras.",
"Definition 3.1.7 The $\\pi $ -$\\emph {adic completion}$ of $\\mathcal {D}^{\\lambda }_n$ is $\\widehat{\\mathcal {D}^{\\lambda }_n} := \\varprojlim \\mathcal {D}^{\\lambda }_n /\\pi ^a\\mathcal {D}^{\\lambda }_n$ .",
"Furthermore, set $\\widehat{\\mathcal {D}^{\\lambda }_{n,K}} := \\widehat{\\mathcal {D}^{\\lambda }_n} \\otimes _R K$ .",
"The adjoint action of $\\textbf {G}$ on $\\mathfrak {g}$ extends to an action on $U(\\mathfrak {g})$ by ring automorphisms, which is filtration-preserving and so descends to an action on $\\text{gr } U(\\mathfrak {g}) \\cong S(\\mathfrak {g})$ .",
"Let: $\\psi : S(\\mathfrak {g})^{\\textbf {G}} \\rightarrow S(\\mathfrak {t})$ denote the composition of the inclusion $S(\\mathfrak {g})^{\\textbf {G}} \\rightarrow S(\\mathfrak {g})$ with the projection $S(\\mathfrak {g}) \\rightarrow S(\\mathfrak {t})$ .",
"By [9], the image of $\\psi $ is contained in $S(\\mathfrak {t})^{\\textbf {W}}$ , and $\\psi $ is injective.",
"Since taking $\\textbf {G}$ -invariants is left exact, we have an inclusion $\\text{gr }(U(\\mathfrak {g})^{\\textbf {G}}) \\rightarrow S(\\mathfrak {g})^{\\textbf {G}}$ .",
"Our next proposition gives a description of the associated graded ring of $U(\\mathfrak {g})^{\\textbf {G}}$ .",
"Proposition 3.1.8 The rows of the diagram: Figure: NO_CAPTIONare exact, and each vertical map is an isomorphism.",
"View the diagram as a sequence of complexes $C^{\\bullet } \\rightarrow D^{\\bullet } \\rightarrow E^{\\bullet }$ .",
"Since $\\pi $ generates the maximal ideal $\\mathfrak {m}$ of $R$ by definition, and $R/\\mathfrak {m} = k$ , it is clear that each complex is exact in the left and in the middle.",
"The exactness of $E^{\\bullet }$ follows from the fact that $S(\\mathfrak {t}_k)^{\\textbf {W}_k}$ is a polynomial ring by Theorem REF : since $n > 2$ we may fix homogeneous generators $s_1, \\cdots , s_l$ and lift these generators to homogeneous generators $S_1, \\cdots , S_l$ of the ring $S(\\mathfrak {t})^{\\textbf {W}}$ with $s_i = S_i (\\text{mod } \\mathfrak {m})$ by the proof of [20].",
"Hence the map $S(\\mathfrak {t})^{\\textbf {W}} \\rightarrow S(\\mathfrak {t}_k)^{\\textbf {W}_k}$ is surjective, and the complex $E^{\\bullet }$ is exact.",
"By [9], $\\psi $ is injective, and since $p$ is nonspecial from Definition REF , $\\psi _k$ is an isomorphism by Theorem REF .",
"Thus the composite map of complexes $\\psi ^{\\bullet } \\circ \\iota ^{\\bullet }$ is injective.",
"Set $F^{\\bullet } := \\text{coker}(\\psi ^{\\bullet } \\circ \\iota ^{\\bullet })$ : by definition, the sequence of complexes $0 \\rightarrow C^{\\bullet } \\rightarrow E^{\\bullet } \\rightarrow F^{\\bullet } \\rightarrow 0$ is exact.",
"Since $C^{\\bullet }$ is exact in the left and in the middle, $H^0(C^{\\bullet }) = H^1(C^{\\bullet }) = 0$ .",
"As $E^{\\bullet }$ is exact, taking the long exact sequence of cohomology shows that $H^0(F^{\\bullet }) = H^2(F^{\\bullet }) = 0$ and yields an isomorphism $H^1(F^{\\bullet }) \\cong H^2(C^{\\bullet })$ .",
"Since $K$ is a field of characteristic zero, the map $\\psi _K \\circ \\iota _K: \\text{gr}(U(\\mathfrak {g}_K)^{\\textbf {G}_K}) \\rightarrow S(\\mathfrak {t}_K)^{\\textbf {W}_K}$ is an isomorphism by [9].",
"Hence $F^0 = F^1 = \\text{coker}(\\psi \\circ \\iota )$ is $\\pi $ -torsion.",
"Now $H^0(F^{\\bullet }) = 0$ , and so we have an exact sequence $0 \\rightarrow F^0 \\rightarrow F^1$ .",
"So $F^0 = F^1 = 0$ , and hence $H^1(F^{\\bullet }) = H^2(C^{\\bullet }) = 0$ .",
"It follows that the top row $C^{\\bullet }$ is exact.",
"Hence $\\psi ^{\\bullet } \\circ \\iota ^{\\bullet }: C^{\\bullet } \\rightarrow E^{\\bullet }$ is an isomorphism in all degrees except possibly 2, and so is an isomorphism via the Five Lemma.",
"The result follows from the fact that $\\psi ^{\\bullet }$ and $\\iota ^{\\bullet }$ are both injections.",
"It follows that, since $\\psi \\circ \\iota $ is a graded isomorphism and $p$ is nonspecial, $\\text{gr}(U(\\mathfrak {g})^\\textbf {G})$ is isomorphic to a commutative polynomial algebra over $R$ in $l$ variables by Theorem REF .",
"The commutative polynomial algebra $R[x_1, \\cdots , x_l]$ is a free $R$ -module and hence is flat, and so $(U(\\mathfrak {g})^\\textbf {G})$ is a deformable $R$ -algebra by [2].",
"Furthermore, $\\widehat{U(\\mathfrak {g})^\\textbf {G}_{n,K}}$ is also a commutative polynomial algebra over $R$ in $l$ variables, so the $\\pi $ -adic completion $\\widehat{U(\\mathfrak {g})^\\textbf {G}_{n,K}}$ is a commutative Tate algebra.",
"By [2], we have a commutative square consisting of deformable $R$ -algebras: Figure: NO_CAPTIONWe set: $\\mathcal {U}^{\\lambda }_n := U(\\mathfrak {g}) \\otimes _{(U(\\mathfrak {g})^{\\textbf {G}})_n} R_{\\lambda }, \\\\\\widehat{\\mathcal {U}^{\\lambda }_n} := \\varprojlim \\frac{\\mathcal {U}^{\\lambda }_n}{\\pi ^a \\mathcal {U}^{\\lambda }_n}, \\\\\\widehat{\\mathcal {U}^{\\lambda }_{n,K}} := \\widehat{\\mathcal {U}^{\\lambda }_n} \\otimes _R K.$ By commutativity of the diagram, the map: $U(\\phi )_n \\otimes (j \\circ i)_n: U(\\mathfrak {g})_n \\otimes U(\\mathfrak {t})_n \\rightarrow \\widetilde{\\mathcal {D}_n}$ factors through $U((\\mathfrak {g})^\\textbf {G})_n$ , and we obtain the algebra homomorphisms: $\\phi ^{\\lambda }_n: \\mathcal {U}^{\\lambda }_n \\rightarrow \\mathcal {D}^{\\lambda }_n, \\\\\\widehat{\\phi ^{\\lambda }_n}: \\widehat{\\mathcal {U}^{\\lambda }_n} \\rightarrow \\widehat{\\mathcal {D}^{\\lambda }_n}, \\\\\\widehat{\\phi ^{\\lambda }_{n,K}}: \\widehat{\\mathcal {U}^{\\lambda }_{n,K}} \\rightarrow \\widehat{\\mathcal {D}^{\\lambda }_{n,K}}.$ Theorem 3.1.9 (a) $\\widehat{\\mathcal {U}^{\\lambda }_{n,K}} \\cong \\widehat{U(\\mathfrak {g})_{n,K}} \\otimes _{\\widehat{U(\\mathfrak {g})^\\textbf {G}_{n,K}}} K_{\\lambda }$ is an almost commutative affinoid $K$ -algebra.",
"(b) The map $\\widehat{\\phi ^{\\lambda }_{n,K}}: \\widehat{\\mathcal {U}^{\\lambda }_{n,K}} \\rightarrow \\Gamma (\\mathcal {B}, \\widehat{\\mathcal {D}^{\\lambda }_{n,K}})$ is an isomorphism of complete doubly filtered $K$ -algebras.",
"(c) There is an isomorphism $S(\\mathfrak {g}_k) \\otimes _{S(\\mathfrak {g}_k)^{\\textbf {G}_k}} k \\cong \\text{Gr }(\\widehat{\\mathcal {U}^{\\lambda }_{n,K}})$ .",
"(a): This is identical to the proof given in [2].",
"(b): Let $ \\lbrace U_1, \\cdots , U_m \\rbrace $ be an open cover of $\\mathcal {B}$ by open affines that trivialise the torsor $\\xi $ , which exists by [2].",
"The special fibre $\\mathcal {B}_k$ is covered by the special fibres $U_{i,k}$ .",
"It suffices to show that the complex: $C^{\\bullet }: 0 \\rightarrow \\widehat{\\mathcal {U}_{n,K}} \\rightarrow \\bigoplus _{i=1}^m \\widehat{\\mathcal {D}^{\\lambda }_{n,K}}(U_i) \\rightarrow \\bigoplus _{i<j} \\widehat{\\mathcal {D}^{\\lambda }_{n,K}}(U_i \\cap U_j)$ is exact.",
"Clearly, $C^{\\bullet }$ is a complex in the category of complete doubly-filtered $K$ -algebras, and so it suffices to show that the associated graded complex $\\text{Gr}(C^{\\bullet })$ is exact.",
"By [2], there is a commutative diagram with exact rows: Figure: NO_CAPTIONVia the identification $\\text{gr}(U(\\mathfrak {g})) = S(\\mathfrak {g})$ , Proposition REF induces a commutative square: Figure: NO_CAPTIONwhere the horizontal maps are isomorphisms and the vertical maps are inclusions.",
"Since $\\text{Gr}(K_{\\lambda })$ is the trivial $\\text{Gr}(\\widehat{U(\\mathfrak {g})^{\\textbf {G}}_{n,K}})$ -module $k$ , we have a natural surjection: $S(\\mathfrak {g}_k) \\otimes _{S(\\mathfrak {g}_k)^{{\\textbf {G}}_k}} k \\cong \\text{Gr}(\\widehat{U(\\mathfrak {g})_{n,K}} \\otimes _{\\text{Gr}(\\widehat{U(\\mathfrak {g})^{\\textbf {G}}_{n,K}})} \\text{Gr}(K_{\\lambda })) \\rightarrow \\text{Gr}(\\widehat{\\mathcal {U}^{\\lambda }_{n,K}}).$ This surjection fits into the commutative diagram: Figure: NO_CAPTIONThe bottom row is $\\text{Gr}(C^{\\bullet })$ by definition, and the top row is induced by the moment map $T^*\\mathcal {B}_k \\rightarrow \\mathfrak {g}_k^*$ .",
"To see this, note that by Lemma REF , we have an identification $\\widetilde{\\mathcal {N}^*} \\rightarrow T^*\\mathcal {B}$ under our assumptions on $p$ , and so exactness of the top row is equivalent to the existence of an isomorphism: $S(\\mathfrak {g}) \\otimes _{S(\\mathfrak {g})^{\\textbf {G}}} k \\cong \\Gamma (\\widetilde{\\mathcal {N}^*}, \\mathcal {O}_{\\widetilde{\\mathcal {N}^*}}).$ By Theorem REF , $\\mathcal {N}^*$ is a normal variety and, by Theorem REF , the map $\\gamma : T^*\\mathcal {B} \\rightarrow \\mathcal {N}^*$ is a resolution of singularities.",
"It follows, by Corollary REF , that there is an isomorphism of global sections: $\\gamma ^*: \\Gamma (\\mathcal {N}^*, \\mathcal {O}_{\\mathcal {N}^*}) \\rightarrow \\Gamma (T^*\\mathcal {B}, \\mathcal {O}_{T^*\\mathcal {B}}).$ Recall from the proof of Theorem REF that $\\mathcal {O}(\\mathcal {N}^*) = S(\\mathfrak {g}) \\otimes _{S(\\mathfrak {g})^{\\textbf {G}}} k$ .",
"Putting these isomorphisms together, we see that $S(\\mathfrak {g}) \\otimes _{S(\\mathfrak {g})^{\\textbf {G}}} k \\cong \\Gamma (\\widetilde{\\mathcal {N}^*}, \\mathcal {O}_{\\widetilde{\\mathcal {N}^*}})$ .",
"Now the second and third vertical arrows are isomorphisms by [2], which shows that $\\text{Gr}(C^{\\bullet })$ is exact.",
"(c) This is immediate, since one can also show that the first vertical arrow in the above diagram is an isomorphism via the Five Lemma.",
"Definition 3.1.10 For each $\\lambda \\in \\text{Hom}_R(\\pi ^n\\mathfrak {h}, R)$ , we define a functor: $\\text{Loc}^{\\lambda }: \\widehat{U(\\mathfrak {g})^{\\lambda }_{n,K}}-\\text{mod} \\rightarrow \\widehat{\\mathcal {D}^{\\lambda }_{n,K}}-\\text{mod}$ given by $M \\mapsto \\widehat{\\mathcal {D}^{\\lambda }_{n,K}} \\otimes _{\\widehat{\\mathcal {U}^{\\lambda }_{n,K}}} M$ ."
],
[
"Modules over completed enveloping algebras",
"The adjoint action of $\\textbf {G}$ on $\\mathfrak {g}$ induces an action of $\\textbf {G}$ on $U(\\mathfrak {g})$ by algebra automorphisms.",
"Composing the inclusion $U(\\mathfrak {g})^{\\textbf {G}} \\rightarrow U(\\mathfrak {g})$ with the projection $U(\\mathfrak {g}) \\rightarrow U(\\mathfrak {t})$ defined by the direct sum decomposition $\\mathfrak {g} = \\mathfrak {n} \\oplus \\mathfrak {t} \\oplus \\mathfrak {n}^+$ yields the $\\emph {Harish-Chandra}$ $\\emph {homomorphism}$ : $\\phi : U(\\mathfrak {g})^{\\textbf {G}} \\rightarrow U(\\mathfrak {t})$ This is a morphism of deformable $R$ -algebras, so by applying the deformation and $\\pi $ -adic completion functors, one obtains the $\\emph {deformed Harish}$ -$\\emph {Chandra homomorphism}$ : $\\widehat{\\phi _{n,K}}: \\widehat{U(\\mathfrak {g})^\\textbf {G}_{n,K}} \\rightarrow \\widehat{U(\\mathfrak {t})_{n,K}}$ which we will denote via the shorthand $\\widehat{\\phi }: Z \\rightarrow \\widetilde{Z}$ .",
"We have an action of the Weyl group $\\textbf {W}$ on the dual Cartan subalgebra $\\mathfrak {t}^*_K$ via the shifted dot-action: $w \\bullet \\lambda = w(\\lambda + \\rho ^{\\prime }) - \\rho ^{\\prime }$ where $\\rho ^{\\prime }$ is equal the half-sum of the T-roots on $\\mathfrak {n}^+$ .",
"Viewing $U(\\mathfrak {t})_K$ as an algebra of polynomial functions on $\\mathfrak {t}^*_K$ , we obtain a dot-action of $\\textbf {W}$ on $U(\\mathfrak {t})_K$ .",
"This action preserves the $R$ -subalgebra $U(\\mathfrak {t})_n$ of $U(\\mathfrak {t})_K$ and so extends naturally to an action of $\\textbf {W}$ on $\\widetilde{Z}$ .",
"Theorem 3.2.1 Suppose that that $\\textbf {G} = PGL_n$ , $p|n$ , and $n > 2$ .",
"Then: (a) set $A := \\widehat{U(\\mathfrak {g})_{n,K}}$ .",
"The algebra $Z$ is contained in the centre of $A$ .",
"(b) the map $\\widehat{\\phi }$ is injective, and its image is the ring of invariants $\\widetilde{Z}^{\\textbf {W}}$ .",
"(c) the algebra $\\widetilde{Z}$ is free of rank $|\\textbf {W}|$ as a module over $\\widetilde{Z}^{\\textbf {W}}$ .",
"(d) $\\widetilde{Z}^{\\textbf {W}}$ is isomorphic to a Tate algebra $K \\langle S_1, \\cdots , S_l \\rangle $ as complete doubly filtered $K$ -algebras.",
"(a): The algebra $U(\\mathfrak {g})^{\\textbf {G}}_K$ is central in $U(\\mathfrak {g})_K$ via [15].",
"Since $U(\\mathfrak {g})_K$ is dense in $A$ , it is also contained in the centre of $A$ .",
"But $U(\\mathfrak {g})^{\\textbf {G}}_K$ is also dense in $Z$ , and so $Z$ is central in $A$ .",
"(b): By the Harish-Chandra homomorphism (see [9]), $\\phi $ sends $U(\\mathfrak {g})^{\\textbf {G}}_K$ onto $U(\\mathfrak {t})^{\\textbf {W}}_K$ , and so $\\widehat{\\phi }(Z)$ is contained in $\\widetilde{Z}^{\\textbf {W}}$ .",
"This is a complete doubly filtered algebra whose associated graded ring $\\text{Gr}(\\widetilde{Z}^{\\textbf {W}})$ can be identified with $S(\\mathfrak {t}_k)^{\\textbf {W}_k}$ .",
"This induces a morphism of complete doubly filtered $K$ -algebras $\\alpha : Z \\rightarrow \\widetilde{Z}^{\\textbf {W}}$ .",
"Its associated graded map $\\text{Gr}(\\alpha ): \\text{Gr}(Z) \\rightarrow \\text{Gr}(\\widetilde{Z}^{\\textbf {W}})$ can be identified with the isomorphism $\\psi _k: S(\\mathfrak {g}_k)^{\\textbf {G}_k} \\rightarrow S(\\mathfrak {t}_k)^{\\textbf {W}_k}$ by Proposition REF .",
"Hence $\\text{Gr}(\\alpha )$ is an isomorphism, and so $\\alpha $ is an isomorphism by completeness.",
"(c): By Theorem REF and Theorem REF , $S(\\mathfrak {t}_k)$ is a free graded $S(\\mathfrak {t}_k)^{\\textbf {W}_k}$ -module of rank $|\\textbf {W}|$ .",
"Hence, by [2], $\\widetilde{Z}$ is finitely generated over $Z$ , and in fact is free of rank $|\\textbf {W}|$ .",
"(d): By Theorem REF and Theorem REF , $S(\\mathfrak {t}_k)^{\\textbf {W}_k}$ is a polynomial algebra in $l$ variables.",
"Fix double lifts $s_1, \\cdots , s_l \\in U(\\mathfrak {t})^{\\textbf {W}}$ of these generators, as in the proof of Proposition REF .",
"Define an $R$ -algebra homomorphism $R[S_1, \\cdots , S_l] \\rightarrow \\widetilde{Z}^{\\textbf {W}}$ which sends $S_i$ to $s_i$ .",
"This extends to an isomorphism $K \\langle S_1, \\cdots , S_l \\rangle \\rightarrow \\widetilde{Z}^{\\textbf {W}}$ of complete doubly filtered $K$ -algebras.",
"We identify the $k$ -points of the scheme $\\mathfrak {g}^* := \\text{Spec}(\\text{Sym}_R \\mathfrak {g})$ with the dual of the $k$ -vector space $\\mathfrak {g}$ , so $\\mathfrak {g}^*(k) = \\mathfrak {g}^*_k$ .",
"Let $G$ denote the $k$ -points of the algebraic group scheme $\\textbf {G}$ .",
"$G$ acts on $\\mathfrak {g}_k$ and $\\mathfrak {g}^*_k$ via the adjoint and coadjoint action respectively.",
"Recall the definition of the enhanced moment map $\\beta : \\widetilde{T^*\\mathcal {B}}(k) \\rightarrow \\mathfrak {g}^*_k$ from Definition REF .",
"Given $y \\in \\mathfrak {g}^*_k$ , write $G.y$ to denote the $G$ -orbit of $y$ under the coadjoint action.",
"We write $\\mathcal {N}$ (resp.",
"$\\mathcal {N}^*$ ) to denote the nilpotent cone (resp.",
"dual nilpotent cone) of the $k$ -vector spaces $\\mathfrak {g}_k$ and $\\mathfrak {g}^*_k$ .",
"Proposition 3.2.2 Suppose $p$ is nonspecial for $G$ .",
"For any $y \\in \\mathcal {N}^*$ , we have $\\text{dim } \\beta ^{-1}(y) = \\text{dim } \\mathcal {B} - \\frac{1}{2} \\text{dim } G.y$ .",
"This is stated for $\\mathcal {N}$ as [18].",
"The result follows by applying the $G$ -equivariant bijection $\\kappa : \\mathcal {N} \\rightarrow \\mathcal {N}^*$ from [18].",
"We now let $\\mathfrak {g}_{\\mathbb {C}}$ denote the complex semisimple Lie algebra with the same root system as $G$ , and let $G_{\\mathbb {C}}$ be the corresponding adjoint algebraic group.",
"By [8], there is a unique non-zero nilpotent $G_{\\mathbb {C}}$ -orbit in $\\mathfrak {g}^*_{\\mathbb {C}}$ , under the coadjoint action, of minimal dimension.",
"Since each coadjoint $G_{\\mathbb {C}}$ -orbit is a symplectic manifold, it follows that each of these dimensions is an even integer.",
"We set: $r := \\frac{1}{2} \\text{min } \\lbrace \\text{dim } G_{\\mathbb {C}} \\cdot y \\mid 0 \\ne y \\in \\mathfrak {g}_{\\mathbb {C}} \\rbrace $ Proposition 3.2.3 For any non-zero $y \\in \\mathcal {N}^*, \\frac{1}{2} \\text{dim } G \\cdot y \\ge r$ , with no restrictions on $(G,p)$ .",
"We will demonstrate that this inequality holds for all split semisimple algebraic groups $G$ defined over an algebraically closed field $k$ of positive characteristic.",
"When the characteristic $p$ is small, we will proceed via a case-by-case calculation of the maximal dimension of the centraliser $Z_G(y)$ of $y \\in \\mathcal {N}^*$ .",
"By Proposition REF , $\\text{dim } \\beta ^{-1}(y) = \\text{dim } \\mathcal {B} - \\frac{1}{2} \\text{dim } G \\cdot y$ .",
"We may assume $y \\in \\mathcal {N}$ and $G$ acts on $\\mathfrak {g}$ via the adjoint action by [18].",
"By [14], we see that: $\\text{dim }\\beta ^{-1}(y) = \\frac{1}{2}(\\text{dim } Z_G(y) - \\text{rk }(G))$ where $Z_G(y)$ denotes the centraliser of $y$ in $G$ .",
"Hence it suffices to demonstrate that the following inequality: $\\text{dim } \\mathcal {B} - \\frac{1}{2}(\\text{dim } Z_G(y) - \\text{rk }(G)) \\ge r$ holds in all types.",
"We evaluate on a case-by-case basis, aiming to find the maximal dimension of the centraliser.",
"We first note that, using the work of [25], we have the following table: Table: NO_CAPTIONBy [21], when $p$ is nonspecial, the dimension of the centraliser is independent of the isogeny type of $G$ .",
"Since $p$ is always nonspecial for a group of type $A$ , it therefore suffices to consider $Z_{\\mathfrak {sl}_n}(y)$ .",
"Since $p$ is good and $SL_n$ is a simply connected algebraic group, by [21], it suffices to consider the centraliser of a non-identity unipotent element in $SL_n$ .",
"Via the identification $GL_n(k) = SL_n(k)Z(GL_n(k))$ , it is sufficient to compute $Z_{GL_n(k)}(u)$ , for some unipotent matrix $u$ .",
"This dimension is bounded above by $n^2$ , the dimension of $GL_n(k)$ as an algebraic group, and so we have the expression: $\\text{dim } \\mathcal {B} - \\frac{1}{2}(\\text{dim } Z_G(y) - \\text{rk }(G)) \\\\\\ge \\frac{1}{2} n(n+1) - \\frac{1}{2}(n^2 - n)\\ge n.$ Hence the inequality is verified in type $A$ .",
"For the remaining classical groups, view $y \\in \\mathcal {N}$ as a nilpotent matrix, which without loss of generality may be taken to be in Jordan normal form.",
"Let $m_1 \\ge \\cdots \\ge m_r$ be the sizes of the Jordan blocks, with $\\sum _{i=1}^r m_i = n$ , the rank of the group.",
"By [13], we have: $\\text{dim } Z_G(y) = \\sum _{i=1}^r (im_i - \\chi _V(m_i))$ where $\\chi _V$ is a function $\\chi _V: \\mathbb {N} \\rightarrow \\mathbb {N}$ .",
"It follows that: $\\text{dim } Z_G(y) \\le \\sum _{i=1}^r im_i = \\sum _{j=1}^n \\sum _{i=j}^r m_i.$ Since $m_1 \\ge \\cdots \\ge m_r$ by construction, the maximum value of this sum is attained when $m_k = 1$ for all $k$ .",
"Hence we obtain the inequality $\\text{dim } Z_G(y) \\le \\frac{1}{2}n(n+1)$ .",
"Using this, it is easy to see that the required inequality holds except possibly in the cases $B_2, B_3, D_4$ and $D_5$ .",
"For these cases, along with all exceptional cases, we directly verify that the inequality holds using the calculations on dimensions of centralisers in [21].",
"This allows us to prove our generalisation of [2]; a result on the minimal dimension of finitely generated modules over $\\pi $ -adically completed enveloping algebras.",
"Definition 3.2.4 Let $A$ be a Noetherian ring.",
"$A$ is $\\emph {Auslander-Gorenstein}$ if the left and right self-injective dimension of $A$ is finite and every finitely generated left or right $A$ -module $M$ satisfies, for $i \\ge 0$ and every submodule $N$ of $\\text{Ext}^i_A(M,A)$ , $\\text{Ext}^j_A(N,A) = 0$ for $j < i$ .",
"In this case, the $\\emph {grade}$ of $M$ is given by: $j_A(M) := \\text{inf} \\lbrace j \\mid \\text{Ext}^j_A(M,A) \\ne 0 \\rbrace $ and the $\\emph {canonical dimension}$ of $M$ is given by: $d_A(M) := \\text{inj.dim}_A(A) - j_A(M).$ By the discussion in [2], the ring $\\widehat{U(\\mathfrak {g})_{n,K}}$ is Auslander-Gorenstein and so it makes sense to define the canonical dimension function: $d: \\lbrace \\text{finitely generated } \\widehat{U(\\mathfrak {g})_{n,K}}-\\text{modules} \\rbrace \\rightarrow \\mathbb {N}.$ Theorem 3.2.5 Suppose $n > 0$ and let $M$ be a finitely generated $\\widehat{U(\\mathfrak {g})_{n,K}}$ -module with $d(M) \\ge 1$ .",
"Then $d(M) \\ge r$ .",
"By [2], we may assume that $M$ is $Z$ -locally finite.",
"We may also assume that $M$ is a $\\widehat{\\mathcal {U}^{\\lambda }_{n,K}}$ -module for some $\\lambda \\in \\mathfrak {h}^*_K$ , by passing to a finite field extension if necessary and applying [2].",
"By Proposition $\\ref {9.3}$ (b), $\\lambda \\circ (i \\circ \\widehat{\\phi }) = (w \\bullet \\lambda ) \\circ (i \\circ \\widehat{\\phi })$ for any $w \\in \\textbf {W}$ .",
"Hence we may assume $\\lambda $ is $\\rho $ -dominant by [2].",
"Hence $\\text{Gr }(M)$ is a $\\text{Gr }(\\widehat{\\mathcal {U}^{\\lambda }_{n.K}}) \\cong S(\\mathfrak {g}_k) \\otimes _{S(\\mathfrak {g}_k)^{\\textbf {G}_k}} k$ -module by Theorem $\\ref {6.10}$ .",
"If $\\mathcal {M} := \\text{Loc}^{\\lambda }(M)$ is the corresponding coherent $\\widehat{\\mathcal {D}^{\\lambda }_{n,K}}$ -module in the sense of Definition REF , then $\\beta (\\text{Ch}(\\mathcal {M})) = \\text{Ch}(M)$ via [2].",
"Let $X$ and $Y$ denote the $k$ -points of the characteristic varieties $\\text{Ch}(\\mathcal {M})$ and $\\text{Ch}(M)$ respectively.",
"Now $\\text{Gr}(M)$ is annihilated by $S^+(\\mathfrak {g}_k)^{\\textbf {G}_k}$ , and so $Y \\subseteq \\mathcal {N}^*$ .",
"We see that the map $\\beta : T^*\\mathcal {B} \\rightarrow \\mathfrak {g}$ maps $X$ onto $Y$ .",
"Let $f: X \\rightarrow Y$ be the restriction of $\\beta $ to $X$ .",
"By [2], since $\\text{dim } Y = d(M) \\ge 1$ we can find a non-zero smooth point $y \\in Y$ .",
"By surjectivity, we have a smooth point $x \\in f^{-1}(y)$ .",
"The induced differential $df_x: T_{X,x} \\rightarrow T_{Y,y}$ on Zariski tangent spaces yields the inequality: $\\text{dim } Y + \\text{dim } f^{-1}(y) \\ge \\text{dim } T_{X,x}$ By [2], $\\text{dim } T_{X,x} \\ge \\text{dim } \\mathcal {B}$ .",
"Hence: $d(M) = \\text{dim } Y \\ge \\text{dim } \\mathcal {B} - \\text{dim } \\beta ^{-1}(y)$ By Proposition $\\ref {9.8}$ and Proposition $\\ref {9.9}$ , the RHS equals $r$ .",
"$\\textbf {Proof of Theorem C:}$ This follows from Theorem REF and [2] in the split semisimple case.",
"We may then apply the same argument as in [1] to remove the split hypothesis on the Lie algebra."
]
] | 1906.04460 |
[
[
"Coherent transfer of quantum information in silicon using resonant SWAP\n gates"
],
[
"Abstract Solid state quantum processors based on spins in silicon quantum dots are emerging as a powerful platform for quantum information processing.",
"High fidelity single- and two-qubit gates have recently been demonstrated and large extendable qubit arrays are now routinely fabricated.",
"However, two-qubit gates are mediated through nearest-neighbor exchange interactions, which require direct wavefunction overlap.",
"This limits the overall connectivity of these devices and is a major hurdle to realizing error correction, quantum random access memory, and multi-qubit quantum algorithms.",
"To extend the connectivity, qubits can be shuttled around a device using quantum SWAP gates, but phase coherent SWAPs have not yet been realized in silicon devices.",
"Here, we demonstrate a new single-step resonant SWAP gate.",
"We first use the gate to efficiently initialize and readout our double quantum dot.",
"We then show that the gate can move spin eigenstates in 100 ns with average fidelity $\\bar{F}_{SWAP}^p$ = 98%.",
"Finally, the transfer of arbitrary two-qubit product states is benchmarked using state tomography and Clifford randomized benchmarking, yielding an average fidelity of $\\bar{F}_{SWAP}^c$ = 84% for gate operation times of ~300 ns.",
"Through coherent spin transport, our resonant SWAP gate enables the coupling of non-adjacent qubits, thus paving the way to large scale experiments using silicon spin qubits."
],
[
"methods",
"Beyond the calibration required for the projection-SWAP, there are three additional constraints that must be satisfied to achieve high fidelity coherent-SWAP gates.",
"First, the resonant SWAP pulse must remain phase coherent with the qubits in their doubly-rotating reference frame between calibrations (i.e.",
"for hours).",
"Second, because of the constraint that the exchange interaction is always positive, the time-averaged exchange pulse necessarily has some static component, which leads to evolution under an Ising interaction [25].",
"These Ising phases must be calibrated out.",
"Finally, voltage pulses on any gate generally displace both electrons by some small amount.",
"This movement induces phase shifts in both qubits, since they are located in a large magnetic field gradient.",
"These phase shifts must be compensated for.",
"To satisfy these additional tuning requirements, we first ensure that our RF exchange pulse remains phase coherent.",
"Each qubit's reference frame is defined by the microwave signal generator controlling it, so by mixing together the local oscillators of these signal generators, we obtain a beat frequency that is phase locked to the doubly-rotating two-qubit reference frame.",
"We then amplitude modulate this signal to generate our exchange pulses.",
"A detailed schematic is shown in the supplementary information [25].",
"To calibrate for the single and two-qubit Ising phases, we use state tomography on both qubits before and after applying a SWAP gate.",
"In these measurements, we vary the input states to distinguish between errors caused by two-qubit Ising phases, and the single qubit phase shifts.",
"We choose a SWAP time and amplitude such that the Ising phases cancel out, which for this particular configuration occurs for a 302 ns SWAP gate.",
"The single qubit phase shifts were measured to be 180$^{\\circ }$ for $Q_3$ and 140$^{\\circ }$ for $Q_4$ .",
"By superimposing the SWAP pulse with a dc exchange pulse, one can compensate for the Ising phases at arbitrary SWAP lengths, leading to faster operation [25]."
],
[
"Author Contributions",
"AJS, MJG, and JRP designed and planned the experiments, AJS fabricated the device and performed the measurements, MJG provided theory support, LFE and MB provided the isotopically enriched heterostructure, AJS, MJG, and JRP wrote the manuscript with input from all the authors."
],
[
"Competing Interests",
"The authors declare no competing financial interests."
],
[
"Data Availability",
"The data supporting the findings of this study are available within the paper and its Supplementary Information [25].",
"The data are also available from the authors upon reasonable request.",
"Funded by Army Research Office grant No.",
"W911NF-15-1-0149, DARPA grant No.",
"D18AC0025, and the Gordon and Betty Moore Foundation's EPiQS Initiative through Grant GBMF4535.",
"Devices were fabricated in the Princeton University Quantum Device Nanofabrication Laboratory.",
"The authors acknowledge the use of Princeton’s Imaging and Analysis Center, which is partially supported by the Princeton Center for Complex Materials, a National Science Foundation MRSEC program (DMR-1420541)"
]
] | 1906.04512 |
[
[
"Approximate Gradient Descent Convergence Dynamics for Adaptive Control\n on Heterogeneous Networks"
],
[
"Abstract Adaptive control is a classical control method for complex cyber-physical systems, including transportation networks.",
"In this work, we analyze the convergence properties of such methods on exemplar graphs, both theoretically and numerically.",
"We first illustrate a limitation of the standard backpressure algorithm for scheduling optimization, and prove that a re-scaling of the model state can lead to an improvement in the overall system optimality by a factor of at most $\\mathcal{O}(k)$ depending on the network parameters, where $k$ characterizes the network heterogeneity.",
"We exhaustively describe the associated transient and steady-state regimes, and derive convergence properties within this generalized class of backpressure algorithms.",
"Extensive simulations are conducted on both a synthetic network and on a more realistic large-scale network modeled on the Manhattan grid on which theoretical results are verified."
],
[
"Introduction",
"We consider the scheduling problem on queuing networks, and specifically on urban road networks.",
"Concretely, the problem consists of the allocation of time slots to traffic lights at intersections.",
"While the routing policy may impact the stability of the scheduling solution [2], the routing problem is often considered decoupled [15] and is not addressed here.",
"The computational complexity of the scheduling problem on large-scale road networks has motivated the search for efficient decentralized algorithms only requiring local knowledge of network properties and efficient in the absence of coordination.",
"Such decentralized approaches have proven quite efficient in practice [18], [27], have connections with fluid dynamic models [4], and are amenable to agent-based learning methods such as reinforcement learning [16].",
"In the context of communication networks, the backpressure algorithm [21] provides a throughput-maximizing control policy, i.e.",
"a policy that guarantees that given any feasible flow, the maximal network queue size is asymptotically bounded.",
"Furthermore, the backpressure policy requires only evaluation of queue size on neighboring road links.",
"Several properties of the backpressure algorithm make it appealing for adaptive control of dynamical road networks.",
"First, the backpressure solution is a policy, which by definition is able to handle variability of the network state.",
"Second it can be implemented and deployed in a fully decentralized manner since it requires only local information.",
"Lastly, it comes with theoretical guarantees on the queue size, and has been shown to perform very well in practice.",
"In the context of intelligent transport systems, significant research efforts have been dedicated to extensions of the backpressure algorithm in recent years, in particular to address the specifics of traffic light scheduling on road networks [23].",
"The case of unknown routing rates was also investigated [8], as well as the case of queues with finite capacity [9], see also [11] and [12].",
"In-depth theoretical analysis of the backpressure properties, such as its behavior under heavy load conditions [3], or its link with game theory [22] have also been investigated [17], [14].",
"Similar results exist for the closely related max weight algorithm [20] and an adapted backpressure algorithm [24] has been devised in this context.",
"The backpressure algorithm was also shown to be a greedy gradient descent over the quadratic potential [26].",
"In this context, acceleration methods have been proposed [29].",
"We refer to [13] for application to wireless sensor networks.",
"One of the most explicit drawbacks of the backpressure algorithm occurs at a network scale: in steady state, for certain model networks, queue sizes strictly decrease from the origin to the destination along every possible path [5], [28], meaning that commuters traveling over longer paths incur longer queues.",
"In this body of work, adjustments have been proposed via the consideration of an additional design cost explicitly accounting for path lengths, hence attempting to compensate that drawback.",
"Other limitations have been investigated for specific network configurations [19].",
"In this work, we propose to analyze the convergence dynamics of the backpressure algorithm.",
"Specifically, we are interested in a fine-grained analysis of the backpressure algorithm in different regimes, and associated convergence properties.",
"We first explicitly re-cast the backpressure algorithm as a more general approximate gradient descent method, and show that in that class of methods, significant performance gaps exist depending on the choice of parameters.",
"These general conclusions are derived on the case of a fundamental building block for network flow analysis, namely the $2\\times 1$ network including two upstream links connected to one downstream link.",
"We then conduct an in-depth theoretical analysis of the $2\\times 1$ network, and verify these results in simulation.",
"We confirm experimentally that the results from the theoretical analysis obtained on a simple network apply to realistic networks such as the Manhattan grid.",
"The main contributions of this work include: illustration of arbitrarily large performance gaps in the backpressure class of adaptive control algorithms, theoretical identification and characterization of transient and stationary regimes of the $2 \\times 1$ network under backpressure algorithms, numerical validation of theoretical properties and illustration on realistic networks such as the Manhattan grid.",
"The rest of this article is organized as follows.",
"We first introduce notations and formulate the problem considered.",
"We then present our main results on the convergence dynamics on the $2\\times 1$ network.",
"We subsequently analyze the relative convergence of two instances of approximate gradient descent algorithms.",
"Finally, we present detailed numerical results of the algorithm performance, and conclude."
],
[
"Network model",
"We consider a discrete-time network of queues with $q_{l,m}(t) \\in \\mathbb {R^{+}}$ denoting the (continuous) number of vehicles queuing at location $l$ at time $t$ with the intention of traveling to the downstream link $m$ next.",
"In the transportation context, each $q_{l,m}(\\cdot )$ represents a distinct queue of vehicles waiting to cross an intersection with segregated movements (e.g.",
"turn left, go straight, turn right).",
"We also note $q$ the vector of $q_{l,m}$ and omit the time dependency for compactness.",
"Queuing networks can also model public transport and multi-modal networks [10], although the emergence of mobile data often requires hybrid approaches [1].",
"The outflow $s_{l,m}(t)$ of queue $q_{l,m}(t)$ at time $t$ is the maximum number of vehicles able to cross the intersection within a time slot, defined as the minimum of the queue size and the queue capacity, assumed static: $s_{l,m}(t) = \\min (q_{l,m}(t),c_{l,m}).$ When this minimum $s_{l,m}(t)$ is reached at the queue size $q_{l,m}(t)$ , the intersection is in unsaturated regime, and when the minimum is reached at the queue capacity $c_{l,m} \\in \\mathbb {R^{+}_{*}}$ , the intersection is in saturated regime.",
"Given initial conditions $q_{l,m}(t=0)$ for the queues, and prescribed source and sink flows $e_{l,m}(t)$ specifying the number of vehicles entering and leaving the network, the conservation of vehicles reads: $q_{l,m}(t+1) = & \\, q_{l,m}(t) + r_{l,m}(t)\\sum \\limits _k u_{k,l}(t) s_{k,l}(t) \\nonumber \\\\& - u_{l,m}(t)\\,s_{l,m}(t) + e_{l,m}(t)$ where $r_{l,m}(t) \\in [0,1]$ is the proportion of vehicles reaching node $l$ intending to visit node $m$ next, and such that $\\sum _{m}r_{l,m}(t) = 1$ , and $u_{l,m}(t) \\in \\lbrace 0,1\\rbrace $ is a control variable specifying whether queue $q_{l,m}(t)$ is activated, i.e.",
"has green light, at time $t$ .",
"For each intersection, the activation set follows standard constraints encoding compatible movements, (e.g.",
"in the case of left-hand driving, turn right movements can be activated simultaneously, but not go straight and turn left movements).",
"Given an objective function $V(q)$ satisfying Lyapunov properties, the scheduling problem is concerned with the design of an activation policy $u_{l,m}(\\cdot )$ with good properties with respect to $V(\\cdot )$ ."
],
[
"Backpressure algorithm",
"In this section we recall some existing results.",
"The backpressure algorithm [21] provides maximal throughput stability in the sense that if the inputs flows are feasible in expectation, then the queue sizes are asymptotically bounded.",
"Definition 1 The backpressure policy is the solution $u$ to the maximization problem : $\\max \\limits _{u} \\sum \\limits _{l,m}{\\left( q_{l,m} - \\sum \\limits _k{q_{m,k} r_{m,k}} \\right) c_{l,m} u_{l,m}}.$ It can be shown, e.g.",
"see [25], that this objective function leads to activating at each decision point the traffic movement maximizing the difference between its upstream queue and its downstream queue, hence the term “backpressure”.",
"We first show that the back pressure policy, although arising from a local greedy formulation, corresponds to an approximate gradient descent step.",
"Proposition 1 The backpressure algorithm (REF ) is an approximate gradient descent step update on the objective function $V(q) = \\frac{1}{2}\\sum \\limits _{l,m} q^2_{l,m} = \\frac{1}{2}q^{T}q$ .",
"If we note $\\delta (t+1)=q(t+1)-q(t)$ , the one-step temporal difference in the objective function reads: $V(q(t+1)) - V(q(t)) = \\delta (t+1)^T q(t) +\\frac{1}{2}\\delta (t+1)^T\\delta (t+1).$ Expanding $\\delta (t+1)$ using the conservation equation (REF ) we can re-write $\\delta (t+1)^T q(t)$ as: $\\delta (t+1)^T q(t) = \\\\ E^T q - \\sum \\limits _{l,m} \\left( q_{l,m} - \\sum \\limits _k q_{m,k} r_{m,k} \\right) u_{l,m} s_{l,m}$ where E is the vector of $e_{l,m}$ , and the time-dependence is omitted on the right-hand side.",
"In the saturated regime, the first term on the right-hand side of equation (REF ) dominates, and a steepest gradient descent step on the approximate temporal difference $\\delta (t+1)^T q(t)$ reads: $\\arg \\min _u \\delta (t+1)^T q(t) = \\\\ \\arg \\max _u \\sum \\limits _{l,m} \\left( q_{l,m} - \\sum \\limits _k q_{m,k} r_{m,k} \\right) u_{l,m} s_{l,m}.$ Approximating the throughput (REF ) as $s_{l,m} \\approx c_{l,m}$ leads to the definition of the backpressure (REF ), which corresponds to making the approximation that the queues are in the saturated regime.",
"Motivated by expression (REF ), in the following we define the priority of a queue as: $p_{l,m} = \\left( q_{l,m} - \\sum \\limits _k{q_{m,k} r_{m,k}} \\right) c_{l,m}.$ This view of backpressure as a general one step update for an approximate gradient descent in the context of adaptive control motivates us to consider a generalization of the objective function via re-scaling.",
"Specifically, given $\\gamma _{l,m} > 0$ , we consider a generalized objective function $V(q) = \\frac{1}{2}\\sum \\limits _{l,m} \\gamma _{l,m} q^2_{l,m}$ associated with the generalized priorities: $p_{l,m} = \\left( \\gamma _{l,m} q_{l,m} - \\sum \\limits _k{\\gamma _{m,k} q_{m,k} r_{m,k}} \\right) c_{l,m}.$ We now illustrate that this re-scaling can impact the performance of the approximate gradient descent method by an arbitrary factor depending on the network heterogeneity.",
"We focus the analysis on the comparison between two values of $\\gamma _{l,m}$ , the case of $\\gamma _{l,m} = 1$ which corresponds to the classical backpressure, and the case of $\\gamma _{l,m} = 1/c_{l,m}$ which corresponds to a variant of the backpressure algorithm where time spent in the queue is the quantity to be optimized (since $q/c$ is the steady-state saturated regime approximation of time spent in the queue)."
],
[
"Heterogeneous Flows",
"We now define a simple but fundamental example and show the limitations of the backpressure algorithm on that case.",
"Consider a simple network with 2 upstream queues $q_{1,3}$ , $q_{2,3}$ and 1 downstream queue $q_{3,4}$ , where heterogeneity between the upstream queues is parameterized by a factor $k$ .",
"This topology corresponds to the classical merge junction in traffic engineering [6], see [7] for the underlying mathematical theory of network fluid-dynamics model.",
"Given a reference capacity $c$ , link capacities are defined as $c_{2,3} = k \\, c_{1,3} = kc$ .",
"Link inflows are defined as $f_{2,3} = k \\, f_{1,3} = k\\eta c$ .",
"Without loss of generality, we also assume that the downstream queue is constant since the main object of this study is the competing dynamics of upstream queues given an arbitrary downstream queues.",
"For stability, we also make the classical assumption that the inflow is lower than the uniform capacity, i.e.",
"$\\eta _i = \\frac{f_i}{c_i} \\le 0.5$ .",
"Figure: 2×12\\times 1 parametric junction with two upstream queues and one downstream queue.We now prove that the classical backpressure algorithm (REF ) can perform arbitrary poorly for heterogeneous networks.",
"We present the results for the case $q_{3,4}(t) =0$ .",
"Proposition 2 Consider the $2 \\times 1$ network from Figure REF , with $\\eta = 0.5$ and $q_{3,4}(t) = 0$ .",
"A basic scheduling alternating activation of each upstream queue leads to: $q_{1,3}(t) \\approx \\eta c \\quad \\text{and} \\quad q_{2,3}(t) \\approx k \\eta c$ while the backpressure activation rule (REF ) yields: $q_{1,3}(t) \\ge k^2 \\eta c \\quad \\text{and} \\quad q_{2,3}(t) \\ge k \\eta c$ First, given that in this discrete time setting each vehicle spends at least one time step in the queue, we have $q_{l,m} \\ge f_{l,m}$ .",
"Here, since the demand is feasible, an alternating schedule would result in $q_{1,3}(t) \\approx f_{1,3}$ , and similarly for $q_{2,3}$ , which proves (REF ).",
"Second, on this example, the backpressure priorities from equation (REF ) read $p_{l,m}(t) = c_{l,m} q_{l,m}(t)$ .",
"Since $q_{l,m}(t) \\ge f_{l,m}$ , we have $p_{2,3}(t) = c_{2,3} q_{2,3}(t) \\ge k^2 c^2 \\eta $ .",
"The queue $q_{1,3}$ is activated only if $p_{1,3}(t) = c_{1,3} q_{1,3}(t) > p_{2,3}(t) \\ge k^2 c^2 \\eta $ , or equivalently $q_{1,3}(t) > k^2 c \\eta $ .",
"When $q_{1,3}$ goes below that value, queue $q_{2,3}$ is activated because it has higher priority.",
"Hence the minimal queue size over time under backpressure is $k$ times the queue size under the alternating schedule.",
"A similar analysis can be conducted with non-zero $q_{3,4}$ , the result is obtained for a sufficient upstream queue heterogeneity compared to the downstream queue value.",
"In the rest of the article, we investigate the convergence dynamics of the approximate gradient descent dynamics under the generalized backpressure priorities (REF )."
],
[
"Stability Domain",
"We first characterize the asymptotic convergence of the parametric $2\\times 1$ network from Figure REF .",
"To simplify the notations on this network, we index the upstream queues by $i \\in \\lbrace 1,2\\rbrace $ ."
],
[
"Regime Types",
"For $\\eta < 0.5$ , the demand is feasible, capacities exceed input flows, which means that at least one queue is activated more often than needed to process the input flow, which means $s_{u,i}(t) < c_i$ .",
"For the $2\\times 1$ junction, either only one upstream queue is in the unsaturated regime, and we call this network regime R1, or both upstream queues are in the unsaturated regime, and we call this network regime $R2$ .",
"Due to space limitation, we focus on the regime $R1$ which exhibits more complex behavior (indeed under the condition $\\eta < 0.5$ regime $R2$ can be proven to be a transient regime evolving into $R1$ eventually) and serves as illustrative example of a limitation of the classical backpressure algorithm in the example of previous section.",
"With a slight abuse of notations, we use the saturation state, s for saturated and u for unsaturated, to index the queue (e.g.",
"$(u,s)=(1,2)$ when queue 1 is unsaturated and queue 2 is in the saturated state)."
],
[
"A re-scaled backpressure algorithm",
"A discussed earlier, a queue is bounded below by a $q_{i,min}$ which is the value of its input flow: $q_{i,min} = f_i \\le q_i(t).$ The expressions of the associated minimal priority $p_{i,min}$ can be derived by instantiating equation (REF ) with the priority value (REF ): $p_{i,min} = p_i(q_i = q_{i,min}) \\le p_i(t)$ In order for a queue to be activated, its priority must be at least greater than the minimal priority of every competing queue.",
"We call $p_{act}$ this a-priori minimal priority to be activated: $p_{act} = \\max \\limits _j p_{j,min},$ and the expression associated $q_{i,act}$ follows from (REF ) as: $\\begin{split}q_{i,act} = q_i(p_i = p_{act}) = \\frac{\\gamma _0}{\\gamma _i} Q + \\frac{1}{\\gamma _i c_i} p_{act}.\\end{split} $ In regime $R1$ one queue is saturated and the other queue is unsaturated.",
"It follows from (REF ) that: $q_{s,act} = \\frac{\\gamma _0}{\\gamma _s} Q + \\frac{\\gamma _j f_j - \\gamma _0 Q}{\\gamma _s c_s} c_j \\ge c_s$ where $j$ is such that $p_j = p_{act}$ .",
"If $j = s$ , meaning that queue $j$ is in the saturated state, then equation (REF ) simplifies to $\\eta \\ge 1 $ , which is impossible by assumption.",
"Hence the queue $j$ must be in the unsaturated state when reaching its activation priority, and the other queue is in the saturated state: $p_{act} = p_{u, min} = p_u(q_u=f_u)$ We can now express sufficient conditions for each of the queues to be in saturated or unsaturated state, depending on the exogenous network parameters.",
"Proposition 3 In the $R1$ regime, the following states can exist: $(u,s) = (1,2)$ when the following conditions are satisfied: for $\\gamma = \\mathbf {1}_n$ : $ Q \\ge \\frac{k^2 - \\eta }{k-1}c $ for $\\gamma = [\\frac{1}{c_i}]_{i \\in \\lbrace 1,...,n\\rbrace }$ : $ Q \\ge \\frac{k - \\eta }{k-1}(k+1)c$ $(u,s) = (2,1)$ when the following conditions are satisfied: for $\\gamma = \\mathbf {1}_n$ : $ Q \\le \\frac{k^2\\eta - 1}{k-1}c $ for $\\gamma = [\\frac{1}{c_i}]_{i \\in \\lbrace 1,...,n\\rbrace }$ : $Q \\le \\frac{k\\eta - 1}{k-1}(k+1)c$ The bounds follow from instantiating equation (REF ) on the cases $(u,s) = (1,2)$ and $(u,s) = (2,1)$ .",
"We represent the different phases characterized by Proposition REF in Figure REF .",
"Figure: Saturated and unsaturated states for each link in R1R1 regime.",
"When kk is large compared to QQ (bottom right part of the chart), i.e.",
"in case of significant heterogeneity, the smaller capacity queue (named 0) is saturated, while for larger QQ values (top left part of the chart), i.e.",
"for high network load and resulting high coupling between high-demand links, the larger capacity queue (named 1) is saturated.We observe that with the proposed non-uniform weights $\\gamma _{l,m}=1/c_{l,m}$ , the saturation region decreases, which suggests a better utilization of the network capacity."
],
[
"Convergence properties",
"In this section we characterize the transient and stationary phases of the system, and provide theoretical results on the impact of the weights $\\gamma $ on the limit of the gradient descent."
],
[
"Transient state",
"In regime $R1$ , under the assumption that $\\eta < 0.5$ , we first prove that in general, there exists a transient regime during which the maximal priority decreases.",
"Definition 2 The rolling min-max over 2 time slots reads: $\\tilde{p}_{max}(t) = \\min \\limits _{v \\in \\lbrace t-1, t\\rbrace } \\max \\limits _{i \\in \\lbrace 1,2\\rbrace } p_i(v)$ Lemma REF provides conditions to ensure the overall decrease in queue size during the transient phase.",
"Lemma 1 For the $2 \\times 1$ network, with $\\eta < 0.5$ , we have: $\\forall t, \\tilde{p}_{max}(t+2) \\le \\tilde{p}_{max}(t).$ Furthermore, with $i^* = \\arg \\max p_i(t)$ and $j$ the other queue, and with $\\Delta p(t) = p_{i^*}(t) - p_j(t) \\ge 0$ , we have $\\forall t$ : $\\begin{array}{c}q_{i^*}(t) > 2f_{i^*}\\text{ and }\\Big (q_j(t) > f_j \\text{ or } \\Delta p(t) > 0\\Big )\\\\ \\Rightarrow \\tilde{p}_{max}(t+2) < \\tilde{p}_{max}(t)\\end{array}$ The technical proof can be found in the supplemental material.",
"We can now use the specificity of the R1 regime to describe the dynamics of the $2 \\times 1$ network: Corollary 1 If $p_{max}(t) = p_{s}$ , i.e.",
"$i^*(t) = s$ , the saturated queue is activated for the time slot $t$ and: $q_u(t) > q_{u,min}\\Rightarrow \\tilde{p}_{max}(t+2) < p_{max}(t)$ and $\\left\\lbrace \\begin{array}{l} q_u(t) = q_{u,min} \\\\ \\Delta p(t) = 0 \\end{array} \\right.\\Rightarrow \\left\\lbrace \\begin{array}{l} i^*(t) = s \\\\ i^*(t+1) = i^*(t+2) = u \\end{array} \\right.$ In other words $q_s$ is activated at most once consecutively and it is followed by two activations of $q_u$ .",
"The proof is obtained via a disjunction over the values of $q_u(t)$ , details included in the supplemental material.",
"Corollary REF states first that the end of the transient state is related to the time when the unsaturated queue takes its minimal value.",
"Equation (REF ) describes the queuing process in steady state: the saturated queue cannot be activated more than once consecutively.",
"This implies that the unsaturated queue remains very close to its minimum value."
],
[
"Characterization of the steady state",
"We now generalize the result from Corollary REF .",
"Proposition 4 The steady state is reached after a finite time $t_0$ and is characterized by: $\\forall t \\ge t_0,\\tilde{p}_{max}(t) = p_{act}$ where $p_{act}$ is defined by (REF ).",
"First we prove that if queue $s$ is regularly activated then $\\tilde{p}_{max}(t)$ strictly decreases overtime.",
"It eventually reaches the minimal value $p_{act}$ .",
"Lemma REF states that $\\tilde{p}_{max}(t)$ is non-increasing therefore bounded above by $P$ .",
"The input flow within one time slot is bounded (actually equal to $f_i$ ) so between two time slots, $p_{max}(t)$ remains bounded above with $p_{max}(t) \\le P + \\max \\limits _i f_i$ .",
"In the $2\\times 1$ network, $p_i$ is affine of $q_i$ , so queues are also bounded above.",
"Hence there exists an interval such that every queue is activated at least once within this interval otherwise the constants input flows would make it diverge.",
"Let us consider such an interval and a time $t_s$ in that interval at which $q_s$ is activated.",
"Equation (REF ) states that $\\tilde{p}_{max}$ is non increasing.",
"Corollary REF states that while $q_u(t) > q_{u,min}$ , $\\tilde{p}_{max}$ is decreasing strictly.",
"Let us consider the other case, for which $q_u(t_s) = q_{u,min}$ , corresponding to $p_u(t_s) = p_{act}$ according to (REF ).",
"We have that $p_s(t_s) = p_{max}(t_s)$ : if $p_s(t_s) > p_{act}$ , equivalently $\\Delta p(t_s) > 0$ and (REF ) yields $\\tilde{p}_{max}(t)$ strictly decreasing, else, $\\Delta p(t_s) = 0$ , $p_{max}(t) = p_{act}$ , so $\\forall t > t_s, \\tilde{p}_{max}(t) = p_{act}$ which is the lower bound.",
"In any case, either $\\tilde{p}_{max}(t)$ decreases or has reached its lower bound after the activation of $q_s$ , which means that in finite time, $\\tilde{p}_{max}(t)$ is arbitrarily close to $p_{act}$ .",
"It follows that $\\tilde{p}_{max}(t)$ being constant bounds the size of the queues below and above.",
"Experimentally, these bounds are quite tight, see Figure REF .",
"Theorem 1 In the $R1$ regime, given $\\eta < 0.5$ , the $2\\times 1$ network converges to a steady state where under a generalized backpressure algorithm with $\\gamma > 0$ : for the unsaturated queue $q_u$ : $f_u \\le q_u(t) \\le 2f_u$ and for the saturated queue: $q_{s,act} + (f_s - c_s) \\le q_s(t) \\le q_{s,act} + f_s $ with $q_{s,act} = \\frac{\\gamma _0}{\\gamma _s}(1 - \\frac{c_u}{c_s}) Q + \\frac{\\gamma _u c_u}{\\gamma _s c_s} f_u$ The proof consists of expanding the results from Proposition REF .",
"For the unsaturated queue we obtain: $p_u(t) \\le \\tilde{p}_{max}(t) + \\gamma _u f_u c_u = p_{act} + \\gamma _u f_u c_u$ Equation (REF ) states that $p_{act}= p_u(q_u=f_u)$ Consequently the queues associated with the priorities from equation (REF ) read: $q_{u}(t) \\le 2f_u$ and the lower bound corresponds to (REF ).",
"For the saturated queue: $\\begin{array}{rl}p_s(t) &\\le \\tilde{p}_{max}(t) + \\gamma _s f_s c_s \\\\&\\ = p_{act} + \\gamma _s f_s c_s \\\\&\\ = \\underbrace{(\\gamma _u f_u - \\gamma _0 Q) c_u}_{p_{act}\\ (cf.",
"(\\ref {eqn:PactWithO}))} + \\gamma _s f_s c_s \\\\&\\ = p_{s,max}\\end{array}$ We define as well the upper bound $q_{s,max}$ of $q_s$ s.t.",
"$p_{s,max} = p_s(q_s = q_{s,max}) = (\\gamma _s q_{s,max} - \\gamma _0 Q)$ .",
"Consequently: $\\begin{array}{rl}q_{s,max} = f_s + \\frac{\\gamma _0}{\\gamma _s}(1 - \\frac{c_u}{c_s}) Q + \\frac{\\gamma _u c_u}{\\gamma _s c_s} f_u\\end{array}$ For the minimum $q_{s,min}$ , we note that it is reached after an activation done with $p_s(t) = p_{act}$ (defined as the minimal priority to be activated): $\\begin{array}{rl}p_{s,min} &= p_{act} + \\gamma _s (f_s - c_s) c_s \\\\&= p_{s,max} - \\gamma _s c_s c_s\\end{array}$ This corresponds to associated queue values $q_{s,min} = q_{s,max} - c_s$ and the result follows.",
"We illustrate the results of this section using a numerical simulation reported in Figure REF .",
"Figure: Convergence to the steady state for queues (top) and priorities (bottom).",
"As predicted by the theory, the priority decreases (here until time step 25) and then oscillates around a constant activation priority.After a transient state corresponding to priority values higher than $p_{act}$ , the system stabilizes to a periodic state.",
"Priorities oscillate around $p_{act}$ and $\\tilde{p}_{max}(t)$ is constant, as expected from the theory."
],
[
"Total time spent in the network",
"We now derive the total time spent in the network, assuming that the average queue size is the average of the bounds from Theorem REF .",
"Definition 3 The average queue size of $\\bar{q}_s$ is approximated by: $\\bar{q}_s = q_{act} + f_s - \\frac{1}{2}c_s$ The queue size translates directly into time spent given our implicit choice of a unit time step.",
"We can now compare the classical backpressure algorithm and the proposed backpressure algorithm based on total time spent in the network.",
"Theorem 2 In regime $R1$ , the ratio of total time spent in the network as the heterogeneity increases is: with $(u,s) = (1,2)$ : $\\frac{\\bar{q}_{s,classical}}{\\bar{q}_{s,proposed}} \\sim _{k \\rightarrow \\infty } 1$ with $(u,s) = (2,1)$ : $\\frac{\\bar{q}_{s,classical}}{\\bar{q}_{s,proposed}} \\sim _{k \\rightarrow \\infty } k$ We expand the expression of total time spent from Definition REF using the detailed values from Theorem REF ."
],
[
"Numerical Results",
"We first introduce the experimental setup, then go over results of benchmark experiments, and finally analyze performance on scenarios mimicking realistic conditions."
],
[
"Experimental setup",
"We consider a heterogeneous Manhattan grid wherein some links have high capacity (major arterial roads) and other links have low capacity (secondary arterial roads).",
"At each junction, 3 distinct movements are authorized (left, straight, right) with the straight movement having double capacity.",
"The average demand for each origin node, destination node pair is drawn from an exponential distribution.",
"For every origin-destination pair, the path minimizing the travel-time at the speed limit is computed, and flow is assigned accordingly.",
"Routing rates at each junction for the aggregate flow are computed from the full assignment, by computing the proportion of flow using each movement compared to the incoming flow.",
"Input flow is drawn from a Poisson law with mean defined for each origin-destination pair as explained above.",
"Performance metrics such as time spent in the network are computed by summing over time the sizes of queues."
],
[
"Benchmark experiments",
"In this section we perform controlled experiments to investigate and validate various properties of the proposed algorithm, compared to the classical backpressure algorithm.",
"We consider a $10\\times 10$ Manhattan grid with a major arterial every 5 blocks, and a time step of 30 seconds.",
"First we analyze the impact of the parameter $\\rho $ characterizing the magnitude of the demand.",
"Figure REF displays the ratio of time spent in the network for 500 time steps.",
"Figure: Ratio of time spent in the network for the two algorithms for increasing demand.",
"Every point represents 300 simulations and displays the deviation of the sample for classical backpressure (“bp”) and proposed algorithm (“new”).These results confirm the theoretical results on the existence of three regimes depending on the values taken by $k,Q$ : for low demand (low $\\rho $ values), the flow is not really constrained so both algorithms have similar performance, for demand around and slightly above capacity there is little available supply and the new algorithm improves on the classical backpressure, and for higher demand the network is too saturated to leave room for optimization and both algorithms have similar performance.",
"In this setting, the network is unstable for $\\rho > 2$ .",
"The improvement is at most $25\\%$ .",
"Second we analyze the impact of the parameter $h$ characterizing the network heterogeneity, with $1/h$ being the density of major arterial roads; $h = 0$ corresponds to no major arterial, $h = 1$ corresponds to all roads being major arterial, 2 to one over 2, etc.",
"The results of simulations are presented in Figure REF .",
"Figure: Ratio of time spent in the network for the two algorithms for different distances between parallel highways.",
"Every point represents 300 simulations, for classical backpressure (“bp”) and proposed algorithm (“new”).Low values of $h$ correspond to a homogeneous network and the experiments confirm ($h = 0,1,2$ ) that the proposed algorithm has no significant impact on traffic compared to the classical backpressure.",
"For higher $h$ the network is more heterogeneous and as expected the proposed algorithm has better relative performance.",
"We now analyze the impact of the ratio of the capacities between major arterial roads and secondary arterial roads.",
"The results of the simulations are displayed in Figure REF .",
"Figure: Ratio of time spent in the network for the two algorithms for increasing capacity heterogeneity.",
"Every point represents 300 simulations, for classical backpressure (“bp”) and proposed algorithm (“new”).For a homogeneous network (capacity ratio close to 1), the performance of both algorithms is similar.",
"The comparative performance of the new algorithm with respect to the regular backpressure increases as the capacity ratio increases.",
"The higher the capacity ratio the higher the flow on the major arterials (because their attractivity increases) so the more flow heterogeneity there is between major arterial and other roads.",
"Figure: Log-ratio of the average queue size for the proposed algorithm versus the classical algorithm, for every link in the network, including major arterials (highways) and secondary arterials (other roads).These cases correspond to the lower part of Figure REF for the single junction case where the proposed algorithm was proven theoretically to have better performance.",
"In order to understand how the proposed algorithm achieves better performances we compare the average size of each queue for a heterogeneous network in Figure REF .",
"The results highlight that the proposed algorithm results in a moderate queue size increase on major arterials but yields significant queue size reduction on secondary arterial roads, meaning that the proposed algorithm is better able to take advantage of the network heterogeneity.",
"In the next section we further explore the algorithm performance in realistic scenarios."
],
[
"Peak hour scenario",
"In this section, we analyze a scenario modeled after a peak hour.",
"The specific network considered is a Manhattan network with a $50\\times 10$ grid with a major arterial road every 4 blocks.",
"The factor $\\rho $ varies over time in a triangular shape from 0 to 3 to model a demand temporally exceeding the network capacity ($\\rho > 2$ corresponds to an instable network).",
"Figure: Cumulative time spent in the network for both algorithms (the unscaled value of ρ\\rho is also represented).Figure REF illustrates that the excess demand causes the cumulative time spent in the network to increase in all cases.",
"However the increase is less important in the case of the proposed algorithm, in particular when the demand returns to a low value, and subsequently.",
"This can be explained by the fact that in the first part of the peak period, the network is capacity-constrained, hence no algorithm is given sufficient freedom to optimize.",
"However in the second phase of the peak time, and subsequently, the proposed algorithm is able to take better advantage of available capacity."
],
[
"Incident scenario",
"The second scenario that we consider is an incident modeled as a link with zero capacity for a one hour period in a Manhattan network.",
"The network size and simulation parameters are chosen such that the boundary links are never impacted, and the incident happens after the network loading period, and clears before the end of the simulation.",
"We first analyze in Figure REF the extent to which queues are smoothed in space via the backpressure effect of preserving already large queues from further inflow.",
"Figure: Maximum queue size: as a function of distance to incident (at the intersection directly connected to the incident link, 1 hop and 2 hops upstream, as well as 1 hop downstream) for the classical backpressure (“bp”), the proposed algorithm (“new”), and the fixed cycle policy (“fixed”).For both the classical backpressure and the proposed algorithm, the queues at the incident location are much lower compared to the fixed cycle policy, which in practice reduces the chances of grid-lock.",
"This is achieved at the cost of having slightly longer queues upstream of the incident.",
"We now consider the cumulative queue length across the network as a function of time in Figure REF .",
"Figure: Queue size in the vicinity of the incident link: under the classical backpressure policy (“bp”), the proposed control policy (“new”), and a fixed cycle policy (“fix”) as a benchmark.",
"The incident start and end times are indicated with vertical dashed lines.We observe that both the classical backpressure and the proposed algorithm outperform the fixed cycle policy, and the proposed algorithm slightly improves on the classical backpressure, with the difference between all methods increasing as the network gets more saturated, since the key benefit of the adaptive scheduling algorithms lies in efficiently using available link capacities around saturated conditions."
],
[
"Conclusion",
"In this work we investigated the problem of the convergence properties of an approximate gradient descent method for network adaptive control.",
"Using a fundamental $2\\times 1$ network, we proved that different regimes exist depending on exogenous parameters such as the magnitude of the demand with respect to the network capacity, and the heterogeneity of the flow across competing links.",
"We characterized each of these regimes theoretically, and verified in simulation on realistic network the expected theoretical properties.",
"As part of this analysis, we also showed that appropriate calibration of the weights in the objective function can significantly improve the asymptotic objective value, by up to $\\mathcal {O}(k)$ with $k$ the ratio of competing queues capacity."
]
] | 1906.04388 |
[
[
"Perfect spin filtering by symmetry in molecular junctions"
],
[
"Abstract Obtaining highly spin-polarized currents in molecular junctions is crucial and desirable for nanoscale spintronics devices.",
"Motivated by our recent symmetry-based theoretical argument for complete blocking of one spin conductance channel in atomic-scale junctions [A. Smogunov and Y. J. Dappe, Nano Lett.",
"15, 3552 (2015)], we explore the generality of the proposed mechanism and the degree of achieved spin-polarized current for various ferromagnetic electrodes (Ni, Co, Fe) and two different molecules, quaterthiophene and p-quaterphenyl.",
"A simple analysis of the spin-resolved local density of states of a free electrode allowed us to identify the Fe(110) as the most optimal electrode, providing perfect spin filtering and high conductance at the same time.",
"These results are confirmed by $ab$ $initio$ quantum transport calculations and are similar to those reported previously for model junctions.",
"It is found, moreover, that the distortion of the p-quaterphenyl molecule plays an important role, reducing significantly the overall conductance."
],
[
"Perfect spin filtering by symmetry in molecular junctions Dongzhe Li Service de Physique de l'Etat Condensé (SPEC), CEA, CNRS, Université Paris-Saclay, CEA Saclay 91191 Gif-sur-Yvette Cedex, France Department of Physics, University of Konstanz, 78457 Konstanz, Germany Yannick J. Dappe Service de Physique de l'Etat Condensé (SPEC), CEA, CNRS, Université Paris-Saclay, CEA Saclay 91191 Gif-sur-Yvette Cedex, France Alexander Smogunov [email protected] Service de Physique de l'Etat Condensé (SPEC), CEA, CNRS, Université Paris-Saclay, CEA Saclay 91191 Gif-sur-Yvette Cedex, France Obtaining highly spin-polarized currents in molecular junctions is crucial and desirable for nanoscale spintronics devices.",
"Motivated by our recent symmetry-based theoretical argument for complete blocking of one spin conductance channel in atomic-scale junctions [A. Smogunov and Y. J. Dappe, Nano Lett.",
"15, 3552 (2015)], we explore the generality of the proposed mechanism and the degree of achieved spin-polarized current for various ferromagnetic electrodes (Ni, Co, Fe) and two different molecules, quaterthiophene and p-quaterphenyl.",
"A simple analysis of the spin-resolved local density of states of a free electrode allowed us to identify the Fe(110) as the most optimal electrode, providing perfect spin filtering and high conductance at the same time.",
"These results are confirmed by $ab$ $initio$ quantum transport calculations and are similar to those reported previously for model junctions.",
"It is found, moreover, that the distortion of the p-quaterphenyl molecule plays an important role, reducing significantly the overall conductance.",
"72.25.-b, 75.47.-m, 31.15.E-, 31.15.at Molecular spintronics is a very promising and emerging field, whose aim is to manipulate the spin degree of freedom in molecular based devices [1], [2].",
"Such devices should possess a large spin-relaxation length which is important for using spin degree of freedom in transport properties.",
"In particular, due to low spin-orbit coupling in organic molecules, the electron spin-relaxation length will be rather large, making such organic-based devices very promising for spintronics applications.",
"The property of high interest in spintronics is the spin-filtering or the spin polarization of (zero bias) electric current, $\\text{P}=(G_{\\uparrow }-G_{\\downarrow })/(G_{\\uparrow }+G_{\\downarrow })$ , where $G_{\\uparrow }$ and $G_{\\downarrow }$ are the conductances of majority and minority spin channels, respectively.",
"Another important property is the magneto-resistance (MR) - the strong change in electrical conductance $G$ between parallel and antiparallel magnetic alignments of two ferromagnetic electrodes - $\\text{MR}=(G_{\\text{P}}-G_{\\text{AP}})/G_{\\text{AP}}$ .",
"In this context, achieving as large as possible P (ideally, 100%) and MR (ideally, infinite) represents a crucial issue.",
"Unexpected large MR of up to 300% and large spin-dependent transport lengths have been reported in spin valves using Alq$_3$ [3], [4].",
"The MR of about 60%, 67% and 80% have been reported for Co$|$ C$_{60}$$|$ Co [5], Fe$|$ C$_{70}$$|$ Fe [6] and Ni$|$ C$_{60}$$|$ Ni [7] magnetic junctions, respectively.",
"At the single molecular scale, due to strong hybridization between molecular orbitals and ferromagnetic electrode states, it has been found that the giant MR (GMR) can reach rather large values, ranging from 50% to 80% [8], [9], [10], in the contact regime.",
"In the tunneling regime, a tunneling MR (TMR) value of up to 100% has been found in the case of single C$_{60}$ molecule deposited on the Chromium surface [11].",
"Moreover, a selective and efficient spin injection at the ferromagnetic-organic interface has also been reported by locally controlling the inversion of the spin polarization close to the Fermi level [12].",
"This effect, however, was found to depend strongly on the details of nanocontact configuration as well as on the molecule-electrode coupling.",
"Quite generally, the conductance in ferromagnetic nanocontacts is dominated by weakly polarized $s$ orbitals resulting in a partial spin-polarized current [13], [14], [15].",
"Therefore, blocking transport via $s$ orbitals and promoting transport via $d$ (or $p$ ) orbitals seems to be a good strategy to obtain high spin-polarizations.",
"Very recently, a 100% spin-polarized currents have been reported experimentally in the nickel oxide atomic junctions formed between two nickel electrodes [16] by the break-junction setup.",
"The current work is motivated by our recent study of spin-polarized electron transport through a special class of $\\pi $ -conjugated molecules bridging two Ni electrodes [17].",
"Due to symmetry mismatch between molecular orbitals and conductance channels of the Ni electrodes, the electron transport was carried by only $d$ -orbitals while the $s$ -conductance channels were fully blocked at the electrode-molecule junction.",
"As a consequence, a perfect 100% spin-polarization ($G_{\\downarrow } \\approx 0.65 G_0, G_{\\uparrow } = 0$ ) and an infinite MR have been found for ideal Ni electrodes (represented by semi-infinite Ni chains), while rather moderate values, $G_{\\downarrow } \\approx 0.19 G_0, G_{\\uparrow } \\approx 0.02 G_0$ , were found for more realistic Ni(111) electrodes.",
"The purpose of this Rapid Communication is two-fold: i) to identify molecular bridges (the electrodes and the molecule) employing in full our symmetry argument for perfect spin-filtering; ii) to demonstrate the generality of our symmetry-mismatch mechanism.",
"It will be shown that the Fe(110) electrode is the optimal one, providing highest spin-polarization of incoming conductance channels while polythiophene molecules are probably the best ones due to their highest occupied molecular orbital (HOMO) placed very closely to the Fermi level.",
"Figure: (Color online) (a) The model geometry of ferromagnetic||molecule||ferromagnetic nanojunction used in the paper.Spin-polarized projected density of states (PDOS) on d zx,zy d_{zx, zy} orbitals of one of (equivalent) apex atoms of the electrodes for(b) fcc-Ni, of (001) (top) and (111) (bottom) orientations;(c) hcp-Co, of fcc (top) and hcp (bottom) positions of the pyramid-like tip;(d) bcc-Fe, of (001) (top) and (110) (bottom) orientations.The insets show the atomic structures of corresponding electrodes which are terminatedwith only one apex atom.",
"The pyramids contain either 4 [Ni(111), hcp-Co, and Fe(110)] or 5 [Ni(001) and Fe(001)] atoms.The d zx d_{zx} and d zy d_{zy} orbitals are degenerate for all presented electrodes except for Fe(110) due to symmetry reason.Spin up (down) results are plotted by solid (dashed) lines.DFT-based spin-dependent quantum transport.",
"The spin-polarized $\\textit {ab initio}$ calculations were carried out using the plane wave electronic structure package QUANTUM ESPRESSO [18] in the framework of the density functional theory (DFT).",
"We used the local density approximation (LDA) with Perdew-Zunger parametrization [19] of the exchange-correlation functional with ultra-soft pseudopotentials (USPP) to describe electron-ions interaction.",
"The molecular junctions were simulated by a supercell as the one shown in Fig.REF (a).",
"Details concerning the calculations can be found in Ref.",
"[20].",
"For the calculations of ballistic transport across the junction, the unit cell of Fig.REF (a) was considered as a scattering region joint perfectly on both sides to semi-infinite electrodes.",
"Electron transmission was then evaluated using the scattering states approach as implemented in the PWCOND code [21].",
"Ballistic conductance (at the infinitesimal voltage) is given by Landauer-Büttiker formula, $G = G_{0}[T_{\\uparrow }(E_{\\text{F}})+T_{\\downarrow }(E_{\\text{F}})]$ , where $G_0 = e^2/h$ is the conductance quantum per spin.",
"A $(8 \\times 8)$ $k$ -points mesh in the $xy$ plane was found to be enough to obtain well converged transmission functions.",
"Notice that the use of the LDA for exchange-correlation potential in a DFT scheme can overestimate the transmission due to the underestimation of band gap [22], [23].",
"However, such effects may affect quantitatively the electron transport properties in our molecular junction, but the main physical trends will remain unaffected due to our robust symmetry argument.",
"Moreover, as we will see later, the conductance is provided by the HOMO molecular orbital whose position with respect to the Fermi level we believe is described quite well by our DFT calculations.",
"The key idea of symmetry argument is to block the electrode $s$ -channels (both of the majority and minority spins) at the Fermi energy by using $\\pi $ -conjugated molecules having only $\\pi $ -states available around the $E_F$ .",
"The remaining four transport $d$ -channels are all of the minority spin (the majority spin $d$ -states are all occupied and lie well below the $E_F$ ) which would result in a fully spin-polarized current across such molecular junctions.",
"In this context, the spin-polarized density of states at the electrode apex atom plays a crucial role, providing the information on the amount and spin-polarization of states available for the transport.",
"We start therefore by analysing the projected density of states (PDOS) at the electrode apex atom for the free electrodes of different ferromagnetic metals in order to select the best candidate as a spin injection material.",
"Electronic structure of ferromagnetic electrodes.",
"The ferromagnetic electrodes are modeled by five atomic layers containing 16 atoms in each atomic plane and terminated by a pyramid-like tip.",
"Note that the pyramids contain 4 or 5 atoms for cubic or hexagonal electrodes, respectively (see insets in the Fig.",
"REF ).",
"The two bottom layers were fixed while the other three layers and the pyramid were relaxed until the atomic forces are less than 1 meV/Å.",
"We emphasize that it is enough to look at only $d_{zx, zy}$ states of the apex atom (Fig.",
"REF ) since the other apex $d$ -states have no overlap with out-of-plane molecular $\\pi $ -states around the Fermi energy.",
"Note that the molecule is assumed to be aligned with the transport direction $Z$ but otherwise can rotate around it.",
"In the case if it lies exactly in the $YZ$ plane, for example, the molecular states will hybridize with only $d_{zx}$ -states, which are of appropriate symmetry (odd with respect to the molecule $YZ$ plane).",
"We notice also that, by symmetry, the $d_{zx}$ and $d_{zy}$ orbitals are degenerated for all the presented electrodes except for the Fe(110) electrode.",
"As expected, a quite general feature has been found for all the ferromagnetic electrodes – the apex DOS around the Fermi energy is dominated by partially filled minority $d$ -states.",
"For Ni(001) electrodes (Fig.",
"REF (b), upper panel), the spin-down PDOS peak lies in the vicinity of the Fermi energy while the spin-up PDOS is very small but not negligible and starts growing significantly at only $E - E_{\\text{F}} < -0.45$ eV.",
"In the case of Ni(111) (Fig.",
"REF (b), down panel), a relatively smaller [compared to the Ni(001)] spin-down PDOS is found at the Fermi energy with the maximum around $0.25$ eV above the Fermi energy.",
"For hcp-Co electrodes as shown in Fig.",
"REF (c), two different adsorption sites, namely fcc and hcp, for the pyramid were considered.",
"We found that the hcp site is slightly more favorable with respect to the fcc one, with an energy gain of about 17.84 meV.",
"Compared to the fcc-Ni electrode, the apex PDOS for spin down polarization has relatively smaller amplitude at the Fermi energy while the spin up states are almost absent.",
"For the case of bcc-Fe electrode we have considered two crystallographic orientations, (001) and (110), shown in Fig.",
"REF (d) on upper and lower panels, respectively.",
"The apex atom PDOS of the Fe(001) electrode exhibits a similar spin polarisation as for Ni(111) even though the peaks are rather different.",
"On the other hand, almost perfect behavior of the PDOS has been found for the Fe(110) orientation.",
"Namely, rather smooth and large PDOS for spin down states in a quite large energy window [$-$ 0.75 eV, $+$ 0.5 eV] with a maximum around the Fermi energy and almost zero PDOS around the Fermi energy for spin up states.",
"In addition, the non-negligible PDOS for spin up states start about $-0.8$ eV with respect to the Fermi energy.",
"We conclude, therefore, that among all the electrodes considered, the Fe(110) looks as the optimal one for providing highly spin-polarized incoming current.",
"It is expected also to improve considerably over the Ni(111) electrode considered in our previous work.",
"[17] Figure: Electronic and transport properties of Fe(110)||quaterthiophene||Fe(110) nanojunction:(a) spin-dependent PDOS on d zx d_{zx} orbitals of the Fe apex atom,(b) spin-resolved density of states on the quaterthiophene molecule,(c) spin-resolved transmission function T(E)T(E) as a function of the electron energyEE in the parallel magnetic configuration.Spin up (down) results are plotted by solid (dashed) lines.Figure: (Color online) Electronic and transport properties of p-tetraphenyl molecule suspended between two Fe(110) electrodes.Three molecular conformations were considered: flat molecule (a), distorted molecule with 30 ∘ 30^{\\circ } rotation between adjacent phenyl cycles (b),and distorted molecule with 90 ∘ ^{\\circ } rotation (c).Top panels present model geometries of molecular junctions.Middle panels show the charge-density isosurface plots for the HOMO of the free molecule.Note that the isosurfaces of positive and negative isovalues are shown in red and blue, respectively.Down panels report the spin-resolved PDOS on the molecule, the spin up (down) curves presented by solid (dashed) lines.At the bottom, the conductances for both spin channels are indicated.Fe(110)$|$ quaterthiophene$|$ Fe(110).",
"After having selected the best electrode, which in our case turns out to be the Fe(110), we will now confirm our findings by calculating explicitely the spin-polarized transmission functions.",
"We first consider the quaterthiophene molecule which contains four cycles of thiophene (standing perpendicular to the surface), suspended between two Fe(110) electrodes, as shown in Fig.",
"REF (a) as a geometric model.",
"Experimentally, two well-established approaches, namely the scanning tunneling microscope (STM) tip manipulation and the mechanically controllable break junction (MCBJ) technique, are used in the fabrication of such molecular junctions.",
"For instance, recent experiments showed that such molecular junction could be well established by using the STM tip to pick up one end of an individual polythiophene on the surface and subsequently lifting it up [24], [25].",
"In addition, the stable and highly conductive molecular junctions were successfully formed by MCBJ technique [26], [27].",
"By symmetry, the $d_{zx}$ -orbital of the Fe apex has non-zero overlap with the out-of-plane $\\pi $ -states of the molecule (if the $YZ$ plane is choosen to correspond to the molecule plane).",
"The PDOS of the Fe apex, as shown in Fig.",
"REF (a), presents a very similar feature compared to the case of the free electrode (see Fig.",
"REF (d), down panel).",
"However, a slight splitting of the majority spin states near the Fermi level, as well as a small peak at about $-0.13$ eV for the majority spin states arises due to the hybridization with the molecule.",
"Clearly, the PDOS of the Fe apex atom is still dominated by the minority spin states near the Fermi level.",
"The transport properties of molecular junctions are determined by molecular orbitals located near the Fermi energy.",
"In Fig.",
"REF (b) we present the DOS projected on the molecule, where one can clearly see that the HOMO is very close to the Fermi energy, meaning that the HOMO is the main transport channel.",
"The electronic levels of insulating and semiconducting nanostructure are not described accurately since the DFT is limited to ground states.",
"However, the HOMO described by DFT is equal to the exact ionization potential if we know the exact exchange-correlation potential [28], [29].",
"To address properly the quantum transport in molecular junctions ($i.e.$ the molecule is weakly coupled to the electrodes) one requires here a dynamical treatment of the electron-electron interactions which is beyond the standard DFT mean-field method such as $GW$ approximation.",
"M. Strange et al [23] has shown by fully self-consistent $GW$ scheme quantum transport in molecular junctions that the $GW$ is better than DFT for energy alignment of the molecule, however, they found that the difference between $GW$ quasiparticle energies and Kohn-Sham DFT eigenvalues significantly larger for lowest unoccupied molecular orbital (LUMO) than the HOMO.",
"Therefore, the DFT-based quantum transport presented in this work may quantitatively different compared to beyond DFT mean-filed technique, but the main physical trends will remain unaffected.",
"The spin-up resonance located at about $-0.13$ eV is very sharp due to the lack of appropriate symmetry states in the electrode for the majority spin near the Fermi energy.",
"On the contrary, the HOMO state for spin-down channel has significantly broader structure (going from $-0.5$ to $0.13$ eV roughly) which reflects the increased hybridization with electrode states.",
"A similar effect (but less pronounced) has been also reported for the same molecular junction bridging two Ni(111) electrodes [17].",
"As a consequence, the remarquable difference of the conductance for the two spin channels is observed as shown in Fig.",
"REF (c).",
"For spin-down electrons, the significant and broad transmission peak is observed.",
"It is located between $-0.1$ eV and $0.1$ eV, with a maximum close to the Fermi energy.",
"At the same time, the transmission coefficient in the spin-up channel is nearly zero at the Fermi level.",
"For that reason, the transport is fully due to spin-down electrons – we find $G_{\\downarrow } \\approx 0.70 G_0$ and $G_{\\uparrow } \\approx 0.004 G_0$ , for spin down and up polarizations, respectively.",
"Fe(110)$|$ p-tetraphenyl$|$ Fe(110).",
"In order to understand the effect of a molecule on the spin filtering efficiency, we now investigate another molecular junction made of p-tetraphenyl connecting two Fe(110) electrodes, as shown in Fig.",
"REF .",
"This molecule was chosen because it has much smaller band gap of 0.4 eV in the infinite chain configuration compare to the poly-thiophene chain of 1.4 eV.",
"Therefore, we thought that the corresponding molecular junction will show higher spin down conductance.",
"We start by discussing the flat p-tetraphenyl junction as presented in Fig.",
"REF (a).",
"From the charge-density isosurface plots of the HOMO level, we can clearly see the out-of-plane character of the HOMO orbital.",
"Compared to the quaterthiophene junction discussed in the previous section, the HOMO was found, however, to be shifted by about $-0.48$ eV with respect to the Fermi energy, which results in the lower spin down conductance, $G_{\\downarrow } \\approx 0.09 G_0$ .",
"The conductance in the spin up channel is again almost zero, indicating a 100% spin-filtering.",
"We next investigate the distortion effect of the molecule on its transport property.",
"Among three configurations considered (see Fig.",
"REF ), we have found that the one with phenyl rings rotated consequenly by 30$^{\\circ }$ around the $Z$ axis [Fig.",
"REF (b)] is the most favorable, explicitely, $0.12$ and $0.27$ eV lower in energy than the flat molecule [Fig.",
"REF (a)] and the one with two parts rotated by 90$^{\\circ }$ [Fig.",
"REF (c)], respectively.",
"Interestingly, the HOMO level moves away from the Fermi energy when the degree of the distortion is increased, i.e.",
"the HOMO moves to about $-0.52$ and $-0.75$ eV for the distortions of 30$^{\\circ }$ and 90$^{\\circ }$ , respectively.",
"As a result, a much smaller conductance of $G_{\\downarrow } \\approx 0.04 G_0$ is found for 30$^{\\circ }$ distortion compared to the flat molecule.",
"Interestingly, for the 90$^{\\circ }$ distortion, we have found almost zero conductance in both spin channels.",
"This is attributed to two reasons: i) the 2-fold degenerate HOMOs are strongly localized on either right or left sides [Fig.",
"REF (c)] and are thus completely decoupled from the corresponding electrode, which can be seen as the breaking of the conjugation; ii) the HOMO is positioned very far from the Fermi energy.",
"In this case, the molecular junction can be seen as two uncoupled molecules sandwiched between the electrodes, which does not allow the electric current to pass.",
"Summary.",
"We present ab initio quantum transport calculations of spin-polarized electron transport through a special class of $\\pi $ -conjugated molecules bridging two ferromagnetic electrodes.",
"By analysing systematically the PDOS on the apex atom of free ferromagnetic electrodes for different materials as well as for different crystallographic orientations, we selected the Fe(110) as an optimal electrode for efficient spin injection.",
"Here, a perfect DOS of appropriate symmetry is obtained in spin down channel – large and smooth around the Fermi energy – while spin up contribution is negligibly small.",
"A perfect spin filtering has been found for both quaterthiophene and p-tetraphenyl molecular junctions bridging Fe(110) electrodes which confirms the generality of our recently proposed symmetry mechanism to obtain fully-polarized currents in single molecule nanojunctions[17].",
"Interestingly, a rather high spin down conductance, $G_{\\downarrow } \\approx 0.70 G_0$ , is found for a quaterthiophene junction while much smaller value, $G_{\\downarrow } \\approx 0.09 G_0$ , for a p-tetraphenyl one.",
"Therefore, the quaterthiophene (and, more generally, poly-thiophene molecules) seems to be the best candidate for spin-filtering property, which is directly related to a very close position of its HOMO to the Fermi energy, $E=-0.13$ eV.",
"In the case of p-tetraphenyl, the conductance was found to decrease when the phenyl rings are rotated ones with respect to the others and gets completely cut in the orthogonal configuration.",
"Finally, it should be emphasized that no anti-parallel magnetic alignment of two electrodes were studied explicitely in the paper, however, we can argue that the perfectly spin-polarized conductances reported here should directly result also in ideally infinite ratios of MR as has been verified in our previous study[17].",
"We believe that our results will be important for future electronics and digital information technologies based on hybrid metal/organics components.",
"This work was performed using computation resources from GENCI-[TGCC] project (Grant No.",
"2015097416 and 2016097416)."
]
] | 1906.04412 |
[
[
"Spectral fluctuation analysis of ionospheric inhomogeneities over\n Brazilian territory Part II: E-F valley region plasma instabilities"
],
[
"Abstract The turbulent-like process associated with ionospheric instabilities, has been identified as the main nonlinear process that drives the irregularities observed in the different ionospheric regions.",
"In this complementary study, as proposed in the first article of this two-paper series [Fornari et al., Adv.",
"Space Res.",
"58, 2016], we performed the detrended fluctuation analysis of the equatorial E-F valley region electron density fluctuations measured from an in situ experiment performed over the Brazilian territory.",
"The spectral consistency with the K41 turbulent universality class is analyzed for E-F valley region from the DFA spectra for four electron density time series.",
"A complementary detrended fluctuation analysis for four time series of the F-layer electric field is also presented.",
"Consistent with the results obtained for the F region, the analysis for the E-F valley region also shows a very high spectral variation ($\\gg50\\%$).",
"Thus, the spectral analysis performed in both parts of the series suggest that a process such as the homogeneous turbulence K41 ($\\beta =-5/3\\pm 2\\%$) is inappropriate to describe both the fluctuations of electron density and the electric field associated with the main ionospheric instabilities."
],
[
"Introduction",
"The characteristic features of in situ ionospheric plasma density fluctuation data may provide important information on the structural processes associated with ionospheric irregularities [16].",
"In Part I of this work, [4] analyzed in situ F region electric field fluctuation data using the detrended fluctuation analysis (DFA) [19] technique to verify the wide variation in the spectral indices reported in earlier rocket experiments based on power spectral density (PSD) method.",
"The results show that the high variability of the spectral indices is not due to the statistical limitation of the data, and does not constitute a K41 type of universality class The K41 is a theoretical framework for turbulence proposed by Kolmogorov in 1941, which forms a basis to understand the behavior of homogeneous multiplicative energy cascade from turbulent-like processes.",
"Here the turbulent energy spectrum follows a precise power law behavior with index $-5/3\\pm 2\\%$ in the inertial range [5].",
"Therefore, the K41 spectrum represents a universality class for homogeneous turbulent processes.. As shown, in the first part of this study, PSD although widely used, falls short to characterize turbulence spectra from in situ ionospheric plasma density fluctuation measurements.",
"Many studies have shown that the power spectra of these fluctuations exhibit two or three different spectral exponents indicating the scaling complexity of the process involved [11], [28].",
"In general, the spectral indices that have been reported show deviation from the K41 theory but do not elucidate the statistical properties of the energy cascading that is supposed to drive the ionospheric turbulence [11].",
"In this context, the DFA proposed by [19] is a potential method that could render insights into the statistical properties of the turbulence phenomena.",
"As envisioned in Part I, here (Part II) the DFA is applied to in situ E-F valley region (hereafter, valley region) data.",
"The valley region is located between the top of the E region and the base of the F region.",
"The valley region, specifically the equatorial ones, hosts a variety of plasma irregularities both during the day, the so-called 150 km echoes [14], [23], and at dusk-nighttime [1].",
"This region is still a less explored area of research compared to the F region given the technical limitations in observing it.",
"It can be studied by using powerful incoherent and coherent scatter radar and in situ experiments.",
"Various studies have been reported on the correlation between the valley region irregularities and the equatorial plasma instabilities in the F region: Radar observations revealed that (i) the valley region irregularities are often found when the equatorial spread F (ESF) occurred after the sunset and that their spatial structures and temporal variations have resemblance with the ESF, and (ii) the valley region irregularities are a result of the coupling between the unstable equatorial F region and the underlying low-latitude valley and the E region [30], [31], [18], [33], [15], [12].",
"Studies based on in situ data found that electric field and gravity waves may play a key role in the generation of these structures (in the valley regions) and that the structures are produced by the generalized Rayleigh-Taylor instability mechanism at the base of the F region [31], [21], [25], [16], [24].",
"[24] reported the presence of wave-like structures in valley region data obtained from an experiment over Brazil, and the same data is used for the present analysis.",
"In literature, the DFA is applied to study ionospheric irregularities, but we could not find its application to in situ valley region data.",
"This work presents the first instance of application of the DFA to in situ E-F valley region electron density fluctuation data.",
"The paper is organized as follows.",
"Section 2 describes the data along with the electron density vertical profile.",
"The DFA is presented in Section 3 followed by the concluding remarks in Section 4."
],
[
"In situ valley region data",
"The vertical profile of electron density was obtained from a conical Langmuir probe on-board a two-stage VS-30 Orion sounding rocket experiment launched from an equatorial rocket launching station, Alcântara ($2.24^{\\circ }$ S, $44.4^{\\circ }$ W, dip latitude $5.5^{\\circ }$ S), on December 8, 2012, at 19:00 LT, under quiet geomagnetic conditions.",
"During the $\\sim $ 11 min flight, rocket trajectory was in the north-northeast direction towards the magnetic equator, ranging $\\sim $ 384 km horizontally with an apogee covering typical F region altitudes of $\\sim $ 428 km.",
"The conical Langmuir probe worked both in swept and constant bias modes.",
"The probe sensor potential was swept from -1 V to +2.5 V linearly in about 1.5 s, during which the electron kinetic temperature was determined from the collected probe current.",
"Then, the potential was maintained at +2.5 V (constant bias mode) for 1 s, during which the collected probe current was used to estimate electron density and its fluctuations, in each experiment cycle.",
"This work utilizes the electron density fluctuation data obtained from the conical Langmuir probe.",
"Fig.",
"REF shows variations in the vertically distributed electron density in the downleg (descent of the rocket) trajectory of the flight.",
"At the time of launch, the ground-based equipment detected conditions favorable for the generation of plasma bubbles in the F region.",
"[24] reported the presence of several small- and medium-scale plasma irregularities in the valley region (120-300 km) during both ascent and descent, which were more prominent during the descent of the rocket.",
"In the downleg profile, the average electron density observed was around $9\\times 10^9$ $m^{-3}$ , equivalent to $1/10$ th of the E region maximum, and then, it gradually increased after 300 km, where the broad base of F region was detected.",
"These observations are consistent with the work reported by [32], which stated that under quiet conditions, the electron concentration in the valley around midnight is about $1/10$ th of the E region maximum, and width of the valley is very wide compared to the disturbed nights.",
"[20] reported observing a deep valley region above 120 km, i.e., 120-140 km, where the electron density fell by two orders of magnitude in their experiment (to a few hundreds per cubic centimeters).",
"Fig.",
"REF (left panel) shows the selected time series from the downleg electron density profile around average heights of 143, 205, and 263.9 km from the valley region and around 316.9 km, just above the wide base of the F region."
],
[
"Fluctuation analysis, results and interpretation",
"The DFA proposed by [19] could render insights into the statistical properties of turbulence phenomena.",
"Originally proposed to detect long-range correlations in DNA sequences and in data influenced by trends, the DFA is widely used in many branches of sciences - medicine, physics, finance and social sciences - to understand the complexity of systems through its scaling exponent that characterizes fractal dynamics of the system [9], [29].",
"The robustness of DFA can be attributed to some of its interesting features.",
"For instance, [3] investigated the influence of the length of a time series in quantifying the correlation behavior using techniques like autocorrelation analysis, Hurst exponent, and DFA.",
"The comparison study revealed that the DFA is practically unaffected by the length of time series, contrary to that observed from the results of Hurst analysis or autocorrelation analysis.",
"Another interesting feature has been reported by [2] who altered time series by excluding parts of it, stitching the rest and subjecting it to the DFA.",
"The study revealed that even with the removal of 50% of the time series, the scaling behavior of positively correlated signals is unaltered, implying that time series need not be continuous.",
"[7] established an equivalence relation between the PSD exponent, $\\beta $ , and the DFA exponent, $\\alpha $ , given by $\\beta \\equiv 2\\alpha -1$ .",
"[13] showed that this relationship is valid for the higher order DFA subject to the constraint $0<\\alpha <m+1$ , where $m$ is the order of detrending polynomial in the DFA.",
"The DFA involves obtaining cumulative sum of the mean subtracted time series followed by dividing it into non-overlapping segments $(s)$ , referred to as scales.",
"Further, these segments are detrended using the linear least squares or higher order polynomial ($m$ ) method and the variance is calculated.",
"Depending on the detrending order, $m$ , the analysis is referred to as DFA$m$ .",
"Averaging the root mean square over the segments $(s)$ gives the fluctuation function, $F(s)$ .",
"Linear fit to the fluctuation function profile yields the scaling exponent $\\alpha $ .",
"Implementation procedure can be found in Part I of this paper [4].",
"In this work, four time series of electron density fluctuations from the downleg profiles corresponding to the valley region are selected.",
"The selected time series correspond to the mean heights of $143, 205, 263.9,$ and $316.9$ km (please see left panel in Fig.",
"REF ).",
"The selected time series are subjected to DFA.",
"Scales are varied from 10 to $N/4$ with a factor of $2^{\\frac{1}{8}}$ , where $N$ is the length of time series [6].",
"The fluctuation function computed from DFA is plotted as a function of scales for all the selected time series (right panel in Fig.",
"REF ) on a log-log scale.",
"The profiles of fluctuation function for all the chosen cases exhibit long-range correlation with a crossover.",
"Crossover refers to a change in the scaling exponent for different scale ranges, and it usually arises due to a change in the correlation properties over different spatial or temporal scales, or from trends in the data.",
"The exponents $\\alpha 1$ and $\\alpha 2$ are obtained from the linear fit of $F(s)$ , where $\\alpha 1$ refers to smaller scales and $\\alpha 2$ refers to larger scales.",
"Our analysis reveals $\\alpha 1$ to be in the range of $0.28$ to $1.76$ and $\\alpha 2$ in the range of $0.67$ to $1.5$ .",
"For mean heights corresponding to 143 and 205 km, we observe $\\alpha 1$ is smaller than $\\alpha 2$ , contrary to the observation for mean heights corresponding to $263.9$ and $316.9$ km.",
"In order to be sure that the obtained crossover is intrinsic to the data and not an artifact, we investigated the time series with higher order DFAs, of the order 1-5.",
"For this investigation, we used the methodology prescribed by [8] to identify false crossovers.",
"Artificial crossover exhibits similar characteristic length with identical scaling.",
"Fig.",
"REF presents the analysis for downleg time series corresponding to the mean height of 143 km with DFA of 1st to 5th order.",
"The crossover exponents are listed in Table REF .",
"It can be observed that as the order of detrending increases, crossover point moves towards larger scales and have different scaling exponents.",
"This investigation confirms that the obtained crossover is an intrinsic property of electron density fluctuation data in the valley region.",
"The PSD exponent, $\\beta $ , is calculated using the equivalence relationship given above, and the standard deviation $\\sigma _m$ (in $\\%$ ) is determined.",
"The computed DFA exponents in our analysis show a wide range of $\\beta $ from $-0.98$ to $-2.14$ with $\\sigma _m = 58\\%$ .",
"Table REF summarizes the variations in the $\\beta $ exponent obtained from the previous equivalent studies [22], [10], [17], [26], [27] and compares with the present work.",
"All studies reported in Table REF are based on electron density fluctuation data obtained through rocket experiments.",
"It is observed that the computed standard deviation $\\sigma _m\\gg 50\\%$ , which affirms that the underlying mechanism for instabilities differs from the K41 homogeneous turbulence, given the accepted deviation is $\\sigma _m\\le 2\\%$ [5].",
"We also performed the DFA on in situ electric field fluctuation data from the F region obtained from an earlier experiment conducted on December 18, 1995, at 21:17 LT, under quiet geomagnetic conditions from the same equatorial launching station Alcântara (2.24$^{\\circ }$ S, 44.4$^{\\circ }$ W, dip latitude 5.5$^{\\circ }$ S) [4].",
"The rocket flight traversed through similar altitudes of 200-300 km.",
"This data indicated the presence of a large plasma bubble at an altitude of $\\sim $ 280 km.",
"Fig.",
"REF presents the time series and the corresponding DFA.",
"The data from aforementioned experiments is selected for the altitudes of 200-300 km.",
"In valley region data, small-to-medium scale plasma irregularities [24] are found, while F-region data shows medium-to-large scale plasma irregularities (Fig.",
"2 in [16], 2003).",
"Hence, it will be interesting to compare the scaling exponents of plasma densities around similar altitudes for these two different regions.",
"Fig.",
"REF shows the scaling exponent plotted as a function of height for the valley region (left panel) and the F region (right panel).",
"For this plot, we have used a single linear fit for valley region data.",
"The shaded horizontal bar in the plot represents the exponent value, $\\alpha =1.33\\pm 2\\%$ , for the homogeneous turbulence described by the K41 theory.",
"The range of $\\alpha $ exponents for the F region is higher than that of the valley region, which may be due to different scaling present in these regions.",
"Wide variations of the scaling exponent from the K41 theory are observed for both regions."
],
[
"Concluding remarks",
"In this paper, the complementary in situ E-F valley region irregularities are studied using the DFA.",
"This study is important as studies of the equatorial E-F valley region at nighttime are scarce.",
"Our analysis shows that the E-F valley region electron density fluctuations exhibit long-range correlation with crossovers that are intrinsic to the data for all the chosen altitudes.",
"The F region irregularities obtained from an earlier experiment are also analyzed using the DFA and similar results in terms of long-range correlations are obtained for all the chosen altitudes.",
"The PSD exponent $\\beta $ is computed for the current data and compared with earlier similar experiments.",
"The results show $\\sigma _m\\gg 50\\%$ .",
"These observations along with the profile of $\\alpha $ with respect to the height indicate that scaling exponents show wide variation from the K41 theory, for both the E-F valley and F regions.",
"This implies that the turbulent-like ionospheric fluctuations as a whole cannot be described by the K41 homogeneous energy cascade theory.",
"Given this scenario and considering the different mechanisms responsible for the plasma instability along different ionospheric regions, it is necessary to investigate the model for non-homogeneous turbulence that will help to understand the observed high spectral variability.",
"A future study that emerges naturally in this scenario is to look for multifractal signature from the data analyzed here.",
"This investigation is in progress and will be published in an upcoming paper."
],
[
"Acknowledgement",
"The authors are grateful to the Institute of Aeronautics and Space (IAE/DCTA) and Alcântara Launch Center (CLA) for providing sounding rocket and launch operation, respectively.",
"NJ acknowledges the financial assistance received from CAPES.",
"RRR is grateful to FAPESP sponsored by Process No.",
"$2014/11156-4$ .",
"S.Savio acknowledges the financial support from China-Brazil Joint Laboratory for Space Weather, National Space Science Center, Chinese Academy of Science.",
"F.C.",
"de Meneses acknowledges the financial support given by the National Council of Science and Technology of Mexico (CONACYT), CAS-TWAS Fellowship for Postdoctoral and Visiting Scholars from Developing Countries under Grant no.",
"201377GB0001, and the Brazilian Council for Scientific and Technological Development (CNPq) under Grant number $312704/2015-1$ .",
"S.Stephany thanks CNPq for grant 307460/2015-0"
]
] | 1906.04316 |
[
[
"Enhancing Battle Maps through Flow Graphs"
],
[
"Abstract So-called battle maps are an appropriate way to visually summarize the flow of battles as they happen in many team-based combat games.",
"Such maps can be a valuable tool for retrospective analysis of battles for the purpose of training or for providing a summary representation for spectators.",
"In this paper an extension to the battle map algorithm previously proposed by the author and which addresses a shortcoming in the depiction of troop movements is described.",
"The extension does not require alteration of the original algorithm and can easily be added as an intermediate step before rendering.",
"The extension is illustrated using gameplay data from the team-based multiplayer game World of Tanks."
],
[
"Introduction",
"Information visualization has become an indispensable element for analyzing behavioral data of players as reflected in the increasing efforts and literature on this topic (see [2]).",
"In this context, visualization is usually considered a means for developers to analyze the collected data in order to inform the further development or to drive business decisions.",
"However, visualizations can also be targeted towards players.",
"Visualizations within games historically served mainly to convey the in-game status to players [3].",
"Recently, fueled by developers making in-game data publicly available, visualizations for the purpose of training with the goal to improve players' skills and performance (cf.",
"[3], [4]) have also gained in popularity.",
"Moreover, visualizations can also benefit spectators of esports events – on-site or, similar to traditional sport events [5], when streamed or broadcast.",
"Such training visualizations and visualizations for enhancing the spectator experience of video games have, however, received comparatively less attention among academia despite visualizations being part of games since the very early days.",
"Notable examples in the space of esports include the work of Block et al.",
"[6] who generated audience-facing summary visualizations from live and historic match data and from Charleer et al.",
"[7] who proposed dashboards superimposed over the game stream and which show live statistics about the current match status.",
"Common to both works is their focus on presenting data about the current state of the game rather than a synthesized summary of the complete match.",
"Training visualizations, on the other hand, are mostly produced by the player community itself, creating tools such as Scelight [8] for analyzing StarCraft II matches.",
"Among video games, team-based games – as team sports – pose additional requirements on visualization as these need to reflect the collaborative behaviour of multiple players (cf. [9]).",
"At the same time, players and viewers of such games can benefit strongly from visualizations that facilitate the understanding of player activity and their coordination.",
"Wallner and Kriglstein [10] investigated three different summary representations for retrospective analysis of team-based combat games for the purpose of training, including battle maps.",
"These maps, inspired from depictions of battles by historians and military planers, provide a concise overview of unit movements and encounters.",
"This short paper reports on new progress with respect to automatically creating battle maps from in-game data by extending the algorithm proposed in previous work of the author [1] to further enhance their readability.",
"This is accomplished by merging individual troop movements to better highlight the splitting and merging of troops and their total strength."
],
[
"Related Work",
"Aggregation of movement data takes on a crucial role in geographic and traffic data visualization (e.g., [11], [12]).",
"While approaches in these domains are certainly relevant for the aggregation of player movements, they are too numerous to cover here in detail.",
"We will therefore restrict the discussion to works specifically conducted within games research.",
"Examples in this space include the work of Wallner et al.",
"[13] who also made use of the same territory subdivision technique [14] as the battle map algorithm [1] extended in this work.",
"However, instead of using the derived cells to infer abstracted trajectories the cells are directly used to show the amount of movement between them.",
"Moura et al.",
"[15] also showed player flows between areas but instead of deriving the areas from the movement data itself the areas are defined based on the level geometry (e.g., rooms).",
"Canossa et al.",
"[16], on the other hand, make use of heatmaps to convey in which regions movement has occurred, thus showing rather the amount of movement that occurred without indication of the direction.",
"More closely related to our work, Miller and Crowcroft [17] used similar techniques to detect group movements in World of Warcraft.",
"Avatar movement was characterized using hotspots which, in turn, where detected by subdividing the environment into regular cells and assessing how much time the avatars spend in these cells (hotspots are also part of the battle map algorithm, but these are derived through the clustering of combat points).",
"Two avatars that move between two hotspots and maintaining a certain distance are then considered to move together.",
"Affiliation to the same group is thus defined through the geospatial locations while the battle map defines it through the similarity of semantic trajectories."
],
[
"Algorithm",
"In previous work, Wallner [1] proposed an algorithm for automatically deriving battle maps from tracked gameplay data.",
"It can be briefly summarized as follows: First, the territory of the game map is subdivided into small cells.",
"Next, trajectories of all involved units are simplified by replacing them with semantic trajectories, that is, sequences of visited locations (i.e.",
"landmarks) based on the recorded geospatial positions and the derived cells.",
"These semantic trajectories are then partitioned into sets according to their origin and destination, with trajectories within each set being further grouped based on their similarity to each other (since the same destination can be reached from the same origin by taking different routes).",
"For each group a representative trajectory is calculated that represents the overall movement of the group.",
"Finally, these representative trajectories are used for visualizing the group movements.",
"Resulting battle maps obtained using this algorithm and by processing replay files from the game World of Tanks [18] are depicted in Figure REF (left).",
"Shortly summarized, World of Tanks is an online warfare game where two teams of players compete against each other for certain objectives, with each player controlling a single tank.",
"Once a player's tank is destroyed the player cannot re-enter the match.",
"The hatched arrows in Figure REF show troop movements, while gradient shaded arrows show long distance attacks.",
"Sites of combat are enclosed within a white curve and bases and spawn points are shown using icons.",
"While an evaluation [10] attested the visualization on overall good readability it also revealed that judging troop movements correctly can be prone to errors, most likely due to the overlapping movement arrows.",
"For example, in Figure REF (top, left) the troops of the red team leaving the base in the lower right corner all move along the same path and are thus occluded until they branch off.",
"This makes judging the actual amount of troops during the first segments of the path more difficult.",
"The increased width of the arrows with accumulated travel distance may also affect the perception of troop strengths.",
"This paper addresses these issues by proposing an intermediate step before rendering as detailed in the following.",
"As no adjustments to the steps before need to be made it can be easily integrated into the original algorithm."
],
[
"Extension",
"The basic idea is to replace the individual representative trajectories with a graph structure in order to reduce occlusions and better show the splitting and merging of troops.",
"The approach is similar to that of a flow tree [19] which connects a single origin with multiple destinations.",
"However, since the merging of troops should also be visible we use a graph structure (henceforth referred to as flow graph) – more specifically a weighted directed acyclic graph – instead of a tree.",
"It should be observed that the graph is acyclic because it summarizes unidirectional movement from one source to multiple destinations.",
"If troops would, for example, backtrack this would be captured by another flow graph.",
"Also, these flow graphs are constructed separately for each team and for each origin.",
"Given a number of representative trajectories all starting at the same origin $o$ and represented by a sequence of landmarks $o=l_1 \\rightarrow l_2 \\rightarrow l_3 \\rightarrow \\ldots \\rightarrow l_k$ with each landmark associated with a location $p_i$ , first all landmarks within the enclosure of the last landmark $l_k$ (i.e.",
"destination) are removed from the sequence.",
"Next all transitions between landmarks $l_i \\rightarrow l_{i+1}$ , are added as directed edges $(l_i, l_{i+1})$ to a directed acyclic graph $G$ where the nodes represent the landmarks occurring in the representative trajectories.",
"If an edge already exists than the weight of the edge is incremented.",
"This results in a graph summarizing the transitions between landmarks.",
"Next, to ensure a smooth representation of the flow of movement, a cubic Hermite spline is derived for each edge $(l_0, l_1)$ in a flow graph starting from its root node.",
"The tangent $t_0$ at the starting position of the spline is set to the principal movement direction at $l_0$ , that is, the average direction of all incoming and outgoing movement at $l_0$ .",
"This way all the outgoing flow at a node will leave it along the same direction.",
"If troops split up at the node than the curve starts in the 'middle' of the outgoing directions while also accounting for the directions along which the units arrived at a particular node.",
"Similarly, the tangent at the end position is set to the average direction at $l_1$ .",
"Note, that this ensures $C_1$ continuity.",
"To avoid that splines originating from the same node would all start at the same location, the outgoing edges of $l_0$ are sorted clockwise around $p_0$ and the starting points of the corresponding splines are displaced along the normal of $t_0$ based on the edge weights.",
"Similarly, the incoming edges at $l_1$ are sorted clockwise around $p_1$ and the ending points of the associated splines are offset along the normal of $t_1$ .",
"This avoids occlusions of the splines at the nodes and reduces crossings of the outgoing and incoming flow.",
"Once all edges are processed, this process yields a piecewise cubic spline representation of the flow graph which can be used for rendering it in a visually appealing way.",
"The width (normalized with respect to the maximum troop size) reflects the number of units but in contrast to [10] is kept constant over traveled distance do not make the impression that the troop strength changes.",
"Lastly, labels are added to offer exact values on the number of units.",
"However, as the flow does not branch at each node but rather only at a few nodes, labels are not added to each edge (i.e.",
"spline) but instead only to segments between two nodes where each node has more than one outgoing edge."
],
[
"Results and Discussion",
"Figure REF compares results obtained without (left) and with (right) the extension by the example of two matches played on two different maps of World of Tanks [18].",
"All other parameters of the algorithm such as the similarity of paths required to group them together were kept constant.",
"The images on the top show a battle fought on the map Himmelsdorf, a confined urban map composed of squares and narrow streets.",
"The most visible difference is with respect to the red team where without the extension the arrows of troops leaving the base at the lower right corner overlap.",
"As such the number of units might be perceived to be lower than it is actually the case.",
"In contrast, with the extension the width of the arrow better accentuates the total troop strength (e.g., immediately after leaving the base) and also how the troops split up.",
"It is thus likely to give a better impression of troop strength over the course of movement.",
"In contrast, Figure REF (bottom) shows a match played on Cliff, a more open map which allows for more variation in unit movement, with troops splitting up and then reuniting again.",
"Since troops after merging at a location – as it is the case twice for the blue team – are represented by a single but thicker arrow, the merging and the resulting troop strength are visually emphasized compared to the result without the extension.",
"It should be noted that the time dimension of troop movements needs to be taken into account when merging individual representative trajectories into a flow graph.",
"Merging movements which took place at very different points in time would create a false impression of the actual tactical movements of units.",
"In the above examples, this was implicitly considered as the representative trajectories reflect movements which originated and ended at a location within a certain time span.",
"Another approach could be to only merge movements if the units where present at roughly the same time at the semantic area of interest.",
"Future work may also focus on splitting activity into time slots to produce multi-stage battle maps which could improve perception of the time dimension further.",
"Finally, follow up work will need to focus on evaluating the extension to assess if the proposed changes lead to the anticipated improvement in readability."
],
[
"Conclusions",
"This paper proposed a way to merge individual troop movements into combined flows in order to improve the readability of troop movements, their splitting and merging, and their overall strength in the context of battle maps.",
"Such maps have shown to be a valuable tool for players to reflect on their gameplay [10].",
"In addition, they could also be a promising asset in the context of esports.",
"For example, by offering a summary visualization for the viewers or as an aid for shoutcasters.",
"While this paper focused on the algorithmic aspects, further work will need to evaluate the proposed technique to ascertain the expected improvement."
]
] | 1906.04435 |
[
[
"Normalization of Hamiltonian and nonlinear stability of triangular\n equilibrium points in the photogravitational restricted three body problem\n with P-R drag in non-resonance case"
],
[
"Abstract Normal forms of Hamiltonian are very important to analyze the nonlinear stability of a dynamical system in the vicinity of invariant objects.",
"This paper presents the normalization of Hamiltonian and the analysis of nonlinear stability of triangular equilibrium points in non-resonance case, in the photogravitational restricted three body problem under the influence of radiation pressures and P-R drags of the radiating primaries.",
"The Hamiltonian of the system is normalized up to fourth order through Lie transform method and then to apply the Arnold-Moser theorem, Birkhoff normal form of the Hamiltonian is computed followed by nonlinear stability of the equilibrium points is examined.",
"Similar to the case of classical problem, we have found that in the presence of assumed perturbations, there always exists one value of mass parameter within the stability range at which the discriminant $D_4$ vanish, consequently, Arnold-Moser theorem fails, which infer that triangular equilibrium points are unstable in nonlinear sense within the stability range.",
"Present analysis is limited up to linear effect of the perturbations, which will be helpful to study the more generalized problem."
],
[
"Introduction",
"Since the time of Poincaré, invariant objects are very much important to understand the behavior of a dynamical system, especially, phase space.",
"Moreover, there are many possible approaches to find the invariant objects, whereas the normal forms (truncated) are very useful because these can give integrable approximations to the dynamics under appropriate hypothesis [15].",
"Because of the approximation of true dynamics by the normal forms, invariant objects of the initial system get approximated also, accordingly [36], [17].",
"The approximate first integrals are those quantities, which are almost preserved through the system's flow.",
"This shows that the surface levels by the flow are almost invariant.",
"Some informations about the dynamics can be obtained through this property.",
"To minimize the overflow and complexity in the computations, an appropriate approach is to use of power series or Fourier sires, or a combination of both to represent the object.",
"Because in many cases they needed only a few numbers of terms to maintain the good accuracy.",
"Some other approach can also be found in [10], [16], in which trigonometric series is used.",
"The normal forms of the Hamiltonian system up to some finite order is necessary to study the nonlinear stability of the equilibrium points using Arnold-Moser theorem in non-resonance case.",
"They also help to know the behavior of dynamics in the neighborhood of the invariant objects.",
"Many researchers have described the different method to find the normal forms of the Hamiltonian of the dynamical system [29], [2], [8], [39], [6], [15], [19].",
"In the normal forms, the central idea is to find suitable transforms of the phase co-ordinates, which can convert the Hamiltonian system in its simplest form up to a finite order of accuracy.",
"Normalization of Hamiltonian is obtained to change the Hamiltonian into its simplest form using the method of Lie transforms [6], [15].",
"Because of radiating primary in the present problem under the analysis, force due to radiation pressure came into existence [35], [32], which acts in opposite direction to the gravitational attraction force of the primary.",
"Concept of Poynting-Roberston drag is came into the picture when, [30] investigated the effect of radiation pressure on the moving particle in interplanetary space and [34] modified the Poynting's theory through the principle of relativity.",
"In the analysis of Roberston, he considered only first order terms in the expression related to the ratio of velocity of the particle to that of the light.",
"The radiation force is expressed as $\\vec{F}&=&F_p\\left(\\frac{\\vec{R}}{R}-\\frac{\\vec{V}.\\vec{R}\\vec{R}}{c R^2}-\\frac{\\vec{V}}{c}\\right),$ where $F_p$ is the radiation pressure force due to radiating primary; $\\vec{R}$ is the position vector of the particle relative to the radiating primary; $\\vec{V}$ is the velocity of the particle; and $c$ is the speed of the light.",
"First term of the equation (REF ) denotes the radiation pressure, second term represents the Doppler shift due to the motion of the particle, whereas third term corresponds to the absorption and subsequent re-emission part of induced radiation.",
"The combined form of the last two terms of the equation (REF ) known as Poynting-Robertson (P-R) drag.",
"[4] analyzed the photogravitational restricted three body problem (RTBP) with P-R drag under the frame of Sun-planet-particle system and found that non-collinear (triangular) equilibrium points are unstable.",
"Effect of P-R drag including radiation pressure is described by [35].",
"A similar analysis is presented by [28] and [31] to observed the effect of P-R drag in the context of existence and stability of the equilibrium points.",
"[20] examined the nonlinear stability in the generalized photogravitational RTBP with P-R drag of first primary and oblateness of secondary and found that triangular equilibrium points are unstable, whereas [27] investigated about the stability of non-collinear equilibrium points in the photogravitational elliptic RTBP with P-R drag.",
"[21] and [18] have analyzed the effect of radiation pressure force on the existence and linear stability of the equilibrium points in the generalized photogravitational Chermnykh-like problem with a disc.",
"They found that the effect of perturbation factors are significant.",
"In literature, many researchers have analyzed the photogravitational RTBP in nonlinear sense by considering one or two perturbations at a time [24], [13], [38], [22], [1] but very few of them have considered the problem under the combined influence of few perturbations [20], [19].",
"[14] have discussed about the nonlinear stability of out of plane equilibrium points in the RTBP with oblate primary and found that $L_6$ point is stable in nonlinear sense.",
"[33] have obtained diagonalized form of the Hamiltonian with P-R drag.",
"[19] have studied nonlinear stability of triangular equilibrium points in the Chermnykh-like problem, in the presence of radiation pressure, oblateness and a disc.",
"They found that these perturbations affect the numerical results significantly.",
"Due to above reasons in addition to wide applications of the RTBP in mission design, we are motivated to study the problem under the influence of the radiation pressures and P-R drags of both primary and secondary.",
"In the present study, we are interested to compute the fourth order normalized Hamiltonian and utilizing them to analyze the nonlinear stability of triangular equilibrium points using Arnold-Moser theorem in non-resonance case.",
"Because of both primary and secondary radiating, the problem under analysis includes the four perturbing parameters in the form of mass reduction factors $q_1,\\,q_2$ due to the radiation pressures of the primaries and P-R drags $W_1,\\,W_2$ of both the primaries, respectively.",
"The paper is organized as follows: In Section-, we have formulated the problem and found the equations of motion.",
"Section- presents the second order normalized Hamiltonian of the problem under analysis.",
"Nonlinear stability analysis is discussed in Section-.",
"Section- is devoted to Birkhoff normal form and application of Arnold-Moser theorem in non-resonance case.",
"Results are concluded in Section-.",
"For algebraic and numerical computations, Mathematica®[40] software package is used.",
"The results of this study may be used to describe more generalized problem under the influence of other perturbations such as albedo, solar wind drag, Stokes drag etc.",
"[12], [37]."
],
[
"Mathematical Formulation",
"We consider the photogravitational restricted three body problem with P-R drag, which consists of motion of an infinitesimal mass under the influence of gravitational field and radiation effect of two massive and radiating bodies of masses $m_1$ and $m_2,\\, (m_1>m_2)$ , respectively, called primaries.",
"Forces, which govern the motion of infinitesimal mass are gravitational attractions, radiation pressures and P-R drags of both the primaries, respectively.",
"It is assumed that gravitational effect of infinitesimal mass on the system is negligible.",
"Units are normalized such as units of mass and distance are taken as the sum of the masses of both the primaries and separation distance between them, respectively, whereas unit of time is the time period of the rotating frame.",
"We suppose that the coordinate of the primaries are $(-\\mu ,\\,0),\\, (1-\\mu ,\\, 0),$ respectively and that of infinitesimal mass is $(x,\\,y)$ , then the equations of motion [33] are $\\ddot{x}-2\\dot{y}&=&\\frac{\\partial U}{\\partial x},\\\\\\ddot{y}-2\\dot{x}&=&\\frac{\\partial U}{\\partial y},$ where $\\frac{\\partial U}{\\partial x}&=&x-\\frac{q_1(1-\\mu )(x+\\mu )}{r_{1}^{3}}-\\nonumber \\\\&&\\frac{q_2(1-\\mu )(x+\\mu -1)}{r_{2}^{3}}-\\nonumber \\\\&&\\frac{W_1S_1}{r_{1}^2}-\\frac{W_2S_2}{r_{2}^2},\\\\\\frac{\\partial U}{\\partial y}&=&y-\\frac{q_1(1-\\mu )y}{r_{1}^{3}}-\\frac{q_2(1-\\mu )y}{r_{2}^{3}}\\nonumber \\\\&&-\\frac{W_1S_3}{r_{1}^2}-\\frac{W_2S_4}{r_{2}^2},$ and further $S_1&=&\\frac{(x+\\mu )\\lbrace (x+\\mu )\\dot{x}+y\\dot{y}\\rbrace }{r_{1}^2}+\\dot{x}-y,\\\\S_2&=&\\frac{(x+\\mu -1)\\lbrace (x+\\mu -1)\\dot{x}+y\\dot{y}\\rbrace }{r_{2}^2}+\\dot{x}-y,\\\\S_3&=&\\frac{y\\lbrace (x+\\mu )\\dot{x}+y\\dot{y}\\rbrace }{r_{1}^2}+\\dot{y}+x+\\mu ,\\\\S_4&=&\\frac{y\\lbrace (x+\\mu -1)\\dot{x}+y\\dot{y}\\rbrace }{r_{2}^2}+\\dot{y}+x+\\mu -1$ with $q_i=1-F_{pi}/F_{gi},\\,i=1,2$ as mass reduction factors of both the primaries, respectively; $F_{pi},\\,F_{gi},\\, i=1,2$ -are the forces of radiation pressure and gravitational attraction of the respective primaries; $W_1=[(1-q_1)(1-\\mu )]/c_d$ and $W_2=[(1-q_2)\\mu ]/{c_d}$ as P-R drags of both the primaries, respectively; $c_d$ -is the speed of light in non-dimensional form; $r_1,\\,r_2$ - are distances of infinitesimal mass from the first and second primary, which are given as $r_{1}^2=(x+\\mu )^2+y^2,\\quad r_{2}^2=(x+\\mu -1)^2+y^2.$ The co-ordinates $(x_0,\\,\\pm y_0)$ of triangular equilibrium points $L_{4,5}$ are obtained on similar basis as in [33].",
"To overcome the complexity in the analysis, co-ordinates $x_0$ and $y_0$ are linearized with respected to $W_1,\\,W_2,\\epsilon _1,\\,\\epsilon _2$ , keeping in mind that the perturbing parameters lie in $(0,\\,1)$ so, take $q_1=1-\\epsilon _1,\\,q_2=1-\\epsilon _2$ , where $\\epsilon _1,\\,\\epsilon _2$ are very small.",
"The linearized co-ordinates $x_0$ and $y_0$ are $x_0&=&\\frac{1}{2}-\\mu -\\frac{4W_1(2-\\mu )}{3\\sqrt{3}}-\\nonumber \\\\&&\\frac{4W_2(1+2\\mu )}{3\\sqrt{3}}-\\frac{\\epsilon _1}{3}+\\frac{\\epsilon _2}{3},\\\\y_0&=&\\pm \\left[\\frac{\\sqrt{3}}{2}+\\frac{4W_1(2-3\\mu )}{9}+\\right.\\nonumber \\\\&&\\left.\\frac{4W_2(1-3\\mu )}{9}-\\frac{\\epsilon _1}{3\\sqrt{3}}-\\frac{\\epsilon _2}{3\\sqrt{3}}\\right].$ The plus sign corresponds to $L_4$ , whereas minus sign corresponds to $L_5$ .",
"The Hamiltonian function of the problem is written as $H&=&p_x\\dot{x}+p_y\\dot{y}-\\frac{\\dot{x}^2+\\dot{y}^2}{2}-\\frac{x^2+y^2}{2}-\\nonumber \\\\&&x\\dot{y}+\\dot{x}y-\\frac{(1-\\mu )q_1}{r_1}-\\frac{\\mu q_2}{r_2}-\\nonumber \\\\&&W_1S_5-W_2S_6,$ where $S_5&=&\\frac{(x+\\mu )\\dot{x}+y\\dot{y}}{2r_{1}^2}-\\arctan {\\left(\\frac{y}{x+\\mu }\\right)},\\\\S_6&=&\\frac{(x+\\mu -1)\\dot{x}+y\\dot{y}}{2r_{1}^2}-\\arctan {\\left(\\frac{y}{x+\\mu -1}\\right)}.$ The conjugate momenta $p_x,\\,p_y$ corresponding to generalized co-ordinate $x,\\,y$ respectively, are given as $p_x&=&\\dot{x}-y+\\frac{W_1(x+\\mu )}{2r_{1}^2}+\\frac{W_2(x+\\mu -1)}{2r_{2}^2},\\\\p_y&=&\\dot{y}+x+\\frac{W_1 y}{2r_{1}^2}+\\frac{W_2 y}{2r_{2}^2}.$"
],
[
"Second Order Normal Form of the Hamiltonian",
"In the present analysis only the stability of $L_4$ is analyzed, because the dynamics of $L_5$ is similar to that of $L_4$ .",
"Only first order terms in the perturbing parameters $W_1,\\,W_2,\\,q_1,\\,q_2$ are considered for simplifying the complex calculations involved in the problem through out the analysis.",
"The second order normal form of the Hamiltonian of the problem under analysis is obtained in [33] and for self sufficiency of this paper, we have taken some necessary expressions in appropriate form there to use under this section.",
"Shifting the origin to the triangular equilibrium point $L_4$ using simple transformations as $&& x^*=x-x_0,\\, y^*=y-y_0,\\,\\\\&&{p_x}^*=p_x+y_0,\\, {p_y}^*=p_y-x_0.$ Substituting these variables in Hamiltonian (REF ), we get new Hamiltonian $H^*$ .",
"Now, expanding the new Hamiltonian using Taylor's series about the origin, which is now, the triangular equilibrium point, $H^*$ can be written as $H^*=H_{0}^*+H_{1}^*+H_{2}^*+H_{3}^*+\\dots +H_{n}^*+\\dots ,$ where $H_{n}^*=\\sum H_{ijkl}{x^*}^i{y^*}^j{{p_x}^*}^k{{p_y}^*}^l,$ such that $i+j+k+l=n$ .",
"Since, the origin is the triangular equilibrium point, $H_{1}^*$ must vanish, whereas $H_{0}^*$ is constant hence, it can be dropped out as it is irrelevant to the dynamics.",
"The quadratic Hamiltonian $H_{2}^*$ , which is to be normalized first and then to be used for higher order normalization, is given as $H_{2}^*&=&\\frac{{{p_x}*}^2+{{p_y}^*}^2}{2}+y^*{p_x}^*-x^*{p_y}^*\\nonumber \\\\&&+E{x^*}^2+Gx^*y^*+F{y^*}^2,$ where $E&=&\\frac{1}{8}+\\frac{4W_1}{\\sqrt{3}}+\\frac{2W_1}{\\sqrt{3}}+\\frac{\\epsilon _1}{4}-\\frac{\\epsilon _2}{2},\\\\F&=&-\\frac{5}{8}-\\frac{4W_1}{\\sqrt{3}}-\\frac{2W_1}{\\sqrt{3}}-\\frac{\\epsilon _1}{4}+\\frac{\\epsilon _2}{2},\\\\G&=&-\\gamma \\left(1-\\frac{32W_1}{9\\sqrt{3}}-\\frac{16W_1}{9\\sqrt{3}}\\right.\\nonumber \\\\&&\\left.-\\frac{2\\epsilon _1}{9}+\\frac{4\\epsilon _2}{9}\\right),\\\\\\text{with}\\quad \\gamma &=&\\frac{3\\sqrt{3}}{4}(1-2\\mu ).$ In the present study, the problem is dealt with four perturbation parameters in the form of P-R drag and radiation pressure of both the primaries.",
"Hence, the coefficient $H_{ijkl}$ for $i,\\,j,\\,k,\\,l=0,\\,1,\\,2,\\,3,\\,4$ such that $i+j+k+l=4$ in (REF ) can be bifurcated into five parts such as $H_{ijkl1},\\,H_{ijkl2},\\,H_{ijkl3},\\,H_{ijkl4}$ , and $H_{ijkl5}$ , which corresponds to the terms in classical case, terms with P-R drag of first primary $W_1$ , P-R drag of second primary $W_2$ , radiation pressure of first primary $\\epsilon _1=1-q_1$ and radiation pressure of second primary $\\epsilon _2=1-q_2$ , respectively.",
"Thus, $ H_{ijkl}&=&H_{ijkl1}+H_{ijkl2}+H_{ijkl3}\\nonumber \\\\&&+H_{ijkl4}+H_{ijkl5}.$ It is noted that if there is no perturbations in the system, i.e.",
"$W_1=W_2=\\epsilon _1=\\epsilon _2=0$ , then $H_{ijkl}=H_{ijkl1}$ , which is nothing but the coefficient of the Hamiltonian in classical case.",
"Hamiltonian equations of motion of the infinitesimal mass in matrix form is written as $\\begin{bmatrix}\\dot{x^*}\\\\\\dot{y^*}\\\\\\dot{{p_x}^*}\\\\\\dot{{p_y}^*}\\end{bmatrix}=\\begin{bmatrix}0&1&1&0\\\\-1&0&0&1\\\\-2E&-G&0&1\\\\-G&-2F&-1&0\\end{bmatrix}\\begin{bmatrix}x^*\\\\y^*\\\\{p_x}^*\\\\{p_y}^*\\end{bmatrix}.$ The characteristic equation of the system (REF ) is $ &&\\lambda ^4+2(E+F+1)\\lambda ^2+(4EF-G^2-\\nonumber \\\\&&2E-2F+1)=0.$ Solving the simplified discriminant of the characteristic equation (REF ) as $(E+F+1)^2-(4EF-G^2-2E-2F+1)=0,$ we have the value of critical mass ratio $0<\\mu _c\\le (1/2)$ as $\\mu _c&=&0.0385209+0.0823761 W_1+0.0823761 W_2\\nonumber \\\\&&+0.0178349 \\epsilon _1-0.356699 \\epsilon _2,$ which is similar to that of [20] and [18] and agree with the classical value $\\mu _c=0.0385209$ .",
"Figure: Variation of critical mass ratio μ c \\mu _c with respect to (a) W 1 W_1, (b) W 2 W_2, (c) ϵ 1 \\epsilon _1 and (d) ϵ 2 \\epsilon _2.Figure (REF )(a-d) shows the variations of critical mass ratio $\\mu _c$ with respect to perturbing parameters $W_1,\\,W_2,\\, \\epsilon _1$ and $\\epsilon _2,$ respectively.",
"We observed that the effects of the perturbations in question are significant.",
"As, system will be stable when four roots of the characteristic equation (REF ) are pure imaginary, which is possible when the mass parameter $\\mu $ satisfy the condition $0<\\mu <\\mu _c$ .",
"Since, we are analyzing the nonlinear stability within the range of linear stability $0<\\mu <\\mu _c$ , it is obvious to assume that roots of the characteristic equation (REF ) are pure imaginary.",
"Suppose, the roots of the characteristic equation (REF ) are $\\pm i\\omega _1$ and $\\pm i\\omega _2$ , where $\\omega _1,\\,\\omega _2$ can be obtained by solving the equation $ &&\\omega ^4-2(E+F+1)\\omega ^2+(4EF-G^2-\\nonumber \\\\&&2E-2F+1)=0.$ Motion corresponds to frequencies $\\omega _{1},\\,\\omega _2\\in \\mathbb {R}$ are known as long and short periodic motion of infinitesimal mass at $L_4$ with periods of $2\\pi /\\omega _1$ and $2\\pi /\\omega _2$ , respectively.",
"Frequencies $\\omega _{1},\\,\\omega _2$ corresponding to the long and short periodic motion are related to each other by the means of relations $ \\omega _{1}^2+\\omega _{2}^2&=&2E+2F+2,\\\\\\omega _{1}^2\\omega _{2}^2&=&4EF-G^2-2E-2F+1.$ Substituting the values of $E,\\,F$ and $G$ from equations (REF -), we get $ \\omega _{1}^2+\\omega _{2}^2&=&1,\\\\\\omega _{1}^2\\omega _{2}^2&=&\\frac{27}{16}\\gamma ^2-4\\sqrt{3}W_1-2\\sqrt{3}W_2-\\nonumber \\\\&&\\frac{2\\epsilon _1}{3}+\\frac{3\\epsilon _2}{4},$ where the values of $\\omega _1$ and $\\omega _2$ are $\\omega _1=\\sqrt{-1+\\sqrt{1-4\\delta }},\\quad \\omega _2=\\sqrt{-1-\\sqrt{1-4\\delta }},$ with $\\delta =\\frac{27}{16}-\\gamma ^2-4\\sqrt{3}W_1-2\\sqrt{3}W_2-\\frac{3\\epsilon _1}{4}+\\frac{3\\epsilon _2}{2}.$ The real normalized Hamiltonian of the Hamiltonian (REF ) up to second order is given as [33] $H_2=\\omega _1\\frac{\\bf {x^2+p_{x}^2}}{2}+\\omega _2\\frac{\\bf {y^2+p_{y}^2}}{2},$ which is complexified by using the co-ordinate transformations $\\bf {x}&=&\\frac{X+iP_X}{\\sqrt{2}},\\\\\\bf {y}&=&\\frac{-Y+iP_Y}{\\sqrt{2}},\\\\\\bf {p_x}&=&\\frac{iX+P_X}{\\sqrt{2}},\\\\\\bf {p_y}&=&\\frac{iY-P_Y}{\\sqrt{2}}$ and changed as $H_2=i\\omega _1 XP_X-i\\omega _2YP_Y,$ Finally, symplectic matrix $\\bf {C}$ of the symplectic transformations, which are used to obtain the complex normal form of Hamiltonian is given as [33] $&&{\\bf {C}}=\\begin{bmatrix}s_{ij}\\end{bmatrix},\\,1\\le i,j\\le 4$ with $&&s_{11}=0=s_{12},\\,s_{13}=\\frac{1-2F+\\omega _{1}^2}{\\sqrt{d(\\omega _1)}},\\\\&& s_{14}=\\frac{1-2F+\\omega _{1}^2}{\\sqrt{d(\\omega _2)}},\\,s_{21}=\\frac{2\\omega _1}{\\sqrt{d(\\omega _1)}},\\\\&& s_{22}=\\frac{2\\omega _1}{\\sqrt{d(\\omega _2)}},\\, s_{23}=\\frac{G}{\\sqrt{d(\\omega _1)}},\\\\&&s_{24}=\\frac{G}{\\sqrt{d(\\omega _2)}},\\, s_{31}=\\frac{\\omega _{1}^3-(2F+1)\\omega _1}{\\sqrt{d(\\omega _1)}},\\\\&& s_{32}=\\frac{\\omega _{2}^3-(2F+1)\\omega _2}{\\sqrt{d(\\omega _2)}},\\, s_{33}=\\frac{-G}{\\sqrt{d(\\omega _1)}},\\\\&& s_{34}= \\frac{-G}{\\sqrt{d(\\omega _2)}},\\, s_{41}=\\frac{G\\omega _1}{\\sqrt{d(\\omega _1)}},\\, s_{42}=\\frac{G\\omega _2}{\\sqrt{d(\\omega _2)}},\\\\&& s_{43}=\\frac{1-2F-\\omega _{1}^2}{\\sqrt{d(\\omega _1)}},\\, s_{44}=\\frac{1-2F-\\omega _{2}^2}{\\sqrt{d(\\omega _2)}},$ where $d(\\omega _i)$ for $i=1,\\,2$ is obtained from the following equation $ d(\\omega )&=&\\omega \\left[\\omega ^4-(2E+6F)\\omega ^2+\\right.\\nonumber \\\\&&\\left.",
"(4EF+4F^2-2E+2F-2)\\right].$"
],
[
"Nonlinear Stability in Non-resonance Case",
"Nonlinear stability of the equilibrium points can be described in two cases, one as resonance case and other as non-resonance case.",
"For resonance case, the nonlinear stability is studied through the theorems of [23] as in [11] and for non-resonance case, it is analyzed through the Arnold-Moser theorem.",
"In the present analysis the nonlinear stability of the perturbed triangular equilibrium point in non-resonance case will be studied through Arnold-Moser theorem [25], [26], which is described as follows: Consider the Hamiltonian expressed in action variables $ I_1,\\, I_2$ and angles variables $\\phi _1,\\, \\phi _2$ as, $ &&K=K_2+ K_4 + \\dots +K_{2m}+K_{2m+1}, $ in which: (i) $K_{2m}$ is homogeneous polynomial of degree $m$ in action variables $I_1,\\,I_2$ and $K_{2m+1}$ is higher degree polynomial than $m$ (ii) $K_2=\\omega _1 I_1-\\omega _2 I_2$ with $\\omega _{1,2}$ as positive constants (iii) $K_4=-(AI_{1}^2+BI_1I_2+CI_{2}^2)$ , where $A,\\,B,\\,C$ are constants to be determined.",
"Since, $K_2,\\,K_4,\\,\\dots ,K_{2m}$ are functions of $I_1$ and $I_2$ , the Hamiltonian (REF ) follows the Birkhoff normal form [2] up to the terms $m$ .",
"This can be obtained with some non-resonance condition on the frequencies $\\omega _{1},\\,\\omega _2$ .",
"To state the Arnold-Moser theorem, we assume that $K$ is in the required form.",
"Arnold-Moser Theorem: The origin is stable for the system whose Hamiltonian is (REF ) provided for some $\\nu ,\\,\\,2\\le \\nu \\le m$ , $D_{2\\nu } = K_{2\\nu }(\\omega _2,\\omega _1) \\ne 0$.",
"Since, for Arnold-Moser theorem, Birkhoff normal form of the Hamiltonian is necessary and for Birkhoff normal form, assumption of non-resonance on frequencies is required.",
"The non-resonance condition of frequencies as in [9], [19] is that if $\\omega _{1},\\,\\omega _2$ are frequencies of infinitesimal mass in linear dynamics and $\\sigma \\in \\mathbb {Z}$ such that $\\sigma \\ge 2$ , then $\\sigma _1\\omega _1+\\sigma _2\\omega _2\\ne 0 $ for all $\\sigma _{1},\\,\\sigma _2\\in \\mathbb {Z}$ satisfying $|\\sigma _1|+|\\sigma _2|\\le 2\\sigma $ .",
"This is also, called as condition of irrationality, which insures that there exists a symplectic normalizing transformation which transform the Hamiltonian (REF ) in the form of Hamiltonian (REF ).",
"Coefficients of the normalized Hamiltonian are independent on the integer $\\sigma $ as well as to the transformation obtained.",
"In specific $\\text{det}\\begin{vmatrix}\\frac{\\partial ^2K}{\\partial I_{1}^2}&\\frac{\\partial ^2K}{\\partial I_1\\partial I_{2}}&\\frac{\\partial K}{\\partial I_{1}}\\\\\\frac{\\partial ^2K}{\\partial I_2\\partial I_{1}}&\\frac{\\partial ^2K}{\\partial I_{2}^2}&\\frac{\\partial K}{\\partial I_{2}}\\\\\\frac{\\partial K}{\\partial I_{1}}&\\frac{\\partial K}{\\partial I_{2}}&0\\end{vmatrix}_{I_1,I_2=0}$ is invariant of the Hamiltonian (REF ) with respect to the symplectic transformation considered.",
"The nonlinear stability of perturbed triangular equilibrium points is analyzed through the Arnold-Moser theorem under these conditions.",
"In classical case frequencies $\\omega _{1},\\,\\omega _2$ satisfy the condition $0<\\omega _2<(1/\\sqrt{2})<\\omega _1<1$ .",
"Therefore, if $\\sigma =2$ , then irrationality condition (REF ) fails for following pairs of integers $\\sigma _1=1,\\, \\sigma _2=-2$ , $\\sigma _1=-1,\\, \\sigma _2=2$ , $\\sigma _1=1,\\, \\sigma _2=-3$ and $\\sigma _1=-1,\\, \\sigma _2=3$ .",
"First, two pairs of integers with condition (REF ) yield $(\\omega _1/\\omega _2)=(1/2)$ and last two pairs of integers give $(\\omega _1/\\omega _2)=(1/3)$ , which are also known as second and third order resonance of the frequencies respectively.",
"If $(\\omega _1/\\omega _2)=(1/2)$ or $\\omega _1=2\\omega _2$ , then from equations (REF -), we get $ \\frac{4}{25}=\\frac{27}{16}\\gamma ^2-4\\sqrt{3}W_1-2\\sqrt{3}W_2-\\frac{2\\epsilon _1}{3}+\\frac{3\\epsilon _2}{4}.$ Simplifying equation (REF ), we have a quadratic equation in $\\mu $ as $&&\\frac{27}{16}\\mu ^2-\\frac{27}{16}\\mu +\\left(\\sqrt{3}W_1+\\frac{\\sqrt{3}W_2}{2}+\\right.\\nonumber \\\\&&\\left.\\frac{3\\epsilon _1}{16}-\\frac{3\\epsilon _2}{8}+\\frac{1}{25}\\right)=0.$ The solution $\\mu =\\mu _{c1}$ of equation (REF ) within the stability range $0<\\mu <\\mu _c$ is $\\mu _{c1}&=&0.0242939+1.078820 W_1+0.539409 W_2\\nonumber \\\\&&+0.116785 \\epsilon _1-0.233571 \\epsilon _2.$ This means, Arnold-Moser theorem fails at $\\mu _{c1}\\in (0,\\,\\mu _c)$ .",
"If $(\\omega _1/\\omega _2)=(1/3)$ or $\\omega _1=3\\omega _2$ , then proceeding on similar basis, we find that Arnold-Moser theorem fails at $\\mu =\\mu _{c2}$ , where $\\mu _{c2}&=&0.013516+1.054920 W_1+0.527459 W_2\\nonumber \\\\&&+0.114198 \\epsilon _1-0.228396 \\epsilon _2.$ Equations (REF -REF ) are similar to that of the results in [9], [18] and agree with classical result in the absence of perturbing parameters.",
"To see the effects of perturbing parameters on $\\mu _{c1}$ and $\\mu _{c2}$ , its numerical values are computed and presented in Table-REF .",
"From Table-REF , it is clear that the values of $\\mu _{c1}$ and $\\mu _{c2}$ are very much affected from radiation pressures and P-R drags of the primaries.",
"Table: μ c1 \\mu _{c1} and μ c2 \\mu _{c2} at different values of perturbing parameters."
],
[
"Fourth order Normalized Hamiltonian",
"Since, Birkhoff's normal form up to fourth order of the Hamiltonian is necessary to apply the Arnold-Moser theorem, which is computed from second order normalized Hamiltonian (REF ) using Lie transform method described in [5], [7], [15], [3], [19].",
"As, in the paper of [7] as well as in the book of [3], higher order normalized Hamiltonian is $K&=&K_2+K_3+K_4+\\dots +K_n+\\dots ,$ where $K_n&=&\\sum {K_{ijkl}X^iY^j{P_X}^k{P_Y}^l}$ such that $i+j+k+l=n.$ Quadratic part of $K$ is $K_2=H_2$ , whereas $K_n$ through the nth step of Lie transform is given as $K_n&=&\\frac{1}{n}\\lbrace H_2,\\,G_n\\rbrace +(\\text{known terms}),$ where Lie bracket of normalized quadratic Hamiltonian $H_2$ and generating function $G_n$ is defined as $\\lbrace H_2,\\,G_n\\rbrace &=&\\frac{\\partial H_2}{\\partial X}\\frac{\\partial G_n}{\\partial P_X}-\\frac{\\partial H_2}{\\partial P_X}\\frac{\\partial G_n}{\\partial X}\\nonumber \\\\&&+\\frac{\\partial H_2}{\\partial Y}\\frac{\\partial G_n}{\\partial P_Y}-\\frac{\\partial H_2}{\\partial P_Y}\\frac{\\partial G_n}{\\partial Y}.$ Using $H_2$ from equation (REF ), it reduces to $\\lbrace H_2,\\,G_n\\rbrace &=&\\mathbf {i}\\omega _1\\left(P_X\\frac{\\partial G_n}{\\partial P_X}-X\\frac{\\partial G_n}{\\partial X}\\right)\\nonumber \\\\&&+\\mathbf {i}\\omega _2\\left(P_Y\\frac{\\partial G_n}{\\partial P_Y}-Y\\frac{\\partial G_n}{\\partial Y}\\right).$ The choice of generating function $G_n$ is such that the above partial differential operator on $G_n$ remove large possible number of terms from the expression of $K_n$ .",
"As, each terms of the $K_n$ is of the form $\\alpha X^iY^j{P_X}^k{P_Y}^l$ , where $\\alpha $ is constant, we can assume terms in $G_n$ of the form $\\beta X^iY^j{P_X}^k{P_Y}^l$ , where constant $\\beta $ is to be determined.",
"Therefore, we obtain that $\\frac{\\lbrace H_2,\\,G_n\\rbrace }{n}=\\frac{\\mathbf {i}\\beta }{n}\\left[(k-i)\\omega _1-(l-j)\\right]X^iY^j{P_X}^k{P_Y}^l,$ and hence, $\\beta =\\frac{\\mathbf {i}\\alpha }{\\left[(k-i)\\omega _1-(l-j)\\right]},\\,\\,i+j+k+l=n.$ This shows that even in the non-resonance case, the term of the form $X^iY^j{P_X}^i{P_Y}^j$ in $K_n$ can not be deleted because of vanishing denominator in (REF ) at $i=k,\\,j=l$ , whereas in the resonance case some additional non-removable terms occur while solving the generating function $G_n$ .",
"Hence, in non-resonance case, the Hamiltonian of the present problem can be written in the form of (REF ), in which $K_2&=&i\\omega _1XP_X -i\\omega _2YP_Y,\\\\K_3&=&0,\\\\K_4&=&\\frac{AX^2P_{X}^2+BXP_XYP_Y+CY^2P_{Y}^2}{2},$ where $A=2K_{2020},\\,B=2K_{1111},$ and $C=2K_{0202}$ .",
"Using action variables $I_1=iXP_X$ and $I_2=iYP_Y$ in equations (REF -), we get $K_2&=&\\omega _1I_1 -\\omega _2I_2,\\\\K_3&=&0,,\\\\K_4&=&-\\left(AI_{1}^2+BI_I1_2+CI_{2}^2\\right).$ Thus, normalized Hamiltonian up to fourth order is $K(I_1,\\,I_2)&=&K_2+K_3+K_4\\nonumber \\\\&=&\\omega _1I_1 -\\omega _2I_2-K_{2020}I_{1}^2+\\nonumber \\\\&&K_{1111}I_I1_2+K_{0202}I_{2}^2,$ which agree with that of [9], [20], [19].",
"Figure: Zero (μ 0 )(\\mu _0) of the determinant D 4 D_4 within the stability range 0<μ<μ c 0<\\mu <\\mu _c at: (a) W 1 =W 2 =ϵ 1 =ϵ 2 =0W_1=W_2=\\epsilon _1=\\epsilon _2=0 (classical case); (b) W 1 =0.015,W 2 =0.005,ϵ 1 =0.003,ϵ 2 =0.03W_1=0.015,\\,W_2=0.005,\\,\\epsilon _1=0.003,\\,\\epsilon _2=0.03 (perturbed case).Figure: Zero (μ 0 )(\\mu _0) of the determinant D 4 D_4 within the stability range 0<μ<μ c 0<\\mu <\\mu _c at: (a) W 1 =0.015,W 2 =ϵ 1 =ϵ 2 =0W_1=0.015,\\,W_2=\\epsilon _1=\\epsilon _2=0 (only in presence of P-R drag of first primary); (b) Zoom of specified region of figure (a).Figure: Zero (μ 0 )(\\mu _0) of the determinant D 4 D_4 within the stability range 0<μ<μ c 0<\\mu <\\mu _c at: (a) W 2 =0.005,W 1 =ϵ 1 =ϵ 2 =0W_2=0.005,\\,W_1=\\epsilon _1=\\epsilon _2=0 (only in presence of P-R drag of second primary); (b) Zoom of specified region of figure (a).Figure: Zero (μ 0 )(\\mu _0) of the determinant D 4 D_4 within the stability range 0<μ<μ c 0<\\mu <\\mu _c at: ϵ 1 =1-q 1 =0.003,W 1 =W 2 =ϵ 2 =0\\epsilon _1=1-q_1=0.003,\\,W_1=W_2=\\epsilon _2=0 (only in presence of radiation pressure of first primary).Figure: Zero (μ 0 )(\\mu _0) of the determinant D 4 D_4 within the stability range 0<μ<μ c 0<\\mu <\\mu _c at ϵ 2 =1-q 2 =0.03,W 1 =W 2 =ϵ 1 =0\\epsilon _2=1-q_2=0.03,\\,W_1=W_2=\\epsilon _1=0 (only in presence of radiation pressure of second primary).Form equations (REF ), it is clear that fourth order normalized Hamiltonian is the function of only action variables $I_1,\\, I_2$ , which shows that these are in Birkhoff normal form.",
"The coefficients $K_{ijkl}$ used in the equation ( or ) can be written into 5 parts such as $K_{ijkl1}$ , $K_{ijkl2}$ , $K_{ijkl3}$ , $K_{ijkl4}$ and $K_{ijkl5}$ for $i,\\,j,\\,k,\\,l=0,\\,1,\\,2,\\,3,\\,4$ such that $i+j+k+l=4$ .",
"These coefficients corresponds to the term of classical part, terms with P-R drags $W_1$ and $W_2$ of first and second primary, radiation pressures $\\epsilon _1=1-q_1$ and $\\epsilon _2=1-q_2$ of first and second primary, respectively.",
"In the absence of perturbing parameters i.e.",
"for $W_1=W_1=\\epsilon _1=\\epsilon _2=0$ , $K_{ijkl}={K{ijkl1}}$ .",
"Therefore, $K_{2020},\\,K_{1111}$ and $K_{0202}$ become $K_{2020}&=&K_{20201}+K_{20202}+K_{20203}+\\nonumber \\\\&&K_{20204}+K_{20205},\\\\K_{1111}&=&K_{11111}+K_{11112}+K_{11113}+\\nonumber \\\\&&K_{11114}+K_{11115},\\\\K_{0202}&=&K_{02021}+K_{02022}+K_{02023}+\\nonumber \\\\&&K_{02024}+K_{02025}.$ The algebraic expressions of above 15 coefficients on right hand sides of equations (REF -) are too complicated and huge to be placed here hence, we avoid to present in the paper.",
"These are utilized to compute the determinant $D_{4}=K_4(\\omega _2,\\,\\omega _1)$ for applying the Arnold-Moser theorem.",
"For the simplicity, $D_4$ is expressed as $ D_4&=&\\left(\\frac{A_1}{B_1}\\right) +\\left(\\frac{A_2}{B_2}\\right) W_1+\\left(\\frac{A_3}{B_3}\\right) W_2+\\nonumber \\\\&&\\left(\\frac{A_4}{B_4}\\right) \\epsilon _1+\\left(\\frac{A_5}{B_5}\\right) \\epsilon _2,$ where $A_i,\\, B_i,\\, i=1,2,3,4,5$ are numerator and denominator of the coefficients, which correspond to classical part, P-R drags $W_1$ and $W_2$ of the primaries, radiation pressure $\\epsilon _1=1-q_1$ and $\\epsilon _2=1-q_2$ of the primaries, respectively.",
"On simplification, we found that $A_1&=&-35+541\\omega _{1}^2\\omega _{2}^2-644\\omega _{1}^4\\omega _{2}^4,\\quad \\quad \\quad \\quad $ $A_2&=&26244\\left(2262-653b\\right)-\\nonumber \\\\&&27\\left(5292162-4787719 b\\right)\\omega _{1}^2\\omega _{2}^2\\nonumber \\\\&&-2\\left(402982614-10430203 b\\right)\\omega _{1}^4\\omega _{2}^4\\nonumber \\\\&&+32\\left(12457908-1490819b\\right)\\omega _{1}^6\\omega _{2}^6\\nonumber \\\\&&+1024\\left(67581+1634b\\right)\\omega _{1}^8\\omega _{2}^8,$ $A_3&=&-78732\\left(416-241\\sqrt{3} b\\right)-\\nonumber \\\\&&27\\left(3181248+4414649sqrt{3} b\\right)\\omega _{1}^2\\omega _{2}^2\\nonumber \\\\&&-6\\left(72610776+10390609\\sqrt{3} b\\right)\\omega _{1}^4\\omega _{2}^4\\nonumber \\\\&&+32\\left(6808752-212191\\sqrt{3} b\\right)\\omega _{1}^6\\omega _{2}^6\\nonumber \\\\&&+1024\\left(36828+997\\sqrt{3} b\\right)\\omega _{1}^8\\omega _{2}^8,$ $A_4&=&8748\\left(195\\sqrt{3}-584 b\\right)-\\nonumber \\\\&&27\\left(465795\\sqrt{3}-1556744 b\\right)\\omega _{1}^2\\omega _{2}^2\\nonumber \\\\&&+2\\left(10722915\\sqrt{3}-11609036 b\\right)\\omega _{1}^4\\omega _{2}^4\\nonumber \\\\&&+32\\left(200970\\sqrt{3}-103079 b\\right)\\omega _{1}^6\\omega _{2}^6\\nonumber \\\\&&-512\\left(2565\\sqrt{3}-3217 b\\right)\\omega _{1}^8\\omega _{2}^8,$ $A_5&=&-8748\\left(507\\sqrt{3}-688 b\\right)+\\nonumber \\\\&&27\\left(315819\\sqrt{3}-1533728 b\\right)\\omega _{1}^2\\omega _{2}^2\\nonumber \\\\&&+2\\left(3085838\\sqrt{3}+759212 b\\right)\\omega _{1}^4\\omega _{2}^4\\nonumber \\\\&&-32\\left(941526\\sqrt{3}-283835 b\\right)\\omega _{1}^6\\omega _{2}^6\\nonumber \\\\&&-512\\left(10251\\sqrt{3}+877 b\\right)\\omega _{1}^8\\omega _{2}^8,$ $B_1&=&8\\left(1-4\\omega _{1}^2\\omega _{2}^2\\right)\\left(4-25\\omega _{1}^2\\omega _{2}^2\\right),\\\\B_2&=&864ab,\\\\B_3&=&B_2,\\\\B_4&=&1152ab,\\\\B_5&=&576ab,$ $a&=&\\left[\\omega _{1}^2\\omega _{2}^2\\left(1-4\\omega _{1}^2\\omega _{2}^2\\right)\\right.\\\\&&\\left.\\left(4-25\\omega _{1}^2\\omega _{2}^2\\right)\\left(117+16\\omega _{1}^2\\omega _{2}^2\\right)\\right],\\\\b&=&\\sqrt{\\left(27-16\\omega _{1}^2\\omega _{2}^2\\right)}.$ In the absence of perturbing parameters, $D_4&=&\\frac{-35+541\\omega _{1}^2\\omega _{2}^2-644\\omega _{1}^4\\omega _{2}^4}{8\\left(1-4\\omega _{1}^2\\omega _{2}^2\\right)\\left(4-25\\omega _{1}^2\\omega _{2}^2\\right)}, $ which agree with the classical result [9], [25], [20], [19].",
"In order to analyze the nonlinear stability of triangular equilibrium points in non-resonance case using Arnold-Moser theorem, we plot the determinant $D_4$ with respect to the mass parameter $\\mu $ to insure the value of $D_4=K_4(\\omega _2,\\omega _1)$ .",
"From figures (REF -REF ), it is clear that within the linear stability range $0<\\mu <\\mu _c$ , there exists one value of mass parameter $\\mu =\\mu _0$ , called the zero of $D_4$ , at which $D_4$ vanish in each case.",
"Thus, Arnold-Moser theorem fails, which insure that in non-resonance case, triangular equilibrium points of the problem under analysis are unstable in nonlinear sense within the linear stability range $0<\\mu <\\mu _c$ .",
"To see the effect of perturbing parameters, we have computed values of the zero $(\\mu _0)$ of $D_4$ and critical mass ratio $(\\mu _c)$ at different values of perturbing parameters $W_1,\\,W_2,\\,\\epsilon _1,\\,\\epsilon _2$ and results are placed in Table-REF .",
"From Table-REF , it is noticed that on increase in the values of $W_1,\\,W_1,\\,\\epsilon _1$ , value of critical mass $\\mu _c$ increases but the value of $\\mu _0$ is nonzero in each case.",
"On the other hand, on increase in the value of $\\epsilon _2$ , $\\mu _c$ decreases with nonzero $\\mu _0$ .",
"Thus, from the figures (REF -REF ) as well as from the Table-REF , it is clear that radiation pressure and P-R drag of both the primaries affect the linear stability range of the problem significantly.",
"The nonzero value of the zero $(\\mu _0)$ of the determinant $D_4$ in the Arnold-Moser theorem under non-resonance case, insure the instability of triangular equilibrium points, within the range of stability $0<\\mu <\\mu _c$ .",
"Table: Zero (μ 0 )(\\mu _0) of D 4 D_4 and critical mass ratio μ c \\mu _c at different values of perturbing parameters."
],
[
"Conclusions",
"We have considered the photogravitational restricted three body problem in the presence of radiation pressure force and P-R drag of both the massive bodies, which are radiating in nature.",
"Analysis of nonlinear stability of the triangular equilibrium points is performed in non-resonance case using Arnold-Moser theorem under the influence of four perturbing parameters in the form of P-R drags $W_1, \\, W_2$ and mass reduction factors $q_1,\\,q_2$ , of both the primaries.",
"First, we have normalized the Hamiltonian of the problem up to order four using Lie transform method and then Birkhoff normal form of the Hamiltonian constructed, which is necessary to apply the Arnold-Moser theorem in non-resonance case.",
"The determinant $D_4$ of the Arnold-Moser theorem is computed analytically under the consideration of only linear order terms of perturbing parameters, which agree with that of [9], [25], [20], [19] in the absence of perturbing parameters.",
"To apply the Arnold-Moser theorem in non-resonance case, we have plotted the determinant $D_4$ with respect to the mass parameter $\\mu $ within the stability range $0<\\mu <\\mu _c$ .",
"It is observed that in presence as well as in absence of perturbing parameters, there exist a nonzero value of $\\mu =\\mu _0$ at which $D_4$ vanish (figures (REF -REF )), which insure that triangular equilibrium points are unstable in nonlinear sense.",
"The effect of perturbing parameters are also analyzed and it is found that on increasing the values of $W_1,\\,W_1,\\,\\epsilon _1$ , critical mass ratio $\\mu _c$ increases, with the existence of nonzero $\\mu _0$ in each case , whereas on increasing the value of $\\epsilon _2$ , $\\mu _c$ decreases with the existence of nonzero $\\mu _0$ (Figure (REF ) and Table-REF ).",
"A similar trend is also seen in case of $\\mu _{c1}$ and $\\mu _{c2}$ (Table-REF ).",
"Thus, we conclude that due to radiation pressure and P-R drag of both the primaries, the linear stability range of the problem get changed, significantly.",
"Also, due to existence of nonzero value of the zero $(\\mu _0)$ of the determinant $D_4$ in the Arnold-Moser theorem under non-resonance case, within the range of stability $0<\\mu <\\mu _c$ , triangular equilibrium points are unstable in nonlinear sense.",
"Present analysis is limited up to first order terms of the perturbing parameter, which may be extended to higher order inclusion of the terms.",
"The results obtained can help to analyze the more generalized problem under the influence of other perturbations such as albedo, solar wind drag, Stokes drag etc.",
"We all are thankful to the Inter-University Center for Astronomy and Astrophysics (IUCAA), Pune for providing references through its library and computation facility in addition to local hospitality.",
"First author is also thankful to UGC, New Delhi for providing financial support through UGC Start-up Research Grant No.-F.30-356/2017(BSR)."
]
] | 1906.04482 |
[
[
"Dependence of outer boundary condition on protoneutron star\n asteroseismology with gravitational-wave signatures"
],
[
"Abstract To obtain the eigenfrequencies of a protoneutron star (PNS) in the postbounce phase of core-collapse supernovae (CCSNe), we perform a linear perturbation analysis of the angle-averaged PNS profiles using results from a general relativistic CCSN simulation of a $15 M_{\\odot}$ star.",
"In this work, we investigate how the choice of the outer boundary condition could affect the PNS oscillation modes in the linear analysis.",
"By changing the density at the outer boundary of the PNS surface in a parametric manner, we show that the eigenfrequencies strongly depend on the surface density.",
"By comparing with the gravitational wave (GW) signatures obtained in the hydrodynamics simulation, the so-called surface $g$-mode of the PNS can be well ascribed to the fundamental oscillations of the PNS.",
"The frequency of the fundamental oscillations can be fitted by a function of the mass and radius of the PNS similar to the case of cold neutron stars.",
"In the case that the position of the outer boundary is chosen to cover not only the PNS but also the surrounding postshock region, we obtain the eigenfrequencies close to the modulation frequencies of the standing accretion-shock instability (SASI).",
"However, we point out that these oscillation modes are unlikely to have the same physical origin of the SASI modes seen in the hydrodynamics simulation.",
"We discuss possible limitations of applying the angle-averaged, linear perturbation analysis to extract the full ingredients of the CCSN GW signatures."
],
[
"Introduction",
"Success of direct observations of gravitational waves (GWs) from the compact binary mergers ushered in a new era of GW astronomy.",
"Up to now, GWs from five binary black hole (BH) mergers, i.e., GW150914 [1], GW151226 [2], GW170104 [3], GW170608 [4], and GW170814 [5], and one binary neutron star (NS) merger, i.e., GW170817 [6], have been detected by LIGO (Laser Interferometer Gravitational-wave Observatory) Scientific Collaboration and Virgo Collaboration.",
"In the event of GW170817 [7], the electromagnetic-wave counterpart has been detected, which opens yet another new era of multi-messenger astronomy.",
"In addition to the advanced LIGO and advanced Virgo, KAGRA will be operational in the coming years [8].",
"Furthermore, the third-generation detectors have been proposed such as Einstein Telescope and Cosmic Explorer [9], [10].",
"At such high level of precision, these detectors are sensitive enough to a wide variety of compact objects.",
"Next to the primary targets of the compact binary coalescence, other intriguing sources include core-collapse supernovae (CCSNe) [11], which mark the catastrophic end of massive stars and produce all these compact objects.",
"In order to study the GW signatures from CCSNe, numerical simulations have been done extensively (e.g., [13], [14], [15], [18], [16], [12], [19], [17]).",
"The most distinct GW emission process commonly seen in recent self-consistent three-dimensional (3D) models is associated with the excitation of core/protoneutron star (PNS) oscillatory modes [15], [20], [45].",
"This is supported by the evidence that the Brunt-Väisälä frequency estimated at the PNS surface is in good accordance with that of the strongest GW component.",
"The typical GW frequency of the surface $g$ -mode is approximately expressed by $GM_{\\rm PNS}/R^2_{\\rm PNS}$ [13], [14], [15], [12] with $G$ the gravitational constant, $M_{\\rm PNS}$ and $R_{\\rm PNS}$ the mass and radius of the PNS, respectively.",
"In the postbounce phase, the PNS mass increases with time due to the mass accretion and the PNS radius decreases with time due to the mass accretion onto the PNS and neutrino cooling.",
"Accordingly, the typical GW frequency of the surface $g$ -mode increases with time after bounce [13], [12], which is roughly in the range of $\\sim 500-1000$ Hz.",
"These oscillations are excited because the PNS surface is chimed by the mass motions.",
"Recent studies indicate that the dominant excitation process may be sensitive to the spacial dimension in the hydrodynamics simulations.",
"In axisymmetric two-dimensional (2D) models, the mass accretion from above the PNS, where the mass accretion activity to the PNS is influenced by the growth of neutrino-driven convection and the standing accretion-shock instability (SASI) [21], [22], are the main excitation process of the PNS surface oscillations [12], [13].",
"While Ref.",
"[16] showed in the 3D models that the PNS convection could also significantly contribute to the postbounce GW emission.",
"In addition to the PNS oscillations, recent 3D CCSN models have shown another remarkable GW signatures whose frequencies are close to the modulation frequency of the SASI motion, i.e., $\\sim 100$ Hz [15], [16], [17], [20].",
"Thus, the detection of the GWs with $\\sim 100$ Hz separately from those with $\\sim 500-1000$ Hz may provide a probe into the SASI activity in the pre-explosion supernova core [15], [16].",
"The hydrodynamics modeling is really powerful to clarify the inner-workings of the forming compact objects, while linear perturbation approaches are also valuable to understand the physics behind the numerical results obtained by simulations.",
"Given angle-averaged profiles obtained in hydrodynamics models, oscillation spectra are determined by a linear analysis.",
"Then, if one could find a correlation between the properties of the background model and the resultant oscillation spectra, one can extract the information about the background model through observations of the spectra.",
"This technique is known as asteroseismology, which has been extensively investigated in the context of cold NSs.",
"With this technique, it has been suggested that the properties of the NSs such as the mass ($M$ ), radius ($R$ ), and EOS, would be constrained with GW asteroseismology, where one would get an information about the source object with the GW spectra (e.g., [23], [24], [25], [26], [27], [28], [29], [30]).",
"Compared to a lot of studies with the linear perturbation analysis on cold NSs, similar studies on PNSs are very few [31], [32], [33], [34], [35], [36], [37], [38].",
"The paucity of the perturbative studies on the PNSs may come from the difficulty for preparing for the background model of the PNSs.",
"That is, unlike the case of nearly hydrostatic cold NSs, one also needs the time dependent radial distributions of the electron fraction and, e.g., the entropy per baryon for constructing the finite temperature PNS models.",
"However, these time dependent spatial profiles are determined only via the self-consistent CCSN simulations which are computationally expensive.",
"Among the recent studies to tackle with this problem [31], [32], [33], [34], [35], [36], [37], [38], we have found that the frequencies of the fundamental $(f)$ and the space-time $(w)$ modes [33], [35] can be respectively expressed as a function of the average density and compactness of the PNSs almost independently of the EOS of PNSs, in a similar way to the case of cold NSs [23], [24].",
"In this context, a universal relation of the CCSN GW spectra is recently reported in Ref.",
"[39].",
"Up to now, two representative ways have been proposed for constructing the background PNS models for determining the eigenfrequencies in the linear perturbation analysis.",
"They differ in the definition of the PNS surface.",
"One is the PNS model, in which the surface density is fixed as a specific value, for example, of $\\sim 10^{10}$ g cm$^{-3}$ [33], [35], [37].",
"In this case, one can impose the boundary condition similarly as taken in the stellar oscillation analysis and can classify the stellar oscillations.",
"However, unlike the usual cold NS case, the low density matter still hovers and the accretion shock also exists outside this density region, whose influences might not be negligible.",
"Therefore the PNS model covering up to the shock radius is also proposed [36], [38].",
"With the boundary condition imposed at the shock, one can investigate the global oscillations inside the whole postshock region, although the eigenvalue problem to solve is significantly different from that with the PNS model with the fixed surface density.",
"In this study, we calculate the eigenfrequencies in the PNS models by the linear perturbation analysis with the two different boundary conditions, i.e., either at the PNS surface with a fixed specific density or at the shock radius, with an attempt to identify the excitation mechanism of the GW signatures seen in the numerical simulation.",
"This paper is organized as follows.",
"Section starts with a brief summary of the PNS models employed in this work.",
"In Section , we describe the linear perturbation analysis to solve the eigenvalue problem.",
"Section presents our results and the comparison with the GW signal computed in the numerical simulations.",
"We summarize our results and discuss their implications in Section .",
"Unless otherwise mentioned, we adopt geometric units in the following, $c=G=1$ , where $c$ denotes the speed of light, and the metric signature is $(-,+,+,+)$ .",
"The time is measured after bounce ($T_{\\rm pb} = 0)$"
],
[
"PNS Models",
"The line element is expressed as $ds^2=-\\alpha ^2dt^2+\\gamma _{ij}(dx^i+\\beta ^idt)(dx^j+\\beta ^jdt), $ where $\\alpha $ , $\\beta ^i$ , and $\\gamma _{ij}$ are the lapse, shift vector, and three metric, respectively.",
"To prepare the background of PNS models, the metric functions $\\alpha $ , $\\beta ^i$ , and $\\gamma _{ij}$ from hydrodynamics simulations, which are not spherically symmetric, are transformed into the spherically symmetric properties, assuming that the hydrodynamic background at each time step is also static and spherically symmetric.",
"In this procedure, all variables defined on the Cartesian coordinates in numerical relativity simulations are transformed into those in polar coordinates by spatially linear interpolation at each time step.",
"Then, the space-time in the isotropic coordinates can be rewritten as $ds^2 =&-\\alpha ^2 dt^2+ \\gamma _{\\hat{r}\\hat{r}}(d\\hat{r}^2+\\hat{r}^2d\\theta ^2+\\hat{r}^2\\rm {sin}^2\\theta d\\phi ^2),$ where $\\hat{r}$ denotes the isotropic radius $\\hat{r}=\\sqrt{x^2+y^2+z^2}$ .",
"In the calculations of stellar oscillations, we adopt the following spherically symmetric space-time $ds^2 =-e^{2\\Phi } dt^2 + e^{2\\Lambda } dr^2 + r^2\\left(d\\theta ^2 + \\sin ^2\\theta d\\phi ^2\\right), $ as a background space-time, where $\\Phi $ and $\\Lambda $ are functions of only $r$ .",
"We remark that the metric expressed by Eq.",
"(REF ) is similar to the Schwarzschild metric and is given by the coordinate transformation from the isotropic coordinates, i.e.",
"Eqs.",
"(REF ) or (REF ).",
"Additionally, the metric function $\\Lambda $ is associated with the mass function $m$ in such a way that $e^{-2\\Lambda }=1-2m/r$ .",
"Then, the background four-velocity of the fluid element is given by $u^\\mu =(e^{-\\Phi },0,0,0)$ .",
"Comparing Eqs.",
"(REF ) and (REF ), the conversion relation is expressed as followings $& &&e^{2\\Phi } =\\alpha ^2,&& \\\\& &&r^2=\\gamma _{\\hat{r}\\hat{r}}\\hat{r}^2,&& \\\\&\\text{and}&&&& \\nonumber \\\\& &&e^{2\\Lambda }dr^2=\\gamma _{\\hat{r}\\hat{r}}d\\hat{r}^2.&& $ From these, one can deduce the following relations $dr=&\\left(\\gamma _{\\hat{r}\\hat{r}}+\\frac{\\hat{r}}{2}\\frac{\\partial \\gamma _{\\hat{r}\\hat{r}}}{\\partial \\hat{r}}\\right)\\frac{\\hat{r}}{r}d\\hat{r},\\\\m=&\\left[1-\\frac{(\\gamma _{\\hat{r}\\hat{r}}+\\hat{r}\\partial _{\\hat{r}}\\gamma _{\\hat{r}\\hat{r}}/2)^2}{\\gamma _{\\hat{r}\\hat{r}}^2}\\right] \\frac{\\gamma _{\\hat{r}\\hat{r}}^{1/2}}{2}\\hat{r}.",
"$ In this study, instead of using Eq.",
"(), we evaluate the enclosed gravitational mass $m$ within $\\hat{r}$ and use a simple conversion relation $r=\\hat{r}(1+m/2\\hat{r})^2$ , from isotropic to Schwarzschild coordinates.",
"Although this simple conversion relation can originally be applied to the exterior of the object, we employ it as it can suppress the high frequency structural noise that appears when using Eq.",
"() without some appropriate smoothing.",
"Since we use the spatial derivative of $\\Lambda $ that is a function of $m$ in the following seismology analysis, spurious noise should be suppressed.",
"We consider that the difference between the correct, i.e.",
"Eq.",
"(), and simple evaluations is not so significant.",
"The highest values of $\\exp {(2\\Lambda )}=(1-2m/r)^{-1}$ appear at $\\hat{r}\\sim 1.3\\times 10^6$ cm and they differ approximately 1 % between both evaluations.",
"Figure: (Spherically-averaged) radial profiles of the rest mass density (ρ\\rho ), entropy per baryon (ss), and electron faction (Y e Y_e) at 48, 148, 248, and 348 ms after core bounce for a 3D-GR model of SFHx in .Figure: Similar to Figure 1, but for the time evolution of the PNS gravitational mass (left panel) and radius (right panel) as a function of the postbounce time.",
"The different lines correspond to the different definitions of the PNS model, i.e., ρ s =5×10 9 \\rho _s=5\\times 10^9 (filled-circle), 10 10 10^{10} (diamond), 10 11 10^{11} g cm -3 ^{-3} (square), and at the shock (open-circle).In the present study, we especially focus on the numerical results constructed with SFHx EOS [42].",
"The initial hydrodynamic profile is taken from a $15M_\\odot $ progenitor model [41] in the simulation [15].",
"In Fig.",
"REF , we show the radial profiles of the rest mass density $\\rho $ , entropy per baryon $s$ , and electron fraction $Y_e$ at several time snapshots after bounce.",
"From this figure, one can observe that the profiles at 248 ms is almost the same as that at 348 ms. On these background properties, we consider the specific oscillations in PNS at each time step.",
"As PNS models, we consider two different approaches, i.e., 1) as in Ref.",
"[37], the position, where the rest mass density is equivalent to be $\\rho _s=5\\times 10^9$ , $10^{10}$ , and $10^{11}$ g cm$^{-3}$ , is considered as the stellar surface of a background PNS, or 2) the domain inside the shock radius is adopted for calculating the frequencies of stellar oscillations as in Refs.",
"[36], [38].",
"Here, we define the position of the shock radius, where the entropy per baryon becomes $s=7$ $k_{\\rm B}$ baryon$^{-1}$ at the outermost radial position with excluding obviously infalling unshocked stellar mantles.",
"In Fig.",
"REF , we show the time evolution of the PNS gravitational mass and radius, which are determined with different definitions of the PNS surface, as a function of the postbounce time $T_{\\rm pb}$ .",
"One can observe that the gravitational masses after $\\sim 150$ ms are almost independent from the definition of PNS surface, while the PNS radius still depends on the surface density.",
"In the right panel of Fig.",
"REF , we also show the shock radius, which does not change monotonically with time due to the vigorous SASI motion [15]."
],
[
"Perturbation equations in the Cowling approximation",
"In this paper, we simply assume the relativistic Cowling approximation [43], i.e., the metric perturbations are neglected during the stellar oscillations, where the oscillation frequencies can be discussed qualitatively but the damping of oscillations (or the imaginary part of complex frequencies) due to the GW emission can not be calculated.",
"We remark that our perturbation formalism is basically the same as in Ref.",
"[37] with $\\delta \\hat{\\alpha }=0$ , noting that this should be improved in our future work as in [37].",
"The Lagrangian displacement vector of fluid element $\\xi ^i$ for the polar type oscillations is given by $\\xi ^i(t,r,\\theta ,\\phi ) = \\left(e^{-\\Lambda }W, -V\\partial _\\theta , -\\frac{V}{\\sin ^2\\theta }\\partial _\\phi \\right)\\frac{1}{r^2}Y_{\\ell k}(\\theta ,\\phi ),$ where $W$ and $V$ are a function of $t$ and $r$ , while $Y_{\\ell k}(\\theta ,\\phi )$ denotes the spherical harmonics with the azimuthal quantum number $\\ell $ and the magnetic quantum number $k$ .",
"With $\\xi ^i$ , one can obtain the perturbed four-velocity $\\delta u^\\mu $ as $\\delta u^\\mu = \\left(0, e^{-\\Lambda }\\dot{W}, -\\dot{V}\\partial _\\theta , -\\frac{\\dot{V}}{\\sin ^2\\theta }\\partial _\\phi \\right)\\frac{1}{r^2}e^{-\\Phi }Y_{\\ell k}, $ where the dot denotes the partial derivative with respect to $t$ .",
"In addition, one should add the perturbations of the baryon number density $n_{\\rm b}$ , the pressure $p$ , and the energy density $\\varepsilon $ .",
"From the baryon number conservation with the Cowling approximation, one can obtain the relation as $\\frac{\\Delta n_{\\rm b}}{n_{\\rm b}} = -\\left[e^{-\\Lambda }W^{\\prime } + \\ell (\\ell +1)V\\right]\\frac{1}{r^2}Y_{\\ell k}, $ where $\\Delta n_{\\rm b}$ is the Lagrangian perturbation of the baryon number density and the prime denotes the partial derivative with respect to $r$ .",
"Assuming the adiabatic perturbations, the Lagrangian perturbations of the pressure ($\\Delta p$ ) and $\\Delta n_{\\rm b}$ are related to the adiabatic index $\\Gamma _1$ via $\\Gamma _1 \\equiv \\left(\\frac{\\partial \\ln p}{\\partial \\ln n_{\\rm b}}\\right)_s = \\frac{n_{\\rm b}}{p}\\frac{\\Delta p}{\\Delta n_{\\rm b}},$ while one can get the additional equation from the energy conservation law (or the first law of thermodynamics), i.e., $\\Delta \\varepsilon = (\\varepsilon + p)\\frac{\\Delta n_{\\rm b}}{n_{\\rm b}}, $ where $\\Delta \\varepsilon $ denotes the Lagrangian perturbation of the energy density.",
"Since the Lagrangian perturbation of a property $x$ , i.e., $\\Delta x$ , is associated with the Eulerian perturbation ($\\delta x$ ) in the linear analysis, such as $\\Delta x = \\delta x + \\xi ^i\\partial _i x$ , by combining Eqs.",
"(REF ) and (REF ), one can obtain that $\\delta p = c_s^2 \\delta \\varepsilon + p\\Gamma _1 {\\cal A}\\xi ^r, $ where $c_s$ is the sound velocity and ${\\cal A}$ is the relativistic Schwarzschild discriminant given by $c_s^2 \\equiv \\left(\\frac{\\partial p}{\\partial \\varepsilon }\\right)_s= \\frac{\\Delta p}{\\Delta \\varepsilon }= \\frac{p\\Gamma _1}{\\varepsilon + p}, \\\\{\\cal A}(r) \\equiv \\frac{\\varepsilon ^{\\prime }}{\\varepsilon + p} - \\frac{p^{\\prime }}{p\\Gamma _1}= \\frac{1}{\\varepsilon + p}\\left(\\varepsilon ^{\\prime } - \\frac{p^{\\prime }}{c_s^2}\\right).$ We remark that ${\\cal A}$ is a little different from that introduced in Ref.",
"[43], where the factor $e^{-\\Lambda }$ is also included in the discriminant.",
"We also remark that $c_s$ is determined from the adopted EOS independently of the stellar structure, while ${\\cal A}$ is determined only with the stellar structure.",
"With this discriminant ${\\cal A}$ , the relativistic Brunt-Väisälä frequency, $f_{\\rm BV}$ , is given by $f_{\\rm BV} = {\\rm sgn}({\\cal N}^2)\\sqrt{|{\\cal N}^2|} / 2\\pi ,$ where ${\\cal N}^2$ is given by [43] ${\\cal N}(r)^2 = -\\Phi ^{\\prime }e^{2\\Phi -2\\Lambda }{\\cal A}(r).$ We remark that the region with ${\\cal A}>0$ (${\\cal A}<0$ ), which corresponds to ${\\cal N}^2<0$ (${\\cal N}^2>0$ ), is stable (unstable) with respect to the convection.",
"The radial profiles of ${\\cal A}$ and $f_{\\rm BV}$ at $T_{\\rm pb}=48$ , 148, 248, and 348 ms are shown in Fig.",
"REF .",
"From this figure, one can see that most regions of PNS seem to be stable with respect to the convection, although this may be an original feature in the PNS model obtained in Ref.",
"[15] in which the 3D hydrodynamic motion rapidly washes out the negative entropy gradient (see the middle panel in Fig.",
"REF ).",
"One can also see that the absolute value of $f_{\\rm BV}$ becomes larger in the earlier phases after core bounce.",
"Now, $\\delta p$ and $\\delta \\varepsilon $ are generally expressed as $\\delta p(t,r)Y_{\\ell k}$ and $\\delta \\varepsilon (t,r)Y_{\\ell k}$ , respectively.",
"Thus, one obtains the following equations for any $\\ell $ -th perturbations, $\\delta \\varepsilon = -\\frac{\\varepsilon + p}{r^2}\\left[e^{-\\Lambda }W^{\\prime } + \\ell (\\ell +1)V\\right] - \\frac{\\varepsilon ^{\\prime }}{r^2}e^{-\\Lambda }W, \\\\\\delta p = c_s^2\\delta \\varepsilon + \\frac{p\\Gamma _1 {\\cal A}}{r^2}e^{-\\Lambda }W, $ where Eq.",
"(REF ) comes from Eqs.",
"(REF ) and (REF ), while Eq.",
"() comes from Eq.",
"(REF ).",
"Figure: (Spherically averaged) radial profiles of the relativistic Schwarzschild discriminant 𝒜{\\cal A} (left panel) and Brunt-Väisälä frequency (right panel) at T pb =48T_{\\rm pb}=48, 148, 248, and 348ms.",
"The inset in the right panel is just a zoom-up, focusing on the frequency range below 2 kHz in the absolute value.In addition, the perturbed energy-momentum conservation law, i.e., $\\nabla _\\nu \\delta T^{\\mu \\nu }=0$ , gives us the following equations, $\\frac{\\varepsilon + p}{r^2}e^{-2\\Phi }\\ddot{W} + e^{-\\Lambda }\\delta p^{\\prime } + \\Phi ^{\\prime } e^{-\\Lambda }\\left(\\delta \\varepsilon + \\delta p\\right) = 0, \\\\\\delta p = (\\varepsilon + p)e^{-2\\Phi }\\ddot{V}, $ which correspond to the $r$ - and $\\theta $ -components of the perturbed energy-momentum conservation law.",
"We remark that the $t$ -component of the perturbed energy-momentum conservation law is exactly the same as Eq.",
"(REF ).",
"By combining Eqs.",
"(REF ) – () and assuming that $W(t,r)=e^{i\\omega t}W(r)$ and $V(t,r)=e^{i\\omega t}V(r)$ , one can get the perturbation equations for $W$ and $V$ as $W^{\\prime } = \\frac{1}{c_s^2}\\left(\\Phi ^{\\prime }W + \\omega ^2 r^2e^{-2\\Phi +\\Lambda }V\\right) - \\ell (\\ell +1)e^{\\Lambda }V, \\\\V^{\\prime } = -\\frac{1}{r^2}e^{\\Lambda }W + 2\\Phi ^{\\prime } V - {\\cal A}\\left(\\frac{1}{\\omega ^2r^2}\\Phi ^{\\prime }e^{2\\Phi -\\Lambda }W + V\\right).$ In order to solve this equation system, one has to impose appropriate boundary conditions.",
"The regularity condition should be imposed at the stellar center, i.e., $W = W_0 r^{\\ell +1} \\ \\ {\\rm and}\\ \\ V=-\\frac{W_0}{\\ell }r^{\\ell },$ where $W_0$ is constant.",
"The boundary condition is that the Lagrangian perturbation of pressure should be zero at the surface of PNS, i.e., $\\Phi ^{\\prime }e^{-\\Lambda }W + \\omega ^2 r^2 e^{-2\\Phi }V = 0,$ for the case that the PNS surface is determined by the critical density, while it is that the radial displacement should be zero at the shock radius, i.e., $W=0$ , for the case that the oscillations are considered in the domain inside the shock radius.",
"At last, the problem to solve becomes the eigenvalue problem with respect to $\\omega $ .",
"Once the eigenfrequency $\\omega $ is determined, it is connected to the oscillation frequency, $f$ , via $f=\\omega /2\\pi $ ."
],
[
"Asteroseismology of PNS",
"Recently, the sophisticated time-frequency analysis [44] showed that the various GW signatures with wide frequency ranges can be extracted from the GW spectrogram for the 3D-GR model (SFHx) employed in this work [15].",
"For instance, as shown in Fig.",
"REF , they found the sequences of A, B, C, C#, and D. The sequence A has been observed in the several previous studies, which is considered as “the surface $g$ -mode\" [13], [14], [15] of the PNS.",
"The sequences B and D could come from the mass accretion influenced by SASI [15].",
"It is noteworthy that the low frequency component B has been also reported in other recent 3D studies [17], [45].",
"The excitation mechanism of the sequence C (and C#) is still unclear.",
"In this paper, we attempt to compare the eigenfrequencies derived from the perturbation analysis with the GW frequencies obtained from the hydrodynamics simulations as in Fig.",
"REF .",
"As a baseline, we mainly focus on the sequence A in this work.",
"Figure: The characteristic GW frequencies extracted by the time-frequency analysis for the 3D model SFHx in .Figure: Eigenfrequencies in the PNS model with ρ s =10 11 \\rho _s=10^{11} g cm -3 ^{-3}.",
"In particular, the ff and p i p_i for i=1-6i=1-6 are explicitly shown with the open-squares together with the dotted lines, where the double squares denote the cases that the node number in the eigenfunction is different from the standard definition (see Fig.",
").",
"For reference, the various excited GW frequencies derived from the simulation data are also shown with the red lines.Figure: The eigenfunctions of WW for the ff-modes are shown in the left panel, where the amplitude of WW is normalized by the amplitude at the stellar surface and is shifted a little for easily distinguishing the lines.",
"The radial-dependent pulsation energy density EE given by Eq.",
"() is shown in the right panel, where the amplitudes are normalized appropriately."
],
[
"PNS surface determined by the fixed density",
"First, we consider the PNS model, whose surface density ($\\rho _s$ ) is changed in a parametric manner.",
"In Fig.",
"REF , we show the eigenfrequencies determined in the PNS model with $\\rho _s=10^{11}$ g cm$^{-3}$ .",
"Among many eigenfrequencies, we identify the $f$ and $p_i$ -modes for $i=1-6$ , which are shown with open squares (and double squares) connected by thin-dotted lines.",
"In addition, for reference the various excited GW frequencies derived from the simulation data are shown in Fig.",
"REF .",
"The eigenfunctions of $W$ for the $f$ -modes with various time steps are shown in the left panel of Fig.",
"REF , where the amplitude of $W$ is normalized by the value at the PNS surface and is shifted upward in order to distinguish each line easily.",
"From this figure, one can observe that the eigenfunctions of $W$ at any time step become as the standard definition of $f$ -mode, i.e., the eigenfunction monotonically increases outward without any nodes.",
"However, in fact, the eigenfunctions at 108 and 128 ms are different from the standard definition, where the node number is more than one.",
"Even so, since the both eigenfunctions are very similar to the other $f$ -mode eigenfunctions, we identify these modes as $f$ -modes.",
"For example, the eigenfunction of $|W|$ at 128 ms is shown in Fig.",
"REF .",
"In the similar way, by checking the shape of the eigenfunctions and by counting the node number, we identify the $p_1$ - and $p_2$ -modes as shown in Fig.",
"REF , where the eigenfunctions of $W$ for the $p_1$ -modes at each time step are shown in Fig.",
"REF .",
"Again, the double squares denote the eigenfrequencies that are different from the standard definition by the node number.",
"Figure: The details of the eigenfunction of |W||W| for the ff-modes at 128ms for the PNS model with ρ s =10 11 \\rho _s=10^{11} g/cm 3 ^3 is shown, which is an example of the specific case shown with the double squares in Fig.",
".",
"This sample has one node in the eigenfunction shown by the arrow, but the shape of eigenfunction is almost the same as that of the other ff-mode shown in the left panel of Fig.",
".Figure: The eigenfunctions of WW for the p 1 p_1-modes are shown for the PNS model with ρ s =10 11 \\rho _s=10^{11} g/cm 3 ^3, where the amplitude is normalized by the surface amplitude and is shifted a little.Figure: For the PNS model with ρ s =10 11 \\rho _s=10^{11} g/m 3 ^3, the ff-mode frequencies are shown as a function of the square root of the PNS average density.",
"The solid line denotes the fitting formula given by Eq.",
"(), while the dashed line is the analytical formula of the ff-mode frequency for the star with uniform incompressible fluid given by Eq.",
"().In the right panel of Fig.",
"REF , we also show the radial dependent pulsation energy density in the $f$ -mode oscillation at each time step, where in the same way as in Refs.",
"[36], [37], the Newtonian pulsation energy density at each radial position can be estimated with our variables as $E(r) \\sim \\frac{\\omega ^2\\varepsilon }{r^4} \\left[W^2 + \\ell (\\ell +1)r^2V^2\\right].",
"$ One can observe that the amplitude of the $f$ -mode eigenfunction becomes maximum at the stellar surface.",
"The pulsation energy density, however, becomes maximum at around $80-90 \\%$ of the PNS radius.",
"From Fig.",
"REF we can see a good agreement of the sequence A (which is referred as the surface $g$ -mode [13], [14], [15]) with the $f$ -mode oscillations in the PNS, when we take the specific surface density of $10^{11}$ g cm$^{-3}$ .",
"In Fig.",
"REF , the filled squares correspond to the eigenfrequencies, which we can not unambiguously identify as either $f$ -, $p_1$ -, or $p_2$ -modes (e.g., Fig.",
"REF ).",
"These modes are left unidentified in this work.",
"Figure: Same as in Fig.",
", but for the PNS models with ρ s =5×10 9 \\rho _s=5\\times 10^9 (left panel) and 10 10 10^{10} g cm -3 ^{-3} (right panel).",
"For the PNS model with ρ s =10 10 \\rho _s=10^{10} g/cm 3 ^3, as an example of the eigenfunction of which eigenmode is leftunidentified, we will consider the frequency at 108 ms shown by the asterisk, discussed with Fig.",
".Figure: The ff-mode frequencies from the PNS models with different definition of the surface density are shown as a function of the corresponding PNS average density, where the squares, diamonds, and circles correspond to the results with ρ s =10 11 \\rho _s=10^{11}, 10 10 10^{10}, and 5×10 9 5\\times 10^{9} g cm -3 ^{-3}, respectively.Figure: Same as in the left panel of Fig.",
", but for the PNS models with ρ s =5×10 9 \\rho _s=5\\times 10^9 g/cm 3 ^3.Figure: An example of the eigenfunction WW, of which eigenmode is left unidentified (from the radial node number of WW).This corresponds to the frequency shown by the asterisk in the right panel of Fig.",
", which is close to the sequence A.The identification of the sequence A with the $f$ -mode oscillation indicates that one could extract the PNS properties by observing the $f$ -mode originated GWs.",
"In practice, it is well-known that, since the $f$ -mode is associated with the acoustic waves, its frequency can be characterized with the stellar average density almost independently of the adopted EOS (see [23], [24] for cold NSs and also [33], [35] even for the PNS models).",
"In Fig.",
"REF , we show the $f$ -mode frequency from the PNS model for each time step as a function of the corresponding square root of the PNS average density.",
"From this figure, as in the previous studies, one can observe that the $f$ -mode frequencies can be expressed as a linear function of the square root of the PNS average density.",
"Additionally, with this data, we obtain the fitting formula expressing the $f$ -mode frequency, i.e., $f_f\\ {\\rm (Hz)} = -87.34 + 4080.78 \\left(\\frac{M_{\\rm PNS}}{1.4M_\\odot }\\right)^{1/2}\\left(\\frac{R_{\\rm PNS}}{10\\ {\\rm km}}\\right)^{-3/2},$ where $M_{\\rm PNS}$ and $R_{\\rm PNS}$ denotes the PNS (gravitational) mass and radius, respectively.",
"The resultant fitting formula is also shown in Fig.",
"REF with the thick-solid line.",
"With this fitting formula, one may know the time evolution of the PNS average density from the observation of the gravitational waves.",
"We remark that the $\\ell $ -th $f$ -mode frequency for the star with uniform incompressible fluid has been derived analytically as $f_f^{(\\rm a)} = \\frac{1}{2\\pi }\\sqrt{\\frac{2M_{\\rm PNS}}{R_{\\rm PNS}^3}\\frac{2\\ell (\\ell -1)}{2\\ell +1}}, $ which is known as a Kelvin $f$ -mode.",
"The expected $\\ell =2$ frequency is also shown in Fig.",
"REF with dashed line, but it seems that this formula assuming the incompressible fluid is not suitable for expressing the $f$ -mode frequencies for the PNS models.",
"In the similar way, we determine the eigenfunctions in the PNS models with $\\rho _s=5 \\times 10^9$ and $10^{10}$ g/cm$^3$ , which are shown in Fig.",
"REF .",
"From this figure together with Fig.",
"REF , we find that the eigenfrequencies depend on the selection of the surface density of the PNS model.",
"In fact, the frequencies of $f$ and $p_i$ -modes decrease, as $\\rho _s$ decreases.",
"This tendency may be understood as a result of the decrease of the average density of the PNS, as $\\rho _s$ decreases, because the $f$ and $p_i$ -modes are a kind of acoustic waves, whose frequencies can be characterized by the average density of the PNS.",
"In fact, as shown in Fig.",
"REF , the $f$ -mode frequencies can be expressed well as a function of the PNS average density, even though the definition of the surface density is different.",
"The dependence of the eigenfrequencies on the surface density seems to be consistent with Ref.",
"[37] as least in the early postbounce phase.",
"On the other hand, in the phase later than $\\sim 500$ ms after bounce, Morozova et al.",
"[37] showed that the eigenfrequencies are almost independent from the selection of the surface densities of the PNS models.",
"This could be because the density gradient in the vicinity of the background PNS models becomes steeper in the later phase, making the average density less sensitive to the choice of the surface densities.",
"Thus, although the GW signal (the sequence A) obtained in the 3D numerical simulation [15] is well ascribed to the $f$ -mode oscillations in the PNS model with $\\rho _s=10^{11}$ g/cm$^3$ , this result may not be universal at least in the early postbounce phase, i.e., one may have to select a specific surface density to identify the GW signal.",
"In such a case, it could be more difficult to extract the PNS information from direct GW observations.",
"Additionally, for the PNS model with $\\rho _s=5 \\times 10^9$ g/cm$^3$ , the eigenfunction of $W$ and the radial dependent pulsation energy density at each time step are shown in Fig.",
"REF .",
"We remark that eigenfunction of $W$ and the radial dependent pulsation energy density for the PNS model with $\\rho _s=10^{10}$ g/cm$^3$ are more or less similar to those for the PNS model with $\\rho _s=5 \\times 10^9$ g/cm$^3$ .",
"The eigenfunctions of $W$ look similar to those shown in Fig.",
"REF , but one can see the difference in the radial dependent pulsation energy density.",
"From this figure, it seems that the oscillations around the stellar surface become more important in the PNS model with lower $\\rho _s$ .",
"Furthermore, as an example of the eigenmode that could not be identified as a specific mode, we show the eigenfunction of $W$ for the PNS model with $\\rho _s=10^{10}$ g/cm$^3$ at 108 ms, which is shown with the asterisk in the right panel of Fig.",
"REF .",
"Obviously, this eigenfunction is satisfied the boundary condition but the shape of eigenfunction is apparently different from the other $f$ - or $p_1$ mode.",
"We note that the lower frequencies, such as sequences B or D in Fig.",
"REF , are not excited in the PNS models with the specific surface density irrespective of its value.",
"In addition, some of the eigenfrequencies lower than the $f$ -mode in Figs.",
"REF and REF could be considered as $g$ -mode oscillations.",
"However these modes are left unidentified because of the lack of the clear node structure in the eigenfunctions as mentioned above.",
"Finally, the GW signal (the sequence A) is compared with the $f$ -mode frequencies calculated in this study with different surface density and the surface $g$ -mode with the formula proposed in Ref.",
"[13], i.e., $f_{\\rm peak} = \\frac{1}{2\\pi }\\frac{M_{\\rm PNS}}{R_{\\rm PNS}^2}\\sqrt{\\frac{1.1 m_n}{\\langle E_{{\\bar{\\nu }}_e}\\rangle }}\\left(1-\\frac{M_{\\rm PNS}}{R_{\\rm PNS}}\\right)^2, $ where $\\langle E_{{\\bar{\\nu }}_e}\\rangle $ denotes the mean energy of electron antineutrinos and $m_n$ is the neutron mass.",
"We remark that the Brunt-Väisälä frequencies estimated at the PNS surface is original “surface $g$ -mode\", with which Eq.",
"(REF ) is approximately derived.",
"Those frequencies are shown in Fig.",
"REF , where the left and right panels correspond to the results of $f$ -mode frequencies obtained in the linear analysis and surface $g$ -mode frequencies calculated with Eq.",
"(REF ), respectively, for the PNS models with $\\rho _s=10^{11}$ (circles), $10^{10}$ (squares), and $5\\times 10^{9}$ g/cm$^3$ (diamonds).",
"From this figure, one can observe that the both frequencies strongly depend on the surface density, but agree well with the GW signal of the sequence of A for the PNS model with $\\rho _s=10^{11}$ g/cm$^3$ .",
"Even so, since the surface $g$ -mode (or the Brunt-Väisälä frequency at the PNS surface) is the local value while $f$ -mode is the global oscillations of PNS, it may be more natural that the GW signal (sequence of A) is considered as a result of the $f$ -mode oscillations.",
"Figure: The GW signal (the sequence of A) is compared with the ff-mode GW of PNS in the left panel and with the surface gg-mode calculated with Eq.",
"() in the right panel, where the circles, squares, and diamonds denote the PNS models constructed with ρ s =10 11 \\rho _s=10^{11}, 10 10 10^{10}, and 5×10 9 5\\times 10^{9} g/cm 3 ^3, respectively."
],
[
"PNS inside the shock radius",
"Next, we consider the oscillations inside the shock radius.",
"In this case, as mentioned before, the boundary condition at the shock radius is that the radial component of the Lagrangian displacement should be zero.",
"That is, the eigenfunction of $W$ is always zero at the shock radius, where the standard classification of the eigenmode may not be adopted.",
"Intrinsically, the eigenvalue problem to solve with this PNS model is significantly different from that with the PNS model whose surface density is fixed.",
"Anyway, as an advantage of this PNS model, the ambiguity for selecting the position of boundary disappears, while the spherical symmetric model may not be a good assumption in the region whose density is very low, because the matter motion is not neglected in such a region.",
"Furthermore the excitation of GWs in the numerical simulation may come from such an oscillation inside the whole shocked region, although there are currently a few studies [36], [38] examining this effect.",
"We try to determine the eigenfrequencies with the PNS model inside the shock radius.",
"The resultant eigenfrequencies are shown in Fig.",
"REF , where the same modes are connected with dotted lines.",
"From this figure, one can observe that even lower frequencies can be excited with the PNS model inside the shock radius, which is different feature compared to the results with the PNS model whose surface density is fixed.",
"In fact, in some time interval, it seems that the eigenfrequencies are excited close to the sequences B and D. Figure: The eigenfrequencies calculated in this study are shown with marks, while the excited GW frequencies in the numerical simulation are again shown with various red lines.",
"The double circles are the lowest and the second lowest eigenfrequencies at 268 ms, which are focused in Fig.",
".",
"The open circles are the examples, of which eigenfunctions will be discussed in Fig.",
".",
"The modes, whose eigenfunctions are the similar to each other, are connected with the dotted lines.Figure: The left and right panels correspond to the eigenfunction WW and pulsation energy density given by Eq.",
"() for the lowest (denoted with 1) and the second lowest (denoted with 2) eigenfrequencies at 268 ms after core bounce, respectively.",
"In the both panels, the vertical axis is normalized appropriately.In order to see the oscillation behavior for such eigenfrequencies, we especially focus on the lowest and the second lowest eigenfrequencies at 268 ms after core bounce, which are denoted with the double circles in Fig.",
"REF .",
"The corresponding eigenfunction of $W$ and the radial dependent pulsation energy density are shown in Fig.",
"REF , where the solid and dashed lines correspond to the results with the lowest and the second lowest eigenfrequencies, respectively.",
"One can see that the eigenfunctions are very similar to the standard classification of stellar oscillation except for the behavior close to the stellar surface, i.e., the lowest and the second lowest eigenfrequencies may correspond to the $f$ - and $p_1$ -modes.",
"In addition, in the similar fashion to the results with the PNS model whose surface density is fixed, the amplitude of eigenfunction $W$ and the radial dependent pulsation energy density become significant on the outer part of the oscillation region.",
"However, it has been reported that the excitation of the GW signal according to the sequence B (or maybe also D) effectively comes from the inner part of the PNS, such as $\\sim 20$ km [15], [16].",
"Thus, although the lower eigenfrequencies obtained via the eigenvalue problem inside the shock radius appear close to the sequences B and D, these frequencies may not physically correspond to the excitation of gravitational wave signal in the sequences B and D. We remark that in our model we found only $f$ - and $p_i$ -mode like frequencies, while not only $f$ - and $p_i$ -mode like frequencies but also $g_i$ -mode like frequencies are found in the previous similar analysis [36], [38].",
"This discrepancy may come from the different PNS models obtained by the numerical simulations.",
"We need further studies changing the PNS models to draw a robust conclusion on this choice of the boundary condition.",
"Figure: Examples of the eigenfunctions of |W||W| with the frequencies shown by the open circles in Fig.",
".There are a few possible reasons to explain the discrepancy between the the sequence B (and D) and the eigenmodes obtained by the linear analysis.",
"The first and perhaps the main reason is that, in our perturbation analysis using the static background model, the restoring force against the perturbations is assumed to be the acoustic mode.",
"On the contrary, the SASI that is considered to be the emission mechanism of the component B [15], [16] is sustained by the cycle of the fluid advection and the acoustic mode [22].",
"It may thus not be suitable to use the static background model that completely omits the fluid advection.",
"As the second reason, the background model is actually far from spherical symmetry particularly in the non-linear SASI phase ($T_{\\rm pb}\\gtrsim 150$ ms, [15]).",
"These facts would make it even harder to extract the proper eigenmodes for the corresponding GW components.",
"We also remark that one can not clearly find a specific correspondence between the eigenfrequencies and the sequence A on Fig.",
"REF .",
"In practice, we show examples of the eigenfunction of $|W|$ in Fig.",
"REF with the frequencies shown by the open circles in Fig.",
"REF , which are close to the sequence A, but one can not straightforwardly identify these modes as the same eigenmode by checking the shape of the eigenfunction (or the radial node numbers).",
"Anyway, in this study we have made a linear analysis with only one result obtained by the numerical simulation in Ref.",
"[15], i.e., our result may not be always acceptable for any PNS models.",
"In order to make a robust statement for the CCSN GW signals, we have to make more systematical analyses somewhere by adopting various results of different numerical simulations."
],
[
"Conclusion and Discussions",
"In an attempt to obtain the eigenfrequencies of a PNS in the postbounce phase of CCSNe, we have performed a linear perturbation analysis of the angle-averaged PNS profiles using results from a general relativistic CCSN simulation of a $15 M_{\\odot }$ star.",
"Particularly, we paid attention to how the choice of the outer boundary condition could affect the PNS oscillation modes in the linear analysis.",
"By changing the density at the outer boundary of the PNS surface in a parametric manner, we showed that the eigenfrequencies strongly depend on the surface density.",
"By comparing with the GW signals from the hydrodynamics model, it was shown that the so-called surface $g$ -mode of the PNS can be well ascribed to the fundamental oscillations of the PNS.",
"The best match was obtained when the PNS surface is chosen at $10^{11}$ g/cm$^3$ .",
"We pointed out that the frequency of the fundamental oscillations can be fitted by a function of the mass and radius of the PNS similar to the case of cold NSs.",
"In the case that the position of the outer boundary is chosen to cover not only the PNS but also the surrounding postshock region, we obtained the eigenfrequencies close to the modulation frequencies of the SASI.",
"On the other hand, our results suggested that these oscillation modes are unlikely to have the same physical origin of the SASI modes obtained in the hydrodynamics simulation.",
"We have discussed possible limitations of applying the angle-averaged, linear perturbation analysis to extract the full facets of the CCSN GW signatures.",
"In order to identify the GW signatures in the spectrograms more in a systematic manner, one may need to conduct a more detailed linear analysis as in Ref.",
"[39].",
"In this study we adopted the relativistic Cowling approximation, which could be applicable to the early postbounce phase because the stellar compactness is not so large and the relativistic effect may not be so significant.",
"To apply the similar analysis to the late postbounce phase or to the very massive progenitor stars leading to a BH formation as reported in Ref.",
"[46], we need to perform the linear analysis taking into account the metric perturbation, which we shall leave for the future work.",
"Towards the observation of the most remarkable spectral GW signature (i.e., the ramp-up $f$ -mode) in the laser interferometers, dedicated data analysis schemes (e.g., [47]) need to be further developed.",
"This study was supported in part by the Grants-in-Aid for the Scientific Research of Japan Society for the Promotion of Science (JSPS, Nos.",
"JP17K05458, JP26707013, JP17H01130, JP17K14306, JP18H01212 ), the Ministry of Education, Science and Culture of Japan (MEXT, Nos.",
"JP15H00789, JP15H01039, JP15KK0173, JP17H05206 JP17H06357, JP17H06364) by the Central Research Institute of Fukuoka University (Nos.171042,177103) and the Research Institute of Explosive Stellar Phenomena (REISEP), and JICFuS as a priority issue to be tackled by using Post `K' Computer.",
"TK was supported by the European Research Council (ERC; FP7) under ERC Starting Grant EUROPIUM-677912."
]
] | 1906.04354 |
[
[
"Lightweight and Efficient Neural Natural Language Processing with\n Quaternion Networks"
],
[
"Abstract Many state-of-the-art neural models for NLP are heavily parameterized and thus memory inefficient.",
"This paper proposes a series of lightweight and memory efficient neural architectures for a potpourri of natural language processing (NLP) tasks.",
"To this end, our models exploit computation using Quaternion algebra and hypercomplex spaces, enabling not only expressive inter-component interactions but also significantly ($75\\%$) reduced parameter size due to lesser degrees of freedom in the Hamilton product.",
"We propose Quaternion variants of models, giving rise to new architectures such as the Quaternion attention Model and Quaternion Transformer.",
"Extensive experiments on a battery of NLP tasks demonstrates the utility of proposed Quaternion-inspired models, enabling up to $75\\%$ reduction in parameter size without significant loss in performance."
],
[
"Introduction",
"Neural network architectures such as Transformers [37], [9] and attention networks [24], [30], [3] are dominant solutions in natural language processing (NLP) research today.",
"Many of these architectures are primarily concerned with learning useful feature representations from data in which providing a strong architectural inductive bias is known to be extremely helpful for obtaining stellar results.",
"Unfortunately, many of these models are known to be heavily parameterized, with state-of-the-art models easily containing millions or billions of parameters [37], [28], [10], [29].",
"This renders practical deployment challenging.",
"As such, the enabling of efficient and lightweight adaptations of these models, without significantly degrading performance, would certainly have a positive impact on many real world applications.",
"To this end, this paper explores a new way to improve/maintain the performance of these neural architectures while substantially reducing the parameter cost (compression of up to 75%).",
"In order to achieve this, we move beyond real space, exploring computation in Quaternion space (i.e., hypercomplex numbers) as an inductive bias.",
"Hypercomplex numbers comprise of a real and three imaginary components (e.g., $i,j,k$ ) in which inter-dependencies between these components are encoded naturally during training via the Hamilton product $\\otimes $ .",
"Hamilton products have fewer degrees of freedom, enabling up to four times compression of model size.",
"Technical details are deferred to subsequent sections.",
"While Quaternion connectionist architectures have been considered in various deep learning application areas such as speech recognition [23], kinematics/human motion [25] and computer vision [11], our work is the first hypercomplex inductive bias designed for a wide spread of NLP tasks.",
"Other fields have motivated the usage of Quaternions primarily due to their natural 3 or 4 dimensional input features (e.g., RGB scenes or 3D human poses) [23], [25].",
"In a similar vein, we can similarly motivate this by considering the multi-sense nature of natural language [15], [19], [12].",
"In this case, having multiple embeddings or components per token is well-aligned with this motivation.",
"Latent interactions between components may also enjoy additional benefits, especially pertaining to applications which require learning pairwise affinity scores [24], [30].",
"Intuitively, instead of regular (real) dot products, Hamilton products $\\otimes $ extensively learn representations by matching across multiple (inter-latent) components in hypercomplex space.",
"Alternatively, the effectiveness of multi-view and multi-headed [37] approaches may also explain the suitability of Quaternion spaces in NLP models.",
"The added advantage to multi-headed approaches is that Quaternion spaces explicitly encodes latent interactions between these components or heads via the Hamilton product which intuitively increases the expressiveness of the model.",
"Conversely, multi-headed embeddings are generally independently produced.",
"To this end, we propose two Quaternion-inspired neural architectures, namely, the Quaternion attention model and the Quaternion Transformer.",
"In this paper, we devise and formulate a new attention (and self-attention) mechanism in Quaternion space using Hamilton products.",
"Transformation layers are aptly replaced with Quaternion feed-forward networks, yielding substantial improvements in parameter size (of up to $75\\%$ compression) while achieving comparable (and occasionally better) performance."
],
[
"Contributions",
"All in all, we make the following major contributions: We propose Quaternion neural models for NLP.",
"More concretely, we propose a novel Quaternion attention model and Quaternion Transformer for a wide range of NLP tasks.",
"To the best of our knowledge, this is the first formulation of hypercomplex Attention and Quaternion models for NLP.",
"We evaluate our Quaternion NLP models on a wide range of diverse NLP tasks such as pairwise text classification (natural language inference, question answering, paraphrase identification, dialogue prediction), neural machine translation (NMT), sentiment analysis, mathematical language understanding (MLU), and subject-verb agreement (SVA).",
"Our experimental results show that Quaternion models achieve comparable or better performance to their real-valued counterparts with up to a 75% reduction in parameter costs.",
"The key advantage is that these models are expressive (due to Hamiltons) and also parameter efficient.",
"Moreover, our Quaternion components are self-contained and play well with real-valued counterparts."
],
[
"Background on Quaternion Algebra",
"This section introduces the necessary background for this paper.",
"We introduce Quaternion algebra along with Hamilton products, which form the crux of our proposed approaches."
],
[
"Quaternion",
"A Quaternion $Q \\in \\mathbb {H}$ is a hypercomplex number with three imaginary components as follows: $Q = r + x\\mathbf {i}+ y\\mathbf {j}+ z\\mathbf {k},$ where $\\textbf {ijk}= \\mathbf {i}^2=\\mathbf {j}^2=\\mathbf {k}^2=-1$ and noncommutative multiplication rules apply: $\\mathbf {ij} = \\mathbf {k}, \\mathbf {jk} = \\mathbf {i}, \\mathbf {ki} = \\mathbf {j}, \\mathbf {ji} = -\\mathbf {k}, \\mathbf {kj} = -\\mathbf {i}, \\mathbf {ik} = -\\mathbf {j}$ .",
"In (REF ), $r$ is the real value and similarly, $x,y,z$ are real numbers that represent the imaginary components of the Quaternion vector $Q$ .",
"Operations on Quaternions are defined in the following."
],
[
"Addition",
"The addition of two Quaternions is defined as: $Q+P=Q_r + P_r + (Q_x + P_x)\\mathbf {i}\\\\+ (Q_y + P_y)\\mathbf {j}+ (Q_z + P_z)\\mathbf {k},$ where $Q$ and $P$ with subscripts denote the real value and imaginary components of Quaternion $Q$ and $P$ .",
"Subtraction follows this same principle analogously but flipping $+$ with $-$ ."
],
[
"Scalar Multiplication",
"Scalar $\\alpha $ multiplies across all components, i.e., $\\alpha Q = \\alpha r + \\alpha x \\mathbf {i}+ \\alpha y \\mathbf {j}+\\alpha z \\mathbf {k}.$"
],
[
"Conjugate",
"The conjugate of $Q$ is defined as: $Q^* = r - x\\mathbf {i}- y\\mathbf {j}- z\\mathbf {k}.$"
],
[
"Norm",
"The unit Quaternion $Q^{\\triangleleft }$ is defined as: $Q^{\\triangleleft } = \\frac{Q}{\\sqrt{r^2 + x^2 + y^2 + z^2}}.$"
],
[
"Hamilton Product",
"The Hamilton product, which represents the multiplication of two Quaternions $Q$ and $P$ , is defined as: $Q \\otimes P &= (Q_rP_r - Q_xP_x - Q_yP_y - Q_zP_z) \\nonumber \\\\ &+ (Q_xP_r + Q_rP_x - Q_zP_y + Q_yP_z)\\:\\mathbf {i}\\nonumber \\\\ &+ (Q_yP_r + Q_zP_x + Q_rP_y - Q_xP_z)\\:\\mathbf {j}\\nonumber \\\\ &+ (Q_zP_r - Q_yP_x + Q_xP_y + Q_rP_z)\\: \\mathbf {k},$ which intuitively encourages inter-latent interaction between all the four components of $Q$ and $P$ .",
"In this work, we use Hamilton products extensively for vector and matrix transformations that live at the heart of attention models for NLP."
],
[
"Quaternion Models of Language",
"In this section, we propose Quaternion neural models for language processing tasks.",
"We begin by introducing the building blocks, such as Quaternion feed-forward, Quaternion attention, and Quaternion Transformers."
],
[
"Quaternion Feed-Forward",
"A Quaternion feed-forward layer is similar to a feed-forward layer in real space, while the former operates in hypercomplex space where Hamilton product is used.",
"Denote by $W \\in \\mathbb {H}$ the weight parameter of a Quaternion feed-forward layer and let $Q \\in \\mathbb {H}$ be the layer input.",
"The linear output of the layer is the Hamilton product of two Quaternions: $W \\otimes Q$ ."
],
[
"Saving Parameters? How and Why",
"In lieu of the fact that it might not be completely obvious at first glance why Quaternion models result in models with smaller parameterization, we dedicate the following to address this.",
"For the sake of parameterization comparison, let us express the Hamilton product $W \\otimes Q$ in a Quaternion feed-forward layer in the form of matrix multiplication, which is used in real-space feed-forward.",
"Recall the definition of Hamilton product in (REF ).",
"Putting aside the Quaterion unit basis $[1, \\mathbf {i}, \\mathbf {j}, \\mathbf {k}]^\\top $ , $W \\otimes Q$ can be expressed as: $\\begin{bmatrix}W_r & -W_x & -W_y & -W_z \\\\W_x & W_r & -W_z & W_y \\\\W_y & W_z & W_r & -W_x \\\\W_z & -W_y & W_x & W_r \\\\\\end{bmatrix}\\begin{bmatrix}r \\\\x\\\\y\\\\z \\\\\\end{bmatrix},$ where $W = W_r + W_x\\mathbf {i}+ W_y\\mathbf {j}+ W_z\\mathbf {k}$ and $Q$ is defined in (REF ).",
"Figure: 4 weight parameter variables (W r ,W x ,W y ,W z W_r, W_x, W_y, W_z) are used in 16 pairwise connections between components of the input and output Quaternions.We highlight that, there are only 4 distinct parameter variable elements (4 degrees of freedom), namely $W_r, W_x, W_y, W_z$ , in the weight matrix (left) of (REF ), as illustrated by Figure REF ; while in real-space feed-forward, all the elements of the weight matrix are different parameter variables ($4 \\times 4 = 16$ degrees of freedom).",
"In other words, the degrees of freedom in Quaternion feed-forward is only a quarter of those in its real-space counterpart, resulting in a 75% reduction in parameterization.",
"Such a parameterization reduction can also be explained by weight sharing [23], [22]."
],
[
"Nonlinearity",
"Nonlinearity can be added to a Quaternion feed-forward layer and component-wise activation is adopted [22]: $\\alpha (Q) = \\alpha (r) + \\alpha (x)\\mathbf {i}+ \\alpha (y)\\mathbf {j}+ + \\alpha (z)\\mathbf {k},$ where $Q$ is defined in (REF ) and $\\alpha (.",
")$ is a nonlinear function such as tanh or ReLU."
],
[
"Quaternion Attention",
"Next, we propose a Quaternion attention model to compute attention and alignment between two sequences.",
"Let $A \\in \\mathbb {H}^{\\ell _a \\times d}$ and $B \\in \\mathbb {H}^{\\ell _b \\times d}$ be input word sequences, where $\\ell _a, \\ell _b$ are numbers of tokens in each sequence and $d$ is the dimension of each input vector.",
"We first compute: $E = A \\otimes B^{\\top },$ where $E \\in \\mathbb {H}^{\\ell _a \\times \\ell _b}$ .",
"We apply Softmax(.)",
"to $E$ component-wise: $G &= \\text{ComponentSoftmax}(E) \\\\B^{\\prime } &= G_{R}B_{R} + G_{X}B_{X}\\mathbf {i}+ G_YB_{Y}\\mathbf {j}+ G_{Z}B_{Z}\\mathbf {k},$ where $G$ and $B$ with subscripts represent the real and imaginary components of $G$ and $B$ .",
"Similarly, we perform the same on $A$ which is described as follows: $F &= \\text{ComponentSoftmax}(E^{\\top }) \\\\A^{\\prime } &= F_{R}A_{R} + F_{X}A_{X}\\mathbf {i}+ F_YA_{Y}\\mathbf {j}+ F_{Z}A_{Z}\\mathbf {k},$ where $A^{\\prime }$ is the aligned representation of $B$ and $B^{\\prime }$ is the aligned representation of $A$ .",
"Next, given $A^{\\prime } \\in \\mathbb {R}^{\\ell _b \\times d}, B^{\\prime } \\in \\mathbb {R}^{\\ell _A \\times d}$ we then compute and compare the learned alignments: $C_1 = \\sum \\text{QFFN}([A_i^{\\prime };B_i, A_i^{\\prime } \\otimes B_i; A_i^{\\prime } - B_i]) \\\\C_2 = \\sum \\text{QFFN}([B_i^{\\prime };A_i, B_i^{\\prime } \\otimes A_i; B_i^{\\prime } - A_i]),$ where QFFN(.)",
"is a Quaternion feed-forward layer with nonlinearity and $[;]$ is the component-wise contatentation operator.",
"$i$ refers to word positional indices and $\\sum $ over words in the sequence.",
"Both outputs $C_1,C_2$ are then passed $Y = \\text{QFFN}([C_1; C_2; C_1 \\otimes C_2; C_1 - C_2]),$ where $Y \\in \\mathbb {H}$ is a Quaternion valued output.",
"In order to train our model end-to-end with real-valued losses, we concatenate each component and pass into a final linear layer for classification."
],
[
"Quaternion Transformer",
"This section describes our Quaternion adaptation of Transformer networks.",
"Transformer [37] can be considered state-of-the-art across many NLP tasks.",
"Transformer networks are characterized by stacked layers of linear transforms along with its signature self-attention mechanism.",
"For the sake of brevity, we outline the specific changes we make to the Transformer model."
],
[
"Quaternion Self-Attention",
"The standard self-attention mechanism considers the following: $A = \\text{softmax}(\\frac{QK^\\top }{\\sqrt{d_k}})V,$ where $Q,K,V$ are traditionally learned via linear transforms from the input $X$ .",
"The key idea here is that we replace this linear transform with a Quaternion transform.",
"$Q = W_{q} \\otimes X ; K = W_{k} \\otimes X ; V = W_{v} \\otimes X,$ where $\\otimes $ is the Hamilton product and $X$ is the input Quaternion representation of the layer.",
"In this case, since computation is performed in Quaternion space, the parameters of $W$ is effectively reduced by 75%.",
"Similarly, the computation of self-attention also relies on Hamilton products.",
"The revised Quaternion self-attention is defined as follows: $A = \\text{ComponentSoftmax}(\\frac{Q \\otimes K}{\\sqrt{d_k}})V.$ Note that in (REF ), $Q \\otimes K$ returns four $\\ell \\times \\ell $ matrices (attention weights) for each component $(r,i,j,k)$ .",
"Softmax is applied component-wise, along with multiplication with $V$ which is multiplied in similar fashion to the Quaternion attention model.",
"Note that the Hamilton product in the self-attention itself does not change the parameter size of the network."
],
[
"Quaternion Transformer Block",
"Aside from the linear transformations for forming query, key, and values.",
"Tranformers also contain position feed-forward networks with ReLU activations.",
"Similarly, we replace the feed-forward connections (FFNs) with Quaternion FFNs.",
"We denote this as Quaternion Transformer (full) while denoting the model that only uses Quaternion FFNs in the self-attention as (partial).",
"Finally, the remainder of the Transformer networks remain identical to the original design [37] in the sense that component-wise functions are applied unless specified above."
],
[
"Embedding Layers",
"In the case where the word embedding layer is trained from scratch (i.e., using Byte-pair encoding in machine translation), we treat each embedding to be the concatenation of its four components.",
"In the case where pre-trained embeddings such as GloVe [26] are used, a nonlinear transform is used to project the embeddings into Quaternion space."
],
[
"Connection to Real Components",
"A vast majority of neural components in the deep learning arsenal operate in real space.",
"As such, it would be beneficial for our Quaternion-inspired components to interface seamlessly with these components.",
"If input to a Quaternion module (such as Quaternion FFN or attention modules), we simply treat the real-valued input as a concatenation of components $r,x,y,z$ .",
"Similarly, the output of the Quaternion module, if passed to a real-valued layer, is treated as a $[r;x;y;z]$ , where $[;]$ is the concatenation operator."
],
[
"Output layer and Loss Functions",
"To train our model, we simply concatenate all $r,i,j,k$ components into a single vector at the final output layer.",
"For example, for classification, the final Softmax output is defined as following: $Y = \\text{Softmax}(W([r;x;y;z]) + b),$ where $Y \\in \\mathbb {R}^{|C|}$ where $|C|$ is the number of classes and $x,y,z$ are the imaginary components.",
"Similarly for sequence loss (for sequence transduction problems), the same can be also done."
],
[
"Parameter Initialization",
"It is intuitive that specialized initialization schemes ought to be devised for Quaternion representations and their modules [23], [22].",
"$w = |w|(\\cos (\\theta ) + q^{\\triangleleft }_{imag} \\sin (\\theta ),$ where $q^{\\triangleleft }_{imag}$ is the normalized imaginary constructed from uniform randomly sampling from $[0,1]$ .",
"$\\theta $ is randomly and uniformly sampled from $[-\\pi ,\\pi ]$ .",
"However, our early experiments show that, at least within the context of NLP applications, this initialization performed comparable or worse than the standard Glorot initialization.",
"Hence, we opt to initialize all components independently with Glorot initialization.",
"Table: Experimental results on pairwise text classification and ranking tasks.",
"Q-Att achieves comparable or competitive results compared with DeAtt with approximately one third of the parameter cost."
],
[
"Experiments",
"This section describes our experimental setup across multiple diverse NLP tasks.",
"All experiments were run on NVIDIA Titan X hardware."
],
[
"Our Models",
"On pairwise text classification, we benchmark Quaternion attention model (Q-Att), testing the ability of Quaternion models on pairwise representation learning.",
"On all the other tasks, such as machine translation and subject-verb agreement, we evaluate Quaternion Transformers.",
"We evaluate two variations of Transformers, full and partial.",
"The full setting converts all linear transformations into Quaternion space and is approximately $25\\%$ of the actual Transformer size.",
"The second setting (partial) only reduces the linear transforms at the self-attention mechanism.",
"Tensor2Tensorhttps://github.com/tensorflow/tensor2tensor.",
"is used for Transformer benchmarks, which uses its default Hyperparameters and encoding for all experiments."
],
[
"Pairwise Text Classification",
"We evaluate our proposed Quaternion attention (Q-Att) model on pairwise text classification tasks.",
"This task involves predicting a label or ranking score for sentence pairs.",
"We use a total of seven data sets from problem domains such as: Natural language inference (NLI) - This task is concerned with determining if two sentences entail or contradict each other.",
"We use SNLI [4], SciTail [13], MNLI [40] as benchmark data sets.",
"Question answering (QA) - This task involves learning to rank question-answer pairs.",
"We use WikiQA [41] which comprises of QA pairs from Bing Search.",
"Paraphrase detection - This task involves detecting if two sentences are paraphrases of each other.",
"We use Tweets [14] data set and the Quora paraphrase data set [38].",
"Dialogue response selection - This is a response selection (RS) task that tries to select the best response given a message.",
"We use the Ubuntu dialogue corpus, UDC [17]."
],
[
"Implementation Details",
"We implement Q-Att in TensorFlow [1], along with the Decomposable Attention baseline [24].",
"Both models optimize the cross entropy loss (e.g., binary cross entropy for ranking tasks such as WikiQA and Ubuntu).",
"Models are optimized with Adam with the learning rate tuned amongst $\\lbrace 0.001, 0.0003\\rbrace $ and the batch size tuned amongst $\\lbrace 32,64\\rbrace $ .",
"Embeddings are initialized with GloVe [26].",
"For Q-Att, we use an additional transform layer to project the pre-trained embeddings into Quaternion space.",
"The measures used are generally the accuracy measure (for NLI and Paraphrase tasks) and ranking measures (MAP/MRR/Top-1) for ranking tasks (WikiQA and Ubuntu).",
"Table: Experimental results on sentiment analysis on IMDb and Stanford Sentiment Treebank (SST) data sets.",
"Evaluation measure is accuracy."
],
[
"Baselines and Comparison",
"We use the Decomposable Attention model as a baseline, adding $[a_i;b_i;a_i\\odot b_i; a_i-b_i]$ before the compareThis follows the matching function of [5].",
"layers since we found this simple modification to increase performance.",
"This also enables fair comparison with our variation of Quaternion attention which uses Hamilton product over Element-wise multiplication.",
"We denote this as DeAtt.",
"We evaluate at a fixed representation size of $d=200$ (equivalent to $d=50$ in Quaternion space).",
"We also include comparisons at equal parameterization ($d=50$ and approximately $200K$ parameters) to observe the effect of Quaternion representations.",
"We selection of DeAtt is owing to simplicity and ease of comparison.",
"We defer the prospect of Quaternion variations of more advanced models [5], [34] to future work."
],
[
"Results",
"Table REF reports results on seven different and diverse data sets.",
"We observe that a tiny Q-Att model ($d=50$ ) achieves comparable (or occasionally marginally better or worse) performance compared to DeAtt ($d=200$ ), gaining a $68\\%$ parameter savings.",
"The results actually improve on certain data sets (2/7) and are comparable (often less than a percentage point difference) compared with the $d=200$ DeAtt model.",
"Moreover, we scaled the parameter size of the DeAtt model to be similar to the Q-Att model and found that the performance degrades quite significantly (about $2\\%-3\\%$ lower on all data sets).",
"This demonstrates the quality and benefit of learning with Quaternion space."
],
[
"Sentiment Analysis",
"We evaluate on the task of document-level sentiment analysis which is a binary classification problem."
],
[
"Implementation Details",
"We compare our proposed Quaternion Transformer against the vanilla Transformer.",
"In this experiment, we use the tiny Transformer setting in Tensor2Tensor with a vocab size of $8K$ .",
"We use two data sets, namely IMDb [18] and Stanford Sentiment Treebank (SST) [31].",
"Table: Experimental results on neural machine translation (NMT).",
"Results of Transformer Base on EN-VI (IWSLT 2015), EN-RO (WMT 2016) and EN-ET (WMT 2018).",
"Parameter size excludes word embeddings.",
"Our proposed Quaternion Transformer achieves comparable or higher performance with only 67.9%67.9\\% parameter costs of the base Transformer model."
],
[
"Results",
"Table REF reports results the sentiment classification task on IMDb and SST.",
"We observe that both the full and partial variation of Quaternion Transformers outperform the base Transformer.",
"We observe that Quaternion Transformer (partial) obtains a $+1.0\\%$ lead over the vanilla Transformer on IMDb and $+2.5\\%$ on SST.",
"This is while having a $24.5\\%$ saving in parameter cost.",
"Finally the full Quaternion version leads by $+1.3\\%/1.6\\%$ gains on IMDb and SST respectively while maintaining a 75% reduction in parameter cost.",
"This supports our core hypothesis of improving accuracy while saving parameter costs."
],
[
"Neural Machine Translation",
"We evaluate our proposed Quaternion Transformer against vanilla Transformer on three data sets on this neural machine translation (NMT) task.",
"More concretely, we evaluate on IWSLT 2015 English Vietnamese (En-Vi), WMT 2016 English-Romanian (En-Ro) and WMT 2018 English-Estonian (En-Et).",
"We also include results on the standard WMT EN-DE English-German results."
],
[
"Implementation Details",
"We implement models in Tensor2Tensor and trained for $50k$ steps for both models.",
"We use the default base single GPU hyperparameter setting for both models and average checkpointing.",
"Note that our goal is not to obtain state-of-the-art models but to fairly and systematically evaluate both vanilla and Quaternion Transformers."
],
[
"Results",
"Table REF reports the results on neural machine translation.",
"On the IWSLT'15 En-Vi data set, the partial adaptation of the Quaternion Transformer outperforms (+2.5%) the base Transformer with a $32\\%$ reduction in parameter cost.",
"On the other hand, the full adaptation comes close $(-0.4\\%)$ with a $75\\%$ reduction in paramter cost.",
"On the WMT'16 En-Ro data set, Quaternion Transformers do not outperform the base Transformer.",
"We observe a $-0.1\\%$ degrade in performance on the partial adaptation and $-4.3\\%$ degrade on the full adaptation of the Quaternion Transformer.",
"However, we note that the drop in performance with respect to parameter savings is still quite decent, e.g., saving 32% parameters for a drop of only $0.1$ BLEU points.",
"The full adaptation loses out comparatively.",
"On the WMT'18 En-Et dataset, the partial adaptation achieves the best result with $32\\%$ less parameters.",
"The full adaptation, comparatively, only loses by $1.0$ BLEU score from the original Transformer yet saving $75\\%$ parameters.",
"Table: Experimental results on mathematical language understanding (MLU).",
"Both Quaternion models outperform the base Transformer model with up to 75%75\\% parameter savings."
],
[
"WMT English-German",
"Notably, Quaternion Transformer achieves a BLEU score of 26.42/25.14 for partial/full settings respectively on the standard WMT 2014 En-De benchmark.",
"This is using a single GPU trained for $1M$ steps with a batch size of 8192.",
"We note that results do not differ much from other single GPU runs (i.e., 26.07 BLEU) on this dataset [20]."
],
[
"Mathematical Language Understanding",
"We include evaluations on a newly released mathematical language understanding (MLU) data set [39].",
"This data set is a character-level transduction task that aims to test a model's the compositional reasoning capabilities.",
"For example, given an input $x=85, y=-523, x * y$ the model strives to decode an output of $-44455$ .",
"Several variations of these problems exist, mainly switching and introduction of new mathematical operators."
],
[
"Implementation Details",
"We train Quaternion Transformer for $100K$ steps using the default Tensor2Tensor setting following the original work [39].",
"We use the tiny hyperparameter setting.",
"Similar to NMT, we report both full and partial adaptations of Quaternion Transformers.",
"Baselines are reported from the original work as well, which includes comparisons from Universal Transformers [9] and Adaptive Computation Time (ACT) Universal Transformers.",
"The evaluation measure is accuracy per sequence, which counts a generated sequence as correct if and only if the entire sequence is an exact match."
],
[
"Results",
"Table REF reports our experimental results on the MLU data set.",
"We observe a modest $+7.8\\%$ accuracy gain when using the Quaternion Transformer (partial) while saving $24.5\\%$ parameter costs.",
"Quaternion Transformer outperforms Universal Transformer and marginally is outperformed by Adaptive Computation Universal Transformer (ACT U-Transformer) by $0.5\\%$ .",
"On the other hand, a full Quaternion Transformer still outperforms the base Transformer ($+2.8\\%$ ) with $75\\%$ parameter saving."
],
[
"Subject Verb Agreement",
"Additionally, we compare our Quaternion Transformer on the subject-verb agreement task [16].",
"The task is a binary classification problem, determining if a sentence, e.g., `The keys to the cabinet _____ .'",
"follows by a plural/singular."
],
[
"Implementation",
"We use the Tensor2Tensor framework, training Transformer and Quaternion Transformer with the tiny hyperparameter setting with $10k$ steps."
],
[
"Results",
"Table REF reports the results on the SVA task.",
"Results show that Quaternion Transformers perform equally (or better) than vanilla Transformers.",
"On this task, the partial adaptation performs better, improving Transformers by $+0.7\\%$ accuracy while saving $25\\%$ parameters.",
"Table: Experimental results on subject-verb agreement (SVA) number prediction task."
],
[
"Related Work",
"The goal of learning effective representations lives at the heart of deep learning research.",
"While most neural architectures for NLP have mainly explored the usage of real-valued representations [37], [3], [24], there have also been emerging interest in complex [7], [2], [11] and hypercomplex representations [23], [22], [11].",
"Notably, progress on Quaternion and hypercomplex representations for deep learning is still in its infancy and consequently, most works on this topic are very recent.",
"Gaudet and Maida proposed deep Quaternion networks for image classification, introducing basic tools such as Quaternion batch normalization or Quaternion initialization [11].",
"In a similar vein, Quaternion RNNs and CNNs were proposed for speech recognition [22], [23].",
"In parallel Zhu et al.",
"proposed Quaternion CNNs and applied them to image classification and denoising tasks [44].",
"Comminiello et al.",
"proposed Quaternion CNNs for sound detection [6].",
"[42] proposed Quaternion embeddings of knowledge graphs.",
"[43] proposed Quaternion representations for collaborative filtering.",
"A common theme is that Quaternion representations are helpful and provide utility over real-valued representations.",
"The interest in non-real spaces can be attributed to several factors.",
"Firstly, complex weight matrices used to parameterize RNNs help to combat vanishing gradients [2].",
"On the other hand, complex spaces are also intuitively linked to associative composition, along with holographic reduced representations [27], [21], [33].",
"Asymmetry has also demonstrated utility in domains such as relational learning [36], [21] and question answering [32].",
"Complex networks [35], in general, have also demonstrated promise over real networks.",
"In a similar vein, the hypercomplex Hamilton product provides a greater extent of expressiveness, similar to the complex Hermitian product, albeit with a 4-fold increase in interactions between real and imaginary components.",
"In the case of Quaternion representations, due to parameter saving in the Hamilton product, models also enjoy a 75% reduction in parameter size.",
"Our work draws important links to multi-head [37] or multi-sense [15], [19] representations that are highly popular in NLP research.",
"Intuitively, the four-component structure of Quaternion representations can also be interpreted as some kind of multi-headed architecture.",
"The key difference is that the basic operators (e.g., Hamilton product) provides an inductive bias that encourages interactions between these components.",
"Notably, the idea of splitting vectors has also been explored [8], which is in similar spirit to breaking a vector into four components."
],
[
"Conclusion",
"This paper advocates for lightweight and efficient neural NLP via Quaternion representations.",
"More concretely, we proposed two models - Quaternion attention model and Quaternion Transformer.",
"We evaluate these models on eight different NLP tasks and a total of thirteen data sets.",
"Across all data sets the Quaternion model achieves comparable performance while reducing parameter size.",
"All in all, we demonstrated the utility and benefits of incorporating Quaternion algebra in state-of-the-art neural models.",
"We believe that this direction paves the way for more efficient and effective representation learning in NLP.",
"Our Tensor2Tensor implementation of Quaternion Transformers will be released at https://github.com/vanzytay/QuaternionTransformers."
],
[
"Acknowledgements",
"The authors thank the anonymous reviewers of ACL 2019 for their time, feedback and comments."
]
] | 1906.04393 |
[
[
"Rearrangement operations on unrooted phylogenetic networks"
],
[
"Abstract Rearrangement operations transform a phylogenetic tree into another one and hence induce a metric on the space of phylogenetic trees.",
"Popular operations for unrooted phylogenetic trees are NNI (nearest neighbour interchange), SPR (subtree prune and regraft), and TBR (tree bisection and reconnection).",
"Recently, these operations have been extended to unrooted phylogenetic networks, which are generalisations of phylogenetic trees that can model reticulated evolutionary relationships.",
"Here, we study global and local properties of spaces of phylogenetic networks under these three operations.",
"In particular, we prove connectedness and asymptotic bounds on the diameters of spaces of different classes of phylogenetic networks, including tree-based and level-k networks.",
"We also examine the behaviour of shortest TBR-sequence between two phylogenetic networks in a class, and whether the TBR-distance changes if intermediate networks from other classes are allowed: for example, the space of phylogenetic trees is an isometric subgraph of the space of phylogenetic networks under TBR.",
"Lastly, we show that computing the TBR-distance and the PR-distance of two phylogenetic networks is NP-hard."
],
[
"Introduction",
"Phylogenetic trees and networks are leaf-labelled graphs that are used to visualise and study the evolutionary history of taxa like species, genes, or languages.",
"While phylogenetic trees are used to model tree-like evolutionary histories, the more general phylogenetic networks can be used for taxa whose past includes reticulate events like hybridisation or horizontal gene transfer [29], [17], [30].",
"Such reticulate events arise in all domains of life [31], [26], [23], [33].",
"In some cases, it can be useful to distinguish between rooted and unrooted phylogenetic networks.",
"In a rooted phylogenetic network, the edges are directed from a designated root towards the leaves.",
"Hence, it models evolution along the passing of time.",
"An unrooted phylogenetic network, on the other hand, has undirected edges and thus represent evolutionary relatedness of the taxa.",
"In some cases, unrooted phylogenetic networks can be thought of as rooted phylogenetic networks in which the orientation of the edges has been disregarded.",
"Such unrooted phylogenetic networks are called proper [19], [8].",
"Here we focus on unrooted, binary, proper phylogenetic networks, where binary means that all vertices except for the leaves have degree three.",
"The set of phylogenetic networks on the same taxa can be partitioned into tiers that contain all networks of the same size.",
"A rearrangement operation transforms a phylogenetic tree into another tree by making a small graph theoretical change.",
"An operation that works locally within the tree is the NNI (nearest neighbour interchange) operation, which changes the order of the four edges incident to an edge $e$ .",
"See for example the NNI from $T_1$ to $T_2$ in fig:utrees:rearrangementOps.",
"Two further popular rearrangement operations are the SPR (subtree prune and regraft) operation, which as the name suggests prunes (cuts) an edge and then regrafts (attaches) the resulting half edge again, and the TBR (tree bisection and reconnection) operation, which first removes an edge and then adds a new one to reconnect the resulting two smaller trees.",
"See, for example, the SPR from $T_2$ to $T_3$ and the TBR from $T_3$ to $T_4$ in fig:utrees:rearrangementOps.",
"The set of phylogenetic trees on a fixed set of taxa together with a rearrangement operation yields a graph where the vertices are the trees and two trees are adjacent if they can be transformed into each other with the operation.",
"We call this a space of phylogenetic trees.",
"This construction also induces a metric on phylogenetic trees as the distance of two trees is then given as the distance in this space, that is, the minimum number of applications of the operation that are necessary to transform one tree into the other [28].",
"However, computing the distance of two trees under NNI, SPR, and TBR is NP-hard [6], [13], [1].",
"Nevertheless, both the space of phylogenetic trees and a metric on them are of importance for the many inference methods for phylogenetic trees that rely on local search strategies [11], [27].",
"Figure: The three rearrangement operations on unrooted phylogenetic trees: The NNI from T 1 T_1 to T 2 T_2 changes the order of the four edges incident to ee; the SPR from T 2 T_2 to T 3 T_3 prunes the edge e ' e^{\\prime }, and then regrafts it again; and the TBR from T 3 T_3 to T 4 T_4 first removes the edge e '' e^{\\prime \\prime }, and then reconnects the resulting two trees with a new edge.",
"Note that every NNI is also an SPR and every SPR is also a TBR but not vice versa.Recently, these rearrangement operations have been generalised to phylogenetic networks, both for unrooted networks [14], [16], [9] and for rooted networks[2], [9], [12], [21].",
"For unrooted networks, Huber et al.",
"[14] first generalised NNI to level-1 networks, which are phylogenetic networks where all cycles are vertex disjoint.",
"This generalisation includes a horizontal move that changes the topology of the network, like an NNI on a tree, and vertical moves that add or remove a triangle to change the size of the network.",
"Among other results, they then showed that the space of level-1 networks and its tiers are connected under NNI [14].",
"Note that connectedness implies that the distance between any two networks in such a space is finite and that NNI thus induces a metric.",
"This NNI operation was then extended by Huber et al.",
"[16] to work for general unrooted phylogenetic networks.",
"Again, connectedness of the space was proven.",
"Later, Francis et al.",
"[9] gave lower and upper bounds on the diameter (the maximum distance) of the space of unrooted phylogenetic network of a fixed size under NNI.",
"They also showed that SPR and TBR can straightforwardly be generalised to phylogenetic networks, that the connectedness under NNI implies connectedness under SPR and TBR, and they gave bounds on the diameters.",
"These bounds for SPR were made asymptotically tight by Janssen et al. [19].",
"Here, we improve these bounds on the diameter under TBR.",
"There are several generalisations of SPR on rooted phylogenetic trees to rooted phylogenetic networks for which connectedness and diameters have been obtained [2], [9], [12], [19], [18].",
"For example, Bordewich et al.",
"[2] introduced SNPR (subnet prune and regraft), a generalisation of SPR that includes vertical moves, which add or remove an edge.",
"They then proved connectedness under SNPR for the space of rooted phylogenetic networks and for special classes of phylogenetic networks including tree-based networks.",
"Roughly speaking, these are networks that have a spanning tree that is the subdivision of a phylogenetic tree on the same taxa [10], [8].",
"Furthermore, Bordewich et al.",
"[2] gave several bounds on the SNPR-distance of two phylogenetic networks.",
"Further bounds and a characterisation of the SNPR-distance of a tree and a network were recently proven by Klawitter and Linz [20].",
"Here, we show that these bounds and characterisation on the SNPR-distance of rooted phylogenetic networks are analogous to the TBR-distance of two unrooted phylogenetic networks.",
"In this paper, we study spaces of unrooted phylogenetic networks under NNI, PR (prune and regraft), and TBR.",
"Here, the PR and the TBR operation are the generalisation of SPR and TBR on trees, respectively, where vertical moves add or remove an edge like the vertical moves of the SNPR operation in the rooted case.",
"After the preliminary section, we examine the relation of NNI, PR, and TBR; in particular, how a sequence using one of these operations can be transformed into a sequence using another operation (sec:relations).",
"We then study properties of shortest paths under TBR in sec:paths.",
"This includes the translation of the results from Bordewich et al.",
"[2] and Klawitter and Linz [20] on the SNPR-distance of rooted phylogenetic networks to the TBR-distance of unrooted phylogenetic networks.",
"Next, we consider the connectedness and diameters of spaces of phylogenetic networks for different classes of phylogenetic networks, including tree-based networks and level-$k$ networks (sec:connectedness).",
"A subspace of phylogenetic networks (e.g., the space of tree-based networks) is an isometric subgraph of a larger space of phylogenetic networks if, roughly speaking, the distance of two networks is the same in the smaller and the larger space.",
"In sec:isometric we study such isometric relations and answer a question by Francis et al.",
"[9] by showing that the space of phylogenetic trees is an isometric subgraph of the space of phylogenetic networks under TBR.",
"We use this result in sec:complexity to show that computing the TBR-distance is NP-hard.",
"In the same section, we also show that computing the PR-distance is NP-hard."
],
[
"Preliminaries",
"This section provides notation and terminology used in the remainder of the paper.",
"In particular, we define phylogenetic networks and special classes thereof, and rearrangement operations and how they induce distances.",
"Throughout this paper, $X={1, 2,\\ldots , n}$ denotes a finite set of taxa."
],
[
"Phylogenetic networks.",
"An unrooted, binary phylogenetic network $N$ on a set of taxa $X$ is an undirected multigraph such that the leaves are bijectively labelled with $X$ and all non-leaf vertices have degree three.",
"It is called proper if every cut-edge separates two labelled leaves [8], and improper otherwise.",
"This property implies that every edge lies on a path that connects two leaves.",
"More importantly, a network can be rooted at any leaf if and only if it is proper [19].",
"If not mentioned otherwise, we assume that a phylogenetic network is proper.",
"Furthermore, note that our definition of a phylogenetic network permits the existence of parallel edges in $N$ , i.e., we allow that two distinct edges join the same pair of vertices.",
"An unrooted, binary phylogenetic tree $T$ on $X$ is an unrooted, binary phylogenetic network on $X$ that is a tree.",
"Let $u\\mathcal {N}_n$ denote the set of all unrooted, binary proper phylogenetic networks on $X$ and let $u\\mathcal {T}_n$ denote the set of all unrooted, binary phylogenetic trees on $X$ , where $X = {1, 2,\\ldots , n}$ .",
"To ease reading, we refer to an unrooted, binary proper phylogenetic network (resp.",
"unrooted, binary phylogenetic tree) on $X$ simply as phylogenetic network or network (resp.",
"phylogenetic tree or tree).",
"fig:unets:treeAndNetwork shows an example of a tree $T \\in u\\mathcal {T}_{6}$ , a network in $N \\in u\\mathcal {N}_{6}$ , and an improper network $M$ .",
"Figure: An unrooted, binary phylogenetic tree T∈u𝒯 6 T \\in u\\mathcal {T}_{6} and an unrooted, binary proper phylogenetic network N∈u𝒩 6 N \\in u\\mathcal {N}_{6}.",
"The unrooted, binary phylogenetic network MM is improper since the cut-edge ee does not lie on a path that connects two leaves.An edge of a network $N$ is an external edge if it is incident to a leaf, and an internal edge otherwise.",
"A cherry ${a, b}$ of $N$ is a pair of leaves $a$ and $b$ in $N$ that are adjacent to the same vertex.",
"For example, each network in fig:unets:treeAndNetwork contains the cherry ${1, 5}$ ."
],
[
"Tiers.",
"We say a network $N = (V, E)$ has reticulation number $r$ for $r = {E} - ({V} - 1)$ , that is, the number of edges that have to be deleted from $N$ to obtain a spanning tree of $N$ .",
"For example, the network $N$ in fig:unets:treeAndNetwork has reticulation number three.",
"Note that a phylogenetic tree is a phylogenetic network with reticulation number zero.",
"Let $u\\mathcal {N}_{n,r}$ denote tier $r$ of $u\\mathcal {N}_n$ , the set of networks in $u\\mathcal {N}_n$ that have reticulation number $r$ .",
"In graph theory the value ${E} - ({V} - 1)$ of a connected graph is also called the cyclomatic number of the graph [7]."
],
[
"Embedding.",
"Let $G$ be an undirected graph.",
"Subdividing an edge ${u, v}$ of $G$ consists of replacing ${u, v}$ by a path form $u$ to $v$ that contains at least one edge.",
"A subdivision $G^*$ of $G$ is a graph that can be obtained from $G$ by subdividing edges of $G$ .",
"If $G$ has no degree two vertices, there exists a canonical embedding of vertices of $G$ to vertices of $G^*$ and of edges of $G$ to paths of $G^*$ .",
"Let $N \\in u\\mathcal {N}_n$ .",
"We say $G$ has an embedding into $N$ if there exists a subdivision $G^*$ of $G$ that is a subgraph of $N$ such that the embedding maps each labelled vertex of $G^*$ to a labelled vertex of $N$ with the same label."
],
[
"Displaying.",
"Let $T \\in u\\mathcal {T}_n$ and $N \\in u\\mathcal {N}_n$ .",
"We say $N$ displays $T$ if $T$ has an embedding into $N$ .",
"For example, in fig:unets:treeAndNetwork the tree $T$ is displayed by both networks $N$ and $M$ .",
"Let $D(N)$ be the set of trees in $u\\mathcal {T}_n$ that are displayed by $N$ .",
"This notion can be extended to trees with fewer leaves, and to networks.",
"For this, let $M$ be a phylogenetic network on $Y \\subseteq X = {1, \\ldots , n}$ .",
"We say $N$ displays $M$ if $M$ has an embedding into $N$ .",
"Let $P = {M_1,\\ldots , M_k}$ be a set of phylogenetic networks $M_i$ on $Y_i \\subseteq X = {1, \\ldots , n}$ .",
"Then let $u\\mathcal {N}_n(P)$ denote the subset of networks in $u\\mathcal {N}_n$ that display each network in $P$ ."
],
[
"Tree-based networks.",
"A phylogenetic network $N \\in u\\mathcal {N}_n$ is a tree-based network if there is a tree $T \\in u\\mathcal {T}_n$ that has an embedding into $N$ as a spanning tree.",
"In other words, there exists a subdivision $T^*$ of $T$ that is a spanning tree of $N$ .",
"The tree $T$ is then called a base tree of $N$ .",
"Let $u\\mathcal {TB}_n$ denote the set of tree-based networks in $u\\mathcal {N}_n$ .",
"For $T \\in u\\mathcal {T}_n$ , let $u\\mathcal {TB}_n(T)$ denote the set of tree-based networks in $u\\mathcal {TB}_n$ with base tree $T$ ."
],
[
"Level-$k$ networks.",
"A blob $B$ of a network $N \\in u\\mathcal {N}_n$ is a nontrivial two-connected component of $N$ .",
"The level of $B$ is the minimum number of edges that have to be removed from $B$ to make it acyclic.",
"The level of $N$ is the maximum level of all blobs of $N$ .",
"If the level of $N$ is at most $k$ , then $N$ is called a level-$k$ network.",
"Let $u\\mathcal {LV}\\text{-}k_{n}$ denote the set of level-$k$ networks in $u\\mathcal {N}_n$ ."
],
[
"$r$ -Burl.",
"An $r$ -burl is a specific type of blob that we define recursively: a 1-burl is the blob consisting of a pair of parallel edges; an $r$ -burl is the blob obtained by placing a pair of parallel edges on one of the parallel edges of an $r-1$ -burl for all $r>1$ .",
"See for example the network $M$ in fig:unets:handcuffed."
],
[
"$r$ -Handcuffed trees and caterpillars.",
"Let $T \\in u\\mathcal {N}_n$ and let $a$ and $b$ be two leaves of $T$ .",
"Let $e$ and $f$ be the edges incident to $a$ and $b$ , respectively.",
"Subdivide $e$ and $f$ with vertices ${u_1, \\ldots , u_r}$ and ${v_1, \\ldots , v_r}$ , respectively, and add the edges ${u_1, v_1}, \\ldots , {u_r, v_r}$ .",
"The resulting network is an $r$ -handcuffed tree $N \\in u\\mathcal {N}_n$ with base tree $T$ on the handcuffed leaves ${a, b}$ .",
"Note that $N$ has reticulation number $r$ .",
"If the tree $T$ is a caterpillar and $a$ and $b$ form a cherry of $T$ , then the resulting network $N$ is an $r$ -handcuffed caterpillar.",
"Furthermore, we call an $r$ -handcuffed caterpillar sorted if it is handcuffed on the leafs 1 and 2 and the leafs from 3 to $n$ have a non-decreasing distance to leaf 1.",
"See fig:unets:handcuffed for an example.",
"Figure: A network MM with a 3-burl and a sorted 3-handcuffed caterpillar NN."
],
[
"Suboperations.",
"To define rearrangement operations on phylogenetic networks, we first define several suboperations.",
"Let $G$ be an undirected graph.",
"A degree-two vertex $v$ of $G$ with adjacent vertices $u$ and $w$ gets suppressed by deleting $v$ and its incident edges, and adding the edge ${u, w}$ .",
"The reverse of this suppression is the subdivision of ${u, w}$ with vertex $v$ .",
"Let $N \\in u\\mathcal {N}_n$ be a network, and ${u, v}$ an edge of $N$ .",
"Then ${u, v}$ gets removed by deleting ${u, v}$ from $N$ and suppressing any resulting degree-two vertices.",
"We say ${u, v}$ gets pruned at $u$ by transforming it into the half edge ${\\cdot , v}$ and suppressing $u$ if it becomes a degree-two vertex.",
"Note that otherwise $u$ is a leaf.",
"In reverse, we say that a half edge ${\\cdot , v}$ gets regrafted to an edge ${x, y}$ by transforming it into the edge ${u, v}$ where $u$ is a new vertex subdividing ${x, y}$ ."
],
[
"TBR.",
"A TBR operation is the rearrangement operation that transforms a network $N\\in u\\mathcal {N}_n$ into another network $N^{\\prime } \\in u\\mathcal {N}_n$ in one of the following four ways: [leftmargin=*,label=(TBR$^-$ )] (TBR$^0$ ) Remove an internal edge $e$ of $N$ , subdivide an edge of the resulting graph with a new vertex $u$ , subdivide an edge of the resulting graph with a new vertex $v$ , and add the edge ${u, v}$ ; or, prune an external edge $e = {u, v}$ of $N$ that is incident to leaf $v$ at $u$ , regraft ${\\cdot , v}$ to an edge of the resulting graph.",
"(TBR$^+$ ) Subdivide an edge of $N$ with a new vertex $u$ , subdivide an edge of the resulting graph with a new vertex $v$ , and add the edge $e = {u, v}$ .",
"(TBR$^-$ ) Remove an edge $e$ of $N$ .",
"The TBR operation is known on unrooted phylogenetic trees as tree bisection and reconnection.",
"Since in general networks are not trees and a TBR on a network does not necessarily bisect it, we use TBR now as a word on its own.",
"For the reader who would however like to have an expansion of TBR we suggest \"total branch relocation\".",
"We welcome other suggestions.",
"Note that a TBR$^0$ can also be seen as the operation that prunes the edge $e = {u, v}$ at both $u$ and $v$ and then regrafts both ends.",
"Hence, we say that a TBR$^0$ moves the edge $e$ .",
"Furthermore, we say that a TBR$^+$ adds the edge $e$ and that a TBR$^-$ removes the edge $e$ .",
"These operations are illustrated in fig:unets:TBR.",
"Note that a TBR$^0$ has an inverse TBR$^0$ and that a TBR$^+$ has an inverse TBR$^-$ , and that furthermore a TBR$^+$ increases the reticulation number by one and a TBR$^-$ decreases it by one.",
"Since a TBR operation has to yield a phylogenetic network, there are some restrictions on the edges that can be moved or removed.",
"Firstly, if removing an edge by a TBR$^0$ yields a disconnected graph, then in order to obtain a phylogenetic network an edge has to be added between the two connected components.",
"Similarly, a TBR$^-$ cannot remove a cut-edge.",
"Secondly, the suppression of a vertex when removing an edge may not yield a loop ${u, u}$ .",
"Thirdly, removing or moving an edge cannot create a cut-edge that does not separate two leaves.",
"Otherwise the network would not be proper.",
"Figure: Illustration of the TBR operation.The network N 2 N_2 can be obtained from N 1 N_1 by a TBR 0 ^0 that moves the edge u,v{u, v} and the network N 3 N_3 can be obtained from N 2 N_2 by a TBR + ^+ that adds the edge u ' ,v ' {u^{\\prime }, v^{\\prime }}.",
"Each operation has its corresponding TBR 0 ^0 and TBR - ^- operation, respectively, that reverses the rearrangement.The TBR$^0$ operation equals the well known TBR (tree bisection and reconnection) operation on unrooted phylogenetic trees [1].",
"The TBR operation on trees has recently been generalised to TBR$^0$ on improper unrooted phylogenetic networks by Francis et al.",
"[9]."
],
[
"PR.",
"A PR (prune and regraft) operation is the rearrangement operation that transforms a network $N \\in u\\mathcal {N}_n$ into another network $N^{\\prime } \\in u\\mathcal {N}_n$ with a PR$^+$ $=$ TBR$^+$ , a PR$^-$ $=$ TBR$^-$ , or a PR$^0$ that prunes and regrafts an edge $e$ only at one endpoint, instead of at both like a TBR$^0$ .",
"Like for TBR, we the say that the PR$^{0/+/-}$ moves/adds/removes the edge $e$ in $N$ .",
"The PR operation is a generalisation of the well-known SPR (subtree prune and regraft) operation on unrooted phylogenetic trees [1].",
"Like for TBR, the generalisation of SPR to PR$^0$ for networks has been introduced by Francis et al.",
"[9]."
],
[
"NNI.",
"An NNI (nearest neighbour interchange) operation is a rearrangement operation that transforms a network $N\\in u\\mathcal {N}_n$ into another network $N^{\\prime } \\in u\\mathcal {N}_n$ in one of the following three ways: [leftmargin=*,label=(NNI$^-$ )] (NNI$^0$ ) Let $e= \\lbrace u, v\\rbrace $ be an internal edge of $N$ .",
"Prune an edge $f$ ($f \\ne e$ ) at $u$ , and regraft it to an edge $f^{\\prime }$ ($f^{\\prime } \\ne e$ ) that is incident to $v$ .",
"(NNI$^+$ ) Subdivide two adjacent edges with new vertices $u^{\\prime }$ and $v^{\\prime }$ , respectively, and add the edge $\\lbrace u^{\\prime }, v^{\\prime }\\rbrace $ .",
"(NNI$^-$ ) If $N$ contains a triangle, remove an edge of the triangle.",
"These operations are illustrated in fig:unets:NNI.",
"We say that an NNI$^0$ moves the edge $f$ .",
"Alternatively, we call the edge $e$ of an NNI$^0$ the axis of the operation, as the operation can also be defined as pruning $f$ at $u$ , and $f^{\\prime \\prime }\\ne f^{\\prime }$ at $v$ , and regrafting $f$ at $v$ and $f^{\\prime \\prime }$ at $u$ .",
"The NNI operation has been introduced on trees by Robinson [25] and generalised to networks by Huber et al.",
"[14], [16].",
"Figure: Illustration of the NNI operation.The network N 2 N_2 (resp.",
"N 3 N_3) can be obtained from N 1 N_1 (resp.",
"N 2 N_2) by an NNI 0 ^0 with the axis {u,v}\\lbrace u, v\\rbrace ; alternatively, N 2 N_2 can be obtained from N 1 N_1 using the NNI 0 ^0 of {1,u}\\lbrace 1,u\\rbrace to the triangle, and N 3 N_3 from N 2 N_2 by moving {1,u}\\lbrace 1,u\\rbrace to the bottom edge of the square.",
"The labels are inherited naturally following the first interpretation of the NNI 0 ^0 moves.The network N 4 N_4 can be obtained from N 3 N_3 by an NNI + ^+ that extends xx into a triangle.Each operation has its corresponding NNI 0 ^0 and NNI - ^- operation, respectively, that reverses the transformation."
],
[
"Sequences and distances.",
"Let $N, N^{\\prime } \\in u\\mathcal {N}_n$ be two networks.",
"A TBR-sequence from $N$ to $N^{\\prime }$ is a sequence $\\sigma = (N = N_0, N_1, N_2, \\ldots , N_k = N^{\\prime }) $ of phylogenetic networks such that $N_i$ can be obtained from $N_{i-1}$ by a single TBR for each $i \\in {1, 2, ..., k}$ .",
"The length of $\\sigma $ is $k$ .",
"The TBR-distance $\\operatorname{d_{\\textup {TBR}}}(N, N^{\\prime })$ between $N$ and $N^{\\prime }$ is the length of a shortest TBR-sequence from $N$ to $N^{\\prime }$ , or infinite if no such sequence exists.",
"Let $\\mathcal {C}_n$ be a class of phylogenetic networks.",
"The TBR-distance on $\\mathcal {C}_n$ is defined like on $u\\mathcal {N}_n$ but with the restriction that every network in a shortest TBR-sequence has to be in $\\mathcal {C}_n$ .",
"The class $\\mathcal {C}_n$ is connected under TBR if, for all pairs $N, N^{\\prime } \\in \\mathcal {C}_n$ , there exists a TBR-sequence $\\sigma $ from $N$ to $N^{\\prime }$ such that each network in $\\sigma $ is in $\\mathcal {C}_n$ .",
"Hence, for the TBR-distance to be a metric on $\\mathcal {C}_n$ , the class has to be connected under TBR and the TBR operation has to be reversible.",
"We already noted above that the latter holds for TBR (and NNI and PR).",
"For a connected class $\\mathcal {C}_n$ , the diameter is the maximum distance between two of its networks under its metric.",
"The definition for NNI and PR are analogous.",
"Let $\\mathcal {C}_n^{\\prime }$ be a subclass of $\\mathcal {C}_n$ .",
"Then $\\mathcal {C}_n^{\\prime }$ is an isometric subgraph of a $\\mathcal {C}_n$ under, say, TBR if for every $N, N^{\\prime } \\in \\mathcal {C}_n^{\\prime }$ the TBR-distance of $N$ and $N^{\\prime }$ in $\\mathcal {C}_n^{\\prime }$ equals the TBR-distance of $N$ and $N^{\\prime }$ in $\\mathcal {C}_n$ ."
],
[
"Relations of rearrangement operations",
"On trees, it is well known that every NNI is also an SPR, which, in turn, is also a TBR.",
"We observe that the same holds for the generalisations of these operations as defined above.",
"Observation 3.1 Let $N \\in u\\mathcal {N}_n$ .",
"Then, on $N$ , every NNI is a PR and every PR is a TBR.",
"For the reverse direction, we first show that every TBR can be mimicked by at most two PR like in $u\\mathcal {T}_n$ .",
"Then we show how to substitute a PR with an NNI-sequence.",
"Lemma 3.2 Let $N, N^{\\prime } \\in u\\mathcal {N}_n$ such that $\\operatorname{d_{\\textup {TBR}}}(N, N^{\\prime }) = 1$ .",
"Then $1 \\le \\operatorname{d_{\\textup {PR}}}(N, N^{\\prime }) \\le 2$ , where a TBR$^0$ may be replaced by two PR$^0$ .",
"If $N^{\\prime }$ can be obtained from $N$ by a TBR$^+$ or TBR$^-$ , then by the definition of PR$^+$ and PR$^-$ it follows that $\\operatorname{d_{\\textup {PR}}}(N, N^{\\prime }) = 1$ .",
"If $N^{\\prime }$ can be obtained from $N$ by a TBR$^0$ that is also a PR$^0$ , the statement follows.",
"Assume therefore that $N^{\\prime }$ can be obtained from $N$ by a TBR$^0$ that moves the edge $e = {u, v}$ of $N$ to $e^{\\prime } = {x, y}$ of $N^{\\prime }$ .",
"Let $G$ be the graph obtained from $N$ by removing $e$ , or equivalently the graph obtained from $N^{\\prime }$ by removing $e^{\\prime }$ .",
"If $e$ is a cut-edge, then so is $e^{\\prime }$ , and without loss of generality $u$ and $x$ as well as $v$ and $y$ subdivide an edge in the same connected components of $G$ .",
"Furthermore, if $u$ subdivides an edge of a pendant blob in $G$ , then so does $x$ .",
"Otherwise $N^{\\prime }$ would not be proper.",
"Therefore, the PR$^0$ that prunes $e$ at $u$ and regrafts it to obtain $x$ yields a phylogenetic network $N^{\\prime \\prime }$ .",
"The choices of $u$ and $x$ ensure that $N^{\\prime \\prime }$ is connected and proper.",
"There is then a PR$^0$ from $N^{\\prime \\prime }$ to $N^{\\prime }$ that prunes ${x, v}$ at $v$ and regrafts it at $y$ to obtain $N^{\\prime }$ .",
"Hence, $\\operatorname{d_{\\textup {PR}}}(N, N^{\\prime }) \\le 2$ .",
"Corollary 3.3 Let $N, N^{\\prime } \\in u\\mathcal {N}_n$ .",
"Then $\\operatorname{d_{\\textup {TBR}}}(N, N^{\\prime }) \\le \\operatorname{d_{\\textup {PR}}}(N, N^{\\prime }) \\le 2 \\operatorname{d_{\\textup {TBR}}}(N, N^{\\prime })$ .",
"Lemma 3.4 Let $N, N^{\\prime } \\in u\\mathcal {N}_{n,r}$ such that there is a PR$^0$ that transforms $N$ into $N^{\\prime }$ .",
"Let $e$ be the edge of $N$ pruned by this PR$^0$ .",
"Then there exists an NNI$^0$ -sequence from $N$ to $N^{\\prime }$ that only moves $e$ and whose length is in $\\mathcal {O}(n +r)$ .",
"Moreover, if neither $N$ nor $N^{\\prime }$ contains parallel edges, then neither does any intermediate networks in the NNI-sequence.",
"Assume that $N$ can be transformed into $N^{\\prime }$ by pruning the edge $e = \\lbrace u, v\\rbrace $ at $u$ and regrafting it to $f = \\lbrace x, y\\rbrace $ .",
"Note that there is then a (shortest) path $P = (u = v_0, v_1, v_2, \\ldots , v_k = x)$ from $u$ to $x$ in $N \\setminus \\lbrace e\\rbrace $ , since otherwise $N^{\\prime }$ would be disconnected.",
"Without loss of generality, assume that $P$ does not contain $y$ .",
"Furthermore, assume for now that $P$ does not contain $v$ .",
"The idea is now to move $e$ along $P$ to $f$ with NNI$^0$ .",
"In particular, we show how to construct a sequence $\\sigma = (N = N_0, N_1, \\ldots , N_{k} = N^{\\prime })$ such that either $N_{i+1}$ can be obtained from $N_{i}$ by an NNI$^0$ or $N_{i+1} = N_{i}$ , and such that $N_i$ contains the edge $e_i = \\lbrace v_i, v\\rbrace $ .",
"This process is illustrated in fig:unets:PRZtoNNIZ.",
"Assume we have constructed the sequence up to $N_i$ .",
"Let $g = \\lbrace v_{i+1}, w\\rbrace $ with $w \\ne v$ be the edge incident to $v_{i+1}$ that is not on $P$ .",
"Obtain $N_{i+1}$ from $N_i$ by swapping $e_i$ and $g$ with an NNI$^0$ on the axis $\\lbrace v_{i}, v_{i+1}\\rbrace $ .",
"Note that this preserves the path $P$ and that $N_{i+1}$ may only contain a parallel edge if $N$ or $N^{\\prime }$ contains parallel edges.",
"As a result, we get $N_k = N^{\\prime }$ .",
"Figure: How to mimic the PR 0 ^0 that prunes the edge {u,v}\\lbrace u, v\\rbrace at uu and regrafts to {x,y}\\lbrace x, y\\rbrace with NNI 0 ^0 operations that move uu of {u,v}\\lbrace u, v\\rbrace along the path P=(u=v 0 ,v 1 ,v 2 =x)P = (u = v_0, v_1, v_2 = x) (for the proof of clm:PRZtoNNIZ).",
"Labels follow the definition of NNI 0 ^0 along an axis.It remains to show that every network in $\\sigma $ is proper.",
"Assume otherwise and let $N_{i+1}$ be the first improper network in $\\sigma $ .",
"Then $N_{i+1}$ contains a cut-edge $e_c$ that separates a blob $B$ from all leaves.",
"We claim that $e_c$ is part of $P$ .",
"Indeed, the pruning of the NNI$^0$ from $N_i$ to $N_{i+1}$ has to create $B$ and the regrafting cannot be to $B$ , so it has to pass along $e_c$ (fig:unets:PRZtoNNIZ:properness).",
"However, as $P$ is a path, the moving edge cannot pass $e_c$ again, so all networks $N_j$ for $j > i$ including $N^{\\prime }$ are improper; a contradiction.",
"Hence, all intermediate networks $N_i$ are proper and thus $\\sigma $ is an NNI$^0$ -sequence from $N$ to $N^{\\prime }$ .",
"Figure: How an NNI 0 ^0 in the proof of clm:PRZtoNNIZ may result an improper network where e c e_c separates a blob BB from all leaves.",
"The moving edge {v,v i }\\lbrace v,v_i\\rbrace of N i N_i becomes the moving edge {v,v i+1 }\\lbrace v,v_{i+1}\\rbrace of N i+1 N_{i+1}.",
"Labels follow the definition of NNI 0 ^0 along an axis.Next, assume that $P$ contains $v_i = v$ .",
"Then first apply the process above to move $v$ of $\\lbrace u, v\\rbrace $ along $P^{\\prime } = (v = v_i, v_{i+1}, \\ldots , v_k)$ to $v_k$ .",
"In the resulting network, apply the process above to move $u$ of $\\lbrace u, v\\rbrace = \\lbrace u, v_k\\rbrace $ along $P^{\\prime \\prime } = (u = v_0, v_1, \\ldots , v_i)$ to $v_i$ .",
"The process again avoids the creation of a network $N_j$ with parallel edges, if neither $N$ nor $N^{\\prime }$ contains parallel edges.",
"Furthermore, from fig:unets:PRZtoNNIZ:properness we get that if $\\sigma $ would contain improper network then $u$ would be contained in the blob $B$ .",
"However, then ${u, v}$ and $e_c$ would be edges from $B$ to the rest of the network; again a contradiction.",
"Lastly, note that the length of $P$ is in $\\mathcal {O}(n + r)$ since $N$ contains only $2n + 3r - 1$ edges.",
"Hence, the length of $\\sigma $ is also in $\\mathcal {O}(n +r)$ .",
"Lemma 3.5 Let $n \\ge 3$ .",
"Let $N, N^{\\prime } \\in u\\mathcal {N}_n$ such that there is a PR$^-$ that transforms $N$ into $N^{\\prime }$ .",
"Let $e$ be the edge of $N$ removed by this PR$^-$ .",
"Let $N$ have reticulation number $r$ .",
"Then, there is an NNI$^0$ -sequence followed by one NNI$^-$ that transforms $N$ and $N^{\\prime }$ by only moving and removing $e$ and whose length is in $\\mathcal {O}(n + r)$ .",
"Moreover, if neither $N$ nor $N^{\\prime }$ contains parallel edges, then neither do the intermediate networks in the NNI-sequence.",
"Assume the PR$^-$ removes $e = {u, v}$ from $N$ to obtain $N^{\\prime }$ .",
"If $e$ is part of a triangle, the PR$^-$ move is an NNI$^-$ move.",
"If $e$ is a parallel edge, then move either $u$ or $v$ with an NNI$^0$ to obtain a network with a triangle that contains $e$ .",
"Then the previous case applies.",
"So assume otherwise, namely that $e$ is not part of a triangle or a pair of parallel edges.",
"Then move $u$ with an NNI$^0$ -sequence closer to $v$ to form a triangle as follows.",
"Because removing $e$ in $N$ yields the proper network $N^{\\prime }$ , it follows that $N \\setminus {e}$ contains a shortest path $P$ from $u$ to $v$ .",
"Since $e$ is not part of a triangle, this path must contain at least two nodes other than $u$ and $v$ .",
"Let ${x, y}$ and ${y, v}$ be the last two edges on $P$ .",
"Consider the PR$^0$ that prunes $\\lbrace u, v\\rbrace $ at $u$ and regrafts it to ${x, y}$ .",
"Note that this creates a triangle on the vertices $y$ , $u$ and $v$ .",
"By clm:PRZtoNNIZ we can replace this PR$^0$ with an NNI$^0$ -sequence.",
"Lastly, we can remove $\\lbrace u, v\\rbrace $ with an NNI$^-$ to obtain $N^{\\prime }$ .",
"The bound on the length of the NNI-sequence as well as the second statement follow from clm:PRZtoNNIZ.",
"To conclude this section, we note that all previous results combined show that we can replace a TBR-sequence with a PR-sequence, which we can further replace with an NNI-sequence.",
"For several connectedness results in sec:connectedness this allows us to focus on TBR and then derive results for NNI and PR."
],
[
"Shortest paths",
"In this section, we focus on bounds on the distance between two specified networks.",
"We restrict to the TBR-distance in $u\\mathcal {N}_n$ and in $u\\mathcal {N}_{n,r}$ , and study the structure of shortest sequences of moves.",
"We make several observations about these sequences in general, and some about shortest sequences between two networks that have certain structure in common, e.g., common displayed networks.",
"Hence, we get bounds on the TBR-distance between two networks, and we uncover properties of the spaces of phylogenetic networks which allow for reductions of the search space.",
"For example, if $N$ and $N^{\\prime }$ have reticulation number $r$ , no shortest path from $N$ to $N^{\\prime }$ contains a network with a reticulation number less than $r$ .",
"The proof of this statement relies on the following observation about the order in which TBR$^0$ and TBR$^+$ operations can occur in a shortest path.",
"Observation 4.1 Let $N, N^{\\prime } \\in u\\mathcal {N}_{n,r}$ such that there exists a TBR-sequence $\\sigma _0 = (N, M, N^{\\prime })$ that uses a TBR$^+$ and a TBR$^-$ .",
"Then there is a TBR$^0$ that transforms $N$ into $N^{\\prime }$ .",
"Rephrasing clm:unets:TBR:PMtoZ, a TBR$^+$ followed by a TBR$^-$ , or vice versa, can be replaced by a TBR$^0$ .",
"This case can thus not occur in a shortest TBR-sequence.",
"Next, we look at a TBR$^0$ followed by a TBR$^+$ .",
"Lemma 4.2 Let $N, N^{\\prime } \\in u\\mathcal {N}_n$ with reticulation number $r$ and $r+1$ such that there exists a shortest TBR-sequence $\\sigma _0 = (N, M, N^{\\prime })$ that starts with a TBR$^0$ .",
"Then there is a TBR-sequence $\\sigma _{+} = (N, M^{\\prime }, N^{\\prime })$ that starts with a TBR$^+$ .",
"Note that the TBR$^0$ from $N$ to $M$ of $\\sigma _{0}$ can be replaced with a sequence consisting of a TBR$^+$ followed by a TBR$^-$ .",
"This TBR$^-$ and the TBR$^+$ from $M$ to $N^{\\prime }$ can now be combined to a TBR$^0$ , which gives us a sequence $\\sigma _{+}$ .",
"Let $N, N^{\\prime } \\in u\\mathcal {N}_{n,r}$ and consider a shortest TBR-sequences from $N$ to $N^{\\prime }$ that contains TBR$^+$ and TBR$^-$ operations.",
"If the reverse statement of clm:unets:TBR:ZPtoPZ would also hold, then we could shuffle the sequence such that consecutive TBR$^+$ and TBR$^-$ can be replaced with a TBR$^0$ .",
"This would imply that $u\\mathcal {N}_{n,r}$ is an isometric subgraph of $u\\mathcal {N}_n$ under TBR.",
"However, we now show that the reverse statement of clm:unets:TBR:ZPtoPZ does not hold in general, and, hence, adjacent operations of different types in a shortest TBR-sequence cannot always be swapped.",
"Lemma 4.3 Let $n \\ge 4$ and $r \\ge 2$ .",
"Let $N, N^{\\prime } \\in u\\mathcal {N}_n$ with reticulation number $r$ and $r+1$ such that there exists a shortest TBR-sequence $\\sigma _+ = (N, M^{\\prime }, N^{\\prime })$ that starts with a TBR$^+$ .",
"Then it is not guaranteed that there is a TBR-sequence $\\sigma _{0} =(N, M, N^{\\prime })$ that starts with a TBR$^0$ .",
"We claim that the networks $N$ and $N^{\\prime }$ in fig:unets:PZbutnoZP are a pair of networks for which no TBR-sequence $\\sigma _{0} =(N, M, N^{\\prime })$ exists that starts with a TBR$^0$ .",
"The two networks $M_1$ and $M_2$ in fig:unets:PZbutnoZP are the only two TBR$^-$ neighbours of $N^{\\prime }$ .",
"However, it is easy to check that the TBR$^0$ -distance of $N$ and $M_i$ , $i \\in {1, 2}$ , is at least two.",
"Hence, a shortest TBR sequence from $N$ to $N^{\\prime }$ that starts with a TBR$^0$ has length three and so $\\sigma _{0}$ cannot exist.",
"Note that we can add an edge to each of the pair of parallel edges to obtain an example without parallel edges.",
"Moreover, the example can be extended to higher $n$ and $r$ by adding extra leaves between leaf 3 and 4, and replacing a pair of parallel edges by a chain of parallel edges in each network.",
"Figure: Two networks N,N ' ∈u𝒩 n N, N^{\\prime } \\in u\\mathcal {N}_n with TBR-distance two such that there exist a shortest TBR-sequence from NN to N ' N^{\\prime } starting with a TBR + ^+ move (to M ' M^{\\prime }).",
"However, there is no shortest TBR-sequence starting with a TBR 0 ^0, since the networks M 1 M_1 and M 2 M_2, which are the TBR - ^- neighbours of N ' N^{\\prime }, have TBR 0 ^0-distance at least two to NN.Note that the TBR$^0$ used in fig:unets:PZbutnoZP to prove clm:unets:TBR:PZtoZP:notInNets is a PR$^0$ .",
"Hence, the statement of clm:unets:TBR:PZtoZP:notInNets also holds for PR.",
"On the positive side, if one of the two networks is a tree, then we can swap the TBR$^+$ with the TBR$^0$ .",
"Lemma 4.4 Let $T \\in u\\mathcal {T}_n$ and $N \\in u\\mathcal {N}_n$ with reticulation number one such that there exists a shortest TBR-sequence $\\sigma _{+} = (T, N^{\\prime }, N)$ that starts with a TBR$^+$ .",
"Then there is a TBR-sequence $\\sigma _{0} =(T, T^{\\prime }, N)$ that starts with a TBR$^0$ .",
"We show how to obtain $\\sigma _{0}$ from $\\sigma _{+}$ .",
"Suppose that $N^{\\prime }$ is obtained from $T$ by adding the edge $f$ and that $N$ is obtained from $N^{\\prime }$ by removing $e^{\\prime }$ and adding $e$ .",
"Note that $f$ is an edge of the cycle $C$ in $N^{\\prime }$ .",
"Furthermore, $e^{\\prime }$ and $f$ are distinct.",
"Indeed, otherwise there would be a shorter TBR-sequence from $T$ to $N$ that simply adds $e$ to $T$ .",
"Assume for now that $e^{\\prime }$ is an edge of $C$ in $N^{\\prime }$ .",
"Then, $e^{\\prime }$ can be removed with a TBR$^-$ from $N^{\\prime }$ to obtain a tree $T^{\\prime }$ .",
"Hence, the TBR$^+$ from $T$ to $N^{\\prime }$ and the TBR$^-$ from $N^{\\prime }$ to $T^{\\prime }$ can be merged into a TBR$^0$ from $T$ to $T^{\\prime }$ .",
"Furthermore, the edge $e$ can then be added to $T^{\\prime }$ with a TBR$^+$ to obtain $N$ .",
"This yields the sequence $\\sigma _{0}$ .",
"Next, assume that $e^{\\prime }$ is not an edge of $C$ in $N^{\\prime }$ .",
"Then, $e^{\\prime }$ is a cut-edge in $N^{\\prime }$ and $e$ is a cut-edge in $N$ .",
"Let $\\bar{e}$ be the edge of $T$ that equals $e^{\\prime }$ , if it exists, or the edge that gets subdivided by $f$ into $e^{\\prime }$ and another edge.",
"Let $\\bar{f}$ be the edge of $N$ defined as follows: it is equal to $f$ itself if $f$ is not touched by the TBR$^0$ move from $N^{\\prime }$ to $N$ ; it is the extension of $f$ if one of its endpoints is suppressed by this move; it is one of the two edges obtained by subdividing $f$ .",
"Now let $T^{\\prime }$ be a tree obtained by removing $\\bar{f}$ from $N$ .",
"Then, there is a TBR$^0$ from $T$ to $T^{\\prime }$ that moves $\\bar{e}$ to $\\bar{e}^{\\prime }$ and furthermore a TBR$^+$ that adds $\\bar{f}$ to $T^{\\prime }$ and yields $N$ .",
"We obtain again $\\sigma _{0}$ .",
"An example is given in fig:unets:PZtoZP:trees.",
"Figure: There is a TBR-sequence from TT to NN that first adds ff with a TBR + ^+ and then moves e ' e^{\\prime } to ee with a TBR 0 ^0.",
"From this, a TBR-sequence can be derived that moves e ¯\\bar{e} to e ¯ ' \\bar{e}^{\\prime } with a TBR 0 ^0 and then adds f ¯\\bar{f} with a TBR + ^+.Next, we look at shortest paths between a tree and a network.",
"First, we show that if a network displays a tree, then there is a simple TBR$^-$ -sequence from the network to the tree.",
"Recall that $D(N)$ is the set of trees in $u\\mathcal {T}_n$ displayed by $N \\in u\\mathcal {N}_n$ .",
"This result is the unrooted analogous to Lemma 7.4 by Bordewich et al.",
"[2] on rooted phylogenetic networks.",
"Lemma 4.5 Let $N \\in u\\mathcal {N}_{n,r}$ and $T \\in u\\mathcal {T}_n$ .",
"Then $T \\in D(N)$ if and only if $\\operatorname{d_{\\textup {TBR}}}(T, N) = r$ , that is, iff there exists a TBR$^-$ -sequence of length $r$ from $N$ to $T$ .",
"Note that $\\operatorname{d_{\\textup {TBR}}}(T, N) \\ge r$ , since a TBR can reduce the reticulation number by at most one.",
"Furthermore, if we apply a sequence of $r$ TBR$^-$ moves on $N$ , we arrive at a tree that is displayed by $N$ .",
"Hence, if $T \\notin D(N)$ , then $\\operatorname{d_{\\textup {TBR}}}(T,N) > r$ .",
"We now use induction on $r$ to show that $\\operatorname{d_{\\textup {TBR}}}(T, N) \\le r$ if $T\\in D(N)$ .",
"If $r = 0$ , then $T = N$ and the inequality holds.",
"Now suppose that $r > 0$ and that the statement holds whenever a network with a reticulation number less than $r$ displays $T$ .",
"Fix an embedding of $T$ into $N$ and colour all edges of $N$ not covered by this embedding green.",
"Note that removing a green edge with a TBR$^-$ might result in an improper network or a loop.",
"Therefore, we have to show that there is always at least one edge that can be removed such that the resulting graph is a phylogenetic network.",
"For this, consider the subgraph $H$ of $N$ induced by the green edges.",
"If $H$ contains a component consisting of a single green edge $e$ , then removing $e$ from $N$ with a TBR$^-$ yields a network $N^{\\prime }$ .",
"If $H$ contains a tree component $S$ , then it is easy to see that removing an external edge of $S$ from $N$ with a TBR$^-$ yields a network $N^{\\prime }$ .",
"Otherwise, as $N$ is proper, a component $S$ displays a tree $T_S$ whose external edges cover exactly the external edges of $S$ .",
"We can then apply the same case distinction to the edges of $S$ not covered by $T_S$ and either directly find an edge to remove or find further trees that cover the smaller remaining components.",
"Since $S$ is finite, we eventually find an edge to remove.",
"The induction hypothesis then applies to $N^{\\prime }$ .",
"This concludes the proof.",
"Note that the proof of clm:unets:TBR:pathDown also works if $T$ is a network displayed by $N$ .",
"Hence, we get the following corollary.",
"Corollary 4.6 Let $N \\in u\\mathcal {N}_{n,r}$ and let $N^{\\prime } \\in u\\mathcal {N}_{n,r^{\\prime }}$ such that $N^{\\prime }$ is displayed by $N$ .",
"Then $\\operatorname{d_{\\textup {TBR}}}(N^{\\prime }, N) = r - r^{\\prime }$ , that is, there exists a TBR$^-$ -sequence of length $r-r^{\\prime }$ from $N$ to $N^{\\prime }$ .",
"clm:unets:TBR:pathDown and clm:unets:TBR:pathDown:nets now allow us to construct TBR-sequences between networks that go down tiers and then come up again.",
"In fact, for rooted networks this can sometimes be necessary as Klawitter and Linz have shown [20].",
"However, we now show that this is never necessary for TBR on unrooted networks.",
"Lemma 4.7 Let $N, N^{\\prime } \\in u\\mathcal {N}_n$ .",
"Then in no shortest TBR-sequence from $N$ to $N^{\\prime }$ does a TBR$^-$ precede a TBR$^+$ .",
"Consider a minimal counterexample with $N, N^{\\prime } \\in u\\mathcal {N}_n$ such that there exists a shortest TBR-sequence $\\sigma $ from $N$ to $N^{\\prime }$ that uses exactly one TBR$^-$ and TBR$^+$ and that starts with this TBR$^-$ .",
"If $\\sigma $ uses TBR$^0$ operations between the TBR$^-$ and the TBR$^+$ , then, by clm:unets:TBR:ZPtoPZ, we can swap the TBR$^+$ forward until it directly follows the TBR$^-$ .",
"However, then we can obtain a TBR-sequence shorter than $\\sigma $ by combining the TBR$^-$ and the TBR$^+$ into a TBR$^0$ by clm:unets:TBR:PMtoZ; a contradiction.",
"Combining clm:unets:TBR:pathDown,clm:unets:TBR:pathDown:nets,clm:unets:TBR:ZPtoPZ, we easily derive the following two corollaries about short sequences that do not go down tiers before going back up again.",
"Corollary 4.8 Let $N, N^{\\prime } \\in u\\mathcal {N}_n$ with reticulation number $r$ and $r^{\\prime }$ , with $r\\ge r^{\\prime }$ .",
"Then $\\operatorname{d_{\\textup {TBR}}}(N, N^{\\prime }) \\le \\min {\\operatorname{d_{\\textup {TBR}}}(T, T^{\\prime }) \\colon T \\in D(N), T^{\\prime } \\in D(N^{\\prime })} + r \\text{.",
"}$ Corollary 4.9 Let $N, N^{\\prime } \\in u\\mathcal {N}_n$ with reticulation number $r$ and $r^{\\prime }$ , and $r\\ge r^{\\prime }$ .",
"Let $T \\in u\\mathcal {T}_n$ such that $T \\in D(N), D(N^{\\prime })$ .",
"Then $\\operatorname{d_{\\textup {TBR}}}(N, N^{\\prime }) \\le r \\text{.",
"}$ Both clm:unets:TBR:distanceViaDisplayedTrees,clm:unets:TBR:distanceSharedDisplayedTrees can easily be proven by first finding a sequence that goes down to tier 0 and back up to tier $r$ , and then combining the $r^{\\prime }$ TBR$^-$ with $r^{\\prime }$ TBR$^+$ into $r^{\\prime }$ TBR$^0$ using clm:unets:TBR:ZPtoPZ.",
"The following lemma is the unrooted analogue to Proposition 7.7 by Bordewichet al. [2].",
"We closely follow their proof.",
"Lemma 4.10 Let $N, N^{\\prime } \\in u\\mathcal {N}_n$ such that $\\operatorname{d_{\\textup {TBR}}}(N, N^{\\prime }) = k$ .",
"Let $T \\in D(N)$ .",
"Then there exists a $T^{\\prime } \\in D(N)$ such that $\\operatorname{d_{\\textup {TBR}}}(T, T^{\\prime }) \\le k \\text{.",
"}$ The proof is by induction on $k$ .",
"If $k = 0$ , then the statement trivially holds.",
"Suppose that $k = 1$ .",
"If $T \\in D(N^{\\prime })$ , then set $T^{\\prime } = T$ , and we have $\\operatorname{d_{\\textup {TBR}}}(T, T^{\\prime }) = 0 \\le 1$ .",
"So assume otherwise, namely that $T \\notin D(N^{\\prime })$ .",
"Note that that if $N^{\\prime }$ has been obtained from $N$ by a TBR$^+$ , then $N^{\\prime }$ displays $T$ .",
"Therefore, distinguish whether $N^{\\prime }$ has been obtained from $N$ by a TBR$^0$ or TBR$^-$ $\\sigma $ .",
"Suppose that $N^{\\prime }$ has been obtained from $N$ by a TBR$^0$ that moves the edge $e = {u, v}$ of $N$ .",
"Fix an embedding $S$ of $T$ into $N$ .",
"Since $N^{\\prime }$ does not display $T$ , the edge $e$ is covered by $S$ .",
"Let $\\bar{e}$ be the edge of $T$ that gets mapped to the path of $S$ that covers $e$ .",
"Let $S_1$ and $S_2$ be the subgraphs of $S \\setminus {e}$ .",
"Note that $S_1, S_2$ have embeddings into $N$ and $N^{\\prime }$ .",
"Now, if in $N$ there exists a path $P$ from the embedding of $S_1$ to the embedding of $S_2$ that avoids $e$ , then the graph consisting of $P$ , $S_1$ , and $S_2$ is a tree $T^{\\prime }$ displayed by $N^{\\prime }$ .",
"Otherwise $e$ is a cut-edge of $N$ and the TBR$^0$ moves $e$ to an edge $e^{\\prime }$ connecting the two components of $N \\setminus {e}$ .",
"Then in $N^{\\prime }$ there is path $P$ from the embedding of $S_1$ to the embedding of $S_2$ in $N^{\\prime }$ .",
"Together they form an embedding of a tree $T^{\\prime }$ displayed by $N^{\\prime }$ .",
"In both cases $T^{\\prime }$ can also be obtained from $T$ by moving $\\bar{e}$ to where $P$ attaches to $S_1$ and $S_2$ .",
"If $N^{\\prime }$ is obtained from $N$ by a TBR$^-$ , then the first case has to apply.",
"Now suppose that $k \\ge 2$ and that the hypothesis holds for any two networks with TBR-distance at most $k-1$ .",
"Let $N^{\\prime \\prime } \\in u\\mathcal {N}_n$ such that $\\operatorname{d_{\\textup {TBR}}}(N, N^{\\prime \\prime }) = k-1$ and $\\operatorname{d_{\\textup {TBR}}}(N^{\\prime \\prime }, N^{\\prime }) = 1$ .",
"Thus by induction there are trees $T^{\\prime \\prime }$ and $T^{\\prime }$ such that $T^{\\prime \\prime } \\in D(N^{\\prime \\prime })$ with $\\operatorname{d_{\\textup {TBR}}}(T, T^{\\prime \\prime }) \\le k-1$ and $T^{\\prime } \\in D(N^{\\prime })$ with $\\operatorname{d_{\\textup {TBR}}}(T^{\\prime \\prime }, T^{\\prime }) \\le 1$ .",
"It follows that $\\operatorname{d_{\\textup {TBR}}}(T, T^{\\prime }) \\le k$ , thereby completing the proof of the lemma.",
"By setting one of the two networks in the previous lemma to be a phylogenetic tree and noting that the roles of $N$ and $N^{\\prime }$ are interchangeable, the next two corollaries are immediate consequences of clm:unets:TBR:pathDown,clm:unets:TBR:existingCloseDisplayedTree.",
"Corollary 4.11 Let $T \\in u\\mathcal {T}_n$ , $N \\in u\\mathcal {N}_{n,r}$ such that $\\operatorname{d_{\\textup {TBR}}}(T, N) = k$ .",
"Then for every $T^{\\prime } \\in D(N)$ $\\operatorname{d_{\\textup {TBR}}}(T, T^{\\prime }) \\le k \\text{.",
"}$ Corollary 4.12 Let $N \\in u\\mathcal {N}_{n,r}$ and let $T, T^{\\prime } \\in D(N)$ .",
"Then $\\operatorname{d_{\\textup {TBR}}}(T, T^{\\prime }) \\le r \\text{.",
"}$ The following theorem is the unrooted analogous of Theorem 7 by Klawitter and Linz [20] and their proof can be applied straightforward by swapping SNPR and rooted networks with TBR and unrooted networks, and by using clm:unets:TBR:pathDown,clm:unets:TBR:existingCloseDisplayedTree and clm:unets:TBR:treesIsometric.",
"Theorem 4.13 Let $T \\in u\\mathcal {T}_n$ and let $N \\in u\\mathcal {N}_{n,r}$ .",
"Then $\\operatorname{d_{\\textup {TBR}}}(T, N) = \\min \\limits _{T^{\\prime } \\in D(N)} \\operatorname{d_{\\textup {TBR}}}(T, T^{\\prime }) + r \\text{.",
"}$"
],
[
"Connectedness and diameters",
"Whereas in the previous section we studied the maximum distance between two given networks, here, we focus on global connectivity properties of several classes of phylogenetic networks under NNI, PR, and TBR.",
"These results imply that these operations induce metrics on these spaces.",
"For each connected metric space, we can ask about its diameter.",
"Since a class of phylogenetic networks that contains networks with unbounded reticulation number naturally has an unbounded diameter, this questions is mainly of interest for the tiers of a class.",
"First, we recall some known results from unrooted phylogenetic trees.",
"Theorem 5.1 (Li et al.",
"[22], Ding et al.",
"[5]) The space $u\\mathcal {T}_n$ is connected under NNI$^0$ with the diameter in $\\Theta (n \\log n)$ , PR$^0$ with the diameter in $n - \\Theta (\\sqrt{n})$ , and TBR$^0$ with the diameter in $n - \\Theta (\\sqrt{n})$ ."
],
[
"Network space",
"Huber et al.",
"[16] proved that the space of phylogenetic networks that includes improper networks is connected under NNI.",
"We reprove this for our definition of $u\\mathcal {N}_n$ , but first look at the tiers of this space.",
"Theorem 5.2 Let $n \\ge 0$ , $r \\ge 0$ , and $m = n + r$ .",
"Then $u\\mathcal {N}_{n,r}$ is connected under NNI with the diameter in $\\Theta (m \\log m)$ .",
"Let $N \\in u\\mathcal {N}_{n,r}$ and let $T \\in u\\mathcal {T}_n$ be a tree displayed by $N$ .",
"We show that $N$ can be transformed into a sorted $r$ -handcuffed caterpillar $N^*$ with $\\mathcal {O}(m \\log m)$ NNI.",
"Our process is as follows and illustrated in fig:unets:NNIdiam:process.",
"Step 1.",
"Transform $N$ into a network $N_T$ that is tree-based on $T$ .",
"Step 2.",
"Transform $N_T$ into handcuffed tree $N_H$ on the leafs 1 and 2.",
"Step 3.",
"Transform $N_H$ into a sorted handcuffed caterpillar $N^*$ .",
"Figure: The process used in the proof of clm:unets:NNIconnected:tier.We transform a network NN into a tree-based network N T N_T, then into a handcuffed tree N H N_H, and finally into a sorted handcuffed caterpillar N * N^*.We now describe this process in detail.",
"For Step 1, we show how to construct an NNI$^0$ -sequence $\\sigma $ from $N$ to $N_T$ , and we give a bound on the length of $\\sigma $ .",
"Let $S$ be an embedding of $T$ into $N$ , that is, $S$ is a subdivision of $T$ and a subgraph of $N$ .",
"Colour all edges of $N$ used by $S$ black and all other edges green.",
"Note that this yields green, connected subgraphs $G_1, \\ldots , G_l$ of $N$ ; more precisely, the $G_i$ are the connected components of the graph induced by the green edges of $N$ .",
"Note that each $G_i$ has at least two vertices in $S$ , since otherwise $N$ would not be proper.",
"Furthermore, if each $G_i$ consists of a single edge, then $N$ is tree-based on $T$ .",
"Assuming otherwise, we show how to break the $G_i$ apart.",
"First, if there is a triangle on vertices $v_1, u, v_2$ where $v_1$ and $v_2$ are adjacent vertices in $S$ and $u$ is their neighbour in $G_i$ , then change the embedding of $S$ (and $T$ ) so that it takes the path $v_1, u, v_2$ instead of $v_1, v_2$ (see fig:unets:NNIdiam:tbaseda).",
"Otherwise, there is an edge ${v, u}$ where $v$ is in $S$ and the other vertices adjacent to $u$ are not adjacent to $v$ .",
"Let ${u, w_1}$ and ${u, w_2}$ be the other edges incident to $u$ .",
"Apply an NNI$^0$ to move ${u, w_1}$ to $S$ as in fig:unets:NNIdiam:tbasedb.",
"Note that each such NNI$^0$ decreases the number of vertices in green subgraphs and increases the number of vertices in $S$ .",
"Furthermore, the resulting networks is clearly proper.",
"Therefore, repeat these cases until all $G_i$ consist of single edges.",
"Let the resulting graph be $N_T$ .",
"Since there are at most $2(r-1)$ vertices in all green subgraphs that are not in $S$ , the number of required NNI$^0$ for Step 1 is at most $ 2(r-1)\\text{.",
"}$ Figure: Transformation and NNI 0 ^0 used in Step 1 to obtain a tree-based network N T N_T.In Step 2 we transform $N_T$ into a handcuffed tree $N_H$ on the leaves 1 and 2.",
"Let $M = {{u_1, v_1}, {u_2, v_2}, \\ldots , {u_r, v_r}}$ be the set of green edges in $N_T$ , that is, the edges that are not in the embedding $S$ of $T$ into $N_T$ .",
"Without loss of generality, assume that for $i \\in {1, \\ldots , r}$ the distance between $u_i$ and leaf 1 in $S$ is at most the distance of $v_i$ to leaf 1 in $S$ .",
"The idea is to sweep along the edges of $S$ to move the $u_i$ towards leaf 1 and then do the same for the $v_i$ towards leaf 2.",
"For an edge $e$ of $T$ , let $P_e$ be the path of $S$ corresponding to $e$ .",
"Let $e_1$ be the edge of $T$ incident to leaf 1.",
"Impose directions on the edges of $T$ towards leaf 1.",
"Do the same for the edges of $S$ accordingly.",
"This gives a partial order $\\preceq $ on the edges of $T$ with $e_1$ as maximum.",
"Let $\\prec $ be a linear extension of $\\preceq $ on the edges of $T$ .",
"Let $e = (x, y)$ be the minimum of $\\prec $ .",
"Let $P_e = (x, \\ldots , y)$ be the corresponding path in $S$ .",
"From $x$ to $y$ along $P_e$ , proceed as follows.",
"If there is an edge $(u_i, v_l)$ in $P_e$ , then swap $u_i$ and $v_l$ with an NNI$^0$ .",
"If there is an edge $(u_i, u_j)$ in $P_e$ then move the $u_j$ endpoint of the green edge incident to $u_j$ onto the green edge incident to $u_i$ with an NNI$^0$ .",
"Otherwise, if there is an edge $(u_i, y)$ in $P_e$ , then move $u_i$ beyond $y$ .",
"This is illustrated in fig:unets:NNIdiam:sweep.",
"Informally speaking, we stack $u_j$ onto $u_i$ so they can move together towards $e_1$ .",
"Repeat this process for each edge in the order given by $\\prec $ .",
"For the last edge $e_1$ , ignore case (iii).",
"Next “unpack” the stacked $u_i$ 's on $e_1$ .",
"We now count the number of NNI$^0$ needed.",
"Firstly, each $v_l$ is swapped at most once with a $u_i$ .",
"Secondly, each $u_j$ is moving to and from a green edge at most once.",
"Furthermore, each vertex of $S$ corresponding to a vertex of $T$ is swapped at most twice.",
"Hence, the total number of NNI$^0$ required is at most $ 3r + 2n \\text{.",
"}$ Figure: NNI 0 ^0 used in Step 2to obtain a hand-cuffed tree N H N_H.",
"The label of the moving endpoint follows this endpoint to its regrafting point.Repeat this process for the $v_i$ towards leaf 2.",
"Since the $v_i$ do not have to be swapped with $u_j$ , the total number of NNI$^0$ required for this is at most $ 2r + 2n \\text{.",
"}$ Note that the resulting network may not yet be a handcuffed tree as the order of the $u_i$ and $v_j$ may be different.",
"Hence, lastly in Step 2, to obtain $N_H$ sort the edges with the mergesort-like algorithm by Li et al. [22].",
"They show that the required number of NNI$^0$ for this is at most $ r (1 + \\log r) \\text{.",
"}$ For Step 3, consider the path $P$ in $S$ from leaf 1 to 2.",
"If $P$ contains only one pendant subtree, then $N_H$ is handcuffed on the cherry ${1, 2}$ .",
"Otherwise, use NNI$^0$ to reduce it to one pendant subtree.",
"This takes at most $n$ NNI$^0$ .",
"Next, transform the pendant subtree of $P$ into a caterpillar to obtain a handcuffed caterpillar, again with at most $n$ NNI$^0$ .",
"Lastly, sort the leaves with the algorithm from Li et al.",
"[22] to obtain the sorted handcuffed caterpillar $N^*$ .",
"The required number of NNI$^0$ to get from $N_H$ to $N^*$ is at most $ 2n + n \\log n \\text{.",
"}$ Since we can transform any network $N\\in u\\mathcal {N}_{n,r}$ into $N^*$ , it follows that $u\\mathcal {N}_{n,r}$ is connected under $\\textup {NNI} $ .",
"Furthermore, adding eq:unets:NNI:diam1eq:unets:NNI:diam5 up and multiplying the result by two shows that the diameter of $u\\mathcal {N}_{n,r}$ under NNI$^0$ is at most $ 2(6n + 8r + n \\log n + r \\log r) \\in \\mathcal {O}((n + r) \\log (n + r)) \\text{.",
"}$ Francis et al.",
"[9] gave the lower bound $\\Omega (m \\log m)$ on the diameter of tier $r$ of the space that allows improper networks under NNI$^0_{\\text{improper}}$ (NNI$^0$ without the properness condition).",
"Their proof consists of two parts: a lower bound on the total number of networks in a tier ${u\\mathcal {N}_{n,r}}$ , and upper bounds on the number of networks that can be reached from one network for each fixed number of NNI$^0_{\\text{improper}}$ .",
"The diameter of $u\\mathcal {N}_{n,r}$ is at least the smallest number of moves needed for which previously mentioned upper bound is greater than the lower bound on ${u\\mathcal {N}_{n,r}}$ .",
"Our version of NNI$^0$ is stricter than theirs as we do not allow improper networks.",
"Hence, the number of networks that can be reached with a fixed number of NNI$^0$ is at most the number of networks that can be reached with the same number of NNI$^0_{\\text{improper}}$ .",
"Furthermore, their lower bound on ${u\\mathcal {N}_{n,r}}$ is found by counting the number of Echidna networks, a class of networks only containing proper networks.",
"Combining these two observations, we see that their lower bound for the diameter of $u\\mathcal {N}_{n,r}$ under NNI$^0_{\\text{improper}}$ is also a lower bound for $u\\mathcal {N}_{n,r}$ under NNI$^0$ .",
"From clm:unets:NNIconnected:tier we get the following corollary.",
"Corollary 5.3 The space $u\\mathcal {N}_n$ is connected under NNI with unbounded diameter.",
"Since, by clm:NNIisPRisTBR, every NNI is also a PR and TBR, the statements in clm:unets:NNIconnected:tier and clm:unets:NNI:connected also hold for PR and TBR.",
"This observation has been made before by Francis et al.",
"[9] for tiers of the space of networks that allow improper networks.",
"Corollary 5.4 The spaces $u\\mathcal {N}_n$ and $u\\mathcal {N}_{n,r}$ are connected under the PR and TBR operation.",
"We now look at the diameters of $u\\mathcal {N}_{n,r}$ under PR and TBR.",
"Theorem 5.5 Let $n \\ge 0$ , $r \\ge 0$ .",
"Then the diameter of $u\\mathcal {N}_{n,r}$ under PR$^0$ is in $\\Theta (n + r)$ with the upper bound $n + 2r$ .",
"The asymptotic lower bound was proven by Francis et al. [9].",
"Concerning an upper bound, Janssen et al.",
"[19] showed that the distance of two improper networks $M$ and $M^{\\prime }$ under PR is at most $n + \\frac{8}{3}r$ , of which $\\frac{2}{3}r$ PR$^0$ moves are used to transform $M$ and $M^{\\prime }$ into proper networks $N$ and $N^{\\prime }$ .",
"Hence, the PR-distance of $N$ and $N^{\\prime }$ is at most $n + 2r$ .",
"Theorem 5.6 Let $n \\ge 0$ , $r \\ge 0$ .",
"Then the diameter of $u\\mathcal {N}_{n,r}$ under TBR is in $\\Theta (n + r)$ with the upper bound $n - 3 - {\\frac{\\sqrt{n - 2} - 1}{2}} + r\\text{.",
"}$ Like for PR, the lower bound was proven by Francis et al. [9].",
"In clm:unets:TBR:distanceViaDisplayedTrees we show that the TBR-distance of two networks $N$ and $N^{\\prime } \\in u\\mathcal {N}_{n,r}$ that display a tree $T$ and $T^{\\prime } \\in u\\mathcal {T}_n$ , respectively, is at most $\\operatorname{d_{\\textup {TBR}}}(T, T^{\\prime }) + r$ .",
"Since $\\operatorname{d_{\\textup {TBR}}}(T, T^{\\prime }) \\le n - 3 - {\\frac{\\sqrt{n - 2} - 1}{2}}$ by Theorem 1.1 of Ding et al.",
"[5] it follows that $\\operatorname{d_{\\textup {TBR}}}(N, N^{\\prime }) \\le n - 3 - {\\frac{\\sqrt{n - 2} - 1}{2}} + r$ ."
],
[
"Networks displaying networks",
"Bordewich [3] and Mark et al.",
"[24] showed that the space of rooted phylogenetic trees that display a set of triplets (trees on three leaves) is connected under NNI.",
"Furthermore, Bordewich et al.",
"[2] showed that the space of rooted phylogenetic networks that display a set of rooted phylogenetic trees is connected.",
"We give a general result for unrooted phylogenetic networks that display a set of networks.",
"For this, we will use clm:unets:TBR:pathDown, which, as we recall, guarantees that if a network $N \\in u\\mathcal {N}_{n,r}$ displays a tree $T \\in u\\mathcal {T}_n$ , then there is a sequence of $r$ TBR$^-$ from $N$ to $T$ .",
"Proposition 5.7 Let $P = {P_1, ..., P_k}$ be a set of $k$ phylogenetic networks $P_i$ on $Y_i \\subseteq X = {1, \\ldots , n}$ .",
"Then $u\\mathcal {N}_n(P)$ is connected under NNI, PR, and TBR.",
"Define the network $N_P \\in u\\mathcal {N}_n(P)$ as follows.",
"Let $P_0 \\in u\\mathcal {T}_n$ be the caterpillar where the leaves are ordered from 1 to $n$ ; that is, $P_0$ contains a path $(v_2, v_3, \\ldots , v_{n-1})$ such that leaf $i$ is incident to $v_i$ , leaf 1 is incident to $v_2$ , and leaf $n$ is incident to $v_{n-1}$ .",
"Let $e_i$ be the edge incident to leaf $i$ in $P_0$ .",
"Subdivide $e_i$ with $k$ vertices $u_i^1, \\ldots , u_i^k$ .",
"Now, for $P_j \\in P$ , $j \\in {1, \\ldots , k}$ , identify leaf $i$ of $P_j$ with $u_i^j$ of $P_0$ and remove its label $i$ .",
"Finally, in the resulting network suppress any degree two vertex.",
"This is necessary if one or more of the $P_j$ have fewer than $n$ leaves.",
"The resulting network $N_P$ now displays all networks in $P$ .",
"An example is given in fig:unets:CanonicalDisplayingNetwork.",
"Figure: The canonical network N P ∈u𝒩 5 N_P \\in u\\mathcal {N}_{5} that displays the set of phylogenetic networks P=(P 1 ,P 2 )P = (P_1, P_2) with the underlying caterpillar P 0 P_0.Let $N \\in u\\mathcal {N}_n(P)$ .",
"Construct a TBR-sequence from $N$ to $N_P$ by, roughly speaking, building a copy of $N_P$ attached to $N$ , and then removing the original parts of $N$ .",
"First, add $P_0$ to $N$ by adding an edge $e = {v_1, v_2}$ from the edge incident to leaf 1 to the edge incident to leaf 2 with a TBR$^+$ .",
"Then add another edge from $e$ to the edge incident to leaf 3, and so on up to leaf $n$ .",
"Colour all newly added edges and the edges incident to the leaves blue, and all other edges red.",
"Note that the blue edges now give an embedding of $P_0$ into the current network.",
"Now, ignoring all red edges, it is straight forward to add the $P_j$ , $j \\in \\lbrace 1, \\ldots , k\\rbrace $ one after the other with TBR$^+$ such that the resulting network displays $N_P$ .",
"For example, one could start by adding a tree displayed by $P_j$ and then adding any other edges.",
"The first part works similar to the construction of $P_0$ and the second part is possible by clm:unets:TBR:pathDown.",
"Lastly, remove all red edges with TBR$^-$ such that every intermediate network is proper.",
"This is again possible by clm:unets:TBR:pathDown and yields the network $N_P$ .",
"Note that in the first two stages the red edges (plus external edges) display $P$ and in the last phase the non-red edges display $P$ .",
"Since we only used TBR$^+$ and TBR$^-$ operations, the statement also holds for PR.",
"For NNI, by clm:PRMtoNNIM we can replace each of these operations that add or remove an edge $e$ by NNI-sequences that only move and remove or add the edge $e$ .",
"Hence, the statement also holds for NNI.",
"For the following corollary, note that a quartet is an unrooted binary tree on four leaves and a quarnet is an unrooted binary, level-1 network on four leaves [15].",
"Corollary 5.8 Let $X = {1, ..., n}$ .",
"Let $P$ be a set of phylogenetic trees on $X$ , a set of quartets on $X$ , or a set of quarnets on $X$ .",
"Then $u\\mathcal {N}_n(P)$ is connected under NNI, PR, and TBR."
],
[
"Tree-based networks",
"A related but more restrictive concept to displaying a tree is being tree-based.",
"So, next, we consider the class of tree-based networks.",
"We start with the tiers of $u\\mathcal {TB}_n(T)$ , which is the set of tree-based networks that have the tree $T$ as base tree.",
"Theorem 5.9 Let $T \\in u\\mathcal {T}_n$ .",
"Then the space $u\\mathcal {TB}_{n,r}(T)$ is connected under TBR with the diameter being between ${\\frac{r}{3}}$ and $r$ , PR with the diameter being between ${\\frac{r}{2}}$ and $2r$ , and NNI with the diameter being in $\\mathcal {O}(r(n + r))$ .",
"We start with the proof for TBR.",
"Let $N, N^{\\prime } \\in u\\mathcal {TB}_{n,r}(T)$ .",
"Consider embeddings of $T$ into $N$ and $N^{\\prime }$ .",
"Let $S = {e_1, \\ldots , e_r}$ and $S^{\\prime } = {e_1^{\\prime }, \\ldots , e_r^{\\prime }}$ be the set of all edges not covered by this embedding of $T$ in $N$ and in $N^{\\prime }$ .",
"Since $N$ is tree-based, $S$ and $S^{\\prime }$ consist of vertex-disjoint edges.",
"Following the embeddings of $T$ into $N$ and $N^{\\prime }$ , it is straightforward to move each edge $e_i$ with a TBR$^0$ from $N$ to where $e_i^{\\prime }$ is in $N^{\\prime }$ .",
"In total, this requires at most $r$ TBR$^0$ .",
"Since every intermediate network is clearly in $u\\mathcal {TB}_{n,r}(T)$ , this gives connectedness of $u\\mathcal {TB}_{n,r}(T)$ and an upper bound of $r$ on the diameter.",
"For the lower bound, consider a network $M$ with $r$ pairs of parallel edges and $M^{\\prime }$ without any.",
"Observe that a TBR$^0$ can break at most three pairs of parallel edges and that only if a pair of parallel edges is removed and attached to two other pairs of parallel edge.",
"Hence, for these particular $N$ and $N^{\\prime }$ we have that $\\operatorname{d_{\\textup {TBR}}}(N, N^{\\prime }) \\ge {\\frac{r}{3}}$ .",
"The constructed TBR$^0$ -sequence for $N$ to $N^{\\prime }$ above can be converted straightforwardly into a PR$^0$ -sequence from $N$ to $N^{\\prime }$ of length at most $2r$ .",
"For the lower bound, let $M$ and $M^{\\prime }$ be as above and note that a PR can break at most two pairs of parallel edges.",
"Hence, $\\operatorname{d_{\\textup {PR}}}(M, M^{\\prime }) \\ge {\\frac{r}{2}}$ .",
"By clm:PRZtoNNIZ, the PR-sequence can be used to construct an NNI-sequence from $N$ to $N^{\\prime }$ that only moves the edges $e_i$ along paths of the embedding of $T$ .",
"Since the PR-sequence has length at most $2r$ and each PR can be replaced by an NNI sequence of length at most $\\mathcal {O}(n + r)$ , this gives the upper bound of $\\mathcal {O}(r(n + r))$ on the diameter of $u\\mathcal {TB}_{n,r}(T)$ under NNI.",
"We use clm:unets:tbased:connectedness to prove connectedness of other spaces of tree-based networks.",
"Theorem 5.10 Let $T \\in u\\mathcal {T}_n$ .",
"Then the spaces $u\\mathcal {TB}_n(T)$ , $u\\mathcal {TB}_{n,r}$ , and $u\\mathcal {TB}_n$ are each connected under TBR, PR, and NNI.",
"Moreover, the diameter of $u\\mathcal {TB}_{n,r}$ is in $\\Theta (n + r)$ under TBR and PR and in $\\mathcal {O}(n \\log n + r(n + r))$ under NNI.",
"Assume without loss of generality that $T$ has the cherry ${1, 2}$ .",
"First, let $N$ and $N^{\\prime }$ be in tiers $r$ and $r^{\\prime }$ of $u\\mathcal {TB}_n(T)$ , respectively, such that they are $r$ - and $r^{\\prime }$ -handcuffed on the cherry ${1, 2}$ .",
"Then $\\operatorname{d_{\\textup {NNI}}}(N, N^{\\prime }) = {r^{\\prime } - r}$ , as we can decrease the number of handcuffs with NNI$^-$ .",
"Since, by clm:unets:tbased:connectedness, the tiers of $u\\mathcal {TB}_{n,r}(T)$ are connected, the connectedness of $u\\mathcal {TB}_n(T)$ follows.",
"Second, let $N, N^{\\prime } \\in u\\mathcal {TB}_{n,r}$ be tree-based networks on $T$ and $T^{\\prime }$ respectively, and with an $r$ -burl on the edge incident to leaf 1.",
"Ignoring the burls, by clm:utrees:diameter, $N$ can be transformed into $N^{\\prime }$ by transforming $T$ into $T^{\\prime }$ with $\\mathcal {O}(n \\log n)$ NNI$^0$ or with $\\mathcal {O}(n)$ PR$^0$ or TBR$^0$ .",
"With clm:unets:tbased:connectedness, the connectedness of $u\\mathcal {TB}_{n,r}$ and the upper bounds on the diameter follow.",
"The lower bound on the diameter under PR and TBR also follows from clm:utrees:diameter and clm:unets:tbased:connectedness, Lastly, the connectedness of $u\\mathcal {TB}_n$ follows similarly from the connectedness of $u\\mathcal {T}_n$ and $u\\mathcal {TB}_{n,r}$ ."
],
[
"Level-$k$ networks",
"To conclude this section, we prove the connectedness of the space of level-$k$ networks.",
"Theorem 5.11 Let $n \\ge 2$ and $k \\ge 1$ .",
"Then, the space $u\\mathcal {LV}\\text{-}k_{n}$ is connected under TBR and PR with unbounded diameter.",
"Let $N \\in u\\mathcal {LV}\\text{-}k_{n}$ and $T \\in u\\mathcal {T}_n$ .",
"We show that $N$ can be transformed into the network $M \\in u\\mathcal {LV}\\text{-}k_{n}$ that can be obtained from $T$ by adding a $k$ -burl to the edge incident to leaf 1.",
"First, create a $k$ -burl in $N$ on the edge incident to leaf 1.",
"This can be done using $k$ PR$^+$ .",
"Next, using clm:unets:TBR:pathDown remove all other blobs.",
"This gives a network $M^{\\prime }$ which consists of a tree $T^{\\prime }$ with a $k$ -burl at leaf 1.",
"There is a PR$^0$ -sequence from $T^{\\prime }$ to $T$ , which is easily converted into a sequence from $M^{\\prime }$ to $M$ .",
"This proves the connectedness of $u\\mathcal {LV}\\text{-}k_{n}$ under PR and also TBR.",
"Lastly, note that the diameter is unbounded because the number of possible reticulations in a level-$k$ network is unbounded.",
"Note that an NNI$^+$ cannot directly create a pair of parallel edges.",
"We may instead add a triangle with an NNI$^+$ and then use an NNI$^0$ to transform it into a pair of parallel edges.",
"However, adding the triangle within a level-$k$ blob of a level-$k$ network, then adding the triangle would increase the level.",
"Therefore, to prove connectedness of level-$k$ networks under NNI we use the same idea as for PR but are more careful to not increase the level.",
"Theorem 5.12 Let $n \\ge 3$ and $k \\ge 1$ .",
"Then, the space $u\\mathcal {LV}\\text{-}k_{n}$ is connected under NNI with unbounded diameter.",
"Let $N \\in u\\mathcal {LV}\\text{-}k_{n}$ and let $T \\in u\\mathcal {T}_n$ .",
"Like in the proof of clm:unets:lvlk:connectedness:TBR, we want to transform $N$ into a network $M$ obtained from $T$ by adding a $k$ -burl to the edge incident to leaf 1.",
"Let $B$ be a level-$k$ blob of $N$ .",
"Assume that $N$ contains another blob $B^{\\prime }$ .",
"By clm:unets:TBR:pathDown there is a PR$^+$ -sequence that can remove $B^{\\prime }$ .",
"Use clm:PRMtoNNIM to substitute this sequence with an NNI-sequence that reduces $B^{\\prime }$ to a level-1 blob.",
"Note that this can be done locally within blob $B^{\\prime }$ and its incident edges.",
"Therefore, this process does not increase the level of a network along this sequence.",
"If $B^{\\prime }$ is now a cycle of size at least three, then we can shrink it to a triangle, if necessary, and remove it with an NNI$^-$ .",
"If $B^{\\prime }$ is a pair of parallel edges and one of its vertices is incident to a degree three vertex $v$ that is not part of a level-$k$ blob, then use an NNI$^0$ to increase the size of $B^{\\prime }$ into a triangle by including $v$ or merge it with the blob containing $v$ .",
"Next, either remove the resulting triangle, or repeat the process above to remove the new blob.",
"Otherwise, ignore $B^{\\prime }$ for now and continue with another blob of the current network that is neither $B^{\\prime }$ nor $B$ .",
"When this process terminates, we arrive at a network that has only blob $B$ , and, potentially, pairs of parallel edges that are incident to both $B$ and a leaf.",
"That is the case since a pair of parallel edges incident to a degree three vertex not in $B$ could be removed with an NNI$^0$ and an NNI$^-$ .",
"If the edge incident to leaf 1 contains a pair of parallel edges or is incident to a degree three vertex not in $B$ , then use $k-1$ NNI$^+$ and NNI$^0$ (or $k$ in the latter case) to create a $k$ -burl next to leaf 1.",
"Otherwise, if $B$ is incident to three or more cut-edges, then one of them is not incident to leaf 1 and can be moved to the edge incident to leaf 1 with an NNI$^0$ -sequence.",
"If $B$ is incident to two or fewer cut-edges, there is a vertex incident to three cut edges (since $n \\ge 3$ ) and one of them can be moved to the edge incident to leaf 1 with an NNI$^0$ -sequence.",
"Then apply the first case again to create a $k$ -burl.",
"Finally, remove $B$ and any remaining pair of parallel edges.",
"This gives a network $M^{\\prime }$ which consists of a tree $T^{\\prime }$ with a $k$ -burl at leaf 1.",
"There is an NNI$^0$ -sequence from $T^{\\prime }$ to $T$ , which is easily converted into a sequence from $M^{\\prime }$ to $M$ .",
"Lastly, note that the diameter is unbounded because for each $r\\ge 0$ , there is a level-$k$ network with $r$ reticulations."
],
[
"Isometric relations between spaces",
"Recall that a space $\\mathcal {C}_n$ is an isometric subgraph of $u\\mathcal {N}_n$ under a rearrangement operation, say TBR, if the TBR-distance of two networks in $\\mathcal {C}_n$ is the same as their TBR-distance in $u\\mathcal {N}_n$ .",
"In this section, we investigate this question for $u\\mathcal {T}_n$ under TBR, and for tree-based networks and level-k networks under TBR and PR.",
"We start with $u\\mathcal {T}_n$ .",
"The proof of the following theorem follows the proof by Bordewich et al.",
"[2] for their equivalent statement for SNPR on rooted phylogenetic trees and networks closely.",
"Theorem 6.1 The space $u\\mathcal {T}_n$ is an isometric subgraph of $u\\mathcal {N}_n$ under TBR.",
"Moreover, every shortest TBR-sequence from $T \\in u\\mathcal {T}_n$ to $T^{\\prime } \\in u\\mathcal {T}_n$ only uses TBR$^0$ .",
"Let $\\operatorname{d}_\\mathcal {T}$ and $\\operatorname{d}_\\mathcal {N}$ be the TBR-distance in $u\\mathcal {T}_n$ and $u\\mathcal {N}_n$ respectively.",
"To prove the statement, it suffices to show that $\\operatorname{d}_\\mathcal {T}(T, T^{\\prime }) = \\operatorname{d}_\\mathcal {N}(T, T^{\\prime })$ for every pair $T, T^{\\prime } \\in u\\mathcal {T}_n$ .",
"Note that $\\operatorname{d}_\\mathcal {T}(T, T^{\\prime }) \\ge \\operatorname{d}_\\mathcal {N}(T, T^{\\prime })$ holds by definition.",
"To prove the converse, let $\\sigma = (T = N_0, N_1, \\ldots , N_k = T^{\\prime })$ be a shortest TBR-sequence from $T$ to $T^{\\prime }$ .",
"Consider the following colouring of the edges of each $N_i$ , for $i \\in {0, \\ldots , k}$ .",
"Colour all edges of $T = N_0$ blue.",
"For $i \\in {1, \\ldots , k}$ preserve the colouring of $N_{i-1}$ to a colouring of $N_i$ for all edges except those affected by the TBR.",
"In particular, an edge that gets added or moved is coloured red, an edge resulting from a vertex suppression is coloured blue if the two merged edges were blue and red otherwise, and the edges resulting from an edge subdivision are coloured like the subdivided edge.",
"Let $F_i$ be the graph obtained from $N_i$ by removing all red edges.",
"We claim that $F_i$ is a forest with at most $k + 1$ components.",
"Since $F_0 = T$ , the statement holds for $i = 0$ .",
"If $N_i$ is obtained from $N_{i-1}$ by a TBR$^+$ , then $F_i = F_{i-1}$ .",
"If $N_i$ is obtained from $N_{i-1}$ by a TBR$^0$ or TBR$^-$ , then at most one component gets split.",
"Note that $F_k$ is a so-called agreement forest for $T$ and $T^{\\prime }$ and thus $\\operatorname{d}_\\mathcal {T}(T, T^{\\prime }) \\le k = \\operatorname{d}_\\mathcal {N}(T, T^{\\prime })$ by Theorem 2.13 by Allen and Steel [1].",
"Furthermore, if $\\sigma $ would use a TBR$^+$ , then the forest $F_k$ would contain at most $k$ components.",
"However, then $\\operatorname{d}_\\mathcal {T}(T, T^{\\prime }) < k$ ; a contradiction.",
"Francis et al.",
"[9] gave the example in fig:unets:NNI:tierNonIsometric to show that the tiers $u\\mathcal {N}_{n,r}$ for $n \\ge 5$ and $r > 0$ are not isometric subgraphs of $u\\mathcal {N}_n$ under NNI.",
"Their question of whether tier zero, $u\\mathcal {T}_n$ , is an isometric subgraph of $u\\mathcal {N}_n$ under NNI remains open.",
"Lemma 6.2 Let $n \\ge 5$ and $r \\ge 0$ .",
"Then the space $u\\mathcal {N}_{n,r}$ is not an isometric subgraph of $u\\mathcal {N}_n$ under NNI.",
"Figure: An NNI-sequence from NN to N ' N^{\\prime } using an NNI + ^+ that adds ff, an NNI 0 ^0 that moves ee, and an NNI - ^- that removes e ' e^{\\prime }.",
"A shortest NNI 0 ^0-sequence from NN to N ' N^{\\prime } has length three.Lemma 6.3 For $n=4$ and $r=13$ the space $u\\mathcal {N}_{n,r}$ is not an isometric subgraph of $u\\mathcal {N}_n$ under PR.",
"For the networks $N$ and $N^{\\prime }$ in $u\\mathcal {N}_{n,r}$ shown in fig:unets:PR:nonIsometric there is a length three PR-sequence that traverses tier $r+1$ , for example, like the depicted sequence $\\sigma = (N = N_0, N_1, N_2,$ $N_3 = N^{\\prime })$ .",
"To prove the statement we show that every PR$^0$ -sequence from $N$ to $N^{\\prime }$ has length at least four.",
"The networks $N$ and $N^{\\prime }$ contain the highlighted (sub)blobs $B_1$ , $B_2$ , (resp.",
"$B_1^{\\prime }$ and $B_2^{\\prime }$ ), $B_3$ , and $B_4$ .",
"Observe that the edges between $B_1$ and $B_2$ and between $B_3$ and $B_4$ may only be pruned from a blob by a PR$^0$ if they get regrafted to the same blob again.",
"Otherwise the resulting network is improper.",
"Note that to derive $B_1^{\\prime }$ from $B_1$ an edge has to be regrafted to the “top” of $B_1$ and the edge to $B_2$ has to be pruned.",
"By the first observation, combining these into one PR$^0$ cannot build the connection to $B_3$ .",
"The same applies for the transformation of $B_2$ into $B_2^{\\prime }$ and its connection to $B_4$ .",
"Therefore, we either need four PR$^0$ to derive $B_1^{\\prime }$ and $B_2^{\\prime }$ or two PR$^0$ plus two PR$^0$ to build the connections to $B_3$ and $B_4$ .",
"In conclusion, at least four PR$^0$ are required to transform $N$ into $N^{\\prime }$ , which concludes this proof.",
"By replacing a leaf with a tree, and adding more pairs of parallel edges to edge leading to 4, this example can be made to work for $n\\ge 4$ and $r\\ge 13$ .",
"Figure: A length three PR-sequence from NN to N ' N^{\\prime } that uses a PR + ^+, which adds ff, a PR 0 ^0, which moves ee, and a PR - ^-, which removes e ' e^{\\prime }.A PR 0 ^0-sequence from NN to N ' N^{\\prime } has length at least four.Theorem 6.4 For $n \\ge 6$ the space $u\\mathcal {TB}_n$ is not an isometric subgraph of $u\\mathcal {N}_n$ under TBR and PR.",
"Let $N$ be the network in fig:unets:tbased:nonIsometric.",
"Let $N^{\\prime }$ be the network derived from $N$ by swapping the labels 1 and 2.",
"Note that $\\operatorname{d_{\\textup {TBR}}}(N, N^{\\prime }) = \\operatorname{d_{\\textup {PR}}}(N, N^{\\prime }) = 2$ , since, from $N$ to $N^{\\prime }$ , we can move leaf 2 next to leaf 1 and then move leaf 1 to where leaf 2 was.",
"However, then the network in the middle is not tree-based, since the blob derived from the Petersen graph has no Hamiltonian path if the two pendent edges of the blob are next to each other [8].",
"We claim that there is no other length two TBR-sequence from $N$ to $N^{\\prime }$ .",
"For this proof we call a blob derived from the Petersen graph a Petersen blob.",
"Figure: A tree-based network on the left and a Hamiltonian path through a blob derived from the Petersen graph on the right.First, note that the TBR$^0$ -sequence of $N$ and $N^{\\prime }$ is at least two and there is thus no TBR-sequence that consists of a TBR$^-$ and a TBR$^+$ .",
"Otherwise, these two operations could be merged into a single TBR$^0$ by clm:unets:TBR:PMtoZ.",
"Note that we can only move leaf 1 or 2 by pruning an incident edge if we do not affect the split 1 versus 2, 3 or break the tree-based property.",
"Therefore, they either have to be swapped using edges of the Petersen blobs or the $(4, 5, 6)$ -chain has to be reversed and leaf 3 moved to the other Petersen blob.",
"However, it is straightforward to check that neither can be done with two TBR$^0$ .",
"In particular, we can look at what edge the first TBR$^0$ might move and then check whether a second TBR$^0$ can arrive at $N^{\\prime }$ .",
"If the first TBR$^0$ breaks a Petersen blob, the problem is that the second TBR$^0$ has to restore it.",
"We then find that this does not allows us to make the initially planned changes to arrive at $N^{\\prime }$ .",
"On the other hand, if we avoid breaking the Petersen blob and reverse the $(4, 5, 6)$ -chain, then leaf 3 is still on the wrong side; and if we move leaf 3 to the other Petersen blob, then not enough TBR$^0$ moves remain to reverse the chain.",
"Since there is no other length two TBR$^0$ -sequence there is also no other length two PR-sequence.",
"Theorem 6.5 For $n\\ge 5$ and large enough $k$ , the space $u\\mathcal {LV}\\text{-}k_{n}$ is not an isometric subgraph of $u\\mathcal {N}_n$ under TBR and PR.",
"For even $k$ , the networks $N$ and $N^{\\prime }$ in fig:unets:lvlk:nonIsometric have TBR- and PR-distance two via the network $M$ .",
"However, note that in $M$ the blobs of size $\\frac{k}{2} + 1$ a $\\frac{k}{2}$ are merged into a blob of size $k + 1$ .",
"Therefore, $M$ is not a level-$k$ network.",
"We claim that there is no TBR- or PR-sequence of length two that does not go through a level-$(k+1)$ network like $M$ .",
"An example for odd $k$ can be derived from this.",
"Figure: For even kk, a PR 0 ^0-sequence from a level-kk network NN to a level-kk network N ' N^{\\prime } (hidden reticulations of the blob-parts given inside, at least two leaves ommited: in B 1 B_1 and in B 3 B_3).",
"However, the network MM in the middle is a level-(k+1)(k+1) but not a level-kk network.It is easy to see that the TBR-distance of $N$ and $N^{\\prime }$ is at least two and there is thus no TBR-sequence that consists of a TBR$^-$ and a TBR$^+$ .",
"Otherwise, these two operations could be merged into a single TBR$^0$ by clm:unets:TBR:PMtoZ.",
"We thus have to prove that there is no length two TBR$^0$ -sequence from $N$ to $N^{\\prime }$ that avoids a level-$(k+1)$ network.",
"Note that it requires two TBR$^0$ (or PR$^0$ ) to connect $B_2$ and $B_3$ into $B_2^{\\prime }$ .",
"Similarly, it requires either two prunings from the upper five-cycle of $B_2$ to obtain the triangle $B_3^{\\prime }$ or one pruning within that cycle.",
"However, in the latter option this would not contribute to connecting $B_2$ and $B_3$ and hence overall at least three operations would be needed.",
"Therefore we have to combine the two operations necessary to create $B_2^{\\prime }$ and to create $B_3^{\\prime }$ , which however gives us a sequence like the one shown in fig:unets:lvlk:nonIsometric.",
"Note that the results of this section that show that the spaces of tree-based networks and level-$k$ networks are not isometric subgraphs of the space of all networks also hold if we restrict these spaces to a particular tier $r$ (for large enough $r$ )."
],
[
"Computational complexity",
"In this section, we consider the computational complexity of computing the TBR-distance and the PR-distance.",
"First, we recall the known results on phylogenetic trees.",
"Theorem 7.1 ([6], [13], [1]) Computing the distance of two trees in $u\\mathcal {T}_n$ is NP-hard for the NNI-distance, the SPR-distance, and the TBR-distance.",
"In clm:unets:TBR:treesIsometric, we have shown that $u\\mathcal {T}_n$ is an isometric subgraph of $u\\mathcal {N}_n$ under TBR.",
"Hence, with clm:unets:distancesNPhard, we get the following corollary.",
"Corollary 7.2 Computing the TBR-distance of two arbitrary networks in $u\\mathcal {N}_n$ is NP-hard.",
"We can use the same two theorems to prove that computing the TBR-distance in tiers is also hard.",
"Theorem 7.3 Computing the TBR-distance of two arbitrary networks in $u\\mathcal {N}_{n,r}$ is NP-hard.",
"We (linear-time) reduce the NP-hard problem of computing the TBR-distance of two trees in $u\\mathcal {T}_n$ to computing the TBR-distance of two networks in $u\\mathcal {N}_{n+1,r}$ .",
"For this, let $T, T^{\\prime } \\in u\\mathcal {T}_n$ .",
"Let $e$ be the edge incident to leaf $n$ of $T$ .",
"Obtain $S$ from $T$ by subdividing $e$ with a new vertex $u$ and adding the edge $\\lbrace u, v\\rbrace $ where $v$ is a new vertex labelled $n+1$ .",
"Next, add $r$ handcuffs to the cherry ${n, n+1}$ to obtain the network $N \\in u\\mathcal {N}_{n+1,r}$ .",
"Analogously obtain $N^{\\prime }$ from $T^{\\prime }$ .",
"The equality $\\operatorname{d_{\\textup {TBR}}}(T, T^{\\prime }) = \\operatorname{d_{\\textup {TBR}}}(N, N^{\\prime })$ follows from clm:unets:TBR:existingCloseDisplayedTree, and the fact that networks handcuffed at a cherry display exactly one tree.",
"More precisely, a TBR-sequence between $T$ and $T^{\\prime }$ induces a TBR-sequence of the same length between $N$ and $N^{\\prime }$ , hence $\\operatorname{d_{\\textup {TBR}}}(T, T^{\\prime }) \\ge \\operatorname{d_{\\textup {TBR}}}(N, N^{\\prime })$ .",
"Conversely, by clm:unets:TBR:existingCloseDisplayedTree and the fact that $D(N)=\\lbrace T\\rbrace $ and $D(N^{\\prime })=\\lbrace T^{\\prime }\\rbrace $ , it follows that $\\operatorname{d_{\\textup {TBR}}}(T, T^{\\prime }) \\le \\operatorname{d_{\\textup {TBR}}}(N, N^{\\prime })$ .",
"Since computing the TBR-distance in $u\\mathcal {T}_n$ is NP-hard, the statement follows.",
"To prove that computing the PR-distance is hard, we use a different reduction.",
"Van Iersel et al.",
"prove that deciding whether a tree is displayed by a (not necessarily proper) phylogenetic network (Unrooted Tree Containment; UTC) is NP-hard [32].",
"Combining this with clm:unets:TBR:pathDown, we arrive at our result.",
"Theorem 7.4 Computing the PR-distance of two arbitrary networks in $u\\mathcal {N}_n$ is NP-hard.",
"We reduce from UTC to the problem of computing the PR-distance of two networks in $u\\mathcal {N}_n$ .",
"Let $(N,T)$ with $N$ a (not necessarily proper) network and $T\\in u\\mathcal {T}_n$ be an arbitrary instance of UTC.",
"We obtain an instance $(N^{\\prime },T^{\\prime },r^{\\prime })$ of the PR-distance decision problem as follows: remove all cut-edges of $N$ that do not separate two labelled leaves, and let $N^{\\prime \\prime }$ be the connected component containing all the leaves; now, let $N^{\\prime }$ be the proper network obtained from $N^{\\prime \\prime }$ by suppressing all degree two nodes.",
"The instance of the PR-distance decision problem consists of $N^{\\prime }$ , $T^{\\prime }=T$ , and the reticulation number $r^{\\prime }$ of $N^{\\prime }$ .",
"As we can compute in polynomial time whether a cut edge separates two labelled leaves, the reduction is polynomial time.",
"Because a displayed tree uses only cut-edges that separate two labelled leaves, $T$ is displayed by $N$ if and only if it is displayed by $N^{\\prime }$ .",
"By clm:unets:TBR:pathDown, $T$ is a displayed tree of $N$ , if and only if $\\operatorname{d_{\\textup {PR}}}(N^{\\prime },T^{\\prime })\\le r$ , which concludes the proof.",
"Unlike for the hardness proof of TBR-distance, we cannot readily adapt this proof to the PR-distance in $u\\mathcal {N}_{n,r}$ .",
"For this purpose, we need to learn more about the structure of PR-space."
],
[
"Concluding remarks",
"In this paper, we investigated basic properties of spaces of unrooted phylogenetic networks and their metrics under the rearrangement operations NNI, PR, and TBR.",
"We have proven connectedness and bounds on diameters for different classes of phylogenetic networks, including networks that display a particular set of trees, tree-based networks, and level-$k$ networks.",
"Although these parameters have been studied before for classes of rooted phylogenetic network [2], this is the first paper that studies these properties for classes of unrooted phylogenetic networks besides the space of all networks.",
"A summary of our results is shown in tbl:unets:connectedness.",
"To see the improvements in diameter bounds, we compare our results to previously found bounds: For the space of phylogenetic trees $u\\mathcal {T}_n$ it was known that the diameter is asymptotically linearithmic and linear in the size of the trees under NNI and SPR/TBR [22], [5], respectively.",
"Here, we have shown that the diameter under NNI is also asymptotically linearithmic for higher tiers of phylogenetic networks.",
"Whether this also holds in the rooted case is still open.",
"We have further (re)proven the asymptotic linear diameter for PR and TBR of these tiers and, in particular, improved the upper bound on the diameter under TBR to $n - 3 - {\\frac{\\sqrt{n - 2} - 1}{2}} + r$ from the previously best bound $n + 2r$ [19].",
"Table: Connectedness and diameters, if bounded, for the various classes and rearrangement operations.",
"Here m=n+rm = n + r, PP is a set of phylogenetic networks, and T∈u𝒯 n T \\in u\\mathcal {T}_n.To uncover local structures of network spaces, we looked at properties of shortest sequences of moves between two networks.",
"Here we found that shortest TBR-sequences between networks in the same tier never traverse lower tiers, and shortest TBR-sequences between trees also never traverse higher tiers.",
"This implies that $u\\mathcal {T}_n$ is an isometric subgraph of $u\\mathcal {N}_n$ , and that computing the TBR-distance between two networks in $u\\mathcal {N}_n$ is NP-hard.",
"This answers a question by Francis et al. [9].",
"We have attempted to prove similar results for other subspaces and rearrangement moves.",
"However, for higher tiers, we have not been able to prove that shortest TBR-sequences never traverse higher tiers.",
"To answer this question we may need to utilise agreement graphs such as frequently used for phylogenetic trees [1], [4] and, more recently, also for rooted phylogenetic networks [20], [21].",
"Concerning NNI and PR we gave counterexamples to prove that higher tiers are not isometric subgraphs of $u\\mathcal {N}_n$ .",
"The questions whether $u\\mathcal {T}_n$ is isometrically embedded in $u\\mathcal {N}_n$ under PR and NNI remains open.",
"Answering these questions positively would also provide an answer to the question whether computing the shortest NNI-distance between two networks is NP-hard, and clues toward proving whether the PR-distance between two networks in the same tier is NP-hard.",
"Further negative results that we have shown are that the spaces of tree-based networks and level-$k$ are not isometric subgraphs of the space of all phylogenetic networks.",
"Throughout this paper, we have restricted our attention to proper networks.",
"We could also have chosen to use unrooted networks without the properness condition.",
"This definition, which is mathematically more elegant, is used in most other papers, so it seems to be the obvious choice.",
"However, it is not natural to have cut-edges that do not separate leaves: such networks carry no biological meaning.",
"It is desirable that networks are rootable and thus have an evolutionary interpretation.",
"Unrooted phylogenetic networks are rootable if they have at most one blob with one cut-edge.",
"While using this in the definition of an unrooted phylogenetic network could therefore be sufficient, we go one step further, and ask that there is no such blob.",
"This makes a network rootable at any leaf (i.e., with any taxon as out-group), which gives a stronger biological interpretation and usability.",
"The fact that our definition of unrooted phylogenetic networks is mathematically more restrictive, means that any positive result we have proven is likely also true when using a less restrictive definition.",
"That is, connectedness for those definitions follows easily by finding sequences to proper networks, like done by Jansen et al. [19].",
"As we may be able to find short sequences for this purpose, the diameter results will likely also still hold.",
"This means that whatever definitions may be used in practice, with minor additional arguments, our results provide the theoretical background necessary to justify local search operations.",
"[1]AcknowledgmentsAcknowledgments"
],
[
"Acknowledgements",
"The first author was supported by the Netherlands Organization for Scientific Research (NWO) Vidi grant 639.072.602.",
"The second author thanks the New Zealand Marsden Fund for their financial support.",
"[1]Referencesreferences"
]
] | 1906.04468 |
[
[
"Phylogenetic correlations can suffice to infer protein partners from\n sequences"
],
[
"Abstract Determining which proteins interact together is crucial to a systems-level understanding of the cell.",
"Recently, algorithms based on Direct Coupling Analysis (DCA) pairwise maximum-entropy models have allowed to identify interaction partners among paralogous proteins from sequence data.",
"This success of DCA at predicting protein-protein interactions could be mainly based on its known ability to identify pairs of residues that are in contact in the three-dimensional structure of protein complexes and that coevolve to remain physicochemically complementary.",
"However, interacting proteins possess similar evolutionary histories.",
"What is the role of purely phylogenetic correlations in the performance of DCA-based methods to infer interaction partners?",
"To address this question, we employ controlled synthetic data that only involve phylogeny and no interactions or contacts.",
"We find that DCA accurately identifies the pairs of synthetic sequences that share evolutionary history.",
"While phylogenetic correlations confound the identification of contacting residues by DCA, they are thus useful to predict interacting partners among paralogs.",
"We find that DCA performs as well as phylogenetic methods to this end, and slightly better than them with large and accurate training sets.",
"Employing DCA or phylogenetic methods within an Iterative Pairing Algorithm (IPA) allows to predict pairs of evolutionary partners without a training set.",
"We demonstrate the ability of these various methods to correctly predict pairings among real paralogous proteins with genome proximity but no known physical interaction, illustrating the importance of phylogenetic correlations in natural data.",
"However, for physically interacting and strongly coevolving proteins, DCA and mutual information outperform phylogenetic methods.",
"We discuss how to distinguish physically interacting proteins from those only sharing evolutionary history."
],
[
"Abstract",
"Determining which proteins interact together is crucial to a systems-level understanding of the cell.",
"Recently, algorithms based on Direct Coupling Analysis (DCA) pairwise maximum-entropy models have allowed to identify interaction partners among paralogous proteins from sequence data.",
"This success of DCA at predicting protein-protein interactions could be mainly based on its known ability to identify pairs of residues that are in contact in the three-dimensional structure of protein complexes and that coevolve to remain physicochemically complementary.",
"However, interacting proteins possess similar evolutionary histories.",
"What is the role of purely phylogenetic correlations in the performance of DCA-based methods to infer interaction partners?",
"To address this question, we employ controlled synthetic data that only involve phylogeny and no interactions or contacts.",
"We find that DCA accurately identifies the pairs of synthetic sequences that share evolutionary history.",
"While phylogenetic correlations confound the identification of contacting residues by DCA, they are thus useful to predict interacting partners among paralogs.",
"We find that DCA performs as well as phylogenetic methods to this end, and slightly better than them with large and accurate training sets.",
"Employing DCA or phylogenetic methods within an Iterative Pairing Algorithm (IPA) allows to predict pairs of evolutionary partners without a training set.",
"We further demonstrate the ability of these various methods to correctly predict pairings among real paralogous proteins with genome proximity but no known direct physical interaction, illustrating the importance of phylogenetic correlations in natural data.",
"However, for physically interacting and strongly coevolving proteins, DCA and mutual information outperform phylogenetic methods.",
"We finally discuss how to distinguish physically interacting proteins from proteins that only share a common evolutionary history.",
"Many biologically important protein-protein interactions are conserved over evolutionary time scales.",
"This leads to two different signals that can be used to computationally predict interactions between protein families and to identify specific interaction partners.",
"First, the shared evolutionary history leads to highly similar phylogenetic relationships between interacting proteins of the two families.",
"Second, the need to keep the interaction surfaces of partner proteins biophysically compatible causes a correlated amino-acid usage of interface residues.",
"Employing simulated data, we show that the shared history alone can be used to detect partner proteins.",
"Similar accuracies are achieved by algorithms comparing phylogenetic relationships and by methods based on Direct Coupling Analysis (DCA), which are primarily known for their ability to detect the second type of signal.",
"Using natural sequence data, we show that in cases with shared evolutionary history but without known physical interactions, both methods work with similar accuracy, while for some physically interacting systems, DCA and mutual information outperform phylogenetic methods.",
"We propose methods allowing both to predict interactions between protein families and to find interacting partners among paralogs.",
"The vast majority of cellular processes are carried out by interacting proteins.",
"Functional interactions between proteins allow multi-protein complexes to properly assemble, and ensure the specificity of signal transduction pathways.",
"Hence, mapping functional protein-protein interactions is a crucial question in biology.",
"Since high-throughput experiments remain challenging [1], an attractive alternative is to exploit the growing amount of sequence data in order to identify functional protein-protein interaction partners.",
"The amino-acid sequences of interacting proteins are correlated, both because of evolutionary constraints arising from the need to maintain physico-chemical complementarity among contacting amino-acids, and because of shared evolutionary history.",
"The first type of correlations has recently received substantial interest, both within single proteins and across protein partners.",
"Global statistical models built from the observed sequence correlations using the maximum entropy principle [2], [3], [4], [5], and assuming pairwise interactions, known as Direct Coupling Analysis (DCA), have been used with success to determine three-dimensional protein structures from sequences [6], [7], [8], to analyze mutational effects [9], [10], [11], [12] and conformational changes [13], [14], to find residue contacts between known interaction partners [5], [15], [16], [17], [18], [19], [20], [21], and most recently to predict interaction partners among paralogs from sequence data [22], [23].",
"Similar global statistical models have also revealed functional relationships in other contexts [24], [25].",
"The success of DCA-based approaches at predicting protein-protein interactions [22], [23] could originate only from correlations between residues that are in direct contact in the three-dimensional protein complex structure, thus needing to maintain physico-chemical complementarity.",
"However, additional correlations arise in protein sequences due to their common evolutionary history, i.e.",
"phylogeny [26], [27], [28], even in the absence of structural constraints.",
"Functionally related protein families [29], especially interacting ones [30] tend to have similar phylogenies, and methods directly based on phylogeny and on sequence similarity [31], [32], [33], [34], [35], in particular the Mirrortree method [31], [34], [35] allow to predict which protein families interact.",
"The similarities in the phylogenies of interacting protein families can arise from the coevolution of residues in structural contact, but also from more global shared evolutionary pressures, resulting in similar evolutionary rates [36], [37], [38], [39], [40], and from shared evolutionary history unrelated to constraints, including common timing of speciation and gene duplication events [39].",
"While being detrimental to the identification of contacting residues by DCA [5], [6], [28], these additional sources of signal can aid the identification of interaction partners.",
"Accordingly, a method based on mutual information (MI) was recently shown to slightly outperform the DCA-based one [41].",
"MI includes all types of statistical dependence between the sequences of interacting partners.",
"To what extent do purely phylogenetic correlations contribute to the prediction of interaction partners from sequences by DCA?",
"If sequences only share a common evolutionary history, i.e.",
"in the absence of functional constraints, how do DCA-based methods compare to phylogenetic methods?",
"Answering these questions is important to understand the reasons of the success of DCA-based methods, and will open the way to developing new methods that combine useful information from both phylogeny and contacts.",
"To address these questions, we generate controlled synthetic data that only involve phylogeny, in the absence of functional constraints.",
"Our DCA-based method correctly identifies pairs of synthetic “sequences” that share evolutionary history, even without any training set, thanks to an Iterative Pairing Algorithm (IPA).",
"(Strikingly, this high predictive power is obtained in the absence of real couplings from interactions, from purely phylogenetic correlations.)",
"On this synthetic dataset, we find that the DCA-based IPA and a phylogeny-based IPA reach similar performances, with DCA slightly outperforming the phylogenetic method for large training sets.",
"We then show examples of natural proteins without known direct physical interactions but with shared evolutionary history that can be accurately paired by our various methods, thus illustrating the importance of phylogenetic correlations in real data.",
"For a pair of actually interacting and strongly coevolving protein families, we find that DCA and MI substantially outperform phylogenetic methods.",
"Finally, we propose methods to predict protein-protein interactions from the level of protein families to that of paralogs.",
"We generate controlled synthetic data where “sequences” are modeled as strings of binary variables (bits) taking values 0 or 1.",
"In real protein sequences, each site can feature 21 states (20 amino acids, plus the alignment gap), but binary models where the consensus or reference amino acid is denoted by 0 and mutant states by 1 retain all conceptual ingredients, and have proved useful to identify sectors of collectively correlated amino acids [27], [42], as well as to predict fitness landscapes from sets of closely related proteins [43].",
"Our synthetic sequences are evolved along a phylogenetic tree represented by a branching process with random mutations, in the absence of any constraint stemming from interactions or function.",
"Hence, all correlations in this synthetic data arise from shared evolutionary history (and finite-size noise).",
"The data generation process is illustrated in Fig.",
"REF .",
"Figure: Construction of a synthetic dataset of chains sharing evolutionary history.",
"Starting from a random ancestral chain AB of bits whose two halves A and B are shaded in blue and red, a series of nn duplication and mutation steps (“generations”, here n=3n=3) are performed (bold: mutated bits; here 2 bits per chain are mutated at each step), resulting in 2 n =82^n=8 chains.",
"Species are then constructed randomly, here with m=2m=2 chains per species.",
"Some species are considered as the training set (green), and the other ones constitute the testing set (pink), where the pairings between each chain A and each chain B will be blinded.Specifically, we consider perfect binary trees for simplicity.",
"The ancestral chain, composed of uniformly randomly distributed bits, is duplicated, giving rise to two chains, and mutations are performed independently in the two duplicate chains: each mutation changes the state of one uniformly randomly chosen bit of the chain.",
"Then, the new chains are duplicated again, and so on.",
"We employ two different models for the occurrence of mutations.",
"In the simplest model, a fixed number of mutations per total chain length is performed along each branch of the tree, i.e.",
"between two duplication steps.",
"In a more realistic model, the number of mutations per branch is drawn in a Poisson distribution with fixed mean.",
"After a given number $n$ of duplication steps (“generations”, representing ancestry in terms of speciation or gene duplication events), a final dataset of $2^n$ chains is obtained (see Fig.",
"REF ).",
"In practice, we take $n\\le 12$ to have tractable datasets of a few thousands of final chains.",
"Throughout, we perform inference using these final chains, which correspond to contemporary sequences in a natural dataset.",
"In each of these chains, the state of each bit is uniformly distributed.",
"However, correlations exist between chains due to their shared evolutionary history.",
"The strength of these correlations depends on the number of mutations per branch and on how close the chains are along the phylogenetic tree.",
"In order to introduce the notion of species in a minimal way, we randomly group chains into sets of equal size $m$ , each representing a species.",
"The $m$ different chains within a species can then be thought of as paralogs, i.e.",
"homologs sharing common ancestry and present in the same genome.",
"In reality, different correlations are expected between the paralogs present in a given species and the orthologs present across species.",
"Later on, we therefore also consider another type of phylogeny that accounts for these effects, and assess the robustness of our conclusions to this variant.",
"Note that the present minimal model with random species is realistic in the case where exchange between species (i.e.",
"horizontal gene transfer) is sufficiently frequent.",
"We finally cut each chain of the final dataset in two halves of equal length.",
"These halves, denoted by chain A and chain B, thus represent a pair of proteins that possess the same evolutionary history.",
"Next, we blind the pairings for the chains A and B from some species (testing set) and ask whether DCA-, MI- and similarity-based methods are able to pair each A chain with its “evolutionary partner”, namely with the B chain that possesses the same evolutionary history, starting from the known pairs (training set).",
"We test several inference methods to predict pairings between chains A and B in our synthetic datasets.",
"For each of them, performance is assessed both with a training set and without a training set.",
"In the first case, the parameters defining scores are computed using the training set and employed to pair data in the testing set [4], [15].",
"In the second case, we employ the Iterative Pairing Algorithm (IPA) developed in Refs.",
"[22], [41] to bootstrap the predictions starting from initial random within-species pairings.",
"Below, we present the various inference methods assuming that there is a training set.",
"The extension to the training set-free case is then performed exactly as described in Refs.",
"[22], [41].",
"Matlab implementations of the MI-IPA, the DCA-IPA and the Mirrortree-IPA on our standard HK-RR dataset are freely available at https://doi.org/10.5281/zenodo.1421781, https://doi.org/10.5281/zenodo.1421861 and https://doi.org/10.5281/zenodo.3377592 respectively."
],
[
"Training set statistics.",
"To describe the statistics of a training set of synthetic paired chains AB, of total length $2L$ (where $L$ is the length of a chain A or B), we employ the empirical one-site frequencies of each state $\\sigma _i\\in \\lbrace 0,1\\rbrace $ at each site $i\\in \\lbrace 1,\\dots ,2L\\rbrace $ , denoted by $f_i(\\sigma _i)$ , and the two-site frequencies of occurrence of each ordered pair of states $(\\sigma _i,\\sigma _j)$ at each ordered pair of sites $(i,j)$ , denoted by $f_{ij}(\\sigma _i,\\sigma _j)$ .",
"Correlations are then computed as $C_{ij}(\\sigma _i,\\sigma _j)=f_{ij}(\\sigma _i,\\sigma _j)-f_i(\\sigma _i)f_j(\\sigma _j)$ ."
],
[
"Pseudocount.",
"When dealing with real protein sequences, pseudocounts are often introduced to avoid mathematical issues such as divergences due to amino-acid pairs that never appear, both with DCA [5], [15], [6], [7] and with MI [41].",
"Introducing a pseudocount weight $\\Lambda $ , which effectively corresponds to adding a fraction $\\Lambda $ of chains with uniformly distributed states, the corrected one-body frequencies read $\\tilde{f}_i(\\sigma _i)=\\Lambda /2+(1-\\Lambda )f_i(\\sigma _i)$ .",
"Similarly, the corrected two-body frequencies read $\\tilde{f}_{ij}(\\sigma _i,\\sigma _j)=\\Lambda /4+(1-\\Lambda )f_{ij}(\\sigma _i,\\sigma _j)\\textrm { if }i\\ne j$ and $\\tilde{f}_{ii}(\\sigma _i,\\sigma _j)=\\delta _{\\sigma _i\\sigma _j}\\Lambda /2 +(1-\\Lambda )f_{ii}(\\sigma _i,\\sigma _j)= \\delta _{\\sigma _i\\sigma _j}\\tilde{f}_i(\\sigma _i)$ , where $\\delta _{\\sigma _i\\sigma _j}=1$ if $\\sigma _i=\\sigma _j$ and 0 otherwise.",
"We investigated the impact of varying $\\Lambda $ on the performance of pairing prediction on our synthetic data using DCA and MI.",
"Fig.",
"REF shows that small nonzero values of $\\Lambda $ perform best for MI while larger ones improve DCA performance.",
"Therefore, in what follows, we always took $\\Lambda =0.015$ for MI and $\\Lambda =0.5$ for DCA, which is the typical value used when applying DCA to real proteins [7], [6]."
],
[
"DCA-based method.",
"In DCA [5], [7], [6], [44], one starts from the empirical covariances $C_{ij}(\\sigma _i,\\sigma _j)$ between all pairs of sites $(i,j)$ , computed on the training set.",
"Importantly, here, we are considering paired chains AB, and $i$ and $j$ range from 1 to the total length $2L$ of such a chain.",
"DCA is based on building a global statistical model from these covariances (and the one-body frequencies) [5], [7], [6], [44], through the maximum entropy principle [2].",
"This results in a $2L$ -body probability distribution $P$ of observing a given sequence $(\\sigma _1,\\dots ,\\sigma _{2L})$ that reads $P(\\sigma _1,\\dots ,\\sigma _{2L})=\\exp \\left[\\sum _{i<j}e_{ij}(\\sigma _i,\\sigma _j)+\\sum _{i=1}^{2L} h_i(\\sigma _i)\\right]/Z$ , where $Z$ is a normalization constant: this corresponds to the Boltzmann distribution associated to a Potts model with couplings $e_{ij}(\\sigma _i,\\sigma _j)$ and fields $h_i(\\sigma _i)$ [44].",
"Inferring the couplings and the fields that appropriately reproduce the empirical covariances is a difficult problem, known as an inverse statistical physics problem [45].",
"Note that these parameters are not all independent due to the gauge degree of freedom, so one can set e.g.",
"$h_i(0)=0$ and $e_{ij}(0,\\sigma _j)=e_{ij}(\\sigma _i,0)=0$ for all $i,j$ and $\\sigma _i,\\sigma _j$ , thus leaving only $h_i(1)$ and $e_{ij}(1,1)$ to determine.",
"Within the mean-field approximation, which will be employed throughout, these coupling strengths can be approximated by $e_{ij}(1,1)=-C^{-1}_{ij}(1,1)$ [46], [7], [6].",
"We then transform to the zero-sum (or Ising) gauge, yielding $e_{ij}(0,1)=e_{ij}(1,0)=-e_{ij}(0,0)=-e_{ij}(1,1)=C^{-1}_{ij}(1,1)/4$ .",
"The interest of this gauge is that it attributes the smallest possible fraction of the energy to the couplings, and the largest possible fraction to the fields [5], [47].",
"Note that a fully equivalent approach is to consider sequences of Ising spins instead of bits, and to employ an Ising model.",
"Here, we have chosen the Potts model formalism for consistency with protein sequence analysis by DCA.",
"The effective interaction energy $E_{AB}$ of each possible pair AB in the testing set, constructed by concatenating a chain A and a chain B, can then be assessed via $E_{AB}=-\\sum _{i=1}^{L}\\sum _{j=L+1}^{2L} e_{ij}(\\sigma _i^A,\\sigma _j^B)\\,.$ In real proteins, approximately minimizing such a score has proved successful at predicting interacting partners [15], [22].",
"Note that we only sum over inter-chain pairs (i.e.",
"pairs of sites involving one site in A and one in B) because we are interested in interactions between A and B.",
"Importantly, DCA was designed to infer actual interactions between contacting amino acids through the couplings $e_{ij}$ [5], [7], [6], [44].",
"By contrast, in the present synthetic data, there are no such interactions, and all correlations have their origin in phylogeny (or finite-size noise).",
"Nevertheless, the DCA-based interaction energy in Eq.",
"REF contains information about these correlations, and we will investigate how well it captures them."
],
[
"MI-based method.",
"Our method based on Mutual Information (MI) was introduced in [41].",
"As with DCA, we start by describing the statistics of the training set, which is composed of complete chains AB.",
"For this, we employ the single-site frequencies $\\tilde{f}_i(\\sigma _i)$ and the two-site frequencies $\\tilde{f}_{ij}(\\sigma _i,\\sigma _j)$ (see above).",
"The pointwise mutual information (PMI) of a pair of states $(\\sigma _i,\\sigma _j)$ at a pair of sites $(i,j)$ is defined as [48], [49], [50]: $\\textrm {PMI}_{ij}(\\sigma _i,\\sigma _j)=\\log \\left[\\frac{\\tilde{f}_{ij}(\\sigma _i,\\sigma _j)}{\\tilde{f}_i(\\sigma _i)\\tilde{f}_j(\\sigma _j)}\\right]\\,.$ Averaging this quantity over all possible pairs of states yields an estimate of the mutual information (MI) between sites $i$ and $j$ [51]: $\\textrm {MI}_{ij}=\\sum _{\\sigma _i,\\sigma _j}\\tilde{f}_{ij}(\\sigma _i,\\sigma _j)\\,\\textrm {PMI}_{ij}(\\sigma _i,\\sigma _j)$ .",
"Next, we define a pairing score $S_\\mathrm {AB}$ for each possible pair AB of chains from the testing set as the sum of the PMIs of the inter-chain pairs of sites of this concatenated chain AB (i.e.",
"those that involve one site in chain A and one site in chain B): $S_\\mathrm {AB}=\\sum _{i=1}^{L}\\sum _{j=L+1}^{2L}\\textrm {PMI}_{ij}(\\sigma _i^A,\\sigma _j^B)\\,.$ In real proteins, approximately maximizing such a score has proved successful at predicting interacting partners, slightly outperforming DCA [41]."
],
[
"Mirrortree-based method.",
"Methods based only on phylogeny and sequence similarity have been developed to predict protein-protein interactions.",
"In particular, the Mirrortree method quantifies the similarities of distance matrices between the proteins of two families to determine whether they interact [31], [34], [35], and has allowed the successful prediction of protein-protein interactions.",
"This method generally relies on finding one ortholog of the proteins of interest in each species and does not address the question of which paralog of family A interacts with which paralog of family B.",
"However, related approaches have tackled this problem [52], [53], [54], [55], [33], [56], which was subsequently directly addressed by the DCA- and MI-based methods of Refs.",
"[22], [23], [41].",
"We introduce an approach close to the original Mirrortree algorithm [31], [34], [35] that addresses the paralog pairing problem.",
"Specifically, let $\\lbrace A_1B_1,\\dots , A_MB_M\\rbrace $ be the training set, which contains $M$ known pairs of chains.",
"For each chain A of the testing set, we compute the vector $d_A=(d(A,A_1),\\dots , d(A,A_M))$ of Hamming distances between A and each chain A of the training set.",
"We also compute an analogous vector $d_B$ for each chain B of the training set.",
"Next, we define a pairing score $M_{AB}$ for each possible pair AB of chains A and B from the testing set as the Pearson correlation $\\rho $ between $d_A$ and $d_B$ : $M_{AB}=\\rho (d_A,\\,d_B)\\,.$ This score thus assigns high values to pairs AB that have highly similar phylogenetic relationships to the training set, hinting towards substantial shared evolutionary history between A and B.",
"It can be used for predicting partnerships exactly as the DCA- and MI-based scores in Eqs.",
"REF and REF .",
"Note that one then aims to maximize $M_{AB}$ , just like $S_{AB}$ , while DCA effective interaction energies $E_{AB}$ should be minimized."
],
[
"Other methods based on sequence similarity.",
"Because there are many ways to exploit sequence similarity in order to assess shared evolutionary history, we also consider variants beyond our Mirrortree-based method.",
"Specifically, we present results obtained using orthology between pairs, defined as reciprocal best hits in terms of Hamming distances, as well as results obtained by simply employing the Hamming distance of each possible AB pair of the testing set to its closest AB pair in the training set as a pairing score.",
"These methods are detailed and studied in Fig.",
"REF ."
],
[
"Pairing prediction.",
"We employed two different approaches to predict pairings from each of the three scores defined in Eqs.",
"REF , REF and REF .",
"In the first approach, for each chain A, we simply picked the chain B within the same species that optimizes the pairing score.",
"Note that this simple method is asymmetric and allows multiple chains A to be matched with the same chain B.",
"In the second approach, we used the Hungarian algorithm (also known as the Munkres algorithm) [57], [58], [59] to find the one-to-one association of each chain A with a chain B that optimizes the sum of the pairing scores within each species."
],
[
"DCA accurately identifies pairs of chains that only share a common evolutionary history",
"First, we set out to assess whether DCA can identify pairs of chains AB that only share a common evolutionary history.",
"For this, we generated chains of bits employing a branching process with random mutations, in the absence of any interaction or functional constraint (see Methods and Fig.",
"REF ).",
"The only correlations present among these chains thus arise from shared evolutionary history (and finite-size noise).",
"We first ask whether DCA pairing scores (Eq.",
"REF ) learned on a training set of complete chains allow to correctly predict pairs of evolutionary partners in a testing set of chains separated into half chains A and B.",
"Specifically, we generated data using a phylogenetic tree of 10 generations, with 5 mutations per branch, out of 100 bits in each complete chain AB, thus yielding 1024 chains AB.",
"Given the relatively small number of mutations per branch, many of the resulting chains AB possess substantial similarities arising from their common evolutionary history.",
"Specifically, Fig.",
"REF A shows the histogram of Hamming distances between all B chains in the dataset, featuring a typical fraction 0.3 of sites with different states.",
"The degree of similarity between two given chains arises from their relatedness along the phylogenetic tree used to generate the data.",
"We ordered the chains employing this phylogenetic tree, so that sister chains are closest to one another etc.",
"(see Fig.",
"REF A).",
"Figure: Pairs of chains with common evolutionary history have small DCA effective interaction energies.",
"A: Chains are numbered according to the phylogenetic tree representing the branching process used for data generation (see Fig. ).",
"The same numbering is employed for chains A and for chains B that possess the same evolutionary history.",
"B: Matrix of DCA effective interaction energies E AB E_{AB} (Eq. )",
"for all pairs AB made from a chain A and a chain B of the testing set, numbered according to phylogeny as illustrated in panel A.",
"Data was generated using a tree of 10 generations, with exactly 5 mutations per branch, out of 200 bits in each chain AB, thus yielding 1024 chains AB.",
"Next, 75% of them were randomly selected to form the training set employed to build the DCA model, while the remaining 25% constitute the testing set.Next, we randomly picked 75% of the chains to form a substantial training set, and inferred a DCA model from this training set (see Methods).",
"We employed the inferred couplings $e_{ij}(\\sigma _i,\\sigma _j)$ to compute effective interaction energies $E_{AB}$ (Eq.",
"REF ) between all chains A and all chains B of the remaining 25% of the dataset, which constitutes our testing set.",
"The effective interaction energies obtained are shown in Fig.",
"REF B.",
"Importantly, the diagonal of the matrix, corresponding to actual evolutionary partners, features small energies.",
"Furthermore, a nested block structure is apparent in the matrix, reflecting the phylogenetic tree (recall that chains A and B are both ordered according to the tree as shown in Fig.",
"REF A).",
"Specifically, for 22% of chains A in the testing set, the smallest DCA effective interaction energy $E_{AB}$ is obtained with their evolutionary partner (corresponding to the diagonal in Fig.",
"REF B).",
"In Fig.",
"REF , we further demonstrate that those chains B that have smaller $E_{AB}$ with a chain A than its evolutionary partner B are very similar to that chain B and strongly related to it.",
"Furthermore, if the dataset is divided into random species with 4 chains AB each (see Methods), the smallest $E_{AB}$ for a chain A within its species accurately identifies its evolutionary partner B for 93% of chains A of the testing set.",
"Hence, with a large training set, DCA is able to learn phylogenetic correlations, and to identify evolutionary partners.",
"Recall that the usual goal of DCA is to infer couplings $e_{ij}(\\sigma _i,\\sigma _j)$ stemming from actual interactions, which do not exist in our synthetic data.",
"Let us investigate the robustness of the ability of DCA to identify pairs of evolutionary partners, and compare it to other methods.",
"First, we ask how large a training set is necessary to learn the correlations arising from phylogeny.",
"Fig.",
"REF A shows that a sufficiently large training set is required for DCA to accurately identify evolutionary partners within each species, in line with previous results about DCA-based predictions of protein-protein interactions [22], [23] and three-dimensional protein structures [5], [6], [7] from real sequences.",
"Furthermore, similar trends are observed both when employing the Mutual Information (MI) based score $S_{AB}$ (Eq.",
"REF ), consistently with [41], and when using the Mirrortree-inspired score $M_{AB}$ (Eq.",
"REF ) that only relies on sequence similarity.",
"All these methods predict pairings much better than the chance expectation (yellow) and reach very high fractions of true positives for training sets larger than $\\sim $ 100 pairs AB (see Fig.",
"REF A).",
"Better performance is obtained when pairings are predicted using the Hungarian algorithm, which finds a global optimal one-to-one matching within each species (see Methods) than when simply picking for each chain A the optimal partner B within its species [41].",
"Fig.",
"REF A further shows that with the first approach, the Mirrortree method performs better for small training sets, while DCA and MI outperform it for larger datasets.",
"These differences become almost negligible when using the Hungarian algorithm.",
"Since the pairing task becomes harder when the number of pairs per species increases, we next studied how performance is affected by this important parameter.",
"Fig.",
"REF B, which employs a substantial training set, shows that the performance of all three pairing scores decays as species contain more pairs AB, as expected.",
"However, this decay is far slower than for the chance expectation (yellow), which highlights the robustness of our methods.",
"Here, DCA reaches the highest performance, followed by MI and then by Mirrortree, in line with the results obtained on Fig.",
"REF A for large training sets.",
"The good performance of the Mirrortree approach, which just relies on sequence similarities, arises from the fact that a possible pair AB that is very similar to correct pairs tends to be a correct pair too, as evidenced in Fig.",
"REF .",
"Indeed, pairs that are very similar to correct pairs tend to be their close “relatives” along the tree.",
"Other variants based on sequence similarity can thus be constructed.",
"In Fig.",
"REF , we present two such variants: one employs as a pairing score the smallest Hamming distance from a possible pair AB of the testing set to its closest neighbor in the training set, and the second one is based on the notion of orthologous pairs.",
"Both of them perform very well with large training sets, but are less robust than our other methods to decreasing training set size.",
"Figure: Performance of pairing prediction versus training set size and number of pairs per species.A: Fraction of pairs correctly identified (TP fraction) versus training set size, for DCA-, MI-, and Mirrortree-based methods.",
"The three pairing scores corresponding to each of these three methods are employed in two ways: either within each species we find the chain B with optimal pairing score with each chain A (dashed lines), or within each species we employ the Hungarian matching algorithm to find the one-to-one pairing of chains A and B that optimizes the sum of the pairing scores (solid lines).",
"Each species comprises 4 chains AB.",
"B: Fraction of pairs correctly identified (TP fraction) versus number of pairs per species, employing the same methods (and same colors) as in panel A, and a training set of 50% of the total dataset.",
"In both panels, data was generated using a tree of 10 generations, with exactly 5 mutations per branch, out of 200 bits in each chain AB, thus yielding 1024 chains AB.",
"Species were built randomly, and some of them were chosen randomly to build the training set, the remaining ones making up the testing set.",
"Yellow curves show the chance expectation, i.e.",
"the average TP fraction obtained for random within-species pairings.",
"Results are averaged over 100 replicates in panel A and 20 replicates in panel B, each corresponding to a different realization of the branching process used for data generation.",
"The standard deviation of the TP fraction is 2-3% for large training sets with 4 chains per species (panel A).Because the ability of our methods to predict pairings relies on the shared evolutionary history of chains A and B, it is crucial to understand how the mutation rate affects performance.",
"Let $\\mu =n\\mu _0$ denote the average total number of mutations between the ancestral complete chain AB and any complete chain AB at a leaf of the phylogenetic tree, where $n$ represents the number of generations and $\\mu _0$ the average number of mutations per generation (see Fig.",
"REF ).",
"If the maximum number of differences between two complete final chains AB, namely $2\\mu $ , becomes larger than the total length $2L$ of a complete chain AB, then correlations are lost between these two least-related chains AB.",
"Thus, we expect the performance of our pairing prediction methods to decay for $\\mu \\gtrsim L$ .",
"This constitutes a lower bound of the actual number $\\mu $ of mutations causing performance to substantially drop, because (i) we have considered the two least-related chains along the trees, and chains that diverged upon later duplication steps are more correlated, and (ii) since each mutation affects a random site, and each site can mutate several times, some sites may never mutate even when $\\mu \\gtrsim L$ , and thus some correlations can survive in this regime even between the most distant sequences.",
"Similarly, if $\\mu _0\\gtrsim L$ , i.e.",
"$\\mu \\gtrsim nL$ , then even sister complete final chains AB lose correlation, which gives an upper bound for the number of mutations causing performance to drop.",
"Fig.",
"REF shows heatmaps of the performance of DCA- and Mirrortree-based pairing predictions versus the total number $\\mu $ of mutations per chain AB and the single chain length $L$ , with a substantial training set.",
"For both methods, performance is very good and robust over a large range of values of $L$ and $\\mu $ .",
"In addition, in both cases, a clear transition between good and poor performance is visible as $\\mu $ is increased at each $L$ .",
"We observe that this transition occurs along a line, such that good performance is obtained for $\\mu \\lesssim 3.6\\,L - 72$ .",
"This linear behavior and its slope are consistent with our predictions above.",
"We also observe that performance drops if there are extremely few mutations, because chains are too conserved and remain almost all the same, and if chains are too short, because there is too much redundancy.",
"Another important parameter is the total number $n$ of generations in the phylogenetic tree, which sets the total number of chains ($2^n$ ).",
"We found that varying $n$ from 8 to 12 yielded no significant change the heatmaps of Fig.",
"REF .",
"Finally, the DCA- and Mirrortree-based methods perform very similarly over the whole range of parameters studied in Fig.",
"REF : specifically, the mean difference of the TP fractions obtained using the two methods is $4\\times 10^{-3}$ .",
"Despite their conceptual differences, these methods rely on learning phylogenetic correlations from the training set, and thus have similar dependences on evolutionary parameters such as the mutation rate.",
"Figure: Performance of DCA- and Mirrortree-based predictions for various data parameters.",
"The fraction of pairs AB correctly predicted (TP fraction) is shown versus the average total number of mutations μ\\mu per chain AB and the length LL of a single chain A or B for DCA (panel A) and Mirrortree (panel B).",
"The Hungarian algorithm was employed to predict pairings.",
"For each μ\\mu and LL, data was generated using a tree of n=10n=10 generations, thus yielding 1024 chains AB, and random species, each comprising 4 chains AB, were constructed.",
"Half of the species were chosen to form a training set of 512 pairs, and predictions were made on the remaining species, which form the testing set.",
"Here the chance expectation of TP fraction, obtained for random within-species pairings, is 0.25.In Refs.",
"[22], [41], it was shown that DCA- and MI-based approaches allow to predict interacting partners among the paralogs of actual interacting proteins from their sequences without any training set, i.e.",
"without any prior knowledge of interacting pairs, thanks to an Iterative Pairing Algorithm (IPA).",
"In this approach, at the first iteration, (usually poor) predictions are made employing pairing scores learned on random within-species pairings.",
"At each subsequent iteration $n > 1$ , the predictions from the previous iteration that are deemed most reliable [22], [41] are progressively incorporated.",
"Specifically, pairing scores are re-learned on the $(n-1)N_\\mathrm {increment}$ top-ranked predicted pairs from the previous iteration, where $N_\\mathrm {increment}$ represents the increment step: hence, the number of predicted pairs that are employed to make the next predictions increases by $N_\\mathrm {increment}$ at each iteration (see Refs.",
"[22], [41] for details, and Ref.",
"[23] for alternative iterative approaches).",
"This iterative strategy gradually improves predictive power and has yielded accurate predictions of interacting partners in ubiquitous prokaryotic protein families [22], [41].",
"Here, we employed the IPA on our synthetic data where correlations arise only from shared evolutionary history.",
"For comparison, we also developed and studied a variant of the IPA based on the Mirrortree approach, which employs the pairing score $M_{AB}$ (Eq.",
"REF ), instead of the DCA-based effective interaction energy $E_{AB}$ (Eq.",
"REF ) [22] or MI score $S_{AB}$ (Eq.",
"REF ) [41].",
"Fig REF A shows the TP fraction obtained for different values of the increment step $N_\\mathrm {increment}$ , both for the DCA-IPA and for the Mirrortree-IPA, at the first and last iterations.",
"Overall, it shows that the iterative approach allows to make very accurate pairing predictions in the absence of a training set.",
"With both algorithms, a strong improvement of predictive power is observed at the last iteration, compared to the first iteration and to the random expectation.",
"Furthermore, the iterative method performs best for small increment steps $N_\\mathrm {increment}$ , which highlights the interest of the iterative approach.",
"We emphasize that the high final TP fractions are attained without any prior knowledge of pairings.",
"As discussed in Refs.",
"[22], [41], an important ingredient for the IPA to bootstrap its way toward high predictive power is that among pairs AB comprising a chain A and a chain B from the same species, correct pairs of partners possess more neighbors in terms of sequence similarity, quantified by the Hamming distance, than incorrect pairs.",
"Ref.",
"[22] called this the Anna Karenina effect, referring to the first sentence of Tolstoy's novel.",
"We studied the Anna Karenina effect in our synthetic dataset.",
"Fig.",
"REF shows that correct pairs have closer correct neighbors than incorrect pairs: for instance, employing a threshold Hamming distance of 0.15 to define neighbors (see Fig.",
"REF A), correct pairs have 6.2 neighbors on average, of which 90% are correct pairs, while incorrect pairs have 0.63 neighbors on average, of which 33% are correct pairs (see Fig.",
"REF B).",
"In this case, correct pairs thus have almost 10 times more neighbors than incorrect ones, demonstrating a strong Anna Karenina effect.",
"This favors correct pairs in the IPA, especially at early iterations [22].",
"Figure: Pairing prediction without any training set.",
"A: The fraction of pairs AB correctly predicted (TP fraction) is shown versus the increment step N increment N_\\textrm {increment} of the iterative process, for the DCA-IPA and the Mirrortree-IPA, at the first and last iterations.",
"Data was generated using a tree of 10 generations, with 20 mutations per branch on average, out of 200 bits in each chain AB, thus yielding 1024 chains AB, and random species with 4 pairs AB each were constructed.",
"B: TP fraction versus number of pairs per species, for the DCA-IPA and the Mirrortree-IPA, at the first and last iterations, as well as for the Switch-IPA, which uses the Mirrortree pairing score for the first half of iterations and then switches to the DCA pairing score (the first iteration is thus the same for the Switch-IPA as for the Mirrortree-IPA).",
"An increment step N increment =100N_\\textrm {increment}=100 was used.",
"Data was generated using a tree of 10 generations, with 5 mutations per branch on average, out of 200 bits in each chain AB, thus yielding 1024 chains AB, and random species with the same number of chains AB each were constructed.",
"Colors are the same as in panel A, with the addition of the Switch-IPA.",
"In both panels, predictions were made without any training set, and the first iteration employed random within-species pairings to compute the initial pairing scores.",
"The Hungarian algorithm was employed to predict pairings.",
"Results are averaged over 20 replicates in panel A and 100 in panel B, each corresponding to a different realization of the branching process used for data generation.",
"Error bars represent 95% confidence intervals (in many cases, markers are larger than error bars).For the parameters used in Fig.",
"REF A, the Mirrortree-IPA performs slightly better than the DCA-IPA, and this difference exists right from the first iteration.",
"How does the performance of these two methods depend on parameters characterizing the dataset?",
"Fig.",
"REF shows heatmaps of the performance of the DCA-IPA and of the Mirrortree-IPA as a function of the total number $\\mu $ of mutations per chain AB and of the single chain length $L$ , without any training set, at the first and last iterations.",
"It shows that the Mirrortree-IPA performs better than the DCA-IPA at the first iteration, especially as mutation rates become larger (Fig.",
"REF A and C).",
"Recall that at the first iteration, pairing scores are calculated on random within-species pairings, where most pairs (75% on average for species with 4 pairs each) are incorrect.",
"Taken together with our earlier observation that Mirrortree outperforms DCA for small training sets (see Fig.",
"REF A), it means that DCA requires a substantial and accurate training set to properly learn correlations and reach good performance, as is already known in the case of real protein sequences [5], [6], [7].",
"Nevertheless, the IPA allows DCA to robustly reach high predictive power at the last iteration over a broad range of values of $\\mu $ and $L$ (Fig.",
"REF B).",
"This range is almost as large as for the Mirrortree-IPA (Fig.",
"REF D), despite the initial lower performance of DCA.",
"At the last iteration, there is only a rather narrow band, close to the transition line from large to small TP fractions, where the DCA-IPA is outperformed by the Mirrortree-IPA.",
"Note that the parameters employed in Fig.",
"REF A are in this region.",
"Importantly, the parameter range where the DCA-IPA and the Mirrortree-IPA perform well without any training set is very similar to that obtained in Fig.",
"REF for DCA and Mirrortree predictions from a large training set.",
"This result illustrates the power of the iterative approach, which truly allows to bypass the need for a training set.",
"In Fig.",
"REF B, we further investigate the impact of the number of pairs AB per species on performance of the DCA-IPA and of the Mirrortree-IPA without any training set.",
"Very good performance is obtained for species comprising up to 8 pairs AB, and then we observe a decay, which is steeper than with a large training set (Fig.",
"REF ).",
"We further observe that the DCA-IPA reaches higher performance than the Mirrortree-IPA for small numbers of pairs per species, while the opposite is true for larger numbers of pairs per species.",
"Again, the Mirrortree-IPA performs better than the DCA-IPA at the first iteration, and this might explain the decreased final performance of the DCA-IPA for large numbers of pairs per species: confronted with many pairing possibilities, the DCA-IPA may not be able to recover from the large amount of noise in the initial matches.",
"An interesting question is whether one can improve predictive power by combining DCA and Mirrotree approaches.",
"It is all the more attractive that the predictions made using the two scores are almost independent (Fig.",
"REF A), in contrast to those made using DCA and MI [41] (Fig.",
"REF B).",
"Because Mirrortree performs better than DCA at the first iteration while DCA becomes better for larger and more accurate training sets, we devised a version of the IPA that uses the Mirrortree pairing score for the first half of iterations and then switches to the DCA pairing score.",
"The final TP fractions obtained with this Switch-IPA are shown in Fig.",
"REF B.",
"We find that it performs as well as the best among the Mirrortree-IPA and the DCA-IPA, which should make it more broadly applicable.",
"However, it does not yield further improvements.",
"So far, we have considered a minimal model for species, where chains are randomly grouped into sets of equal size $m$ , each representing a species.",
"Such a model would be realistic in the case where exchange between species (i.e.",
"horizontal gene transfer) is very frequent.",
"But in other evolutionary regimes, different correlations are expected between the chains present in a given species (paralogs) and between the most closely related chains across species (orthologs), due to the fact that species are evolutionary units.",
"In practice, paralogous and orthologous pairs respectively arise from duplication and speciation events.",
"In a duplication event, a chain from a given species gives rise to 2 paralogous chains within this species.",
"Note that loss events can also occur, thus decreasing the number of chains within a species.",
"In a speciation event, all chains are duplicated to give rise to 2 distinct species.",
"In order to assess the robustness of our results to these effects, we now consider a phylogeny model that explicitly accounts for duplication-loss and speciation events, without any exchange within species (see Fig.",
"REF A).",
"For simplicity, we assume that duplication and loss always happen together, so that the number of pairs per species remains constant.",
"Fig.",
"REF B shows the performance of DCA and Mirrortree scores at predicting pairs of evolutionary partners from a substantial training set, versus the fraction of species that undergo a duplication-loss event upon speciation.",
"Overall, performance is good, but it decreases when the frequency of duplication-loss events increases.",
"When there are no duplication-losses, this model features $m$ distinct phylogenies ($m=4$ in Fig.",
"REF ), one for each ancestral chain, and similarities between the chains of the testing set and of the training set that belong to the same phylogeny allow to predict evolutionary partners.",
"Conversely, when there are many duplication-losses, chains from one single phylogeny will end up fixing in each species, analogously to asexual birth-death population genetics models at fixed population size where all individuals are descended from a single ancestor after a sufficient time [60].",
"Moreover, chains resulting from recent duplication events will be very hard to distinguish, resulting in pairing ambiguities.",
"Fig.",
"REF B also shows that DCA outperforms Mirrortree, consistently with other cases with a substantial training set.",
"Finally, in Fig.",
"REF C, we consider the case without a training set.",
"We find that the first iteration of the Mirrortree-IPA performs much better than that of the DCA-IPA, while things are more even after the iterative process, consistently with our previous results.",
"Furthermore, Fig.",
"REF C shows that in the absence of duplication-loss, final performance is close to the random expectation of 0.25, which stands in contrast with the case with a training set.",
"This is because in the absence of a training set, incorrect pairs comprising A and B chains from different phylogenies cannot be distinguished from those coming from the same phylogeny.",
"Duplication-loss events allow to break this symmetry thanks to the fixation process whereby possible cross-phylogeny pairs become rare within each species.",
"However, when duplication-loss events are too frequent, performance decays again, for the same reason as with a training set: pairs resulting from recent duplications are hard to distinguish.",
"This tradeoff yields an optimal performance for an intermediate frequency of duplication-loss events.",
"Overall, our DCA and Mirrortree-based pairing predictions are robust to modifications of the phylogeny used to generate the data, as long as evolutionary correlations exist.",
"Since the various methods presented here allow to reliably pair synthetic chains on the basis of shared evolutionary history only, it should also be the case for real proteins.",
"While it is difficult to be certain that two real protein families share evolutionary history but do not bear common functional evolutionary constraints, we chose two pairs of protein families that are generally encoded in close proximity on prokaryotic genomes but that do not have known direct physical interactions [61], namely the Escherichia coli protein pairs LOLC-MACA and ACRE-ENVR and their homologs.",
"(Note that ENVR has regulatory roles on ACRE expression [62].)",
"Indeed, we expect proteins encoded close to one another to share common evolutionary history, because they tend to be horizontally transferred together, and to have similar levels of expression and evolutionary rates [36], [40].",
"Fig.",
"REF A and B shows that the DCA-, MI- and Mirrortree-IPA are all able to reliably pair LOLC-MACA and ACRE-ENVR homologs that are encoded close to one another on genomes, despite the absence of known direct physical interactions between these protein families.",
"We further compared these results to three pairs of protein families with known direct physical interactions [41].",
"Fig.",
"REF C, D and E shows that the Mirrortree-IPA performs less well than the DCA and MI-based versions in these cases, especially for the dataset of histidine kinases and response regulators (Fig.",
"REF C).",
"These proteins feature a strongly coevolving interaction interface [63], diversified across many paralogs per species (average number of pairs per species in our dataset: $\\left<m\\right>=11$ ) to avoid unwanted crosstalk.",
"This argues in the favor of the DCA-IPA and the MI-IPA rather than Mirrortree-based methods in order to predict partnership among physically interacting partners.",
"Interestingly, the MI-IPA slightly outperforms the DCA-IPA for all the real datasets in Fig.",
"REF , while this is not the case for synthetic data comprising only phylogenetic correlations (see Fig.",
"REF ).",
"Figure: Pairing predictions for real pairs of protein families.",
"The final fraction of protein pairs correctly predicted (TP fraction) obtained without a training set by the MI-IPA, the DCA-IPA and the Mirrortree-IPA is shown versus the increment step N increment N_{\\mathrm {increment}} of the iterative process.",
"A, B: Pairs of protein families with no known direct physical interaction but that are encoded closely on the genome; datasets include ∼2000\\sim 2000 homologous pairs.",
"C, D, E: Pairs of protein families with known direct physical interactions; for large families (C and D), datasets of ∼5000\\sim 5000 pairs comprising only full species were extracted from the larger complete datasets, while in E, the full dataset of ∼2000\\sim 2000 pairs was used.",
"In all panels, the mean number 〈m〉\\langle m\\rangle of pairs per species is indicated, and yellow lines represent the average TP fraction obtained for random within-species pairings.",
"All results are averaged over 50 replicates that differ in their initial random within-species pairings.",
"Datasets were constructed as described in , , starting from the P2CS database , for histidine kinase-response regulator (HK-RR) dataset (C) and using a method adapted from that relies on finding homologs of Escherichia coli protein pairs in all other cases.We have shown that the IPA, which aims to find partners among the paralogs of two protein families, accurately identifies pairs of proteins even if they only share a common evolutionary history, in the absence of direct interactions.",
"In order to computationally predict protein-protein interactions from sequence data, it is in addition necessary to distinguish the pairs of protein families that physically interact from those that only share evolutionary history.",
"First, interacting protein families often share more evolutionary history than non-interacting ones.",
"This idea is at the heart of the Mirrortree approach [31], [34], [35], which is usually implemented on alignments of orthologs, thus allowing to predict which protein families interact, but not addressing the paralog pairing problem.",
"We have applied the original Mirrortree [31], [34] and pMirrortree [35] methods to the pairs of protein families studied in Fig.",
"REF , focusing on the orthologs of the reference E. coli protein pairs (see Table REF ).",
"The two pairs without known physical interactions (LOLC-MACA and ACRE-ENVR) feature the smallest Mirrortree scores among the five pairs considered, and they also have non-significant pMirrortree scores, while two out of the three pairs with known direct physical interactions (HK-RR and XDHA-XDHC) possess significant pMirrortree scores.",
"These rather encouraging results confirm the power of the Mirrortree method, and support the hypothesis that our two pairs without known physical interactions are really non-interacting.",
"Another way to distinguish physically interacting pairs of protein families from non-interacting ones is to leverage the DCA scores of amino-acid pairs.",
"Indeed, strong DCA scores tend to correspond to contacting amino-acid pairs [5], [7], [6], and thus, their presence can reveal actual interacting partners [17], [21], [22].",
"Fig.",
"REF demonstrates that outliers in DCA scores [22] exist for the three pairs with known direct physical interactions studied in Fig.",
"REF (HK-RR, MALG-MALK and XDHA-XDHC), while they are absent for the two pairs without known physical interactions (LOLC-MACA and ACRE-ENVR).",
"Therefore, in these examples, outliers in DCA scores successfully allow to distinguish pairs of protein families that physically interact from those that only share a similar evolutionary history.",
"Importantly, the results on Fig.",
"REF were obtained on the pairs of protein sequences predicted by the IPA, which means that this method can be directly combined with the IPA.",
"Note that the absence of outliers in DCA scores in the pairs without known physical interactions hints that phylogenetic correlations result in multiple small DCA couplings, while direct physical interactions yield few large DCA couplings.",
"We also checked that our phylogeny-only synthetic data (with the same parameters as in Fig.",
"REF ) does not yield outliers in DCA scores.",
"Therefore, combining either traditional Mirrortree approaches or the study of outliers in DCA scores with the IPA can allow both to predict interacting pairs of protein families and to solve the paralog pairing problem, starting from two independent multiple sequence alignments of the protein families.",
"Figure: Outliers in DCA score provide evidence of protein-protein interactions.",
"DCA scores (APC-corrected Frobenius norms , ) were evaluated for each pair of amino-acid sites at the final IPA iteration.",
"Their averages over 500 IPA replicates that differ in their initial random pairings of sequences are shown ranked by decreasing value.",
"Datasets correspond to the same protein family pairs as in Fig.",
", and the IPA was run with N increment =50N_\\mathrm {increment} = 50.",
"Outliers in DCA score appear for all three pairs of protein families with known direct physical interactions (BASS-BASR, MALG-MALK, XDHA-XDHC), but not for the other ones (LOLC-MACA, ACRE-ENVR).Recently, methods relying on pairwise maximum entropy DCA models, originally employed to identify amino acids in contact in the three-dimensional structure of proteins and multi-protein complexes, have allowed to reliably predict interacting partners among the paralogs of several ubiquitous prokaryotic protein families, starting from sequences only [22], [23].",
"An important motivation for these methods is that the need to maintain the physico-chemical complementarity of contacting amino acids induces correlations in amino-acid usage between the sequences of interacting partners [63], [5].",
"However, correlations between the sequences of interacting partners can also arise from their shared evolutionary history [39].",
"In the present work, employing controlled synthetic data, we demonstrated that DCA is able to accurately identify partners that only share evolutionary history, in the absence of functional and structural constraints.",
"This result holds even in the absence of a training set, thanks to the Iterative Pairing Algorithm (IPA) [22].",
"Because our controlled synthetic data only comprises signal from phylogeny, we compared our DCA-based approach to methods that explicitly rely on phylogeny, through sequence similarity.",
"Specifically, we proposed a method based on the Mirrortree approach [31], [34], [35] to predict pairs among the paralogs of two protein families.",
"We obtained similar performances, with DCA slightly outperforming Mirrortree when a substantial correct training set is available.",
"We also considered a method based on Mutual Information (MI) [41], yielding similar performances for our synthetic data, with DCA often slightly outperforming MI, while MI tends to outperform DCA for natural sequence data [41].",
"The robustness of MI to finite dataset size effects [41] and its ability to quantify statistical dependence whatever its origin might make it most appropriate for complex natural data.",
"Finally, we applied the DCA-IPA, the MI-IPA and the Mirrortree-IPA to natural sequence data from several pairs of prokaryotic protein families, with or without known direct physical interactions, but always with genome proximity and thus significant shared evolutionary history.",
"We obtained accurate predictions for all these datasets.",
"Therefore, correlations from evolutionary history can play an important part in the performance of these algorithms in the case of natural sequence data.",
"Interestingly, we found that the Mirrortree-IPA performs significantly less well than the MI-IPA and DCA-IPA on the histidine kinase-response regulator dataset, which features large numbers of paralogs per species and is known to possess strongly coevolving contacts [63], [5], and to which DCA is thus particularly well-suited.",
"This points to the complementarity of these methods.",
"In addition, we showed that pairs of protein families with known direct physical interactions can be distinguished from those without known direct physical interactions, either by employing the original Mirrortree approach involving orthologs only [31], [34], [35], or by studying the amino-acid pairs that are outliers in DCA score [22].",
"Hence, combining these methods with the IPA allows to both predict interacting protein families and to find interacting partners among paralogs.",
"The ability of DCA to identify evolutionary partners in the absence of functional and structural constraints can be viewed as surprising.",
"Indeed, DCA models are mainly known for their ability to identify structural contacts, and they emphasize small-eigenvalue modes of the covariance matrix of sequences [66], [67], as illustrated by the inversion of the covariance matrix involved in the mean-field approximation (see Methods), while important signal from phylogeny lies in the large-eigenvalue modes of the covariance matrix [26], [27], [28].",
"In addition, phylogenetic correlations are often considered deleterious to structure prediction [3], [5], [6], [28], which is one of the major applications of DCA.",
"Nevertheless, a DCA model is fundamentally a global statistical model that aims to faithfully reproduce the empirical pairwise correlations observed in the data [44], [45].",
"As such, it should also encode phylogenetic correlations, thus rationalizing our result.",
"Our findings demonstrate that DCA models capture coevolution in a broad sense, not limited to contacting pairs of residues, but also including shared evolutionary history unrelated to functional constraints.",
"Importantly, while phylogenetic correlations are often viewed as a confounding factor for structure prediction [3], [5], [6], [28], our work shows that they are actually useful in order to address the paralog pairing problem by DCA.",
"This is consistent with previous work directly exploiting phylogenetic correlations to predict protein-protein interactions [29], [30], [32], [34], [35].",
"Interesting further directions include analyzing the contributions of phylogeny topology and evolutionary rates [36], [37], [38], [39], [40] in the phylogenetic signal that is useful for DCA-based pairing predictions, and addressing how signals from phylogeny and from amino-acid contacts combine together.",
"Here, our natural data examples were pairs of proteins colocalized on genomes, allowing us to easily evaluate performance via the fraction of correctly predicted protein pairs.",
"However, computational interaction prediction and paralog pairing methods are particularly important in order to discover interaction partners that are not encoded in close genomic locations.",
"This corresponds to the general case in eukaryotes, which makes this problem extremely relevant.",
"In prokaryotes too, important interactions between very different cellular processes exist across operons [68].",
"It will be of great interest to apply the IPA to such cases.",
"One restrictive assumption we have made so far is that interactions are one-to-one, which is appropriate for strongly specific protein-protein interactions, but not for cases involving promiscuity, crosstalk, or non-interacting orphan proteins.",
"Hence, an important direction for future work is to generalize the IPA to allow for multiple partners.",
"AFB thanks Ned S. Wingreen and Yaakov Kleeorin for inspiring discussions.",
"The authors thank Institut de Biologie Paris-Seine (IBPS) at Sorbonne Université for funding via a Collaborative Grant (Action Incitative) to AFB and MW.",
"MW also acknowledges funding by the EU H2020 research and innovation program MSCA-RISE-2016 under grant agreement No.",
"734439 InferNet."
]
] | 1906.04266 |
[
[
"Structure of chaotic eigenstates and their entanglement entropy"
],
[
"Abstract We consider a chaotic many-body system (i.e., one that satisfies the eigenstate thermalization hypothesis) that is split into two subsystems, with an interaction along their mutual boundary, and study the entanglement properties of an energy eigenstate with nonzero energy density.",
"When the two subsystems have nearly equal volumes, we find a universal correction to the entanglement entropy that is proportional to the square root of the system's heat capacity (or a sum of capacities, if there are conserved quantities in addition to energy).",
"This phenomenon was first noted by Vidmar and Rigol in a specific system; our analysis shows that it is generic, and expresses it in terms of thermodynamic properties of the system.",
"Our conclusions are based on a refined version of a model of a chaotic eigenstate originally due to Deutsch, and analyzed more recently by Lu and Grover."
],
[
"Introduction",
"Consider a macroscopic system of volume $V$ partitioned into two spatial subsystems 1 and 2 with volumes $V_1$ and $V_2 = V - V_1$ .",
"We assume, without loss of generality, that $V_1 \\le V_2$ .",
"We also assume that the hamiltonian of the system is a sum of local terms, and so can be partitioned as $H = H_1 + H_2 + H_{12},$ where $H_a$ acts nontrivially only in region $a$ ($a=1,2$ ) and all terms coupling the two subsystems are contained in $H_{12}$ .",
"We further assume that the system obeys the eigenstate thermalization hypothesis (ETH) [1], [2], [3] for matrix elements of local observables between energy eigenstates corresponding to nonzero energy densities.",
"To simplify notation, we take all energies to be in units of a fundamental energy scale (e.g., the coefficient of an exchange term in a spin chain) and all lengths, areas, and volumes to be in units of a fundamental length (e.g., the lattice spacing).",
"We also set $k_B = 1$ throughout.",
"Let ${E}$ denote an eigenstate of $H$ with energy $E$ , with nonzero energy density $E/V$ .",
"For notational convenience, and again without loss of generality, we shift $H_{12}$ by a constant (if necessary) so that $\\langle E | H_{12} | E \\rangle = 0.$ We can write ${E}$ in a basis of tensor products of the eigenstates of $H_1$ and $H_2$ , ${E} = \\sum _{i,J} M_{iJ} {i}_1\\otimes {J}_2.$ Deutsch [4] conjectured that the coefficient matrix $M_{iJ}$ can be treated as a random matrix with a narrow bandwidth that keeps the sum of the subsystem energies $E_{1i}+E_{2J}$ close to the total system energy $E$ , and using this conjecture showed that the entanglement entropy of the smaller subsystem equals its thermodynamic entropy.",
"More recently, Lu and Grover [5] used this ansatz to calculate the Rényi entropies of the subsystem.",
"Other related work on entanglement entropy at nonzero energy density in chaotic systems includes Refs.",
"[6], [7], [8], [9], [10]; for a review of basic concepts, see Ref. [11].",
"In this work, we refine the original conjecture by characterizing the coefficient matrix more completely.",
"We further show that at or very near $V_1=V_2$ , there is an extra contribution to the entanglement entropy that scales like $\\sqrt{V}$ .",
"Specifically, for $V_1=V_2$ exactly, we find that the entanglement entropy is given by $S_{\\text{ent}} = \\frac{1}{2} S - \\sqrt{\\frac{C}{2\\pi }} +O(A),$ where $S$ is the thermodynamic entropy of the system at energy $E$ , $C$ is its heat capacity, and $A$ is the area of the boundary between the two subsystems.",
"When the system is far from a critical point (which we assume for simplicity), both $S$ and $C$ typically scale like the volume $V$ of the system.",
"We do not compute the coefficient of the $O(A)$ term, since it depends on details of the hamiltonian.",
"The $\\sqrt{C}$ and $O(A)$ terms are distinguished by their scaling with system size (except in $d=2$ spatial dimensions).",
"They are also distinguished by the fact that the latter depends on a property of the boundary between the two subsystems, while the former depends on a property of the system as a whole.",
"The extension of Eq.",
"(REF ) to $V_1 \\ne V_2$ is given in Eq.",
"(REF ) below; the $\\sqrt{C}$ correction remains significant for $| V_1 - V_2 | \\lesssim \\sqrt{V}$ .",
"A contribution to $S_{\\text{ent}}$ scaling like $\\sqrt{V}$ was found previously by Vidmar and Rigol [12] in a study of a one-dimensional system with one conserved quantum number.",
"Our explanation for the appearance of such a term is essentially the same as theirs, but our formula applies more generally to any system that obeys ETH, and relates the correction to thermodynamic properties of the system.",
"Furthermore, we generalize our result to systems with any finite number of conserved quantities in addition to energy.",
"In such cases, $C$ in Eq.",
"(REF ) becomes the sum of all entries in a matrix of capacities; see Sec. .",
"The rest of this paper is organized as follows.",
"In the rest of the Introduction, we summarize all of our key results in more precise language, for the simplest case in which only energy is conserved.",
"Sections – elaborate on the derivation of the summarized results.",
"The generalization to systems with additional conserved quantities is discussed in the second half of Sec.",
", and full details are provided in the [app:cons]Appendix.",
"Section has our concluding discussion.",
"$\\bullet $ Structure of the coefficient matrix.",
"Assuming, in line with Refs.",
"[4], [5], that $M_{iJ}$ has the general structure of a random matrix that is sharply banded in total energy, and neglecting any dependence of $M_{iJ}$ on the energy difference $E_{1i} - E_{2J}$ , we show that it takes the form $M_{iJ} = e^{-S(E_{1i} + E_{2J})/2} F(E_{1i} + E_{2J} - E)^{1/2} \\, C_{iJ} ,$ where $S(E)$ is the thermodynamic entropy of the full system at energy $E$ (equal to the logarithm of the density of states, and assumed to be a monotonically increasing function of energy, so that temperature is nonnegative), $F(\\varepsilon )$ is a window function centered on $\\varepsilon =0$ with a width $\\Delta $ equal to the quantum uncertainty in the interaction hamiltonian, $\\Delta = \\sqrt{\\langle E | H_{12}^2 | E \\rangle } ,$ and $C_{iJ}$ is a matrix of coefficients which, when averaged over narrow bands of energies of each subsystem near $E_1$ and $E_2$ (but with each band still containing many subsystem energy eigenstates), obeys $\\overline{C_{iJ}}=0, \\quad \\ \\overline{C^*_{iJ}C_{i^{\\prime } J^{\\prime }}}=\\delta _{ii^{\\prime }}\\delta _{JJ^{\\prime }},$ where the overbar denotes the dual narrow-band energy averaging.",
"Furthermore, for a system in two or more spatial dimensions, the window function is a gaussian, $F(\\varepsilon ) = \\frac{e^{-\\varepsilon ^2\\!/2\\Delta ^2}}{\\sqrt{2\\pi } \\Delta } .$ In two or more spatial dimensions, where $H_{12}$ is a sum of local terms along the boundary between the two subsystems, we show that $\\Delta \\sim \\sqrt{A}$ , where $A$ is the area of the boundary.",
"For a one-dimensional system, $\\Delta $ is an order-one quantity (in terms of its scaling with system size).",
"$\\bullet $ Structure of the reduced density matrix.",
"The reduced density matrix $\\rho _1 _2 |E\\rangle \\!\\langle E|$ of subsystem 1 takes the form (1)ij = e-S(E)+S2(E-E1) [ ij + e-S2(E-E1)/2 e-2/82 Rij ] , where $E_1 (E_{1i} + E_{1j})/2$ and $\\omega E_{1i} - E_{1j}$ , $S_a(E_a)$ is the thermodynamic entropy of subsystem $a$ at energy $E_a$ ($a=1,2$ ), and the $R_{ij}$ are $O(1)$ numbers that vary erratically.",
"We have dropped terms of order $\\Delta ^2\\sim A$ and smaller in the exponents.",
"The diagonal term is in agreement with Lu and Grover [5], and matches the “subsystem ETH” ansatz of Dymarsky et al. [13].",
"The off-diagonal term, though exponentially smaller than the diagonal term, alters the spectrum of eigenvalues of $\\rho _1$ at energies $E_1>E_1^*$ , where $E_1^*$ is the solution to $S_1(E_1^*) = S_2(E-E_1^*).$ For $E_1>E_1^*$ , the density of states of subsystem 2 is smaller than the density of states of subsystem 1.",
"However, the nonzero eigenvalues of $\\rho _1$ are the same as those of $\\rho _2 _1 |E\\rangle \\!\\langle E|$ .",
"This effect occurs locally in energy.",
"Hence, in the energy interval $[E_1, E_1 + dE_1]$ for $E_1>E^*_1$ , $\\rho _1$ has approximately $e^{S_2(E-E_1)} dE_1$ nonzero eigenvalues, and each of these nonzero eigenvalues is approximately equal to $e^{-S(E)} e^{S_1(E_1)}$ .",
"$\\bullet $ Correction to the entanglement entropy.",
"From the discussion above, it follows that $\\rho _1^n = \\frac{\\int dE_1 \\, e^{S_{\\text{min}}(E_1)} \\bigl [e^{-S(E)} e^{S_{\\text{max}}(E_1)} \\bigr ]^n}{e^{-S(E)} \\int dE_1 \\,e^{S_{\\text{min}}(E_1) +S_{\\text{max}}(E_1)}},$ where Smax(E1) [ S1(E1), S2(E-E1) ], Smin(E1) [ S1(E1), S2(E-E1) ] .",
"The denominator in Eq.",
"(REF ) is the numerator with $n=1$ , and itself equals one up to small corrections; see Sec. .",
"The entanglement entropy is the $n\\rightarrow 1$ limit of the $n$ th Rényi entropy, $S_{\\text{ent}}(E) = \\lim _{n\\rightarrow 1}S_{\\text{Ren},n}(E),$ where $S_{\\text{Ren},n}(E) \\frac{1}{1-n} \\log \\rho _1^n .$ From Eqs.",
"(REF )–(REF ), we get $S_{\\text{ent}}(E) = \\frac{\\int {E_1} e^{S_1(E_1) + S_2(E-E_1)} [S(E) - S_{\\text{max}}(E_1)]}{\\int {E_1} e^{S_1(E_1) + S_2(E-E_1)}} .$ After performing the integrals over $E_1$ by Laplace's method, we find $S_{\\text{ent}}(E) = \\min (\\bar{S}_1, \\bar{S}_2) - \\sqrt{\\frac{2K}{\\pi }} \\, \\Phi \\!\\left(\\frac{\\bar{S}_2 - \\bar{S}_1}{\\sqrt{8 K}}\\right)+O(A),$ where $\\bar{S}_1 S_1(\\bar{E}_1)$ and $\\bar{S}_2 S_2(E-\\bar{E}_1)$ are the subsystem entropies at the stationary point $\\bar{E}_1$ , given by $S^{\\prime }_1(\\bar{E}_1)=S^{\\prime }_2(E-\\bar{E}_1),$ $K C_1 C_2/(C_1+C_2)$ is the harmonic mean of the subsystem heat capacities $C_a -\\beta ^2/\\bar{S}^{\\prime \\prime }_a$ at constant volume and inverse temperature $\\beta S^{\\prime }_1(\\bar{E}_1)$ , and we have defined the function (x) -+dye-y2 (|y-x|-|x|) = ( x x - x ) + e-x2 , where $x$ is the error function; see Fig.",
"REF .",
"Since $\\Phi (x)$ decays to zero exponentially from $\\Phi (0)=1$ , this correction is negligible for $|\\bar{S}_2-\\bar{S}_1|\\gg \\sqrt{K}$ .",
"Figure: A plot of the function Φ(x)\\Phi (x) defined in Eq.",
"(), which parameterizes the correction to the entanglement entropy, Eq.",
"().For a uniform system with $V_1=fV$ , $V_2=(1-f)V$ , and $f \\le \\frac{1}{2}$ , we have $\\bar{S}_1=fS(E)$ , $\\bar{S}_2=(1-f)S(E)$ , $C_1=fC$ , $C_2=(1-f)C$ , and $K=f(1-f)C$ , where $C-\\beta ^2/S^{\\prime \\prime }(E)$ is the heat capacity of the full system.",
"The heat capacity $C$ scales like the volume of the system, so $|\\bar{S}_2-\\bar{S}_1| \\gg \\sqrt{K}$ is equivalent to $|\\frac{1}{2}-f| \\gg 1/\\sqrt{V}$ .",
"For $f=\\frac{1}{2}$ exactly, we recover Eq.",
"(REF ).",
"$\\bullet $ Correction to the Rényi entropy for ${n<1}$ .",
"Evaluating Eq.",
"(REF ) by Laplace's method, and then evaluating the leading terms in Eq.",
"(REF ), we find $S_{\\text{Ren},n}(E) = \\frac{\\left[S_1({\\cal E}_1) + n S_2(E-{\\cal E}_1) - n S(E)\\right]}{1-n},$ where ${\\cal E}_1 \\min (\\bar{E}_1,E^*_1),$ $E^*_1$ is the solution to Eq.",
"(REF ), and $\\bar{E}_1$ is the solution to $S^{\\prime }_1(\\bar{E}_1)=n S^{\\prime }_2(E-\\bar{E}_1).$ For $\\bar{E}_1< E^*_1$ , Eq.",
"(REF ) coincides with the result of Ref. [5].",
"For $n>1$ , the convexity of the entropy function (equivalently, positivity of the temperature and the heat capacity) guarantees that $\\bar{E}_1< E^*_1$ .",
"However, for $n<1$ , it is possible to have $E^*_1<\\bar{E}_1$ , and then Eq.",
"(REF ) differs from the result of Ref. [5].",
"In particular, for a uniform system split exactly in half, $E^*_1<\\bar{E}_1$ for all $n<1$ , and then $S_{\\text{Ren},n<1}(E) =S(E)/2$ , up to subleading corrections."
],
[
"Envelope function of the coefficient matrix\n",
"We first establish a useful identity.",
"In the limit of $\\Delta \\rightarrow 0$ , we can ignore the energy of the interaction.",
"Then we can compute the density of states $e^{S(E)}$ of the total system at energy $E$ by dividing the energy between the two subsystems, and taking the product of the number of states of each subsystem.",
"This yields $e^{S(E)} = \\int _0^E dE_1\\,e^{S_1(E_1)}e^{S_2(E-E_1)}.$ Note that Eq.",
"(REF ) shows that the denominator in Eq.",
"(REF ) equals one in the $\\Delta \\rightarrow 0$ limit.",
"Next we warm up by computing $\\langle E|E\\rangle =1$ .",
"From Eqs.",
"(REF ) and (REF ), we have $\\langle E|E\\rangle = \\sum _{iJ} e^{-S(E_{1i} + E_{2J})} F(E_{1i} + E_{2J} - E) \\, |C_{iJ}|^2.$ The sums over $i$ and $J$ implement the narrow-band averaging of Eq.",
"(REF ), and can then be replaced by integrals over $E_1$ and $E_2$ with factors of the densities of states, yielding E|E= 0dE1 eS1(E1)0dE2 eS2(E2) e-S(E1 + E2) F(E1 + E2 - E) .",
"In the limit $\\Delta \\rightarrow 0$ , $F(\\varepsilon )\\rightarrow \\delta (\\varepsilon )$ , the Dirac delta function.",
"In this limit we have E|E= e-S(E)0dE1 eS1(E1)eS2(E-E1) =1, where the final result follows from Eq.",
"(REF ).",
"For finite $\\Delta $ , we take $F(\\varepsilon )$ to have the gaussian form of Eq.",
"(REF ), although we only need that $F(\\varepsilon )$ be sharply peaked at $\\varepsilon =0$ with width $\\Delta $ .",
"We then evaluate the integrals in Eq.",
"() by Laplace's method.",
"The conditions for a stationary point of the exponent are S1'(E1) = S'(E1 + E2) + (E1 + E2 - E)/2 , S2'(E2) = S'(E1 + E2) + (E1 + E2 - E)/2 .",
"For small $\\Delta $ , the solution is $E_1=\\bar{E}_1$ , $E_2=\\bar{E}_2$ , where E1+E2 = E, S1'(E1)=S2'(E2) = S'(E) , where $\\beta $ is again the inverse temperature of the system as a whole and of each subsystem.",
"Next we Taylor-expand the entropies about the stationary point, S(E1 + E2) = S(E) + (E1 + E2 - E) - 12 (2/C) (E1 + E2 - E)2 + , S1(E1) = S1 + (E1 - E1) - 12 (2/C1) (E1 - E1)2 + , S2(E2) = S2 + (E2 - E2) - 12 (2/C2) (E2 - E2)2 + .",
"To leading order in the system volume, Eq.",
"(REF ) implies $\\bar{S}_1+\\bar{S}_2 = S(E).$ Thus the constant and linear terms all cancel in the combination $S_1(E_1)+S_2(E_2)-S(E_1+E_2)$ that appears in Eq. ().",
"This cancellation is why it was necessary to have $S(E_1+E_2)$ in the exponent in Eq.",
"(REF ) rather than $S(E)$ .",
"The remaining quadratic terms yield gaussian integrals that give an $O(1)$ result for the integral in Eq. ().",
"Adjusting the $O(1)$ terms in the entropies is then necessary to yield the final result of $\\langle E|E\\rangle =1$ .",
"The heat capacities in Eqs.",
"()–() are proportional to the volumes of the corresponding macroscopic regions, whereas $\\Delta ^2 \\propto A$ (the area of the 1–2 boundary), as will be demonstrated below.",
"Consequently $1/\\Delta ^2 \\gg \\beta ^2/C, \\beta ^2/C_1, \\beta ^2/C_2$ , and hence the distribution of $E_1+E_2$ is controlled by $F(\\varepsilon )$ .",
"Then we have the following generalization of Eq.",
"(), $\\langle E|(H_1+H_2-E)^n|E\\rangle \\approx \\int _{-\\infty }^{+\\infty }d\\varepsilon \\, F(\\varepsilon )\\varepsilon ^n.$ We emphasize that the derivation of Eq.",
"(REF ) from Eqs.",
"(REF ) and (REF ) does not rely on the precise form of $F(\\varepsilon )$ ; it relies only on $F(\\varepsilon )$ being sharply peaked at $\\varepsilon = 0$ with width $\\Delta $ that satisfies $\\beta ^2 \\Delta ^2 \\ll C_1, C_2$ .",
"For $n=1,2$ in Eq.",
"(REF ), we can replace $H_1+H_2-E$ with $H_1+H_2-H$ , since $H$ will always appear next to either the ket or bra form of its eigenstate.",
"Then using $H_1+H_2-H=-H_{12}$ , we find for $n=1$ that $\\int _{-\\infty }^{+\\infty }d\\varepsilon \\,F(\\varepsilon )\\varepsilon = -\\langle E|H_{12}|E\\rangle = 0,$ where the second equality follows from our shift of $H_{12}$ .",
"For $n=2$ , we get $\\int _{-\\infty }^{+\\infty }d\\varepsilon \\,F(\\varepsilon )\\varepsilon ^2 = \\langle E|H_{12}^2|E\\rangle .$ The left-hand side equals $\\Delta ^2$ by definition, and so Eq.",
"(REF ) verifies Eq.",
"(REF ).",
"In two or more spatial dimensions, our assumption on the locality of $H$ implies that $H_{12}$ is a sum of local terms on the boundary $B$ between regions 1 and 2, $H_{12} = \\sum _{x\\in B} h_x .$ We then have $\\langle E|H_{12}^2|E\\rangle = \\sum _{x,y \\in B } \\langle E|h_x h_y|E\\rangle .$ Assuming that ETH holds for the bilocal operator $h_x h_y$ , the eigenstate expectation value can be replaced by a thermal expectation value at inverse temperature $\\beta $ .",
"We further assume that this thermal correlation function decays rapidly for $|x-y|\\gg \\xi $ to the disconnected form $\\langle h_x\\rangle \\langle h_y\\rangle $ , where $\\xi $ is an appropriate correlation length Note that the eigenstate expectation value $\\langle E | h_x h_y | E \\rangle $ must in general differ from the thermal expectation value $\\langle h_x h_y \\rangle $ by $O(1/V)$ , even when $|x-y| \\gg \\xi $ ; this is needed to recover $\\langle E | (H - E)^2 | E \\rangle = 0$ .",
"However, this difference only contributes an $O(A/V)$ correction to Eq.",
"(REF ), and hence can be neglected.. Summing the disconnected form over $x$ and/or $y$ yields zero, by Eq.",
"(REF ).",
"Hence the double sum in Eq.",
"(REF ) effectively becomes a single sum over the boundary, yielding $\\Delta ^2 = \\langle E|H_{12}^2|E\\rangle \\sim A \\, \\xi ^{d-1} \\langle h_x^2\\rangle ,$ where $h_x$ is any one term in $H_{12}$ , and the angle brackets denote either the eigenstate or thermal average, which are equal by ETH.",
"Equation (REF ) shows that $\\Delta ^2\\sim A$ , the boundary area.",
"We can now generalize this argument to higher powers of $H_{12}$ , again assuming rapid decay of $\\langle h_x h_y \\cdots \\rangle $ whenever an index or group of indices is separated by more than $\\xi $ from the others.",
"The multiple sum over $x,y,\\ldots $ will then yield approximately zero for odd powers, and be dominated by the factorization into correlated pairs for even powers More precisely, the third and higher cumulants of $H_{12}/\\Delta $ in the state $| E \\rangle $ are suppressed relative to the variance, $\\langle E | (H_{12}/\\Delta )^2 | E \\rangle \\equiv 1$ , by powers of $1/\\sqrt{A}$ ..",
"This then yields, in accord with the usual combinatorics of Wick's theorem, $\\langle E|H_{12}^{2n}|E\\rangle \\approx (2n-1)!!",
"\\,\\Delta ^{2n},$ characteristic of a gaussian distribution.",
"Returning to Eq.",
"(REF ), and using $H_1+H_2-E=H-E-H_{12},$ we have -+ d F()2n (H - E - H12)2n H122n - H12 (H-E) H122n-2 + , The first term is given by Eq.",
"(REF ), and we would like to show that the remaining terms can be neglected.",
"From Eqs.",
"(REF ) and (REF ), we see that we effectively have $H_{12}\\sim \\sqrt{A}$ , so the terms in Eq.",
"() with factors of $H-E$ will be suppressed unless $H-E\\sim \\sqrt{A}$ as well.",
"In each of these terms, $H$ acts on a state of the form $H_{12}^k|E\\rangle $ .",
"We have $H_{12}=\\sum _x h_x$ , and each $h_x$ is an $O(1)$ operator that can change the energy only by an $O(1)$ amount.",
"Hence, acting with $k$ such operators can change the energy by at most an $O(k)$ amount, which is $O(1)$ in terms of its scaling with $A$ .",
"Summing over $x$ can increase the coefficient of the normalized state, but does not increase the maximum change in energy.",
"Hence $H-E \\sim O(1)$ , and so the terms with one or more factors of $H-E$ in Eq.",
"() can be neglected.",
"We conclude that, up to corrections suppressed by powers of $\\xi ^{d-1}/A$ or $A/V$ , $\\int _{-\\infty }^{+\\infty } d\\varepsilon \\, F(\\varepsilon )\\varepsilon ^{2n} = (2n-1)!!",
"\\,\\Delta ^{2n},$ and therefore that $F(\\varepsilon )$ is a gaussian with width $\\Delta $ , Eq.",
"(REF ).",
"In one spatial dimension, $H_{12}$ is a single term, rather than a sum of $O(A)$ terms.",
"Hence the combinatoric analysis that led to Eq.",
"(REF ) does not apply, and so we cannot conclude that the shape of $F(\\varepsilon )$ is gaussian.",
"However, Eqs.",
"(REF ) and (REF ) are still valid, and so $F(\\varepsilon )$ is still sharply peaked at $\\varepsilon =0$ with a width $\\Delta $ that is given by Eq.",
"(REF ).",
"All of our results for the corrections to the entanglement entropy, including Eqs.",
"(REF ) and (REF ), and their generalizations to multiple conserved quantities via Eqs.",
"()–(REF ) below, only depend on this sharply-peaked nature of $F(\\varepsilon )$ , and not on the details of its shape, and so hold for all dimensions, including $d=1$ ."
],
[
"Reduced density matrix",
"From Eq.",
"(REF ), the reduced density matrix of subsystem 1 is $(\\rho _1)_{ij} = \\sum _K M_{iK}M^*_{jK}.$ Using Eqs.",
"(REF ), (REF ), and (), assuming $\\Delta ^2\\ll C/\\beta ^2$ , and neglecting prefactors, we have (1)ij = e-S(E) - 2/82 K [ e-(E2K + E1 - E) e-(E2K + E1 - E)2/22 CiK CjK* ], where $E_1 (E_{1i} + E_{1j})/2$ and $\\omega E_{1i} - E_{1j}$ .",
"Taking the statistical average and using Eq.",
"(REF ), only the diagonal term survives, hence $\\omega =0$ , and we get (1)ij = e-S(E) K [ e-(E2K + E1 - E) e-(E2K + E1 - E)2/22 ] ij.",
"We again replace the sum over $K$ with an integral over $E_2$ weighted by the density of states of subsystem 2, which yields (1)ij = e-S(E) 0dE2 eS2(E2) e-(E2 + E1 - E) e-(E2 + E1 - E)2/22 ij.",
"The last exponential factor, arising from the window function $F(\\varepsilon )$ , forces $E_2$ to be close to $E-E_1$ .",
"Expanding $S_2(E_2)$ about this point, we have $S_2(E_2) = S_2(E-E_1) +\\beta _{21}(E_2+E_1-E) + \\cdots ,$ where $\\beta _{21}S^{\\prime }_2(E-E_1)$ is the inverse temperature of subsystem 2 when its energy is $E-E_1$ .",
"Performing the integral over $E_2$ in Eq.",
"() then yields $\\overline{(\\rho _1)_{ij}} =e^{-S(E) +S_2(E-E_1)+\\Delta ^2(\\beta -\\beta _{21})^2/2} \\delta _{ij}.$ Since $\\Delta ^2\\sim A$ , the last term in the exponent is smaller than the first two, which scale like volume.",
"Additionally, we expect other terms of $O(A)$ to arise from finer structure in the $C_{iJ}$ coefficients that we have neglected.",
"These are necessary to produce the usual “area law” for the entanglement entropy of the ground state (for a review, see Ref.",
"[16]), and we expect such correlations to persist at nonzero energy density.",
"To estimate the size of the fluctuating off-diagonal elements of $\\rho _1$ , we compute the statistical average of the absolute square of $(\\rho _1)_{ij}$ , $i\\ne j$ .",
"We neglect any statistical correlations in the $C_{iK}$ coefficients, and assume that $\\overline{C_{iK}C^*_{jK}C^*_{iL}C_{jL}} =\\delta _{KL}.$ Then we have |(1)ij|2 = e-2S(E) - 2/42 K [ e-2(E2K + E1 - E) e-(E2K + E1 - E)2/2 ].",
"Following the same steps that led to Eq.",
"(REF ), we get $\\overline{|(\\rho _1)_{ij}|^2} =e^{-2S(E) +S_2(E-E_1) - \\omega ^2\\!/4\\Delta ^2 + \\Delta ^2(\\beta -\\beta _{21}/2)^2} .$ As in the case of the diagonal components, we expect additional terms of $O(A)$ to arise from neglected correlations in the $C_{iK}$ coefficients.",
"In the limit that we neglect $O(A)$ corrections, Eqs.",
"(REF ) and (REF ) together yield Eq.",
"()."
],
[
"Corrections to the entanglement and Rényi entropies",
"In the limit that we neglect all subleading corrections, we evaluate the numerator of Eq.",
"(REF ) by Laplace's method, which simply yields the maximum value of the integrand.",
"This gives Eq.",
"(REF ) for the $n$ th Rényi entropy.",
"We consider subleading corrections only in the case of the entanglement entropy, $n=1$ .",
"In this case we must evaluate Eq.",
"(REF ).",
"We have $S_{\\text{max}} = {\\textstyle \\frac{1}{2}} (S_1+S_2) + {\\textstyle \\frac{1}{2}} |S_1-S_2|.$ Using Eqs () and ()–(REF ), and changing the integration variable from $E_1$ to $u\\beta (E_1-\\bar{E}_1),$ we get S1+S2 = S - u2/2K, S1-S2 = S1-S2 + 2u +O(u2/K), where again $K:=C_1 C_2/(C_1+C_2)$ .",
"The factor of $e^{-u^2/2K}$ in the integrand is peaked well away from the lower limit of integration, which can therefore be extended to $-\\infty $ .",
"When performing the gaussian integral, values of $u^2$ larger than $K$ are exponentially suppressed; thus the $O(u^2/K)$ term in Eq.",
"() gives only an $O(1)$ contribution, and can be neglected.",
"Putting all of this together, Eq.",
"(REF ) becomes $S_{\\text{ent}} =\\tfrac{1}{2} S- \\frac{\\int _{-\\infty }^{+\\infty } {u} e^{-u^2/2K} {\\frac{1}{2}(\\bar{S}_2 - \\bar{S}_1) - u}}{\\int _{-\\infty }^{+\\infty } {u} e^{-u^2/2K}} .$ Making a final rescaling of $u\\rightarrow \\sqrt{2K}y$ , we get Eq.",
"(REF ).",
"Note that, if we are interested in infinite temperature ($\\beta = 0$ ), then we should also take $C_j \\rightarrow 0$ so that $\\beta ^2/C_j$ remains finite and nonzero.",
"In this limit, $C\\rightarrow 0$ , and so the correction in Eq.",
"(REF ) vanishes.",
"We also note that the precise distribution of eigenvalues of $\\rho _1$ near $E_1$ makes at most an $O(1)$ correction to the entanglement entropy.",
"For example, the Marčenko-Pastur law [17] gives the Page correction $-e^{S_{\\text{min}}}/2e^{S_{\\text{max}}}$ to the entanglement entropy of a random state [18], [19].",
"This can be neglected.",
"We can also generalize to the case of a system with another conserved quantity, such as particle number.",
"In the most general case, there are $m$ conserved quantities $Q^a$ ($a=1,\\ldots ,m$ ), including energy, which we take to be $Q^1$ .",
"A quantum state is then labeled by the values of all $m$ quantities.",
"We can then repeat our entire analysis (see the [app:cons]Appendix for details).",
"The thermodynamic entropy of the full system as a function of the $Q^a$ 's (near the values that label the state) takes the form S(Q+Q) = S(Q) + a Qa - 12(C-1)ab a Qa b Qb + , with $\\lambda ^1 \\equiv \\beta $ .",
"This generalizes Eq.",
"(); similar generalizations apply to the subsystem entropies.",
"We then ultimately arrive at Eq.",
"(REF ) with $u \\lambda ^a \\delta Q^a$ and K a,b=1m [ (C1-1 + C2-1)-1 ]ab = f(1-f) a,b=1m Cab , where $\\mathbf {C}_1$ and $\\mathbf {C}_2$ are the capacity matrices for the two subsystems, and the second equality holds for a uniform system with capacity matrix $\\mathbf {C}$ and with $f= V_1/V$ .",
"Equation (REF ) then holds with $K$ given by Eq.",
"(), and Eq.",
"(REF ) holds with $C = \\sum _{a,b=1}^m \\mathbf {C}_{ab} .$ We can now reproduce the results of Ref.",
"[12] for a system with a conserved particle number.",
"There the system was studied near infinite temperature, so that the thermodynamic entropy was taken to be effectively independent of system energy.",
"Hence the problem reduces to the case of a single conserved quantity, the filling fraction $n$ .",
"Then the thermodynamic entropy of the system takes the form $S(n) = -L \\, [n\\ln n + (1-n)\\ln (1-n)],$ where $L$ is the linear volume of the one-dimensional system; this is Eq.",
"(13) in Ref. [12].",
"In the notation of our Eq.",
"(), with a single $Q$ that we identify as $n$ , we have = S'(n), 2 C-1 = -S”(n), which yields $C = L \\, n(1-n)\\left[\\ln \\left(\\frac{1-n}{n}\\right)\\right]^2.$ When used in Eq.",
"(REF ), this reproduces Eq.",
"(17) of Ref.",
"[12] (with $L_A=L/2$ )."
],
[
"Conclusions",
"We have reconsidered the ansatz of Refs.",
"[4], [5] for an energy eigenstate of a chaotic many-body system that, by assumption, obeys the eigenstate thermalization hypothesis for local observables.",
"This ansatz expresses the energy eigenstate of the full system in the basis of energy eigenstates of two subsystems, each contiguous in space, that interact along their mutual boundary, and is specified by Eqs.",
"(REF ), (REF ), and (REF ).",
"One of the results of this paper is that the width $\\Delta $ of the energy window function $F(\\varepsilon )$ is given by Eq.",
"(REF ) in terms of the subsystem interaction hamiltonian, and that (in two or more spatial dimensions) $F(\\varepsilon )$ has the gaussian form of Eq.",
"(REF ).",
"We further showed that the ansatz for the energy eigenstate leads to a reduced density matrix that takes the form of Eq. ().",
"The off-diagonal elements, though exponentially small, are relevant to the calculation of Rényi entropies when the fraction of the energy in the smaller subsystem is large enough to give it a larger entropy than the larger subsystem; this modifies the results of Ref.",
"[5] for $n<1$ .",
"In the case of equal or nearly equal volume for the two subsystems, there is a correction to the entanglement entropy (corresponding to Rényi index $n=1$ ) that scales like the square-root of the system volume.",
"In the case of equal subsystem volumes, this correction, displayed in Eq.",
"(REF ), is $\\Delta S_\\text{ent}=-\\sqrt{C/2\\pi }$ , where $C$ is the heat capacity of the whole system.",
"Such a correction was previously found in a specific system by Vidmar and Rigol [12]; our analysis is more general and shows that the effect is generic.",
"We also extended our results to the case of multiple conserved quantities.",
"The correction to the entanglement entropy at equal subsystem volumes is the same, but with $C$ now given by a sum of the elements of a matrix of capacities.",
"We believe that our work further illuminates the role of the entanglement and Rényi entropies of a subsystem as quantities worthy of study that encode key features of the physical properties of the system as a whole.",
"We thank Eugenio Bianchi, Tarun Grover, Tsung-Cheng Lu, Marcos Rigol, and Lev Vidmar for helpful discussions.",
"This work was supported in part by the Microsoft Corporation Station Q (C.M.",
")."
],
[
"Appendix: Multiple conserved quantities",
"In the most general case, there are $m$ conserved quantities $Q^a$ ($a=1,\\ldots ,m$ ), including energy, which we take to be $Q^1$ .",
"We assume that each $Q^a$ is a sum of local terms, and so can be partitioned as in Eq.",
"(REF ), $Q^a = Q^a_1 + Q^a_2 + Q^a_{12} ,$ with QaQb = 0 , Qa1Qb1 = 0 , Qa2Qb2 = 0 .",
"Let ${q}$ denote a simultaneous eigenstate of the $Q^a$ , with eigenvalues $q^a$ .",
"Without loss of generality, we shift each $Q^a_{12}$ so that $\\langle q | Q^a_{12} | q \\rangle = 0.$ We can write ${q}$ in a basis of tensor products of the eigenstates of $Q^a_1$ and $Q^a_2$ , ${q} = \\sum _{i,J} M_{iJ} {i}_1\\otimes {J}_2,$ where $Q^a_1 {i}_1 = q^a_{1i} {i}_1$ and $Q^a_2 {J}_2 = q^a_{2J} {J}_2$ .",
"The thermodynamic entropy of the full system is now a function $S(q) \\equiv S(q^1,\\dots ,q^m)$ of all the $q^a$ 's.",
"Its Taylor expansion, about the values that label the state, takes the form S(q+q) = S(q) + a qa - 12 (C-1)ab a qa b qb + , with $\\lambda ^1 \\equiv \\beta $ .",
"This generalizes Eq. ().",
"We assume that $\\lambda ^a > 0$ and that the capacity matrix $\\mathbf {C}$ is positive definite; this generalizes positivity of temperature and heat capacity.",
"Similar generalizations apply to the subsystem entropies.",
"We can then repeat our entire analysis.",
"The coefficient matrix $M_{iJ}$ takes the form $M_{iJ} = e^{-S(q_{1i} + q_{2J})/2} F(q_{1i} + q_{2J} - q)^{1/2} \\, C_{iJ} .$ Here $F(z)$ is a window function centered on $z_a=0$ with second moments given by $\\int [m]{z} F(z) \\, z_a z_b = \\mathbf {D}_{ab} \\langle q | Q_{12}^a Q_{12}^b | q \\rangle .$ Equations (REF ) and (REF ) together imply $\\langle q | {Q^a_{12}}{Q^b_{12}} | q \\rangle = 0 ,$ so $\\mathbf {D}_{ab}$ is symmetric, as it needs to be for the equality in Eq.",
"(REF ) to make sense.",
"The $C_{iJ}$ coefficients obey Eq.",
"(REF ), with the averaging now over narrow bands of all components of $q_1$ and $q_2$ .",
"For a system in two or more spatial dimensions, the window function is a multivariate gaussian, $F(z) = \\frac{e^{- z \\cdot \\mathbf {D}^{-1} z/2}}{(2\\pi )^{m/2} \\sqrt{\\det \\mathbf {D}}} .$ The matrix elements of $\\mathbf {D}$ scale like $\\mathbf {D}_{ab} \\sim A$ , where $A$ is the area of the boundary between regions 1 and 2.",
"When $m = 1$ , $\\mathbf {D}$ reduces to $\\Delta ^2$ .",
"The reduced density matrix $\\rho _1 _2 |q\\rangle \\!\\langle q|$ of subsystem 1 takes the form (1)ij = e-S(q)+S2(q-q1) [ ij + e-S2(q-q1)/2 e- w D-1 w/8 Rij ] , where $q^a_1 (q^a_{1i} + q^a_{1j})/2$ and $w^a q^a_{1i} - q^a_{1j}$ , the $R_{ij}$ are $O(1)$ numbers that vary erratically, and we have dropped terms of order $\\mathbf {D}_{ab} \\sim A$ in the exponents.",
"We adopt the notation q1 q1* if S1(q1) < S2(q-q1) , q1 q1* if S1(q1) > S2(q-q1) , q1 q1* if S1(q1) = S2(q-q1) .",
"In other words, $\\succ $ is the order on $q_1$ induced by the function $S_1(q_1) - S_2(q-q_1)$ , and $q_1^*$ is some point at which this function vanishes.",
"The off-diagonal term in Eq.",
"(REF ), though exponentially smaller than the diagonal term, is relevant for $q_1 \\succ q_1^*$ .",
"In the small box $[q^1_1, q^1_1 + dq^1_1] \\times \\cdots \\times [q^m_1, q^m_1 + dq^m_1]$ for $q_1 \\succ q^*_1$ , $\\rho _1$ has approximately $e^{S_2(q-q_1)} dq^1_1 \\cdots dq^m_1$ nonzero eigenvalues, each one approximately equal to $e^{-S(E)} e^{S_1(q_1)}$ .",
"Equation (REF ) for the Rényi entropy generalizes to $S_{\\text{Ren},n}(q) = \\frac{\\left[S_1(\\mathcal {Q}_1) + n S_2(q-\\mathcal {Q}_1) - n S(q)\\right]}{1-n} ,$ where $\\mathcal {Q}_1$ is the point at which $S_1(q_1) + n S_2(q-q_1)$ attains its maximal value in the region $q_1 \\precsim q_1^*$ .",
"For $n>1$ , the convexity of the entropy function guarantees that $\\mathcal {Q}_1\\prec q^*_1$ .",
"In this case, $\\mathcal {Q}_1 = \\bar{q}_1$ , the solution to $\\nabla S_1(\\bar{q}_1) = n \\nabla S_2(q-\\bar{q}_1) .$ However, for $n<1$ , it is possible to have $\\mathcal {Q}_1 \\sim q_1^* \\prec \\bar{q}_1$ .",
"In particular, for a uniform system split exactly in half, $q^*_1\\prec \\bar{q}_1$ for all $n<1$ , and then $S_{\\text{Ren},n<1}(q) =S(q)/2$ , up to subleading corrections.",
"For the entanglement entropy, we repeat the steps in Sec.",
"and arrive at the generalization of Eq.",
"(REF ), $S_{\\text{ent}} =\\tfrac{1}{2} S- \\frac{\\int [m]{v} e^{- v \\cdot \\mathbf {K}^{-1} v/2} {\\frac{1}{2}(\\bar{S}_2 - \\bar{S}_1) - r \\cdot v}}{\\int [m]{v} e^{-v \\cdot \\mathbf {K}^{-1} v /2}} ,$ where $\\mathbf {K}^{-1} \\mathbf {C}_1^{-1} + \\mathbf {C}_2^{-1}$ , $\\mathbf {C}_1$ and $\\mathbf {C}_2$ are the capacity matrices for the two subsystems, $r (1,1,\\dots ,1)$ , $v^a \\lambda ^a (q^a_1 - \\bar{q}^a_1)$ , $\\bar{S}_1 S_1(\\bar{q}_1)$ , $\\bar{S}_2 S_2(q-\\bar{q}_1)$ , and $\\bar{q}_1$ is the solution to $\\nabla S_1(\\bar{q}_1) = \\nabla S_2(q-\\bar{q}_1) .$ The integral in Eq.",
"(REF ) is of the form I = [m]v e- v K-1 v/2 f(r v) = u [m]v e- v K-1 v/2 (u - r v) f(u) = u k2 [m]v e- v K-1 v/2 + ik(u - r v) f(u) .",
"Performing all the gaussian integrals, $I \\propto \\int {u} e^{-u^2 / (2 \\, r \\cdot \\mathbf {K} r)} f(u) .$ Thus, Eq.",
"(REF ) reduces to Eq.",
"(REF ) with $K = r \\cdot \\mathbf {K} \\, r= \\sum _{a,b=1}^m \\mathbf {K}_{ab} ,$ which is equivalent to Eq.",
"()."
]
] | 1906.04295 |
[
[
"Transport in disordered systems: the single big jump approach"
],
[
"Abstract In a growing number of strongly disordered and dense systems, the dynamics of a particle pulled by an external force field exhibits super-diffusion.",
"In the context of glass forming systems, super cooled glasses and contamination spreading in porous medium it was suggested to model this behavior with a biased continuous time random walk.",
"Here we analyze the plume of particles far lagging behind the mean, with the single big jump principle.",
"Revealing the mechanism of the anomaly, we show how a single trapping time, the largest one, is responsible for the rare fluctuations in the system.",
"These non typical fluctuations still control the behavior of the mean square displacement, which is the most basic quantifier of the dynamics in many experimental setups.",
"We show how the initial conditions, describing either stationary state or non-equilibrium case, persist for ever in the sense that the rare fluctuations are sensitive to the initial preparation.",
"To describe the fluctuations of the largest trapping time, we modify Fr\\'{e}chet's law from extreme value statistics, taking into consideration the fact that the large fluctuations are very different from those observed for independent and identically distributed random variables."
],
[
"Introduction",
"Diffusion and transport in a vast number of weakly disordered systems follows Gaussian statistics.",
"As a consequence, the packet of the spreading particles is symmetrically spread with respect to (w.r.t.)",
"the mean $\\langle x(t)\\rangle $ .",
"In contrast, for strongly disordered systems, the packet is found to be non-Gaussian and non-symmetric [1], [2].",
"Starting on $x=0$ , the slowest particles are trapped by the disorder, resulting in a plume of particles far lagging behind the mean $\\langle x(t)\\rangle $ , i.e., the fluctuations are large and break symmetry (see Fig.",
"REF ).",
"Deep energetic and entropic traps, which hinder the motion are expected to lead to a slow down of the diffusion.",
"The most frequently used quantifier of diffusion processes is clearly the mean square displacement (MSD).",
"However, in the presence of deep traps, the MSD exhibits super-diffusion.",
"This is not an indication for a fast process, instead it is due to the very slow particles far lagging behind the mean, which lead to very large fluctuations of displacements.",
"Thus slow dynamics of a minority of particles leads to enhanced fluctuations and symmetry breaking w.r.t.",
"$\\langle x(t)\\rangle $ .",
"Such processes are widespread, in particular many works focused on the surprising discovery of the super-diffusion in dense environments [3], [4], [5], [6], [7], [8].",
"This was originally investigated in the context of diffusion in disordered material [1], [9], [10], [3], [2], [11], [12], contamination spreading in porous medium [13], [14], [15], [16], simulation of biased particles in glass forming systems [4] and super cooled liquids [6], pulled by a constant force.",
"Figure: The density of positions of particles for an ordinary CTRW model.",
"The spreading packet is non-Gaussian.The left plume of particles is due to the long trapping times, which implies that some particles are moving by far slower if compared with the mean 〈x(t)〉\\langle x(t)\\rangle .",
"Somewhat paradoxically, these slow particles lead to super-diffusion as the MSD grows like t 3-α t^{3-\\alpha } .In this work we show how rare events in this process are determined by the largest trapping times.",
"In turn, it controls the behavior of the MSD.",
"The typical fluctuations are defined for x∼〈x(t)〉x\\sim \\langle x(t)\\rangle , i.e., close to the peak of the packet, while we focus on the rare fluctuations shown by the red solid line.",
"The parameters are a=5a=5, σ=1\\sigma =1, and α=1.5\\alpha =1.5; see Eqs.",
"() and ().Figure: Two trajectories of particles ending at small and large xx when t=1000t=1000.For the case where x(t=1000)x(t=1000) is near the original position, we see a very long waiting time, as the particle is trapped for a time of the order of tt.",
"In contrast, when x(t=1000)≃〈x(t)〉x(t=1000)\\simeq \\langle x(t)\\rangle , the trapping times are relatively short and comparable with each other.",
"The inset shows the trajectory of the particle at a short time.The parameters are the same as in Fig.",
".Here we investigate the spreading of the packet of particles, using the biased continuous time random walk (CTRW) [17], [18], [19].",
"Our goal is to characterize precisely the mechanism leading to the large fluctuations.",
"We promote the idea of the single big jump principle.",
"This means that one and only one trapping time is responsible for the rare fluctuations.",
"Thus in this work we show the relation between the theory of extreme value statistics and the anomalous transport.",
"For that we need to modify the well-known Fréchet law [20], [21] which describes extreme events for uncorrelated systems.",
"Similarly we present an analysis of the far tail of the spreading of the packet of particles, showing the deviations from the Lévy statistics describing the bulk statistics.",
"This is done for both non-stationary and equilibrium initial conditions.",
"While the typical fluctuations in our systems are not sensitive to the initial conditions, the rare fluctuations do, and this we believe is a general theme for systems with fat-tailed statistics.",
"We will relate the position of the random walker $x(t)$ and the longest trapping interval $\\tau _{\\max }$ .",
"The typical fluctuations of both observables were considered previously, and were shown to behave as if they are composed of independent and identically distributed (IID) events, namely the Lévy stable law and Fréchet's law hold for typical fluctuations (Eqs.",
"(REF ) and (REF ) below).",
"We show below how these laws must be corrected when dealing with the far tail.",
"In turn standard Cramer's theorem from large deviation theory [22], which identifies the large fluctuations with the accumulation of many small steps, fails in this case studied here.",
"More precisely, we claim below that one can obtain two limiting laws both for $x(t)$ and $\\tau _{\\max }$ , the first is the just mentioned Lévy, Fréchet laws and the second is an infinite density, i.e., a non-normalized state.",
"What is the principle of big jump?",
"Many works have focus on the dominance of one big jump in a stochastic process.",
"For example consider the activation process of a particle over a barrier, modeled with an over-damped Langevin equation.",
"If the noise is non-correlated and Gaussian, this escape is achieved by many small displacements, accumulating to give the rare escape from the well.",
"On the other hand, if the noise is of the Lévy type, one event giving rise to a large fluctuation dominates the escape [23].",
"Similar ideas hold for the analysis of random partition functions and were used in the study of the Sinai model [24], [25].",
"In the context of a run-and-tumble model and combination phenomenon these insights are well understood [26], [27].",
"Roughly speaking, one can see that the largest summand is of the order of the total sum, a theme which is already known.",
"To be more specific consider $N$ random variables $\\lbrace \\vartheta _1,\\vartheta _2,\\cdots ,\\vartheta _N\\rbrace $ .",
"Let $\\vartheta _{\\max }$ be the maximum of the set and $S_{N}=\\sum _{i=1}^N\\vartheta _i$ is the sum.",
"The dominance effect, found for example if $\\vartheta _i$ are IID random variables drawn from a fat-tailed distribution, is the claim that $S_N$ and $\\vartheta _{\\max }$ are of the same order [28].",
"More exactly, $S_N$ and $\\vartheta _{\\max }$ scale with $N$ the same way.",
"A more profound case is when the distribution of $\\vartheta _{\\max }$ is the same as that of $S_N$ , besides a trivial constant and in a limit to be specified later.",
"This is what we and others refer to as the principle of big jump.",
"This statement was shown to be valid for sub-exponential IID variables [28] and see also [29], [30].",
"In the IID case, the statement is valid for any $N$ , so the limit $N\\rightarrow \\infty $ is not at all required.",
"Here our aim is show how the big jump principle holds for diffusion in disorder systems using the CTRW model.",
"We will modify the principle to discuss the largest trapping time and its relation to the position of the random walker, so the principle discussed below is very different if compared to the original, in particular we depart form the IID case.",
"In [31], [32], we promoted a rate method to the big jump approach which was used to predict non-analytical behaviors of the far tail of Lévy walk process and the so called quenched Lévy-Lonentz gas model.",
"In these works, the very basic approach is different from what we have here; see Eq.",
"(REF ) below.",
"Further the connection to the modified Fréchet law, and the difference between stationary and non-equilibrium initial conditions are discussed here for the first time.",
"Figure: Illustrations of an ordinary CTRW (top) when the process begins at time t=0t=0 and x=0x=0, and an equilibrium CTRW (bottom).The bottom panel describes an ongoing equilibrium process, i.e., a stationary case were the dynamics started long before the start of the observation (see the blue dashed line for an illustration).The t i t_i corresponds to the time when the ii-th event occurs, and the backward recurrence time is B t =t-t 4 B_t=t-t_4.",
"The only difference between these two processes is the statistics of the waiting time of the first step.",
"However due to the disorder, in particular the power-law trapping time distribution, this difference crucially influences the rare events and also the behavior of the MSD.The organization of the paper is as follows.",
"In Sec.",
", we outline the single big principle and give the corresponding definitions.",
"Non-equilibrium and equilibrium initial conditions are investigated in Secs.",
"and , respectively.",
"Finally, we conclude the manuscript with a discussion.",
"We also present simulation results confirming the theoretical predications.",
"We consider two types of biased CTRWs [17], [33], [18], [34], [19], the first is initiated at time $t=0$ while the second is an equilibrium process.",
"These two models, differ in the first trapping time statistics, but otherwise they are identical.",
"Let $\\phi (\\tau )$ be the probability density function (PDF) of all the sojourn times while $h(\\tau )$ is the PDF of the first one.",
"It should be emphasized that the correct choice of $h(\\tau )$ depends on the initial conditions.",
"For the widely investigated non-equilibrium initial condition, we assign $h_{{\\rm or}}(\\tau )=\\phi (\\tau )$ [35].",
"This time process is sometimes called an ordinary renewal process, hence we use the subscript `or' to denote this type of initial condition.",
"While in equilibrium situation we use [36], [37], [38], [39] $h_{{\\rm eq}}(\\tau )=\\frac{\\int _\\tau ^\\infty \\phi (y)dy}{\\langle \\tau \\rangle },$ where $\\langle \\tau \\rangle =\\int _{0}^\\infty \\tau \\phi (\\tau )d\\tau $ is the mean trapping time.",
"We will soon explain the physical meaning of these processes.",
"We are interested in the position of the random walker $x(t)$ , which starts at $x=0$ when $t=0$ .",
"After waiting for time $\\tau _1$ , drawn from $h(\\tau )$ , the particle makes a spatial jump.",
"The PDF of jump size $\\chi $ , is Gaussian $f(\\chi )=\\frac{1}{\\sqrt{2\\sigma ^2\\pi }}\\exp \\left[-\\frac{(\\chi -a)^2}{2\\sigma ^2}\\right],$ where $a>0$ is the average size of the jumps.",
"Physically this is determined by an external constant force field that induces a net drift.",
"From Eq.",
"(REF ) the Fourier transform of $f(\\chi )$ is $\\widetilde{f}(k)=\\exp (ika-\\sigma ^2 k^2/2)$ .",
"This yields $\\widetilde{f}(k)\\sim 1+ika-\\frac{\\sigma ^2+a^2}{2}k^2$ with $k\\rightarrow 0$ .",
"After the jump, say to $x_1$ , the particle will pause for time $\\tau _2$ , whose statistical properties are drawn from $\\phi (\\tau )$ .",
"Then the process is renewed.",
"We consider the widely applicable case, where the PDF of trapping times is $\\phi (\\tau )=\\left\\lbrace \\begin{split}&0, & \\hbox{$\\tau <\\tau _0$;} \\\\&\\alpha \\frac{\\tau _0^\\alpha }{\\tau ^{1+\\alpha }}, & \\hbox{$\\tau \\ge \\tau _0$}\\end{split}\\right.$ with $1<\\alpha <2$ .",
"As well-known such a fat-tailed distribution yields a wide range of anomalous behaviors.",
"See [10], [17] for review on CTRW and further discussion on physical systems below.",
"From the Abelian theorem, the Laplace $\\tau \\rightarrow s$ transform of $\\phi (\\tau )$ is $\\widehat{\\phi }(s)\\sim 1-\\langle \\tau \\rangle s+ b_\\alpha s^\\alpha $ with $b_\\alpha =(\\tau _0)^\\alpha |\\Gamma (1-\\alpha )|$ and $s\\rightarrow 0$ .",
"The leading term is the normalization condition.",
"We focus on $1<\\alpha <2$ , where the mean $\\langle \\tau \\rangle $ of the waiting time is finite, but not the variance.",
"The term $s^\\alpha $ comes from the long tail of the waiting times (and it is responsible for the deviations from the normal behavior).",
"Specific values of $\\alpha $ for a range of physical systems and models are given in [10], [40].",
"For an equilibrium initial condition the rate of performing a jump is stationary in the sense that for any time $t$ the average number of jumps is $\\langle N(t)\\rangle =\\frac{t}{\\langle \\tau \\rangle },$ so the effective rate $1/\\langle \\tau \\rangle $ is a constant.",
"In contrast, for the ordinary renewal process we have in the long time limit $\\langle N(t)\\rangle \\sim t/\\langle \\tau \\rangle $ , hence for short times the two processes are not identical.",
"Since the mean $\\langle \\tau \\rangle $ is finite, one would expect naively that statistical laws for the two processes will be identical in the long time limit.",
"While this is correct for some observables, for others this is false.",
"The prominent example is the MSD.",
"In particular, for the calculation of the rare events one must make the distinction between the two models; see below.",
"Equilibrium initial condition is found when the particle is inserted in the medium long before the process begins.",
"More specifically when the process starts at some time $-t_a$ before the measurement begins at time $t=0$ , and in the limit $t_a\\rightarrow \\infty $ .",
"All along we consider the displacement of the particle compared to its initial position, namely we assign $x(0)=0$ .",
"For a schematic presentation of the random processes see Fig.",
"REF .",
"Non-equilibrium initial conditions are found when the processes are initiated at time $t=0$ .",
"For example in Scher Montroll theory [41], a flash of light excites charge carriers at time $t=0$ and then the process of diffusion begins, then we have an ordinary process.",
"Mathematically, these two models merely differ by the statistics of the waiting time of the first step, and hence it is interesting to compare them, to see if this seemingly small modification of the model is important or not in the long time limit.",
"For a Poisson process the two models are identical.",
"In contrast, for heavy-tailed processes under investigation, we find from Eqs.",
"(REF ) and (REF ) that $h_{{\\rm eq}}(\\tau )\\sim \\frac{(\\tau _0)^\\alpha }{\\langle \\tau \\rangle }\\tau ^{-\\alpha }.$ As $1<\\alpha <2$ we see that the average time for the first waiting time diverges (but not for the second etc).",
"This means that in a stationary state the process is slower if compared to the ordinary case, hence we expect that in this case particles will be lagging even more behind the mean displacement.",
"Let us discuss the applicability of the CTRW model.",
"As mentioned Scher and Montroll showed how this theory describes diffusion of charge carriers in disordered medium.",
"In some experiments, one can find $\\alpha =T/T_g$ where $T$ is the temperature and $T_g$ is the measure of the disorder.",
"This is also the case for Bouchaud trap model describing glassy dynamics [10].",
"In the context of contamination spreading biased CTRW is used with $\\alpha =1.73$ [16].",
"Based on numerical simulations, Winter and Schroer showed the super diffusive behavior and related the dynamics to the biased CTRW [4], [6].",
"In these systems one expects that at very long times we find normal diffusion.",
"There are also many examples of CTRW without bias [17], [40], [42], [43].",
"It is interesting to add a bias in these systems to compare the effect discussed here."
],
[
"Main results: the big jump principle",
"Transport and diffusion processes, either normal or anomalous, are composed of a large number of displacements.",
"Hence statistical laws, like the central limit theorem, are useful tools describing universal aspects of the phenomenon.",
"In our case a single event is controlling the statistics of the spreading packet at its tail.",
"Let $\\lbrace \\tau _1,\\tau _2,\\cdots ,\\tau _N,B_t\\rbrace $ be the set of the waiting times between jump events, and $\\sum _{i=1}^{N}\\tau _i+B_t=t$ is the measurement time.",
"Here $B_t$ , called the backward recurrence time, is the time elapsing between the moment of last jump $t_N=\\sum _{i=1}^N\\tau _i$ and the measurement time $t$ .",
"$N$ is the random number of jumps in $(0,t)$ [44].",
"We define the largest waiting time according to $\\tau _{\\max }=\\max \\lbrace \\tau _1,\\tau _2,\\cdots ,\\tau _N,B_t\\rbrace .$ One main conclusion of this manuscript is that the statistics of $\\tau _{\\max }$ determines the fluctuations of the position $x(t)$ of the biased random walker.",
"This holds for rare fluctuations of $x(t)$ , that still control the behavior of the most typical observable in the field: the MSD.",
"Due to the fat tailed distribution of the trapping time $\\phi (\\tau )$ , and using basic arguments from extreme value statistics of IID random variables, one expects that the typical fluctuations scale like $\\tau _{\\max }\\propto t^{1/\\alpha }$ , while for a thin tailed distribution of waiting time, e.g., $\\phi (\\tau )=\\exp (-\\tau )$ , we have $\\tau _{\\max }\\propto \\log (t)$ [20].",
"For the latter example `$\\propto $ ' means that $\\tau _{\\max }$ is of the order of $\\log (t)$ and similarly for the former case.",
"While we are not dealing with IID random variables, the constraint is weak in the sense that it does not modify the typical fluctuations, see below and Ref.",
"[28], [45].",
"Note that all these scalings, i.e., $\\tau _{\\max }\\propto t^{1/\\alpha }$ and $\\tau _{\\max }\\propto \\log (t)$ , describe typical fluctuations, sometimes called bulk fluctuations.",
"These fluctuations are described by normalized densities, specified by Fréchet's law and the Gumbel law.",
"On the contrary, here we focus on rare fluctuations, that is to say, both $\\tau _{\\max }$ and $t$ are comparable.",
"When Eq.",
"(REF ) holds, for the biased CTRW we will demonstrate that for small $x$ , i.e., the left plume in Fig.",
"REF $x \\doteqdot \\frac{t-\\tau _{\\max }}{\\langle \\tau \\rangle } a,$ where “$\\doteqdot $ ” indicates that the random variables on both sides follow the same distribution.",
"However, the PDFs describing the position of the particle $x$ when $x$ is not small and of $\\tau _{\\max }$ are far from being identical, indeed they will be calculated below.",
"The meaning of small $x$ and large $\\tau _{\\max }$ will soon become clear when we formulate the problem more precisely.",
"For now based on Figs.",
"REF and REF , we see Eq.",
"(REF ) works well when $x\\ll \\langle x(t)\\rangle $ and $\\tau _{\\max } \\simeq t$ .",
"For example when $x<50\\ll \\langle x(t)\\rangle \\simeq 1667$ in the bottom panel of Fig.",
"REF , or for the trajectory on the left panel of Fig.",
"REF , where $\\tau _{\\max }=988$ , when $t=1000$ and then $x\\simeq 40\\ll \\langle x(t)\\rangle \\simeq 1667$ .",
"Eq.",
"(REF ) means that the distribution of $x\\ll \\langle x(t)\\rangle $ is the same as the average size of the jumps $a$ , times the typical number of jumps made in $(0,t-\\tau _{\\max })$ , which is the time `free' of the longest waiting time.",
"A correlation plot based on Eq.",
"(REF ) is demonstrated numerically in Fig.",
"REF .",
"Using simulations of the ordinary CTRW process, we generate trajectories and search for positions of the random walkers at time $t$ and record $\\tau _{\\max }$ .",
"Then we plot the random entries observing that for small $x$ , there is a perfect correlation as predicated by Eq.",
"(REF ).",
"Such correlation plots indicate that Eq.",
"(REF ) is working.",
"We call this the principle of big jump, and it is valid for both stationary and ordinary processes.",
"Here the big jump means large trapping time, see further discussion on the term big jump and its origin in the discussion and summary.",
"Now we will analytically derive Eq.",
"(REF ) and discuss its consequence.",
"For that we obtain the distribution of $\\tau _{\\max }$ and then of $x$ .",
"Remark 1 Our main results in this manuscript are Eqs.",
"(REF ), (REF ), (REF ), and (REF ) which give explicit formulas for the PDF of $x$ and $\\tau _{\\max }$ for the two types of processes under investigation.",
"In [31] we promoted a rate formalism to treat similar problems, e.g.",
"the Lévy walk.",
"Here the focus is on the exact calculation of the statistics of rare events both for $\\tau _{\\max }$ and $x$ , and on the relation between these two random variables, i.e., Eq.",
"(REF )."
],
[
"The statistics of $\\tau _{\\max }$",
"Let us proceed to derive the general formulas describing the statistics of the longest waiting times which are valid for both the ordinary and equilibrium renewal processes.",
"The case of an ordinary renewal theory, was considered previously by Godréche, Majumdar and Schehr in Ref.",
"[45].",
"They investigated the typical fluctuations of $\\tau _{\\max }$ , and these as explained below exhibit behavior identical to a classical case of extreme value statistics, namely Fréchet's law holds for typical fluctuations.",
"Here our goal is very different, we aim to obtain the rare events, namely investigate the behavior when $\\tau _{\\max }$ is of the order $t$ .",
"In this case the fluctuations greatly differ from the IID case.",
"We define the probability that $\\tau _{\\max }$ is smaller than $L$ $\\begin{split}F(t,L) &=Prob[\\tau _{\\max }\\le L].\\end{split}$ The corresponding PDF is $P_{\\tau _{\\max }}(t,L)$ and as usual $F(t,L)=\\int _0^LP_{\\tau _{\\max }}(t,y)dy$ .",
"Clearly the probability depends on the measurement time $t$ and this dependence is especially important for fat-tailed waiting time PDFs.",
"It is helpful to introduce the joint probability distribution of $\\tau _{\\max }$ and the number of renewals $N$ $\\begin{split}F_{n}(t,L) & =Prob(\\tau _{\\max }\\le L,N=n) \\\\&=\\int _0^Ld\\tau _1 \\int _0^Ld\\tau _2\\cdots \\int _0^Ld B_t P_{\\tau _{\\max }}(\\tau _1,\\tau _2,\\cdots ,\\tau _n,B_t)\\\\&=\\int _0^Lh(\\tau _1) d\\tau _1\\int _0^L\\phi (\\tau _2)d\\tau _2\\cdots \\int _0^L\\Phi (B_t)dB_t\\delta \\left(t-\\left(\\sum _{1}^{n}\\tau _j+B_t\\right)\\right).\\end{split}$ Here $h(\\cdot )$ in the third line of Eq.",
"(REF ) is governed by the process we investigate, and $\\Phi (t)$ determined by the type of the process and the number of renewals $\\displaystyle \\Phi (t)=\\left\\lbrace \\begin{array}{ll}\\displaystyle \\int _t^\\infty h(\\tau )d\\tau , & \\hbox{$n=0$, equilibrium process;} \\\\\\displaystyle \\int _t^\\infty \\phi (\\tau )d\\tau , & \\hbox{otherwise.",
"}\\end{array}\\right.$ Taking the Laplace transform w.r.t.",
"$t$ , we find $\\widehat{F}_{n}(s,L)=\\left\\lbrace \\begin{array}{ll}\\displaystyle \\int _0^L\\exp (-s\\tau _1)\\int _{\\tau _1}^\\infty h(\\tau )d\\tau d\\tau _1, & \\hbox{$n=0$;} \\\\\\displaystyle \\int _0^L\\exp (-s\\tau )h(\\tau _1) d\\tau _1\\left(\\int _0^L\\exp (-s\\tau )\\phi (\\tau )d\\tau \\right)^{n-1}\\int _0^L\\exp (-sB_t)\\int _{B_t}^\\infty \\phi (\\tau )d\\tau dB_t, & \\hbox{$n\\ge 1$.",
"}\\end{array}\\right.$ The case $n=0$ corresponds to realizations with no renewals during the time interval $(0,t)$ .",
"One can check that $\\sum _{n=0}^\\infty \\widehat{F}_{n}(s,L\\rightarrow \\infty )=1/s$ .",
"This means that the density of $\\tau _{\\max }$ is normalized.",
"The sum of $n$ from zero to infinity gives $\\widehat{F}(s,L)=\\int _0^L\\exp (-s\\tau _1)\\int _{\\tau _1}^\\infty h(\\tau )d\\tau d\\tau _1+ \\int _0^L\\exp (-s\\tau _1)h(\\tau _1) d\\tau _1\\frac{\\int _0^L\\exp (-sB_t)\\int _{B_t}^\\infty \\phi (y)dyd{B_t}}{1-\\int _0^L\\exp (-s\\tau )\\phi (\\tau )d\\tau }.$ The first term is related to the survival probability and the second term corresponds to the probability that at least one renewal happened in $(0,t)$ .",
"For the equilibrium renewal process, we insert Eq.",
"(REF ) into Eq.",
"(REF ) while for the ordinary case we use $h_{{\\rm or}}(\\tau )=\\phi (\\tau )$ .",
"Below, from Eq.",
"(REF ) we will calculate the far tail of the distribution of $\\tau _{\\max }$ for the two different processes, i.e., the ordinary process and the equilibrium one, and prove that Eq.",
"(REF ) is valid for both cases.",
"Here we consider the ordinary renewal process and the ordinary CTRW to build the relation between the rare events of positions and the largest waiting times.",
"Figure: Random variables τ i \\tau _i for an ordinary renewal process with Eq.",
"() and α=1.5\\alpha =1.5.",
"The observation time tt is 1000 and i=100i=100, 200, ⋯\\cdots correspond to the 100-th, 200-th, ⋯\\cdots waiting time of the fractional renewal process, respectively.",
"Clearly in our case the number of renewals, shown in the top-right corner of the subplots, is a random variable; see inset.",
"Due to thefat-tailed trapping times, the fluctuations of NN is large which comes from the large fluctuations of waiting times."
],
[
"The rare fluctuations of $\\tau _{\\max }$",
"The aim is to investigate the PDF of $\\tau _{\\max }$ for the non-equilibrium process which is denoted as $P_{{\\rm or},\\tau _{\\max }}(t,L)$ .",
"We first treat the problem heuristically to calculate the typical fluctuations.",
"Let $\\langle N\\rangle =t/\\langle \\tau \\rangle $ be the average number of renewals in the long time limit.",
"For simplification, we neglect $B_t$ in Eq.",
"(REF ) and ignore the constraint $\\sum _{i=1}^N\\tau _i+B_t=t$ , further we replace the random $N$ with $\\langle N\\rangle $ .",
"This means that we treat this problem as if the waiting times are IID random variables, an approximation which turns out not sufficient in our case, still ignoring the correlation [45] $\\begin{split}Prob(\\tau _{\\max }<L)&=Prob^{N}(\\tau _i<L)\\\\&\\simeq \\left[1-\\left(\\frac{\\tau _0}{L}\\right)^\\alpha \\right]^N\\\\&\\sim \\exp \\left[-\\langle N\\rangle \\left(\\frac{\\tau _0}{L}\\right)^\\alpha \\right].\\end{split}$ This is the well-known Fréchet distribution [21].",
"A closer look reveals a drawback of this treatment of the typical fluctuations, since within this approximation the PDF of $\\tau _{\\max }$ is $P_{\\tau _{\\max }}(t,L)\\sim \\alpha \\langle N\\rangle (\\tau _0)^\\alpha /L^{1+\\alpha }$ , for $L\\rightarrow \\infty $ .",
"However in our setting $\\tau _{\\max }\\le t$ .",
"This means that we must modify Fréchet's law at its tail, in other words, the constraint that the sum of all the waiting times and the backward recurrence time is equal to the measurement time $t$ , comes into play when $\\tau _{\\max }\\propto t$ , as expected.",
"Note that the number of renewals in our case is a random variable; see Fig.",
"REF .",
"Now we use an exact solution of the problem to calculate the rare events.",
"Considering the non-equilibrium renewal process, we insert $h(\\tau )=\\phi (\\tau )$ into Eq.",
"(REF ) to get [45] $\\int _{L}^{\\infty } \\widehat{P}_{{\\rm or},{\\tau _{\\max }}}(z)dz=\\frac{1}{s}\\frac{1}{1+\\widehat{G}(s,L)}$ with $\\widehat{G}(s,L)=\\frac{s\\exp (sL)}{p_0(L)}\\int _0^{L}p_0(t)\\exp (-st)dt$ and the survival probability $p_0(t)=\\int _t^\\infty \\phi (\\tau )d\\tau \\simeq \\left(\\frac{\\tau _0}{t}\\right)^\\alpha .$ We are interested in the limit $s\\rightarrow 0$ (corresponding to long measurement time) and $L\\rightarrow \\infty $ in such a way that $sL$ remains a constant.",
"As mentioned the typical fluctuations are described by Fréchet's law Eq.",
"(REF ) and here instead we consider the rare fluctuations.",
"Using Eq.",
"(REF ), for $L\\rightarrow \\infty $ , Eq.",
"(REF ) becomes $\\begin{split}\\widehat{G}(s,L) &\\sim \\frac{\\exp (sL)sL^\\alpha \\langle \\tau \\rangle }{(\\tau _0)^\\alpha },\\end{split}$ where we have used the limit $\\lim _{L\\rightarrow \\infty }\\int _0^{L} p_0(t)\\exp (-st)dt=\\frac{1-\\widehat{\\phi }(s)}{s}\\sim \\langle \\tau \\rangle $ with $s\\rightarrow 0$ .",
"From Eq.",
"(REF ) we see that $\\widehat{G}(s,L)$ is large for $sL\\rightarrow \\rm {constant}$ and $\\alpha >1$ .",
"According to Eq.",
"(REF ), we find $\\frac{\\partial \\widehat{G}(s,L) }{\\partial L}\\sim s\\widehat{G}(s,L)+\\frac{\\alpha }{L} \\widehat{G}(s,L)+\\cdots .$ Note that Eq.",
"(REF ) can also be derived directly from Eq.",
"(REF ).",
"Utilizing Eq.",
"(REF ) and $F(t,L)=\\int _0^{L}P_{{\\rm or},\\tau _{\\max }}(t,y)dy,$ and after some simple rearrangements $\\widehat{P}_{{\\rm or},{\\tau _{\\max }}}(s,L)=\\frac{1}{s}\\frac{\\frac{\\partial \\widehat{G}(s,L)}{\\partial L}}{[1+\\widehat{G}(s,L)]^2},$ where we used the relation that $P_{{\\rm or},\\tau _{\\max }}(t,L)$ is the derivative of Eq.",
"(REF ) w.r.t.",
"$L$ .",
"Combining Eqs.",
"(REF ) and (REF ), we have $\\widehat{P}_{{\\rm or},{\\tau _{\\max }}}(s,L)\\sim \\frac{1}{\\widehat{G}(s,L)}+\\frac{\\alpha }{sL\\widehat{G}(s,L)}+\\cdots .$ Note that the first two terms on the right-hand side of Eq.",
"(REF ), namely $1/\\widehat{G}(s,L)$ and $\\alpha /(\\widehat{G}(s,L)sL)$ , are comparable when $sL\\rightarrow \\rm {constant}$ .",
"Hence from Eqs.",
"(REF ) and (REF ), we get $\\widehat{P}_{{\\rm or},{\\tau _{\\max }}}(s,L)\\sim \\frac{(\\tau _0)^\\alpha }{\\langle \\tau \\rangle }\\frac{\\exp (-sL)}{sL^\\alpha }\\left(1+\\frac{\\alpha }{sL} \\right).$ Taking the inverse Laplace transform $s\\rightarrow t$ of Eq.",
"(REF ) gives our second main result with the scaling $L\\propto t$ $P_{{\\rm or},{\\tau _{\\max }}}(t,L)\\sim \\frac{(\\tau _0)^\\alpha }{t^\\alpha \\langle \\tau \\rangle } \\left[\\alpha \\left(\\frac{L}{t}\\right)^{-\\alpha -1}-(\\alpha -1)\\left(\\frac{L}{t}\\right)^{-\\alpha }\\right]$ with $0\\le L\\le t$ .",
"Theoretical predication of Eq.",
"(REF ) is compared with numerical simulations in Fig.",
"REF .",
"As explained before, Eq.",
"(REF ) describing the far tail of the distribution of $\\tau _{\\max }$ is a modification of Fréchet's law.",
"Figure: Scaled PDF of the longest time interval t α P or ,τ max (t,L)t^{\\alpha }P_{{\\rm or},{\\tau _{\\max }}}(t,L) versus L/tL/t.",
"The red solid lines predicated by Eq.",
"() or equivalently Eq.",
"(), describe the rare fluctuations showing the behavior of LL when it is of the order of tt.The simulations, plotted by the symbols, are generated by averaging 10 6 10^6 trajectories with α=3/2\\alpha =3/2 and τ 0 =1\\tau _0=1.",
"The figure clearly shows a perfect agreement of the simulations compared with the theoretical result Eq.",
"(), which has a sharp cutoff at the tail of density at τ max /t→1\\tau _{\\max }/t\\rightarrow 1 (see the red dash-doted lines).",
"This is very different if compared with typical fluctuations calculated with Fréchet's distribution Eq.",
"(), which clearly does not describe wellthe far tail (see inset).According to Eq.",
"(REF ), we find $\\lim _{t\\rightarrow \\infty }\\langle \\tau \\rangle \\left(\\frac{t}{\\tau _0}\\right)^\\alpha P_{{\\rm or},{\\tau _{\\max }}}(t,L)= \\mathcal {I}_{{\\rm or},\\alpha }\\left(\\frac{L}{t}\\right),$ where $\\mathcal {I}_{{\\rm or},\\alpha }(y)=\\alpha y^{-\\alpha -1}-(\\alpha -1)y^{-\\alpha }$ with $0<y<1$ .",
"This scaling solution describes the far tail of the distribution, where Fréchet's law does not work.",
"In fact, these two laws are related as the $y^{-\\alpha -1}$ term matches the far tail of the Fréchet law, as it should.",
"Since $0<y<1$ implies $\\tau _{\\max }<t$ , moments of $\\tau _{\\max }$ are computed w.r.t.",
"this scaling solution.",
"In contrast, the Fréchet law gives diverging variance of $\\tau _{\\max }$ , which is certainly not a possibility since $\\tau _{\\max }$ is bounded.",
"The expression in Eq.",
"(REF ) is an infinite density describing a non-normalising limiting law.",
"More exactly $\\mathcal {I}_{{\\rm or},\\alpha }(\\cdot )$ is not normalizable, the moments of order $q>\\alpha $ of $\\tau _{\\max }$ are calculated w.r.t.",
"this non-normalised state.",
"More details on infinite densities see Refs.",
"[46], [47], [48], [49], [50]"
],
[
"The rare fluctuations of the position",
"We now investigate the distribution of $x$ proving the validity of the big jump principle Eq.",
"(REF ).",
"Let $P_{{\\rm or}}(x,t)$ be the PDF of finding the walker on $x$ at time $t$ .",
"The starting point is the well-known Montroll-Weiss equation, which gives Fourier-Laplace transform of the $P_{{\\rm or}}(x,t)$ [10], [17] $\\widetilde{\\widehat{P}}_{{\\rm or}}(k,s)=\\frac{1-\\widehat{\\phi }(s)}{s}\\frac{1}{1-\\widetilde{f}(k)\\widehat{\\phi }(s)}$ with $\\widetilde{\\widehat{P}}_{{\\rm or}}(k,s)=\\int _{-\\infty }^{\\infty }\\int _0^{\\infty }\\exp (ikx-st)P_{{\\rm or}}(x,t)dtdx$ .",
"Here $\\widetilde{f}(k)$ is the Fourier transform of the jump length PDF $f(\\chi )$ , and $\\widehat{\\phi }(s)$ is the Laplace transform of waiting time PDF.",
"The long wave length approximation, i.e., the small $s$ and $k$ limit, is routinely applied to investigate the long time limit of $P_{{\\rm or}}(x,t)$ .",
"However, how to choose the limit of $k\\rightarrow 0$ and $s\\rightarrow 0$ is actually slightly subtle.",
"Utilizing Eqs.",
"(REF ) and (REF ), and assuming that the ratio $|s^\\alpha |/|k|$ is fixed, we get $\\widetilde{\\widehat{P}}_{{\\rm or}}(k,s)\\sim \\frac{\\langle \\tau \\rangle }{-ika+s\\langle \\tau \\rangle -(\\tau _0)^\\alpha |\\Gamma (1-\\alpha )| s^\\alpha },$ Inverting, we then find a known limit theorem [51], [52] $P_{{\\rm or}}(x,t)\\sim \\frac{1}{a(t/\\overline{t})^{1/\\alpha }}L_{\\alpha ,1}\\left(\\frac{x-at/\\langle \\tau \\rangle }{a(t/\\overline{t})^{1/\\alpha }}\\right),$ where $\\overline{t}=\\langle \\tau \\rangle ^{1+\\alpha }/((\\tau _0)^\\alpha |\\Gamma (1-\\alpha )|)$ , $L_{\\alpha ,1}(\\cdot )$ is the non-symmetrical Lévy stable law with characteristic function $\\exp [(ik)^\\alpha ]$ , and $a>0$ .",
"This central limit theorem, just like Fréchet's law, has its limitations.",
"As a stand alone formula, it predicates $\\langle x^2(t)\\rangle =\\infty $ , since the second moment of the Lévy distribution does not exist.",
"This means that we must consider a different method to describe the far tail.",
"To proceed we reanalyze Eq.",
"(REF ) but now fixing $|s|/|k|$ .",
"This is a large deviation approach since such a scaling implies a ballistic scaling behavior of $x$ and $t$ , unlike $x-at/\\langle \\tau \\rangle \\propto t^{1/\\alpha }$ in Eq.",
"(REF ).",
"The strategy we use now, i.e., the determination of $P_{{\\rm or}}(x,t)$ for $x\\propto t$ , is similar to the approach in the previous section where we calculated $P_{{\\rm or},\\tau _{\\max }}(t,L)$ .",
"The obvious difference is that there we start with Eq.",
"(REF ), while here with the Montroll-Weiss Eq.",
"(REF ).",
"More specifically in the Sec.",
"REF we assume that $sL\\propto \\rm {constant}$ , while here $|s|$ and $|k|$ are small and of the same order, where $s$ and $k$ are Laplace pair and Fourier pair of $t$ and $x$ , respectively.",
"We restart from Eq.",
"(REF ), which gives $\\widetilde{\\widehat{P}}_{{\\rm or}}(k,s)\\sim \\underbrace{\\frac{\\langle \\tau \\rangle }{\\langle \\tau \\rangle s-ika}}_{{ {\\rm leading}}}+\\underbrace{\\frac{ika (\\tau _0)^\\alpha |\\Gamma (1-\\alpha )| s^{\\alpha -1}}{(s\\langle \\tau \\rangle -ika)^2}}_{{ {\\rm correction}}}+\\cdots .$ The derivation of Eq.",
"(REF ) is given in Appendix .",
"The inversion of the leading term is trivial, but it yields a delta function $\\delta (x-at/\\langle \\tau \\rangle )$ .",
"Mathematically we choose a scaling that shrinks the density to an uninteresting object.",
"Luckily, the correction term is important as it describes the far tail.",
"So for $x\\ne at/\\langle \\tau \\rangle $ , we have $P_{{\\rm or}}(x,t)\\sim \\mathcal {F}^{-1}_{k\\rightarrow x}\\mathcal {L}^{-1}_{s\\rightarrow t}\\left[\\frac{a(\\tau _0)^\\alpha |\\Gamma (1-\\alpha )| iks^{\\alpha -1}}{(s\\langle \\tau \\rangle -ika)^2}\\right]$ with $\\mathcal {F}^{-1}_{k\\rightarrow x}$ and $\\mathcal {L}^{-1}_{s\\rightarrow t}$ being the inverse Fourier and the inverse Laplace transforms, respectively.",
"We first perform the inverse Laplace transform using the convolution theorem and the pairs $\\left\\lbrace \\begin{array}{ll}\\displaystyle \\mathcal {L}^{-1}_{s\\rightarrow t}[s^{\\alpha -1}]=\\frac{t^{-\\alpha }}{\\Gamma (1-\\alpha )}, & \\hbox{~} \\\\\\displaystyle \\mathcal {L}^{-1}_{s\\rightarrow t}\\left[\\frac{1}{(s-ika/\\langle \\tau \\rangle )^2}\\right]=t\\exp \\left(ika\\frac{t}{\\langle \\tau \\rangle }\\right) & \\hbox{~}\\end{array}\\right.$ and find $P_{{\\rm or}}(x,t)\\sim \\mathcal {F}^{-1}_{k\\rightarrow x}\\left[-ik\\frac{a(\\tau _0)^\\alpha }{\\langle \\tau \\rangle ^2}\\int _0^t \\frac{y\\exp \\left(\\frac{ikay}{\\langle \\tau \\rangle }\\right)}{(t-y)^{\\alpha }} dy\\right].$ The inverse Fourier transform of $\\exp (ikay/\\langle \\tau \\rangle )$ is a delta function and the $ik$ in front of this expression is the spatial derivative in $x$ space, hence we get $P_{{\\rm or}}(x,t)\\sim \\frac{(\\tau _0)^\\alpha }{\\langle \\tau \\rangle }\\frac{\\partial }{\\partial x}\\int _0^t\\frac{y\\delta \\left(y-\\frac{x\\langle \\tau \\rangle }{a}\\right)}{(t-y)^{\\alpha }}dy.$ Then after simple rearrangements $P_{{\\rm or}}(x,t)\\sim \\frac{(\\tau _0)^\\alpha }{at^{\\alpha }}\\mathcal {I}_{{\\rm or},\\alpha }(\\xi )$ with $0<\\xi <1$ , $\\xi =1-(x/a)/(t/\\langle \\tau \\rangle )$ , and $\\mathcal {I}_{{\\rm or},\\alpha }(\\cdot )$ being defined by Eq.",
"(REF ).",
"As Fig.",
"REF demonstrates, this equation describes the far tail of the density of the spreading packet, and it is complementary to the Lévy law Eq.",
"(REF ).",
"The MSD of the process is obtained w.r.t.",
"integration over the formula Eq.",
"(REF ) and in that sense this equation “cures” the drawback of the Lévy density.",
"More importantly is the fact that the distribution of $\\tau _{\\max }$ Eq.",
"(REF ) and $x$ Eq.",
"(REF ) have the same structure, beyond a trivial Jacobian.",
"In other words, given the fact that these observables have the same distribution, we have proven the single big jump principle Eq.",
"(REF ) for the ordinary processes.",
"The statistics of one waiting time, $\\tau _{\\max }$ , determines the fluctuations at small $x$ .",
"And since Eq.",
"(REF ) gives the MSD, which is used in most experimental, theoretical and numerical works to characterize the fluctuations, we see that the MSD is directly related to the single big jump principle and extreme value statistics.",
"One should note that low-order moments like $\\langle |x-\\langle x\\rangle |^q\\rangle $ with $q<\\alpha $ are finite w.r.t.",
"the Lévy density, and these are given by integration w.r.t.",
"Eq.",
"(REF ).",
"Figure: Scaled PDF P or (x,t)P_{{\\rm or}}(x,t) is compared with the prediction of the single big jump principle and the Lévy central limit theorem describing rare and typical fluctuations.The parameters are a=1a=1, σ=1\\sigma =1, α=1.5\\alpha =1.5, τ 0 =0.1\\tau _0=0.1, and for the simulation we used 5×10 6 5\\times 10^6 trajectories.The inset exhibits a comparison among typical fluctuations Eq.",
"(), rare fluctuations Eq.",
"(), and simulations.Clearly our theory performs perfectly, while Eq.",
"() over shoots (see inset) and extentsto the positive infinity.",
"In reality there is a clear cutoff at x=0x=0 being exclusively revealed by the single big jump principleanalysis (see the marked red dash-doted lines).Remark 2 We now study the case of CTRW in two dimensions and focus on an ordinary process.",
"The joint length PDF is $f(\\chi _x,\\chi _y)=f_x(\\chi _x)f_y(\\chi _y)$ where $f_x(\\chi _x)$ is the same as in Eq.",
"(REF ) and $f_y(\\chi _y)=\\exp (-(\\chi _y)^2/2(\\sigma _y)^2)/(\\sqrt{2\\pi (\\sigma _y)^2})$ with $\\sigma _y$ being a constant.",
"This means that the drift is only in $x$ direction.",
"Similar to our previous calculations, we use the Montroll-Weiss equation and find $P_{\\rm {or}}(x,y,t)\\sim \\frac{(\\tau _0)^\\alpha }{at^{\\alpha }}\\mathcal {I}_{{\\rm or},\\alpha }(\\xi )\\delta (y).$ The marginal density $P_{{\\rm or}}(x,t)$ is the same as the one dimensional case Eq.",
"(REF ).",
"Note that $\\tau _{\\max }$ is of the order $t$ (for the far tail), so in the $y$ direction the particles are practically frozen.",
"Hence we get a delta function since there is no drift in the $y$ direction."
],
[
"The equilibrium case",
"Up to now we have considered the case when a physical clock was started immediately at the beginning of the process, i.e., an ordinary CTRW.",
"Here we consider the equilibrium initial condition.",
"We note that for $0<\\alpha <1$ , i.e., when the average trapping time diverges, this is related to Aging CTRW [53], [54], [33], [40], [34] which is used as a tool to describe complex systems ranging from Anderson insulator to colloidal suspensions and it was first introduced by Monthus and Bouchaud to illustrate the diffusion in glasses [55].",
"In contrast, when $1<\\alpha <2$ and Eq.",
"(REF ) holds, we have a stationary process.",
"Then as mentioned already, the mean waiting time for the first event is infinite; see Eq.",
"(REF ).",
"In practice, if we start the process at time $-t_a$ and $t_a$ is large but finite the averaged first waiting time observed after time $t_a$ will increase with $t_a$ , and when $t_a$ tends to infinity it will diverge.",
"Here we focus on the statistics of particles with an equilibrium condition, i.e., $t_a\\rightarrow \\infty $ ."
],
[
"The rare fluctuations of the position",
"In Fourier-Laplace space, the density of spreading particles is given by [33] $\\begin{split}\\widetilde{\\widehat{P}}_{\\rm {eq}}(k,s) & =\\frac{1-\\widehat{h}_{{\\rm eq}}(s)}{s}+\\frac{(1-\\widehat{\\phi }(s))\\widehat{h}_{\\rm {eq}}(s)\\widetilde{f}(k)}{s(1-\\widehat{\\phi }(s)\\widetilde{f}(k))}\\end{split}$ This equation is a modification of the Montroll-Weiss equation, taking into consideration the equilibrium initial state.",
"Using the Laplace transform of Eq.",
"(REF ), we have $\\begin{split}\\widetilde{\\widehat{P}}_{\\rm {eq}}(k,s)=\\frac{\\langle \\tau \\rangle s-1+\\widetilde{\\phi }(s)}{\\langle \\tau \\rangle s^2}+\\frac{(1-\\widehat{\\phi }(s))^2 \\widetilde{f}(k)}{\\langle \\tau \\rangle s^2(1-\\widehat{\\phi }(s)\\widetilde{f}(k))}.\\end{split}$ The first term on the right-hand side is $k$ independent, hence its inverse Fourier transform gives a delta function on the initial condition $x=0$ describing non-moving particles.",
"This population of motionless particles is non negligible in the sense that they contribute to the MSD; see Eq.",
"(REF ).",
"Based on Eq.",
"(REF ), we consider typical fluctuations, i.e., $k,s\\rightarrow 0$ and $|k|\\propto |s^\\alpha |$ $\\begin{split}\\widetilde{\\widehat{P}}_{{\\rm eq }}(k,s) &\\sim \\frac{(1-\\widehat{\\phi }(s))^2}{\\langle \\tau \\rangle s^2}\\frac{1}{1-\\widehat{\\phi }(s)\\widetilde{f}(k)} \\\\& \\sim \\frac{\\langle \\tau \\rangle }{\\langle \\tau \\rangle s-ika-b_\\alpha s^\\alpha },\\end{split}$ where we used the asymptotic behaviors of $\\widetilde{\\phi }(s)$ and $\\widehat{f}(k)$ .",
"The inverse Laplace-Fourier transform of Eq.",
"(REF ) yields $P_{\\rm {eq}}(x,t)\\sim \\frac{1}{a(t/\\overline{t})^{1/\\alpha }}L_{\\alpha ,1}\\left(\\frac{x-at/\\langle \\tau \\rangle }{a(t/\\overline{t})^{1/\\alpha }}\\right),$ According to Eq.",
"(REF ), the typical fluctuations are the same as the one of the ordinary case; see Eq.",
"(REF ) and the dashed lines in Fig.",
"REF .",
"That is, the bulk fluctuations do not depend on the initial state.",
"On the other hand, the MSDs of both cases are different, this means that the far tail of $P_{\\rm {eq}}(x,t)$ should be modified compared with the ordinary case.",
"As mentioned before, the normalized density Eq.",
"(REF ) gives an unphysical infinite MSD due to the slowly decaying tail of asymmetric Lévy distribution.",
"This means that we expect modifications of this limiting law at the far tail.",
"For the rare events of the equilibrium CTRW, i.e., both $s$ and $k$ are small and comparable, inserting $\\widehat{\\phi }(s)$ and $\\widetilde{f}(k)$ into Eq.",
"(REF ) gives $\\begin{split}\\widetilde{\\widehat{P}}_{\\rm {eq}}(k,s) &\\sim \\frac{b_\\alpha }{\\langle \\tau \\rangle s^{2-\\alpha }}+\\frac{\\langle \\tau \\rangle -2b_\\alpha s^{\\alpha -1}}{\\langle \\tau \\rangle s-ika-b_\\alpha s^\\alpha }.\\end{split}$ Rewriting the second term of the right-hand side of Eq.",
"(REF ) as $\\frac{\\langle \\tau \\rangle -2b_\\alpha s^{\\alpha -1}}{\\langle \\tau \\rangle s-ika-b_\\alpha s^\\alpha }\\sim \\frac{\\langle \\tau \\rangle -b_\\alpha s^{\\alpha -1}}{\\langle \\tau \\rangle s-ika}+\\frac{b_\\alpha s^{\\alpha -1}ika}{(\\langle \\tau \\rangle s-ika)^2}$ and using the relation $\\mathcal {F}^{-1}\\mathcal {L}^{-1}\\left[\\frac{ s^{\\alpha -1}}{\\langle \\tau \\rangle s-ika}\\right]=\\frac{\\left(t-\\frac{\\langle \\tau \\rangle }{a}x\\right)^{-\\alpha }}{a\\Gamma (1-\\alpha )},$ we get the main results of this section describing the packet when $x$ is of the order of $t$ $P_{\\rm {eq}}(x,t)\\sim \\frac{(\\tau _0)^\\alpha t^{1-\\alpha }}{\\langle \\tau \\rangle (\\alpha -1)}\\delta (x)+\\frac{(\\tau _0)^\\alpha }{at^\\alpha }\\mathcal {I}_{{\\rm eq},\\alpha }(\\xi ),$ where the non-normalised state function reads $\\mathcal {I}_{{\\rm eq},\\alpha }(\\xi )=\\alpha \\xi ^{-\\alpha -1}+(2-\\alpha )\\xi ^{-\\alpha }$ and $\\xi =1-(x/a)/(t/\\langle \\tau \\rangle )$ .",
"Comparing with Eq.",
"(REF ), we see that the infinite densities for the equilibrium and non-equilibrium processes are different.",
"This indicates that initial conditions influence the statistics at small position even when the measurement time is long $t\\gg \\langle \\tau \\rangle $ .",
"The rare fluctuations for the equilibrium case are larger if compared with the ordinary process, in particular they include a delta function contribution; see the data marked in a red circle in Fig.",
"REF .",
"This means that particles not moving at all contribute to the rare events.",
"Note that Eq.",
"(REF ) can be matched to the far tail of the Lévy distribution Eq.",
"(REF ), as it should.",
"Figure: Scaled PDF of the position versus 1-(x/a)/(t/〈τ〉)1-(x/a)/(t/\\langle \\tau \\rangle ).The symbols are simulation results obtained from 3×10 6 3\\times 10^6 realizations.",
"The red solid line calculated from Eq.",
"() describing the behavior when x∝tx\\propto t is consistent with the far tail of simulation results; see inset.We also plot the theory of an ordinary process Eq.",
"(), showing that it clearly fails, and it under estimates the rare fluctuations described by the equilibrium theory.Notice that the delta like contribution circled in red, describes non-moving particles at x=0x=0.Here α=1.5\\alpha =1.5, a=1a=1, σ=1\\sigma =1, t a =10 4 t_a=10^4, t=1000t=1000, and τ 0 =0.1\\tau _0=0.1.We further check that the MSD is determined by the rare fluctuations Eq.",
"(REF ) resulting in a different MSD compared with the ordinary process.",
"Using the random variable $\\eta =(x-at/\\langle \\tau \\rangle )/(at/\\langle \\tau \\rangle )$ with $-1<\\eta <0$ , from Eq.",
"(REF ) we get $\\langle \\eta ^2\\rangle _{{\\rm eq}} \\sim \\frac{2b_\\alpha t^{1-\\alpha }}{\\langle \\tau \\rangle \\Gamma (4-\\alpha )};$ see Appendix .",
"Similarly, $\\langle \\eta ^2\\rangle _{{\\rm or}}$ is also obtained according to Eq.",
"(REF ).",
"Utilizing Eqs.",
"(REF ) and (REF ), $\\langle x^2\\rangle -\\langle x\\rangle ^2 \\sim \\displaystyle \\left\\lbrace \\begin{array}{ll}\\displaystyle \\frac{2a^2b_\\alpha (\\alpha -1) t^{3-\\alpha }}{\\langle \\tau \\rangle ^3 \\Gamma (4-\\alpha )} , & \\hbox{ordinary;} \\\\\\displaystyle \\frac{2a^2b_\\alpha t^{3-\\alpha }}{\\langle \\tau \\rangle ^3 \\Gamma (4-\\alpha )}, & \\hbox{equilibrium.",
"}\\end{array}\\right.$ Though the MSDs for both cases grow as power law $t^{3-\\alpha }$ , the MSD for the equilibrium case is larger than the ordinary one.",
"Since the mean of the first waiting time following Eq.",
"(REF ) is infinite, the probability of particles experiencing a long trapping time increases rapidly compared with an ordinary situation.",
"In turn, this considerably yields inactive particles which are trapped on the origin for the whole observation time $t$ far lagging behind the mean.",
"Hence, the MSD for the equilibrium process has a deep relationship with the motionless particles; see Eqs.",
"(REF ).",
"It is interesting to find that the MSDs for both cases are determined by the far tail of the packet described by the infinite densities.",
"As expected, when $\\alpha \\rightarrow 2$ , these two processes show normal diffusion with no difference, so then the initial condition is unimportant.",
"Figure: Simulations of the distribution of ξ\\xi with the scaling ξ=τ max /t\\xi =\\tau _{\\max }/t and α=1.5\\alpha =1.5 compared to the analytical predication obtained from Eq. ().",
"Clearly both the typical fluctuations Eq.",
"() and the rare events Eq.",
"() with an non-equilibrium condition do not work at the far tail of the distribution; see inset."
],
[
"The rare fluctuations of $\\tau _{\\max }$",
"After calculating $P(x,t)$ for small $x$ , the next aim is to deal with the far tail of the PDF $\\tau _{\\max }$ when $\\tau _{\\max }$ and $t$ are comparable.",
"From Eq.",
"(REF ), we have $\\begin{split}\\frac{1}{s}-\\widehat{F}_{{\\rm eq}}(s,L) & =\\frac{1}{s(1+\\widehat{G}(s,L))} \\\\& +\\frac{1-\\int _0^{L}\\exp (-s\\tau _1)h_{{\\rm eq}}(\\tau _1)d\\tau _1}{s\\left(1+\\frac{1}{\\widehat{G}(s,L)}\\right)}\\\\&-\\int _0^{L}\\exp (-sB_t)\\int _{B_t}^\\infty h_{{\\rm eq}}(\\tau )d\\tau dB_t,\\end{split}$ where $\\widehat{G}(s,L)$ is defined in Eq.",
"(REF ).",
"It gives the PDF by the derivative $\\widehat{P}_{{\\rm eq},{\\tau _{\\max }}}(s,L)=-\\frac{\\partial [\\frac{1}{s}-\\widehat{F}_{\\rm {eq}}(s,L)]}{\\partial L}.$ Note that $\\frac{\\partial }{\\partial L}\\frac{1-\\int _0^{L}\\exp (-s\\tau _1)h_{{\\rm eq}}(\\tau _1)d\\tau _1}{s(1+1/\\widehat{G}(s,L))}\\sim \\frac{\\exp (-sL)}{-s}h_{{\\rm eq}}(L)$ since $\\widehat{G}(s,L)$ is large with $L\\propto 1/s$ .",
"Using Eqs.",
"(REF ) and (REF ), $\\widehat{P}_{{\\rm eq},\\tau _{\\max }}(s,L)$ reduces to $\\begin{split}\\widehat{P}_{{\\rm eq},{\\tau _{\\max }}}(s,L) &\\sim \\frac{1}{\\widehat{G}(s,L)}+\\frac{\\alpha }{sL\\widehat{G}(s,L)}\\\\&~~~ +\\exp (-sL) \\int _L^\\infty h_{{\\rm eq}}(\\tau )d\\tau \\\\&~~~+\\frac{\\exp (-sL)}{s}h_{{\\rm eq}}(L).\\end{split}$ Note that Eq.",
"(REF ) is a uniform approximation in Laplace space which is effective for numerous $L$ and large $t$ .",
"More exactly, within this approximation, we have the only condition that the observation time $t$ is large enough without considering the scaling between $t$ and $L$ .",
"For the typical fluctuations, the leading term of Eq.",
"(REF ) is the same as the ordinary process.",
"Thus $\\int _0^L P_{{\\rm eq},{\\tau _{\\max }}}(t,y)dy\\sim \\exp \\left[-\\frac{t}{\\langle \\tau \\rangle } \\left(\\frac{\\tau _0}{L}\\right)^\\alpha \\right].$ We see that the typical fluctuations of the longest time interval of both the equilibrium and the ordinary renewal processes are the same and independent of the initial conditions, describing the behavior when $L^\\alpha $ is of the order of $t$ .",
"Next we turn our attention to the case when $L\\propto t$ .",
"Restart from Eq.",
"(REF ), the inverse Laplace transform gives our main result describing the far tail of the density $\\begin{split}P_{{\\rm eq},{\\tau _{\\max }}}(t,L) & \\sim \\frac{(\\tau _0)^\\alpha }{t^\\alpha \\langle \\tau \\rangle }\\mathcal {I}_{{\\rm eq},\\alpha }(y)+\\delta (t-L) \\\\& \\times \\int _{L}^\\infty h_{{\\rm eq}}(\\tau )d\\tau +\\theta (t-L)h_{{\\rm eq}}(L)\\end{split}$ with $L\\le t$ .",
"Utilizing Eqs.",
"(REF ) and (REF ), we have $P_{{\\rm eq},{\\tau _{\\max }}}(t,L)\\sim \\frac{(\\tau _0)^\\alpha }{t^\\alpha \\langle \\tau \\rangle }\\mathcal {I}_{{\\rm eq},\\alpha }\\left(\\frac{L}{t}\\right)+\\delta (t-L)\\frac{(\\tau _0)^\\alpha L^{1-\\alpha } }{(\\alpha -1)\\langle \\tau \\rangle };$ see Fig.",
"REF .",
"From Eqs.",
"(REF ) and (REF ), it can be seen that the principle Eq.",
"(REF ) is also valid for the equilibrium case.",
"Though the typical fluctuations of $\\tau _{\\max }$ for equilibrium and ordinary process show no difference, their far tails are distinct from each other [see Eqs.",
"(REF ) and (REF )]."
],
[
"Discussion and summary",
"We have related the theory of extreme value statistics and the fluctuations of a particle diffusing in a disordered system with traps.",
"As mentioned, the observation of a non-Gaussian packet $P(x,t)$ and super-diffusive MSD is widely reported [1], [9], [10], [13], [14], [3], [4], [5], [6], [2], [15], [7], [8], [11], [12].",
"Here we showed that a modification of Fréchet's law is required to fully characterize these fluctuations.",
"The largest waiting time $\\tau _{\\max }$ is clearly shorter than the observation time $t$ , namely the sum $\\sum _{i=1}^N\\tau _i+B_t$ is constrained, hence naturally we have deviations from the Fréchet law.",
"In other words, the theory of IID random variables, completely fails to describe the phenomenon of the far tail of the packet.",
"More profound is the observation that the statistics of $\\tau _{\\max }$ determines the far tail of $P(x,t)$ for the ordinary and equilibrium processes.",
"One trapping event, the longest of the lot, controls the statistics of large deviations, and this is very different if compared with standard large deviation theory [22], where many small jumps in the same direction control the statistics.",
"Our work is related to the so called single big jump principle, which was originally formulated for $N$ IID random variables $\\lbrace \\vartheta _1,\\vartheta _2,\\cdots , \\vartheta _N\\rbrace $ [28].",
"It states that $\\sum _{i=1}^N\\vartheta _i\\doteqdot \\max \\lbrace \\vartheta _1,\\vartheta _2, \\cdots , \\vartheta _N\\rbrace $ when the distribution of $\\vartheta _i$ is sub-exponential, and for large maximum.",
"Note that in the CTRW model considered in this manuscript we do not have any large spatial jump, instead we have long sticking events where the particles do not move.",
"More importantly, in our case the waiting times are constrained by the total measurement time and hence correlated, and their number $N$ fluctuates.",
"Hence the situation encountered here is simply different (though related) to the original one.",
"Thus one aspect of our work was to modify the principle as we did in Eq.",
"(REF ) and then describe the rare events with new Eqs.",
"(REF ), (REF ), (REF ), and (REF ).",
"This allowed us to connect the big jump theory to infinite densities.",
"The solutions describing the far tails of the distributions of $x$ and $\\tau _{\\max }$ are non-normalizable, still with proper scaling they are the limits of the perfectly normalised probability densities.",
"For example in Eq.",
"(REF ), we multiply the normalized density $P_{{\\rm or},{\\tau _{\\max }}}(t,L)$ by $\\langle \\tau \\rangle (t/\\tau _0)^\\alpha $ and then get the infinite density $\\mathcal {I}_{{\\rm or},\\alpha }(L/t)$ .",
"The variance of $\\tau _{\\max }$ and the super-diffusive MSD are calculated with these non-normalised states, meaning that these quantifiers of the anomaly are sensitive to rare events.",
"We showed that the initial condition is an important factor controlling the behavior of the far tail of distribution of interest.",
"We calculated these for the stationary and ordinary renewal processes, showing that for the stationary process motionless particles give an important contribution to the description of the rare fluctuations and the MSD.",
"On the one hand this implies that the far tails are non-universal in their shapes.",
"This can therefore be used to characterize the nature of the underlying process.",
"As for universality, this shows up in the principle of big jump Eq.",
"(REF ), as the relation between the trapping time and the position $x$ , is independent of the underlying process.",
"We note that the surprising super-diffusion of a biased tracer in a crowded medium was also found based on a many body theory [5], [56], [8], diffusion of contamination in disordered systems, and for numerical simulations of glass forming systems [4], [6] where it is interesting to check the relation of the dynamics and the big jump principle.",
"The investigation of the single big-jump principle in the context of other models of random walks in random environments is of great interest.",
"For example the biased quenched trap model, exhibits typical fluctuations which are the same as those found for the biased CTRW [10], [9], [57], [58].",
"Will this repeat for the rare events is still unknown.",
"Recently the case of $N$ IID random variables constrained to have a given sum was investigated, and under certain conditions the Fréchet law was found [59], [60], [61].",
"From the constraint it is clear that the far tail of the distribution of maximum cannot be modeled with the Fréchet law since there is a cutoff at the far tail.",
"It would be of interest to investigate the far tail of this model (there $N$ was fixed while here $N$ is random) and see if the non-normalized density is found here as well.",
"The authors would like to thank the anonymous reviewers for their helpful and constructive comments.",
"EB acknowledges the Israel Science Foundation's grant 1898/17."
],
[
"\nCalculation of Eq. (",
"We now present the detailed derivation of Eq.",
"(REF ) in the main text starting from the Montroll-Weiss Equation (REF ).",
"Here we are interested in the case $x-at/\\langle \\tau \\rangle \\propto at/\\langle \\tau \\rangle $ instead of $x-at/\\langle \\tau \\rangle \\propto at^{1/\\alpha }$ describing the typical fluctuations (see the main text).",
"In Fourier-Laplace space, this corresponds to $|s|\\propto |k|$ .",
"Plugging Eqs.",
"(REF ) and (REF ) into Eq.",
"(REF ) leads to $\\begin{split}\\widetilde{\\widehat{P}}_{\\rm or}(k,s) & \\sim \\frac{1}{\\langle \\tau \\rangle s-ika-(\\tau _0)^\\alpha |\\Gamma (1-\\alpha )|s^\\alpha }(\\langle \\tau \\rangle -(\\tau _0)^\\alpha |\\Gamma (1-\\alpha )|s^{\\alpha -1}) \\\\& =\\frac{1}{(\\langle \\tau \\rangle s-ika)(1-\\frac{(\\tau _0)^\\alpha |\\Gamma (1-\\alpha )|s^\\alpha }{\\langle \\tau \\rangle s-ika})}(\\langle \\tau \\rangle -(\\tau _0)^\\alpha |\\Gamma (1-\\alpha )|s^{\\alpha -1}),\\end{split}$ where we use that $\\langle \\tau \\rangle s$ and $ika$ are comparable, and neglect the term $k^2$ since $k^2\\ll |s|$ , $|k|$ .",
"Using $1/(1-y)\\simeq 1+y$ with $y\\rightarrow 0$ , and $s^{2\\alpha -1}/(\\langle \\tau \\rangle s-ika)\\propto s^{2\\alpha -2}\\ll s^{\\alpha -1}$ , Eq.",
"(REF ) reduces to $\\widetilde{\\widehat{ P}}_{\\rm or}(k,s)\\sim \\frac{1}{\\langle \\tau \\rangle s-ika}\\left(\\langle \\tau \\rangle -b_\\alpha s^{\\alpha -1}+\\frac{(\\tau _0)^{\\alpha }|\\Gamma (1-\\alpha )|\\langle \\tau \\rangle }{\\langle \\tau \\rangle s-ika}s^\\alpha \\right).$ Regrouping, we have $\\widetilde{\\widehat{ P}}_{\\rm or}(k,s)\\sim \\frac{1}{\\langle \\tau \\rangle s-ika}\\left(\\langle \\tau \\rangle +\\frac{iks^{\\alpha -1}a(\\tau _0)^{\\alpha }|\\Gamma (1-\\alpha )|}{\\langle \\tau \\rangle s-ika}\\right),$ which gives Eq.",
"(REF ) in the main text."
],
[
"Moments of the position for equilibrium case",
"We further consider the moments of the position for an equilibrium situation by using [18] $\\langle \\widehat{x}^q(s)\\rangle =(-i)^q\\frac{\\partial ^q \\widetilde{\\widehat{P}}_{\\rm {eq }}(k,s)}{\\partial k^q}\\Big |_{k=0}$ to check our theoretical result Eq.",
"(REF ).",
"For $q=1$ , using Eqs.",
"(REF ) and (REF ), we have $\\langle \\widehat{x}(s)\\rangle _{{\\rm eq}}=\\frac{a}{\\langle \\tau \\rangle s^2},$ from which yields $\\langle x(t)\\rangle _{{\\rm eq}}=a\\frac{t}{\\langle \\tau \\rangle }.$ This is the exact result growing linearly with time $t$ ; see also Eq.",
"(REF ).",
"Note that for an ordinary process, the asymptotic behavior of $\\langle x(t)\\rangle $ is $at/\\langle \\tau \\rangle $ .",
"When $q=2$ , from Eq.",
"(REF ) the second moment of $x(t)$ is $\\langle x^2(t)\\rangle _{{\\rm eq}}\\sim \\frac{a^2 t^2}{\\langle \\tau \\rangle ^2}+\\frac{2a^2b_\\alpha t^{3-\\alpha }}{\\langle \\tau \\rangle ^3\\Gamma (4-\\alpha )}.$ Utilizing Eqs.",
"(REF ) and (REF ), the MSD is $\\begin{split}\\langle (x(t)-\\langle x(t)\\rangle )^2 \\rangle _{{\\rm eq}} & =\\langle x^2(t)\\rangle _{{\\rm eq}}-\\langle x(t)\\rangle ^2_{{\\rm eq}} \\\\&\\sim \\frac{2a^2b_\\alpha t^{3-\\alpha }}{\\langle \\tau \\rangle ^3 \\Gamma (4-\\alpha )}.\\end{split}$ It gives that the process shows super-diffusion, increasing faster than the ordinary process.",
"As expected, Eq.",
"(REF ) is consistent with Eq.",
"(REF ) obtained from the infinite density Eq.",
"(REF ).",
"We further consider how motionless particles contribute to the MSD.",
"Taking the inverse Laplace-Fourier transform on the first term on the right-hand side of Eq.",
"(REF ) gives $\\int _t^\\infty h_{{\\rm eq}}(\\tau )d\\tau \\delta (x)$ .",
"From Eq.",
"(REF ) one can show that $\\begin{split}&\\langle (x-\\langle x\\rangle _{{\\rm eq}})^2\\rangle _{\\rm {eq}}\\\\&\\ge \\int _{-\\infty }^\\infty (x-\\langle x\\rangle _{\\rm eq} )^2 \\int _t^\\infty h_{{\\rm eq}}(\\tau )d\\tau \\delta (x)dx\\\\&=\\frac{a^2(\\tau _0)^\\alpha t^{3-\\alpha }}{\\langle \\tau \\rangle ^3 (\\alpha -1)},\\end{split}$ where we used the relation Eq.",
"(REF ).",
"Since the MSD grows like $t^{3-\\alpha }$ , clearly this term describing non-moving particles controls the leading term of the MSD Eq.",
"(REF ).",
"While for the non-equilibrium case we get a contribution of motionless particles to the MSD which increases like $t^{2-\\alpha }$ and is negligible."
],
[
"The calculation of MSDs using the infinite densities",
"In principle, the MSDs can be calculated according to Eq.",
"(REF ).",
"However, in the long time limit it is easy to calculate MSDs based on the non-normalized density.",
"This method is also valid for high-order moments [49].",
"From Eq.",
"(REF ) the scaling behavior of $\\xi =1-(x/a)/(t/\\langle \\tau \\rangle )$ gives $P_{\\rm {or}}(\\xi ,t)\\sim \\frac{(\\tau _0)^\\alpha t^{1-\\alpha }}{\\langle \\tau \\rangle }\\mathcal {I}_{\\rm {or},\\alpha }(\\xi ),$ where $0<\\xi <1$ .",
"The second moment of $\\xi $ is $\\begin{split}\\langle \\xi ^2\\rangle _{\\rm {or}}& \\sim \\int _0^1\\xi ^2 P_{\\rm {or}}(\\xi ,t) d\\xi \\\\& =\\frac{2(\\tau _0)^\\alpha t^{1-\\alpha }}{\\langle \\tau \\rangle (2-\\alpha )(3-\\alpha )}.\\end{split}$ and $\\langle x(t)\\rangle \\sim at/\\langle \\tau \\rangle $ .",
"Using the relation $\\langle \\xi ^2\\rangle _{\\rm {or}}=\\langle (1-\\frac{x/a}{t/\\langle \\tau \\rangle })^2\\rangle _{\\rm {or}}$ , we have $\\begin{split}\\left\\langle \\left(x-\\frac{at}{\\langle \\tau \\rangle }\\right)^2\\right\\rangle _{\\rm {or}} & =\\left\\langle \\left(a\\xi \\frac{t}{\\langle \\tau \\rangle }\\right)^2\\right\\rangle _{\\rm {or}} \\\\& =\\left(a\\frac{t}{\\langle \\tau \\rangle }\\right)^2\\langle (\\xi )^2\\rangle _{\\rm {or}}\\\\&\\sim \\frac{2a^2(\\tau _0)^\\alpha t^{3-\\alpha }}{\\langle \\tau \\rangle ^3(2-\\alpha )(3-\\alpha )}\\\\&=\\frac{2a^2b_\\alpha (\\alpha -1) t^{3-\\alpha }}{\\langle \\tau \\rangle ^3 \\Gamma (4-\\alpha )}.\\end{split}$ This means that the MSD of the non-equilibrium case is determined by the far tail of the density, i.e., the infinite density.",
"Similarly, the MSD of an equilibrium process follows $\\begin{split}\\left\\langle \\left(x-\\frac{at}{\\langle \\tau \\rangle }\\right)^2\\right\\rangle _{\\rm {eq}} &\\sim \\int _{-\\infty }^{\\infty }\\left(x-\\frac{at}{\\langle \\tau \\rangle }\\right)^2\\frac{(\\tau _0)^\\alpha t^{1-\\alpha }}{\\langle \\tau \\rangle (\\alpha -1)}\\delta (x)dx\\\\&~~~~+\\left(a\\frac{t}{\\langle \\tau \\rangle }\\right)^2\\langle \\xi ^2\\rangle _{\\rm eq}\\\\&\\sim \\frac{2a^2b_\\alpha t^{3-\\alpha }}{\\langle \\tau \\rangle ^3 \\Gamma (4-\\alpha )}\\end{split}$ with $\\langle \\xi ^2\\rangle _{\\rm eq}=\\int _0^1\\xi ^2 \\frac{(\\tau _0)^\\alpha }{t^{\\alpha -1}\\langle \\tau \\rangle }\\mathcal {I}_{{\\rm eq},\\alpha }(\\xi )d\\xi $ .",
"Here we want to stress that in the case of integrable observables, one can use the non-normalized state described by the infinite density."
]
] | 1906.04249 |
[
[
"Resistance distance-based graph invariants and spanning trees of graphs\n derived from the strong product of $P_2$ and $C_n$"
],
[
"Abstract Let $G_n$ be a graph obtained by the strong product of $P_2$ and $C_n$, where $n\\geqslant3$.",
"In this paper, explicit expressions for the Kirchhoff index, multiplicative degree-Kirchhoff index and number of spanning trees of $G_n$ are determined, respectively.",
"It is surprising to find that the Kirchhoff (resp.",
"multiplicative degree-Kirchhoff) index of $G_n$ is almost one-sixth of its Wiener (resp.",
"Gutman) index.",
"Moreover, let $\\mathcal{G}^r_n$ be the set of subgraphs obtained from $G_n$ by deleting any $r$ vertical edges of $G_n$, where $0\\leqslant r\\leqslant n$.",
"Explicit formulas for the Kirchhoff index and the number of spanning trees for any graph $G^r_n\\in \\mathcal{G}^r_{n}$ are completely established, respectively.",
"Finally, it is interesting to see that the Kirchhoff index of $G^r_n$ is almost one-sixth of its Wiener index."
],
[
"Introduction",
"Let $G$ be a simple connected graph with vertex set $V(G)$ and edge set $E(G)$ .",
"Then $|V(G)|$ and $|E(G)|$ are called the order and the size of $G$ , respectively.",
"For a simple graph $G$ of order $n$ , its adjacent matrix $A(G)=(a_{ij})_{n\\times n}$ is a $(0,1)$ -matrix with $a_{ij}=1$ if and only if two vertices $i$ and $j$ are adjacent in $G$ , and the matrix $D(G):=\\textrm {diag}(d_{1},d_{2},\\ldots ,d_{n})$ is called the diagonal matrix of vertex degrees, where $d_{i}$ is the degree of vertex $i$ in $G$ for $1\\leqslant i\\leqslant n$ .",
"Then the Laplacian matrix of $G$ is defined as ${L}(G)=D(G)-A(G)$ , whereas the normalized Laplacian matrix of $G$ is defined to be $\\mathcal {L}(G)=D(G)^{-\\frac{1}{2}}L(G)D(G)^{-\\frac{1}{2}}$ .",
"It should be stressed that ${(d_{i})}^{-\\frac{1}{2}}=0$ for $d_i=0$ [6].",
"Hence, we have $&\\hspace{42.67912pt}(\\mathcal {L}(G))_{ij}=\\left\\lbrace \\begin{array}{ll}{1,} &\\textrm {if i=j};\\\\{-\\frac{1}{\\sqrt{d_{i}d_{j}}}}, &\\textrm {if i\\ne j and v_i\\sim v_j};\\\\{0,} & \\textrm {otherwise}.\\\\\\end{array}\\right.",
"&$ Let $d_{ij}$ denote the distance between two vertices $i$ and $j$ in $G$ .",
"The Wiener index of $G$ , introduced in [32], is defined as $W(G)=\\sum _{i<j}d_{ij}$ .",
"Later, Gutman [11] gave a weighted version of the Wiener index which is defined as $\\textrm {Gut}(G)=\\sum _{i<j}d_id_jd_{ij}$ and now known as Gutman index.",
"If $G$ is an $n$ -vertex tree, Gutman [11] proved that $\\textrm {Gut}(G)=4W(G)-(2n-1)(n-1)$ .",
"In 1993, Klein and Randić [18] proposed the concept of resistance distance by considering the electronic network.",
"The resistance distance $r_{ij}$ is the effective resistance between two vertices $i$ and $j$ after every edge of a graph $G$ was putted one unit resistor.",
"This parameter is intrinsic to both graph theory and mathematical chemistry.",
"Similar to the definition of the Wiener index, Klein and Randić [18] defined $K\\!f(G):=\\sum _{i<j}r_{ij}$ to be the Kirchhoff index of $G$ .",
"They also found that $K\\!f(G)\\leqslant {W}(G)$ with equality if and only if $G$ is a tree.",
"For a connected graph $G$ of order $n$ , Klein [16] and Lov$\\acute{a}$ sz [23], independently, obtained that ${K\\!f}(G)=n\\sum ^{n}_{i=2}\\frac{1}{\\rho _i},$ where $0=\\rho _1<\\rho _2\\leqslant \\cdots \\leqslant \\rho _n$ $(n\\geqslant 2)$ are the eigenvalues of $L(G)$ .",
"In 2007, Chen and Zhang [5] proposed a weighted version of the Kirchhoff index, which is defined as $K\\!f^{*}(G)=\\sum _{i<j}d_{i}d_{j}r_{ij}$ .",
"This index is now known as multiplicative degree-Kirchhoff index.",
"Another weighted form of Kirchhoff index is the addictive degree-Kirchhoff index [10].",
"Multiplicative degree-Kirchhoff index is closely related to the spectrum of the normalized Laplacian matrix $\\mathcal {L}(G)$ .",
"For a connected graph $G$ of order $n$ and size $m$ , Chen and Zhang [5] showed that ${K\\!f}^{*}(G)=2m\\sum ^{n}_{i=2}\\frac{1}{\\lambda _i},$ where $0=\\lambda _1<\\lambda _2\\leqslant \\cdots \\leqslant \\lambda _n$ $(n\\geqslant 2)$ are the eigenvalues of $\\mathcal {L}(G)$ .",
"Some techniques to determine the Kirchhoff index and multiplicative degree-Kirchhoff index were given in [2], [3], [7], [8], [25].",
"Some other topics on the Kirchhoff index and the multiplicative degree-Kirchhoff index of a graph may be referred to [26], [30], [34], [36], [37] and references therein.",
"In the last decades, many researchers are devoted to give closed formulas for the Kirchhoff index and the multiplicative degree-Kirchhoff index of graphs with special structures, such as cycles [17], ladder graphs [9], ladder-like chains [4], [21], liner phenylenes [19], [29], [38], linear polyomino chains [15], [33], linear pentagonal chains [12], [31], linear hexagonal chains [14], [20], [33], linear octagonal-quadrilateral networks [22], linear crossed chains [27], [28], and composite graphs [36],.",
"Given two graphs $G$ and $H$ , the strong product of $G$ and $H$ , denoted by $G\\boxtimes H$ , is the graph with vertex set $V(G)\\times V(H)$ , where two distinct vertices $(u_1,v_1)$ and $(u_2,v_2)$ are adjacent whenever $u_1$ and $u_2$ are equal or adjacent in $G$ , and $v_1$ and $v_2$ are equal or adjacent in $H$ .",
"Pan and Li [28] determined the Kirchhoff index, multiplicative degree-Kirchhoff index and number of spanning trees of graph $P_2\\boxtimes P_n$ .",
"Motivated by [28], we consider the graph $P_2\\boxtimes C_n$ , where $n\\geqslant 3$ .",
"Let $G_n=P_2\\boxtimes C_n$ , where the graph $G_n$ is depicted in Figure 1.",
"Obviously, $|V{(G_n)}|=2n$ and $|E{(G_n)}|=5n$ .",
"Let $E^{\\prime }$ be the set of vertical edges of $G_n$ , where $E^{\\prime }=\\lbrace ii^{\\prime }:i=1,2,\\dots ,n\\rbrace $ .",
"Let $\\mathcal {G}^r_n$ be the set of subgraphs obtained from $G_n$ by deleting $r$ vertical edges of $G_n$ , where $0\\leqslant r\\leqslant n$ .",
"It is easy to obtain that $\\mathcal {G}^0_n=\\lbrace G_n\\rbrace $ .",
"Figure: Graph G n G_n with some labelled vertices.In this paper, explicit expressions for the Kirchhoff index, the multiplicative degree-Kirchhoff index and the number of spanning trees of $G_n$ are determined, respectively.",
"It is nice to find that the Kirchhoff (resp.",
"multiplicative degree-Kirchhoff) index of $G_n$ is almost one-sixth of its Wiener (resp.",
"Gutman) index.",
"Additionally, for any graph $G^r_n\\in \\mathcal {G}^r_{n}$ , its Kirchhoff index and number of spanning trees are completely determined, respectively.",
"Moreover, the Kirchhoff index of $G^r_n$ is shown to be almost one-sixth of its Wiener index."
],
[
"Preliminaries",
"For convenience, let $V_1=\\lbrace 1,2,\\ldots ,n\\rbrace $ and $V_2=\\lbrace 1^\\prime ,2^\\prime ,\\ldots ,n^\\prime \\rbrace $ .",
"Then $L(G_n)$ and $\\mathcal {L}(G_n)$ can be written as below: $&\\hspace{28.45274pt}{L}(G_n)=\\left(\\begin{array}{cc}{L}_{{11}} & {L}_{{12}} \\\\{L}_{{21}} & {L}_{{22}}\\end{array}\\right),&&\\mathcal {L}(G_n)=\\left(\\begin{array}{cc}\\mathcal {L}_{{11}} & \\mathcal {L}_{{12}} \\\\\\mathcal {L}_{{21}} & \\mathcal {L}_{{22}}\\end{array}\\right),&$ where ${L}_{{ij}}$ (resp.",
"$\\mathcal {L}_{{ij}}$ ) denotes the submatrix whose rows correspond to vertices in $V_i$ and columns correspond to vertices in $V_j$ .",
"By the construction of $G_n$ , one can easily verify that $L_{11}=L_{22}$ , ${L}_{12}=L_{21}$ , $\\mathcal {L}_{11}=\\mathcal {L}_{22}$ and $\\mathcal {L}_{12}=\\mathcal {L}_{21}$ .",
"Let $T=\\left(\\begin{array}{cc}\\frac{1}{\\sqrt{2}}I_{n} & \\frac{1}{\\sqrt{2}}I_{n} \\\\\\frac{1}{\\sqrt{2}}I_{n} & -\\frac{1}{\\sqrt{2}}I_{n}\\end{array}\\right).$ Then $&\\hspace{28.45274pt}T{L}(G_n)T=\\left(\\begin{array}{cc}{L}_{A} & 0 \\\\0 & {L}_{S}\\end{array}\\right),&&T\\mathcal {L}(G_n)T=\\left(\\begin{array}{cc}\\mathcal {L}_{A} & 0 \\\\0 & \\mathcal {L}_{S}\\end{array}\\right),&$ where ${L}_A={L}_{{11}}+{L}_{{12}}$ , ${L}_S={L}_{{11}}-{L}_{{12}}$ , $\\mathcal {L}_A=\\mathcal {L}_{{11}}+\\mathcal {L}_{{12}}$ and $\\mathcal {L}_S=\\mathcal {L}_{{11}}-\\mathcal {L}_{{12}}$ .",
"Let $\\Phi (B):=\\det (xI-B)$ be the characteristic polynomial of a square matrix $B$ , where $I$ is an unit matrix with the same order as that of $B$ .",
"Similar to the decomposition theorem obtained in [13], [27], we can obtain the decomposition theorem of $G_n$ as below.",
"Here, we omit the proof.",
"Lemma 2.1 Let ${L}_A$ , ${L}_S$ , $\\mathcal {L}_A$ and $\\mathcal {L}_S$ be defined as above.",
"Then $&\\hspace{28.45274pt}\\Phi (L(G_n))={\\Phi ({L}_A)}\\cdot {\\Phi ({L}_S)},& &\\Phi (\\mathcal {L}(G_n))={\\Phi (\\mathcal {L}_A)}\\cdot {\\Phi (\\mathcal {L}_S)}.&$ Lemma 2.2 [18] Let $G$ be an $n$ -vertex connected graph.",
"Then $K\\!f(G)=n\\sum ^n_{i=2}\\frac{1}{\\rho _i}$ .",
"Lemma 2.3 [5] Let $G$ be an $n$ -vertex connected graph with $m$ edges.",
"Then $K\\!f^{*}(G)=2m\\sum ^n_{i=2}\\frac{1}{\\lambda _i}$ .",
"Lemma 2.4 [6] Let $G$ be an $n$ -vertex connected graph.",
"Then $\\tau (G)=\\frac{1}{n}\\prod ^n_{i=2}\\rho _i$ , where $\\tau (G)$ is the number of spanning trees of $G$ ."
],
[
"Resistance distance-based graph invariants and number of spanning trees of $G_n$",
"In this section, we will determine the Kirchhoff index, multiplicative degree-Kirchhoff index and number of spanning trees of $G_n$ .",
"The Laplacian matrix $L(C_n)$ of $C_n$ is very important in this section.",
"It was proved in [1] that the eigenvalues of $L(C_n)$ are $4\\sin ^2(\\frac{\\pi i}{n})$ , where $i=1,2,\\ldots ,n$ .",
"In the rest of this paper, let $\\alpha _i:=4\\sin ^2(\\frac{\\pi i}{n})$ .",
"Then $\\alpha _n=0$ and $\\alpha _i>0$ for $i=1,2,\\ldots ,n-1$ ."
],
[
"Kirchhoff index and number of spanning trees of $G_n$ ",
"In this subsection, we will give the explicit formulas for the Kirchhoff index and number of spanning trees of $G_n$ .",
"Moreover, we will prove that the Kirchhoff index of $G_n$ is almost one-sixth of its Wiener index.",
"First, one can see that ${L}_{{11}}=\\left(\\begin{array}{ccccccc}5& -1& 0 & 0& \\dots & 0& -1\\\\-1& 5& -1 & 0& \\dots & 0& 0\\\\0& -1 & 5& -1 & \\dots & 0& 0\\\\0& 0& -1 & 5& \\dots & 0& 0 \\\\\\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\0& 0& 0& 0 & \\dots & 5& -1\\\\-1& 0& 0& 0 & \\dots & -1& 5\\end{array}\\right)_{n\\times {n}}$ and ${L}_{{12}}=\\left(\\begin{array}{ccccccc}-1& -1& 0 & 0& \\dots & 0& -1\\\\-1& -1& -1 & 0& \\dots & 0& 0\\\\0& -1 & -1& -1 & \\dots & 0& 0\\\\0& 0& -1 & -1& \\dots & 0& 0 \\\\\\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\0& 0& 0& 0 & \\dots & -1& -1\\\\-1& 0& 0& 0 & \\dots & -1& -1\\end{array}\\right)_{n\\times {n}}.$ Since $L_{A}=L_{11}+L_{12}$ and $L_{S}=L_{11}-L_{12}$ , then ${L}_{{A}}=\\left(\\begin{array}{ccccccc}4& -2& 0 & 0& \\dots & 0& -2\\\\-2& 4& -2 & 0& \\dots & 0& 0\\\\0& -2 & 4& -2 & \\dots & 0& 0\\\\0& 0& -2 & 4& \\dots & 0& 0 \\\\\\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\0& 0& 0& 0 & \\dots & 4& -2\\\\-2& 0& 0& 0 & \\dots & -2& 4\\end{array}\\right)_{n\\times {n}}$ and ${L}_{{S}}=\\left(\\begin{array}{ccccccc}6& 0& 0 & 0& \\dots & 0& 0\\\\0& 6& 0 & 0& \\dots & 0& 0\\\\0& 0 & 6& 0 & \\dots & 0& 0\\\\0& 0& 0 & 6& \\dots & 0& 0 \\\\\\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\0& 0& 0& 0 & \\dots & 6& 0\\\\0& 0& 0& 0 & \\dots & 0& 6\\end{array}\\right)_{n\\times {n}}.$ Note that ${L}_A=2L(C_n)$ and ${L}_S$ is a diagonal matrix.",
"By Lemma REF , $2\\alpha _1,2\\alpha _2,\\ldots ,2\\alpha _n,6,6,\\ldots ,6$ are all the eigenvalues of ${L}{(G_n)}$ , i.e.",
"$0,2\\alpha _1,2\\alpha _2,\\ldots ,2\\alpha _{n-1},6,6,\\ldots ,6$ are all the eigenvalues of ${L}{(G_n)}$ .",
"Then we can get the following theorem.",
"Theorem 3.1 For $n\\ge 3$ , let $G_n=P_2\\boxtimes C_n$ .",
"Then $(1)$ $K\\!f(G_n)=\\frac{n^3+4n^2-n}{12}.$ $(2)$ $\\tau (G_n)=n\\cdot 2^{2n-2}\\cdot 3^n.$ $(3)$ $\\lim _{n\\rightarrow \\infty }\\frac{K\\!f(G_n)}{W(G_n)}=\\frac{1}{6}.$ $(1)$ Note that $|V(G_n)|=2n$ and $K\\!f(C_n)=\\frac{n^3-n}{12}$ (see [35]).",
"By Lemma REF , we have $\\hspace{42.67912pt}K\\!f(G_n)&=2n\\bigg (\\sum ^{n-1}_{i=1}\\frac{1}{2\\alpha _i}+\\frac{n}{6}\\bigg )\\nonumber &\\\\&=n\\sum ^{n-1}_{i=1}\\frac{1}{\\alpha _i}+\\frac{n^2}{3}\\nonumber &\\\\&=K\\!f(C_n)+\\frac{n^2}{3}\\nonumber &\\\\&=\\frac{n^3+4n^2-n}{12}.&$ $(2)$ It follows from Lemma REF that $\\hspace{42.67912pt}\\tau (G_{n})&=\\frac{1}{2n}\\prod ^{n-1}_{i=1}(2\\alpha _i)\\cdot 6^n\\nonumber &\\\\&=2^{n-2}\\cdot 6^n\\cdot \\frac{1}{n}\\prod ^{n-1}_{i=1}\\alpha _i\\nonumber &\\\\&=2^{n-2}\\cdot 6^n\\cdot \\tau (C_n)\\nonumber &\\\\&=n\\cdot 2^{2n-2}\\cdot 3^{n}.&$ $(3)$ We now calculate the value of $W(G_n)$ .",
"If $n$ is odd, for each vertex $i$ in $G_n$ , we have $f(i,n)=1+4\\sum ^{\\frac{n-1}{2}}_{k=1}{k}=\\frac{n^2+1}{2}.$ Table: Kirchhoff index and number of spanning trees of graphs from G 3 G_3 to G 11 G_{11}.Then $W(G_n)=\\frac{2\\sum ^{n}_{i=1}f(i,n)}{2}=\\sum ^{n}_{i=1}f(i,n)=\\frac{n^3+n}{2}.$ If $n$ is even, for each vertex $i$ in $G_n$ , we have $f(i,n)=1+4\\sum ^{\\frac{n-2}{2}}_{k=1}{k}+2\\cdot \\frac{n}{2}=\\frac{n^2+2}{2}.$ Then $W(G_n)=\\frac{2\\sum ^{n}_{i=1}f(i,n)}{2}=\\sum ^{n}_{i=1}f(i,n)=\\frac{n^3+2n}{2}.$ Note that $K\\!f(G_n)=\\frac{n^3+4n^2-n}{12}$ .",
"Hence, we have $\\lim _{n\\rightarrow \\infty }\\frac{K\\!f(G_n)}{W(G_n)}=\\frac{1}{6}$ as desired.",
"Kirchhoff index and number of spanning trees of graphs from $G_3$ to $G_{11}$ are listed in Table 1, respectively."
],
[
"Multiplicative degree-Kirchhoff index of $G_n$",
"In this subsection, we will determine the multiplicative degree-Kirchhoff index of $G_n$ and show that the multiplicative degree-Kirchhoff index of $G_n$ is almost one-sixth of its Gutman index.",
"Note that $\\mathcal {L}_{{11}}=\\left(\\begin{array}{ccccccc}1& -\\frac{1}{5}& 0 & 0& \\dots & 0& -\\frac{1}{5}\\\\-\\frac{1}{5}& 1& -\\frac{1}{5}& 0& \\dots & 0& 0\\\\0& -\\frac{1}{5}& 1& -\\frac{1}{5} & \\dots & 0& 0\\\\0& 0& -\\frac{1}{5} & 1& \\dots & 0& 0 \\\\\\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\0& 0& 0& 0 & \\dots & 1& -\\frac{1}{5}\\\\-\\frac{1}{5}& 0& 0& 0 & \\dots & -\\frac{1}{5}& 1\\end{array}\\right)_{n\\times {n}}$ and $\\mathcal {L}_{{12}}=\\left(\\begin{array}{ccccccc}-\\frac{1}{5}& -\\frac{1}{5}& 0 & 0& \\dots & 0& -\\frac{1}{5}\\\\-\\frac{1}{5}& -\\frac{1}{5}& -\\frac{1}{5}& 0& \\dots & 0& 0\\\\0& -\\frac{1}{5}& -\\frac{1}{5}& -\\frac{1}{5} & \\dots & 0& 0\\\\0& 0& -\\frac{1}{5} & -\\frac{1}{5}& \\dots & 0& 0 \\\\\\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\0& 0& 0& 0 & \\dots & -\\frac{1}{5}& -\\frac{1}{5}\\\\-\\frac{1}{5}& 0& 0& 0 & \\dots & -\\frac{1}{5}& -\\frac{1}{5}\\end{array}\\right)_{n\\times {n}}.$ Table: Multiplicative degree-Kirchhoff index of graphs from G 3 G_3 to G 15 G_{15}.Since $\\mathcal {L}_A=\\mathcal {L}_{{11}}+\\mathcal {L}_{{12}}$ and $\\mathcal {L}_S=\\mathcal {L}_{{11}}-\\mathcal {L}_{{12}}$ , then we have $\\mathcal {L}_{A}=\\left(\\begin{array}{ccccccc}\\frac{4}{5}& -\\frac{2}{5}& 0 & 0& \\dots & 0& -\\frac{2}{5}\\\\-\\frac{2}{5}& \\frac{4}{5}& -\\frac{2}{5}& 0& \\dots & 0& 0\\\\0& -\\frac{2}{5}& \\frac{4}{5}& -\\frac{2}{5} & \\dots & 0& 0\\\\0& 0& -\\frac{2}{5} & \\frac{4}{5}& \\dots & 0& 0 \\\\\\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\0& 0& 0& 0 & \\dots & \\frac{4}{5}& -\\frac{2}{5}\\\\-\\frac{2}{5}& 0& 0& 0 & \\dots & -\\frac{2}{5}& \\frac{4}{5}\\end{array}\\right)_{n\\times {n}}$ and $\\mathcal {L}_{S}=\\left(\\begin{array}{ccccccc}\\frac{6}{5}& 0& 0 & 0& \\dots & 0& 0\\\\0& \\frac{6}{5}& 0& 0& \\dots & 0& 0\\\\0& 0& \\frac{6}{5}& 0 & \\dots & 0& 0\\\\0& 0& 0& \\frac{6}{5}& \\dots & 0& 0 \\\\\\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\0& 0& 0& 0 & \\dots & \\frac{6}{5}& 0\\\\0& 0& 0& 0 & \\dots & 0& \\frac{6}{5}\\end{array}\\right)_{n\\times {n}}.$ Note that $\\mathcal {L}_A=\\frac{2}{5}L(C_n)$ and $\\mathcal {L}_S$ is a diagonal matrix.",
"By Lemma REF , $\\frac{2}{5}\\alpha _1,\\frac{2}{5}\\alpha _2,\\ldots ,\\frac{2}{5}\\alpha _n,\\frac{6}{5},\\frac{6}{5},\\ldots ,\\frac{6}{5}$ are all the eigenvalues of $\\mathcal {L}{(G_n)}$ , i.e.",
"$0,\\frac{2}{5}\\alpha _1,\\frac{2}{5}\\alpha _2,\\ldots ,\\frac{2}{5}\\alpha _{n-1},\\frac{6}{5},\\frac{6}{5},\\ldots ,\\frac{6}{5}$ are all the eigenvalues of $\\mathcal {L}{(G_n)}$ .",
"Then we can get the following theorem.",
"Theorem 3.2 For $n\\ge 3$ , let $G_n=P_2\\boxtimes C_n$ .",
"Then $(1)$ $K\\!f^{*}(G_n)=\\frac{25n^3+100n^2-25n}{12}.$ $(2)$ $\\lim _{n\\rightarrow \\infty }\\frac{K\\!f^{*}(G_n)}{Gut(G_n)}=\\frac{1}{6}.$ $(1)$ Since $|E(G_n)|=5n$ and $K\\!f(C_n)=\\frac{n^3-n}{12}$ , by Lemma REF , we have $\\hspace{42.67912pt}K\\!f^*(G_n)&=10n\\bigg (\\sum ^{n-1}_{i=1}\\frac{5}{2\\alpha _i}+\\frac{5n}{6}\\bigg )\\nonumber &\\\\&=25n\\sum ^{n-1}_{i=1}\\frac{1}{\\alpha _i}+\\frac{25n^2}{3}\\nonumber &\\\\&=25K\\!f(C_n)+\\frac{25n^2}{3}\\nonumber &\\\\&=\\frac{25n^3+100n^2-25n}{12}.&$ $(2)$ It is easy to calculate that $\\textrm {Gut}(G_n)=25W(G_n)$ since $G_n$ is regular.",
"Note that $K\\!f^{*}(G_n)=\\frac{25n^3+100n^2-25n}{12}$ .",
"Together with Theorem REF , we have $\\lim _{n\\rightarrow \\infty }\\frac{K\\!f^{*}(G_n)}{\\textrm {Gut}(G_n)}=\\frac{1}{6}$ as desired.",
"Multiplicative degree-Kirchhoff index of graphs from $G_3$ to $G_{15}$ are listed in Table 2."
],
[
"Kirchhoff index and number of spanning trees of $G^r_n$",
"In this section, we will determine the Kirchhoff index and number of spanning trees for any graph $G^r_n\\in \\mathcal {G}^r_{n}$ .",
"Moreover, we will show that the Kirchhoff index of $G^r_n$ is nearly one-sixth of its Wiener index.",
"Similar to Lemma REF , we can obtain that the spectrum of $L(G^r_n)$ consists of the eigenvalues of both $L_A(G^r_n)$ and $L_S(G^r_n)$ .",
"Let $d_i$ be the degree of vertex $i$ in $G^r_n$ .",
"Then $d_i=4$ or $d_i=5$ in $G^r_n$ .",
"It is routine to check that ${L}_{{11}}(G^r_n)=\\left(\\begin{array}{ccccccc}d_1& -1& 0 & 0& \\dots & 0& -1\\\\-1& d_2& -1 & 0& \\dots & 0& 0\\\\0& -1 & d_3& -1 & \\dots & 0& 0\\\\0& 0& -1 & d_4& \\dots & 0& 0 \\\\\\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\0& 0& 0& 0 & \\dots & d_{n-1}& -1\\\\-1& 0& 0& 0 & \\dots & -1& d_n\\end{array}\\right)_{n\\times {n}}$ and ${L}_{{12}}(G^r_n)=\\left(\\begin{array}{ccccccc}t_1& -1& 0 & 0& \\dots & 0& -1\\\\-1& t_2& -1 & 0& \\dots & 0& 0\\\\0& -1 & t_3& -1 & \\dots & 0& 0\\\\0& 0& -1 & t_4& \\dots & 0& 0 \\\\\\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\0& 0& 0& 0 & \\dots & t_{n-1}& -1\\\\-1& 0& 0& 0 & \\dots & -1& t_n\\end{array}\\right)_{n\\times {n}}.$ where $t_{i}=0$ if $d_{i}=4$ and $t_{i}=-1$ if $d_{i}=5$ .",
"Then for any graph $G^r_n\\in \\mathcal {G}^r_{n}$ , $d_{i}+t_i=4$ holds for all $1\\leqslant i\\leqslant n$ .",
"Since $L_A(G^r_n)=L_{11}(G^r_n)+L_{12}(G^r_n)$ and $L_S(G^r_n)=L_{11}(G^r_n)-L_{12}(G^r_n)$ , then ${L}_{{A}}(G^r_n)=\\left(\\begin{array}{ccccccc}4& -2& 0 & 0& \\dots & 0& -2\\\\-2& 4& -2 & 0& \\dots & 0& 0\\\\0& -2 & 4& -2 & \\dots & 0& 0\\\\0& 0& -2 & 4& \\dots & 0& 0 \\\\\\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\0& 0& 0& 0 & \\dots & 4& -2\\\\-2& 0& 0& 0 & \\dots & -2& 4\\end{array}\\right)_{n\\times {n}}$ and ${L}_{{S}}(G^r_n)=\\left(\\begin{array}{ccccccc}s_1& 0& 0 & 0& \\dots & 0& 0\\\\0& s_2& 0 & 0& \\dots & 0& 0\\\\0& 0 & s_3& 0 & \\dots & 0& 0\\\\0& 0& 0 & s_4& \\dots & 0& 0 \\\\\\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\0& 0& 0& 0 & \\dots & s_{n-1}& 0\\\\0& 0& 0& 0 & \\dots & 0& s_n\\end{array}\\right)_{n\\times {n}}.$ where $s_{i}=4$ if $d_{i}=4$ and $s_{i}=6$ if $d_{i}=5$ .",
"Note that ${L}_A(G^r_n)=2L(C_n)$ and ${L}_S$ is a diagonal matrix.",
"Then, $2\\alpha _1,2\\alpha _2,\\ldots ,2\\alpha _n,s_1,s_2,\\ldots ,s_n$ are all the eigenvalues of ${L}{(G^r_n)}$ , i.e.",
"$0,2\\alpha _1,2\\alpha _2,\\ldots ,2\\alpha _{n-1},s_1,s_2,\\ldots ,s_n$ are all the eigenvalues of ${L}{(G^r_n)}$ .",
"Next, we can get the following theorem.",
"Theorem 4.1 For any graph $G^r_{n}\\in \\mathcal {G}^r_{n}$ with $n\\geqslant 3$ , we have $(1)$ $K\\!f(G^r_n)=\\frac{n^3+4n^2+(2r-1)n}{12}.$ $(2)$ $\\tau (G^r_n)=n\\cdot 2^{2n+r-2}\\cdot 3^{n-r}.$ $(3)$ $\\lim _{n\\rightarrow \\infty }\\frac{K\\!f(G^r_n)}{W(G^r_n)}=\\frac{1}{6}.$ For any graph $G^r_{n}\\in \\mathcal {G}^r_{n}$ with $n\\ge 3$ , without loss of generality, assume that $s_1=s_2=\\cdots =s_r=4$ and $s_{r+1}=s_{r+2}=\\cdots =s_n=6$ .",
"$(1)$ Note that $|V(G^r_n)|=2n$ and $K\\!f(C_n)=\\frac{n^3-n}{12}$ .",
"By Lemma REF , we have $\\hspace{42.67912pt}K\\!f(G^r_n)&=2n\\bigg (\\sum ^{n-1}_{i=1}\\frac{1}{2\\alpha _i}+\\frac{n-r}{6}+\\frac{r}{4}\\bigg )\\nonumber &\\\\&=n\\sum ^{n-1}_{i=1}\\frac{1}{\\alpha _i}+\\frac{n(n-r)}{3}+\\frac{nr}{2}\\nonumber &\\\\&=K\\!f(C_n)+\\frac{2n^2+nr}{6}\\nonumber &\\\\&=\\frac{n^3+4n^2+(2r-1)n}{12}.&$ $(2)$ By Lemma REF , we obtain $\\hspace{42.67912pt}\\tau (G^r_{n})&=\\frac{1}{2n}\\prod ^{n-1}_{i=1}(2\\alpha _i)\\cdot 4^r\\cdot 6^{n-r}\\nonumber &\\\\&=2^{2n+r-2}\\cdot 3^{n-r}\\cdot \\frac{1}{n}\\prod ^{n-1}_{i=1}\\alpha _i\\nonumber &\\\\&=2^{2n+r-2}\\cdot 3^{n-r}\\cdot \\tau (C_n)\\nonumber &\\\\&=n\\cdot 2^{2n+r-2}\\cdot 3^{n-r}.&$ $(3)$ It is easy to calculate that $W(G^r_n)=W(G_n)+r$ .",
"Since $K\\!f(G^r_n)=\\frac{n^3+4n^2+(2r-1)n}{12}$ , thus we have $\\lim _{n\\rightarrow \\infty }\\frac{K\\!f(G^r_n)}{W(G^r_n)}=\\frac{1}{6}$ .",
"Remark 1 If $r=0$ , then $\\mathcal {G}^0_n=\\lbrace G_n\\rbrace $ .",
"One can see that Theorem REF is a corollary of Theorem REF ."
],
[
"Concluding remarks",
"In this paper, we first establish the explicit expressions for the Kirchhoff index, multiplicative degree-Kirchhoff index and number of spanning trees of $G_n$ , where $G_n=P_2\\boxtimes C_n$ and $n\\geqslant 3$ .",
"We find that the Kirchhoff (resp.",
"multiplicative degree-Kirchhoff) index is almost one-sixth of its Wiener (resp.",
"Gutman) index.",
"Later, we construct a family of graphs obtained from $G_n$ by deleting any $r$ vertical edges of $G_n$ , and show that the Kirchhoff indices of these graphs are almost one-sixth of their Gutman indices.",
"It would be interesting to determine their multiplicative degree-Kirchhoff indices and Gutman indices.",
"We will do it in the near future.",
"Motivated by the construction $P_2\\boxtimes P_n$ in [28] and $P_2\\boxtimes C_n$ in this paper, we propose the following question for further study: Question 5.1 For a simple connected graph $G$ , how can we determine the Laplacian spectrum and normalized Laplacian specteum of the graph $P_2\\boxtimes {G}$ ?"
],
[
"Acknowledgements",
"The authors would like to express their sincere gratitude to all the referees for their careful reading and insightful suggestions."
]
] | 1906.04339 |
[
[
"No signs of star formation being regulated in the most luminous quasars\n at z~2 with ALMA"
],
[
"Abstract We present ALMA Band~7 observations at $850\\mu$m of 20 luminous ($\\log\\, L_{\\rm bol}>46.9$ [erg s$^{-1}$]) unobscured quasars at $z\\sim2$.",
"We detect continuum emission for 19/20 quasars.",
"After subtracting an AGN contribution, we measure the total far-IR luminosity for 18 quasars, assuming a modified blackbody model, and attribute the emission as indicative of the star formation rate (SFR).",
"Our sample can be characterized with a log-normal SFR distribution having a mean of 140 $M_\\odot$ yr$^{-1}$ and a dispersion of 0.5 dex.",
"Based on an inference of their stellar masses, the SFRs are similar, in both the mean and dispersion, with star-forming main-sequence galaxies at the equivalent epoch.",
"Thus, there is no evidence for a systematic enhancement or suppression (i.e., regulation or quenching) of star formation in the hosts of the most luminous quasars at $z\\sim2$.",
"These results are consistent with the Magneticum cosmological simulation, while in disagreement with a widely recognized phenomenological model that predicts higher SFRs than observed here based on the high bolometric luminosities of this sample.",
"Furthermore, there is only a weak relation between SFR and accretion rate onto their supermassive black holes both for average and individual measurements.",
"We interpret these results as indicative of star formation and quasar accretion being fed from the available gas reservoir(s) in their host with a disconnect due to their different physical sizes, temporal scales, and means of gas processing."
],
[
"Introduction",
"It is well established that in an average sense the growth of supermassive black holes (SMBHs) is closely related to the evolution of galaxies and the build-up of their stellar component.",
"This is demonstrated by the tight relation between the SMBH mass and bulge stellar velocity dispersion or bulge mass in the local Universe , , , the close match between the cosmic evolution of the black hole accretion rate density and star formation rate density , , , and by the correlation between average SMBH and stellar growth in star-forming galaxies , , .",
"Understanding the interplay between SMBH accretion and star formation is essential for our picture of galaxy formation, particularly since feedback effects due to quasars are likely in play.",
"Theoretical models of galaxy evolution suggest star formation and black hole growth to be linked via a common supply of cold/molecular gas and triggered via major mergers , , .",
"These models generally require strong AGN feedback which self regulates black hole growth and quenches star formation (SF) in massive galaxies , , .",
"AGN winds and outflows are promising feedback mechanisms , , .",
"Such outflows have been observed in recent years in ionized gas , , , , , and molecular gas , , , , .",
"However, the demographics and the impact of such outflows are still not well understood; thus the importance and details of AGN feedback remains an open issue.",
"The most luminous quasars ($L_{\\rm bol}>10^{46}$ erg s$^{-1}$ ) should be particularly effective at impacting the ISM hence star formation , , , , thus it might be expected that a significant fraction of this population is undergoing quenching or have recently been quenched, leading to SFRs below the MS. On the other side, intense episodes of SF and of black hole growth would be fed by the same gas reservoir and potentially triggered by major mergers, so extremely luminous AGN activity would coincide with intense SF in their host galaxies [5], either in a model where the effect of AGN feedback is slow or delayed or in a scenario without AGN feedback.",
"At least for some very luminous AGN such intense SF is observed , , , , .",
"A further possibility is that there is no fundamental correlation between AGN and SF activity even for extreme AGN luminosities, implying very luminous AGN would have SFRs consistent with the SF main sequence.",
"In principle also a mix of these scenarios is possible, which would lead to a broad distribution of SFR.",
"Several studies have investigated the SF properties of luminous AGN at $z>1$ using Herschel far-IR photometry , , , , , , , , , , , , , Spitzer/IRS or sub-mm observations , , , , .",
"Based on individual detections, they typically find high SFRs, sometimes exceeding 1000 $M_\\odot $ /yr.",
"However, due to the modest SFR sensitivities achieved and the large fraction of non-detections in these observations, these sources are likely biased and do not represent the typical population.",
"While they demonstrate that intense star formation can exist in the hosts of luminous quasars, they do not provide information on the intrinsic SFR-distribution in this luminosity regime.",
"For their non-detection these studies have to rely on stacking.",
"However, this approach can be significantly biased if the intrinsic FIR luminosity distribution is highly skewed, so linear means are dominated by a few high-luminosity galaxies.",
"In fact, recent work based on deep 850$\\mu $ m observations with ALMA , and SCUBA-2 found intrinsic SFR distributions that are consistent with a log-normal rather than a normal distribution.",
"For their sample of moderate-luminosity AGN hosts they found a significant fraction (up to 50%) having SFRs below the MS, suggesting different SFR-distributions between star forming galaxies and moderate luminosity AGN, while their linear means are still consistent.",
"This might be an indication for the suppression of SF due to AGN feedback.",
"i.e.",
"quenching.",
"Based on the comparison of a model with and without AGN feedback in the EAGLE hydrodynamical cosmological simulation , recently argued that a broad width of the intrinsic SFR-distribution, in particular at high stellar mass ($>2\\times 10^{10} M_\\odot $ ), is a signature of AGN feedback at work on SF in their host galaxies.",
"Here, we present ALMA observations to establish the intrinsic SFR-distribution of 20 luminous quasars at $z\\sim 2$ , around the peak of AGN and SF activity.",
"Deriving a SFR for very luminous AGN from short-wavelength continuum is very challenging, due to the significant AGN contribution at almost all wavelengths.",
"In luminous quasars, the AGN contribution is dominant below 20$\\mu $ m and still significant around 60$\\mu $ m, which requires a careful spectral energy distribution (SED) decomposition of the AGN and SF component , .",
"Another major challenge for both Herschel imaging and single-dish sub-mm observations is confusion , , .",
"At wavelengths longer than $\\sim 100\\mu $ m the AGN SED falls off rapidly , , i.e.",
"in the sub-mm regime AGN contamination is minimized, and in most cases negligible.",
"Radio emission will also be contaminated by AGN contribution and thus does not serve as pure SFR tracer , , .",
"Therefore, continuum observations in ALMA Band-7 at $850\\mu $ m observed frame provide a unique, extremely sensitive SFR tracer for very luminous quasars, which does not suffer from source confusion and minimizes contamination from the AGN thus is a cleaner indicator of the SFR in the hosts of rapidly-growing SMBHs.",
"In Section we describe our sample selection, the ALMA observations, archival Herschel data and measurements of SMBH mass and bolometric luminosity.",
"We present our results on the AGN SED and their star formations rates in Section .",
"In Section , we present the star formation rate distribution derived from our sample and compare it to previous observations and theoretical models.",
"Our conclusions are given in Section .",
"Throughout this paper we use a Hubble constant of $H_0 = 70$ km s$^{-1}$ Mpc$^{-1}$ and cosmological density parameters $\\Omega _\\mathrm {m} = 0.3$ and $\\Omega _\\Lambda = 0.7$ .",
"We assume a initial mass function for estimates of stellar mass and star formation rate.",
"Our study is focused on the extreme luminosity end of optically-selected, unobscured (i.e.",
"broad line/type-1) quasars.",
"Given the orientation scenario to unify obscured and unobscured AGN [6], the restriction to unobscured quasars would a priori not introduce a bias.",
"However, within an evolution framework of black hole activity , [4], we are specifically targeting AGN after their dust enshrouded blowout phase, in which they emerge as luminous unobscured quasars.",
"We discuss the possible consequences of this selection further in Section .",
"We draw our sample from the Sloan Digital Sky Survey (SDSS) DR7 quasar catalog , , with $\\delta <+15$ deg.",
"We focus on a narrow range in redshift $1.9<z<2.1$ to avoid any uncertainties due to redshift evolution within our sample.",
"This redshift is of special importance since it corresponds to the peak epoch of star formation and AGN activity.",
"Furthermore, it is the highest redshift for which reliable black hole masses based on the broad MgII line can be derived from the optical SDSS spectra.",
"We select the most luminous quasars within this redshift range, based on the bolometric luminosity given in , $\\log L_\\mathrm {bol,S11}> 47.3$ [erg s$^{-1}$ ].",
"This bolometric luminosity is based on either $L_{3000}$ or $L_{1350}$ using a constant bolometric correction factor of 5.15 and 3.81, respectively .",
"In section REF , we present a reevaluation of the bolometric luminosity for our sample.",
"For this we use a different bolometric correction factor.",
"This choice lowers on average the bolometric correction used throughout the paper by a factor of 1.6, compared to those in .",
"Thus, our selection corresponds to $\\log L_{\\rm bol}>46.9$ [erg s$^{-1}$ ], for the bolometric correction discussed in section REF .",
"These selection criteria result in an initial sample of 62 quasars.",
"We further removed radio-detected quasars, based on the Faint Images of the Radio Sky at Twenty Centimeter survey to avoid contamination by AGN synchrotron emission, i.e all our targets have a 1.4 GHz flux below the FIRST detection limit of $\\sim 1.0$ mJy and are classified as radio-quiet.",
"This cut removed 17 objects, 6 are classified as radio-loud, 6 as radio-quiet but FIRST-detected and 5 quasars are outside of the FIRST footprint.",
"Out of a total sample of 45 quasars meeting our selection criteria we randomly selected 20 targets for ALMA observations, where we usually grouped two targets together to share a phase calibrator to minimize the overheads.",
"We did not put any restrictions on the sample in respect to the availability of Herschel FIR data, in order to be able to draw the rare target population of the most luminous quasars within our redshift window from the full SDSS sky area available to ALMA.",
"For 5 quasars in the sample Herschel coverage of the objects is available in the archive.",
"We further verified that none of our sources are gravitationally lensed .",
"Figure: Black hole mass – luminosity plane of SDSS quasars at redshift 1.9<z<2.11.9<z<2.1 and at δ<+15deg\\delta <+15\\deg (gray circles).",
"We highlight our ALMA sample of luminous quasars in red and show our luminosity threshold at logL bol >46.9\\log L_\\mathrm {bol}>46.9 as red solid line.",
"The solid black, dashed black and dotted black lines indicate Eddington ratios of 1, 0.1 and 0.01.We show the location of our sample in the SMBH mass-luminosity plane in Figure REF in relation to the general SDSS DR7 quasar population at the same redshift (using black hole masses and bolometric luminosities from section REF ).",
"By our selection, the sample constitutes the most luminous sources in the SDSS DR7 quasar catalog, with $\\log L_{\\rm bol}>46.9$ [erg s$^{-1}$ ], black hole masses $\\log M_{\\rm {BH}}>9.2$ [$M_\\odot $ ] and Eddington ratios above 10%."
],
[
"ALMA Observations",
"Our sample of 20 luminous SDSS quasars was observed with ALMA during Cycle 5 (2017.1.00102.S; PI: A. Schulze) in May 2018 in Band 7.",
"The representative frequency was set to 350.5 GHz (854$\\mu $ m) with four base bands, each with a band width of 1875 MHz.",
"The high frequency spectral window is chosen to be free of strong atmospheric absorption and to optimize our sensitivity, since the spectral energy distribution of our targets is strongly falling towards lower frequencies.",
"The observations were carried out using 40 antennas in the 12 m array with baselines 15m – 313m (configuration C43-2).",
"The average major beam size achieved is $\\sim 0.9$ , corresponding to 7.5 kpc at $z=2$ (slightly smaller than our 1$$ request).",
"We did not aim to spatially resolve the emission but rather ensure a measure of the total $850\\mu $ m continuum emission from the host galaxy.",
"The maximum recoverable scale is $\\sim 6.5$ , thus spatially-extended SF emission in the host galaxy is fully recovered.",
"Our achieved spatial resolution avoids confusion with potentially close companions, in contrast to single dish observations which typically achieve $\\gtrsim 13$ .",
"The requested sensitivity was 0.12 mJy beam$^{-1}$ , a level that probes SFRs $\\sim $ 0.2 dex below the MS. For a conservative lower limit on the stellar mass of their host galaxy, $M_*$ = $10^{10.5}\\,M_\\odot $ typical for luminous quasars at $z\\sim 2$ , a galaxy on the MS would have a SFR of $61\\,M_\\odot $ yr$^{-1}$ based on the relation from at $z=2.0$ .",
"We aimed to be sensitive to most galaxies on the MS, thus we set our detection threshold 0.2 dex below this value at $38\\,M_\\odot $ yr$^{-1}$ , corresponding to a FIR luminosity ($8-1000\\mu $ m) of $\\log L_\\mathrm {IR}=45.2$ [erg s$^{-1}$ ] using the relation by .",
"This gives a sensitivity of 0.12 mJy beam$^{-1}$ at 345 GHz for a 3$\\sigma $ detection.",
"Our achieved sensitivities are always below this value, with a mean around $0.09$ mJy beam$^{-1}$ .",
"To measure fluxes, we processed the ALMA data ourselves by reproducing the observatory calibration with their custom-made script based on Common Astronomy Software Application package .",
"We converted the data into uvfits format to perform further analysis with the IRAM GILDAS tool working on the uv-space (visibility) data.",
"We measured the 850$\\mu $ m fluxes and galaxy sizes by fitting the sources with models directly in the uv space.",
"Gaussian models were used when the emission was found to be resolved at more than 3$\\sigma $ , while point source models were fit otherwise.",
"Given the large synthesised beam of the ALMA observations and the typical sizes recovered for the resolved galaxies (Table REF ) we do not expect substantial flux underestimate for galaxies fit with point source models.",
"For more details on the method and for discussions on the advantages of the uv space analysis rather than imaging and cleaning the products we refer to and , respectively.",
"This approach ensures not to loose flux, also for resolved sources, and generally returns the highest SNR flux measurement from the data, contrary to e.g.",
"using aperture photometry.",
"The typical flux error for our sample is $\\sim 0.1$ mJy, with the values for each object reported together with the flux values in Table REF .",
"We show the Band 7 continuum images of all 20 QSO targets for visualisation purposes in Fig.",
"REF .",
"These images are based on the delivered data products using the ALMA pipeline (CASA version 5.1.1), applying a shallow clean.",
"In total 19 of the 20 targets are detected at $850\\mu $ m at more than $3\\sigma $ at the optical QSO position, 16 of them at more than $5\\sigma $ .",
"The remaining source, SDSS J1225+0206, has a $1\\sigma $ flux measurement of 0.1 mJy, just above our sensitivity limit and is considered a non-detection.",
"While we are extracting sources blindly, we ensure that the flux is not boosted by noise for faint sources.",
"For all detections their position is less than $\\sim 1/2$ of the beam away from the QSO position, typically $<0.2$ .",
"7 of the 20 quasars are spatially resolved, while the rest are un-resolved given the $\\sim 0.9$ beam.",
"For our sample, we found 4 additional sources within the ALMA beamOne source each in the beam of SDSS J1225+0206, SDSS J1228+0522, SDSS J2246$-$ 0049 and SDSS J2313+0034.",
"None of them is detected in the optical SDSS images.",
"A detailed discussion of these sources is beyond the scope of this paper and will be presented elsewhere.",
"Here, we briefly discuss the implications on the multiplicity of AGN in sub-mm observations.",
"The majority of our sample (80%) have a unique source within the ALMA beam.",
"Only 20% have multiple sources.",
"This is consistent with the recent results of for FIR bright quasars.",
"They found multiple sources in $\\sim 30$ % of their sample.",
"This supports their suggestion that on average the majority of optically bright quasars is not triggered by early-stage mergers, but the results of other processes, and extends it the quasar population over a broader range of SFR.",
"This suggestion is also consistent with quasar host galaxy studies at $z\\sim 2$ with HST , .",
"reported on ALMA observations of 6 luminous quasars at $z\\sim 4.8$ and found three of them to have a companion source in the ALMA beam.",
"Recent observations of a larger sample by this group lead to a multiplicity fraction around $\\sim 30\\%$ , more in line with our results ."
],
[
"Herschel data",
"Out of our sample, 5 objects are covered by Herschel maps in the Herschel Science Archive.",
"Three of them have been observed as part of H-ATLAS , and the other two fall into fields targeting nearby galaxies.",
"Four of them have PACS and SPIRE data, while the remaining quasar only has SPIRE coverage.",
"We retrieved the raw Herschel data from the archive and reprocessed them following the method in , but with the latest calibration in Herschel Interactive Processing Environment (HIPE) version 14.",
"None of the targets are detected in either the PACS or SPIRE bands.",
"The typical $3\\sigma $ upper limits are $\\sim 40$ mJy beam$^{-1}$ at 350$\\mu $ m, corresponding to a luminosity of $\\sim 10^{46}$ erg s$^{-1}$ at $z\\sim 2$ .",
"Thus, as shown in section REF , the Herschel upper limits are not constraining for the SED of our objects.",
"Therefore, we do not further consider these Herschel upper limits in the following analysis.",
"Figure: ALMA Band 7 continuum images for the quasar sample.",
"The color scale gives the flux in mJy.",
"We show the beam size as white ellipse in the lower left corner.Table: Sample."
],
[
"Optical spectral properties, black hole masses and bolometric luminosities",
"Optical spectra, covering the MgII and CIV broad line region, are available for all 20 objects from SDSS DR7 [1].",
"In addition, BOSS spectra are available for 9 quasars in SDSS DR14 [2], which provide a wider wavelength coverage and an improved quality especially around MgII.",
"Thereby, the use of the new BOSS spectra improves the spectral measurements of MgII and $L_{3000}$ , which are essential for the black hole mass estimate and bolometric luminosity.",
"Furthermore, we are interested in shape measurements (e.g., asymmetry) of the CIV line profile, which are not provided in the SDSS DR7 black hole mass catalog .",
"This made it necessary to perform a dedicated, consistent and visually verified spectral modeling of the latest available spectra for this sample and to re-calculate the black hole masses and bolometric luminosities in a consistent manner.",
"For this, we use the primary spectra given in the SDSS DR14 quasar catalog .",
"We use the improved redshifts from for our study.",
"We perform spectral model fits independently to the MgII and CIV line regions, including a power law continuum, iron emission template and a multi-Gaussian model for the broad emission lines.",
"Details on the line fitting are given in Appendix , and we show the optical spectra and the best-fit models in Figures REF and REF .",
"We measure the full width at half maximum (FWHM) of the broad MgII line from the multi-Gaussian fit and the continuum luminosity at 3000Å $L_{3000}$ from the power law continuum and report these in Table REF .",
"Black hole mass estimates can be obtained using the virial method , .",
"Under the assumption of virialized motion of the broad line region (BLR) gas and using established empirical scaling relations between continuum luminosity and BLR size , , this method allows an estimate of the mass of the SMBH, with a typical uncertainty of $\\sim $ 0.3 dex.",
"While the broad H$\\beta $ line is generally considered as the most reliable black hole mass estimator, the Mgii line has also been shown to provide robust black hole masses , , , , .",
"Since H$\\beta $ is not available for our sources, we base our black hole mass estimates on Mgii.",
"Specifically, we use the relation from $M_{\\rm {BH}}(\\rm {MgII})= 10^{6.74} \\left( \\frac{L_{3000}}{10^{44}\\,\\mathrm {erg\\,s}^{-1}}\\right)^{0.62} \\left( \\frac{\\mathrm {FWHM}}{1000\\,\\mathrm {km\\,s}^{-1} }\\right)^2 M_\\odot $ While we also observe and fit the CIV line, this line is known to be a poor SMBH mass estimator, especially for very luminous AGN , , , due to a non-virial component, potentially associated with an outflow, thus we do not use this line as a black hole mass estimator.",
"We rather use it as a tracer of AGN winds, indicated by (1) the blue shift between the centroid of the Mgii line, which is known to be close to the systemic redshift , and CIV; (2) the line asymmetry of the CIV line.",
"We further see the presence of intrinsic absorption features associated with the CIV line in a subset of our targets.",
"Our measure of the bolometric luminosity is based on applying a bolometric correction factor $f_{\\rm bol}$ to the continuum luminosity $L_{3000}$ , i.e.",
"$L_{\\rm bol}=f_{\\rm bol} L_{3000}$ .",
"A commonly adopted value is a constant factor $f_{\\rm bol}=5.15$ .",
"However, this value most likely overestimates $L_{\\rm bol}$ , especially for luminous quasars , .",
"Our estimate does include emission from the dust torus, which represents reprocessed emission and thus should be excluded.",
"Furthermore, the AGN SED is luminosity dependent, which should lead to a decrease in $f_{\\rm bol, 3000}$ with increasing UV-luminosity.",
"Therefore, we use the luminosity-dependent $f_{\\rm bol}$ prescription for $L_{3000}$ by , based on the bolometric corrections by , which considers these effects.",
"At $L_{3000}>10^{46}$ erg s$^{-1}$ $f_{\\rm bol}$ flattens at a value of $\\sim 3.2$ .",
"Since all quasars in our sample have $L_{3000}$ above this value we adopt a constant bolometric correction factor $f_{\\rm bol}=3.2$ for our sample.",
"Note that our bolometric luminosities are systematically lower by a factor 1.6 compared to the use of the commonly adopted value of $f_{\\rm bol}=5.15$ .",
"The Eddington ratio is given by $\\lambda _{\\rm Edd}=L_{\\rm {bol}}/L_{\\rm {Edd}}$ , where $L_{\\rm {Edd}}\\cong 1.3\\times 10^{38} (M_{\\rm {BH}}/ M_\\odot )$ erg s$^{-1}$ is the Eddington luminosity for the object, given its black hole mass.",
"We show the location of our sample in the SMBH mass-luminosity plane in Figure REF in relation to the general SDSS DR7 quasar population at the same redshift.",
"With this selection, our sample constitutes the most luminous sources in the SDSS DR7 quasar catalog, with $\\log L_{\\rm bol}>46.9$ [erg s$^{-1}$ ], black hole masses $\\log M_{\\rm {BH}}>9.2$ [$M_\\odot $ ] and Eddington ratios above 10%.",
"Figure: Spectral energy distribution (SED) and SED decomposition results for our luminous quasar sample.",
"The black circles show the optical to mid-IR photometry from SDSS, 2MASS and WISE.",
"The ALMA data point at 850μ850\\mu m is shown as a black square.",
"The solid lines give the best fit AGN SED template to the optical to mid-IR photometry, based on the three templates provided in : 1) a normal quasar (red line), 2) a WDD quasar (magenta line) and 3) a HDD quasar (green line).",
"The best-fit modified black body to the AGN-subtracted ALMA flux is shown by the blue dashed line.",
"The thin black line represents the total (AGN+SF) SED.Figure: Left panel: Examples of empirical AGN SED templates proposed in the literature, normalized at 20μ\\mu m. These are the template by , , ,and the extended version of the SED by , as presented in .",
"Right panel: Comparison of the standard AGN SED template by to the AGN SEDs templates of warm-dust-deficient (WDD, magenta dashed line) quasars and hot-dust-deficient (HDD, green dotted-dashed line) quasars, as presented by .",
"All templates are normalized at 1μ1\\mu m."
],
[
"AGN spectral energy distribution",
"Due to the high luminosity of our sample, the quasar emission will dominate the total luminosity at almost all wavelengths.",
"This is particularly true for the UV, optical, mid-IR, and even around $60\\mu $ m in the FIR.",
"However, at $850\\mu $ m, the quasar emission is minimized and most likely sub-dominant or even fully negligible.",
"Thus, we investigate the SEDs of our sample to assess the potential AGN contribution at $850\\mu $ m. We make use of multi-wavelength photometry, as provided in the SDSS DR14 quasar catalog .",
"This includes optical photometry in $ugriz$ from SDSS [3], $JHK$ near-IR photometry from the Two Micron All Sky Survey and mid-IR data from the Wide-Field Infrared Survey at $3.4, 4.6, 12$ and $22 \\mu $ m. We omit the $u$ -band data, since at the redshift of our objects it is severely contaminated by Ly$\\alpha $ emission and Ly$\\alpha $ forest absorption.",
"We do not consider the Herschel upper limits for the fit.",
"We note that they are always significantly above our best-fit SED.",
"We also do not include the ALMA data in the fit.",
"We show the photometric data points for our sample in Figure REF .",
"Several different AGN SED templates have been used in previous studies to decompose the AGN and the host galaxy emission.",
"In Figure REF we show a few commonly adopted AGN SED templates , , , , each normalized at 20$\\mu $ m. The SED template by covers the UV-mid-IR range and does not fully extend into the FIR.",
"The AGN templates by (their high luminosity template), (as provided in ) and (as provided in ) are largely consistent at $\\lambda _{\\rm rest}>20\\mu $ m.Contrary, the SED template proposed by would predict a significant AGN contribution from cold dust at $\\lambda _{\\rm rest}>100\\mu $ m. However, their results have been challenged more recently , , .",
"Our ALMA observations of extremely luminous quasars seem to support the concerns raised by these authors.",
"In 18/20 cases our measured 850$\\mu $ m flux is below the expectation from the template by , when the latter is anchored to the quasar photometry at $\\lambda _{\\rm rest}=60\\mu $ m. Thus, we do not further consider their SED template in our study.",
"In the following, we use the AGN template by , since it provides the widest wavelength coverage from the UV to the FIR.",
"This enables us to use the full UV to mid-IR photometry available to constrain the AGN SED.",
"The template is based on the original AGN SED by , removing the star formation contribution from that SED.",
"In the UV to mid-IR regime it is consistent with the more recent quasar SED by .",
"While the SED for the majority of luminous quasars is well represented by this SED template, demonstrated that there are significant sub-populations ($30-40$ %) whose SED deviates from the standard SED, primarily in the mid-IR due to a dust-deficiency.",
"characterize them as 1) hot-dust-deficient (HDD) quasars ($15-23$ %), showing very weak emission from the NIR all the way to the FIR, and 2) warm-dust-deficient (WDD) quasars ($14-17$ %), with similar NIR emission as normal quasars, but relatively weak emission in the mid-IR to FIR , , , .",
"We show their respective SED templates in the right panel of Figure REF .",
"To characterize the AGN SED for our sample, we use a simple approach by fitting with a library of SEDs consisting of the three AGN SED temples provided in : 1) the standard AGN template by , 2) a warm-dust-deficient (WDD) quasar template and 3) a hot-dust-deficient (HDD) quasar template.",
"We consider these templates to broadly cover the diversity of expected UV to mid-IR SED shapes for our sample.",
"For each template, the absolute normalization is the only free parameter in our $\\chi ^2$ minimization routine.",
"We then choose the SED template with the smallest $\\chi ^2$ value as the best fit for each object in our sample.",
"In all cases the difference in $\\chi ^2$ is significant ($\\Delta \\chi ^2_{\\mathrm {reduced}}>>1$ ).",
"We do not consider the host galaxy contribution to the UV to mid-IR emission, since for our very luminous quasars it is clearly sub-dominant over the full wavelength range.",
"The best fit results for the AGN SED are shown in Figure REF .",
"From the 20 luminous quasars in our sample, 9 are fitted with a standard SED, 8 with a WDD SED and 3 with a HDD SED.",
"The latter corresponds to a fraction of 15%, consistent with previous work , , .",
"The fraction of 40% modeled as WDD in our sample is higher than reported for low-$z$ PG quasars .",
"However, they also find a strong increase of the WDD fraction with luminosity, generally consistent with our results.",
"This is also consistent with the observation of an anti-correlation between the mid-IR to optical luminosity ratio with luminosity , , , .",
"In the context of this paper, we mainly use the best fit AGN SED to estimate and subtract the AGN contribution to the ALMA 850$\\mu $ m continuum flux.",
"Table: ALMA measurements"
],
[
"FIR luminosities and Star formation rates",
"In this section, we use our ALMA continuum measurements to estimate the total FIR luminosity from star formation (SF; integrated over $8-1000\\mu $ m), hereafter $L_{\\rm IR}$ , and the SFR in the quasar host galaxies.",
"Our measured ALMA Band-7 continuum fluxes and the corresponding luminosity are listed in Table REF .",
"We subtract the AGN contribution from the AGN SED best-fit from this luminosity and use this AGN-subtracted luminosity throughout to probe the emission due to SF.",
"We verified that our qualitative conclusions do not change if we would use the continuum fluxes without subtracting the AGN contribution instead.",
"For SDSS J2313+0034, the most luminous quasar in the sample, the expected AGN contribution is higher than the ALMA measurement, and we set an upper limit to $L_{850\\mu {\\rm m}}$ due to star formation at the ALMA measurement.",
"For the rest of the sample we find an AGN contribution to $L_{850\\mu \\mathrm {m}}$ in the range of $3-59\\%$ , with a median value of 14%.",
"The AGN contribution shows a positive correlation with the quasar luminosity, with a Spearman rank order coefficient of $r_S=0.53$ .",
"We also verified that our results do not depend on the specific choice of AGN template used to account for the AGN contribution.",
"For our three adopted AGN templates we find a mean difference between them of 0.08 dex in $\\log L_{850\\mu \\mathrm {m},{\\rm CD}}$ or SFR.",
"Estimates of total $L_{\\rm IR}$ and SFR from a single measurement on the Rayleigh-Jeans-tail of the cold dust emission bears reasonably high uncertainties, due to the unknown dust temperature and dust mass.",
"We use two independent approaches to estimate $L_{\\rm IR}$ to alleviate such uncertainties.",
"First, we assume a modified blackbody (MBB) spectrum to represent the thermal dust emission, presumably heated by SF: $L_{\\rm mbb}(\\nu ) = N_{\\rm mbb} \\frac{ \\nu ^{3+\\beta } }{e^{h \\nu / k_b T_{\\rm d}}-1}$ We fix the emissivity index to $\\beta =1.6$ and the dust temperature to $T_{\\rm d}=47$ K and determine the normalization $N_{\\rm mbb}$ from the ALMA continuum flux measurement.",
"These values are widely adopted in previous studies of high-$z$ quasar host galaxies , , , .",
"found that these assumptions provide results in good agreement with Herschel observations for a representative sample of luminous type-1 quasars at $z\\sim 4.8$ studied with ALMA.",
"This adopted value for $\\beta $ is also consistent with studies of local luminous and ultraluminous infrared galaxies .",
"Other AGN host studies with detections at several wavelengths, in particular close to the peak of the blackbody emission using e.g.",
"Herschel, are able to fit for $T_{\\rm d}$ and typically report temperatures around $30-50$ K , , , , , .",
"We use this temperature range to derive lower and upper limits on $L_{\\rm IR}$ .",
"We consider this uncertainty, introduced by the unknown dust temperature, to be the dominating source of uncertainty on $L_{\\rm IR}$ .",
"Figure: Observed SFR distribution for our ALMA sample (gray histogram).",
"The black line shows a kernel density estimate of the SFR distribution, using a Gaussian kernel with 0.25 dex bandwidth and the gray dashed line represents the best fit log-normal distribution to the data as discussed in the text.",
"Left panel: SFRs derived assuming a MS galaxy cold dust template .",
"Middle panel: SFR based on modified blackbody with T d =47T_{\\rm d}=47 K and β=1.6\\beta =1.6.",
"Right panel: SFR based on modified blackbody with β=1.6\\beta =1.6 and T d T_{\\rm d} randomly drawn from a Gaussian distribution of 6 K dispersion around T d =47T_{\\rm d}=47 K. In all panels, we indicate the location of the main sequence at z=2z=2 and logM * =11.0\\log M_\\ast =11.0 as vertical blue dashed line.",
"The range of SFR consistent with the SF-MS, given the range of M * M_\\ast expected for our sample and the dispersion of the MS of 0.3 dex, is shown as blue dotted lines.The second approach uses a characteristic IR SED template of MS galaxies at $z\\sim 2$ as provided in .",
"This template is based on the SED library presented in , derived by fitting the theoretical template library of dust models by to Herschel observations of distant galaxies.",
"In addition, we also test the template for starburst galaxies by .",
"It has been shown that using a modified blackbody fit or SED template libraries provide generally consistent results for $L_{\\rm IR}$ .",
"We convert $L_{\\rm IR}$ into a SFR using the relation by , corrected to a initial mass function, following SFR$/M_\\odot \\, {\\rm yr}^{-1} = L_{\\rm IR}/ 10^{10} L_\\odot $ .",
"We provide $L_{\\rm IR}$ and SFR derived from both approaches in Table REF .",
"We find a tight correlation between the $L_{\\rm IR}$ estimates based on the MBB assumption and those from the MS galaxy template.",
"However, the latter are lower by 0.27 dex, consistent with the assumption of a lower dust temperature in the MBB model of $T_{\\rm d}\\sim 40$ K. This is consistent with , who find dust temperatures around $30-35$ K for modified blackbody fits with $\\beta =1.5$ to their mean $\\left< U \\right>$ models.",
"A dust temperature of 40 K is also within our adopted uncertainty on $T_{\\rm d}=30-50$ K. The $z\\sim 2$ starburst model by is in good agreement with the MBB $L_{\\rm IR}$ values, being higher by only $0.02$ dex.",
"Alternatively, assuming a higher dust temperature of $T_{\\rm d}=60$ K would increase the $L_{\\rm IR}$ estimates by 0.43 dex.",
"Since several studies on luminous quasars hosts report dust temperatures closer to 47 K , , , in the following we use the $L_{\\rm IR}$ values from the MBB model as default values, but discuss the consequences of adopting the lower $L_{\\rm IR}$ values from the MS SED template where appropriate."
],
[
"The intrinsic SFR distribution of luminous quasars",
"Based on the MBB model, we find a broad range of SFRs for our sample.",
"We list their values in Table REF .",
"The SFRs span a range of $35-1513\\, M_\\odot $ yr$^{-1}$ , with a median value of $180\\, M_\\odot $ yr$^{-1}$ .",
"The strongest star formation is detected for SDSS J1236+0500, the only object in our sample with a SFR exceeding 1000 M$_\\odot $ yr$^{-1}$ .",
"Thanks to our high detection rate of 90%, we are able to construct the intrinsic SFR distribution of luminous broad-line quasars at $z\\sim 2$ from our sample.",
"In Figure REF , we show the SFR distribution based on the MBB model (middle panel) and the MS SED template model (left panel).",
"For the two SF non-detections, for simplicity we assume they are located in the bin below our SFR sensitivity limit.",
"Assuming a single effective temperature in the modified blackbody fit for all objects is obviously a simplification, as there will be a distribution of dust temperatures.",
"This assumption does not significantly affect the mean of the SFR distribution, but its shape.",
"To investigate the effect of an underlying dust temperature distribution, we perform a Monte Carlo simulation.",
"We assign a value for $T_{\\rm d}$ , randomly drawn from a Gaussian distribution with mean of 47 K and a dispersion of 6 K. These values are consistent with the results by for high SFR host galaxies of luminous quasars at similar redshift.",
"For each AGN in our sample, we obtain $L_{\\rm IR}$ and SFR from a modified blackbody of this randomly drawn temperature.",
"We derive the SFR distribution from 1000 random realizations.",
"The resulting SFR distribution is shown in the right panel of Figure REF .",
"Table: SFR log-normal distribution parametersDue to the tight correlation between the SFRs from the MBB fit and the MS galaxy template fit, the shape of the SFR distributions derived from these models are consistent, but shifted by 0.27 dex.",
"The main difference in the random $T_{\\rm d}$ case is a slight broadening of the SFR distribution, compared to the adopted single temperature case.",
"The observed SFR distribution in all three cases is fully consistent with a log-normal distribution, according to a Shapiro-Wilk test for normality.",
"Therefore, we adopt a log-normal function to parametrize the SFR distribution, consistent with results for moderate-luminosity AGN , .",
"We derive the mean, standard deviation and their confidence intervals for a log-normal SFR distribution using a Bayesian approach .",
"We list the parameters of the log-normal SFR distribution for the three cases in Table REF and show it with dashed gray lines in Figure REF .",
"Since we do not have stellar mass estimates for our sample, but only $M_{\\rm {BH}}$ , it is not straightforward to accurately locate our objects in respect to the SF-MS.",
"Nevertheless, the typical stellar mass and stellar mass distribution for luminous quasars at $z\\sim 2$ can be obtained from the study by .",
"They observed a sample of 19 luminous quasars with massive SMBHs with the Hubble Space Telescope (HST) to study the underlying host galaxy, reporting a median stellar mass of $\\log M_\\ast \\,/ M_\\odot =11.1$ , which we here adopt as the average $M_\\ast $ of our sample.",
"For the MS relation by at $z=2$ , this mass corresponds to $\\log {\\rm SFR}=2.24$ for a galaxy on the MS.",
"Adopting instead the MS relation by gives an almost identical result.",
"For an expected $M_\\ast $ range of our sample of $\\log M_\\ast \\,=10.5-11.5$ , we find $\\log {\\rm SFR}=1.78-2.54$ .",
"We indicate the position of the MS based on these assumptions in Figure REF as vertical blue dashed line, with the range of the MS, including a dispersion of 0.3 dex, is indicated by the blue dotted lines.",
"Adopting the MBB SFRs, we find the bulk of the population to be consistent with the MS of star formation.",
"If we assume a MS SFR of $\\log {\\rm SFR}=2.24$ (at $\\log M_\\ast \\,=11.1$ ) with dispersion of 0.3 dex, 10 quasar hosts are on the main sequence, 3 above and 7 below.",
"Allowing for a broader range of $M_\\ast $ , only one object, SDSS J1236+0500, is clearly in the starburst regime.",
"The two SF non-detections could be located below the MS, but their upper limits are still consistent with SFRs on the MS. Interestingly, one of them is SDSS J2313+0034, the most luminous quasar in our sample.",
"We discuss any trends with AGN luminosity, $M_{\\rm {BH}}$ and $\\lambda _{\\rm Edd}$ in Section .",
"For the case of a $T_{\\rm d}$ distribution, we find largely consistent results, with an increase of the number of quasar hosts above and below the MS. For the SFR derived from the SED template, the sample mean is below the mean of the SF-MS, while the majority is still consistent with the MS within its scatter.",
"Five quasar host are clearly located below the MS, while only SDSS J2313+0034 is unambiguously elevated from the MS.",
"The width of the SFR distribution is broader than the typical width of the MS at this mass of $\\sim 0.25-0.3$ dex , .",
"This could hint at a SFR distribution of the most luminous AGN being not consistent with the SF-MS, similar to recent results for moderate luminosity AGN , .",
"However, given the uncertainty in the stellar masses of our sample, it is possible that this broadening is a consequence of a broader underlying $M_\\ast $ distribution.",
"To summarize, the SFRs of luminous $z\\sim 2$ quasar host galaxies are consistent with the SF-MS.",
"Although, a fraction of the sample might be located off the MS since the SFR distribution is slightly broader than the SF MS, with possibly $5-15$ % in the starburst regime and $0-35$ % located below the MS.",
"The latter population might be in the process of quenching star formation in the host galaxy, while the AGN is still actively accreting at a high rate.",
"Even so, there appears to be no statistical difference between the SFR distributions of quasar hosts and typical SF galaxies of a given stellar mass.",
"However, direct stellar mass or dynamical mass measurements will be needed to reliably pinpoint our sample with respect to the SF-MS.",
"Figure: L IR L_{\\rm IR} due to star formation as a function of AGN L bol L_{\\rm bol} for the ALMA sample studied in this work (black circles - individual measurements) in comparison to previous results in the literature.",
"Left panel: comparison with the mean <L IR >< L_{\\rm IR}> for AGN at 1.5<z<2.51.5<z<2.5 from stacked Herschel data from and .",
"We indicate our sample median as open black square and the linear mean (corresponding to a stack) as open red square.Right panel: comparison to individually measured L IR L_{\\rm IR} for AGN at 1.8<z<2.51.8<z<2.5 for moderate luminosity AGN , luminous and hyper-luminous quasars .",
"The open symbols with arrows show upper limits and the open red triangle shows the stack for the Herschel non-detections in .",
"In both panels, the model by is shown by the red dashed line (see Section )."
],
[
"Comparison with previous work",
"The connection between SF and AGN activity is a very active field of research, thanks to the legacy of Herschel and now ALMA.",
"We discuss here our ALMA results for luminous $z\\sim 2$ quasars in the broader context of AGN-SF studies.",
"These broadly fall into two categories 1) studies of the average SFR trends by stacking of X-ray or optically-selected AGN; and 2) studies of AGN individually detected by Herschel or in the sub-mm.",
"In the left panel of Figure REF we compare our results to the average SFR trends at $1.5<z<2.0$ , reported in the studies by and , based on Herschel stacks.",
"The former work focused on moderate-luminosity X-ray selected AGN, while the latter work studied luminous optically-selected unobscured quasars from SDSSWe used the X-ray bolometric correction from to convert X-ray luminosity to bolometric luminosity in .",
"We rescaled $L_{\\rm bol}$ given in following our discussion in section REF ..",
"They consistently report an average SFR which does not show a dependence on the AGN luminosity, beyond a weak increase expected due to the different stellar mass distributions of their host galaxies.",
"The red open square shows the linear mean for our ALMA sample, corresponding to a stacking approach.",
"Our result is fully consistent with the highest luminosity bin by .",
"It thus falls well into the trend discussed in that study, namely that on average AGN are hosted by normal star-forming galaxies, irrespective of their instantaneous AGN luminosity.",
"The SFR distribution discussed in section REF agrees with this interpretation.",
"In contrast to the studies on stacked sources, our work in addition provides information on individual objects and on the SFR distribution.",
"This complements recent work on moderate-luminosity AGN , , combining observations from Herschel and ALMA, shown in the right panel of Figure REF .",
"In the figure we restrict the redshift range for this and all other samples shown to $1.8<z<2.5$ , to approximately match our redshift range.",
"Within this range, the sample by includes 36 AGN with 10 of them having a SFR measurement while the rest are only upper limits.",
"Taking into account the upper limits, there is a broad distribution of SFR, without a significant dependence on AGN luminosity.",
"Our observations continue these trends to the highest AGN luminosities.",
"There appears to be a mild increase in the SFR distribution, consistent with the results on stacked data.",
"However, such an increase is expected under the assumption that AGN host galaxies are predominantly located on the SF-MS.",
"The most luminous AGN will on average have higher $M_{\\rm {BH}}$ compared to moderate-luminosity AGN.",
"Likewise, assuming the $M_{\\rm {BH}}-M_\\ast \\,$ relation, they will on average have more massive host galaxies.",
"In Figure REF , we also show SFRs measured from Herschel for individual luminous quasars over $1.8<z<2.5$ in the studies by and .",
"Here, we only show a sub-sample from these studies within the given redshift range.",
"Given the sensitivity of the Herschel data their detections naturally occupy the regime of high SFR, many of them most likely located in the starburst regime.",
"Following our discussion in section REF , we classify AGN hosts with SFR$>690\\, M_\\odot /$ yr as a likely starburst.",
"In our sample only 1/20 quasars (5%) fall into this regime, while this is the case for 9/44 (at $1.8<z<2.5$ ) in the sample (20%).",
"This apparent discrepancy might be caused by systematic differences in the sample selection or in the estimation of SFR or simply due to still limited statistics.",
"On the contrary, in the sub-mm SCUBA sample of luminous quasars by 2/34 (6%) of their targets within $1.8<z<2.5$ are detected at $850\\mu $ m, with SFR$>1000\\, M_\\odot /$ yr, more in line with our results.",
"Larger samples will be required to also robustly constrain the wings of the SFR distribution.",
"Figure: Individual measurements of SFR for luminous AGN with logL bol >46.9\\log L_{\\rm bol}>46.9 at 1.8<z<51.8<z<5 as a function of redshift.",
"We compare our ALMA results (black circles) to measurements based on single-dish sub-mm observations and the Herschel studies by , , and .",
"presented ALMA observations of 28 SDSS quasars over $2<z<4$ , selected by Herschel to be FIR bright and thus likely have extreme SFR.",
"Their sample is very complimentary to ours.",
"While we focus on the most luminous quasars and sample a broad range of SFR, they originally select their sample as having the highest SFRs and cover a wider range in AGN luminosity.",
"Their study specifically probes the high SFR tail of the AGN SFR distribution, while the goal of our study is to establish the full SFR distribution at high $L_{\\mathrm {bol}}$ .",
"We show the subset of their study with comparable AGN luminosities further below.",
"Recently, presented results on the SFRs in 25 powerful high-redshift radio galaxies at $1<z<5.2$ , with a median at $z\\sim 2.4$ , using sensitive ALMA observation in combination with previous Herschel data.",
"Their sample of luminous AGN hosted by massive galaxies ($\\log \\, M_\\ast \\,\\sim 11.3$ ) tends to show SFR on or below the MS, with a median SFR of $110\\,M_\\odot /$ yr, interpreted by them as representing galaxies on the way to quenching.",
"Their sample is comparable to ours in terms of AGN luminosity and expected host stellar mass, but while we focus on radio-quiet, unobscured AGN, they study radio-luminous, obscured AGN.",
"Nevertheless, their results on typically low SFR in luminous AGN is consistent with our work.",
"In Figure REF , we plot individual measurements of SFR for luminous quasars, with $\\log L_{\\rm bol}>46.9$ over a broader range of redshift, $1.8<z<5$ .",
"We restrict the original samples to these constraints.",
"The samples include the Herschel studies of luminous optically-selected quasars at $z=2-3.5$ and $z\\sim 4.8$ by and , respectively.",
"We also show the study by of hyper-luminous quasars from the WISSH survey with archival Herschel detections, and the SCUBA $850\\mu $ m observations of a heterogenous sample of luminous quasars at $1.5<z<3$ by .",
"For the latter study, we estimate SFR from their reported $850\\mu $ m fluxes in the same way as for our sample.",
"Given the high $850\\mu $ m fluxes of their detections, AGN contamination is negligible for these objects.",
"Due to their relatively low sensitivity at $850\\mu $ m of $\\sim 2.5$ mJy, only quasar hosts with SFR$> 1000\\, M_\\odot /$ yr are detected at $>3\\sigma $ (in total 9/57 targets in their full sample).",
"Furthermore, we show the 5 quasars in the recent study by with $L_{\\mathrm {bol}}>46.9$ .",
"Again, we estimate SFR from their $850\\mu m$ ALMA flux using a modified black body model, consistent to our work.",
"Given their sensitivity limit, the 5 studies shown in Figure REF report quasar hosts with SFR$> 300\\, M_\\odot /$ yr, while only 3 of our targets reach as high SFR.",
"This demonstrates how our work complements those previous studies.",
"While they are based on a larger parent sample and thus are more sensitive to the rare population of high SFR AGN hosts, we fill in the moderate SFR regime with individual detections, although only over a narrow range in redshift.",
"Similarly deep ALMA studies at other redshifts will be needed to robustly assess the evolution of the SFR distribution with redshift.",
"Potential evidence for such an evolution has been discussed in .",
"They argue that the SFR in the most luminous quasars peaks around $z=4-5$ and declines towards lower redshift.",
"We consider this suggestion here by comparing the fraction of quasar hosts above a certain SFR threshold, e.g.",
"at SFR$> 1000\\, M_\\odot /$ yr.",
"While in our sample at $<z>=2.0$ , 5% of the sample are above this SFR cut, the fraction is 16% in at a mean redshift of $<z>=2.3$ , 17% (12/70 with $\\log L_{\\rm bol}>46.9$ ) in at $<z>=2.7$ , and 23% (10/44) in at $<z>=4.8$ .",
"This might indicate a decline in the fraction of strong starbursts in the most luminous quasars from $z\\sim 5$ to $z\\sim 2$ .",
"Larger, homogeneously-selected quasar samples over a broad range of redshift will be required to verify this hypothesis.",
"Our work and those discussed above focus on optically-selected, type-1 (unobscured) AGN, i.e.",
"heavily reddened or obscured quasars are not included in our study.",
"The intrinsic type-2 AGN fraction is a function of luminosity, and is low at the high luminosities studied here , .",
"Since these type-2 AGN are likely unified by orientation, excluding them they will not affect our conclusions.",
"However, reddened quasars are usually associated with a special evolutionary phase, representing young quasars in the transition phase between strongly star forming, FIR/sub-mm bright galaxies and normal unobscured quasars , .",
"Their space density, especially at the luminous end, has been suggested to be comparable to that of unobscured quasars .",
"Sub-mm studies with ALMA found typically much higher SFR in such a reddened quasar population than what we report here , supporting the transition population hypothesis for reddened quasars.",
"However, current sample sizes are still too small to draw firm conclusions."
],
[
"Comparison with models",
"We now discuss if these results for the SFR distribution of luminous quasars are consistent with current models, specifically in respect to two models: 1) the phenomenological model by , and 2) results from a large cosmic volume hydrodynamical simulation .",
"The simple model by is mainly designed to simultaneously explain different observations on the AGN-SF connection.",
"It successfully explains the flat relation of <SFR> at moderate AGN luminosities (Figure REF ), but also predicts the emergence of a tight relation at the highest $L_{\\rm AGN}$ , where the AGN tend to have massive SMBHs and a narrow range of $\\lambda _{\\rm Edd}$ .",
"In Figure REF , we show the prediction of the model for luminous AGN ($\\log L_{\\rm bol}>46.9$ ) at $z=2$ as red dashed-dotted line.",
"Their SFR distribution has a mean of 2.58 and a dispersion of 0.31.",
"Our results indicate a mean SFR $\\sim 0.4$ dex lower, suggesting a weaker correlation between SFR and AGN activity as expected in their default model.",
"Furthermore, our results show a broader distribution compared to the model.",
"With the green dashed line in Figure REF , we show the predicted SFR distribution from a cosmological, hydrodynamical simulation using the Magneticum Pathfinder simulation set , which is based on the GADGET3 code.",
"Specifically, we use a large size cosmological simulation with a co-moving box size of (500Mpc)$^3$ , simulated with an initial particle number of $2\\times 1564^3$ .",
"This box size is larger than for example the EAGLE or the Illustris , simulations.",
"Therefore, in contrast to those, the Magneticum simulation enables us to probe the rare population of the most luminous AGN.",
"This simulation is able to reproduce the AGN population up to $z=3$ .",
"In particular, at redshift $z=2$ it matches well the observed AGN luminosity function and the active black hole mass function .",
"We extract SFRs and SMBH mass accretion rates from the $z=2$ snapshot of the simulation and convert the latter to $L_{\\rm bol}$ assuming a radiative efficiency of $\\eta =0.1$ .",
"Figure REF shows the distribution of SFR for the 77 AGN in the simulation box with $\\log L_{\\rm bol}>46.9$ .",
"We find a generally good agreement between the Magneticum simulation and our results.",
"While the mean $\\log $ SFR of 2.33 is somewhat higher and the dispersion of 0.4 dex lower than our result, a Kolmogorov-Smirnov test gives p-values of $p_{\\rm KS}=0.14$ and $p_{\\rm KS}=0.46$ for the single $T_{\\rm d}$ MBB model and the MBB model with $T_{\\rm d}$ distribution, respectively.",
"Thus, our sample is consistent with being drawn from the same distribution as the Magneticum simulation result, or, in other words, both the observed and simulated samples are representative of galaxies within the star-forming main sequence.",
"While recent work demonstrated that current cosmological simulations are consistent with observations of moderate-luminosity AGN on the AGN-SF connection , , we show that this is also the case for cosmological simulations including the most luminous AGN.",
"However, a larger sample of very luminous AGN both in numerical simulations and in observations are required to obtain more stringent constraints on the comparison between theory and observation at the most luminous end, especially if feedback effects are more subtle than usually assumed.",
"We note that our work as well as the Magneticum simulation show a broader distribution that the simple model.",
"Recently, argued that such a broad SFR distribution could be an indication for the presence of AGN feedback, which will lead to a broadening of the SFR distribution compared to a non-AGN feedback scenario.",
"This is based on their study of the AGN-SF connection in the Eagle simulation set, where they could compare a simulation box run with and without AGN feedback.",
"The explored Magneticum simulation box includes an AGN feedback prescription.",
"The agreement of our observations with this simulation may be indicative of slight effects of AGN feedback.",
"Unfortunately, the Magneticum simulation set does not include boxes without an AGN feedback implementation that could serve as a no-feedback reference, while the Eagle simulation boxes are too small to include the luminous AGN population in significant numbers.",
"A more detailed discussion of the full AGN-SF distribution from the Magneticum simulation suite and its comparison to observations is beyond the scope of this paper and will be presented elsewhere (Hirschmann et al., in preparation).",
"Figure: Observed SFR distribution, as in Figure , for the single dust temperature MBB model (left panel) and the random T d T_{\\rm d} MBB model (right panel).In addition, we compare the observations to the SFR distribution predicted from the phenomenological model by and from the large scale Magneticum cosmological simulation (green dashed line)."
],
[
"Conclusions",
"Based on ALMA Band 7 observations at $850\\mu $ m of 20 luminous ($\\log \\, L_{\\rm bol}>46.9$ [erg s$^{-1}$ ]) unobscured quasars at $z\\sim 2$ , our results indicate that SMBHs at high Eddington rates reside in typical star-forming galaxies without any conspicuous evidence for a relation between accretion and star formation or, in particular, with the latter being enhanced or suppressed in a systematic fashion.",
"This brings into question the role of feedback from SMBHs in regulating the growth of the massive galaxy population.",
"However, a picture may be emerging where star formation and accretion are supplied by gas (possibly the result of a single inflow event) from reservoirs within their host galaxy but drawn on different physical and temporal scales with each having their own efficiency with respect to gas processing.",
"To recap some of the details, this work expands upon previous studies of AGNs which relied on stacking of Herschel data or only FIR bright Herschel- and ALMA-detected objects .",
"Thanks to the high sensitivity and high spatial resolution of ALMA, we are able to detect the FIR continuum (i.e., SFR) for individual AGN for the vast majority of our targets (19/20), filling in a unique regime in the $L_{\\rm AGN}-$ SFR plane.",
"We estimate the AGN contribution to the sub-mm flux by fitting AGN templates to the UV to mid-IR photometry and correct the sub-mm fluxes for this contamination.",
"The key findings are the following.",
"We find a broad distribution of SFR, consistent with a log-normal distribution with a mean of 140 $M_\\odot $ yr$^{-1}$ and a dispersion of 0.5 dex.",
"The SFR distribution is largely consistent with that of the star-forming MS.",
"Although, this relies on an inference of the stellar masses of their host galaxies from a related luminous quasar sample at $z\\sim 2$ that is equivalent to using the $M_{BH}-M_{stellar}$ relation to infer their stellar masses.",
"The SFR distribution is both broader and shifted to lower SFR than the simple phenomenological model by , but consistent with results from a large scale cosmological hydrodynamical simulation that may hint at subtle effects of quasar feedback.",
"Comparing our results with previous work at higher redshift, we find tentative evidence for an increase of the fraction of AGN showing intense starburst activity (SFR$>1000\\, M_\\odot $ yr$^{-1}$ ) with increasing redshift from $z\\sim 2$ to $z\\sim 5$ , in line with the suggestion by .",
"Although, we cannot make any statistically significant claims given issues with selection and sample size.",
"We do not find any statistically significant correlation between SFR and AGN properties (Section ), namely $L_{\\rm bol}$ , $M_{\\rm {BH}}$ and $\\lambda _{\\rm Edd}$ , Civ blueshift, equivalent width and line asymmetry.",
"However, we caution that at least part of this lack of correlation could be caused by the restriction of our sample to high luminosities, thus covering a restricted range in $M_{\\rm {BH}}$ , and $\\lambda _{\\rm Edd}$ .",
"This work clearly demonstrates the unique capabilities of ALMA to determine the dust properties of luminous high-z quasars and infer their SFR distribution free of contamination from the quasar itself.",
"It is now imperative to establish the intrinsic SFR distributions for quasars across all cosmic epochs, especially those at the earliest times , , ."
],
[
"Acknowledgements",
"A.S. is supported by the EACOA fellowship.",
"We are grateful for constructive discussions with Renyue Cen.",
"This paper makes use of the following ALMA data: ADS/JAO.ALMA#2017.1.00102.S.",
"ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile.",
"The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.",
"MH acknowledges financial support from the Carlsberg Foundation via a Semper Ardens grant (CF15-0384)."
]
] | 1906.04290 |
[
[
"Window Based BFT Blockchain Consensus"
],
[
"Abstract There is surge of interest to the blockchain technology not only in the scientific community but in the business community as well.",
"Proof of Work (PoW) and Byzantine Fault Tolerant (BFT) are the two main classes of consensus protocols that are used in the blockchain consensus layer.",
"PoW is highly scalable but very slow with about 7 (transactions/second) performance.",
"BFT based protocols are highly efficient but their scalability are limited to only tens of nodes.",
"One of the main reasons for the BFT limitation is the quadratic $O(n^2)$ communication complexity of BFT based protocols for $n$ nodes that requires $n \\times n$ broadcasting.",
"In this paper, we present the {\\em Musch} protocol which is BFT based and provides communication complexity $O(f n + n)$ for $f$ failures and $n$ nodes, where $f < n/3$, without compromising the latency.",
"Hence, the performance adjusts to $f$ such that for constant $f$ the communication complexity is linear.",
"Musch achieves this by introducing the notion of exponentially increasing windows of nodes to which complains are reported, instead of broadcasting to all the nodes.",
"To our knowledge, this is the first BFT-based blockchain protocol which efficiently addresses simultaneously the issues of communication complexity and latency under the presence of failures."
],
[
"Introduction",
"Consensus is used to agree on a new block to be appended to the chain by the nodes in the network.",
"A blockchain is compromised of two main components:a cryptographic engine and a consensus engine.",
"The main performance and scalability bottleneck of a blockchain also lies in these components.",
"Here we only focus on improving consensus component of blockchains.",
"As already mentioned, PoW-based protocols are highly scalable.",
"In Bitcoin [1], which is one of the most successful implementation of blockchain technology, typically the number of nodes (replicas) are usually large in the range of thousands [1], [2].",
"PoW involves the calculation of a number based on the hash value of a block adjusted by a difficulty level.",
"Solving this cryptographic puzzle by nodes (miners) limits the rate of the block generation as solving the puzzle is CPU intensive.",
"Bitcoin uses PoW but the number of transactions per second can reach up to just 7 transactions per second [2].",
"The block generation rate is approximately 10 minutes [1].",
"Additionally, the power utilized by Bitcoin mining in 2014 was between 0.1-10 GW and was comparable to Ireland's electricity consumption at that time [3].",
"Different solutions were proposed, for example, Ethereum [4] uses faster PoW, BitcoinNG [5] uses two types of blocks, namely, key blocks and micro-blocks, and has achieved $10 \\times $ more throughput in comparison with Bitcoin.",
"But all these solutions fall well short of matching the throughput offered by leading credit-card companies (2000 on average and 10000 maximum transactions per second).",
"On other the hand, BFT-based [6] protocols guarantee consensus in the presence of malicious (Byzantine) nodes, which can fail in arbitrary ways including crashes, software bugs and even coordinated malicious attacks.",
"Typically, BFT-based algorithms execute in epochs, where in each epoch the correct (non-malicious) nodes achieve agreement for a set of proposed transactions.",
"In each epoch there is a primary node that helps to reach agreement.",
"The consensus is achieved during each epoch and an entry or a set of entries are added to the log.",
"In case the primary is found to be Byzantine,a view change (select new primary) takes effect to provide liveness.",
"These protocols have shown the ability to achieve throughput of tens of thousand transactions per second [7], [8].",
"However, their scalability has been tested with a very small number of nodes $n$ , usually 10 to 20 nodes, due to the requirement for $n \\times n$ broadcast [2], that is, they have quadratic communication complexity.",
"To address the scalability issues in BFT protocols, we introduce the Musch blockchain protocol.",
"Musch is BFT-based and achieves $O(fn + n)$ communication complexity in an epoch, where $f$ is the actual number of Byzantine nodes ($f < n/3$ ).",
"For small (i.e.",
"constant) $f$ the communication complexity is linear, and hence, Musch has scalable performance.",
"Musch does not need to know the actual value of $f$ since it automatically adjusts to the actual number of nodes that exhibit faulty behavior in each epoch.",
"At the same time, the latency is comparable with other efficient BFT-based protocols [7], [9].",
"The performance of our algorithm is based on a novel mechanism of communication with a set of window nodes.",
"Nodes reach sliding windows moving over node IDs to recover from faults during consensus.",
"If a replica does not receive expected messages from the primary, it complains to the window nodes from which it recovers updates (see Fig.",
"REF ).",
"Initially, the window consists of only one node.",
"If the complainer replica doesn't receive a valid response from any of the the window nodes, it considers the next window of double size to which it sends the complaint.",
"The last window size is no more than $2f$ which guarantees to have a correct node within the last window.",
"This gives $O(fn + n)$ communication complexity.",
"In this way, Musch avoids $n \\times n$ broadcasts while guaranteeing consistency.",
"Table: Characteristics of state of Art BFT protocols.",
"The actual faulty nodes is ff, while f ' f^{\\prime } is an upper bound, f≤f ' f \\le f^{\\prime }.Table REF compares Musch with other state of art BFT-based protocols such as PBFT [9], FastBFT [10], and Aliph [8].",
"We compared the communication complexity measured as number of total exchanged messages during an epoch.",
"Our algorithm's performance depends solely on $fn$ , while the other algorithms have quadratic communication complexity.",
"Hence, our algorithm has an advantage when $f$ is asymptotically smaller than $n$ , resulting in less than quadratic communication complexity.",
"When $f$ is a constant our algorithm is optimal.",
"Additionally, we also compared the critical path length, as the number of one-way message latency it takes for a client request to be processed and the response is received by the client.",
"Note that the total number of nodes is $n = 3f^{\\prime }+1$ , where $f^{\\prime }$ is a conservative upper bound on the number of faulty nodes.",
"The actual faults are bounded by $f \\le f^{\\prime }$ .",
"Our algorithm does not need to know $f$ .",
"SBFT [11] has also tried to address the issue of scalability and have tested their protocol with 100 replicas, while achieving $10 \\times $ better performance than Ethereum.",
"In normal mode when $f=0$ , SBFT's message complexity will be $O(nc)$ ($c$ is the number of collectors) as compared to Musch's $O(n)$ .",
"The actual value of Byzantine nodes ($f$ ) has to be known (to choose correct $c$ such that $c\\ge f$ ) for the system to avoid the fall-back protocol.",
"But in practice it is impossible to know the actual value of $f$ (avoiding fallback is not possible for small $c$ ).",
"Fall-back mode executes efficient PBFT with $O(n^2)$ complexity.",
"This $O(n^2)$ complexity causes additional latency and performance degradation in SBFT."
],
[
"Paper Outline",
"We continue with this paper as follows.",
"In Section , we give the model of the distributed system.",
"In Section , we present our algorithm.",
"Protocol checkpoints are presented in Section .",
"We give the correctness analysis in Section , and the communication complexity bound analysis in Section ."
],
[
"System Model",
"Like other BFT-based state machine replication protocols Musch also assumes an adversarial failure model.",
"Under this model, servers and even clients may deviate from their normal behavior in arbitrary ways, which includes hardware failures, software bugs, or malicious intent.",
"Our protocol can tolerate up to $f^{\\prime }$ number of Byzantine replicas where the total number of replicas in the network $n = 3f^{\\prime }+1$ .",
"Replica ID is an integer from the replica set $\\lbrace 1, \\ldots , n\\rbrace $ that identifies each replica.",
"The actual number of Byzantine replicas in the network is denoted by $f$ , and at any moment during execution $0 \\le f \\le f^{\\prime }$ .",
"If $f=0$ then the execution is fault-free.",
"However, $f$ may not be known.",
"Our algorithm's communication complexity adapts to any value of $f$ .",
"Figure: Windows of nodes"
],
[
"Protocol",
"Our proposed protocol uses echo broadcast [12], where the primary proposes a block of transactions, and replicas respond by sending back signed hashes of the block.",
"We assume strong adversarial coordinated attacks by various malicious replicas.",
"However, replicas will not be able to break collision resistant hashes, encryption, or signatures.",
"We assume that all messages sent by replicas and the primary are signed.",
"For example if primary $p$ proposes a block of transactions $\\langle B \\rangle _p$ to the replica $i$ , we assume that it has been signed by primary $p$ .",
"Any unsigned message will be discarded.",
"To avoid repetition of message and signatures, Musch also uses signature aggregation [13] to use a single collective signature instead of appending all replica signatures, to keep signature size constant.",
"As the primary $p$ receives message $M_i$ with their respective signatures $\\sigma _i$ from each replica $i$ , the primary then uses these received signatures to generate an aggregated signature $\\sigma $ .",
"The aggregated signature can be verified by replicas given the messages $M_1,M_2,\\ldots , M_y$ where $y \\le n$ , the aggregated signature $\\sigma $ , and public keys $PK_1,PK_2,\\ldots ,PK_y$ .",
"Like other BFT-based protocols [9], [14], [15] each replica $i$ knows the public keys of other replicas in the network.",
"In Section REF we explain how to use the IDs to define the windows that we use in the algorithm.",
"It is not possible to ensure the safety and liveness of consensus algorithms in asynchronous systems where even a single replica can crash fail [16].",
"Musch's safety holds in asynchronous environments.",
"But to circumvent this impossibility for liveness, Musch assumes partial synchrony [17].",
"This partial synchrony is achieved by using arbitrarily large unknown but fixed worst case global stabilization delays.",
"During normal operation, Musch guarantees that at least $2f^{\\prime }+1$ replicas in each epoch are consistent (out of the $n = 3f^{\\prime }+1$ ).",
"Let $T$ be the maximum round-trip message delay in the network.",
"In our algorithm, at any moment of time, the suffix of the execution histories between any two replicas differ by at most the maximum number of blocks that can be committed during a time period of $O(T \\log f^{\\prime })$ .",
"Thus, any inconsistency is limited to only a small period of time.",
"Musch executes in epochs.",
"An epoch is a slot of time in which $2f^{\\prime }+1$ replicas receive block $B$ proposed by the primary $p$ and agree to commit it.",
"Thus, during each epoch a block is generated and added to the chain.",
"Since $p$ is responsible for aggregating replica signatures for block agreement, if less than $2f^{\\prime }+1$ replica signatures are collected then a view change will be triggered and the primary will be changed.",
"It should be noted that $p$ is also responsible for collecting transactions from clients, ordering the transactions, and sending them to the replicas."
],
[
"Normal Operation",
"As shown in Algorithm REF , the primary $p$ collects a set of transactions from the clients into an ordered list of transactions $L^a$ (which it will propose in a candidate block) with a sequence number $s$ , view number $v$ , hash $d=hash(L^a)$ , and hash history $h_s = Hash(h_{s-1},d)$ into candidate block $B= \\langle \\langle ORDER, s, v, d, h_s \\rangle _p , L^a \\rangle $ .",
"Primary $p$ then proposes (broadcasts) the candidate block $B$ to each replica $i$ .",
"As shown in Algorithm REF , upon receipt of $B$ each replica $i$ validates the information, and then replica $i$ responds to the primary $p$ with the willingness to accept the block in a message $H_i=\\langle RESPONSE, s, v,d,i \\rangle _i$ to the primary.",
"The primary collects at least $2f^{\\prime }+1$ responses from the replicas, aggregates them to $H$ , and generates a compressed aggregated signature $\\sigma $ [13].",
"Then, the primary broadcasts $\\langle COMMIT, H \\rangle _{\\sigma }$ .",
"Upon receipt, each replica $i$ verifies $2f^{\\prime }+1$ signatures and the candidate block $B$ commits.",
"If verified successfully each replica $i$ responds to the client with the reply message $\\langle REPLY,s, v, c , r ,t, i \\rangle _{\\sigma _i}$ , where $c$ is the client, $t$ is the timestamp and $r$ is the result of execution.",
"Upon receipt of $f^{\\prime }+1$ valid $REPLY$ messages (which might take $2f^{\\prime }+1$ messages to receive) a client accepts the result.",
"Assuming a continuous creation of blocks, the primary starts the new epoch immediately after the old epoch finishes.",
"Let $T$ be the maximum delivery delay of a message in the network.",
"According to the protocol, in the epoch of the new block $B$ with sequence number $s$ there will be two messages that replica $i$ expects to receive from the primary: (i) the $ORDER$ type message for block $B$ with sequence $s$ within $\\Delta _2 = T$ time from the end of the previous epoch, and then (ii) the $COMMIT$ type message for block $s$ within $\\Delta _3 = 2T$ time since the receipt of the $ORDER$ message.",
"Therefore, the maximum time for an epoch for a replica $i$ is $\\Delta _1 = \\Delta _2 + \\Delta _3$ .",
"A replica $i$ goes into recovery mode at time $\\Delta _1$ if either of the two expected messages is not received.",
"Uponupondoend Checkcheck alwaysthatend [t] Primary $p$ Latest committed block sequence number is $s_p$ receipt of transactions from a set of clients $C$ Create a block $B$ with sequence number $s_p + 1$ Broadcast $B$ to replicas receipt of $2f^{\\prime }+1$ hashes $H_i$ of $B$ from replicas Aggregate the hashes into $H$ Commit ($B$ ,$H$ ) Broadcast $H$ to replicas Send $REPLY$ to client set $C$ [t] Replica $i$ Normal Execution Latest committed block sequence number is $s_i$ receipt of block $B$ from primary $p$ with sequence number $s$ Calculate hash $H_i$ of block $B$ Send $H_i$ to primary $p$ receipt of aggregated hash $H$ for block $B$ from primary $H$ is signed by at least $2f^{\\prime }+1$ replicasCommit $(B,H)$ Send $REPLY$ to each client $c$ Special Cases at any time no receipt of expected $s_i+1$ block $B$ or respective hash $H$ within a timeout period Execute Algorithm REF with parameter $Complain$ receipt of a block $B$ with sequence $s > s_i + 1$ Execute Algorithm REF with parameter $Complain$ receipt of valid set of complains $S$ with $f^{\\prime }+1$ complainers Execute Algorithm REF initiate view change"
],
[
"Recovery Mode",
"In BFT protocols, when a replica detects an error it broadcasts complaints to all replicas in the network.",
"In contrast to this, a replica in Musch during a failure event will only complain to a subset of replicas in the network called window nodes.",
"If $i$ did not receive a response from the current window then the replica complains to the next window of double size until it receive response from at least one correct replica.",
"The window sequences are fixed, $W_1, W_2, \\ldots , W_{k^{\\prime }}$ , where $k^{\\prime } = \\lceil \\lg (f^{\\prime } + 1) \\rceil $ .",
"Suppose the replica IDs are taken from the set $\\lbrace 1, \\ldots , n\\rbrace $ and sorted in ascending order (see Fig.",
"REF ).",
"The window $W_1$ consists of a single node with the smallest ID, Window $W_2$ consists of two replicas with the next IDs in order, Window $W_3$ consists of four replicas with the next higher IDs, and so on.",
"Therefore the window $W_j$ consists of $2^{j-1}$ replicas, whose IDs are ranked between $2^{j-1}, \\ldots , 2^{j}-1$ .",
"During the execution of the algorithm, the maximum window that will be contacted is actually $k = \\lceil \\lg (f + 1) \\rceil $ , where $k \\le k^{\\prime }$ , since this guarantees that at least one correct node will be encountered among all the window nodes from $W_1$ up to $W_k$ .",
"ParamParameters [t] Fault Recovery in Replica $i$ $Complain$ from $i$ Let $l$ be the block sequence number in $Complain$ for which $i$ has not received either $B$ or $H$ $j=1$ window index current window is $W_j$ $i \\in W_j$ all window nodes prior to $W_j$ are faulty Broadcast $COMPLAIN$ message to replicas $i$ is in a later window than $W_j$ or $i$ is not a window node at all Send $COMPLAIN$ to all nodes in window $W_j$ there is no commit by a certain timeout $j = j + 1$ increase window Goto Line REF listen for responses Let $l^{\\prime } \\ge l$ be the expected sequence number of blocks in the time period since $Complain$ issued receipt of blocks and respective hashes up to at least $l^{\\prime }$ Commit all received pairs of block and hash $(B,H)$ [t] Window Node $i$ receipt of $COMPLAIN$ or $PROOF$ message from replica $j$ $COMPLAIN$ from $j$ is valid Add $COMPLAIN$ by distinct complainer to the set of complains $S$ distinct number of complainers in $S$ is at least $f^{\\prime }+1$ Broadcast $S$ Execute Algorithm REF Reset $S$ to empty Let $l$ be the sequence number of block requested in $COMPLAIN$ $i$ has the $l$ th block and its hash Send all blocks and respective hashes starting from sequence $l$ up to the latest to replica $j$ $PROOF$ is valid Broadcast $PROOF$ to replicas Execute Algorithm REF Reset $S$ to empty Algorithm REF describes how a replica complains to the window(s), and Algorithm REF shows the respective reactions from the window nodes.",
"As shown in Algorithm REF , if replica $i$ complains that it didn't receive expected message ($ORDER$ or $COMMIT$ ) from $p$ during normal operation, it sends the complaint in the form of $\\langle COMPLAIN, s, v, d \\rangle _i$ , where $d$ and $s$ belongs to the last committed block in the chain of $i$ .",
"If it complains to a window $W_j$ , this message is sent to all nodes in $W_j$ which will then know that replica $i$ does not have $ORDER$ or $COMMIT$ messages after block $s$ .",
"If replica $i$ has received a message from the primary that proves the maliciousness of $p$ , then it attaches the proof in its complaint $\\langle COMPLAIN, PROOF \\rangle _i$ to $W_j$ .",
"When $i$ enters the recovery mode it first complains to window $W_1$ , which has a single node.",
"If $i$ doesn't get any useful response from $W_1$ then it complains to $W_2$ , which has two nodes, so it informs both nodes.",
"This process can repeat until $i$ contacts all nodes in $W_k$ , the last window.",
"It is guaranteed that replica $i$ will get a response from a correct node in one of these windows.",
"As shown in Algorithm REF , the window nodes respond to complaints by returning the requested information.",
"If they do not have it then they call themselves Algorithm REF as well.",
"If the complainer $i$ is a window node itself, it will stop until it reaches its own window size and will broadcast the complaint.",
"Upon broadcast it is guaranteed that it will receive response.",
"The response can be either receipt of missing messages or a view change.",
"If replica $i$ received the missing messages it will forward it to the complainers that it knows, else it will result in view change (primary will be replaced).",
"Note that regular replicas and window nodes may be complaining at the same time and probably for the same reason.",
"A regular node will have to wait for the window nodes to first obtain a response.",
"It is important to coordinate the actions of the windows nodes and the regular replicas to receive the responses efficiently without message replication.",
"For a regular replica $i$ , the timeout period for waiting a response from the window $W_j$ is at most $\\Lambda _j = j3T + 6T$ .",
"As it takes $\\Delta _1=3T$ to detect timeout for the current epoch, then it takes at most $j3T$ to receive a message from the previous window and send the message back to the replica.",
"In case window $j$ does not receive a message from window $j-1$ , it will broadcast its complaint and it is guaranteed that it will receive a response, which it will send back to replica $i$ ($3T$ ).",
"From the start time $t$ of the current epoch, if $i$ does not get a response within $t + \\Lambda _j$ then it will contact the next window $W_{j+1}$ ."
],
[
"View Change",
"A view change can be triggered if a correct window node $i \\in W_l$ receives at least $f^{\\prime }+1$ distinct replica complaints (against primary $p$ ) as shown in Algorithm REF .",
"This guarantees that at least one of the complaints is coming from a correct replica.",
"Another reason for view change can be the receipt of an explicit $PROOF$ against $p$ by window node $i$ .",
"Once view change is triggered, window node $i$ broadcasts the set of $COMPLAIN$ or $PROOF$ messages it has received to all replicas (Algorithm REF ).",
"Without loss of generality, consider the case where window node $i$ has sent $PROOF$ to all replicas (the same mechanism also applies to other sets of $COMPLAIN$ messages).",
"Upon receipt of $PROOF$ a replica $j$ increments its view number ($v=v+1$ ) and assigns new primary $p^{\\prime }$ (namely, $p^{\\prime } = v \\mod {n}$ ) (Algorithm REF ).",
"Replica $j$ then adds its most recent block hash $d$ and block number $s$ in the message along with $PROOF$ in a message $\\langle VIEWCHANGE, PROOF , s,d,j \\rangle _j$ and sends it to the new primary $p^{\\prime }$ (Algorithm REF ).",
"Upon receipt of at least $2f^{\\prime }+1$ view change messages from different replicas, $p^{\\prime }$ stores them into set $Q$ .",
"Then, $p^{\\prime }$ broadcasts $\\langle Q \\rangle _{\\sigma }$ , where $\\sigma $ is an aggregated signature for all replicas involved in $Q$ (Algorithm REF ).",
"Upon receipt of this message, each replica recovers the latest block history.",
"Assume $s^{\\prime }$ is the highest block number committed so far in the chain.",
"The block $s^{\\prime }$ must have been committed by at least $2f^{\\prime }+1$ replicas, and since $Q$ has size at least $2f^{\\prime }+1$ , it must be that $f^{\\prime }+1$ replicas in $Q$ have also committed $s^{\\prime }$ , one of which is a correct node.",
"Thus, every replica upon receipt of $Q$ can figure out that the latest committed valid block number is $s^{\\prime }$ .",
"Once $s^{\\prime }$ is known, a replica $i$ will check if block with sequence $s^{\\prime }$ is the latest block in its history $h_i$ , and if it is, $i$ sends a confirmation message $s^{\\prime }_i$ to $p^{\\prime }$ (Algorithm REF ).",
"In this case, at least $f^{\\prime }+1$ correct replicas know the latest block of $p^{\\prime }$ ($s^{\\prime }_{p^{\\prime }}$ ).",
"If $ s^{\\prime }$ is same as $s^{\\prime }_{p^{\\prime }}$ then $p^{\\prime }$ begins updating all other replicas that have fallen behind (Algorithm REF ).",
"$p^{\\prime }$ will not send any block generated earlier than the water mark $H$ (Section ).",
"If $p^{\\prime }$ does not have $s^{\\prime }$ as its latest block then at least $f^{\\prime }+1$ correct replicas know about it and they send missing blocks and their respective $COMMIT$ messages to $p^{\\prime }$ and then $p^{\\prime }$ updates other replicas as described above.",
"Once $p^{\\prime }$ has updated other replicas it will wait to receive at least $2f^{\\prime }+1$ correct replicas have sent confirmation $\\ s^{\\prime }_i $ (Algorithm REF ).",
"Since there are at least $2f^{\\prime }+1$ correct replicas, $p^{\\prime }$ signs the latest block in their histories that $p^{\\prime }$ has received using an aggregated signature $V \\leftarrow \\bigcup _i \\langle s^{\\prime }_i \\rangle _{\\sigma }$ and broadcasts it to the replicas.",
"Upon receipt of $V$ each replica is now ready for the new epoch of the next block and is waiting to receive an $ORDER$ message from the new primary $p^{\\prime }$ (Algorithm REF ).",
"In case a replica does not receive expected messages ($Q$ , $V$ or blocks and their hashes within a certain expected time), then it issues a new complaint which is processed similar to the other types of complaints as described above.",
"During the view change process there may be some clients who send their request but it will not be processed because replicas are busy.",
"To address this as we mentioned earlier the client $c$ will broadcast its request after epoch time $\\Delta _1$ , if it did not receive the response from $p$ .",
"In such case, all replicas receive the request $T_c$ and forward it to the $p$ .",
"Upon receipt of $2f^{\\prime }+1$ such forwarded requests, $p$ considers $T_c$ to be included in the $ORDER$ message as soon as possible.",
"$p$ will have to propose those backlogged requests before proposing the new requests it receives.",
"If it proposes a request that has not been seen by $2f^{\\prime }+1$ replicas (of which $f^{\\prime }+1$ replicas are correct/honest replicas) proposing the backlogged transactions then the replicas can send a complaint which will result in a view change.",
"[t] Replica $i$ View Change Select new primary $p^{\\prime }=v \\mod {n}$ Send $VIEWCHANGE$ containing latest local block number $s_i$ to $p^{\\prime }$ Receive aggregated $VIEWCHANGE$ s $Q$ from $p^{\\prime }$ $Q$ contains at least $2f^{\\prime }+1$ $VIEWCHANGE$ s Get the latest block number ($s^{\\prime }$ ) that has been signed by at least $f^{\\prime }+1$ replicas in $Q$ latest block $s_i$ in replica $i$ is same as $s^{\\prime }$ Replica has not lost any block Receive messages (blocks and their respective hashes) up to $s^{\\prime }$ from $p^{\\prime }$ before timeout Once updated ($s^{\\prime }_i = s^{\\prime }$ ) send $s^{\\prime }_i$ to $p^{\\prime }$ Receive $V$ from $p^{\\prime }$ containing aggregated histories of at least $2f^{\\prime }+1$ replicas [!htp] New Primary View Change Receive $VIEWCHANGE$ messages from replicas Aggregate at least $2f^{\\prime }+1$ $VIEWCHANGE$ messages into $Q$ Broadcast $Q$ to replicas Get the latest block number ($s^{\\prime }$ ) that has been signed by at least $f^{\\prime }+1$ replicas in $Q$ latest block in $p^{\\prime }$ is same as $s^{\\prime }$ New primary $p^{\\prime }$ has not lost any block Receive messages (blocks and their respective hashes) up to $s^{\\prime }$ from $f^{\\prime }+1$ replicas that are up to date Send messages with missing blocks and hashes to all replicas $i$ who have fallen behind, $s_i < s^{\\prime }$ , where $s_i$ should not be less than latest water mark Once received updated $s^{\\prime }_i$ from each replica $i$ , where $s^{\\prime }_i = s^{\\prime }$ , aggregate $s^{\\prime }_i$ into $V$ Broadcast $V$"
],
[
"Checkpoints",
"As an optimization to the protocol, we use checkpoints to improve on the number of messages exchanged during view change.",
"Checkpoints are typically used as a way to truncate the log in other BFT-based protocols [9].",
"In addition to that, we can also use it to prevent malicious replicas from downloading older messages from a new primary $p^{\\prime }$ and delaying the completion of the view change process.",
"As we know from Section REF , some correct replicas might miss messages and go into recovery mode.",
"These replicas need to download those missing messages.",
"But malicious replicas might try to download very old blocks and delay the view change process.",
"To bound this we use checkpoints.",
"To maintain the safety condition it is required that at least $2f^{\\prime }+1$ replicas agree on the checkpoint.",
"The checkpoint is created after a constant number of blocks (e.g., sequence number divisible by 200).",
"In Musch, replicas can agree on checkpoints during block agreement (checkpoint number to be added to the $RESPONSE$ message).",
"A checkpoint that is agreed upon by $2f^{\\prime }+1$ replicas of which at least $f^{\\prime }+1$ are honest is called a stable checkpoint.",
"Checkpoints have low and high watermarks.",
"Low watermark $h$ is the last stable checkpoint and the high water mark $H$ is the sum of low water mark and $k$ number of blocks($H=k+h$ ), where $k$ is large enough (i.e.",
"$k=400$ ).",
"If a replica wants to download a block older than $H$ , $p^{\\prime }$ will ignore the download request and might think that the replica is maliciously trying to delay the view change process."
],
[
"Correctness Analysis",
"In this section we provide proof of correctness and analysis of the Musch protocol.",
"Before we proceed, it is important to define transaction completion and protocol correctness for the Musch protocol.",
"We say that a transaction $T_c$ issued by a client $c$ is considered to be completed by $c$ if $c$ receives at least $f^{\\prime }+1$ valid $\\langle REPLY,s, v, c , r ,t, i \\rangle _{\\sigma _i}$ messages.",
"It is guaranteed that upon receipt of $2f^{\\prime }+1$ $REPLY$ messages from different replicas at least $f^{\\prime }+1$ of them are valid.",
"We will prove that Musch satisfies the following correctness criteria: Definition 1 (Liveness) Every transaction proposed by the correct client will eventually be completed in finite time.",
"Definition 2 (Safety) A system is safe if a correct primary proposes a block of ordered transactions with block number $s$ and it is committed by at least $2f^{\\prime }+1$ replicas, then any block that has been committed earlier will have smaller block number ($s^{\\prime } < s$ ) in the chain.",
"Thus, block $B_{s^{\\prime }}$ will be the prefix of block $B_s$ in the chain.",
"Additionally the order of transactions within the block will remain identical in all correct replicas (due to Merkle treeMerkle trees are hash-based data structures in which each leaf node is hash of a data block and each non leaf node is hash of its children.",
"It is mainly used for efficient data verification.",
")."
],
[
"Safety",
"Lemma 1 Any two committed blocks $B_{s^{\\prime }}$ and $B_{s}$ must have a different block number.",
"Consider committed blocks $B_{s^{\\prime }}$ and $B_s$ .",
"At least a set of $2f^{\\prime }+1$ replicas $S_1$ have agreed to all transactions with $B_{s^{\\prime }}$ and have committed it.",
"Similarly, at least a set of $2f^{\\prime }+1$ replicas have agreed for the transactions in block $B_{s}$ and committed it.",
"Since there are $3f^{\\prime }+1$ replicas, there is at least one correct replica (out of the at least $f^{\\prime }+1$ replicas in $S_1 \\cap S_2$ ) that committed both for $B_{s^{\\prime }}$ and $B_{s}$ .",
"But a correct replica only commits one block with a specific block number.",
"Thus, both blocks must have different numbers.",
"The same mechanism applies during recovery mode.",
"Lemma 2 If block $B_{s^{\\prime }}$ commits earlier than block $B_{s}$ , then $B_{s^{\\prime }}$ has a smaller block number than $B_{s}$ .",
"As per Lemma REF , at least one correct replica $k$ has committed both $B_{s^{\\prime }}$ and $B_{s}$ .",
"Suppose, that $B_{s^{\\prime }}$ gets a block number $s^{\\prime }$ which is smaller than the block number $s$ of $B_s$ , that is $s^{\\prime }< s$ ($s \\ne s^{\\prime }$ from Lemma REF ).",
"A correct Replica $k$ will only accept $B_s$ if $B_s$ is consistent with its local history (only if $s>s^{\\prime }$ ).",
"Lemma 3 Musch is safe during view change.",
"During a view change (Algorithms REF and REF ), all replicas including the new primary $p^{\\prime }$ retrieve the latest history and block number $s^{\\prime }$ as at least $f^{\\prime }+ 1$ replicas will agree on the latest block number $s^{\\prime }$ , which includes a correct replica that knows $s^{\\prime }$ .",
"All correct replicas know the latest block $s^{\\prime }_{p^{\\prime }}$ in the history of $p^{\\prime }$ from $\\langle Q\\rangle _{\\sigma }$ .",
"If $s^{\\prime }_{p^{\\prime }}=s^{\\prime }$ then $p^{\\prime }$ begins updating all other replicas that have fallen behind in history, in other words it updates all the replicas that do not have blocks and respective $COMMIT$ messages up to $s^{\\prime }$ .",
"If $p^{\\prime }$ does not have $s^{\\prime }$ as its latest block then at least $f^{\\prime }+1$ correct replicas know about it (from $Q$ ) and they send missing blocks and their respective $COMMIT$ messages to $p^{\\prime }$ and then $p^{\\prime }$ updates other replicas as described above.",
"Once $p^{\\prime }$ updated (receive blocks and $COMMIT$ s up to $s^{\\prime }$ ) other replicas it will take $T$ timeout period to receive $s_i $ (update confirmation) from at least $2f^{\\prime }+1$ replicas.",
"Then, $p^{\\prime }$ signs all their histories using an aggregated signature $V \\leftarrow \\bigcup _i \\langle s_i \\rangle _{\\sigma }$ and broadcasts it to the replicas.",
"Upon receipt of $V$ each replica is now ready for the new epoch of the next block and is waiting to receive an $ORDER$ message from the new primary $p^{\\prime }$ .",
"Theorem 4 (Safety) Musch is safe.",
"Lemma REF guarantees safety when the new primary $p^{\\prime }$ is correct.",
"If $p^{\\prime }$ is not correct, safety will be guaranteed when eventually a correct primary will be chosen.",
"Therefore, based on Lemmas REF , REF and REF , Musch is safe when replicas are either in normal, recovery, or view change mode."
],
[
"Liveness",
"In this section we provide a proof for liveness of Musch.",
"Lemma 5 Musch satisfies liveness when the primary is correct.",
"Consider a correct primary $p$ that executes Algorithm REF , and also the replicas that execute Algorithm REF .",
"Primary $p$ receives at least $2f^{\\prime }+1$ correct $RESPONSE$ messages from replicas, aggregates and signs them using an aggregation signature $\\sigma $ .",
"It then broadcasts the signed $COMMIT$ message to all replicas.",
"Upon receipt of the $COMMIT$ message each replica will commit the block.",
"The primary $p$ along with all correct replicas also forwards a reply message to each client $\\langle REPLY,s, v, c , r ,t, i \\rangle $ and clients will mark the transaction as completed.",
"Lemma 6 If there are $f^{\\prime }+1$ complaints, or there is a complaint with a proof of maliciousness against the primary, then a view change will occur.",
"Algorithm REF guarantees that, in the worst case, a replica $i$ can find a window node $W_{k}$ to complain, where, $k = \\lceil \\lg (f + 1) \\rceil $ and $W_{k}$ contains at least one correct replica, since $W_k$ contains at least $2^{\\lg (f+1)} = f+1$ nodes.",
"Observe that once a replica $i$ has found a honest window node, it is guaranteed that the honest node will reply to its valid complaint either by sending back blocks and $COMMIT$ s or if the number of complaints are greater than $f^{\\prime }$ , then the window node will broadcast all complaints to the network causing a view change (Algorithm REF ).",
"If a replica $j \\in W_{k}$ receives at least $f^{\\prime }+1$ complaints from other replicas it triggers a view change according to the Algorithm REF .",
"Since $f^{\\prime }+1$ complaints are received, this guarantees that at least one honest replica has complained.",
"Similarly, $j$ may receive an explicit proof that the primary $p$ is faulty ($p$ 's history is incorrect, or it has proposed an invalid transaction, etc.).",
"In such a case only one complaint is needed to prove that $p$ is malicious and a view change will be triggered.",
"Lemma 7 If a transaction is not completed then a view change will occur.",
"If a transaction does not complete after sufficient time $\\Delta _1$ , then the client $c$ broadcasts its transaction $T_c$ to the replicas.",
"Upon receipt of $T_c$ , the replicas check if they have already committed a block that contains $T_c$ .",
"If they did, each replica $i$ will send $\\langle ACK, T_c\\rangle _i$ to the client and upon receipt of $2f^{\\prime }+1$ $ACK$ messages the client will consider the transaction as complete.",
"If primary $p$ has not proposed the transaction $T_c$ , then each replica will forward $T_c$ to $p$ and will expect that $p$ will include it in the next $ORDER$ message (during normal operation).",
"If $p$ does not include it in the next $ORDER$ message, then replicas will start complaining, which will result in a view change (if at least $f^{\\prime }+1$ replicas complain, from Lemma REF ).",
"Another case that can prevent a request from being committed is when replicas receive a $COMMIT$ message signed by less than $2f^{\\prime }+1$ replicas.",
"In this case, this can be used as proof against $p$ and a complaint can be made, which will result in a view change (Lemma REF ).",
"Lemma 8 Musch satisfies liveness even if a client request is received during a view change.",
"During the view change process, there may be some clients who send their request for transaction $T_c$ but it will not be processed because replicas are busy with the view change.",
"To address this, as mentioned earlier the client $c$ will broadcast its request after epoch timeout $\\Delta _1$ , if it did not receive a response from $p$ .",
"In such a case, all replicas receive the request $T_c$ and forward it to the new primary $p^{\\prime }$ .",
"Upon receipt of $2f^{\\prime }+1$ such forwarded requests the $p^{\\prime }$ considers $T_c$ to be included in the $ORDER$ message as soon as possible.",
"The new primary $p^{\\prime }$ will have to propose those backlogged client requests during the view change, before proposing the new requests it receives.",
"If it proposes a request that has not been seen by $2f^{\\prime }+1$ replicas (of which $f^{\\prime }+1$ replicas are correct/honest replicas), proposing the backlogged transactions then the replicas can start complaints, which will result in a new view change (Lemma REF ).",
"Theorem 9 (Liveness) Musch satisfies liveness and all correct transactions will be completed eventually.",
"Based on Lemmas REF , REF and REF , any correct transaction request by a client will be completed within a finite period of time."
],
[
"Communication Complexity",
"In communication complexity, we count all messages that cause a reaction in our algorithm and we refer to these as effective messages.",
"In contrast, there are ineffective messages, which have sources that have been identified as malicious, and so the recipient can ignore these messages.",
"We will measure the number of effective messages exchanged in an epoch, and we will consider worst cases scenarios, with or without view change.",
"In other words, we consider worst-case performance attacks when malicious replicas attempt to increase the communication of the protocol by causing messages to be sent from correct replicas.",
"In the communication complexity we consider separately the messages sent between clients and replicas, and those sent only between replicas."
],
[
"Client-Replica Communication Complexity",
"If a client sends a transaction to the primary $p$ , and does not receive a response from the primary $p$ within $\\Delta _1$ , then the client broadcasts to the primary $p$ (a broadcast involves $n$ messages).",
"Upon receipt of a broadcast from a client, if replica $i$ has already processed the client's transaction it will answer to the client with an acknowledgement.",
"If not, the replica $i$ will forward the client's request to the primary, forcing it to process it as soon as possible.",
"The liveness property of our algorithm, Theorem REF , will guarantee that eventually at least $2f^{\\prime }+1 = O(n)$ of the replicas will send acknowledgements to the client.",
"Therefore, we get the following result: Lemma 10 For each transaction sent by a client, at most $O(n)$ messages will be exchanged between the client and the replicas in order to process the transaction (i.e., include the transaction in a block)."
],
[
"Replica-Replica Communication Complexity",
"In this section we analyze the communication complexity of the consensus engine of our protocol, which includes the primary $p$ and the replicas (in total $n$ nodes).",
"A malicious primary $p$ and malicious replicas both can try to increase the communication complexity."
],
[
"Messages caused by malicious primary",
"Let $R_c$ be the set of replicas that complain.",
"First, we examine the case when the nodes in $R_c$ did not receive the block or $COMMIT$ message and they complain.",
"A malicious primary $p$ can afford not to send such messages up to at most $f^{\\prime }$ replicas, without getting caught as being malicious; that is, $|R_c| \\le f^{\\prime }$ .",
"In this case, each of the complainers in $R_c$ may have to communicate with up to $2f+1$ window nodes, since this guarantees a window that has at least one correct window node.",
"This gives at most $(2f+1)|R_c|$ messages.",
"In the worst case, out of the $2f+1$ window replicas at most $f+1$ will be the honest ones that will broadcast to all $n$ replicas and will receive their response, to be forwarded to the complainers $R_c$ , giving at most $2(f+1)n + (f+1)|R_c|$ additional messages.",
"The total communication complexity in this case will be (since $|R_c| \\le f^{\\prime } < n/3$ ): $\\begin{split}(2f+1)|R_c| +2 (f+1) n + (f+1) |R_c| \\\\ \\le (5f + 4)n = O(fn+n).\\end{split}$"
],
[
"Messages caused by malicious replicas",
"Suppose the set of complainers $R_c$ are malicious, thus, $|R_c| \\le f$ .",
"Window nodes do not respond to repetitive complains from the same replica (non-effective messages), which prevents malicious replicas from increasing the communication complexity.",
"Nevertheless, each window node may respond once to each malicious request.",
"A window node $j$ can respond to a complain message in the following ways: If window node $j$ has the appropriate response to the complain (i.e.",
"it has the block or $COMMIT$ ) it will send it back to the replica that complained.",
"At most $2f^{\\prime }+1$ window nodes will be accessed by each replica in $R_c$ , since this is the bound on the total number of window nodes.",
"Therefore, in this case, the number of messages are at most: $\\begin{split}2 (2f^{\\prime }+1)|R_c| \\le (4f^{\\prime }+2)f \\\\ < (4n/3 + 2)f = O(fn+n).\\end{split}$ If window node $j$ does not have the appropriate response (block or $COMMIT$ ), then $j$ itself is also executing the window protocol from smaller to larger windows, and when it eventually points to its own window, it will broadcast the complaint to get a response from other replicas (acting as a regular window node).",
"This scenario can only happen if all the previous windows are populated by $f$ faulty nodes.",
"The number of complaints from $R_c$ to up to $2f^{\\prime }+1$ window nodes are bounded by $(2f^{\\prime }+1)|R_c|$ .",
"Similarly, the respective responses are bounded by $(2f^{\\prime }+1)|R_c|$ .",
"For calculating the messages from the broadcasts, out of the $2f^{\\prime }+1$ total window nodes, at most $2f+1$ window nodes will react to the received complaints with broadcasts, since the first encountered window of size at least $f+1$ will respond to any complaint from a valid node.",
"Thus, each of the up to $2f+1$ windows nodes broadcasts to all replicas, causing $(2f+1)n$ additional messages.",
"Therefore, in this case, the number of messages are at most: $\\begin{split}2(2f^{\\prime }+1)|R_c|+ (2f+1)n \\\\ < (4n/3 + 2) f + (2f+1)n = O(fn+n).\\end{split}$"
],
[
"View Change Communication Complexity",
"When a correct window node receives $f^{\\prime }+1$ complaints it will broadcast all of them to all replicas ($n$ messages).",
"There are at most $f+1$ window nodes that will broadcast (since those window nodes could be correct in the last accessed window size), resulting to at most $(f+1)n$ messages.",
"Upon receipt of the broadcast message, each replica begins the view change process.",
"The replica sends back a $VIEWCHANGE$ message to the new primary $p^{\\prime }$ which also includes its history ($n$ messages).",
"The new primary $p^{\\prime }$ aggregates all $VIEWCHANGE$ messages into $\\langle Q \\rangle _{\\sigma _p^{\\prime }}$ and broadcasts ($n$ messages).",
"Upon receipt each replica extracts the most recent block as described in Section REF .",
"Therefore, the number of messages from this part of the algorithm is at most: $(f+1)n+n+n= fn + 3n.$ During this, at least $f^{\\prime }+1$ correct replicas have the latest committed block $s^{\\prime }$ , and this block is chosen as the starting point for the next epoch, which will build another block ($s^{\\prime }+1$ ) over it.",
"All other replicas that have block number less than $s^{\\prime }$ as their latest block have to download all the blocks up to $s^{\\prime }$ from $p^{\\prime }$ .",
"If $p^{\\prime }$ does not have $s^{\\prime }$ as its latest block, then $f^{\\prime }+1$ replicas that have it will bring $p^{\\prime }$ up to date.",
"Thus, if $f^{\\prime }+1$ replicas have $s^{\\prime }$ as their latest block, then, at most $2f^{\\prime }$ replicas in the worst case get (download) messages up from the high water mark in checkpoint $H$ to $s^{\\prime }$ .",
"Let $e$ be the number of committed blocks from $H$ to $s^{\\prime }$ .",
"For each committed block we need two messages, first the block itself and the second is the $COMMIT$ message.",
"Thus, we have: $2e (2f^{\\prime }+f^{\\prime }) = 6ef^{\\prime }.$ Assuming frequent checkpoints (say every a fixed number of blocks), we can assume that $e$ is a constant.",
"From Equations REF and REF we have for the total number of messages in view change: $fn + 3n + 6e(n/3) = O(fn+n).$"
],
[
"Overall Messages",
"Combining Equations REF , REF , REF , we obtain $O(fn+n)$ communication complexity in a single epoch for the communication complexity between replicas.",
"From Equation REF the communication complexity is also $O(fn+n)$ during view change.",
"Therefore, we have the following result: Lemma 11 The number of messages exchanged between replicas in an epoch or during view change are $O(fn+n)$ .",
"Combining Lemmas REF and REF we obtain the main result for the communication complexity: Theorem 12 (Communication complexity) For $\\tau $ initiated transactions in an epoch, the communication complexity is $O(\\tau n + fn)$ .",
"For constant $\\tau $ , the communication complexity is $O(fn+n)$ ."
],
[
"Conclusions",
"In this paper we proposed Musch, a BFT-based consensus protocol, in an effort to avoid excessive messages and improve the scalability of blockchain algorithms.",
"Through the use of windows, the algorithm adapts to the actual number of faulty nodes $f$ , and in this way it avoids unnecessary messages.",
"This improvement does not sacrifice on the latency, since our algorithm still uses a small number of communication rounds.",
"For future work, it would be interesting to investigate whether we can decrease the message complexity further, i.e.",
"to $O(n)$ under $f$ faults, by introducing an intelligent scheme to detect faulty nodes and foil attempts to increase message complexity."
]
] | 1906.04381 |
[
[
"Equitable factorizations of edge-connected graphs"
],
[
"Abstract In this paper, we show that every $(3k-3)$-edge-connected graph $G$, under a certain condition on whose degrees, can be edge-decomposed into $k$ factors $G_1,\\ldots, G_k$ such that for each vertex $v\\in V(G_i)$, $|d_{G_i}(v)-d_G(v)/k|< 1$, where $1\\le i\\le k$.",
"As application, we deduce that every $6$-edge-connected graph $G$ can be edge-decomposed into three factors $G_1$, $G_2$, and $G_3$ such that for each vertex $v\\in V(G_i)$, $|d_{G_i}(v)-d_{G}(v)/3|< 1$, unless $G$ has exactly one vertex $z$ with $d_G(z) \\stackrel{3}{\\not\\equiv}0$.",
"Next, we show that every odd-$(3k-2)$-edge-connected graph $G$ can be edge-decomposed into $k$ factors $G_1,\\ldots, G_k$ such that for each vertex $v\\in V(G_i)$, $d_{G_i}(v)$ and $d_G(v)$ have the same parity and $|d_{G_i}(v)-d_G(v)/k|< 2$, where $k$ is an odd positive integer and $1\\le i\\le k$.",
"Finally, we give a sufficient edge-connectivity condition for a graph $G$ to have a parity factor $F$ with specified odd-degree vertices such that for each vertex $v$, $| d_{F}(v)-\\varepsilon d_G(v)|< 2$, where $\\varepsilon $ is a real number with $0< \\varepsilon < 1$."
],
[
"Introduction",
"In this article, graphs may have loops and multiple edges.",
"Let $G$ be a graph.",
"The vertex set, the edge set, and the maximum degree of $G$ are denoted by $V(G)$ , $E(G)$ , and, $\\Delta (G)$ , respectively.",
"We denote by $d_G(v)$ the degree of a vertex $v$ in the graph $G$ , whether $G$ is directed or not.",
"Also the out-degree and in-degree of $v$ in a directed graph $G$ are denoted by $d_G^+(v)$ and $d_G^-(v)$ .",
"An orientation of $G$ is said to be $p$ -orientation, if for each vertex $v$ , $d_G^+(v)\\stackrel{k}{\\equiv }p(v) $ , where $p:V(G)\\rightarrow Z_k$ is a mapping and $Z_k$ is the cyclic group of order $k$ .",
"For any two disjoint vertex sets $A$ and $B$ , we denote by $d_G(A,B)$ the number of edges with one end in $A$ and the other one in $B$ .",
"Also, we denote by $d_G(A)$ the number of edges of $G$ with exactly one end in $A$ .",
"Note that $d_G(\\lbrace v\\rbrace )$ denotes the number of non-loop edges incident with $v$ , while $d_G(v)$ denotes the degree of $v$ .",
"We denote by $G[A]$ the induced subgraph of $G$ with the vertex set $A$ containing precisely those edges of $G$ whose ends lie in $A$ .",
"Let $g$ and $f$ be two integer-valued functions on $V(G)$ .",
"A parity $(g,f)$ -factor of $G$ refers to a spanning subgraph $F$ such that for each vertex $v$ , $g(v)\\le d_F(v)\\le f(v)$ , and also $d_F(v)$ , $g(v)$ , and $f(v)$ have the same parity.",
"An $f$ -parity factor refers to a spanning subgraph $F$ such that for each vertex $v$ , $d_F(v)$ and $f(v)$ have the same parity.",
"A graph $G$ is called $m$ -tree-connected, if it contains $m$ edge-disjoint spanning trees.",
"A graph $G$ is termed essentially $\\lambda $ -edge-connected, if all edges of any edge cut of size strictly less than $\\lambda $ are incident to a vertex.",
"A graph $G$ is called odd-$\\lambda $ -edge-connected, if $d_G(A)\\ge \\lambda $ , for every vertex $A$ with $d_G(A)$ odd.",
"Note that $2\\lambda $ -edge-connected graphs are odd-$(2\\lambda +1)$ -edge-connected.",
"A factorization $G_1,\\ldots , G_k$ of $G$ is called equitable factorization, if for each vertex $v$ , $|d_{G_i}(v)-d_G(v)/k|<1$ , where $1\\le i\\le k$ .",
"Throughout this article, all variables $k$ are positive and integer.",
"In 1971 de Werra observed that bipartite graphs admit equitable factorizations.",
"Theorem 1.1 .",
"([27]) Every bipartite graph $G$ can be edge-decomposed into $k$ factors $G_1,\\ldots , G_k$ such that for each $v\\in V(G_i)$ , $ | d_{G_i}(v)-d_G(v)/k|< 1 .$ In 1994 Hilton and de Werra developed Theorem REF to simple graphs by replacing the following weaker version.",
"In 2008 Hilton [13] conjectured that for those simple graphs for which whose vertices with degree divisible by $k$ form an induced forest, the upper bound can be replaced by that of Theorem REF .",
"Later, this conjecture was confirmed by Zhang and Liu (2011) [28].",
"Theorem 1.2 .",
"([15]) Every simple graph $G$ can be edge-decomposed into $k$ factors $G_1,\\ldots , G_k$ such that for each $v\\in V(G_i)$ , $ | d_{G_i}(v)-d_G(v)/k|\\le 1 .$ Recently, Thomassen (2019) made another attempt for developing Theorem REF to highly edge-connected regular graphs and concluded the following result.",
"An intresting consequence of it says that every 6-edge-connected $3r$ -regular graph can be edge-decomposed into three $r$ -regular factors.",
"Theorem 1.3 .",
"([26]) Let $k$ and $r$ be two odd positive integers.",
"If $G$ is an odd-$(3k-2)$ -edge-connected $kr$ -regular graph, then it can be edge-decomposed into $r$ -regular factors.",
"In this paper, we generalize Thomassen's result in several ways together with developing Theorem REF to highly edge-connected graphs as the next theorem.",
"In addition, we provide a common version for Theorem REF and whose even-regular version by replacing another concept of edge-connectivity.",
"Theorem 1.4 .",
"Let $G$ be a $(3k-3)$ -edge-connected graph, where $k$ is a positive integer.",
"If there is a vertex set $Z$ with $|E(G)|\\stackrel{k}{\\equiv }\\sum _{v\\in Z}d_G(v)$ , then $G$ can be edge-decomposed into $k$ factors $G_1,\\ldots , G_k$ such that for each $v\\in V(G_i)$ , $|d_{G_i}(v)-d_G(v)/k|< 1 .$ In 1982 Hilton constructed the following parity version of Theorem REF on the existence of even factorizations with bounded degrees of even graphs.",
"Theorem 1.5 .",
"([14]) Every graph $G$ with even degrees can be edge-decomposed into $k$ even factors $G_1,\\ldots , G_k$ such that for each vertex $v$ , $|d_{G_i}(v)-d_{G}(v)/k|<2.$ In this paper, we provide two extensions of Hilton's result in highly odd-edge-connected graphs and highly edge-connected even graphs by proving the following results.",
"An interesting consequence of them says that every $(3k-3)$ -edge-connected graph of even order whose degrees lie in the set $\\lbrace 3k, 3k+2,...,5k\\rbrace $ can be edge-decomposed into $k$ factors whose degrees lie in the set $\\lbrace 3,5\\rbrace $ .",
"Note that the next theorem can also be considered as an extension of Theorem REF .",
"Theorem 1.6 .",
"Let $G$ be a graph and let $k$ be an odd positive integer.",
"If $G$ is odd-$(3k-2)$ -edge-connected, then it can be edge-decomposed into $k$ factors $G_1,\\ldots , G_k$ such that for each $v\\in V(G_i)$ , $ d_{G_i}(v)\\stackrel{2}{\\equiv } d_G(v)$ and $|d_{G_i}(v)-d_{G}(v)/k|<2.$ Theorem 1.7 .",
"Let $G$ be a graph with even degrees, let $f:V(G)\\rightarrow Z_2$ be a mapping with $ \\sum _{v\\in V(G)}f(v)\\stackrel{2}{\\equiv }0$ , and let $k$ be an even positive integer.",
"If for every vertex set $X$ with $ \\sum _{v\\in X}f(v)$ odd, $d_G(X)\\ge 3k-2$ , then $G$ can be edge-decomposed into $f$ -parity factors $G_1,\\ldots , G_k$ such that for each $v\\in V(G_i)$ , $|d_{G_i}(v)-d_{G}(v)/k|<2.$ In the remainder of this paper, we turn our attention to the existence of a parity factor whose degrees are close to $\\varepsilon $ of the corresponding degrees in the main graph, and we come up with the following results on even graphs.",
"Theorem 1.8 .",
"Let $\\varepsilon $ be a real number with $0\\le \\varepsilon \\le 1$ .",
"If $G$ is a graph with even degrees, then it admits an even factor $F$ such that for each vertex $v$ , $ | d_{F}(v)-\\varepsilon d_G(v)| < 2.$ Theorem 1.9 .",
"Let $\\varepsilon _1,\\ldots ,\\varepsilon _k$ be $k$ positive real numbers with $\\varepsilon _1+\\cdots +\\varepsilon _k=1$ .",
"If $G$ is a graph with even degrees, then it can be edge-decomposed into $k$ even factors $G_1,\\ldots , G_k$ such that for each $v\\in V(G_i)$ , $|d_{G_i}(v)-\\varepsilon _i d_G(v)|<6.$"
],
[
"Preliminary results on directed graphs",
"The following theorem provides a relationship between orientations and equitable factorizations of graphs which is motivated by Theorem 2 in [26].",
"This result plays an important role in the next section.",
"Theorem 2.1 .",
"Every directed graph $G$ can be edge-decomposed into $k$ factors $G_1,\\ldots , G_k$ such that for each $v\\in V(G_i)$ that $1\\le i\\le k$ , $ | d^+_{G_i}(v)-d^+_G(v)/k|< 1 \\text{ and } | d^-_{G_i}(v)- d^-_G(v)/k|< 1.$ In particular, $ | d_{G_i}(v)- d_G(v)/k|< 1$ , when $ d^+_G(v)$ or $d^-_G(v)$ is divisible by $k$ .",
"Furthermore, one can impose at least one of the following conditions arbitrary: $|\\,|E(G_i)|-|E(G)|/k\\,|<1$ , where $1\\le i\\le k$ .",
"If $u\\in V(G)$ and $d_G^+(u)\\stackrel{k}{\\equiv }d_G^-(u)$ , then $d_{G_i}(u)\\stackrel{2}{\\equiv }(d_G^+(u)-d_G^-(u))/k$ , where $1\\le i\\le k$ .",
"Proof.",
"The proof presented here is inspired by the proof of Theorem 2 in [26].",
"First split every vertex $v$ into two vertices $v^+$ and $v^-$ such that $v^+$ whose incident edges were directed away from $v$ in $G$ and $v^-$ whose incident edges were directed toward $v$ in $G$ .",
"In this construction, every loop in $G$ incident with $v$ is transformed into an edge between $v^+$ and $v^-$ .",
"Now, split every vertex in the new graph into vertices with degrees divisible by $k$ , except possibly one vertex with degree less than $k$ .",
"Call the resulting (loopless) bipartite graph $G_0$ .",
"Since $G_0$ has maximum degree at most $k$ , it admits a proper edge-coloring with at most $k$ colors $c_1\\ldots , c_k$ (to prove this, it is enough to consider a $k$ -regular supergraph of it and apply König Theorem [19]).",
"Let $G_i$ be the factor of $G$ consisting of the edges of $G$ corresponding to the edges with color $c_i$ in $G_0$ .",
"Note that for every vertex $v$ , at least $\\lfloor d^+_G(v)/k\\rfloor $ (resp.",
"$\\lfloor d^-_G(v)/k\\rfloor $ ) edges with color $c_i$ are incident with the vertex $v^+$ (resp.",
"$v^-$ ) in $G_0$ .",
"Moreover, at most $\\lceil d^+_G(v)/k\\rceil $ (resp.",
"$\\lceil d^-_G(v)/k\\rceil $ ) edges with color $c_i$ are incident with the vertex $v^+$ (resp.",
"$v^-$ ) in $G_0$ .",
"This completes the proof of the first part.",
"First we are going to prove item (i).",
"Let us consider such a proper edge-coloring with the minimum $\\sum _{1\\le i\\le k}|m_i-m/k|$ , where $m_i=|E(G_i)|$ and $m=|E(G)|$ .",
"We claim that $|m_i-m/k|<1$ for every color $c_i$ .",
"Suppose, to the contrary, that there is a color $c_i$ with $|m_i-m/k|\\ge 1$ .",
"We may assume that $m_i\\ge m/k+1$ ; as the proof of the case $m_i\\le m/k-1$ is similar.",
"Thus there is another color $c_j$ with $c_j<m/k$ so that $m_i> m_j+1$ .",
"Let $H$ be the factor of $G_0$ consisting of the edges that are colored with $c_i$ or $c_j$ .",
"According to the proper edge coloring of $G_0$ , the graph $H$ must be the union of some paths and cycles.",
"Since all cycles have even size and $m_i> m_j$ , it is easy to check that there is a path $P$ such that whose edges are colored alternatively by $c_i$ and $c_j$ , and also whose end edges are colored with $c_i$ .",
"Now, it is enough to exchange the colors of the edges of $P$ to find another proper edge-coloring satisfying $m^{\\prime }_i=m_i-1$ and $m^{\\prime }_j=m_j+1$ where $m^{\\prime }_i$ and $m^{\\prime }_j$ are the number of edges having these new colors.",
"Since $|m^{\\prime }_j-m/k|+|m^{\\prime }_i-m/k|< (|m_j-m/k|+1)+(|m_i-m/k|-1)$ , one can easily derive at a contradiction.",
"Therefore, $G_1,\\ldots , G_k$ are the desired factors we are looking for.",
"Now, we are going to prove item (ii).",
"Let us add some artificial edges to the graph $G_0$ .",
"For every vertex $u$ with $d_G^+(u)\\stackrel{k}{\\equiv }d_G^-(u)\\stackrel{k}{\\lnot \\equiv }0$ , denote by $u_0^+$ and $u_0^-$ the two vertices in $G_0$ having the same degree strictly less than $k$ obtained from splitting of $u^+$ and $u^-$ .",
"Add some new artificial edges between these two vertices to construct these vertices with degree $k$ .",
"Apply this method for all such vertices $u$ described above.",
"Consider a proper edge-coloring with at most $k$ colors $c_1\\ldots , c_k$ for the new resulting bipartite graph $G^{\\prime }_0$ .",
"Similarly, we define $G_i$ to be the factor of $G$ consisting of the edges of $G$ corresponding to the edges with color $c_i$ in $G_0$ .",
"Note that for every vertex $u$ with $d_G^+(u)\\stackrel{k}{\\equiv }d_G^-(u)\\stackrel{k}{\\lnot \\equiv }0$ , either one edge between $u_0^+$ and $u_0^-$ in $G^{\\prime }_0$ (may be an artificial edge) is colored with $c_i$ or two edges incident with $u_0^+$ and $u_0^-$ in $G_0$ are colored with $c_i$ .",
"Thus for every vertex $u$ with $d_G^+(u)\\stackrel{k}{\\equiv }d_G^-(u)$ , we must have $d^-_{G_i}(u)- d^-_G(u)/k = d^+_{G_i}(u)-d^+_G(u)/k $ and so $d_{G_i}(u)\\stackrel{2}{\\equiv }d^+_{G_i}(u) -d^-_{G_i}(u)=(d_G^+(u)-d_G^-(u))/k$ .",
"Therefore, $G_1,\\ldots , G_k$ are again the desired factors we are looking for.",
"$\\Box $ Corollary 2.2 .",
"([2]) Every graph $G$ can be edge-decomposed into $k$ factors $G_1,\\ldots , G_k$ such that for each factor $G_i$ , $|\\,|E(G_i)|-|E(G)|/k\\, |<1$ , and for each $v\\in V(G_i)$ , $\\lfloor \\frac{d_G(v)}{2k}\\rfloor +\\lfloor \\frac{d_G(v)+1}{2k}\\rfloor \\le d_{G_i}(v)\\le \\lceil \\frac{d_G(v)}{2k}\\rceil +\\lceil \\frac{d_G(v)-1}{2k}\\rceil .$ Proof.",
"First, consider an orientation for $G$ such that for each vertex $v$ , $|d^+_G(v)-d^-_G(v)|\\le 1$ .",
"Next, apply Theorem REF (i) to this directed graph.",
"$\\Box $"
],
[
"Factorizations of edge-connected graphs",
"In this section, we are going to give a sufficient condition for the existence of equitable factorizations in highly edge-connected graphs.",
"For this purpose, let us recall the following lemma from [12], [22].",
"Lemma 3.1 .",
"([12], [22]) Let $G$ be a graph, let $k$ be an integer, $k\\ge 2$ , and let $p:V(G)\\rightarrow Z_k$ be a mapping with $|E(G)| \\stackrel{k}{\\equiv } \\sum _{v\\in V(G)}p(v)$ .",
"If $G$ is $(3k-3)$ -edge-connected or $(2k-2)$ -tree-connected, then it admits a $p$ -orientation modulo $k$ .",
"Furthermore, the result holds for loopless $(2k-1)$ -edge-connected essentially $(3k-3)$ -edge-connected graphs provided that $p(v)=0$ or $p(v)\\stackrel{k}{\\equiv }d_G(v)$ for each vertex $v$ .",
"Before stating the main result, we begin with the following theorem that only needs replacing a weaker condition on a single arbitrary vertex.",
"Theorem 3.2 .",
"Let $G$ be a graph with $z\\in V(G)$ and let $k$ be a positive integer.",
"If $G$ is $(3k-3)$ -edge-connected or $(2k-2)$ -tree-connected, then it can be edge-decomposed into $k$ factors $G_1,\\ldots , G_k$ satisfying $||E(G_i)|-|E(G)|/k|<1$ such that for each $v\\in V(G_i)$ , $ | d_{G_i}(v)-d_G(v)/k| < {\\left\\lbrace \\begin{array}{ll}2 &\\text{when $v= z$};\\\\1, &\\text{otherwise}.\\end{array}\\right.",
"}$ Furthermore, for the vertex $z$ , we can also have $| d_{G_i}(z)-d_{G_j}(z)|\\le 2$ .",
"Proof.",
"According to Lemma REF , the graph $G$ admits an orientation such that for each $v\\in V(G)\\setminus z $ , $d^+_G(v)\\stackrel{k}{\\equiv }0$ , and $d^+_G(z)\\stackrel{k}{\\equiv }|E(G)|$ .",
"Now, it is enough to apply Theorem REF (i).",
"$\\Box $ Now, we are reedy to state the main result of this section which is a strengthened version of Theorem REF .",
"Theorem 3.3 .",
"Let $G$ be a graph and let $k$ be a positive integer.",
"Assume that $G$ is $(3k-3)$ -edge-connected or $(2k-2)$ -tree-connected.",
"If there is a vertex set $Z$ with $|E(G)|\\stackrel{k}{\\equiv }\\sum _{v\\in Z}d_G(v)$ , then $G$ can be edge-decomposed into $k$ factors $G_1,\\ldots , G_k$ satisfying $||E(G_i)|-|E(G)|/k|<1$ such that for each $v\\in V(G_i)$ , $|d_{G_i}(v)-d_G(v)/k|< 1 .$ Furthermore, the result holds for loopless $(2k-1)$ -edge-connected essentially $(3k-3)$ -edge-connected graphs.",
"Proof.",
"For each $v\\in Z$ , define $p(v)=d_G(v)$ , and for each $v\\in V(G)\\setminus Z$ , define $p(v)=0$ .",
"By the assumption, we have $|E(G)|\\stackrel{k}{\\equiv }\\sum _{v\\in V(G)}p(v)$ .",
"Thus by Lemma REF , the graph $G$ admits a $p$ -orientation modulo $k$ so that for each $v\\in Z$ , $d^-_G(v)$ is visible by $k$ and for each $v\\in V(G)\\setminus Z$ , $d^+_G(v)$ is divisible by $k$ .",
"Hence the assertion follows from Theorem REF (i).",
"$\\Box $ Graphs with size divisible by $k$ are natural candidates for graphs satisfying the assumptions of Theorem REF .",
"We examine them to deduce the following corollary.",
"Corollary 3.4 .",
"Let $G$ be a graph of size divisible by $k$ .",
"If $G$ is $(3k-3)$ -edge-connected or $(2k-2)$ -tree-connected, then it can be edge-decomposed into $k$ factors $G_1,\\ldots , G_k$ with the same size such that for each $v\\in V(G_i)$ , $|d_{G_i}(v)-d_G(v)/k|< 1 .$ Proof.",
"Apply Theorem REF with $Z=\\emptyset $ .",
"Note that $|E(G)|$ is divisible by $k$ .",
"$\\Box $ The next corollary gives a criterion for the existence of a 3-equitable factorization in edge-connected graphs.",
"Corollary 3.5 .",
"Let $G$ be a 6-edge-connected graph.",
"Then $G$ has not exactly one vertex $z$ with $d_G(z) \\stackrel{3}{\\lnot \\equiv }0$ if and only if it can be edge-decomposed into three factors $G_1$ , $G_2$ , and $G_3$ such that for each $v\\in V(G_i)$ , $|d_{G_i}(v)-d_G(v)/3|< 1 .$ Furthermore, the result holds for loopless 5-edge-connected essentially 6-edge-connected graphs.",
"Proof.",
"Set $Z_0$ to be the set of all vertices of $G$ such that whose degrees are not divisible by 3.",
"First assume that $|Z_0|\\ne 1$ .",
"If $|Z_0|=0$ , then $2|E(G)|$ must be divisible by 3 and so $|E(G)|$ must be divisible by 3.",
"Therefore, it is easy to check that there is a subset $Z$ of $Z_0$ such that $|E(G)|\\stackrel{3}{\\equiv }\\sum _{v\\in Z}d_G(v)$ whether $|Z_0|\\ge 2$ or not.",
"Hence by Theorem REF , the graph $G$ has the desired factorization.",
"Now, assume $Z_0=\\lbrace z\\rbrace $ .",
"Suppose, to the contrary, that $G$ can be edge-decomposed into three factors $G_1$ , $G_2$ , and $G_3$ such that for each $v\\in V(G_i)$ , $|d_{G_i}(v)-d_G(v)/3|< 1$ .",
"We may therefore assume that $d_{G_1}(z)=\\lfloor d_G(z)/3\\rfloor $ and $d_{G_2}(z)=\\lceil d_G(z)/3\\rceil $ .",
"On the other hand, for all vertices $v\\in V(G)\\setminus \\lbrace z\\rbrace $ , $d_{G_1}(v)=d_{G_2}(v)=d_G(v)/3$ which implies that $d_{G_1}(z)$ and $d_{G_2}(z)$ have the same parity, because of the handshaking lemma.",
"This is contradiction, as desired.",
"$\\Box $ As we observed above, graphs with a number of vertices whose degrees are not divisible by $k$ are other natural candidates for graphs satisfying the assumptions of Theorem REF .",
"By employing the following lemma, we examine a special case of them to imply the next corollary.",
"Lemma 3.6 .",
"(Chowla [6]) Let $k$ be an integer number with $k\\ge 2$ and let $x_1, \\ldots , x_{k-1}\\in Z_k\\setminus \\lbrace 0\\rbrace $ be $k-1$ integer numbers coprime with $k$ (not necessarily distinct).",
"If $m\\in Z_k$ , then there is a subset $Z\\subseteq \\lbrace 1,\\ldots , k-1\\rbrace $ such that $m\\stackrel{k}{\\equiv }\\sum _{i\\in Z}x_i$ .",
"Corollary 3.7 .",
"Let $G$ be a $(3k-3)$ -edge-connected graph where $k$ is a prime number.",
"If $G$ contains at least $k-1$ vertices whose degree are coprime with $k$ , then it can be edge-decomposed into $k$ factors $G_1,\\ldots , G_k$ satisfying $||E(G_i)|-|E(G)|/k|<1$ such that for each $v\\in V(G_i)$ , $|d_{G_i}(v)-d_G(v)/k|< 1.$ Proof.",
"Let $Z_0$ be the set of all vertices of $G$ such that whose degrees are coprime with $k$ .",
"Since $|Z_0|\\ge k-1$ , there is a subset $Z$ of $Z_0$ such that $|E(G)|\\stackrel{k}{\\equiv }\\sum _{v\\in Z}d_G(v)$ according to Lemma REF .",
"Hence the assertion follows from Theorem REF immediately.",
"$\\Box $ It is perhaps surprising that the lower bound of $k-1$ in the above-mentioned corollary is best possible according to the following observation.",
"It remains to decide whether the statement of Corollary REF holds if $G$ contains at least $k-1$ vertices whose degree are not divisible by $k$ .",
"The statement is obviously true for all prime numbers $k$ .",
"Observation 3.8 .",
"For every integer $k$ with $k\\ge 2$ , there are infinitely many highly edge-connected graphs $G$ having at least $k-2$ vertices whose degrees are coprime with $k$ , while $G$ cannot be edge-decomposed into $k$ factors $G_1,\\ldots , G_k$ such that for each $v\\in V(G_i)$ , $|d_{G_i}(v)-d_G(v)/k|< 1$ .",
"Proof.",
"Let $r$ be an arbitrary odd positive integer.",
"Choose a $kr$ -edge-connected graph $G$ of odd order in which whose degrees are $rk$ except for $k-2$ vertices having degree $kr+1$ (for $k$ even, one can consider a $kr$ -edge-connected $kr$ -regular large graph of odd order and insert a new perfect matching of size $k/2-1$ to it).",
"If $G$ has the desired factorization, then according to the vertex degree, there must be a factor $G_i$ such that for all vertices $v$ , $d_{G_i}(v)=\\lfloor d_G(v)/k\\rfloor $ .",
"This implies that $G$ an $r$ -regular factor which is not possible, because of the handshaking lemma.",
"Hence the assertion holds.",
"$\\Box $"
],
[
"Parity factorizations of odd-edge-connected graphs",
"In this section, we are going to develop each of Theorems REF and REF in two ways based on Theorem REF .",
"For this purpose, we first need the following lemma which improves the edge-connectivity needed in Lemma REF for a special case.",
"Lemma 4.1 .",
"([22]) Let $k$ be an odd positive integer.",
"If $G$ is an odd-$(3k-2)$ -edge-connected graph, then it admits an orientation such that for each vertex $v$ , $d_G^+(v)\\stackrel{k}{\\equiv }d_G^-(v)$ .",
"The following theorem generalizes Theorem REF to non-Eulerian graphs for the case that $k$ is odd.",
"This result can also be considered as an improvement of a result due to Shu, Zhang, and Zhang (2012) [24] who proved this result for odd-$(2k-1)$ -edge-connected graphs without considering the restriction on vertex degrees.",
"Theorem 4.2 .",
"Let $k$ be an odd positive integer.",
"If $G$ is an odd-$(3k-2)$ -edge-connected or $(2k-2)$ -tree-connected graph, then it can be edge-decomposed into $k$ factors $G_1,\\ldots , G_k$ such that for each $v\\in V(G_i)$ , $ d_{G_i}(v)\\stackrel{2}{\\equiv } d_G(v)$ and $|d_{G_i}(v)-d_{G}(v)/k|<2.$ Proof.",
"According to Lemmas REF and REF , the edge-connectivity condition implies the graph $G$ admits an orientation such that for each vertex $v$ , $d_G^+(v)\\stackrel{k}{\\equiv }d_G^-(v)$ .",
"By Theorem REF (ii), the graph $G$ can be edge-decomposed into $k$ factors $G_1,\\ldots , G_k$ such that for each $v\\in V(G_i)$ , $| d^-_{G_i}(v)- d^-_G(v)/k|< 1$ and $| d^+_{G_i}(v)-d^+_G(v)/k|< 1$ , and also $d_{G_i}(v)\\stackrel{2}{\\equiv }(d_G^+(v)-d_G^-(v))/k$ .",
"Therefore, $|d_{G_i}(v)-d_{G}(v)/k|\\le |d^+_{G_i}(v)-d^+_{G}(v)/k|+|d^-_{G_i}(v)-d^-_{G}(v)/k|<2$ , and all $d_{G_i}(v)$ have the same parity for which $1\\le i\\le k$ .",
"Since $d_G(v)=\\sum _{1\\le j\\le k}d_{G_j}(v)$ and $k$ is odd, $d_{G_i}(v)$ and $d_{G}(v)$ must have the same parity.",
"Hence the theorem holds.",
"$\\Box $ The edge-connectivity needed in Lemma REF can also be improved for the following special case.",
"We are going to apply it to deduce the next result on Eulerian graphs.",
"Lemma 4.3 .",
"([12]) Let $k$ be an even positive integer, let $G$ be an Eulerian graph, and let $Q\\subseteq V(G)$ with $|Q|$ even.",
"If $d_G(X)\\ge 3k-2$ for every $X\\subseteq V(G)$ with $|X\\cap Q|$ odd, then $G$ admits an orientation such that for each $v\\in V(G)\\setminus Q$ , $d_G^+(v)=d_G^-(v)$ , and for each $v\\in Q$ , $|d_G^+(v)-d_G^-(v)|=k$ .",
"The next theorem strengthens Theorem REF by imposing a new parity restriction on degrees of the desire factors for graphs with higher edge-connectivity.",
"Theorem 4.4 .",
"Let $k$ be an even positive integer, let $G$ be an Eulerian graph, and let $f:V(G)\\rightarrow Z_2$ be a mapping with $ \\sum _{v\\in V(G)}f(v)$ even.",
"If $d_G(X)\\ge 3k-2$ for every vertex set $X$ with $ \\sum _{v\\in X}f(v)$ odd, or $G$ is $(2k-2)$ -tree-connected, then $G$ can be edge-decomposed into $f$ -parity factors $G_1,\\ldots , G_k$ such that for each $v\\in V(G_i)$ , $|d_{G_i}(v)-d_{G}(v)/k|<2.$ Proof.",
"According to Lemmas REF and REF , the edge-connectivity condition implies the graph $G$ admits an orientation such that for each vertex $v$ , $d_G^+(v)=d_G^-(v)$ when $f(v)$ is even, and $|d_G^+(v) -d_G^-(v)|=k$ when $f(v)$ is odd.",
"Thus by Theorem REF (ii), the graph $G$ can be edge-decomposed into $k$ factors $G_1,\\ldots , G_k$ such that for each $v\\in V(G_i)$ , $| d^-_{G_i}(v)- d^-_G(v)/k|< 1$ and $| d^+_{G_i}(v)-d^+_G(v)/k|< 1$ , and also $d_{G_i}(v)\\stackrel{2}{\\equiv }(d_G^+(v)-d_G^-(v))/k$ .",
"Therefore, $|d_{G_i}(v)-d_{G}(v)/k|\\le |d^+_{G_i}(v)-d^+_{G}(v)/k|+|d^-_{G_i}(v)-d^-_{G}(v)/k|<2$ and $d_{G_i}(v)\\stackrel{2}{\\equiv }f(v)$ .",
"Hence the theorem holds.",
"$\\Box $ An attractive application of Theorems REF and REF is given in the following corollary.",
"Corollary 4.5 .",
"Let $k$ and $r$ be two positive integers.",
"Let $G$ be a graph whose degrees lie in the set $\\lbrace rk, rk+2,...,rk+2k\\rbrace $ for which $r|V(G)|$ is even.",
"If $G$ is $(3k-3)$ -edge-connected or $(2k-2)$ -tree-connected, then it can be edge-decomposed into $k$ $\\lbrace r,r+2\\rbrace $ -factors.",
"Proof.",
"If $k$ is odd, then by Theorem REF , the graph $G$ can be edge-decomposed into $k$ factors $G_1,\\ldots , G_k$ such that for each $v\\in V(G_i)$ , $ d_{G_i}(v)\\stackrel{2}{\\equiv } d_G(v)\\stackrel{2}{\\equiv }r$ and $r-2\\le d_{G}(v)/k -2<d_{G_i}(v)<d_{G}(v)/k+2\\le r+4.$ If $k$ is even, then by applying Theorem REF with $f(v)=r$ , these factors can similarly be found such that for each $v\\in V(G_i)$ , $ d_{G_i}(v)\\stackrel{2}{\\equiv } r$ and $r-2<d_{G_i}(v)< r+4.$ This completes the proof.",
"$\\Box $"
],
[
"Graphs with degrees divisible by $k$ : regular factorizations",
"In this subsection, we restrict out attention to graphs with degrees divisible by $k$ and derive some results based on the following reformulation of Theorems REF and REF on this family of graphs.",
"Theorem 4.6 .",
"Let $k$ be a positive integer and let $G$ be a graph with size and degrees divisible by $k$ .",
"Take $Q$ to be the set of all vertices $v$ with $d_G(v)/k$ odd.",
"If for every vertex X with $|X\\cap Q|$ odd, $d_G(X)\\ge 3k-2,$ then $G$ can be edge-decomposed into $k$ factors $G_1,\\ldots , G_k$ such that for each $v\\in V(G_i)$ , $d_{G_i}(v)=d_G(v)/k$ .",
"Proof.",
"To show a a directive proof, we first consider an orientation for $G$ such that out-degree of each vertex is divisible by $k$ using Lemmas REF and REF .",
"Next, it is enough to apply Theorem REF (i).",
"$\\Box $ The following corollary partially confirms Conjecture 2 in [26] by giving a supplement for Theorem REF .",
"Corollary 4.7 .",
"Let $r$ be an odd positive integer.",
"If $G$ is a $kr$ -regular graph $G$ of even order satisfying $d_G(A)\\ge 3k-2$ , for every vertex set $A$ with $|A|$ odd, then it can be edge-decomposed into $r$ -factors.",
"Proof.",
"Since $G$ has even order, its size must be divisible by $k$ .",
"Note also that for each vertex $v$ , $d_G(v)/k$ is odd.",
"Thus the assertion follows from Theorem REF immediately, $\\Box $ The following corollary gives a supplement for Theorem 3 in [26].",
"Corollary 4.8 .",
"Let $r$ be a positive integer and let $r_1,\\ldots , r_m$ be $m$ positive integers satisfying $r=r_1+\\cdots +r_m$ and $r_i\\ge r/k-1\\ge 2$ in which $k$ is positive divisor of $r$ with $r/k$ odd.",
"If $G$ is an $r$ -regular graph of even order and for every vertex $X$ with $|X|$ odd, $d_G(X)\\ge 3k-2,$ then it can be edge-decomposed into factors $G_1,\\ldots , G_m$ such that every graph $G_i$ is $r_i$ -regular.",
"Proof.",
"Apply Corollary REF along with a similar argument stated in the proof of Theorem 3 in [26].",
"$\\Box $ The following corollary is a counterpart of Corollary REF and replaces a weaker edge-connectivity condition compared to Theorem 4 in [26].",
"The proof technique shows a worthwhile application of this kind of odd-edge-connectivity for working with supergraphs.",
"Corollary 4.9 .",
"Let $r$ be a positive integer and let $r_1,\\ldots , r_m$ be $m$ positive integers satisfying $r=r_1+\\cdots +r_m$ and $r_i\\ge r/k-1\\ge 2$ in which $k$ is positive divisor of $r$ with $r/k$ odd.",
"If $G$ is a graph with $r|V(G)|$ even satisfying $\\Delta (G)\\le r$ and for every vertex X with $|X|$ odd, $d_G(X)\\ge 3k-3,$ then it can be edge-decomposed into factors $G_1,\\ldots , G_m$ satisfying $\\Delta (G_i)\\le r_i$ for each $i$ with $1\\le i \\le m$ .",
"Proof.",
"Add some edges to $G$ , as long as possible, such that the resulting graph $G^{\\prime }$ has maximum degree at most $r$ .",
"Since adding loops is possible and $r|V(G)|$ is even, the graph $G^{\\prime }$ must be $r$ -regular.",
"Obviously, for every vertex set $X$ with $|X|$ odd, we still have $d_{G^{\\prime }}(X)\\ge 3k-3$ .",
"In addition, since $d_{G^{\\prime }}(X)$ and $k$ have the same parity, it implies that $d_{G^{\\prime }}(X)\\ge 3k-2$ .",
"Thus by Corollary REF , the graph $G^{\\prime }$ can be edge-decomposed into factors $G^{\\prime }_1,\\ldots , G^{\\prime }_m$ such that every graph $G^{\\prime }_i$ is $r_i$ -regular.",
"Now, it is enough to induce this factorization for $G$ to complete the proof.",
"$\\Box $"
],
[
"A sufficient edge-connectivity condition for the existence of a parity factor",
"In 1956 Hoffman made the following theorem on the the existence factors in bipartite graphs whose degrees are close to $\\varepsilon $ of the corresponding degrees in the main graph.",
"Theorem 5.1 .",
"([16]) Let $\\varepsilon $ be a real number with $0\\le \\varepsilon \\le 1$ .",
"If $G$ is a bipartite graph, then it has a factor $F$ such that for each vertex $v$ , $ | d_{F}(v)-\\varepsilon d_G(v)|< 1 .$ In 1983 Kano and Saito generlized Theorem REF to general graphs as the next version.",
"Theorem 5.2 .",
"([18]) Let $\\varepsilon $ be a real number with $0\\le \\varepsilon \\le 1$ .",
"If $G$ is a graph, then it has a factor $F$ such that for each vertex $v$ , $| d_{F}(v)-\\varepsilon d_G(v)|\\le 1 .$ In 2007 Correa and Goemans [8] formulated the following factorization version of Theorem REF for bipartite graphs.",
"Later, Correa and Matamala (2008) [7] remarked that it is possible to generalize their result to general graphs by replacing Theorem REF in their proof, and Feige and Singh (2008) [10] introduced an interesting alternative proof for this theorem using linear algebraic techniques.",
"Theorem 5.3 .",
"([7], [8]) Let $\\varepsilon _1,\\ldots ,\\varepsilon _k$ be $k$ nonnegative real numbers with $\\varepsilon _1+\\cdots +\\varepsilon _k=1$ .",
"If $G$ is a graph, then it can be edge-decomposed into $k$ factors $G_1,\\ldots , G_k$ such that for each $v\\in V(G_i)$ , $|d_{G_i}(v)-\\varepsilon _i d_G(v)|<3.$ In this section, we shall prove the parity versions of Theorems REF and REF which were mentioned in the introduction as Theorems REF and REF .",
"Besides them, we also generalize a recent result in [12].",
"Our proofs are based on the following well-known lemma due to Lovász (1972).",
"Lemma 5.4 .",
"([21]) Let $G$ be a graph and let $g$ and $f$ be two integer-valued functions on $V(G)$ satisfying $g(v)\\le f(v)$ and $g(v)\\stackrel{2}{\\equiv }f(v)$ for each $v\\in V(G)$ .",
"Then $G$ has a parity $(g,f)$ -factor if and only if for all disjoint subsets $A$ and $B$ of $V(G)$ , $\\omega _{f}(G, A,B)<2+ \\sum _{v\\in A} f(v)+\\sum _{v\\in B} (d_{G}(v)-g(v))-d_G(A,B),$ where $\\omega _{f}(G, A,B)$ denotes the number of components $G[X]$ of $G\\setminus (A\\cup B)$ satisfying $\\sum _{v\\in X}f(v)\\stackrel{2}{\\lnot \\equiv }d_G(X,B)$ ."
],
[
"Parity $\\varepsilon $ -factors",
"The following theorem provides a parity version for Theorem REF on edge-connected graphs.",
"Note that some edge-connected versions of that theorem were former studied in [3], [9], [17].",
"Theorem 5.5 .",
"Let $G$ be a connected graph, let $f:V(G)\\rightarrow Z_2$ be a mapping with $ \\sum _{v\\in V(G)}f(v)\\stackrel{2}{\\equiv }0$ , and let $\\varepsilon $ be a real number with $0< \\varepsilon < 1$ .",
"If for every nonempty proper subset $X$ of $V(G)$ , $d_G(X)\\ge {\\left\\lbrace \\begin{array}{ll}1/\\varepsilon , &\\text{when $\\sum _{v\\in X}f(v)$ is odd};\\\\1/(1-\\varepsilon ), &\\text{when $\\sum _{v\\in X}(d_G(v)-f(v))$ is odd},\\end{array}\\right.",
"}$ then $G$ has an $f$ -parity factor $F$ such that for each vertex $v$ , $ \\lfloor \\varepsilon d_G(v) \\rfloor -1\\le d_{F}(v)\\le \\lceil \\varepsilon d_G(v) \\rceil +1.$ Furthermore, for a given arbitrary vertex $z$ , we can arbitrary have $d_F(z)\\ge \\varepsilon d_G(z)$ or $d_F(z)\\le \\varepsilon d_G(z)$ .",
"Proof.",
"For each vertex $v$ , let us define $g_0(v)\\in \\lbrace \\lfloor \\varepsilon d_G(v)\\rfloor -1,\\lfloor \\varepsilon d_G(v)\\rfloor \\rbrace $ and $f_0(v)\\in \\lbrace \\lceil \\varepsilon d_G(v)\\rceil ,\\lceil \\varepsilon d_G(v)\\rceil +1 \\rbrace $ such that $g_0(v)\\stackrel{2}{\\equiv }f_0(v)\\stackrel{2}{\\equiv } f(v)$ .",
"If $g(z)<\\varepsilon d_G(z)$ and our goal is to impose that $d_F(z)\\ge \\varepsilon d_G(z)$ , we will replace $g_0(z)$ by $g_0(z)+2$ , and if $f(z)>\\varepsilon d_G(z)$ and our goal is to impose that $d_F(z)\\le \\varepsilon d_G(z)$ , we will replace $f_0(z)$ by $f_0(z)-2$ .",
"Let $A$ and $B$ be two disjoint vertex subsets of $V(G)$ with $A\\cup B\\ne \\emptyset $ .",
"By the definition of $g$ and $f$ , we must have ${\\sum _{v\\in A} \\varepsilon d_G(v)+\\sum _{v\\in B} (1-\\varepsilon ) d_G(v)< 2+ \\sum _{v\\in A} f_0(v)+\\sum _{v\\in B} (d_{G}(v)-g_0(v)),}$ whether $z\\in A\\cup B$ or not.",
"Take $P$ to be the collection of all vertex sets $X$ such that $G[X]$ is a component of $G\\setminus (A\\cup B)$ satisfying $\\sum _{v\\in X}f(v)\\stackrel{2}{\\lnot \\equiv } d_G(X,B)$ .",
"It is easy to check that ${\\sum _{X\\in P}(\\varepsilon d_G(X,A)+(1-\\varepsilon ) d_G(X,B))\\le \\sum _{v\\in A} \\varepsilon d_G(v)+\\sum _{v\\in B} (1-\\varepsilon ) d_G(v)\\, -d_G(A,B).",
"}$ Define $P_c=\\lbrace X\\in P: d_G(X,A)>0 \\text{ and }d_G(X,B)>0\\rbrace $ .",
"Obviously, $|P_c|= \\sum _{X\\in P_c}(\\varepsilon +(1-\\varepsilon ))\\le \\sum _{X\\in P_c}(\\varepsilon d_G(X,A)+(1-\\varepsilon ) d_G(X,B)).$ Set $P_a=\\lbrace X\\in P: d_G(X,A)=0\\rbrace $ and $P_b=\\lbrace X\\in P: d_G(X,B)=0\\rbrace $ .",
"According to the definition, for every $X\\in P_a$ , we have $\\sum _{v\\in X}f(v)\\stackrel{2}{\\lnot \\equiv }d_G(X,B)=d_G(X)\\stackrel{2}{\\equiv }\\sum _{v\\in X}d_G(v),$ which implies that $\\sum _{v\\in X}(d_G(v)-f(v))$ is odd.",
"Similarly, for every $X\\in P_b$ , $\\sum _{v\\in X}f(v)$ must be odd.",
"Thus by the assumption, we must have $|P_a|\\le \\sum _{X\\in P_a}(1-\\varepsilon ) d_G(X) =\\sum _{X\\in P_a}(\\varepsilon d_G(X,A)+(1-\\varepsilon ) d_G(X,B)),$ and also $|P_b|\\le \\sum _{X\\in P_b}\\varepsilon d_G(X) =\\sum _{X\\in P_b}(\\varepsilon d_G(X,A)+(1-\\varepsilon ) d_G(X,B)).$ Therefore, ${|P|=|P_a|+|P_b|+|P_c|\\le \\sum _{X\\in P}(\\varepsilon d_G(X,A)+(1-\\varepsilon ) d_G(X,B)).",
"}$ According to Relations (REF ), (REF ), and (REF ), one can conclude that $\\omega _{f}(G, A,B)=|P|<2+ \\sum _{v\\in A} f_0(v)+\\sum _{v\\in B} (d_{G}(v)-g_0(v))-d_G(A,B).$ When both of the sets $A$ and $B$ are empty, the above-mentioned inequality must automatically hold, because $\\omega _{f}(G, \\emptyset ,\\emptyset )=0$ .",
"Thus the assertion follows from Lemma REF .",
"$\\Box $ Remark 5.6 .",
"Note that if we had replaced the weaker condition $|d_F(z)-\\varepsilon d_G(z)|\\le 1$ for the vertex $z$ , we could impose this condition for another vertex as well.",
"In particular, we could impose this condition for three arbitrary vertices provided that $\\varepsilon =1/2$ .",
"The following corollary is an improved version of the main result in [5] by replacing an odd edge-connectivity condition.",
"Note that $r$ -regular bipartite graphs are in the class of odd-$r$ -edge-connected graphs.",
"Corollary 5.7 .",
"([4], [5]) Let $r$ and and $r_0$ be two odd positive integers with $r\\ge r_0$ .",
"If $G$ is an odd-$\\lceil r/r_0\\rceil $ -edge-connected $r$ -regular graph, then it contains an $r_0$ -factor.",
"Proof.",
"Apply Theorem REF with $\\varepsilon =r_0/r$ and $f(v)=r_0$ (mod 2).",
"$\\Box $ Corollary 5.8 .",
"([12]) Let $G$ be a graph and let $f:V(G)\\rightarrow Z_2$ be a mapping with $ \\sum _{v\\in V(G)}f(v)\\stackrel{2}{\\equiv }0$ .",
"If $G$ is 2-edge-connected, then it has an $f$ -parity factor $F$ such that for each vertex $v$ , $ \\lfloor \\frac{d_G(v)}{2} \\rfloor -1\\le d_{F}(v)\\le \\lceil \\frac{d_G(v)}{2} \\rceil +1.$ Furthermore, for a given arbitrary vertex $z$ , we can arbitrary have $d_F(z)\\ge d_G(z)/2$ or $d_F(z)\\le d_G(z)/2$ .",
"Proof.",
"Apply Theorem REF with $\\varepsilon =1/2$ .",
"$\\Box $ Corollary 5.9 .",
"Let $G$ be a graph and let $f:V(G)\\rightarrow Z_2$ be a mapping with $ \\sum _{v\\in V(G)}f(v)\\stackrel{2}{\\equiv }0$ .",
"If $G$ is 3-edge-connected, then it has an $f$ -parity factor $F$ such that for each vertex $v$ , $ \\lfloor \\frac{2\\, d_G(v) }{3}\\rfloor -1\\le d_{F}(v)\\le \\lceil \\frac{2\\, d_G(v) }{3} \\rceil +1.$ Proof.",
"Apply Theorem REF with $\\varepsilon =2/3$ .",
"$\\Box $ An immediate consequence of the following corollary was appeared in [20] which says that every 2-edge-connected graph with minimum degree at least three admits a factor whose degrees are positive and even.",
"Corollary 5.10 .",
"Let $\\varepsilon $ be a real number with $0\\le \\varepsilon \\le 2/3$ .",
"If $G$ is a 2-edge-connected graph, then it has an even factor $F$ such that for each vertex $v$ , $ | d_{F}(v)-\\varepsilon d_G(v)| < 2.$ The next corollary provides an improved version for a result in [23] due to Lu, Wang, and Lin (2015) who proved that every $2m$ -edge-connected graph with minimum degree at least $2m + 1$ contains an even factor with minimum degree at least $2m$ .",
"Corollary 5.11 .",
"Let $G$ be a 2-edge-connected graph and let $\\varepsilon $ be a real number with $2/3\\le \\varepsilon < 1$ .",
"If $G$ is odd-$\\lceil 1/(1-\\varepsilon )\\rceil $ -edge-connected, then it has an even factor $F$ such that for each vertex $v$ , $ | d_{F}(v)-\\varepsilon d_G(v)|<2.$ When the main graph has no odd edge-cuts, one can derive the following simpler version of Corollaries REF and REF .",
"Corollary 5.12 .",
"Let $\\varepsilon $ be a real number with $0\\le \\varepsilon \\le 1$ .",
"If $G$ is a graph with even degrees, then it admits an even factor $F$ such that for each vertex $v$ , $ | d_{F}(v)-\\varepsilon d_G(v)| < 2.$ Here, we introduce a simple inductive proof for Theorem 8 in based on Corollary REF .",
"Corollary 5.13 .",
"([14]) Every graph $G$ with even degrees can be edge-decomposed into $k$ even factors $G_1,\\ldots , G_k$ such that for each vertex $v$ , $|d_{G_i}(v)-d_{G}(v)/k|<2.$ Proof.",
"By induction on $k$ .",
"We may assume that $k\\ge 2$ as the assertion trivially holds when $k=1$ .",
"According to Corollary REF , the graph can be edge-decomposed into two even factors $G^{\\prime }$ and $G_{k}$ such that for each vertex $v$ , $|d_{G_k}(v)-\\frac{1}{k}d_{G}(v)|<2$ .",
"By the induction hypothesis, the graph $G^{\\prime }$ can be edge-decomposed into $k-1$ even factors $G_1,\\ldots , G_{k-1}$ such that for each vertex $v$ , $|d_{G_i}(v)-\\frac{1}{k-1}d_{G^{\\prime }}(v)|<2.$ We claim that these are the desired factors we are looking for.",
"Let $v\\in V(G)$ .",
"Since $d_{G^{\\prime }}(v)\\le \\frac{k-1}{k}d_{G}(v)+2-\\frac{1}{k}$ , we must have $d_{G_i}(v) \\le \\frac{1}{k-1}(d_{G^{\\prime }}(v)-1)+2\\le \\frac{1}{k}(d_G(v)+1)+2.$ Since $d_G(v)$ is even, $(d_G(v)+1)/k$ must not be an even integer number, which implies that $d_{G_i}(v) \\le \\frac{1}{k}d_G(v)+2$ .",
"Since $d_{G^{\\prime }}(v)$ is even, if $\\frac{1}{k}d_{G}(v)$ is an even integer number, then $d_{G^{\\prime }}(v)\\le \\frac{k-1}{k}d_{G}(v)$ .",
"Therefore, since $d_{G_i}(v)$ is even, one can conclude that $d_{G_i}(v) < \\frac{1}{k}d_G(v)+2$ whether $\\frac{1}{k}d_{G}(v)$ is an integer number or not.",
"Similarly, we have $d_{G_i}(v) > \\frac{1}{k}d_{G}(v)-2$ .",
"Hence the assertion holds.",
"$\\Box $ By replacing Corollary REF in the proof of Theorem REF , one can formulate a parity version of it as the following theorem.",
"It would be interesting to determine the sharp upper bound on vertex degrees to make another generalization for Corollary REF .",
"Theorem 5.14 .",
"Let $\\varepsilon _1,\\ldots ,\\varepsilon _k$ be $k$ nonnegative real numbers with $\\varepsilon _1+\\cdots +\\varepsilon _k=1$ .",
"If $G$ is a graph with even degrees, then it can be edge-decomposed into $k$ even factors $G_1,\\ldots , G_k$ such that for each $v\\in V(G_i)$ , $|d_{G_i}(v)-\\varepsilon _i d_G(v)|<6.$ Proof.",
"Apply Corollary REF along with the same arguments stated in the proof of Theorem 4 in [8].",
"$\\Box $"
]
] | 1906.04325 |
[
[
"Semantic-guided Encoder Feature Learning for Blurry Boundary Delineation"
],
[
"Abstract Encoder-decoder architectures are widely adopted for medical image segmentation tasks.",
"With the lateral skip connection, the models can obtain and fuse both semantic and resolution information in deep layers to achieve more accurate segmentation performance.",
"However, in many applications (e.g., blurry boundary images), these models often cannot precisely locate complex boundaries and segment tiny isolated parts.",
"To solve this challenging problem, we firstly analyze why simple skip connections are not enough to help accurately locate indistinct boundaries and argue that it is due to the fuzzy information in the skip connection provided in the encoder layers.",
"Then we propose a semantic-guided encoder feature learning strategy to learn both high resolution and rich semantic encoder features so that we can more accurately locate the blurry boundaries, which can also enhance the network by selectively learning discriminative features.",
"Besides, we further propose a soft contour constraint mechanism to model the blurry boundary detection.",
"Experimental results on real clinical datasets show that our proposed method can achieve state-of-the-art segmentation accuracy, especially for the blurry regions.",
"Further analysis also indicates that our proposed network components indeed contribute to the improvement of performance.",
"Experiments on additional datasets validate the generalization ability of our proposed method."
],
[
"Introduction",
" Automatic image segmentation is an essential step for many medical image analysis applications, include computer-aided radiation therapy, disease diagnosis and treatment effect evaluation.",
"One of the major challenges for this task is the blurry nature of medical images (e.g., CT, MR and, microscopic images), which can often result in low-contrast and even vanishing boundaries, as shown in Fig.",
"REF .",
"Many encoder-decoder based networks have been proposed for semantic segmentation [5], [8], [10] and achieved very promising performances on various tasks.",
"UNet [8], a typical encoder-decoder architecture which combines shallow and deep features with a skip connection is widely used in many image segmentation tasks.",
"Some works are proposed to enhance the UNet [9], [6], [12].",
"However, Heller et al.",
"[2] found that the deep segmentation models are robust on the non-boundary regions, but not very robust to boundaries.",
"Actually, these models usually fail to properly segment the blurry boundaries, especially for the case with extremely low tissue contrast.",
"For example, prostate boundaries in MR or CT pelvic images are often blurry.",
"To solve this challenge, we argue high resolution with rich semantic based feature learning is desired.",
"Figure: Illustration of the blurry and vanishing boundarieswithin pelvic MRI images, together with overlaid ground truth contour and the typical feature maps in the encoder layer of a conventional UNet.",
"(a) and (b) are the two typical slices of two subjects, in which boundaries of bladder and rectum are relatively clear, but prostate is blurry.Besides the variants of UNet, to better delineate the boundaries, Ravishankar et al.",
"[7] proposed a multi-task network to robustly segment the organs by jointly regressing the boundaries and foreground.",
"Zhu et al.",
"[11] proposed a boundary-weighted domain adaptive neural network to accurately extract the boundaries of MRI prostate.",
"However, all these methods do not consider the fact that the voxels around blurry boundaries are highly similar.",
"Thus, we should not directly classify or regress the voxels to be on the boundary or not.",
"In this paper, we propose a novel semantic-guided encoder feature learning mechanism to improve the skip connection in previous encoder-decoder architectures, so that we can work better for low-contrast medical image segmentation.",
"The design of our network is mainly based on the idea of explicitly utilizing high-resolution semantic information to compensate for the deficiency on inaccurate boundary delineation of the existing encoder-decoder networks.",
"Specifically, we propose to concatenate the low-layer (encoder) feature maps and the high-layer (decoder) feature maps, and then design a channel-wise attention and spatial-wise attention to help learn (which can also be viewed as a kind of feature selection) the high-resolution semantic encoder feature maps.",
"With these better learned encoder feature maps, we further concatenate (or element-wisely add) it to the corresponding decoder layers in encoder-decoder framework.",
"Moreover, we propose using soft label to indicate the probability of a voxel being on the boundary.",
"Accordingly, a soft cross-entropy loss is proposed as a metric for the blurry boundary delineation problem."
],
[
"Method",
" The architecture of our proposed framework is presented in Fig.",
"REF , in which an encoder-decoder architecture is introduced with three tasks (segmentation, clear boundary detection, and blurry boundary detection).",
"The proposed semantic-guided encoder feature learning module (SGM) is further highlighted in Fig.",
"REF .",
"In the following subsections, we will analyze the deficiency of the skip connection in the current encoder-decoder framework.",
"Then, we introduce the proposed semantic-guided encoder feature learning strategy.",
"Moreover, we will describe the soft contour constraint for blurry boundary delineation.",
"Finally, we describe the implementation details.",
"Figure: Illustration of the architecture of our proposed method, which consists of a semantic-guided module (SGM).",
"(a) means a segmentation branch, and (b) and (c) indicate boundary detection branches."
],
[
"Analysis of Skip Connection in Encoder-Decoder Architecture",
"In the classical encoder-decoder architecture [8], shallow and deep features are usually complementary to each other.",
"For example, shallow features are rich in resolution but insufficient in semantic information, while deep features are highly semantic but lack of spatial details.",
"The skip connection proposed in UNet [8] is supposed to provide high-resolution information from the shallow (encoder) layers to the deep (decoder) layers, so that we can improve localization precision without losing classification accuracy.",
"However, the raw (simple) skip connection has several drawbacks.",
"a) It would bring `noise' (unnecessary information) to the deep layers which will definitely affect the concatenation of feature maps, as shown in the visualized encoder feature maps in Fig.",
"REF .",
"b) The huge gap between shallow and deep features will decrease the power of this combination.",
"c) Moreover, for the clear boundaries (e.g., bladder and rectum), the encoder feature maps can provide sufficiently precise localization information as shown in Fig.",
"REF , which can thus work well with the raw skip connection.",
"But the blurry boundary (e.g., prostate) cannot be well described in the encoder feature maps as shown in Fig.",
"REF , which thus cannot provide accurate localization information with simple skip connection.",
"Therefore, it is highly desired to select discriminative features, not simply inhibiting indiscriminative features from shallow layers; in other words, we should learn high resolution semantic features from the encoder.",
"To achieve such an effect, Roy et al.",
"[9] proposed concurrent spatial-and-channel squeeze and excitation module to boost meaningful features and suppress weak ones.",
"Oktay et al.",
"[6] proposed gated attention mechanism to select the salient part of the feature maps to further improve the UNet.",
"However, in both works, the feature learning process is actually conducted in an implicit manner which will limit the learning efficiency."
],
[
"Semantic-guided Encoder Feature Learning",
" To overcome the above mentioned problems, we propose to explicitly learn the high resolution semantic features (which are also more discriminative) from shallow (encoder) layers with semantic guidance from deep (decoder) layers.",
"The key idea is to encode semantic concept from deep layer features to guide the learning of shallow features.",
"As shown in Fig.",
"REF , our semantic-guided feature learning module (i.e., SG module or SGM) is designed to selectively enhance or suppress the features of shallow layer at each stage so that we can enhance the consistency between shallow and deep layers without losing resolution information.",
"Besides the widely-used channel-wise encoder, we have also designed the spatial-wise encoder as described below.",
"Figure: Illustration of the proposed semantic-guided module (SGM), as shown in ( a ).",
"The pink blocks represent the features of shallow layers, while the red ones represent the features of deep layers.",
"Different from direct skip connection in UNet, we propose using semantic concept from deep layers to guide feature learning in the corresponding shallow layers, for which a channel-wise encoder and a spatial-wise encoder are both proposed, as shown in (b).",
"`GAP' means Global Average Pooling.We consider the feature maps of a certain encoder layer (i.e., shallow features) to be $S = \\left\\lbrace {{s_1},{s_2},...,{s_K}} \\right\\rbrace $ , where ${s_i} \\in {R^{H \\times W \\times T}}$ .",
"We also assume the up-sampled feature maps in the corresponding decoder layer (deep features) to be $D = \\left\\lbrace {{d_1},{d_2},...,{d_K}} \\right\\rbrace $ , where ${d_i} \\in {R^{H \\times W \\times T}}$ .",
"We concatenate the two group of feature maps together and thus result in a bank of high-resolution and rich-semantic mixed feature maps as shown in Eq.",
"REF .",
"$F = \\left\\lbrace {{s_1},{s_2},...,{s_K},{d_1},{d_2},...,{d_K}} \\right\\rbrace .$ Channel-wise Encoding: With a global average pooling layer, we obtain a vector $Q = \\left\\lbrace {{q_1},{q_2},...,{q_K},...{q_{2K}}} \\right\\rbrace $ , where $q_k$ is a scalar and corresponds to the averaging value of the k-th feature maps in $F$ .",
"Then, two successive fully connected layer are adopted to fuse the resolution and semantic information: $Z = {W_1}\\left( {\\mathop {\\rm ReLU} \\left( {{W_2}Q} \\right)} \\right)$ , with ${W_1} \\in { R^{K \\times K}}$ and ${W_2} \\in { R^{2K \\times K}}$ .",
"This encodes the channel-wise dependencies by considering both shallow and deep features.",
"We apply a sigmoid activation function to map the neurons to probabilities so that we can formulate as a channel-wise importance descriptor, which can be described as $\\sigma \\left( Z \\right)$ .",
"Thus, the semantic-guided channel-wise encoded feature maps are formulated as Eq.",
"REF .",
"$SGCF = \\left\\lbrace {\\sigma \\left( {{z_1}} \\right){s_1},\\sigma \\left( {{z_2}} \\right){s_2},...,\\sigma \\left( {{z_K}} \\right){s_K}} \\right\\rbrace $ Note that the weight $\\sigma \\left( {{z_k}} \\right)$ before the shallow feature map $s_k$ can be viewed as an indicator of how important this specific feature map is.",
"Thus, we argue this channel-wise encoding is actually a semantic-guided feature selection process in a channel-wise manner, which is able to ignore less meaningful feature maps and emphasize the meaningful ones.",
"In other words, it can help remove the `noise' and retain the useful information.",
"More importantly, since $\\sigma \\left( Z \\right)$ has taken both high resolution and rich semantic information into account, it has more discriminative capacity than the case of only considering shallow layer information in [9].",
"Spatial-wise Encoding: Now we come to consider the spatial-wise importance to achieve better fine-grained image segmentation.",
"Based on the concatenated feature maps $F$ , we apply a $2K \\times 1 \\times 1 \\times 1$ convolution to squeeze the channels.",
"Therefore, we can obtain a one-channel output feature map $U$ , where $U \\in {R^{H \\times W \\times T}}$ .",
"We directly apply sigmoid function to acquire a probability map for $U$ .",
"Similarly, the semantic-guided spatial-wise encoded shallow feature maps can be described in Eq.",
"REF .",
"$SGSF = \\left\\lbrace {\\sigma \\left( U \\right) \\otimes {s_1},\\sigma \\left( U \\right) \\otimes {s_2},...,\\sigma \\left( U \\right) \\otimes {s_K}} \\right\\rbrace $ Since $\\sigma \\left({U_{h,w,t}}\\right)$ corresponds to the relative importance of a spatial information at $\\left( {h,w,t} \\right)$ of a given shallow layer feature map, it can help select more important features to relevant spatial locations and also ignore the irrelevant ones.",
"Moreover, $\\sigma \\left( U \\right)$ is a fusion of both resolution and rich semantic information, thus it can provide a better localization capacity even for the blurry boundary regions which cannot done by [9].",
"As a result, we view this spatial-wise encoding as a semantic-guided recalibration process.",
"Combination of Encoded Feature Maps: Now we can formulate both channel-wise and spatial-wise encoding by a simple element-wise addition operation, as shown in eq.",
"REF .",
"$SGF = {\\rm {SGCF}} + {\\rm {SGSF}}$ This $SGF$ considers both channel-wise encoded and spatial-wise encoded information, thus, it contains not only the discriminative (semantic) features, but also more accurate localization information.",
"Final Combination with Deep-Layer Feature Maps: To this end, we can simply complete the concatenation operation or element-wise addition operation.",
"Instead of using the shallow feature maps $S$ , we use the channel-wise and spatial-wise encoded shallow feature maps $SGF$ to combine with the deep-layer feature maps $D$ (through concatenation or element-wise addition).",
"Compared with the raw skip connection in UNet, our encoded shallow feature maps $SGF$ has same resolution but much more semantic and precise localization information (especially for the blurry regions), and thus can make the combination more reasonable.",
"At the same time, since the operations in the encoder are mostly $1 \\times 1 \\times 1$ convolution, the number of parameters just increases a little bit.",
"To further increase the model's discriminative capacity, we also adopt the multi-scale deep supervision strategy as in [10] after feature fusion at each stage."
],
[
"Boundary Delineation with Soft Contour Constraint",
" In mammal visual system [3], contour delineation closely correlates with object segmentation.",
"To incorporate the knowledge to improve the segmentation accuracy, we integrate the task of contour detection with the task of segmentation, assuming that introducing a task of contour detection can help guide the network to concentrate more on the boundaries of organ regions, thus helping overcome the adverse effect of low tissue contrast.",
"In this paper, as shown in Fig.",
"REF , two boundary detection tasks are added to the end of the network as auxiliary guidance.",
"To extract the contour for training, we first delineate the boundaries of different organs by performing Canny detector on the ground-truth segmentation.",
"For the organs with clear boundaries (i.e., bladder and rectum in our case), we model the problem as a classification problem.",
"However, due to the potential sample imbalance problem, we propose using focal loss to alleviate such an issue, as shown in Eq.",
"REF .",
"${L_{cboundary}} = - \\sum \\nolimits _h {\\sum \\nolimits _w {\\sum \\nolimits _t {\\sum \\nolimits _{c \\in csets} {{I_{\\left\\lbrace {{Y_{h,w,t}},c} \\right\\rbrace }}{{\\left( {1 - \\hat{p}\\left( {{X_{h,w,t}};\\theta } \\right)} \\right)}^\\gamma }\\left( {1 - \\hat{p}\\left( {{X_{h,w,t}};\\theta } \\right)} \\right)} } } }$ Note that, for the regions with blurry boundaries (i.e., prostate in our case), the voxels near the boundaries look almost same.",
"As a result, it will be more reasonable to assign soft labels (instead of hard labels) around the ground-truth boundaries.",
"Thus, we can formulate the blurry-boundary delineation task as a soft classification problem, which estimates the probability of each voxel being on the organ boundaries.",
"Then, for these blurry boundaries, we further exert a Gaussian filter (with a bandwidth of $\\delta $ , i.e., empirically set to 3 in our study) on the obtained boundary map.",
"In other words, for each voxel, we generate an approximate probability belonging the blur boundary of an organ.",
"Hence, we can formulate soft classification as a soft cross-entropy loss function as defined in Eq.",
"REF .",
"${L_{bboundary}} = - \\sum \\nolimits _h {\\sum \\nolimits _w {\\sum \\nolimits _t {{p_{h,w,t}}\\left( {1 - \\hat{p}\\left( {{X_{h,w,t}};\\theta } \\right)} \\right)} } }$"
],
[
"Implementation Details",
" Pytorchhttps://github.com/pytorch/pytorch is adopted to implement our proposed method shown in Fig.",
"REF .",
"The code can be obtained by this linkhttps://github.com/ginobilinie/SemGuidedSeg.",
"We adopt Adam algorithm to optimize the network.",
"The input size of the segmentation network is $144 \\times 144 \\times 16$ .",
"The network weights are initialized by the Xavier algorithm, and weight decay is set to be 1e-4.",
"For the network biases, we initialize them to 0.",
"The learning rate for the network is initialized to 2e-3, followed by decreasing the learning rate 10 times every 2 epochs during the training until 1e-7.",
"Four Titan X GPUs are utilized to train the networks."
],
[
"Experiments and Results",
" Our pelvic dataset consists of 50 prostate cancer patients from a cancer hospital, each with one T2-weighted MR image and corresponding manually-annotated label map by medical experts.",
"In particular, the prostate, bladder and rectum in all these MRI scans have been manually segmented, which serve as the ground truth for evaluating our segmentation method.",
"All these images were acquired with 3T MRI scanners.",
"The image size is mostly $256\\times 256\\times \\left( {120 \\sim 176} \\right)$ , and the voxel size is $1\\times 1\\times 1~\\text{mm}^3$ .",
"A typical example of the MR image and its corresponding label map are given in Fig.",
"REF .",
"Five-fold cross validation is used to evaluate our method.",
"Specifically, in each fold of cross validation, we randomly chose 35 subjects as training set, 5 subjects as validation set, and the remaining 10 subjects as testing set.",
"Unless explicitly mentioned, all the reported performance by default is evaluated on the testing set.",
"As for evaluation metrics, we utilize Dice Similarity Coefficient (DSC) and Average Surface Distance (ASD) to measure the agreement between the manually and automatically segmented label maps."
],
[
"Comparison with State-of-the-art Methods",
" To demonstrate the advantage of our proposed method, we also compare our method with other three widely-used methods on the same dataset as shown in Table REF : 1) SSAE [1], 2) UNet [8], 3) SResSegNet [10].",
"Figure: Visualization of pelvic organ segmentation results by four methods.",
"In (a) and (b), orange, silver and pink contours indicate the manual ground-truth segmentations, while yellow, red and cyan ones indicate automatic segmentations.",
"(a) Clear boundary case, (b) blurry boundary case, and (c) 3D renderings of segmentations.Table: DSC and ASD on the pelvic dataset by four different methods.Table REF quantitatively compares our method with three state-of-the-art segmentation methods.",
"We can see that our method achieves better accuracy than the other state-of-the-art methods in terms of both DSC and ASD.",
"It is worth noting that our proposed method can achieve much better performance for the blurry-boundary organ (i.e., prostate), which indicates the effectiveness of our proposed network components for blurry boundary delineation.",
"We also visualize some typical segmentation results in Fig.",
"REF , which further show the superiority of our proposed method, especially for the blurry regions around the prostate."
],
[
"Impact of Each Proposed Component",
" As our method consists of several novel proposed components, we conduct empirical studies below to analyze them.",
"Impact of Proposed SG Module: As mentioned in Sec.",
"REF , we propose a semantic-guided encoder feature learning module to learn more discriminative features in shallow layers.",
"The effectiveness of the SG module is further confirmed by the improved performance, e.g., 2.40%, 4.41% and 2.8% performance improvements in terms of DSC for bladder, prostate, and rectum, respectively, compared with the UNet with multi-scale deep supervision.",
"Relationship with Similar Work: Several previous work are proposed to use attention mechanism [9], [6] to enhance the encoder-decoder networks.",
"However, our work is different from them mainly in the follow way: We propose to use highly semantic information from the decoder to explicitly guide the building of attention mechanism, so that we can efficiently learn the encoder features.",
"To further compare them, we visually present the three typical learned feature maps (selected by clustering) of a certain layer (i.e., combined layer) in different networks at a certain training iteration (i.e., 4 epochs).",
"The methods include FCN [5], UNet [8], UNet with concurrence SE module [9] (ConSEUNet), attention-UNet [6] (AttUNet) and our proposed one(SGUNet).",
"The visualized maps are in Fig.",
"REF .",
"Figure: (a-e): Visualization of three typical learned feature maps of a certain layer by five different networks.",
"(f) and (g) are the corresponding input MRI and the MRI overlaid by the ground-truth contours.",
"(h) is the performance gain with different strategies towards the UNet with multi-scale deep supervision.Fig.",
"REF (a-e) indicates that the raw encoder-decoder network (i.e., FCN and UNet) cannot handle well for the blurry boundary cases.",
"The attention based networks can generate higher semantic maps with better localization information.",
"Among them, our proposed method can learn more precise boundaries due to explicit semantic guidance.",
"Also, our proposed method have a faster convergence compared to other methods.",
"Besides, the quantitative analysis in Fig.",
"REF (h) is consistent with the the conclusion of qualitative analysis.",
"Impact of Soft Contour Constraint: As introduced in Sec.",
"REF , we apply a hard contour constraint for clear-boundary organs while a soft contour constraint for the blurry-boundary organs.",
"Since hard contour constraint is a widely adopted strategy, we directly compare our proposed soft contour constraint with the case of using hard constraint.",
"With soft constraint on the prostate, we can achieve a slight performance gain such as 0.2% in terms of DSC; but we can achieve more performance gain in terms of ASD (0.8%), which is mainly because the soft contour constraint can help more accurately locate the blurry boundaries."
],
[
"Validation on Extra Dataset",
" To show the generalization ability of our proposed algorithm, we conduct additional experiments on the PROMISE12-challenge dataset [4].",
"This dataset contains 50 labeled subjects where only prostate was annotated.",
"We can achieve a high DSC ($0.92$ ), small ASD ($1.57$ ) in average based on five-fold cross validation.",
"As for the extra 30 subjects' testing dataset whose ground-truth label maps are hidden from us, the performance of our proposed algorithm is still very competitive (we are ranking in the top 6 among 290 submission with an overall score of $89.46$ .",
"The details can be available via this linkhttps://promise12.grand-challenge.org/evaluation/results/) compared to the state-of-the-art methods on the 30 subjects' testing dataset [10], [11].",
"These experimental results indicate a very good generalization capability of our proposed algorithm."
],
[
"Conclusion",
" In this paper, we have presented a novel semantic-guided encoder feature learning strategy to learn both highly semantic and rich resolution information features, so that we can better deal with the blurry-boundary delineation problem.",
"In particular, our SG module can improve the raw skip connection of the raw encoder-decoder models by enhancing the discriminative features while compressing the less informative features.",
"Furthermore, we propose a soft contour constraint to model the blurry-boundary detection, while an ordinary hard contour constraint to model the clear-boundary detection; this strategy is validated effective to help boundary localization and alleviate inter-class errors.",
"By integrating all these proposed components into the network, our final proposed framework has achieved sufficient improvement compared to other methods, in terms of both accuracy and robustness, also on the extra dataset."
]
] | 1906.04306 |
[
[
"Bell's Theorem and Locally-Mediated Reformulations of Quantum Mechanics"
],
[
"Abstract Bell's Theorem rules out many potential reformulations of quantum mechanics, but within a generalized framework, it does not exclude all \"locally-mediated\" models.",
"Such models describe the correlations between entangled particles as mediated by intermediate parameters which track the particle world-lines and respect Lorentz covariance.",
"These locally-mediated models require the relaxation of an arrow-of-time assumption which is typically taken for granted.",
"Specifically, some of the mediating parameters in these models must functionally depend on measurement settings in their future, i.e., on input parameters associated with later times.",
"This option (often called \"retrocausal\") has been repeatedly pointed out in the literature, but the exploration of explicit locally-mediated toy-models capable of describing specific entanglement phenomena has begun only in the past decade.",
"A brief survey of such models is included here.",
"These models provide a continuous and consistent description of events associated with spacetime locations, with aspects that are solved \"all-at-once\" rather than unfolding from the past to the future.",
"The tension between quantum mechanics and relativity which is usually associated with Bell's Theorem does not occur here.",
"Unlike conventional quantum models, the number of parameters needed to specify the state of a system does not grow exponentially with the number of entangled particles.",
"The promise of generalizing such models to account for all quantum phenomena is identified as a grand challenge."
],
[
"Introduction",
"Bell's Theorem places a strong restriction on reformulations of Quantum Mechanics (QM): any mathematical model which produces the same output predictions as QM, given the same inputs, must violate Local CausalityBoldface is used for mathematical conditions explicitly defined in the following sections.",
"[16].",
"In this sense, QM is “nonlocal,” but locality is not a simple yes/no question; QM is still “local” according to the operational definition, as it does not allow signaling at a distance, or outside the future lightcone.",
"Thus, QM may even be compatible with a generalized non-operational definition of “locality,” not as strict as Local Causality, but in spirit with Einstein's arguments against action-at-a-distance.",
"This Colloquium will examine a category of potential reformulations of QM which are as “local” as allowed by Bell's Theorem.",
"In order to assess a model's “locality,” even at the operational level of inputs and outputs, the model must define its spacetime-based parameters [those which bell1976b called “local beables”].",
"Such parameters are mathematical variables which are clearly associated with a specific time and place, such as the local values of classical fields, $Q(\\mathbf {x},t)$ .",
"This association allows concepts of “locality” to be meaningfully applied.",
"Conventional QM utilizes inputs corresponding to values that can be controlled by experimental physicists, including the settings of preparation and measurement devices, and it predicts the probability distribution of the values of measureable outputs.",
"Reformulations of QM must be operationally equivalent, utilizing these same inputs and outputs, and providing the same predictions.",
"At minimum, this requires the use of spacetime-based parameters for the inputs and the outputs.",
"These input and output parameters do not continuously span the intermediate spacetime regions where preparations and measurements are not performed; non-operational definitions of “locality” (such as Local Causality) concern parameters associated with these intermediate regions.",
"If a model has no spacetime-based parameters associated with these regions, it will be nonlocal according to these definitions, i.e., it will have unmediated action-at-a-distance.",
"Many physicists see Bell's Theorem as a reason not to introduce such mediating parameters [see, e.g., mermin1986].",
"It is difficult to map an entangled configuration-space wavefunction $\\psi (\\mathbf {x}_1,\\mathbf {x}_2,t)$ onto spacetime-based parameters $Q(\\mathbf {x},t)$ [74], [101], and even if there were such a mapping, Bell's Theorem tells us that no reformulation could possibly conform to Local Causality.",
"However, such a viewpoint presumes that there is no other type of “locality” worth saving, no subset of assumptions inside of Local Causality which might be beneficial for some future reformulation of QM.",
"In fact, there exists a category of quantum reformulations for which an essential aspect of “locality” can be retained [see, e.g., costa1953,price1997,argaman2010].",
"These models utilize spacetime-based parameters associated with intermediate regions between preparations and measurements, allowing these models to be “locally mediated”, in the sense that correlations cannot be introduced or altered except via intermediate spacetime-based mediators.",
"This condition will be explicitly defined in the next section, using the term Continuous Action to contrast with the phrase “action-at-a-distance”.",
"We are most interested in cases where this local mediation is always restricted to time-like or light-like worldlines, allowing those models to also respect Lorentz covariance.",
"Such locally-mediated reformulations of QM must violate a certain time-asymmetric assumption inherent to Local Causality.",
"Specifically, the relevant assumption presumes that no model parameter associated with time $t$ can be dependent upon model inputs associated with times greater than $t$ .",
"The most “local” reformulations of QM—those with Continuous Action—violate this assumption, and are therefore future-input dependent.",
"While some may view such “retrocausality” as unreasonable, it is emphasized that only models with the same predictions as QM are of interest here, with no signaling into the past [see, e.g., price1997].",
"If one considers that inputs to a model also include boundary conditions, it is evident that future-input dependent models are ubiquitous throughout physics.",
"For example, models employing the stationary action principle fall in this category – mathematical inputs constrain both initial and final parameters, and the model determines the classical history at intermediate times.",
"Any calculation of a closed-timelike-curve in general relativity requires a similar all-at-once analysis.",
"Quantum future-input dependent models, such as the Transactional Interpretation [35] and the Two-State-Vector Formalism [2], have also been developed, motivated primarily by time-symmetry rather than locality.",
"Attempts to develop a locally-mediated account of quantum entanglement using future-input dependence have been promoted by a number of forward-thinking authors, beginning even before Bell's work [10], [11], [12], [82], [103], [85].",
"Still, mathematical future-input dependent models reproducing the QM predictions for entangled particles, while explicitly maintaining local mediation, have been put forward mainly in the last decade.",
"One purpose of this Colloquium is to survey the admittedly modest achievements of this recent and still-developing line of enquiry, and to indicate some intriguing directions for further exploration.",
"A formal development of these arguments will also result in a useful categorization of all the ways in which a reformulation of QM can violate Local Causality (we say that such models are “Bell-compatible”).",
"This is accomplished by presenting the assumptions of Bell's Theorem in terms consistent with the recently developed framework of “causal models” [81], which emphasizes the role of QM's controllable inputs.",
"Bell himself spoke of the special importance of inputs, calling them “free external variables in addition to those internal to and conditioned by the theory” [19].",
"Unfortunately, the mathematics of causal models was not well-developed during his lifetime, and bell1976b,bell1981,bell1990 adopted a neutral notation, e.g., $\\lbrace A | a, \\lambda \\rbrace $ for the probability distribution of an output $A$ given an input $a$ and an internal parameter $\\lambda $ .",
"Instead, we will denote this by $p_a(A|\\lambda )$ , emphasizing the input status of $a$ , and allowing a clear categorization of Bell-compatible reformulations of QM (while setting aside extraneous issues such as “superdeterminism”).",
"Developing reformulations of existing theories has historically been very useful – think of the advances of Lagrangian and Hamiltonian classical mechanics.",
"In quantum theory, the path integral has similarly led to new insights, and there is no indication that this strategy of seeking further reformulations has run its course [44].",
"In particular, an alternative quantum model with parameters restored to functions on spacetime, instead of a multi-dimensional configuration space (or a Hilbert space), would have significant advantages.",
"Such a model would have a natural interpretation, with one allowed combination of the spacetime-based parameters corresponding to physical reality, and all other combinations being mere possibilities (as in classical statistical mechanics).",
"As a result, the number of parameters describing an actual system would grow linearly (rather than exponentially) with the extent of that system.",
"This would substantially lessen the disconnect between quantum theory and our linearly-scaling classical theories of relativistic spacetime.",
"Note that a successful reformulation of QM in terms of spacetime-based parameters would certainly not imply that quantum theory was incorrect.",
"Quantum states could still represent our best possible knowledge about measurable aspects of those parameters, given accessible information.",
"In this case, quantum states could be viewed as states of knowledge, a popular perspective in the field of quantum information [30], [99], [61].",
"The next section carefully walks through Bell's Theorem, identifying all the assumptions leading to the contradiction with quantum phenomena.",
"Section III then categorizes Bell-compatible reformulations of QM.",
"Several examples of locally-mediated toy models are detailed in Section IV; those who would like to look at a concrete mathematical model, rather than follow general reasoning, are referred to the model of Section IV.B, for which a detailed derivation is given in the appendix.",
"Section V discusses the approach and indicates avenues for further development.",
"Alternative approaches are briefly discussed in Section VI.",
"Section VII provides the conclusion, encouraging future development of locally-mediated reformulations of QM."
],
[
"Bell's Theorem",
"Our first task is to prove Bell's Theorem.",
"Starting with a certain set of natural assumptions, we will give a mathematical proof of a Bell inequality – specifically, the Clauser-Horne-Shimony-Holt (CHSH) inequality [34].",
"This inequality can be tested operationally (without reference to any underlying theory), and it is experimentally violated, just as predicted by QM.",
"It follows that for any model of these phenomena, at least one of the assumptions which lead to Bell's Theorem must be violated.",
"All such “Bell-compatible” models can then be usefully categorized in terms of which assumptions are relaxed.",
"The analysis is divided into the following Subsections: the first defines the framework rules for the models to be discussed, the second lists the relevant reasonable-but-optional assumptions that could characterize such models, the third provides some historical context, and the fourth provides a derivation of the Theorem."
],
[
"Framework: spacetime-based models",
"In all of physics, one uses mathematical models to generate falsifiable predictions which can be compared with empirical observations.",
"The sort of models that accomplish this are essentially functions that take some parameters as inputs and generate other parameters as outputs.We use the term “parameter” instead of “variable”, as the latter sometimes implies a time-dependent quantity, while inputs and outputs are generally localized in time, as well as space.",
"We are therefore interested in models which come with well-defined input parameters (“inputs” for short), which will be denoted by the set $I$ , and also well-defined output parameters (“outputs”), denoted by $O$ .",
"Models can have other parameters in addition to the inputs and outputs, and the set of these will be denoted by $U$ .",
"We will often discuss the set of all non-input parameters $Q$ (the union of $O$ and $U$ ).",
"Parameters here are not limited to simple scalars—vectors, or more complicated mathematical constructs such as functions may be utilized.",
"As discussed in the Introduction, we are interested in models of spacetime-based parameters, each associated with a particular location in ordinary spacetime.",
"Examples of such parameters include the values of physical fields, such as $\\mathbf {E}(\\mathbf {x},t)$ in classical electromagnetism and $g^{\\mu \\nu }(x^\\gamma )$ in general relativity.",
"Other examples include instrument settings and measurement results, which are associated with definite regions rather than points in spacetime.",
"These parameters correspond to what bell1976b called “local beables” (pronounced be-ables).",
"Unless otherwise noted, our use of the term “parameters” will be restricted to spacetime-based parameters, including the sets $I$ and $Q$ .",
"Of course, some models employ additional mathematical entities which are not spacetime-based.",
"For example, for $N>1$ , the $N$ -particle configuration-space wavefunction in QM is comprised of values that do not correspond to particular locations in spacetime.",
"For the purposes of Bell's analysis and the below discussion, non-spacetime-based parameters such as configuration-space wavefunctions are simply omitted from $Q$ , even if they are mathematically utilized in a given model.",
"It is also possible to construct non-localized parameters out of spacetime-based parameters, such as the total energy of an extended system, but such values are not to be included as elements of $I$ or $Q$ .",
"Deterministic models are those for which specification of all inputs $I$ , including boundary conditions and external forces (if present) always exactly determines the non-input parameters $Q$ .",
"Stochastic models do not predict unique values for $Q$ , but for any full set of inputs, the model assigns a probability for every possible combination of non-input parameters.",
"Thus, a fully-specified mathematical model can always be written as $P_I(Q)$ , a unique joint probability distribution function for the set of non-input parameters, given specific values for the inputs.In many cases, “distribution functional” rather than “function” should be used here, as $Q$ itself typically includes fields, functions of spacetime.",
"Similarly, when $Q$ is continuous, $P_I(Q)$ denotes probability densities rather than probabilities.",
"For deterministic models, these distributions are $\\delta $ -functions, but the analysis is not limited to such cases.",
"This definition, which suffices for the present purposes, is minimal in the sense that it does not include the details regarding how one parameter is deduced from another within the model, nor the physical interpretation of a model.",
"According to the standard rules for probabilities, the full joint probability distribution $P_I(Q)$ of all the non-input parameters of a model can be used to generate marginal distributions, $P_I(Q_1)$ , for any subset $Q_1 \\subset Q$ .",
"It also generates conditional probabilities, $P_I(Q_1|q_2)$ , where $q_2$ are specific values of parameters in another subset, $Q_2$ .",
"In some cases, a model may predict that $Q_1$ and $Q_2$ are statistically independent, meaning that $P_I(Q_1,Q_2) = P_I(Q_1) P_I(Q_2)$ .",
"When statistical independence holds, knowledge of the values of parameters in $Q_2$ does not inform the marginal $P_I(Q_1)$ , as represented by the condition $P_{I}(Q_1|Q_2)=P_{I}(Q_1)$ .",
"Two models that use identical input and output sets $I$ and $O$ (when applied to a given system) and that also have the same marginal output probabilities $P_I(O)$ are said to be in agreement: they always yield the same joint probability of the output parameters for a given set of inputs.",
"Note that agreement does not preclude different predictions at the level of $P_I(Q)$ .",
"Two models can even be in agreement if they utilize different parameters $U$ in $Q$ .",
"For example, in Classical Electromagnetism (CEM), one can change the gauge condition on parameters corresponding to the electromagnetic potentials without changing observable model predictions.",
"The discussion below treats any such parameter-changing reformulations as different models, because they might generally have different properties at the level of non-observable parameters.In the CEM example, the use of Coulomb-gauge potentials as parameters will generally not respect the Local Causality condition defined below, because those potentials can change instantaneously over all space.",
"(As we shall see, even changing the associated spacetime location of a parameter can significantly change the model.)",
"In the following we will focus on models which are in agreement with QM, at least for a specific setup under consideration.",
"Such models are guaranteed to share the empirical success of QM, but are strictly constrained by Bell's Theorem."
],
[
"Physical assumptions",
"The following properties may or may not hold for any specific mathematical model, allowing for a categorization of models into classes and sub-classes.",
"In order to maintain an appropriate scope, we here define only key properties which play significant roles in the discussion to follow, with a few more given in Section REF .",
"For example, the need to formally define relativistic covariance of models does not arise here, although the lightcones of Minkowski spacetime do play a role."
],
[
"Continuous Action (",
"Instead of beginning with Bell's approach to defining locality, we first define a weaker condition, Continuous Action (CA), that encodes the spirit of no-action-at-a-distance without requiring any lightcone structure from relativity, or even a distinction between past and future.",
"As shown in Figure 1(a), consider spacetime regions $\\bf {1}$ and $\\bf {2}$ , with $\\bf {1}$ completely surrounded by a screening region $\\bf {S}$ .",
"This is not merely a spatial region; $\\bf {S}$ spans the past and future of $\\bf {1}$ as well as its spatial extent.",
"We will denote the set of all inputs in regions $\\bf {1}$ and $\\bf {2}$ by $I_1$ and $I_2$ , respectively.",
"If there are any additional inputs, besides $I_1$ and $I_2$ , their values are assumed to be fixed in the definitions below.",
"The non-input parameters in each region are denoted by the corresponding $Q_1$ , $Q_2$ , $Q_S$ .",
"Loosely speaking, a mathematical model violates CA if it has unmediated “action-at-a-distance”, i.e., if changes in $\\bf {2}$ can be associated with changes in $\\bf {1}$ without being also associated with changes within $\\bf {S}$ .",
"For example, a CA-respecting model of a light switch in $\\bf {2}$ correlated with a lamp in $\\bf {1}$ must include a description of the mediating parameters (e.g., the currents in the wires) in the intermediate screening region $\\bf {S}$ .",
"In such a model, knowledge of the values of all the parameters in $\\bf {S}$ makes additional information regarding $\\bf {2}$ redundant for the purpose of predicting what happens in region $\\bf {1}$ .",
"Figure: The screening regions 𝐒\\bf {S} used in different assumptions of “locality”.",
"Given all modeled parameters in the screening region, a screened model will assign the same probabilities for parameters in region 1, regardless of additional knowledge of parameters in region 2.",
"Figure 1a shows the most general case of Continuous Action (CA); Figure 1b breaks time-symmetry by adding the No Future-Input Dependence (NFID) assumption, and Figure 1c references the lightcones according to Bell's Screening Assumption (BSA).Mathematically, CA corresponds to the condition $P_{I_1,I_2}(Q_1|Q_2,Q_S) = P_{I_1}(Q_1|Q_S),$ for all combinations of the parameters in the regions depicted in Figure 1(a).",
"This equation says that $P_{I_1}(Q_1|Q_S)$ is both statistically independent of $Q_2$ , and functionally independent of $I_2$ .",
"When this occurs, we say that $\\bf {S}$ “screens” $\\bf {1}$ from $\\bf {2}$ .The word “shields” is often used in the literature, including bell1990, instead of “screens.” For CA models this equality is required to hold for all simply connected, non-overlapping regions $\\bf {1}$ , $\\bf {2}$ , and $\\bf {S}$ , for which $\\bf {S}$ completely separates $\\bf {1}$ from $\\bf {2}$ and is nowhere vanishingly thin.",
"As there is no essential difference between regions $\\bf {1}$ and $\\bf {2}$ , a model with CA also must have $\\bf {S}$ screen $\\bf {2}$ from $\\bf {1}$ .",
"Readers familiar with probabilistic modeling will notice that the role of the screening region $\\bf {S}$ in CA is analogous to that of a “Markov blanket,” a term coined by pearl1988.",
"We avoid using this terminology not only because of required minor adjustments (discretizing the model in spacetime onto a set of nodes; properly representing the role of inputs), but primarily because many physicists might be misled—Markov's name would likely be immediately associated with Markov processes, which propagate step by step from the past to the future, subject to a particularly strong arrow of time.",
"This is exactly the opposite of our purpose here—generalizing “no action at a distance” to situations in which time symmetry is not broken at all, or is broken in a much weaker manner.",
"Restricting attention to Markov processes would be an additional assumption—limiting attention to directed acyclic graphs with the directions of all edges determined by temporal order—one that will now be formally defined."
],
[
"No Future-Input Dependence (",
"There is a well-known tension between the time-symmetric equations characteristic of fundamental physical theories and the time-asymmetric manner in which models are utilized.",
"For example, if one takes wavefunction “collapse” to be physically meaningful, this process defines a preferred direction of time, breaking the time symmetry evident in unitary evolution.",
"More generally, a preferred direction of time is commonly chosen by limiting attention to models in which all events up to a time $t^{\\prime }$ can be evaluated without regards to events in the future of $t^{\\prime }$ : No Future-Input Dependence (NFID) holds for a mathematical model $P_I(Q)$ if, for any time $t^{\\prime }$ included in the relevant spacetime region, there exists a restricted model $P^{\\prime }_{I^{\\prime }}(Q^{\\prime })$ , where $I^{\\prime }$ is the set of all inputs belonging to times up to $t^{\\prime }$ and $Q^{\\prime }$ is the set of all non-input parameters up to $t^{\\prime }$ , such that $P_I(Q^{\\prime })=P^{\\prime }_{I^{\\prime }}(Q^{\\prime })$ for all possible values of the parameters in $I$ and $Q^{\\prime }$ .",
"In other words, NFID means the marginal $P_I(Q^{\\prime })$ is functionally independent of future inputs.",
"When combined with CA, the assumption of NFID implies that there is no need to consider any parts of the screening region $\\bf {S}$ that lie in the future of both regions 1 and 2.",
"As shown in Figure 1(b), if CA holds for $P$ and $P^{\\prime }$ of a model respecting NFID, then the smaller region ${\\bf S^{\\prime }}$ also screens ${\\bf 1}$ from ${\\bf 2}$ , $P_{I_1,I_2}(Q_1|Q_2,Q_{S^{\\prime }}) = P_{I_1}(Q_1|Q_{S^{\\prime }})$ ."
],
[
"Bell's Screening Assumption (",
"If one accepts both of the above assumptions (CA and NFID), and is furthermore interested in modeling only screening regions that remain applicable in all reference frames, it becomes appropriate to ignore any portion of $\\bf {S}$ that is spacelike separated from bothNote that it is not sufficient to restrict the screening region to lie in the past lightcone of region 1; it must completely screen 1 from the overlap of the past lightcones of 1 and 2 [see, e.g., note 7 in bell1986].",
"regions 1 and 2.",
"This leads to the smaller region $\\bf {S^{\\prime \\prime }}$ shown in Figure 1(c).",
"bell1990 proposed that this smaller region, $\\bf {S^{\\prime \\prime }}$ , should screen region 1 from region 2: $P_{I_1,I_2}(Q_1|Q_2,Q_{S^{\\prime \\prime }}) = P_{I_1}(Q_1|Q_{S^{\\prime \\prime }}).$ It is important to note that this screening condition does not imply that parameters in 1 are independent of parameter values in 2—merely that the latter values are redundant, given the specification of all model parameters in $\\bf {S^{\\prime \\prime }}$ .",
"We will call Eqn.",
"(REF ) Bell's Screening Assumption (BSA)."
],
[
"Local Causality",
"Models which conform to both BSA and NFID are unable to describe certain quantum phenomena, as Bell's Theorem establishes, and will be proved in Section II.D below.",
"We will define this important combination of assumptions as Local Causality.The freedom in choosing the region $\\bf {S}$ in the definition of CA is reflected in the definitions used for “Local Causality” (or “Einstein Locality,” or “Local Realism”).",
"The present Figure 1 resembles Figure 6.4 of bell1990, the “screening region” was effectively the entire past of $\\bf 1$ and $\\bf 2$ in bell1981, and bell1976b used the overlap of their past lightcones.",
"This affects the identification of $\\lambda $ in the separability condition below, Eqn.",
"(REF ), but the subsequent derivation is unchanged.",
"This definition may cause some initial confusion, because Local Causality is often identified with Eqn.",
"(REF ) in the literature, which is formally just BSA.",
"However, in essentially all cases in which this is done, the authors are presupposing NFID, either explicitly or implicitly, and the addition of this assumption turns BSA into Local Causality.",
"bell1990 himself introduced BSA after clearly assuming the past-to-future causal structure associated with NFID [see Figure 6.3 there], and used the term Local Causality to convey this combination, often using the shorter “locality” as a synonym.",
"The reader should be cautioned about interpreting the phrase “Local Causality” as being the simple conjunction of “locality” and “causality”.",
"There are many different meanings that could be ascribed to both of these words (we have already seen three different notions of locality in Fig.",
"1 above).",
"All that is needed in the present analysis is that Local Causality means the well-defined assumptions NFID and BSA.",
"An important condition which follows from NFID (or from Local Causality), but not BSA alone, can be derived by applying it to the $S^{\\prime \\prime }$ region from Figure 1(c).",
"Requiring the probabilities of parameters to be independent of future inputs, and choosing $S^{\\prime \\prime }$ to lie entirely in the past of all of regions $\\bf {1}$ and $\\bf {2}$ (in some reference frame where NFID holds), one obtains the functional independence relation $P_{I_1,I_2}(Q_{S^{\\prime \\prime }}) = P(Q_{S^{\\prime \\prime }}) .$ A variant of this condition will play an important role in the proof of Bell's Theorem below."
],
[
"Historical interlude",
"At this point it is appropriate to emphasize how natural it is to assume that all of the above conditions, summarized by Local Causality, should hold in any detailed model describing real physics.",
"It is convenient to do so by referencing Einstein and Bohr.",
"At the 1927 Solvay conference, Einstein noted that there were two possible “conceptions” of the single-particle quantum wavefunction $\\psi (x,t)$ [8].",
"If viewed as a set of spacetime-based parameters $Q(x,t)$ , the requirement that only a single particle is eventually measured implies some form of wavefunction collapse that, for Einstein, “implies to my mind a contradiction with the postulate of relativity.” Instead, he advocated a conception where “one does not describe the process solely by the Schrödinger wave,\" effectively pointing out the possibility that additional hidden parameters could indicate the particle's actual location.",
"In 1935, Einstein, Podolsky and Rosen (EPR) extended this analysis to a two-particle system (of a type to be analyzed below), and reached the logical conclusion that violations of Local Causality could only be avoided by adding new hidden parameters.",
"If known, these new parameters would allow one to determine the outcomes in more detail than is possible within QM.",
"EPR concluded that QM gave an incomplete description.",
"EPR did not use the formal mathematical language of Bell's analysis.",
"Instead, they implied the existence of spacetime-based parameters $Q$ that encoded “an element of physical reality” (italics in original, here and below), and deduced that hidden $Q$ 's must be present in a complete theory, because in some cases it was possible to “predict with certainty ... the value of a physical quantity,” such as position or momentum, “without in any way disturbing a system.” bohr1935 responded quickly to EPR, defending the completeness of QM on the basis of the notion of complementarity he had developed earlier (in connection with the quantum uncertainty principle).",
"He advocated “a radical revision of our attitude towards the problem of physical reality,” and argued that the phrase “without in any way disturbing a system” used by EPR “contains an ambiguity.” Bohr considered in detail a situation in which the properties of a particle can be discerned by first allowing it to pass through a slit in a diaphragm, and later making a “free choice” of measuring either the momentum or the position of the diaphragm.",
"(It is remarkable, especially in the context of the present work, that the guarantee for “without in any way disturbing” was spatial separation for EPR, but temporal order for Bohr.)",
"“Of course there is $\\dots $ no question of a mechanical disturbance $\\dots $ during the last critical stage of the measuring procedure,” he wrote.",
"But as one can only measure either the position or the momentum of the diaphragm, “even at this stage” there still might be “an influence on $\\dots $ the possible types of predictions regarding the future behavior of the system.” Bohr thus advocatedWe believe it is appropriate to interpret Bohr in this manner, but acknowledge that it is probably impossible to uncontroversially translate his writing into the formal language introduced later.",
"accepting some violations of Local Causality which are present in the formalism of QM, while at the same time excluding other violations—those corresponding to a “mechanical disturbance”.",
"(Similarly, his notion of “completeness” clearly differs from that of EPR.)",
"Most physicists simply adopted Bohr's complementarity, either in its original form or a variant [21], and continued to develop and apply QM to a variety of physical systems [64], [68].",
"But Einstein was not convinced.",
"Summarizing the situation in 1948, he wrote [26]: [T]hose physicists who regard the descriptive methods of quantum mechanics as definitive in principle would $\\dots $ drop the requirement $\\dots $ for the independent existence of the physical reality present in different parts of space.",
"$\\dots $ [W]hen I consider the physical phenomena known to me, and especially those which are being so successfully encompassed by quantum mechanics, I still cannot find any fact anywhere which would make it appear likely that [that] requirement will have to be abandoned.",
"I am therefore inclined to believe that the description of quantum mechanics $\\dots $ has to be regarded as an incomplete and indirect description of reality, to be replaced at some later date by a more complete and direct one.",
"Here Einstein is essentially advocating for models to be built from spacetime-based parameters $Q$ , while offering the opinion that other physicists had prematurely abandoned this possibility.",
"But there was indeed a “fact” that he was not aware of, a theorem that would be proved by Bell in 1964 (sadly, after both Einstein and Bohr had passed away).",
"We now turn to Bell's Theorem, and the fact that all models in agreement with QM must violate the package of assumptions that is Local Causality.",
"Subsequently, we will address the question of whether the hidden-parameter models advocated by Einstein should still be pursued, even given the necessary violation of Local Causality."
],
[
"Statement and proof",
"Bell's Theorem demonstrates that: No model conforming with Local Causality can be in agreement with QM.",
"It is to be emphasized that the disagreement is not only with the predictions of QM, but also with the results of empirical observations – experiments which have been performed.",
"The proof below is based on the CHSH inequality [34], which concerns a particular application of QM to the experimental scenario shown in Figure 2.",
"Specifically, a source emits a pair of particles, and these are later analyzed and detected in spacelike-separated regions $\\bf {1}$ and $\\bf {2}$ .",
"Figure: The essential geometry of a Bell-type experiment.",
"The parameters a,b,ca,b,c are inputs; the arrows indicate that their values come from outside the model.",
"The parameters AA and BB are observable outputs.",
"λ\\lambda is the set of all localized model parameters in the region Λ\\mathbf {\\Lambda }, which screens regions 1\\mathbf {1} and 2\\mathbf {2} from the overlap of their backward lightcones.Mathematical models describing such situations will have an input parameter $c$ specifying the particular settings/arrangement of the common source of the two particles.",
"Additional input parameters $a$ and $b$ specify the settings/arrangement of the detectors in region $\\bf {1}$ and region $\\bf {2}$ respectively.",
"The results of the experiment are the output parameters $A$ in region $\\bf {1}$ and $B$ in region $\\bf {2}$ .",
"The set of all the model's spacetime-based parameters in region $\\bf {\\Lambda }$ is denoted by $\\lambda $ .",
"The parameters $a$ , $b$ and $c$ are inputs, and $A$ and $B$ are outputs, just as in QM.",
"The set $\\lambda $ can be quite general, with possibilities ranging from complex combinations of functions and operators to the simplest possibility: the empty set, for the case with no spacetime-based parameters associated with the region $\\bf {\\Lambda }$ .",
"The proof of the CHSH inequality, following bell1976b,bell1981,bell1990 and peres1978, proceeds in the next two subsections by only assuming Local Causality, without making any reference to QM.",
"The third subsection then proves Bell's Theorem by comparing this Bell inequality with quantum theory and experiments.",
"(A disadvantage of Bell's original 1964 proof is discussed in Section REF .)"
],
[
"Bell's separability condition",
"Any mathematical model capable of producing predictions for the setup of Figure 2 will provide a joint probability distribution $P_{a,b,c}(A,B,\\lambda )$ .",
"The marginal distribution $P_{a,b,c}(A,B)$ can be compared with experiment and with QM.",
"Models will generically also have other parameters, located between the designated regions, but these are not necessary for the main argument.",
"Also not included in $\\lambda $ are non-spacetime-based entities, such as multiparticle wavefunctions, which may be utilized in some models.",
"From the assumption of Local Causality, specifically from BSA, Eqn.",
"(REF ), it follows that $P_{a,b,c}(A|\\lambda ,B) = P_{a,c}(A| \\lambda ),$ because $\\bf {\\Lambda }$ screens $\\bf {1}$ from $\\bf {2}$ , in the sense that the necessary $S^{\\prime \\prime }$ region can be chosen to be fully contained in $\\bf {\\Lambda }$ .",
"Similarly, $P_{a,b,c}(B|\\lambda ,A) = P_{b,c}(B|\\lambda )$ .",
"It also follows from NFID that any model-generated probabilities of $\\lambda $ must be independent of the settings $a$ and $b$ , because those settings lie in the future of $\\lambda $ .",
"In equation form, following (REF ), this reads $P_{a,b,c}(\\lambda )=P_c(\\lambda ).$ This is often known as “measurement independence,” a term that unfortunately obscures the input nature of the measurement settings.",
"It is clearer to call this condition $\\mathbf {\\lambda }$ -independence, as the equation specifies that $\\lambda $ is independent of the inputs $a,b$ , via a direct application of NFID.A different perspective results if one denies free-input-parameter status to the measurement settings, treating $a$ and $b$ as stochastic variables instead of inputs.",
"This is the “superdeterministic” scenario, to be discussed in Section REF , which allows a version of Eqn.",
"(REF ) to be considered as a “no conspiracy”, “freedom of choice”, or even a “free will” condition.",
"Basic probability theory provides a product expression for the joint conditional probability: $P_{a,b,c}(A,B|\\lambda )=P_{a,b,c}(A|B,\\lambda )P_{a,b,c}(B|\\lambda )$ .",
"Since $\\lambda $ is hidden, the observable joint probability is found by summing or integrating this over all possible values of $\\lambda $ .",
"Applying BSA and NFID, by substituting Eqns.",
"(REF ) and (REF ), yields Bell's “Separability Condition”: $P_{a,b,c}(A,B) = \\int d\\lambda \\; P_{c}(\\lambda ) P_{a,c}(A|\\lambda ) P_{b,c}(B|\\lambda ),$ where the integral is understood as a sum if $\\lambda $ is discrete, or a functional integral if $\\lambda $ is a function.",
"This must hold for every applicable model respecting Local Causality."
],
[
"A Bell inequality",
"From Bell's Separability Condition, Eqn.",
"(REF ), one can derive the CHSH inequality [34], a generalized versionSee, e.g., bell1971 for details.",
"of Bell's original inequality [16].",
"It applies to models for which the output parameters in regions $\\bf {1}$ and $\\bf {2}$ , i.e., the outcomes $A$ and $B$ , have two possible values.The proof can be generalized to measurements with continuous results, provided their ranges are restricted, $|A|,|B| \\le 1$ .",
"Assigning $\\pm 1$ to the outcome values on each side, the product $AB$ must then also be $\\pm 1$ .",
"Its expectation value for given inputs, i.e., the correlator of the outcomes, is denoted: $\\langle A B \\rangle _{a,b,c}\\equiv \\sum _{A,B} A \\, B \\, P_{a,b,c}(A,B).$ The CHSH inequality restricts the values of a combination of correlators, which involves two of the possible settings of the input parameter $a$ in region $\\bf {1}$ , labelled $a$ and $a^{\\prime }$ , and two possibilities for the input setting in region $\\bf {2}$ , labelled $b$ and $b^{\\prime }$ .",
"The source input setting $c$ is held constant while the four possible combinations of inputs are manipulated, and will be suppressed from here on (we will later consider only particular Bell states, for which only one value of $c$ is relevant).",
"It is customary to transfer the primes to the $A$ and $B$ parameters, so that, e.g., $\\langle A^{\\prime } B \\rangle $ stands for $\\langle A B \\rangle _{a^{\\prime },b}$ .",
"With this notation, the CHSH inequality concerns the combination $\\langle A B \\rangle + \\langle A^{\\prime } B \\rangle + \\langle A B^{\\prime } \\rangle - \\langle A^{\\prime } B^{\\prime } \\rangle $ .",
"It is easiest to evaluate this combination by sampling the probability distributions in Eqn.",
"(REF ) many ($N$ ) times, in the style of a Monte-Carlo simulation.The proof here follows peres1978.",
"The discussion of the mathematical model rather than the modelled physical experiments avoids the need for any additional assumptions, such as “counterfactual definiteness” (the assumption that when $A_n$ is measured, it is legitimate to discuss $A^{\\prime }_n$ as well).",
"Denoting the $n$ th value sampled from $P(\\lambda )$ by $\\lambda _n$ , we have $A_n$ sampled from $P_a(A|\\lambda _n)$ and $A^{\\prime }_n$ from $P_{a^{\\prime }}(A|\\lambda _n)$ , and similarly for $B_n, B^{\\prime }_n$ .",
"The large $N$ limit is implied, so that, e.g., $\\langle A^{\\prime } B \\rangle = \\frac{1}{N} \\sum _n A^{\\prime }_n B_n \\,$ .",
"The above combination of correlators is then obtained by averaging over $(A_n + A^{\\prime }_n) B_n + (A_n - A^{\\prime }_n) B^{\\prime }_n$ , and it follows from $A_n, A^{\\prime }_n = \\pm 1$ that for each $n$ one of the parentheses must vanish.",
"The averaged combination therefore is of absolute magnitude 2 for each $n$ , and the combination of correlators cannot be larger in magnitude: $\\left| \\langle A B \\rangle + \\langle A^{\\prime } B \\rangle + \\langle A B^{\\prime } \\rangle - \\langle A^{\\prime } B^{\\prime } \\rangle \\right| \\le 2.$ This is the CHSH inequality."
],
[
"Contradiction with QM and experiment",
"When the Bell inequalities were first derived, they were shown to be in conflict with the predictions of QM.",
"Now, they are known to be in direct conflict with actual experiments [56], [47], [94], [89], independent of the formalism of QM, demonstrating the failure of Local Causality.",
"It is simple to demonstrate that at least some QM predictions violate the CHSH inequality, Eqn.",
"(REF ).",
"Consider two photons entangled in a spin-zero Bell state, as in several of the early experiments [32], [7].",
"(Equivalently, two spin-$1/2$ particles can be analyzed.)",
"Suppose each photon encounters a polarizing beamsplitter, with outputs directed onto two single-photon detectors.",
"The two beamsplitters are aligned at angles $a$ and $b$ in regions $\\bf {1}$ and $\\bf {2}$ respectively (these are the measurement settings, defined modulo $\\pi $ ).",
"For the outcome parameters $A$ and $B$ , assign a value of $+1$ when the detectors imply a measured polarization aligned with the setting, and $-1$ for a measurement of the perpendicular polarization.",
"The predictions of QM are then given by the probabilities $p_{a,b}(A,B) = \\frac{1}{4} \\left[ 1 + A B \\cos (2a-2b) \\right].$ The expectation value of the product $AB$ is therefore $\\langle A B \\rangle =\\cos (2a-2b)$ .",
"For certain combinations of settings, this violates the CHSH inequality by a wide margin.",
"The largest violation obtains for $a=0, a^{\\prime }=\\frac{\\pi }{4}, b=-b^{\\prime }=\\frac{\\pi }{8}$ , for which the left hand side of (REF ) is $2\\sqrt{2}$ (each of the four terms contributes $+1/\\sqrt{2}$ ).cirelson1980 has shown that this is the maximal value achievable in QM, while popescu1994 have devised a synthetic model which reaches even higher values, up to 4, the maximum possible.",
"These non-classical correlations between the two photons served historically as an early and striking example of the much wider family of phenomena associated with quantum entanglement [see, e.g., brunner2014,streltsov2017].",
"The observed violations of the inequalities are by impressive margins, greatly exceeding the experimental accuracy.",
"Indeed, as an empirical test of a mathematical model or a class of models, the confidence with which the CHSH inequality is rejected approaches the certainty of a mathematical proof.",
"For example, the experimental results of giustina2015 boast a value less than $3.7 \\cdot 10^{-31}$ for the probability that the results could be obtained under the assumption of Local Causality, according to the standard statistical analysis.",
"Furthermore, this result belongs to the recent generation of “loophole-free” experiments (those cited above), which are free from all of the simplifying assumptions which were necessary for Bell tests with earlier technology.",
"The observations not only violate the CHSH inequality—the quantitative results follow the predictions of QM in fine detail.",
"We now turn to models that can be consistent with these experiments, Bell's Theorem notwithstanding.",
"The upshot of Bell's Theorem is that there is no longer any hope of finding a reformulation of QM which respects Local Causality.",
"But the use of spacetime-based parameters has not been ruled out altogether, and the motivations for using them remain intact.",
"Furthermore, given such parameters, there are still live options for saving CA (Continuous Action), the generalized form of “locality” defined in Section II.B.1.",
"In this sense, Bell's Theorem does not necessarily imply unmediated action-at-a-distance.",
"The rest of this Colloquium is dedicated to an analysis of the possibility of reformulating QM in a “locally-mediated” manner, consistent with both CA and Lorentz covariance.",
"Recall that a “reformulation” here means a model in agreement with QM, with the same inputs $I$ , the same outputs $O$ , and the same model-generated joint probabilities $P_I(O)$ .",
"In preparation for this, the first subsection below proposes a categorization scheme for all models in agreement with QM, and the second clarifies relevant issues of causation and signaling."
],
[
"Categories of Bell-compatible reformulations of QM",
"As stated, Bell's Theorem dictates that no model in agreement with QM can respect Local Causality, which is the conjunction of two assumptions: NFID (No Future-Input Dependence) and BSA (Bell's Screening Assumption).",
"Reformulations of QM must thus violate at least one of these in some non-trivial manner, such that the CHSH inequality can also be violated.",
"Of these two assumptions, we argue that the primary one for categorization purposes should be NFID, because it is often taken for granted, and because the motivation for BSA in Section II included NFID, as depicted in Figure 1.",
"A useful secondary categorization is the CA condition, indicating whether or not action-at-a-distance is implied by a given model.",
"Bell's Theorem thus requires all models in agreement with QM to fall in one of the following categories: Type I: Respect NFID Type IA: Respect CA (Must Violate BSA) Type IB: Violate CA Type II: Violate NFID Type IIA: Respect CA (May Violate BSA) Type IIB: Violate CA (Models which violate CA necessarily also violate BSA.)",
"For convenience, the different Types of models are also identified in Table I.",
"Table: Categories of possible reformulations of QM (and the sections in which they will be discussed).",
"The columns identify whether or not a model conforms with the CA locality condition, and the rows refer to the NFID arrow-of-time condition.",
"Bell's Theorem rules out the subset of Type IA models which conform also to the stricter BSA (Bell's Screening Assumption) locality rule (see Fig.",
"1 above).",
"In the following, we will focus on locally-mediated models, which are of Type IIA.From the definition of NFID, Type I models allow for the calculation of all spacetime-based parameters in temporal order, using inputs that enter into the calculation in that same order.",
"But because of the necessary BSA violation, such models cannot adhere to the light-cone-constrained Cauchy problem typically found in classical physics.",
"Type IA models would have to avoid CA-violation using faster-than-light mediators, bypassing the screening region $\\bf {S^{\\prime \\prime }}$ of Figure 1(c) but passing through the larger screening region $\\bf {S^{\\prime }}$ of Figure 1(b).",
"Such models have not been formally developed, but have been promoted by various authors, including bell1981 himself.",
"In order for this not to violate NFID in a different reference frame, one might propose a special frame in which the model uniquely applies, at the expense of Lorentz covariance.",
"Moreover, to maintain agreement with QM in all cases, it is necessary that the mediating signal should always pass through $\\bf {S^{\\prime }}$ even if this region is blocked, say by a brick wall.",
"For these reasons, we judge such models to be of less interest, and our use of the term “locally-mediated” will exclude such faster-than-light unblockable mediators.",
"The Type IB category includes the standard Schrödinger-picture QM itself, as well as less conventional approaches such as de Broglie-Bohm guiding-waves [24].",
"Such models utilize mathematical intermediaries $R$ to connect distant spacetime-based parameters.",
"Recall that the model parameters ($I$ , $Q$ ) are defined as associated with particular places and times.",
"Values in $R$ might be associated with multiple spacetime locations in some non-separable manner, and do not generally have a form such as $R(\\mathbf {x},t)$ .The efforts of norsen2010,stoica2019 aim to overcome this, which could lead to Type IA models.",
"The most prominent example of a parameter $R$ is the many-body wavefunction, which for an entanglement setup is defined on configuration space.",
"Such an account involves no parameters ${\\lambda }$ in the relevant space-time regions, directly violating both BSA and CA [Eqn.",
"(REF ) becomes a simple product, with ${\\lambda }$ representing a constant, the empty set].",
"The role of the wavefunction in producing the predictions of QM might be described as an abstract mathematical object connecting events in spacetime.",
"Type II models violate the NFID assumption, so they are not temporally-sequential calculations.",
"It is natural to call such models Future-Input Dependent, or FID models.",
"With well-known examples such as the stationary action principle, it is clear that FID models cannot be trivially dismissed, and yet they are rarely brought up in discussions of Bell's Theorem.One factor which surely contributed to this is that Bell himself did not mention the FID possibility in any of his publications [14], see Section REF .",
"This omission continued in several major reviews, goldstein2011 and shimony2017, although the latter has recently been updated with a recognition of retrocausation [70].",
"Type IIA models are particularly interesting because they do not involve action-at-a-distance, in the sense that the screening condition of CA is respected.",
"Furthermore, since NFID is already violated, Bell's Theorem does not rule out the possibility of even retaining the stricter BSA locality condition.",
"As we shall see, in a Type IIA model, the mediation between spacelike regions can take place entirely on timelike worldlines.",
"Instead of the unblockable faster-than-light mediators of Type IA models, all relevant parameters in Type IIA models can be associated with the actual particle histories, allowing Lorentz covariance to be preserved.",
"This in turn means that it is possible to build models without any abstract mathematical structures $R$ “mediating” events in conventional spacetime.",
"Of course, one could still use such structures if desired – say, by retaining the conventional QM configuration-space wavefunction in a model.",
"This would fall into the category of Type IIB models which violate the intuitive NFID condition while restoring neither CA nor BSA.",
"The next two sections will be devoted, respectively, to a review of the specific achievements of the Type IIA toy-models which have already been developed, and to a discussion of the drawbacks and promise of this category of models.",
"The other categories, as well as approaches which do not fall within the framework used here, will be discussed in Section .",
"Before this review, some additional clarifications are necessary, to which we turn next."
],
[
"Causality and locality",
"The failure of Local Causality implied by Bell's Theorem leads naturally to the question: In what sense, if at all, does Local Causality correspond to assumptions of locality and causality?",
"Before continuing, it is necessary to clarify these issues."
],
[
"Cause and effect",
"The definition of NFID in Section REF uses the distinction between input- and non-input-parameters, rather than the words “cause” and “effect.” Nevertheless, the NFID condition is closely related to a definition of causality which arises naturally within the modern account of “interventionist” causation, where causes are identified as interventions [81], [120].",
"If the input parameters in question are deemed to be controllable parameters, then it is appropriate to identify them as causes, according to this account.",
"QM itself clearly adopts this connection between inputs and controllable parameters: the mathematical formalism of QM is a procedure for making operational predictions for observations, given the values of the controllable inputs.",
"As our goal is to discuss models in agreement with QM, it is natural for us to adopt this approach.",
"Such models limit the inputs $I$ to the parameters that QM tells us can be externally controlled.",
"Given this connection between “controllable inputs” and “causes”, one can identify different possible causal structures.",
"In models that respect NFID, non-input parameters are typically functionally dependent on past inputs, but are always functionally independent of future inputs.",
"This “forward-causal” structure is clearly what Bell had in mind when he used the terms “causality” and “causal structure,” with the controllable inputs called “free variables” or “free elements” [19], [20].There is an early exception: bell1964 used “causality” to imply “complete causality,” i.e., determinism.",
"FID models, on the other hand, do not have a forward-causal structure.",
"In other words, they cannot generally compute a given parameter $q(t^{\\prime })$ (or its probability distribution) without specifying certain inputs in the future of $t^{\\prime }$ .",
"In the framework of interventionist causality, if those future inputs are controllable, the FID models are “retrocausal”.The word “retrocausal” conventionally implies there are some future causes of some past parameters, not a purely-reverse-causal structure.",
"Some FID models, such as classical action principles, are not retrocausal.",
"In those cases, the final boundary constraints are required mathematical inputs, but not controllable inputs, and so are not considered causes.",
"Analysis of the causal structure of such a theory requires inverting the functional relation between some of the inputs and some of the outputs, so that a different model is obtained—a model in which all inputs are controllable.",
"Although it makes sense to refer to the action principle itself as a reformulation of Newton's equations, it is only after this inversion that one obtains a model fully in agreement with the standard operational description of classical mechanics, which uses the controllable initial conditions as inputs.",
"At the time of Bell's work, the interventionist approach to causation had not yet been well-developed.",
"An older approach was taken for granted, dictating that if two parameters exhibit cause-effect correlations, it is appropriate to refer to the one earlier in time as a cause, and the later one as an effect, regardless of which one can be externally controlled.",
"This is one topic where one's definition of causation directly impacts the types of mathematical models that one views as acceptable.",
"Applied to the $\\mathbf {\\lambda }$ -independence condition, any violation of Eqn.",
"(REF ) would be viewed as retrocausal in the framework of interventionist causation, an instance of FID.",
"But if one instead assumed that $\\lambda $ was the cause of the settings $a,b$ , because $\\lambda $ occurs before $a,b$ were chosen, one would have to conclude that the settings were effects, and could not be treated as free inputs (see footnote REF above and Section REF below).",
"The model would then not be in agreement with QM."
],
[
"Signals",
"Just as QM restricts the inputs $I$ to be controllable, it also specifies that the outputs $O$ are observable.",
"If $I$ is controllable and $O$ is observable, $P_I(O)$ summarizes all possible signals.",
"And as QM does not allow signals to be sent back in time, it follows that for models in agreement with QM the outputs $O$ cannot depend on future inputs.",
"We shall call this requirement signal causality, or explicitly, $P_I(O^{\\prime })=P^{\\prime }_{I^{\\prime }}(O^{\\prime }),$ where the primed sets of parameters are all those associated with times up to $t^{\\prime }$ , as in the similar Eqn.",
"(REF ).",
"Comparison with Eqn.",
"(REF ) indicates that any violation of NFID in a model in agreement with QM must be at the level of unobservable (hidden) parameters $U$ in $Q$ .",
"Such an FID model would be retrocausal (at a hidden level), but would not violate signal causality.If one demanded not only that “causes” are identified with controllable inputs but also that “effects” are identified with observable outputs, one would be led to take Eqn.",
"(REF ) as representing the causal arrow of time.",
"However, the term retrocausal in the literature does not signify violations of signal causality.",
"We use the more technical term NFID, which explicitly focuses on inputs, in order to minimize confusion.",
"Motivated by special relativity, it is natural to formulate a stronger restriction on signaling.",
"This condition, called signal locality, limits signals to traveling no faster than light, so that signals associated with a particular controlled input are limited to outputs in its future lightcone.",
"For outputs $O_1$ localized in region $\\bf {1}$ , the relevant inputs $I^{\\prime \\prime }$ should thus lie in the past lightcone of $\\bf {1}$ , and the signal locality requirement corresponds to the existence of a restricted model $P^{\\prime \\prime }$ such that $P_I(O_1)=P^{\\prime \\prime }_{I^{\\prime \\prime }}(O_1).$ This condition also holds in QM, and must thus be maintained for any model in agreement with QM.",
"As indicated in the introduction, these signal-based definitions of “locality” and “causality” are operational, in the sense of involving only controllable inputs and observable outputs.",
"Bell's Theorem states that models in agreement with QM must violate either a distinct notion of “locality” (BSA), or a distinct notion of “causality” (NFID) which are not defined operationally, as they refer to hidden model parameters, not signals.",
"Because of these different definitions, models can be local (or causal) in one sense, but not in another.The literature on Bell's Theorem involves quite a few additional “locality” conditions [see, e.g., wiseman2014], but these are not needed for the present discussion."
],
[
"Locally-Mediated Models of Entanglement (Type ",
"This section will discuss reformulations of QM which fall in Type IIA, meaning that they are “locally mediated” as discussed above.",
"These models conform to CA (Continuous Action) and are FID (Future-Input Dependent), allowing for compatibility with Bell's Theorem without a necessary conflict with Lorentz covariance.",
"As noted, such models are underrepresented in the literature on Bell's Theorem, so this section and the next will provide a rather thorough discussion.",
"The essential strategy behind Type IIA models of entanglement is to allow a violation of $\\mathbf {\\lambda }$ -independence, Eqn.",
"(REF ), such that $P_{a,b}(\\lambda )$ is not independent of the input settings $a,b$ .",
"The relevant $\\lambda $ lies in the past light-cones of $a,b$ , so that such models are technically “retrocausal” as defined in Section REF .",
"But as noted there, if agreement with QM is to be maintained, any correlations with future settings must be sequestered in hidden variables, not observable outputs.",
"By restricting attention to models in agreement with QM, there is no possibility of signals being sent back in time, and thus no concern of generating paradoxes.",
"These and other concerns with such models will be further discussed in the next section.",
"The promise of Type IIA models is that, in any given case, there exist parameters $\\lambda $ that can act as local mediators of the actual correlations.",
"It is always simple to find a distribution of shared parameters $\\lambda $ that will produce a given correlation for particular measurement settings; Bell showed that the problem was getting the same $P(\\lambda )$ distribution to consistently work for all measurement settings.",
"But for models $P_{a,b}(\\lambda )$ where the distributions can be different for different settings, Bell's consistency problem disappears.",
"This means that it is possible to retain BSA (Bell's Screening Assumption) in some FID models, or at least the weaker locality condition CA.",
"At the current stage of development of Type IIA models, there are none which are applicable to a wide range of quantum phenomena.",
"Existing models aim at reproducing merely the known correlations for the Bell state, Eqn.",
"(REF ).",
"Several will be presented below, with schematic models in the fist subsection, and a model providing a more detailed description in the second."
],
[
"Schematic models",
"Although the idea of using future-input dependence to explain entanglement had been around for a long time [10], [11], [12], [35], [82], [103], [85], [86], explicit Type IIA mathematical models of entanglement have appeared in the literature mostly in the last decade.",
"One notable exception is pegg1982, a description which could be simplifiedOne complication is that in its original form, the intermediate state appears to be output-dependent, rather than dependent on the future input setting.",
"and expressed in a manner quite similar to that of argaman2010, the model presented next.",
"Consider again the correlations between the polarizations of a pair of entangled photons.",
"Using the terminology of Section REF , where $a$ and $b$ represent the angle settings of polarizers, the spin-zero Bell state correlations can be obtained from the following toy-model.",
"First, take the two photons to both be initially polarized at an angle $\\lambda $ , distributed according to $\\begin{split}P_{a,b}(\\lambda ) = \\frac{1}{4} \\left\\lbrace \\frac{}{} \\delta (a-\\lambda ) +\\delta \\left(a+\\frac{\\pi }{2}-\\lambda \\right) +\\right.",
"\\quad \\\\ \\left.",
"\\;\\;\\;\\delta (b-\\lambda ) + \\delta \\left(b+\\frac{\\pi }{2}-\\lambda \\right) \\right\\rbrace .\\end{split}$ Here $\\lambda \\in [0,\\pi )$ and the $\\delta $ -functions are modulo $\\pi $ .",
"In this model, the initial polarization $\\lambda $ is thus somehow constrained by the future settings to be either $a$ , $a+\\pi /2$ , $b$ or $b+\\pi /2$ with equal probabilities, i.e., to be aligned with one of the detectors.",
"Next, apply Malus' law to obtain the results of the single-photon measurements, $A$ and $B$ [i.e., $P_a(A=1|\\lambda )=\\cos ^2(a-\\lambda )$ , etc.].",
"Combining these, using Eqn.",
"(REF ), reproduces the QM probabilities for the spin-zero Bell state, Eqn.",
"(REF ).",
"While this model is clearly very schematic, it demonstrates that only mediation along the spacetime paths of the particles is required."
],
[
"The Hall model",
"A number of additional Type IIA schematic models follow a similar strategy.",
"They consist of two components: (i) a specification of the sample space of the hidden variables and their distributions $P_{a,b}(\\lambda )$ , and (ii) models for the measurement outcomes, $p_{a,[\\lambda ]}(A)$ and $p_{b,[\\lambda ]}(B)$ , such that the combination of (i) and (ii) per Eqn.",
"(REF ) is in agreement with QM for a specific setup of interest.",
"(The notation $[\\lambda ]$ emphasizes that while $\\lambda $ is an input to the second component, it is not an external input.)",
"For want of space, we will provide the details of just one additional example, that of hall2010.",
"The version adapted to photon polarizations [5] has: $P_{a,b}(\\lambda ) = \\frac{1}{\\pi } \\,\\frac{1 + \\grave{A} \\grave{B} \\cos (2a-2b)}{1 + \\grave{A} \\grave{B} (1-z)} ,$ where $\\grave{A} = {\\rm sign}[\\cos (2a-2\\lambda )]$ , $\\grave{B} = {\\rm sign}[\\cos (2b-2\\lambda )]$ and $z=\\frac{2}{\\pi } |2a-2b|$ are abbreviations.",
"This model is deterministic in the sense that $A$ is fully determined by $a$ and $\\lambda $ through $p_{a,[\\lambda ]}(A)=\\delta _{A,\\grave{A}}$ , with the same expression relating $B$ to $b$ and $\\lambda $ .",
"It reproduces the results of QM for the Bell state, Eqn.",
"(REF ).",
"Here, knowledge of $\\lambda $ provides only a very rough idea of what $a$ and $b$ are.",
"When properly quantified, the information about $a$ and $b$ which can be gleaned from the past parameter $\\lambda $ amounts to less than 0.07 bits per entangled pair [52].",
"In this sense, one may view the toy-model of Eqn.",
"(REF ) as a dramatic improvement over that of Eqn.",
"(REF )."
],
[
"Additional toy-models",
"A large number of additional schematic entanglement models exist in the literature, the majority of which are Type IB models.",
"The original bell1964 work contained such a model for “illustration” purposes, and many others were developed over the years, relying on different “resources”: communication, shared randomness and/or nonlocal boxes [see, e.g., degorre2005, and references therein].",
"Each of these models proposes a novel distribution which may be denoted by $P_{a,b}(\\lambda )$ , and a way in which that $\\lambda $ generates the output statistics of QM, per steps (i) and (ii) above.",
"Simply associating $\\lambda $ with the world-lines of the entangled particles, rather than with the time of the measurements, can then change the Type IB model into a Type IIA model.",
"An example is given by barrett2011, who modified the model of toner2003,degorre2005 by “moving” $\\lambda $ to the past.",
"Other changes in the spacetime location of $\\lambda $ can affect the assessment of the NFID and CA conditions, leading to a new model of a different Type, even if the distribution $P_{a,b}(\\lambda )$ is unchanged.",
"For Type IIA models, taking $\\lambda $ to be associated with the emission event at the source but not with the particle worldlines—as arguably done in the machine-learning-generated models of weinstein2017,weinstein2018—formally results in a Type IIB model, with no local mediators.",
"But such models are easily transformed back into Type IIA, simply by copying $\\lambda $ onto mediators along both worldlines.",
"Alternatively, $\\lambda $ might be associated with the time of the measurements, rather than the emission or the worldlines; resulting in a model of Type IB.",
"Reinterpreting Eqn.",
"(REF ) in this manner leads to precisely the model of dilorenzo2012.",
"Yet another example is provided by the work of sen2018, who began with the model of brans1988 (itself obtained by associating the parameters of standard QM with the past) and explicitly transformed it into a Bohmian-style FID model.",
"The schematic Type IIA models above show both promise and limitations.",
"On the positive side, they all serve as proof-of-principle examples, indicating that Bell inequalities can be violated without also introducing action-at-a-distance, and they provide a variety of points of departure for future development.",
"With the mediating parameters $\\lambda $ associated with the particle worldlines, other advantages quickly become evident.",
"For example, a recent application [92] of a FID model to entanglement in accelerating reference frames indicates a nearly-trivial reconciliation of quantum phenomena and general relativity, for a case that is even quite difficult for quantum field theory.",
"On the negative side, however, these models all simply assert the connection between the settings and $\\lambda $ , without a proposed mechanism or explanation.",
"One natural justification for such a connection would be an appeal to time symmetry: one could argue that the symmetry exhibited by micro-scale phenomena implies an equal role for both past and future.",
"This would make the future settings $(a,b)$ just as important as the initial state preparation $(c)$ when modeling $\\lambda $ .",
"But this justification seems inapplicable because these schematic models do not possess time symmetry in any sense.",
"We now turn to a Bell-compatible FID reformulation which restores microscopic time symmetry, and does so in a manner that provides an account of both the $P_{a,b}(\\lambda )$ distribution and the outcome probabilities."
],
[
"The Schulman Lévy-flight model",
"Conventional QM is typically viewed as time-symmetric, but its intermediate calculations are notably time-asymmetric.",
"For example, consider the polarization of a photon which is known to have passed through two consecutive polarizers set at angles $\\theta _1$ and $\\theta _2$ .",
"The conventional description associates the angle $\\theta _1$ with the polarization of the photon between the two polarizers, but time-reversal symmetry implies that $\\theta _2$ should be just as relevant to the intermediate description.",
"Any time-symmetric account of the intermediate photon should therefore take both angles into account, and would be a Type IIA model.",
"Such a time-symmetric model has been developed by schulman1997,schulman2012, using a time-varying polarization angle $q(t)$ .Schulman's discussion of spin-$1/2$ particles is here adapted to photons.",
"Schulman considered the possibility that $q(t)$ could be perturbed by microscopic rotations $dq$ (“kicks”) so that $q(t)$ evolves from $\\theta _1$ to $\\theta _2$ (or $\\theta _2 + n \\pi $ ) between the polarizers, without requiring a collapse at the last instant.",
"If the magnitude of each microscopic kick is normally distributed (or has a finite second moment) one would obtain diffusive behavior, which is inappropriate.",
"However, if $q(t)$ describes a Lévy flight, e.g., if the magnitudes of the kicks are distributed according to the Cauchy (Lorentzian) distribution, $\\propto d\\gamma / [{(dq)^2+(d\\gamma )^2}]$ with a small width $d\\gamma $ , the net rotation $\\Delta q$ has a similar probability distribution: $P(\\Delta q) = \\frac{1}{\\pi } \\frac{\\gamma }{(\\Delta q)^2+\\gamma ^2},$ where $\\gamma $ is the sum of all the $d\\gamma $ widths of all the kicks along the path.",
"With $q(t)$ constrained to $\\theta _1$ at the time of the initial polarizer, $t_{\\rm i}$ , and to $\\theta _2$ at $t_{\\rm f}$ , the final time, $q(t)$ provides an appealing time-symmetric description of the dynamics (constrained by initial and final boundaries).",
"Moreover, and this is the main point of Schulman's derivation, the model correctly predicts the outcome probabilities for a single photon in the limit $\\gamma \\rightarrow 0$ , if the measurement acts as a boundary constraint corresponding to discrete possibilities, requiring the photon polarization to either be aligned or perpendicular to the polarizer angle (either $\\theta _2$ or $\\theta _2 + \\pi /2$ ).",
"Adding all the equivalent contributions corresponding to $\\theta _2 + n\\pi $ per Eqn.",
"(REF ) gives a result $\\propto 1/\\sin ^2(\\theta _1-\\theta _2)$ .",
"Comparing this to the other possible outcome, summing over $\\theta _2 + (n+\\frac{1}{2})\\pi $ , reproduces Malus' law upon normalization: the probability for a photon of initial polarization $\\theta _1$ to align with a polarizer oriented at $\\theta _2$ is $\\cos ^2(\\theta _1-\\theta _2)$ .",
"A detailed derivation can be found in the Appendix.",
"Note that for small $\\gamma $ the path $q(t)$ is very close to being a constant, but the initial and final requirements enforce at least one significant “kick”, with a distribution $\\propto d\\gamma / (dq)^2$ .",
"In the $\\gamma \\rightarrow 0$ limit, paths with a single kick dominate.",
"There is thus an event which corresponds to “collapse” in this description (unless $\\theta _1=\\theta _2$ or $\\theta _1=\\theta _2+\\pi /2$ ), but it happens at an arbitrary time between preparation and measurement, rather than at the time of the measurement, and thus respects time symmetry.",
"This model can be trivially extended to the case of two maximally entangled photons, by combining two copies of the single-particle model, $q_1(t)$ and $q_2(t)$ , and constraining their unknown initial polarization angles to be identical, $q_1(t_{\\rm i}) = q_2(t_{\\rm i})$ , [111], [3].",
"Identifying this initial polarization as the hidden parameter $\\lambda $ reproduces precisely the probability distribution of the simplistic toy-model above, Eqn.",
"(REF ).",
"This follows because the overwhelmingly most probable scenario is to have only one significant kick in the combination of the two paths, and this in turn requires $\\lambda $ to match one of the two future settings.",
"In this model, the screening region $S^{\\prime \\prime }$ from Figure 1(c) contains the parameters $q_1(t)$ .",
"No inputs on the other arm of the experiment can affect the probability of the measured outcome $q_1(t_{\\rm f})$ without also affecting the earlier values $q_1(t)$ , conforming to BSA.",
"(The earlier schematic models also respect BSA for similar reasons.)",
"The mechanism by which the correlations are enforced is NFID-violating: the future settings $(a,b)$ constrain the full histories $q_1(t)$ and $q_2(t)$ , including the possible initial value of the hidden parameter, $\\lambda $ .",
"This explicitly violates NFID and Eqn.",
"(REF ), violating Local Causality.",
"All locality conditions from Figure 1 are thus preserved.",
"The Schulman Type IIA model supplies a future boundary mechanism to explain the future-input dependence [an account missing from component (i) of the above schematic models, as noted there], and the very same mechanism provides the correct outcome probabilities [to explain component (ii)].",
"Indeed, this two-particle toy-model is currently the most sophisticated example of how a model can yield the correct Bell-state correlations while retaining the BSA (or the CA) condition of locality.",
"It demonstrates a spacetime-based mediation of the correlations involved in entanglement, via a mechanism that uses the entire history rather than instantaneous “states”.",
"By assigning probabilities to histories rather than states, this approach avoids the tension between entanglement and relativistic covariance.",
"It demonstrates how Type IIA models need not conflict with relativity (as noted already in Section REF ), because all of the mediation is by parameters that reside on timelike or lightlike worldlines.",
"If the relevant parameters $\\lambda $ reside on the classical worldlines of the entangled particles, this looks essentially similar in every reference frame, no matter which particle is measured “first.” It is striking that the same set of rules is applicable to both one-photon and two-photon setups, as explained above, and is also valid if additional measurements are considered.",
"For a single photon, it provides the appropriate Malus-law probabilities for any number of sequential polarization measurements.It thus provides a “natural” mechanism or explanation for violations of Leggett-Garg inequalities [59]; these “beyond-Bell” inequalities facilitate experimental demonstrations of additional surprising quantum phenomena.",
"Again, the Type IIA approach describes a relationship between microscopic hidden variables and macroscopic observable results which appear quite perplexing from a “macro-realistic” NFID-assuming point of view.",
"For the two-photon entanglement setup, as the hidden parameter $\\lambda $ is associated with the photon polarization, it is natural to ask whether an additional measurement of this polarization along the path of the photons could shed light on the mechanisms involved.",
"QM itself describes how this would fail—after such a measurement, the two photons will no longer be entangled.",
"The Schulman model successfully describes this: the additional measurement would be associated with another boundary constraint, changing the entire history of the experiment, and requiring two “significant kicks” instead of one, reproducing again the often-perplexing results of standard QM.",
"The two-particle Schulman model can also be trivially generalized from the spin-zero state to any maximally entangled two-qubit state by performing polarization rotations on one of the two photons.The strategy of reducing a two-particle entanglement problem to two single-particle problems can be extended to all maximally-entangled bipartite states [115].",
"Further generalizations to scenarios with several particles [22], [78] might no longer respect BSA if some of the correlating parameters are localized on connected zigzagging worldlines (an entanglement-swapping setup), but would continue to respect CA and would still be Type IIA.",
"The challenge of extending this type of model to partially-entangled states remains an open problem."
],
[
"Discussion",
"In this section, we first address existing criticism of the Type IIA approach, and then discuss its potential and directions for future exploration."
],
[
"Objections to Type ",
"Despite the availability of the simple models presented above, much of the contemporary discussion of Bell's Theorem fails to recognize such a possibility.",
"For example, in the recent round of loophole-free experiments [56], [47], [94], [89], not one article mentioned the possibility of Type II or FID (Future-Input Dependent) models.",
"In the rare case where experimental papers mention a retrocausal option, it is typically relegated to a mere footnote [53], [88].",
"With this lack of attention, there are few published concerns about Type II models in the recent literature, although a number of “intuitive” objections are likely to occur to most physicists upon first encountering these models.",
"The most common such concerns will be addressed first, followed by a discussion of specific formal arguments which have appeared in the literature.",
"One common objection to FID models is that they violate some unwritten principle of “causality”.",
"Formalizing this objection is difficult, but one evident concern is that such models might lead to logical difficulties with time-travel paradoxes.",
"But time-travel paradoxes require communication with the past, with at least some level of observable signal, and this is forbidden in models in agreement with QM which conform to signal causality, Eqn.",
"(REF ) above.",
"For any FID model in agreement with QM, the future-input dependence is always at the level of the hidden parameters, $\\lambda $ , and as there is no protocol for observing the values of these parameters (without changing the whole setup), such models do not allow retro-signaling.",
"Another common concern is that FID models imply future inputs must “exist” to constrain hidden parameters in the past, and some find this block-universe view problematic [98], [58].",
"But it appears ill-advised to avoid developing a theory for such reasons—it would have been a pity, for example, if Newton were to avoid developing the Law of Universal Gravitation because he perceived its nonlocality to be unacceptable.",
"Furthermore, treating future events as valid model parameters and analyzing entire spacetime regions “all at once” is common in physics, e.g., in general relativity and with Wick rotations.",
"And in any case, one can always wait until the whole relevant spacetime region is in the past, and perform the model analysis retrospectively.",
"We set aside this objection as an essentially anthropocentric restriction on mathematical models [112].",
"As a related objection, some might take the view that because QM conforms to signal causality, and so do all other established physical theories, there should never be any reason to consider FID reformulations of QM.",
"However, as we have seen above, the failure of Local Causality provides just such a reason.",
"Bell's Theorem does not formally tell us whether it is the “locality” (BSA) or “causality” (NFID) aspects of our models which require adjustment, so we should seriously consider both options rather than simply choosing the one we take to be more plausible.",
"And again, such a restriction is routinely ignored by physicists in practice.",
"Histories approaches such as griffiths2001, and path integrals in general, encourage one to consider the past and future together as a single structure, violating the spirit of NFID.",
"In Heisenberg-picture QM, measurement operators are often evolved back in time to the previous measurement.",
"And some analyses of “delayed-choice” experiments, such as that of bohr1935 briefly described in Section REF , allow one to make incompatible inferences about past events for different future measurement choices.",
"If those past events are parameterized, this also violates NFID."
],
[
"Formal objections",
"An early technical argument against FID models is due to maudlin1994.",
"Adapting it to the above Bell-state setup, consider the case where one measurement is performed early enough so that the result $A$ can be sent ahead of the other particle (say, via a laser signal) to the other measurement device.",
"This output parameter $A$ could then be used to determine the other setting $b$ , via some algorithm $b=f(A)$ .",
"The challenge is one of self-consistency: if one uses a model that requires $b$ as an input to generate $\\lambda $ , and then uses $\\lambda $ to generate the outcomes $A,B$ , the function $f(A)$ might be found to disagree with the value of $b$ utilized in the calculation.",
"This is of particular concern for the schematic models designed with one experiment in mind (such as those in Section REF above), because this is an essentially different experimental configuration.",
"But it is unreasonable to expect precisely the same model, with the same inputs and outputs, to apply to this new configuration.",
"In this version of Maudlin's challenge, the setting parameter $b$ is no longer an input to the model (it cannot be freely set), so an analysis of this new experiment would require a Type IIA model of the form $P_a(Q)$ , rather than the original $P_{a,b}(Q)$ .",
"The Schulman model of Section REF is general enough to handle this new configuration, because the boundary constraints imposed by the future measurements are still enforced in the global solution, no matter whether the settings are free inputs or calculated parameters.",
"So long as the solution is calculated “all at once”—assigning probabilities to entire histories rather than states—every intermediate solution is self-consistent, by definition [23], [63], [111].In general, the agreement-with-QM status of the original $P_{a,b}(Q)$ guarantees through signal causality that its operational version, $P_{a,b}(O)$ , can be restricted to times up to the first measurement, yielding $P^{\\prime }_a(A)$ ; subsequently, the full applicable model can be reconstructed: $P_a(Q) = \\sum _A P^{\\prime }_a(A) P_{a,f(A)}(Q|A)$ .",
"A more recent objection, that applies even to all-at-once accounts, has appeared in wood2015—although, notably, this stands as an objection to all accounts of entanglement phenomena, not specifically Type IIA models.",
"The essential point is that causal channels are typically accompanied by signal channels, absent some special “fine tuning” of the underlying model.",
"Such fine-tuning would require additional explanation.",
"In any causal account of entanglement, such as the faster-than-light option of Type IA models, signal locality (the inability to send a spacelike signal) must be the result of some perfect cancellation in the marginal probabilities.",
"This is said to be “fine-tuned” because even a slight deviation would lead to spacelike signaling.",
"For example, in Quantum Field Theory, it is the perfect commutativity of spacelike-separated operators which guarantees the necessary “fine-tuning.” The situation might appear to lead to an additional challenge to Type II models, with causal channels into the past, because another fine-tuning argument can be applied to signal causality (the inability to send signals into the past).",
"But a more careful analysis reveals that the fine-tuning objection is not significantly worse for Type II models than it is for Type I models, because spacelike signaling violates signal causality in some reference frame.",
"Further analysis of the Schulman model has revealed that the appearance of signal locality follows from a basic symmetry [3], providing just the sort of explanation (from symmetry) that is most often used to explain fine-tunings in high-energy physics.",
"A more comprehensive explanation of both signal locality and signal causality has also recently been proposed by adlam2018a.",
"Clearly, finding mathematical/physical principles underlying these signaling restrictions will be an important challenge for future reformulations of QM.",
"There is also a flip-side to the Wood-Spekkens fine-tuning argument.",
"If an underlying physics model indeed breaks time symmetry according to the NFID condition, it would take a very finely balanced restriction to make microscopic physics look as time-symmetric as it does.",
"leifer2017a weigh this argument against the Wood-Spekkens fine-tuning argument, and propose that the time symmetry argument is stronger.",
"The examples of Section demonstrate that a number of Type IIA models can successfully account for the Bell-state correlations.",
"Thus, Bell's Theorem cannot be said to stand in the way of a locally-mediated reformulation of QM.",
"In particular, the Schulman model admirably achieves a description in agreement with QM which conforms to CA (Continuous Action), employing exclusively spacetime-based parameters with local, time-symmetric interconnections, which pose no difficulties for Lorentz covariance.",
"A further dramatic advantage of such models relates to the exponential complexity of quantum states.",
"By using only spacetime-based parameters $Q$ , the model $P_I(Q)$ has an evident physical interpretation: it specifies the probability of each possible set of events in spacetime $Q$ , while only one particular configuration actually occurs.",
"This is analogous to the Liouville equation in classical mechanics, where the statistical distributions can be exponentially complex, but only one phase-space configuration is taken to represent an actual physical system (even when we do not know which).",
"The complexity of this actual configuration scales linearly with the number of particles or the size of the modeled spacetime region.Time plays a different role in the context of the Liouville equation, as within classical dynamics the configuration at one time determines the whole path.",
"The Schulman model provides a simple example of such linear scaling, in that the parameters required for a two-particle experiment are merely two copies of the single-particle case.",
"The exponential growth of the conventional wavefunction $\\psi (t)$ with particle number might lead one to think that achieving such linear scaling would be impossible, especially if one viewed the information contained in $\\psi (t)$ as some physical entity which had to be translated into parameters $Q(t)$ (the subset of $Q$ pertaining to a time $t$ ).",
"But note that $\\psi (t)$ contains information about all possible measurement outcomes which might occur, for all possible future measurement settings.",
"In an FID model, $Q(t)$ can be a function of those future settings, and therefore need only inform the outcomes for the actual future measurement, vastly reducing the required number of parameters [for further analysis, see wharton2014].",
"Beyond Bell state correlations, there are plenty of other quantum phenomena that must be addressed to approach a full reformulation of QM.",
"Single-particle interference appears challenging, but may be resolved in a Type IIA model by adopting a field-based rather than a particle-based viewpoint [114].Particle-like phenomena could arise from the discreteness of measurement interactions (the detector “clicks”), enforced by boundary constraints, not by discreteness of the parameters.",
"Recent Type IIA models have tackled other issues, including position measurements of entangled particles [93], and formal relativistic covariance [110], [55], [104].",
"Presumably more models will be developed in the near future, addressing additional issues such as 3-particle and partial entanglement phenomena.",
"There are many avenues which could be pursued in searches for such models.",
"Existing reformulations, such as Stochastic Mechanics [71], [72] and Stochastic Quantization [37], could perhaps provide excellent starting points.",
"There are also some recent efforts which first evaluate the probabilities for the outcomes, $P_I(O)$ (using one of the standard methods of QM), and then define additional mediating parameters so that overall the resulting model is Type IIA [both sutherland2017 and drummond2019 can be read in this manner].",
"While such approaches may claim applicability to a wide range of quantum phenomena, in our view, additional development is necessary for these models to fulfill their promise, such that the mediating parameters explain the outputs rather than the other way around.",
"There are quite a number of additional results in the literature which should guide the development of Type IIA locally-mediated models.",
"Many of these have been developed in the context of locality [for a review, see brunner2014; a recent example is carmi2018].",
"A potentially important result which explicitly questions the arrow of time has recently been proven by shrapnel2017.",
"By dropping the usual NFID assumption, their analysis indicates that such models must be “contextual,” meaning that distinct hidden-parameter accounts would be required for situations not distinguished by standard QM.",
"While it is not unreasonable to expect the details of intermediate hidden parameters to depend on the detailed intermediate context, it still raises the question of why standard QM cannot distinguish these differences.",
"This might indicate the development of models with inherent hidden symmetries, where this contextuality could seem more natural.",
"Eventually, Type IIA models must also provide a satisfactory treatment of quantum measurements, but at the present stage of development this goal is not yet clearly in sight.",
"Still, the Type IIA Schulman model improves upon standard Schrödinger-picture-with-collapse QM in two ways.",
"First, measurements do not correspond to any sudden collapse, so they look more like an ordinary interaction (the collapse-like event occurs somewhere between preparation and measurement).",
"Second, there is no confusion about whether the size of the relevant configuration space should expand (as in a QM interaction) or be reduced (as in a QM measurement), because nothing lives in configuration space; all parameters are associated with spacetime.",
"A future Type IIA theory should provide an explanation for why the interaction between some large systems (measurement devices) and some smaller systems (such as the measured particles) can be described effectively by imposing boundary constraints on the smaller systems.",
"It is worth noting that such behavior is evident near large conductors in electromagnetism and thermal reservoirs in classical thermodynamics.",
"It is also well-known that smaller systems exhibit an evident time-symmetry in a way that larger, thermodynamic systems do not.",
"Understanding this is particularly important if time-symmetry is used as justification for introducing FID, because this symmetry must somehow give way to the asymmetric signal causality at larger, observable scales.",
"Taking the Schulman model as an illustrative example, the only time-asymmetry enters via a subtle distinction between photon preparations and photon measurements.",
"Both of these have controllable settings, but preparations have an additional point of control: the initial polarization is also treated as an input.",
"(For the case of entanglement, the initial correlation between two polarizations is an input.)",
"In contrast, the measurement does not allow this same level of control; one can input the final polarizer angle, but not the measured polarization (the latter is an output, not an input).",
"This empirically-based distinction between full control at preparations and mere setting control at measurements provides the symmetry-breaking mechanism which leads to the appearance of signal causality in the model.",
"Everything else about the model respects time symmetry – most notably, the intermediate account between preparation and measurement.",
"It is possible to attribute this distinction between preparation and measurement to the involvement of macroscopic “agents,” who have control of some quantities but not others [86].",
"Alternatively, one may attempt to include a description of the measurement process itself in the mathematical model.",
"Due to the observation that quantum measurements must have irreversibly recorded results [see, e.g.",
"miller1996], one should not expect a completely time-symmetric model to achieve this.",
"Future research into this issue may look in detail at the effects of a thermal environment, which could be included in a Type IIA description.",
"In both classical and quantum cases, such treatments break time symmetry by fixing the initial states of the environment (averaging them over a known thermal distribution), while leaving its final states to be computed [see, e.g., feynman1963].",
"For an appropriate interaction between the system's degrees of freedom and the environment, the information regarding the values of some of the parameters pertaining to the “measured” system, $Q_M$ , are effectively amplified and copied many times in the final state of the environment [see, e.g., zurek2003].",
"An intriguing possibility, called “lenient causality” in argaman2018, is that the time-symmetry-breaking in models of this type could impose signal causality for parameters such as $Q_M$ , without leading to NFID for the microscopic parameters.Better still, it could lead to a condition such as Information Causality [79], which is known to essentially guarantee compliance with the Tsirelson bound [31]."
],
[
"Alternatives and Misconceptions",
"While the previous two sections have discussed Type IIA (locally-mediated) models in detail, there are many other models in the literature which can reproduce the experimentally observed CHSH-violations, including of course the existing formulations of QM.",
"This section will briefly discuss each of the general possibilities, giving references to some of the approaches not reviewed here.",
"These either violate Local Causality in some different way, as categorized in Section REF , or else they fall outside our framework, i.e., they are not in agreement with QM as defined above.",
"Note that a specific approach can lead to a variety of models, and that a model must be fully specified in order to allow for a clear categorization.",
"(For example, as we have already seen in Section REF for the schematic models, a change in the spacetime location associated with $\\lambda $ is sufficient to change the Type of the model.)",
"Below, we devote a subsection to each of the possibilities, and a final subsection to some misconceptions that might lead one to mistakenly believe there were additional categories of models."
],
[
"Type ",
"Type I models have no parameters that are dependent upon future inputs.",
"In Section such models are categorized into Type IA which would have faster-than-light mediators, and Type IB in which distant regions can directly influence each other via non-spacetime-based (mathematical) intermediaries, such as the configuration-space wavefunction of conventional Schrödinger-picture QM.",
"The many-body wavefunction also enforces distant correlations in other Type IB approaches, including Bohmian mechanics [24] and spontaneous-collapse models [45] (which only achieve full agreement with QM in an appropriate limit).",
"Development of such models continues, e.g., with so-called “flash” models, which have parameters in spacetime (the flashes), but no intermediate screening parameters [106].",
"As noted in Section , no representative Type IA model has been formally developed [norsen2010 might be the closest].",
"spekkens2015 has noted that one can convert standard QM into a corresponding Type IA model by introducing “local copies” of the wavefunction $|\\psi (t)\\rangle $ at every point in space with time coordinate $t$ ; “collapse” due to a distant measurement would then instantaneously update all of these new spacetime-based parameters.",
"Information is thus transferred from one region to another at an infinite speed, bypassing the $S^{\\prime \\prime }$ region of Figure 1(c), while passing through the upper boundary of the region $S^{\\prime }$ in Figure 1(b).",
"Whichever Type I technique one uses to enforce correlations across spacelike separations, such a connection makes it difficult to achieve Lorentz covariance, even when signal locality is satisfied.",
"In such models, when entanglement correlations between regions $\\mathbf {1}$ and $\\mathbf {2}$ are described, some observers see $\\mathbf {1}$ affecting $\\mathbf {2}$ , while other observers see $\\mathbf {2}$ affecting $\\mathbf {1}$ .",
"These descriptions do not properly transform into each other under Lorentz transformations,The requirement here is not only that individual parameters transform covariantly, but that the overall description of which events affect which be consistent among different frames; see, e.g., gisin2010.",
"motivating the possibility of omitting them altogether, resulting in a purely operational model, with just $P_I(O)$ .",
"Despite these difficulties, it is clear that Type I models are overwhelmingly represented in the relevant discussions in the literature."
],
[
"Type ",
"While the previous sections have focused on Type IIA models with spacetime-based mediators, other Future-Input Dependent models can include non-spacetime-based entities, directly linking distant regions.",
"Such Type IIB models often use configuration-space wavefunctions, in addition to their spacetime-based parameters.",
"Many of the above concerns about Type I models (failure of Lorentz covariance, non-local influences, etc.)",
"are therefore applicable to Type IIB models as well.",
"One popular Type IIB model is the Two-State-Vector Formalism introduced by aharonov1991, which essentially doubles the state space of conventional QM.",
"For single-particle cases, it adds to the ordinary wavefunction $\\psi (\\mathbf {x},t)$ another wavefunction $\\phi ({\\mathbf {x},t})$ , a solution of the Schrödinger Equation which is determined by the setting and the outcome of the next strong measurement on the particle (essentially a future boundary constraint).Another similar example is the Transactional Interpretation [35], [36], where the individual “confirmation” waves correspond to $\\phi $ .",
"While these are naturally interpreted as spacetime-based parameters, for entanglement scenarios the relevant state vectors are conventional configuration-space wavefunctions, $\\psi (\\mathbf {x}_1,\\mathbf {x}_2,t)$ and $\\phi (\\mathbf {x}_1,\\mathbf {x}_2,t)$ , and these entangled two-particle wavefunctions cannot easily be mapped onto spacetime-based fields.",
"These wavefunctions are not spacetime-based, but are at least time-based parameters, and in this generalized sense, they exhibit a violation of the essential ideas behind NFID.",
"Having departed from spacetime, they no longer have any localized screening parameters, and so violate the CA locality condition [see also vaidman2013].",
"It is therefore fair to categorize such a model as Type IIB."
],
[
"Models outside the framework",
"Various approaches in the literature raise more exotic possibilities, essentially claiming to not fall under any of the 4 model Types listed in Section REF .",
"These approaches depart from our framework (Section REF ), either by violating the rules of probability theory, or by dropping aspects of the requirement of agreement with QM.",
"The latter models risk losing the empirical content of QM, i.e., the comparison of $P_I(O)$ to experiment.",
"In order to still claim some form of agreement with QM, the $P_I(O)$ predictions must be recovered, at least at an effective level.",
"At that effective level, such models always fall within one of the Section REF model Types.",
"One example is the Many Worlds Interpretation [42], sometimes claimed as a way to avoid Bell's Theorem because all possible measurement outcomes are represented in a never-collapsed wavefunction.",
"In this approach, the measurement problem is avoided by removing the Born rule from the fundamental description, but then the empirical success of QM, the $P_I(O)$ , is removed as well [66].",
"Proponents of Many Worlds would argue that at an effective level, a version of the Born rule is still applicable, but the result is then a Type IB effective model, in the same category as conventional QM.",
"The deviation from our model framework which appears most frequently in the recent literature [perhaps because it was discussed repeatedly by bell1981,bell1990,bell1977] is “superdeterminism,” which retains the implicit NFID assumption while considering violations of the “$\\mathbf {\\lambda }$ -independence” assumption, Eqn.",
"(REF ).",
"This cannot be done within our framework (Section REF ), which treats the measurement settings $(a,b)$ as input parameters, corresponding to the mathematical concept of free variables.",
"But if the settings are treated as statistical parameters, $\\mathbf {\\lambda }$ -independence becomes $P_{c}(\\lambda |a,b)=P_c(\\lambda ),$ where $c$ encodes the free preparation setting, still treated as an input.",
"This is a statistical-independence relation, and permits a Bayesian inversion to an equation sometimes known as the “no conspiracies” assumption: $P_{c}(a,b|\\lambda )=P_c(a,b).$ Violations of this condition can then be pursued, by expanding $\\lambda $ (or using additional variables) to include the systems that choose the measurement settings.",
"This approach has been seriously considered in the literature [e.g., thooft2016], despite the fact that it is only coherent if it makes sense to talk about the probabilities of the settings, $a$ and $b$ .",
"But such probabilities cannot be defined without creating a conflict with standard QM, where $a$ and $b$ are free inputs.Note also that the original suggestion, shimony1976, aimed only to emphasize the importance of the free-variable assumption, and argued that scientific exploration necessarily involves the assumption that “hidden conspiracies of this sort do not occur.” The reply of bell1977 observed that even if the settings were chosen by a mechanical pseudorandom generator which could be included in an enlarged model, they would still be “effectively free for the purpose at hand.” Indeed, in the explicit superdeterministic toy-models which have been proposed for the Bell-state correlations, the relevant hidden variables [$\\lambda _0$ in brans1988, and $\\mu $ in hall2016] are simply copies of the measurement setting parameters $a$ and $b$ , transferred to earlier times.",
"The other elements of these models prescribe a specific form of $P_{a,b}(\\lambda )$ and a role for $\\lambda $ in generating the outputs, as discussed in Section REF .",
"In practice, therefore, explicit superdeterminstic models which agree with QM are forced to treat the future settings $a$ and $b$ as free inputs.",
"Once this is acknowledged, the model again falls within the framework, and its Type can be identified.",
"There are additional well-established methods which can be more spacetime oriented, but do not meet the probability rules of our framework.",
"For example, path-integral accounts of QM utilize spacetime-localized paths.",
"It might be tempting to think that each path might be represented by a set of parameters $Q$ , but the path integral cannot be parsed into normalized probabilities $P_I(Q)$ where only one path $Q$ can be taken to exist.Introducing an FID viewpoint, along with a different parsing of $Q$ , might potentially resolve this problem [113].",
"Similarly, Quantum Field Theory (QFT) can be viewed as assigning a complex amplitude to all possible field configurations in spacetime, but each of these configurations cannot be assigned a probability.",
"A further example is given by the consistent histories approach [e.g., griffiths2011], where the probability rules for the intermediate description $P_I(Q)$ are changed, while those for the outputs, $P_I(O)$ , are not.",
"These approaches represent directions which are, in a sense, more radical than the search for Type IIA models."
],
[
"Misconceptions",
"It has often been claimed that Bell's Theorem is based on additional assumptions not identified above, including determinism and realism.Once the discussion is cast purely in terms of mathematical models as done here, assumptions of “realism” can play no role [see norsen2007 for a discussion in a wider context].",
"Note that when “realism” is taken to imply that systems have properties prior to measurements, the NFID assumption is again being taken for granted, assuming not only that the systems have “objective” properties, but also that these properties are independent of the settings of future measurements.",
"These erroneous claims are already well-addressed in the literature [73], [75], [77], [66], [67], but some clarifications will be repeated here in order to alert the reader to some of the many controversies in the literature.",
"Bell did not originally present his proof as outlined in Section REF ; this unified approach only came later.",
"The EPR paper (see Section REF ), had already demonstrated that certain perfect correlations between distant measurements clearly violate Local Causality, unless one adds deterministic hidden parameters.",
"bell1964 built upon this result, and showed that even with deterministic hidden parameters Local Causality could not be saved, as other predictions of QM could not be obtained.",
"Unfortunately, the argumentation of EPR contained several additional elements which made it appear paradoxical even before Bell's work, and the notion that bohr1935 had refuted it was widespread [see, e.g., clauser1969].",
"As bell1964 did not go through the EPR part of the argument in any detail [118], [76], many have concluded that the implications could be avoided by not postulating hidden parameters in the first place, or by not requiring them to be “deterministic” (or “realistic”, or “counterfactual definite”, etc.).",
"But such moves do not save Local Causality, for the reasons given in the EPR paper.",
"Bell himself later wrote: “It is remarkably difficult to get this point across, that determinism is not a presupposition of the analysis” (Bell, 1981; emphasis in original).The original derivation of the CHSH inequality [34] simply assumed deterministic hidden parameters, without using the EPR argument.",
"It was rapidly understood that the same inequality also holds for indeterministic local hidden-variable models [see footnote 10 of bell1971, clauser1974, or the unified type of proof as in Section REF ], but this is often ignored.",
"It is hoped that the explicit discussion of the framework and assumptions in the present work will help alleviate such difficulties."
],
[
"Conclusions",
"This Colloquium began by carefully framing the assumptions that lead to Bell's Theorem, in terms of input-parameters $I$ and non-input parameters $Q$ , both associated with locations in space and time.",
"By defining a model in terms of the probabilities $P_I(Q)$ which it generates, Bell's Theorem indicates that any such model which is in agreement with QM must violate one of the original assumptions, one of the components of Local Causality.",
"This allows a natural categorization of all possible reformulations of QM, as described in Section REF .",
"To the extent we require the parameters in our mathematical models to correspond to physical events, this Local Causality violation is quite significant.",
"Einstein described the physical justification for Local Causality in a 1948 letter [26]: If one asks what, irrespective of quantum mechanics, is characteristic of the world of ideas of physics, one is first of all struck by the following: the concepts of physics relate to a real outside world...",
"It is further characteristic of these physical objects that they are thought of as arranged in a space-time continuum.",
"An essential aspect of this arrangement of things in physics is that they lay claim, at a certain time, to an existence independent of one another, provided these objects `are situated in different parts of space'.",
"The following idea characterizes the relative independence of objects far apart in space (A and B): external influence on A has no direct influence on B ...",
"But Bell showed that this line of thinking leads to limitations on distant correlations which are in direct conflict with QM.",
"The outcomes of spatially-separated experiments are correlated in a manner which cannot be explained only in terms of common past inputs.",
"Still, it does not follow that our only option is to throw out the entirety of Einstein's analysis, giving up on “physical objects... arranged in a space-time continuum”.",
"At least one of the assumptions that make up Local Causality needs to go, but spacetime-associated parameters might still be retained.",
"Indeed, if they are not retained to some extent, all concepts of “locality” lose their usual meaning.",
"One concept of locality in particular, Continuous Action, is defined above in a time-neutral manner that prevents unmediated “action-at-a-distance”.",
"Even given Bell's Theorem, this definition of locality can be retained in two different styles of quantum models, categorized as Type IA and Type IIA.The different Types are identified in Table I of Section III.A.",
"The former would require faster-than-light mediating parameters, so only the latter is compatible with Lorentz covariance.",
"The price for retaining Lorentz covariance while forbidding action-at-a-distance is the violation of an assumption arguably unrelated to locality: the premise that a model's parameters should not functionally depend on inputs associated with the future of those parameters, or No Future-Input Dependence.",
"Without this assumption [or its corollary, Eqn.",
"(REF ), $\\mathbf {\\lambda }$ -independence] Bell's Theorem cannot be derived.",
"This analysis therefore motivates Type IIA models with Future-Input Dependence and Continuous Action as the most “local” models compatible with QM.",
"Very roughly, these models would violate Local Causality by violating our intuition of “causality” rather than our intuition of “locality”.See Section III.B for clarification of these issues.",
"Einstein saw no reason to relax either one of these, and Bohr effectively relaxed both, taking an operational view which keeps only the signal causality and signal locality conditions.",
"Bell and his followers took the “causality” condition for granted, without realizing that an alternative exists,When prompted to consider the failure of $\\mathbf {\\lambda }$ -independence, Eqn.",
"(REF ), which they called measurement-independence, they always considered the conspiratorial superdeterministic option, discussed in section VI.C.",
"and as a result studied Type IA and Type IB models.",
"Others took an operational approach which drops both the “causality” and the “locality” requirements for the internal (hidden) variables, resulting in development of Type IIB models.",
"As analyzed in Section IV, Type IIA models of quantum entanglement have effective connections associated only with the particle world-lines, either within the lightcones or on the lightcones for photons (i.e., there are no direct space-like connections).",
"Dropping the No Future-Input Dependence assumption allows these to be effective two-way connections.",
"Using this strategy, Einstein's “independence of objects far apart in space” can be softened without requiring connections which violate the spirit of relativity.",
"In particular, this view accommodates entanglement scenarios by allowing an external influence on A to have an indirect influence on B, via mediating events in the intersection of their past lightcones, without raising any difficulties with Lorentz covariance.",
"As discussed in Section REF , this need not lead to logical inconsistencies, or deviations from conventional QM predictions.",
"Physics models with explicit Future-Input Dependence have been developed already in the context of classical Electrodynamics [116], [117], and their relevance to Bell-like scenarios was pointed out even before Bell's Theorem emerged [10], and then repeatedly since [e.g., pegg1982,price1997].",
"Despite this, the development of explicit Type IIA models of entanglement phenomena has only recently begun in earnest, and is currently limited to a few particular applications, most notably the Bell state correlations which typically serve to demonstrate the issue of Bell's Theorem.",
"The detailed discussion of the proof-of-principle examples of such models in Section is hoped to introduce these possibilities to a wider audience, and Section REF indicates several possible avenues for future developments.",
"This would include describing more complicated entanglement scenarios and developing a treatment of quantum measurements as interactions between small and large systems.",
"It is emphasized that while Future-Input Dependent (or “retrocausal”) models of QM can have an underlying structure that is as time-symmetric as classical physics, all such models must have a mechanism to recover the time-asymmetric condition of signal causality.",
"Two possibilities for such a mechanism have been suggested above.",
"The first emphasizes the role of time-asymmetric “agents” employing the theory: they select which parameters of a theory to use as inputs of a specific model and which as outputs.See price1997 for further discussion.",
"The second considers the possibility of a time-symmetry-breaking physical principle (perhaps due to the low entropy of the big bang), with possibly relatively minor effects on the mathemtical model, e.g., a specification of initial conditions.",
"As a result of this mild symmetry breaking, irreversibility could appear in the thermodynamic limit, and with it, signal causality.It is interesting to note that bell1990 already asked: “Could it be that causal structure emerges only in something like a `thermodynamic' approximation?” But his tentative answer was negative, possibly due to his taking NFID for granted.",
"A successful Type IIA reformulation of QM would employ only spacetime-based parameters and would associate conventional probabilities with each fully-specified configuration.",
"An appropriate interpretation would take only one of these possibilities to actually occur in Nature.",
"In other words, the number of parameters describing a system would grow only linearly with its extent.",
"This stands as a dramatic advantage over existing approaches, where the number of necessary parameters scales exponentially with the number of particles in the system.",
"Combined with Lorentz covariance, this could greatly alleviate the disconnect between quantum theory and general relativity.",
"Such a reformulation would also shed light on an unresolved issue in quantum foundations—how to interpret the conventional wavefunction $\\psi $ and the collapse postulate.",
"Although $\\psi $ is not included in the underlying model, it could still represent available knowledge about the actual parameters—a viewpoint that has become known as “$\\psi $ -epistemic” [99].",
"Such states of incomplete knowledge naturally reside in configuration space (as in classical statistical mechanics), as they have to represent a large number of possible correlations.",
"Unitary evolution of these states would then correspond to time-evolving the available information, in analogy to Liouville dynamics.",
"Learning additional information about future settings and future outcomes would then lead to a Bayesian updating of $\\psi $ , corresponding to a (non-physical) collapse.",
"This is essentially the style of model advocated by Einstein, where the actual state of the system was not $\\psi $ , but rather something more fundamental [54].",
"While the present work is focused on Bell's Theorem, additional lines of research are also converging on the promise of Future-Input Dependent models.",
"As discussed above, leifer2017a motivate such models via time symmetry.",
"Another argument is motivated by the much-discussed Pusey-Barrett-Rudolph (PBR) Theorem [87], recently reviewed by leifer2014, and yet another relies on arguments concerning the complexity achievable with quantum computation [6].",
"One of Leifer's conclusions exactly matches ours, promoting the development of “retrocausal $\\dots $ models that posit a deeper reality underlying quantum theory that does not include the quantum state.” The spacetime-associated parameters $Q$ in Future-Input Dependent models would mathematically represent this “deeper reality”.",
"Fully realizing this goal remains an open challenge.",
"Acknowledgements: The authors warmly thank A. Briggs, P. Drummond, J. Finkelstein, S. Friederich, R. Sutherland, and several anonymous Referees for useful discussions and comments on a draft of the manuscript.",
"This work is supported in part by the Fetzer Franklin Fund of the John E. Fetzer Memorial Trust."
],
[
"Appendix: Derivation of the Schulman Model",
"Schulman's original single-particle model applies to a single spin-$1/2$ particle; here we convert it to a photon polarization problem.",
"The photon's classical trajectory is known, and it has a real (hidden) polarization direction $q(t)$ everywhere on its trajectory.",
"The photon is prepared and measured by passing through two polarization cubes, with the first set at an angle $\\theta _1$ and the second set at $\\theta _2$ .",
"The initial polarization is constrained, $q(t_1)=\\theta _1$ , as is usual for initial boundary conditions.",
"Schulman enforced a similar final boundary condition at measurement, where the final polarization was constrained to be either $q(t_2)=\\theta _2$ or $q(t_2)=\\theta _2+\\pi /2$ .",
"This final constraint is controllable (modulo $\\pi /2$ ) and the model is FID.",
"The time-asymmetry (modulo $\\pi /2$ at the output, but modulo $\\pi $ at the input) is external: an experimenter can choose to block a photon with an unwanted input polarization, but does not know the output polarization until it is too late to interfere.",
"Otherwise, everything in this model is fully time-symmetric.",
"Such two-time-boundary problems can only be solved “all-at-once,” with probabilities assigned to entire histories, $q(t)$ , not instantaneous states.",
"(One can extract the latter probabilities from the former.)",
"Defining a net rotation $\\Delta q\\equiv \\int ^{t_2}_{t_1} \\frac{ d q(t)}{dt} dt$ (which is permitted to be larger than $2\\pi $ for multiple rotations), the convolution of Schulman's proposed Cauchy kicks imply the probability assignment of Eqn.",
"(REF ): $P(\\Delta q)\\propto \\frac{1}{(\\Delta q)^2+\\gamma ^2}.$ Remarkably, this distribution recovers Malus' Law as $\\gamma \\rightarrow 0$ .",
"Seeing this requires adding the probabilities for all the rotations which end at the same polarization angle (modulo $\\pi $ ), and normalization.",
"The evaluation requires summing over all the possibilities of getting from $\\theta _1$ to $\\theta _2$ (mod $\\pi $ ), allowing for rotations through angles larger than $\\pi $ in both directions.",
"The sum, $\\sum _{n=-\\infty }^{\\infty } \\frac{1}{( \\Delta \\theta + n\\,\\pi )^{2}}$ with $\\Delta \\theta = \\theta _1 - \\theta _2$ , can be calculated, similarly to Euler's solution of the Basel problem ($\\sum _{n=1 }^{\\infty } \\frac{1}{n^{2}}$ ), by equating two different families of polynomial approximations to the same function, in this case, $f(x)=\\sin (\\Delta \\theta +x) \\sin (\\Delta \\theta -x)$ .",
"One family is the Taylor expansion, and as $f(x)=\\frac{1}{2} \\left( \\cos (2x) - \\cos (2 \\, \\Delta \\theta ) \\right)$ , the coefficient of $x^2$ is $-1$ , yielding $f(x) = \\sin ^2(\\Delta \\theta ) - x^2 +O(x^4)$ .",
"The other polynomial approximation scheme is obtained by multiplying the value of the function at $x=0$ by a factor of $(1-x/z_k)$ for each of the zeros $z_k$ (roots) of the original function (a specific approximation is obtained by including only roots up to a certain absolute magnitude).",
"Treating the roots in pairs, $z_n=-z^{\\prime }_n=\\Delta \\theta +n\\pi $ , gives $f(x) = \\sin ^2(\\Delta \\theta ) \\,\\Pi _{n=-\\infty }^\\infty \\left( 1 - \\frac{x^2}{(\\Delta \\theta + n\\pi )^2} \\right)$ , and expanding only up to terms quadratic in $x$ gives the necessary sum: $\\sum _{n=-\\infty }^{\\infty } \\frac{1}{( \\Delta \\theta + n\\,\\pi )^{2}} =\\frac{1}{ \\sin ^{2} \\left( \\Delta \\theta \\right) } .$ Normalizing the probabilities for either $q(t_2)=\\theta _2$ or $q(t_2)=\\theta _2+\\pi /2$ , is achieved by simply multiplying by the product of the corresponding denominators on the right-hand side of Eqn.",
"(REF ), yielding Malus' Law: $p=\\cos ^2(\\Delta \\theta )$ , as required for Section REF .",
"Remarkably, Schulman also used this idea to prove the Born rule, in the sense of showing that probabilities $\\propto |\\psi |^x$ are compatible with this idea of multiple kicks only for $x=2$ (whether or not the Cauchy-Lorentz distribution is used for each kick)."
]
] | 1906.04313 |
[
[
"Three product formulas for ratios of tiling counts of hexagons with\n collinear holes"
],
[
"Abstract Rosengren found an explicit formula for a certain weighted enumeration of lozenge tilings of a hexagon with an arbitrary triangular hole.",
"He pointed out that a certain ratio corresponding to two such regions has a nice product formula.",
"In this paper, we generalize this to hexagons with arbitrary collinear holes.",
"It turns out that, by using same approach, we can also generalize Ciucu's work on the number and the number of centrally symmetric tilings of a hexagon with a fern removed from its center.",
"This proves a recent conjecture of Ciucu."
],
[
"Introduction",
"Enumeration of lozenge tilings of a region on a triangular lattice has been studied for many decades.",
"In particular, people are interested in regions whose number of lozenge tilings is expressed as a simple product formula.",
"One such region is a hexagonal region with a triangular hole in the center.",
"Many works have been done on this topic by Ciucu [2], Ciucu et al.",
"[6], and Okada and Krattenthaler [10].",
"Later, Rosengren [11] found a formula for a weighted enumeration of lozenge tilings of a hexagon with an arbitrary triangular hole.",
"He pointed out that the ratio between numbers of lozenge tilings of two such regions whose holes have symmetric position with respect to the center has a nice product formula.",
"In this paper, we give a conceptual explanation of the symmetry, which enables us to generalize the result to hexagons with arbitrary collinear triangular holes.",
"In his paper, Ciucu [3] defined a new structure, called a fern, which is an arbitrary string of triangles of alternating orientations that touch at corners and are lined up along a common axis.",
"He considered a hexagon with a fern removed from its center and proved that the ratio of the number of lozenge tilings of two such regions is given by a simple prodcut formula.",
"Later, Ciucu [5] also proved that the same kind of ratio for centrally symmetric lozenge tilings also has a simple product formula.",
"In particular, he pointed out that for hexagons with a fern removed from the center, the ratio of centrally symmetric lozenge tilings is the square root of the ratio of the total number of tilings.",
"Ciucu also conjectured in [5] (See also [4]) that this square root phenomenon holds more generally, when any finite number of collinear ferns are removed in a centrally symmetric way.",
"In this current paper, we prove Ciucu's conjecture, and we extend it further."
],
[
"Statement of Main Results",
"Any hexagon on a triangular lattice has a property that difference between two parallel sides is equal for all 3 pairs.",
"Thus, we can assume that the side lengths of the hexagon are a, b+k, c, a+k, b, c+k in clockwise order, where a is a length of a top side.",
"Also, without loss of generality, we can assume that k is non-negative and a southeastern side of a hexagon (=c) is longer than or equal to a side length of a southwestern side (=b).",
"Note that this hexagonal region has k more up-pointing unit-triangles than down-pointing unit-triangles.",
"Since every lozenge consists of one up-pointing unit-triangle and one down-pointing unit-triangle, to be completely tiled by lozenges, we have to remove k more up-pointing unit-triangles than down-pointing unit-triangles from the hexagon.",
"There are many ways to do that, but let's consider a following case.",
"Let's call a set of triangles on a triangular lattice is collinear or lined up if horizontal side of all triangles are on a same line.",
"Now, let's consider any horizontal line passing through the hexagon.",
"Suppose the line is l-th horizontal line from bottom side of the hexagon.",
"Note that the length of the horizontal line depends on the size of l: Let's denote the length of the line by L(l).",
"Then we have $L(l)=a+k-l+min(b,l)+min(c,l)$ .",
"For any subsets $X=\\lbrace x_1, ..., x_{m+k}\\rbrace $ and $Y=\\lbrace y_1, ..., y_m\\rbrace $ of $[L(l)]:=\\lbrace 1,2,...,L(l)\\rbrace $ , let $H_{a,b,c}^{k,l}(X:Y)$ be the region obtained from the hexagon of side length $a$ , $b+k$ , $c$ , $a+k$ , $b$ , $c+k$ in clockwise order from top by removing up-pointing unit-triangles whose labels of horizontal sides form a set $X=\\lbrace x_1, x_2, ..., x_{m+k}\\rbrace $ , and down-pointing unit-triangles whose labels of horizontal sides form a set $Y=\\lbrace y_1, y_2, ..., y_m\\rbrace $ on the l-th horizontal line from the bottom, where labeling on the horizontal line is $1,2,...,\\textit {L(l)}$ from left to right.",
"Let's call the horizontal line as a baseline of removed triangles.",
"Similarly, let $\\overline{H}_{a,b,c}^{k,l}(X:Y)$ be a same kind of region, except that labeling on the horizontal line is 1, 2, ..., $L(l)$ from right to left.",
"Also, for any region R on a triangular lattice, let M(R) be a number of lozenge tilings of the region.",
"First theorem expresses a ratio of numbers of lozenge tilings of two such region as a simple product formula.",
"Theorem 2.1 Let a, b, c, k, l, m be any non-negative integers such that $b \\le c$ , $0 \\le l \\le b+c$ and $m \\le min(b, l, b+c-l)$ .",
"Also, let $X=\\lbrace x_1, x_2, ..., x_{m+k}\\rbrace $ and $Y=\\lbrace y_1, y_2, ..., y_m\\rbrace $ be subsets of $[L(l)]=\\lbrace 1, 2, ..., L(l)\\rbrace $ .",
"Then $\\begin{aligned}&\\frac{M({H_{a,b,c}^{k,l}(X:Y)})}{M({\\overline{H}_{a,b,c}^{k,b+c-l}(X:Y)})}\\\\&=\\frac{H(k+l)H(b+c-l)}{H(l)H(b+c+k-l)}\\\\&\\cdot \\frac{\\prod _{i=1}^{m+k}(x_i-b+max(b, l))_{(b-l)}\\cdot (a+k+min(b, l)+1-x_i)_{(c-l)}}{\\prod _{j=1}^{m}(y_i-b+max(b, l))_{(b-l)}\\cdot (a+k+min(b, l)+1-y_j)_{(c-l)}}\\end{aligned}$ where the hyperfactorial H(n) is defined by $H(n):=0!1!\\cdot \\cdot \\cdot (n-1)!$ Figure: Two regions H 4,3,7 2,4 ({2,4,6,9}:{4,8})H_{4,3,7}^{2,4}(\\lbrace 2,4,6,9\\rbrace :\\lbrace 4,8\\rbrace )(left) and H 4,3,7 2,6 ({2,4,6,9}:{4,8})H_{4,3,7}^{2,6}(\\lbrace 2,4,6,9\\rbrace :\\lbrace 4,8\\rbrace )(right)To state next results, we need to recall a result of Cohn, Larsen and Propp [8], which is a lozenge tilings interpretation of a classical result of Gelfand and Tsetlin [9].",
"Recall that $\\Delta (S):=\\prod _{s_1<s_2, s_1,s_2\\in S}{(s_2-s_1)}$ and $\\Delta (S,T):=\\prod _{s \\in S ,t \\in T}{|t-s|}$ for any finite sets S and T. Proposition 2.2 For any non-negative integers $m, n$ and any subset $S=\\lbrace s_1, s_2,...,s_n\\rbrace \\subset [m+n]:=\\lbrace 1, 2,..., m+n\\rbrace $ , let $T_{m,n}(S)$ be the region on a triangular lattice obtained from the trapezoid of side lengths $m$ , $n$ , $m+n$ , $n$ clockwise from the top by removing the up-pointing unit-triangles whose bottoms sides are labeled by elements of a set $S=\\lbrace s_1, s_2,...,s_n\\rbrace $ , where bottom side of the trapezoid is labeled by $1, 2, ..., m+n$ from left to right.",
"Then $M(T_{m,n}(S))=\\frac{\\Delta (S)}{\\Delta ([n])}=\\frac{\\Delta (S)}{H(n)}$ For any finite subset of integers $S=\\lbrace s_1, s_2,..., s_n\\rbrace $ , where elements are written in increasing order, let $T(S)$ be a region obtained by translating a region $T_{s_n-(s_1-1)-n,n}(s_1-(s_1-1), s_2-(s_1-1),...,s_n-(s_1-1))$ by $(s_1-1)$ units to the right and $s(S):=M(T(S))=\\frac{\\Delta (S)}{H(n)}$ .",
"A region on a triangular lattice is called balanced if it contains same number of up-pointing unit-triangles and down-pointing unit-triangles.",
"Geometrically, $T(S)$ is the balanced region that can be obtained from a trapezoid of bottom length $(s_n-s_1+1)$ by deleting up-pointing unit-triangles whose labels are $s_1$ , $s_2$ ,..., $s_n$ on bottom, where bottom line is labeled by $s_1$ , $(s_1+1)$ ,..., $(s_n-1)$ , $s_n$ (See Figure 2.2).",
"Figure: A region T({-3, 0, 1, 3, 7})In his paper, Ciucu [3] defined a new structure, called a fern, which is an arbitrary string of triangles of alternating orientations that touch at corners and are lined up.",
"For non-negative integers $a_1$ ,...,$a_k$ , a fern $F(a_1,...,a_k)$ is a string of k lattice triangles lined up along a horizontal lattice line, touching at their vertices, alternately oriented up and down and having sizes $a_1$ ,...,$a_k$ from left to right (with the leftmost oriented up).",
"We call the horizontal lattice line as a baseline of the fern.",
"Figure: Budded fern F(2,1,-1,-1,1,-1:2,0,1,0,1)F(2,1,-1,-1,1,-1:2,0,1,0,1) (top left) and its baseline representation (top right), corresponding budded bowtie (bottom left) and its baseline representation (bottom right).",
"A red point is a turning pointNow, let's give additional structure to fern by adding buds (triangles) on the baseline, and we will call this new structure as a budded fern.",
"To label this new structure, we remove all unit-triangles from the budded fern, excepts unit-triangles whose horizontal side is on the baseline.",
"We call it as a baseline representation of the budded fern.",
"Then we count numbers of consecutive up-pointing unit-triangles, down-pointing unit-triangles and vertical unit-lozenges on the baseline.",
"When we count these numbers, up-pointing (or down-pointing) unit-triangle which is contained in a vertical unit-lozenge is not considered as an up-pointing (or down-pointing) unit-triangle.",
"If an up-pointing unit-triangle and a down-pointing unit-triangle are adjacent, we think as if there are 0 vertical lozenges between them.",
"Now, we line up these numbers from left to right, and put - to numbers that represent numbers of down-pointing unit-triangles.",
"Then, by allowing $a^k_1=0$ and $a^k_{r_k}=0$ , we get a sequence of integers $a^k_1$ , $w^k_1$ , $a^k_2$ , $w^k_2$ ,..., $a^k_{r_k-1}$ , $w^k_{r_k-1}$ , $a^k_{r_k}$ , where $a^k_i$ represent a (signed) number of consecutive up-pointing (or down-pointing) unit-triangles, and $w^k_i$ represent a number of consecutive vertical unit-lozenges.",
"Let $A_k$ represents a sequence $(a^k_1, a^k_2,,..., a^k_{r_k})$ and $W_k$ represents a sequence $(w^k_1, w^k_2,,..., w^k_{r_k-1})$ .",
"Then we denote the original budded fern as $F(A_k:W_k)$ , and its baseline representation as $F_{br}(A_k:W_k)$ .",
"Let $L^k_i$ be a leftmost vertex of $a^k_i$ consecutive triangles, and $R^k_i$ be a rightmost vertex of the consecutive triangles.",
"Also, let $I^k:=\\lbrace i\\in [r_k]|a^k_i> 0\\rbrace $ , $J^k:=\\lbrace i\\in [r_k]|a^k_i< 0\\rbrace $ , $p_k:=\\sum _{i\\in I^k} a^k_i$ and $n_k:=-\\sum _{i\\in J^k} a^k_i$ .",
"From this budded fern, we will construct a corresponding budded bowtie as follows.",
"From a baseline representation of the budded fern, we move up-pointing unit-triangles to left, and down-pointing unit-triangles to right along the baseline, fixing vertical lozenges.",
"Then we call a right vertex of a right-most up-poiting unit-triangle which is not a part of a vertical lozenge as a turning point of a new structure and denote it by $T^k$ .",
"Then we put vertical lozenges between consecutive up-pointing (or down-pointing) unit-triangles as much as possible.",
"Then we get a bowtie (possibly a slipped bowtie) with some triangles attached.",
"We call this as a budded bowtie and denote it and its baseline representation by $B(A_k:W_k)$ and $B_{br}(A_k:W_k)$ , respectively.",
"Also, let $u_k$ be a smallest positive integer such that $p_k \\le |a^k_1|+|a^k_2|+...+|a^k_{u_k}|$ , and $v_k \\in [a^k_{u_k}]$ be a positive integer such that $|a^k_1|+|a^k_2|+...+|a^k_{u_k-1}|+v_k=p_k$ .",
"When we say a budded fern $F(A_k:W_k)$ , we equip with corresponding sequences $A_k$ , $W_k$ , sets $I^k$ , $J^k$ , indices $r_k$ , $p_k$ , $n_k$ , $u_k$ , $v_k$ and vertices $L_1^k, L_2^k,..., L_{r_k}^k$ , $R_1^k, R_2^k,..., R_{r_k}^k$ , $T^k$ .",
"Now, let $H_{a,b,c}^{k,l}(F(A_1:W_1),..., F(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1})$ be a region obtained from the hexagon of side length $a$ , $b+k$ , $c$ , $a+k$ , $b$ , $c+k$ in clockwise order from top by removing budded ferns $F(A_1:W_1),..., F(A_n:W_n)$ on a $l$ -th horizontal line from the bottom so that a distance between a leftmost vertex on the horizontal line and a leftmost vertex of $F(A_1:W_1)$ is $m_1$ , a distance between a rightmost vertex on the horizontal line and a rightmost vertex of $F(A_t:W_t)$ is $m_{t+1}$ , and a distance between two adjacent budded ferns $F(A_i:W_i)$ and $F(A_{i+1}:W_{i+1})$ is $m_{i+1}$ for all $i\\in [t-1]$ .",
"We can similarly think of a region $H_{a,b,c}^{k,l}(F_{br}(A_1:W_1),..., F_{br}(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1})$ .",
"From the region, we label the $l$ -th horizontal line from the bottom by 1,2,...,$L(l)$ from left to right.",
"Let $X^1$ be a set of labels whose corresponding segment is a side of an up-pointing unit-triangular hole, but not a side of an down-pointing unit-triangular hole.",
"Similarly, let $X^2$ be a set of labels of segments whose corresponding segment is a side of an down-pointing unit-triangular hole, but not a side of an up-pointing unit-triangular hole and $W$ be a set of label of segments whose corresponding segment is a side of both up-pointing and down-pointing unit-triangular holes.",
"Similarly, let $H_{a,b,c}^{k,l}(B(A_1:W_1),..., B(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1})$ be a region obtained from the hexagon of side length $a$ , $b+k$ , $c$ , $a+k$ , $b$ , $c+k$ in clockwise order from top by removing budded bowties $B(A_1:W_1),..., B(A_t:W_t)$ from a $l$ -th horizontal line, where positions of removed budded bowties on the horizontal line is exactly same as positions of corresponding budded ferns.",
"Again, we can think of a region $H_{a,b,c}^{k,l}(B_{br}(A_1:W_1),..., B_{br}(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1})$ and we can define sets $Y^1$ and $Y^2$ from this region as we defined sets $X^1$ and $X^2$ from $H_{a,b,c}^{k,l}(F_{br}(A_1:W_1),..., F_{br}(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1})$ .",
"Note that we have $X^1\\cup X^2=Y^1\\cup Y^2$ .",
"For any point and a line on the triangular lattice, let distance between a point and a line be a shortest length of a path from the point to the extension of the line along lattice.",
"Especially, for any lattice point $E$ in a hexagon, let $d_{NW}(E)$ be a distance between the point $E$ and a northwestern side of the hexagon.",
"Similarly, we can define $d_{SW}(E)$ , $d_{NE}(E)$ and $d_{SE}(E)$ to be distances between a point $E$ and a southwestern side, northeastern side and southeastern side of the hexagon, respectively.",
"Next theorem expresses a ratio of numbers of lozenge tilings of the two regions as a simple product formula.",
"Figure: An example of regions (from left to right): H 12,8,15 3,13 (F(2,-1,1:1,1),F(-1,1,2,-1:0,1,0):3,6,2)H_{12,8,15}^{3,13}(F(2,-1,1:1,1), F(-1,1,2,-1:0,1,0):3,6,2) and H 12,8,15 3,13 (B(2,-1,1:1,1),B(-1,1,2,-1:0,1,0):3,6,2)H_{12,8,15}^{3,13}(B(2,-1,1 : 1,1), B(-1,1,2,-1 : 0,1,0) : 3, 6, 2)Theorem 2.3 Let $a, b, c, k, l, m_1,...,m_{t+1}$ be any non-negative integers and $F(A_1:W_1)$ ,..., $F(A_t:W_t)$ be any budded ferns.",
"Let $p:=\\sum _{i=1}^{t}{p_i}$ , $n:=\\sum _{i=1}^{t}{n_i}$ , $w:=\\sum _{i=1}^{t}{\\sum _{j=1}^{r_i-1}w_j^i}$ and $m:=\\sum _{i=1}^{t+1}m_i$ .",
"Suppose indices satisfy following conditions: 1) $p=n+k$ , 2) $p+n+w+m=L(l)$ , 3) $n+w\\le min(b,l,b+c-l)$ .",
"Then we have $\\begin{aligned}&\\frac{M(H_{a,b,c}^{k,l}(F(A_1:W_1), F(A_2:W_2),..., F(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))}{M(H_{a,b,c}^{k,l}(B(A_1:W_1), B(A_2:W_2),..., B(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))}\\\\&=\\frac{s(X^1)s(X^2)}{s(Y^1)s(Y^2)}\\\\&\\begin{aligned}\\cdot \\prod _{i=1}^{t}\\Bigg [&\\frac{H(d_{SW}(T^i))H(d_{NW}(L^i_{u_i}))H(d_{SE}(L^i_{u_i}))H(d_{NE}(T^i))}{H(d_{SW}(L^i_{u_i}))H(d_{NW}(T^i))H(d_{SE}(T^i))H(d_{NE}(L^i_{u_i}))}\\\\&\\cdot \\prod _{j < u_i, j \\in J_i}\\frac{H(d_{SW}(R^i_j))H(d_{NW}(L^i_j))H(d_{SE}(L^i_j))H(d_{NE}(R^i_j))}{H(d_{SW}(L^i_j))H(d_{NW}(R^i_j))H(d_{SE}(R^i_j))H(d_{NE}(L^i_j))} \\\\&\\cdot \\prod _{j \\ge u_i, j \\in I_i}\\frac{H(d_{SW}(L^i_j))H(d_{NW}(R^i_j))H(d_{SE}(R^i_j))H(d_{NE}(L^i_j))}{H(d_{SW}(R^i_j))H(d_{NW}(L^i_j))H(d_{SE}(L^i_j))H(d_{NE}(R^i_j))} \\Bigg ]\\end{aligned}\\end{aligned}$ Now, let's consider a case when a region $H_{a,b,c,k,l}(F(A_1:W_1),..., F(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1})$ is centrally symmetric, that is invariant under $180^{\\circ }$ rotation with respect to a center of a hexagon.",
"To satisfy this condition, the region should satisfy following conditions: (1) $b$ and $c$ have same parity (2) $k=0$ and $l=\\frac{b+c}{2}$ (3) $m_s=m_{t+2-s}$ for all $s\\in [t+1]$ and $r_i=r_{t+1-i}$ for all $i\\in [t]$ (4) $a^i_{j}=-a^{t+1-i}_{r_i+1-j}$ and $w^i_u=w^{t+1-i}_{r_i-u}$ for all $i\\in [t]$ , $j\\in [r_i]$ , $u\\in [r_i-1]$ When these conditions hold, a region $H_{a,b,c}^{0,\\frac{b+c}{2}}(F(A_1:W_1),..., F(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1})$ and a corresponding region $H_{a,b,c}^{0,\\frac{b+c}{2}}(B(A_1:W_1),..., B(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1})$ are centrally symmetric, so we can compare their number of centrally symmetric lozenge tilings.",
"Let $M_\\odot (G)$ be a number of centrally symmetric lozenge tiling of a region G on a triangular lattice.",
"The last theorem expresses a ratio of numbers of centrally symmetric lozenge tilings of the two regions as a simple product formula.",
"Figure: Centrally symmetric regions (from left to right) H 10,11,13 0,12 (F(-1,1,-2,0:1,0,1),F(0,2,-1,1:1,0,1):2,5,2)H_{10,11,13}^{0,12}(F(-1,1,-2,0:1,0,1), F(0,2,-1,1:1,0,1):2,5,2) and H 10,11,13 0,12 (B(-1,1,-2,0:1,0,1),B(0,2,-1,1:1,0,1):2,5,2)H_{10,11,13}^{0,12}(B(-1,1,-2,0:1,0,1), B(0,2,-1,1:1,0,1):2,5,2)Theorem 2.4 Let $a,b,c, m_1,..., m_{t+1}$ be any non-negative integers and $F(A_1:W_1),..., F(A_t:W_t)$ be any budded ferns that satisfy all four conditions stated above.",
"Let $p:=\\sum _{i=1}^{t}{p_i}$ , $n:=\\sum _{i=1}^{t}{n_i}$ , $w:=\\sum _{i=1}^{t}{\\sum _{j=1}^{r_i-1}w_j^i}$ and $m:=\\sum _{i=1}^{t+1}m_i$ .",
"Suppose indices satisfy following additional conditions: 1) $p+n+w+m=a+b$ 2) $p+w=n+w\\le b$ .",
"Then we have $\\begin{aligned}&\\frac{M_\\odot (H_{a,b,c}^{0,\\frac{b+c}{2}}(F(A_1:W_1),..., F(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))}{M_\\odot (H_{a,b,c}^{0,\\frac{b+c}{2}}(B(A_1:W_1),..., B(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))}\\\\&=\\sqrt{\\frac{M(H_{a,b,c}^{0,\\frac{b+c}{2}}(F(A_1:W_1),..., F(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))}{M(H_{a,b,c}^{0,\\frac{b+c}{2}}(B(A_1:W_1),..., B(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))}}\\\\&=\\frac{s(X^1)}{s(Y^1)}\\cdot \\prod _{i=1}^{t}\\Bigg [\\frac{H(d_{SE}(L^i_{u_i}))H(d_{NE}(T^i))}{H(d_{SE}(T^i))H(d_{NE}(L^i_{u_i}))}\\\\&\\cdot \\prod _{j<u_i, j\\in J_i}\\frac{H(d_{SE}(L^i_j))H(d_{NE}(R^i_j))}{H(d_{SE}(R^i_j))H(d_{NE}(L^i_j))}\\prod _{j\\ge u_i, j\\in I_i}\\frac{H(d_{SE}(R^i_j))H(d_{NE}(L^i_j))}{H(d_{SE}(L^i_j))H(d_{NE}(R^i_j))}\\Bigg ]\\end{aligned}$"
],
[
"Proof of the main results",
"A region on a triangular lattice is called balanced if it contains same number of up-pointing and down-pointing unit-triangles.",
"Let's recall a useful result which is implicit in work of Ciucu [1] (See also Ciucu and Lai [7]).",
"Lemma 3.1 (Region-splitting Lemma).",
"Let $R$ be a balanced region on a triangular lattice.",
"Assume that a subregion $S$ of $R$ satisfies the following two conditions: (1) (Seperating Condition) There is only one type of unit-triangle (either up-pointing or down-pointing) running along each side of the border between $S$ and $R-S$ (2) (Balancing Condition) $S$ is balanced.",
"Then $M(R)=M(S)M(R-S)$ To prove theorems in this paper, we need to simplify expressions that involves $\\Delta $ .",
"For this purpose, let's recall a property of $\\Delta $ : Let $X=\\lbrace x+1,x+2_,...,x+m\\rbrace $ and $Y=\\lbrace y+1,y+2,...,y+n\\rbrace $ be two sets of consecutive integers such that $x+m<y+1$ .",
"Then $\\begin{aligned}\\Delta (X,Y)=\\prod _{i=1}^{m}(y-x-m+i)_n&=\\prod _{i=1}^{m}\\frac{(y-x+n-m+i-1)!}{(y-x-m+i-1)!",
"}\\\\&=\\frac{H(y-x-m)H(y-x+n)}{H(y-x)H(y-x+n-m)}\\end{aligned}$ The crucial idea of this paper is the following: Each of our three main results involves the ratio of the number of tilings of two regions.",
"For each of these two regions, we partition the set of lozenge tilings of each region according to the positions of the vertical lozenges that are bisectecd by the baseline.",
"The partition classes obtained for the numerator and denominator are naturally paired up.",
"Then, using Proposition 2.2. and Lemma 3.1., we verify that the ratio of the number of tilings in the corresponding partition classes does not depend on the choice of partition class (i.e.",
"it is the same for all classes of the partition).",
"Proof of Theorem 2.1.",
"Let's first consider a case when $b < l \\le c$ .",
"From any lozenge tiling of $H_{a,b,c}^{k,l}(X:Y)$ , we will generate a pair of lozenge tiling of two trapezoidal regions with some dent on top (or bottom).",
"If we focus on lozenges below the baseline, then the lozenges form a pentagonal region that has $b$ down-pointing unit-triangle dents on top.",
"Among $b$ dents, $m$ of them are from the region $H_{a,b,c}^{k,l}(X:Y)$ itself, namely down-pointing unit-triangles whose bases are labeled by $y_1$ , $y_2$ ,..., $y_m$ , and remaining $(b-m)$ of them are down-pointing unit-triangles whose labels of their bases are from $[L(l)]\\setminus (X\\cup Y)=[a+b+k]\\setminus (X\\cup Y)$ .",
"Let $Z:=\\lbrace z_1, z_2,..., z_{b-m}\\rbrace \\subset [L(l)]\\setminus (X\\cup Y)$ be a set of labels of bases of remaining $(b-m)$ dents, and let $B:=\\lbrace -|b-l|+1, -|b-l|+2,..., -1, 0\\rbrace $ .",
"Then, we can easily see that there is a natural bijection between a set of lozenge tilings of the pentagonal region having $b$ down-pointing unit-triangle dents on top and a set of lozenge tilings of a region $T(B \\cup Z\\cup Y)$ .",
"So from a lozenge tiling of $H_{a,b,c}^{k,l}(X:Y)$ , we generate a lozenge tiling of a region $T(B \\cup Z\\cup Y)$ .",
"Figure: Correspondence between a lozenge tiling and a pair of trapezoid regions with dents (when b<l≤cb<l\\le c)Now we return to the lozenge tiling of $H_{a,b,c}^{k,l}(X:Y)$ , and focus on lozenges above the baseline.",
"Again, they from a pentagonal region that have $(b+k)$ up-pointing unit-triangle dents on bottom.",
"Among $(b+k)$ dents, $(m+k)$ of them are from the region $H_{a,b,c}^{k,l}(X:Y)$ itself, namely up-pointing unit-triangles whose bases are labeled by $x_1$ , $x_2$ ,..., $x_{m+k}$ and remaining $(b-m)$ of them are up-pointing unit-triangles whose labels form a set $Z$ .",
"Let $C:=\\lbrace L(l)+1,L(l)+2,...,L(l)+|c-l|\\rbrace $ .",
"Then same observation allow us to see that there is a bijection between a set of lozenge tilings of the pentagonal region having $(b+k)$ up-pointing unit-triangle dents on bottom and a set of lozenge tilings of a region $T(Z\\cup X\\cup C)$ .",
"Thus, we generate a lozenge tiling of a region $T(Z\\cup X\\cup C)$ from a lozenge tiling of $H_{a,b,c}^{k,l}(X:Y)$ .",
"Hence, from a lozenge tiling of $H_{a,b,c}^{k,l}(X:Y)$ , we generate a pair of lozenge tiling of a region $T(B \\cup Z\\cup Y)$ and $T(Z\\cup X\\cup C)$ and this correspondence is reversible (See Figure 3.1.).",
"Now, we partiton a set of lozenge tiling of the region $H_{a,b,c}^{k,l}(X:Y)$ by a set $Z:=\\lbrace z_1, z_2,..., z_{b-m}\\rbrace \\subset [L(l)]\\setminus (X\\cup Y)$ which represents labels of position of vertical lozenges on the baseline.",
"Number of lozenge tilings of the region $H_{a,b,c}^{k,l}(X:Y)$ with $(b-m)$ vertical lozenges on the baseline whose labels of position form a set $Z:=\\lbrace z_1, z_2,..., z_{b-m}\\rbrace \\subset [L(l)]\\setminus (X\\cup Y)$ is $H_{a,b,c}^{k,l}(Z\\cup X:Z\\cup Y)$ .",
"Also, by Lemma 3.1., $M(H_{a,b,c}^{k,l}(Z\\cup X:Z\\cup Y))$ is just a product of number of lozenge tilings of two pentagonal regions with unit-triangular dents on top (or bottom).",
"However, numbers of lozenge tilings of two pentagonal region is same as number of lozenge tilings of regions $T(B \\cup Z\\cup Y)$ and $T(Z\\cup X\\cup C)$ , respectively.",
"Thus, we have $\\begin{aligned}&H_{a,b,c}^{k,l}(X:Y)\\\\&=\\sum _{Z=\\lbrace z_1, z_2,..., z_{b-m}\\rbrace \\subseteq [L(l)]\\setminus (X\\cup Y)}H_{a,b,c}^{k,l}(X\\cup Z:Y\\cup Z)\\\\&=\\sum _{Z \\subseteq [L(l)]\\setminus (X\\cup Y)}M(T(X\\cup {Z}\\cup {C}))M(T(B\\cup {Y}\\cup {Z}))\\\\&=\\sum _{Z \\subseteq [L(l)]\\setminus (X\\cup Y)}s(X\\cup {Z}\\cup {C})s(B\\cup {Y}\\cup {Z})\\\\&=\\sum _{Z\\subseteq [L(l)]\\setminus ({{X}\\cup {Y}})}{\\frac{\\Delta (X\\cup {Z}\\cup {C})}{H(b+c+k-l)}}\\cdot {\\frac{\\Delta (B\\cup {Y}\\cup {Z})}{H(l)}}\\\\&=\\frac{1}{H(l)\\cdot H(b+c+k-l)}\\cdot \\sum _{Z\\subseteq [L(l)]\\setminus ({{X}\\cup {Y}})}{\\Delta (X\\cup {Z}\\cup {C})\\cdot \\Delta (B\\cup {Y}\\cup {Z})}\\end{aligned}$ A lozenge tiling of ${\\overline{H}_{a,b,c}^{k,b+c-l}(x_1,x_2,...,x_{m+k}:y_1,y_2,...,y_m)}$ can be also analyzed in a similar way and we can express a number of lozenge tiling of it as follows: $\\begin{aligned}&M({\\overline{H}_{a,b,c}^{k,b+c-l}(X:Y)})\\\\&=\\frac{1}{H(k+l)\\cdot H(b+c-l)}\\cdot \\sum _{Z\\subseteq [L(l)]\\setminus ({{X}\\cup {Y}})}{\\Delta (B\\cup {X}\\cup {Z})\\cdot \\Delta (Y\\cup {Z}\\cup {C})}\\end{aligned}$ where the sum is taken over all $(b-m)$ elements subset $Z\\subseteq [L(l)]\\setminus ({{X}\\cup {Y}})$ .",
"However, for any $(b-m)$ elements subset $Z\\subseteq [L(l)]\\setminus ({{X}\\cup {Y}})$ , $\\begin{aligned}\\frac{\\Delta (X\\cup {Z}\\cup {C})\\cdot \\Delta (B\\cup {Y}\\cup {Z})}{\\Delta (B\\cup {X}\\cup {Z})\\cdot \\Delta (Y\\cup {Z}\\cup {C})}&=\\frac{\\Delta (X)\\Delta (Z)\\Delta (C)\\Delta (X,Z)\\Delta (X,C)\\Delta (Z,C)}{\\Delta (B)\\Delta (X)\\Delta (Z)\\Delta (B,X)\\Delta (B,Z)\\Delta (X,Z)}\\\\&~~~\\cdot \\frac{\\Delta (B)\\Delta (Y)\\Delta (Z)\\Delta (B,Y)\\Delta (B,Z)\\Delta (Y,Z)}{\\Delta (Y)\\Delta (Z)\\Delta (C)\\Delta (Y,Z)\\Delta (Y,C)\\Delta (Z,C)}\\\\&=\\frac{\\Delta (X,C)\\Delta (B,Y)}{\\Delta (B,X)\\Delta (Y,C)}\\end{aligned}$ Note that this ratio does not depend on a choice of a set Z.",
"Hence, by combining (3.3), (3,4) and (3.5), we have $\\begin{aligned}&\\frac{M({H_{a,b,c}^{k,l}(X:Y)})}{M({\\overline{H}_{a,b,c}^{k,b+c-l}(X:Y)})}\\\\&=\\frac{H(k+l)H(b+c-l)}{H(l)H(b+c+k-l)}\\cdot \\frac{\\sum _{Z\\subseteq [a+b+k]\\setminus ({{X}\\cup {Y}})}{\\Delta (X\\cup {Z}\\cup {C})\\cdot \\Delta (B\\cup {Y}\\cup {Z})}}{\\sum _{Z\\subseteq [a+b+k]\\setminus ({{X}\\cup {Y}})}{\\Delta (B\\cup {X}\\cup {Z})\\cdot \\Delta (Y\\cup {Z}\\cup {C})}}\\\\&=\\frac{H(k+l)H(b+c-l)}{H(l)H(b+c+k-l)}\\cdot \\frac{\\Delta (X,C)\\Delta (B,Y)}{\\Delta (B,X)\\Delta (Y,C)}\\\\&=\\frac{H(k+l)H(b+c-l)}{H(l)H(b+c+k-l)}\\cdot \\frac{\\prod _{i=1}^{m+k}{(a+b+k+1-x_i)_{(c-l)}}\\cdot \\prod _{j=1}^{m}{(y_j)_{(l-b)}}}{\\prod _{i=1}^{m+k}{(x_i)_{(l-b)}}\\cdot \\prod _{j=1}^{m}{(a+b+k+1-y_j)_{(c-l)}}}\\\\&=\\frac{H(k+l)H(b+c-l)}{H(l)H(b+c+k-l)}\\cdot \\frac{\\prod _{i=1}^{m+k}(x_i+l-b)_{(b-l)}(a+b+k+1-x_i)_{(c-l)}}{\\prod _{j=1}^{m}(y_i+l-b)_{(b-l)}(a+b+k+1-y_j)_{(c-l)}}\\\\&=\\frac{H(k+l)H(b+c-l)}{H(l)H(b+c+k-l)}\\\\&~~~\\cdot \\frac{\\prod _{i=1}^{m+k}(x_i-b+max(b, l))_{(b-l)}(a+k+min(b, l)+1-x_i)_{(c-l)}}{\\prod _{j=1}^{m}(y_i-b+max(b,l))_{(b-l)}(a+k+min(b, l)+1-y_j)_{(c-l)}}\\end{aligned}$ Now let's consider a case when $l \\le b$ Similar observation enable us to observe that $M(H_{a,b,c,k,l}(X:Y))$ can be written as a sum of $M(H_{a,b,c}^{k,l}(X\\cup Z:Y\\cup Z))$ , where $Z=\\lbrace z_1, z_2,...,z_{l-m}\\rbrace \\subset [L(l)]\\setminus (X\\cup Y)=[a+k+l]\\setminus (X\\cup Y)$ represents a set of labels of positions of vertical lozenges on the baseline.",
"Also, by Lemma and 3.1. and same argument as we used in previous case, $M(H_{a,b,c}^{k,l}(X\\cup Z:Y\\cup Z))$ is equal to a product of $M(T(B\\cup {X}\\cup {Z}\\cup {C}))$ and $M(T(Y\\cup {Z}))$ , where $B=\\lbrace -|b-l|+1, -|b-l|+2,..., -1, 0\\rbrace $ and $C=\\lbrace L(l)+1, L(l)+2,..., L(l)+|c-l|\\rbrace $ .",
"Hence Figure: Correspondence between a lozenge tiling and a pair of trapezoid regions with dents (when l≤bl\\le b)$H_{a,b,c}^{k,l}(Z\\cup X:Z\\cup Y)=\\frac{\\Delta (B\\cup {X}\\cup {Z}\\cup {C})}{H(b+c+k-l)}\\cdot \\frac{\\Delta (Y\\cup {Z})}{H(l)}$ If we sum over every $(l-m)$ element set $Z\\subseteq [L(l)]\\setminus ({{X}\\cup {Y}})$ , then we have a representation of number of lozenge tiling of $H_{a,b,c}^{k,l}(X:Y)$ as follows: $\\begin{aligned}&M(H_{a,b,c}^{k,l}(X:Y))\\\\&=\\frac{1}{H(l)\\cdot H(b+c+k-l)}\\cdot \\sum _{Z\\subseteq [L(l)]\\setminus ({{X}\\cup {Y}})}{\\Delta (B\\cup {X}\\cup {Z}\\cup {C})\\cdot \\Delta (Y\\cup {Z})}\\end{aligned}$ By same observation, we can represent a number of lozenge tiling of a hexagon $\\overline{H}_{a,b,c}^{k,b+c-l}(X:Y)$ as follows: $\\begin{aligned}&M(\\overline{H}_{a,b,c}^{k,b+c-l}(X:Y))\\\\&=\\frac{1}{H(k+l)\\cdot H(b+c-l)}\\cdot \\sum _{Z\\subseteq [L(l)]\\setminus ({{X}\\cup {Y}})}{\\Delta (X\\cup {Z})\\cdot \\Delta (B\\cup {Y}\\cup {Z}\\cup {C})}\\end{aligned}$ Now, we observe a ratio $\\frac{\\Delta (B\\cup {X}\\cup {Z}\\cup {C})\\cdot \\Delta (Y\\cup {Z})}{\\Delta (X\\cup {Z})\\cdot \\Delta (B\\cup {Y}\\cup {Z}\\cup {C})}$ for any subset $Z\\subseteq [L(l)]\\setminus ({{X}\\cup {Y}})$ with $(l-m)$ elements: $\\begin{aligned}&\\frac{\\Delta (B\\cup {X}\\cup {Z}\\cup {C})\\cdot \\Delta (Y\\cup {Z})}{\\Delta (X\\cup {Z})\\cdot \\Delta (B\\cup {Y}\\cup {Z}\\cup {C})}\\\\&=\\frac{\\Delta (B)\\Delta (X)\\Delta (Z)\\Delta (C)\\Delta (B,X)\\Delta (B,Z)\\Delta (B,C)\\Delta (X,Z)\\Delta (X,C)\\Delta (Z,C)}{\\Delta (X)\\Delta (Z)\\Delta (X,Z)}\\\\&\\cdot \\frac{\\Delta (Y)\\Delta (Z)\\Delta (Y,Z)}{\\Delta (B)\\Delta (Y)\\Delta (Z)\\Delta (C)\\Delta (B,Y)\\Delta (B,Z)\\Delta (B,C)\\Delta (Y,Z)\\Delta (Y,C)\\Delta (Z,C)}\\\\&=\\frac{\\Delta (B,X)\\Delta (X,C)}{\\Delta (B,Y)\\Delta (Y,C)}\\\\&=\\frac{\\prod _{i=1}^{m+k}{(x_i)_{(b-l)}\\cdot (a+b+k+1-x_i)}_{(c-l)}}{\\prod _{j=1}^{m}{(y_j)_{(b-l)}\\cdot (a+b+k+1-y_j)_{(c-l)}}}\\end{aligned}$ Note that this ratio does not depend on a choice of a set Z.",
"Hence, by combining (3.8), (3.9) and (3.10), we have $\\begin{aligned}&\\frac{M({H_{a,b,c}^{k,l}(X:Y)})}{M({\\overline{H}_{a,b,c}^{k,b+c-l}(X:Y)})}\\\\&=\\frac{H(k+l)H(b+c-l)}{H(l)H(b+c+k-l)}\\cdot \\frac{\\sum _{Z\\subseteq [a+k+l]\\setminus ({{X}\\cup {Y}})}{\\Delta (B\\cup {X}\\cup {Z}\\cup {C})\\cdot \\Delta (Y\\cup {Z})}}{\\sum _{Z\\subseteq [a+k+l]\\setminus ({{X}\\cup {Y}})}{\\Delta (X\\cup {Z})\\cdot \\Delta (B\\cup {Y}\\cup {Z}\\cup {C})}}\\\\&=\\frac{H(k+l)H(b+c-l)}{H(l)H(b+c+k-l)}\\cdot \\frac{\\prod _{i=1}^{m+k}{(x_i)_{(b-l)}\\cdot (a+b+k+1-x_i)}_{(c-l)}}{\\prod _{j=1}^{m}{(y_j)_{(b-l)}\\cdot (a+b+k+1-y_j)_{(c-l)}}}\\\\&=\\frac{H(k+l)H(b+c-l)}{H(l)H(b+c+k-l)}\\\\&~~~\\cdot \\frac{\\prod _{i=1}^{m+k}{(x_i-b+max(b, l))_{(b-l)}\\cdot (a+k+max(b, l)+1-x_i)}_{(c-l)}}{\\prod _{j=1}^{m}{(y_j-b+max(b, l))_{(b-l)}\\cdot (a+k+max(b, l)+1-y_j)_{(c-l)}}}\\end{aligned}$ The case when $c < l \\le b+c$ can be proved similarly as we did for the case when $l \\le b$ .",
"Hence, the theorem has been proved.",
"$\\square $ Figure: NO_CAPTIONProof of Theorem 2.3.",
"Again, let's consider a case when $b < l \\le c$ first.",
"If we compare two regions $H_{a,b,c}^{k,l}(F(A_1:W_1),..., F(A_t:W_t) : m_1, ..., m_{t+1})$ and $H_{a,b,c}^{k,l}(F_{br}(A_1:W_1),..., F_{br}(A_t:W_t) : m_1,..., m_{t+1})$ , two regions are different by equilateral triangles with zig-zag horizontal boundary.",
"However, in any lozenge tiling of a region $H_{a,b,c}^{k,l}(F_{br}(A_1:W_1),..., F_{br}(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1})$ , those regions are forced to be tiled by vertical lozenges (See Figure 3.3).",
"Hence, two regions have same number of lozenge tiling.",
"Similarly, two regions $H_{a,b,c}^{k,l}(B(A_1:W_1),..., B(A_t:W_t) : m_1,..., m_{t+1})$ and $H_{a,b,c}^{k,l}(B_{br}(A_1:W_1),..., B_{br}(A_t:W_t) : m_1,..., m_{t+1})$ have same number of lozenge tilings.",
"Thus we have $\\begin{aligned}&\\frac{M(H_{a,b,c}^{k,l}(F(A_1:W_1),..., F(A_t:W_t) : m_1,..., m_{t+1}))}{M(H_{a,b,c}^{k,l}(B(A_1:W_1),..., B(A_t:W_t) : m_1,..., m_{t+1}))}\\\\&=\\frac{M(H_{a,b,c}^{k,l}(F_{br}(A_1:W_1),..., F_{br}(A_t:W_t) : m_1,..., m_{t+1}))}{M(H_{a,b,c}^{k,l}(B_{br}(A_1:W_1),..., B_{br}(A_t:W_t) : m_1,..., m_{t+1}))}\\end{aligned}$ For $i\\in [t]$ , $j\\in [r_i]$ , let $X^i_j=\\lbrace d_{NW}(L^i_j)+1, d_{NW}(L^i_j)+2,..., d_{NW}(R^i_j) (=d_{NW}(L^i_j)+a^i_j)\\rbrace $ , $V_i=\\lbrace d_{NW}(L^i_j)+1, d_{NW}(L^i_j)+2,..., d_{NW}(T^i) (=d_{NW}(L^i_j)+v_i)\\rbrace $ and $\\overline{V_i}=X^i_{u_i}\\setminus V_i=\\lbrace d_{NW}(T^i)+1,..., d_{NW}(R^i_j)\\rbrace $ .",
"Then $X^1=\\cup _{i=1}^{t}\\cup _{j\\in I_i}X^i_j$ , $X^2=\\cup _{i=1}^{t}\\cup _{j\\in J_i}X^i_j$ , $Y^1=\\cup _{i=1}^{t}((\\cup _{j=1}^{u_i-1}X^i_j)\\cup V_i)$ and $Y^2=\\cup _{i=1}^{t}(\\overline{V_i}\\cup (\\cup _{j=u_i+1}^{r_i}X^i_j))$ .",
"By same observation as we did in the Theorem 1, Lozenge tiling of a hexagonal region $H_{a,b,c}^{k,l}(F_{br}(A_1:W_1),..., F_{br}(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1})$ can be partitioned according to $(b-n-w)$ vertical unit-lozenges that are bisected by the $l$ -th horizontal line.",
"Let $H_{a,b,c}^{k,l}(F_{br}(A_1:W_1),..., F_{br}(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1} : z_1,..., z_{b-n-w})$ be a region obtained from $H_{a,b,c}^{k,l}(F_{br}(A_1:W_1),..., F_{br}(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1})$ by removing $(b-n-w)$ unit-lozenges that are bisected by segments on $l$ -th horizontal line whose labels are elements of a set $Z=\\lbrace z_1,z_2,...,z_{b-n-w}\\rbrace $ .",
"Then, by same argument as we used in proof of Theorem 2.1., we have $\\begin{aligned}&M(H_{a,b,c}^{k,l}(F_{br}(A_1:W_1),..., F_{br}(A_t:W_t) : m_1,..., m_{t+1} : z_1,..., z_{b-n-w}))\\\\&=s(Z\\cup {X^1}\\cup W\\cup C)\\cdot s(B\\cup Z\\cup {X^2}\\cup W)\\\\&=\\frac{\\Delta (Z\\cup {X^1}\\cup W\\cup C)}{H(b+c+k-l)}\\cdot \\frac{\\Delta (B\\cup Z\\cup {X^2}\\cup W)}{H(l)}\\end{aligned}$ where $B=\\lbrace -|b-l|+1,..., -1, 0\\rbrace $ and $C=\\lbrace L(l)+1, L(l)+2,..., L(l)+|c-l|\\rbrace $ .",
"If we sum over every $(b-n-w)$ element set $Z\\subset [L(l)]\\setminus (X^1\\cup X^2\\cup W)$ , then we have a representation of number of lozenge tilings of $H_{a,b,c}^{k,l}(F_{br}(A_1:W_1),..., F_{br}(A_t:W_t) : m_1,..., m_{t+1})$ as follows: $\\begin{aligned}&M(H_{a,b,c}^{k,l}(F_{br}(A_1:W_1),..., F_{br}(A_t:W_t) : m_1,..., m_{t+1}))\\\\&=\\sum _{Z}\\frac{\\Delta (Z\\cup {X^1}\\cup W\\cup C)\\cdot \\Delta (B\\cup Z\\cup {X^2}\\cup W)}{H(l)\\cdot H(b+c+k-l)}\\\\&=\\frac{\\sum _{Z}\\Delta (Z\\cup {X^1}\\cup W\\cup C)\\cdot \\Delta (B\\cup Z\\cup {X^2}\\cup W)}{H(l)\\cdot H(b+c+k-l)}\\\\\\end{aligned}$ Similarly, a number of lozenge tilings of $H_{a,b,c}^{k,l}(B_{br}(A_1:W_1),..., B_{br}(A_t:W_t) : m_1,..., m_{t+1})$ can be expressed as follow: $\\begin{aligned}&M(H_{a,b,c}^{k,l}(B_{br}(A_1:W_1),..., B_{br}(A_t:W_t) : m_1,..., m_{t+1}))\\\\&=\\sum _{Z}\\frac{\\Delta (Z\\cup {Y^1}\\cup W\\cup C)\\cdot \\Delta (B\\cup Z\\cup {Y^2}\\cup W)}{H(l)\\cdot H(b+c+k-l)}\\\\&=\\frac{\\sum _{Z}\\Delta (Z\\cup {Y^1}\\cup W\\cup C)\\cdot \\Delta (B\\cup Z\\cup {Y^2}\\cup W)}{H(l)\\cdot H(b+c+k-l)}\\\\\\end{aligned}$ Now, let's observe a ratio $\\frac{\\Delta (Z\\cup {X^1}\\cup W\\cup C)\\cdot \\Delta (B\\cup Z\\cup {X^2}\\cup W)}{\\Delta (Z\\cup {Y^1}\\cup W\\cup C)\\cdot \\Delta (B\\cup Z\\cup {Y^2}\\cup W)}$ for any set $Z\\subset [L(l)]\\setminus (X^1\\cup X^2\\cup W)$ with ($b-n-w$ ) elements: $\\begin{aligned}&\\frac{\\Delta (Z\\cup {X^1}\\cup W\\cup C)\\cdot \\Delta (B\\cup Z\\cup {X^2}\\cup W)}{\\Delta (Z\\cup {Y^1}\\cup W\\cup C)\\cdot \\Delta (B\\cup Z\\cup {Y^2}\\cup W)}\\\\&=\\frac{\\Delta (Z)\\Delta (X^1)\\Delta (W)\\Delta (C)\\Delta (Z,X^1)\\Delta (Z,W)\\Delta (Z,C)\\Delta (X^1,W)\\Delta (X^1,C)\\Delta (W,C)}{\\Delta (Z)\\Delta (Y^1)\\Delta (W)\\Delta (C)\\Delta (Z,Y^1)\\Delta (Z,W)\\Delta (Z,C)\\Delta (Y^1,W)\\Delta (Y^1,C)\\Delta (W,C)}\\\\&\\cdot \\frac{\\Delta (B)\\Delta (Z)\\Delta (X^2)\\Delta (W)\\Delta (B,Z)\\Delta (B,X^2)\\Delta (B,W)\\Delta (Z,X^2)\\Delta (Z,W)\\Delta (X^2,W)}{\\Delta (B)\\Delta (Z)\\Delta (Y^2)\\Delta (W)\\Delta (B,Z)\\Delta (B,Y^2)\\Delta (B,W)\\Delta (Z,Y^2)\\Delta (Z,W)\\Delta (Y^2,W)}\\\\&=\\frac{\\Delta (X^1)\\Delta (X^2)\\Delta (X^1,C)\\Delta (B,X^2)}{\\Delta (Y^1)\\Delta (Y^2)\\Delta (Y^1,C)\\Delta (B,Y^2)}\\\\&=\\frac{s(X^1)s(X^2)}{s(Y^1)s(Y^2)}\\cdot \\frac{\\Delta (X^1,C)}{\\Delta (Y^1,C)}\\cdot \\frac{\\Delta (B,X^2)}{\\Delta (B,Y^2)}\\end{aligned}$ In above simplification, we use a fact that $X^1 \\cup X^2=Y^1 \\cup Y^2$ , which implies $\\Delta (Z,X^1)\\Delta (Z,X^2)=\\Delta (Z,Y^1)\\Delta (Z,Y^2)$ and $\\Delta (X^1,W)\\Delta (X^2,W)=\\Delta (Y^1,W)\\Delta (Y^2,W)$ .",
"Note that what we get does not depend on a choice of a set $Z$ .",
"Hence, by (3.14), (3.15), (3.16) and (3.17), we have $\\begin{aligned}&\\frac{M(H_{a,b,c}^{k,l}(F(A_1:W_1),..., F(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))}{M(H_{a,b,c}^{k,l}(B(A_1:W_1),..., B(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))}\\\\&=\\frac{s(X^1)s(X^2)}{s(Y^1)s(Y^2)}\\cdot \\frac{\\Delta (X^1,C)}{\\Delta (Y^1,C)}\\cdot \\frac{\\Delta (B,X^2)}{\\Delta (B,Y^2)}\\end{aligned}$ Since $X^1=\\cup _{i=1}^{t}\\cup _{j\\in I_i}X^i_j$ and $Y^1=\\cup _{i=1}^{t}((\\cup _{j=1}^{u_i-1}X^i_j)\\cup V_i)$ , $\\begin{aligned}\\frac{\\Delta (X^1,C)}{\\Delta (Y^1,C)}&=\\frac{\\prod _{i=1}^{t}\\prod _{j\\in I_i}\\Delta (X^i_j, C)}{\\prod _{i=1}^{t}((\\prod _{j=1}^{u_i-1}\\Delta (X^i_j, C))\\cdot \\Delta (V_i, C))}\\\\&=\\prod _{i=1}^{t}\\Bigg [\\frac{1}{\\Delta (V_i, C)}\\prod _{j<u_i, j\\in J_i}\\frac{1}{\\Delta (X^i_j, C)}\\prod _{j\\ge u_i, j\\in I_i}\\Delta (X^i_j, C)\\Bigg ]\\end{aligned}$ However, by (3.2), we have $\\begin{aligned}\\Delta (V_i, C)&=\\frac{H(L(l)-d_{NW}(L^i_{u_i})-v_i)H(L(l)-d_{NW}(L^i_{u_i})+|c-l|)}{H(L(l)-d_{NW}(L^i_{u_i}))H(L(l)-d_{NW}(L^i_{u_i})+|c-l|-v_i)}\\\\&=\\frac{H(d_{SE}(T^i))H(d_{NE}(L^i_{u_i}))}{H(d_{SE}(L^i_{u_i}))H(d_{NE}(T^i))}\\end{aligned}$ and $\\begin{aligned}\\Delta (X^i_j, C)&=\\frac{H(L(l)-d_{NW}(L^i_{u_i})-a^i_j)H(L(l)-d_{NW}(L^i_{u_i})+|c-l|)}{H(L(l)-d_{NW}(L^i_{u_i}))H(L(l)-d_{NW}(L^i_{u_i})+|c-l|-a^i_j)}\\\\&=\\frac{H(d_{SE}(R^i_j))H(d_{NE}(L^i_j))}{H(d_{SE}(L^i_j))H(d_{NE}(R^i_j))}\\end{aligned}$ Hence, by (3.18), (3.19) and (3.20), we have $\\begin{aligned}\\frac{\\Delta (X^1,C)}{\\Delta (Y^1,C)}=&\\prod _{i=1}^{t}\\Bigg [\\frac{H(d_{SE}(L^i_{u_i}))H(d_{NE}(T^i))}{H(d_{SE}(T^i))H(d_{NE}(L^i_{u_i}))}\\\\&\\cdot \\prod _{j<u_i, j\\in J_i}\\frac{H(d_{SE}(L^i_j))H(d_{NE}(R^i_j))}{H(d_{SE}(R^i_j))H(d_{NE}(L^i_j))}\\prod _{j\\ge u_i, j\\in I_i}\\frac{H(d_{SE}(R^i_j))H(d_{NE}(L^i_j))}{H(d_{SE}(L^i_j))H(d_{NE}(R^i_j))}\\Bigg ]\\end{aligned}$ Also, since $X^2=\\cup _{i=1}^{t}\\cup _{j\\in J_i}X^i_j$ and $Y^2=\\cup _{i=1}^{t}(\\overline{V_i} \\cup (\\cup _{j=u_i+1}^{r_i}X^i_j))$ , $\\begin{aligned}\\frac{\\Delta (B,X^2)}{\\Delta (B,Y^2)}&=\\frac{\\prod _{i=1}^{t}\\prod _{j\\in J_i}\\Delta (B, X^i_j)}{\\prod _{i=1}^{t}(\\Delta (B, \\overline{V_i})\\cdot \\prod _{j=u_i+1}^{r_i}\\Delta (B, X^i_j))}\\\\&=\\prod _{i=1}^{t}\\Bigg [\\frac{\\Delta (B, X^i_{u_i})}{\\Delta (B, \\overline{V_i})}\\prod _{j<u_i, j\\in J_i}\\Delta (B, X^i_j)\\prod _{j\\ge u_i, j\\in I_i}\\frac{1}{\\Delta (B, X^i_j)}\\Bigg ]\\\\&=\\prod _{i=1}^{t}\\Bigg [\\Delta (B, V_i)\\prod _{j<u_i, j\\in J_i}\\Delta (B, X^i_j)\\prod _{j\\ge u_i, j\\in I_i}\\frac{1}{\\Delta (B, X^i_j)}\\Bigg ]\\end{aligned}$ Again, by (3.2), we have $\\begin{aligned}\\Delta (B, V_i)&=\\frac{H(d_{NW}(L^i_{u_i}))H(d_{NW}(L^i_{u_i})+|l-b|+v_i)}{H(d_{NW}(L^i_{u_i})+|l-b|)H(d_{NW}(L^i_{u_i})+v_i)}\\\\&=\\frac{H(d_{NW}(L^i_{u_i}))H(d_{SW}(T^i))}{H(d_{SW}(L^i_{u_i}))H(d_{NW}(T^i))}\\\\&=\\frac{H(d_{NW}(L^i_{u_i}))H(d_{SW}(T^i))}{H(d_{NW}(T^i))H(d_{SW}(L^i_{u_i}))}\\end{aligned}$ and $\\begin{aligned}\\Delta (B, X^i_j)&=\\frac{H(d_{NW}(L^i_{u_i}))H(d_{NW}(L^i_{u_i})+|l-b|+a^i_j)}{H(d_{NW}(L^i_{u_i})+|l-b|)H(d_{NW}(L^i_{u_i})+a^i_j)}\\\\&=\\frac{H(d_{NW}(L^i_j))H(d_{SW}(R^i_j))}{H(d_{SW}(L^i_j))H(d_{NW}(R^i_j))}\\\\&=\\frac{H(d_{NW}(L^i_j))H(d_{SW}(R^i_j))}{H(d_{NW}(R^i_j))H(d_{SW}(L^i_j))}\\end{aligned}$ Hence, by (3.22), (3.23) and (3.24), we have $\\begin{aligned}\\frac{\\Delta (B,X^2)}{\\Delta (B,Y^2)}=&\\prod _{i=1}^{t}\\Bigg [\\frac{H(d_{NW}(L^i_{u_i}))H(d_{SW}(T^i))}{H(d_{NW}(T^i))H(d_{SW}(L^i_{u_i}))}\\\\&\\cdot \\prod _{j<u_i, j\\in J_i}\\frac{H(d_{NW}(L^i_j))H(d_{SW}(R^i_j))}{H(d_{NW}(R^i_j))H(d_{SW}(L^i_j))}\\prod _{j\\ge u_i, j\\in I_i}\\frac{H(d_{NW}(R^i_j))H(d_{SW}(L^i_j))}{H(d_{NW}(L^i_j))H(d_{SW}(R^i_j))}\\Bigg ]\\end{aligned}$ Thus, by (3.17), (3.21) and (3.25), $\\begin{aligned}&\\frac{M(H_{a,b,c}^{k,l}(F(A_1:W_1),..., F(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))}{M(H_{a,b,c}^{k,l}(B(A_1:W_1),..., B(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))}\\\\&=\\frac{s(X^1)s(X^2)}{s(Y^1)s(Y^2)}\\\\&\\cdot \\prod _{i=1}^{t}\\Bigg [\\frac{H(d_{SE}(L^i_{u_i}))H(d_{NE}(T^i))H(d_{NW}(L^i_{u_i}))H(d_{SW}(T^i))}{H(d_{SE}(T^i))H(d_{NE}(L^i_{u_i}))H(d_{NW}(T^i))H(d_{SW}(L^i_{u_i}))}\\\\&\\cdot \\prod _{j<u_i, j\\in J_i}\\frac{H(d_{SE}(L^i_j))H(d_{NE}(R^i_j))H(d_{NW}(L^i_j))H(d_{SW}(R^i_j))}{H(d_{SE}(R^i_j))H(d_{NE}(L^i_j))H(d_{NW}(R^i_j))H(d_{SW}(L^i_j))}\\\\&\\cdot \\prod _{j\\ge u_i, j\\in I_i}\\frac{H(d_{SE}(R^i_j))H(d_{NE}(L^i_j))H(d_{NW}(R^i_j))H(d_{SW}(L^i_j))}{H(d_{SE}(L^i_j))H(d_{NE}(R^i_j))H(d_{NW}(L^i_j))H(d_{SW}(R^i_j))}\\Bigg ]\\end{aligned}$ Now, let's consider a case when $l \\le b$ .",
"For $i\\in [t]$ , $j\\in [r_i]$ , let $X^i_j=\\lbrace d_{SW}(L^i_j)+1, d_{SW}(L^i_j)+2,..., d_{SW}(R^i_j) (=d_{SW}(L^i_j)+a^i_j)\\rbrace $ , $V_i=\\lbrace d_{SW}(L^i_j)+1, d_{SW}(L^i_j)+2,..., d_{SW}(T^i) (=d_{SW}(L^i_j)+v_i)\\rbrace $ and $\\overline{V_i}=X^i_{u_i}\\setminus V_i=\\lbrace d_{SW}(T^i)+1,..., d_{SW}(R^i_j)\\rbrace $ .",
"Then $X^1=\\cup _{i=1}^{t}\\cup _{j\\in I_i}X^i_j$ , $X^2=\\cup _{i=1}^{t}\\cup _{j\\in J_i}X^i_j$ , $Y^1=\\cup _{i=1}^{t}((\\cup _{j=1}^{u_i-1}X^i_j)\\cup V_i)$ and $Y^2=\\cup _{i=1}^{t}(\\overline{V_i}\\cup (\\cup _{j=u_i+1}^{r_i}X^i_j))$ .",
"By same argument, the ratio can be expressed as follows: $\\begin{aligned}&\\frac{M(H_{a,b,c}^{k,l}(F(A_1:W_1),..., F(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))}{M(H_{a,b,c}^{k,l}(B(A_1:W_1),..., B(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))}\\\\&=\\frac{s(X^1)s(X^2)}{s(Y^1)s(Y^2)}\\cdot \\frac{\\Delta (X^1,C)}{\\Delta (Y^1,C)}\\cdot \\frac{\\Delta (B,X^1)}{\\Delta (B,Y^1)}\\end{aligned}$ where $B=\\lbrace -|b-l|+1,..., -1, 0\\rbrace $ and $C=\\lbrace L(l)+1, L(l)+2,..., L(l)+|c-l|\\rbrace $ .",
"However, we know that $\\begin{aligned}\\frac{\\Delta (X^1,C)}{\\Delta (Y^1,C)}&=\\frac{\\prod _{i=1}^{t}\\prod _{j\\in I_i}\\Delta (X^i_j, C)}{\\prod _{i=1}^{t}((\\prod _{j=1}^{u_i-1}\\Delta (X^i_j, C))\\cdot \\Delta (V_i, C))}\\\\&=\\prod _{i=1}^{t}\\Bigg [\\frac{1}{\\Delta (V_i, C)}\\prod _{j<u_i, j\\in J_i}\\frac{1}{\\Delta (X^i_j, C)}\\prod _{j\\ge u_i, j\\in I_i}\\Delta (X^i_j, C)\\Bigg ]\\\\\\end{aligned}$ Also, we have $\\begin{aligned}\\frac{\\Delta (B,X^1)}{\\Delta (B,Y^1)}&=\\frac{\\prod _{i=1}^{t}\\prod _{j\\in I_i}\\Delta (B, X^i_j)}{\\prod _{i=1}^{t}((\\prod _{j=1}^{u_i-1}\\Delta (B, X^i_j))\\cdot \\Delta (B, V_i))}\\\\&=\\prod _{i=1}^{t}\\Bigg [\\frac{1}{\\Delta (B, V_i)}\\prod _{j<u_i, j\\in J_i}\\Delta (B, X^i_j)\\prod _{j\\ge u_i, j\\in I_i}\\frac{1}{\\Delta (B, X^i_j)}\\Bigg ]\\\\\\end{aligned}$ However, by (3.2), we have $\\begin{aligned}\\Delta (B, X^i_j)&=\\frac{H(d_{SW}(L^i_j))H(d_{SW}(L^i_j)+|b-l|+a^i_j)}{H(d_{SW}(L^i_j)+|b-l|)H(d_{SW}(L^i_j)+a^i_j)}\\\\&=\\frac{H(d_{SW}(L^i_j))H(d_{NW}(R^i_j))}{H(d_{SW}(R^i_j))H(d_{NW}(L^i_j))}\\\\\\end{aligned}$ $\\begin{aligned}\\Delta (X^i_j, C)&=\\frac{H(L(l)-d_{SW}(L^i_j)-a^i_j)H(L(l)-d_{SW}(L^i_j)+|c-l|)}{H(L(l)-d_{SW}(L^i_j))H(L(l)-d_{SW}(L^i_j)+|c-l|-a^i_j)}\\\\&=\\frac{H(d_{SE}(R^i_j))H(d_{NE}(L^i_j))}{H(d_{SE}(L^i_j))H(d_{NE}(R^i_j))}\\\\\\end{aligned}$ and similarly $\\begin{aligned}\\Delta (B, V_i)&=\\frac{H(d_{SW}(L^i_{u_i}))H(d_{NW}(T^i))}{H(d_{SW}(T^i))H(d_{NW}(L^i_{u_i}))}, \\Delta (V_i, C)=\\frac{H(d_{SE}(T_i))H(d_{NE}(L^i_{u_i}))}{H(d_{SE}(L^i_{u_i}))H(d_{NE}(T_i))}\\end{aligned}$ Thus, by (3.27)-(3.32), we have $\\begin{aligned}&\\frac{M(H_{a,b,c}^{k,l}(F(A_1:W_1),..., F(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))}{M(H_{a,b,c}^{k,l}(B(A_1:W_1),..., B(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))}\\\\&=\\frac{s(X^1)s(X^2)}{s(Y^1)s(Y^2)}\\\\&\\cdot \\prod _{i=1}^{t}\\Bigg [\\frac{H(d_{SE}(L^i_{u_i}))H(d_{NE}(T^i))H(d_{NW}(L^i_{u_i}))H(d_{SW}(T^i))}{H(d_{SE}(T^i))H(d_{NE}(L^i_{u_i}))H(d_{NW}(T^i))H(d_{SW}(L^i_{u_i}))}\\\\&\\cdot \\prod _{j<u_i, j\\in J_i}\\frac{H(d_{SE}(L^i_j))H(d_{NE}(R^i_j))H(d_{NW}(L^i_j))H(d_{SW}(R^i_j))}{H(d_{SE}(R^i_j))H(d_{NE}(L^i_j))H(d_{NW}(R^i_j))H(d_{SW}(L^i_j))}\\\\&\\cdot \\prod _{j\\ge u_i, j\\in I_i}\\frac{H(d_{SE}(R^i_j))H(d_{NE}(L^i_j))H(d_{NW}(R^i_j))H(d_{SW}(L^i_j))}{H(d_{SE}(L^i_j))H(d_{NE}(R^i_j))H(d_{NW}(L^i_j))H(d_{SW}(R^i_j))}\\Bigg ]\\end{aligned}$ The case when $c <l$ can be proved similarly as we did for the case when $l \\le b$ .",
"Hence, the Theorem 2 has been proved.",
"$\\square $ Figure: Cyclically symmetric lozenge tiling of a region H 10,11,13 0,12 (F(-1,1,-2,0:1,0,1),F(0,2,-1,1:1,0,1):2,5,2)H_{10,11,13}^{0,12}(F(-1,1,-2,0:1,0,1), F(0,2,-1,1:1,0,1) : 2, 5, 2)Proof of Theorem 2.4.",
"Let's use the same notation as we used in the proof of Theorem 2.3.",
"Like proof of previous theorems, we label the baseline by $1, 2,..., L(\\frac{b+c}{2})$ from left to right.",
"Note that in this case, sets $X^1$ , $X^2$ , $Y^1$ , $Y^2$ and $W$ satisfy $X^2=\\lbrace L(\\frac{b+c}{2})+1-x|x\\in X^1\\rbrace $ , $Y^2=\\lbrace L(\\frac{b+c}{2})+1-y|y\\in Y^1\\rbrace $ and $W=\\lbrace L(\\frac{b+c}{2})+1-w|w\\in W\\rbrace $ because the region is centrally symmetric.",
"Crucial observation is that a centrally symmetric lozenge tiling of the region is uniquely determined by lozenges below (or above) the horizontal line (See Figure 3.4) .",
"Hence, by combining this observation and same argument that we have used in the proof of previous theorems, we have $\\begin{aligned}&M_\\odot (H_{a,b,c}^{0,\\frac{b+c}{2}}(F(A_1:W_1),..., F(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))\\\\&=\\sum _{Z}\\frac{\\Delta (Z\\cup X^1 \\cup W \\cup C)}{H(\\frac{b+c}{2})}\\\\&=\\frac{\\sum _{Z}\\Delta (Z\\cup X^1\\cup W \\cup C)}{H(\\frac{b+c}{2})}\\end{aligned}$ where the sum is taken over all sets $Z\\subset [L(\\frac{b+c}{2})]\\setminus (X^1\\cup X^2\\cup W)$ with $(b-n-w)$ elements that satisfies $Z=\\lbrace L(\\frac{b+c}{2})+1-z|z\\in Z\\rbrace $ Similarly, number of centrally symmetric lozenge tiling of the region $H_{a,b,c}^{0,\\frac{b+c}{2}}(B(A_1:W_1),..., B(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1})$ can be written as follows: $\\begin{aligned}&M_\\odot (H_{a,b,c}^{0,\\frac{b+c}{2}}(B(A_1:W_1),..., B(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))\\\\&=\\sum _{Z}\\frac{\\Delta (Z\\cup Y^1\\cup W \\cup C)}{H(\\frac{b+c}{2})}\\\\&=\\frac{\\sum _{Z}\\Delta (Z\\cup Y^1\\cup W \\cup C)}{H(\\frac{b+c}{2})}\\end{aligned}$ Again, the sum is taken over all sets $Z\\subset [L(\\frac{b+c}{2})]\\setminus (X^1\\cup X^2\\cup W)$ with $(b-n-w)$ elements that satisfies $Z=\\lbrace z\\in Z | L(\\frac{b+c}{2})+1-z\\rbrace $ .",
"For such $Z$ , we have $\\begin{aligned}\\Delta (Z, X^2)&=\\prod _{z\\in Z, x_2\\in X^2}|z-x_2|\\\\&=\\prod _{z\\in Z, x_1\\in X^1}|(L(\\frac{b+c}{2})+1-z)-(L(\\frac{b+c}{2})+1-x_1)|\\\\&=\\prod _{z\\in Z, x_1\\in X^1}|x_1-z|\\\\&=\\Delta (Z, X^1)\\end{aligned}$ Similarly, we also have $\\Delta (Z, Y^2)=\\Delta (Z, Y^1)$ .",
"Hence we have $\\begin{aligned}\\Delta (Z, X^1)=\\sqrt{\\Delta (Z, X^1)\\Delta (Z, X^2)}&=\\sqrt{\\Delta (Z, X^1\\cup X^2)}\\\\&=\\sqrt{\\Delta (Z, Y^1\\cup Y^2)}\\\\&=\\Delta (Z, Y^1)\\end{aligned}$ By same reasoning, we have $\\Delta (X^1, W)=\\Delta (Y^1, W)$ .",
"Now, we observe a ratio $\\frac{\\Delta (Z\\cup X^1\\cup W \\cup C)}{\\Delta (Z\\cup Y^1\\cup W \\cup C)}$ for any set $Z$ : $\\begin{aligned}&\\frac{\\Delta (Z\\cup X^1\\cup W \\cup C)}{\\Delta (Z\\cup Y^1\\cup W \\cup C)}\\\\&=\\frac{\\Delta (Z)\\Delta (X^1)\\Delta (W)\\Delta (C)}{\\Delta (Z)\\Delta (Y^1)\\Delta (W)\\Delta (C)}\\\\&\\cdot \\frac{\\Delta (Z,X^1)\\Delta (Z,W)\\Delta (Z,C)\\Delta (X^1,W)\\Delta (X^1,C)\\Delta (W,C)}{\\Delta (Z,Y^1)\\Delta (Z,W)\\Delta (Z,C)\\Delta (Y^1,W)\\Delta (Y^1,C)\\Delta (W,C)}\\\\&=\\frac{s(X^1)}{s(Y^1)}\\cdot \\frac{\\Delta (X^1,C)}{\\Delta (Y^1,C)}\\end{aligned}$ Since this ratio does not depend on a choice of a set $Z$ , by (3.35), (3.36) and (3.40), we have $\\begin{aligned}&\\frac{M_\\odot (H_{a,b,c}^{0,\\frac{b+c}{2}}(F(A_1:W_1),..., F(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))}{M_\\odot (H_{a,b,c}^{0,\\frac{b+c}{2}}(B(A_1:W_1),..., B(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))}\\\\&=\\frac{s(X^1)}{s(Y^1)}\\cdot \\frac{\\Delta (X^1,C)}{\\Delta (Y^1,C)}\\end{aligned}$ However, as we have seen in the proof of the Theorem 2.3., $\\begin{aligned}&\\frac{M(H_{a,b,c}^{0,\\frac{b+c}{2}}(F(A_1:W_1),..., F(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))}{M(H_{a,b,c}^{0,\\frac{b+c}{2}}(B(A_1:W_1),..., B(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))}\\\\&=\\frac{s(X^1)s(X^2)}{s(Y^1)s(Y^2)}\\cdot \\frac{\\Delta (X^1,C)}{\\Delta (Y^1,C)}\\cdot \\frac{\\Delta (B,X^2)}{\\Delta (B,Y^2)}\\end{aligned}$ Since our region is centrally symmetric, we have $\\begin{aligned}s(X^1)&=\\frac{1}{H(p)}\\prod _{x<y, x,y\\in X^1}(y-x)\\\\&=\\frac{1}{H(n)}\\prod _{x<y, x,y\\in X^1}((L(\\frac{b+c}{2})+1-x)-(L(\\frac{b+c}{2})+1-y))\\\\&=\\frac{1}{H(n)}\\prod _{y^{\\prime }<x^{\\prime }, x^{\\prime },y^{\\prime }\\in X^2}(x^{\\prime }-y^{\\prime })\\\\&=s(X^2)\\end{aligned}$ Similarly, $s(Y^1)=s(Y^2)$ , $\\Delta (B,X^2)=\\Delta (X^1,C)$ and $\\Delta (B,Y^2)=\\Delta (Y^1,C)$ Hence we have $\\begin{aligned}&\\frac{M_\\odot (H_{a,b,c}^{0,\\frac{b+c}{2}}(F(A_1:W_1),..., F(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))}{M_\\odot (H_{a,b,c}^{0,\\frac{b+c}{2}}(B(A_1:W_1),..., B(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))}\\\\&=\\frac{s(X^1)}{s(Y^1)}\\cdot \\frac{\\Delta (X^1,C)}{\\Delta (Y^1,C)}\\\\&=\\sqrt{\\frac{s(X^1)s(X^2)}{s(Y^1)s(Y^2)}\\cdot \\frac{\\Delta (X^1,C)}{\\Delta (Y^1,C)}\\cdot \\frac{\\Delta (B,X^2)}{\\Delta (B,Y^2)}}\\\\&=\\sqrt{\\frac{M(H_{a,b,c}^{0,\\frac{b+c}{2}}(F(A_1:W_1),..., F(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))}{M(H_{a,b,c}^{0,\\frac{b+c}{2}}(B(A_1:W_1),..., B(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))}}\\end{aligned}$ Also, by (3.21), $\\begin{aligned}\\frac{\\Delta (X^1,C)}{\\Delta (Y^1,C)}=&\\prod _{i=1}^{t}\\Bigg [\\frac{H(d_{SE}(L^i_{u_i})H(d_{NE}(T^i)}{H(d_{SE}(T^i)H(d_{NE}(L^i_{u_i})}\\\\&\\cdot \\prod _{j<u_i, j\\in J_i}\\frac{H(d_{SE}(L^i_j)H(d_{NE}(R^i_j)}{H(d_{SE}(R^i_j)H(d_{NE}(L^i_j)}\\prod _{j\\ge u_i, j\\in I_i}\\frac{H(d_{SE}(R^i_j)H(d_{NE}(L^i_j)}{H(d_{SE}(L^i_j)H(d_{NE}(R^i_j)}\\Bigg ]\\end{aligned}$ Hence, by (3.43) and (3.44), we have $\\begin{aligned}&\\frac{M_\\odot (H_{a,b,c}^{0,\\frac{b+c}{2}}(F(A_1:W_1),..., F(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))}{M_\\odot (H_{a,b,c}^{0,\\frac{b+c}{2}}(B(A_1:W_1),..., B(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))}\\\\&=\\sqrt{\\frac{M(H_{a,b,c}^{0,\\frac{b+c}{2}}(F(A_1:W_1),..., F(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))}{M(H_{a,b,c}^{0,\\frac{b+c}{2}}(B(A_1:W_1),..., B(A_t:W_t) : m_1, m_2,..., m_t, m_{t+1}))}}\\\\&=\\frac{s(X^1)}{s(Y^1)}\\cdot \\prod _{i=1}^{t}\\Bigg [\\frac{H(d_{SE}(L^i_{u_i}))H(d_{NE}(T^i))}{H(d_{SE}(T^i))H(d_{NE}(L^i_{u_i}))}\\\\&\\cdot \\prod _{j<u_i, j\\in J_i}\\frac{H(d_{SE}(L^i_j))H(d_{NE}(R^i_j))}{H(d_{SE}(R^i_j))H(d_{NE}(L^i_j))}\\prod _{j\\ge u_i, j\\in I_i}\\frac{H(d_{SE}(R^i_j))H(d_{NE}(L^i_j))}{H(d_{SE}(L^i_j))H(d_{NE}(R^i_j))}\\Bigg ]\\end{aligned}$"
],
[
"Acknowledgement",
"The author would like to thank to his advisor, Professor Mihai Ciucu for his encouragement and useful discussions.",
"The geometric interpretation of terms in formulas which unifies the results is due to him.",
"Also, the author thanks Jeff Taylor for installing software and frequent helpful assistance."
]
] | 1906.04532 |
[
[
"Generative Adversarial Networks are Special Cases of Artificial\n Curiosity (1990) and also Closely Related to Predictability Minimization\n (1991)"
],
[
"Abstract I review unsupervised or self-supervised neural networks playing minimax games in game-theoretic settings: (i) Artificial Curiosity (AC, 1990) is based on two such networks.",
"One network learns to generate a probability distribution over outputs, the other learns to predict effects of the outputs.",
"Each network minimizes the objective function maximized by the other.",
"(ii) Generative Adversarial Networks (GANs, 2010-2014) are an application of AC where the effect of an output is 1 if the output is in a given set, and 0 otherwise.",
"(iii) Predictability Minimization (PM, 1990s) models data distributions through a neural encoder that maximizes the objective function minimized by a neural predictor of the code components.",
"I correct a previously published claim that PM is not based on a minimax game."
],
[
"Introduction",
"Computer science has a rich history of problem solving through computational procedures seeking to minimize an objective function maximized by another procedure.",
"For example, chess programs date back to 1945 [104], and for many decades have successfully used a recursive minimax procedure with continually shrinking look-ahead, e.g., [100].",
"Game theory of adversarial players originated in 1944 [46].",
"In the field of machine learning, early adversarial settings include reinforcement learners playing against themselves [60] (1959), or the evolution of parasites in predator-prey games, e.g., [26], [87] (1990).",
"In 1990, a new type of adversarial technique was introduced in the field of unsupervised or self-supervised artificial neural networks (NNs) [61], [65] (Sec.",
").",
"Here a single agent has two separate learning NNs.",
"Without a teacher, and without external reward for achieving user-defined goals, the first NN somehow generates outputs.",
"The second NN learns to predict consequences or properties of the generated outputs, minimizing its errors, typically by gradient descent.",
"However, the first NN maximizes the objective function minimized by the second NN, effectively trying to generate data from which the second NN can still learn to improve its predictions.",
"This survey will review such unsupervised minimax techniques, and relate them to each other.",
"Sec.",
"focuses on unsupervised Reinforcement Learning (RL) through Artificial Curiosity (since 1990).",
"Here the prediction errors are (intrinsic) reward signals maximized by an RL controller.",
"Sec.",
"points out that Generative Adversarial Networks (GANs, 2010-2014) and its variants are special cases of this approach.",
"Sec.",
"discusses a more sophisticated adversarial approach of 1997.",
"Sec.",
"addresses unsupervised encoding of data through Predictability Minimization (PM, 1991), where the predictor's error is maximized by the encoder's feature extractors.",
"Sec.",
"addresses issues of convergence.",
"For historical accuracy, I will sometimes refer not only to peer-reviewed publications but also to technical reports, many of which turned into reviewed journal articles later."
],
[
"Adversarial Artificial Curiosity (AC, 1990)",
"In 1990, unsupervised or self-supervised adversarial NNs were used to implement curiosity [61], [65] in the general context of exploration in RL [31], [92], [101] (see Sec.",
"6 of [79] for a survey of deep RL).",
"The goal was to overcome drawbacks of traditional reward-maximizing RL machines which use naive strategies (such as random action selection) to explore their environments.",
"The basic idea is: An RL agent with a predictive NN world model maximizes intrinsic reward obtained for provoking situations where the error-minimizing world model still has high error and can learn more.",
"I will refer to this approach as Adversarial Artificial Curiosity (AC) of 1990, or AC1990 for short, to distinguish it from our later types of Artificial Curiosity since 1991 (Sec.",
").",
"In what follows, let $m,n,q$ denote positive integer constants.",
"In the AC context, the first NN is often called the controller C. C may interact with an environment through sequences of interactions called trials or episodes.",
"During the execution of a single interaction of any given trial, C generates an output vector $x \\in \\mathbb {R}^n$ .",
"This may influence an environment, which produces a reaction to $x$ in form of an observation $y \\in \\mathbb {R}^q$ .",
"In turn, $y$ may affect C's inputs during the next interaction if there is any.",
"In the first variant of AC1990 [61], [65], C is recurrent, and thus a general purpose computer.",
"Some of C's adaptive recurrent units are mean and variance-generating Gaussian units, such that C can become a generative model that produces a probability distribution over outputs—see Section “Explicit Random Actions versus Imported Randomness” [61] (see also [66], [102]).",
"(What these stochastic units do can be equivalently accomplished by having C perceive pseudorandom numbers or noise, like the generator NNs of GANs [20]; Sec.",
").",
"To compute an output action during an interaction, C updates all its NN unit activations for several discrete time steps in a row—see Section “More Network Ticks than Environmental Ticks” [61].",
"In principle, this allows for computing highly nonlinear, stochastic mappings from environmental inputs (if there are any) and/or from internal “noise” to outputs.",
"The second NN is called the world model M [61], [62], [66], [23].",
"In the first variant of AC1990 [61], [65], M is also recurrent, for reasons of generality.",
"M receives C's outputs $x \\in \\mathbb {R}^n$ as inputs and predicts their visible environmental effects or consequences $y \\in \\mathbb {R}^q$ .",
"According to AC1990, M minimizes its prediction errors by gradient descent, thus becoming a better predictor.",
"In absence of external reward, however, the adversarial C tries to find actions that maximize the errors of M: M's errors are the intrinsic rewards of C. Hence C maximizes the errors that M minimizes.",
"The loss of M is the gain of C. Without external reward, C is thus intrinsically motivated to invent novel action sequences or experiments that yield data that M still finds surprising, until the data becomes familiar and boring.",
"The 1990 paper [61] describes gradient-based learning methods for both C and M. In particular, backpropagation [40], [41] through the model M down into the controller C (whose outputs are inputs to M) is used to compute weight changes for C, generalizing previous work on feedforward networks [99], [98], [47], [30].",
"This is closely related to how the code generator NN of Predictability Minimization (Sec. )",
"can be trained by backpropagation through its predictor NN [64], [67], [82], and to how the GAN generator NN (Sec. )",
"can be trained by backpropagation through its discriminator NN [50], [20].",
"Furthermore, the concept of backpropagation through random number generators [102] is used to derive error signals even for those units of C that are stochastic [61].",
"However, the original AC1990 paper points out that the basic ideas of AC are not limited to particular learning algorithms—see Section “Implementing Dynamic Curiosity and Boredom” [61].",
"Compare more recent summaries and later variants / extensions of AC1990's simple but powerful exploration principle [74], [77], which inspired much later work, e.g., [89], [52], [77]; compare [53], [54], [8].",
"See also related work of 1993 [39], [38].",
"To summarize, unsupervised or self-supervised minimax-based neural networks of the previous millennium (now often called CM systems [80]) were both adversarial and generative (using terminology of 2014 [20], Sec.",
"), stochastically generating outputs yielding experimental data, not only for stationary patterns but also for pattern sequences, even for the general case of RL, and even for recurrent NN-based RL in partially observable environments [61], [65]."
],
[
"A Special Case of AC1990: Generative Adversarial Networks",
"Let us now consider a special case of a curious CM system as in Sec.",
"above, where each sequence of interactions of the CM system with its environment (each trial) is limited to a single interaction, like in bandit problems [59], [19], [3], [2].",
"The environment contains a representation of a user-given training set $X$ of patterns $ \\in \\mathbb {R}^n$ .",
"$X$ is not directly visible to C and M, but its properties are probed by AC1990 through C's outputs or actions or experiments.",
"In the beginning of any given trial, the activations of all units in C are reset.",
"C is blind (there is no input from the environment).",
"Using its internal stochastic units [61], [66] (Sec.",
"), C then computes a single output $x \\in \\mathbb {R}^n$ .",
"In a pre-wired fraction of all cases, $x$ is replaced by a randomly selected “real” pattern $\\in X$ (the simple default exploration policy of traditional RL chooses a random action in a fixed fraction of all cases [31], [92], [101]).",
"This ensures that M will see both “fake” and “real” patterns.",
"The environment will react to output $x$ and return as its effect a binary observation $y \\in \\mathbb {R}$ , where $y=1$ if $x \\in X$ , and $y=0$ otherwise.",
"As always in AC1990-like systems, M now takes C's output $x$ as an input, and predicts its environmental effect $y$ , in that case a single bit of information, 1 or 0.",
"As always, M learns by minimizing its prediction errors.",
"However, as always in absence of external reward, the adversarial C is learning to generate outputs that maximize the error minimized by M. M's loss is C's negative loss.",
"Since the stochastic C is trained to maximize the objective function minimized by M, C is motivated to produce a distribution over more and more realistic patterns, e.g., images.",
"Since 2014, this particular application of the AC principle (1990) has been called a Generative Adversarial Network (GAN) [20].",
"M was called the discriminator, C was called the generator.",
"GANs and related approaches are now widely used and studied, e.g., [56], [13], [28], [51], [1], [18], [42], [7], [95]."
],
[
"Additional comments on AC1990 & GANs & Actor-Critic",
"The first variant of AC1990 [61], [65], [57] generalized to the case of recurrent NNs a well-known way [99], [98], [47], [49], [30], [83] of using a differentiable world model M to approximate gradients for C's parameters even when environmental rewards are non-differentiable functions of C's actions.",
"In the simple differentiable GAN environment above, however, there are no such complications, since the rewards of C (the 1-dimensional errors of M) are differentiable functions of C's outputs.",
"That is, standard backpropagation [40] can directly compute the gradients of C's parameters with respect to C's rewards, like in Predictability Minimization (1991) [64], [67], [68], [82], [86], [72] (Sec.",
").",
"Unlike the first variant of AC1990 [61], [65], most current GAN applications use more limited feedforward NNs rather than recurrent NNs to implement M and C. The stochastic units of C are typically implemented by feeding noise sampled from a given probability distribution into C's inputs [20].In the GAN-like AC1990 setup of Sec.",
", real patterns (say, images) are produced in a pre-wired fraction of all cases.",
"However, one could easily give C the freedom to decide by itself to focus on particular real images $\\in X$ that M finds still difficult to process.",
"For example, one could employ the following procedure: once C has generated a fake image $\\hat{x} \\in \\mathbb {R}^n$ , and the activation of a special hidden unit of C is above a given threshold, say, 0.5, then $\\hat{x}$ is replaced by the pattern in $X$ most similar to $\\hat{x}$ , according to some similarity measure.",
"In this case, C is not only motivated to learn to generate almost realistic fake images that are still hard to classify by M, but also to address and focus on those real images that are still hard on M. This may be useful as C sometimes may find it easier to fool M by sending it a particular real image, rather than a fake image.",
"To my knowledge, however, this is rarely done with standard GANs.",
"Actor-Critic methods [35], [93] are less closely related to GANs as they do not embody explicit minimax games.",
"Nevertheless, a GAN can be seen as a modified Actor-Critic with a blind C in a stateless MDP [55].",
"This in turn yields another connection between AC1990 and Actor-Critic (compare also Section \"Augmenting the Algorithm by Temporal Difference Methods” [61])."
],
[
"A closely related special case of AC1990: Conditional GANs (2010)",
"Unlike AC1990 [61] and the GAN of 2014 [20], the GAN of 2010 [50] (now known as a conditional GAN or cGAN [45]) does not have an internal source of randomness.",
"Instead, such cGANs depend on sufficiently diverse inputs from the environment.",
"cGANs are also special cases of the AC principle (1990): cGAN-like additional environmental inputs just mean that the controller C of AC1990 is not blind any more like in the example above with the GAN of 2014 [20].",
"Like the first version of AC1990 [61], the cGAN of 2010 [50] minimaxed Least Squares errors.",
"This was later called LSGAN [43]."
],
[
"AC1990 and StyleGANs (2019)",
"The GAN of 2014 [20] perceives noise vectors (typically sampled from a Gaussian) in its input layer and maps them to outputs.",
"The more general StyleGAN [33], however, allows for noise injection in deeper hidden layers as well, to implement all sorts of hierarchically structured probability distributions.",
"Note that this kind of additional probabilistic expressiveness was already present in the mean and variance-generating Gaussian units of the recurrent generator network C of AC1990 [61] (Sec.",
")."
],
[
"Summary: GANs and cGANs etc. are simple instances of AC1990",
"cGANs (2010) and GANs (2014) are quite different from certain earlier adversarial machine learning settings [60], [26] (1959-1990) which neither involved unsupervised neural networks nor were about modeling data nor used gradient descent (see Sec.",
").",
"However, GANs and cGANs are very closely related to AC1990.",
"GANs are essentially an application of the Adversarial Artificial Curiosity principle of 1990 (Sec. )",
"where the generator network C is blind and the environment simply returns whether C's current output is in a given set.",
"As always, C maximizes the function minimized by M (Sec.",
").",
"Same for cGANS, except that in this case C is not blind any more (Sec.",
"REF ).",
"Similar for StyleGANs (Sec.",
"REF )."
],
[
"The generality of AC1990",
"It should be emphasized though that AC1990 has much broader applicability [89], [52], [77], [8] than the GAN-like special cases above.",
"In particular, C may sequentially interact with the environment for a long time, producing a sequence of environment-manipulating outputs resulting in complex environmental constructs.",
"For example, C may trigger actions that generate brush strokes on a canvas, incrementally refining a painting over time, e.g., [22], [17], [103], [27], [48].",
"Similarly, M may sequentially predict many other aspects of the environment besides the single bit of information in the GAN-like setup above.",
"General AC1990 is about unsupervised or self-supervised RL agents that actively shape their observation streams through their own actions, setting themselves their own goals through intrinsic rewards, exploring the world by inventing their own action sequences or experiments, to discover novel, previously unknown predictability in the data generated by the experiments.",
"Not only the 1990s but also recent years saw successful applications of this simple principle (and variants thereof) in sequential settings, e.g., [54], [8].",
"Since the GAN-like environment above is restricted to a teacher-given set $X$ of patterns and a procedure deciding whether a given pattern is in $X$ , the teacher will find it rather easy to evaluate the quality of C's $X$ -imitating behavior.",
"In this sense the GAN setting is “more” supervised than certain other applications of AC1990, which may be “highly\" unsupervised in the sense that C may have much more freedom when it comes to selecting environment-affecting actions."
],
[
"Improvements of AC1990",
"Numerous improvements of the original AC1990 [61], [65] are summarized in more recent surveys [74], [77].",
"Let us focus here on a first important improvement of 1991.",
"The errors of AC1990's M (to be minimized) are the rewards of its C (to be maximized, Sec.",
").",
"This makes for a fine exploration strategy in many deterministic environments.",
"In stochastic environments, however, this might fail.",
"C might learn to focus on those parts of the environment where M can always get high prediction errors due to randomness, or due to computational limitations of M. For example, an agent controlled by C might get stuck in front of a TV screen showing highly unpredictable white noise, e.g., [77] (see also [8]).",
"Therefore, as pointed out in 1991, in stochastic environments, C's reward should not be the errors of M, but (an approximation of) the first derivative of M's errors across subsequent training iterations, that is, M's improvements [63], [75].",
"As a consequence, despite M's high errors in front of the noisy TV screen above, C won't get rewarded for getting stuck there, simply because M's errors won't improve.",
"Both the totally predictable and the fundamentally unpredictable will get boring.",
"This insight led to lots of follow-up work [77].",
"For example, one particular RL approach for AC in stochastic environments was published in 1995 [91].",
"A simple M learned to predict or estimate the probabilities of the environment's possible responses, given C's actions.",
"After each interaction with the environment, C's reward was the KL-Divergence [37] between M's estimated probability distributions before and after the resulting new experience (the information gain) [91].",
"(This was later also called Bayesian Surprise [29]; compare earlier work on information gain and its maximization without NNs [88], [16].)",
"AC1990's above-mentioned limitations in probabilistic environments, however, are not an issue in the simple GAN-like setup of Sec.",
", because there the environmental reactions are totally deterministic: For each image-generating action of C, there is a unique deterministic binary response from the environment stating whether the generated image is in $X$ or not.",
"Hence it is not obvious that above-mentioned improvements of AC1990 hold promise also for GANs."
],
[
"AC1997: Adversarial Brains Bet on Outcomes of Probabilistic Programs",
"Of particular interest in the context of the present paper is one more advanced adversarial approach to curious exploration of 1997 [70], [71], [73], referred to as AC1997.",
"AC1997 is about generating computational experiments in form of programs whose execution may change both an external environment and the RL agent's internal state.",
"An experiment has a binary outcome: either a particular effect happens, or it doesn't.",
"Experiments are collectively proposed by two reward-maximizing adversarial policies.",
"Both can predict and bet on experimental outcomes before they happen.",
"Once such an outcome is actually observed, the winner will get a positive reward proportional to the bet, and the loser a negative reward of equal magnitude.",
"So each policy is motivated to create experiments whose yes/no outcomes surprise the other policy.",
"The latter in turn is motivated to learn something about the world that it did not yet know, such that it is not outwitted again.",
"More precisely, a single RL agent has two dueling, reward-maximizing policies called the left brain and the right brain.",
"Each brain is a modifiable probability distribution over programs running on a general purpose computer.",
"Experiments are programs sampled in a collaborative way that is influenced by both brains.",
"Each experiment specifies how to execute an instruction sequence (which may affect both the environment and the agent's internal state), and how to compute the outcome of the experiment through instructions implementing a computable function (possibly resulting in an internal binary yes/no classification) of the observation sequence triggered by the experiment.",
"The modifiable parameters of both brains are instruction probabilities.",
"They can be accessed and manipulated through programs that include subsequences of special self-referential policy-modifying instructions [69], [84].",
"Both brains may also trigger the execution of certain bet instructions whose effect is to predict experimental outcomes before they are observed.",
"If their predictions or hypotheses differ, they may agree to execute the experiment to determine which brain was right, and the surprised loser will pay an intrinsic reward (the real-valued bet, e.g., 1.0) to the winner in a zero sum game.",
"That is, each brain is intrinsically motivated to outwit or surprise the other by proposing an experiment such that the other agrees on the experimental protocol but disagrees on the predicted outcome.",
"This outcome is typically an internal computable abstraction of complex spatio-temporal events generated through the execution the self-invented experiment.",
"This motivates the unsupervised or self-supervised two brain system to focus on \"interesting\" computational questions, losing interest in \"boring\" computations (potentially involving the environment) whose outcomes are consistently predictable by both brains, as well as computations whose outcomes are currently still hard to predict by either brain.",
"Again, in the absence of external reward, each brain maximizes the value function minimised by the other.",
"Using the meta-learning Success-Story RL algorithm [69], [84], AC1997 learns when to learn and what to learn [70], [71], [73].",
"AC1997 will also minimize the computational cost of learning new skills, provided both brains receive a small negative reward for each computational step, which introduces a bias towards simple still surprising experiments (reflecting simple still unsolved problems).",
"This may facilitate hierarchical construction of more and more complex experiments, including those yielding external reward (if there is any).",
"In fact, AC1997's artificial creativity may not only drive artificial scientists and artists, e.g., [76], but can also accelerate the intake of external reward, e.g., [70], [73], intuitively because a better understanding of the world can help to solve certain problems faster.",
"Other RL or evolutionary algorithms could also be applied to such two-brain systems implemented as two interacting (possibly recurrent) RL NNs or other computers.",
"However, certain issues such as catastrophic forgetting are presumably better addressed by the later PowerPlay framework (2011) [78], [90], which offers an asymptotically optimal way of finding the simplest yet unsolved problem in a (potentially infinite) set of formalizable problems with computable solutions, and adding its solution to the repertoire of a more and more general, curious problem solver.",
"Compare also the One Big Net For Everything [81] which offers a simplified, less strict NN version of PowerPlay.",
"How does AC1997 relate to GANs?",
"AC1997 is similar to standard GANs in the sense that both are unsupervised generative adversarial minimax players and focus on experiments with a binary outcome: 1 or 0, yes or no, hypothesis true or false.",
"However, for GANs the experimental protocol is prewired and always the same: It simply tests whether a recently generated pattern is in a given set or not (Sec.",
").",
"One can restrict AC1997 to such simple settings by limiting its domain and the nature of the instructions in its programming language, such that possible bets of both brains are limited to binary yes/no outcomes of GAN-like experiments.",
"In general, however, the adversarial brains of AC1997 can invent essentially arbitrary computational questions or problems by themselves, generating programs that interact with the environment in any computable way that will yield binary results on which both brains can bet.",
"A bit like a pure scientist deriving internal joy signals from inventing experiments that yield discoveries of initially surprising but learnable and then reliably repeatable predictabilities."
],
[
"Predictability Minimization (PM)",
"An important NN task is to learn the statistics of given data such as images.",
"To achieve this, the principles of gradient descent/ascent were used in yet another type of unsupervised minimax game where one NN minimizes the objective function maximized by another.",
"This duel between two unsupervised adversarial NNs was introduced in the 1990s in a series of papers [64], [67], [68], [82], [86], [72].",
"It was called Predictability Minimization (PM).",
"PM's goal is to achieve an important goal of unsupervised learning, namely, an ideal, disentangled, factorial code [5], [4] of given data, where the code components are statistically independent of each other.",
"That is, the codes are distributed like the data, and the probability of a given data pattern is simply the product of the probabilities of its code components.",
"Such codes may facilitate subsequent downstream learning [82], [86], [72].",
"PM requires an encoder network with initially random weights.",
"It maps data samples $x \\in \\mathbb {R}^n$ (such as images) to codes $y \\in [0,1]^m$ represented across $m$ so-called code units.",
"In what follows, integer indices $i,j$ range over $1,\\ldots ,m$ .",
"The $i$ -th component of $y$ is called $y_i \\in [0,1]$ .",
"A separate predictor network is trained by gradient descent to predict each $y_i$ from the remaining components $y_j (j \\ne i)$ .",
"The encoder, however, is trained to maximize the same objective function (e.g., mean squared error) minimized by the predictor.",
"Compare the text near Equation 2 in the 1996 paper [82]: “The clue is: the code units are trained (in our experiments by online backprop) to maximize essentially the same objective function the predictors try to minimize;\" or Equation 3 in Sec.",
"4.1 of the 1999 paper [72]: “But the code units try to maximize the same objective function the predictors try to minimize.\"",
"Why should the end result of this fight between predictor and encoder be a disentangled factorial code?",
"Using gradient descent, to maximize the prediction errors, the code unit activations $y_j$ run away from their real-valued predictions in $[0,1]$ , that is, they are forced towards the corners of the unit interval, and tend to become binary, either 0 or 1.",
"And according to a proof of 1992 [12], [68],It should be mentioned that the above-mentioned proof [12], [68] is limited to binary factorial codes.",
"There is no proof that PM is a universal method for approximating all kinds of non-binary distributions (most of which are incomputable anyway).",
"Nevertheless, it is well-known that binary Bernoulli distributions can approximate at least Gaussians and other distributions, that is, with enough binary code units one should get at least arbitrarily close approximations of broad classes of distributions.",
"In the PM papers of the 1990s, however, this was not studied in detail.",
"the encoder's objective function is maximized when the $i$ -th code unit maximizes its variance (thus maximizing the information it conveys about the input data) while simultaneously minimizing the deviation between its (unconditional) expected activations $E(y_i)$ and its predictor-modeled, conditional expected activations $E(y_i \\mid \\lbrace y_j, j \\ne i \\rbrace )$ , given the other code units.",
"See also conjecture 6.4.1 and Sec.",
"6.9.3 of the thesis [68].",
"That is, the code units are motivated to extract informative yet mutually independent binary features from the data.",
"PM's inherent class of probability distributions is the set of multivariate binomial distributions.",
"In the ideal case, PM has indeed learned to create a binary factorial code of the data.",
"That is, in response to some input pattern, each $y_i$ is either 0 or 1, and the predictor has learned the conditional expected value $E(y_i \\mid \\lbrace y_j, j \\ne i \\rbrace )$ .",
"Since the code is both binary and factorial, this value is equal to the code unit's unconditional probability $P(y_i=1)$ of being on (e.g., [67], Equation in Sec.",
"2).",
"E.g., if some code unit's prediction is 0.25, then the probability of this code unit being on is 1/4.",
"The first toy experiments with PM [64] were conducted nearly three decades ago when compute was about a million times more expensive than today.",
"When it had become about 10 times cheaper 5 years later, it was shown that simple semi-linear PM variants applied to images automatically generate feature detectors well-known from neuroscience, such as on-center-off-surround detectors, off-center-on-surround detectors, orientation-sensitive bar detectors, etc [82], [86]."
],
[
"Is it true that PM is NOT a minimax game?",
"The NIPS 2014 GAN paper [20] states that PM differs from GANs in the sense that PM is NOT based on a minimax game with a value function that one agent seeks to maximize and the other seeks to minimise.",
"It states that for GANs \"the competition between the networks is the sole training criterion, and is sufficient on its own to train the network,\" while PM \"is only a regularizer that encourages the hidden units of a neural network to be statistically independent while they accomplish some other task; it is not a primary training criterion\" [20].",
"But this claim is incorrect, since PM is indeed a pure minimax game, too, e.g., [82], Equation 2.",
"There is no \"other task.\"",
"In particular, PM was also trained [64], [67], [68], [82], [86], [72] (also on images [82], [86]) such that \"the competition between the networks is the sole training criterion, and is sufficient on its own to train the network.\""
],
[
"Learning generative models through PM variants",
"One of the variants in the first peer-reviewed PM paper ([67] e.g., Sec 4.3, 4.4) had an optional decoder (called reconstructor) attached to the code such that data can be reconstructed from its code.",
"Let's assume that PM has indeed found an ideal factorial code of the data.",
"Since the codes are distributed like the data, with the decoder, we could immediately use the system as a generative model, by randomly activating each binary code unit according to its unconditional probability (which for all training patterns is now equal to the activation of its prediction—see Sec.",
"), and sampling output data through the decoder.Note that even one-dimensional data may have a complex distribution whose binary factorial code (Sec. )",
"may require many dimensions.",
"PM's goal is the discovery of such a code, with an a priori unknown number of components.",
"For example, if there are 8 input patterns, each represented by a single real-valued number between 0 and 1, each occurring with probability 1/8, then there is an ideal binary factorial code across 3 binary code units, each active with probability 1/2.",
"Through a decoder on top of the 3-dimensional code of the 1-dimensional data we could resample the original data distribution, by randomly activating each of the 3 binary code units with probability 50% (these probabilities are actually directly visible as predictor activations).",
"With an accurate decoder, the sampled data must obey the statistics of the original distribution, by definition of factorial codes.",
"However, to my knowledge, this straight-forward application as a generative model was never explicitly mentioned in any PM paper, and the decoder (as well as additional, optional local variance maximization for the code units) was actually omitted in several PM papers after 1993 [82], [86], [72] which focused on unsupervised learning of disentangled internal representations, to facilitate subsequent downstream learning [82], [86], [72].",
"Nevertheless, generative models producing data through stochastic outputs of minimax-trained NNs were described in 1990 [61], [65] (see Sec.",
"on Adversarial Artificial Curiosity) and 2014 [20] (Sec.",
").",
"Compare also the concept of Adversarial Autoencoders [42]."
],
[
"Learning factorial codes through GAN variants",
"PM variants could easily be used as GAN-like generative models (Sec.",
"REF ).",
"In turn, GAN variants could easily be used to learn factorial codes like PM.",
"If we take a GAN generator network trained on random input codes with independent components, and attach a traditional encoder network to its output layer, and train this encoder to map the output patterns back to their original random codes, then in the ideal case this encoder will become a factorial code generator that can also be applied to the original data.",
"This was not done by the GANs of 2014 [20].",
"However, compare InfoGANs [9] and related work [42], [14], [15]."
],
[
"Relation between PM and GANs and their variants",
"Both PM and GANs are unsupervised learning techniques that model the statistics of given data.",
"Both employ gradient-based adversarial nets that play a minimax game to achieve their goals.",
"While PM tries to make easily decoded, random-looking, factorial codes of the data, GANs try to make decoded data directly from random codes.",
"In this sense, the inputs of PM's encoders are like the outputs of GAN's decoders, while the outputs of PM's encoders are like the inputs of GAN's decoders.",
"In another sense, the outputs of PM's encoders are like the outputs of GAN's decoders because both are shaped by the adversarial loss.",
"Effectively, GANs are trying to approximate the true data distribution through some other distribution of a given type (e.g.",
"Gaussian, binomial, etc).",
"Likewise, PM is trying to approximate it through a multivariate factorial binomial distribution, whose nature is also given in advance (see Footnote 2).",
"While other post-PM methods such as the Information Bottleneck Method [94] based on rate distortion theory [11], [10], Variational Autoencoders [34], [58], Noise-Contrastive Estimation [21], and Self-Supervised Boosting [96] also exhibit certain relationships to PM, none of them employs gradient-based adversarial NNs in a PM-like minimax game.",
"GANs do.",
"A certain duality between PM variants with attached decoders (Sec.",
"REF ) and GAN variants with attached encoders (Sec.",
"REF ) can be illustrated through the following work flow pipelines (view them as very similar 4 step cycles by identifying their beginnings and ends—see Fig.",
"REF ): Figure: Symmetric work flows of PM and GAN variants.",
"Both PM and GANs model given data distributions in unsupervised fashion.",
"PM uses gradient-based minimax or adversarial training to learn an encoder of the data, such that the codes are distributed like the data, and the probability of a given pattern can be read off its code as the product of the predictor-modeled probabilities of the code components (Sec.",
").",
"GANs, however, use gradient-based minimax or adversarial training to directly learn a decoder of given codes (Sec.",
").",
"In turn, to decode its codes again, PM can learn a non-adversarial traditional decoder (omitted in most PM papers after 1992—see Sec.",
").",
"Similarly, to encode the data again, GAN variants can learn a non-adversarial traditional encoder (absent in the 2014 GAN paper but compare InfoGANs—see Sec.",
").While PM's minimax procedure starts from the data and learns a factorial code in form of a multivariate binomial distribution, GAN's minimax procedure starts from the codes (distributed according to any user-given distribution), and learns to make data distributed like the original data.",
"Pipeline of PM variants with standard decoders: data $\\rightarrow $ minimax-trained encoder $\\rightarrow $ code $\\rightarrow $ traditional decoder (often omitted) $\\rightarrow $ data Pipeline of GAN variants with standard encoders (compare InfoGANs): code $\\rightarrow $ minimax-trained decoder $\\rightarrow $ data $\\rightarrow $ traditional encoder $\\rightarrow $ code It will be interesting to study experimentally whether the GAN pipeline above is easier to train than PM to make factorial codes or useful approximations thereof."
],
[
"Convergence of Unsupervised Minimax",
"The 2014 GAN paper [20] has a comment on convergence under the greatly simplifying assumption that one can directly optimize the relevant functions implemented by the two adversaries, without depending on suboptimal local search techniques such as gradient descent.",
"In practice, however, gradient descent is almost always the method of choice.",
"So what's really needed is an analysis of what happens when backpropagation [40], [41], [97] is used for both adversarial networks.",
"Fortunately, there are some relevant results.",
"Convergence can be shown for both GANs and PM through two-time scale stochastic approximation [6], [36], [32].",
"In fact, Hochreiter's group used this technique to demonstrate convergence for GANs [25], [24]; the proof is directly transferrable to the case of PM.",
"Of course, such proofs show only convergence to exponentially stable equilibria, not necessarily to global optima.",
"Compare, e.g., [44]."
],
[
"Conclusion",
"The notion of Unsupervised Minimax refers to unsupervised or self-supervised adaptive modules (typically neural networks or NNs) playing a zero sum game.",
"The first NN somehow learns to generate data.",
"The second NN learns to predict properties of the generated data, minimizing its error, typically by gradient descent.",
"The first NN maximizes the objective function minimized by the second NN, trying to produce outputs that are hard on the second NN.",
"Examples are provided by Adversarial Artificial Curiosity (AC since 1990, Sec.",
"), Predictability Minimization (PM since 1991, Sec.",
"), Generative Adversarial Networks (GANs since 2014; conditional GANs since 2010, Sec.",
").",
"This is very different from certain earlier adversarial machine learning settings which neither involved unsupervised NNs nor were about modeling data nor used gradient descent (see Sec.",
", REF ).",
"GANs and cGANs are applications of the AC principle (1990) where the environment simply returns whether the current output of the first NN is in a given set (Sec.",
").",
"GANs are also closely related to PM, because both GANs and PM model the statistics of given data distributions through gradient-based adversarial nets that play a minimax game (Sec.",
").",
"The present paper clarifies some of the previously published confusion surrounding these issues.",
"AC's generality (Sec.",
"REF ) extends GAN-like unsupervised minimax to sequential problems, not only for plain pattern generation and classification, but even for RL problems in partially observable environments.",
"In turn, the large body of recent GAN-related insights might help to improve training procedures of certain AC systems."
],
[
"Acknowledgments",
"Thanks to Paulo Rauber, Joachim Buhmann, Sepp Hochreiter, Sjoerd van Steenkiste, David Ha, Róbert Csordás, Louis Kirsch, and several anonymous reviewers, for useful comments on a draft of this paper.",
"This work was partially funded by a European Research Council Advanced Grant (ERC no: 742870)."
]
] | 1906.04493 |
[
[
"Single Image Blind Deblurring Using Multi-Scale Latent Structure Prior"
],
[
"Abstract Blind image deblurring is a challenging problem in computer vision, which aims to restore both the blur kernel and the latent sharp image from only a blurry observation.",
"Inspired by the prevalent self-example prior in image super-resolution, in this paper, we observe that a coarse enough image down-sampled from a blurry observation is approximately a low-resolution version of the latent sharp image.",
"We prove this phenomenon theoretically and define the coarse enough image as a latent structure prior of the unknown sharp image.",
"Starting from this prior, we propose to restore sharp images from the coarsest scale to the finest scale on a blurry image pyramid, and progressively update the prior image using the newly restored sharp image.",
"These coarse-to-fine priors are referred to as \\textit{Multi-Scale Latent Structures} (MSLS).",
"Leveraging the MSLS prior, our algorithm comprises two phases: 1) we first preliminarily restore sharp images in the coarse scales; 2) we then apply a refinement process in the finest scale to obtain the final deblurred image.",
"In each scale, to achieve lower computational complexity, we alternately perform a sharp image reconstruction with fast local self-example matching, an accelerated kernel estimation with error compensation, and a fast non-blind image deblurring, instead of computing any computationally expensive non-convex priors.",
"We further extend the proposed algorithm to solve more challenging non-uniform blind image deblurring problem.",
"Extensive experiments demonstrate that our algorithm achieves competitive results against the state-of-the-art methods with much faster running speed."
],
[
"Introduction",
"Blur is one of the most common artifacts of digital images.",
"It happens mainly because the camera does not locate in focus, or the camera is held unsteadily, or the objects in the scene are moving quickly over the period of exposure time, resulting in that image sensor accumulates light not only from a single point but also from neighbouring regions.",
"Consequently, a blurry image is captured with unclear edges and details, which greatly degrade the visual quality of the image.",
"As a fundamental degradation model, the blur process is usually assumed to be shift-invariant, and can be represented as the convolution of a sharp image and a blur kernel, ${\\mathbf {b}}={\\mathbf {k}}\\otimes {\\mathbf {x}}+{\\mathbf {n}},$ where ${\\mathbf {b}}$ is the observed blurry image, ${\\mathbf {x}}$ is the latent sharp image, ${\\mathbf {k}}$ is the blur kernel, $\\otimes $ is the convolution operation, and ${\\mathbf {n}}$ is the additive Gaussian noise.",
"The blind image deblurring problem is to recover both the latent sharp image ${\\mathbf {x}}$ and the blur kernel ${\\mathbf {k}}$ , given only the observed blurry image ${\\mathbf {b}}$ [1], [2].",
"It is a highly ill-posed problem because the solution is not only unstable but also non-unique.",
"As a convolution in the spatial domain is equivalent to the point-wise product in the frequency domain, the observation ${\\mathbf {b}}$ can be factorized into infinite feasible solution pairs of ${\\mathbf {x}}$ and ${\\mathbf {k}}$ .",
"It is difficult to achieve the unique correct pair without sufficient prior knowledge about ${\\mathbf {x}}$ and ${\\mathbf {k}}$ .",
"Previous methods [3], [4], [5], [6], [7], [8], [9], [10], [11], [12] treated the blind image deblurring as a joint optimization problem and introduced sophisticated image priors, which can penalize image blurriness and promote image sharpness, such as the mixture of Gaussian functions that fits the heavy-tailed prior of natural images [3], [4], normalized sparse prior [5], framelet based prior [6], $l_0$ -norm based priors [7], [8], color line prior [9], dark channel prior [10], low-rank prior [11] and graph based prior [12], etc.",
"However, these image priors are usually computationally expensive, resulting in complicated optimization algorithms.",
"In this paper, we propose a novel multi-scale image prior to tackle the challenging blind image deblurring problem, inspired by the well-known self-example prior in image super-resolution (SR) [13], [14], [15].",
"In image SR, self-example prior is obtained through image down-sampling.",
"An observed image is down-sampled to generate a coarse-scale image, and the self-example prior is modeled as mappings from patches in the coarse-scale image to the corresponding patches in the fine-scale observed image.",
"Although the self-example prior has been successfully applied in image SR, when an observed image is blurred by an arbitrary blur kernel, such as a motion blur, there is no straightforward conclusion that the mappings can still model the image deblurring process.",
"In [16], Michaeli and Irani first analyzed the effect of image down-sampling in the blind image deblurring problem.",
"They found that image down-sampling increased the internal patch recurrence.",
"With this observation, they proposed to promote the internal patch recurrence for blind image deblurring.",
"However, iterative promotion of patch recurrence requires enormous complicated patch matchings [17], which are extremely slow in practice.",
"Different from [16], in this paper, we focus on the extreme case of image down-sampling.",
"We keep down-sampling an observed blurry image to a very coarse level and observe that: the coarse enough image down-sampled from a blurry observation is approximately a low-resolution version of the latent sharp image.",
"We theoretically verify this phenomenon on 2D signals under a general down-sampling operator, and define the coarse enough image as a latent structure prior of the latent sharp image.",
"We further introduce a multi-scale strategy to work in tandem with latent structure prior on an image pyramid.",
"Starting from the coarsest scale, we employ the prior image to blindly restore a finer scale sharp image.",
"The newly restored sharp image is then employed as the latent structure prior of the next finer scale.",
"The strategy works gradually from the coarsest scale to the finest scale.",
"Thus, we name our prior Multi-Scale Latent Structure (MSLS) prior.",
"Tailored to the MSLS prior, we propose an efficient algorithm to solve the blind image deblurring problem, which is much faster than the aforementioned methods [3], [4], [5], [6], [7], [8], [9], [10], [11], [16], [12].",
"The proposed algorithm comprises two phases, i.e., we first preliminarily restore sharp images gradually from the coarsest scale to the finest scale, and then perform a refinement in the finest scale.",
"In each scale, we introduce a joint optimization to restore both the latent sharp image and the blur kernel.",
"Specifically, to achieve lower complexity, we alternately apply a sharp image reconstruction with fast local self-example matching inspired by image SR [15], an accelerated kernel estimation with error compensation, and a fast non-blind image deblurring.",
"Furthermore, beyond uniform image blur, we extend the proposed algorithm to solve a more challenging non-uniform blind image deblurring problem.",
"Experimental results demonstrate that in both cases our algorithm achieves the state-of-the-art deblurring performance.",
"The contributions of this paper can be summarized as follows: 1) We introduce a Multi-Scale Latent Structure (MSLS) prior of latent sharp image for deblurring.",
"We theoretically verify this prior under a general down-sampling operation.",
"2) We propose a powerful and efficient blind image deblurring algorithm based on the MSLS prior, which achieves the state-of-the-art deblurring performance and is much faster than the previous methods with computationally expensive image priors.",
"3) We further demonstrate that the proposed algorithm can be extended to tackle more complex non-uniform motion blur, apart from simply uniform blur.",
"The outline of the paper is as follows.",
"In Section , we review related work in image deblurring.",
"In Section , we provide a theoretical analysis of our observation and introduce the MSLS prior.",
"In Section , we present the proposed blind deblurring algorithm in detail.",
"Experiments and final conclusion are provided in Section and Section , respectively."
],
[
"Related Work",
"Image deblurring is an ill-posed inverse problem, which is to recover a latent sharp image ${\\mathbf {x}}$ from a blurry observation ${\\mathbf {b}}$ .",
"Based on whether the blur kernel ${\\mathbf {k}}$ is known or not, the problem has been divided into two categories, i.e., non-blind image deblurring and blind image deblurring."
],
[
"Non-blind Image Deblurring",
"For non-blind image deblurring, the blur kernel is given and the problem is to recover the latent sharp image from the blurry observation with the blur kernel.",
"Non-blind image deblurring is a very unstable process and is easily disturbed by noise, even a small amount of noise will lead to severe distortions in the estimation.",
"This phenomenon is called ill-posedness.",
"Many researches have been done to deal with the ill-posed problem including classical filtering methods [18], [19], [20] and regularization based methods [21], [22], [23], [24], [25], [26].",
"In the regularization based methods, regularization terms are usually proposed based on the prior of the latent sharp image, such as Total Variation (TV) prior [21] or sparse priors [23], [24].",
"By combining an $l_2$ -norm or an $l_1$ -norm [27] constrained data fidelity term with regularization terms, image deblurring can be modeled as an optimization problem.",
"A latent sharp image can be recovered by solving the optimization problem.",
"Besides Gaussian noise, non-blind deblurring is also disturbed by outliers, e.g., pixel saturation.",
"Interested readers can refer to [28], [29], [30] for more details."
],
[
"Blind Image Deblurring",
"For blind image deblurring, the blur kernel is unknown and the problem is to recover both the latent sharp image and the blur kernel from only a blurry observation.",
"The blind image deblurring is much more challenging than the non-blind problem because the solution for the problem is not only unstable but also non-unique.",
"In order to deal with this highly ill-posed blind image deblurring problem, there have been many pioneering works which attempted to approximate the problem in different perspectives.",
"In early works, blind image deblurring has been processed like image enhancing.",
"Image diffusion and shock filtering methods [31], [32], [33] were proposed to enhance the edges of blurry images blurred by out-of-focus or Gaussian kernel.",
"Another kind of approximation is to introduce parameterized forms of blur kernels or to impose specific constraints, such as centrosymmetry, on the blur kernels [34], [35], [36], [37], [38].",
"These methods can handle specific blur kernels, but cannot handle general motion blur which is usually irregular.",
"There are also numerous works focusing on the multi-image deblurring [39], [40], [41], [42], [43], [44].",
"Multiple images can provide more data constraints for kernel estimation and help to solve the blind image deblurring problem.",
"Nevertheless, multiple relevant images in their assumptions are sometimes unavailable, which limits their practicability.",
"Recently, with the fast progress of regularization and optimization theory, many sophisticated image priors [3], [4], [5], [6], [7], [8], [9], [11], [10], [12] were proposed to handle the single image blind image deblurring problem with general blur kernels, such as the mixture of Gaussians prior that fits the heavy-tailed prior of natural images [3], [4], normalized sparse prior [5], framelet based prior [6], $l_0$ -norm based priors [8], [7], color line prior [9], dark channel prior [10], low rank prior [11] and graph based prior [12], etc.",
"The priors promote image sharpness and penalize image blurriness, which work as a regularizer in the optimization model guiding the solver to converge to the latent sharp image.",
"Nevertheless, these image priors are usually non-convex, which result in very computationally expensive optimization algorithms."
],
[
"Observation and Multi-scale Latent Structure Prior",
"In this section, we introduce in detail the proposed multi-scale latent structure prior."
],
[
"Observation and Theoretical Analysis",
"Michaeli and Irani [16] first analyzed the effect of image down-sampling in the blind image deblurring problem and found that image down-sampling increased the internal patch recurrence.",
"Differently, in this work, we focus on the extreme case of image down-sampling and down-sample a blurry image to a very coarse level.",
"We observe that a coarse enough image down-sampled from a blurry observation is approximately a low-resolution version of the latent sharp image.",
"We provide a theoretical proof on 2D signals to verify this observation under a general down-sampling operation.",
"In the proof, we assume a noise-free blurred image.",
"We empirically demonstrate that the observation is also robust with noise, by deblurring noisy and blurry images in the experiments in Section .",
"Claim 1 A coarse enough image down-sampled from a blurry observation is approximately a low-resolution version of the latent sharp image.",
"Proof: According to the degradation model of (REF ), a noise-free blurry image ${\\mathbf {b}}$ can be represented as ${\\mathbf {b}}[i,j]={\\mathbf {k}}\\otimes {\\mathbf {x}}=\\iint {\\mathbf {k}}(u,v){\\mathbf {x}}(i-u,j-v)dudv,$ where ${\\mathbf {x}}$ is the latent sharp image and ${\\mathbf {k}}$ is the blur kernel.",
"When ${\\mathbf {b}}$ is $\\alpha $ -times down-sampled, its down-sampled version is given by ${\\mathbf {b}}_\\alpha [i, j]&=\\left({\\mathbf {b}}\\otimes {\\mathbf {h}}\\right)\\downarrow =\\left({\\mathbf {k}}\\otimes {\\mathbf {x}}\\otimes {\\mathbf {h}}\\right)\\downarrow \\\\&= \\left({\\mathbf {k}}\\otimes ({\\mathbf {x}}\\otimes {\\mathbf {h}})\\right)\\downarrow = \\left({\\mathbf {k}}\\otimes {\\mathbf {x}}_h\\right)\\downarrow \\\\&=\\iint {\\mathbf {k}}(u,v){\\mathbf {x}}_h(\\alpha i-u,\\alpha j-v)dudv,$ where ${\\mathbf {b}}_\\alpha [i,j]$ is a down-sampled blurry image, ${\\mathbf {h}}$ is a general low-pass filter to avoid aliasing, ${\\mathbf {x}}_h$ is a filtered band-limited image, $\\downarrow $ represents the pixel extraction process and $\\alpha $ is the down-sampling factor.",
"By replacing $u$ , $v$ with $\\alpha t$ , $\\alpha s$ , (REF ) can be further written as ${\\mathbf {b}}_\\alpha [i,j]&=\\iint \\alpha ^2{\\mathbf {k}}(\\alpha t, \\alpha s)\\cdot {\\mathbf {x}}_h(\\alpha i-\\alpha t,\\alpha j-\\alpha s) dtds \\\\&= \\iint {\\mathbf {k}}_\\alpha (t,s)\\cdot {\\mathbf {x}}_\\alpha (i-t,j-s)dtds.$ where ${\\mathbf {x}}_\\alpha $ and ${\\mathbf {k}}_\\alpha $ are the down-sampled sharp image and the down-sampled blur kernel, respectively.",
"From (REF ), the down-sampled blur kernel is: ${\\mathbf {k}}_\\alpha (u,v)=\\alpha ^2 {\\mathbf {k}}(\\alpha u, \\alpha v).$ The normalization condition $\\iint {\\mathbf {k}}_\\alpha = 1$ is satisfied if the assumption $\\iint {\\mathbf {k}}=1$ is imposed.",
"It follows that the support of down-sampled blur kernel ${\\mathbf {k}}_\\alpha $ is $\\alpha $ times smaller than that of blur kernel ${\\mathbf {k}}$ in both horizontal and vertical directions.",
"Figure: 1D illustration of down-sampling a discrete gaussian kernel in the spatial and frequency domain.",
"(a) spatial domain.",
"(b) frequency domain in [-π,π][-\\pi , \\pi ].We further transform ${\\mathbf {k}}_\\alpha $ into the frequency domain: ${\\mathbf {K}}_\\alpha (\\tilde{u},\\tilde{v})&={\\mathcal {F}}\\left({\\mathbf {k}}_\\alpha (u,v)\\right)=\\alpha ^2{\\mathcal {F}}\\left({\\mathbf {k}}(\\alpha u, \\alpha v)\\right) \\\\&=\\alpha ^2 \\left( \\frac{1}{\\alpha ^2}{\\mathbf {K}}(\\tilde{u}/\\alpha , \\tilde{v}/\\alpha )\\right) \\\\&={\\mathbf {K}}(\\tilde{u}/\\alpha , \\tilde{v}/\\alpha )$ where ${\\mathcal {F}}(\\cdot )$ denotes the Fourier transform and ${\\mathbf {K}}(\\tilde{u},\\tilde{v})={\\mathcal {F}}({\\mathbf {k}}(u,v))$ .",
"${\\mathbf {K}}_\\alpha $ is expanded by $\\alpha $ in the frequency domain.",
"Though ${\\mathbf {K}}_\\alpha $ is expanded, ${\\mathbf {x}}_h$ and ${\\mathbf {x}}_\\alpha $ are low-pass filtered and band-limited.",
"Thus, aliasing is avoided while down-sampling but only the information within the band-limit is preserved.",
"As the down-sampling becomes deeper, i.e., $\\alpha $ is larger, ${\\mathbf {K}}_\\alpha $ in the band-limit tends to constant 1 and its corresponding spatial response will converges to Dirac delta function $\\delta $ [45].",
"The proof so far works in the continuous domain, i.e., ${\\mathbf {k}}$ and ${\\mathbf {x}}$ are two continuous signals.",
"For discrete signals in practise, we sample the continuous signals and do normalization on ${\\mathbf {k}}$ afterwards to ensure $\\sum _{uv} {\\mathbf {k}}_\\alpha (u,v) =1$ .",
"Approximately, ${\\mathbf {k}}_\\alpha (u,v)$ will converge to a delta kernel in the discrete domain: $\\delta (i,j)={\\left\\lbrace \\begin{array}{ll}1, & i=0\\ and\\ j=0,\\\\0, & i\\ne 0\\ or\\ j\\ne 0.\\end{array}\\right.",
"}$ which satisfies $\\sum _i\\sum _j \\delta (i,j)=1$ .",
"An illustrative experiment in the discrete domain is shown in Fig. 1.",
"Finally, we have ${\\mathbf {b}}_\\alpha [i,j] \\rightarrow {\\mathbf {x}}_\\alpha \\otimes \\delta ={\\mathbf {x}}_\\alpha ,$ i.e., the down-sampled blurry image converges to the low-resolution version of the latent sharp image."
],
[
"Multi-Scale Latent Structure Prior",
"Claim 1 is very useful for blind image deblurring, since it gets rid of the effect of the unknown blur kernel and provides a low-resolution prior of the latent sharp image, referred as the latent structure prior in our algorithm.",
"Based on Claim 1, we further introduce a multi-scale strategy to work in tandem with latent structure prior on an image pyramid.",
"Starting from the coarsest scale, the prior image is employed to blindly restore a finer scale sharp image.",
"The newly restored sharp image is then employed as the latent structure prior of the next finer scale.",
"The strategy works gradually from the coarsest scale to the finest scale, thus we name the coarse-to-fine prior—Multi-Scale Latent Structure (MSLS) prior.",
"In the next section, we will elaborate the blind image deblurring algorithm tailored to the proposed MSLS prior."
],
[
"Blind Image Deblurring Algorithm",
"In this section, we introduce a joint optimization scheme to estimate both the latent sharp image and the blur kernel based on the proposed MSLS prior.",
"The algorithm comprises two phases, i.e., preliminary restoration in coarse scales in Sec.REF and refined restoration in the finest scale in Sec.REF .",
"We further extend the proposed algorithm to solve non-uniform blind image deblurring problem in Sec.REF ."
],
[
"Preliminary Restoration in Coarse Scales",
"Blind image deblurring is modeled to solve the following optimization problem, $\\operatornamewithlimits{\\arg \\min }_{{\\mathbf {k}},{\\mathbf {x}}} \\Phi ({\\mathbf {k}}\\otimes {\\mathbf {x}}-{\\mathbf {b}})+\\lambda _1\\Psi _1({\\mathbf {k}})+\\lambda _2\\Psi _2({\\mathbf {x}})$ where $\\Phi ({\\mathbf {k}}\\otimes {\\mathbf {x}}-{\\mathbf {b}})$ is the data fidelity term, $\\Psi _1({\\mathbf {k}})$ and $\\Psi _2({\\mathbf {x}})$ are the regularizers of blur kernel ${\\mathbf {k}}$ and latent sharp image ${\\mathbf {x}}$ , respectively.",
"Previous methods, such as [3], [4], [5], [6], [7], [8], [9], [10], [11], [16], [12], introduced sophisticated image priors, which are usually either non-convex or computationally expensive.",
"To tackle the challenging problem mentioned above, we take advantage of the proposed MSLS prior and introduce an efficient local self-example matching strategy to substitute for complex non-convex regularization of ${\\mathbf {x}}$ in sharp image reconstruction.",
"In each coarse scale, our algorithm comprises three steps, as sketched in Algorithm 1.",
"[htb] Preliminary Restoration in Each Coarse Scale [1] Blurry image ${\\mathbf {b}}$ in current scale, prior image ${\\mathbf {x}}_{pr}$ .",
"Kernel size $h\\times h$ .",
"Estimated kernel ${\\hat{\\mathbf {k}}}$ and the latent sharp image ${\\mathbf {x}}_{l}$ .",
"Initialize latent image ${\\mathbf {x}}_{l} = {\\mathbf {b}}$ and ${\\hat{\\mathbf {k}}}= \\delta $ .",
"for iter $=$ $1 \\rightarrow max\\_iteration$ do (a) Estimate ${\\hat{\\mathbf {x}}}$ using ${\\mathbf {x}}_{l}$ and ${\\mathbf {x}}_{pr}$ .",
"(b) Minimize (REF ) to estimate ${\\hat{\\mathbf {k}}}$ , given ${\\hat{\\mathbf {x}}}$ .",
"(c) Minimize (REF ) to update ${\\mathbf {x}}_{l}$ , given ${\\hat{\\mathbf {k}}}$ .",
"endfor ${\\hat{\\mathbf {k}}}$ and ${\\mathbf {x}}_{l}$ ; Specifically, in step (a), we blindly reconstruct a sharp image with fast local self-example matching.",
"Step (b) and (c) correspond to a novel kernel estimation with error compensation and a non-blind image deblurring, which are both convex optimization and can be efficiently solved with Fast Fourier Transform (FFT) acceleration.",
"The prior image is first initialized as the coarsest scale image ${\\mathbf {x}}_\\alpha $ .",
"After solving one scale, we update the prior image with the newly deblurred result and continue to restore the next finer scale.",
"The diagram of preliminary restoration is illustrated in Fig.REF .",
"In the following, we introduce in detail these three steps: Figure: The diagram of preliminary restoration in coarse scales.",
"(a) sharp image reconstruction.",
"(b) kernel estimation.",
"(c) non-blind deblurring.",
"We use scale factor β=log 2 3\\beta =\\log _23 for multi-scale image pyramid construction.",
"In the coarsest scale, ⌊β n ⌋=α\\lfloor \\beta ^n\\rfloor = \\alpha leads to a low-resolution sharp image.Figure: A single-scale illustrative experiment of sharp image reconstruction using local self-example matching strategy.",
"The latent sharp image is blurred by a 7×\\times 7 motion blur kernel.",
"The blurry image is down-sampled until the blur kernel is approximately a delta function.",
"According to Claim 1, the coarse enough image is a prior of the latent sharp image.",
"The corresponding kernels are shown on the right.",
"Patches in the blurry image find their NN patches in corresponding local areas of the prior image.",
"An example of matched patches is shown on the left.",
"All sharp NN patches are fused together to reconstruct the sharp image.Step (a): Sharp Image Reconstruction with Local Self-Example Matching: The finer scale latent sharp image is reconstructed with a fast local self-example matching strategy, inspired by [15].",
"The latent sharp image ${\\mathbf {x}}_l$ is first initialized as the blurry image ${\\mathbf {b}}$ .",
"We divide the latent image ${\\mathbf {x}}_l$ into overlapped small patches (usually of size $5\\times 5$ pixels with $50\\%$ overlap).",
"Then, each patch $i$ finds its nearest neighbor (NN) in the corresponding local area of the coarser-scale sharp prior image ${\\mathbf {x}}_{pr}$ , measured by (REF ), $d(i,j)=\\Vert \\mathbf {P}_i{\\mathbf {x}}_l-\\mathbf {P}_j{\\mathbf {x}}_{pr}\\Vert _2,\\ \\ j\\in \\mathcal {N}(i{^{\\prime }})$ where $\\mathbf {P}_i$ is the patch extraction operation at position $i$ .",
"$i{^{\\prime }}$ is a position on the sharp prior image ${\\mathbf {x}}_{pr}$ .",
"If ${\\mathbf {x}}_l$ is down-sampled to the size of ${\\mathbf {x}}_{pr}$ , the position $i$ on ${\\mathbf {x}}_l$ will be projected on the position $i^{\\prime }$ on ${\\mathbf {x}}_{pr}$ .",
"$\\mathcal {N}(i{^{\\prime }})$ represents the local neighborhood of the position $i{^{\\prime }}$ .",
"$d(i,j)$ denotes the distance between the patch $i$ and the patch $j$ .",
"The prior image ${\\mathbf {x}}_{pr}$ is $\\beta $ -times smaller than ${\\mathbf {x}}_l$ , as shown in Fig.",
"REF .",
"Finally, we fuse the searched NN patches together by computing weighted average in the overlapped areas, in order to avoid blocking artifacts.",
"The weights are defined by Hamming windowhttps://en.wikipedia.org/wiki/Window_function.",
"The 2D Hamming window function is defined as follows: $w(i)=\\theta &-\\gamma \\cos \\left(\\frac{2\\pi i}{N-1}\\right) \\\\W(i,j)&=w(i)\\cdot w(j)$ where $\\theta =0.54$ and $\\gamma =0.46$ are fixed parameters, $N$ is the size of window, $w$ and $W$ are 1D and 2D weights of Hamming window, $i$ and $j$ are the indices.",
"An illustrative example of a single-scale sharp image reconstruction is shown in Fig.",
"REF .",
"In our algorithm, local patch matching works not as an approximation of the global patch matching but a better choice for sharp image reconstruction, which avoids the patch over-fitting and considerably reduces the computational complexity of the NN search: In motion blurred images with different depths of field, the blurry patch in the foreground tends to find an over-fitting blurry NN in the background area of the prior image, if the searching distance is not constrained.",
"This misleads the sharp image reconstruction and causes the incorrect kernel estimation in the followed step.",
"We report a single-scale illustrative experiment in Fig.",
"REF .",
"When the search range is constrained by the locality property, more wanted sharp patches are found and the kernel estimation result is also improved.",
"Figure: An single-scale illustrative experiment to demonstrate the efficacy of locality.",
"(a) Square 1 is in the sharp foreground and square 2 is in the blurry background that is first blurred by a Gaussian kernel with σ=2\\sigma =2.",
"Then, the image is blurred by a 7×\\times 7 motion blur kernel.",
"(b) The patch in the blurry image finds its NN patch in the prior image.",
"For the global patch matching, the NN patch is in the background square 2, which is an over-fitting blurry patch.",
"For the local patch matching, the NN patch is in the foreground square 1, which is a sharp patch.",
"The local searching method leads to a more appropriate kernel estimation."
],
[
"Accelerate Searching",
"The computational complexity of local patch matching is $O(C)$ , where $C$ is a constant that accounts for the search range.",
"The search range is set to $10\\times 10$ for $5\\times 5$ sized patches in our algorithm.",
"There are $(10-2\\times \\lfloor \\frac{5}{2}\\rfloor )\\times (10-2\\times \\lfloor \\frac{5}{2}\\rfloor )=36$ times mean square error (MSE) computations at each location.",
"Step (b): Kernel Estimation with Error Compensation: Given the sharp image reconstructed with MSLS prior, we estimate the blur kernel afterwards.",
"Since the finer-scale sharp image is reconstructed from the coarser-scale prior image, there is an inevitable error between the reconstructed image and the ground-truth.",
"In this step, we propose a new image blur model (REF ) for our kernel estimation, in order to compensate for the error: ${\\mathbf {b}}={\\mathbf {k}}\\otimes ({\\hat{\\mathbf {x}}}+{\\mathbf {x}}_c)+{\\mathbf {n}}.$ where ${\\hat{\\mathbf {x}}}$ is the reconstructed image and ${\\mathbf {x}}_c$ is the compensatory layer.",
"With the adjustable ${\\mathbf {x}}_c$ layer, ${\\hat{\\mathbf {x}}}+{\\mathbf {x}}_c$ can be equivalent to the latent sharp image ${\\mathbf {x}}$ and the estimated kernel ${\\hat{\\mathbf {k}}}$ can be more accurate.",
"Given ${\\hat{\\mathbf {x}}}$ and model (REF ), computing ${\\hat{\\mathbf {k}}}$ and ${\\mathbf {x}}_c$ results in solving an optimization problem (REF ).",
"The model (REF ) is used as a fidelity term to constrain that the ${\\hat{\\mathbf {k}}}$ and ${\\hat{\\mathbf {x}}}_c$ follow the image convolution process.",
"Instead of directly using (REF ), we transform it into gradient domain, in order to reduce ringing artifacts [46].",
"The regularizer of ${\\mathbf {k}}$ is penalized by an $l_2$ -norm and the regularizer of ${\\mathbf {x}}_c$ is penalized by an $l_1$ -norm to ensure the spatial sparsity of ${\\mathbf {x}}_c$ : $\\operatornamewithlimits{\\arg \\min }_{{\\mathbf {k}},{\\mathbf {x}}_c} \\frac{1}{2}\\Vert {\\mathbf {k}}\\otimes \\nabla ({\\hat{\\mathbf {x}}}+{\\mathbf {x}}_c)-\\nabla {\\mathbf {b}}\\Vert _2^2+\\lambda _1\\Vert {\\mathbf {k}}\\Vert _2^2+\\lambda _2\\Vert {\\mathbf {x}}_c\\Vert _1$ where $\\lambda _1$ and $\\lambda _2$ are two trade-off parameters to control the regularization strength.",
"At first glance, (REF ) is a non-convex optimization problem, since there exists a ${\\mathbf {k}}\\otimes \\nabla {\\mathbf {x}}_c$ .",
"Here we apply a variable substitution to make it convex, i.e., replacing ${\\mathbf {k}}\\otimes \\nabla {\\mathbf {x}}_c$ with a new variable ${\\mathbf {v}}$ .",
"Note that ${\\mathbf {x}}_c$ is assumed sparse and the size of ${\\mathbf {k}}$ is much smaller than the size of ${\\mathbf {x}}_c$ , so ${\\mathbf {k}}\\otimes \\nabla {\\mathbf {x}}_c$ can also be considered sparse.",
"We thus substitute $\\Vert {\\mathbf {v}}\\Vert _1$ for $\\Vert {\\mathbf {x}}_c\\Vert _1$ .",
"Now the objective function (REF ) can be reformulated as a convex optimization problem (REF ): $\\operatornamewithlimits{\\arg \\min }_{{\\mathbf {k}},{\\mathbf {v}}} \\frac{1}{2}\\Vert {\\mathbf {k}}\\otimes \\nabla {\\hat{\\mathbf {x}}}+{\\mathbf {v}}-\\nabla {\\mathbf {b}}\\Vert _2^2+\\lambda _1\\Vert {\\mathbf {k}}\\Vert _2^2+\\lambda _2\\Vert {\\mathbf {v}}\\Vert _1$ We alternately compute ${\\mathbf {k}}$ and ${\\mathbf {v}}$ by fixing one and addressing another.",
"The $\\nabla = \\lbrace \\nabla _x, \\nabla _y\\rbrace $ are derivative operators in horizontal and vertical directions respectively.",
"The ${\\mathbf {v}}= \\lbrace {\\mathbf {v}}_x, {\\mathbf {v}}_y\\rbrace $ are the corresponding layers of horizontal and vertical directions and are initialized to zeros.",
"In (REF ), terms related to ${\\mathbf {k}}$ are all quadratic, so a closed-form solution can be derived for ${\\mathbf {k}}$ .",
"Moreover, to fast solve ${\\mathbf {k}}$ , we compute the solution in the frequency domain, ${\\mathbf {k}}=\\mathcal {F}^{-1}\\left(\\frac{\\overline{\\mathcal {F}(\\nabla {\\hat{\\mathbf {x}}})}\\mathcal {F}(\\nabla {\\mathbf {b}})-\\overline{\\mathcal {F}(\\nabla {\\hat{\\mathbf {x}}})}\\mathcal {F}({\\mathbf {v}})}{\\overline{\\mathcal {F}(\\nabla {\\hat{\\mathbf {x}}})}\\mathcal {F}(\\nabla {\\hat{\\mathbf {x}}})+2\\lambda _1}\\right)$ where $\\mathcal {F}(\\cdot )$ and $\\mathcal {F}^{-1}(\\cdot )$ are Fourier and inverse Fourier transform implemented by FFT.",
"The $\\overline{\\mathcal {F}(\\cdot )}$ means the complex conjugate of fourier transformed values.",
"As each element in ${\\mathbf {v}}$ is independent, we use proximal mapping [47] to solve the $l_1$ -norm optimization problem.",
"The solver works as a fast soft-thresholding: $v_i=sgn(z_i)\\cdot \\max \\left\\lbrace 0,|z_i|-\\lambda _2\\right\\rbrace $ where $i$ means the $i$ -th pixel in ${\\mathbf {v}}_x$ or ${\\mathbf {v}}_y$ , $z_i=\\nabla b_i-({\\mathbf {k}}\\otimes \\nabla {\\hat{\\mathbf {x}}})_i$ .",
"An illustrative deblurring example is illustrated in Fig.",
"REF .",
"Fig.",
"REF is a blurry image.",
"Fig.",
"REF and Fig.",
"REF show two blind deblurred results without and with the compensatory layer ${\\mathbf {x}}_c$ .",
"Fig.",
"REF , Fig.",
"REF and Fig.",
"REF are their close-ups of the license plate, respectively.",
"Solving with ${\\mathbf {x}}_c$ achieves clearer deblurred result.",
"The corresponding estimated kernel is also much better, in which less outliers appear beside the true trajectory.",
"We further analyze the role of the auxiliary variable ${\\mathbf {v}}$ in optimization of (REF ).",
"Fig.",
"REF is the close-up of the license plate in ${\\hat{\\mathbf {x}}}$ , which is reconstructed from the coarser-scale prior image.",
"It can be found that ${\\hat{\\mathbf {x}}}$ contains sharp edges but loses some details, e.g., the license number.",
"To overcome this issue, we introduce the ${\\mathbf {x}}_c$ to compensate for the error.",
"In this case, ideally, ${\\mathbf {x}}_c$ should be the license number.",
"To solve both ${\\mathbf {k}}$ and ${\\mathbf {x}}_c$ in (REF ), we substitute a new variable ${\\mathbf {v}}$ for ${\\mathbf {k}}\\otimes \\nabla {\\mathbf {x}}_c$ and solve (REF ).",
"Fig.",
"REF and Fig.",
"REF are the ${\\mathbf {v}}_x$ and ${\\mathbf {v}}_y$ obtained by solving (REF ).",
"It can be found that the restored ${\\mathbf {v}}_x$ and ${\\mathbf {v}}_y$ are the derivatives of the license number $\\nabla {\\mathbf {x}}_c$ convolving with the kernel ${\\mathbf {k}}$ , which demonstrates that the reconstructed result matches the theoretical analysis.",
"Figure: (a) A blurry image with size 384×\\times 406 and kernel size 27×\\times 27.",
"(b) Deblurred result with only the MSLS prior.",
"(c) Deblurred result with both the MSLS prior and the compensatory layer.",
"(d) Close-up of the license plate in (a).",
"(e) Close-up of the license plate in (b) and corresponding kernel estimation.",
"(f) Close-up of the license plate in (c) and corresponding kernel estimation.",
"(g) Close-up of the license plate in 𝐱 ^{\\hat{\\mathbf {x}}}.",
"(h) Close-up of the license plate in 𝐯 x {\\mathbf {v}}_x.",
"(i) Close-up of the license plate in 𝐯 y {\\mathbf {v}}_y.Step (c): Non-blind Image Deblurring: After the kernel ${\\hat{\\mathbf {k}}}$ is computed, we conduct the non-blind deblurring to update the latent image ${\\mathbf {x}}_l$ .",
"We apply the Total Variation (TV) regularized non-blind deblurring algorithm, which is formulated as follows: $\\operatornamewithlimits{\\arg \\min }_{\\mathbf {x}}\\frac{1}{2}\\Vert {\\hat{\\mathbf {k}}}\\otimes {\\mathbf {x}}-{\\mathbf {b}}\\Vert _2^2+\\mu \\Vert \\nabla {\\mathbf {x}}\\Vert _1$ where $\\mu $ is a regularization parameter.",
"We apply the alternating direction method of multipliers (ADMM) [48], [49] to lead to a fast solver in the frequency domain.",
"Preliminary restoration can fast estimate sharp images in the coarse scales but possibly results in artifacts.",
"In order to achieve more accurate restoration for complex cases, we propose to perform a refined restoration in the finest scale, inspired by [50].",
"Given the preliminary restored image as initialization, the diagram of refined restoration in the finest scale is shown in Fig.REF .",
"Since the refined restoration is employed in the finest scale, there is no prior image updated from coarse scales.",
"Thus, we update the prior image by filtering and down-sampling the restored image in each iteration.",
"We use the edge-preserving guided filter [51] to reduce the possible artifacts without introducing any extra blur.",
"Besides, we also employ higher-order derivatives in the data fidelity term to add more constraints for accurate kernel estimation [52]: $\\operatornamewithlimits{\\arg \\min }_{{\\mathbf {k}},{\\mathbf {v}}} \\frac{1}{2}\\Vert {\\mathbf {k}}\\otimes \\nabla _*{\\hat{\\mathbf {x}}}+{\\mathbf {v}}_*-&\\nabla _*{\\mathbf {b}}\\Vert _2^2 \\\\+&\\lambda _3\\Vert {\\mathbf {k}}\\Vert _2^2+\\lambda _4\\Vert {\\mathbf {v}}_*\\Vert _1.$ where $\\nabla _*$$=$ {$\\nabla _x$ , $\\nabla _y$ , $\\nabla _{xx}$ , $\\nabla _{yy}$ , $\\nabla _{xy}$ } and ${\\mathbf {v}}_*$$=$ {${\\mathbf {v}}_x$ , ${\\mathbf {v}}_y$ , ${\\mathbf {v}}_{xx}$ , ${\\mathbf {v}}_{yy}$ , ${\\mathbf {v}}_{xy}$ }.",
"The algorithm is outlined in Algorithm 2.",
"[htb] Refined Restoration in the Finest Scale [1] Blurry image ${\\mathbf {b}}$ and preliminary estimation $\\tilde{{\\mathbf {x}}}$ .",
"Kernel size $h\\times h$ .",
"Refined kernel ${\\hat{\\mathbf {k}}}$ , the latent sharp image ${\\mathbf {x}}_l$ .",
"Initialize latent image ${\\mathbf {x}}_l = \\tilde{{\\mathbf {x}}}$ .",
"for iter $=$ $1 \\rightarrow max\\_iteration$ do (a) Update prior ${\\mathbf {x}}_{pr}$ by filtering and down-sampling ${\\mathbf {x}}_l$ .",
"(b) Estimate ${\\hat{\\mathbf {x}}}$ from ${\\mathbf {x}}_l$ and ${\\mathbf {x}}_{pr}$ .",
"(c) Minimize (REF ) to update ${\\hat{\\mathbf {k}}}$ , given ${\\hat{\\mathbf {x}}}$ .",
"(d) Minimize (REF ) to update ${\\mathbf {x}}_l$ , given ${\\hat{\\mathbf {k}}}$ .",
"endfor ${\\hat{\\mathbf {k}}}$ , ${\\mathbf {x}}_l$ ; Fig.",
"REF illustrates an example of our refined restoration for a challenging blur kernel with fine details.",
"Through preliminary estimation and refinement, the kernel becomes more accurate and the artifacts are reduced in the deblurred image.",
"As a comparison, Fig.",
"REF shows the result of [16], which converges to a sub-optimal solution.",
"Figure: (a) A close-up of a blurry image with image size 972×\\times 966 and kernel size 69×\\times 69.",
"(b) Preliminary restored result before refinement.",
"(c) Refined result.",
"(d) Restoration result from Michaeli & Irani ."
],
[
"Extension to Non-Uniform Deblurring",
"The non-uniform image deblurring is a more challenging problem than the uniform one, which includes complicated camera motions, such as rotation and translation.",
"Specifically, by representing the camera motion as a more general projective transformation [53], [54], the blur process can be modeled as: ${\\mathbf {b}}=\\sum _i k_i{\\mathbf {H}}_i{\\mathbf {x}}+{\\mathbf {n}}$ where ${\\mathbf {x}}$ and ${\\mathbf {b}}$ are vectors of the latent sharp image and the blurry observation, respectively.",
"${\\mathbf {H}}_i$ is the $i$ -th projective transformation matrix of the camera, $k_i$ is the weight of the $i$ -th transformation in the blur kernel ${\\mathbf {k}}$ , ${\\mathbf {n}}$ is the additive noise.",
"(REF ) is a linear transformation, which can be further represented as: ${\\mathbf {b}}={\\mathbf {A}}_{\\mathbf {k}}{\\mathbf {x}}+{\\mathbf {n}}={\\mathbf {B}}_{\\mathbf {x}}{\\mathbf {k}}+{\\mathbf {n}}$ where matrix ${\\mathbf {A}}_{\\mathbf {k}}=\\sum _i k_i{\\mathbf {H}}_i$ and $col({\\mathbf {B}}_{\\mathbf {x}})_i={\\mathbf {H}}_i{\\mathbf {x}}$ .",
"$col(\\cdot )_i$ denotes the $i$ -th column.",
"Our algorithm can be extended to non-uniform deblurring by assuming that non-uniform image blur is locally uniform.",
"Based on this assumption, the MSLS prior still works and we apply the same sharp image reconstruction with local self-example matching.",
"For the kernel estimation in each scale, we replace the convolution model (REF ) with (REF ) and modify (REF ) to the following formulation: $\\operatornamewithlimits{\\arg \\min }_{{\\mathbf {k}},{\\mathbf {x}}_c} \\frac{1}{2}\\Vert {\\mathbf {B}}_{\\nabla ({\\hat{\\mathbf {x}}}+{\\mathbf {x}}_c)}{\\mathbf {k}}-&\\nabla {\\mathbf {b}}\\Vert _2^2 \\\\+&\\lambda _1\\Vert {\\mathbf {k}}\\Vert _2^2+\\lambda _2\\Vert {\\mathbf {x}}_c\\Vert _1.$ where $col({\\mathbf {B}}_{\\nabla ({\\hat{\\mathbf {x}}}+{\\mathbf {x}}_c)})_i={\\mathbf {H}}_i(\\nabla ({\\hat{\\mathbf {x}}}+{\\mathbf {x}}_c))$ .",
"We replace ${\\mathbf {B}}_{\\nabla {\\mathbf {x}}_c}{\\mathbf {k}}$ with ${\\mathbf {v}}$ and replace $\\Vert {\\mathbf {x}}_c\\Vert _1$ with $\\Vert {\\mathbf {v}}\\Vert _1$ like (REF ).",
"Then, to solve ${\\mathbf {k}}$ results in a linear equation: $({\\mathbf {B}}_{\\nabla {\\hat{\\mathbf {x}}}}^T{\\mathbf {B}}_{\\nabla {\\hat{\\mathbf {x}}}}+2\\lambda _1\\mathbf {I}){\\mathbf {k}}={\\mathbf {B}}_{\\nabla {\\hat{\\mathbf {x}}}}^T\\nabla {\\mathbf {b}}-{\\mathbf {B}}_{\\nabla {\\hat{\\mathbf {x}}}}^T{\\mathbf {v}}.$ where $col({\\mathbf {B}}_{\\nabla {\\mathbf {x}}_c})_i={\\mathbf {H}}_i(\\nabla {\\mathbf {x}}_c)$ , $col({\\mathbf {B}}_{\\nabla {\\hat{\\mathbf {x}}}})_i={\\mathbf {H}}_i(\\nabla {\\hat{\\mathbf {x}}})$ , and $\\mathbf {I}$ is an identity matrix.",
"Unlike the fast solver in the frequency domain (REF ), we use conjugate gradient method to solve (REF ), because non-uniform blur no longer follows the convolution theorem in the frequency domain.",
"We apply the same soft-thresholding to solve ${\\mathbf {v}}$ by replacing $z_i=\\nabla b_i-({\\mathbf {k}}\\otimes \\nabla {\\hat{\\mathbf {x}}})_i$ with $z_i=\\nabla b_i-({\\mathbf {B}}_{\\nabla {\\hat{\\mathbf {x}}}}{\\mathbf {k}})_i$ in (REF ).",
"For TV regularized non-blind deblurring, (REF ) becomes: $\\operatornamewithlimits{\\arg \\min }_{\\mathbf {x}}\\frac{1}{2}\\Vert {\\mathbf {A}}_{\\hat{\\mathbf {k}}}{\\mathbf {x}}-{\\mathbf {b}}\\Vert _2^2+\\mu \\Vert \\nabla {\\mathbf {x}}\\Vert _1,$ which can also be solved with ADMM.",
"Similar modification can be done in the refined restoration of the finest scale."
],
[
"Experiments and Discussions",
"In this section, we have done extensive experiments to demonstrate the performance of our algorithm on the artificial dataset of Sun et al.",
"[55], real uniformly blurred images and non-uniformly blurred images.",
"Our experimental platform is a Windows 7 desktop computer with Intel i5 CPU.",
"For uniform deblurring, 8G memory is more than enough.",
"For non-uniform deblurring, we increase the memory to 24G.",
"We use Matlab R2014b to run and test all the matlab codes.",
"We tune the parameters on the dataset of Sun et al.",
"and find that they generate satisfactory results for all other images.",
"The parameters are set as follows.",
"That is, $\\lambda _1=\\lambda _3=5$ , $\\lambda _2=\\lambda _4=0.05$ , $\\mu =0.01$ .",
"Besides, the down-sampling factor $\\beta $ is set to $\\log _23$ for the image pyramid construction.",
"We downsample a blurry image scale by scale until its corresponding kernel is a delta function of size $1\\times 1$ .",
"The $max\\_iteration = 3$ in the preliminary restoration and the refined restoration as a trade-off between accuracy and speed."
],
[
"Artificial Blurry Dataset",
"In this experiment, we compare our algorithm with previous blind image deblurring algorithms on a large dataset introduced by Sun et al.",
"[55].",
"This dataset includes 640 blurry images (typically $1024\\times 768$ ), which are made by performing convolutions between 80 sharp nature images and 8 blur kernels offered by [4].",
"Each image is then added $1\\%$ white Gaussian noise.",
"The blur kernels are assumed unknown and the kernel sizes are set to $51\\times 51$ for all test cases.",
"In Fig.",
"REF and REF , we show the visual results deblurred by our algorithm and eight competing algorithms, i.e., Krishnan et al.",
"[5], Lai et al.",
"[9], Pan et al.",
"[10], Michaeli & Irani [16], Cho & Lee [46], Xu & Jia [50], Sun et al.",
"[55], Cho et al.",
"[56].",
"For Pan et al.",
"[10], the results are generated with their code.",
"For other algorithms, the results are provided by their authors or by Sun et al.",
"in the dataset.",
"As shown in Fig.",
"REF and REF , our algorithm performs visually better than other algorithms.",
"The edges and details are well restored and the artifacts in our results are much less than those in others.",
"Figure: Blurry images and deblurred results with zoomed regions.",
"(a) Blurry images and ground-truth kernels.",
"(b) Cho et al.",
".",
"(c) Cho & Lee .",
"(d) Krishnan et al.",
".",
"(e) Lai et al.",
".",
"(f) Ours.Figure: Blurry images and deblurred results with zoomed regions.",
"(a) Blurry images and ground-truth kernels.",
"(b) Xu & Jia.",
"(c) Sun et al.",
".",
"(d) Michaeli & Irani.",
"(e) Pan et al.",
".",
"(f) Ours.Besides the visual quality assessment, we also measure the quality of the deblurred results using error ratio, which was introduced by [2].",
"All the algorithms estimate blur kernels and use the same non-blind deblurring algorithm to restore the latent sharp images.",
"The error ratios are then computed by $r=\\frac{\\Vert {\\mathbf {x}}-{\\mathbf {x}}_{{\\hat{\\mathbf {k}}}}\\Vert ^2}{\\Vert {\\mathbf {x}}-{\\mathbf {x}}_{{\\mathbf {k}}}\\Vert ^2}$ where ${\\mathbf {x}}_{{\\hat{\\mathbf {k}}}}$ is the deblurred result computed using the estimated kernel ${\\hat{\\mathbf {k}}}$ , ${\\mathbf {x}}_{{\\mathbf {k}}}$ is the deblurred result computed using the ground-truth kernel ${\\mathbf {k}}$ , ${\\mathbf {x}}$ is the ground-truth sharp image.",
"If $r=1$ , it means that the blindly deblurred result is as good as the result restored using the ground-truth kernel.",
"The smaller $r$ is, the better the blindly deblurred result is.",
"We also agree with [16] that the visual quality is satisfactory when $r\\le 5$ and consider it as the threshold to decide the success.",
"For all the competing methods except [10], of which the results are provided by their authors or by Sun et al., they estimated the blur kernels and used non-blind deblurring algorithm [25] to restore sharp images.",
"For Pan et al.",
"[10], we generate deblurred images with their code.",
"Their code estimates the blur kernels and uses the non-blind method proposed in [8] to restore sharp images.",
"In order to fairly compare with all the algorithms, we run our algorithms on both settings (Using [25] and [8] for the last-step non-blind deblurring, respectively).",
"Fig.",
"REF reports the cumulative performance of error ratio.",
"Each curve represents the fractions of 640 images that can be deblurred under different error ratios.",
"For error ratios of results restored using [25] (solid lines), the curves of Sun et al.",
"[55] and Lai et al.",
"[9] are slightly higher than ours when $error\\ ratio<2.2$ .",
"Since all results in this range are very well recovered, the visual differences between their results and ours are hardly noticed.",
"When $error\\ ratio\\ge 2.2$ , our curve achieves the highest and the improvement is obvious.",
"For error ratios of results restored using [8] (dash lines), the two curves are very close at the beginning and our curve soon becomes higher than that of Pan et al.",
"[10].",
"Figure: Cumulative distribution of error ratios.",
"Each curve represents the fractions of 640 images that can be deblurred under different error ratios.In Table REF , we further report three statistical measures, i.e., mean error ratio, worst error ratio and success rate, to further compare all the blind deblurring algorithms.",
"According to the table, the mean errors of our algorithm are the smallest and the success rates of our algorithm are the highest among all the algorithms on both settings.",
"Although our worst error ratios are larger than Michaeli & Irani [16] and Pan et al.",
"[10], our second worst results are better on both settings.",
"Table: Quantitative comparison of all methods over the entire dataset (640 blurry images).In Fig.",
"REF , we show an example of an unsatisfactory blind image deblurring result.",
"In our approach, we require that there are objects with profiles, of which the sizes are much larger than that of the blur kernel.",
"However, in this case, smooth sky accounts for most of the image.",
"Only a windmill and foothills in the foreground provide profiles.",
"The small windmills in the background will be removed after down-sampling.",
"Thus, there aren't enough constraints in the data fidelity term for the complex blur kernel restoration, leading to the unsatisfactory blind deblurring result.",
"Figure: An example of an unsatisfactory blind image deblurring result.",
"(a) A blurred image and its corresponding true blur kernel.",
"(b) The deblurred result from our algorithm."
],
[
"Real Blurry Images",
"In the second part of this section, we apply our algorithms on real blurry images and compare with algorithms, of which the implementations are available, i.e., Krishnan et al.",
"[5], Levin et al.",
"[4], Michaeli & Irani [16] and Pan et al.",
"[10].",
"In Fig.",
"REF , REF , REF and REF , we blindly deblur an image with different depths of field, a human portrait with a complex background, an image full of humans, and an image with very severe motion blur, respectively.",
"Our algorithm estimates the blur kernels robustly and results in less artifacts in the recovered images.",
"Table REF reports the running time of the competing algorithms and ours on real blurry images in Fig.",
"REF , REF , REF and REF .",
"All the algorithms run on the Matlab platform.",
"According to Table REF , our algorithm runs much faster than other competing algorithms.",
"The efficiency of our algorithm can be further improved with C++ and GPU implementation.",
"Figure: Flower.",
"Image Size: 618×464618\\times 464, Kernel Size: 69×6969\\times 69.",
"(a) Blurry Image.",
"(b) Krishnan et al.",
".",
"(c) Levin et al.",
".",
"(d) Michaeli & Irani .",
"(e) Pan et al.",
".",
"(f) Ours.Figure: Picasso.",
"Image Size: 800×532800\\times 532, Kernel Size: 69×6969\\times 69.",
"(a) Blurry Image.",
"(b) Krishnan et al.",
".",
"(c) Levin et al.",
".",
"(d) Michaeli & Irani .",
"(e) Pan et al.",
".",
"(f) Ours.Figure: Pietro.",
"Image Size: 800×600800\\times 600, Kernel Size: 69×6969\\times 69.",
"(a) Blurry Image.",
"(b) Krishnan et al.",
".",
"(c) Levin et al.",
".",
"(d) Michaeli & Irani .",
"(e) Pan et al.",
".",
"(f) Ours.Figure: Roma.",
"Image Size: 1229×8251229\\times 825, Kernel Size: 95×9595\\times 95.",
"(a) Blurry Image.",
"(b) Krishnan et al.",
".",
"(c) Levin et al.",
".",
"(d) Michaeli & Irani .",
"(e) Pan et al.",
".",
"(f) Ours.Table: Running time of different algorithms on real images."
],
[
"Non-Uniform Blurry Images",
"In Fig.",
"REF and REF , we demonstrate the performance of our algorithm on the non-uniform blind deblurring problem.",
"Directly using uniform deblurring algorithm fails to restore the correct sharp image, since the existing blur does not follow the convolution model.",
"as shown in Fig.",
"REF .",
"Extended to non-uniform blur model (REF ), our algorithm can tackle the spatial-variant blurry cases.",
"We compare our algorithm with recent non-uniform blind deblurring algorithms [57], [53], [7], [8], of which the results are available.",
"Our results are visually comparable or better than those of the competing methods.",
"Figure: Elephant.",
"Image Size: 601×401601\\times 401.",
"(a) Blurry Image.",
"(b) Our uniform deblurring.",
"(c) Harmeling et al.",
".",
"(d) Hirsch et al.",
".",
"(e) Our non-uniform deblurring.",
"(f) Estimated non-uniform kernel.Figure: Butchershop.",
"Image Size: 601×401601\\times 401.",
"(a) Blurry Image.",
"(b) Harmeling et al.",
".",
"(c) Xu et al.",
"(d) Pan et al.",
".",
"(e) Our non-uniform deblurring.",
"(f) Estimated non-uniform kernel."
],
[
"Conclusion",
"In this paper, we propose a Multi-Scale Latent Structure prior, which is derived from down-sampling a blurry image.",
"With this prior, we design an efficient algorithm to jointly restore both the latent sharp image and the blur kernel from a single blurry observation.",
"The qualitative and quantitative experiments demonstrate that our algorithm is competitive against the state-of-the-art methods in blindly recovering uniform and non-uniform blurry images.",
"Limitation: In our approach, we require that there are objects with profiles, of which the sizes are much larger than that of the blur kernel.",
"If the precondition is disobeyed, objects will be removed after down-sampling.",
"In this case, the data fidelity term of the optimization function fails to work and will converge to an uncertain kernel.",
"Nevertheless, to the best of our knowledge, it is also difficult for other blind deblurring algorithms to deal with the blurry images, in which all objects are smaller than blur kernels.",
"Recently, there have been methods focusing on specific object deblurring problems, for example, text or face deblurring.",
"These methods can work better by considering the properties of specific objects.",
"In the future, we would like to incorporate the advantage of these techniques with our algorithm to make a more robust unified framework.",
"Besides, although the parameter kernel size in our algorithm is not sensitive and can be set relatively large, it is still more preferred to have it automatically set according to the individuality of the blurry images.",
"In the future, we would like to investigate how to infer the optimal kernel size from a blurry image to make our algorithm more practical."
]
] | 1906.04442 |
[
[
"A ruled quintic surface in $PG(6,q)$"
],
[
"Abstract In this article we look at a scroll of $PG(6,q)$ that uses a projectivity to rule a conic and a twisted cubic.",
"We show this scroll is a ruled quintic surface $\\mathcal V^5_2$, and study its geometric properties.",
"The motivation in studying this scroll lies in its relationship with an $\\mathbb F_q$-subplane of $PG(2,q^3)$ via the Bruck-Bose representation."
],
[
"Introduction",
"In this article we consider a scroll of ${\\rm PG}(6,q)$ that rules a conic and a twisted cubic according to a projectivity.",
"The motivation in studying this scroll lies in its relationship with an $\\mathbb {F}_{q}$ -subplane of ${\\rm PG}(2,q^3)$ via the Bruck-Bose representation as described in Section .",
"In ${\\rm PG}(6,q)$ , let $ be a non-degenerate conic in a plane $$, $ is called the conic directrix.",
"Let ${\\cal N}_3$ be a twisted cubic in a 3-space $\\Pi _3$ with $\\alpha \\cap \\Pi _3=\\emptyset $ , ${\\cal N}_3$ is called the twisted cubic directrix.",
"Let $\\phi $ be a projectivity from the points of $ to the points of $ N3$.By this we mean that if we write the points of $ and ${\\cal N}_3$ using a non-homogeneous parameter, so $\\lbrace C_\\theta =(1,\\theta ,\\theta ^2)\\,|\\,\\theta \\in \\mathbb {F}_{q}\\cup \\lbrace \\infty \\rbrace \\rbrace $ , and ${\\cal N}_3=\\lbrace N_\\epsilon =(1,\\epsilon ,\\epsilon ^2,\\epsilon ^3)\\,|\\,\\epsilon \\in \\mathbb {F}_{q}\\cup \\lbrace \\infty \\rbrace \\rbrace $ , then $\\phi \\in {\\rm PGL}(2,q)$ is a projectivity mapping $(1,\\theta )$ to $(1,\\epsilon )$ .",
"Let $ be the set of points of $ PG(6,q)$ lying on the $ q+1$ lines joining each point of $ to the corresponding point (under $\\phi $ ) of ${\\cal N}_3$ .",
"These $q+1$ lines are called the generators of $.",
"As the two subspaces $$ and $ 3$ are disjoint, $ is not contained in a 5-space.",
"We note that this construction generalises the ruled cubic surface $3_2$ in ${\\rm PG}(4,q)$ , a variety that has been well studied, see [8].",
"We will be working with normal rational curves in ${\\rm PG}(6,q)$ , if ${\\cal N}$ is a normal rational curve that generates an $i$ -dimensional space, then we call ${\\cal N}$ an $i$ -dim nrc, and often use the notation ${\\cal N}_i$ .",
"See [6] for details on normal rational curves.",
"As we will be looking at 5-dim nrcs contained in $,we will assume $ q6$ throughout.$ This article studies the geometric structure of $.In Section~\\ref {sec-simple}, we show that $ is a variety ${{\\mathcal {V}}^5_2}$ of order 5 and dimension 2, and that all such scrolls are projectively equivalent.",
"Further, we show that $ contains exactly $ q+1$ lines and one non-degenerate conic.In Section~\\ref {sec-BB}, we describe the Bruck-Bose representation of $ PG(2,q3)$ in $ PG(6,q)$, and discuss how $ corresponds to an $\\mathbb {F}_{q}$ -subplane of ${\\rm PG}(2,q^3)$ .",
"We use the Bruck-Bose setting to show that $ contains exactly $ q2$ twisted cubics, and that each can act as a directrix of $ .",
"In Section , we count the number of 4 and 5-dim nrcs contained in $.",
"Further,we determine how 5-spaces meet $ , and count the number of 5-spaces of each intersection type.",
"The main result is Theorem REF .",
"In Section , we determine how 5-spaces meet $ in relation to the regular 2-spread in the Bruck-Bose setting.$"
],
[
"Simple properties of $$",
"Theorem 2.1 Let $ be a scroll of $ PG(6,q)$ that rules a conic and a twisted cubic according to a projectivity.",
"Then$ is a variety of dimension 2 and order 5, denoted ${{\\mathcal {V}}^5_2}$ and called a ruled quintic surface.",
"Further, any two ruled quintic surfaces are projectively equivalent.",
"Proof Let $ be a scroll of $ PG(6,q)$ with conic directrix $ in a plane $\\alpha $ , twisted cubic directrix ${\\cal N}_3$ in a 3-space $\\Pi _3$ , and ruled by a projectivity as described in Section 1.",
"The group of collineations of ${\\rm PG}(6,q)$ is transitive on planes; and transitive on 3-spaces.",
"Further, all non-degenerate conics in a projective plane are projectively equivalent; and all twisted cubics in a 3-space are projectively equivalent.",
"Hence without loss of generality, we can coordinatise $ as follows.",
"Let $$ be the plane which is the intersection of the four hyperplanes$ x0=0, x1=0, x2=0, x3=0.$Let $ be the non-degenerate conic in $\\alpha $ with points $C_\\theta =(0,0,0,0,1,\\theta ,\\theta ^2)$ for $\\theta \\in \\mathbb {F}_q\\cup \\lbrace \\infty \\rbrace $ .",
"Note that the points of $ are the exact intersection of $$ with the quadric of equation $ x52=x4x6.$Let $ 3$ be the 3-space which is the intersection of the three hyperplanes$ x4=0, x5=0, x6=0.$Let $ N3$ be the twisted cubic in $ 3$ with points $ N=(1,,2,3,0,0,0)$ for $ Fq{}$.",
"Note that the points of $ N3$ are the exact intersection of $ 3$ with the three quadrics with equations $ x12=x0x2, x22=x1x3, x0x3=x1x2.$A projectivity in $ PGL(2,q)$ is uniquely determined by the image of three points, sowithout loss of generality, let $ have generator lines $\\ell _\\theta =\\lbrace V_{\\theta ,t}=N_\\theta + t C_\\theta ,\\ t\\in \\mathbb {F}_q\\cup \\lbrace \\infty \\rbrace \\rbrace $ for $\\theta \\in \\mathbb {F}_{q}\\cup \\lbrace \\infty \\rbrace $ .",
"That is, $V_{\\theta ,t}=(1,\\theta ,\\theta ^2,\\theta ^3,t,t\\theta ,t\\theta ^2)$ .",
"Equivalently, $ consists of the points $ Vx,y,z=(x3, x2y, xy2, y3, zx2, zxy, zy2)$ for $ x,yFq$, not both 0, $ zFq{}.$It is straightforward to verify that the pointset of $ is the exact intersection of the following ten quadrics, $x_{0}x_{5} =x_{1}x_{4},\\ x_0x_6= x_1x_5=x_2x_4,\\ x_1x_6=x_2x_5=x_3x_4,\\ x_2x_6=x_3x_5,\\ x_1^2=x_0x_2,\\ x_2^2=x_1x_3,\\ x_5^2=x_4x_6,\\ x_0x_3=x_1x_2.$ Hence the points of $ form a variety.$ We follow [7] to calculate the dimension and order of $.The following map defines an algebraic one to one correspondence between the plane $$ of $ PG(3,q)$ with points $ (x,y,z,0)$, $ x,y,zFq$ not all 0, and the points of $ .",
"$\\begin{array}{rccl}\\sigma : &\\pi &\\longrightarrow && (x,y,z,0) & \\longmapsto & (x^3,x^2y,xy^2,y^3,x^2z,xyz,y^2z).\\end{array}$ Thus $ is an absolutelyirreducible variety of dimension two and so we are justifiedin calling it a surface.Now consider a generic 4-space of $ PG(6,q)$ with equation given by the two hyperplanes$ 1: a0x0++a6x6=0$ and$ 2: b0x0++b6x6=0$, $ ai,biFq$.The point $ Vx,y,z=(x3,x2y,xy2,y3,x2z,xyz,y2z)$ lies on $ 1$ if $ a0x3+a1x2y+ a2xy2+a3y3+a4x2z+a5xyz+a6y2z=0$.",
"This corresponds to a cubic $ K$ in the plane $$, moreover $ K$ contains the point $ P=(0,0,1,0)$, and $ P$ is a double point of $ K$.Similarly $ Vx,y,z2$ corresponds to a cubic in $$ with a double point $ (0,0,1,0)$.",
"Two cubics in a plane meet generically in nine points.",
"As $ (0,0,1,0)$ lies in the kernel of $$, in $ PG(6,q)$ the 4-space $ 12$ meets $ in five points, and so $ has order 5.",
"\\hbox{}\\hfill $$ $ Theorem 2.2 Let ${{\\mathcal {V}}^5_2}$ be a ruled quintic surface in ${\\rm PG}(6,q)$ .",
"No two generators of ${{\\mathcal {V}}^5_2}$ lie in a plane.",
"No three generators of ${{\\mathcal {V}}^5_2}$ lie in a 4-space.",
"No four generators of ${{\\mathcal {V}}^5_2}$ lie in a 5-space.",
"Proof Let ${{\\mathcal {V}}^5_2}$ be a ruled quintic surface of ${\\rm PG}(6,q)$ with conic directrix $ in a plane $$, and twisted cubic directrix $ N3$ lying in a 3-space $ 3$.Suppose two generator lines $ 0,1$ of $ V52$ lie in a plane.Let $ m$ be the line in $$ joining the distinct points $ 0$, $ 1$.",
"Let $ m'$ be the line in $ 3$ joining the distinct points $ 03$, $ 13$.",
"The lines $ m,m'$ lie in the plane $ 0,1$ and so meet in a point, contradicting $$, $ 3$ being disjoint.Hence the generator lines of $ V52$ are pairwise skew.$ For part 2, suppose a 4-space $\\Pi _4$ contains three distinct generators of ${{\\mathcal {V}}^5_2}$ .",
"As distinct generators meet $ in distinct points,$ 4$ contains three distinct points of $ , and so contains the plane $\\alpha $ .",
"Further, distinct generators meet ${\\cal N}_3$ in distinct points, hence $\\Pi _4$ contains three points of ${\\cal N}_3$ , and so $\\Pi _4\\cap \\Pi _3$ has dimension at least two.",
"Hence $\\langle \\Pi _4,\\Pi _3\\rangle $ has dimension at most $4+3-2=5$ .",
"However, ${{\\mathcal {V}}^5_2}\\subseteq \\langle \\Pi _4,\\Pi _3\\rangle $ , a contradiction as ${{\\mathcal {V}}^5_2}$ is not contained in a 5-space.",
"For part 3, suppose a 5-space $\\Pi _5$ contains four distinct generators of ${{\\mathcal {V}}^5_2}$ .",
"Distinct generators meet $\\Pi _3$ in distinct points of ${\\cal N}_3$ , so $\\Pi _5$ contains four points of ${\\cal N}_3$ , which do not lie in a plane.",
"Hence $\\Pi _5$ contains $\\Pi _3$ .",
"Similarly $\\Pi _5$ contains $\\alpha $ , and so $\\Pi _5$ contains ${{\\mathcal {V}}^5_2}$ , a contradiction as ${{\\mathcal {V}}^5_2}$ is not contained in a 5-space.",
"$\\square $ Corollary 2.3 No two generators of ${{\\mathcal {V}}^5_2}$ lie in a 3-space containing $\\alpha $ .",
"Proof Suppose a 3-space $\\Pi _3$ contained $\\alpha $ and two generators of ${{\\mathcal {V}}^5_2}$ .",
"Let $P$ be a point of ${{\\mathcal {V}}^5_2}$ not in $\\Pi _3$ , and let $\\ell $ be the generator of ${{\\mathcal {V}}^5_2}$ through $P$ .",
"Then $\\Pi _4=\\langle \\Pi _3,P\\rangle $ contains two distinct points of $\\ell $ , namely $P$ and $\\ell \\cap , and so $ 4$ contains $$.",
"That is, $ 4$ is a 4-space containing three generators, contradicting Theorem~\\ref {x-gen}.",
"\\hbox{}\\hfill $$ $ We now show that the only lines on ${{\\mathcal {V}}^5_2}$ are the generators, and the only non-degenerate conic on ${{\\mathcal {V}}^5_2}$ is the conic directrix.",
"We show later in Theorem REF that there are exactly $q^2$ twisted cubics on ${{\\mathcal {V}}^5_2}$ , and that each is a directrix.",
"Theorem 2.4 Let ${{\\mathcal {V}}^5_2}$ be a ruled quintic surface in ${\\rm PG}(6,q)$ .",
"A line of ${\\rm PG}(6,q)$ meets ${{\\mathcal {V}}^5_2}$ in 0, 1, 2 or $q+1$ points.",
"Further, ${{\\mathcal {V}}^5_2}$ contains exactly $q+1$ lines, namely the generator lines.",
"Proof Let ${{\\mathcal {V}}^5_2}$ be a ruled quintic surface of ${\\rm PG}(6,q)$ with conic directrix $ lying in a plane $$, and twisted cubic directrix $ N3$ lying in the 3-space $ 3$.",
"Let $ m$ be a line of $ PG(6,q)$ that is not a generator of $ V52$, andsuppose $ m$ meets $ V52$ in three points $ P,Q,R$.",
"As $ m$ is not a generator of $ V52$, the points $ P,Q,R$ lie on distinct generator lines denoted $ P,Q,R$ respectively.As $ is a non-degenerate conic, $m$ is not a line of $\\alpha $ and so at most one of the points $P,Q,R$ lie in $.Suppose firstly that $ P,Q,R.",
"Then $\\langle \\alpha ,m\\rangle $ is a 3- or 4-space that contains the three generators $\\ell _P,\\ell _Q,\\ell _R$ , contradicting Theorem REF .",
"Now suppose $P\\in and $ Q,R.",
"Then $\\Sigma _3=\\langle \\alpha ,m\\rangle $ is a 3-space which contains the two generator lines $\\ell _Q,\\ell _R$ .",
"So $\\Sigma _3\\cap \\Pi _3$ contains the distinct points $\\ell _R\\cap {\\cal N}_3$ , $\\ell _Q\\cap {\\cal N}_3$ , and so has dimension at least one.",
"Hence $\\langle \\Sigma _3,\\Pi _3\\rangle $ has dimension at most $3+3-1=5$ , a contradiction as ${{\\mathcal {V}}^5_2}\\subset \\langle \\Sigma _3,\\Pi _3\\rangle $ , but ${{\\mathcal {V}}^5_2}$ is not contained in a 5-space.",
"Hence a line of ${\\rm PG}(6,q)$ is either a generator line of ${{\\mathcal {V}}^5_2}$ , or meets ${{\\mathcal {V}}^5_2}$ in 0, 1 or 2 points.",
"$\\square $ Theorem 2.5 The ruled quintic surface ${{\\mathcal {V}}^5_2}$ contains exactly one non-degenerate conic.",
"Proof Let ${{\\mathcal {V}}^5_2}$ be a ruled quintic surface with conic directrix $ in a plane $$.",
"Suppose $ V52$ contains another non-degenerate conic $$ in a plane $ '$.",
"If $$ contains two points on a generator $$ of $ V52$, then $ 'V52$ contains $$ and $$.However, by the proof of Theorem~\\ref {the-nine-quad}, $ V52$ is the intersection of quadrics, and the configuration $$ is not contained in any planar quadric.",
"Hence $$ contains exactly one point on each generator of $ V52$.",
"We consider the three cases where $ '$ is either empty, a point or a line.Suppose $ '=$, then $ ,'$ is a 5-space that contains $ and $$ , and so contains two distinct points on each generator of ${{\\mathcal {V}}^5_2}$ .",
"Hence $\\langle \\alpha ,\\alpha ^{\\prime }\\rangle $ contains each generator of ${{\\mathcal {V}}^5_2}$ and so contains ${{\\mathcal {V}}^5_2}$ , a contradiction as ${{\\mathcal {V}}^5_2}$ is not contained in a 5-space.",
"Suppose $\\alpha \\cap \\alpha ^{\\prime }$ is a point $P$ , then $\\langle \\alpha ,\\alpha ^{\\prime }\\rangle $ is a 4-space that contains at least $q$ generators of ${{\\mathcal {V}}^5_2}$ , contradicting Theorem REF as $q\\ge 6$ .",
"Finally, suppose $\\alpha \\cap \\alpha ^{\\prime }$ is a line, then $\\langle \\alpha ,\\alpha ^{\\prime }\\rangle $ is a 3-space that contains at least $q-1$ generators, contradicting Theorem REF as $q\\ge 6$ .",
"So ${{\\mathcal {V}}^5_2}$ contains exactly one non-degenerate conic.",
"$\\square $ We aim to classify how 5-spaces meet ${{\\mathcal {V}}^5_2}$ , so we begin with a simple description.",
"Remark 2.6 Let $\\Pi _5$ be a 5-space, then $\\Pi _5\\cap {{\\mathcal {V}}^5_2}$ contains a set of $q+1$ points, one on each generator.",
"Lemma 2.7 A 5-space meets ${{\\mathcal {V}}^5_2}$ in either (a) a 5-dim nrc, (b) a 4-dim nrc and 0 or 1 generators, (c) a 3-dim nrc and 0, 1 or 2 generators, (d) the conic directrix and 0, 1, 2 or 3 generators.",
"Proof Using properties of varieties (see for example [7]) we have ${{\\mathcal {V}}^5_2}\\cap 1_5=5_1$ , that is, the variety ${{\\mathcal {V}}^5_2}$ meets a 5-space $1_5$ in a curve of degree five.",
"Denote this curve of ${\\rm PG}(6,q)$ by $\\mathcal {K}$ .",
"The degree of $\\mathcal {K}$ can be partitioned as $5=4+1=3+2=3+1+1=2+2+1=2+1+1+1=1+1+1+1+1.$ By Theorem REF , the only lines on ${{\\mathcal {V}}^5_2}$ are the generators.",
"By Theorem REF , $\\mathcal {K}$ does not contain more than 3 generators.",
"By Remark REF , $\\mathcal {K}$ contains at least one point on each generator.",
"Hence $\\mathcal {K}$ is not empty, and is not the union of 1, 2 or 3 generators, so the partition $1+1+1+1+1$ for the degree of $\\mathcal {K}$ does not occur.",
"Suppose that the degree of $\\mathcal {K}$ is partitioned as either (a) $2+2+1$ or (b) $2+1+1+1$ .",
"By Remark REF , $\\mathcal {K}$ contains a point on each generator, so $\\mathcal {K}$ contains an irreducible conic.",
"By Theorem REF , this conic is the conic directrix $ of $ V52$, and case (a) does not occur.Hence $ K$ consists of $ and 0, 1, 2 or 3 generators of ${{\\mathcal {V}}^5_2}$ .",
"Suppose that the degree of $\\mathcal {K}$ is partitioned as $3+1+1$ .",
"So $\\mathcal {K}$ consists of at most 2 generators, and an irreducible cubic $\\mathcal {K}^{\\prime }$ .",
"By Remark REF , $\\mathcal {K}$ contains a point on each generator, so $\\mathcal {K}^{\\prime }$ contains a point on at least $q-1$ generators.",
"If $\\mathcal {K}^{\\prime }$ generates a 3-space, then it is a 3-dim nrc of ${\\rm PG}(6,q)$ .",
"If not, $\\mathcal {K}^{\\prime }$ is an irreducible cubic contained in a plane $\\Pi _2$ .",
"By the proof of Theorem REF , $\\mathcal {K}^{\\prime }$ is contained in a quadric, so $\\mathcal {K}^{\\prime }$ is not an irreducible planar cubic.",
"Thus $\\mathcal {K}^{\\prime }$ is a 3-dim nrc of ${\\rm PG}(6,q)$ .",
"Hence $\\mathcal {K}$ consists of a 3-dim nrc and 0, 1 or 2 generators of ${{\\mathcal {V}}^5_2}$ .",
"Suppose that the degree of $\\mathcal {K}$ is partitioned as $2+3$ .",
"By Remark REF , $\\mathcal {K}$ contains a point on each generator.",
"As argued above, $\\mathcal {K}$ does not contain an irreducible planar cubic.",
"Suppose $\\mathcal {K}$ contained both an irreducible conic $ and a twisted cubic $ N3$, then there is at least one generator $$ that meet $ and ${\\cal N}_3$ in distinct points.",
"In this case $\\ell $ lies in the 5-space and so lies in $\\mathcal {K}$ , a contradiction.",
"So $\\mathcal {K}$ is not the union of an irreducible conic and a twisted cubic.",
"Suppose that the degree of $\\mathcal {K}$ is partitioned as $4+1$ .",
"So $\\mathcal {K}$ consists of at most 1 generator, and an irreducible quartic $\\mathcal {K}^{\\prime }$ .",
"By Remark REF , $\\mathcal {K}$ contains a point on each generator, so $\\mathcal {K}^{\\prime }$ contains a point on at least $q$ generators.",
"If $\\mathcal {K}^{\\prime }$ generates a 4-space, then it is a 4-dim nrc of ${\\rm PG}(6,q)$ .",
"If not, $\\mathcal {K}^{\\prime }$ is an irreducible quartic contained in a 3-space $\\Pi _3$ .",
"Let $\\ell ,m$ be two generators not in $\\mathcal {K}$ , then by Remark REF they meet $\\mathcal {K}^{\\prime }$ .",
"So $\\langle \\Pi _3,\\ell ,m\\rangle $ has dimension at most 5, and meets ${{\\mathcal {V}}^5_2}$ in a irreducible quartic and 2 lines, which is a curve of degree 6, a contradiction.",
"Thus $\\mathcal {K}^{\\prime }$ is a 4-dim nrc of ${\\rm PG}(6,q)$ .",
"That is, $\\mathcal {K}$ consists of a 4-dim nrc and 0 or 1 generators of ${{\\mathcal {V}}^5_2}$ .",
"Suppose the curve $\\mathcal {K}$ is irreducible.",
"By Remark REF , $\\mathcal {K}$ contains a point on each generator.",
"So either $\\mathcal {K}$ is a 5-dim nrc of ${\\rm PG}(6,q)$ , or $\\mathcal {K}$ lies in a 4-space.",
"Suppose $\\mathcal {K}$ lies in a 4-space $\\Pi _4$ , and let $\\ell $ be a generator, then $\\langle \\Pi _4,\\ell \\rangle $ has dimension at most 5 and meets ${{\\mathcal {V}}^5_2}$ in a curve of degree 6, a contradiction.",
"So $\\mathcal {K}$ is a 5-dim nrc of ${\\rm PG}(6,q)$ .",
"$\\square $ Corollary 2.8 Let $\\Pi _r$ be an $r$ -space, $r=3,4,5$ that contains an $r$ -dim nrc of ${{\\mathcal {V}}^5_2}$ .",
"Then $\\Pi _r$ contains 0 generators of ${{\\mathcal {V}}^5_2}$ .",
"Proof First suppose $r=3$ , by Lemma REF , a 5-space containing a twisted cubic ${\\cal N}_3$ of ${{\\mathcal {V}}^5_2}$ contains at most two generators of ${{\\mathcal {V}}^5_2}$ .",
"Hence a 4-space containing ${\\cal N}_3$ contains at most one generator of ${{\\mathcal {V}}^5_2}$ .",
"Hence the 3-space $\\Pi _3$ containing ${\\cal N}_3$ contains no generator of ${{\\mathcal {V}}^5_2}$ .",
"If $r=4$ , by Lemma REF , a 5-space containing a 4-dim nrc ${\\cal N}_4$ of ${{\\mathcal {V}}^5_2}$ contains at most one generator of ${{\\mathcal {V}}^5_2}$ .",
"Hence the 4-space $\\Pi _4$ containing ${\\cal N}_4$ contains no generators of ${{\\mathcal {V}}^5_2}$ .",
"If $r=5$ , then by Lemma REF , $\\Pi _5$ contains 0 generators of ${{\\mathcal {V}}^5_2}$ .",
"$\\square $ Theorem 2.9 Let ${\\cal N}_r$ be an $r$ -dim nrc lying on ${{\\mathcal {V}}^5_2}$ , $r=3,4,5$ .",
"Then ${\\cal N}_r$ contains exactly one point on each generator of ${{\\mathcal {V}}^5_2}$ .",
"Proof Let ${\\cal N}_r$ be an $r$ -dim nrc lying on ${{\\mathcal {V}}^5_2}$ , $r=3,4,5$ , and denote the $r$ -space containing ${\\cal N}_r$ by $\\Pi _r$ .",
"If $\\Pi _r$ contained 2 points of a generator of ${{\\mathcal {V}}^5_2}$ , then it contains the whole generator, so by Corollary REF , the $q+1$ points of ${\\cal N}_r$ are one on each generator of ${{\\mathcal {V}}^5_2}$ .",
"$\\square $"
],
[
"${{\\mathcal {V}}^5_2}$ and {{formula:2a8dfaee-3d42-4cc6-881a-cb9e60b86fa6}} -subplanes of {{formula:35543a06-f67a-4eef-9bba-7830f33da0a7}}",
"To study ${{\\mathcal {V}}^5_2}$ in more detail, we use the linear representation of ${\\rm PG}(2,q^3)$ in ${\\rm PG}(6,q)$ developed independently by André and Bruck and Bose in [1], [4], [5].",
"Let $\\mathcal {S}$ be a regular 2-spread of ${\\rm PG}(6,q)$ in a 5-space $\\Sigma _\\infty $ .",
"Let $\\mathcal {I}$ be the incidence structure with: points the points of ${\\rm PG}(6,q)\\backslash \\Sigma _\\infty $ ; lines the 3-spaces of ${\\rm PG}(6,q)$ that contain a plane of $\\mathcal {S}$ and are not in $\\Sigma _\\infty $ ; and incidence is inclusion.",
"Then $\\mathcal {I}$ is isomorphic to ${\\rm AG}(2,q^3)$ .",
"We can uniquely complete $\\mathcal {I}$ to ${\\rm PG}(2,q^3)$ , the points on $\\ell _\\infty $ correspond to the planes of $\\mathcal {S}$ .",
"We call this the Bruck-Bose representation of ${\\rm PG}(2,q^3)$ in ${\\rm PG}(6,q)$ , see [2] for a detailed discussion on this representation.",
"Of particular interest is the relationship between the ruled quintic surface of ${\\rm PG}(6,q)$ and the $\\mathbb {F}_{q}$ -subplanes of ${\\rm PG}(2,q^3)$ .",
"To describe this relationship, we need to use the cubic extension of ${\\rm PG}(6,q)$ to ${\\rm PG}(6,q^3)$ .",
"The regular 2-spread $\\mathcal {S}$ has a unique set of three conjugate transversal lines in this cubic extension, denoted $g,g^q,g^{q^2}$ , which meet each extended plane of $\\mathcal {S}$ , for more details on regular spreads and transversals, see [6].",
"An $r$ -space $\\Pi _r$ of ${\\rm PG}(6,q)$ lies in a unique $r$ -space of ${\\rm PG}(6,q^3)$ , denoted $\\Pi _r^{^{\\mbox{\\tiny \\ding {73}}}}$ .",
"A nrc ${\\cal N}$ of ${\\rm PG}(6,q)$ lies in a unique nrc of ${\\rm PG}(6,q^3)$ , denoted ${\\cal N}^{^{\\mbox{\\tiny \\ding {73}}}}$ .",
"Let ${{\\mathcal {V}}^5_2}$ be a ruled quintic surface with conic directrix $, twisted cubic directrix $ N3$, and associated projectivity $$.",
"Thenwe can extend $ V52$ to a unique ruled quintic surface $ V52✩$ of $ PG(6,q3)$with conic directrix $✩$, twisted cubic directrix $ N3✩$, with the same associated projectivity, that is, extend $$ from acting on $ PG(1,q)$ to acting on $ PG(1,q3)$.",
"We need the following characterisations.$ Result 3.1 [2], [3] Let $\\mathcal {S}$ be a regular 2-spread in a 5-space $\\Sigma _\\infty $ in ${\\rm PG}(6,q)$ and consider the Bruck-Bose plane ${\\rm PG}(2,q^3)$ .",
"An $\\mathbb {F}_{q}$ -subline of ${\\rm PG}(2,q^3)$ that meets $\\ell _\\infty $ in a point corresponds in ${\\rm PG}(6,q)$ to a line not in $\\Sigma _\\infty $ .",
"An $\\mathbb {F}_{q}$ -subline of ${\\rm PG}(2,q^3)$ that is disjoint from $\\ell _\\infty $ corresponds in ${\\rm PG}(6,q)$ to a twisted cubic ${\\cal N}_3$ lying in a 3-space about a plane of $\\mathcal {S}$ , such that the extension ${\\cal N}_3^{^{\\mbox{\\tiny \\ding {73}}}}$ to ${\\rm PG}(6,q^3)$ meets each transversal of $\\mathcal {S}$ in a point.",
"An $\\mathbb {F}_{q}$ -subplane of ${\\rm PG}(2,q^3)$ tangent to $\\ell _\\infty $ at the point $T$ corresponds in ${\\rm PG}(6,q)$ to a ruled quintic surface ${{\\mathcal {V}}^5_2}$ with conic directrix in the spread plane corresponding to $T$ , such that in the cubic extension ${\\rm PG}(6,q^3)$ , the transversals $g,g^q,g^{q^2}$ of $\\mathcal {S}$ are generators of ${{{\\mathcal {V}}^5_2}}{^{\\mbox{\\tiny \\ding {73}}}}$ .",
"Moreover, the converse of each is true.",
"We use this characterisation to show that ${{\\mathcal {V}}^5_2}$ contains exactly $q^2$ twisted cubics.",
"Theorem 3.2 The ruled quintic surface ${{\\mathcal {V}}^5_2}$ contains exactly $q^2$ twisted cubics, each is a directrix of ${{\\mathcal {V}}^5_2}$ .",
"Proof By Theorem REF , all ruled quintic surfaces are projectively equivalent.",
"So without loss of generality, we can position a ruled quintic surface so that it corresponds to an $\\mathbb {F}_{q}$ -subplane of ${\\rm PG}(2,q^3)$ which we denote by ${\\mathcal {B}}$ .",
"That is, by Result REF , $\\mathcal {S}$ is a regular 2-spread in a hyperplane $\\Sigma _\\infty $ , ${{\\mathcal {V}}^5_2}\\cap \\Sigma _\\infty $ is the conic directrix $ of $ V52$, $ lies in a plane of $\\mathcal {S}$ , and in the cubic extension ${\\rm PG}(6,q^3)$ , the transversals $g,g^q,g^{q^2}$ of $\\mathcal {S}$ are generators of ${{{\\mathcal {V}}^5_2}}^{^{\\mbox{\\tiny \\ding {73}}}}$ .",
"Let ${\\cal N}_3$ be a twisted cubic contained in ${{\\mathcal {V}}^5_2}$ , and denote the 3-space containing ${\\cal N}_3$ by $\\Pi _3$ .",
"As ${{\\mathcal {V}}^5_2}\\cap \\Sigma _\\infty =, $ 3$ meets $$ in a plane, we show this is a plane of $ S$.In $ PG(6,q3)$, $ V52✩$ is a ruled quintic surface that contains the twisted cubic $ N3✩$, moreover, the transversals $ g,gq,gq2$ of $ S$ are generators of $ V52✩$.",
"So by Theorem~\\ref {one-pt-gen}, $ N3✩$ contains one point on each of $ g,gq$ and $ gq2$.",
"Hence the 3-space $ 3✩$ contains an extended plane of $ S$, and so $ 3$ meets $$ in a plane of $ S$.Hence $ 3=$, further, by Theorem~\\ref {one-pt-gen}, $ N3$ contains one point on each generator of $ V52$, thus $ N3$ is a directrix of $ V52$.$ By Result REF , ${\\cal N}_3$ corresponds in ${\\rm PG}(2,q^3)$ to an $\\mathbb {F}_{q}$ -subline of ${\\mathcal {B}}$ disjoint from $\\ell _\\infty $ .",
"Conversely, every $\\mathbb {F}_{q}$ -subline of ${\\mathcal {B}}$ disjoint from $\\ell _\\infty $ corresponds to a twisted cubic on ${{\\mathcal {V}}^5_2}$ .",
"Thus the twisted cubics in ${{\\mathcal {V}}^5_2}$ are in 1-1 correspondence with the $\\mathbb {F}_{q}$ -sublines of ${\\mathcal {B}}$ that are disjoint from $\\ell _\\infty $ .",
"As there are $q^2$ such $\\mathbb {F}_{q}$ -sublines, there are $q^2$ twisted cubics on ${{\\mathcal {V}}^5_2}$ .",
"$\\square $ Suppose we position ${{\\mathcal {V}}^5_2}$ so that it corresponds via the Bruck-Bose representation to a tangent $\\mathbb {F}_{q}$ -subplane ${\\mathcal {B}}$ of ${\\rm PG}(2,q^3)$ .",
"So we have a regular 2-spread $\\mathcal {S}$ in a hyperplane $\\Sigma _\\infty $ , and the conic directrix of ${{\\mathcal {V}}^5_2}$ lies in a plane $\\alpha \\in \\mathcal {S}$ .",
"We define the splash of ${\\mathcal {B}}$ to be the set of $q^2+1$ points on $\\ell _\\infty $ that lie on an extended line of ${\\mathcal {B}}$ .",
"The splash of ${{\\mathcal {V}}^5_2}$ is defined to be the corresponding set of $q^2+1$ planes of $\\mathcal {S}$ .",
"We denote the splash of ${{\\mathcal {V}}^5_2}$ by $\\mathbb {S}$ .",
"Note that $\\alpha $ is a plane of $\\mathbb {S}$ .",
"We show that the remaining $q^2$ planes of $\\mathbb {S}$ are related to the $q^2$ twisted cubics of ${{\\mathcal {V}}^5_2}$ .",
"Corollary 3.3 Let $\\mathcal {S}$ be a regular 2-spread in a hyperplane $\\Sigma _\\infty $ of ${\\rm PG}(6,q)$ .",
"Without loss of generality, we can position ${{\\mathcal {V}}^5_2}$ so that it corresponds via the Bruck-Bose representation to a tangent $\\mathbb {F}_{q}$ -subplane of ${\\rm PG}(2,q^3)$ .",
"Then the conic directrix of ${{\\mathcal {V}}^5_2}$ lies in a plane $\\alpha \\in \\mathcal {S}$ , the $q^2$ 3-spaces containing a twisted cubic of ${{\\mathcal {V}}^5_2}$ meet $\\Sigma _\\infty $ in distinct planes of $\\mathcal {S}$ , and these planes together with $\\alpha $ form the splash $\\mathbb {S}$ of ${{\\mathcal {V}}^5_2}$ .",
"Proof By Theorem REF , all ruled quintic surfaces are projectively equivalent, so without loss of generality, let ${{\\mathcal {V}}^5_2}$ be positioned so that it corresponds to an $\\mathbb {F}_{q}$ -subplane ${\\mathcal {B}}$ of ${\\rm PG}(2,q^3)$ which is tangent to $\\ell _\\infty $ .",
"Let $b$ be an $\\mathbb {F}_{q}$ -subline of ${\\mathcal {B}}$ disjoint from $\\ell _\\infty $ , so the extension of $b$ meets $\\ell _\\infty $ in a point $R$ which lies in the splash of ${\\mathcal {B}}$ .",
"By Result REF , $b$ corresponds in ${\\rm PG}(6,q)$ to a twisted cubic of ${{\\mathcal {V}}^5_2}$ which lies in a 3-space that meets $\\Sigma _\\infty $ in the plane of $\\mathbb {S}$ corresponding to the point $R$ .",
"$\\square $ Using this Bruck-Bose setting, we describe the 3-spaces of ${\\rm PG}(6,q)$ that contain a plane of the regular 2-spread $\\mathcal {S}$ .",
"Corollary 3.4 Position ${{\\mathcal {V}}^5_2}$ as in Corollary REF , so $\\mathcal {S}$ is a regular 2-spread in the hyperplane $\\Sigma _\\infty $ , and the conic directrix of ${{\\mathcal {V}}^5_2}$ lies in a plane $\\alpha $ contained in the splash $\\mathbb {S}\\subset \\mathcal {S}$ of ${{\\mathcal {V}}^5_2}$ .",
"Let $\\beta \\in \\mathbb {S}\\backslash \\alpha $ , then there exists a unique 3-space containing $\\beta $ that meets ${{\\mathcal {V}}^5_2}$ in a twisted cubic.",
"The remaining 3-spaces containing $\\beta $ (and not in $\\Sigma _\\infty $ ) meet ${{\\mathcal {V}}^5_2}$ in 0 or 1 point.",
"Let $\\gamma \\in \\mathcal {S}\\backslash \\mathbb {S}$ , then each 3-space containing $\\gamma $ and not in $\\Sigma _\\infty $ meets ${{\\mathcal {V}}^5_2}$ in 0 or 1 point.",
"Proof By Corollary REF , we can position ${{\\mathcal {V}}^5_2}$ so that it corresponds to an $\\mathbb {F}_{q}$ -subplane ${\\mathcal {B}}$ of ${\\rm PG}(2,q^3)$ which is tangent to $\\ell _\\infty $ .",
"The 3-spaces that contain a plane of $\\mathcal {S}$ (and do not lie in $\\Sigma _\\infty $ ) correspond to lines of ${\\rm PG}(2,q^3)$ .",
"Each point on $\\ell _\\infty $ not in ${\\mathcal {B}}$ but in the splash of ${\\mathcal {B}}$ lies on a unique line that meets ${\\mathcal {B}}$ in an $\\mathbb {F}_{q}$ -subline.",
"By Result REF , this corresponds to a twisted cubic in ${{\\mathcal {V}}^5_2}$ .",
"The remaining lines meet ${\\mathcal {B}}$ in 0 or 1 point, so the remaining 3-spaces meet ${{\\mathcal {V}}^5_2}$ in 0 or 1 point.",
"$\\square $ As ${{\\mathcal {V}}^5_2}$ corresponds to an $\\mathbb {F}_{q}$ -subplane, we have the following result.",
"Theorem 3.5 Let ${{\\mathcal {V}}^5_2}$ be a ruled quintic surface in ${\\rm PG}(6,q)$ .",
"Two twisted cubics on ${{\\mathcal {V}}^5_2}$ meet in a unique point.",
"Let $P,Q$ be points lying on different generators of ${{\\mathcal {V}}^5_2}$ , and not in the conic directrix.",
"Then $P,Q$ lie on a unique twisted cubic of ${{\\mathcal {V}}^5_2}$ .",
"Proof Without loss of generality let ${{\\mathcal {V}}^5_2}$ be positioned as described in Corollary REF .",
"So the conic directrix lies in a plane $\\alpha $ contained in a regular 2-spread $\\mathcal {S}$ in $\\Sigma _\\infty $ , and ${{\\mathcal {V}}^5_2}$ corresponds to a $\\mathbb {F}_{q}$ -subplane ${\\mathcal {B}}$ of ${\\rm PG}(2,q^3)$ tangent to $\\ell _\\infty $ .",
"Let ${\\cal N}_1,{\\cal N}_2$ be two twisted cubics contained in ${{\\mathcal {V}}^5_2}$ .",
"By Result REF , they correspond in ${\\rm PG}(2,q^3)$ to two $\\mathbb {F}_{q}$ -sublines of ${\\mathcal {B}}$ not containing ${\\mathcal {B}}\\cap \\ell _\\infty $ , and so meet in a unique affine point $P$ .",
"This corresponds to a unique point $P\\in {{\\mathcal {V}}^5_2}\\backslash \\alpha $ lying in both ${\\cal N}_1$ and ${\\cal N}_2$ , proving part 1.",
"For part 2, let $P,Q$ be points lying on distinct generators of ${{\\mathcal {V}}^5_2}$ , $P,Q\\notin .",
"If the line $ PQ$ met $$, then$ ,P,Q$ is a 3-space that contains $$ and the generators of $ V52$ containing $ P$ and $ Q$, contradicting Corollary~\\ref {2-gen-alpha}.Hence the line $ PQ$ is skew to $$.",
"In $ PG(2,q3)$, $ P,Q$ correspond to two affine points in the tangent $ Fq$-subplane $ B$, so they lie on a unique $ Fq$-subline $ b$ of $ B$.By Result~\\ref {FFA-result}, the generators of $ V52$ correspond to the $ Fq$-sublines of $ B$ through the point $ B$.",
"As $ PQ$ is skew to $$, we have $ b=$.Hence by Result~\\ref {FFA-result}, in $ PG(6,q)$, $ P,Q$ lie on a unique twisted cubic of $ V52$.",
"\\hbox{}\\hfill $$ $"
],
[
"5-spaces meeting ${{\\mathcal {V}}^5_2}$",
"In this section we determine how 5-spaces meet ${{\\mathcal {V}}^5_2}$ and count the different intersection types.",
"A series of lemmas is used to prove the main result which is stated in Theorem REF .",
"Lemma 4.1 Let ${{\\mathcal {V}}^5_2}$ be a ruled quintic surface of ${\\rm PG}(6,q)$ with conic directrix $.",
"Of the$ q3+q2+q+1$ $ 5$-spaces of $ PG(6,q)$ containing $ , $r_i$ meet ${{\\mathcal {V}}^5_2}$ in precisely $ and $ i$ generators, where$$r_3=\\frac{q^3-q}{6},\\quad r_2=q^2+q,\\quad r_1=\\frac{q^3}{2}+\\frac{q}{2}+1,\\quad r_0=\\frac{q^3-q}{3}.$$$ Proof Let ${{\\mathcal {V}}^5_2}$ be a ruled quintic surface of ${\\rm PG}(6,q)$ with conic directrix $ lying in a plane $$.",
"By Lemma~\\ref {5-space-quintic}, a 5-space containing $ contains at most three generator lines of ${{\\mathcal {V}}^5_2}$ .",
"By Theorem REF , three generators of ${{\\mathcal {V}}^5_2}$ lie in a unique 5-space.",
"Hence there are $r_3={q+1\\atopwithdelims ()3}$ 5-spaces that contain three generators of ${{\\mathcal {V}}^5_2}$ .",
"Such a 5-space contains three points of $, and so contains $ and $\\alpha $ .",
"Denote the generator lines of ${{\\mathcal {V}}^5_2}$ by $\\ell _0,\\ldots ,\\ell _q$ and consider two generators, $\\ell _0,\\ell _1$ say.",
"By Corollary REF , $\\Sigma _4=\\langle \\alpha ,\\ell _0,\\ell _1\\rangle $ is a 4-space.",
"By Theorem REF , $\\langle \\Sigma _4,\\ell _i\\rangle $ $i=2,\\ldots , q$ are distinct 5-spaces.",
"That is, $q-1$ of the 5-spaces about $\\Sigma _4$ contain 3 generators, and hence the remaining two contain $\\ell _0$ , $\\ell _1$ and no further generator of ${{\\mathcal {V}}^5_2}$ .",
"Hence by Lemma REF , $q-1$ of the 5-spaces about $\\Sigma _4$ meet ${{\\mathcal {V}}^5_2}$ in exactly $ and 3 generators; and the remaining two 5-spaces about $ 4$ meet $ V52$ in exactly $ and two generators.",
"There are ${q+1\\atopwithdelims ()2}$ choices for $\\Sigma _4$ , hence the number of 5-spaces that meet ${{\\mathcal {V}}^5_2}$ in precisely $ and two generators is $$r_2=2\\times {q+1\\atopwithdelims ()2}=(q+1)q.$$Next, let $ r1$ be the number of $ 5$-spaces that meet $ V52$ in precisely $ and one generator.",
"We count in two ways ordered pairs $(\\ell ,\\Pi _5)$ where $\\ell $ is a generator of ${{\\mathcal {V}}^5_2}$ , and $\\Pi _5$ is a 5-space that contains $\\ell $ and $\\alpha $ , giving $(q+1)(q^2+q+1)=3r_3\\ +\\ 2r_2\\ +\\ r_1.$ Hence $r_1=q^3/2+q/2+1$ .",
"Finally, the number of 5-spaces containing $ and zero generators is $ r0=(q3+q2+q+1) - r3-r2-r1=(q3-q)/3$, as required.",
"\\hbox{}\\hfill $$ $ Lemma 4.2 Let ${{\\mathcal {V}}^5_2}$ be a ruled quintic surface of ${\\rm PG}(6,q)$ and let ${\\cal N}_3$ be a twisted cubic directrix of ${{\\mathcal {V}}^5_2}$ .",
"Of the $q^2+q+1$ 5-spaces of ${\\rm PG}(6,q)$ containing ${\\cal N}_3$ , $s_i$ meet ${{\\mathcal {V}}^5_2}$ in precisely ${\\cal N}_3$ and $i$ generators, where $s_2=\\frac{q^2+q}{2},\\quad s_1=q+1,\\quad \\quad s_0=\\frac{q^2-q}{2}.$ The total number of 5-spaces that meet ${{\\mathcal {V}}^5_2}$ in a twisted cubic and $i$ generators is $q^2s_i$ , $i=0,1,2$ .",
"Proof Let ${{\\mathcal {V}}^5_2}$ be a ruled quintic surface of ${\\rm PG}(6,q)$ with a twisted cubic directrix ${\\cal N}_3$ lying in the 3-space $\\Pi _3$ .",
"By Lemma REF , a 5-space containing ${\\cal N}_3$ contains at most two generators of ${{\\mathcal {V}}^5_2}$ , so the number of 5-spaces that contain $\\Pi _3$ and exactly two generator lines is $s_2={q+1\\atopwithdelims ()2}$ .",
"Let $\\ell $ be a generator of ${{\\mathcal {V}}^5_2}$ and consider the 4-space $\\Pi _4=\\langle \\Pi _3,\\ell \\rangle $ .",
"For each generator $m\\ne \\ell $ , $\\langle \\Pi _4,m\\rangle $ is a 5-space about $\\Pi _4$ that meets ${{\\mathcal {V}}^5_2}$ in ${\\cal N}_3$ , $\\ell $ and $m$ , and in no further point by Lemma REF .",
"This accounts for $q$ of the 5-spaces containing $\\Pi _4$ .",
"Hence the remaining 5-space containing $\\Pi _4$ meets ${{\\mathcal {V}}^5_2}$ in exactly ${\\cal N}_3$ and $\\ell $ .",
"That is, exactly one of the 5-spaces about $\\Pi _4=\\langle \\Pi _3,\\ell \\rangle $ meets ${{\\mathcal {V}}^5_2}$ in precisely ${\\cal N}_3$ and $\\ell $ .",
"There are $q+1$ choices for the generator $\\ell $ , hence $s_1=q+1$ .",
"Finally $s_0=(q^2+q+1)-s_2-s_1=(q^2-q)/2$ , as required.",
"For part 2, by Lemma REF , ${{\\mathcal {V}}^5_2}$ contains $q^2$ twisted cubics, so the total number of 5-spaces meeting ${{\\mathcal {V}}^5_2}$ in a twisted cubic and $i$ generators is $q^2s_i$ , $i=0,1,2$ .",
"$\\square $ The next result looks at properties of 4-dim nrcs contained in ${{\\mathcal {V}}^5_2}$ .",
"In particular, we show that there are no 5-spaces that meet ${{\\mathcal {V}}^5_2}$ in a 4-dim nrc and 0 generator lines.",
"Lemma 4.3 Let ${{\\mathcal {V}}^5_2}$ be a ruled quintic surface of ${\\rm PG}(6,q)$ with conic directrix $ in the plane $$, and let$ N4$ be a $ 4$-dim nrc contained in $ V52$.\\begin{enumerate}\\item The q+1 5-spaces containing {\\cal N}_4 each contain a distinct generator line of {{\\mathcal {V}}^5_2}.\\item The 4-space containing {\\cal N}_4 meets \\alpha in a point P, and either P={\\cal N}_4or q is even and P is the nucleus of .\\end{enumerate}$ Proof Let ${{\\mathcal {V}}^5_2}$ be a ruled quintic surface in ${\\rm PG}(6,q)$ with conic directrix $ lying in a plane $$.Let $ N4$ be a 4-dim nrc contained in $ V52$, so $ N4$ lies in a 4-space which we denote $ 4$.By Corollary~\\ref {3-4-no-gen}, $ 4$ does not contain a generator of $ V52$.By Lemma~\\ref {5-space-quintic}, a 5-space containing $ N4$ can contain at most one generator of $ V52$.",
"Hence each of the $ q+1$ 5-spaces containing $ N4$ contains a distinct generator.",
"In particular, if we label the points of $ by $Q_0,\\ldots ,Q_{q}$ , and the generator through $Q_i$ by $\\ell _{Q_i}$ , then the $q+1$ 5-spaces containing ${\\cal N}_4$ are $\\Sigma _i=\\langle \\Pi _4,\\ell _{Q_i}\\rangle $ , $i=0,\\ldots ,q$ .",
"If $\\Pi _4$ met the plane $\\alpha $ in a line, then $\\langle \\Pi _4,\\alpha \\rangle $ is a 5-space whose intersection with ${{\\mathcal {V}}^5_2}$ contains ${\\cal N}_4$ and $, contradicting Lemma~\\ref {5-space-quintic}.",
"Hence $ 4$ meets $$ in a point $ P$.There are three possibilities for the point $ P=4$, namely $ P; $q$ even and $P$ the nucleus of $; or $ q$ even, $ P, and $P$ not the nucleus of $.$ Case 1, suppose $P\\in .For $ i=0,...,q$, the 5-space $ i=4,Qi$ meets $$ in a line $ mi$.",
"Label $ so that $P=Q_0$ , so the line $m_0$ is the tangent to $ at $ P$, and $ mi$, $ i=1,...,q$, is the secant line $ PQi$.",
"We now show that $ P=Q0$ is a point of $ N4$.Let $ i{1,...,q}$, then by Lemma~\\ref {5-space-quintic}, $ i$ meets $ V52$ in precisely $ N4Qi$, and $ iV52$ is the two points $ P,Qi$.",
"As $ PQi$ we have $ PN4$.",
"That is, $ P=N4$.$ Case 2, suppose $q$ is even and $P=\\Pi _4\\cap \\alpha $ is the nucleus of $.For $ i=0,...,q$, the 5-space $ i=4,Qi$ meets $$ in the tangent to $ through $Q_i$ .",
"In this case, ${\\cal N}_4=\\emptyset $ .",
"Case 3, suppose $P=\\Pi _4\\cap \\alpha $ is not in $, and $ P$ is not the nucleus of $ .",
"Now $P$ lies on some secant $m=QR$ of $, for some points $ Q,R.",
"The intersection of the 5-space $\\langle \\Pi _4,m\\rangle $ with ${{\\mathcal {V}}^5_2}$ contains ${\\cal N}_4$ and two points $R,Q$ of $.As $ R,Q$ lie on distinct generators and are not in $ N4$, this contradicts Lemma~\\ref {5-space-quintic}.Hence this case cannot occur.",
"\\hbox{}\\hfill $$ $ We can now describe how a nrc of ${{\\mathcal {V}}^5_2}$ meets the conic directrix, and note that Theorem REF shows that each possibility in part 3 can occur.",
"Corollary 4.4 Let ${{\\mathcal {V}}^5_2}$ be a ruled quintic surface of ${\\rm PG}(6,q)$ with conic directrix $.\\begin{enumerate}\\item A twisted cubic {\\cal N}_3\\subseteq {{\\mathcal {V}}^5_2} contains 0 points of .\\item A 4-dim nrc{\\cal N}_4\\subseteq {{\\mathcal {V}}^5_2} either (i) contains 1 point of ; or (ii) contains 0 points of , in which case q is even and the 4-space containing {\\cal N}_4 contains the nucleus of .\\item A 5-dim nrc{\\cal N}_5\\subseteq {{\\mathcal {V}}^5_2} contains 0 or 1 or 2 points of .\\end{enumerate}$ Proof Let ${{\\mathcal {V}}^5_2}$ be a ruled quintic surface of ${\\rm PG}(6,q)$ with conic directrix $ in a plane $$.",
"Let $ N3$ be a twisted cubic of $ V52$, so byLemma~\\ref {tc-direct}, $ N3$ is a directrix of $ V52$, and so is disjoint from $$, proving part 1.",
"Next let $ N4$ be a 4-dim nrc on $ V52$, and let $ 4$ be the 4-space containing $ N4$.By Lemma~\\ref {5contains4}, $ 4$ is a point $ P$, and either $ P=N4$; or $ q$ is even and $ P$ is the nucleus of $ , and so $P\\notin {{\\mathcal {V}}^5_2}$ , hence $P\\notin {\\cal N}_4$ , proving part 2.",
"Let $\\Pi _5$ be a 5-space containing a 5-dim nrc of ${{\\mathcal {V}}^5_2}$ .",
"By Lemma REF , $\\Pi _5$ cannot contain $\\alpha $ .",
"Hence $\\Pi _5$ meets $\\alpha $ in a line, and so contains at most two points of $, proving part 3.",
"\\hbox{}\\hfill $$ $ We now use the Bruck-Bose setting to count the 4-dim nrcs contained in ${{\\mathcal {V}}^5_2}$ .",
"Lemma 4.5 Let $\\mathcal {S}$ be a regular 2-spread in a 5-space $\\Sigma _\\infty $ in ${\\rm PG}(6,q)$ .",
"Position ${{\\mathcal {V}}^5_2}$ as in Corollary REF , so ${{\\mathcal {V}}^5_2}$ has splash $\\mathbb {S}\\subset \\mathcal {S}$ .",
"Then a 5-space/4-space about a plane $\\beta \\in \\mathbb {S}$ cannot contain a 4-dim nrc of ${{\\mathcal {V}}^5_2}$ .",
"Proof Position ${{\\mathcal {V}}^5_2}$ as described in Corollary REF , so $\\mathcal {S}$ is a regular 2-spread in a 5-space $\\Sigma _\\infty $ , the conic directrix of ${{\\mathcal {V}}^5_2}$ lies in a plane $\\alpha \\in \\mathcal {S}$ , and $\\mathbb {S}\\subset \\mathcal {S}$ denotes the splash of ${{\\mathcal {V}}^5_2}$ .",
"By Lemma REF , a 4-space containing $\\alpha $ cannot contain a 4-dim nrc of ${{\\mathcal {V}}^5_2}$ .",
"Let $\\beta \\in \\mathbb {S}\\backslash \\alpha $ , then by Corollary REF , $\\beta $ lies in exactly one 3-space that contains a twisted cubic of ${{\\mathcal {V}}^5_2}$ , denote these by $\\Pi _3$ and ${\\cal N}_3$ respectively.",
"By Theorem REF , ${\\cal N}_3$ is a directrix of ${{\\mathcal {V}}^5_2}$ , and so $\\Pi _3$ is disjoint from $\\alpha $ .",
"So if $\\ell _P$ is a generator of ${{\\mathcal {V}}^5_2}$ , then $\\Pi _4=\\langle \\Pi _3,\\ell _P\\rangle $ is a 4-space and $\\Pi _4\\cap \\alpha $ is the point $P=\\ell _P\\cap .",
"Let $$ be a line of $$ through $ P$ and let $ 5=3,$.",
"If $$ is tangent to $ , then $\\Pi _5\\cap {{\\mathcal {V}}^5_2}$ is exactly ${\\cal N}_3\\cup \\ell _P$ .",
"If $\\ell $ is a secant of $, so $ {P,Q}$, then $ 5V52$ consists of $ N3,P$ and the generator $ Q$ through $ Q$.Varying $ P$ and $$, we get all the 5-spaces that contain $$ and contain 1 or 2 generators of $ V52$.",
"That is, each 5-space containing $$ and 1 or 2 generators of $ V52$ also contains $ N3$.",
"The remaining 5-spaces about $$ hence contain 0 generators of $ V52$ and meet $$ in an exterior line of $ .",
"Hence by Lemma REF , none of the 5-spaces about $\\beta $ contain a 4-dim nrc of ${{\\mathcal {V}}^5_2}$ .",
"$\\square $ Lemma 4.6 The number of 4-dim nrcs contained in ${{\\mathcal {V}}^5_2}$ is $q^4-q^2$ .",
"The number of 5-spaces that meet ${{\\mathcal {V}}^5_2}$ in a 4-dim nrc and one generator is $q^5+q^4-q^3-q^2$ .",
"Proof Without loss of generality position ${{\\mathcal {V}}^5_2}$ as described in Corollary REF .",
"That is, let $\\mathcal {S}$ be a regular 2-spread in a 5-space $\\Sigma _\\infty $ , let the conic directrix of ${{\\mathcal {V}}^5_2}$ lie in a plane $\\alpha \\in \\mathcal {S}$ , and let $\\mathbb {S}\\subset \\mathcal {S}$ be the splash of ${{\\mathcal {V}}^5_2}$ .",
"Straightforward counting shows that a 5-space distinct from $\\Sigma _\\infty $ contains a unique spread plane.",
"If this plane is in the splash $\\mathbb {S}$ , then by Lemma REF , the 5-space does not contain a 4-dim nrc of ${{\\mathcal {V}}^5_2}$ .",
"So a 5-space containing a 4-dim nrc of ${{\\mathcal {V}}^5_2}$ contains a unique plane of $\\mathcal {S}\\backslash \\mathbb {S}$ .",
"Consider a plane $\\gamma \\in \\mathcal {S}\\backslash \\mathbb {S}$ .",
"Let $P\\in , let $ P$ be the generator of $ V52$ through $ P$, and consider the 4-space $ 4=,P$.Suppose first that $ 4$ contains two generators of $ V52$, then there is a 5-space $ 5$ containing $$ and two generators.",
"By Lemma~\\ref {5-space-quintic}, $ 5$ contains either $ or a twisted cubic of ${{\\mathcal {V}}^5_2}$ .",
"A 5-space distinct from $\\Sigma _\\infty $ cannot contain two planes of $\\mathcal {S}$ , so $\\Pi _5$ does not contain $.",
"Moreover, by Corollary~\\ref {remark-BB}, $ 5$ does not contain a twisted cubic of $ V52$.",
"Hence $ 4$ contains exactly one generator of $ V52$.If every generator of $ V52$ contained at least one point of $ 4$, then the intersection of $ 4$ with $ V52$ contains at least $ P$ and $ q$ further points, one on each generator.",
"By Lemma~\\ref {5-space-quintic} and Corollary~\\ref {3-4-no-gen}, the only possibility is that $ 4V52$ contains a twisted cubic, which is not possible by Corollary~\\ref {remark-BB}.Hence there is at least one generator which is disjoint from $ 4$, denote this $ Q$.",
"Label the points of $ Q$ by $ X0,...,Xq$, then the $ q+1$ 5-spaces containing $ 4$ are $ i=,P,Xi$.",
"For each $ i=0,...,q$, the intersection of $ i$ with $ V52$ contains the generator $ P$ and the point $ Xi$.",
"ByCorollary~\\ref {remark-BB}, $ i$does not contain a twisted cubic of $ V52$.",
"Hence byLemma~\\ref {5-space-quintic}, $ iV52$ is $ P$ and a 4-dim nrc.$ That is, there are $(q+1)^2$ 5-spaces containing $\\gamma $ and one generator of ${{\\mathcal {V}}^5_2}$ , each contains a 4-dim nrc of ${{\\mathcal {V}}^5_2}$ .",
"Further, if $\\Pi _5$ is a 5-space containing $\\gamma $ and zero generators of ${{\\mathcal {V}}^5_2}$ , then by Lemma REF , $\\Pi _5$ does not contain a 4-dim nrc of ${{\\mathcal {V}}^5_2}$ .",
"Hence as there are $q^3-q^2$ choices for $\\gamma $ , there are $(q+1)^2\\times (q^3-q^2)=q^5+q^4-q^3-q^2 $ 5-spaces that meet ${{\\mathcal {V}}^5_2}$ in one generator and a 4-dim nrc.",
"By Lemma REF , every 4-dim nrc in ${{\\mathcal {V}}^5_2}$ lies in $q+1$ such 5-spaces.",
"Hence the number of 4-dim nrcs contained in ${{\\mathcal {V}}^5_2}$ is $(q^5+q^4-q^3-q^2)/(q+1)$ as required.",
"$\\square $ We now count the number of 5-dim nrcs contained in ${{\\mathcal {V}}^5_2}$ .",
"Lemma 4.7 The number of 5-spaces meeting ${{\\mathcal {V}}^5_2}$ in a 5-dim nrc is $q^6-q^4$ .",
"Proof We show that the number of 5-spaces meeting ${{\\mathcal {V}}^5_2}$ in a 5-dim nrc is $q^6-q^4$ by counting in two ways the number $x$ of incident pairs $(A,\\Pi _5)$ where $A$ is a point of ${{\\mathcal {V}}^5_2}$ , and $\\Pi _5$ is a 5-space containing $A$ .",
"The number of ways to choose a point $A$ of ${{\\mathcal {V}}^5_2}$ is $(q+1)^2.$ The point $A$ lies in $q^5+q^4+q^3+q^2+q+1$ 5-spaces.",
"So $x= (q+1)^2\\times (q^5+q^4+q^3+q^2+q+1)=q^7+3q^6+4q^5+4q^4+4q^3+4q^2+3q+1.$ Alternatively, we count the 5-spaces first; there are several possibilities for $\\Pi _5$ .",
"By Lemmas REF , $\\Pi _5\\cap {{\\mathcal {V}}^5_2}$ is either empty, or contains an $r$ -dim nrc, for some $r\\in \\lbrace 2,\\ldots ,5\\rbrace $ .",
"Let $n_r$ be the number of pairs $(A,\\Pi _5)$ with $A\\in {{\\mathcal {V}}^5_2}\\cap \\Pi _5$ and $\\Pi _5$ containing an $r$ -dim nrc of ${{\\mathcal {V}}^5_2}$ .",
"Note that $x=n_2+n_3+n_4+n_5.$ We now calculate $n_2$ , $n_3$ and $n_4$ , and then use (REF ) to determine the number of 5-spaces meeting ${{\\mathcal {V}}^5_2}$ in a 5-dim nrc.",
"Consider a 5-space $\\Pi _5$ that contains the conic directrix $, so by Lemma~\\ref {5containsC}, $ 5$ contains 0, 1, 2 or 3 generators of $ V52$, and the number of 5-spaces meeting $ V52$ in exactly the conic directrix and $ i$ generators is $ ri$.",
"In this case the number of ways to pick a point of $ 5V52$ is $ iq+q+1$.Hence the total number of pairs $ (A,5)$ with $ 5$ containing the conic directrix is$$n_2=\\sum _{i=0}^3 r_i(iq+q+1)= 2q^4+4q^3+4q^2+3q+1.$$$ Consider a 5-space $\\Pi _5$ that contains a twisted cubic, then by Lemma REF , $\\Pi _5$ contains 0, 1 or 2 generators of ${{\\mathcal {V}}^5_2}$ , and the number of 5-spaces meeting ${{\\mathcal {V}}^5_2}$ in a given twisted cubic and $i$ generators is $s_i$ .",
"In this case the number of ways to pick $A$ in ${{\\mathcal {V}}^5_2}\\cap \\Pi _5$ is $iq+q+1$ .",
"Hence the number of pairs $(A,\\Pi _5)$ with $\\Pi _5$ containing a twisted cubic of ${{\\mathcal {V}}^5_2}$ is $n_3=q^2\\sum _{i=0}^2 s_i(iq+q+1)= 2q^5+4q^4+3q^3+q^2.$ Consider a 5-space $\\Pi _5$ that contains a 4-dim nrc of ${{\\mathcal {V}}^5_2}$ .",
"By Lemma REF , $\\Pi _5$ contains 1 generator of ${{\\mathcal {V}}^5_2}$ .",
"By Lemma REF , the number of 5-spaces meeting ${{\\mathcal {V}}^5_2}$ in exactly a 4-dim nrc and one generator is $q^5+q^4-q^3-q^2$ .",
"The number of ways to pick $A$ in ${{\\mathcal {V}}^5_2}\\cap \\Pi _5$ is $2q+1$ .",
"So $n_4= (q^5+q^4-q^3-q^2) \\times (2q+1)=2q^6+3q^5-q^4-3q^3-q^2.$ Denote the number of 5-spaces containing a 5-dim nrc of ${{\\mathcal {V}}^5_2}$ by $y$ .",
"Then the number of pairs $(A,\\Pi _5)$ with $\\Pi _5$ containing a 5-dim nrc of ${{\\mathcal {V}}^5_2}$ is $n_5=y\\times (q+1).$ Substituting the calculated values for $x,n_2,n_3,n_4,n_5$ into (REF ) and rearranging gives $y=q^6-q^4$ as required.",
"$\\square $ Summarising the preceding lemmas gives the following theorem describing ${{\\mathcal {V}}^5_2}$ .",
"Theorem 4.8 Let ${{\\mathcal {V}}^5_2}$ be the ruled quintic surface in ${\\rm PG}(6,q)$ , $q\\ge 6$ .",
"${{\\mathcal {V}}^5_2}$ contains exactly Table: NO_CAPTION A 5-space meets ${{\\mathcal {V}}^5_2}$ in one of the following configurations Table: NO_CAPTION"
],
[
"5-spaces and the Bruck-Bose spread",
"Let $\\mathcal {S}$ be a regular 2-spread in a 5-space $\\Sigma _\\infty $ in ${\\rm PG}(6,q)$ , and position ${{\\mathcal {V}}^5_2}$ so that it corresponds to a tangent $\\mathbb {F}_{q}$ -subplane of ${\\rm PG}(2,q^3)$ .",
"So ${{\\mathcal {V}}^5_2}$ has splash $\\mathbb {S}\\subset \\mathcal {S}$ , the conic directrix $ lies in a plane $ S$, and each of the $ q2$ 3-spaces containing a twisted cubic directrix of $ V52$ meets $$ in a distinct plane of $ S$.In Corollary~\\ref {tc-splash-3}, we looked at how 3-spaces containing a plane of $ S$ meet $ V52$.",
"In Lemma~\\ref {part-of-next-lemma}, we looked at how 4-spaces containing a plane of $ S$ meet $ V52$.",
"Next we look at how5-spaces containing a plane of $ S$ meet $ V52$.",
"Note thatstraightforward counting shows that a 5-space distinct from $$ contains a unique plane $$ of $ S$, and meets every other plane of $ S$ in a line.",
"If $ =$, then Lemma~\\ref {5containsC} describes the possible intersections with $ V52$.",
"The next theorem describes the possible intersections with $ V52$ for the remaining cases $ S$ and $ SS$.$ Theorem 5.1 Position ${{\\mathcal {V}}^5_2}$ as in Corollary REF , so $\\mathcal {S}$ is a regular 2-spread in a hyperplane $\\Sigma _\\infty $ , the conic directrix $ lies in a plane $ S$ and $ V52$ has splash $ SS$.",
"Let $$ be a line of $$ with $ |=i$ and let $ S$, $$.",
"Then the $ q$ 5-spaces containing $ ,$ and distinct from $$ meet $ V52$ as follows.\\begin{enumerate}\\item If \\pi \\in \\mathbb {S}\\backslash \\alpha , then q-1 meet {{\\mathcal {V}}^5_2} in a 5-dim nrc, and 1 meets {{\\mathcal {V}}^5_2} in a twisted cubic and i generators.\\item If \\pi \\in \\mathcal {S}\\backslash \\mathbb {S}, then q-i meet {{\\mathcal {V}}^5_2} in a 5-dim nrc, and i meet {{\\mathcal {V}}^5_2} in a 4-dim nrc and 1 generator.\\end{enumerate}$ Proof By [2], the group of collineations of ${\\rm PG}(6,q)$ fixing $\\mathcal {S}$ and ${{\\mathcal {V}}^5_2}$ is transitive on the planes of $\\mathbb {S}\\backslash \\alpha $ and on the planes of $\\mathcal {S}\\backslash \\mathbb {S}$ .",
"As this group fixes the conic directrix $, it is transitive on the lines of $$ tangent to $ , the lines of $\\alpha $ secant to $ andthe lines of $$ exterior to $ .",
"So without loss of generality let $\\ell _0$ be a line of $\\alpha $ exterior to $, let $ 1$ be a line of $$ tangent to $ , let $\\ell _2$ be a line of $\\alpha $ secant to $, let $$ be a plane in $ S$, and let $$ be a plane of $ SS$.",
"For $ i=0,1,2$, label the 4-spaces$ 4,i=,i$ and $ 4,i=,i$.By Corollary~\\ref {tc-splash-3}, as $ S$, there is a unique twisted cubic of $ V52$ that lies in a 3-space about $$, denote this 3-space by $ 3$.",
"Hence for $ i=0,1,2$, there is a unique 5-space containing $ 4,i$ whose intersection with $ V52$ contains a twisted cubic, namely the 5-space $ 3,i$.$ First consider the line $\\ell _0$ which is exterior to $.A 5-space meeting $$ in $ 0$ contains 0 points of $ , and so contains 0 generators of ${{\\mathcal {V}}^5_2}$ .",
"The 4-space $\\Sigma _{4,0}=\\langle \\beta ,\\ell _0\\rangle $ lies in $q$ 5-spaces distinct from $\\Sigma _\\infty $ , each containing 0 generators of ${{\\mathcal {V}}^5_2}$ .",
"Exactly one of these 5-spaces, namely $\\langle \\Pi _3,\\ell _0\\rangle $ , contains a twisted cubic of ${{\\mathcal {V}}^5_2}$ .",
"The remaining $q-1$ 5-spaces about $\\Sigma _{4,0}$ contain 0 generators, and do not contain a conic or twisted cubic of ${{\\mathcal {V}}^5_2}$ , so by Theorem REF , they meet ${{\\mathcal {V}}^5_2}$ in a 5-dim nrc, proving part 1 for $i=0$ .",
"For part 2, let $\\Pi _5\\ne \\Sigma _\\infty $ be any 5-space containing $\\Pi _{4,0}=\\langle \\gamma ,\\ell _0\\rangle $ .",
"As $\\gamma \\notin \\mathbb {S}$ , by Corollary REF , $\\Pi _5$ cannot contain a twisted cubic of ${{\\mathcal {V}}^5_2}$ .",
"As $\\Pi _5$ contains 0 generator lines of ${{\\mathcal {V}}^5_2}$ and does not contain a conic or twisted cubic of ${{\\mathcal {V}}^5_2}$ , by Theorem REF , $\\Pi _5$ meets ${{\\mathcal {V}}^5_2}$ in a 5-dim nrc.",
"That is, the $q$ 5-spaces (distinct from $\\Sigma _\\infty $ ) containing $\\Pi _{4,0}$ meet ${{\\mathcal {V}}^5_2}$ in a 5-dim nrc, proving part 2 for $i=0$ .",
"Next consider the line $\\ell _1$ which is tangent to $.",
"Let $ P=1 and denote the generator of ${{\\mathcal {V}}^5_2}$ through $P$ by $\\ell _P$ .",
"A 5-space meeting $\\alpha $ in a tangent line contains 1 point of $, and so contains at most one generator of $ V52$.",
"So exactly one 5-space contains $ 4,1$ and a generator, namely the 5-space $ 4,1,P$.Consider the 5-space $ 3,1$, it contains $ P$ and a twisted cubic of $ V52$ which by Corollary~\\ref {N-r-on-V} is disjoint from $$, hence $ 3,1$ contains the generator $ P$.",
"That is,$ 3,1$ contains $$, $ 1$, $ P$ and so$ 3,1=4,1,P$.",
"That is, the intersection of $ 4,1,P$ with $ V52$ is a twisted cubic and one generator.Let $ 5$ be one of the remaining $ q-1$ 5-spaces (distinct from $$) that contains $ 4,1$, so $ 5$ contains 0 generators of $ V52$ and does not contain a conic or twisted cubic of $ V52$.",
"So by Theorem~\\ref {count-nrc}, $ 5$ meets $ V52$ in a 5-dim nrc,proving part 1 for $ i=1$.",
"For part 2, we consider $ 4,1=,1$.",
"By Corollary~\\ref {remark-BB}, as $ S$, no 5-space containing $ 4,1$ contains a twisted cubic of $ V52$.",
"The 5-space $ 4,1,P$ contains one generator of $ V52$, so by Theorem~\\ref {count-nrc}, it meets $ V52$ in exactly a 4-dim nrc and the generator $ P$.",
"Let $ 5$ be one of the remaining $ q-1$ 5-spaces containing $ 4,1$, then $ 5$ contains 0 generators of $ V52$.",
"So by Theorem~\\ref {count-nrc}, $ 5$ meets $ V52$ in a 5-dim nrc,proving part 2 for $ i=1$.$ Finally, consider the line $\\ell _2$ which is secant to $.",
"Let $ 2={P,Q}$ and let $ P,Q$ be the generators of $ V52$ through $ P,Q$ respectively.",
"The intersection of the 5-space $ 3,2$ and $ V52$ contains a twisted cubic, and $ P$ and $ Q$.",
"By Corollary~\\ref {N-r-on-V}, this twisted cubic is disjoint from $$, so $ 3,2$ contains the two generators $ P,Q$.",
"Thus$ 3,2=4,2,P=4,2,Q=4,2,P,Q$.The remaining $ q-1$ 5-spaces (distinct from $$) about $ 4,2$ contain 0 generators and two points of $ .",
"By Lemma REF they cannot contain a 4-dim nrc of ${{\\mathcal {V}}^5_2}$ .",
"So by Theorem REF , they meet ${{\\mathcal {V}}^5_2}$ in a 5-dim nrc, proving part 1 for $i=2$ .",
"For part 2, let $\\Pi _5\\ne \\Sigma _\\infty $ be a 5-space containing $\\Pi _{4,2}=\\langle \\gamma ,\\ell _2\\rangle $ .",
"By Corollary REF , $\\Pi _5$ does not contain a twisted cubic of ${{\\mathcal {V}}^5_2}$ , as $\\gamma \\notin \\mathbb {S}$ .",
"So by Theorem REF , $\\Pi _5$ contains at most one generator of ${{\\mathcal {V}}^5_2}$ .",
"Hence $\\langle \\Pi _{4,2},\\ell _P\\rangle $ , $\\langle \\Pi _{4,2},\\ell _Q\\rangle $ are distinct 5-spaces about $\\Pi _{4,2}$ , and by Theorem REF , they each meet ${{\\mathcal {V}}^5_2}$ in a 4-dim nrc and one generator.",
"Let $\\Sigma _5\\ne \\Sigma _\\infty $ be one of the remaining $q-2$ 5-spaces about $\\Pi _{4,2}$ .",
"Then $\\Sigma _5$ contains 0 generators of ${{\\mathcal {V}}^5_2}$ , and so by Theorem REF , meets ${{\\mathcal {V}}^5_2}$ in a 5-dim nrc, proving part 2 for $i=2$ .",
"$\\square $ Let $\\mathcal {S}$ be a regular 2-spread in a 5-space $\\Sigma _\\infty $ in ${\\rm PG}(6,q)$ , and position ${{\\mathcal {V}}^5_2}$ so that it corresponds to a tangent $\\mathbb {F}_{q}$ -subplane of ${\\rm PG}(2,q^3)$ .",
"So ${{\\mathcal {V}}^5_2}$ has splash $\\mathbb {S}\\subset \\mathcal {S}$ , the conic directrix $ lies in a plane $ S$, and each of the $ q2$ 3-spaces containing a twisted cubic directrix of $ V52$ meets $$ in a distinct plane of $ S$.In Corollary~\\ref {tc-splash-3}, we looked at how 3-spaces containing a plane of $ S$ meet $ V52$.",
"In Lemma~\\ref {part-of-next-lemma}, we looked at how 4-spaces containing a plane of $ S$ meet $ V52$.",
"Next we look at how5-spaces containing a plane of $ S$ meet $ V52$.",
"Note thatstraightforward counting shows that a 5-space distinct from $$ contains a unique plane $$ of $ S$, and meets every other plane of $ S$ in a line.",
"If $ =$, then Lemma~\\ref {5containsC} describes the possible intersections with $ V52$.",
"The next theorem describes the possible intersections with $ V52$ for the remaining cases $ S$ and $ SS$.$ Theorem 5.1 Position ${{\\mathcal {V}}^5_2}$ as in Corollary REF , so $\\mathcal {S}$ is a regular 2-spread in a hyperplane $\\Sigma _\\infty $ , the conic directrix $ lies in a plane $ S$ and $ V52$ has splash $ SS$.",
"Let $$ be a line of $$ with $ |=i$ and let $ S$, $$.",
"Then the $ q$ 5-spaces containing $ ,$ and distinct from $$ meet $ V52$ as follows.\\begin{enumerate}\\item If \\pi \\in \\mathbb {S}\\backslash \\alpha , then q-1 meet {{\\mathcal {V}}^5_2} in a 5-dim nrc, and 1 meets {{\\mathcal {V}}^5_2} in a twisted cubic and i generators.\\item If \\pi \\in \\mathcal {S}\\backslash \\mathbb {S}, then q-i meet {{\\mathcal {V}}^5_2} in a 5-dim nrc, and i meet {{\\mathcal {V}}^5_2} in a 4-dim nrc and 1 generator.\\end{enumerate}$ Proof By [2], the group of collineations of ${\\rm PG}(6,q)$ fixing $\\mathcal {S}$ and ${{\\mathcal {V}}^5_2}$ is transitive on the planes of $\\mathbb {S}\\backslash \\alpha $ and on the planes of $\\mathcal {S}\\backslash \\mathbb {S}$ .",
"As this group fixes the conic directrix $, it is transitive on the lines of $$ tangent to $ , the lines of $\\alpha $ secant to $ andthe lines of $$ exterior to $ .",
"So without loss of generality let $\\ell _0$ be a line of $\\alpha $ exterior to $, let $ 1$ be a line of $$ tangent to $ , let $\\ell _2$ be a line of $\\alpha $ secant to $, let $$ be a plane in $ S$, and let $$ be a plane of $ SS$.",
"For $ i=0,1,2$, label the 4-spaces$ 4,i=,i$ and $ 4,i=,i$.By Corollary~\\ref {tc-splash-3}, as $ S$, there is a unique twisted cubic of $ V52$ that lies in a 3-space about $$, denote this 3-space by $ 3$.",
"Hence for $ i=0,1,2$, there is a unique 5-space containing $ 4,i$ whose intersection with $ V52$ contains a twisted cubic, namely the 5-space $ 3,i$.$ First consider the line $\\ell _0$ which is exterior to $.A 5-space meeting $$ in $ 0$ contains 0 points of $ , and so contains 0 generators of ${{\\mathcal {V}}^5_2}$ .",
"The 4-space $\\Sigma _{4,0}=\\langle \\beta ,\\ell _0\\rangle $ lies in $q$ 5-spaces distinct from $\\Sigma _\\infty $ , each containing 0 generators of ${{\\mathcal {V}}^5_2}$ .",
"Exactly one of these 5-spaces, namely $\\langle \\Pi _3,\\ell _0\\rangle $ , contains a twisted cubic of ${{\\mathcal {V}}^5_2}$ .",
"The remaining $q-1$ 5-spaces about $\\Sigma _{4,0}$ contain 0 generators, and do not contain a conic or twisted cubic of ${{\\mathcal {V}}^5_2}$ , so by Theorem REF , they meet ${{\\mathcal {V}}^5_2}$ in a 5-dim nrc, proving part 1 for $i=0$ .",
"For part 2, let $\\Pi _5\\ne \\Sigma _\\infty $ be any 5-space containing $\\Pi _{4,0}=\\langle \\gamma ,\\ell _0\\rangle $ .",
"As $\\gamma \\notin \\mathbb {S}$ , by Corollary REF , $\\Pi _5$ cannot contain a twisted cubic of ${{\\mathcal {V}}^5_2}$ .",
"As $\\Pi _5$ contains 0 generator lines of ${{\\mathcal {V}}^5_2}$ and does not contain a conic or twisted cubic of ${{\\mathcal {V}}^5_2}$ , by Theorem REF , $\\Pi _5$ meets ${{\\mathcal {V}}^5_2}$ in a 5-dim nrc.",
"That is, the $q$ 5-spaces (distinct from $\\Sigma _\\infty $ ) containing $\\Pi _{4,0}$ meet ${{\\mathcal {V}}^5_2}$ in a 5-dim nrc, proving part 2 for $i=0$ .",
"Next consider the line $\\ell _1$ which is tangent to $.",
"Let $ P=1 and denote the generator of ${{\\mathcal {V}}^5_2}$ through $P$ by $\\ell _P$ .",
"A 5-space meeting $\\alpha $ in a tangent line contains 1 point of $, and so contains at most one generator of $ V52$.",
"So exactly one 5-space contains $ 4,1$ and a generator, namely the 5-space $ 4,1,P$.Consider the 5-space $ 3,1$, it contains $ P$ and a twisted cubic of $ V52$ which by Corollary~\\ref {N-r-on-V} is disjoint from $$, hence $ 3,1$ contains the generator $ P$.",
"That is,$ 3,1$ contains $$, $ 1$, $ P$ and so$ 3,1=4,1,P$.",
"That is, the intersection of $ 4,1,P$ with $ V52$ is a twisted cubic and one generator.Let $ 5$ be one of the remaining $ q-1$ 5-spaces (distinct from $$) that contains $ 4,1$, so $ 5$ contains 0 generators of $ V52$ and does not contain a conic or twisted cubic of $ V52$.",
"So by Theorem~\\ref {count-nrc}, $ 5$ meets $ V52$ in a 5-dim nrc,proving part 1 for $ i=1$.",
"For part 2, we consider $ 4,1=,1$.",
"By Corollary~\\ref {remark-BB}, as $ S$, no 5-space containing $ 4,1$ contains a twisted cubic of $ V52$.",
"The 5-space $ 4,1,P$ contains one generator of $ V52$, so by Theorem~\\ref {count-nrc}, it meets $ V52$ in exactly a 4-dim nrc and the generator $ P$.",
"Let $ 5$ be one of the remaining $ q-1$ 5-spaces containing $ 4,1$, then $ 5$ contains 0 generators of $ V52$.",
"So by Theorem~\\ref {count-nrc}, $ 5$ meets $ V52$ in a 5-dim nrc,proving part 2 for $ i=1$.$ Finally, consider the line $\\ell _2$ which is secant to $.",
"Let $ 2={P,Q}$ and let $ P,Q$ be the generators of $ V52$ through $ P,Q$ respectively.",
"The intersection of the 5-space $ 3,2$ and $ V52$ contains a twisted cubic, and $ P$ and $ Q$.",
"By Corollary~\\ref {N-r-on-V}, this twisted cubic is disjoint from $$, so $ 3,2$ contains the two generators $ P,Q$.",
"Thus$ 3,2=4,2,P=4,2,Q=4,2,P,Q$.The remaining $ q-1$ 5-spaces (distinct from $$) about $ 4,2$ contain 0 generators and two points of $ .",
"By Lemma REF they cannot contain a 4-dim nrc of ${{\\mathcal {V}}^5_2}$ .",
"So by Theorem REF , they meet ${{\\mathcal {V}}^5_2}$ in a 5-dim nrc, proving part 1 for $i=2$ .",
"For part 2, let $\\Pi _5\\ne \\Sigma _\\infty $ be a 5-space containing $\\Pi _{4,2}=\\langle \\gamma ,\\ell _2\\rangle $ .",
"By Corollary REF , $\\Pi _5$ does not contain a twisted cubic of ${{\\mathcal {V}}^5_2}$ , as $\\gamma \\notin \\mathbb {S}$ .",
"So by Theorem REF , $\\Pi _5$ contains at most one generator of ${{\\mathcal {V}}^5_2}$ .",
"Hence $\\langle \\Pi _{4,2},\\ell _P\\rangle $ , $\\langle \\Pi _{4,2},\\ell _Q\\rangle $ are distinct 5-spaces about $\\Pi _{4,2}$ , and by Theorem REF , they each meet ${{\\mathcal {V}}^5_2}$ in a 4-dim nrc and one generator.",
"Let $\\Sigma _5\\ne \\Sigma _\\infty $ be one of the remaining $q-2$ 5-spaces about $\\Pi _{4,2}$ .",
"Then $\\Sigma _5$ contains 0 generators of ${{\\mathcal {V}}^5_2}$ , and so by Theorem REF , meets ${{\\mathcal {V}}^5_2}$ in a 5-dim nrc, proving part 2 for $i=2$ .",
"$\\square $"
]
] | 1906.04319 |
[
[
"k-Means Aperture Optimization Applied to Kepler K2 Time Series\n Photometry of Titan"
],
[
"Abstract Motivated by the Kepler K2 time series of Titan, we present an aperture optimization technique for extracting photometry of saturated moving targets with high temporally- and spatially-varying backgrounds.",
"Our approach uses $k$-means clustering to identify interleaved families of images with similar Point-Spread Function and saturation properties, optimizes apertures for each family independently, then merges the time series through a normalization procedure.",
"By applying $k$-means aperture optimization to the K2 Titan data, we achieve $\\leq$0.33% photometric scatter in spite of background levels varying from 15% to 60% of the target's flux.",
"We find no compelling evidence for signals attributable to atmospheric variation on the timescales sampled by these observations.",
"We explore other potential applications of the $k$-means aperture optimization technique, including testing its performance on a saturated K2 eclipsing binary star.",
"We conclude with a discussion of the potential for future continuous high-precision photometry campaigns for revealing the dynamical properties of Titan's atmosphere."
],
[
"Introduction",
"Saturn's largest moon Titan has a cold, dense ($\\sim $ 1.5 bar, 94 K) N$_{2}$ ($\\sim $ 98%) and CH$_{4}$ ($\\sim $ 2%) atmosphere that is rendered nearly opaque by photochemically-generated organic aerosols composed of hydrocarbons and nitriles (e.g., West et al.",
"2014).",
"Titan's atmosphere drives the only known extant non-terrestrial hydrological cycle, including clouds, rain, fluvial channels, and lakes/seas (see e.g., Hörst 2017).",
"The precipitable column of methane in Titan's atmosphere is far larger than that of water in the Earth's atmosphere (Atreya et al.",
"2006).",
"Titan's atmospheric dynamics and chemistry are connected and complex: the atmosphere is highly stratified (Strobel et al.",
"2009), at altitude it rotates much faster than the moon's surface (e.g., Lebonnois et al.",
"2014), it supports both short- and long-lived storm systems (e.g., Griffith et al.",
"2014), and some of its molecular constituents result from exogenic sources (Hörst et al.",
"2008).",
"From a photochemical standpoint, Titan's atmosphere is a frozen analog to the early Earth's, and understanding the processes that govern its atmosphere are of great astrobiological interest.",
"For more information about Titan's atmosphere, see the recent review by Hörst (2017).",
"Earth-based monitoring programs have used variability of Titan's brightness in and out of a methane spectral absorption band to identify the onset of storms on Titan and trigger subsequent follow-up from large observing facilities (see e.g., Bouchez and Brown, 2005).",
"In late 2016, the Kepler Space Telescope conducted a similar monitoring campaign of Titan during its K2 mission.",
"This campaign lacked the spectral information of previous efforts, but had the potential to deliver substantially higher photometric precision and better sampling of dynamically-relevant timescales.",
"In 2001, transient clouds were witnessed daily over a 16-night Palomar observing program, and they varied on timescales as short as 3 hours (Bouchez and Brown, 2005), a timescale very well sampled by the continuous K2 Long Cadence (LC) 29.4 minute image cadence.",
"In the following sections, we describe the K2 observations of Titan and the challenges inherent in analyzing them, describe our methods for modeling scattered light from Saturn and for optimizing photometric apertures for extracting the Titan lightcurve, and present the extracted lightcurve.",
"We conclude with a brief discussion of these results and the potential applications of our techniques to existing and future datasets, including a demonstration application on K2 data of a saturated eclipsing binary star.",
"A repository containing a Python implementation of the $k$ -means aperture optimization algorithms as described in this paper is available onlinehttps://github.com/alex-parker/kmao."
],
[
"Image data from K2 are delivered as 1D pixel masks and collections of these masks must be assembled to reconstruct a 2D image.",
"We assemble the Titan K2 image dataset from the delivered pixel masks using the same approach as Ryan et al.",
"(2017).",
"An example image from the sequence is shown in Figure 1.",
"The Titan K2 dataset includes more than 400 Long Cadence (LC) images of the field that Titan and Saturn move through but which precede the arrival of Titan and Saturn into the field.",
"We leverage this sample of sky-only data to define stable World Coordinate System (WCS) parameters for the images of interest using the following procedure.",
"For each image in the sequence containing Titan, we find the image from the preceding sequence not containing Titan that minimizes the sum of the squares of the difference for pixels that (1) do not contain substantial Titan or Saturn flux, and (2) have been high-pass filtered to remove low-frequency sky variations.",
"We use a list of eight stars linked to the STScI GSC 2.3 (see Table 1) to determine WCS parameters for the matched comparison image, and apply these parameters to the Titan/Saturn image of interest.",
"Finally, we subtract the matched stars-only image from the Titan/Saturn image to remove starlight contamination.",
"All told, 182 images were identified which contained Titan sufficiently far from the chip or mask boundary and sufficiently far from the saturation effects of Saturn to be considered for further photometric analysis.",
"These images span the UTC range 2016-Dec-03 17:37:31.0 to 2016-Dec-07 10:52:45.3.",
"Table: NO_CAPTION"
],
[
"Saturn Scattered Light",
"The chief challenge of extracting accurate photometry of Titan is removing the contribution from Saturn's scattered light.",
"We tested several models of the scattered light, including constructing a PSF model and scaling it to fit to data in an annulus around Titan, building a 3D (Saturn-relative sky plane position and time) flux interpolant from which to predict flux at any point in a given Titan aperture at a given time, and others.",
"The selected model performed far better and more stably than these other methods without introducing substantial model complexity.",
"We first translate the image (x,y) pixel coordinates into a scaled polar coordinate system centered on Saturn's location in the focal plane.",
"The coordinates are defined as ${r_{s} = \\sqrt{(x-x_{s})^2 + (y-y_{s})^2}\\cr \\theta _{s} = s \\times r_{s} \\times \\arctan _{2}(y-y_{s}, x-x_{s}),}$ where $s$ is a scaling factor designed to prioritize either the radial behavior or azimuthal behavior in the scattered light profiles.",
"We determined by brute force that $s=2$ was optimal for the portion of the Saturn scattered light field that Titan passed through in these observations, weakly prioritizing radial behavior.",
"We then convert the raw Kepler flux into $log_{10}$ (flux), and build a thin-plate Radial Basis Function interpolant (using scipy.interpolate.Rbf with an $r^2 \\log (r)$ basis function; c.f.",
"Millman & Aivazis 2011) in the Saturn-radial coordinate system and interpolating $\\log _{10}$ (flux).",
"This interpolant is built on data drawn from a 2-pixel wide boundary around any proposed Titan aperture, and we then use it to predict $log_{10}$ (flux) across that aperture.",
"Of all tested methods, this approach produced the lowest scatter for Titan's flux extracted from any given static aperture, the lowest scatter after $k$ -means aperture optimization (see following section), and the lowest Spearman's rank correlation coefficient between the predicted sky values in the aperture and the extracted Titan lightcurve.",
"Finally, visual inspection of the predicted scattered light field over the photometric aperture showed smooth boundaries and qualitatively accurate reproduction of discrete features (e.g., diffraction spikes) of the Saturn PSF as they swept across the aperture.",
"Since this approach predicts the flux inside an aperture based on an extrapolation of the flux observed outside the aperture, it grows less accurate the larger the aperture is.",
"This drives us to a preference for as small an aperture as possible, and the selection of this aperture is described in the following section."
],
[
"$K$ -Means Aperture Optimization",
"Defining the optimum photometry aperture for a Kepler time series is critical for achieving high precision measurements, and substantial effort has gone into determining these for stationary targets (e.g., Smith et al.",
"2016).",
"For a moving, saturated target, however, a single aperture must be made quite large to encompass a large fraction of the PSF and the saturation charge bleed spikes.",
"Depending on the sub-pixel position and integrated motion of the target over a 30 minute LC exposure, the shape and extent of the charge bleed spikes can vary substantially.",
"Pixel-Level Decorrelation (PLD; Luger et al.",
"2016, Luger et al.",
"2018) can deliver high precision on saturated, relatively stationary targets by incorporating the PLD vectors from neighboring stars to recover information lost due to saturation, but this technique leverages the relative stability of a network of stationary sources has not been generalized to moving sources.",
"Sky plane motion decorrelates pixel values from one frame to the next independently of the spacecraft pointing.",
"To circumvent this, other K2 studies of bright solar system targets have adopted extremely large moving apertures to capture as much of the PSF as possible (e.g., Neptune by Simon et al.",
"2016).",
"For Titan, adopting a single large aperture would result in substantial contamination from Saturn's time-varying PSF.",
"Instead, we would like to construct an aperture that is as small as possible on each image to limit the contribution of Saturn scattered light.",
"However, without ultra-accurate PSF models, a frame-by-frame varying aperture would be prohibitively difficult to calibrate.",
"The methods presented here are a hybrid of these two end-member approaches that results in a workable compromise between aperture losses and background contamination.",
"We optimize a small set of $k$ apertures, each applied to a unique subset of images with similar properties, which enable the use of compact apertures by enabling internal calibration of the aperture correction for $k-1$ of the apertures with respect to a single reference aperture.",
"This allows different apertures to be applied to images where, for example, the target falls near the center of a pixel versus images where it falls near a pixel boundary, enabling separate optimization of the apertures applied to the differing resultant charge bleed spikes, PSF sampling, and so forth.",
"These independent apertures also naturally detrend pointing-based systematics, as pointing-related processes express themselves in detectable ways on the image plane during a single exposure.",
"We define these apertures by applying $k$ -means clustering on a feature set derived from the target image at each epoch.",
"$K$ -means clustering assigns membership of a set of $n$ samples of into $k$ groups of equal variance in an $N$ -dimensional feature space, where each sample is assigned membership to the cluster with the nearest mean in that feature space (see, e.g., MacQueen 1967).",
"In our case, we extract an ordered 1D list of pixel values from the image of a target drawn from a constant large aperture on the focal plane, normalize this list to its mean, and consider this normalized list to be our feature set for performing $k$ -means clustering.",
"This process creates $k$ clusters of images with similar properties – effectively, similar motion blur, saturation, and sub-pixel target positioning, recovered directly from the images rather than from a prediction.",
"Because the assignment of an image to a cluster is determined only by its normalized pixel values, clusters are populated by images that are relatively uniformly sampled from throughout the time series.",
"This property of interleaved sampling enables the use of a finite-difference based scheme to robustly calibrate the aperture corrections for $k-1$ of the apertures with respect to a reference aperture.",
"The full process for defining these apertures follows: Determine accurate WCS parameters for each image and subtract an image of the background starfield.",
"Using the updated star-based WCS parameters and a prediction of the target's equatorial position as seen from K2 in each frame, co-register the target images to the nearest integer pixel.",
"Determine and subtract a preliminary scattered light model (e.g., as described in Section 3) from each image.",
"For each of the $n$ images, flatten the array of $N$ pixel values from a large aperture around the target into a 1D list of $N$ features.",
"Stack these feature arrays into a 2D ($n \\times N$ ) array of features extracted from the image set.",
"Perform $k$ -means clustering on this 2D feature array to identify $k$ sub-groups of images with similar properties.",
"Stack each cluster of similar target images to generate a cluster average image.",
"Trim pixels from an initial large aperture by iteratively removing the pixel with the lowest estimated signal-to-noise ratio until the aperture contains 99% of the total target flux estimated from the original large aperture.",
"Dilate this trimmed aperture by one pixel.",
"Extract target photometry using the dilated, trimmed apertures applied to the clusters of target images from which each was derived.",
"This results in $k$ independent lightcurves sampled at epochs that are interleaved in time.",
"To merge them into a single record, we identify the cluster with the largest number of members and define it as our reference lightcurve.",
"We then determine and apply multiplicative aperture corrections for each other cluster in order to bring the full set into agreement with itself by minimizing the point-to-point scatter $\\sigma _p$ , defined as: ${\\sigma _{p} = 0.8166 \\times \\sqrt{\\frac{\\sum _{i=2}^{n-1}(d - \\bar{d})^2}{n-1}}\\mbox{ , where:}\\cr d = (y_{i} - \\frac{1}{2}(y_{i-1}+y_{i+1})}$ This is a scaled L2 norm on the second-order finite difference.",
"Intuitively, this represents a high-frequency estimate of the standard deviation of measurements around an unknown underlying signal.",
"This formulation assumes a continuous, uniformly-sampled sequence that is sorted in the independent variable (time, in our case).",
"The leading factor 0.8166 scales $\\sigma _{p}$ to match the standard deviation of a sequence if the underlying signal is constant in time and the data are normally distributed; this scaling factor is included merely for convenience of comparing to other estimates of intrinsic scatter.",
"We adopt $\\sigma _{p}$ as our objective function to for minimization to determine aperture corrections, similar to the use of Total Variation (TV, the L1 norm on the first-order finite difference) by White et al.",
"(2017) in halophot.",
"However, while halophot uses the TV objective function to optimize a set of weights on a fixed set of pixels in an aperture (spatial weighting), we are optimizing a set of aperture corrections that vary in time.",
"Objective functions based on first-order finite differences can introduce complex dependencies on the behavior of the underlying signal; for example, for a linear trend with fixed-amplitude Gaussian noise, an L1 or L2 norm on first-order difference increases with decreasing slope, while $\\sigma _{p}$ remains constant because of the second-order finite difference used to define it.",
"Under the assumption that within a time series there is a fixed, uncorrelated noise spectrum on top of a well-sampled underlying signal, we adopt $\\sigma _{p}$ as our objective function because it is relatively insensitive to the properties of the underlying signal, instead responding to the amplitude of the high-frequency scatter around the underlying signal.",
"For $k$ apertures, there are $k-1$ aperture corrections to determine and apply.",
"We determine the set of $k-1$ aperture corrections that minimizes $\\sigma _p$ using simplex optimization.",
"The relatively simple step (v) in the aperture definition process could be replaced by more complex single-aperture optimization approaches, such as those presented in Smith et al.",
"(2016).",
"We investigated watershed segmentation (e.g., Lund et al.",
"2015) but found that it was not suitable given the rapidly changing, complex topology of the sky flux.",
"An aperture-by-aperture optimization of a set of non-binary aperture weights á la halophot may provide an avenue for future improvement.",
"Regardless of the aperture optimization procedure, the rest of the $k$ -means aperture optimization process would remain identical."
],
[
"Application to the Titan Lightcurve",
"For the Titan ($K_p\\sim 8.8$ magnitude) dataset of 182 images, we found that $k=5$ apertures was ideal for $k$ -means aperture optimization.",
"Using fewer than five apertures resulted in poor photometric performance, while using more apertures resulted in very small incremental improvements that did not merit the additional free parameters introduced.",
"All attempts to optimize a single aperture resulted in poorer performance, with final lightcurve scatter at least two times worse than the nominal 5-aperture solution, and typically showing strong correlation with the background sky flux.",
"The five optimized apertures, the cluster-averaged Titan images for each, and the number of unique images ascribed to each cluster are shown in Figure 1.",
"Note that with $k=5$ clusters, the smallest number of images in this dataset assigned to any cluster is 16.",
"When determining an ideal value for $k$ , it is important to consider the the smallest cluster size, as a smaller set of images assigned to any given cluster may introduce a risk of overfitting.",
"The extracted and merged lightcurves are illustrated in Figures 2 & 3.",
"The sky lightcurves clearly illustrate one of the chief challenges of this dataset — the rapidly-varying sky that contributes up to 60% of the total flux measured in a given aperture.",
"The final overall scatter in the lightcurve is $\\sigma \\sim 0.33\\%$ , while the point-to-point scatter is $\\sigma _p \\sim 0.19\\%$ .",
"There is a weak secular trend in the Titan lightcurve.",
"A least-squares line fit to the lightcurve drops by $\\sim 0.5\\%$ over the 3.7-day course of the observations.",
"A Pearson Rank Correlation Coefficient indicates that this trend is significant at $p\\le 0.03$ .",
"This secular trend could be due all or in part to either (a) a systematic error in, for example, the estimation of Saturn's scattered light in the Titan aperture, or (b) the 15.945-day rotation of Titan's surface contributing a low-amplitude rotational variation visible through narrow windows of atmospheric transparency in the long-wavelength tail of Kepler's bandpass.",
"Otherwise, the largest apparent feature in the lightcurve occurs at about the mid-time, where a small ($\\sim 0.5\\%$ ) apparent brightening occurs over $\\sim 8$ hours, then fades on the same timescale.",
"The timescale of this brightening is similar to the timescale between K2 roll-position correction thruster firing events, which could indicate a subtle pointing-related systematic still present in our data.",
"However, a Lomb-Scargle periodogram (Lomb 1976, Scargle 1982) of Titan's lightcurve does not reveal substantial power near the 4 cycle/day peak of thruster firing events (Figure 4), suggesting that the relatively short-period variability in Titan's lightcurve is not in large part attributable to pointing-related systematics.",
"Removing the weak linear correlation between the pointing offsets and Titan's flux (Figure 4) results in negligible change in the final Titan lightcurve."
],
[
"Comparison to Cassini Observations",
"During the period of the K2 observations (December 3rd and December 7th 2016), there were no observations of Titan from the Cassini Spacecraft.",
"However, clouds were observed prior to (November 14th and November 29th 2016) and after (December 18th and December 30th 2016) the K2 observations in Cassini Imaging Science Subsystem (ISS) and Visible and Infrared Mapping Spectrometer (VIMS) measurements, indicating that this was a period of relatively frequent cloud activity (Turtle et al.",
"2018).",
"In the ISS observations these clouds were reported as “streaks” and were not the large (presumably convective) CH4 cloud systems seen in previous Earth-based observations (e.g., Bouchez & Brown 2005).",
"These cloud features are low contrast in the broad visible Kepler bandpass.",
"An example from March 21, 2017 is given in Figure 5.",
"The broad-band visible image shows extremely low contrast, while the image targeting a narrow NIR band of atmospheric transparency shows a high-contrast cloud streak.",
"The average contrast of the cloud streak with its surroundings is $\\sim 2\\%$ in the broadband image and $\\sim 10\\%$ in the narrowband NIR image.",
"If we adopt $2\\%$ as a cloud streak contrast in the Kepler bandpass, a streak would need to subtend $25\\%$ of the area of Titan's disk before its contribution to the broadband reflected flux would be comparable to the $\\sim 0.5\\%$ amplitude features in the K2 lightcurve of Titan.",
"For comparison, the cloud streak seen in Figure 5 subtends approximately 8% of Titan's illuminated disk.",
"Higher contrast broadband atmospheric features would be detectable with commensurately smaller areal coverage, but it is unlikely that cloud streaks similar to those seen by Cassini before and after the K2 observations would have been detectable in the lightcurve."
],
[
"Discussion and Future Applications",
"The $k$ -means aperture optimization approach may hold further utility in other situations where a moving source is subject to significant background contamination, particularly if the source is saturated.",
"Observations of the planetary satellites (including Titan), planets, and sun-grazing comets by solar monitoring instruments may be one such example.",
"Extracting a lightcurve from the K2 dataset of Triton or other planetary satellites may also benefit from similar photometric techniques.",
"With further effort, $k$ -means aperture optimization may provide a useful complement to existing methods for K2 stellar photometry, particularly in the bright star regime.",
"The pointing drift, periodic thruster firings, and reaction wheel desaturation events that distinguish the two-wheel K2 mission from the Kepler prime mission introduce time variable point-spread functions and substantial focal plane motion even for targets stationary in the sky plane.",
"These introduce substantial systematics in the photometric time series of stationary targets if not carefully accounted for.",
"This challenge has led to the development of a suite of photometry pipelines that mitigate these effects through a variety of means, including decorrelation against spacecraft pointing with piecewise-linear models (e.g., Vanderburg & Johnson 2014) or via Gaussian process models (Aigrain et al.",
"2015, 2016), or by PLD (Luger et al.",
"2018).",
"$K$ -means aperture optimization represents a novel means of mitigating image-plane effects like those introduced in K2, so we performed a test to compare the performance of the basic algorithm on an $K_p=9.2$ magnitude eclipsing binary star in K2 Campaign 0 data (EPIC 202063160); using the same prescriptions as with Titan, but with $k=30$ apertures over 1,641 images rather than $k=5$ over 182 images.",
"We illustrate the results in Figure 6. k-Means aperture optimization delivers a lightcurve with approximately twice the scatter of EVEREST 2.0 (Luger et al.",
"2018).",
"With further refinements made to the aperture optimization step, it is possible that $k$ -means aperture optimization will be competitive with the state of the art for saturated stars in $K2$ , providing an alternative pathway to precision photometry under conditions where other methods are prohibited.",
"Long-term monitoring of Titan's variability from a stable space-based platform holds great promise for building a comprehensive picture of the power spectrum of variability in its complex atmosphere.",
"While the K2 lightcurve of Titan shows no compelling evidence for variability that is concretely attributable to the atmosphere, observations of similar precision in more appropriate spectral bandpasses would be able to track far smaller atmospheric changes than those that could have been revealed by the current K2 lightcurve."
],
[
"Acknowledgements",
"The analysis in this work was supported by the NASA grant 15-K2-GO4_2-100.",
"This research has made use of NASA's Astrophysics Data System Bibliographic Services.",
"This research made use of numerous community-developed software packages, including Astropy (Robitaille et al.",
"2013, Price-Whelan et al.",
"2018), Numpy (Oliphant 2006), Matplotlib (Hunter 2007), SciPy (Jones et al.",
"2001), scikit-learn (Pedregosa et al.",
"2011), and scikit-image (van der Walt et al.",
"2014).",
"We wish to thank the referee for a very thorough and constructive review."
]
] | 1906.04220 |
[
[
"Synthesis of Computable Regular Functions of Infinite Words"
],
[
"Abstract Regular functions from infinite words to infinite words can be equivalently specified by MSO-transducers, streaming $\\omega$-string transducers as well as deterministic two-way transducers with look-ahead.",
"In their one-way restriction, the latter transducers define the class of rational functions.",
"Even though regular functions are robustly characterised by several finite-state devices, even the subclass of rational functions may contain functions which are not computable (by a Turing machine with infinite input).",
"This paper proposes a decision procedure for the following synthesis problem: given a regular function $f$ (equivalently specified by one of the aforementioned transducer model), is $f$ computable and if it is, synthesize a Turing machine computing it.",
"For regular functions, we show that computability is equivalent to continuity, and therefore the problem boils down to deciding continuity.",
"We establish a generic characterisation of continuity for functions preserving regular languages under inverse image (such as regular functions).",
"We exploit this characterisation to show the decidability of continuity (and hence computability) of rational and regular functions.",
"For rational functions, we show that this can be done in $\\mathsf{NLogSpace}$ (it was already known to be in $\\mathsf{PTime}$ by Prieur).",
"In a similar fashion, we also effectively characterise uniform continuity of regular functions, and relate it to the notion of uniform computability, which offers stronger efficiency guarantees."
],
[
"Introduction",
"The notions of computability and continuity have been central in computability theory, as well as in real and functional analysis.",
"Computability for discrete sets like natural numbers, finite words, finite graphs and so on have been extensively studied over the last seven to eight decades, through several models of computation including Turing machines working on finite words.",
"Computability notions have been extended to infinite objects, like infinite sequences of natural numbers, motivated by real analysis, or computation of functions of real numbers.",
"An infinite word $\\alpha $ over a finite alphabet $\\Sigma $ is a function $\\alpha : \\mathbb {N} \\rightarrow \\Sigma $ and is written as $\\alpha =\\alpha (0)\\alpha (1) \\dots $ The set of infinite words over $\\Sigma $ is denoted by $\\Sigma ^\\omega $ .",
"Computability of functions over infinite words In this paper, we are interested in functions from infinite words to infinite words.",
"The model of computation we consider for infinite words is a deterministic multitape machine with 3 tapes : a read-only one-way tape holding the input, a two-way working tape with no restrictions and a write-only one-way output tape.",
"All three tapes hold infinite words.",
"A function $f$ is computable if there exists such a machine $M$ such that, if its input tape is fed with an infinite word $u$ in the domain of $f$ , then $M$ outputs longer and longer prefixes of $f(x)$ when reading longer and longer prefixes of $x$ .",
"This machine model has been defined for instance in [17].",
"Not all functions are computable.",
"For instance, assuming an effective enumeration $M_1, M_2, \\dots $ of Turing machines (on finite word inputs), the function $f_H$ defined as $f_H(a^{\\omega })=b_1b_2b_3 \\dots $ where $b_i \\in \\lbrace 0,1\\rbrace $ is such that $b_i=1$ iff $M_i$ halts on input $\\epsilon $ , is not computable, otherwise the halting problem would be decidable.",
"This raises a natural decision problem: given a (finite) specification of a function $f$ from infinite words to infinite words, is $f$ computable ?",
"In the domain of program synthesis, this could be rephrased as “Is $f$ implementable ?”, since the notion of computability gives a natural answer to the question of what it means to be an implementation for such a function.",
"Even though the notion of computability for functions of infinite words require “physically unrealisable” infinite input, its makes sense, for instance, in a streaming scenario where the input is received as a non-terminating stream of symbols.",
"In this context, it is important to have the guarantee that infinitely often, one can output some finite part of the output.",
"Computability and continuity It turns out that there is some connection between computability and continuity.",
"It is known that computable functions are continuous for the Cantor topology, that is when words are close to each other if they share a long common prefix.",
"We give an intuition for this here.",
"A function $f: \\Sigma ^{\\omega } \\rightarrow \\Gamma ^{\\omega }$ is continuous if whenever a sequence $x_1, x_2, \\dots $ of infinite words in the domain of $f$ converges to an infinite word $x$ in the domain of $f$ , the sequence $f(x_1), \\dots , f(x_n), \\dots $ converges.",
"The notion of convergence is quantified using a metric between two infinite words.",
"For $p, q \\in \\Sigma ^{\\omega }$ , the distance between $p, q$ is defined as 0 if $p=q$ , or as $2^{-n}$ , where $n$ is the length of the longest common prefix of $p, q$ .",
"Assume $x_1, x_2, \\dots $ in the domain of $f$ converges to $x$ , then, up to taking a subsequence, we can assume that the $x_i$ share longer and longer common prefixes with $x$ .",
"As $f$ is computable by a deterministic machine, it behaves the same on these prefixes, and therefore it outputs words on $x_i$ which gets closer as $i$ tends to infinity, hence they converge as well.",
"This shows that computable functions are continuous.",
"However, in the reverse direction, not all continuous functions are computable, a counter-example being the function $f_H$ defined before, which is continuous since it is defined on a single input.",
"Regular functions In this paper, we study regular functions, which form a very well-behaved class.",
"Regular functions on infinite words are captured by streaming $\\omega $ -string transducers (SST), deterministic two-way Muller transducers with look around (2DMTla), and also by MSO-transducers à la Courcelle [1].",
"We propose the model of deterministic two-way transducers with a prophetic Büchi look-ahead (2DFTpla) and show that they are equivalent to 2DMTla.",
"This kind of transducers is defined by a deterministic two-way automaton without accepting states, extended with output words on the transitions, and which can consult another automaton, called the look-ahead automaton, to check whether an infinite suffix satisfies some regular property.",
"We assume this automaton to be a prophetic Büchi automaton [4], because this class has interesting properties while capturing all regular languages of infinite words.",
"The look-ahead is necessary to capture regular functions : the regular function $j(wab^{\\omega })=wwb^{\\omega }$ , and $j(b^{\\omega })=b^{\\omega }$ Figure: NO_CAPTIONcannot be captured by a two-way deterministic transducer without look-around, not even a non-deterministic transducer.",
"The reason is that one needs a non-deterministic choice to identify the last occurrence of $a$ and after this choice, check that there are only $b$ symbols on the input.",
"However, two passes over $w$ are necessary, and so those non-deterministic choices must be done at the same input position, which is impossible to ensure with finite memory.",
"Consider the function defined for all $w\\in \\lbrace a,b\\rbrace ^\\omega $ by $f(w) = a^\\omega $ if there are infinitely many $a$ s in $w$ , by $f(w) =b^\\omega $ otherwise.",
"This function can be realised by a non-deterministic two-way (even one-way) transducer without look-ahead which guesses whether there are infinitely many $a$ or not.",
"The restriction of $f$ to the regular language of words which contain infinitely many $a$ 's is however definable by a deterministic one-way transducer.",
"Another, less trivial example, is given by the function $g(u_1 \\# u_2 \\dots u_n \\# \\dots )=u_1u_1 u_2u_2 \\dots $ defined for all inputs which contain infinitely many $\\#$ , and realisable by a deterministic two-way transducer without look-ahead, which performs two passes for each factor $u_i$ in between two $\\#$ symbols.",
"Contributions We first show that for regular functions, computability and continuity coincide.",
"To the best of our knowledge, this connection was not made before.",
"We then prove that continuity (and hence computability) is decidable for regular functions given by deterministic Büchi two-way transducers with look-ahead.",
"Using our techniques, we also get almost for free the decidability of uniform continuity, although we do not connect it to any notion of computability.",
"For rational functions (functions defined by non-deterministic one-way Büchi transducers), we also get that continuity and uniform continuity are decidable in NLogSpace.",
"Deciding continuity and uniform continuity for rational functions was already known to be decidable in PTime, from Prieur [12].",
"Our proof technique relies on a characterisation of non-continuity by the existence of pairs of sequences of words which have a nice regular structure.",
"This characterisation works for any function $f$ which preserves regular languages of infinite words under inverse image.",
"In particular, it applies to regular functions, which are known to have this property.",
"Using this characterisation, we derive a decidability test for continuity of rational functions based on a transducer structural pattern which, using the pattern logic of [7], is decidable in NLogSpace.",
"For regular functions, the decidability test is based on a thorough study of the form of output words produced by idempotent loops in two-way transducers, which was done in [2] and used in another context.",
"We leave as open the question of which transducer model captures continuous regular functions and conjecture it is the model of deterministic two-way transducers without look-ahead.",
"Related work To the best of our knowledge, our results are new and the notion of continuity has not been extensively studied in the transducers literature.",
"We have mentioned the work by Prieur [12] which is the closest to ours.",
"Another related result in the context of reactive synthesis has been obtained in [8]: they show that for any binary relation $R$ of infinite words defined by a one-way letter-to-letter Büchi transducer, it is decidable whether there exists a total continuous function $f$ such that for all input $x\\in \\Sigma ^\\omega $ , $(x,f(x))\\in R$ .",
"Notions of continuity have been defined for rational functions in [3] but they are different from the classical notion we take in this paper and have been defined for finite words.",
"Structure of the Paper Section introduces the model of transducers used in the paper.",
"Section establishes a connection between computability and continuity for regular functions.",
"Section proves a characterisation of continuity of functions preserving regular languages under inverse image.",
"Finally, Section studies the decidability of continuity.",
"Most of the proofs have been omitted but can be found in Appendix."
],
[
"Preliminaries",
"Given a finite set $\\Sigma $ , we denote by $\\Sigma ^*$ (resp.",
"$\\Sigma ^\\omega $ ) the set of finite (resp.",
"infinite) words over $\\Sigma $ , and by $\\Sigma ^\\infty $ the set of finite and infinite words.",
"We denote by $|u|\\in \\mathbb {N}\\cup \\lbrace \\infty \\rbrace $ the length of $u\\in \\Sigma ^\\infty $ (in particular $|u|=\\infty $ if $u\\in \\Sigma ^\\omega $ ).",
"For a word $w=a_1a_2a_3\\dots $ , $w[{:}j]$ denotes the prefix $a_1a_2 \\dots a_j$ of $w$ .",
"$w[j]$ denotes $a_j$ , the $j^{th}$ symbol of $w$ .",
"$w[j{:}]$ denotes the suffix $a_{j+1} a_{j+2} \\dots $ of $w$ .",
"For a word $w$ and $i\\le j$ , $w[i:j]$ denotes the factor of $w$ with positions from $i$ to $j$ , both included.",
"For two words $u,v\\in \\Sigma ^\\infty $ , $u \\preceq v$ (resp.",
"$u\\prec v$ ) denotes that $u$ is a prefix (resp.",
"strict prefix) of $v$ (in particular if $u,v\\in \\Sigma ^\\omega $ , $u\\preceq v$ iff $u=v$ ).",
"For $u \\in \\Sigma ^*$ , let ${\\uparrow }u$ denote the set of words $w \\in \\Sigma ^\\infty $ having $u$ as prefix i.e.",
"$u \\preceq w$ .",
"Given two words $u,v\\in \\Sigma ^\\infty $ , we say that there exists a mismatch, denoted $\\textsf {mismatch}(u,v)$ , between $u$ and $v$ , if there exists a position $i\\le |u|,|v|$ such that $u[i]\\ne v[i]$ .",
"A Büchi automaton is a tuple $B=(Q, \\Sigma ,\\delta , Q_0, F)$ consisting of a finite set of states $Q$ , a finite alphabet $\\Sigma $ , a set $Q_0 \\subseteq Q$ of initial states, a set $F \\subseteq Q$ of accepting states, and a transition function $\\delta : Q \\times \\Sigma \\rightarrow 2^Q$ .",
"A run $\\rho $ on a word $w=a_1a_2 \\dots \\in \\Sigma ^{\\omega }$ starting in a state $q_1$ in $B$ is an infinite sequence $q_1 \\stackrel{a_1}{\\rightarrow } q_2 \\stackrel{a_2}{\\rightarrow } \\dots $ such that $q_{i+1} \\in \\delta (q_i, a_i)$ .",
"Let $\\mathsf {Inf}(\\rho )$ denote the set of states visited infinitely often along $\\rho $ .",
"The run $\\rho $ is a final run iff $\\mathsf {Inf}(\\rho ) \\cap F \\ne \\emptyset $ .",
"A run is accepting if it is final and starts from an initial state.",
"A word $w\\in \\Sigma ^\\omega $ is accepted ($w \\in L(B)$ ) iff it has an accepting run.",
"A language of $\\omega $ -words $L$ is called regular if $L = L(B)$ for some Büchi automaton $B$ .",
"An automaton is co-deterministic if any two final runs on any word $w$ are the same [4].",
"Likewise, an automaton is co-complete if every word has at least one final run.",
"A prophetic automaton $P=(Q_P, \\Sigma , \\delta _P,Q_0, F_P)$ is a Büchi automaton which is co-deterministic and co-complete.",
"Equivalently, a Büchi automaton is prophetic iff each word admits a unique final run.",
"The states of the prophetic automaton partition $\\Sigma ^{\\omega }$ : each state $q$ defines a set of words $w$ such that $w$ has a final run starting from $q$ .",
"For any state $q$ , let $L(P,q)$ be the set of words having a final run starting at $q$ .",
"Then $\\Sigma ^{\\omega }=\\uplus _{q \\in Q_P} L(P,q)$ .",
"It is known [4] that prophetic automata capture $\\omega $ -regular languages."
],
[
"Transducers",
"We recall the definitions of one-way and two-way transducers over infinite words.",
"A one-way transducer $\\mathcal {A}$ is a tuple $(Q, \\Sigma , \\Gamma , \\delta , Q_0, F)$ where $Q$ is a finite set of states, $Q_0, F$ respectively are sets of initial and accepting states; $\\Sigma , \\Gamma $ respectively are the input and output alphabets; $\\delta \\subseteq (Q \\times \\Sigma \\times Q \\times \\Gamma ^*)$ is the transition relation.",
"$\\mathcal {A}$ has the Büchi acceptance condition.",
"A transition in $\\delta $ of the form $(q, a, q^{\\prime }, \\gamma )$ represents that from state $q$ , on reading a symbol $a$ , the transducer moves to state $q^{\\prime }$ , producing the output $\\gamma $ .",
"Runs, final runs and accepting runs are defined exactly as in Büchi automata, with the addition that each transition produces some output $\\in \\Gamma ^*$ .",
"The output produced by an accepting run $\\rho $ , denoted $\\mathsf {out}(\\rho )$ , is obtained by concatenating the outputs generated by transitions along $\\rho $ .",
"Let $\\mathsf {dom}(\\mathcal {A})$ represent the language accepted by the underlying automaton of $\\mathcal {A}$ , ignoring the outputs.",
"The relation computed by $\\mathcal {A}$ is defined as $[\\!",
"[ \\mathcal {A} ]\\!]",
"= \\lbrace (u, v) \\in \\Sigma ^\\omega \\times \\Gamma ^{\\omega } | u \\in \\mathsf {dom}(\\mathcal {A}),\\rho ~\\text{is an accepting run of}~u,\\mathsf {out}(\\rho ) = v \\rbrace $We assume that final runs always produce infinite words, which can be enforced by a Büchi condition.. We say that $\\mathcal {A}$ is functional if $[\\!",
"[ \\mathcal {A} ]\\!",
"]$ is a function.",
"A relation (function) is rational iff it is recognized by a one-way (functional) transducer.",
"Two-way transducers extend one-way transducers and two-way finite state automata.",
"A two-way transducer is a two-way automaton with outputs.",
"Let $\\Sigma _{\\vdash }=\\Sigma \\uplus \\lbrace \\vdash \\rbrace $ .",
"A deterministic Büchi two-way transducer (2DBT) is given as $\\mathcal {B}= (Q, \\Sigma , \\Gamma , \\delta , q_0, F)$ where $Q$ is a finite set of states, $q_0$ is the unique initial state, and $F \\subseteq Q$ is a set of accepting states, $\\Sigma $ and $\\Gamma $ are finite input and output alphabets respectively, and the transition function has type $\\delta : Q \\times \\Sigma \\rightarrow Q \\times \\Gamma ^* \\times \\lbrace 1, -1\\rbrace $ .",
"A two-way transducer stores its input $\\vdash a_1 a_2\\dots $ on a two-way tape, and each index of the input can be read multiple times.",
"A configuration of a two-way transducer is a tuple $(q, i) \\in Q \\times \\mathbb {N}$ where $q \\in Q$ is a state and $i \\in \\mathbb {N}$ is the current position on the input tape.",
"The position is an integer representing the gap between consecutive symbols.",
"Thus, at $\\vdash $ , the position is 0, between $\\vdash $ and $a_1$ , the position is 1, between $a_i$ and $a_{i+1}$ , the position is $i+1$ and so on.",
"Given $w=a_1a_2 \\dots $ , from a configuration $(q,i)$ , on a transition $\\delta (q,a_i)=(q^{\\prime },\\gamma ,d)$ , $d \\in \\lbrace 1,-1\\rbrace $ , we obtain the configuration $(q^{\\prime }, i+d)$ and the output $\\gamma $ is appended to the output produced so far.",
"This transition is denoted as $(q,i) \\stackrel{a_i/\\gamma }{\\longrightarrow } (q^{\\prime }, i+d)$ .",
"A run $\\rho $ of a 2DBT is a sequence of transitions $(q_0,i_0=0) \\stackrel{a_{i_0}/\\gamma _1}{\\longrightarrow } (q_1, i_1)\\stackrel{a_{i_1}/\\gamma _2}{\\longrightarrow } \\dots $ .",
"The output of $\\rho $ , denoted $\\mathsf {out}(\\rho )$ is then $\\gamma _1 \\gamma _2 \\dots $ .",
"The run $\\rho $ reads the whole word $w$ if $\\mathsf {sup}\\lbrace i_n \\mid 0\\le n <|\\rho |\\rbrace = \\infty $ .",
"The output $[\\!",
"[ \\mathcal {B} ]\\!",
"](w)$ of a word $w$ on run $\\rho $ is defined only when $\\mathsf {sup}\\lbrace i_n \\mid 0\\le n <|\\rho |\\rbrace = \\infty $ , $\\inf (\\rho ) \\cap F \\ne \\emptyset $ , and equals $\\mathsf {out}(\\rho )$ .",
"In [1], regular functions are shown to be those definable by a two-way deterministic transducer with Muller acceptance condition, along with a regular look-around (2DMTla).",
"Formally, a 2DMTla is a tuple $(\\mathcal {T}, A, B)$ where $\\mathcal {T}$ is a deterministic two-way automaton with outputs, equipped with Muller acceptance condition, $A$ is a look-ahead automaton and $B$ is a look-behind automaton.",
"The look-ahead is a one-way automaton with the Muller acceptance condition and the look-behind is a DFA.",
"In this paper, we propose an alternative machine model for regular functions, namely, 2DFTpla.",
"A 2DFTpla is a deterministic two-way automaton with outputs, along with a look-ahead given by a prophetic automaton [4].",
"Formally, a 2DFTpla is a pair $(\\mathcal {T}, A)$ where $A=(Q_A, \\Sigma , \\delta _A, S_A, F_A)$ is a prophetic look-ahead Büchi automaton and $\\mathcal {T}= (Q, \\Sigma , \\Gamma , \\delta ,$ $ q_0)$ is a two-way transducer s.t.",
"$\\Sigma $ and $\\Gamma $ are finite input and output alphabets, $Q$ is a finite set of states, $q_0 \\in Q$ is a unique initial state, $\\delta : Q \\times \\Sigma \\times Q_A \\rightarrow Q \\times \\Gamma ^* \\times \\lbrace -1,+1\\rbrace $ is a partial transition function.",
"$\\mathcal {T}$ has no acceptance condition : every infinite run in $\\mathcal {T}$ is a final run.",
"The 2DFTpla is deterministic in the sense that for every word $w=a_1a_2a_3 \\dots \\in \\Sigma ^{\\omega }$ , every input position $i \\in \\mathbb {N}$ , and state $q \\in Q$ , there is a unique state $p \\in Q_A$ such that $a_i a_{i+1} \\dots \\in L(A,p)$ .",
"Given $w=a_1a_2 \\dots $ , from a configuration $(q,i)$ , on a transition $\\delta (q,a_i,p)=(q^{\\prime },\\gamma ,d)$ , $d \\in \\lbrace 1,-1\\rbrace $ , such that $a_{i} a_{i+1} a_{i+2} \\dots \\in L(A,p)$ , we obtain the configuration $(q^{\\prime }, i+d)$ and the output $\\gamma $ is appended to the output produced so far.",
"This transition is denoted as $(q,i) \\stackrel{a_i,p/\\gamma }{\\longrightarrow } (q^{\\prime }, i+d)$ .",
"A run $\\rho $ of a 2DFTpla $(\\mathcal {T},A)$ is a sequence of transitions $(q_0,i_0=0) \\stackrel{a_{i_0},p_1/\\gamma _1}{\\longrightarrow } (q_1, i_1)\\stackrel{a_{i_1},p_2/\\gamma _2}{\\longrightarrow } \\dots $ .",
"The output of $\\rho $ , denoted $\\mathsf {out}(\\rho )$ is then $\\gamma _1 \\gamma _2 \\dots $ .",
"The run $\\rho $ reads the whole word $w$ if $\\mathsf {sup}\\lbrace i_n \\mid 0\\le n <|\\rho |\\rbrace = \\infty $ .",
"The output $[\\!",
"[ (\\mathcal {T},A) ]\\!",
"](w)$ of a word $w$ on run $\\rho $ is defined as in 2DMTla, ($\\mathsf {sup}\\lbrace i_n \\mid 0\\le n <|\\rho |\\rbrace = \\infty $ , $\\inf (\\rho ) \\cap F_A \\ne \\emptyset $ ) and is equal to $\\mathsf {out}(\\rho )$ .",
"It is known that 2DFTpla is strictly more expressive than 2DBT [1], moreover, 2DFTpla are equivalent to 2DMTla, and capture all regular functions.",
"A function $f:\\Sigma ^\\omega \\rightarrow \\Gamma ^\\omega $ is regular iff it is 2DFTpla definable.",
"Figure: NO_CAPTIONConsider the function $j: \\Sigma ^\\omega \\rightarrow \\Gamma ^\\omega $ such that $j(uab^\\omega ) = uub^\\omega $ for $u \\in \\Sigma ^*$ and $j(b^\\omega ) = b^\\omega $ , depicted in the figure.",
"The 2DFTpla is on the left, $P$ is on the right.",
"The transitions are decorated as $\\alpha , p|\\gamma , d$ where $\\alpha \\in \\lbrace a,b\\rbrace $ , $p$ is a state of $P$ , $\\gamma $ is the output and $d$ is the direction.",
"In transitions not using the look-ahead information, the decoration is simply $\\alpha |\\gamma , d$ .",
"$\\mathcal {L}(P, p_1) = ~{\\vdash }\\Sigma ^{*}ab^{\\omega }$ , $\\mathcal {L}(P, p_2) = ~{\\vdash }b^\\omega $ , $\\mathcal {L}(P, p_3) = \\Sigma ^+ab^\\omega $ , $\\mathcal {L}(P, p_4) = ab^\\omega $ .",
"Recall that, as mentioned in the introduction $j$ cannot be realised by a non-deterministic two-way transducer w/o look-ahead."
],
[
"Computability and Continuity for Regular functions",
"We first define the notions of continuity and computability for functions of $\\omega $ -words.",
"Given two words $u, v \\in \\Sigma ^{\\omega }$ , their distance is defined as $d(u,v)=0$ if $u=v$ , and $2^{-|u \\wedge v|}$ if $u \\ne v$ .",
"$u \\wedge v$ is the longest common prefix of $u$ and $v$ .",
"Next, we define the notion of continuity.",
"We interchangeably use the following two equivalent notions [13] for continuity.",
"[Continuity] A function $f:\\Sigma ^\\omega \\rightarrow \\Gamma ^\\omega $ is continuous at $x\\in \\mathsf {dom}(f)$ if (equivalently) (a) for all $(x_n)_{n \\in \\mathbb {N}}$ converging to $x$ , where $x_i \\in \\mathsf {dom}(f)$ for all $i \\in \\mathbb {N}$ , $(f(x_n))_{n \\in \\mathbb {N}}$ converges.",
"(b) $\\forall i \\ge 0$ $\\exists j \\ge 0$ $\\forall y \\in \\mathsf {dom}(f)$ , $|x \\wedge y| \\ge j \\Rightarrow |f(x) \\wedge f(y)| \\ge i$ A function is continuous if it is continuous at any $x\\in \\mathsf {dom}(f)$ .",
"[Computability] A function $f: \\Sigma ^{\\omega } \\rightarrow \\Gamma ^{\\omega }$ is computable if there exists a deterministic multitape machine $M$ computing it in the following sense.",
"$M$ has a read-only one-way input tape, a two-way working tape, and a write-only one-way output tape.",
"All tapes have a left delimiter $\\vdash $ and are infinite to the right.",
"Let $x \\in \\mathsf {dom}(f)$ .",
"For any $j \\in \\mathbb {N}$ , let $M(x,j)$ denote the output produced by $M$ till the time it moves to the right of position $j$ , onto position $j+1$ (or $\\epsilon $ if this move never happens).",
"$f$ is computable by $M$ if for all $x \\in \\mathsf {dom}(f)$ such that $y = f(x)$ , $\\forall i \\ge 0$ , $\\exists j \\ge 0$ such that $y[:i] \\preceq M(x,j)$ .",
"Example.",
"As a first example, consider $g : \\lbrace a,b\\rbrace ^{\\omega } {\\rightarrow }\\lbrace c,d\\rbrace ^{\\omega }$ defined by $g(a^{\\omega })=a^{\\omega }, g(a^nc^{\\omega })=a^{2n}c^{\\omega }$ , and $g(a^nd^{\\omega })=a^n d^{\\omega }$ for all $n \\ge 0$ .",
"It is easy to verify that $g$ is continuous.",
"$g$ is also computable via the machine $M$ described as follows.",
"As long as $M$ reads an $a$ , it outputs an $a$ and writes an $a$ on its working tape, and keeps moving right.",
"When it sees a $d$ , it continues moving right, with output $d$ .",
"It is easy to see that $\\forall j \\ge 0$ , $\\exists i=j $ such that $y[:j] \\preceq M(x,i)$ .",
"If it sees a $c$ after the $a$ 's, then it outputs an $\\epsilon $ , writes a $c$ on the working tape, and moves back on the working tape till the beginning.",
"Then it reads each $a$ on the working tape, outputs an $a$ , and moves right, till it reads $c$ on the working tape.",
"Then on, it keeps on reading and producing $c$ .",
"Consider now the function $f$ defined as $f(a^{\\omega })=c^{\\omega }$ , $f(a^nb^{\\omega })=d^{\\omega }$ for all $n \\ge 0$ .",
"$f$ is not continuous : for $x = a^\\omega $ , $i = 1$ , for all $j$ , $\\exists y_j = a^jb^\\omega $ such that $|x \\wedge y_j| \\ge j$ , but $|f(x) \\wedge f(y_j)| < i $ .",
"$f$ is not computable as well.",
"Indeed, when reading a sequence of $a$ s, if the machine outputs $c$ , then maybe after that sequence there is an infinite sequence of $b$ and this output was wrong.",
"The machine would have to know if a $b$ occurs in the future.",
"As announced, continuity and computability coincide for regular functions: A regular function $f: \\Sigma ^\\omega \\rightarrow \\Gamma ^\\omega $ is computable if and only if it is continuous.",
"[Sketch of Proof] If $f$ is computable by some machine $M$ , then it is not difficult to see that it is continuous.",
"Intuitively, the longer the prefix of input $x\\in \\mathsf {dom}(f)$ is processed by $M$ , the longer the output $M$ produces on that prefix, which converges to $f(x)$ , according to the definition of continuity.",
"The converse direction is less trivial.",
"Suppose that $f$ is continuous.",
"We design the machine $M_f$ , represented as Algorithm , which is shown to compute $f$ .",
"This machine processes longer and longer prefixes $x[{:}i]$ of its input $x$ (for loop at line 2), and tests (line 3) whether a symbol $\\gamma $ can be safely appended to the output.",
"This is the case if for any accepting continuation $x^{\\prime }$ of $x[{:}i]$ , i.e.",
"$y = x[{:}i]x^{\\prime }\\in \\mathsf {dom}(f)$ , the word $\\textsf {out}.\\gamma $ , where $\\textsf {out}$ is the output produced so far, is a prefix of $f(y)$ .",
"Since $f(y)$ is infinite, this is equivalent to saying there is no mismatch between $f(y)$ and $\\textsf {out}.\\gamma $ .",
"Since $f$ is continuous, it can be shown that infinitely often the test at line 3 holds true and since $\\textsf {out}$ is invariantly a prefix of $f(x)$ , we get the result.",
"If $f$ is given by a 2DFTpla , we show that the test at line 3, which we call in the following the mismatch problem is decidable (Lemma below), concluding the proof.",
"See Appendix REF for details.",
"$x \\in \\Sigma ^\\omega $ $\\mathsf {out}$ := $\\epsilon $ ; redthis is written on the working tape $i = 0$ to $+\\infty $ $\\exists \\gamma \\in \\Gamma $ s.t $\\forall y \\in {\\uparrow }x[{:}i] \\cap \\mathsf {dom}(f)$ , $\\lnot \\mathsf {mismatch}($$\\mathsf {out}$ .$\\gamma , f(y))$ $\\mathsf {out}$ := $\\mathsf {out}$ .$\\gamma $ ; redappend to the working tape output $\\gamma $ ; redthis is written on the output tape Algorithm describing $M_f$ .",
"The $\\mathsf {mismatch}$ problem for 2DFTpla .",
"Given $u \\in \\Sigma ^*$ and $v \\in \\Gamma ^*$ , and a 2DFTpla $(\\mathcal {T}, P)$ realising $f$ , is there $y \\in \\Sigma ^{\\omega }$ such that $uy \\in \\mathsf {dom}(f)$ and $\\mathsf {mismatch}(v, f(uy))$ ?",
"The $\\mathsf {mismatch}$ problem for 2DFTpla is $\\textsc {PSpace}$ -complete.",
"[Sketch of proof] This lemma is proved in two steps.",
"First, we show that the problem can be reduced to the same problem, but for a transducer without look-ahead, modulo annotating input words with look-ahead states (i.e.",
"words over alphabet $\\Sigma \\times Q_P$ ).",
"In particular for any 2DFTpla , one can construct in PTime, an equivalent 2DBT working on valid annotated words, where valid means that the annotation is correct (position $i$ is annotated with state $p$ iff the suffix of the input starting at position $i$ belongs to the look-ahead language $L(P,p)$ ).",
"In a second step, we show how to decide the mismatch problem on 2DBT $\\mathcal {A}$ .",
"Since the output $v$ is given as input for the mismatch problem, we know that the mismatch, if it exists, occurs within the first $|v|$ positions of the output of $f(y)$ .",
"Therefore, we are able to construct a two-way automaton $A_{u,v,f}$ working on $\\omega $ -words whose language is non-empty iff there exists $y\\in \\Sigma ^\\omega $ such that $uy\\in \\mathsf {dom}(f)$ and there is a mismatch between $v$ and $f(uy)$ .",
"The automaton $A_{u,v,f}$ simulates the behaviour of $\\mathcal {A}$ by counting, up to $|v|$ , the length of the prefixes of $f(y)$ produced by $\\mathcal {A}$ , and accepts whenever it finds a position $i\\le |v|$ such that $v[i]\\ne f(y)[i]$ , and rejects otherwise.",
"The hardness is obtained by reduction of the intersection problem for $n$ DFAs.",
"Appendix REF has the full proof."
],
[
"Topology preliminaries",
"A regular word (sometimes called ultimately periodic) over $\\Sigma $ is a word of the form $uv^\\omega $ with $u\\in \\Sigma ^*$ and $v\\in \\Sigma ^+$ .",
"The set of regular words is denoted by $\\mathsf {Rat}(\\Sigma )$ .",
"The topological closure of a language $L\\subseteq \\Sigma ^\\omega $ , denoted by $\\bar{L}$ , is the smallest language containing it and closed under taking the limit of converging sequences, i.e.",
"$\\bar{L} = \\left\\lbrace x\\mid \\ \\forall u\\prec x,\\exists y,\\ uy\\in L\\right\\rbrace $ .",
"A language is closed if it is equal to its closure.",
"Let $L\\subseteq D\\subseteq \\Sigma ^\\omega $ .",
"The language $L$ is closed for $D$ if $L=\\bar{L}\\cap D$ .",
"A sequence of words $\\left(x_n\\right)_{n\\in \\mathbb {N}}$ is called regular if there exists $u,v,w\\in \\Sigma ^*$ , $z\\in \\Sigma ^+$ such that for all $n\\in \\mathbb {N}$ , $x_n=uv^nwz^\\omega $ .",
"The following results are folklore or easy-to-get results shown in Appendix REF : Let $L\\subseteq D\\subseteq \\Sigma ^\\omega $ be regular languages.",
"Then $\\bar{L}$ is regular, $L\\subseteq \\overline{L\\cap \\mathsf {Rat}(\\Sigma )}$ (i.e.",
"the regular words are dense in a regular language), any regular word of $\\bar{L}$ is the limit of a regular sequence of $L$ , $L$ is closed for $D$ iff $L\\cap \\mathsf {Rat}(\\Sigma )$ is closed for $D\\cap \\mathsf {Rat}(\\Sigma )$ ."
],
[
"Characterizations of continuity and uniform continuity\nfor regularity-preserving functions",
"We now give a characterisation of continuity and uniform continuity for functions preserving regular languages under inverse image, called regularity-preserving functions [10].",
"This characterisation will be useful later on to get decidability of continuity and uniform continuity for rational and regular functions.",
"We recall the definition of uniform continuity: [Uniform continuity] A function $f: \\Sigma ^{\\omega } \\rightarrow \\Gamma ^{\\omega }$ is uniformly continuous if: $\\forall i \\ge 0$ $\\exists j \\ge 0$ s.t.",
"$\\forall x,y \\in \\mathsf {dom}(f)$ , $|x \\wedge y| \\ge j \\Rightarrow |f(x) \\wedge f(y)| \\ge i$ .",
"For totally bounded metric spaces, uniform continuity coincides with another notion of continuity, Cauchy continuity, which is usually weaker.",
"Cauchy continuity is a more local notion than uniform continuity and will suit us more in the following.",
"A Cauchy continuous function is a function which maps Cauchy sequences (here converging sequences since we deal with complete spaces) to Cauchy sequences.",
"[Cauchy continuity] Let $f:\\Sigma ^\\omega \\rightarrow \\Gamma ^\\omega $ be a function.",
"We say that $f$ is Cauchy continuous at $x$ if for any sequence $\\left(x_n\\right)_{n\\in \\mathbb {N}}$ of $\\mathsf {dom}(f)^\\omega $ converging to $x\\in \\Sigma ^{\\omega }$ , the sequence $\\left(f(x_n)\\right)_{n\\in \\mathbb {N}}$ converges.",
"Moreover, $f$ is Cauchy continuous if and only if it is at any point.",
"The following is a standard result shown in Appendix REF : $f: \\Sigma ^\\omega \\rightarrow \\Gamma ^\\omega $ is uniformly continuous iff it is Cauchy continuous.",
"Notice that a function is continuous if and only if it is Cauchy continuous at any point of its domain.",
"Similarly, a function is Cauchy continuous if and only if it is Cauchy continuous at any point of the topological closure of its domain.",
"The following notions define particular sequences and pairs of sequences of words: Let $f:\\Sigma ^\\omega \\rightarrow \\Gamma ^\\omega $ be a partial function.",
"Let $\\left(x_n\\right)_{n\\in \\mathbb {N}}$ be a sequence of $\\mathsf {dom}(f)^\\omega $ converging to $x\\in \\Sigma ^\\omega $ , such that $\\left(f(x_n)\\right)_{n\\in \\mathbb {N}}$ is not convergent.",
"Such a sequence is called a bad sequence at $x$ for $f$ .",
"Let $\\left(x_n\\right)_{n\\in \\mathbb {N}}$ and $\\left(x_n^{\\prime }\\right)_{n\\in \\mathbb {N}}$ be two sequences of $\\mathsf {dom}(f)^\\omega $ both converging to $x\\in \\Sigma ^\\omega $ , such that either $\\left(f(x_n)\\right)_{n\\in \\mathbb {N}}$ is not convergent, $\\left(f(x_n^{\\prime })\\right)_{n\\in \\mathbb {N}}$ is not convergent, or $\\lim _nf(x_n)\\ne \\lim _nf(x_n^{\\prime })$ .",
"Such a pair of sequences is called a bad pair of sequences at $x$ for $f$ .",
"A bad pair is called regular if both its sequences are regular sequences.",
"A function is Cauchy continuous if and only if it has no bad sequence.",
"A function is continuous if it has no bad sequence at any point of its domain.",
"Similarly a function has a bad sequence at a point $x$ if and only if it has a bad pair at the same point.",
"We now show that for regularity-preserving functions, continuity and Cauchy continuity can be characterized by the behavior of regular sequences only.",
"Let $f:\\Sigma ^\\omega \\rightarrow \\Gamma ^\\omega $ be a regularity-preserving function.",
"If $f$ has a bad sequence at some point $x$ then it has a regular bad pair at some point $z$ .",
"Moreover $x\\in \\mathsf {dom}(f) \\Leftrightarrow z\\in \\mathsf {dom}(f) $ .",
"Let $f:\\Sigma ^\\omega \\rightarrow \\Gamma ^\\omega $ be a function preserving regularity.",
"Let us assume that $f$ has a bad sequence $\\left(x_n\\right)_{n\\in \\mathbb {N}}$ at some point $x$ .",
"By compactness of $\\Gamma ^\\omega $ we can extract two subsequences $\\left(x_n\\right)_{n\\in \\mathbb {N}}$ and $\\left(x_n^{\\prime }\\right)_{n\\in \\mathbb {N}}$ of $\\mathsf {dom}(f)^\\omega $ both converging to $x$ , such that $y=\\lim _nf(x_n)\\ne \\lim _nf(x_n^{\\prime })=y^{\\prime }$ .",
"Let $i=|y\\wedge y^{\\prime }|$ , and let $B_y=\\left\\lbrace z\\mid \\ |y\\wedge z|>i\\right\\rbrace =\\uparrow y[:i+1]$ and $B_{y^{\\prime }}=\\left\\lbrace z\\mid \\ |y^{\\prime }\\wedge z|>i\\right\\rbrace =\\uparrow y^{\\prime }[:i+1]$ .",
"By definition we have $B_y\\cap B_{y^{\\prime }}=\\varnothing $ , and moreover both sets $B_{y},B_{y^{\\prime }}$ are regular.",
"Up to extracting subsequences, we can assume that for all $n$ , $x_n\\in f^{-1}(B_y)$ and $x_n^{\\prime }\\in f^{-1}(B_{y^{\\prime }})$ .",
"This means that $x\\in \\overline{f^{-1}(B_{y})}\\cap \\overline{f^{-1}(B_{y^{\\prime }})}$ .",
"Since $f$ is regularity-preserving, and from Proposition REF .REF , the set $\\overline{f^{-1}(B_{y})}\\cap \\overline{f^{-1}(B_{y^{\\prime }})}$ is regular, and non-empty.",
"Hence there exists a regular word $z\\in \\overline{f^{-1}(B_{y^{\\prime }})}\\cap \\overline{f^{-1}(B_{y^{\\prime }})}$ .",
"Moreover, since $\\mathsf {dom}(f)$ is also regular, we can choose $z$ so that $x\\in \\mathsf {dom}(f) \\Leftrightarrow z\\in \\mathsf {dom}(f) $ .",
"Since $z\\in \\overline{f^{-1}(B_{y})}$ , there is a sequence $\\left(z_n\\right)_{n\\in \\mathbb {N}}$ of words in $f^{-1}(B_{y})$ which converges to $z$ .",
"Furthermore, since $f^{-1}(B_{y})$ is regular and since $z$ is a regular word, we can assume, from Proposition REF .REF , that the sequence $\\left(z_n\\right)_{n\\in \\mathbb {N}}$ is regular.",
"Similarly, there is a regular sequence $\\left(z_n^{\\prime }\\right)_{n\\in \\mathbb {N}}$ in $f^{-1}(B_{y^{\\prime }})$ which converges to $z$ .",
"If either sequence $\\left(f(z_n)\\right)_{n\\in \\mathbb {N}}$ or $\\left(f(z_n^{\\prime })\\right)_{n\\in \\mathbb {N}}$ is not convergent then we are done.",
"If both sequences are convergent, then $\\lim _n f(z_n) \\in B_y$ and $\\lim _n f(z_n^{\\prime }) \\in B_{y^{\\prime }}$ (because $B_y$ and $B_{y^{\\prime }}$ are both closed), which means that $|\\lim _n f(z_n)\\wedge \\lim _n f(z_n^{\\prime })|\\le i$ , hence the pair $\\left(\\left(f(z_n)\\right)_{n\\in \\mathbb {N}},\\left(f(z_n^{\\prime })\\right)_{n\\in \\mathbb {N}}\\right)$ is bad and regular.",
"Let us introduce a notion that will make dealing with regular bad pairs a bit easier.",
"We say that a pair of sequences is synchronized if it is of the form: $\\left(\\left(uv^nwz^\\omega \\right)_n,\\left(uv^nw^{\\prime }z^{\\prime \\omega }\\right)_n\\right)$ Note that a synchronized pair is in particular regular.",
"By taking subsequences, it is not difficult to turn a regular bad pair into a synchronised bad pair (shown in Appendix REF ): Let $f:\\Sigma ^\\omega \\rightarrow \\Gamma ^\\omega $ be a function with a regular bad pair at some point $x$ .",
"Then $f$ has a synchronized bad pair at $x$ .",
"Finally, as a consequence of the previous lemmas, we get the following characterisation: A function $f:\\Sigma ^\\omega \\rightarrow \\Gamma ^\\omega $ preserving regularity (by inverse) is continuous (resp.",
"uniformly continuous) iff it has no synchronised bad pair at any point of its domain (resp.",
"it has no synchronised bad pair)."
],
[
"Deciding Continuity and Uniform Continuity",
"In this section, we show how to decide continuity and uniform continuity for rational and regular functions."
],
[
"Rational case",
"We exhibit structural patterns which are shown to be satisfied by a one-way Büchi transducer iff the rational function it defines is not continuous (resp.",
"not uniformly continuous).",
"We express those patterns in the pattern logic for transducers defined in [7], which is based on existential run quantifiers of the form $\\exists \\pi :p\\xrightarrow{} q$ where $\\pi $ is a run variable, $p,q$ are state variables and $u,v$ word variables, and which intuitively means that there exists a run $\\pi $ from state $p$ to state $q$ on input $u$ , producing output $v$ .",
"The two patterns are given in Figure REF .",
"Figure: Patterns characterising non-continuity andnon-uniform continuity of rational functionsA one-way transducer is called trim if all its states appear in some accepting run.",
"Any one-way Büchi transducer can be trimmed in polynomial time.",
"A trim one-way Büchi transducer defines a non continuous (resp.",
"non uniformly continuous) function if and only if it satisfies the formula $\\phi _{\\text{cont}}$ (resp.",
"$\\phi _{\\text{u-cont}}$ ).",
"[Sketch of Proof] Showing that the patterns of Figure REF induce non-continuity and non-uniform continuity, respectively is quite simple.",
"Indeed, the first pattern is a witness that $\\left(uv^nwz\\right)_{n\\in \\mathbb {N}}$ is a bad sequence at a point $uv^\\omega $ of its domain, for $z$ a word with a final run from $r_2$ , which entails non-continuity by Remark REF .",
"Similarly, the pattern of $\\phi _{\\text{u-cont}}$ witnesses that the pair $\\left(\\left(uv^nwz\\right)_{n\\in \\mathbb {N}},\\left(uv^nw^{\\prime }z^{\\prime }\\right)_{n\\in \\mathbb {N}}\\right)$ is synchronised and bad (with $z,z^{\\prime }$ words that have a final run from $r_1,r_2$ , respectively), which entails non-uniform continuity by Coro.",
"REF .",
"In order to show the other direction, we make use of the characterization of Coro.",
"REF .",
"From a synchronized bad pair, we are able to find a pair of runs with a synchronized loop, such that iterating the loop does not affect the existing mismatch between the outputs of the two runs, which is in essence what the pattern formulas of Figure REF state.",
"The full proof is available in Appendix REF .",
"Deciding if a one way Büchi transducer defines a continuous (resp.",
"uniformly continuous) function can be done in $\\textsc {NLogSpace}$ .",
"From Lemma REF , non continuity (resp.",
"non uniform continuity) is equivalent to the existence of some patterns.",
"According to [7], such patterns are $\\textsc {NLogSpace}$ decidable."
],
[
"Regular case",
"The case of regular functions is more intricate.",
"To get decidability, we have to exploit the form of the output words produced by particular loops of any run of a two-way transducer, called idempotent loops.",
"Idempotent loops always exist for sufficiently long inputs and indeed have a nice structure which allows one to characterise the form of the output words produced when iterating such loops [2].",
"The definition of idempotent loops is quite technical and we refer the reader to [2] for a detailed definition.",
"Moreover, we have abstracted the main property of idempotent loops, which is a key result in our context, and for which it is not necessary to know the precise definition of idempotency.",
"So, given a deterministic two-way transducer $T$ on finite words (we need the notion only for finite words) and an input word $u_1u_2u_3$ , we will say that $u_2$ is idempotent in $(u_1,u_2,u_3)$ (or just idempotent when $u_1,u_3$ are clear from the context), if in the run $r$ of $T$ on $u_1u_2u_3$ , the restriction of $r$ to $u_2$ (which is a sequence of possibly disconnected runs on $u_2$ ) is idempotent in the sense of [2].",
"Given a language of $\\omega $ -words $L\\subseteq \\Sigma ^\\omega $ , we denote by $\\text{Pref}(L)$ the set of finite prefixes of words in $L$ , i.e.",
"$\\text{Pref}(L) = \\lbrace u\\in \\Sigma ^*\\mid \\exists v\\in L\\cdot u\\preceq v\\rbrace $ .",
"In order to deal with look-aheads more easily, we remove look-aheads by considering words annotated with look-ahead information.",
"Given a 2DFTpla $(\\mathcal {T},P)$ over alphabet $\\Sigma $ and with a set of look-ahead states $Q_P$ realizing a function $f$ , we define $\\widetilde{\\mathcal {T}}$ , a 2DBT over $\\Sigma \\times Q_P$ which simulates $(\\mathcal {T},P)$ over words annotated with look-ahead states, and which accepts only words with a correct look-ahead annotation with respect to $P$ (the formal definition can be found in Appendix REF ).",
"We denote by $\\tilde{f}$ the function it realises, in particular for all words $u\\in \\mathsf {dom}(f)$ , there exists a unique annotated word $\\tilde{u}\\in \\mathsf {dom}(\\tilde{f})$ such that $\\tilde{f}(\\tilde{u}) =f(u)$ .",
"For any annotated word $\\tilde{u}$ , $\\pi (\\tilde{u}) = u$ stands for its $\\Sigma $ -projection.",
"From $\\widetilde{\\mathcal {T}}$ , we define $\\mathcal {T}_*$ , a deterministic two-way transducer over $(\\Sigma \\times Q_P)^*$ , which just simulates $\\widetilde{\\mathcal {T}}$ and accepts words in $\\text{Pref}(\\mathsf {dom}(\\tilde{f}))$ .",
"In particular, $\\widetilde{\\mathcal {T}}$ behaves as $\\mathcal {T}$ until it reaches the right border of its input.",
"Let $f_*$ be the function realized by $\\mathcal {T}_*$ .",
"We have that, for any infinite word $x\\in \\mathsf {dom}(\\tilde{f})$ , $\\tilde{f}(x)=\\lim _{u\\prec x}f_*(u)$ .",
"The following lemma is a first characterisation of non-continuity which we can get by exploiting the existence of synchronised bad pairs.",
"Let $f : \\Sigma ^\\omega \\rightarrow \\Gamma ^\\omega $ be a regular function defined by some deterministic transducer $\\mathcal {T}$ with look-ahead and let $Q_P$ be the set of look-ahead states.",
"Then $f$ is not continuous (resp.",
"uniformly continuous) iff there exist finite words $u_1,u^{\\prime }_1,u_2,u^{\\prime }_2,u_3,u^{\\prime }_3\\in (\\Sigma \\times Q_P)^*$ such that $u_1u_2u_3, u_1^{\\prime }u_2^{\\prime }u_3^{\\prime }\\in \\mathsf {dom}(f_*)$ and $\\pi (u_1) = \\pi (u^{\\prime }_1)$ , $\\pi (u_2) = \\pi (u^{\\prime }_2)$ , and $x=\\pi (u_1)\\pi (u_2)^\\omega \\in \\mathsf {dom}(f)$ (resp.",
"$x\\in \\Sigma ^\\omega $ ) $u_2$ and $u^{\\prime }_2$ are idempotent in $(u_1,u_2,u_3)$ and $(u_1^{\\prime },u_2^{\\prime },u_3^{\\prime })$ respectively (for $\\mathcal {T}_*$ ) there exists $i$ such that for all $n\\ge 1$ , $f_{*}(u_1u_2^nu_3)[i]\\ne f_{*}(u^{\\prime }_1{u^{\\prime }_2}^nu^{\\prime }_3)[i]$ .",
"[Sketch of Proof] The proof of this result is similar to the one of Lemma REF .",
"The easy direction is to show that the existence of words $u_1,u^{\\prime }_1,u_2,u^{\\prime }_2,u_3,u^{\\prime }_3$ as above is enough to exhibit non-continuity (resp.",
"non uniform continuity).",
"In the other direction, as for the rational case, we start from the result of Lemma REF which states that it suffices to check for synchronized bad pair to decide continuity/uniform continuity.",
"Like in the rational case, we successively extract subsequences of the synchronized bad pair and at each step we need to preserve synchronicity as well as badness.",
"The main idea is that if we iterate enough times the loop in the synchronized bad pair, we will end up with synchronized idempotent loops.",
"The more detailed version is available in Appendix REF .",
"Given a deterministic two-way transducer $\\mathcal {T}$ and words $u_1,u_2,u_3\\in \\Sigma ^*$ such that $u_1u_2u_3\\in \\text{Pref}(\\mathsf {dom}(\\mathcal {T}))$ and $u_2$ is idempotent for $\\mathcal {T}$ , we say that $u_2$ is producing in $(u_1,u_2,u_3)$ if the run of $\\mathcal {T}$ on $u_1u_2u_3$ produces something when reading at least one symbol of $u_2$ , at some point in the run.",
"If $u_2$ is producing, then $|f_*(u_1u_2^iu_3)|<|f_*(u_1u_2^{i+1}u_3)|$ for all $i\\ge 1$ .",
"Our goal is now to give another characterisation of (non) continuity, which replaces the quantification on $n$ in Lemma REF (property 3) by a property which does not need iteration, and therefore which is more amenable to an algorithmic check.",
"It is based on the following key result.",
"Let $\\Sigma $ be an alphabet such that $\\#\\notin \\Sigma $ .",
"Let $f : \\Sigma ^\\omega \\rightarrow \\Gamma ^\\omega $ be a regular function defined by some deterministic two-way transducer $T$ .",
"There exists a function $\\rho _{T} : (\\Sigma ^*)^3\\rightarrow \\Gamma ^*$ defined on all tuples $(u_1,u_2,u_3)$ such that $u_2$ is idempotent and $u_1u_2u_3\\in \\text{Pref}(\\mathsf {dom}(f))$ , and which satisfies the following conditions: if $u_2$ is producing in $(u_1,u_2,u_3)$ , then $\\rho _{T}(u_1,u_2,u_3)\\prec \\rho _{T}(u_1u_2,u_2,u_2u_3)$ for all $n\\ge 1$ , $\\rho _{T}(u_1,u_2,u_3)\\preceq f_*(u_1u_2^nu_3)$ for all $n\\ge 1$ , $\\rho _{T}(u_1,u_2,u_3)=f_*(u_1u_2^nu_3)$ if $u_2$ is not producing in $(u_1,u_2,u_3)$ the finite word function $\\rho ^{\\prime }_{T} : u_1\\#u_2\\# u_3\\mapsto \\rho _{T}(u_1,u_2,u_3)$ is effectively regular.",
"The proof of Lemma REF is based on a thorough study of the form of the output words produced by idempotent loops, which heavily relies on results from [2].",
"The whole proof, which requires technical notions, can be found in Appendix REF .",
"We give a new characterisation of continuity based on the function $\\rho _{T}$ .",
"In contrast to Lemma REF , this characterisation states that we do not need to iterate the loop to check the existence of a mismatch for all iterations, as we just need to inspect $\\rho _{T}(u_1,u_2,u_3)$ as defined in Lemma REF .",
"Let $f : \\Sigma ^\\omega \\rightarrow \\Gamma ^\\omega $ be a function defined by some deterministic two-way transducer $\\mathcal {T}$ with look-ahead and let $Q_P$ be the set of look-ahead states.",
"$f$ is not continuous (resp.",
"not uniformly continuous) iff there exist $u_1,u^{\\prime }_1,u_2,u^{\\prime }_2,u_3,u^{\\prime }_3\\in (\\Sigma \\times Q_P)^*$ s.t.",
"$u_1u_2u_3, u_1^{\\prime }u_2^{\\prime }u_3^{\\prime }\\in \\mathsf {dom}(f_*)$ and $\\pi (u_1) = \\pi (u^{\\prime }_1)$ , $\\pi (u_2) = \\pi (u^{\\prime }_2)$ , and $x=\\pi (u_1)\\pi (u_2)^\\omega \\in \\mathsf {dom}(f)$ (resp.",
"$x\\in \\Sigma ^\\omega $ ) $u_2$ and $u^{\\prime }_2$ are idempotent in $(u_1,u_2,u_3)$ and $(u_1^{\\prime },u_2^{\\prime },u_3^{\\prime })$ respectively (for $\\widetilde{\\mathcal {T}}$ ) there is a mismatch between $\\rho _{\\widetilde{\\mathcal {T}}}(u_1,u_2,u_3)$ and $\\rho _{\\widetilde{\\mathcal {T}}}(u^{\\prime }_1,u^{\\prime }_2,u^{\\prime }_3)$ .",
"[Sketch of proof] We show how to replace condition 3 of Lemma REF by condition 3 of this lemma.",
"One direction is easy: if $\\rho _{\\widetilde{\\mathcal {T}}}(u_1,u_2,u_3)[i]\\ne \\rho _{\\widetilde{\\mathcal {T}}}(u^{\\prime }_1,u^{\\prime }_2,u^{\\prime }_3)[i]$ for some $i$ , then by Condition 2 of Lemma REF , we get the result.",
"Conversely, assume there is $i$ such that $f_*(u_1u_2^nu_3)[i]\\ne f_*(u^{\\prime }_1(u^{\\prime }_2)^nu^{\\prime }_3)[i]$ for all $n\\ge 1$ and $u_2,u^{\\prime }_2$ are both producing (the other cases are similar and done in Appendix).",
"By Condition 1 of Lemma REF , $\\rho _{\\widetilde{\\mathcal {T}}}(u_1,u_2,u_3)\\prec \\rho _{\\widetilde{\\mathcal {T}}}(u_1u_2,u_2,u_2u_3)\\prec \\dots \\prec \\rho _{\\widetilde{\\mathcal {T}}}(u_1u_2^k,u_2,u_2^ku_3)$ for all $k\\ge 1$ , and similarly for the $u^{\\prime }_i$ .",
"Therefore, by taking $k$ large enough, $\\rho _{\\widetilde{\\mathcal {T}}}(u_1u_2^k,u_2,u_2^ku_3)$ and $\\rho _{\\widetilde{\\mathcal {T}}}(u^{\\prime }_1{u^{\\prime }_2}^k,u^{\\prime }_2,{u^{\\prime }_2}^ku^{\\prime }_3)$ , have length at least $i$ .",
"By Condition 2, $x = \\rho _{\\widetilde{\\mathcal {T}}}(u_1u_2^k,u_2,u_2^ku_3)\\preceq f_*(u_1u_2^{n}u_3)$ and $x^{\\prime } = \\rho _{\\widetilde{\\mathcal {T}}}(u^{\\prime }_1{u^{\\prime }_2}^k,u^{\\prime }_2,{u^{\\prime }_2}^ku^{\\prime }_3)\\preceq f_*(u^{\\prime }_1{u^{\\prime }_2}^{n}u^{\\prime }_3)$ for all $n\\ge 2k+1$ , from which get $x[i]\\ne x^{\\prime }[i]$ .",
"Finally, we show how to decide continuity by reduction to the emptiness problem of finite-visit two-way Parikh automata [9], [5].",
"Continuity and uniform continuity are decidable for regular functions.",
"[Sketch] The proof is based on Lemma REF .",
"First, we encode words $u_1,u^{\\prime }_1,u_2,u^{\\prime }_2$ as words over the alphabet $(\\Sigma \\times {Q_P}^2)^*$ to hard-code condition 1 of the lemma.",
"In particular, we define the language $L$ of words of the form $w_1\\# w_2 \\# u_3 \\# u^{\\prime }_3$ such that $w_1,w_2\\in (\\Sigma \\times P^2)^*$ represent $u_1,u^{\\prime }_1,u_2,u^{\\prime }_2$ and such that conditions 1-2-3 of the lemma are satisfied.",
"Condition 2 and condition $\\pi (u_1)\\pi (u_2)^\\omega \\in \\mathsf {dom}(f)$ are simple because they are regular properties of words, the domain of $f$ being regular.",
"For condition 4, we need counters to identify positions $i$ and $j$ such that $\\rho _{\\widetilde{\\mathcal {T}}}(u_1,u_2,u_3)[i]\\ne \\rho _{\\widetilde{\\mathcal {T}}}(u_1,u_2,u_3)[j]$ , and later on check that $i=j$ .",
"In particular, we rely on the model of two-way Parikh automata which extend two-way automata with counters which can be only incremented, and an accepting semi-linear condition on the counters.",
"If such automata visit any input position a bounded number of times, their emptiness is decidable [9], [5].",
"We show that $L$ is definable by such an automaton which simulates the transducer obtained by Lemma REF .",
"(4) and which is finite-visit."
],
[
"Discussion and Future Work",
"Although we also study the notion of uniform continuity, we do not make any connection with a computational model.",
"The notion of effectively uniformly continuous functions $f$ , well known in the field of computable analysis, seems to be a good candidate notion.",
"Additionally to being computable in the sense of this paper, it also requires the existence of a function $m:\\mathbb {N}\\rightarrow \\mathbb {N}$ , called a modulus of continuity, which is computable and such that for any words $x,y$ , we have $|f(x)\\wedge f(y)|\\ge m(|x\\wedge y|)$ .",
"Effective uniform continuity is arguably a more useful notion than simple computability, since it tells you how far into the input you should look in order to produce a close enough approximation of the output.",
"In contrast, for a computable function, one might need to look arbitrarily far into the input in order to produce a single letter of the output, which does not sound very practical.",
"Another interesting direction we already mentioned is finding a transducer model which captures exactly the computable regular functions.",
"We conjecture that 2DFT characterize computable regular functions but we have no proof of it yet."
],
[
"Section ",
"[Proof of Theorem REF ] We show that a function is 2DMTla definable iff it is 2DFTpla definable.",
"Given a 2DFTpla $(\\mathcal {A}, P)$ , where $\\mathcal {A}= (Q, \\Sigma , \\Gamma , \\delta _{\\mathcal {A}}, q_0)$ and $P=(Q_P, \\Sigma , \\delta _P, Q_0, F)$ , we construct a 2DMTla $(\\mathcal {T}, A, B)$ as follows: For every state $p \\in Q_P$ , we construct an equivalent Muller automaton $A_p$ with initial state $s_p$ s.t.",
"$\\mathcal {L}(P, p) = \\mathcal {L}(A_p)$ .",
"The Muller look-ahead automaton $A$ used in the 2DMTla $(\\mathcal {T}, A, B)$ is the disjoint union of the Muller automata $A_p$ for all $p \\in Q_P$ .",
"$\\mathcal {T}= (Q, \\Sigma , \\Gamma , \\delta _{\\mathcal {T}}, q_0, 2^Q\\backslash \\emptyset )$ , and $\\delta _{\\mathcal {T}}$ is obtained by modifying the transition function $\\delta _{\\mathcal {A}}(q, a, p) = (q^{\\prime }, \\gamma , d)$ as $\\delta _{\\mathcal {T}}(q, a, s_p) = (q^{\\prime }, \\gamma , d)$ .",
"Since the language accepted by two distinct states of a prophetic automaton are disjoint, the language accepted by $A_p$ and $A_p^{\\prime }$ are disjoint for $p \\ne p^{\\prime }$ .",
"The look-behind automaton $B$ accepts all of $\\Sigma ^*$ .",
"It is easy to see that $(\\mathcal {T}, A, B)$ is deterministic : on any position $i$ of the input word $a_1a_2 \\dots $ , and any state $q \\in Q$ , there is a unique $A_p$ accepting $a_{i+1}a_{i+2} \\dots $ .",
"The domain of $(\\mathcal {T}, A, B)$ is the same as that of $(\\mathcal {A}, P)$ , since the accepting states of $A$ are the union of the accepting states of all the $A_p, p \\in Q_P$ .",
"Since all the transitions $\\delta _{\\mathcal {T}}$ have the same outputs as in $\\delta _{\\mathcal {A}}$ for each $(q,a,p)$ , the function computed by $(\\mathcal {A}, P)$ is the same as that computed by $(\\mathcal {T}, A, B)$ .",
"For the other direction, given a 2DMTla, it is easy to remove the look-behind by a product construction [6].",
"Assume we start with such a modified 2DMTla $(\\mathcal {T}, A)$ with no look-behind.",
"Let $\\mathcal {T}=(Q_{\\mathcal {T}}, \\Sigma , \\Gamma , \\delta _{\\mathcal {T}}, s_{\\mathcal {T}}, \\mathcal {F}_{\\mathcal {T}})$ and $A=(Q_A,\\Sigma , \\delta _A, s_A,\\mathcal {F}_A)$ where $\\mathcal {F}_{\\mathcal {T}}, \\mathcal {F}_A$ respectively are the Muller sets corresponding to $\\mathcal {T}$ and $A$ .",
"We describe how to obtain a corresponding 2DFTpla $(\\mathcal {A}, P)$ with $P$ , a prophetic Büchi look-ahead automaton.",
"$\\mathcal {A}$ is described as $(Q_{\\mathcal {T}}, \\Sigma , \\Gamma , \\delta , p_0)$ .",
"The prophetic look-ahead automaton is described as follows.",
"Corresponding to each state $q \\in Q_A$ , let $P_q$ be a prophetic automaton such that $\\mathcal {L}(A, q) = \\mathcal {L}(P_q)$ (this is possible since prophetic automata capture $\\omega $ -regular languages [4]).",
"Since $\\mathcal {A}$ has no accepting condition, we also have a prophetic look-ahead automaton to capture $\\mathsf {dom}(\\mathcal {T})$ .",
"The Muller acceptance of $\\mathcal {T}$ can be translated to a Büchi acceptance condition, and let $P_{\\mathsf {dom}(\\mathcal {T})}$ represent the prophetic automaton such that $\\mathcal {L}(P_{\\mathsf {dom}(\\mathcal {T})})=\\mathsf {dom}(\\mathcal {T})$ .",
"Thanks to the fact that prophetic automata are closed under synchronized product, the prophetic automaton $P$ we need, is the product of $P_q$ for all $q \\in Q_A$ and $P_{\\mathsf {dom}(\\mathcal {T})}$ .",
"Assuming an enumeration $q_1, \\dots , q_n$ of $Q_A$ , the states of $P$ are $|Q_A|+1$ tuples where the first $|Q_A|$ entries correspond to states of $P_{q_1}, \\dots , P_{q_n}$ , and the last entry is a state of $P_{\\mathsf {dom}(\\mathcal {T})}$ .",
"Using this prophetic automaton $P$ and transitions $\\delta _{\\mathcal {T}}$ , we obtain the transitions $\\delta $ of $\\mathcal {A}$ as follows.",
"Consider a transition $\\delta _{\\mathcal {T}}(p, a, q_i) = (q^{\\prime }, \\gamma , d)$ .",
"Correspondingly in $\\mathcal {A}$ , we have $\\delta (p,a,\\kappa )=(q^{\\prime }, \\gamma , d)$ where $\\kappa $ is a $|Q_A|+1$ tuple of states such that the $i$ th entry of $\\kappa $ is an initial state of $P_{q_i}$ .",
"From the initial state $s_{\\mathcal {T}}$ , on reading $\\vdash $ , if we have $\\delta _{\\mathcal {T}}(p_0, \\vdash , q_i) = (q^{\\prime }, \\gamma , d)$ , then $\\delta (p_0,\\vdash ,\\kappa )=(q^{\\prime }, \\gamma , d)$ such that the $i$ th entry of $\\kappa $ is an initial state of $P_{q_i}$ and the last entry of $\\kappa $ is an initial state of $P_{\\mathsf {dom}(\\mathcal {T})}$ .",
"To see why $(\\mathcal {A}, P)$ is deterministic.",
"For each state $q \\in Q_{\\mathcal {T}}$ , for each position $i$ in the input word $a_1a_2 \\dots a_i \\dots $ , there is a unique state $p \\in Q_A$ such that $a_{i+1}a_{i+2}\\dots $ is accepted by $A$ .",
"By our construction, the language accepted from each $p \\in Q_A$ is captured by the prophetic automaton $P_p$ ; by the property of the prophetic automaton $P$ , we know that for any $q_1, q_2 \\in Q_A$ with $q_1 \\ne q_2$ , $L(P_{q_1}) \\cap L(P_{q_2}) = \\emptyset $ .",
"Thus, in $(\\mathcal {A},P)$ , for each state $q$ of $\\mathcal {A}$ , there is a unique state $p \\in P$ such that the suffix is accepted by $L(P)$ ; further, from the initial state of $\\mathcal {A}$ , from $\\vdash $ , there is a unique state $p \\in P$ which accepts $\\mathsf {dom}((\\mathcal {T},A))$ .",
"Hence $\\mathsf {dom}((\\mathcal {A},P))$ is exactly same as $\\mathsf {dom}((\\mathcal {T},A))$ .",
"For each $\\delta _{\\mathcal {T}}(q,a,q_i)=(q^{\\prime }, \\gamma , d)$ , we have the transition $\\delta (q,a,\\kappa )=(q^{\\prime }, \\gamma , d)$ with the $i$ th entry of $\\kappa $ equal to the initial state of $P_{q_i}$ , which preserves the outputs on checking that the suffix of the input from the present position is in $L(P_{q_i})$ .",
"Notice that the entries $j \\ne i$ of $\\kappa $ are decided uniquely, since there is a unique state in each $P_q$ from where each word has an accepting run.",
"Hence, $(\\mathcal {A},P)$ and $(\\mathcal {T},A)$ capture the same function."
],
[
"Proof of Theorem ",
"The two directions of this equivalence are given by Lemma REF and Lemma REF respectively.",
"If a function $f$ is computable, then it is continuous.",
"Assume that $f$ is computable.",
"We prove the continuity of $f$ .",
"Let $M$ be the machine computing $f$ .",
"Let $x \\in \\mathsf {dom}(f)$ be on the input tape of $M$ .",
"For all $i \\ge 0$ , define $\\mathsf {Pref_{M}}(x, i)$ to be the smallest $j \\ge 0$ such that, when $M$ moves to the right of $x[:j]$ into cell $j+1$ , it has output at least the first $i$ symbols of $f(x)$ .",
"For any $i \\ge 0$ , choose $j = \\mathsf {Pref_{M}}(x, i)$ .",
"Consider any $z\\in \\mathsf {dom}(f)$ such that $|x \\wedge z| \\ge j$ .",
"After reading $j$ symbols of $x$ , the machine $M$ outputs at least $i$ symbols of $f(x)$ .",
"Since $|x \\wedge z| \\ge j$ , the first $j$ symbols of $x$ and $z$ are the same, and $M$ being deterministic, outputs the same $i$ symbols on reading the first $j$ symbols of $z$ as well.",
"These $i$ symbols form the prefix for both $f(x), f(z)$ , and hence, $|f(x) \\wedge f(z)| \\ge i$ .",
"Thus, for every $x \\in \\mathsf {dom}(f)$ and for all $i$ , there exists $j= \\mathsf {Pref_{M}}(x, i)$ such that, for all $z \\in \\mathsf {dom}(f)$ , if $|x \\wedge z| \\ge j$ , then $|f(x) \\wedge f(z)| \\ge i$ implying continuity of $f$ .",
"If a regular function is continuous, then it is computable.",
"Let $f$ be a continuous regular function.",
"We define a machine $M_f$ to compute $f$ .",
"The working of $M_f$ is described in Algorithm .",
"For two words $x, y$ , let $\\mathsf {mismatch}(x,y)$ denote that there is some position $i \\ge 1$ such that $x[i] \\ne y[i]$ .",
"To argue the termination of algorithm on all inputs, we have to decide the test in line number 3.",
"We first show (Lemma REF ) the soundness of the algorithm (that is, $M_f$ indeed computes $f(x)$ ) assuming the decidability of the test in line number 3.",
"Then we show the decidability of the test (Lemma ).",
"The following Lemma proves the soundness of Algorithm and thanks to that, the limit of $\\mathsf {out}$ converges to $f(x)$ .",
"For a continuous function $f$ , $x \\in \\mathsf {dom}(f)$ , $\\mathsf {out}$ is updated infinitely often in Algorithm .",
"Moreover, machine $M_f$ computes $f(x)$ as defined in Definition .",
"Assume that $\\mathsf {out}$ is not updated infinitely often.",
"Then, line 4 is not executed after some iteration $m$ .",
"Let $\\mathsf {out}$$_m$ represent the value of $\\mathsf {out}$ after $m$ iterations, and let the length of $\\mathsf {out}$$_m$ be $\\ell $ .",
"Then, for all $k > m$ , and for all $\\gamma \\in \\Gamma $ , there is an extension $y_k$ of $x[{:}k]$ , $y_k \\in \\mathsf {dom}(f)$ , for which $\\mathsf {mismatch}($$\\mathsf {out}$$_m.\\gamma $ , $f(y_k))$ .",
"This violates the continuity of $f$ as seen below.",
"For $x \\in \\mathsf {dom}(f)$ , choose $i=\\ell +1 $ .",
"For all $j > m$ , the extension $y_j$ is s.t.",
"$|x \\wedge y_j| \\ge j$ and $|f(x) \\wedge f(y_j)| < i$ .",
"For all $j \\le m$ , the extension $y_{m+1}$ is s.t.",
"$|x \\wedge y_{m+1}| \\ge j$ and $|f(x) \\wedge f(y_{m+1})| < i$ .",
"This contradicts the continuity of $f$ , proving that $\\mathsf {out}$ is updated infinitely often.",
"Next we show that, $M_f$ , as described in the algorithm indeed computes $f(x)=y$ .",
"Observe that, in each iteration $i$ , $\\mathsf {out}$ is appended with a symbol $\\gamma $ if $\\mathsf {out}$ .$\\gamma $ has no mismatch with $f(z)$ , for all possible extensions $z$ of $x[{:}i]$ .",
"This gives the invariant that, in each iteration $i$ , $\\mathsf {out}$$_i \\preceq f(x)$ .",
"In the Algorithm , $M_f(x, i)$ denotes $\\mathsf {out}$$_i$ .",
"Assume there exists a position $j$ of $y=f(x)$ such that for all positions $i \\ge 0$ of $x$ , $y[{:}j] \\mathsf {out\\textsubscript {i}}$ .",
"This implies that for all $i \\ge 0$ , either $\\mathsf {mismatch}(y[{:}j], \\mathsf {out\\textsubscript {i}}$ ) or $\\mathsf {out\\textsubscript {i}} \\prec y[{:}j]$ .",
"However, as observed already, we have the invariant $\\mathsf {out}$$_i \\preceq f(x)$ for all $i \\ge 0$ .",
"Hence, $\\mathsf {mismatch}(y[{:}j], \\mathsf {out\\textsubscript {i}})$ is not possible.",
"Since $\\mathsf {out}$ is updated infinitely often, there is a strictly increasing sequence $i_1 <i_2 < \\ldots \\in \\mathbb {N}$ such that $\\mathsf {out\\textsubscript {i_1}} \\prec \\mathsf {out\\textsubscript {i_2}} \\prec \\ldots $ .",
"For all $j \\ge 0$ , there exists $i_k$ s. t. $j < |\\mathsf {out\\textsubscript {k}}|$ , and $y[{:}j] \\preceq \\mathsf {out\\textsubscript {i_k}}=M_f(x,i_k)$ .",
"Thus, the machine $M_f$ described in algorithm computes $f$ ."
],
[
"The mismatch problem: Proof of Lemma ",
"Before we discuss the proof of the decidability of the $\\mathsf {mismatch}$ problem, we set up some notations.",
"Given a 2DFTpla $(\\mathcal {T}, P)$ over input alphabet $\\Sigma $ , and $P=(Q_P, \\Sigma , \\delta _P, S_P, F_P)$ , let $u \\in (\\Sigma \\times Q_P)^\\omega $ be an annotated word over the extended alphabet $\\Sigma \\times Q_P$ .",
"Given $u=(\\vdash ,p_0) (a_1,p_1)(a_2,p_2) \\dots $ , let $u[i]$ represent $(a_i, p_i)$ and $\\pi _{\\Sigma }(u) \\in \\Sigma ^{\\omega }$ and $\\pi _{P}(u) \\in Q_P^{\\omega }$ respectively denote the projections of $u$ to its first and second components respectively.",
"An annotated word $u$ is good if $\\forall i \\ge 1, \\pi _{\\Sigma }(u[i-1{:}]) \\in \\mathcal {L}(P,\\pi _{P}(u[i-1]))$ .",
"That is, the suffix $a_ia_{i+1}a_{i+2} \\dots $ of $\\pi _{\\Sigma }(u)$ has a final run in $P$ starting from the state $p_{i-1}=\\pi _{P}(u[i-1])$ .",
"As a first step, we show that, given $f$ specified as a 2DFTpla $(\\mathcal {T}, P)$ with input alphabet $\\Sigma $ , we can construct a function $\\tilde{f}$ specified as a 2DBT $\\widetilde{\\mathcal {T}}$ over the input alphabet $\\Sigma \\times Q_P$ , such that $\\tilde{f}(u)=f(\\pi _{\\Sigma }(u))$ for all good annotated words $u$ .",
"Elimination of look-ahead, construction of $\\widetilde{\\mathcal {T}}$.",
"Let $f$ be specified as a 2DFTpla $(\\mathcal {T}, P)$ , with $\\mathcal {T}=(Q_{\\mathcal {T}},\\Sigma , \\Gamma , \\delta _{\\mathcal {T}}, s_{\\mathcal {T}},2^{Q}\\backslash \\emptyset )$ and $P=(Q_P, \\Sigma , \\delta _P, S_P, F_P)$ .",
"The state space of $\\widetilde{\\mathcal {T}}$ is $Q_{\\mathcal {T}} \\cup (Q_{\\mathcal {T}} \\times \\Sigma \\times Q_P)\\cup (Q^{\\prime }_{\\mathcal {T}} \\times \\Sigma \\times Q_P)$ , and has initial state $s_{\\mathcal {T}}$ .",
"Given a word $(\\vdash ,p_0) (a_1,p_1)(a_2,p_2) \\dots $ , we start in state $s_{\\mathcal {T}}$ , reading $(\\vdash ,p_0)$ , move to the right in state $(s^{\\prime }_{\\mathcal {T}}, \\vdash , p_0)$ , and output $\\epsilon $ .",
"The states of $Q_{\\mathcal {T}}, (Q^{\\prime }_{\\mathcal {T}} \\times \\Sigma \\times Q_P)$ behave in a deterministic manner : from any state $r \\in Q_{\\mathcal {T}}$ , on reading some $(a_i, p_i)$ , we move right, in state $(r^{\\prime }, a_i, p_i)$ , and output $\\epsilon $ .",
"From any state $(r^{\\prime }, a_i, p_i) \\in Q^{\\prime }_{\\mathcal {T}} \\times \\Sigma \\times Q_P$ , on reading $(a,p)$ , we move left in state $(r,a_i,p_i)$ , and output $\\epsilon $ if $p \\in \\delta _P(p_i,a_i)$ .",
"This step checks the consistency of the annotation : if $(a_i,p_i)$ and $(a_{i+1}, p_{i+1})$ appear consecutively in the annotated word, then it must be that $p_{i+1} \\in \\delta _P(p_i,a_i)$ .",
"From a state $(q,a,p) \\in (Q_{\\mathcal {T}} \\times \\Sigma \\times Q_P)$ , on reading $(a_i,p_i)$ , we mimic the transitions of $(\\mathcal {T}, P)$ : $\\delta _{\\mathcal {T}^{\\prime }}((q,a,p), (a_i,p_i))=(r, \\gamma , d)$ iff $a=a_i, p=p_i$ , and $\\delta _{\\mathcal {T}}(q,a,p)=(r, \\gamma , d)$ .",
"The Büchi acceptance condition is given by the set of states $Q_{\\mathcal {T}} \\times \\Sigma \\times F_P$ .",
"It is easy to see that $\\widetilde{\\mathcal {T}}$ is deterministic : $\\widetilde{\\mathcal {T}}$ has all the transitions of $\\mathcal {T}$ from states of the form $Q_{\\mathcal {T}} \\times \\Sigma \\times Q_P$ .",
"It also has the transitions from $Q_{\\mathcal {T}} \\cup (Q^{\\prime }_{\\mathcal {T}} \\times \\Sigma \\times Q_P)$ which behave deterministically as described above.",
"The determinism of $\\widetilde{\\mathcal {T}}$ follows from the determinism of $(\\mathcal {T},P)$ .",
"Now we show that $\\tilde{f}(u)=f(\\pi _{\\Sigma }(u))$ .",
"Consider any word $w=\\vdash a_1a_2 \\dots \\in \\mathsf {dom}(f)$ .",
"$w$ has a unique accepting run in $(\\mathcal {T},P)$ .",
"By the property of $P$ , at each position $i$ , there is a unique state $p_i$ of $P$ such that $a_{i+1}a_{i+2} \\dots \\in L(P,p_i)$ .",
"Consider the good annotation $(\\vdash , p_0)(a_1,p_1) \\dots $ of $w$ .",
"This word is accepted by $\\widetilde{\\mathcal {T}}$ : we check the consistency of the annotation of every two consecutive symbols, using states from $Q_{\\mathcal {T}} \\cup (Q^{\\prime }_{\\mathcal {T}} \\times \\Sigma \\times Q_P)$ without producing any outputs, and states $(q,a,p) \\in Q_{\\mathcal {T}} \\times \\Sigma \\times Q_P$ mimic the transition $\\delta _{\\mathcal {T}}(q,a,p)$ , producing the same outputs and moving in the same direction.",
"Since $w$ is accepted in $(\\mathcal {T},P)$ , we know that $w \\in L(P)$ .",
"The Büchi acceptance condition of $\\widetilde{\\mathcal {T}}$ checks the same condition for acceptance of the good annotated word; hence $w \\in \\mathsf {dom}((\\mathcal {T},P))$ iff the good annotation of $w$ is in $\\mathsf {dom}(\\widetilde{\\mathcal {T}})$ .",
"By construction, $\\tilde{f}(\\tilde{w})=f(w)$ , where $\\tilde{w}$ is the good annotation of $w$ .",
"The converse direction is done in a similar way, starting from an annotated word $u=(\\vdash , p_0)(a_1,p_1) \\dots $ accepted by $\\widetilde{\\mathcal {T}}$ .",
"The transitions in $\\widetilde{\\mathcal {T}}$ ensure that (i) the annotation is consistent, (ii) the outputs produced are same at each position $i$ , and (iii) the acceptance condition checks that the annotation is good.",
"If there were two consecutive symbols $(a_i,p_i)(a_{i+1},p_{i+1})$ such that $p_{i+1} \\in \\delta _P(a_i,p_i)$ , and $a_{i+2}a_{i+2} \\dots \\notin L(P, p_{i+1})$ , then we will not see a final state of $P$ infinitely often, since all subsequent states of $P$ appearing in the annotation will witness the non-acceptance of $a_{i+1}a_{i+2} \\dots $ .",
"Hence, whenever $u$ is accepted in $\\widetilde{\\mathcal {T}}$ , producing $\\tilde{f}(u)$ , $\\pi _{\\Sigma }(u)$ is accepted in $(\\mathcal {T},P)$ , such that $f(\\pi _{\\Sigma }(u))=\\tilde{f}(u)$ .",
"We work on $\\widetilde{\\mathcal {T}}$ rather than $(\\mathcal {T},P)$ to decide the mismatch problem.",
"By the above construction of $\\widetilde{\\mathcal {T}}$ , for a given $u \\in \\Sigma ^*, v \\in \\Gamma ^*$ , there exists a $y \\in \\Sigma ^{\\omega }$ , s.t.",
"$\\mathsf {mismatch}(v, f(uy))$ iff there is a good annotation $\\widetilde{uy} \\in \\mathsf {dom}(\\widetilde{\\mathcal {T}})$ for which $\\mathsf {mismatch}(v, \\tilde{f}(\\widetilde{uy}))$ .",
"Hence the $\\mathsf {mismatch}$ problem for 2DFTpla $(\\mathcal {T}, P)$ reduces to the $\\mathsf {mismatch}$ problem for 2DBT $\\widetilde{\\mathcal {T}}$ ."
],
[
"The $\\mathsf {mismatch}$ problem for 2",
"We first show the $\\textsc {PSpace}$ -membership.",
"Let $f$ be a function specified as a 2DBT $\\mathcal {T}=(Q, \\Sigma , \\Gamma , \\delta ,q_0,F)$ .",
"Without loss of generality, assume that $\\mathcal {T}$ produces at most one output symbol in each transition.",
"Given $u \\in \\Sigma ^*, v \\in \\Gamma ^*$ , define $L_{\\mathsf {mis}}=\\lbrace uy \\in \\Sigma ^{\\omega } \\mid uy \\in \\mathsf {dom}(f), \\mathsf {mismatch}(v, f(uy)\\rbrace $ .",
"Given $u,v$ as above, we construct a two-way Büchi automaton $\\mathcal {A}$ such that $L(\\mathcal {A})=L_{\\mathsf {mis}} \\ne \\emptyset $ iff there exists $y \\in \\Sigma ^{\\omega }$ s.t.",
"$\\mathsf {mismatch}(v, f(uy))$ and $uy \\in \\mathsf {dom}(\\mathcal {T})$ .",
"The state space of $\\mathcal {A}$ is $Q \\cup \\lbrace 1,2, \\dots , |u|\\rbrace \\cup (Q \\times \\lbrace 1,2,\\dots , |v|\\rbrace ) \\cup \\lbrace \\bot \\rbrace $ .",
"The initial state of $\\mathcal {A}$ is 1, and $F$ is the set of accepting states.",
"The transitions are defined as follows.",
"Given an input $w \\in \\Sigma ^{\\omega }$ , $\\mathcal {A}$ ensures that the first $|u|$ symbols of $w$ satisfy $u=w[{:}|u|]$ .",
"Since $u=a_1 \\dots a_k$ is an input to the $\\mathsf {mismatch}$ problem, this is done by starting in state 1, reading the first symbol of $w$ , checking it if it is same as $a_1$ , and if so, move to the right in state 2, and continue this till we reach the last symbol of $u$ in state $|u|$ .",
"Anytime we find a symbol not in $u$ , $\\mathcal {A}$ enters the state $\\bot $ .",
"On successfully reading the first $|u|$ symbols of $w$ and checking it to be $u$ , $\\mathcal {A}$ comes all the way back to $\\vdash $ , and enters the state $(q_0,1)$ .",
"From state $(q_0,1)$ , $\\mathcal {A}$ mimics $\\mathcal {T}$ , and checks if a mismatch with $v$ is detected in the first $|v|$ output symbols produced.",
"The second component of the state grows till at most $|v|$ while checking for the mismatch.",
"To begin, if the first symbol of $v$ is the same as the first output symbol produced by $\\mathcal {T}$ , then, the second component of the state is incremented from 1 to 2.",
"In general, if the second component is $i$ , and if the $i$ th symbol of $v$ is the same as the $i$ th output symbol produced, then the second component increments to $i+1$ .",
"If no mismatch is detected, and the second component is already $|v|$ , then the trap state $\\bot $ is entered.",
"Formally, for $\\gamma \\in \\Gamma $ , $\\delta ((q,i),a)=((q^{\\prime },i+1),\\gamma , d)$ if $\\delta _{\\mathcal {T}}(q,a)=(q^{\\prime },\\gamma ,d)$ and $\\gamma \\ne v[i]$ , $\\delta ((q,i),a)=(q^{\\prime },\\gamma , d)$ if $\\delta _{\\mathcal {T}}(q,a)=(q^{\\prime },\\gamma ,d)$ and $\\gamma = v[i]$ , $\\delta ((q,|v|),a)=\\bot $ if $\\delta _{\\mathcal {T}}(q,a)=(q^{\\prime },\\gamma ,d)$ and $\\gamma = v[|v|]$ , Once the mismatch is detected, $\\mathcal {A}$ behaves just like $\\mathcal {T}$ , and all transitions of $\\mathcal {T}$ are also present.",
"If $\\mathcal {T}$ accepts $w$ , so does $\\mathcal {A}$ .",
"The size of $\\mathcal {A}$ is polynomial in the size of $\\mathcal {T}$ .",
"From [14], [11], we know that given a two-way Büchi automata with $n$ states, we can construct an equivalent NBA with $\\mathcal {O}(2^{n^2})$ states.",
"This, along with the $\\textsc {NLogSpace}$ complexity of emptiness checking of NBA [16], gives us a $\\textsc {PSpace}$ procedure to test the mismatch for 2DBT.",
"Next, we show $\\textsc {PSpace}$ hardness.",
"We reduce the emptiness problem of the intersection of $n$ DFAs to the mismatch problem for 2DBT.",
"Given $n$ DFAs $A_1, \\dots , A_n$ over $\\Sigma =\\lbrace a,b\\rbrace $ , checking if $\\bigcap _{i=1}^n L(A_i)$ is empty is $\\textsc {PSpace}$ -complete.",
"Consider a function $f: \\Sigma ^* \\#^{\\omega } \\rightarrow \\#^{\\omega } \\cup \\$^{\\omega }$ defined as follows.",
"$f(w\\#^{\\omega })= \\left\\lbrace \\begin{array}{cl} \\#^{\\omega }, ~\\text{if}~w \\in \\bigcap _{i=1}^n L(A_i),\\\\\\$^{\\omega }, ~\\text{if}~w \\notin \\bigcap _{i=1}^n L(A_i)\\end{array}\\right.$ Let $u=\\epsilon , v=\\$$ .",
"Then there exists $y \\in \\Sigma ^*\\#^{\\omega }$ such that $\\mathsf {mismatch}(v,f(y))$ iff $w \\in \\bigcap _{i=1}^n L(A_i)$ .",
"$f$ can be specified as a 2DBT $\\mathcal {A}$ whose size is polynomial in $A_1, \\dots , A_n$ as follows.",
"Given an input $\\vdash w \\#^{\\omega }$ , $\\mathcal {A}$ starts in the initial state of $A_1$ , and checks if $w \\in L(A_1)$ , and if so, moves all the way back to $\\vdash $ .",
"Then from the initial state of $A_2$ , it checks if $w \\in L(A_2)$ , comes back to $\\vdash $ and so on, until it has checked if $w \\in L(A_n)$ .",
"Nothing is output till the checks are complete.",
"If the check on $A_n$ is successful, from the final state of $A_n$ , $\\mathcal {A}$ keeps moving right, and outputs $\\#$ on each input symbol.",
"If $w \\notin L(A_j)$ for some $A_j$ , then from the rejecting state of $A_j$ , $\\mathcal {A}$ continues moving right, and outputs a $\\$$ on each input symbol.",
"Clearly, the description of $\\mathcal {A}$ is polynomial in the sizes of $A_1, \\dots , A_n$ .",
"To summarize, the $\\mathsf {mismatch}$ problem for 2DFTpla is solved as follows.",
"Given $u \\in \\Sigma ^*$ and $v \\in \\Gamma ^*$ , guess a good annotation $\\tilde{u}$ of $u$ .",
"Let $\\mathcal {A}$ be the 2DBT that checks the goodness of the annotation.",
"The size of $\\mathcal {A}$ is polynomial in the size of the 2DFTpla.",
"Check the mismatch w.r.t $\\tilde{u}$ and $v \\in \\Gamma ^*$ .",
"This answers the mismatch w.r.t $u$ and $v$ .",
"Thanks to Lemma , we obtain the $\\textsc {PSpace}$ -completeness of the $\\mathsf {mismatch}$ problem for 2DFTpla."
],
[
"Proof of Proposition ",
" Proof of Proposition REF .REF : Let $L\\subseteq \\Sigma ^{\\omega }$ be regular.",
"Let $uv^\\omega $ be a regular word in $\\bar{L}$ .",
"By regularity of $L$ , there is a power of $v$ , $v^k$ such that for any words, $w,x$ , $wv^kx\\in L \\Leftrightarrow wv^{2k}x\\in L$ .",
"Let us consider the language $K=\\left\\lbrace x\\mid \\ uv^kx\\in L\\right\\rbrace $ which is non-empty since $uv^\\omega \\in \\bar{L}$ .",
"Moreover, $K$ is regular, and thus contains a regular word $wz^\\omega $ .",
"Hence we have that $uv^\\omega $ is the limit of the sequence $\\left(uv^{kn}wz^\\omega \\right)_{n\\in \\mathbb {N}}$ of $L$ .",
"Proof of Proposition REF .REF : Let us assume that $L$ is closed for $D$ , that is $\\bar{L}\\cap D=L$ .",
"From Proposition REF .REF , we have $\\overline{L\\cap \\mathsf {Rat}(\\Sigma )}=\\bar{L}$ .",
"Since $\\bar{L}\\cap D=L$ , we have that $\\bar{L}\\cap D\\cap \\mathsf {Rat}(\\Sigma )=L\\cap \\mathsf {Rat}(\\Sigma )$ , hence $L\\cap \\mathsf {Rat}(\\Sigma )$ is closed for $D\\cap \\mathsf {Rat}(\\Sigma )$ .",
"Conversely, let us assume that $L\\cap \\mathsf {Rat}(\\Sigma )$ is closed for $D\\cap \\mathsf {Rat}(\\Sigma )$ .",
"This means that $\\overline{L\\cap \\mathsf {Rat}(\\Sigma )}\\cap D\\cap \\mathsf {Rat}(\\Sigma )=L\\cap \\mathsf {Rat}(\\Sigma )$ .",
"Let us assume towards a contradiction that there exist $x\\in \\bar{L}\\cap D$ such that $x\\notin L$ .",
"Since $\\bar{L}\\cap D$ and $L$ are regular, this means that we can assume $x\\in \\mathsf {Rat}(\\Sigma )$ .",
"Hence $x\\in \\bar{L}\\cap D\\cap \\mathsf {Rat}(\\Sigma )$ .",
"From Proposition REF .REF , we have $\\bar{L}=\\overline{L\\cap \\mathsf {Rat}(\\Sigma )}$ , hence $x\\in \\overline{L\\cap \\mathsf {Rat}(\\Sigma )}\\cap D\\cap \\mathsf {Rat}(\\Sigma )=L\\cap \\mathsf {Rat}(\\Sigma )$ , which contradicts the fact that $x$ is regular and not in $L$ ."
],
[
"Proof of Proposition ",
"Uniform continuity implies Cauchy continuity for any metric space.",
"Let us show it in our case, for the sake of completeness.",
"Let $f:\\Sigma ^\\omega \\rightarrow \\Gamma ^\\omega $ be a uniformly continuous function and let $\\left(x_n\\right)_{n\\in \\mathbb {N}}$ of $\\mathsf {dom}(f)^\\omega $ be converging to $x$ .",
"Let $i\\ge 0$ , and let $j$ be such that $\\forall x,y \\in \\mathsf {dom}(f)$ , $|x \\wedge y| \\ge j \\Rightarrow |f(x) \\wedge f(y)| \\ge i$ .",
"Since $\\left(x_n\\right)_{n\\in \\mathbb {N}}$ converges, there is an integer $N$ such that for any $m,n\\ge N$ , $|x_m \\wedge x_n| \\ge j$ , and thus $|f(x_m) \\wedge f(x_n)| \\ge i$ .",
"Hence $\\left(f(x_n)\\right)_{n\\in \\mathbb {N}}$ converges (it is a Cauchy sequence and $\\Gamma ^\\omega $ is complete).",
"Let us now assume that $f$ is not uniformly continuous.",
"Then: $\\exists i \\ge 0$ $\\forall j \\ge 0$ , $\\exists x_j,y_j \\in \\mathsf {dom}(f)$ , $|x_j \\wedge y_j| \\ge j$ and $ |f(x_j) \\wedge f(y_j)| < i$ .",
"Let $\\left(x_{j_n}\\right)_{n\\in \\mathbb {N}}$ be a convergent subsequence of $\\left(x_{j}\\right)_{j\\in \\mathbb {N}}$ .",
"We have that $|x_{j_n} \\wedge y_{j_n}| \\ge n$ and $|f(x_{j_n}) \\wedge f(y_{j_n})| < i$ for any $n\\in \\mathbb {N}$ .",
"Let $x^{\\prime }_n=x_{j_n}$ and let $y^{\\prime }_n=y_{j_n}$ .",
"We can now consider $\\left(y^{\\prime }_{n_m}\\right)_{m\\in \\mathbb {N}}$ a convergent subsequence of $\\left(y_n^{\\prime }\\right)_{n\\in \\mathbb {N}}$ .",
"Let $x^{\\prime \\prime }_m=x^{\\prime }_{n_m}$ and let $y^{\\prime \\prime }_m=y^{\\prime }_{n_m}$ .",
"We still have that $|x^{\\prime \\prime }_{m} \\wedge y^{\\prime \\prime }_{m}| \\ge m$ and $|f(x^{\\prime \\prime }_m) \\wedge f(y^{\\prime \\prime }_m)| < i$ .",
"Since $\\left(x^{\\prime \\prime }_n\\right)_{n\\in \\mathbb {N}}$ and $\\left(y^{\\prime \\prime }_n\\right)_{n\\in \\mathbb {N}}$ are both convergent, then they converge both to the same limit $x$ .",
"Let $\\lim _nf(x_n^{\\prime \\prime })=z$ , $\\lim _nf(y_n^{\\prime \\prime })=t$ and for all $n$ , $|f(x^{\\prime \\prime }_n) \\wedge f(y^{\\prime \\prime }_n)| < i$ , which means that $|z \\wedge t| < i$ .",
"Now the sequence alternating between $x^{\\prime \\prime }_n$ and $y^{\\prime \\prime }_n$ converges to $x$ but its image is divergent, hence $f$ is not Cauchy continuous at $x$ ."
],
[
"Proof of Lemma ",
"Let $f:\\Sigma ^\\omega \\rightarrow \\Gamma ^\\omega $ be a function with a regular bad pair $\\left(\\left(x_n\\right)_{n\\in \\mathbb {N}},\\left(x^{\\prime }_n\\right)_{n\\in \\mathbb {N}}\\right)$ at some point $x$ .",
"If one of the image sequences is divergent, let us say $\\left(f(x_n)\\right)_{n\\in \\mathbb {N}}$ , then $\\left(\\left(x_n\\right)_{n\\in \\mathbb {N}},\\left(x^{\\prime }_n\\right)_{n\\in \\mathbb {N}}\\right)$ is a synchronized bad pair at $x$ .",
"Let us assume that both image sequences converge.",
"Let $x_n=uv^nwz^\\omega $ and let $x^{\\prime }_n=u^{\\prime }v^{\\prime n}w^{\\prime }z^{\\prime \\omega }$ Let us first assume that $x^{\\prime }_n$ is constant equal to $x$ .",
"Let $x^{\\prime }=u^{-1}x$ , we have that $v^nx^{\\prime }=x^{\\prime }$ and $uv^nx^{\\prime }=x$ for any $n$ .",
"Thus the pair $\\left(\\left(x_n\\right)_{n\\in \\mathbb {N}},\\left(x\\right)_{n\\in \\mathbb {N}}\\right)$ is a synchronized bad pair.",
"Let us now assume that neither sequence is constant, which means that $|v|,|v^{\\prime }|>0$ .",
"Without loss of generality, let us assume that $|u|\\ge |u^{\\prime }|$ , let $k\\in \\mathbb {N}$ , let $p<|v^{\\prime }|$ be such that $|u^{\\prime }|+k|v^{\\prime }|+p=|u|$ , let $v^{\\prime \\prime }=v^{\\prime }[p+1:|v^{\\prime }|]v^{\\prime }[1:p]$ and let $w^{\\prime \\prime }=v^{\\prime }[p+1:|v^{\\prime }|]w^{\\prime }$ .",
"Then we can write $x^{\\prime \\prime }_n=x^{\\prime }_{n+k+1}=u^{\\prime }(v^{\\prime })^kv^{\\prime }[1:p]\\cdot (v^{\\prime \\prime })^n \\cdot w^{\\prime \\prime }z^{\\prime \\omega }=uv^{\\prime \\prime n}w^{\\prime \\prime }z^{\\prime \\omega }$ .",
"Note that $v^\\omega =v^{\\prime \\prime \\omega }$ , which means that $v^{|v^{\\prime \\prime }|}=v^{\\prime \\prime |v|}$ .",
"Let $y_n=x_{|v^{\\prime \\prime }|n}{=}uv^{|v^{\\prime \\prime }|n}wz^{\\omega }$ and let $y^{\\prime \\prime }_n{=}x^{\\prime \\prime }_{|v|n}{=}u{v^{\\prime \\prime }}^{|v|n}wz^{\\omega }$ .",
"Then the pair $\\left(\\left(y_n\\right)_{n\\in \\mathbb {N}},\\left(y^{\\prime \\prime }_n\\right)_{n\\in \\mathbb {N}}\\right)$ is a synchronized bad pair at $x$ ."
],
[
"Proof of Lemma ",
"Let us consider the continuous case.",
"The uniformly continuous case is similar.",
"Let $f$ be a function realized by a trim transducer $T$ .",
"Let us first show that if the pattern of $\\phi _{\\text{cont}}$ appears, then we can exhibit a bad pair at a point of the domain.",
"Let us assume: $\\begin{array}{lll}\\end{array}\\exists \\pi _1:p_1 \\xrightarrow{} q_1,\\ \\exists \\pi _1^{\\prime }:q_1 \\xrightarrow{} q_1\\\\\\exists \\pi _2:p_2 \\xrightarrow{} q_2,\\ \\exists \\pi _2^{\\prime }:q_2 \\xrightarrow{} q_2,\\ \\exists \\pi _2^{\\prime \\prime }:q_2 \\xrightarrow{} r_2\\\\\\big (\\mathsf {init}(p_1)\\wedge \\mathsf {init}(p_2)\\wedge \\mathsf {final}(q_1)\\big )\\wedge \\\\\\big ( \\mathsf {mismatch}(u_1,u_2)\\vee (v_2=\\epsilon \\wedge \\mathsf {mismatch}(u_1,u_2w_2)) \\big )$ $Then the word $ v$ is in the domain of the function realized by the transducer since it has an accepting run.Let $ zt$ be a word accepted from state $ r2$, which exists by trimness, and let us consider the pair $ ((uv)nN,(uvnwzt)nN)$.We have $ f(uv)=u1v1$ and $ u2limn f(uvnwzt)$.",
"If $ mismatch(u1,u2)$ then the pair is bad, otherwise, we must have $ v2=$ and thus $ u2w2limn f(uvnwzt)$.",
"Since $ mismatch(u1,u2w2)$ we again have that the pair is bad.$ Let us assume that the function $f$ is not continuous at some point $x\\in \\mathsf {dom}(f)$ .",
"Then according to Corollary REF , there is a synchronized bad pair $\\left(\\left(uv^nwz^\\omega \\right)_{n\\in \\mathbb {N}},\\left(uv^nw^{\\prime }z^{\\prime \\omega }\\right)_{n\\in \\mathbb {N}}\\right)$ converging to some point of $\\mathsf {dom}(f)$ .",
"Our goal is to exhibit a pattern as in $\\phi _{\\text{cont}}$ .",
"If $v$ is empty, then the sequences are constant which contradicts the functionality of $T$ .",
"Then either $\\left(\\left(uv^nwz^\\omega \\right)_{n\\in \\mathbb {N}},\\left(uv^\\omega \\right)_{n\\in \\mathbb {N}}\\right)$ is a bad pair or $\\left(\\left(uv^\\omega \\right)_{n\\in \\mathbb {N}},\\left(uv^nw^{\\prime }z^{\\prime \\omega }\\right)_{n\\in \\mathbb {N}}\\right)$ is a bad pair (or both).",
"Without loss of generality, let $\\left(\\left(uv^nwz^\\omega \\right)_{n\\in \\mathbb {N}},\\left(uv^\\omega \\right)_{n\\in \\mathbb {N}}\\right)$ be a bad pair.",
"Let us consider an accepting run $\\rho $ of $T$ over $uv^\\omega $ .",
"Since the run is accepting, there is a final state $q_1$ visited infinitely often.",
"Let $l\\in \\left\\lbrace 1,\\ldots ,|v|\\right\\rbrace $ be such that, the state reached after reading $uv^nv[1:l]$ in the run $\\rho $ is $q_1$ for infinitely many $n$ s. Let $v^{\\prime }=v[l+1:|v|]v[1:l]$ , let $u^{\\prime }=uv[1:l]$ and let $w^{\\prime }=v[l+1:|v|]$ , we have that $\\left(\\left(u^{\\prime }v^{\\prime n}w^{\\prime }z^\\omega \\right)_{n\\in \\mathbb {N}},\\left(u^{\\prime }v^{\\prime \\omega }\\right)_{n\\in \\mathbb {N}}\\right)$ is a bad pair.",
"To simplify notations, we write this pair again as $\\left(\\left(uv^nwz^\\omega \\right)_{n\\in \\mathbb {N}},\\left(uv^\\omega \\right)_{n\\in \\mathbb {N}}\\right)$ .",
"We have gained that on the run over $uv^\\omega $ , the final state $q_1$ is reached infinitely often after reading $v$ factors, and let $k,l$ be a integers such that ${p_1}\\xrightarrow{} q_1\\xrightarrow{}q_1$ , with $p_1$ an initial state.",
"We consider $l$ different sequences, for $r\\in \\left\\lbrace 0,\\ldots ,l-1\\right\\rbrace $ we define the sequence $s_r=\\left(uv^{ln+r}wz^\\omega \\right)_{n\\in \\mathbb {N}}$ .",
"Since $\\left(\\left(uv^nwz^\\omega \\right)_{n\\in \\mathbb {N}},\\left(uv^\\omega \\right)_{n\\in \\mathbb {N}}\\right)$ is a bad pair, there must be a value $r$ such that $\\left(\\left((uv^k)(v^l)^n(v^rw)z^\\omega \\right)_{n\\in \\mathbb {N}},\\left((uv^k)(v^l)^\\omega \\right)_{n\\in \\mathbb {N}}\\right)$ is a bad pair.",
"To simplify notations, again, we rename this pair $\\left(\\left(uv^nwz^\\omega \\right)_{n\\in \\mathbb {N}},\\left(uv^\\omega \\right)_{n\\in \\mathbb {N}}\\right)$ , and we know that ${p_1}\\xrightarrow{} q_1\\xrightarrow{}q_1$ , with $p_1$ initial and $q_1$ final.",
"For any $m$ larger than the number of states $|Q|$ of $T$ , let us consider the run of $T$ over $uv^mwz^\\omega $ .",
"Let $i_m,j_m, k_m$ be such that we have the accepting run ${p_m}\\xrightarrow{} q_m\\xrightarrow{}q_m\\xrightarrow{}$ , with $i_m+j_m+k_m=m$ and $0<j_m\\le |Q|$ .",
"And let $r_m=i_m+k_m \\mod {j}_m$ .",
"Let us consider for some $m$ , the sequence $\\left((uv^{r_m}) (v^{j_m})^n (wz)^\\omega \\right)_{n\\in \\mathbb {N}}$ .",
"By the presence of these loops, each of these sequences has a convergent image.",
"There is actually a finite number of such sequences, since $j_m$ and $r_m$ take bounded values.",
"Since the original pair is bad, this means that there must exist $m$ such that the sequence $\\left(f((uv^{r_m}) (v^{j_m})^n (wz)^\\omega )\\right)_{n\\in \\mathbb {N}}$ does not converge to $f(uv^\\omega )$ .",
"Let $u^{\\prime }=uv^{i_m}$ , $v^{\\prime }=v^{j_m}$ and $w^{\\prime }=v^{k_m}w$ , then $\\left(\\left(u^{\\prime }v^{\\prime n}w^{\\prime }z^\\omega \\right)_{n\\in \\mathbb {N}},\\left(u^{\\prime }v^{\\prime \\omega }\\right)_{n\\in \\mathbb {N}}\\right)$ is a bad pair.",
"Once again we rename the pair $\\left(\\left(uv^nwz^\\omega \\right)_{n\\in \\mathbb {N}},\\left(uv^\\omega \\right)_{n\\in \\mathbb {N}}\\right)$ and now we have both ${p_1}\\xrightarrow{} q_1\\xrightarrow{}q_1$ and ${p_2}\\xrightarrow{} q_2\\xrightarrow{}q_2$ , such that $p_1,p_2$ are initial, $q_1$ is final and $wz^\\omega $ has a final run from $q_2$ .",
"Now we have established the shape of the pattern, we only have left to show the mismatch properties.",
"$\\begin{array}{lll}p_1 \\xrightarrow{} q_1 \\xrightarrow{} q_1\\\\p_2 \\xrightarrow{} q_2 \\xrightarrow{} q_2\\xrightarrow{}\\\\\\end{array}$ Let us first assume that $v_2\\ne \\epsilon $ .",
"Then $\\lim _nf(uv^nwz^\\omega )=u_2v_2^\\omega $ .",
"Since the pair is bad, there exists $k$ such that $\\mathsf {mismatch}(u_1v_1^k, u_2v_2^k)$ .",
"Hence, up to taking $u^{\\prime }=uv^k$ , we have established the pattern: $\\begin{array}{lll}\\exists \\pi _1:p_1 \\xrightarrow{} q_1,\\ \\exists \\pi _1^{\\prime }:q_1 \\xrightarrow{} q_1\\\\\\exists \\pi _2:p_2 \\xrightarrow{} q_2,\\ \\exists \\pi _2^{\\prime }:q_2 \\xrightarrow{} q_2\\\\\\big (\\mathsf {init}(p_1)\\wedge \\mathsf {init}(p_2)\\wedge \\mathsf {final}(q_1)\\big )\\wedge \\big ( \\mathsf {mismatch}(u_1^{\\prime },u_2^{\\prime }) \\big )\\end{array}$ Let us now assume that $v_2= \\epsilon $ .",
"Then $\\lim _nf(uv^nwz^\\omega )=u_2y_2$ .",
"Then there is a prefix $w_2$ of $y_2$ and an integer $k$ such that $\\mathsf {mismatch}(u_1v_1^k, u_2w_2)$ .",
"Hence, up to taking $u^{\\prime }=uv^k$ , and $w^{\\prime }$ a sufficiently long prefix of $wz^\\omega $ we have established the pattern: $\\begin{array}{lll}\\exists \\pi _1:p_1 \\xrightarrow{} q_1,\\ \\exists \\pi _1^{\\prime }:q_1 \\xrightarrow{} q_1\\\\\\exists \\pi _2:p_2 \\xrightarrow{} q_2,\\ \\exists \\pi _2^{\\prime }:q_2 \\xrightarrow{} q_2,\\ \\exists \\pi _2^{\\prime \\prime }:q_2 \\xrightarrow{} r_2\\\\\\big (\\mathsf {init}(p_1)\\wedge \\mathsf {init}(p_2)\\wedge \\mathsf {final}(q_1)\\big )\\wedge \\big ( (v_2=\\epsilon \\wedge \\mathsf {mismatch}(u_1,u_2w_2)) \\big )\\end{array}$ which concludes the proof."
],
[
"Proof of Lemma ",
"Let us assume there are such words satisfying the above properties.",
"Since $f_{*}$ is defined over $\\text{Pref}(\\mathsf {dom}(\\tilde{f}))$ , for any $n$ there are some $u_{4,n},u_{4,n}^{\\prime }$ such that $u_1u_2^nu_{4,n},u_1^{\\prime }u_2^{\\prime n}u_{4,n}^{\\prime }\\in \\mathsf {dom}(\\tilde{f})$ .",
"Moreover, the sequences $\\left(\\pi (u_1u_2^nu_{4,n})\\right)_{n\\in \\mathbb {N}}$ and $\\left(\\pi (u_1^{\\prime }u_2^{\\prime n}u_{4,n}^{\\prime })\\right)_{n\\in \\mathbb {N}}$ both converge to $x$ .",
"However, since there is a mismatch between $f_{*}(u_1u_2^nu_3)$ and $ f_{*}(u^{\\prime }_1{u^{\\prime }_2}^nu^{\\prime }_3)$ at position $i$ , there is also one between $\\tilde{f}(u_1u_2^nu_{4,n})$ and $\\tilde{f}(u_1^{\\prime }u_2^{\\prime n}u_{4,n}^{\\prime })$ at position $i$ .",
"Thus the pair $\\left(\\left(\\pi (u_1u_2^nu_{4,n})\\right)_{n\\in \\mathbb {N}},\\left(\\pi (u_1^{\\prime }u_2^{\\prime n}u_{4,n}^{\\prime })\\right)_{n\\in \\mathbb {N}}\\right)$ is a bad pair.",
"Hence $f$ is not Cauchy continuous at $x$ .",
"Thus if $x\\in \\mathsf {dom}(f)$ , $f$ is not continuous.",
"Let us assume that $f$ is not Cauchy continuous.",
"Then according to Lemma REF , there exists synchronized bad pair $\\left(\\left(uv^nwz^\\omega \\right)_{n\\in \\mathbb {N}},\\left(uv^nw^{\\prime }z^{\\prime \\omega }\\right)_{n\\in \\mathbb {N}}\\right)$ at some point $x$ .",
"Moreover, if $f$ is not continuous, we can even assume $x\\in \\mathsf {dom}(f)$ .",
"Let $m$ be larger than $|Q_P|$ and let us consider the run of $v^nwz^\\omega $ of the look-ahead automaton $P$ of $(\\mathcal {T},P)$ .",
"If we look at the sequence of states reached after reading powers of $v$ , we have $p_1\\xrightarrow{}q_1\\xrightarrow{}q_1\\xrightarrow{}$ , with $j_m+k_m+l_m=m$ and $0<k_m\\le |Q_P|$ .",
"Let $r_m=j_m+l_m \\mod {k}_m$ .",
"There exists only a finite number of sequences $\\left(uv^{r_m}(v^{k_m})^nwz^\\omega \\right)_{n\\in \\mathbb {N}}$ , since $j_m,r_m$ take bounded values.",
"Since the original pair is bad, there exists some $m$ such that $\\left(\\left((uv^{j_m})(v^{k_m})^n(v^{l_m}wz^\\omega )\\right)_{n\\in \\mathbb {N}},\\left((uv^{j_m})v^nw^{\\prime }z^{\\prime \\omega }\\right)_{n\\in \\mathbb {N}}\\right)$ is a bad pair.",
"Thus we have that for some $0\\le r<k_m$ the synchronized pair $\\left(\\left((uv^{j_m})(v^{k_m})^n(v^{l_m}wz^\\omega )\\right)_{n\\in \\mathbb {N}},\\left((uv^{j_m})(v^{j_m})^n(v^{r}w^{\\prime }z^{\\prime \\omega })\\right)_{n\\in \\mathbb {N}}\\right)$ is bad.",
"To simplify notations, we rename this pair $\\left(\\left(uv^nwz^\\omega \\right)_{n\\in \\mathbb {N}},\\left(uv^nw^{\\prime }z^{\\prime \\omega }\\right)_{n\\in \\mathbb {N}}\\right)$ .",
"Moreover we have obtained that $p\\xrightarrow{}q_1\\xrightarrow{}q_1\\xrightarrow{}$ .",
"Doing the same construction for the other sequence we can also assume that $p^{\\prime }\\xrightarrow{}q_1^{\\prime }\\xrightarrow{}q_1^{\\prime }\\xrightarrow{}$ is a final run.",
"Thus there exists $u_1,u_2,u_3$ and $u_1^{\\prime },u_2^{\\prime },u_3^{\\prime }$ the respective labelings of $u,v,wz^\\omega $ and $u,v^{\\prime },w^{\\prime }z^{\\prime \\omega }$ , in their corresponding runs.",
"Since $v$ loops over a state in both runs, we have for any $m$ , that the labelings corresponding to the runs of $uv^nwz^\\omega $ and $uv^nw^{\\prime }z^{\\prime \\omega }$ are $u_1u_2^mu_3$ and $u_1^{\\prime }u_2^{\\prime m}u_3^{\\prime }$ , respectively.",
"Let us first consider the case where one of the two sequences, $\\left(\\tilde{f}(u_1u_2^nu_3)\\right)_{n\\in \\mathbb {N}}$ , $\\left(\\tilde{f}(u_1^{\\prime }u_2^{\\prime m}u_3^{\\prime })\\right)_{n\\in \\mathbb {N}}$ , is not converging.",
"Without loss of generality, we assume it is the former.",
"There exists a large enough number $K$ , such that any sequence crossing sequences larger than $K$ contains an idempotent loop.",
"Let us now consider the (infinite) run of $\\mathcal {T}_*$ over $u_1u_2^mu_3$ , for $m$ larger than $K$ .",
"Then we have for any $m\\ge K$ , the run $C_1\\xrightarrow{}C_2\\xrightarrow{}C_2\\xrightarrow{}$ with $j_m+k_m+l_m=m$ and $0<k_m\\le K$ such that $u_2^{k_m}$ is idempotent in $(u_1u_2^{i_m},u_2^{k_m},u_2^{l_m}u_3)$ .",
"Let $r_m=j_m+l_m \\mod {k}_m$ .",
"First we remark that since we have a loop in the run over $u_1u_2^mu_3$ the sequences $\\left(\\tilde{f}(u_1u_2^{r_m}(u_2^{k_m})^nu_3)\\right)_{n\\in \\mathbb {N}}$ are all converging.",
"(see [2]).",
"By assumption, we can find $m, m^{\\prime }$ such that $\\left(\\tilde{f}(u_1u_2^{r_m}(u_2^{k_m})^nu_3)\\right)_{n\\in \\mathbb {N}}$ and $\\left(\\tilde{f}(u_1u_2^{r_{m^{\\prime }}}(u_2^{k_{m^{\\prime }}})^nu_3)\\right)_{n\\in \\mathbb {N}}$ are converging to different limits, then we extract subsequences and we consider the pair $\\left(\\left(u_1(u_2^{k_mk_{m^{\\prime }}})^nu_2^{r_m}u_3\\right)_{n\\in \\mathbb {N}},\\left(u_1(u_2^{k_mk_{m^{\\prime }}})^nu_2^{r_{m^{\\prime }}}u_3\\right)_{n\\in \\mathbb {N}}\\right)$ which is synchronized and whose images both converge but to different limits.",
"Once more we rename the pair $\\left(\\left(u_1(u_2)^nu_3\\right)_{n\\in \\mathbb {N}},\\left(u_1(u_2)^nu_3^{\\prime }\\right)_{n\\in \\mathbb {N}}\\right)$ .",
"Now we only have left to treat the case when the two images sequences converge to different limits.",
"Up to considering a high enough power of $u_2,u_2^{\\prime }$ , and adding some factors in the prefix and suffix, we can assume that the loops are idempotent.",
"Since the images of the two sequences converge, to different sequences there exists some $i$ such that there is a mismatch between $\\lim _n \\tilde{f}(u_1(u_2)^nu_3)$ and $\\lim _n \\tilde{f}(u_1(u_2^{\\prime })^nu_3^{\\prime })$ at position $i$ .",
"More over since the sequences are converging, there is an integer $N$ such that for all $n\\ge N$ , all images $\\tilde{f}(u_1(u_2)^nu_3)$ agree up to position $i$ , included; and all $\\tilde{f}(u_1(u_2^{\\prime })^nu_3^{\\prime })$ also agree up to position $i$ , included."
],
[
"Proof of Lemma ",
"We first establish Lemma REF for regular functions of finite words, i.e.",
"we show the following lemma: Let $\\Sigma $ be an alphabet such that $\\#\\notin \\Sigma $ .",
"Let $f : \\Sigma ^* \\rightarrow \\Gamma ^*$ be a regular function defined by some deterministic two-way transducer $T$ .",
"There exists a function $\\rho _T : (\\Sigma ^*)^3\\rightarrow \\Gamma ^*$ defined on all tuples $(u_1,u_2,u_3)$ such that $u_2$ is idempotent and $u_1u_2u_3\\in \\mathsf {dom}(f)$ , and which satisfies the following conditions: if $u_2$ is producing in $(u_1,u_2,u_3)$ , then $\\rho _T(u_1,u_2,u_3)\\prec \\rho _T(u_1u_2,u_2,u_2u_3)$ for all $n\\ge 1$ , $\\rho _T(u_1,u_2,u_3)\\preceq f(u_1u_2^nu_3)$ for all $n\\ge 1$ , $\\rho _T(u_1,u_2,u_3)=f(u_1u_2^nu_3)$ if $u_2$ is not producing in $(u_1,u_2,u_3)$ the finite word function $\\rho ^{\\prime }_T : u_1\\#u_2\\# u_3\\mapsto \\rho _T(u_1,u_2,u_3)$ is effectively regular.",
"Before proving it, let us see why it entails Lemma REF : [Proof of Lemma REF ] Instead of $f$ , consider the function $f_* :\\text{Pref}(\\mathsf {dom}(f))\\rightarrow \\Gamma ^*$ on finite words given by the deterministic two-way transducer $T_*$ (which stops whenever it reaches the right border of its input and otherwise behaves as $T$ ).",
"By Lemma REF , we get the existence of a function $\\rho _{T_*}$ defined for all $u_1,u_2,u_3$ such that $u_2$ is idempotent for $T$ and $u_1u_2u_3\\in \\mathsf {dom}(f_*)$ , satisfying: if $u_2$ is producing in $(u_1,u_2,u_3)$ , then $\\rho _{T_*}(u_1,u_2,u_3)\\prec \\rho _{T_*}(u_1u_2,u_2,u_2u_3)$ for all $n\\ge 1$ , $\\rho _{T_*}(u_1,u_2,u_3)\\preceq f_*(u_1u_2^nu_3)$ for all $n\\ge 1$ , $\\rho _{T_*}(u_1,u_2,u_3)=f_*(u_1u_2^nu_3)$ if $u_2$ is not producing in $(u_1,u_2,u_3)$ the finite word function $u_1\\#u_2\\# u_3\\mapsto \\rho _{T_*}(u_1,u_2,u_3)$ is effectively regular.",
"Note that $u_1u_2u_3\\in \\mathsf {dom}(f_*)$ iff $u_1u_2u_3\\in \\text{Pref}(\\mathsf {dom}(f))$ .",
"Also note that $u_2$ is idempotent for $T$ iff it is idempotent for $T_*$ , by definition of $T_*$ .",
"Moreover, $u_2$ is producing in $(u_1,u_2,u_3)$ for $T$ iff it is for $T_*$ .",
"Hence, by taking $\\rho _T =\\rho _{T_*}$ , we get a function satisfying the condition of Lemma REF , concluding the proof.",
"We now turn to the proof of Lemma REF .",
"We let $f :\\Sigma ^*\\rightarrow \\Gamma ^*$ a regular function of finite words given by a deterministic two-way transducer $T$ .",
"In order to define the function $\\rho _T$ , one needs results about the form of the output words produced when iterating an idempotent $u_2$ .",
"Such a study has fortunately been made in [2] however, to introduce the results of [2], one needs a few notations and new notions which we will try to present not too formally but precisely enough.",
"The reader who wants a formal definition of those notions is referred to [2].",
"Let $r$ be a run of $T$ on $u_1u_2u_3$ .",
"It turns out that the run $r$ , when restricted to $u_2$ , defines a sequence of factors of $r$ which follows a particular structure.",
"We denote by $r|_{u_2}$ this sequence.",
"As an example, consider Figure REF inspired from an example given in [2].",
"This figure represents the run $r$ decomposed into $r = r_1r_2\\dots r_{15}$ and for all $i$ , $\\alpha _i = \\mathsf {out}(r_i)$ .",
"Intersection on $u_2$ , the run $r$ defines the sequence of runs $r|_{u_2} = r_2,r_4,r_6,r_8,r_{10},r_{12},r_{14}$ .",
"Factors on $u_2$ can be classified according to four categories: an LR-run is a factor of $r$ entering $u$ from the left and leaving it from the right.",
"An RR-run is a run entering and leaving $u_2$ from the right.",
"We define RL- and LL-run symmetrically.",
"A traversal is either an LR-, RL-, LL- or RR-run.",
"Figure: A run on u 1 u 2 u 3 u_1u_2u_3 where u 2 u_2 is idempotent, and its three componentsThose factors can themselves be grouped into what is called components in [2].",
"Let us define what a component is.",
"An LR-component $C$ on $u_2$ (for $r$ ) is a sequence $l_1,l_2,\\dots l_p, t, r_1, \\dots ,r_p$ such that for all $i\\in \\lbrace 1,\\dots ,k\\rbrace $ , $l_i$ is an LL-run on $u_2$ , $r_i$ is an RR-run on $u_2$ , and $t$ is an LR-run on $u_2$ , such that the last state of $l_i$ is equal to the initial state of $r_i$ , the initial state of $t$ is equal to the last state of $l_p$ , and the last state of $t$ is equal to the initial state of $l_1$ .",
"An RL-component $C$ is defined symmetrically as a sequence $r_1,\\dots r_p, t, l_1, \\dots l_p$ where the $r_i$ are RR-runs, $t$ is an RL-run and the $l_i$ are LL-runs.",
"As an example, on Figure REF , we have highlighted the three components of the idempotent loop on $u_2$ , which are respectively given by the sequences $C_1 = r_2,r_4,r_6$ , $C_2 = r_8,r_{10},r_{12}$ and $C_3 = r_{14}$ .",
"Note that $C_1$ and $C_3$ are LR while $C_2$ is RL.",
"We define the trace of $C$ as the run $\\text{tr}(C) = tl_1r_1l_2r_2\\dots l_pr_p$ and the output of $C$ as $\\mathsf {out}(C) = \\mathsf {out}(\\text{tr}(C))$ .",
"For example, on Figure REF we have $\\text{tr}(C_1) = r_4r_2r_6$ and $\\mathsf {out}(C_1) = \\alpha _4\\alpha _4\\alpha _6$ , $\\text{tr}(C_2) =r_{10}r_8r_{12}$ and $\\text{tr}(C_3) = r_{14}$ .",
"An anchor point is a position in $r$ which is the initial position of either an LR- or an RL-run in $u_2$ .",
"On Figure REF , the anchor point are represented by black dots.",
"Note that if there are $k$ components in $r|_{u_2}$ , there are $k$ anchor points, because each component contains exactly one RL- or LR-run.",
"An idempotent loop in a run has the following nice structure and property, proved in [2]: [[2]] Let $T$ be a deterministic two-way transducer on finite words defining a regular function $f$ .",
"Let $u = u_1u_2u_3\\in \\mathsf {dom}(f)$ such that $u_2$ is idempotent for $T$ and let $r$ be the run of $T$ on $u$ .",
"Then, $r|_{u_2}$ can be decomposed into a sequence of components $C_1,\\dots ,C_{k}$ for some odd $k\\ge 1$ , where the $C_{2i+1}$ are LR-components and the $C_{2i}$ are RL-components.",
"Moreover, the following run is the accepting run of $T$ on $u_1u_2^{n+1}u_3$ , for all $n\\ge 0$ : $\\pi _0 \\text{tr}(C_1)^n \\pi _1 \\text{tr}(C_2)^n \\dots \\text{tr}(C_k)^n \\pi _k$ where: $\\pi _0$ is the prefix of $r$ up to the first anchor point, for all $1\\le i<k$ , $\\pi _i$ is the factor of $r$ between the $i$ th and $(i+1)$ th anchor point, $\\pi _k$ is the suffix of $r$ starting from the last anchor point.",
"As an illustration, the runs $\\pi _i$ are depicted on Figure REF for our running example.",
"In particular we have $\\pi _0 = r_1r_2r_3$ , $\\pi _1 = r_4\\dots r_9$ , $\\pi _2 =r_{10}r_{11}r_{12}r_{13}$ and $\\pi _3 = r_{14}r_{15}$ .",
"Figure: The decomposition of rr into r=π 0 π 1 π 2 π 3 r = \\pi _0\\pi _1\\pi _2\\pi _3On Figure REF , we have iterated the idempotent three times and shown the decomposition of the run of $T$ on $u_1u_2^3u_3$ into the factors $\\pi _i$ and $\\text{tr}(C_i)$ .",
"Figure: Run on u 1 u 2 3 u 3 u_1u_2^3u_3"
],
[
"Definition of $\\rho _T(u_1,u_2,u_3)$ , Proof of Lemma ",
"We are now almost ready to define $\\rho _T(u_1,u_2,u_3)$ .",
"Again, we let $r$ be the run of $T$ on $u_1u_2u_3$ .",
"We say that a component $C$ of $r|_{u_2}$ is empty if $\\mathsf {out}(\\text{tr}(C)) = \\epsilon $ .",
"We first define $\\text{run}_T(u_1,u_2,u_3)$ as the prefix of $r$ up to the first anchor point of a non-empty component (if it exists).",
"If all components are empty, then we let $\\text{run}_T(u_1,u_2,u_3) = r$ .",
"In other words, $\\text{run}_T(u_1,u_2,u_3) = \\pi _0\\dots \\pi _j$ where $j=k$ if all components are empty, or $j+1$ is the first non-empty component.",
"Finally, we let $\\rho _T(u_1,u_2,u_3)=\\text{out}(\\text{run}_T(u_1,u_2,u_3)).$ As an example, on Figure REF , assuming that $\\alpha _{6}\\ne \\epsilon $ , we would have $\\text{run}_T(u_1,u_2,u_3) = \\pi _0$ and $\\rho _T(u_1u_2u_3) = \\alpha _1\\alpha _2\\alpha _3$ .",
"If instead we assume that $\\alpha _2 = \\alpha _4 = \\alpha _6 = \\epsilon $ and $\\alpha _{10}\\ne \\epsilon $ , then $C_2$ is the first non-empty component, and we have $\\text{run}_T(u_1,u_2,u_3) = \\pi _0\\pi _1$ and $\\rho _T(u_1,u_2,u_3) =\\alpha _1\\dots \\alpha _9$ .",
"Now, we explain why $\\rho _T$ satisfies the properties of Lemma REF .",
"We explain why $\\rho _T(u_1,u_2,u_3)\\prec \\rho _T(u_1u_2,u_2,u_2u_3)$ when $u_2$ is producing for $(u_1,u_2,u_3)$ .",
"In particular the latter assumption is equivalent to saying the there is a non-empty component $C$ .",
"So, let $j$ such that $j+1$ is the first non-empty component.",
"We have $\\text{run}_T(u_1,u_2,u_3) = \\pi _0\\dots \\pi _j$ .",
"Now, from Lemma REF , we have that the run $r^{\\prime }$ of $T$ on $u_1u_2^3u_3$ is of the form: $\\pi _0 \\text{tr}(C_{1})^2\\pi _1 \\text{tr}(C_{2})^2\\dots \\pi _j \\text{tr}(C_{j+1})^2\\pi _{j+1}\\dots \\text{tr}(C_k)^2 \\pi _k$ Since $C_{j+1}$ is non-empty, we have $\\mathsf {out}(\\text{tr}(C_{j+1}))\\ne \\epsilon $ .",
"If one applies Lemma REF to $(u_1u_2)u_2(u_2u_3)$ where the second occurence of $u_2$ is considered as the idempotent loop, we get the existence of runs $\\pi ^{\\prime }_0\\pi ^{\\prime }_1\\dots $ such that the run $r^{\\prime }$ of $T$ on $u_1u_2^3u_3$ is of the form $r^{\\prime } = \\pi ^{\\prime }_0\\pi ^{\\prime }_1\\dots \\pi ^{\\prime }_k$ where $\\pi ^{\\prime }_0$ is the prefix of $r^{\\prime }$ up to the first anchor point of the idempotent loop (second occurence of $u_2$ in $u_1u_2^3u_3$ ), for all $i\\in \\lbrace 1,\\dots ,k-1\\rbrace $ , $\\pi ^{\\prime }_i$ is the factor of $r^{\\prime }$ in between two consecutive anchor points, and $\\pi ^{\\prime }_k$ is the suffix of $r^{\\prime }$ from the last anchor point.",
"On Figure REF , we have illustrated the $\\pi ^{\\prime }_i$ on our running example.",
"Figure: Decomposition of the run on u 1 u 2 3 u 3 u_1u_2^3u_3 withrespect to the second occurrence of u 2 u_2A close inspection of the proof of Lemma REF shows that we have the following relationship: $\\pi ^{\\prime }_0 = \\pi _0\\text{tr}(C_0)\\qquad \\pi ^{\\prime }_i =\\text{tr}(C_i)\\pi _i\\text{tr}(C_{i+1}) \\qquad \\pi ^{\\prime }_k = \\text{tr}(C_k)\\pi _k$ for all $i\\in \\lbrace 1,\\dots ,k-1\\rbrace $ .",
"We also have by definition of $\\rho _T$ , $\\rho _T(u_1u_2,u_2,u_2u_3) = \\mathsf {out}(\\pi ^{\\prime }_0\\dots \\pi ^{\\prime }_j) =\\mathsf {out}(\\pi _0\\dots \\pi _j\\text{tr}(C_{j+1})) =\\rho _T(u_1,u_2,u_3)\\mathsf {out}(\\text{tr}(C_{j+1}))$ , from which we can conclude since $\\mathsf {out}(\\text{tr}(C_{j+1}))\\ne \\epsilon $ .",
"Property 2 is quite easy to obtain.",
"Assume that $u_2$ is producing (the case where it is not producing is a consequence of property 3).",
"Let $j+1$ be the first non-empty component.",
"For all $n\\ge 0$ , the run of $T$ on $u_1u_2^{n+1}u_3$ is of the form $\\pi _0 \\text{tr}(C_{1})^n\\pi _1 \\text{tr}(C_{2})^n\\dots \\pi _j \\text{tr}(C_{j+1})^n\\pi _{j+1}\\dots \\text{tr}(C_k)^n \\pi _k$ by Lemma REF .",
"Since all components $C_i$ for $i<j+1$ are empty, one gets, for all $n\\ge 0$ : $\\begin{array}{lllllll}f(u_1u_2^{n+1}u_3) & = & \\mathsf {out}(\\pi _0 \\pi _1 \\dots \\pi _j\\text{tr}(C_{j+1})^n\\pi _{j+1}\\dots \\text{tr}(C_k)^n \\pi _k) \\\\ &= &\\rho _T(u_1,u_2,u_3)\\mathsf {out}(\\text{tr}(C_{j+1})^n\\pi _{j+1}\\dots \\text{tr}(C_k)^n \\pi _k)\\end{array}$ concluding the proof of this property.",
"If $u_2$ is not producing, then all components are empty and for all $n\\ge 0$ , we get $f(u_1u_2^nu_3) = \\mathsf {out}(\\pi _0\\dots \\pi _k) = \\rho _T(u_1,u_2,u_3)$ .",
"Finally, we show that the function $u_1\\#u_2\\#u_3\\mapsto \\rho _T(u_1,u_2,u_3)$ is computable by a non-deterministic two-way transducer.",
"Its domain is regular: the set of idempotent elements for a deterministic two-way transducer being regular (Lemma REF ), and the domain of a regular function a finite words being regular as well.",
"A skeleton is a word over the alphabet $\\lbrace x_{RR}, x_{RL}, x_{LR},x_{LL}\\rbrace $ .",
"An LR-skeleton component is a word in $\\bigcup _kx_{LL}^kx_{LR} x_{RR}^k$ .",
"An RL-skeleton component is a word in $\\bigcup _kx_{RR}^kx_{RL} x_{LL}^k$ .",
"A skeleton $s$ is valid if it can be decomposed into skeleton components $s = c_1\\dots c_n$ alternating between LR-skeleton components and RL-skeleton components, such that $c_1$ and $c_n$ are LR-skeleton components.",
"A run $r$ on $u_1u_2u_3$ with $u_2$ idempotent satisfies a valid skeleton $s$ decomposed into skeleton components $c_1\\dots c_k$ , written $r\\models s$ , if the sequence of components $C_1\\dots C_{k^{\\prime }}$ on $u_2$ satisfies $k=k^{\\prime }$ and if $c_i$ is LR (resp.",
"RL), then $C_i$ is LR (resp.",
"RL).",
"Let $N$ be the number of states of $T$ .",
"Since $T$ is deterministic, any of its accepting run visit any input position at most $N$ times.",
"Therefore, any accepting run of $T$ , on an idempotent $u_2$ , has at most $N$ components, and each component has at most $N$ elements.",
"Therefore, we consider only valid skeletons of length $N^2$ , they are finitely many.",
"For each such valid skeleton $s$ , we consider the language $D_s$ of words $u_1\\#u_2\\#u_3$ such that $u_1u_2u_3\\in \\mathsf {dom}(f)$ , $u_2$ is idempotent and the accepting run $r$ of $T$ on $u_1u_2u_3$ satisfy $s$ .",
"The language $D_s$ is regular.",
"Indeed, it can be defined by a deterministic two-way automaton $A_s$ which simulates $T$ (w/o producing anything).",
"In a first pass, $A_s$ checks that the input is valid, i.e.",
"that $u_1u_2u_3\\in \\mathsf {dom}(f)$ and $u_2$ is idempotent.",
"As said before, the set of such words $u_1\\#u_2\\#u_3$ is regular so there is a one-way automaton defining this language: $A_s$ first starts by running this automaton.",
"If it eventually reaches the end of its input in some accepting state, then $A_s$ comes back to the first position of the word and runs another deterministic two-way automaton $B_s$ that we now explain.",
"At any point, using the $\\#$ symbols, $B_s$ can know where it is, either in $u_1$ , in $u_2$ or $u_3$ .",
"It keeps this information in its state.",
"It also keeps a suffix of $s$ in its state.",
"So, states of $B_s$ are in the set $Q\\times \\lbrace 1,2,3\\rbrace \\times \\lbrace s^{\\prime }\\mid \\exists s^{\\prime \\prime }\\cdot s^{\\prime \\prime }s^{\\prime }=s\\rbrace $ , where $Q$ is the set of states of $T$ .",
"The initial state of $B_s$ is $(q_0,1,s)$ where $q_0$ is the initial state of $T$ .",
"On the first component, $B_s$ behaves as $T$ (ignoring the output) and changes its second component according to whether it is in $u_1,u_2$ or $u_3$ (using the $\\#$ symbols).",
"When $B_s$ is in some state $(q, p, x_{YZ}s^{\\prime })$ where $x_{YZ}\\in \\lbrace x_{RR}, x_{RL}, x_{LR}, x_{LL}\\rbrace $ and $B_s$ enters $u_2$ , it checks that $Y= R$ if it enters $u_2$ from the right (otherwise it rejects), or $Y=L$ if it enters from the left.",
"The next time $B_s$ leaves $u_2$ , if it is from the left, it checks that $Z = L$ and if it is from the right, it checks that $Z = R$ .",
"When leaving $u_2$ , it moves to state $(q^{\\prime },p^{\\prime },s^{\\prime })$ for some $q^{\\prime }\\in Q$ , $p^{\\prime }\\in \\lbrace 1,3\\rbrace $ .",
"Its accepting states are states $(q_f,p,\\epsilon )$ where $q_f$ is accepting for $T$ .",
"Given a valid skeleton $s$ and its decomposition into skeleton components $c_1c_2\\dots c_k$ , a marking of $s$ is a word $s^\\bullet $ of the form $c_1c_2\\dots c_{j-1} \\bullet c_j c_{j+1}\\dots c_k$ for all $j\\in \\lbrace 1,\\dots ,k\\rbrace $ where $\\bullet $ is a new symbol which can be placed only before some skeleton component.",
"Therefore, there are $k$ possible markings of $s$ .",
"The meaning of this marking is to mark the first non-empty component $c_j$ .",
"Given a marked word $s^\\bullet $ , it is not difficult to modify $B_s$ into $B^{\\prime }_{s^\\bullet }$ which behaves as $B_s$ , but with the additional features that $B^{\\prime }_{s^\\bullet }$ also checks that the components before $\\bullet $ are empty, and that the first component after $\\bullet $ is non-empty.",
"Finally, as $B_{s^\\bullet }$ also simulates $T$ (w/o considering the output), we can easily turn it into a two-way transducer $T_{s^\\bullet }$ which also outputs everything $T$ outputs before the first LR- or RL-run of the first non-empty component (which is marked by $\\bullet $ ).",
"The final transducer $T_\\rho $ is obtained as a disjoint union of all transducers $T_{s^\\bullet }$ for all valid skeleton $s$ of length at most $N^2$ and all possible markings $s^\\bullet $ of $s$ .",
"Note that the transducer $T_\\rho $ still defines a function.",
"Indeed, since $T$ is deterministic , for any input $u_1\\#u_2\\#u_3$ such that $u_1u_2u_3\\in \\mathsf {dom}(T)$ and $u_2$ is idempotent, there is only one possible skeleton $s$ for the run of $T$ on $u_1u_2u_3$ and one possible marking.",
"$\\Box $"
],
[
"Proof of Lemma ",
"We first need some result: For all deterministic two-way Büchi transducer $T$ defining a function $f$ , for all $n\\ge 2i+1$ , $\\rho _{T}(u_1u_2^i,u_2,u_2^iu_3)\\preceq f_*(u_1u_2^{n} u_3)$ .",
"If $u_2$ is producing in $(u_1,u_2,u_3)$ then for all $i\\ge 0$ , $\\rho _{T}(u_1u_2^i,u_2,u_2^iu_3)\\prec \\rho _{T}(u_1u_2^{i+1},u_2,u_2^{i+1}u_3)$ .",
"First, note that for any triple $(u_1,u_2,u_3)$ such that $u_2$ is idempotent and $u_1u_2u_3\\in \\text{Pref}(\\mathsf {dom}(f))$ , we have $(u_1u_2^n,u_2,u_2^nu_3)$ which also satisfies those conditions, in particular $u_1u_2^{2n+1}u_3\\in \\text{Pref}(\\mathsf {dom}(f))$ .",
"Hence, if we apply Lemma REF (1) on $(u_1u_2^i,u_2,u_2^iu_3)$ , one gets the second statement of the corollary.",
"For the second statement, we again know by Lemma REF (2) that for all $n\\ge 1$ , $\\rho _{T}(u_1u_2^i,u_2,u_2^iu_3)\\preceq f_*(u_1u_2^{n+2i}u_3)$ , from which we get the first statement of the corollary.",
"[Proof of Lemma REF ] Let $f$ and $u_1,u^{\\prime }_1,u_2,u^{\\prime }_2,u_3,u^{\\prime }_3\\in (\\Sigma \\times Q_P)^*$ satisfying the three conditions of the lemma.",
"Let $i$ such that $\\rho _{{\\widetilde{\\mathcal {T}}}}(u_1,u_2,u_3)[i] \\ne \\rho _{{\\widetilde{\\mathcal {T}}}}(u^{\\prime }_1,u^{\\prime }_2,u^{\\prime }_3)[i]$ .",
"Note that ${\\widetilde{\\mathcal {T}}}$ is a deterministic two-way transducer w/o look-ahead by definition.",
"Let $n\\ge 0$ .",
"We show that for all $n\\ge 1$ , $f_{{*}}(u_1u_2^nu_3)[i]\\ne f_{{*}}(u^{\\prime }_1{u^{\\prime }_2}^nu^{\\prime }_3)[i]$ .",
"This is is due to the fact that $\\rho _{{\\widetilde{\\mathcal {T}}}}(u_1,u_2,u_3)\\preceq f_{*}(u_1u_2^nu_3)$ and $\\rho _{{\\widetilde{\\mathcal {T}}}}(u^{\\prime }_1,u^{\\prime }_2,u^{\\prime }_3)\\preceq f_{*}(u^{\\prime }_1{u^{\\prime }_2}^nu_3^{\\prime })$ , by Lemma REF .",
"All the conditions of Lemma REF are met, and therefore we get that $f$ is not continuous.",
"Conversely, suppose that $f$ is not continuous.",
"By Lemma REF , we can assume the existence of words $u_1,u_2,u_3,u^{\\prime }_1,u^{\\prime }_2,u^{\\prime }_3$ satisfying all the conditions of Lemma REF .",
"In particular, we know that for some $i\\ge 0$ and for all $n\\ge 1$ , $f_{*}(u_1u_2^nu_3)[i]\\ne f_{*}(u^{\\prime }_1{u^{\\prime }_2}^nu^{\\prime }_3)[i]$ .",
"Suppose that $u_2$ is not producing in $(u_1,u_2,u_3)$ , then $f_{*}(u_1u_2^nu_3) = f_{*}(u_1u_2u_3)$ for all $n\\ge 1$ .",
"By Lemma REF (3) we also have $f_{*}(u_1u_2u_3) = \\rho _{\\mathcal {T}_{*}}(u_1,u_2,u_3)$ , hence for all $n\\ge 1$ , $f_{*}(u_1u_2^nu_3)[i]=\\rho _{{\\mathcal {T}}_{*}}(u_1,u_2,u_3)[i]$ .",
"Similarly, if $u^{\\prime }_2$ is not producing in $(u^{\\prime }_1,u^{\\prime }_2,u^{\\prime }_3)$ we get that for all $n\\ge 1$ , $f_{*}(u^{\\prime }_1{u^{\\prime }_2}^nu^{\\prime }_3)[i]=\\rho _{{\\widetilde{\\mathcal {T}}}}(u^{\\prime }_1,u^{\\prime }_2,u^{\\prime }_3)[i]$ .",
"We now consider four cases: $u_2$ is not producing in $(u_1,u_2,u_3)$ and $u^{\\prime }_2$ is not producing in $(u^{\\prime }_1,u^{\\prime }_2,u^{\\prime }_3)$ , then by the latter observation, $\\rho _{{\\widetilde{\\mathcal {T}}}}(u_1,u_2,u_3)[i] =f_{*}(u_1u_2u_3)[i]\\ne f_{*}(u^{\\prime }_1u^{\\prime }_2u^{\\prime }_3)[i] =\\rho _{{\\widetilde{\\mathcal {T}}}}(u^{\\prime }_1,u^{\\prime }_2,u^{\\prime }_3)[i]$ , hence, all the conditions of Lemma REF are met.",
"$u_2$ is producing in $(u_1,u_2,u_3)$ but $u^{\\prime }_2$ is not producing in $(u^{\\prime }_1,u^{\\prime }_2,u^{\\prime }_3)$ .",
"By Corollary REF , one can choose $j$ large enough such that $|\\rho _{{\\widetilde{\\mathcal {T}}}}(u_1u_2^j,u_2,u_2^ju_3)| \\ge i$ and from the same corollary we also get that for all $n\\ge 2j+1$ , $\\rho _{{\\widetilde{\\mathcal {T}}}}(u_1u_2^j,u_2,u_2^ju_3)\\preceq f_{*}(u_1u_2^nu_3)$ , hence $\\rho _{{\\widetilde{\\mathcal {T}}}}(u_1u_2^j,u_2,u_2^ju_3)[i] =f_{*}(u_1u_2^{2j+1}u_3)[i]$ .",
"On the other hand, Lemma REF (3) applied on $(u^{\\prime }_1{u^{\\prime }_2}^{j},u^{\\prime }_2,{u^{\\prime }_2}^ju^{\\prime }_3)$ yields $\\rho _{{\\widetilde{\\mathcal {T}}}}(u^{\\prime }_1{u^{\\prime }_2}^{j},u^{\\prime }_2,{u^{\\prime }_2}^ju^{\\prime }_3) =f_{*}(u^{\\prime }_1{u^{\\prime }_2}^{2j+1}u^{\\prime }_3)$ because $u^{\\prime }_2$ is not producing.",
"By assumption we know that $f_{*}(u^{\\prime }_1{u^{\\prime }_2}^{2j+1}u^{\\prime }_3)[i] \\ne f_{*}(u_1{u_2}^{2j+1}u_3)[i]$ hence there is a mismatch between $\\rho _{{\\widetilde{\\mathcal {T}}}}(u^{\\prime }_1{u^{\\prime }_2}^{j},u^{\\prime }_2,{u^{\\prime }_2}^ju^{\\prime }_3)$ and $\\rho _{{\\widetilde{\\mathcal {T}}}}(u_1{u_2}^{j},u_2,{u_2}^ju_3)$ .",
"We can conclude since by taking the words $u_1u_2^j$ , $u^{\\prime }_1{u^{\\prime }_2}^j$ , $u_2$ , $u^{\\prime }_2$ , $u_2^ju_3$ and ${u^{\\prime }_2}^ju^{\\prime }_3$ all conditions of Lemma REF are met (we have just shown the last condition, but the other conditions are trivially satisfied, because $u_1,u^{\\prime }_1,u_2,u^{\\prime }_2,u_3,u^{\\prime }_3$ satisfy the respective conditions of Lemma REF .",
"$u_2$ is not producing and $u^{\\prime }_2$ is producing.",
"This case is symmetric to the latter case.",
"$u_2$ is producing in $(u_1,u_2,u_3)$ and $u^{\\prime }_2$ is producing in $(u^{\\prime }_1,u^{\\prime }_2,u^{\\prime }_3)$ .",
"By Corollary REF (applied on $\\rho _{{\\widetilde{\\mathcal {T}}}}$ ), we can choose $j$ large enough such that $|\\rho _{{\\widetilde{\\mathcal {T}}}}(u_1u_2^j,u_2,u_2^ju_3)| \\ge i$ and $|\\rho _{{\\widetilde{\\mathcal {T}}}}(u^{\\prime }_1{u^{\\prime }_2}^j,u^{\\prime }_2,{u^{\\prime }_2}^ju^{\\prime }_3)| \\ge i$ .",
"From the same corollary we also get that for all $n\\ge 2j+1$ , $\\rho _{{\\widetilde{\\mathcal {T}}}}(u_1u_2^j,u_2,u_2^ju_3)\\preceq f_{{*}}(u_1u_2^nu_3)$ and $\\rho _{{\\widetilde{\\mathcal {T}}}}(u^{\\prime }_1{u^{\\prime }_2}^j,u^{\\prime }_2,{u^{\\prime }_2}^ju^{\\prime }_3)\\preceq f_{*}(u^{\\prime }_1{u^{\\prime }_2}^nu^{\\prime }_3)$ .",
"Hence, $\\rho _{{\\widetilde{\\mathcal {T}}}}(u_1u_2^j,u_2,u_2^ju_3)[i]\\ne \\rho _{{\\widetilde{\\mathcal {T}}}}(u^{\\prime }_1{u^{\\prime }_2}^j,u^{\\prime }_2,{u^{\\prime }_2}^ju^{\\prime }_3)[i]$ .",
"As before, we can conclude by taking the words $u_1u_2^j$ , $u^{\\prime }_1{u^{\\prime }_2}^j$ , $u_2$ , $u^{\\prime }_2$ , $u_2^ju_3$ and ${u^{\\prime }_2}^ju^{\\prime }_3$ which satisfy all conditions of Lemma REF ."
],
[
"Proof of Theorem ",
"We now want to prove, based on the characterisation of Lemma REF , that continuity is decidable for regular functions.",
"We need two simple lemmas first: For any deterministic two-way transducer $T$ , the set of words $u_1\\# u_2 \\# u_2$ such that $u_2$ is idempotent in $(u_1,u_2,u_3)$ for $T$ is regular.",
"For any regular function $f {:} \\Sigma ^\\omega {\\rightarrow }\\Gamma ^\\omega $ , the set $\\text{Pref}(\\mathsf {dom}(f))$ is regular.",
"Moreover, one can construct a finite automaton recognising it from any transducer defining $f$ .",
"Let $T$ be some deterministic two-way transducer with look-ahead defining $f$ .",
"By ignoring the input, one gets a two-way automaton with look-ahead, $\\mathcal {A}$ recognising the domain of $f$ .",
"The look-ahead can be replaced by universal transitions: any transition of the form $\\delta (q,\\sigma ,p) = (q^{\\prime },d)$ where $q,q^{\\prime }$ are states of $\\mathcal {A}$ , $p$ is a state of the look-ahead automaton, $\\sigma \\in \\Sigma $ and $d\\in \\lbrace -1,+1\\rbrace $ can be replaced by the universal transition $\\delta ^{\\prime }_p(q,\\sigma ) = (p,+1)\\wedge (q^{\\prime },d)$ , and from any state $q$ of $\\mathcal {A}$ and $\\sigma \\in \\Sigma $ , we construct the alternating transition $\\delta ^{\\prime }(q,\\sigma ) = \\bigvee _{p}\\delta ^{\\prime }_p(q,\\sigma )$ .",
"Hence, $\\mathcal {A}$ is equivalent to some alternating two-way Büchi automaton $\\mathcal {A}^{\\prime }$ .",
"Such automata are known to capture $\\omega $ -regular languages [15].",
"Hence $\\mathsf {dom}(f)$ is $\\omega $ -regular, from which one easily gets that $\\text{Pref}(\\mathsf {dom}(f))$ is regular as well.",
"Briefly, $\\mathsf {dom}(f)$ being $\\omega $ -regular, it is definable by some non-deterministic Büchi automaton $B$ .",
"We remove from $B$ any state $q$ whose right language (the set of words $w$ such that there exists an accepting run on $w$ from state $q$ ) is empty.",
"They can be computed in $\\mathsf {PTIME}$ .",
"Therefore, we get a subautomaton $B^{\\prime }$ of $B$ and set all its states to be accepting.",
"Seen as an NFA (over finite words), we get $L(B^{\\prime }) = \\text{Pref}(\\mathsf {dom}(f))$ .",
"[Proof of Theorem REF ] Let $f$ be a regular function given by some deterministic two-way transducer $\\mathcal {T}$ with look-ahead, where $Q_P$ is the set of states of the look-ahead automaton.",
"We show how to decide whether $f$ is not continuous by checking the existence of words $u_1,u^{\\prime }_1,u_2,u^{\\prime }_2,u_3,u^{\\prime }_3\\in (\\Sigma \\times Q_P)^*$ satisfying the conditions of Lemma REF .",
"To do that, we rely on some encoding of those six words as a single word, forming a language which is definable by a finite-visit two-way Parikh automaton, whose emptiness is known to be decidable [9], [5].",
"For any two alphabets $A,B$ and any two words $w_1\\in A^*$ and $w_2\\in B^*$ of same length, we let $w_1\\otimes w_2\\in (A\\times B)^*$ be their convolution, defined by $(w_1\\otimes w_2)[i] =(w_1[i],w_2[i])$ for any position $i$ of $w_1$ .",
"We now define a language $L_{\\mathcal {T}} \\subseteq (\\Sigma \\times Q_P^2)^*\\#(\\Sigma \\times Q_P^2)^*\\# (\\Sigma \\times Q_P)^*\\# (\\Sigma \\times Q_P)^*$ which consists of words of the form $w = (w_1\\otimes a_1 \\otimes a^{\\prime }_1)\\# (w_2\\otimes a_2\\otimes a^{\\prime }_2)\\#(w_3\\otimes a_3)\\# (w^{\\prime }_3\\otimes a^{\\prime }_3)$ such that: $w_1,w_2,w_3,w^{\\prime }_3\\in \\Sigma ^*$ , $a_1,a^{\\prime }_1,a_2,a^{\\prime }_2,a_3,a^{\\prime }_3\\in {Q_P}^*$ $w_2\\otimes a_2$ and $w_2\\otimes a^{\\prime }_2$ are idempotent for ${\\widetilde{\\mathcal {T}}}$ $w_1w_2^\\omega \\in \\mathsf {dom}(f)$ (in case we want to check for continuity) $(w_1\\otimes a_1)(w_2\\otimes a_2)(w_3\\otimes a_3)\\in \\text{Pref}(\\mathsf {dom}(\\tilde{f}))$ and $(w_1\\otimes a^{\\prime }_1)(w_2\\otimes a^{\\prime }_2)(w^{\\prime }_3\\otimes a^{\\prime }_3)\\in \\text{Pref}(\\mathsf {dom}(\\tilde{f}))$ For a word $w$ of this form, we let $u_i(w) = w_i\\otimes a_i$ for all $i=1,2,3$ and $u^{\\prime }_1(w) = w_1\\otimes a^{\\prime }_1$ , $u^{\\prime }_2(w) = w_2\\otimes a^{\\prime }_2$ and $u^{\\prime }_3(w) = w^{\\prime }_3\\otimes a^{\\prime }_3$ .",
"From Lemma REF and Lemma REF , and the fact that the set of words $w_1\\# w_2$ such that $w_1w_2^\\omega \\in \\mathsf {dom}(f)$ is regular (as $\\mathsf {dom}(f)$ is regular), it is possible to show that $L_{\\mathcal {T}}$ is effectively regular.",
"Finally, we restrict $L_{\\mathcal {T}}$ to the language $E_{\\mathcal {T}} = \\lbrace w\\in L_{\\mathcal {T}}\\mid \\rho _{{\\widetilde{\\mathcal {T}}}}(u_1(w),u_2(w),u_3(w))\\text{ and } \\rho _{{\\widetilde{\\mathcal {T}}}}(u^{\\prime }_1(w),u^{\\prime }_2(w),u^{\\prime }_3(w))\\text{ mismatch}.\\rbrace $ Clearly, $E_{\\mathcal {T}}\\ne \\varnothing $ iff $f$ is not continuous by Lemma REF .",
"We show that $E_{\\mathcal {T}}$ is definable by two-way Parikh automaton which is additionally finite-visit.",
"A two-way Parikh automaton $\\mathcal {P}$ of dimension $d$ is a two-way automaton, running on finite words, extended with vectors of natural numberswe include 0 of dimension $d$ .",
"A run on a finite word is accepting if it reaches some accepting state and the sum of the vectors met along the transitions belong to some given semi-linear set (or equivalently satisfy some given Presburger formula).",
"The emptiness problem for two-way Parikh automata is undecidable but decidable when there exists some computable $k$ such that in any accepting run of $\\mathcal {P}$ , any input position is visited at most $k$ times by that run.",
"Our automaton $\\mathcal {P}$ will be of dimension 2.",
"After processing its input and ending in some accepting state, the sum $(x,y)$ of all the vectors met along the way will correspond to two output positions $x$ and $y$ of $\\rho _{{\\widetilde{\\mathcal {T}}}}(u_1(w),u_2(w),u_3(w))$ and $\\rho _{{\\widetilde{\\mathcal {T}}}}(u^{\\prime }_1(w),u^{\\prime }_2(w),u^{\\prime }_3(w))$ such that the label of $x$ differs from the label of $y$ .",
"The Parikh automaton will accept the run only if $x=y$ (note that the latter equation defines a semi-linear set).",
"To do that, we know from Lemma REF that there exist a deterministic two-way transducer $T_\\rho $ (over finite words) which, given any $u_1\\#u_2\\#u_3$ outputs $\\rho _{{\\widetilde{\\mathcal {T}}}}(u_1,u_2,u_3)$ .",
"It is not difficult to turn $T_\\rho $ into some deterministic two-way transducers $H_\\rho $ and $H^{\\prime }_\\rho $ reading words of the form $w\\in L_{\\mathcal {T}}$ and outputting respectively $\\rho _{{\\widetilde{\\mathcal {T}}}}(u_1(w),u_2(w),u_3(w))$ and $\\rho _{{\\widetilde{\\mathcal {T}}}}(u^{\\prime }_1(w),u^{\\prime }_2(w),u^{\\prime }_3(w))$ .",
"The transducer $H_\\rho $ makes a first pass on its input to check that $w\\in L_{\\mathcal {T}}$ (since $L_{\\mathcal {T}}$ is regular it is definable by some finite automaton), and then simulates $T_\\rho $ on the relevant components of the symbols composing the word (for instance, it ignores the third component of any symbol occurring in $w$ ).",
"Similarly, one can construct $H^{\\prime }_\\rho $ .",
"Finally, $\\mathcal {P}$ reads inputs $w\\in L_{\\mathcal {T}}$ (it can check that indeed its input belongs to $L_{\\mathcal {T}}$ during a first pass, as $L_{\\mathcal {T}}$ is regular) and proceeds with two phases.",
"In the first phase, it simulates $H_\\rho $ w/o producing anything but by summing vectors of the form $(\\ell ,0)$ where $\\ell $ is the length of output produced by the current simulated transition of $H_\\rho $ (wlog we assume that $H_\\rho $ outputs at most one symbol at a time, i.e.",
"$\\ell \\in \\lbrace 0,1\\rbrace $ ).",
"Eventually, when $H_\\rho $ triggers a transition producing some $\\sigma \\in \\Gamma $ , $\\mathcal {P}$ non-deterministically decides to increment its first component by 1, one last time, and stores the symbol $\\sigma $ in its state.",
"Then it proceeds to phase 2, which does exactly the same but on the second vector components (vectors of the form $(0,\\ell )$ ) and by simulating $H^{\\prime }_\\rho $ instead of $H_\\rho $ .",
"This is continued till a non-deterministic choice is made of stopping the increment of the second component, and storing in its state, the last symbol $\\beta $ output by $H^{\\prime }_\\rho $ .",
"Finally, its set of accepting states are pairs $(\\sigma ,\\beta )$ such that $\\sigma \\ne \\beta $ and the semi-linear accepting set is defined by the equation $x=y$ .",
"Clearly, $\\mathcal {P}$ is finite-visit because $H_\\rho $ and $H^{\\prime }_\\rho $ , being deterministic two-way transducers, are finite-visit as well, concluding the proof."
]
] | 1906.04199 |
[
[
"Efficient structure learning with automatic sparsity selection for\n causal graph processes"
],
[
"Abstract We propose a novel algorithm for efficiently computing a sparse directed adjacency matrix from a group of time series following a causal graph process.",
"Our solution is scalable for both dense and sparse graphs and automatically selects the LASSO coefficient to obtain an appropriate number of edges in the adjacency matrix.",
"Current state-of-the-art approaches rely on sparse-matrix-computation libraries to scale, and either avoid automatic selection of the LASSO penalty coefficient or rely on the prediction mean squared error, which is not directly related to the correct number of edges.",
"Instead, we propose a cyclical coordinate descent algorithm that employs two new non-parametric error metrics to automatically select the LASSO coefficient.",
"We demonstrate state-of-the-art performance of our algorithm on simulated stochastic block models and a real dataset of stocks from the S\\&P$500$."
],
[
"Introduction",
"Graph structures can be represented with an adjacency matrix containing the edge weights between different nodes.",
"Since the adjacency matrix is usually sparse, common estimation techniques involve a quadratic optimization with a LASSO penalty to impose this sparsity.",
"While the coefficient of the $L_1$ regularization term is easy to choose when we have prior knowledge of the appropriate sparsity level, in most cases we do not and the parametrization becomes non-trivial.",
"In this paper we focus on causal relationships between time series, following the causal graph process (CGP) approach [4], [3] which assumes an autoregressive model, where the coefficients of each time lag is a polynomial function of the adjacency matrix.",
"Using the adjacency matrix instead of the graph Laplacian allows this set-up to consider directed graphs with positive and negative edge weights.",
"[3] simplify the problem to a combination of quadratic minimisations with $L_1$ regularization terms which they then solve with a sparse gradient projection algorithm.",
"They choose this algorithm for its use of sparse-matrix-computation libraries, which is efficient only for highly sparse graphs; its performance deteriorates significantly with more dense graphs.",
"The cyclical coordinate descent algorithm is a widely used optimisation method due to its speed.",
"Its popularity increased after [7] proved its convergence for functions that decompose into a convex part and a non-differentiable but separable term, which makes it perfectly suited for solving quadratic minimizations involving an $L_1$ regularization term.",
"In [9] the efficiency of both a cyclical and a greedy coordinate descent algorithm to solve the LASSO was shown.",
"Subsequently, [8] demonstrated that the cyclical version is more stable and applied it to solve the graphical LASSO problem.",
"[5] extended the LASSO to high-dimensional graphs by performing individual LASSO regularization on each node.",
"This approach inspired [2] to create the graphical-LASSO with the block coordinate descent algorithm.",
"This algorithm computes a sparse estimate of the precision matrix assuming that the data follows a multivariate Normal distribution.",
"The use of CCD to solve this graph-LASSO problem allows for efficiently computing the sparse precision matrix in large environments.",
"However, this algorithm focuses on simultaneous connections not causal relationships.",
"In contrast, [6], [6] proposed different approaches to compute what they define as a graph-Granger causality effect, however their models assumed prior knowledge of the structure of the underlying adjacency matrix and therefore focused on estimating the weights.",
"In the literature there are many proposed methods for selecting the LASSO coefficient.",
"[1], [5], [6], [6] use a function of the probability of error, which depends on selection of a probability.",
"[9], [4], [3] perform an expensive grid search and use the in- and out-of-sample error to select the coefficient.",
"[2], [8] avoid the problem by simply fixing the coefficient to a selected value.",
"These strategies either require the selection of free parameters or rely on the prediction error to find the correct LASSO coefficient.",
"In this work we follow the idea of the graph-LASSO of [2] and the individual LASSO regularization of [5] to propose a new CCD algorithm that solves the causal graph process problem of [4], [3].",
"The CCD approach allows us to leverage the knowledge of the specific structure of the adjacency matrix to optimise the computational steps.",
"In addition, we propose a new metric that uses the prediction quality of each node separately to select the LASSO coefficient.",
"Our solution does not require additional parameters and produces better results than relying solely on the prediction error.",
"Thus, our algorithm computes the directed adjacency matrix with an appropriate number of edges, as well as the polynomial coefficients of the CGP.",
"Furthermore, the quality of the results and speed are not affected by the sparsity level of the underlying problem.",
"Indeed, while the algorithm proposed by [4], [3] scales cubically with the number of nodes, the algorithm we propose scales quadratically while automatically selecting the LASSO coefficient.",
"We show the performance of our solution on simulated CGPs following a stochastic block model with different sizes, levels of sparsity and history lengths.",
"We assess the quality of the estimated adjacency matrix by considering the difference in number of edges, the percentage of correct positives and the percentage of false positives.",
"We highlight the performance of our approach and its limits.",
"We then run the algorithm on a financial dataset and interpret the results.",
"Section introduces signal processing on graphs.",
"We then introduce our novel algorithm based on a coordinate descent algorithm to efficiently estimate the adjacency matrix of the causal graph process in Section .",
"In Section we present a new non-parametric metric to automatically select the LASSO sparsity coefficient.",
"Finally, in Section we present the results on simulated and real datasets."
],
[
"Background to signal processing on graphs",
"There exist many approaches for modelling a graph; in this paper we use a directed adjacency matrix $A$ .",
"An element $A_{i,j} \\in \\mathbb {R}$ of the adjacency matrix corresponds to the weight of the edge from node $j$ to node $i$ .",
"We consider a time dependent graph with $N$ nodes evolving over $K$ time samples.",
"Let $x(k) \\in \\mathbb {R}^N$ be the vector with the value of each node at time $k$ .",
"With this formulation, the graph signal over $K$ time samples is denoted by the matrix $X(K) = [x(0) \\dots x(K - 1) ] \\in \\mathbb {R}^{N \\times K}$ .",
"We assume the graphs to follow a causal process as defined in [4], [3].",
"The causal graph process (CGP) assumes the graph at time $k$ to follow an autoregressive process over $M$ time lags.",
"The current state of the graph, $x(k)$ , is related to the lag $l$ through a graph filter $P_l(A)$ .",
"The graph filter is considered to be a polynomial function over the adjacency matrix with coefficients $C = \\lbrace c_{l,j}\\rbrace $ defined by: $P_l(A) = \\sum _{j=0}^l c_{l,j} A^j$ .",
"Without loss of generality we can fix the coefficients of the first time lag to be $(c_{1,0},c_{1,1}) = (0, 1)$ .",
"Thus, the CGP at $k$ can be expressed as: $x(k) = w(k) + A x(k-1) + \\dots + \\left(c_{M,0}I + \\dots + c_{M,M}A^M \\right) x(k-M) \\; .$ Where $w(k)$ corresponds to Gaussian noise.",
"Hence, the problem of reconstructing the CGP from a group of time series consists of estimating the adjacency matrix $A$ and the polynomial coefficients $C$ .",
"[4], [3] consider the optimisation problem: $(A, c) = \\min _{A, c} \\frac{1}{2} \\sum _{k=M}^{K-1} \\left\\Vert x(k) - \\sum _{l=1}^M P_l(A) x(k-l) \\right\\Vert _2^2 + \\lambda _1 \\Vert A\\Vert _1 + \\lambda _1^c \\Vert C \\Vert _1 \\; ,$ where they include a LASSO penalty for both the adjacency matrix and the polynomial coefficients to enforce sparsity.",
"They decompose this optimisation into different steps: first estimate the coefficients $R_i = P_i(A)$ , then from those coefficients retrieve the adjacency matrix $A$ , which allows the polynomial coefficients $C$ to be obtained via another minimisation.",
"Due to the $L_1$ penalty, the optimisation problem is not convex for $R_1$ and $C$ .",
"For the first step, they perform a block coordinate descent over the matrix coefficients $R_i$ .",
"Each step of the descent is quadratic except for $R_1$ which has the $L_1$ regularisation term.",
"From the obtained matrix coefficients $R_i$ they perform an additional step in the block descent to obtain the adjacency matrix $A$ .",
"With the estimated adjacency matrix, we can reformulate the minimisation of Equation REF in a function of the vector $C$ with an $L_1$ regularisation term.",
"In their paper, they do not go into details on how they perform these minimisations and just specify that they use a sparse gradient projection algorithm.",
"The authors argue in favour of this algorithm because it is particularly efficient, however only in the case of highly sparse problems; otherwise, for a dense graph this algorithm will scale as the cube of the number of nodes which renders it impractical.",
"This motivates our interest in developing a novel cyclical coordinate descent (CCD) algorithm for this problem."
],
[
"Estimating the adjacency matrix with coordinate descent",
"The sparse gradient projection (SGP) algorithm [4], [3] used to solve each step of the block coordinate descent algorithm does not take full advantage of the structure of the problem.",
"We therefore propose an efficient CCD algorithm for estimating the adjacency matrix and the CGP coefficients, and since the $L_1$ regularisation constraint in the optimisation problem for the matrix coefficients only applies to $R_1$ , we consider the two cases $i=1$ and $i>1$ separately.",
"The detailed steps leading to the equations presented in this section are in Appendix REF ."
],
[
"Update equation for $i>1$",
"In the case of $i>1$ , the minimisation of Equation REF on the matrix coefficient $R_i$ simplifies to a quadratic problem, which is well suited for a CCD algorithm looping over the different lags $i$ .",
"Furthermore, since the minimisation is over a matrix, CCD allows us to avoid computing a gradient over a matrix; instead we compute the update equation for matrix $R_i$ directly.",
"We reformulate the loss function to isolate $R_i$ with $S_k^i = x(k) - \\sum _{l \\ne i}^M R_l x(k-l)$ , thus the utility function is: $\\mathcal {L}(R_i) = \\frac{1}{2} \\sum _{k=M}^{K-1} \\left( S_k^i - R_i x(k-i) \\right)^T \\left( S_k^i - R_i x(k-i) \\right) \\;,$ where $S_k^i$ , $x_k$ are vectors of size $N$ .",
"The derivative of $\\mathcal {L}(R_i)$ with respect to $R_i$ is equal to zero if: $R_i = \\left(\\sum _{k=M}^{K-1} S_k^i x(k-i)^T \\right) \\left(\\sum _{k=M}^{K-1} x(k-i) x(k-i)^T \\right)^{-1} \\; .$ which gives us an update equation for CCD.",
"Since the matrix $\\sum _{k=M}^{K-1} x(k-i) x(k-i)^T$ in equation REF is not guaranteed to be non-singular, we can perform a regularisation step by adding noise to its diagonal to compute the inverse.",
"While this inverse step is expensive it does not depend on the matrices $R_i$ .",
"Thus, it can be computed in advance outside of the CCD loop.",
"Hence, the update consists of a vector-vector multiplication followed by a matrix-matrix multiplication of size $N\\times N$ ."
],
[
"CCD for $i=1$",
"For $i=1$ , the optimisation corresponds to Equation REF with the $L_1$ regularisation term.",
"We can follow the methodology of the previous section and derive a matrix update to compute the CCD step.",
"However, in practice this solution produces matrices that are too dense.",
"Hence, we instead constrain the sparsity on each node.",
"This corresponds to running a CCD over the columns of the matrix $R_1$ .",
"Indeed, a column $j$ of the adjacency matrix corresponds to the weight of the edges going from node $j$ to the nodes it influences.",
"To do so, we have to reformulate the loss function as a problem over the column $j$ of $R_1$ .",
"Let us denote by $R_1^{-j}$ the matrix $R_1$ without the column $j$ , $R_1^j$ the column $j$ of the matrix $R_1$ , $x^{-j}(k-1)$ the vector $x(k-1)$ without the term at index $j$ , and $x^j(k-1)$ the value of the $j$ -th term of $x(k-1)$ .",
"Thus, we can reformulate the error term to isolate the column $j$ : $x(k) - \\sum _{i=1}^M R_i x(k-i) = S_k^1 - R_1^{-j} x^{-j}(k-1) - R_1^j x^j(k-1)$ .",
"Therefore, the Lagrangian of the minimisation over the column $j$ of $R_1$ with a LASSO regularisation term follows as: $\\mathcal {L}(R_1^j) = \\frac{1}{2} \\sum _{k=M}^{K-1} \\left\\Vert S_k^1 - R_1^{-j} x^{-j}(k-1) - R_1^j x^j(k-1) \\right\\Vert _2^2 + \\lambda _1 |R_1^j | \\;.$ Due to the non-differentiable $L_1$ term we need to use sub-gradients and thus introduce the soft-thresholding function to obtain the CCD updating equation.",
"We define the soft-threshold function as $S(a,b) = sign(a) ( |a| - b)_+$ , where $sign(a)$ is the sign of $a$ and $(y)_+ = max(0,y)$ .",
"Then, the derivative of the Lagrangian REF is zero if: $R_1^j = \\frac{S \\left( \\sum _{k=M}^{K-1}\\left(S_k^1 - R_1^{-j} x^{-j}(k-1) \\right) x^j(k-1), \\lambda _1 \\right) }{ \\sum _{k=M}^{K-1} (x^j(k-1))^2} \\;.$ The CCD algorithm for $R_1$ will loop by updating each column using Equation REF .",
"As for the update of $R_i$ , the denominator of Equation REF can be computed outside the loop.",
"Thus the complete algorithm consists of a CCD for each lag $i$ with an inner CCD loop over the columns of $R_1$ .",
"Algorithm REF shows the complete CCD algorithm to compute the matrix coefficients $R_i$ of the CGP process.",
"We stop the descent when the first of four criterion is reached: the maximum number of iterations is reached, the $L_1$ norm of the difference between the matrix coefficients $R$ and its previous value is below a threshold $\\epsilon $ , the $L_1$ norm of the difference between the new and previous in-sample MSE is below $\\epsilon $ or the in-sample MSE increases.",
"We then obtain the adjacency matrix by running an extra step of the columns CCD on $R_1$ .",
"CCD algorithm to compute the matrix coefficients $R$ [1] Compute-R$x, M, N, K$ Compute the denominators of Equations REF and REF outside the loop $R = [ zeros(N, N) \\;,\\; \\forall i \\in [1,M] ]$ Initialise coefficients at zero Convergence criterion not met Run CCD over the columns of $R_1$ with Equation REF Run CCD for each lag $i>1$ with matrix update Equation REF Run the CCD over the columns of $R_1$ to obtain the adjacency matrix $A$"
],
[
"Retrieving the polynomial coefficients $c$",
"Once we have retrieved the adjacency matrix $A$ , the next step of the block coordinate descent is to estimate the polynomial coefficients $C$ .",
"To obtain these coefficients we minimise the MSE over the training set with both $L_1$ and $L_2$ regularisation terms.",
"Hence, we can derive the CCD update for each coefficient $C_{i,j}$ .",
"We denote by $\\hat{C}$ and $\\hat{A}$ the estimated values of $C$ and $A$ respectively.",
"Then, let us define $y_k = x_k - \\hat{A}x_{k-1}$ and $w_k = \\sum _{i^{\\prime },j^{\\prime } \\ne i,j} C_{i^{\\prime },j^{\\prime }} \\hat{A}^{j^{\\prime }} x_{k-i^{\\prime }}$ .",
"The error term of Equation REF as a function of $C_{i,j}$ becomes $\\left\\Vert y_k - w_k - C_{i,j} \\hat{A}^j x_{k-i} \\right\\Vert _2^2$ .",
"Thus, taking into account the $L_1$ and $L_2$ regularisation terms on $C$ , the derivative of the Lagrangian with respect to $C_{i,j}$ is zero if: $C_{i,j} = \\frac{S\\left( \\sum _{k=M}^{K} \\left(\\hat{A}^j x(k-i) \\right)^T \\left(y_k - w_k \\right) , (K-M) \\lambda _1^c \\right)}{\\sum _{k=M}^{K} \\left(\\hat{A}^j x(k-i) \\right)^T \\left(\\hat{A}^j x(k-i)\\right) + 2 (K - M ) \\lambda _2^c } \\;.$ Hence, the CCD algorithm for $C$ will loop over each coefficient $C_{i,j}$ and update it with Equation REF .",
"This step completes the block coordinate descent to obtain the CGP process from the observed time series $x$ .",
"However, this CCD algorithm has three parameters: $\\lambda _1$ for LASSO penalty used to obtain $R_1$ , and $\\lambda _1^c$ and $\\lambda _2^c$ for the $L_1$ and $L_2$ regularisation terms used to compute the polynomial coefficients $C$ .",
"In this paper we focus on the estimation of the adjacency matrix $A$ and the choice of the LASSO parameter, while fixing the regularisation parameters of equation REF to $\\lambda _1^c=0.05$ and $\\lambda _2^c=10^3$ , for which the algorithm appears to be reasonably robust, as is evident from the results."
],
[
"Selecting the $L_1$ coefficient",
"We now introduce two new non-parametric metrics to efficiently and automatically select the LASSO coefficient, $\\lambda _1$ , which directly influences the sparsity of the adjacency matrix $A$ .",
"A classic approach for selecting this involves computing the cross-validation error.",
"When applied to time series this corresponds to computing a prediction error over an out-of-sample time window, see for example [9], [4], [3], in which the authors use the estimated CGP variables $\\hat{A}$ and $\\hat{C}$ to make a prediction and use the MSE to assess its quality.",
"This technique implies a direct relationship between the sparsity level of the adjacency matrix and the MSE of the prediction, which is questionable in practice.",
"Figure REF plots the evolution of the prediction MSE over increasing values of $\\lambda _1$ , alongside the percentage difference in the number of edges between the estimated adjacency matrix and the real one computed for a simulated CGP following a Stochastic Block Model (SBM) graph, i.e.",
"the difference in number of edges divided by the total possible number of edges.",
"It is clear that current techniques do not produce good results and we do not want to follow [2], [8] in fixing different values of $\\lambda _1$ producing a sparse and a dense result without knowing which one is correct.",
"Thus, we need a metric that weights the improvement in prediction error against an increased number of edges.",
"In statistical modelling the AIC and BIC criteria aim to avoid over-fitting by including a penalty on the number of input-variables, which works when the number of output-variables to be predicted is not related to the selected set of input-variables.",
"However, in the case of an adjacency matrix the sparsity of each node also impacts the number of nodes predicted; we want sparsity in the adjacency matrix to encourage each node to more accurately predict a small subset of other nodes.",
"Following this idea further we derive two new error metrics."
],
[
"Two new distance measures for selecting the $L_1$ coefficient",
"We want a distance metric that has no parameters and a maximum around the exact number of edges of the adjacency matrix.",
"An appropriately sparse adjacency matrix has few edges but enough to accurately reproduce the whole CGP process.",
"What is important is not the prediction quality obtained by the complete adjacency matrix but rather the prediction quality of each node independently; the adjacency matrix should have few edges connecting nodes with each connection having a low in- and out-of-sample error.",
"For each node we compute the errors of each connection with other nodes.",
"We sum these errors over all edges and compute the average over time divided by the number of edges.",
"From this error, we compute the sum over all nodes to obtain an error metric of the whole graph: $err = \\sum _{j=1}^N \\frac{1}{\\sum _{i=1}^N \\mathbb {I}_{ \\lbrace A_j \\ne 0 \\rbrace }(i) } \\frac{1}{K-M}\\sum _{k=M}^K \\left\\Vert x(k) \\mathbb {I}_{\\lbrace A_j \\ne 0 \\rbrace } - A x_j(k-1) \\right\\Vert _2^2 \\;,$ where $ 1_{A_j \\ne 0} $ corresponds to a vector of zeros with ones only where the lines of the column $A_j$ of the adjacency matrix $A$ are non-zero.",
"Compared to the in-sample MSE of node $j$ the error metric of Equation REF focuses on the error in the nodes it is connected to.",
"Since we divide by the number of edges, the error should increase as sparsity increases, but it should start decreasing once the gain in the prediction quality of each edge offsets the decrease in number.",
"Intuitively this peak should correspond to the underlying sparsity level of the graph under study; before the peak, the model has too many parameters with poor individual prediction quality, whereas after the peak, the model has too few edges with very low individual error.",
"The error metric of Equation REF averages over the number of edges of each node $j$ .",
"Another approach would be to work with the degree of the graph instead, by which we mean the sum of the absolute values of the weights of the edges of node $j$ .",
"Thus we define the error with degree by: $err^d = \\sum _{j=1}^N \\frac{1}{\\sum _{i=1}^N |A_{i,j}| } \\frac{1}{K-M}\\sum _{k=M}^K \\left\\Vert x(k) \\mathbb {I}_{\\lbrace A_j \\ne 0 \\rbrace } - A x_j(k-1) \\right\\Vert _2^2$ We simulate a CGP on a stochastic block model following the methodology and parameters in [4], [3], which we describe in Section .",
"On this simulated graph we test our intuition by comparing the evolution of our error metrics as a function of the sparsity parameter $\\lambda _1$ .",
"Figure REF shows the evolution of these metrics as well as the commonly used in- and out-of-sample MSE and the AIC and BIC criteria.",
"On this figure both metrics, $err$ and $err^d$ , perform as expected while the others do not show any relationship to the number of edges in the adjacency matrix except for the BIC criterion.",
"While this graph represents only one sample of a simulated scenario, this behaviour is consistent throughout the different simulations in Section .",
"Interestingly, our two metrics complement each other; on average, for dense graphics $err$ often produces better results while for sparser ones $err^d$ is often better.",
"In practice we can use the following pragmatic approach for robust results: if both metrics have a peak we take the mean value of the two resulting $\\lambda _1$ , if only one has a peak we take its value for $\\lambda _1$ .",
"The Table REF in Appendix compares the performance of these different metrics for different SBMs.",
"It is interesting to observe that the performance of the two error metrics we propose is on par with the BIC criterion and actually complement is.",
"Indeed, we observe the best results when averaging the selected $\\lambda _1$ of $err$ , $err^d$ and $BIC$ .",
"Since the BIC criterion is widely known, for the rest of this paper we will study the performance of the two new error distances $err$ and $err^d$ that we propose knowing that the resulting adjacency matrix would be better if the BIC criterion was included.",
"Figure: Comparison of the evolution of different metrics as a function of the value of the LASSO coefficient λ 1 \\lambda _1 for a simulated CGP-SBM graph with 200 nodes, 5 clusters, 3 lags and 1040 time points.",
"The left axis indicates the number of different edges between the estimate and the true adjacency matrix.",
"The blue line shows the evolution of that difference as a function of the coefficient λ 1 \\lambda _1, while the red line highlights the zero mark.",
"The different error metrics are rescaled to be between 0 and 1 on the right y-axis.",
"We compare the two metrics proposed in this paper, errerr and err d err^d, with the in- and out-of-sample error, MSE in MSE_{in} and MSE out MSE_{out}, as well as the AIC and BIC criteria, AICAIC and BICBIC"
],
[
"Applications",
"We are interested in two performance metrics: the accuracy of the LASSO coefficient selection, assessed by measuring the difference in number of edges between the real and estimated adjacency matrix; and the quality of the CCD algorithm, assessed by measuring the percentage of true positives and false positives.",
"For the computations we fix two parameters of our algorithm: the maximum iterations $maxIt=50$ and the convergence limit $\\epsilon =0.1$ .",
"For the initialisation, we observe that the algorithm performs better when starting from zeros, i.e.",
"$Rh=[0]$ , than from random matrices.",
"When starting with matrices of zeros, due to the block coordinate structure of the algorithm, the first step corresponds to solving the mean square problem of a CGP with only one lag, then the second step considers a CGP with two lags, whereby we learn on the errors left by the first lag, and so on.",
"Thus, each step of the descent complements the previous lags by iteratively refining previous predictions.",
"We use the same causal graph stochastic block model (CGP-SBM) structure to assess the performance of our algorithm as [3], and hence our results are directly comparable.",
"However, they focus on minimising the MSE and choose the LASSO coefficient that produces the best results.",
"Although our approach results in a higher MSE, we have shown in Section that the MSE is not linked to the sparsity of the graph.",
"Thus, our algorithm obtains a more accurate estimate of the adjacency matrix and its approximate sparsity.",
"The SBM consists of a graph with a set of clusters where the probability of a connection is higher within a cluster than between them.",
"This structure is interesting because it appears in a wide variety of real applications.",
"The parameters to simulate a CGP-SBM are the number of nodes, $N$ , the number of clusters, $Nc$ , the number of lags, $M$ and the number of time points, $K$ .",
"For each simulation we use a burn in of 500 points.",
"For a visual assessment of our proposed algorithm's performance we show in Figure REF the absolute value of the adjacency matrix used to simulate the CGP and the estimation we obtain with our algorithm.",
"We note that it is hard to visually detect the discrepancies.",
"Hence we added in appendix Figure REF to show the matrix of the differences between real and estimated adjacency matrices and, Figure REF to show the non-zero elements of each matrix and the matrix of differences with a black square at each non-zero edge.",
"Figure: Estimated adjacency matrix A ^\\hat{A}, on the right; the true one AA, on the left.",
"Absolute values of the weights are shown for better visualisation, with blacker points representing bigger weights.",
"The estimation was performed on a CGP-SBM graph with N=100N=100, Nc=5Nc=5, M=3M=3 and K=1560K=1560.We also wish to quantitatively assess the accuracy of the estimated adjacency matrix.",
"Hence, we measure the quality of the results by considering different metrics: the difference in the number of edges between $\\hat{A}$ and $A$ as a absolute value and as percentage of the total number of possible edges, $N^2$ ; the percentage of true positive, i.e.",
"the number of edges in $\\hat{A}$ that are also edges in $A$ over the total number of edges in $A$ ; the percentage of false positive, i.e.",
"the number of edges in $\\hat{A}$ that are not in $A$ over the total number of edges in $\\hat{A}$ ; the mean squared error $MSE=\\Vert \\hat{A} - A\\Vert _2^2 / N^2$ .",
"The two metrics measuring the difference between the real and the selected number of edges assess the performance of the selection of the sparsity coefficient $\\lambda _1$ .",
"The true and false positive rates assess the performance of the CGP-CCD algorithm REF to compute the adjacency matrix.",
"Table: Differences between the adjacency matrix AA and its estimate A ^\\hat{A} for different CGP-SBM environments.",
"NBDE: absolute difference in the number of edges between A ^\\hat{A} and AA, and as a percentage of the total number of possible edges N 2 N^2 with NBDE (%).",
"True positive: number of edges in A ^\\hat{A} that are edges in AA over the total number of edges in AA.",
"False positive: number of edges in A ^\\hat{A} that are not in AA over the total number of edges in A ^\\hat{A}.",
"Mean squared error: MSE=∥A ^-A∥ 2 2 /N 2 MSE=\\Vert \\hat{A} - A\\Vert _2^2 / N^2.For assessing the consistency of the performance we simulate different environments and compute the median of the measures obtained over 10 samples.",
"In the simulated environments, the sparsity level, i.e.",
"the number of non-zeros elements in the adjacency matrix, varies between $1.4\\%$ and $3.0\\%$ with an average at $2.1\\%$ .",
"Table REF shows that the results are consistent for different graph sizes, sparsity levels, lags and numbers of time points with a median over all the environments of percentage-difference in number of edges of $0.41\\%$ , true positive rate of $65.9\\%$ and false positive rate of $20.8\\%$ .",
"Interestingly, even when the number of time points was too small to obtain accurate results, i.e.",
"high true positive rate, the percentage of different edges stays small, below $0.5\\%$ .",
"The results in Table REF assume we know the number of time lags of the underlying CGP we are looking for.",
"Since this assumption is unlikely to hold on real datasets we tested the reliability of the performances by using wrong input parameters.",
"We therefore simulated a graph with $M=5$ lags and ran the learning algorithm for $M=3$ lags and vice-versa, in both scenarios the average results were approximately unchanged.",
"Section REF further studies the performance of the algorithm on real datasets with unknown parameters."
],
[
" Computation time complexity",
"All the computations employed Python $2.7$ using Numpy and Scipy-sparse libraries, which can use up to 4 threads.",
"The Stochastic Gradient projection (SGP) method used by [4], [3] is especially efficient for highly sparse environments, which can leverage sparse matrix-vector computation.",
"In our environment, when we have a graph with more than $1\\%$ of non-zero weights a matrix-matrix or matrix-vector multiplication using the sparse function of Scipy-sparse is slower than using the dense functions of Numpy.",
"Thus, for graphs with an adjacency matrix with more than $1\\%$ of non-zero edges we use the dense library.",
"For example, on a CGP-SBM graph with parameters $(N, Nc, M, K) = (200, 5, 3, 2080)$ with $2,4\\%$ of non-zeros edges, our CCD algorithm solves the optimisation of Equation REF to obtain the matrix $R_1$ faster than the SGP algorithm by more than 100-fold.",
"When the graph size increases to $N=500$ , CCD is faster than by more than 350-fold.",
"We perform an empirical time complexity estimation of the complete block coordinate descent algorithm by measuring the evolution of the execution time as a function of each parameter $(N, Nc, M, K)$ individually.",
"Increasing the number of time lags $M$ has a negligible effect on the execution time of the CCD to obtain the adjacency matrix, although the computation of the polynomial coefficients $C$ scales quadratically with the lags $M$ .",
"We observe that the execution time is not affected by the sparsity level $Nc$ .",
"However, it scales linearly as function of the number of time points $K$ and quadratically as a function of the number of nodes $N$ .",
"This quadratic complexity in $N$ can be tempered in different ways however.",
"In the case of highly sparse graphs we can leverage sparse libraries, and an even faster solution for both dense and sparse graph is to perform the computations on a GPU.",
"Indeed, Algorithm REF does not use much memory and can thus be computed entirely in the GPU memory.",
"With a GPU implementation using the library PyTorch, the algorithm has a 20-fold speed-up compared to our CPU implementation with matrix computations parallelised over 4 threads."
],
[
"An application to financial time series",
"We now apply our algorithm to a real dataset of stock prices consisting of the 371 stocks from the S&P500 that have quotes between $2000/01/03$ and $2018/03/27$ .",
"Since we do not know the exact adjacency matrix of this environment, we test the accuracy of the obtained graph by studying how it changed following a known market shock.",
"More specifically we compute two graphs, one before and one after the financial crisis of $2008/2009$ .",
"For both graphs to use the same number of time points, the first uses prices from $2004/11/15$ to $2009/01/01$ and the second from $2009/11/12$ to $2014/01/01$ .",
"We chose to build the graphs with a 4 year time window, $K=1040$ , since in simulations with the same number of nodes it produces good results.",
"The lag was fixed to one week, $M=5$ , since there are documented trading patterns at a weekly frequency.",
"We note that in the time windows studied modifying the lags to $M=3$ or $M=10$ has negligible impact on the results.",
"For both time windows the error metrics peaked at slightly different values, thus we took the mean of the two for estimating the LASSO coefficient.",
"Interestingly, the algorithm selects a much sparser matrix after the crisis with a sparsity level decreasing from $5.1\\%$ to $2.8\\%$ .",
"This points to a more inter-connected market leading up to the crisis.",
"Since the crisis was due to sub-prime issues, one might expect real-estate and financial firms to have many edges in the graph and influence the market in the pre-crisis period, with the importance of these firms decreasing after the crash.",
"Indeed, this aspect is reflected in the estimated adjacency matrix; before the crisis, financial firms represent more than $60\\%$ of the top ten nodes with the highest number of connections, while it decreases to less than $40\\%$ afterwards, including insurance firms.",
"Furthermore, before the crisis the firm with the highest number of connection was GGP Inc., a real estate firm which went on to file for bankruptcy in 2009.",
"While financial and oil firms represented more than $70\\%$ of the top 20 most connected nodes before the crisis, the graph of 2014 is much more diversified with more sectors in the top 20 and none representing more than $30\\%$ .",
"Figure REF in the appendix shows the evolution of the adjacency matrix before and after the crisis, and we can see the shift in importance and the increase in sparsity.",
"Overall, the post-crisis market is sparser and less concentrated than before 2009, with fewer edges linked to financial firms.",
"Since the CGP-CCD algorithm automatically selects the sparsity level, in addition to studying the different connections we can also study the evolution of the sparsity level.",
"Figure REF shows the evolution through time of the sparsity level of the adjacency matrix and of the log realised variance, $log(RV)$ , of the market.",
"We computed the adjacency matrix every 6 months and the corresponding $log(RV)$ at the last date of that time window.",
"We can observe an interesting correlation between the increase in density of the causal graph and the increase in the realised variance."
],
[
"Conclusion",
"We have proposed a novel cyclical coordinate descent algorithm to efficiently infer the directed adjacency matrix and polynomial coefficients of a causal graph process.",
"Compared to the previous state-of-the-art our solution has lower complexity and does not depend on the sparsity level of the graph for scalability.",
"Furthermore, we propose two new error metrics to automatically select the coefficient of the LASSO constraint.",
"Our solution is able to recover approximately the correct number of edges in the directed adjacency matrix of the CGP.",
"The performance of our algorithm is consistent across the different simulated stochastic block model graphs we tested.",
"In addition, we provided an example application to a real-world dataset consisting of stocks from the S&P500, demonstrating results that are in line with economic theory."
],
[
"For $i>1$",
"Recall the Utility function for $i>1$ detailed in Equation REF : $\\mathcal {L}(R_i) = \\frac{1}{2} \\sum _{k=M}^{K-1} \\left( S_k^i - R_i x(k-i) \\right)^T \\left( S_k^i - R_i x(k-i) \\right) \\;,$ With $S_k^i = x(k) - \\sum _{l \\ne i}^M R_l x(k-l)$ .",
"Then, the derivative of this function with respect to the matrix $R_i$ is: $\\frac{\\partial \\mathcal {L}(R_i)}{\\partial R_i} = R_i \\sum _{k=M}^{K-1} x(k-i) x(k-i)^T - \\sum _{k=M}^{K-1} S_k x(k-i)^T \\;.$ This derivative is equal to zero if $R_i$ follows the Equation REF : $R_i = \\left(\\sum _{k=M}^{K-1} S_k^i x(k-i)^T \\right) \\left(\\sum _{k=M}^{K-1} x(k-i) x(k-i)^T \\right)^{-1} \\; .$ However, the matrix denominator $\\sum _{k=M}^{K-1} x(k-i) x(k-i)^T$ is not guaranteed to be positive semi-definite.",
"In case of singularity we can add an $L_2$ regularisation term, $\\lambda _2 \\Vert R_i \\Vert _2^2$ , to the utility function of Equation REF .",
"With this coefficient the updating equation of matrix $R_i$ becomes: $R_i = \\left(\\sum _{k=M}^{K-1} S_k^i x(k-i)^T \\right) \\left(\\sum _{k=M}^{K-1} x(k-i) x(k-i)^T + 2 \\lambda _2 \\mathbb {I}_N \\right)^{-1} \\; .$ With $\\mathbb {I}_N$ the identity matrix of size $N$ .",
"Thus, with this coefficient we add noise to the diagonal of the matrix to make it non-singular.",
"In practice, Since this coefficient adds a bias, we select the lowest value of the coefficient $\\lambda _2$ that makes this matrix non-singular."
],
[
"For $i=1$",
"In the case of $i=1$ we apply the CCD algorithm by iterating over the columns, let us recall the Lagrangian detailed in Equation REF : $\\mathcal {L}(R_1^j) = \\frac{1}{2} \\sum _{k=M}^{K-1} \\left\\Vert S_k^1 - R_1^{-j} x^{-j}(k-1) - R_1^j x^j(k-1) \\right\\Vert _2^2 + \\lambda _1 |R_1^j | \\;.$ which is a non-differentiable function due to the LASSO regularisation term.",
"Thus we compute the sub-gradient of the Lagrangian with respect to the vector of the column $R_1^j$ : $\\frac{\\partial \\mathcal {L}(R_1^j)}{\\partial R_1^j} = R_1^j \\sum _{k=M}^{K-1} \\left(x^j(k-1)\\right)^2 - \\sum _{k=M}^{K-1} \\left[S_k^1 - R_1^{-j} x^{-j}(k-1) \\right] x^j(k-1)^T + \\lambda _1 \\Gamma _j \\;.$ Where $\\Gamma _j(i)=sign(R_1^j(i))$ if $R_1^j(i) \\ne 0$ , else $\\Gamma _j(i) \\in [-1, 1]$ if $R_1^j(i) = 0$ .",
"Which is why we introduce the soft-thresholding function $S(a,b) = sign(a) ( |a| - b)_+$ , where $sign(a)$ is the sign of $a$ and $(y)_+ = max(0,y)$ .",
"Hence, the derivative is equal to zero if $R_1^j$ follows Equation REF : $R_1^j = \\frac{S \\left( \\sum _{k=M}^{K-1}\\left(S_k^1 - R_1^{-j} x^{-j}(k-1) \\right) x^j(k-1), \\lambda _1 \\right) }{ \\sum _{k=M}^{K-1} (x^j(k-1))^2} \\;.$"
],
[
"Figures & Tables",
"In Table REF , in order to compare the different distance measure to select the coefficient $\\lambda _1$ we ran the computations for a simulated CGP-SBM with the same performance metrics as in Table 1.",
"We used a grid of $\\lambda _1 \\in [30, 300]$ with a step of 5; in this window the AIC and $MSE_{in}$ did not converge, hence we report the result obtained for the smallest value $\\lambda _1=30$ .",
"While on a small graph, with 200 nodes, the BIC has results on par with our distance measures $err$ & $err^d$ , for a larger problem with 500 nodes the difference in performance becomes more significant.",
"In these experiments, the BIC criteria is smoother than $err$ & $err^d$ but often overestimates the number of connections compared to $err$ & $err^d$ .",
"Thus, the average of those three metrics $err$ & $err^d$ & BIC gives results for the NBDE with lower variance than the others, for a small decrease in accuracy compare to $err$ & $err^d$ alone.",
"Table: Differences between AA and its estimate A ^\\hat{A} on CGP-SBM graph for different distance metrics.Figure: This figure compares the selected edges of the estimated adjacency matrix A ^\\hat{A}, in the middle, to the true one AA, on the left.",
"The right matrix correspond to the difference these two matrices.",
"Each black square corresponds to an edge with non-zero weight.",
"The estimation was performed on a CGP-SBM graph with N=100N=100, Nc=5Nc=5, M=3M=3 and K=1300K=1300.Figure: This figure compares the estimated adjacency matrix A ^\\hat{A}, on the right, to the true one AA, on the left.",
"The right matrix correspond to the difference these two matrices.",
"To obtain a better visualisation the plots represent the absolute values of the weights, hence the blacker the bigger the weight.",
"The estimation was performed on a CGP-SBM graph with N=100N=100, Nc=5Nc=5, M=3M=3 and K=1300K=1300.Figure: This figures shows the obtained adjacency matrix for a weekly lag M=5M=5 on 467 European stocks computed using data between 06/11/200306/11/2003 and 01/01/200801/01/2008 on the left, and 03/11/200903/11/2009 to 01/01/201401/01/2014 on the right.",
"The stocks are grouped by financial sectors using the BCIS (Bloomberg Industry Classification Standard) sector classification.",
"For a better visualisation we do not show the weight of the edge but instead a black square for every non-zero edge.",
"The sparsity level was selected by taking the average λ 1 \\lambda _1 value obtained with the error metrics errerr and err d err^d.Figure: For 371 stocks from the S&P500, evolution of the sparsity level of the adjacency matrix,left y-axis, and the log(RV)log(RV) on the right y-axis.",
"The coefficient λ 1 \\lambda _1 value obtained with the error metrics errerr and err d err^d.",
"We define the sparsity level by the percentage of non-zeros edges, hence 100.0100.0 corresponds to a fully connected graph.",
"We estimate the variance of the market by the average of the log-Realised-Variance, log(RV)log(RV), of each stock.",
"We computed the log(RV)log(RV) with exponential weighting on the daily log-returns, using a coefficient of 0.990.99 and time window of 40 days."
]
] | 1906.04479 |
[
[
"Bounds on Scott Ranks of Some Polish Metric Spaces"
],
[
"Abstract If $\\mathcal{N}$ is a proper Polish metric space and $\\mathcal{M}$ is any countable dense submetric space of $\\mathcal{N}$, then the Scott rank of $\\mathcal{N}$ in the natural first order language of metric spaces is countable and in fact at most $\\omega_1^{\\mathcal{M}} + 1$, where $\\omega_1^{\\mathcal{M}}$ is the Church-Kleene ordinal of $\\mathcal{M}$ (construed as a subset of $\\omega$) which is the least ordinal with no presentation on $\\omega$ computable from $\\mathcal{M}$.",
"If $\\mathcal{N}$ is a rigid Polish metric space and $\\mathcal{M}$ is any countable dense submetric space, then the Scott rank of $\\mathcal{N}$ is countable and in fact less than $\\omega_1^{\\mathcal{M}}$."
],
[
"Introduction",
"A common task in mathematics is to distinguish different mathematical structures subjected to the restriction of various first order languages.",
"The Scott analysis is a general model theoretic concept that attempts to find an ${L}$ -isomorphism invariant of an ${L}$ -structure $\\mathcal {M}$ , where ${L}$ is a first order language.",
"Informally, if two tuples $\\bar{a}$ and $\\bar{b}$ of $\\mathcal {M}$ of the same length can be distinguished from each other by an infinitary ${L}$ -formula, the Scott analysis would attempt to assign an ordinal that indicates how difficult it is to distinguish these tuples.",
"The Scott rank of tuples can be defined by the back-and-forth relations (see Definition REF ): Let $\\bar{a} = (a_0,...,a_{p - 1})$ and $\\bar{b} = (b_0,...,b_{p - 1})$ be tuples of length $p$ from an ${L}$ -structure $\\mathcal {M}$ .",
"One says that $\\bar{a} \\sim _0 \\bar{b}$ if and only if the map taking $a_i$ to $b_i$ for $i < p$ is a partial ${L}$ -isomorphism of $\\mathcal {M}$ into $\\mathcal {M}$ .",
"Assume $\\sim _\\alpha $ has been defined, one says that $\\bar{a} \\sim _{\\alpha + 1} \\bar{b}$ if and only if for all $c \\in \\mathcal {M}$ , one can always find a $d \\in \\mathcal {M}$ so that the elongated tuples satisfy the relation $\\bar{a}c \\sim _\\alpha \\bar{b}d$ and similarly in the other direction with the role of $\\bar{a}$ and $\\bar{b}$ reversed.",
"Assume that $\\sim _\\beta $ has been defined for all $\\beta < \\alpha $ , then one defines $\\bar{a} \\sim _\\alpha \\bar{b}$ if and only if for all $\\beta < \\alpha $ , $\\bar{a} \\sim _\\beta \\bar{b}$ .",
"If there is an $\\alpha $ so that $\\lnot (\\bar{a} \\sim _\\alpha \\bar{b})$ , then using the wellfoundedness of the class of ordinals, $\\mathrm {SR}(\\bar{a},\\bar{b})$ is defined to be the minimal such ordinal.",
"Otherwise, one will say $\\mathrm {SR}(\\bar{a},\\bar{b}) = \\infty $ .",
"Intuitively, $\\mathrm {SR}(\\bar{a},\\bar{b}) = \\infty $ indicates that the two tuples are indistinguishable by an infinitary ${L}$ -formula.",
"If $\\mathrm {SR}(\\bar{a},\\bar{b}) \\ne \\infty $ , then $\\mathrm {SR}(\\bar{a},\\bar{b})$ is an ordinal measuring how difficult it is to distinguish these two tuples.",
"For instance, $\\mathrm {SR}(\\bar{a},\\bar{b}) = 0$ means $\\lnot (\\bar{a} \\sim _0 \\bar{b})$ .",
"Thus there is an atomic formula that evaluates differently between $\\bar{a}$ and $\\bar{b}$ .",
"$\\mathrm {SR}(\\bar{a},\\bar{b}) = 1$ would mean that atomic formulas can not distinguish $\\bar{a}$ and $\\bar{b}$ , but there is a formula consisting of an existential quantifier over an atomic formula that evaluates differently between $\\bar{a}$ and $\\bar{b}$ .",
"In this way, the Scott ranks of tuples are closely relate to the ranks of an infinitary ${L}$ -formulas that can be used to distinguish tuples.",
"By taking supremum of all possible pairs of tuples of the same length (varying over all possible lengths), one obtains an ordinal for the entire structure $\\mathcal {M}$ , called the Scott rank of $\\mathcal {M}$ .",
"Another useful perspective on the Scott rank of tuples $\\bar{a}$ and $\\bar{b}$ is through a two player game called the Ehrenfeucht-Fraïsse game $\\mathrm {EF}_\\alpha ^{\\mathcal {M},\\bar{a},\\bar{b}}$ , where $\\alpha $ is an ordinal.",
"Player 1 at each turn plays a pair $(\\beta ,x)$ where $\\beta < \\alpha $ is less than any previous ordinals Player 1 has played and $x$ is a element of $\\mathcal {M}$ chosen to elongate the $\\bar{a}$ -side or the $\\bar{b}$ -side.",
"Player 2 then must choose $y \\in \\mathcal {M}$ to elongate the side opposite which Player 1 has chosen.",
"By the wellfoundedness of the class of ordinals, Player 1 must eventually play the ordinal 0.",
"After Player 2 responds, the game ends.",
"One says that Player 2 wins this game if and only if the mapping $\\bar{a}$ to $\\bar{b}$ and the sequence of responses in the game form a partial ${L}$ -isomorphism.",
"Intuitively, Player 1 winning $\\mathrm {EF}_{\\alpha }^{\\mathcal {M},\\bar{a},\\bar{b}}$ indicates that with $\\alpha $ -degree of flexibility, Player 1 can compel Player 2 to make a move that violates the ${L}$ -structure of $\\mathcal {M}$ .",
"There is a close relationship between Player 2 having a winning strategy in $\\mathrm {EF}_\\alpha ^{\\mathcal {M},\\bar{a},\\bar{b}}$ and the back-and-forth relation $\\sim _\\alpha $ .",
"By a cardinality consideration, the Scott rank of a tuple in $\\mathcal {M}$ is less than $|\\mathcal {M}|^+$ , the cardinal successor of $|\\mathcal {M}|$ .",
"Thus the Scott rank of $\\mathcal {M}$ is less than $|\\mathcal {M}|^+$ .",
"In particular, if $\\mathcal {M}$ is countable, then $\\mathrm {SR}(\\mathcal {M}) < \\omega _1$ , the first uncountable ordinal.",
"Moreover, Nadel showed there is a close relationship between the definability of $\\mathcal {M}$ and the bound on its Scott rank.",
"Nadel () showed that $\\mathrm {SR}(\\mathcal {M}) \\le \\omega _1^{\\mathcal {M}} + 1$ when $\\mathcal {M}$ is considered as a subset of $\\omega $ .",
"Here $\\omega _1^\\mathcal {M}$ is the Church-Kleene ordinal relative to $\\mathcal {M}$ which is the least ordinal $\\alpha $ which does not have a wellordering coded as a subset of $\\omega $ of ordertype $\\alpha $ which is (Turing) computable from $\\mathcal {M}$ .",
"It is also the minimal ordinal height of an admissible set containing $\\mathcal {M}$ .",
"An admissible set containing $\\mathcal {M}$ is simply a transitive set containing $\\mathcal {M}$ satisfying a weak set theory axiom system called Kripke-Platek $(\\mathsf {KP})$ set theory.",
"The reader can consider an admissible set containing $\\mathcal {M}$ as essentially a miniature universe of set theory containing $\\mathcal {M}$ .",
"Let $\\mathcal {M}$ be a countable ${L}$ -structure.",
"The back-and-forth process described above can be used to define a countable infinitary ${L}$ -formula $\\psi _\\mathcal {M}$ so that for any countable ${L}$ -structure $\\mathcal {N}$ , $\\mathcal {N}\\models \\psi _\\mathcal {M}$ if and only if $\\mathcal {M}$ and $\\mathcal {N}$ are ${L}$ -isomorphic.",
"The rank of the sentence $\\psi _\\mathcal {M}$ is closely related to the Scott rank of $\\mathcal {M}$ and $\\psi _\\mathcal {M}$ is roughly the conjunction of all the associated distinguishing formulas for all possible pairs of tuples.",
"For more on the classical and effective Scott analysis for countable structures, see , , and [1].",
"A particular instance of the above is the study of isometries of metric spaces.",
"The natural first order language ${U}$ for metric structures consists of two binary relation symbols for each positive rational $q$ whose intended iterpretations are whether two points have distance less than or more than $q$ .",
"(See Definition REF .)",
"A general metric space can have arbitrarily large cardinality and Scott rank.",
"The collection of Polish metric spaces form a very interesting class of metric spaces.",
"Polish metric spaces are complete separable metric spacces.",
"An uncountable Polish metric space $\\mathcal {N}$ must have cardinality $2^{\\aleph _0}$ .",
"A priori, one has $\\mathrm {SR}(\\mathcal {N}) < (2^{\\aleph _0})^+$ .",
"There is some hope of doing better.",
"If $\\mathcal {N}$ is a Polish metric space, there is a countable submetric space $\\mathcal {M}\\subseteq \\mathcal {N}$ whose completion is $\\mathcal {N}$ .",
"By the general theory of Scott analysis mentioned above, $\\mathrm {SR}(\\mathcal {M}) < \\omega _1$ and in fact $\\mathrm {SR}(\\mathcal {M}) < \\omega _1^{\\mathcal {M}} + 1$ since $\\mathcal {M}$ is a countable structure.",
"In some sense, $\\mathcal {M}$ has full metric information of it own completion $\\mathcal {N}$ .",
"A very natural question asked by Fokina, Friedman, Koerwien, and Nies is whether $\\mathcal {M}$ captures the first order metric structure of it own completion $\\mathcal {N}$ well enough to imply that $\\mathrm {SR}(\\mathcal {N})$ is countable.",
"If so, the author asks whether the Polish metric space $\\mathcal {N}$ with countable dense submetric space $\\mathcal {M}$ satisfies the natural analog of Nadel's effective bounding result for countable structures.",
"Question 1.1 (Fokina, Friedman, Koerwien, Nies) Let $\\mathcal {N}$ be a Polish metric space.",
"Is $\\mathrm {SR}(\\mathcal {N}) < \\omega _1$ ?",
"(Chan) If $\\mathcal {N}$ is a Polish metric space and $\\mathcal {M}$ is a countable dense submetric space of $\\mathcal {N}$ , then is $\\mathrm {SR}(\\mathcal {N}) \\le \\omega _1^\\mathcal {M}+ 1$ ?",
"It appears that both questions are still open.",
"(See [7] and [8].)",
"There are partial answers to these questions.",
"Fokina, Friedman, Koerwien, and Nies showed using some results of Gromov that if $\\mathcal {N}$ is a compact Polish metric space then $\\mathrm {SR}(\\mathcal {N}) \\le \\omega + 1$ .",
"(See also Theorem REF .)",
"Doucha [7] showed that although the cardinality of a Polish metric space $\\mathcal {N}$ is $2^{\\aleph _0}$ , $\\mathrm {SR}(\\mathcal {N})$ is less than or equal to $\\omega _1$ , the first uncountable ordinal.",
"Thus Question REF is reduced to whether it is possible that there is a Polish metric space $\\mathcal {N}$ with $\\mathrm {SR}(\\mathcal {N}) = \\omega _1$ .",
"The goal of this paper is to extend a positively answer to Question REF for larger classes of Polish metric spaces by producing effective countable bounds on Scott rank.",
"This paper will pursue this in the direction of admissibility theory and the Barwise and Jensen theory of infinitary logic in countable admissible fragments.",
"The advantage of this approach is that one produce not only some desired objects but also an entire miniature universe (of a weak set theory $\\mathsf {KP}$ ) containing these objects.",
"One can then perform a variety of arguments internally and externally of this model of $\\mathsf {KP}$ and attempt to reflect internal phenomenon to the real world by absoluteness.",
"This approach gives additional insight on the relation between the Scott rank of the Polish metric space $\\mathcal {N}$ and the definability complexity of any of its countable dense submetric space $\\mathcal {M}$ .",
"It also seems to have the benefit of simplifying some technical arguments since the miniature universe of $\\mathsf {KP}$ set theory can absorb some combinatorics.",
"Section provides the basic definitions.",
"The first order language ${U}$ of metric spaces, the back-and-forth relations, the Ehrenfeucht-Fraïssé game, and the notion of Scott ranks of tuples and structures will be defined.",
"Section will give a proof of the following result of Fokina Friedman, Koerwien, and Nies : Theorem REF .",
"(Fokina, Friedman, Koerwien, Nies) If $\\mathcal {M}$ is a compact Polish metric space, then $\\mathrm {SR}(M) \\le \\omega + 1$ .",
"This result serves as a warmup for the later theorems in the paper.",
"It contains the approximation idea but is simpler than the subsequent theorems since it involve only playing a single game and there are no admissible sets or ordinals of illfounded models of $\\mathsf {KP}$ which are externally illfounded.",
"Nies has mentioned to the author that they had originally proved this result using some theory develop by Gromov.",
"The main combinatorial tool for the proof in this paper is to use the König lemma to produce a compact approximation system (see Definition REF ).",
"The König lemma is the statement that every finitely branching tree has an infinite path.",
"This is a natural combinatorial principle to apply in this setting since the König lemma is equivalent to the compactness of a certain closed subsets of ${{}^\\omega \\omega }$ , in it usual topology.",
"(For instance, the weak König lemma is equivalent to the compactness of the Cantor space, ${{}^\\omega 2}$ .)",
"A countable metric space $\\mathcal {M}$ along with all its distance relations can be identified with a subset of $\\omega $ .",
"Let $\\mathcal {C}(\\mathcal {M})$ denote the metric completion of $\\mathcal {M}$ .",
"Note that the elements of the completion of $\\mathcal {M}$ are represented by $\\mathcal {M}$ -Cauchy sequence which are essentially reals, i.e.",
"elements of ${{}^\\omega \\omega }$ .",
"The Ehrenfeucht-Fraïssé game on $\\mathcal {C}(\\mathcal {M})$ requires Player 2 to give perfect responses in the sense that partial isometries need to be produced.",
"Even if Player 1 plays elements of $\\mathcal {M}$ in the Ehrenfeucht-Fraïssé game on $\\mathcal {C}(\\mathcal {M})$ , Player 2 may need to respond with an element of $\\mathcal {C}(\\mathcal {M}) \\setminus \\mathcal {M}$ to maintain the isometry.",
"However allowing the move to be $\\mathcal {M}$ -Cauchy sequences makes the game no longer an integer game.",
"This game can not be absorbed into any countable admissible set.",
"To resolve this, Section defines a new approximation games $G_\\alpha ^{f,\\bar{a},\\bar{b}}$ (see Definition REF ), where all the moves are ordinals and elements of $\\mathcal {M}$ .",
"Instead of playing perfect responses, Player 2 only needs to produce responses whose errors are no more than that prescribed by some $f : \\omega \\rightarrow \\mathbb {Q}^+$ which is a computable function that is strictly decreasing and converging to 0.",
"Using this game, a new rank $R(\\bar{a},\\bar{b})$ for pairs of tuples $(\\bar{a},\\bar{b})$ will be defined.",
"It will be shown that $\\mathrm {SR}(\\bar{a},\\bar{b}) \\le R(\\bar{a},\\bar{b})$ .",
"Thus bounding $R(\\bar{a},\\bar{b})$ will suffice to give a bound on $\\mathrm {SR}(\\bar{a},\\bar{b})$ .",
"A metric space is said to be proper if and only if every closed ball is compact.",
"It will be shown that in proper Polish metric spaces, if $(\\bar{a},\\bar{b})$ is a limit of a sequence of points $\\langle (\\bar{a}_n,\\bar{b}_n) : n \\in \\omega \\rangle $ so that for all $n \\in \\omega $ , $R(\\bar{a}_n,\\bar{b}_n) > \\alpha $ , then $R(\\bar{a},\\bar{b}) > \\alpha $ .",
"The proof of this result requires playing countably infinite many games simultaneously and thinning out to countably infinite many games at each subsequent stage.",
"In contrast, the main argument of [7] involves $\\omega _1$ -many simultaneous games and requires a thinning to uncountable nonstationary subsets of $\\omega _1$ at subsequent stages.",
"Section reviews the basics of admissibility and the theory of infinitary logic in countable admissible fragment including the Jensen's model existence theorem and Barwise compactness.",
"The main technical simplification comes from Fact REF which states that if there is an illfounded model $\\mathcal {A}$ containing $\\mathcal {M}$ and two pairs of tuples of $\\mathcal {M}$ -Cauchy sequences, $\\bar{a}$ and $\\bar{b}$ , so that $\\mathcal {A}$ thinks that $R(\\bar{a},\\bar{b})$ is an $\\mathcal {A}$ -ordinal which externally $V$ thinks is $\\in ^\\mathcal {A}$ -illfounded, then one can find (in $V$ ) an autoisometry of the completion $\\mathcal {C}(\\mathcal {M})$ taking $\\bar{a}$ to $\\bar{b}$ .",
"This is proved by taking Player 2's winning strategy in $\\mathcal {A}$ for the game associated to the ordinal which is externally illfounded and using it to play forever externally in $V$ to produce an autoisometry.",
"Using this result and an application of Jensen's model existence theorem, one can establish Fact REF which asserts that for any pair of tuples $(\\bar{a},\\bar{b})$ of $\\mathcal {M}$ -Cauchy sequences, $\\mathrm {SR}(\\bar{a},\\bar{b}) \\le R(\\bar{a},\\bar{b}) < \\omega _1^{\\mathcal {M}\\oplus \\bar{a}\\oplus \\bar{b}}$ .",
"This also gives Doucha's result that $\\mathrm {SR}(\\mathcal {C}(\\mathcal {M})) \\le \\omega _1$ .",
"Section contains the two main theorems of the paper.",
"A metric space $\\mathcal {N}$ is rigid if and only if there are no nontrivial autoisometry of $\\mathcal {N}$ .",
"Theorem REF .",
"If $\\mathcal {M}$ is a countable metric space so that $\\mathcal {C}(\\mathcal {M})$ is a rigid metric space, then $\\mathrm {SR}(\\mathcal {C}(\\mathcal {M})) < \\omega _1^\\mathcal {M}$ .",
"Usually, in applications of the Jensen's model existence theorem, one can establish the consistency of the relevant theory of an appropriate countable admissible fragment by simply using the real universe as a model.",
"For this theorem, one does not a priori know such an object exists in the real world and so one must establish the consistency of the relevant theory by using Barwise compactness.",
"If one assumes that the completion of a countable metric space is proper, one can prove the Scott rank of the completion is countable and has the analog of Nadel's effective bound: Theorem REF .",
"Let $\\mathcal {M}$ be a metric space on $\\omega $ .",
"Suppose $\\mathcal {C}(\\mathcal {M})$ is a proper Polish metric space.",
"Then $\\mathrm {SR}(\\mathcal {M}) \\le \\omega _1^{\\mathcal {M}} + 1$ .",
"These two theorems extend a positive answer to Question REF (even the effective form) for the class of rigid Polish metric spaces and proper Polish metric spaces.",
"Since an early draft of this paper, Nies and Turetsky () have produced proofs and expanded some of the results here using recursion theoretic methods.",
"The techniques used here to analyze the first order Scott analysis of Polish metric spaces differ in flavor considerably from the classical and effective Scott analysis of countable structures of a countable first order language.",
"The usual technique for finding bounds on Scott ranks for countable structures essentially involves looking at the closure ordinal of an appropriate monotone operator on the countable structure.",
"(See the introduction of [6] for some more details.)",
"[4] developed the Scott analysis for continuous logic for metric structures.",
"(See [4], [3] and [6] for the notation and more information.)",
"[6] showed that if ${L}$ is a recursive language of continuous logic, $\\Omega $ is a weak modulus of continuity with recursive code, $\\mathcal {D}$ is a countable ${L}$ -pre-structure, and $\\bar{\\mathcal {D}}$ is its completion ${L}$ -structure, then $\\mathrm {SR}_\\Omega (\\bar{D}) \\le \\omega _1^{\\mathcal {D}}$ .",
"(The definition of Scott rank in [4] and [6] is slightly different than the definition used in this paper resulting in a bound that differs by 1.",
"See the introduction in [6] for a brief explanation.)",
"[6] proves an effective bound on the continuous Scott rank depending on the countable dense substructure which is analogous to the effective bound in the classical Scott analysis for countable structure.",
"Moreover in [6], the bound is obtained as a closure ordinal of a certain monotone operator on the countable dense substructure which is positive $\\Sigma $ -definable in an appriopriate admissible set; much like the classical case for countable structures.",
"This may suggest that the metric Scott analysis in continuous logic is the correct and fruitful way to generalize the Scott analysis to Polish metric spaces.",
"The author would like to acknowledge Alexander Kechris and André Nies for comments on earlier drafts of this paper."
],
[
"Basics",
"Definition 2.1 The language of a metric space, denoted ${U}$ , is the following: ${U}= \\lbrace \\dot{d}_q, \\dot{d}^q : q \\in \\mathbb {Q}^+\\rbrace $ , where for each $q \\in \\mathbb {Q}^+$ , $\\dot{d}_q$ and $\\dot{d}^q$ are binary relation symbols.",
"If $\\mathcal {M}= (M,d)$ is a metric space on the set $M$ with distance function $d$ , then $\\mathcal {M}$ is given the canonical ${U}$ -structure by defining $(\\dot{d}_q)^\\mathcal {M}(x,y) \\Leftrightarrow d(x,y) < q$ and $(\\dot{d}^q)^\\mathcal {M}(x,y) \\Leftrightarrow d(x,y) > q$ .",
"Fact 2.2 Let $\\mathcal {M}$ and $\\mathcal {N}$ be two ${U}$ -structures which are metric spaces.",
"There is a bijective isometry between $\\mathcal {M}$ and $\\mathcal {N}$ if and only if there is a ${U}$ -isomorphism between $\\mathcal {M}$ and $\\mathcal {N}$ .",
"Definition 2.3 Let ${L}$ be any countable first order language.",
"Let $\\mathcal {M}$ be a ${L}$ -structure.",
"For each ordinal $\\alpha $ , the relation $\\sim _\\alpha $ is defined on tuples from $M$ of the same length as follows: Let $\\bar{a} = (a_0, ..., a_{p -1})$ and $\\bar{b} = (b_0, ..., b_{p - 1})$ , where $p \\in \\omega $ .",
"$\\bar{a} \\sim _0 \\bar{b}$ if and only if the map sending $a_i$ to $b_i$ for all $i < p$ is a partial ${L}$ -isomorphism.",
"$\\bar{a} \\sim _{\\alpha + 1} \\bar{b}$ if and only if for all $a \\in M$ , there exists a $b \\in M$ so that $\\bar{a}a \\sim _\\alpha \\bar{b}b$ and for all $b \\in M$ , there exists an $a \\in M$ so that $\\bar{a}a \\sim _\\alpha \\bar{b}b$ .",
"If $\\beta $ is a limit ordinals, then $\\bar{a} \\sim _\\beta \\bar{b}$ if and only if for all $\\alpha < \\beta $ , $\\bar{a} \\sim _\\alpha \\bar{b}$ .",
"Define $\\mathrm {SR}(\\bar{a},\\bar{b}) = \\min \\lbrace \\mu \\in \\text{ON} : \\lnot (\\bar{a} \\sim _\\mu \\bar{b})\\rbrace $ if this set is nonempty.",
"Otherwise $\\mathrm {SR}(a,b) = \\infty $ .",
"Define $\\mathrm {SR}(\\bar{a}) = \\sup \\lbrace \\mathrm {SR}(\\bar{a},\\bar{b}) : \\bar{b} \\in {}^{|\\bar{a}|} M \\wedge \\mathrm {SR}(\\bar{a},\\bar{b}) \\ne \\infty \\rbrace $ .",
"Finally, $\\mathrm {SR}(\\mathcal {M}) = \\sup \\lbrace \\mathrm {SR}(\\bar{a}) + 1 : \\bar{a} \\in {}^{<\\omega }M\\rbrace $ .",
"Definition 2.4 Let ${L}$ be some countable first order language.",
"Let $\\mathcal {M}$ be a ${L}$ -structure.",
"For some $p \\in \\omega $ , let $\\bar{a} = (a_0,...,a_{p - 1})$ and $\\bar{b} = (b_0,...,b_{p - 1})$ be tuples from $M$ .",
"Let $\\alpha $ be an ordinal.",
"The Ehrenfeucht-Fraïssé game $\\mathrm {EF}^{\\mathcal {M}, \\bar{a},\\bar{b}}_\\alpha $ is defined as follows: If $\\alpha = 0$ , then Player 2 wins if and only if the map $a_i \\mapsto b_i$ for each $i < p$ is a partial ${L}$ -isomorphism.",
"If $\\alpha > 0$ , then Player 1 and 2 play the following: $\\begin{array}{c | c c c c c c c c }\\bar{a} & (\\alpha _0, \\Gamma _0 = c_0) & {} & {} (\\alpha _1, \\Gamma _1 = c_1) & {} & ... & (\\alpha _{k - 1}, \\Gamma _{k - 1} = d_{k - 1}) & {} \\\\\\hline \\bar{b} & {} & \\Lambda _0 = d_0 & {} & {} \\Lambda _1 = d_1 & ... & {} & \\Lambda _{k - 1} = d_{k - 1}\\end{array}$ $\\Gamma $ and $\\Lambda $ are formally either the symbol $c$ or $d$ .",
"If Player 1 lets $\\Gamma $ be $c$ , then Player 2 must let the next $\\Lambda $ be $d$ .",
"If $\\Gamma $ is $d$ , then $\\Lambda $ must be $c$ .",
"Each $\\Gamma _i$ and $\\Lambda _i$ are elements of $M$ .",
"$\\alpha _0 < \\alpha $ and for all $i < k - 1$ , $\\alpha _{i + 1} < \\alpha _i$ .",
"The game ends when Player 1 plays $\\alpha _{k - 1} = 0$ and $\\Gamma _{k - 1}$ and Player 2 responds with $\\Lambda _{k - 1}$ .",
"When the games ends, a sequence $c_0, ..., c_{k - 1}$ and a sequence $d_0, ..., d_{k - 1}$ have been produced.",
"(The sole purpose of the $\\Gamma $ and $\\Lambda $ notation is to indicate whether player 1 played $c_i$ (left side associated with $\\bar{a}$ ) or $d_i$ (right side associated with $\\bar{b}$ ) and similarly for Player 2.)",
"Player 2 wins if and only if the map $a_i \\mapsto b_i$ for $i < p$ and $c_i \\mapsto d_i$ for $i < k$ is a partial ${L}$ -isomorphism.",
"The above diagram is a sample play: Here, $\\Gamma _0 = c_0$ , $\\Lambda _0 = d_0$ , $\\Gamma _1 = c_1$ , and $\\Lambda _1 = d_1$ .",
"This means Player 1 plays first on the left side, Player 2 responds on the right side, Player 1 follows with a play on left side again, and Player 2 responds on the right side, and so forth.",
"Fact 2.5 Let ${L}$ be some countable first order language.",
"Let $\\mathcal {M}$ be an ${L}$ -structure.",
"Let $\\bar{a}$ and $\\bar{b}$ be tuples from $\\mathcal {M}$ of the same length.",
"$\\mathrm {SR}(\\bar{a},\\bar{b}) > \\alpha $ if and only if Player 2 has a winning strategy in $\\mathrm {EF}^{\\mathcal {M},\\bar{a},\\bar{b}}_\\alpha $ ."
],
[
"Bounds for Compact Polish Metric Spaces",
"Fokina, Friedman, Koerwien, and Nies, showed using results of Gromov about metric spaces that $\\mathrm {SR}(M) \\le \\omega + 1$ , when $M$ is a compact Polish metric space.",
"This section will give a proof of this result using König's lemma.",
"This result will require looking at partial maps that are not isometries but have a predetermined error in distances.",
"This argument is a simple approximation idea using a single game which will be a warmup for the later results on proper metric spaces that combines the approximation idea with admissibility, ordinals of admissible sets which are externally illfounded, and infinitely many games.",
"Definition 3.1 Let $M$ be a compact Polish metric space.",
"Let $p \\in \\omega $ and $\\bar{a} = (a_0, ..., a_{p - 1})$ and $\\bar{b} = (b_0, ..., b_{p - 1})$ be tuples of elements from $M$ .",
"Let $(A_n : n \\in \\omega )$ be a sequence of finite subsets of $M$ with the property that for all $n \\in \\omega $ , $A_n \\subseteq A_{n + 1}$ and $\\bigcup _{z \\in A_n} B_{2^{-n}}(z) = M$ .",
"A compact approximation system (for $M$ , $\\bar{a}$ , and $\\bar{b}$ with respect to $(A_n : n \\in \\omega )$ ) is a sequence $(\\varphi _n : n \\in \\omega )$ with the following properties: (i) $\\varphi _n : A_n \\rightarrow A_n$ .",
"(ii) For all $i < p$ and $z \\in A_n$ , $|d(a_i, z) - d(b_i, \\varphi _n(z))| < 2^{-n}$ .",
"(iii) For all $m \\le n$ , for all $y \\in A_m$ and $z \\in A_n$ , $|d(y,z) - d(\\varphi _m(y), \\varphi _n(z)| < 2^{-m} + 2^{-n}.$ (iv) For all $n \\in \\omega $ , $\\bigcup _{z \\in A_n} B_{2^{-(n - 1)}}(\\varphi _n(z)) = M$ .",
"A $k$ -compact approximation system is a sequence $(\\varphi _n : n \\le k)$ satisfying the above properties below $k$ .",
"Lemma 3.2 Suppose $(\\varphi _n : n \\in \\omega )$ is a compact approximation system for $M$ , $\\bar{a}$ , and $\\bar{b}$ with respect to $(A_n : n \\in \\omega )$ , then there is an autoisometry $\\Phi : M \\rightarrow M$ such that for all $i < p$ , $\\Phi (a_i) = b_i$ .",
"Let $x \\in M$ .",
"Let $(x_n : n \\in \\omega )$ be a sequence with the property that for all $n \\in \\omega $ , $x_n \\in A_n$ and $\\lim _{n \\rightarrow \\infty } x_n = x$ .",
"Define $\\Phi (x) = \\lim _{n \\rightarrow \\infty } \\varphi _n(x_n)$ .",
"It remains to show that $\\Phi $ is well-defined and $\\Phi $ is an autoisometry with $\\Phi (\\bar{a}) = \\bar{b}$ .",
"First to show that $(\\varphi _n(x_n) : n \\in \\omega )$ is a Cauchy sequence: Let $m \\le n$ .",
"By (iii) $d(\\varphi _m(x_m), \\varphi _n(x_n)) < d(x_m,x_n) + 2^{-m} + 2^{-n}$ $\\le d(x_m, x) + d(x_n, x) + 2^{-m} + 2^{-n}$ Since $\\lim _{n \\rightarrow \\infty } x_n = x$ , this shows that $(\\varphi _n(x_n) : n \\in \\omega )$ is a Cauchy sequence.",
"Next, to show that $\\Phi (x)$ is independent of the sequence $(x_n : n \\in \\omega )$ which is used to define it: Suppose $(y_n : n \\in \\omega )$ is another sequence with the property that $y_n \\in A_n$ and $\\lim _{n \\rightarrow \\infty } y_n = x$ .",
"Since $x_n,y_n \\in A_n$ , (iii) states $|d(\\varphi _n(x_n), \\varphi _n(y_n)) - d(x_n,y_n)| < 2^{-n} + 2^{-n}$ Therefore $d(\\varphi _n(x_n), \\varphi _n(y_n)) < 2^{-(n - 1)} + d(x_n,y_n)$ $\\le 2^{-(n - 1)} + d(x_n,x) + d(x, y_n)$ Since $\\lim _{n \\rightarrow \\infty } x_n = \\lim _{n \\rightarrow \\infty } y_n = x$ , the above shows that $\\lim _{n \\rightarrow \\infty } d(\\varphi _n(x_n),\\varphi _n(y_n)) = 0$ .",
"So $\\lim _{n \\rightarrow \\infty } \\varphi _n(x_n) = \\lim _{n \\rightarrow \\infty } \\varphi _n(y_n)$ .",
"This shows that $\\Phi $ is a well-defined function.",
"Next, to show that for any $i < p$ , $\\Phi (a_i) = b_i$ : Let $(x_n : n \\in \\omega )$ be a sequence such that $x_n \\in A_n$ and $\\lim _{n \\rightarrow \\infty } x_n = a_i$ .",
"By (ii), $d(\\varphi _n(x_n), b_i) < d(x_n,a_i) + 2^{-n}$ Since $\\lim _{n \\rightarrow \\infty } x_n = a_i$ , this shows that $\\Phi (a_i) = \\lim _{n \\rightarrow \\infty } \\varphi _n(x_n) = b_i$ .",
"Next, to show that $\\Phi $ is an isometry: Suppose $\\lim _{n \\rightarrow \\infty } e_n = e$ and $\\lim _{n \\rightarrow \\infty } f_n = f$ .",
"Then $|d(e,f) - d(e_n,f_n)| = |d(e,f) - d(e,f_n) + d(e,f_n) - d(e_n,f_n)|$ $\\le |d(e,f) - d(e,f_n)| + |d(e,f_n) - d(e_n, f_n)| \\le d(f,f_n) + d(e,e_n)$ Therefore, $\\lim _{n \\rightarrow \\infty } d(e_n,f_n) = d(e,f)$ .",
"Now suppose $x,y \\in M$ .",
"Let $(x_n : n \\in \\omega )$ and $(y_n : n \\in \\omega )$ be such that $x_n,y_n \\in A_n$ and $\\lim _{n \\rightarrow \\infty } x_n = x$ and $\\lim _{n \\rightarrow \\infty } y_n = y$ .",
"By (iii), $|d(\\varphi _n(x_n), \\varphi _n(y_n)) - d(x_n,y_n)| < 2^{-(n - 1)}$ which implies that $\\lim _{n \\rightarrow \\infty }d(\\varphi _n(x_n),\\varphi _n(y_n)) = \\lim _{n \\rightarrow \\infty } d(x_n,y_n)$ .",
"So using the observation of the previous paragraph for the first and third equality, $d(\\Phi (x),\\Phi (y)) = \\lim _{n \\rightarrow \\infty } d(\\varphi _n(x_n), \\varphi _n(y_n)) = \\lim _{n \\rightarrow \\infty } d(x_n,y_n) = d(x,y).$ This shows that $\\Phi $ is an isometry.",
"Finally to show that $\\Phi $ is surjective: Let $y \\in M$ .",
"By (iv), for all $n \\in \\omega $ , $\\bigcup _{z \\in A_n} B_{2^{-(n - 1)}}(\\varphi _n(z)) = M$ .",
"For each $n \\in \\omega $ , let $x_n \\in A_n$ be such that $d(y, \\varphi _n(x_n)) < 2^{-(n - 1)}$ .",
"Observe that $(x_n : n \\in \\omega )$ is a Cauchy sequence.",
"To see this, by (iii), $d(x_m,x_n) < 2^{-m} + 2^{-n} + d(\\varphi _m(x_m), \\varphi _n(x_n))$ $\\le 2^{-m} + 2^{-n} + d(\\varphi _m(x_m), y) + d(y, \\varphi _n(x_n))$ $\\le 3(2^{-m}) + 3(2^{-n})$ Therefore, let $x = \\lim _{n \\rightarrow \\infty } x_n$ .",
"Then $y = \\lim _{n \\rightarrow \\infty } \\varphi _n(x_n) = \\Phi (x)$ .",
"This shows that $\\Phi $ is surjective and completes the proof of the lemma.",
"Lemma 3.3 Let $M$ be a compact Polish metric space.",
"Let $\\bar{a} = (a_0, ..., a_{p - 1})$ and $\\bar{b} = (b_0, ..., b_{p - 1})$ be tuples from $M$ .",
"Let $(A_n : n \\in \\omega )$ be a sequence of finite subsets of $M$ so that for all $n \\in \\omega $ , $A_n \\subseteq A_{n + 1}$ and $\\bigcup _{z \\in A_n} B_{2^{-n}}(z) = M$ .",
"Suppose $\\mathrm {SR}(\\bar{a},\\bar{b}) > \\omega $ , then there is a compact approximation system for $M$ , $\\bar{a}$ , $\\bar{b}$ with respect to $(A_n : n \\in \\omega )$ .",
"Define $J$ to be the tree of all $k$ -compact approximation system for $M$ , $\\bar{a}$ , and $\\bar{b}$ with respect to $(A_n : n \\in \\omega )$ , where $k$ varies over $\\omega $ .",
"$J$ is ordered by $(\\sigma _i : i \\le m) \\preceq _J (\\tau _i : i \\le n)$ if and only if $m \\le n$ and for all $i \\le m$ , $\\sigma _i = \\tau _i$ .",
"As each $A_n$ is finite, $J$ is a finitely branching tree.",
"Any infinite path through $J$ would be a compact approximation system.",
"By König's lemma, $J$ would have an infinite path if $J$ was infinite.",
"As $\\mathrm {SR}(\\bar{a},\\bar{b}) > \\omega $ , fix a winning strategy for Player 2 in $\\mathrm {EF}^{M,\\bar{a},\\bar{b}}_\\omega $ .",
"To show $J$ is infinite, it suffices to show that there is a $k$ -compact approximation system for each $k \\in \\omega $ .",
"Let $L = |A_k|$ .",
"Enumerate $A_k = \\lbrace c_i : i < L\\rbrace $ .",
"Consider the following game of $\\mathrm {EF}^{M,\\bar{a},\\bar{b}}_\\omega $ where Player 1 plays $(L - i, c_i)$ (i.e.",
"Player 1 plays $\\Gamma = c$ ) and Player 2 always responds with the winning strategy: $\\begin{array}{c | c c c c c c c c}\\bar{a} & (L, c_{0}) & {} & (L - 1, c_{1}) & {} & ... & (1, c_{L - 1}) & {} \\\\\\hline \\bar{b} & {} & d_{0} & {} & d_{1} & ... & {} & d_{L - 1}\\end{array}$ (Note that the last ordinal played is 1, which allows player 1 to play one more time.)",
"(Recall that if $n \\le k$ , $A_n \\subseteq A_k$ .)",
"For each $n \\le k$ and $c_i \\in A_n$ , define $\\varphi _n(c_i)$ to be some element of $A_n$ so that $d(d_i, \\varphi _n(c_i)) < 2^{-n}$ , which is possible since $\\bigcup _{z \\in A_n}B_{2^{-n}}(z) = M$ .",
"This completes the definition of $(\\varphi _n : n \\le k)$ .",
"Now to check that $(\\varphi _n : n \\le k)$ is a $k$ -compact approximation system: (i) is clearly true.",
"For (ii): Pick some $i < p$ and $c_j \\in A_n$ , $|d(a_i, c_j) - d(b_i, \\varphi _n(c_j))| = |d(a_i, c_j) - d(b_i,d_j) + d(b_i, d_j) - d(b_i, \\varphi _n(c_j))|$ $\\le |d(a_i,c_j) - d(b_i,d_j)| + |d(b_i,d_j) - d(b_i,\\varphi _n(c_j))|$ $\\le 0 + d(d_j,\\varphi _n(c_j)) < 2^{-n}$ since Player 2 used its winning strategy for $\\mathrm {EF}^{M,\\bar{a},\\bar{b}}_\\omega $ and by the definition of $\\varphi _n(c_j)$ .",
"For (iii): Let $m \\le n \\le k$ , $c_i \\in A_m$ , and $c_j \\in A_n$ .",
"$|d(c_i,c_j) - d(\\varphi _m(c_i), \\varphi _n(c_j))| = |d(c_i,c_j) - d(d_i,d_j) + d(d_i,d_j) - d(\\varphi _m(c_i), \\varphi _n(c_j))|$ $\\le |d(c_i,c_j) - d(d_i,d_j)| + |d(d_i,d_j) + d(\\varphi _m(c_i), \\varphi _n(c_j))|$ $= 0 + |d(d_i,d_j) - d(\\varphi _m(c_i),\\varphi _n(c_j))|$ $= |d(d_i,d_j) - d(\\varphi _m(c_i), d_j) + d(\\varphi _m(c_i),d_j) - d(\\varphi _m(c_i),\\varphi _n(c_j))|$ $\\le |d(d_i,d_j) - d(\\varphi _m(c_i), d_j)| + |d(\\varphi _m(c_i), d_j) - d(\\varphi _m(c_i), \\varphi _n(c_j))|$ $\\le d(d_i,\\varphi _m(c_i)) + d(d_j, \\varphi _n(c_j)) < 2^{-m} + 2^{-n}$ For (iv): Fix $n \\le k$ .",
"Let $R = |A_n|$ .",
"Let $(c_{i_l} : l < R)$ be the subsequence enumerating $A_n \\subseteq A_k$ .",
"Suppose that $\\bigcup _{l < R} B_{2^{-(n - 1)}}(\\varphi _n(c_{i_l})) \\subsetneq M$ .",
"Then there is some $y$ so that for all $l < R$ , $d(y, \\varphi _n(c_{i_l})) \\ge 2^{-(n - 1)}$ .",
"Note that for all $l < R$ , $d(d_{i_l}, y) \\ge 2^{-n}$ : To see this, suppose that there were some $l < R$ so that $d(d_{i_l}, y) < 2^{-n}$ .",
"Then $d(\\varphi _n(c_{i_l}), y) \\le d(\\varphi _n(c_{i_l}), d_{i_l}) + d(d_{i_l}, y) < 2^{-n} + 2^{-n} = 2^{-(n - 1)}$ This is a contradiction.",
"Let $d_L = y$ .",
"Now continue playing the game $\\mathrm {EF}^{M,\\bar{a},\\bar{b}}_\\omega $ one more time as follows: $\\begin{array}{c | c c c c c c c c c}\\bar{a} & (L, c_{0}) & {} & (L - 1, c_{1}) & {} & ... & (1, c_{L - 1}) & {} & {} & c_L \\\\\\hline \\bar{b} & {} & d_{0} & {} & d_{1} & ... & {} & d_{L - 1} & (0,d_L) & {}\\end{array}$ (This means that $\\Gamma = d$ in the last time that Player 1 moves, i.e.",
"Player 1 played on the right side.)",
"Let $c_L$ be the response by Player 2 using its winning strategy.",
"The claim is that the map induced by this play is not a partial isometry.",
"To see this: Since $\\bigcup _{l < R} B_{2^{-n}}(c_{i_l}) = M$ , there is some $l < R$ so that $d(c_L, c_{i_l}) < 2^{-n}$ .",
"Then $d(y, d_{i_l}) = d(d_L, d_{i_l}) < 2^{-n}$ .",
"This contradicts the result of the previous paragraph.",
"This completes the proof of the lemma.",
"As an immediate corollary, one obtains the result of Fokina, Friedman, Koerwien, and Nies on Scott ranks of compact Polish metric spaces.",
"Theorem 3.4 (Fokina, Friedman, Koerwien, and Nies) If $M$ is a compact Polish metric space, then $\\mathrm {SR}(M) \\le \\omega + 1$ .",
"The next few sections will be concerned with finding an effective bound on the Scott rank of proper Polish metric spaces."
],
[
"Games and Ranks",
"For the rest of the paper, let $\\mathcal {M}$ be a countably infinite metric space.",
"By taking a bijection, one may assume that the domain of the metric space $\\mathcal {M}$ is $\\omega $ .",
"By considering the domain of $\\mathcal {M}$ as $\\omega $ , $\\mathcal {M}$ can be coded as a real, i.e.",
"an element of ${{}^\\omega \\omega }$ , by coding all the interpretations of symbols of ${U}$ as relations on $\\omega $ in some fixed way.",
"This section will consider $\\mathcal {M}$ as a metric space; however, in the following section, one will occasionally refer to $\\mathcal {M}$ as a real which codes the structure in the above way.",
"If $\\mathcal {M}$ is a metric space, then $\\mathcal {C}(\\mathcal {M})$ denotes the metric space completion of $\\mathcal {M}$ .",
"Based on the method of constructing a bijective isometry in [7] Lemma 2.3, one defines a new rank on tuples of elements of $\\mathcal {C}(\\mathcal {M})$ depending on whether Player 2 has a winning strategy in some game on $\\mathcal {M}$ (essentially on $\\omega $ ).",
"Since the Ehrenfeucht-Fraïsse game requires the construction of partial ${U}$ -isomorphisms, even if Player 1 always plays elements of $\\mathcal {M}$ , Player 2 generally needs to respond with a $\\mathcal {M}$ -Cauchy sequence (essentially an element of ${{}^\\omega \\omega }$ ).",
"This makes the definability and absoluteness property of Scott rank (in respect to descriptive set theoretic complexity) quite difficult to determine.",
"A priori, it seems quite possible that playing the Ehrenfeucht-Fraïse game in different models of set theory with either more or less Cauchy sequences could affect the outcome of the game.",
"This new game will be played on $\\mathcal {M}$ so any model of set theory containing $\\mathcal {M}$ (which includes the interpretations of the symbols of ${U}$ ) will play these games correctly.",
"The Ehrenfeucht-Fraïsse game asks Player 1 to play perfectly in the sense that it must produce partial isomorphisms; this new game will be ostensibly easier for Player 2 since it demands only the response be appropriately close to Player 1's move.",
"The following convention in variable naming will be used: The variables $a$ , $c$ , and $x$ will denote objects on the left side.",
"The variable $b$ , $d$ and $y$ will denote objects on the right side.",
"Throughout the paper there may be bars or subscripts attached to these variables but they will always denote plays on the sides indicated.",
"Definition 4.1 Let $\\mathcal {M}$ be a metric space on $\\omega $ .",
"Let $\\mathcal {C}(\\mathcal {M})$ be its completion.",
"Let $\\bar{a} = (a_0, ..., a_{p - 1})$ and $\\bar{b} = (b_0, b_1, ..., b_{p - 1})$ be tuples of elements of $\\mathcal {C}(\\mathcal {M})$ .",
"Let $f : \\omega \\rightarrow \\mathbb {Q}^+$ denote a recursive (i.e.",
"computable) strictly decreasing function converging to 0.",
"Let $\\alpha $ be an ordinal.",
"Define the following game $G^{f,\\bar{a},\\bar{b}}_\\alpha $ : $\\begin{array}{c | c c c c c c c c c c c c c }\\bar{a} & (\\alpha _0,c_0) & {} & {} & c_1 & (\\alpha _2,c_2) & {} & {} & c_3 & \\dots & (\\alpha _{k - 1},c_{k - 1}) & {} \\\\\\hline \\bar{b} & {} & d_0 & (\\alpha _1, d_1) & {} & {} & d_2 & (\\alpha _3, d_3) & {} & \\dots & {} & d_{k - 1}\\end{array}$ Player 1 and Player 2 alternatingly play $(\\alpha _0,c_0)$ , $d_0$ , $(\\alpha _1, d_1)$ , $c_1$ , $(\\alpha _2, c_2)$ , $d_2$ , $(\\alpha _3, d_3)$ , $c_3$ , ..., $(\\alpha _{k - 1}, r_{k - 1})$ , $s_{k - 1}$ , where $r = c$ and $s = d$ if $k$ is odd and $r = d$ and $s = c$ if $k$ is even.",
"For all $i < k$ , $\\alpha _i$ is an ordinal less than $\\alpha $ .",
"For all $i < k$ , $c_i$ and $d_i$ are elements of $\\mathcal {M}$ .",
"Since $\\mathcal {M}$ is a metric space on $\\omega $ , $c_i$ and $d_i$ are natural numbers.",
"For all $i < k - 1$ , $\\alpha _{i + 1} < \\alpha _{i}$ .",
"The game ends when Player 1 plays $\\alpha _{k -1} = 0$ and Player 2 responds.",
"Player 2 wins if and only if the following holds: (I) For all $i < p$ and $j < k$ , $|d(a_i, c_j) - d(b_i, d_j)| < f(j)$ .",
"(II) For all $i,j < k$ , $|d(c_i,c_j) - d(d_i,d_j)| < f(i) + f(j)$ .",
"In the above, the distance function $d$ refers to the distance function of $\\mathcal {C}(\\mathcal {M})$ .",
"Definition 4.2 Let $\\mathcal {M}$ be a metric space on $\\omega $ .",
"Let $\\mathrm {REC}$ be the set of all $f : \\omega \\rightarrow \\mathbb {Q}^+$ which are recursive, strictly decreasing, and converge to 0.",
"Let $\\bar{a}$ and $\\bar{b}$ be two tuples of elements from $\\mathcal {C}(\\mathcal {M})$ of the same length.",
"Let $f \\in \\mathrm {REC}$ .",
"Say $\\bar{a} \\sim ^f_\\alpha \\bar{b}$ if and only if Player 2 has a winning strategy in $G_\\alpha ^{f, \\bar{a},\\bar{b}}$ .",
"Define $\\mathrm {R}(\\bar{a},\\bar{b}) = \\min \\lbrace \\mu \\in \\text{ON} : (\\exists f \\in \\text{REC})\\lnot (\\bar{a} \\sim ^f_\\mu \\bar{b})\\rbrace $ if the above set is nonempty.",
"Otherwise let $\\mathrm {R}(\\bar{a},\\bar{b}) = \\infty $ .",
"The use of the class $\\mathrm {REC}$ of recursive, strictly decreasing functions taking values in the positive rational numbers and converging to 0 is merely for convenience.",
"The important property is that these functions used in the following sections is that they are coded in any admissible set.",
"By inspecting the proof, one can find a much smaller class of such functions that would be adequate for the following arguments.",
"Next, the relationship between $\\mathrm {R}$ and $\\mathrm {SR}$ will be determined: Fact 4.3 Let $\\mathcal {M}$ be a metric space on $\\omega $ .",
"Let $\\bar{a}$ and $\\bar{b}$ be two tuples of elements of $\\mathcal {C}(\\mathcal {M})$ of the same length.",
"For all ordinals $\\alpha $ and $f \\in \\mathrm {REC}$ , if $\\bar{a} \\sim _\\alpha \\bar{b}$ , then $\\bar{a} \\sim _\\alpha ^f \\bar{b}$ .",
"Hence $\\mathrm {SR}(\\bar{a},\\bar{b}) \\le \\mathrm {R}(\\bar{a},\\bar{b})$ .",
"Let $\\alpha < \\mathrm {SR}(\\bar{a},\\bar{b})$ .",
"Let $f \\in \\mathrm {REC}$ .",
"A winning strategy for Player 2 in $G^{f,\\bar{a},\\bar{b}}_\\alpha $ will be produced.",
"The idea is that $\\bar{a} \\sim _\\alpha \\bar{b}$ allows Player 2 to find perfect responses (albeit in $\\mathcal {C}(\\mathcal {M})$ not $\\mathcal {M}$ ) in the Ehrenfeucht-Fraïsse game in the sense that the responses form a partial isometry.",
"So Player 2 will respond in the game $G^{f,\\bar{a},\\bar{b}}_\\alpha $ by simply choosing some element of $M$ (i.e.",
"$\\omega $ ) which is sufficiently close to the perfect response given by $\\bar{a} \\sim _\\alpha \\bar{b}$ .",
"The details follows: Consider a play of $G_\\alpha ^{f,\\bar{a},\\bar{b}}$ .",
"Now suppose $(\\alpha _0,c_0)$ , $y_0$ , $d_0$ , $(\\alpha _1, d_1)$ , $x_1$ , $c_1$ , ..., $(\\alpha _{j - 1}, d_{j - 1})$ , $x_{j - 1}$ , and $c_{j - 1}$ has appeared in the construction thus far (assuming $j$ is even) and satisfies the following: For all $i < j$ , (i) $\\bar{a}\\hat{\\ }c_0\\hat{\\ }x_1 \\hat{\\ } c_2 \\hat{\\ } x_3 \\hat{\\ } ...\\hat{\\ } c_{i} \\sim _{\\alpha _i} \\bar{b} \\hat{\\ } y_0\\hat{\\ } d_1 \\hat{\\ } y_2 \\hat{\\ } d_3 \\hat{\\ } ... \\hat{\\ } y_i$ if $i$ is even.",
"If if $i$ is odd, the same holds with the last $c_i$ replaced by $x_i$ and $y_i$ replaced by $d_i$ .",
"(ii) $d(y_i, d_i) < f(i)$ if $i$ is even or $d(x_i,c_i) < f(i)$ if $i$ is odd.",
"Assuming that $\\alpha _{j - 1} \\ne 0$ , suppose Player 1 chooses to play $(\\alpha _j, c_j)$ where $\\alpha _j < \\alpha _{j - 1}$ .",
"Since $\\bar{a}\\hat{\\ }c_0\\hat{\\ }x_1 \\hat{\\ } c_2 \\hat{\\ } x_3 \\hat{\\ } ...\\hat{\\ } y_{j - 1} \\sim _{\\alpha _{j - 1}} \\bar{b} \\hat{\\ } y_0\\hat{\\ } d_1 \\hat{\\ } y_2 \\hat{\\ } d_3 \\hat{\\ } ... \\hat{\\ } d_{j - 1}$ one can find some $y_j$ so that $\\bar{a}\\hat{\\ }c_0\\hat{\\ }x_1 \\hat{\\ } c_2 \\hat{\\ } x_3 \\hat{\\ } ...\\hat{\\ } y_{j - 1} \\hat{\\ } c_j \\sim _{\\alpha _j} \\bar{b} \\hat{\\ } y_0\\hat{\\ } d_1 \\hat{\\ } y_2 \\hat{\\ } d_3 \\hat{\\ } ... \\hat{\\ } y_{j - 1} \\hat{\\ } y_j$ Now let $d_j \\in M$ be chosen so that $d(y_j, d_j) < f(j)$ .",
"Now continue this process as long as Player 1 has not played the ordinal 0.",
"Of course, depending on whether the stage is even or odd, the variable needs to be appropriately changed.",
"At some stage $k$ , Player 1 will have played $\\alpha _{k - 1} = 0$ .",
"After Player 2 responds, the process finishes.",
"The claim is that the following play of $G^{f,\\bar{a},\\bar{b}}_\\alpha $ is winning for Player 2: $\\begin{array}{c | c c c c c c c c c c c c c }\\bar{a} & (\\alpha _0,c_0) & {} & {} & c_1 & (\\alpha _2,c_2) & {} & {} & c_3 & \\dots & (\\alpha _{k - 1},c_{k - 1}) & {} \\\\\\hline \\bar{b} & {} & d_0 & (\\alpha _1, d_1) & {} & {} & d_2 & (\\alpha _3, d_3) & {} & \\dots & {} & d_{k - 1}\\end{array}$ First pick some $i < p$ and $j < k$ .",
"Without loss of generality, suppose $j$ is even.",
"Then $|d(a_i,c_j) - d(b_i,d_j)| = |d(a_i,c_j) - d(b_i,y_j) + d(b_i,y_j) - d(b_i,d_j)|$ Recall $d(a_i,c_j) = d(b_i,y_j)$ hence $= |d(b_i,y_j) - d(b_i, d_j)| \\le d(y_j, d_j) < f(j)$ Now pick some $i,j < k$ .",
"Without loss of generality suppose $i$ is even and $j$ is odd.",
"Then $|d(c_i,c_j) - d(d_i,d_j)| = |d(c_i,c_j) - d(c_i,x_j) + d(c_i,x_j) - d(d_i,d_j)|$ Recall that $d(c_i,x_j) = d(y_i, d_j)$ .",
"Therefore $= |d(c_i,c_j) - d(c_i,x_j) + d(y_i,d_j) - d(d_i,d_j)|$ $\\le |d(c_i,c_j) - d(c_i,x_j)| + |d(y_i,d_j) - d(d_i,d_j)|$ $\\le d(c_j,x_j) + d(y_i, d_i) < f(j) + f(i)$ All the other even and odd combinations are handled similarly.",
"Definition 4.4 A metric space $\\mathcal {N}$ is proper if and only if if $\\lbrace y : d(x,y) \\le r\\rbrace $ is compact for all $x \\in N$ and $r \\in \\mathbb {R}$ .",
"The key property of proper metric spaces that will be used is that every bounded sequence has a convergent subsequence.",
"In the following, suppose $\\mathcal {N}$ is some metric space with distance $d_\\mathcal {N}$ .",
"Then the distance on ${}^k\\mathcal {N}$ is defined as $d_{{}^k\\mathcal {N}}(\\bar{a},\\bar{b}) = \\sum _{i = 0}^{k - 1} d_\\mathcal {N}(a_i,b_i)$ .",
"Now suppose $\\bar{a}$ and $\\bar{b}$ are two tuples of length $p$ of elements of $\\mathcal {C}(\\mathcal {M})$ .",
"The next technical lemma asserts that if $\\alpha $ is an ordinal, $\\mathcal {C}(\\mathcal {M})$ is a proper metric space, and $(\\bar{a},\\bar{b})$ is a limit (in the $(\\mathcal {C}(\\mathcal {M}))^{2p}$ metric) of points of the form $(\\bar{e},\\bar{f})$ so that $\\mathrm {R}(\\bar{e},\\bar{f}) > \\alpha $ , then $\\mathrm {R}(\\bar{a},\\bar{b}) > \\alpha $ .",
"Fact 4.5 Let $\\mathcal {M}$ be a metric space on $\\omega $ .",
"Suppose $\\mathcal {C}(\\mathcal {M})$ is a proper metric space.",
"Let $\\alpha $ be an ordinal.",
"Let $\\bar{a}$ and $\\bar{b}$ be two tuples of elements of $\\mathcal {C}(\\mathcal {M})$ of the same length $p$ .",
"Suppose that $(\\bar{a},\\bar{b})$ is the limit of the sequence $\\langle (\\bar{a}_n,\\bar{b}_n) : n \\in \\omega \\rangle $ in $(\\mathcal {C}(\\mathcal {M}))^{2p}$ so that for all $n \\in \\omega $ , $\\mathrm {R}(\\bar{a}_n,\\bar{b}_n) > \\alpha $ .",
"Then $\\mathrm {R}(\\bar{a},\\bar{b}) > \\alpha $ .",
"Fix $f \\in \\mathrm {REC}$ .",
"Let $g \\in \\mathrm {REC}$ be defined by $g(n) = \\frac{f(n)}{3}$ for all $n \\in \\omega $ .",
"Suppose $\\bar{a}$ and $\\bar{b}$ take the following form: $\\bar{a} = (a_0, ..., a_{p -1})$ and $\\bar{b} = (b_0, ..., b_{p - 1})$ .",
"For each $n \\in \\omega $ , suppose $\\bar{a}_n$ and $\\bar{b}_n$ take the form: $\\bar{a}_n = (a_0^n, ..., a_{p - 1}^n)$ and $\\bar{b}_n = (b_0^n,...,b_{p - 1}^n)$ .",
"It suffices to show that for all ordinals $\\alpha $ , if $\\bar{a}_n \\sim _\\alpha ^g \\bar{b}_n$ for all $n \\in \\omega $ , then $\\bar{a} \\sim _\\alpha ^f \\bar{b}$ .",
"Fix a winning strategy for Player 2 in each game $G^{g,\\bar{a}_n,\\bar{b}_n}_\\alpha $ .",
"In the following proof, when a response from Player 2 in $G^{g,\\bar{a}_n,\\bar{b}_n}_\\alpha $ is required, it is always assumed it is taken from this fixed winning strategy.",
"Now a winning strategy for $G^{f,\\bar{a},\\bar{b}}_\\alpha $ will be described: By refining $\\langle (\\bar{a}_n,\\bar{b}_n) : n \\in \\omega \\rangle $ to a subsequence, one may assume that $d_p((\\bar{a}_n,\\bar{b}_n), (\\bar{a},\\bar{b})) < \\frac{1}{n} \\ \\ \\ (\\star )$ where $d_p$ is the metric on $\\mathcal {C}(\\mathcal {M})^{2p}$ mentioned above which is defined by summing the distance in each coordinate.",
"Let $A_{-1} = \\omega $ .",
"Fix $j$ and suppose the following have been constructed: For all $i < j$ , $A_i$ and $\\alpha _i$ have been defined.",
"If $i$ is even, then $c_i$ , $y_n$ , and $d_i^n$ for each $n \\in A_i$ have been constructed.",
"If $i$ is odd, then $d_i$ , $x_n$ , and $c_i^n$ for each $n \\in A_i$ have been constructed.",
"These objects satisfy the following: (i) For all $i < j$ , $\\min A_i > \\frac{1}{g(i)}$ and $A_i$ is an infinite subset of $\\omega $ .",
"Hence by property $(\\star )$ on the sequence, one has $d((\\bar{a}_n,\\bar{b}_n),(\\bar{a},\\bar{b})) < g(i)$ , for all $n \\in A_i$ .",
"(ii) For all $i < j - 1$ , $A_{i + 1} \\subseteq A_i$ .",
"(iii) If $i$ is even, for all $n \\in A_i$ , $d(y_i, d_i^n) < g(i)$ .",
"If $i$ is odd, for all $n \\in A_i$ , $d(x_i, c_i^n) < g(i)$ .",
"(iv) For each $i < j$ and $n \\in A_i$ , the following is a play in $G^{g,\\bar{a}_n,\\bar{b}_n}_\\alpha $ according to the fixed winning strategy for Player 2: $\\begin{array}{c | c c c c c c c c c c c c c }\\bar{a}_n & (\\alpha _0,c_0) & {} & {} & c_1^n & (\\alpha _2,c_2) & {} & {} & c_3^n & \\dots & (\\alpha _{i},c_{i}) & {} \\\\\\hline \\bar{b}_n & {} & d_0^n & (\\alpha _1, d_1) & {} & {} & d_2^n & (\\alpha _3, d_3) & {} & \\dots & {} & d_{i}^n\\end{array}$ when $i$ is even.",
"A similar diagram when $i$ is odd with the appropriate variable change.",
"Without loss of generality, suppose that $j$ is odd.",
"Suppose that $\\alpha _{j - 1} \\ne 0$ .",
"Suppose Player 1 plays $(\\alpha _j, d_j)$ where $\\alpha _j < \\alpha _{j - 1}$ .",
"For each $n \\in A_{j - 1} \\cap (\\frac{1}{g(j)}, \\infty )$ , let $c_j^n$ be the response of Player 2 in the following play of $G^{g,\\bar{a}_n,\\bar{b}_n}_\\alpha $ according to the fixed winning strategy: $\\begin{array}{c | c c c c c c c c c c c c c }\\bar{a}_n & (\\alpha _0,c_0) & {} & {} & c_1^n & (\\alpha _2,c_2) & {} & {} & c_3^n & \\dots & (\\alpha _{j - 1},c_{j - 1}) & {} & {} & c_j^n \\\\\\hline \\bar{b}_n & {} & d_0^n & (\\alpha _1, d_1) & {} & {} & d_2^n & (\\alpha _3, d_3) & {} & \\dots & {} & d_{j - 1}^n & (\\alpha _j, d_j) & {}\\end{array}$ Claim: $L_j = \\lbrace c_j^n : n \\in A_{j - 1} \\cap (\\frac{1}{g(j)},\\infty )\\rbrace $ is a bounded set To see this: Note that for all $n \\in A_{j - 1}$ , $d(a_0, c_j^n) \\le d(a_0, a_0^n) + d(a_0^n, c_j^n)$ Since Player 2 plays according to its winning strategy in $G^{g,\\bar{a}_n,\\bar{b}_n}$ , $|d(a_0^n, c_j^n) - d(b_0^n, d_j)| < g(j)$ so $ \\le d(a_0, a_0^n) + d(b_0^n, d_j) + g(j) \\le d(a_0,a_0^n) + d(b_0^n, b_0) + d(b_0, d_j) + g(j)$ Since $n > \\frac{1}{g(j)}$ and property $(\\star )$ on the sequence, $d(a_0, a_0^n) < g(j)$ , $d(b_0, b_0^n) < g(j)$ .",
"Thus $< 3 g(j) + d(b_0,d_j) = 3 \\frac{f(j)}{3} + d(b_0, d_j) = f(j) + d(b_0,d_j)$ This shows that for all $n \\in A_{j - 1}$ , $d(a_0,c_j^n) < f(j) + d(b_0,d_j)$ .",
"Hence $L_j$ is bounded.",
"The claim has been established.",
"Since $L_j$ is bounded and $\\mathcal {C}(\\mathcal {M})$ is a proper metric space, the sequence $\\langle c_j^n : n \\in A_{j - 1} \\cap (\\frac{1}{g(j)}, \\infty )\\rangle $ has a convergent subsequence.",
"Let $x_j \\in \\mathcal {C}(\\mathcal {M})$ be a limit point of a convergent subsequence.",
"Let $A_{j} \\subseteq A_{j - 1} \\cap (\\frac{1}{g(j)}, \\infty )$ be an infinite set so that $d(x_j, c_j^n) < g(j)$ for all $n \\in A_j$ .",
"This completes the recursive construction at stage $j$ .",
"Continue this construction until at some point Player 1 plays $\\alpha _{k - 1} = 0$ .",
"Now the claim is that the following is a winning play for Player 2 in $G^{f,\\bar{a},\\bar{b}}_\\alpha $ : $\\begin{array}{c | c c c c c c c c c c c c c }\\bar{a} & (\\alpha _0,c_0) & {} & {} & c_1^{\\min A_1} & (\\alpha _2,c_2) & {} & {} & c_3^{\\min A_3} & \\dots & (\\alpha _{k - 1},c_{k - 1}) & {} \\\\\\hline \\bar{b} & {} & d_0^{\\min A_0} & (\\alpha _1, d_1) & {} & {} & d_2^{\\min A_2} & (\\alpha _3, d_3) & {} & \\dots & {} & d_{k - 1}^{\\min A_{k - 1}}\\end{array}$ Let $l < p$ and $i < k$ .",
"Without loss of generality, suppose $i$ is even: $|d(a_l, c_i) - d(b_l, d_i^{\\min A_i})|$ $= |d(a_l, c_i) - d(a_l^{\\min A_i}, c_i) + d(a^{\\min A_i}_l, c_i) - d(b^{\\min A_i}_l, d_i^{\\min A_i}) + d(b^{\\min A_i}_l, d_i^{\\min A_i}) - d(b_l,d_i^{\\min A_i})|$ $\\le |d(a_l, c_i) - d(a^{\\min A_i}_l, c_i)| + |d(a^{\\min A_i}_l, c_i) - d(b^{\\min A_i}_l, d_i^{\\min A_i})| + |d(b^{\\min A_i}_l, d_i^{\\min A_i}) - d(b_l,d_i^{\\min A_i})|$ $\\le d(a_l, a^{\\min A_i}_l) + |d(a^{\\min A_i}_l, c_i) - d(b^{\\min A_i}_l, d_i^{\\min A_i})| + d(b_{l}^{\\min A_i}, b_l)$ The first and third terms are less than $g(i)$ by (i).",
"The middle term is less that $g(i)$ since these are responses that come from the winning strategy of $G^{g,\\bar{a}_{\\min A_i},\\bar{b}_{\\min A_i}}_\\alpha $ .",
"$\\le g(i) + g(i) + g(i) = 3 g(i) = 3 \\frac{f(i)}{3} = f(i)$ Now let $i < j < k$ .",
"As an example, assume $i$ is even and $j$ is odd: $|d(c_i, c_j^{\\min A_j}) - d(d_i^{\\min A_i}, d_j)|$ $= | d(c_i, c_j^{\\min A_j}) - d(d_i^{\\min A_j}, d_j) + d(d_i^{\\min A_j}, d_j) - d(y_i, d_j) + d(y_i, d_j) - d(d_i^{\\min A_i}, d_j)|$ $\\le | d(c_i, c_j^{\\min A_j}) - d(d_i^{\\min A_j}, d_j)| + |d(d_i^{\\min A_j}, d_j) - d(y_i, d_j)| + |d(y_i, d_j) - d(d_i^{\\min A_i}, d_j)|$ $\\le | d(c_i, c_j^{\\min A_j}) - d(d_i^{\\min A_j}, d_j)| + d(d_i^{\\min A_j}, y_i) + d(y_i, d_i^{\\min A_i})$ The last two terms are less than $g(i)$ since $A_j \\subseteq A_i$ and (iii).",
"The first term is less than $g(i) + g(j)$ since these come from Player 2 winning response in the appropriate play of $G^{g,\\bar{a}_{\\min A_j}, \\bar{b}_{\\min A_j}}_\\alpha $ .",
"$ \\le g(i) + g(j) + g(i) + g(i) = 3 g(i) + g(j) = 3 \\frac{f(i)}{3} + \\frac{f(j)}{3} < f(i) + f(j)$ So the above describes a winning strategy for $G^{f,\\bar{a},\\bar{b}}_\\alpha $ .",
"$\\bar{a} \\sim _\\alpha ^f \\bar{b}$ .",
"This completes the proof.",
"These are all the results in pure metric space theory that will be needed."
],
[
"Admissibility",
"In order to establish bounds on the Scott rank that come from recursion theory or constructibility theory, one needs to look at admissible sets.",
"$\\mathsf {KP}$ is an axiom system in the language $\\lbrace \\dot{\\in }\\rbrace $ where $\\dot{\\in }$ is a binary relation symbols.",
"$\\mathsf {KP}$ is a weak axiom system for set theory: It includes the basic axioms of set theory such as pairing, union, foundation, and others.",
"The more distinguishing axioms schemes are $\\Delta _1$ -separation and $\\Sigma _1$ -replacement.",
"An admissible set is a transitive set $A$ so that $(A,\\in ) \\models \\mathsf {KP}$ .",
"See [2], [5], [9], or for more on admissible sets.",
"As usual in set theory, for emphasis, $V$ will refer to the real universe.",
"Definition 5.1 An ordinal $\\alpha $ is an admissible ordinal if and only if there is an admissible set $\\mathcal {A}$ so that $A \\cap \\text{ON} = \\alpha $ .",
"If $x \\in {{}^\\omega \\omega }$ , then $\\alpha $ is an $x$ -admissible ordinal if and only if there is an admissible set $\\mathcal {A}$ with $x \\in A$ so that $A \\cap \\mathrm {ON} = \\alpha $ .",
"For any $x \\in {{}^\\omega \\omega }$ , $\\omega _1^x$ is the smallest ordinal $\\alpha $ so that $L_\\alpha (x) \\models \\mathsf {KP}$ .",
"Fact 5.2 An ordinal $\\alpha $ is an $x$ -admissible ordinal if and only if $L_\\alpha (x) \\models \\mathsf {KP}$ .",
"If $x \\in {{}^\\omega \\omega }$ , $L_{\\omega _1^x}(x)$ is the smallest admissible set containing $x$ .",
"The reals of $L_{\\omega _1^x}(x)$ are the $x$ -hyperarithmetic elements.",
"$\\omega _1^x$ is the supremum of the $x$ -hyperarithmetic ordinal as well as the supremum of the $x$ -recursive ordinals.",
"An important fact about $\\mathsf {KP}$ is that the well-founded part of any model of $\\mathsf {KP}$ is a model of $\\mathsf {KP}$ : Fact 5.3 (Truncation Lemma) Let $\\mathcal {B}= (B,\\dot{\\in }^\\mathcal {B}) \\models \\mathsf {KP}$ .",
"Let $\\mathrm {WF}(\\mathcal {B})$ be the collection of $\\dot{\\in }^\\mathcal {B}$ well-founded (in $V$ ) elements of $B$ .",
"Then $\\mathrm {WF}(\\mathcal {B}) \\models \\mathsf {KP}$ .",
"Hence the Mostowski collapse of $\\mathrm {WF}(\\mathcal {B})$ is an admissible set.",
"See [2], Lemma II.8.4.",
"Let ${L}$ be any language.",
"Let ${L}_{\\infty ,\\omega }$ denote infinitary logic in the language ${L}$ .",
"If $\\mathcal {A}$ is an admissible set, then ${L}_\\mathcal {A}= ({L}_{\\infty ,\\omega })^\\mathcal {A}$ .",
"This is the admissible fragment of ${L}_{\\infty ,\\omega }$ determined by $\\mathcal {A}$ .",
"${L}_\\mathcal {A}$ is a countable admissible fragment if $\\mathcal {A}$ is a countable admissible set.",
"(See [2] or for more information.)",
"The following will be a useful method of constructing admissible sets: Fact 5.4 (Jensen's model existence theorem) Let $\\mathcal {A}$ be a countable admissible set.",
"Let ${L}$ be a language which is $\\Delta _1$ definable over $\\mathcal {A}$ and contains a binary relation symbol $\\dot{\\in }$ and constant symbols $\\hat{a}$ for each $a \\in A$ .",
"Let $T$ be a consistent theory in the countable admissible fragment ${L}_\\mathcal {A}$ which is $\\Sigma _1$ definable over $\\mathcal {A}$ and contains the following sentences: (I) $\\mathsf {KP}$ (II) For each $a \\in A$ , $(\\forall v)(v \\dot{\\in }\\hat{a} \\Leftrightarrow \\bigvee _{z \\in a} v = \\hat{z})$ .",
"Then there is a $\\mathcal {B}\\models T$ so that $\\mathrm {WF}(\\mathcal {B})$ is transitive, $\\mathcal {A}\\subseteq \\mathcal {B}$ , and $\\mathrm {ON} \\cap B = \\mathrm {ON} \\cap A$ .",
"See [9] Section 4, Lemma 11 and [5].",
"Arguments using some form of this fact appear in the proof of Sacks theorem about countable admissible ordinals by Friedman and Jensen.",
"A similar fact is used in Grilliot's omitting type proof of this theorem of Sacks (see , Theorem 15).",
"Fact 5.5 (Barwise Compactness) Let $\\mathcal {A}$ be a countable admissible set.",
"Let ${L}$ be a language which is $\\Delta _1$ over $\\mathcal {A}$ .",
"Let $T$ be a theory in the countable admissible fragment ${L}_\\mathcal {A}$ which is $\\Sigma _1$ over $\\mathcal {A}$ .",
"If every $F \\subseteq T$ so that $F \\in A$ is consistent, then $T$ is consistent.",
"See [2] Theorem III.5.6, [9], Section 4, Corollary 8, or [5].",
"Now returning back to metric spaces.",
"The following fact expresses how to use an ill-founded ordinal $\\alpha $ to play the game $G_\\alpha ^{f,\\bar{a},\\bar{b}}$ forever in $V$ to produce an ${U}$ -automorphism: Fact 5.6 Let $\\bar{a}$ and $\\bar{b}$ be tuples in $\\mathcal {C}(\\mathcal {M})$ (that is, tuples of $\\mathcal {M}$ -Cauchy sequences) of the same length.",
"If there exists an ill-founded model $\\mathcal {A}$ of $\\mathsf {ZFC - P}$ with $\\mathrm {WF}(\\mathcal {A})$ transitive so that $\\mathcal {M},\\bar{a},\\bar{b} \\in A$ , $\\mathcal {A}\\models \\bar{a} \\sim _\\alpha ^f \\bar{b}$ for some $f \\in \\mathrm {REC}$ , and $\\alpha $ is an ordinal of $\\mathcal {A}$ which is ill-founded (in $V$ ), then there is a ${U}$ -automorphism of $\\mathcal {C}(\\mathcal {M})$ taking $\\bar{a}$ to $\\bar{b}$ .",
"Note that since $\\mathcal {M}$ is a metric space on $\\omega $ , the sets $\\mathcal {M}$ , $\\bar{a}$ , and $\\bar{b}$ belong to $\\mathrm {WF}(\\mathcal {A})$ .",
"In $\\mathcal {A}$ , fix a winning strategy for $G^{f,\\bar{a},\\bar{b}}_\\alpha $ for Player 2.",
"Let $\\Phi : \\omega \\rightarrow \\omega $ be a surjection such that for all $k \\in \\omega $ , $\\Phi ^{-1}(\\lbrace k\\rbrace )$ is infinite.",
"Let $c_{2i} = \\Phi (i)$ .",
"Let $d_{2i + 1} = \\Phi (i)$ .",
"(Recall that $\\mathcal {M}$ is assumed to be a metric space with domain $\\omega $ .)",
"Since $\\alpha $ is ill-founded, externally in $V$ , choose in $V$ an infinite $\\mathcal {A}$ -decreasing sequence of $\\mathcal {A}$ -ordinals $(\\alpha _n)_{n \\in \\omega }$ : that is, for all $n \\in \\omega $ , $\\mathcal {A}\\models \\alpha _{n + 1} < \\alpha _n$ .",
"Using the winning strategy for Player 2, play as follows: $\\begin{array}{c | c c c c c c c c c c c c c }\\bar{a} & (\\alpha _0,c_0) & {} & {} & c_1 & (\\alpha _2,c_2) & {} & {} & c_3 & \\dots & (\\alpha _{k - 1},c_{k - 1}) & {} \\\\\\hline \\bar{b} & {} & d_0 & (\\alpha _1, d_1) & {} & {} & d_2 & (\\alpha _3, d_3) & {} & \\dots & {} & d_{k - 1}\\end{array}$ where for even $i$ , $c_i$ are defined above and for odd $i$ , $d_i$ are defined above.",
"For even $i$ , $d_i$ comes from the response of Player 2.",
"Similarly for odd $i$ , $c_i$ comes from the response of Player 2.",
"Since distance in $\\mathcal {C}(\\mathcal {M})$ can be expressed as a $\\Delta _1$ statement in $\\mathsf {KP}$ , $\\Delta _1$ absoluteness from $\\mathcal {A}$ down to $\\mathrm {WF}(\\mathcal {A})$ and then up into $V$ shows that distance is computed correctly in $\\mathcal {A}$ .",
"Hence in $V$ , Player 2 has not lost any finite play of $G^{f,\\bar{a},\\bar{b}}_\\alpha $ described above.",
"Since $(\\alpha _n)_{n \\in \\omega }$ is infinite decreasing, the game can always be extended.",
"Playing the game forever in $V$ produces a sequence $(c_n)_{n \\in \\omega }$ and $(d_n)_{n \\in \\omega }$ so that each finite portion of the sequence fits into the above play where Player 2 has not lost.",
"Let $\\Psi : \\omega \\rightarrow \\omega $ be defined by $\\Psi (k) = d_k$ .",
"Let $\\Lambda (k) = c_k$ .",
"Now to define a map $\\Xi : \\mathcal {C}(\\mathcal {M}) \\rightarrow \\mathcal {C}(\\mathcal {M})$ : Let $e \\in \\mathcal {C}(\\mathcal {M})$ .",
"Let $e = (e^n)_{n \\in \\omega }$ be some $\\mathcal {M}$ -Cauchy sequence representing $e$ .",
"Let $\\ell : \\omega \\rightarrow \\omega $ be a strictly increasing sequence so that for all $n$ , $\\Lambda (\\ell (n)) = e^n$ .",
"Let $\\Xi (e)$ be the element of $\\mathcal {C}(\\mathcal {M})$ represented by the $\\mathcal {M}$ -Cauchy sequence $(\\Psi (\\ell (n)))_{n \\in \\omega }$ .",
"It straightforward (using argument similar to those of Section ) to check that $\\Xi $ is well-defined, that is, it does not depend on the Cauchy representation of $e$ or the choice of $\\ell $ .",
"Using the definition of $G^{f,\\bar{a},\\bar{b}}_\\alpha $ , one can check that $\\Xi $ is a ${U}$ -homomorphism and that $\\bar{a}$ is mapped to $\\bar{b}$ .",
"By how $\\Phi $ was chosen, one can show that $\\Xi $ is actually surjective.",
"Hence $\\Xi $ is a ${U}$ -automorphism taking $\\bar{a}$ to $\\bar{b}$ .",
"(It should be noted that the fact that $\\Phi ^{-1}(\\lbrace i\\rbrace )$ is infinite for each $i \\in \\omega $ is important for establishing these properties.)",
"In the following, $\\bar{a}$ and $\\bar{b}$ are considered as tuples of $\\mathcal {M}$ -Cauchy sequences.",
"Since $\\mathcal {M}$ is a metric space on $\\omega $ , $\\bar{a}$ and $\\bar{b}$ may be coded as elements of ${{}^\\omega \\omega }$ .",
"Fact 5.7 If there is no ${U}$ -automorphism taking $\\bar{a}$ to $\\bar{b}$ , then $\\mathrm {R}(\\bar{a},\\bar{b}) < \\omega _1^{\\mathcal {M}\\oplus \\bar{a} \\oplus \\bar{b}}$ and in particular, $\\mathrm {SR}(\\bar{a},\\bar{b}) < \\omega _1^{\\mathcal {M}\\oplus \\bar{a} \\oplus \\bar{b}}$ .",
"([7] Doucha) If there is no ${U}$ -automorphism taking $\\bar{a}$ to $\\bar{b}$ , then $\\mathrm {SR}(\\bar{a},\\bar{b})$ is countable.",
"Therefore, the Scott rank of a Polish metric space is at most $\\omega _1$ .",
"Suppose $\\mathrm {R}(\\bar{a},\\bar{b}) \\ge \\omega _1^{\\mathcal {M}\\oplus \\bar{a} \\oplus \\bar{b}}$ .",
"Let $\\mathcal {A}= L_{\\omega _1^{\\mathcal {M}\\oplus \\bar{a} \\oplus \\bar{b}}}(\\mathcal {M}\\oplus \\bar{a} \\oplus \\bar{b})$ .",
"$\\mathcal {A}$ is a countable admissible set.",
"Let ${L}$ be a language consisting of the following: (i) A binary relation symbol $\\dot{\\in }$ .",
"(ii) For each $a \\in A$ , a constant symbol $\\hat{a}$ .",
"${L}$ is a language which is $\\Delta _1$ -definable in $\\mathcal {A}$ .",
"Now let $T$ be the theory in the countable admissible fragment ${L}_\\mathcal {A}$ consisting of sentences indicated below: (I) $\\mathsf {ZFC - P}$ .",
"(II) For each $a \\in A$ , “$(\\forall v)(v \\dot{\\in }\\hat{a} \\Leftrightarrow \\bigvee _{z \\in a} v = \\hat{z})$ ”.",
"(III) For each $\\alpha < \\omega _1^{\\mathcal {M}\\oplus \\bar{a} \\oplus \\bar{b}}$ , “$\\bigwedge _{f \\in \\mathrm {REC}} \\hat{\\bar{a}} \\sim _\\alpha ^f \\hat{\\bar{b}}$ ”.",
"$T$ is $\\Sigma _1$ definable in $\\mathcal {A}$ .",
"$T$ is consistent.",
"To see this, consider the structure $\\mathcal {B}$ defined by: Let its domain be $B = H_{\\aleph _1}$ , the collection hereditarily countable sets.",
"Let $\\dot{\\in }= \\in \\upharpoonright H_{\\aleph _1}$ .",
"For each $a \\in A$ , let $\\hat{a}^{\\mathcal {B}} = a$ .",
"$\\mathcal {B}\\models T$ since it was assumed that $\\mathrm {R}(\\bar{a},\\bar{b}) \\ge \\omega _1^{\\mathcal {M}\\oplus \\bar{a} \\oplus \\bar{b}}$ .",
"By Fact REF , there is a ${L}$ -structure $\\mathcal {B}$ so that $\\mathcal {B}\\models T$ , $\\text{WF}(\\mathcal {B})$ is transitive, $\\text{ON} \\cap B = \\text{ON} \\cap A = \\omega _1^{\\mathcal {M}\\oplus \\bar{a} \\oplus \\bar{b}}$ , and $\\mathcal {A}$ is an end extension of $\\mathcal {B}$ .",
"$\\mathcal {B}$ must be ill-founded since no transitive set of ordinal height $\\omega _1^{\\mathcal {M}\\oplus \\bar{a}\\oplus \\bar{b}}$ containing $\\mathcal {M}\\oplus \\bar{a}\\oplus \\bar{b}$ can be a model of $\\mathsf {ZFC - P}$ .",
"Hence by (III), there must be some illfounded $\\mathcal {B}$ -ordinal $\\beta $ so that for any $f \\in \\text{REC}$ , $\\mathcal {B}\\models \\bar{a} \\sim _\\beta ^f \\bar{b}$ .",
"Fact REF shows that there is a ${U}$ -automorphism taking $\\bar{a}$ to $\\bar{b}$ .",
"Contradiction.",
"Next, it will be shown that if $\\mathrm {R}(\\bar{a},\\bar{b}) \\ge \\omega _1^\\mathcal {M}$ , then $\\mathrm {R}(\\bar{a},\\bar{b})$ is a limit of elements $(\\bar{e},\\bar{f})$ so that there is a ${U}$ -automorphism of $\\mathcal {C}(\\mathcal {M})$ taking $\\bar{e}$ to $\\bar{f}$ .",
"In fact, in [7] Proposition 2.4, it is shown that certain points have the property that every open neighborhood contains a perfect set of $(\\bar{e},\\bar{f})$ so that there is a ${U}$ -automorphism of $\\mathcal {C}(\\mathcal {M})$ taking $\\bar{e}$ to $\\bar{f}$ .",
"It can also be shown that if $\\mathrm {R}(\\bar{a},\\bar{b}) \\ge \\omega _1^\\mathcal {M}$ , then every open set containing $(\\bar{a},\\bar{b})$ has a perfect set of such $(\\bar{e},\\bar{f})$ .",
"However, this fact is not necessary for producing the effective bound on Scott rank.",
"The reader may choose to skip all the comments about perfect sets in the following two results.",
"If one wants the perfect set result, one will need the following effective perfect set theorem: Fact 5.8 (Harrison) Let $r \\in {{}^\\omega \\omega }$ .",
"Suppose $X$ is a $r$ -recursively presented Polish space.",
"If a $\\Sigma _1^1(r)$ set $A$ contains a member which is not $\\Delta _1^1(r)$ (i.e.",
"$r$ -hyperarithmetic), then $A$ contains a perfect subset.",
"See for more on recursively presented Polish space and Theorem 4F.1.",
"Fact 5.9 Let $\\bar{a}$ and $\\bar{b}$ be tuples of elements of $\\mathcal {C}(\\mathcal {M})$ of the same length $p$ .",
"Suppose $\\mathrm {R}(\\bar{a},\\bar{b}) \\ge \\omega _1^\\mathcal {M}$ and there are no ${U}$ -automorphisms of $\\mathcal {C}(\\mathcal {M})$ taking $\\bar{a}$ to $\\bar{b}$ .",
"Then for any $\\bar{n} \\in {}^{2p}M$ and $m \\in \\omega $ , if $(\\bar{a},\\bar{b}) \\in B_\\frac{1}{m}(\\bar{n})$ (the open ball around $\\bar{n}$ of size $\\frac{1}{m}$ in the metric on ${}^{2p}\\mathcal {C}(\\mathcal {M})$ ), then there is some $(\\bar{e},\\bar{f}) \\in B_\\frac{1}{m}(\\bar{n})$ for which there is a ${U}$ -automorphism taking $\\bar{e}$ to $\\bar{f}$ .",
"In fact, there is a perfect set of such $(\\bar{e},\\bar{f})$ .",
"Fix $m \\in \\omega $ and $\\bar{n} \\in {}^{2p}M$ so that $(\\bar{a},\\bar{b}) \\in B_{\\frac{1}{m}}(\\bar{n})$ .",
"Let $\\mathcal {A}= L_{\\omega _1^\\mathcal {M}}(\\mathcal {M})$ .",
"$\\mathcal {A}$ is a countable admissible set.",
"(Some remarks before continuing: Since $\\bar{a}$ and $\\bar{b}$ are tuples of $\\mathcal {M}$ -Cauchy sequences, they are coded by reals.",
"As it will be shown below, $\\bar{a}$ and $\\bar{b}$ can not belong to $\\mathcal {A}$ .",
"Thus one cannot mention $\\bar{a}$ or $\\bar{b}$ in any countable fragment associated to the admissible set $\\mathcal {A}$ .",
"However, $\\bar{n}$ is a tuple of elements of $\\mathcal {M}$ which is essentially a tuple of integers (since $\\mathcal {M}$ was assumed to be a metric space on $\\omega $ ).",
"Thus $\\bar{n}$ belongs to any admissible set.",
"One is permitted to refer to $\\bar{n}$ .",
"Although $\\bar{a}$ and $\\bar{b}$ cannot be mentioned in the theory, these elements will be used to (externally in $V$ ) verify the consistency of the theory.",
"The details follow as the proof resumes below.)",
"Let ${L}$ be a language consisting of the following: (i) A binary relation symbol $\\dot{\\in }$ .",
"(ii) For each $a \\in A$ , a constant symbol $\\hat{a}$ .",
"(iii) Two new constant symbols $\\dot{\\bar{e}}$ and $\\dot{\\bar{f}}$ .",
"${L}$ is a language which is $\\Delta _1$ definable in $\\mathcal {A}$ .",
"Now let $T$ be the theory in the countable admissible fragment ${L}_\\mathcal {A}$ consisting of the sentences indicated below: (I) $\\mathsf {ZFC - P}$ .",
"(II) For each $a \\in A$ , “$(\\forall v)(v \\dot{\\in }\\hat{a} \\Leftrightarrow \\bigvee _{z \\in a} v = \\hat{z})$ ”.",
"(III) “$d_{2p}(\\hat{\\bar{n}}, (\\dot{\\bar{e}}, \\dot{\\bar{f}})) < \\frac{1}{m}$ ” where $d_{2p}$ is the metric on ${}^{2p}\\mathcal {C}(\\mathcal {M})$ .",
"(IV) For each $\\alpha < \\omega _1^{\\mathcal {M}}$ , “$\\bigwedge _{f \\in \\mathrm {REC}} \\dot{\\bar{e}} \\sim _\\alpha ^f \\dot{\\bar{e}}$ ”.",
"If one want the perfect set version of this result, add on the following (V) For all $\\alpha < \\omega _1^{\\mathcal {M}}$ , “$(\\dot{\\bar{e}},\\dot{\\bar{f}}) \\notin L_{\\alpha }(\\hat{\\mathcal {M}})$ ”.",
"In either case, $T$ is $\\Sigma _1$ definable in $\\mathcal {A}$ .",
"$T$ is consistent.",
"To see this: Consider the following structure $\\mathcal {B}$ .",
"Its domain is $B = H_{\\aleph _1}$ .",
"$\\dot{\\in }^{\\mathcal {B}} = \\in \\upharpoonright H_{\\aleph _1}$ .",
"For each $a \\in A$ , let $\\hat{a}^\\mathcal {B}= a$ .",
"Let $\\dot{\\bar{e}}^\\mathcal {B}= \\bar{a}$ and $\\dot{\\bar{f}}^\\mathcal {B}= \\bar{b}$ .",
"$\\mathcal {B}\\models T$ since $\\mathrm {R}(\\bar{a},\\bar{b}) \\ge \\omega _1^\\mathcal {M}$ .",
"For those interested in the perfect set version, to see (V), note that $\\omega _1^{\\mathcal {M}\\oplus \\bar{a}\\oplus \\bar{b}} > \\omega _1^\\mathcal {M}$ .",
"If not, then $\\mathrm {R}(\\bar{a},\\bar{b}) \\ge \\omega _1^{\\mathcal {M}} \\ge \\omega _1^{\\mathcal {M}\\oplus \\bar{a} \\oplus \\bar{b}}$ .",
"By Fact REF , there is a ${U}$ -automorphism taking $\\bar{a}$ to $\\bar{b}$ .",
"This contradicts the assumption on $(\\bar{a},\\bar{b})$ .",
"Since $\\omega _1^{\\mathcal {M}\\oplus \\bar{a} \\oplus \\bar{b}} > \\omega _1^{\\mathcal {M}}$ , $\\mathcal {M}\\oplus \\bar{a} \\oplus \\bar{b}$ can not belong to any admissible set of ordinal height $\\omega _1^\\mathcal {M}$ .",
"In particular, $(\\bar{a},\\bar{b}) \\notin L_{\\omega _1^{\\mathcal {M}}}(\\mathcal {M})$ .",
"This shows the model $\\mathcal {B}$ satisfies (V).",
"By Fact REF , there exists some model $\\mathcal {B}\\models T$ so that $\\mathrm {WF}(\\mathcal {B})$ is transitive and $\\mathrm {ON} \\cap B = \\mathrm {ON} \\cap A$ .",
"Let $\\bar{e} = \\dot{\\bar{e}}^{\\mathcal {M}}$ and $\\bar{f} = \\dot{\\bar{e}}^{\\mathcal {M}}$ .",
"As before, $\\mathcal {B}$ must be ill-founded.",
"In $\\mathcal {B}$ , there is some ill-founded $\\mathcal {B}$ -ordinal $\\alpha $ so that $\\mathcal {B}\\models \\bar{e} \\sim _\\alpha ^f \\bar{f}$ , for any $f \\in \\mathrm {REC}$ .",
"So by Fact REF in $V$ , there is a ${U}$ -automorphism taking $\\bar{e}$ to $\\bar{f}$ .",
"By absoluteness, $(\\bar{e},\\bar{f}) \\in B_\\frac{1}{m}(\\bar{n})$ .",
"Hence $(\\bar{e},\\bar{f})$ is the desired element.",
"For the perfect set version, note that the set of $C$ of elements $(\\bar{u},\\bar{v}) \\in B_\\frac{1}{m}(\\bar{n})$ so that there is a ${U}$ -automorphism taking $\\bar{u}$ to $\\bar{v}$ is a $\\Sigma _1^1(M)$ set.",
"By (V), the element $(\\bar{e},\\bar{f})$ produced above is not in $L_{\\omega _1^{\\mathcal {M}}}(\\mathcal {M})$ , so in particular not ${\\Delta _1^1}(\\mathcal {M})$ .",
"Hence $C$ must contain a perfect subset by Fact REF ."
],
[
"Main Results",
"Definition 6.1 Let ${L}$ be a language and let $\\mathcal {N}$ be an ${L}$ -structure.",
"$\\mathcal {N}$ is rigid if and only if there are no nontrivial ${L}$ -automorphisms of $\\mathcal {N}$ .",
"Theorem 6.2 Let $\\mathcal {M}$ be a metric space on $\\omega $ .",
"Suppose $\\mathcal {C}(\\mathcal {M})$ is a rigid Polish metric space.",
"Then $\\mathrm {SR}(\\mathcal {C}(\\mathcal {M})) < \\omega _1^\\mathcal {M}$ .",
"Suppose $\\mathrm {SR}(\\mathcal {C}(\\mathcal {M})) \\ge \\omega _1^\\mathcal {M}$ .",
"This means for each $\\alpha < \\omega _1^\\mathcal {M}$ , there is some $\\bar{a}_\\alpha $ and $\\bar{b}_\\alpha $ so that $\\bar{a}_\\alpha \\ne \\bar{b}_\\alpha $ (as elements of $\\mathcal {C}(\\mathcal {M})$ ) and $\\bar{a} \\sim _\\alpha ^f \\bar{b}$ for any $f \\in \\mathrm {REC}$ .",
"Note that for $\\alpha \\ne \\beta $ , the length of $\\bar{a}_\\beta $ and $\\bar{a}_\\beta $ may not be the same.",
"Let $\\mathcal {A}$ be $L_{\\omega _1^\\mathcal {M}}(\\mathcal {M})$ .",
"Let ${L}$ be a language consisting of: (i) A binary relation symbol $\\dot{\\in }$ .",
"(ii) For each $a \\in A$ , a constant symbol $\\hat{a}$ .",
"(iii) Three new constant symbols $\\dot{n}$ , $\\dot{e}$ , and $\\dot{f}$ .",
"${L}$ is $\\Delta _1$ definable over $\\mathcal {A}$ .",
"If $n \\in \\omega $ and $r \\in {{}^\\omega \\omega }$ , let $c_n(r)$ denote the element of ${}^n({{}^\\omega \\omega })$ coded by $r$ (under some fixed recursive coding of $n$ -tuples of reals by a single real).",
"Let $T$ be the following theory in the countable admissible fragment ${L}_\\mathcal {A}$ consisting of the sentences indicated below: (I) $\\mathsf {ZFC - P}$ (II) For each $a \\in A$ , “$(\\forall v)(v \\dot{\\in }\\hat{a} \\Leftrightarrow \\bigvee _{z \\in a} v = \\hat{z})$ ”.",
"(III) “$\\dot{n} \\dot{\\in }\\hat{\\omega }$ ”.",
"“$\\dot{e}$ and $\\dot{f}$ are functions from $\\hat{\\omega }$ to $\\hat{\\omega }$ ”.",
"(IV) “$c_{\\dot{n}}(\\hat{e})$ and $c_{\\dot{n}}(\\hat{f})$ are tuples of $\\mathcal {M}$ -Cauchy sequences”.",
"“$c_{\\dot{n}}(\\hat{e}) \\ne c_{\\dot{n}}(\\hat{f})$ as $\\mathcal {M}$ -Cauchy sequences”.",
"(V) For each $\\alpha < \\omega _1^\\mathcal {M}$ , “$\\bigwedge _{f \\in \\mathrm {REC}} c_{\\dot{n}}(\\dot{e}) \\sim _\\alpha ^f c_{\\dot{n}}(\\dot{f})$ ”.",
"$T$ is $\\Sigma _1$ definable over $\\mathcal {A}$ .",
"To see that $T$ is consistent, one needs to use Barwise compactness.",
"Let $F \\subset T$ be such that $F \\in A$ .",
"Since $F \\in \\mathcal {A}$ , there is an $\\alpha < \\omega _1^\\mathcal {M}$ that bounds all the $\\beta $ 's that appear in statements of type (IV).",
"Consider the model $\\mathcal {B}$ defined by: Its domain is $B = H_{\\aleph _1}$ .",
"$\\dot{\\in }^\\mathcal {B}= \\in \\upharpoonright \\mathcal {B}$ .",
"For each $a \\in A$ , $\\hat{a}^\\mathcal {B}= a$ .",
"Let $\\dot{n}^\\mathcal {B}= |\\bar{a}_\\alpha |$ .",
"Let $e$ and $f$ be two reals so that $c_{|\\bar{a}_\\alpha |}(e) = \\bar{a}_\\alpha $ and $c_{|\\bar{a}_{\\alpha }|}(f) = \\bar{b}_\\alpha $ .",
"It is clear that $\\mathcal {B}\\models F$ .",
"By Barwise compactness, $T$ is consistent.",
"Now Fact REF gives a model $\\mathcal {B}\\models T$ so that $\\mathrm {WF}(\\mathcal {B})$ is transitive, $\\mathcal {A}\\subseteq \\mathcal {B}$ , and $\\mathrm {ON} \\cap \\mathcal {B}= \\mathrm {ON} \\cap \\mathcal {A}= \\omega _1^\\mathcal {M}$ .",
"Let $\\bar{e} = c_{\\dot{n}^\\mathcal {B}}(\\dot{e}^\\mathcal {B})$ , $\\bar{f} = c_{\\dot{n}^\\mathcal {B}}(\\dot{f}^\\mathcal {B})$ .",
"Since $\\mathcal {M}$ is a metric space on $\\omega $ , all $\\mathcal {M}$ -Cauchy sequences in $\\mathcal {B}$ belong to $\\mathrm {WF}(\\mathcal {B})$ .",
"Hence $\\bar{e},\\bar{f} \\in \\mathrm {WF}(\\mathcal {B})$ .",
"By $\\Delta _1$ -absoluteness from $\\mathcal {B}$ down to $\\mathrm {WF}(\\mathcal {B})$ and then up to $V$ , one can show that for all $\\alpha < \\omega _1^\\mathcal {M}$ , $\\bar{e} \\sim _\\alpha ^f \\bar{f}$ for all $f \\in \\mathrm {REC}$ .",
"Also by $\\Delta _1$ -absoluteness, $\\bar{e} \\ne \\bar{f}$ .",
"However $\\mathcal {M}\\oplus \\bar{e} \\oplus \\bar{f}$ is in the admissible set $\\mathrm {WF}(\\mathcal {B})$ (by Fact REF ) which has ordinal height $\\omega _1^\\mathcal {M}$ .",
"Hence $\\omega _1^{\\mathcal {M}\\oplus \\bar{e} \\oplus \\bar{f}} = \\omega _1^\\mathcal {M}$ .",
"Fact REF implies there is a ${U}$ -automorphism taking $\\bar{e}$ to $\\bar{f}$ .",
"This contradicts the assumption that $\\mathcal {C}(\\mathcal {M})$ is a rigid metric space.",
"Theorem 6.3 Let $\\mathcal {M}$ be a metric space on $\\omega $ .",
"Suppose $\\mathcal {C}(\\mathcal {M})$ is a proper Polish metric space.",
"Then $\\mathrm {SR}(\\mathcal {M}) \\le \\omega _1^{\\mathcal {M}} + 1$ .",
"Suppose not, then there exists some tuples $\\bar{a}$ and $\\bar{b}$ of elements of $\\mathcal {C}(\\mathcal {M})$ of the same length so that $R(\\bar{a},\\bar{b}) \\ge \\omega _1^\\mathcal {M}$ but there is no ${U}$ -automorphism taking $\\bar{a}$ to $\\bar{b}$ .",
"By Fact REF , there exist a sequence $(\\bar{a}_n,\\bar{b}_n)_{n \\in \\omega }$ so that $(\\bar{a},\\bar{b})$ is its limit and for all $n \\in \\omega $ , there is a ${U}$ -automorphism taking $\\bar{a}_n$ to $\\bar{b}_n$ .",
"Let $\\alpha $ be an ordinal greater than $\\omega _1^{\\mathcal {M}\\oplus \\bar{a} \\oplus \\bar{b}}$ .",
"The existence of these automorphisms implies that $\\bar{a}_n \\sim ^f_\\alpha \\bar{b}_n$ for all $n \\in \\omega $ and any $f \\in \\mathrm {REC}$ .",
"Then Fact REF implies that $\\bar{a} \\sim _\\alpha ^f \\bar{b}$ .",
"However since $\\alpha > \\omega _1^{\\mathcal {M}\\oplus \\bar{a} \\oplus \\bar{b}}$ , Fact REF implies that there is a ${U}$ -automorphism taking $\\bar{a}$ to $\\bar{b}$ .",
"Contradiction."
]
] | 1906.04351 |
[
[
"Examining Untempered Social Media: Analyzing Cascades of Polarized\n Conversations"
],
[
"Abstract Online social media, periodically serves as a platform for cascading polarizing topics of conversation.",
"The inherent community structure present in online social networks (homophily) and the advent of fringe outlets like Gab have created online \"echo chambers\" that amplify the effects of polarization, which fuels detrimental behavior.",
"Recently, in October 2018, Gab made headlines when it was revealed that Robert Bowers, the individual behind the Pittsburgh Synagogue massacre, was an active member of this social media site and used it to express his anti-Semitic views and discuss conspiracy theories.",
"Thus to address the need of automated data-driven analyses of such fringe outlets, this research proposes novel methods to discover topics that are prevalent in Gab and how they cascade within the network.",
"Specifically, using approximately 34 million posts, and 3.7 million cascading conversation threads with close to 300k users; we demonstrate that there are essentially five cascading patterns that manifest in Gab and the most \"viral\" ones begin with an echo-chamber pattern and grow out to the entire network.",
"Also, we empirically show, through two models viz.",
"Susceptible-Infected and Bass, how the cascades structurally evolve from one of the five patterns to the other based on the topic of the conversation with upto 84% accuracy."
],
[
"Introduction",
"Fringe social media sites, such as Gab, 8chan, and PewTube, have become a fertile ground for individuals and groups with far right and extreme far right views to post and share their messages in an unfettered manner and to galvanize supporters for their cause [1].",
"While most mainstream social media like Reddit, Twitter, and Facebook moderate their content and deplatform more extreme users and groups, the emergence of outlets like 8chan and Gab.ai have given radical groups large content delivery networks to broadcast their polarizing messages.",
"These social networks have morphed into alt-right echo chambers [2], [3] and have garnered close to 450,000 users.",
"The echo chamber effect is often amplified via online conversations and interactions that occur on social networks.",
"In recent times, we have observed that such interactions, combined with the exploitation of online social networks [4] is an effective strategy to recruit members, instigate the public, and ultimately culminate in riots and violence as witnessed recently in Charlottesville and Portland.",
"Due to the threat of violence that these groups bring with them, analyzing the online dynamics of conversations and interactions in such social media sites is an important problem that the research community is trying to address [5], [6].",
"In this work, using the well-known propagation mechanism of information cascades [7], [8], we demonstrate how polarized conversations take shape on Gab.",
"By analyzing 34M posts and 3.7M cascades built from conversations on Gab, we show that there are five different classes of conversation cascades, where each type shows varied level of user participation, and responses.",
"By analyzing the types of cascades, we give post-level intuition and several structural properties of these cascades.",
"To emphasize the post level details of the cascades, we present an algorithm to classify hashtags, and eventually cascades into topics.",
"We observe that controversial topics are adopted by users that are more strongly connected and also these topics generate larger cascades.",
"By analyzing structural dynamics of conversation cascades on Gab, we observe that all cascades start with a simple linear pattern and evolve into other patterns when a topic becomes viral and more users join the conversation.",
"We found that the average time and average posts to make an evolution on Gab is 1.5 days and atleast 3 posts respectively on average.",
"We model this evolution of cascades using popular network growth models like Susceptible-Infected model [9] and Bass model [10].",
"Our best model fit give upto 84% accuracy.",
"Essentially, we answer the following research questions with our corresponding contributions in this broad study of Gab: What are the types of cascading behavior in Gab conversations?",
"We study conversation patterns in Gab as cascades and provide metrics to measure structural patterns within conversations.",
"Our analysis show that the rarely occurring cascades get viral even when their user response rates are much lower compared to commonly occurring cascades.",
"Can we characterize user relationship and topics of Gab conversations?",
"We propose an algorithm to cluster topics that circulate in conversation cascades and our results show that the polarizing topics gain lot of traction in Gab conversations.",
"Can we study evolution of Gab conversations using the cascades?",
"With Gab conversations represented as cascades, we give pathways for these cascades to evolve over time.",
"To capture this evolution patterns algorithmically, we use the Susceptible-Infected model [9] and the Bass model [10].",
"We show applicability of different models to different evolution types."
],
[
"Related Work",
"Related work of our research can be split into three sections: [i)] Applications of Gab analysis Cascading behavior in social media Quantitative analysis of topics in social media .",
"The use of social media to study radicalization [11], discrimination [12], and fringe web communities [13] is gaining traction over the past few years.",
"Particularly, past studies highlights the fact that the advent of Gab.com creates a scope to analyze topics like alt-right echo-chamber and hate speech research [1], [2].",
"Most notably, the recent work studied the spread of hate speech among Gab users with the help of repost cascades and friend/follower network [6].",
"Our research adds an extra dimension to existing Gab works, in which we analyze its conversations, provide analysis on growth of topics, and give measures and metrics to study conversation structure and evolution in the perspective of cascades.",
"Cascades in social media accounts for information dissemination [7], which can be applied to variety of applications like fake news [14], viral marketing [15], and emergency management [16].",
"These information cascades are crucial in incorporating machine learning models for variety of applications like: modeling influence propagation in the social media [17], predicting number of reshares using self-exciting point process [18], and modeling network growth patterns as an alternative to sigmoid models [19].",
"In this work, we reframe information cascades as conversation cascades and give novel ideas on defining models for cascade evolution types in Gab conversations.",
"Just like Twitter [20], hashtags on Gab are means to add metadata to posts that highlights topics.",
"Some researchers defined the topic (class) of hashtags manually [21] [22].",
"Lee et.",
"al.",
"[23] classify Twitter Trending Topics into 18 general categories using one bag-of-words text based approach and one network approach.",
"Wang et.",
"al [24] propose Hashtag Graphbased Topic Model (HGTM) to discover topics of tweets.",
"We believe manual labeling is not scalable and unsupervised algorithms are not accurate enough due to the nature of hashtags, thus we propose a supervised semi-automated procedure to classify hashtags.",
"Preliminaries In this section, we briefly describe about the dataset that we use in all experiments, terminologies in our methods and statistics of cascades and topics in the dataset.",
"Dataset Description Figure: Timeseries of frequency posts, replies, and reshares from the origin of gab.com(August 2016) until the forum went down on the last week of October 2018Gab.com/Gab.ai is a social media forum, founded in 2016, provide a forum to connect and share information among users.",
"Even though the description of the forum looks very similar to most popular social media counterparts like Twitter and Facebook, Gab is known to support individual liberty and committed to contribute for free speech in the social media communityhttps://gab.com/.",
"However, Gab has strong restriction policies on posts and users, which are promoting pornography, terrorism and violence.",
"Users of Gab can share information via posts, post replies, and quotes/reshares.",
"We use the Gab data published by data scraping forum pushshift.iohttps://pushshift.io.",
"Figure REF gives an overview of the dataset as timeseries plot for the number of posts, replies, and quotes appeared in Gab between August 2016 and October 2018.",
"The dataset is a comprehensive collection with 34 million posts, replies, quotes posted between the date range of August 2016 and October 2018, about 15,000 groups and about 300,000 public users information.",
"It is evidential from Figure REF that our dataset comprise of 55% posts, 30% replies, and 15% quotes.",
"This data is available with complete set of metadata like time, attachments, likes, dislikes, replies, quotes along with post and user details.",
"Conversation cascades Microblogging conversations have been widely studied in the context of cascades for a wide spectrum of applications like emotion analysis [5], topic modelling [25] and cascade analysis [26].",
"In this work, we give variety of cascade representations for conversations in Gab and give their in-depth structural and temporal analysis, response rates, and longevity.",
"Figure: Possible shallow cascading structures in replies and reshares of posts and number of posts in each cascade type.",
"As predicted, most of the cascades follow simple patterns(type A).",
"Interestingly, many conversations form cascade type B in which many response come directly to the root post and does not evolve into any other cascade types.",
"Small number of conversations follow cascade type C, which split from the root and each branch from root follows linear pattern.",
"Very few conversations follow cascade type D pattern, which has highly nested structure, both at root level and branch level and cascade type E, which initially follows linear pattern and takes non-linear during the evolution time.Table: Basic statistics of depth, volume, number of unique users, and structural virality(Wiener Index) of all cascade types.",
"Overall, the cascade type E achieves higher structural popularity even though the number of unique user participation is low.With available posts, replies and quotes/reshares from Gab, we construct conversation cascades, where each cascade is one complete conversation.",
"Nodes in each cascade represent original post/reply/quote and edges represent reply/quote of post/reply/quote.",
"The formal definition and the construction process of conversation cascades are given below: Cascade construction: A conversation cascade is a directed graph $G_c=(V_c,E_c)$ , where $V_c$ is a set of posts/replies/quotes and $E_c$ is a set of edges connecting posts ordered by time.",
"We represent a node/post in the cascade as $P(v,t)$ , where $v \\in V_c$ is a post appearing in the social media at time $t$ .",
"There exists a directed edge($v^{\\prime },v$ ), when $P(v,t)$ receives a reply/quote $P(v^{\\prime },t^{\\prime })$ , where $t^{\\prime } > t$ .",
"This process continues each time when a user replies/quotes a post.",
"Thus, root node in conversation cascades represents original post and replies and quotes take branches from the root node(original post).",
"Nodes in second or above level in the cascade take branches again if such nodes in turn get any replies or quotes.",
"In total we constructed 1,721,441 cascades from the Gab data, which comprise of 19,220,059 nodes/posts contributed by 173,581 users and 15,476,852 edges.",
"We evidence from these cascades that conversations in Gab follow one of the five patterns as represented in Figure REF .",
"These conversation cascade types give a generic representation of user engagement patterns over time for a post.",
"These cascades are also used to represent linear and non-linear conversation patterns in Gab.",
"Figure REF gives number of cascades in each conversation cascade pattern.",
"In Table REF , we report statistics of cascade depth, volume, number of unique user participation, and structural virality of all cascade types.",
"We calculate the structural virality of a cascade of size $n$ using Wiener Index(WI) [27] given in the Equation REF , where $d_{ij}$ is the shortest distance of nodes $i$ and $j$ .",
"$WI = \\frac{1}{n(n-1) \\sum _{i=1}^{n} \\sum _{j=1}^{n} d_{ij}}$ From Table REF and Figure REF , we find that cascades of category E attains higher volume and depth with very few user participation and it plays a vital role in Gab conversations.",
"Figure: Depth and Size distribution of each cascade typeWe also analyze depth and volume (number of nodes) of each cascade type and give their corresponding distributions in Figures REF and REF respectively.",
"From Figure REF we note that users in Gab have longer conversations, which take more branches after level 2 in the conversation thread(Cascade types D and E).",
"Cascade type B is not shown in the Figure REF because these conversations split at the root node(post) and terminate at immediate next level.",
"In Figure REF we show distribution of volume(number of nodes/replies/quotes) across cascade types.",
"The volume distribution of the given cascades are similar to the depth distribution given in Figure REF (Cascade types D and E have much engaging participation).",
"Interestingly, we find that cascades of type B have more participation, given that these cascades stop at level 1, compared to cascades of type A. Cascades across topics Different topics spread in different ways over the networks in terms of speed, number of participants and the dynamics of their cascades.",
"We analyze how this intuitively accepted notion applies to cascades on Gab and what topics in particular differ significantly from the others.",
"In section we introduce a procedure by which we classify hashtags into different topics.",
"We give a representative set of hashtags and their topics in Table REF .",
"Then we use the labeled hashtags to classify the cascades where those hashtags appear.",
"Our analysis shows that cascades on more controversial topics have different characteristics than other cascades.",
"They tend to result in larger cascades, majority of them being the same cascade type (Type D).",
"Cascades in these polarizing topics are generated by users that are more strongly connected, i. e. have higher tie strength.",
"We specifically identified three topics, “Antisemitism\", “Anti Islam\", and “White Supremacy\" to be noticeably different from other topics in regards to the nature of the cascades in which they are discussed.",
"We define tie strength of user $u_1$ and $u_2$ as the number of times $u_1$ replies to a post from $u_2$ or $u_2$ replies to a post from $u_1$ .",
"Figure REF shows that users who participated in topics of “Antisemitism\", “Anti Islam\", and “White Supremacy\" have higher tie strength and are more strongly connected which supports the theory that more controversial topics are adopted by users with higher tie strength [28].",
"ADLwww.adl.org believes that white supremacist, hateful, antisemitic bigotry are widespread on Gab.",
"Our findings are aligned with this statement and other studies that argue antisemitism and white nationalist topics are openly expressed on Gab and have great similarities in terms of communities who adopt these topics [22] [29].",
"Figure: Average tie strength of users participated in cascades on different topic.",
"Users participating in cascades on polarizing topics like antisemitism, anti Islam, and white supremacy tend to have higher tie strength, i. e. the frequency of interactions between two users.Table: Topic Categories, a set of examples of hashtags, the number of instances, and the average size of cascades on each category Topic Discovery of Cascades We introduce a novel procedure for labeling hashtags with the topics they belong to.",
"We first looked at the top 200 most used hashtags on Gab.",
"These hashtags then were classified by subject matter experts into 6 topics.",
"Unlike other works identified on Twitter [28], our work do not cover a broad range of topics.",
"Majority of posts on Gab are on political and rather controversial topics, thus we went few steps deeper and classified these political topics into more specific topics.",
"We used tagdefwww.tagdef.com, tagsfinderwww.tagsfinder.com, Google, and Twitter to find the meaning of hashtags and assigned them to one of the 6 categories.",
"Excluding hashtags that are too broad to be assigned to a specific topic, for instance #gab, #eu, #music, or #welcome, we labeled 126 hashtags.",
"In the next step, we used our algorithm to label other hashtags used on Gab based on the 126 hashtags that we manually labeled.",
"As shown in Algorithm , the inputs of the method are the network of hashtags, list of topics with the set of hashtags assigned to each topic, and a constant value as threshold.",
"We defined the network of hashtags as $G=\\lbrace V, E\\rbrace $ where $V$ is the set of hashtags used on Gab, and $E$ is the set of weighted edges.",
"An edge between two vertices indicates that the two hashtags have appeared in the same post at least once and the weight of the edge represents the number of co-occurrences.",
"We create this network of hashtags [24], but we added weights to the edges to underscore the importance of the number of co-occurrences between two hashtags.",
"In line 3 of the algorithm, an edge is passed to get_node_with_no_topic() method which returns the vertex that is not already assigned to any topic.",
"The method returns null if both vertices of the edge are already assigned topics or if neither has any topic assigned to it.",
"Because first, we are not interested in labeling hashtags that are already labeled and secondly, we cannot label a node if none of its neighbors is labeled.",
"If the method returns a node $u$ we get the topics that are assigned to that node in step 6, and then for each topic $t$ in topics, we increment an integer property of node $v$ , the other vertex of the edge, that represents $t$ by $w$ , the weight of edge $e$ .",
"After all edges have been traversed and the properties of their respective vertices are updated we move to steps 9 to 13 where for each node $n$ in the graph and each topic $t$ in topics, we get the property $p_t$ .",
"If the value of the $p_t$ is greater than the threshold $\\tau $ , we conclude that node $n$ , and the hashtag that it represents, belongs to topic $t$ .",
"After we label hashtags with one or more topics, then for each cascade if hashtags of a topic $t_1$ appear more than C times, we assign that cascade to $t_1$ .",
"Note that a cascade could belong to more than one category.",
"Table REF shows the 6 topics we identified, some examples of the hashtags in each category, as well as the number of cascades and average size of cascades on each topic.",
"To evaluate our algorithm, we designed a procedure where we randomly picked 42 hashtags from the set of 126 labeled hashtags and labeled the rest of the hashtags in the set using our algorithm.",
"Then we compared the results of our algorithm with our manual labeling.",
"FIgure REF shows the distribution of hashtags among different topics.",
"Figure REF shows the ground truth, i. e. manually labeled hashtags, and figure REF shows the results of our algorithm.",
"Since one hashtag could belong to multiple topics, we used the evaluation metrics of multi-label learning algorithms mentioned in [30].",
"Table REF shows how well our algorithm performs in labeling the hashtags with topics, we use accuracy, recall, precision, $F1$ score, Hamming Loss (HL), and Subset Accuracy (SA).",
"Table: Performance of hashtag labeling algorithm.",
"Our algorithm produced high results for conventional performance metrics of accuracy, recall, and precision.",
"Low HL (Hamming Loss) and high SA (Subset Accuracy) are also strong indicatives that our algorithm performs well in classifying hashtags into topics.Topic Discovery of Hashtags [1] G, $\\mathbf {T}$ and $\\tau $ label_hashtags $e \\in E$ u = get_node_with_no_topics(e) v = $e-u$ $u \\ne \\emptyset $ $topics=$ get_topics(v) $t \\in topics$ inc_node_prop(u,t,w) $n \\in V$ $t \\in T$ $p_t =$ get_property(G,n,t) $p_t > \\tau $ add_hashtag_to_topic(G,n,t) Figure: Accuracy of the hashtag labeling algorithm, shows the ground truth, i. e. manually labeled hashtags, and shows how our algorithm classified the same hashtags.",
"Overall accuracy of the algorithm is 71% as given in Table Cascade Analysis Response rate in cascades Figure: Avg.",
"response rate at each level/depth of the cascade.",
"Higher values represent faster average response time.",
"Simpler cascades(types A and C) have higher response rates than complex cascade structures(types D and E).With our proposed cascade types in Section REF , we analyze response time of each cascade type.",
"With this experiment, we aim to produce results that depict how fast posts in these cascades get responses and help growing/evolving the cascades.",
"We also provide a notion for response rate to compare velocity of responses at all levels in a given cascade type.",
"We define response rate($\\mathcal {R}_c$ ) of a cascade type as the average speed of response(s) of posts at a given level/depth of a cascade type($c$ ).",
"Equation REF gives a formulation to calculate response rate at a given level/depth ($l$ ) for a given cascade ($c$ ) and $\\Delta _l^c$ is an average response time for posts at a given level/depth ($l$ ) for a given cascade type ($c$ ).",
"$\\mathcal {R}_c^l = \\frac{1}{\\Delta _c^l}$ Figure: Overview of cascade evolution as a state diagram.",
"All cascades start as type A and they can evolve to the maximum of type D.Figure: Summary distributions of amount of time and posts taken by a cascade to evolve into another.",
"Sudden spikes in the plot are due to anomalies in the data.",
"30% of cascades in each cascade type require less than 2 minutes and the number of cascade evolution decreases as the number of posts increasesThe distribution of average response rates of all cascade types at each level is given in Figure REF .",
"Interestingly, we find that all cascade types start with almost equal and slower response rate and progress with faster responses over time.",
"Importantly, cascades of type C achieve overall higher response rate at earlier levels, even though they do not grow as larger and deeper as other cascades.",
"Also, cascades of type A follow constant response rate like other cascade types and spikes as depth and volume increase.",
"We also find that response rate for other larger cascades such as type D and E is inversely proportional to the distribution of volume of the cascade.",
"Evolution of cascades Figure: Normalized timeseries distribution of number of cascade evolution for each evolution type.Given cascade types of Gab conversations, we study their growth patterns and evolution.",
"All Gab conversations/cascading patterns, as given in Figure REF , starts with type A and some of them moderately evolve into other cascade types.",
"All possible transformations within the proposed cascade varieties are given in Figure REF .",
"It is notable from this figure that a conversation cascade can reach its maximum potential by transforming to cascade type D and cascades must evolve into other types(B,C,E) before reaching type D. Providing such evolution patterns and number of occurrences of each cascade from Figure REF , we find that there are significant evolution of cascades in our data.",
"With the availability of evolution patterns in Gab conversations, we provide basic analysis such as time REF and number of posts REF required by a cascade to evolve into another.",
"As a summary of this plot, we present Table REF to give minimum, maximum, average, and standard deviation of number of time and posts required to achieve evolve the cascades.",
"Table: Number of posts and time required by a cascade type to evolve into another.",
"Overall evolution in Gab is slow with smaller number of posts and longer time to perform evolution.Figure REF gives a timeseries on all possible cascade evolution in Gab.",
"Modeling this evolution helps to study intensification of conversations in Gab.",
"Conversations in any social media intensifies when it create interests or controversies among users about the topic.",
"We use traditional models, like Susceptible-Infected [9] and Bass [10] models to fit evolution patterns that exist in Gab conversation cascades.",
"We prefer to use these models, because of their ability to fit sigmoid curves which in turn maps exponential growth or exponential fall [19].",
"Figure: SI and Bass model fit for multiple cascade evolution timeseries.",
"SI model fits for simple evolution types, while the Bass model captures the sharp spike that occurs near the end of the timeseries.The goal of our models is to predict a number of new evolution($\\frac{dn(t)}{dt}$ ) given a time $t$ and a cumulative sum of user parameters($\\epsilon $ ).",
"Equation REF gives a modified equation of SI model.",
"We model $\\alpha ,\\gamma < 1$ to restrict complete participation of susceptibles ($N-n(t)$ ) and infected ones($n(t)$ ) because of an assumption that not all of the previous evolution are responsible for the current evolution.",
"$(SI)\\frac{dn(t)}{dt}= \\epsilon \\ *\\ \\beta \\ *\\ n(t)^{\\alpha }\\ *\\ (N-n(t))^{\\gamma }$ where, $\\beta $ is evolution rate of cascades $N$ is total number of the cascades that evolved $n(t)$ is the cumulative sum of cascades evolved at time $t$ .",
"In other words, total infected ones at time $t$ Figure: Error(%) of SI and Bass model to predict the evolution patterns.",
"Our best model fit is 84% accuracy(in A→CA \\rightarrow C cascade evolution)In Equation REF , we give the Bass model with the user parameter($\\epsilon $ ).",
"Although, Bass model is introduced to describe the process of how new business product is taking effect in population, the model has been widely used in understanding diffusion and influence patterns in social networks also.",
"Like the SI model, Bass model also generates S-shaped curve to fit exponential growth patterns.",
"$(Bass)\\frac{dn(t)}{dt}= \\epsilon \\ *\\ m *\\ \\frac{(p+q)^2}{p} \\frac{e^{-(p+q)t}}{(1+\\frac{p}{q}e^{-(p+q)})^2}$ where, $m$ is total number of potential adopters $p$ is the parameter to model external influences $q$ is the parameter to model internal influences Results of both models to map the cascade evolution is given in Figure REF .",
"Each plot in this figure represents their corresponding evolution type, for example Figures REF and REF marks result for the evolution types $A \\rightarrow B$ and $B \\rightarrow C$ respectively.",
"From these results, we note that the performance of the SI model degrades as the evolution types become complex(for example, types $C \\rightarrow $ and $E \\rightarrow D$ ).",
"We evaluate both models performance by the Mean Absolute Percentage Error (MAPE).",
"Error rate of our models are given in Figure REF .",
"Although this evolution problem itself can have its own model, we leave that work to focus in the future.",
"Conclusion and Discussion Online extremism has gained momentum in the past decade due to extensive usage of social media.",
"In this work we have given an extensive study on Gab using its conversations patterns and their related topics.",
"We provide cascade templates for user conversations in Gab as conversation cascades.",
"Dissecting Gab conversations as these cascade types give an intuition on analyzing more viral and responsive cascades.",
"We provided variety of analysis that revolve around these cascades and given models that fits cascade evolution over time.",
"Also, we studied about topics in the form of hashtag co-occurrence and given an algorithm to cluster hashtags into corresponding topics.",
"In future, we plan to incorporate multiple social media forums like Gab, Twitter, and Reddit in the context of polarizing conversations and hate speech.",
"We mainly focus to study information difussion and mutation patterns across online social media during shock events.",
"Given this problem, there are various interesting areas to work in the near future.",
"For example, we can engineer temporal features such as response rate, content features like word or sentence embedding, and features from ground truth network like follower network to model such information flow across platforms.",
"We can embed these features in addition to post level features to predict the amount of hate in a social media."
],
[
"Preliminaries",
"In this section, we briefly describe about the dataset that we use in all experiments, terminologies in our methods and statistics of cascades and topics in the dataset."
],
[
"Dataset Description",
"Gab.com/Gab.ai is a social media forum, founded in 2016, provide a forum to connect and share information among users.",
"Even though the description of the forum looks very similar to most popular social media counterparts like Twitter and Facebook, Gab is known to support individual liberty and committed to contribute for free speech in the social media communityhttps://gab.com/.",
"However, Gab has strong restriction policies on posts and users, which are promoting pornography, terrorism and violence.",
"Users of Gab can share information via posts, post replies, and quotes/reshares.",
"We use the Gab data published by data scraping forum pushshift.iohttps://pushshift.io.",
"Figure REF gives an overview of the dataset as timeseries plot for the number of posts, replies, and quotes appeared in Gab between August 2016 and October 2018.",
"The dataset is a comprehensive collection with 34 million posts, replies, quotes posted between the date range of August 2016 and October 2018, about 15,000 groups and about 300,000 public users information.",
"It is evidential from Figure REF that our dataset comprise of 55% posts, 30% replies, and 15% quotes.",
"This data is available with complete set of metadata like time, attachments, likes, dislikes, replies, quotes along with post and user details."
],
[
"Conversation cascades",
"Microblogging conversations have been widely studied in the context of cascades for a wide spectrum of applications like emotion analysis [5], topic modelling [25] and cascade analysis [26].",
"In this work, we give variety of cascade representations for conversations in Gab and give their in-depth structural and temporal analysis, response rates, and longevity.",
"Figure: Possible shallow cascading structures in replies and reshares of posts and number of posts in each cascade type.",
"As predicted, most of the cascades follow simple patterns(type A).",
"Interestingly, many conversations form cascade type B in which many response come directly to the root post and does not evolve into any other cascade types.",
"Small number of conversations follow cascade type C, which split from the root and each branch from root follows linear pattern.",
"Very few conversations follow cascade type D pattern, which has highly nested structure, both at root level and branch level and cascade type E, which initially follows linear pattern and takes non-linear during the evolution time.Table: Basic statistics of depth, volume, number of unique users, and structural virality(Wiener Index) of all cascade types.",
"Overall, the cascade type E achieves higher structural popularity even though the number of unique user participation is low.With available posts, replies and quotes/reshares from Gab, we construct conversation cascades, where each cascade is one complete conversation.",
"Nodes in each cascade represent original post/reply/quote and edges represent reply/quote of post/reply/quote.",
"The formal definition and the construction process of conversation cascades are given below: Cascade construction: A conversation cascade is a directed graph $G_c=(V_c,E_c)$ , where $V_c$ is a set of posts/replies/quotes and $E_c$ is a set of edges connecting posts ordered by time.",
"We represent a node/post in the cascade as $P(v,t)$ , where $v \\in V_c$ is a post appearing in the social media at time $t$ .",
"There exists a directed edge($v^{\\prime },v$ ), when $P(v,t)$ receives a reply/quote $P(v^{\\prime },t^{\\prime })$ , where $t^{\\prime } > t$ .",
"This process continues each time when a user replies/quotes a post.",
"Thus, root node in conversation cascades represents original post and replies and quotes take branches from the root node(original post).",
"Nodes in second or above level in the cascade take branches again if such nodes in turn get any replies or quotes.",
"In total we constructed 1,721,441 cascades from the Gab data, which comprise of 19,220,059 nodes/posts contributed by 173,581 users and 15,476,852 edges.",
"We evidence from these cascades that conversations in Gab follow one of the five patterns as represented in Figure REF .",
"These conversation cascade types give a generic representation of user engagement patterns over time for a post.",
"These cascades are also used to represent linear and non-linear conversation patterns in Gab.",
"Figure REF gives number of cascades in each conversation cascade pattern.",
"In Table REF , we report statistics of cascade depth, volume, number of unique user participation, and structural virality of all cascade types.",
"We calculate the structural virality of a cascade of size $n$ using Wiener Index(WI) [27] given in the Equation REF , where $d_{ij}$ is the shortest distance of nodes $i$ and $j$ .",
"$WI = \\frac{1}{n(n-1) \\sum _{i=1}^{n} \\sum _{j=1}^{n} d_{ij}}$ From Table REF and Figure REF , we find that cascades of category E attains higher volume and depth with very few user participation and it plays a vital role in Gab conversations.",
"Figure: Depth and Size distribution of each cascade typeWe also analyze depth and volume (number of nodes) of each cascade type and give their corresponding distributions in Figures REF and REF respectively.",
"From Figure REF we note that users in Gab have longer conversations, which take more branches after level 2 in the conversation thread(Cascade types D and E).",
"Cascade type B is not shown in the Figure REF because these conversations split at the root node(post) and terminate at immediate next level.",
"In Figure REF we show distribution of volume(number of nodes/replies/quotes) across cascade types.",
"The volume distribution of the given cascades are similar to the depth distribution given in Figure REF (Cascade types D and E have much engaging participation).",
"Interestingly, we find that cascades of type B have more participation, given that these cascades stop at level 1, compared to cascades of type A."
],
[
"Cascades across topics",
"Different topics spread in different ways over the networks in terms of speed, number of participants and the dynamics of their cascades.",
"We analyze how this intuitively accepted notion applies to cascades on Gab and what topics in particular differ significantly from the others.",
"In section we introduce a procedure by which we classify hashtags into different topics.",
"We give a representative set of hashtags and their topics in Table REF .",
"Then we use the labeled hashtags to classify the cascades where those hashtags appear.",
"Our analysis shows that cascades on more controversial topics have different characteristics than other cascades.",
"They tend to result in larger cascades, majority of them being the same cascade type (Type D).",
"Cascades in these polarizing topics are generated by users that are more strongly connected, i. e. have higher tie strength.",
"We specifically identified three topics, “Antisemitism\", “Anti Islam\", and “White Supremacy\" to be noticeably different from other topics in regards to the nature of the cascades in which they are discussed.",
"We define tie strength of user $u_1$ and $u_2$ as the number of times $u_1$ replies to a post from $u_2$ or $u_2$ replies to a post from $u_1$ .",
"Figure REF shows that users who participated in topics of “Antisemitism\", “Anti Islam\", and “White Supremacy\" have higher tie strength and are more strongly connected which supports the theory that more controversial topics are adopted by users with higher tie strength [28].",
"ADLwww.adl.org believes that white supremacist, hateful, antisemitic bigotry are widespread on Gab.",
"Our findings are aligned with this statement and other studies that argue antisemitism and white nationalist topics are openly expressed on Gab and have great similarities in terms of communities who adopt these topics [22] [29].",
"Figure: Average tie strength of users participated in cascades on different topic.",
"Users participating in cascades on polarizing topics like antisemitism, anti Islam, and white supremacy tend to have higher tie strength, i. e. the frequency of interactions between two users.Table: Topic Categories, a set of examples of hashtags, the number of instances, and the average size of cascades on each category"
],
[
"Topic Discovery of Cascades",
"We introduce a novel procedure for labeling hashtags with the topics they belong to.",
"We first looked at the top 200 most used hashtags on Gab.",
"These hashtags then were classified by subject matter experts into 6 topics.",
"Unlike other works identified on Twitter [28], our work do not cover a broad range of topics.",
"Majority of posts on Gab are on political and rather controversial topics, thus we went few steps deeper and classified these political topics into more specific topics.",
"We used tagdefwww.tagdef.com, tagsfinderwww.tagsfinder.com, Google, and Twitter to find the meaning of hashtags and assigned them to one of the 6 categories.",
"Excluding hashtags that are too broad to be assigned to a specific topic, for instance #gab, #eu, #music, or #welcome, we labeled 126 hashtags.",
"In the next step, we used our algorithm to label other hashtags used on Gab based on the 126 hashtags that we manually labeled.",
"As shown in Algorithm , the inputs of the method are the network of hashtags, list of topics with the set of hashtags assigned to each topic, and a constant value as threshold.",
"We defined the network of hashtags as $G=\\lbrace V, E\\rbrace $ where $V$ is the set of hashtags used on Gab, and $E$ is the set of weighted edges.",
"An edge between two vertices indicates that the two hashtags have appeared in the same post at least once and the weight of the edge represents the number of co-occurrences.",
"We create this network of hashtags [24], but we added weights to the edges to underscore the importance of the number of co-occurrences between two hashtags.",
"In line 3 of the algorithm, an edge is passed to get_node_with_no_topic() method which returns the vertex that is not already assigned to any topic.",
"The method returns null if both vertices of the edge are already assigned topics or if neither has any topic assigned to it.",
"Because first, we are not interested in labeling hashtags that are already labeled and secondly, we cannot label a node if none of its neighbors is labeled.",
"If the method returns a node $u$ we get the topics that are assigned to that node in step 6, and then for each topic $t$ in topics, we increment an integer property of node $v$ , the other vertex of the edge, that represents $t$ by $w$ , the weight of edge $e$ .",
"After all edges have been traversed and the properties of their respective vertices are updated we move to steps 9 to 13 where for each node $n$ in the graph and each topic $t$ in topics, we get the property $p_t$ .",
"If the value of the $p_t$ is greater than the threshold $\\tau $ , we conclude that node $n$ , and the hashtag that it represents, belongs to topic $t$ .",
"After we label hashtags with one or more topics, then for each cascade if hashtags of a topic $t_1$ appear more than C times, we assign that cascade to $t_1$ .",
"Note that a cascade could belong to more than one category.",
"Table REF shows the 6 topics we identified, some examples of the hashtags in each category, as well as the number of cascades and average size of cascades on each topic.",
"To evaluate our algorithm, we designed a procedure where we randomly picked 42 hashtags from the set of 126 labeled hashtags and labeled the rest of the hashtags in the set using our algorithm.",
"Then we compared the results of our algorithm with our manual labeling.",
"FIgure REF shows the distribution of hashtags among different topics.",
"Figure REF shows the ground truth, i. e. manually labeled hashtags, and figure REF shows the results of our algorithm.",
"Since one hashtag could belong to multiple topics, we used the evaluation metrics of multi-label learning algorithms mentioned in [30].",
"Table REF shows how well our algorithm performs in labeling the hashtags with topics, we use accuracy, recall, precision, $F1$ score, Hamming Loss (HL), and Subset Accuracy (SA).",
"Table: Performance of hashtag labeling algorithm.",
"Our algorithm produced high results for conventional performance metrics of accuracy, recall, and precision.",
"Low HL (Hamming Loss) and high SA (Subset Accuracy) are also strong indicatives that our algorithm performs well in classifying hashtags into topics.Topic Discovery of Hashtags [1] G, $\\mathbf {T}$ and $\\tau $ label_hashtags $e \\in E$ u = get_node_with_no_topics(e) v = $e-u$ $u \\ne \\emptyset $ $topics=$ get_topics(v) $t \\in topics$ inc_node_prop(u,t,w) $n \\in V$ $t \\in T$ $p_t =$ get_property(G,n,t) $p_t > \\tau $ add_hashtag_to_topic(G,n,t) Figure: Accuracy of the hashtag labeling algorithm, shows the ground truth, i. e. manually labeled hashtags, and shows how our algorithm classified the same hashtags.",
"Overall accuracy of the algorithm is 71% as given in Table"
],
[
"Response rate in cascades",
"With our proposed cascade types in Section REF , we analyze response time of each cascade type.",
"With this experiment, we aim to produce results that depict how fast posts in these cascades get responses and help growing/evolving the cascades.",
"We also provide a notion for response rate to compare velocity of responses at all levels in a given cascade type.",
"We define response rate($\\mathcal {R}_c$ ) of a cascade type as the average speed of response(s) of posts at a given level/depth of a cascade type($c$ ).",
"Equation REF gives a formulation to calculate response rate at a given level/depth ($l$ ) for a given cascade ($c$ ) and $\\Delta _l^c$ is an average response time for posts at a given level/depth ($l$ ) for a given cascade type ($c$ ).",
"$\\mathcal {R}_c^l = \\frac{1}{\\Delta _c^l}$ Figure: Overview of cascade evolution as a state diagram.",
"All cascades start as type A and they can evolve to the maximum of type D.Figure: Summary distributions of amount of time and posts taken by a cascade to evolve into another.",
"Sudden spikes in the plot are due to anomalies in the data.",
"30% of cascades in each cascade type require less than 2 minutes and the number of cascade evolution decreases as the number of posts increasesThe distribution of average response rates of all cascade types at each level is given in Figure REF .",
"Interestingly, we find that all cascade types start with almost equal and slower response rate and progress with faster responses over time.",
"Importantly, cascades of type C achieve overall higher response rate at earlier levels, even though they do not grow as larger and deeper as other cascades.",
"Also, cascades of type A follow constant response rate like other cascade types and spikes as depth and volume increase.",
"We also find that response rate for other larger cascades such as type D and E is inversely proportional to the distribution of volume of the cascade."
],
[
"Evolution of cascades",
"Given cascade types of Gab conversations, we study their growth patterns and evolution.",
"All Gab conversations/cascading patterns, as given in Figure REF , starts with type A and some of them moderately evolve into other cascade types.",
"All possible transformations within the proposed cascade varieties are given in Figure REF .",
"It is notable from this figure that a conversation cascade can reach its maximum potential by transforming to cascade type D and cascades must evolve into other types(B,C,E) before reaching type D. Providing such evolution patterns and number of occurrences of each cascade from Figure REF , we find that there are significant evolution of cascades in our data.",
"With the availability of evolution patterns in Gab conversations, we provide basic analysis such as time REF and number of posts REF required by a cascade to evolve into another.",
"As a summary of this plot, we present Table REF to give minimum, maximum, average, and standard deviation of number of time and posts required to achieve evolve the cascades.",
"Table: Number of posts and time required by a cascade type to evolve into another.",
"Overall evolution in Gab is slow with smaller number of posts and longer time to perform evolution.Figure REF gives a timeseries on all possible cascade evolution in Gab.",
"Modeling this evolution helps to study intensification of conversations in Gab.",
"Conversations in any social media intensifies when it create interests or controversies among users about the topic.",
"We use traditional models, like Susceptible-Infected [9] and Bass [10] models to fit evolution patterns that exist in Gab conversation cascades.",
"We prefer to use these models, because of their ability to fit sigmoid curves which in turn maps exponential growth or exponential fall [19].",
"Figure: SI and Bass model fit for multiple cascade evolution timeseries.",
"SI model fits for simple evolution types, while the Bass model captures the sharp spike that occurs near the end of the timeseries.The goal of our models is to predict a number of new evolution($\\frac{dn(t)}{dt}$ ) given a time $t$ and a cumulative sum of user parameters($\\epsilon $ ).",
"Equation REF gives a modified equation of SI model.",
"We model $\\alpha ,\\gamma < 1$ to restrict complete participation of susceptibles ($N-n(t)$ ) and infected ones($n(t)$ ) because of an assumption that not all of the previous evolution are responsible for the current evolution.",
"$(SI)\\frac{dn(t)}{dt}= \\epsilon \\ *\\ \\beta \\ *\\ n(t)^{\\alpha }\\ *\\ (N-n(t))^{\\gamma }$ where, $\\beta $ is evolution rate of cascades $N$ is total number of the cascades that evolved $n(t)$ is the cumulative sum of cascades evolved at time $t$ .",
"In other words, total infected ones at time $t$ Figure: Error(%) of SI and Bass model to predict the evolution patterns.",
"Our best model fit is 84% accuracy(in A→CA \\rightarrow C cascade evolution)In Equation REF , we give the Bass model with the user parameter($\\epsilon $ ).",
"Although, Bass model is introduced to describe the process of how new business product is taking effect in population, the model has been widely used in understanding diffusion and influence patterns in social networks also.",
"Like the SI model, Bass model also generates S-shaped curve to fit exponential growth patterns.",
"$(Bass)\\frac{dn(t)}{dt}= \\epsilon \\ *\\ m *\\ \\frac{(p+q)^2}{p} \\frac{e^{-(p+q)t}}{(1+\\frac{p}{q}e^{-(p+q)})^2}$ where, $m$ is total number of potential adopters $p$ is the parameter to model external influences $q$ is the parameter to model internal influences Results of both models to map the cascade evolution is given in Figure REF .",
"Each plot in this figure represents their corresponding evolution type, for example Figures REF and REF marks result for the evolution types $A \\rightarrow B$ and $B \\rightarrow C$ respectively.",
"From these results, we note that the performance of the SI model degrades as the evolution types become complex(for example, types $C \\rightarrow $ and $E \\rightarrow D$ ).",
"We evaluate both models performance by the Mean Absolute Percentage Error (MAPE).",
"Error rate of our models are given in Figure REF .",
"Although this evolution problem itself can have its own model, we leave that work to focus in the future."
],
[
"Conclusion and Discussion",
"Online extremism has gained momentum in the past decade due to extensive usage of social media.",
"In this work we have given an extensive study on Gab using its conversations patterns and their related topics.",
"We provide cascade templates for user conversations in Gab as conversation cascades.",
"Dissecting Gab conversations as these cascade types give an intuition on analyzing more viral and responsive cascades.",
"We provided variety of analysis that revolve around these cascades and given models that fits cascade evolution over time.",
"Also, we studied about topics in the form of hashtag co-occurrence and given an algorithm to cluster hashtags into corresponding topics.",
"In future, we plan to incorporate multiple social media forums like Gab, Twitter, and Reddit in the context of polarizing conversations and hate speech.",
"We mainly focus to study information difussion and mutation patterns across online social media during shock events.",
"Given this problem, there are various interesting areas to work in the near future.",
"For example, we can engineer temporal features such as response rate, content features like word or sentence embedding, and features from ground truth network like follower network to model such information flow across platforms.",
"We can embed these features in addition to post level features to predict the amount of hate in a social media."
]
] | 1906.04261 |
[
[
"DeepcomplexMRI: Exploiting deep residual network for fast parallel MR\n imaging with complex convolution"
],
[
"Abstract This paper proposes a multi-channel image reconstruction method, named DeepcomplexMRI, to accelerate parallel MR imaging with residual complex convolutional neural network.",
"Different from most existing works which rely on the utilization of the coil sensitivities or prior information of predefined transforms, DeepcomplexMRI takes advantage of the availability of a large number of existing multi-channel groudtruth images and uses them as labeled data to train the deep residual convolutional neural network offline.",
"In particular, a complex convolutional network is proposed to take into account the correlation between the real and imaginary parts of MR images.",
"In addition, the k space data consistency is further enforced repeatedly in between layers of the network.",
"The evaluations on in vivo datasets show that the proposed method has the capability to recover the desired multi-channel images.",
"Its comparison with state-of-the-art method also demonstrates that the proposed method can reconstruct the desired MR images more accurately."
],
[
"DeepcomplexMRI: Exploiting deep residual network for fast parallel MR imaging with complex convolution"
]
] | 1906.04359 |
[
[
"Studies on the $B \\to \\kappa \\bar \\kappa$ decays in the perturbative QCD\n approach"
],
[
"Abstract The $B \\to \\kappa \\bar \\kappa$ decays are investigated for the first time in the perturbative QCD formalism based on the $k_T$ factorization theorem, where the light scalar $\\kappa$ is assumed as a two-quark state.",
"Our numerical results and phenomenological analyses on the CP-averaged branching ratios and CP-violating asymmetries show that: (a) the $B_s^0 \\to \\kappa^+ \\kappa^-$ and $B_s^0 \\to \\kappa^0 \\bar \\kappa^0$ decays have large decay rates around ${\\cal O}(10^{-5})$, which could be examined by the upgraded Large Hadron Collider beauty and/or Belle-II experiments in the near future; (b) a large decay rate about $3 \\times 10^{-6}$ appears in the pure annihilation $B_d^0 \\to \\kappa^+ \\kappa^-$ channel, which could provide more evidences to help distinguish different QCD-inspired factorization approaches, even understand the annihilation decay mechanism; (c) the pure penguin modes $B_d^0 \\to \\kappa^0 \\bar \\kappa^0$ and $B_s^0 \\to \\kappa^0 \\bar \\kappa^0$ would provide a promising ground to search for the possible new physics because of their zero direct and mixing-induced CP violations in the standard model.",
"The examinations with good precision from the future experiments will help to further study the perturbative and/or nonperturbative QCD dynamics involved in these considered decay modes."
],
[
"colorlinks,citecolor=nicegreen,linkcolor=nicered GBgbsn Studies on the $B \\rightarrow \\kappa \\bar{\\kappa }$ decays in the perturbative QCD approachResearch exercises for excellent undergraduate students.",
"Liangliang Su(ËÕÁÁÁÁ) Zewen Jiang(½¯ÔóÎÄ) Xin Liu(ÁõÐÂ) [ Corresponding author: ] [email protected] ORCID: 0000-0001-9419-7462 School of Physics and Electronic Engineering, Jiangsu Normal University, Xuzhou 221116, China The $B \\rightarrow \\kappa \\bar{\\kappa }$ decays are investigated for the first time in the perturbative QCD formalism based on the $k_T$ factorization theorem, where the light scalar $\\kappa $ is assumed as a two-quark state.",
"Our numerical results and phenomenological analyses on the CP-averaged branching ratios and CP-violating asymmetries show that: (a) the $B_s^0 \\rightarrow \\kappa ^+ \\kappa ^-$ and $B_s^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0$ decays have large decay rates around ${\\cal O}(10^{-5})$ , which could be examined by the upgraded Large Hadron Collider beauty and/or Belle-II experiments in the near future; (b) a large decay rate about $3 \\times 10^{-6}$ appears in the pure annihilation $B_d^0 \\rightarrow \\kappa ^+ \\kappa ^-$ channel, which could provide more evidences to help distinguish different QCD-inspired factorization approaches, even understand the annihilation decay mechanism; (c) the pure penguin modes $B_d^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0$ and $B_s^0 \\rightarrow \\kappa ^0\\bar{\\kappa }^0$ would provide a promising ground to search for the possible new physics because of their zero direct and mixing-induced CP violations in the standard model.",
"The examinations with good precision from the future experiments will help to further study the perturbative and/or nonperturbative QCD dynamics involved in these considered decay modes.",
"13.25.Hw, 12.38.Bx, 14.40.Nd JSNU-HEP-2019-1 In the conventional quark model, a meson is composed of one quark and one antiquark, i.e., $q\\bar{q}$ , with different coupling of the orbital and spin angular momenta [1], [2], [3].",
"To date, the structure of the $S$ -wave ground state mesons has almost been determined unambiguously, though the $\\eta $ and $\\eta ^\\prime $ ones may contain the component of gluonium( or pseudoscalar glueball) with different extent [4], [5], [6].",
"However, the components of the $P$ -wave mesons are not easily determined.",
"In particular, the description of the inner structure for the light scalar states such as $a_0(980)$ , $\\kappa $ or $K_0^*(800)$ , $\\sigma $ or $f_0(500)$ , and $f_0(980)$ is controversial, e.g., $q \\bar{q}$ , $\\bar{q}\\bar{q} q q$ , meson-meson bound states, etc., and still not well established currently(for a review, see e.g., Refs.",
"[7], [8], [9]).",
"When the light scalar $f_0(980)$ was first observed in the $B \\rightarrow f_0(980) K$ channel, performed by the Belle [10] and BABAR [11] collaborations in 2002 and 2004, respectively, the investigations on the light scalars in the decay productions of the heavy $B$ mesons were naturally considered as a unique insight to explore their underlying structure.",
"With many channels including light scalars of the heavy $B$ meson decays being opened experimentally [12], [9], $B \\rightarrow SP, SV$ (Here, $P$ and $V$ denote the pseudoscalar and vector meson, respectively) decays have been studied extensively at the theoretical aspects with different approaches/methods, for instance, see [13], [14], [15], [16], [17], [18], [19], [20], [21].",
"With the great development of the Large Hadron Collider beauty(LHCb) and Belle-II experiments [22], more and more modes involving one and/or two scalar states in the $B$ meson decays are expected to be measured with good precision in the future.",
"In this work, we will study the charmless hadronic $B \\rightarrow \\kappa \\bar{\\kappa }$ decays (Here, $B$ denotes the nonstrange $B^+$ and $B_{d}^0$ , and strange $B_s^0$ mesons.)",
"for the first time by employing the perturbative QCD(PQCD) approach [23], [24], [25] based on the $k_T$ factorization theorem, where the light scalar $\\kappa $ will be considered as a lowest-lying $q \\bar{q}$ state.",
"Theoretically, the most important part of a nonleptonic decay amplitude is the effective calculation of the hadronic matrix element, in which the essential inputs are the wave functions (or light-cone distribution amplitudes) of the initial and final hadron states that describe the nonperturbative QCD dynamics independent on the processes.",
"The PQCD approach, as one of the presently three popular QCD-inspired factorizations (the other two are QCD factorization approach [26], [27] and soft-collinear effective theory [28], respectively), has the advantages in computing the Feynman amplitudes by conquering the endpoint singularities that exist in the collinear factorization theorem.",
"By keeping the transverse momentum of the valence quark, associated with the Sudakov factors arising from the $k_T$ resummation [29], [30] and threshold resummation [31], the PQCD approach can be well applied to calculate the hadronic matrix element of the nonleptonic $B$ meson decays.",
"Apart from the factorizable emission diagrams, the nonfactorizable emission ones and the annihilation ones can also be perturbatively calculated.",
"Furthermore, even though the origin of the CP violation and the annihilation decay mechanism are currently unknown, the experimental measurements [12], [9] performed by the BABAR, Belle, and LHCb collaborations have confirmed the direct CP-violating asymmetry of the $B \\rightarrow K\\pi $ decays [23], [32] and the large decay rates of the pure annihilation $B_d^0 \\rightarrow K^+ K^-$ and $B_s^0 \\rightarrow \\pi ^+ \\pi ^-$ modes [33], [34] predicted in the PQCD approach.",
"Certainly, the predictions made in the PQCD approach about the branching ratios and CP violations of the $B \\rightarrow PP, PV/VP,$ and $VV$ decays generally agree with the available data within errors.",
"At the quark level, the considered $B \\rightarrow \\kappa \\bar{\\kappa }$ decays are induced by the $\\bar{b} \\rightarrow \\bar{d}$ or $\\bar{b} \\rightarrow \\bar{s}$ transitions, respectively.",
"The weak effective Hamiltonian $H_{\\rm eff}$ for the $B \\rightarrow \\kappa \\bar{\\kappa }$ decays can be written as [35], $H_{\\rm eff}\\, =\\, {G_F\\over \\sqrt{2}}\\left\\lbrace V_{ub}^*V_{uQ} \\left[C_1(\\mu )O_1^{u}(\\mu )+C_2(\\mu )O_2^{u}(\\mu )\\right]- V_{tb}^*V_{tQ} \\sum _{i=3}^{10}C_i(\\mu )O_i(\\mu )\\right\\rbrace \\;,$ with the Fermi constant $G_F=1.16639\\times 10^{-5}{\\rm GeV}^{-2}$ , the light $Q = d, s$ quark, and Wilson coefficients $C_i(\\mu )$ at the renormalization scale $\\mu $ .",
"The local four-quark operators $O_i(i=1,\\cdots ,10)$ are written as current-current(tree) operators ${\\begin{array}{ll}\\displaystyle O_1^{u}\\, =\\,(\\bar{Q}_\\alpha u_\\beta )_{V-A}(\\bar{u}_\\beta b_\\alpha )_{V-A}\\;,& \\displaystyle O_2^{u}\\, =\\, (\\bar{Q}_\\alpha u_\\alpha )_{V-A}(\\bar{u}_\\beta b_\\beta )_{V-A}\\;;\\end{array}}$ QCD penguin operators ${\\begin{array}{ll}\\displaystyle O_3\\, =\\, (\\bar{Q}_\\alpha b_\\alpha )_{V-A}\\sum _{q^{\\prime }}(\\bar{q}^{\\prime }_\\beta q^{\\prime }_\\beta )_{V-A}\\;,& \\displaystyle O_4\\, =\\, (\\bar{Q}_\\alpha b_\\beta )_{V-A}\\sum _{q^{\\prime }}(\\bar{q}^{\\prime }_\\beta q^{\\prime }_\\alpha )_{V-A}\\;,\\\\\\displaystyle O_5\\, =\\, (\\bar{Q}_\\alpha b_\\alpha )_{V-A}\\sum _{q^{\\prime }}(\\bar{q}^{\\prime }_\\beta q^{\\prime }_\\beta )_{V+A}\\;,& \\displaystyle O_6\\, =\\, (\\bar{Q}_\\alpha b_\\beta )_{V-A}\\sum _{q^{\\prime }}(\\bar{q}^{\\prime }_\\beta q^{\\prime }_\\alpha )_{V+A}\\;;\\end{array}}$ electroweak penguin operators ${\\begin{array}{ll}\\displaystyle O_7\\, =\\,\\frac{3}{2}(\\bar{Q}_\\alpha b_\\alpha )_{V-A}\\sum _{q^{\\prime }}e_{q^{\\prime }}(\\bar{q}^{\\prime }_\\beta q^{\\prime }_\\beta )_{V+A}\\;,& \\displaystyle O_8\\, =\\,\\frac{3}{2}(\\bar{Q}_\\alpha b_\\beta )_{V-A}\\sum _{q^{\\prime }}e_{q^{\\prime }}(\\bar{q}^{\\prime }_\\beta q^{\\prime }_\\alpha )_{V+A}\\;,\\\\\\displaystyle O_9\\, =\\,\\frac{3}{2}(\\bar{Q}_\\alpha b_\\alpha )_{V-A}\\sum _{q^{\\prime }}e_{q^{\\prime }}(\\bar{q}^{\\prime }_\\beta q^{\\prime }_\\beta )_{V-A}\\;,& \\displaystyle O_{10}\\, =\\,\\frac{3}{2}(\\bar{Q}_\\alpha b_\\beta )_{V-A}\\sum _{q^{\\prime }}e_{q^{\\prime }}(\\bar{q}^{\\prime }_\\beta q^{\\prime }_\\alpha )_{V-A}\\;.\\end{array}}$ current-current(tree) operators ${\\begin{array}{ll}\\displaystyle O_1^{u}\\, =\\,(\\bar{Q}_\\alpha u_\\beta )_{V-A}(\\bar{u}_\\beta b_\\alpha )_{V-A}\\;,& \\displaystyle O_2^{u}\\, =\\, (\\bar{Q}_\\alpha u_\\alpha )_{V-A}(\\bar{u}_\\beta b_\\beta )_{V-A}\\;;\\end{array}}$ QCD penguin operators ${\\begin{array}{ll}\\displaystyle O_3\\, =\\, (\\bar{Q}_\\alpha b_\\alpha )_{V-A}\\sum _{q^{\\prime }}(\\bar{q}^{\\prime }_\\beta q^{\\prime }_\\beta )_{V-A}\\;,& \\displaystyle O_4\\, =\\, (\\bar{Q}_\\alpha b_\\beta )_{V-A}\\sum _{q^{\\prime }}(\\bar{q}^{\\prime }_\\beta q^{\\prime }_\\alpha )_{V-A}\\;,\\\\\\displaystyle O_5\\, =\\, (\\bar{Q}_\\alpha b_\\alpha )_{V-A}\\sum _{q^{\\prime }}(\\bar{q}^{\\prime }_\\beta q^{\\prime }_\\beta )_{V+A}\\;,& \\displaystyle O_6\\, =\\, (\\bar{Q}_\\alpha b_\\beta )_{V-A}\\sum _{q^{\\prime }}(\\bar{q}^{\\prime }_\\beta q^{\\prime }_\\alpha )_{V+A}\\;;\\end{array}}$ electroweak penguin operators ${\\begin{array}{ll}\\displaystyle O_7\\, =\\,\\frac{3}{2}(\\bar{Q}_\\alpha b_\\alpha )_{V-A}\\sum _{q^{\\prime }}e_{q^{\\prime }}(\\bar{q}^{\\prime }_\\beta q^{\\prime }_\\beta )_{V+A}\\;,& \\displaystyle O_8\\, =\\,\\frac{3}{2}(\\bar{Q}_\\alpha b_\\beta )_{V-A}\\sum _{q^{\\prime }}e_{q^{\\prime }}(\\bar{q}^{\\prime }_\\beta q^{\\prime }_\\alpha )_{V+A}\\;,\\\\\\displaystyle O_9\\, =\\,\\frac{3}{2}(\\bar{Q}_\\alpha b_\\alpha )_{V-A}\\sum _{q^{\\prime }}e_{q^{\\prime }}(\\bar{q}^{\\prime }_\\beta q^{\\prime }_\\beta )_{V-A}\\;,& \\displaystyle O_{10}\\, =\\,\\frac{3}{2}(\\bar{Q}_\\alpha b_\\beta )_{V-A}\\sum _{q^{\\prime }}e_{q^{\\prime }}(\\bar{q}^{\\prime }_\\beta q^{\\prime }_\\alpha )_{V-A}\\;.\\end{array}}$ with the color indices $\\alpha , \\ \\beta $ and the notations $(\\bar{q}^{\\prime }q^{\\prime })_{V\\pm A} = \\bar{q}^{\\prime } \\gamma _\\mu (1\\pm \\gamma _5)q^{\\prime }$ .",
"The index $q^{\\prime }$ in the summation of the above operators runs through $u,\\;d,\\;s$ , $c$ , and $b$ .",
"It is worth mentioning that since we work in the leading order[${\\cal O}(\\alpha _s)$ ] of the PQCD approach, it is consistent to use the leading order Wilson coefficients.",
"For the renormalization group evolution of the Wilson coefficients from higher scale to lower scale, the formulas as given in Refs.",
"[23], [24] will be adopted directly.",
"Figure: Leading order Feynman diagrams for B→κκ ¯B \\rightarrow \\kappa \\bar{\\kappa } decays in the PQCD formalismThe Feynman diagrams of the $B \\rightarrow \\kappa \\bar{\\kappa }$ decays at leading order in the PQCD formalism are illustrated in Fig.",
"REF : Emission topology: Fig.",
"REF (a) and REF (b) describe the factorizable emission diagrams, while Fig.",
"REF (c) and REF (d) describe the nonfactorizable emission ones; Annihilation topology: Fig.",
"REF (e) and REF (f) describe the nonfactorizable annihilation diagrams, while Fig.",
"REF (g) and REF (h) describe the factorizable annihilation ones.",
"Emission topology: Fig.",
"REF (a) and REF (b) describe the factorizable emission diagrams, while Fig.",
"REF (c) and REF (d) describe the nonfactorizable emission ones; Annihilation topology: Fig.",
"REF (e) and REF (f) describe the nonfactorizable annihilation diagrams, while Fig.",
"REF (g) and REF (h) describe the factorizable annihilation ones.",
"In 2013, one of us(X. Liu) with Xiao and Zou ever studied the $B \\rightarrow K_0^*(1430)\\bar{K}_0^*(1430)$ decays in the PQCD approach [36], where the analytic expressions for the factorization formulas and the decay amplitudes were presented explicitly.",
"Therefore, we just need to replace the $K_0^*(1430)$ state in Ref.",
"[36] with the light $\\kappa $ one to obtain easily the corresponding information of the $B \\rightarrow \\kappa \\bar{\\kappa }$ decays in the PQCD approach.",
"Hence, for simplicity, we will not collect the aforementioned formulas in this paper.",
"The interested readers can refer to Ref.",
"[36] for detail.",
"Then, we can turn to the numerical calculations of the CP-averaged branching ratios and CP-violating asymmetries of the $B \\rightarrow \\kappa \\bar{\\kappa }$ decays in the PQCD approach.",
"Before proceeding, some essential comments on the nonperturbative inputs are as follows: (a) For the heavy $B$ mesons, the wave functions and the distribution amplitudes, and the decay constants are same as those utilized in Ref.",
"[36], but with the updated lifetimes $\\tau _{B_d^0} = 1.52$ ps and $\\tau _{B_s^0} = 1.509$ ps, which can be found clearly in the newest Review of Particle Physics [9].",
"(b) For the light scalar $\\kappa $ , the decay constants and the Gegenbauer moments in the distribution amplitudes have been derived at the normalization scale $\\mu =1$ GeV in the QCD sum rule method [37]: the scalar decay constant $\\bar{f}_{\\kappa } = 0.34 \\pm 0.02$ GeV, the vector decay constant $f_\\kappa = \\bar{f}_{\\kappa }/\\mu $ with $\\mu = m_{\\kappa }/(m_s- m_q)$ ($m_{\\kappa }$ , $m_s$ , and $m_q$ stand for the masses of the light scalar $\\kappa $ , the strange quark $s$ , and the nonstrange light quark $u$ and $d$ , respectively.",
"), and the Gegenbauer moments $B_1= -0.92 \\pm 0.11$ and $B_3= 0.15 \\pm 0.09$ .",
"Here, the running current quark masses $m_s = 0.12$ GeV and $m_q = 0.005$ GeV at $\\mu =1$ GeV, which are translated from those in a $\\overline{\\rm MS}$ scale $\\mu \\approx 2$ GeV [9], are adopted in the calculations.",
"Note that the isospin symmetry is assumed in this work.",
"For the light scalar $\\kappa $ mass $m_\\kappa $ , we adopt the value $m_\\kappa = 0.8$ GeV for rough estimations, because this scalar $\\kappa $ has been assumed as the lowest-lying $q \\bar{q}$ state Moreover, as inferred from the newest Review of Particle Physics [9], this state is also with a finite but indefinite width, whose effect, in principle, has to be included to make relevant predictions more precise.",
"Generally speaking, the width effect could result in the enhancement/reduction of the numerical results with different extent [38].",
"However, up to now, to our best knowledge, the essential $S$ -wave $K\\pi $ distribution amplitudes for resonance $\\kappa $ state with the constrained parameters, e.g., Gegenbauer moments, are absent.",
"Therefore, the width effect will be left for future investigations elsewhere.. (c) For the Cabibbo-Kobayashi-Maskawa(CKM) matrix elements, we also adopt the Wolfenstein parametrization at leading order, but with the updated parameters $A=0.836$ , $\\lambda =0.22453$ , $\\bar{\\rho }= 0.122^{+0.018}_{-0.017}$ , and $\\bar{\\eta }= 0.355^{+0.012}_{-0.011}$ [9].",
"(a) For the heavy $B$ mesons, the wave functions and the distribution amplitudes, and the decay constants are same as those utilized in Ref.",
"[36], but with the updated lifetimes $\\tau _{B_d^0} = 1.52$ ps and $\\tau _{B_s^0} = 1.509$ ps, which can be found clearly in the newest Review of Particle Physics [9].",
"(b) For the light scalar $\\kappa $ , the decay constants and the Gegenbauer moments in the distribution amplitudes have been derived at the normalization scale $\\mu =1$ GeV in the QCD sum rule method [37]: the scalar decay constant $\\bar{f}_{\\kappa } = 0.34 \\pm 0.02$ GeV, the vector decay constant $f_\\kappa = \\bar{f}_{\\kappa }/\\mu $ with $\\mu = m_{\\kappa }/(m_s- m_q)$ ($m_{\\kappa }$ , $m_s$ , and $m_q$ stand for the masses of the light scalar $\\kappa $ , the strange quark $s$ , and the nonstrange light quark $u$ and $d$ , respectively.",
"), and the Gegenbauer moments $B_1= -0.92 \\pm 0.11$ and $B_3= 0.15 \\pm 0.09$ .",
"Here, the running current quark masses $m_s = 0.12$ GeV and $m_q = 0.005$ GeV at $\\mu =1$ GeV, which are translated from those in a $\\overline{\\rm MS}$ scale $\\mu \\approx 2$ GeV [9], are adopted in the calculations.",
"Note that the isospin symmetry is assumed in this work.",
"For the light scalar $\\kappa $ mass $m_\\kappa $ , we adopt the value $m_\\kappa = 0.8$ GeV for rough estimations, because this scalar $\\kappa $ has been assumed as the lowest-lying $q \\bar{q}$ state Moreover, as inferred from the newest Review of Particle Physics [9], this state is also with a finite but indefinite width, whose effect, in principle, has to be included to make relevant predictions more precise.",
"Generally speaking, the width effect could result in the enhancement/reduction of the numerical results with different extent [38].",
"However, up to now, to our best knowledge, the essential $S$ -wave $K\\pi $ distribution amplitudes for resonance $\\kappa $ state with the constrained parameters, e.g., Gegenbauer moments, are absent.",
"Therefore, the width effect will be left for future investigations elsewhere.. (c) For the Cabibbo-Kobayashi-Maskawa(CKM) matrix elements, we also adopt the Wolfenstein parametrization at leading order, but with the updated parameters $A=0.836$ , $\\lambda =0.22453$ , $\\bar{\\rho }= 0.122^{+0.018}_{-0.017}$ , and $\\bar{\\eta }= 0.355^{+0.012}_{-0.011}$ [9].",
"Now, we present the numerical results of the $B \\rightarrow \\kappa \\bar{\\kappa }$ decays in the PQCD formalism.",
"Firstly, the PQCD predictions of the CP-averaged branching ratios can be read as follows: $Br(B^+ \\rightarrow \\kappa ^+ \\bar{\\kappa }^0) &=& 5.46^{+0.17}_{-0.06}(\\omega _B)^{+2.25+1.39}_{-1.73-0.37}(B_i)^{+1.41}_{-1.18}(\\bar{f}_{\\kappa })^{+0.13+0.22}_{-0.09-0.19}(\\rm CKM) \\times 10^{-7}\\;;$ and $Br(B_d^0 \\rightarrow \\kappa ^+ \\kappa ^-) &=& 2.86^{+0.19}_{-0.22}(\\omega _B)^{+1.36+0.40}_{-1.00-0.31}(B_i)^{+0.74}_{-0.62}(\\bar{f}_{\\kappa })^{+0.08+0.15}_{-0.07-0.13}(\\rm CKM) \\times 10^{-6}\\;,\\\\Br(B_d^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0) &=& 7.75^{+0.93}_{-1.05}(\\omega _B)^{+4.85+1.53}_{-3.27-0.00}(B_i)^{+1.99}_{-1.67}(\\bar{f}_{\\kappa })^{+0.08+0.27}_{-0.07-0.27}(\\rm CKM) \\times 10^{-7}\\;;$ and $Br(B_s^0 \\rightarrow \\kappa ^+ \\kappa ^-) &=& 1.15^{+0.25}_{-0.29}(\\omega _B)^{+0.83+0.39}_{-0.55-0.00}(B_i)^{+0.29}_{-0.25}(\\bar{f}_{\\kappa })^{+0.00+0.01}_{-0.00-0.01}(\\rm CKM) \\times 10^{-5}\\;,\\\\Br(B_s^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0) &=& 1.55^{+0.24}_{-0.27}(\\omega _B)^{+1.15+0.30}_{-0.75-0.00}(B_i)^{+0.40}_{-0.33}(\\bar{f}_{\\kappa })^{+0.00+0.00}_{-0.00-0.00}(\\rm CKM) \\times 10^{-5}\\;.$ From the Eqs.",
"(REF )-(), one can find the following points: (a) The considered $B \\rightarrow \\kappa \\bar{\\kappa }$ decays have evidently different CP-averaged branching ratios in the PQCD approach, namely, varying from $10^{-7}$ to $10^{-5}$ .",
"Frankly speaking, these numerical results suffer from large theoretical errors mainly induced by the nonperturbative inputs, such as the shape parameter $\\omega _B$ in the $B$ meson distribution amplitude, the scalar decay constant $\\bar{f}_{\\kappa }$ , especially the Gegenbauer moments $B_{i}(i=1,3)$ in the leading twist light-cone distribution amplitude of $\\kappa $ .",
"The uncertainties of the above mentioned parameters need to be constrained by the future precise measurements and/or Lattice QCD or QCD sum rule calculations.",
"(b) The pure annihilation decay of $B_d^0 \\rightarrow \\kappa ^+ \\kappa ^-$ has the same quark structure as that of the measured one $B_d^0 \\rightarrow K^+ K^-$ , whose decay rate predicted in the PQCD approach has been confirmed by the LHCb experiments [39], [40].",
"Therefore, it is expected that the large branching ratio of the $B_d^0 \\rightarrow \\kappa ^+ \\kappa ^-$ mode given in this work could be examined in the LHCb and/or Belle-II experiments.",
"The confirmation of this PQCD result would provide useful hints to understand the inner structure of the light scalar $\\kappa $ .",
"(c) In light of the large $Br(B_d^0 \\rightarrow \\kappa ^+ \\kappa ^-)_{\\rm PQCD}$ while the small $Br(B_d^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0)_{\\rm PQCD}$ , under the assumption of isospin symmetry, it is postulated that a significant cancellation occurred in the $B_d^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0$ decay between the contributions induced by the emission and the annihilation topologies, which, as a matter of fact, can be found clearly from the numerical results for the factorization decay amplitudes presented in Table REF .",
"(d) The decay rates of the $B_s^0 \\rightarrow \\kappa ^+ \\kappa ^-$ and $B_s^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0$ modes indicate a very small contamination induced by the tree annihilation diagrams associated with a CKM-suppressed factor $V_{us} \\sim \\lambda $ in the $\\bar{b} \\rightarrow \\bar{s}$ transition.",
"Meanwhile, relative to $V_{td} \\sim A\\lambda ^3(1-\\rho -{\\it i} \\eta )$ in the $\\bar{b} \\rightarrow \\bar{d}$ transition, the CKM-enhanced factor $V_{ts} \\sim A\\lambda ^2$ involved in these two decays finally resulted in the highly large and close branching ratios around ${\\cal O}(10^{-5})$ .",
"(e) As mentioned above, because of the enhanced factor $r_{\\rm CKM}=|V_{ts}/V_{td}|^2 \\sim 23.6$ [9], the pure penguin modes $B_d^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0$ and $B_s^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0$ have significantly different decay rates, namely, the former one with $7.75^{+5.55}_{-3.83}\\times 10^{-7}$ while the latter one with $1.55^{+1.28}_{-0.86}\\times 10^{-5}$ , respectively, where the errors have been added in quadrature.",
"In light of the large theoretical errors, a precise ratio of these two branching ratios would be more interested, $R_{s/d}^{00}(\\kappa \\bar{\\kappa })&=&\\frac{Br(B_s^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0)}{Br(B_d^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0)} = 20.0^{+1.7}_{-2.4}\\;,$ Similarly, another two interesting ratios $R_{s/d}^{+-}(\\kappa \\bar{\\kappa })$ and $R_{s}^{00/+-}(\\kappa \\bar{\\kappa })$ could be easily obtained, $R_{s/d}^{+-}(\\kappa \\bar{\\kappa })&=&\\frac{Br(B_s^0 \\rightarrow \\kappa ^+ \\bar{\\kappa }^-)}{Br(B_d^0 \\rightarrow \\kappa ^+ \\bar{\\kappa }^-)} = 4.0^{+1.2}_{-1.1}\\;; \\\\R_{s}^{00/+-}(\\kappa \\bar{\\kappa })&=& \\frac{Br(B_s^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0)}{Br(B_s^0 \\rightarrow \\kappa ^+ \\bar{\\kappa }^-)} = 1.3^{+0.1}_{-0.1}\\;,$ It is clearly found that the uncertainties in the above ratios $R_{s/d}^{00}$ , $R_{s/d}^{+-}$ , and $R_{s}^{00/+-}$ are significantly small because the theoretical errors resulted from the hadronic inputs have been cancelled to a great extent.",
"These values are expected to be examined in the future $B$ -physics experiments to help further understand the involved QCD dynamics in depth.",
"(f) In order to understand the contributions arising from different topologies better, the numerical values for the factorization decay amplitudes are presented explicitly in Table REF .",
"One can find the large nonfactorizable emission contributions and the much larger annihilation contributions in the considered $B \\rightarrow \\kappa \\bar{\\kappa }$ decays, especially in the two $B_s^0$ modes.",
"The underlying reason is that the antisymmetric QCD behavior from the only odd terms in the twist-2 distribution amplitude of the light scalar $\\kappa $ [37], $\\phi _{\\kappa }(x,\\mu )&=&\\frac{3}{\\sqrt{6}}x(1-x)\\biggl \\lbrace f_{\\kappa }(\\mu )+\\bar{f}_{\\kappa }(\\mu )\\sum _{m=1}^\\infty B_m(\\mu )C^{3/2}_m(2x-1)\\biggr \\rbrace \\;,$ where $f_{\\kappa }(\\mu )$ and $\\bar{f}_{\\kappa }(\\mu )$ , $B_m(\\mu )$ , and $C_m^{3/2}(t)$ are the vector and scalar decay constants, Gegenbauer moments, and Gegenbauer polynomials, respectively, make the previously destructive interferences become the presently constructive ones between the valence-quark-radiative and valence-antiquark-radiative diagrams in the nonfactorizable emission and annihilation topologies, as already illustrated in Fig.",
"REF .",
"It is worth mentioning that, as can be seen in Table REF , the annihilation diagrams play a dominant role on both of the CP-averaged decay rates and the CP violations of the considered $B \\rightarrow \\kappa \\bar{\\kappa }$ decays in this work.",
"(g) As for the experimental measurements of the predicted large branching ratios, e.g., $Br(B_s^0\\rightarrow \\kappa ^0 \\bar{\\kappa }^0)=1.55^{+1.28}_{-0.86}\\times 10^{-5}$ and $Br(B_s^0 \\rightarrow \\kappa ^+ \\kappa ^-)=1.15^{+0.99}_{-0.67}\\times 10^{-5}$ , we expect the LHCb and/or Belle-II experiments might measure these channels through the Dalitz plot analysis of $B_s^0 \\rightarrow (K\\pi )_\\kappa (K\\pi )_{\\bar{\\kappa }}$ .",
"In principle, the LHCb and Belle-II experiments have the abilities to detect the $B$ meson decay rates with large branching ratios above $10^{-6}$ .",
"Taking $B_s^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0$ mode as an example, the decay rate ${\\cal B}(\\kappa ^0 \\rightarrow K^+ \\pi ^-)$ is $\\frac{2}{3}$ based on the assumption of isospin symmetry in the strong interactions.",
"Therefore, we could obtain a branching ratio ${\\rm BR}(B_s^0 \\rightarrow (K^+ \\pi ^-)_{\\kappa ^0} (K^- \\pi ^+)_{\\bar{\\kappa }^0})\\equiv Br(B_s^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0)\\cdot {\\cal B}(\\kappa ^0 \\rightarrow K^+ \\pi ^-)\\cdot {\\cal B}(\\bar{\\kappa }^0 \\rightarrow K^- \\pi ^+)= 6.89^{+5.69}_{-3.82} \\times 10^{-6}$ .",
"We hope this large value above $10^{-6}$ could be measured by the LHCb and/or Belle-II experiments when the events with high statistics are collected.",
"Certainly, more information of the intermediate state $\\kappa $ demand the studies on the four-body $B_s^0 \\rightarrow K^+ K^- \\pi ^+ \\pi ^-$ decay armed with the $S$ -wave $K\\pi $ distribution amplitudes with well constrained nonperturbative parameters for $\\kappa $ from Lattice QCD and/or experimental measurements.",
"Unfortunately, they are absent currently to our best knowledge theoretically and experimentally.",
"Therefore, this issue has to be left for future studies elsewhere.",
"Table: The factorization decay amplitudes(in units of 10 -3 10^{-3} GeV 3 ^{3}) of the nonleptonic B→κκ ¯B \\rightarrow \\kappa \\bar{\\kappa } decays in the PQCD approach at leading order, where only the central values are quoted for clarifications.",
"(a) The considered $B \\rightarrow \\kappa \\bar{\\kappa }$ decays have evidently different CP-averaged branching ratios in the PQCD approach, namely, varying from $10^{-7}$ to $10^{-5}$ .",
"Frankly speaking, these numerical results suffer from large theoretical errors mainly induced by the nonperturbative inputs, such as the shape parameter $\\omega _B$ in the $B$ meson distribution amplitude, the scalar decay constant $\\bar{f}_{\\kappa }$ , especially the Gegenbauer moments $B_{i}(i=1,3)$ in the leading twist light-cone distribution amplitude of $\\kappa $ .",
"The uncertainties of the above mentioned parameters need to be constrained by the future precise measurements and/or Lattice QCD or QCD sum rule calculations.",
"(b) The pure annihilation decay of $B_d^0 \\rightarrow \\kappa ^+ \\kappa ^-$ has the same quark structure as that of the measured one $B_d^0 \\rightarrow K^+ K^-$ , whose decay rate predicted in the PQCD approach has been confirmed by the LHCb experiments [39], [40].",
"Therefore, it is expected that the large branching ratio of the $B_d^0 \\rightarrow \\kappa ^+ \\kappa ^-$ mode given in this work could be examined in the LHCb and/or Belle-II experiments.",
"The confirmation of this PQCD result would provide useful hints to understand the inner structure of the light scalar $\\kappa $ .",
"(c) In light of the large $Br(B_d^0 \\rightarrow \\kappa ^+ \\kappa ^-)_{\\rm PQCD}$ while the small $Br(B_d^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0)_{\\rm PQCD}$ , under the assumption of isospin symmetry, it is postulated that a significant cancellation occurred in the $B_d^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0$ decay between the contributions induced by the emission and the annihilation topologies, which, as a matter of fact, can be found clearly from the numerical results for the factorization decay amplitudes presented in Table REF .",
"(d) The decay rates of the $B_s^0 \\rightarrow \\kappa ^+ \\kappa ^-$ and $B_s^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0$ modes indicate a very small contamination induced by the tree annihilation diagrams associated with a CKM-suppressed factor $V_{us} \\sim \\lambda $ in the $\\bar{b} \\rightarrow \\bar{s}$ transition.",
"Meanwhile, relative to $V_{td} \\sim A\\lambda ^3(1-\\rho -{\\it i} \\eta )$ in the $\\bar{b} \\rightarrow \\bar{d}$ transition, the CKM-enhanced factor $V_{ts} \\sim A\\lambda ^2$ involved in these two decays finally resulted in the highly large and close branching ratios around ${\\cal O}(10^{-5})$ .",
"(e) As mentioned above, because of the enhanced factor $r_{\\rm CKM}=|V_{ts}/V_{td}|^2 \\sim 23.6$ [9], the pure penguin modes $B_d^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0$ and $B_s^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0$ have significantly different decay rates, namely, the former one with $7.75^{+5.55}_{-3.83}\\times 10^{-7}$ while the latter one with $1.55^{+1.28}_{-0.86}\\times 10^{-5}$ , respectively, where the errors have been added in quadrature.",
"In light of the large theoretical errors, a precise ratio of these two branching ratios would be more interested, $R_{s/d}^{00}(\\kappa \\bar{\\kappa })&=&\\frac{Br(B_s^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0)}{Br(B_d^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0)} = 20.0^{+1.7}_{-2.4}\\;,$ Similarly, another two interesting ratios $R_{s/d}^{+-}(\\kappa \\bar{\\kappa })$ and $R_{s}^{00/+-}(\\kappa \\bar{\\kappa })$ could be easily obtained, $R_{s/d}^{+-}(\\kappa \\bar{\\kappa })&=&\\frac{Br(B_s^0 \\rightarrow \\kappa ^+ \\bar{\\kappa }^-)}{Br(B_d^0 \\rightarrow \\kappa ^+ \\bar{\\kappa }^-)} = 4.0^{+1.2}_{-1.1}\\;; \\\\R_{s}^{00/+-}(\\kappa \\bar{\\kappa })&=& \\frac{Br(B_s^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0)}{Br(B_s^0 \\rightarrow \\kappa ^+ \\bar{\\kappa }^-)} = 1.3^{+0.1}_{-0.1}\\;,$ It is clearly found that the uncertainties in the above ratios $R_{s/d}^{00}$ , $R_{s/d}^{+-}$ , and $R_{s}^{00/+-}$ are significantly small because the theoretical errors resulted from the hadronic inputs have been cancelled to a great extent.",
"These values are expected to be examined in the future $B$ -physics experiments to help further understand the involved QCD dynamics in depth.",
"(f) In order to understand the contributions arising from different topologies better, the numerical values for the factorization decay amplitudes are presented explicitly in Table REF .",
"One can find the large nonfactorizable emission contributions and the much larger annihilation contributions in the considered $B \\rightarrow \\kappa \\bar{\\kappa }$ decays, especially in the two $B_s^0$ modes.",
"The underlying reason is that the antisymmetric QCD behavior from the only odd terms in the twist-2 distribution amplitude of the light scalar $\\kappa $ [37], $\\phi _{\\kappa }(x,\\mu )&=&\\frac{3}{\\sqrt{6}}x(1-x)\\biggl \\lbrace f_{\\kappa }(\\mu )+\\bar{f}_{\\kappa }(\\mu )\\sum _{m=1}^\\infty B_m(\\mu )C^{3/2}_m(2x-1)\\biggr \\rbrace \\;,$ where $f_{\\kappa }(\\mu )$ and $\\bar{f}_{\\kappa }(\\mu )$ , $B_m(\\mu )$ , and $C_m^{3/2}(t)$ are the vector and scalar decay constants, Gegenbauer moments, and Gegenbauer polynomials, respectively, make the previously destructive interferences become the presently constructive ones between the valence-quark-radiative and valence-antiquark-radiative diagrams in the nonfactorizable emission and annihilation topologies, as already illustrated in Fig.",
"REF .",
"It is worth mentioning that, as can be seen in Table REF , the annihilation diagrams play a dominant role on both of the CP-averaged decay rates and the CP violations of the considered $B \\rightarrow \\kappa \\bar{\\kappa }$ decays in this work.",
"(g) As for the experimental measurements of the predicted large branching ratios, e.g., $Br(B_s^0\\rightarrow \\kappa ^0 \\bar{\\kappa }^0)=1.55^{+1.28}_{-0.86}\\times 10^{-5}$ and $Br(B_s^0 \\rightarrow \\kappa ^+ \\kappa ^-)=1.15^{+0.99}_{-0.67}\\times 10^{-5}$ , we expect the LHCb and/or Belle-II experiments might measure these channels through the Dalitz plot analysis of $B_s^0 \\rightarrow (K\\pi )_\\kappa (K\\pi )_{\\bar{\\kappa }}$ .",
"In principle, the LHCb and Belle-II experiments have the abilities to detect the $B$ meson decay rates with large branching ratios above $10^{-6}$ .",
"Taking $B_s^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0$ mode as an example, the decay rate ${\\cal B}(\\kappa ^0 \\rightarrow K^+ \\pi ^-)$ is $\\frac{2}{3}$ based on the assumption of isospin symmetry in the strong interactions.",
"Therefore, we could obtain a branching ratio ${\\rm BR}(B_s^0 \\rightarrow (K^+ \\pi ^-)_{\\kappa ^0} (K^- \\pi ^+)_{\\bar{\\kappa }^0})\\equiv Br(B_s^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0)\\cdot {\\cal B}(\\kappa ^0 \\rightarrow K^+ \\pi ^-)\\cdot {\\cal B}(\\bar{\\kappa }^0 \\rightarrow K^- \\pi ^+)= 6.89^{+5.69}_{-3.82} \\times 10^{-6}$ .",
"We hope this large value above $10^{-6}$ could be measured by the LHCb and/or Belle-II experiments when the events with high statistics are collected.",
"Certainly, more information of the intermediate state $\\kappa $ demand the studies on the four-body $B_s^0 \\rightarrow K^+ K^- \\pi ^+ \\pi ^-$ decay armed with the $S$ -wave $K\\pi $ distribution amplitudes with well constrained nonperturbative parameters for $\\kappa $ from Lattice QCD and/or experimental measurements.",
"Unfortunately, they are absent currently to our best knowledge theoretically and experimentally.",
"Therefore, this issue has to be left for future studies elsewhere.",
"Then, we will discuss the CP-violating asymmetries of the $B \\rightarrow \\kappa \\bar{\\kappa }$ decays in the PQCD approach.",
"The direct and the mixing-induced CP asymmetries ${\\cal A}_{\\rm dir}$ and ${\\cal A}_{\\rm mix}$ are collected as It is worth pointing out that, due to the nonzero ratio $(\\Delta \\Gamma /\\Gamma )_{B_s^0}$ for the $B_s^0-\\bar{B}_s^0$ mixing as expected in the standard model, the third CP asymmetry ${\\cal A}_{\\rm \\Delta \\Gamma _s}$ will appear in the $B_s^0 \\rightarrow \\kappa \\bar{\\kappa }$ decays [36].",
"Here, the quantity $\\Delta \\Gamma $ is the decay width difference of the $B_s$ meson mass eigenstates [41], [42].",
"Moreover, the three quantities describing the CP violations in the $B_s$ meson decays satisfy the relation: $|{\\cal A}_{\\rm dir}|^2+|{\\cal A}_{\\rm mix}|^2+|{\\cal A}_{\\rm \\Delta \\Gamma _s}|^2=1$ .",
"${\\cal A}_{\\rm dir}(B^+ \\rightarrow \\kappa ^+ \\bar{\\kappa }^0) &=& -87.1^{+14.0}_{-7.7}(\\omega _B)^{+1.9+22.3}_{-0.0-8.7}(B_i)^{+0.0}_{-0.0}(\\bar{f}_{\\kappa })^{+0.6+2.1}_{-0.4-1.8}(\\rm CKM)\\times 10^{-2}\\;;$ and ${\\cal A}_{\\rm dir}(B_d^0 \\rightarrow \\kappa ^+ \\kappa ^-) &=&\\hspace{7.11317pt} 15.4^{+0.1}_{-0.7}(\\omega _B)^{+1.4+3.5}_{-1.1-5.5}(B_i)^{+0.0}_{-0.0}(\\bar{f}_{\\kappa })^{+0.1+0.8}_{-0.1-0.7}(\\rm CKM)\\times 10^{-2}\\;, \\\\{\\cal A}_{\\rm mix}(B_d^0 \\rightarrow \\kappa ^+ \\kappa ^-) &=& -80.0^{+0.0}_{-0.3}(\\omega _B)^{+1.6+2.9}_{-1.3-3.6}(B_i)^{+0.0}_{-0.0}(\\bar{f}_{\\kappa })^{+3.5+1.7}_{-3.0-1.6}(\\rm CKM)\\times 10^{-2}\\;; \\\\{\\cal A}_{\\rm dir}(B_d^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0) &\\approx &\\hspace{7.11317pt} 0.0\\;, \\\\{\\cal A}_{\\rm mix}(B_d^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0) &\\approx &\\hspace{7.11317pt} 0.0\\;;$ and ${\\cal A}_{\\rm dir}(B_s^0 \\rightarrow \\kappa ^+ \\kappa ^-) &=& -35.8^{+6.4}_{-9.3}(\\omega _B)^{+2.7+6.0}_{-3.9-0.0}(B_i)^{+0.0}_{-0.0}(\\bar{f}_{\\kappa })^{+1.0+0.2}_{-1.1-0.2}(\\rm CKM) \\times 10^{-2}\\;,\\\\{\\cal A}_{\\rm mix}(B_s^0 \\rightarrow \\kappa ^+ \\kappa ^-) &=&\\hspace{7.11317pt} 12.3^{+3.4}_{-1.5}(\\omega _B)^{+2.8+12.4}_{-3.8-9.5}(B_i)^{+0.0}_{-0.0}(\\bar{f}_{\\kappa })^{+0.4+0.3}_{-0.4-0.3}(\\rm CKM)\\times 10^{-2}\\;,\\\\{\\cal A}_{\\Delta \\Gamma _s}(B_s^0 \\rightarrow \\kappa ^+ \\kappa ^-) &=&\\hspace{7.11317pt} 92.6^{+2.4}_{-4.8}(\\omega _B)^{+0.5+2.8}_{-1.2-1.3}(B_i)^{+0.0}_{-0.0}(\\bar{f}_{\\kappa })^{+0.4+0.0}_{-0.5-0.1}(\\rm CKM) \\times 10^{-2}\\;; \\\\{\\cal A}_{\\rm dir}(B_s^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0) &=&\\hspace{7.11317pt} 0.0\\;,\\\\{\\cal A}_{\\rm mix}(B_s^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0) &=&\\hspace{7.11317pt} 0.0\\;,\\\\{\\cal A}_{\\Delta \\Gamma _s}(B_s^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0) &=&\\hspace{7.11317pt} 1.0\\;.$ in which the definitions of the direct CP violation ${\\cal A}_{\\rm dir}$ , the mixing-induced one ${\\cal A}_{\\rm mix}$ , even the third one ${\\cal A}_{\\rm \\Delta \\Gamma _s}$ arising from the nonnegligible $(\\Delta \\Gamma /\\Gamma )_{B_s^0}$ term are same as those in Ref. [36].",
"From these numerical results of the CP violations of the $B \\rightarrow \\kappa \\bar{\\kappa }$ decays in the PQCD approach, some comments are in order: Generally speaking, these PQCD predictions are not sensitive to the variation of the scalar decay constant $\\bar{f}_{\\kappa }$ as shown in the above Equations.",
"This can be deduced from the tiny vector decay constant $f_{\\kappa }$ in the leading twist light-cone distribution amplitude of the scalar $\\kappa $ meson(See Eq.",
"(REF ) for detail).",
"Furthermore, both of the twist-3 light-cone distribution amplitudes of $\\kappa $ are proportional to the scalar decay constant $\\bar{f}_{\\kappa }$ because of adopting the asymptotic forms for simplicity [36].",
"Based on the definitions, the CP asymmetry is the ratio of the differences of the related branching ratios between $B \\rightarrow \\kappa \\bar{\\kappa }$ and $\\bar{B} \\rightarrow \\bar{\\kappa }\\kappa $ modes to their corresponding summations, then the scalar decay constant $\\bar{f}_{\\kappa }$ will be cancelled naturally.",
"A large direct CP violation for the $B^+ \\rightarrow \\kappa ^+ \\bar{\\kappa }^0$ mode can be observed, $-87.1^{+26.5}_{-11.8}\\%$ , which indicates that the involved penguin contributions are sizable, within large theoretical errors.",
"While, due to the small branching ratio predicted in the PQCD approach, it might not be easily measured in the near future.",
"Both of the $B_d^0 \\rightarrow \\kappa ^+ \\kappa ^-$ and $B_s^0 \\rightarrow \\kappa ^+ \\kappa ^-$ channels exhibit large CP-violating asymmetries, which are expected to be measured with much more possibilities at the LHCb and/or Belle-II experiments because of their large decay rates, namely, $2.86^{+1.62}_{-1.25} \\times 10^{-6}$ and $1.15^{+0.99}_{-0.67}\\times 10^{-5}$ , where the errors have been added in quadrature too.",
"The confirmations from the future measurements on these two modes would provide the evidences not only to support the assumption of the two-quark structure of the light scalar $\\kappa $ in the present work, but also to help distinguish different factorization approaches on clarifying the origin of the strong phase in the heavy meson decays [43], [44].",
"It is interesting to note that the direct and mixing-induced CP violations are naturally zero in both of the pure penguin $B_d^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0$ and $B_s^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0$ decays due to lack of the interferences from the tree contributions in the standard model.",
"Of course, these two channels, especially the latter one with a large branching ratio as $1.55^{+1.28}_{-0.86} \\times 10^{-5}$ , could provide a promising platform to test the possible new physics beyond the standard model.",
"Generally speaking, these PQCD predictions are not sensitive to the variation of the scalar decay constant $\\bar{f}_{\\kappa }$ as shown in the above Equations.",
"This can be deduced from the tiny vector decay constant $f_{\\kappa }$ in the leading twist light-cone distribution amplitude of the scalar $\\kappa $ meson(See Eq.",
"(REF ) for detail).",
"Furthermore, both of the twist-3 light-cone distribution amplitudes of $\\kappa $ are proportional to the scalar decay constant $\\bar{f}_{\\kappa }$ because of adopting the asymptotic forms for simplicity [36].",
"Based on the definitions, the CP asymmetry is the ratio of the differences of the related branching ratios between $B \\rightarrow \\kappa \\bar{\\kappa }$ and $\\bar{B} \\rightarrow \\bar{\\kappa }\\kappa $ modes to their corresponding summations, then the scalar decay constant $\\bar{f}_{\\kappa }$ will be cancelled naturally.",
"A large direct CP violation for the $B^+ \\rightarrow \\kappa ^+ \\bar{\\kappa }^0$ mode can be observed, $-87.1^{+26.5}_{-11.8}\\%$ , which indicates that the involved penguin contributions are sizable, within large theoretical errors.",
"While, due to the small branching ratio predicted in the PQCD approach, it might not be easily measured in the near future.",
"Both of the $B_d^0 \\rightarrow \\kappa ^+ \\kappa ^-$ and $B_s^0 \\rightarrow \\kappa ^+ \\kappa ^-$ channels exhibit large CP-violating asymmetries, which are expected to be measured with much more possibilities at the LHCb and/or Belle-II experiments because of their large decay rates, namely, $2.86^{+1.62}_{-1.25} \\times 10^{-6}$ and $1.15^{+0.99}_{-0.67}\\times 10^{-5}$ , where the errors have been added in quadrature too.",
"The confirmations from the future measurements on these two modes would provide the evidences not only to support the assumption of the two-quark structure of the light scalar $\\kappa $ in the present work, but also to help distinguish different factorization approaches on clarifying the origin of the strong phase in the heavy meson decays [43], [44].",
"It is interesting to note that the direct and mixing-induced CP violations are naturally zero in both of the pure penguin $B_d^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0$ and $B_s^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0$ decays due to lack of the interferences from the tree contributions in the standard model.",
"Of course, these two channels, especially the latter one with a large branching ratio as $1.55^{+1.28}_{-0.86} \\times 10^{-5}$ , could provide a promising platform to test the possible new physics beyond the standard model.",
"In summary, we have studied the CP-averaged branching ratios and the CP-violating asymmetries of the $B \\rightarrow \\kappa \\bar{\\kappa }$ decays in the PQCD approach based on the $k_T$ factorization theorem.",
"The underlying structure of the light scalars are not determined unambiguously yet.",
"Therefore, the light scalar $\\kappa $ was assumed as a lowest-lying $q\\bar{q}$ meson in the present work.",
"It is expected that the productions of the light scalars in the heavy $B$ meson decays could provide many useful information at another different aspect.",
"The predictions in the PQCD approach showed that: (1) The large decay rates above $10^{-6}$ could be found in the $B_d^0 \\rightarrow \\kappa ^+ \\kappa ^-$ , $B_s^0 \\rightarrow \\kappa ^+ \\kappa ^-$ , and $B_s^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0$ channels, which are expected to be measured at the LHCb and/or Belle-II experiments in the near future; (2) The large direct and mixing-induced CP violations could be found in the $B^+ \\rightarrow \\kappa ^+ \\bar{\\kappa }^0$ , $B_d^0 \\rightarrow \\kappa ^+ \\kappa ^-$ , and $B_s^0 \\rightarrow \\kappa ^+ \\kappa ^-$ modes, however, the small branching ratio $Br(B^+ \\rightarrow \\kappa ^+ \\bar{\\kappa }^0)$ might limit its future measurements; (3) The zero direct and mixing-induced CP-violating asymmetries in the standard model of the pure penguin $B_d^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0$ and $B_s^0 \\rightarrow \\kappa ^0 \\bar{\\kappa }^0$ decays would provide a promising platform to search for the possible new physics beyond the standard model once the nonzero CP violations could be detected evidently in these two modes; (4) The QCD dynamics of the light scalar $\\kappa $ is different from that of the $S$ -wave pseudoscalar $K$ and vector $K^*(892)$ mesons, which turned the previously destructive effects into the presently constructive ones in the nonfactorizable emission and annihilation diagrams, consequently led to the large branching ratios.",
"X.L.",
"thanks Prof. Hai-Yang Cheng for valuable discussions.",
"This work is supported in part by the National Natural Science Foundation of China under Grant Nos.",
"11765012 and 11875033, by the Qing Lan Project of Jiangsu Province (No.",
"9212218405), and by the Research Fund of Jiangsu Normal University (No.",
"HB2016004)."
]
] | 1906.04438 |
[
[
"Towards an UV fixed point in CDT gravity"
],
[
"Abstract CDT is an attempt to formulate a non-perturbative lattice theory of quantum gravity.",
"We describe the phase diagram and analyse the phase transition between phase B and phase C (which is the analogue of the de Sitter phase observed for the spherical spatial topology).",
"This transition is accessible to ordinary Monte Carlo simulations when the topology of space is toroidal.",
"We find that the transition is most likely first order, but with unusual properties.",
"The end points of the transition line are candidates for second order phase transition points where an UV continuum limit might exist."
],
[
"Introduction",
"Since the middle of last century physicists have been pursuing the idea of unifying the four fundamental interactions, the strong, the weak, the electromagnetic and the gravitational interactions.",
"The framework of Quantum Field Theory (QFT) unified the first three of them in the so-called Standard Model.",
"Including gravity remains an unsolved problem in a QFT context Going beyond conventional QFT, string theory provides us with a theory unifying the interaction of matter and gravity.",
"Likewise loop quantum gravity uses concepts beyond conventional QFT..",
"Difficulties appear when one tries to formulate a quantum version of Einstein's theory of General Relativity.",
"The naive quantization leads to a perturbatively non-renormalizable theory which cannot be simply included in the unified model of all interactions.",
"The idea of asymptotic safety introduced by Weinberg [1] is an attempt to formulate a non-perturbative QFT of gravity.",
"It assumes that the renormalization group flow in the bare coupling constant space leads to a non-trivial finite-dimensional ultraviolet fixed point around which a new perturbative expansion can be constructed which leads to a predictive quantum theory of gravity.",
"The so-called Exact Renormalization Group program [2], [3], [4], [5], [6] has tried to establish the existence of such a fixed point with a fair amount of success, but relies in the end, despite the name, on truncation of the renormalization group equations.",
"Thus it would be reassuring if other non-perturbative QFT approaches could confirm the exact renormalization group results.",
"Lattice QFT is such a non-perturbative framework and it is well suited to deal precisely with the situation where one identifies fixed points, since these are where one wants to reach continuum physics by scaling the lattice spacing to zero in a way which keeps physics fixed.",
"It has been very successful providing us with results for QCD which are not accessible via perturbation theory.",
"There exists a number of lattice QFT of gravity.",
"One of them, the so-called Dynamical Triangulation (DT) formalism [7], [8], [9], [10], [11], [12] has provided us with a “proof of concept”, in the sense that it has shown us, in the case of two-dimensional quantum gravity[13], [14], [15], [16], that the continuum limit of the lattice theory of gravity coupled to conformal field theories agree with the corresponding continuum theories.",
"Of course there are no propagating gravitational degrees of freedom in two dimensions, but the main issue with the lattice regularization is whether or not diffeomorphism invariance is recovered when the lattice spacing goes to zero.",
"That is the case in the DT formalism, and for the conformal field theories living on the lattice one obtains precisely the non-trivial critical scaling dimensions obtained also in the continuum, i.e.",
"scaling dimensions which are different from the ones in flat spacetime (the so-called KPZ scaling [17], [18], [19]).",
"The DT formalism was extended to higher dimensional gravity [20], [21], [22], [23], [24], [25], [26], [27], but there it was less successful[28], [29].",
"It is not ruled out that the theory can provide us with a successful version of quantum gravity, but if so the formulation has to be more elaborate than the first models (see [30], [31], [32], [33] for recent attempts).",
"However, there is one modification of DT which seems to work in the sense that lattice theory might have a non-trivial continuum limit, the so-called Causal Dynamical Triangulations model (CDT).",
"The model is more constraint than the DT models because one assumes global hyperbolicity, i.e.",
"the existence of a global time foliation.",
"The CDT model of four-dimensional quantum gravity is realized by considering piecewise linear simplicial discretizations of space-time.",
"The simplicial building blocks can be glued together, satisfying the basic topological constraints of global hyperbolicity (as mentioned) and a simplicial manifold structure.",
"The quantum model is now defined using the Feynman path integral formalism, summing over all such geometries with a suitable action to be defined below.",
"The spatial Universe with a fixed topology evolves in proper time.",
"Geometric states at a fixed value of the (discrete) time are triangulated, using regular three-dimensional simplices (tetrahedra) glued along triangular faces in all possible ways, consistent with topology.",
"The common length of the edges of spatial links is assumed to be $a_s$ .",
"Tetrahedra are the bases of four-dimensional $\\lbrace 4,1\\rbrace $ and $\\lbrace 1,4\\rbrace $ simplices with four vertices at time $t$ connected by time links to a vertex at $t\\pm 1$ .",
"All time edges are assumed to have a universal length $a_t$ .",
"To construct a four-dimensional manifold one needs two additional types of four-simplices: $\\lbrace 3,2\\rbrace $ and $\\lbrace 2,3\\rbrace $ (having three vertices at time $t$ and two vertices at $t \\pm 1$ ).",
"The structure described above permits for every configuration the analytic continuation between imaginary $a_t$ (Lorentzian signature) and real $a_t$ (Euclidean signature).",
"Even after Wick rotation the orientation of the time axis is remembered.",
"The spatial and time links may have a different length, and are related by $\\alpha a_s^2=a_t^2$ .",
"The quantum amplitude between the initial and final geometric states separated by the integer time $T$ is a weighted sum over all simplicial manifolds connecting the two states.",
"In the Lorentzian formulation the weight is assumed to be given by a discretized version of the Hilbert-Einstein action.",
"$\\mathcal {Z_{QG}} = \\int \\mathcal {D}_\\mathcal {M}[g]e^{iS_{EH}[g]}$ where $[g]$ denotes an equivalent class of metrics and $\\mathcal {D_\\mathcal {M}}$ [g] is the integration measure over nonequivalent classes of metrics.",
"A piecewise linear manifold where we have specified the length of links defines a geometry without the need to introduce coordinates.",
"In the CDT approach the integration over equivalent classes of metrics is thus replaced by a summation over all triangulations $\\mathcal {T}$ satisfying the constraints.",
"After a Wick rotation the amplitude becomes a partition function $\\mathcal {Z_{CDT}} = \\sum _\\mathcal {T} e^{-S_R[\\mathcal {T]}},$ where $S_R$ is a suitable form of the Einstein-Hilbert action on piecewise linear geometries.",
"There exists such an action, which even has a nice geometric interpretation, the so-called Regge action $S_R$ for piecewise linear geometries [34].",
"In our case it becomes very simple because we have only two kinds of four-simplices which we glue together to form our piecewise linear four-manifold: $S_R = -(K_0 +6 \\Delta ) \\cdot N_0 + K _4 \\cdot (N_{41} + N_{32}) + \\Delta \\cdot N_{41},$ where $N_0$ is the number of vertices in a triangulation $\\mathcal {T}$ , $N_{41}$ and $N_{32}$ are the numbers of $\\lbrace 4,1\\rbrace $ plus $\\lbrace 1,4\\rbrace $ and $\\lbrace 3,2\\rbrace $ plus $\\lbrace 2,3\\rbrace $ simplices, respectively.",
"The action is parametrized by a set of three dimensionless bare coupling constants, $K_0$ , related to the inverse gravitational constant, $K_4$ – the dimensionless cosmological constant and $\\Delta $ – a function of the parameter $\\alpha $ , the ratio of the spatial and time edge lengths (for a detailed discussion we refer to [35] and to the most recent review [36] and for the original literature to [37], [38]).",
"The amplitude is defined for $K_4 > K_4^{\\mathrm {crit}}$ and the limit $K_4 \\rightarrow K_4^{\\mathrm {crit}}$ corresponds to a (discrete) infinite volume limit.",
"In this limit, the properties of the model depend on values of the two remaining coupling constants.",
"The model was extensively studied in the case, where the spatial topology was assumed to be spherical ($S^3$ ) [39], [40], [41], [42], [43], [44], [45].",
"The model could not be solved analytically and the information about its properties was obtained using Monte Carlo simulations.",
"It was found that the model has a surprisingly rich phase structure, with four different phases.",
"The most interesting among the four phases is phase C, where the model dynamically develops a semiclassical background geometry which in some respect is like (Euclidean) de Sitter geometry, i.e.",
"like the geometry of $S^4$ .",
"Both the semiclassical volume distribution and fluctuations around this distribution can be interpreted in terms of a minisuperspace model [46], [47], [48], [49].",
"For increasing $K_0$ phase C is bounded by a first-order phase transition to phase A, where the time correlation between the consecutive slices is absent.",
"For smaller $\\Delta $ phase C has a phase transition to a so-called bifurcation phase, where one observes the appearance of local condensations of geometry around some vertices of the triangulation [50], [51], [52], [53].",
"The phase transition is in this case of second or higher order.",
"For still lower $\\Delta $ the bifurcation phase is linked with the fourth phase, the so-called B phase, where one observes a spontaneous compactification of volume in the time direction, such that effectively all volume condenses in one time slice.",
"The phase transition between the bifurcation phase and the B phase is also of second or higher order [44].",
"The behavior of the model near continuous phase transitions is crucial if one wants to define a physical large-volume limit (a careful discussion of this can be found in [54]).",
"In this respect phase C stands out, the reason being that only in this phase the large scale structure of the average geometry is “observed” (via the Monte Carlo simulations) to be four-dimensional, isotropic and homogeneous, and one can define an infrared semiclassical limit with a correct scaling of the physical volume [42], [46].",
"Via phase C we thus want a renormalization group flow in the bare coupling constant space towards an UV fixed point (the asymptotic safety fixed point), while keeping physical observables fixed.",
"The natural endpoint of such a flow would be a point in the phase diagram where several phases meet.",
"In the early studies it was speculated that there could be a quadruple point, where all four phases meet.",
"Unfortunately the numerical algorithm used was not efficient in this most physically interesting range in the coupling constant space.",
"As a consequence it was not possible to analyze the model in this range.",
"The present article discusses a new formulation of the model, where the spatial topology is assumed to be that of a three-torus ($T^3$ ) [55], [56], [57], rather than that of a three-sphere, which was the topology used in all the former studies.",
"It was found that the four phases in this case are the same as in the spherical model, with the position of phase boundaries shifted a little This may be a finite-size effect.",
"The diagram was determined by analyzing systems with only one volume..",
"The additional, important bonus in this new formulation comes from the fact that the physically interesting region in the bare coupling constant space mentioned above becomes numerically accessible with the standard algorithm used in the earlier studies.",
"We could then observe that the speculative quadruple point, maybe not surprisingly, separates into two triple points, connected by a phase transition line between phase C and the B phase, and not separated by the bifurcation phase (see Fig.",
"REF ).",
"An important point is that we now have access to these triple points directly from phase C and it is thus possible to have a renormalization group flow from the infrared to the potential UV fixed point entirely in the “physical” C phase.",
"Figure: The phase structure of CDT for a fixed number of time slices T=4T = 4 and average lattice volume N ¯ 41 =160k\\bar{N}_{41} = 160\\mathrm {k}.",
"Blue color represents the bifurfactionphase, black color the crumpled phase, green color the C phase andorange color the A phase.The phases of the model were identified for a system with $\\bar{N}_{41} = 160\\mathrm {k}$ , analyzing the structure of geometry at the grid of points in the coupling constant plane shown in Fig.",
"REF , the different phases represented by dots with different colors.",
"In the presented phase diagram the precise position of phase transitions was not determined.",
"This requires a careful study of the infinite volume limit and scaling of the position of phase transition lines with the lattice volume.",
"The most interesting region is the one separating phase C and B where we may observe two triple points.",
"The present paper is the first step in the analysis of this most physically interesting region.",
"We will perform a detailed analysis of the behavior of the model at $K_0=4.0$ in the neighborhood of the phase transition line.",
"We will try to determine the order of the phase transition at this point.",
"We will show that the transition seems to be a first order transition.",
"The results presented in this article show that the most interesting region in the bare parameter space can successfully be analyzed using the standard Monte Carlo algorithm used in the earlier simulations."
],
[
"The phase structure of CDT",
"As mentioned, the phase diagram of the CDT model with a toroidal spatial topology permits us to investigate the properties of the model in an important range of the bare coupling constants, previously inaccessible to numerical measurements.",
"For systems with a spherical spatial topology a detailed analysis of the phase diagram was performed following two lines in the bare coupling constant space.",
"These were the vertical line with varying $\\Delta $ at $K_0 = 2.2$ and the horizontal line at $\\Delta =0.6$ .",
"In the first case it was possible to analyze the phase transition between C and bifurcation phases and between the bifurcation and B phases.",
"In the second case a transition between the C and A phases was studied (see [58] for recent results).",
"The belief coming from the analysis of the spherical case was that if we decrease the value of $\\Delta $ for a fixed value of $K_0$ we necessarily move from C phase to the bifurcation phase and only, for still lower $\\Delta $ , to the B phase.",
"However, changing to toroidal spatial topology we discovered that this is not the case, probably also in the spherical topology.",
"There exists a range of bare coupling constants where C and B phases are directly neighboring.",
"This happens close to the $\\Delta = 0$ line in the range of $K_0$ between, approximately, 3.5 and 4.5.",
"One may expect the existence of two triple points (instead of the previously conjectured quadruple point): one triple point where C, A and B phases meet, and a second triple point where C, bifurcation and B phases meet.",
"Finding the precise location of the triple points may be numerically more difficult than analyzing the generic transition between phase C and B.",
"As a first step in the detailed analysis we have chosen to determine the position and the order of the phase transition between C and B phases along a vertical line at $K_0=4.0$ .",
"This is approximately in the middle between the position of the two triple points.",
"Since the characteristic behavior in the two phases corresponds to different symmetries of the configurations (we have translational symmetry in time in the C phase and a spontaneously breaking of this symmetry in the B phase) we expect a relatively large hysteresis when we cross the phase boundary.",
"We want to find methods which make the hysteresis effect as small as possible.",
"We also expect relatively large finite size effects.",
"An important point in the analysis will be to check how the hysteresis behaves when the system size goes to infinity.",
"The analysis presented in the paper is based on a study of systems with a fixed time period $T=4$ and different (almost) fixed volumes $N_{41}$ .",
"In the earlier studies, it was shown that reducing the period $T$ does not produce significant finite-size effects [58].",
"On the other hand, in particular in the C phase, the average volume per time slice for a fixed total volume gets relatively large, which is very important.",
"In the Monte Carlo simulations we enforce the lattice volume $N_{41}$ to fluctuate around a chosen value $\\bar{N}_{41}$ , so that the measured $\\langle N_{41} \\rangle = \\bar{N}_{41}$ .",
"This is realized by adding to the Regge action (REF ) a volume-fixing term $S_R \\rightarrow S_R + \\epsilon (N_{41}-\\bar{N}_{41})^2.$ In the thermalization process it is essential to fine-tune the value of $K_4$ in such a way that one gets stability of the system volume.",
"This is realized by letting the value of $K_4$ dynamically change by small steps, until the required stable situation is realized.",
"If a value of $K_4$ is too high, we observe that system volume stabilizes below the target value $\\bar{N}_{41}$ .",
"Similarly, if we take it too small, the volume will be too large.",
"Only for $K_4 \\approx K_4^{\\mathrm {crit}}(\\bar{N}_{41})$ fluctuations of volume are centered around $\\bar{N}_{41}$ with the width controlled by $\\epsilon $ .",
"During the thermalization part of the Monte Carlo simulations the algorithm tries to find the optimal value of $K_4$ for a given fixed set of parameters $K_0$ , $\\Delta $ and $\\bar{N}_{41}$ .",
"The whole process of measurements is organized in the following way: We start a sequence of thermalization runs at a set of $\\Delta $ values in the neighborhood of the expected position of the phase transition.",
"The initial configuration of the system is taken to be the small hyper-cubic configuration discussed in reference [55].",
"We choose the target volume $\\bar{N}_{41}$ and let the system size grow towards $\\bar{N}_{41}$ and adapt the $K_4$ value from the guessed initial value.",
"The initial $K_4$ can be chosen either a little below or a little above the guessed critical value.",
"We find that on the grid of $\\Delta $ values we can determine ranges corresponding to the appearance of two different phases, with a relatively sudden jump between the phases.",
"In general the jump is observed between two neighboring values on the grid of $\\Delta $ .",
"The corresponding values of $K_4$ are markedly different in the two phases.",
"Typically the value is smaller for the C phase than for the B phase.",
"We can determine the phase of the system by the measured values of the order parameters (see later for definitions), which are very different in the different phases.",
"The value of $\\Delta $ where the phase transition is observed depends on the initial value of $K_4$ used in the thermalization process.",
"As a consequence, we observe in general two values $\\Delta ^{\\mathrm {crit}}_{low}(N_{41})$ and $\\Delta ^{\\mathrm {crit}}_{high}(N_{41})$ .",
"Both values are determined with the accuracy depending on the grid of $\\Delta $ .",
"We repeat the analysis on a finer grid, which covers the range where we observed phase transitions.",
"We found the most effective procedure is to restart the Monte Carlo evolution from the same small initial configuration as before, but using as the initial values of $K_4$ the ones determined for the C or the B phase from earlier runs in the neighborhood of the transitions, corresponding to $\\Delta ^{\\mathrm {crit}}_{low}(N_{41})$ or $\\Delta ^{\\mathrm {crit}}_{high}(N_{41})$ respectively.",
"A finer grid permits to determine the two positions of the phase transition with better accuracy.",
"The different position of jumps between the two phases (low or high) can be interpreted as the hysteresis effect in a process where we slowly increase the value of the $\\Delta $ parameter or slowly decrease its value.",
"We observe that the size of the hysteresis for a particular choice of $\\bar{N}_{41}$ does not decrease within reasonable thermalization times.",
"By taking a finer grid in $\\Delta $ we can only determine the end points of a hysteresis curve with a better accuracy.",
"We illustrate the situation in Fig.",
"REF .",
"The lines shown were obtained from the measured values of $\\Delta $ and $K_4$ for $\\bar{N}_{41}=160\\mathrm {k}$ .",
"In the range of $\\Delta $ values between $\\Delta ^{\\mathrm {crit}}_{low}(N_{41})$ and $\\Delta ^{\\mathrm {crit}}_{high}(N_{41})$ , depending on the initial value of $K_4$ a system ends either in the B or C phase.",
"This can be interpreted as a range of parameters, where the two phases may coexist.",
"The distribution of the values of the order parameters (to be defined below), characteristic for the two phases, is very narrow.",
"As a consequence, a tunnelling between the two phases is never observed after we have reached a “stable” ensemble of configurations in the thermalization stage.",
"Figure: The plot illustrates the hysteresis measured during simulations for the target volume N ¯ 41 =160k\\bar{N}_{41}=160\\mathrm {k}.",
"The green and blue dots correspond to the location of the phase C side of the phase-transition, while the red and black dots correspond to the location of the phase B side of the phase-transition.",
"The same colors will be used in the next plots, where we compare results for different volumes.The thermalization path chosen above means in practice, that in the beginning, the system grows in a relatively random way from the initially small configuration to the desired target volume $\\bar{N}_{41}$ and then the geometry evolves to a stable range in the configuration space.",
"The first step can be interpreted as a step in the direction typical for the phase A, where correlations between the spatial configurations in the consecutive time slices are small or absent.",
"Only afterwards we reach the domains corresponding to the two phases we study.",
"As a consequence, we expect that the described method will be very well suited to the future analysis of the triple point involving the A phase.",
"The behavior of the pseudo-critical values $K_4^{\\mathrm {crit}}(N_{41})$ is very similar to that of $\\Delta ^{\\mathrm {crit}}(N_{41})$ .",
"This can be seen in Fig.",
"REF , where we show the values of $K_4^{\\mathrm {crit}}(N_{41})$ plotted as a function of $\\Delta ^{\\mathrm {crit}}(N_{41})$ .",
"On both sides of the hysteresis the dependence is approximately linear, which means that values of both pseudo-critical parameters ($K_4^{\\mathrm {crit}}$ and $\\Delta ^{\\mathrm {crit}}$ ) scale in the same way with the lattice volume $\\bar{N}_{41}$ .",
"Extrapolating the lines to a point where they cross permits to determine values for $K_4^{\\mathrm {\\mathrm {crit}}}$ and $\\Delta ^{\\mathrm {\\mathrm {crit}}}$ in the limit $\\bar{N}_{41} \\rightarrow \\infty $ .",
"The fit gives $K_4^{\\mathrm {\\mathrm {crit}}}(\\infty ) = 1.095 \\pm 0.001$ and $\\Delta ^{\\mathrm {\\mathrm {crit}}}(\\infty ) = 0.022 \\pm 0.002 $ .",
"The errors on this and other plots are the estimated statistical errors and include the grid spacing for $\\Delta $ .",
"Figure: The pseudo-critical value K 4 crit (N 41 )K^{\\mathrm {crit}}_4(N_{41}) as a function of Δ crit (N 41 )\\Delta ^{\\mathrm {crit}}(N_{41}).",
"The data points measured for increasing lattice volume N ¯ 41 \\bar{N}_{41} are going from left to right.",
"Center of the black ellipse corresponds to the estimated position of (Δ crit (∞),K 4 crit (∞))(\\Delta ^{\\mathrm {\\mathrm {crit}}}(\\infty ) \\ , \\ K_4^{\\mathrm {\\mathrm {crit}}}(\\infty ) )and its radii corresponds to the estimated uncertainties.",
"Colors of the fits follow the convention used in Fig.",
".Although the size of the hysteresis shrinks with volume $\\bar{N}_{41}$ , the plots indicate that the shrinking process is relatively slow and thus in order to get rid of the hysteresis one should use extremely large lattice volumes, not tractable numerically.",
"The dependence of $\\Delta ^{\\mathrm {crit}}$ on the lattice volume, ranging between $\\bar{N}_{41} = 40\\mathrm {k}$ and $\\bar{N}_{41} = 1600\\mathrm {k}$ is presented in Fig.",
"REF .",
"As it was explained above, the plot contains four sets of data corresponding to the four different points describing the hysteresis (see Fig.",
"REF ).",
"Figure: The pseudo-critical value Δ crit \\Delta ^{\\mathrm {crit}} as a function of N ¯ 41 \\bar{N}_{41}.",
"The solid lines are (one parameter) fits of formula () with fixed common values of γ=1.64\\gamma = 1.64 and Δ crit (∞)=0.022\\Delta ^{\\mathrm {\\mathrm {crit}}}(\\infty ) = 0.022.",
"Colors of the fits follow the convention used in Fig.",
".",
"The dashed line shows a common fit of all data points to the scaling function () with enforced value of γ=1\\gamma = 1 and Δ crit (∞)=0.022\\Delta ^{\\mathrm {crit}}(\\infty )=0.022.The data points can be fitted with the curve $\\Delta ^{\\mathrm {crit}}(\\bar{N}_{41})= \\Delta ^{\\mathrm {crit}}(\\infty ) - A \\cdot \\bar{N}_{41}^{-1/\\gamma }.$ The best fit for the combined sets of data (with fixed $\\Delta ^{\\mathrm {crit}}(\\infty ) = 0.022$ determined above) was obtained for $\\gamma =1.64\\pm 0.18$ .",
"An alternative fit with $\\gamma = 1$ (and the same value of $\\Delta ^{\\mathrm {crit}}(\\infty )$ ) is excluded as can be seen in Fig.",
"REF (the dashed line).",
"The value $\\gamma =1$ would be a strong evidence for a first order transition.",
"The fits were based on data measured for volumes ranging from $\\bar{N}_{41} = {40}\\mathrm {k}$ to $\\bar{N}_{41} = 720\\mathrm {k}$ .",
"The largest volume $\\bar{N}_{41} = $ 1600k was used only for checking consistency with the extrapolations The analogous plot presenting the four sets of the pseudo-critical $K_4^{\\mathrm {crit}}(\\bar{N}_{41})$ values for the same range of volumes is shown in Fig.",
"REF .",
"The experimental points are again well fitted by the formula $K_4^{\\mathrm {crit}}(\\bar{N}_{41}) = K_4^{\\mathrm {crit}}(\\infty ) - B \\cdot \\bar{N}_{41}^{-1/\\gamma },$ where the measured value of $\\gamma = 1.62 \\pm 0.25$ agrees well with the result obtained for $\\Delta ^{\\mathrm {crit}}$ .",
"The fits are represented by curves with different colors, which again follow the convention used in Fig.",
"REF .",
"On the scale used in this plot the green and blue curves practically overlap.",
"Figure: The pseudo-critical value K 4 crit K_4^{\\mathrm {crit}} as a function of N ¯ 41 \\bar{N}_{41}.",
"The solid lines are (one parameter) fits of formula () with fixed common values of γ=1.62\\gamma = 1.62 and K 4 crit (∞)=1.095.K_4^{\\mathrm {\\mathrm {crit}}}(\\infty ) = 1.095.",
"Colors of the fits follow the convention used in Fig.",
"."
],
[
"Order parameters",
"To identify the phases of CDT with toroidal spatial topology we follow methods used in the previous studies.",
"These are based on the analysis of order parameters which have a different behavior in the different phases.",
"We use order parameters which characterize both global and local properties of the simplicial manifolds.",
"The global order parameters were called $\\mathcal {O}_1$ and $\\mathcal {O}_2$ , where $\\mathcal {O}_1 = \\frac{N_0}{N_{41}},\\quad \\mathcal {O}_2 = \\frac{N_{32}}{N_{41}}.$ In each phase the distributions of $N_0$ and $N_{32}$ are very narrow, and practically Gaussian.",
"Phases B and C are characterized by very different average values for the two distributions.",
"The dependence of the order parameters $\\mathcal {O}_1$ and $\\mathcal {O}_2$ on $\\bar{N}_{41}$ at the endpoints of the hysteresis is presented in Fig.",
"REF .",
"The colors follow the convention used in Fig.",
"REF .",
"Figure: The order-parameters 𝒪 1 \\mathcal {O}_1 and 𝒪 2 \\mathcal {O}_2 as a function of N ¯ 41 \\bar{N}_{41} at the endpoints of the hysteresis.",
"The colors correspond to the convention used in Fig.",
".The data presented on the plots correspond for each $\\bar{N}_{41}$ to the four values of the $\\Delta ^{\\mathrm {crit}}(N_{41})$ points, following again the notation of Fig.",
"REF .",
"It is seen that although both pseudo-critical values $K_4^{\\mathrm {crit}}(N_{41})$ and $\\Delta ^{\\mathrm {crit}}(N_{41})$ become very close for increasing $\\bar{N}_{41}$ , this is not the case for the order parameters, which in fact behave in a way similar to that characterizing the first order transition.",
"It means that a transition between the B and C phases becomes very rapid.",
"On the other hand, due to the observed hysteresis, the method used in this analysis chooses a position of measured values for the order parameters slightly away from the true transition point (located inside the hysteresis region) and thus in fact we were not able to perform stable simulations exactly at $K_4^{\\mathrm {crit}}(N_{41})$ and $\\Delta ^{\\mathrm {crit}}(N_{41})$ corresponding to such a transition point We are currently working on the numerical algorithm which would enable tunneling between both sides of the hysteresis region in a single Monte Carlo run and thus enable to define a more precise position of the transition point.. A similar behavior is observed for the set of local order parameters $\\mathcal {O}_3$ and $\\mathcal {O}_4$ defined by $\\mathcal {O}_3 = \\sum _t (n_{t+1}- n_{t})^2,\\quad \\mathcal {O}_4 = \\max o_p .$ Here $n_t$ is the number of tetrahedra shared by $\\lbrace 4,1\\rbrace $ and $\\lbrace 1,4\\rbrace $ four-simplices with bases at time $t$ and $\\sum _t n_t=\\sum _t \\frac{1}{2}N_{41}(t)=\\frac{1}{2}N_{41}$ .",
"$\\max o_p$ is the maximal order of a vertex in a triangulation.",
"The typical behavior of these two order parameters is expected to be different in phases B and C. Phase B is characterized by having a macroscopic fraction of the four-volume concentrated at a single spatial slice corresponding to some time $t$ (in the sense that almost all $\\lbrace 4,1\\rbrace $ and $\\lbrace 1,4\\rbrace $ four-simplices have four vertices at this spatial slice).",
"This is accompanied by the appearance of two singular vertices located at times $t\\pm 1$ and linked to a macroscopic number of four-simplices in a triangulation.",
"As a consequence, in phase B $\\frac{\\mathcal {O}_3}{\\bar{N}_{41}^{2}}$ and $\\frac{\\mathcal {O}_4 }{\\bar{N}_{41}}$ should be of order one.",
"In phase C there is no such degeneracy and for large $\\bar{N}_{41}$ both $\\frac{\\mathcal {O}_3}{\\bar{N}_{41}^{2}}$ and $\\frac{\\mathcal {O}_4 }{\\bar{N}_{41}}$ should approach zero.",
"The behavior of these two order parameters is presented in Fig.",
"REF .",
"Figure: The order-parameters 𝒪 3 N ¯ 41 2 { \\mathcal {O}_3 }{\\bar{N}_{41}^{2}} and 𝒪 4 N ¯ 41 {\\mathcal {O}_4 }{\\bar{N}_{41}} as a function of N ¯ 41 \\bar{N}_{41} at the endpoints of the hysteresis.",
"The colors correspond to the convention used in Fig.",
"."
],
[
"Conclusion and Discussion",
"In the present article we made a detailed study of the phase transition observed between the phase C and the phase B at the value of the dimensionless gravitational coupling constant $K_0 = 4.0$ .",
"The transition appears to be located close to $\\Delta = 0$ .",
"The identification of this region, and the possibility that one can move all the way to the triple points of the phase diagram, staying entirely inside the “physical” C phase, is a good news for the renormalization group program started in [54] (and temporarily put on hold by the discovery of the bifurcation phase).",
"The renormalization group analysis is probably the cleanest way to connect CDT lattice gravity approach to asymptotic safety.",
"The analysis of the relevant coupling constant region was made possible by switching from spherical spatial topology to toroidal spatial topology.",
"In this first study of the interesting region we positioned ourselves in the middle of the B-C phase transition line, between the two triple endpoints and from the analysis of the Monte Carlo data we conclude that the transition is most likely of first order.",
"Since endpoints of phase transition lines often are of higher order, the triple points might well be of second order and one of them could then serve as a UV fixed point for a quantum theory of gravity.",
"We are actively pursuing this line of research.",
"Let us end by some remarks about our quantum gravity model, viewed as a statistical system of four-dimensional geometries.",
"Despite the almost trivial action (REF ), the model has an amazingly rich phase structure, with four different phases, each characterized by very different dominating geometries.",
"In addition, some of the phase transitions have quite unusual characteristics.",
"The transition between phase B and the bifurcation phase is a second order transition [44], but superficially, for a finite volume, it looked like a first order transition.",
"However, analyzing the behavior as a function of the increasing lattice volume the first order nature faded away.",
"Moving towards larger values of $K_0$ , i.e.",
"towards the region we have been investigating in this article, the transition became more and more like a first order transition.",
"With the spherical spatial topology used in [44] one could not get to the region investigated in the present article, but it is natural to conjecture that passing the triple point moving from the bifurcation-B line to the C-B line, the transition changes from second order to first order.",
"However, this first order transition is still somewhat unusual.",
"Firstly, it has kept the characteristics of the second order bifurcation-B transition that the finite size behavior of the pseudo-critical points, given by eqs.",
"(REF ) and (REF ) have non-trivial exponents $\\gamma $ .",
"Secondly, the hysteresis gap goes to zero with increasing volume, which is a non-standard behavior in the case of a first order transition.",
"However, the jumps of the order parameters seem volume independent and that is the main reason that we classify the transition as being a first order transition.",
"The large finite size effects we observe might be related to the global changes of dominant configurations which take place between phase C and phase B, and these global rearrangements might, for finite volumes, have a different “phase-space” in the case of spherical and toroidal topologies.",
"That might explain why our Monte Carlo algorithm can access the B-C transition only in the case of toroidal topology.",
"The statistical theory of geometries is a fascinating area which is almost unexplored for spacetime dimensions larger than two."
],
[
"Acknowledgements",
"DN would like to thank Renate Roll the fruitful discussions and hospitality during his stay at Radboud University in Nijmegen.",
"JGS acknowledges support from the grant UMO-2016/23/ST2/00289 from the National Science centre, Poland.",
"JA acknowledges support from the Danish Research Council grant Quantum Geometry, grant 7014-00066B.",
"AG and DN acknowledges support by the National Science Centre, Poland, under grant no.",
"2015/17/D/ST2/03479."
]
] | 1906.04557 |
[
[
"Identifying Galactic Halo Substructure in 6D Phase-space Using\n $\\sim$13,000 LAMOST K Giants"
],
[
"Abstract We construct a large halo K-giant sample by combining the positions, distances, radial velocities, and metallicities of over 13,000 LAMOST DR5 halo K giants with the Gaia DR2 proper motions, which covers a Galactocentric distance range of 5-120 kpc.",
"Using a position-velocity clustering estimator (the 6Distance), we statistically quantify the presence of position-velocity substructure at high significance: K giants have more close pairs in position-velocity space than a smooth stellar halo.",
"We find that the amount of substructure in the halo increases with increasing distance and metallicity.",
"With a percolation algorithm named friends-of-friends (FoF) to identify groups, we identify members belonging to Sagittarius (Sgr) Streams, Monoceros Ring, Virgo overdensity, Hercules-Aquila Cloud, Orphan Streams and other unknown substructures and find that the Sgr streams account for a large part of grouped stars beyond 20 kpc and enhance the increase of substructure with distance and metallicity.",
"For the first time, we identify spectroscopic members of Monoceros Ring in the south and north Galactic hemisphere, which presents a rotation of about 185 km s^{-1} and mean metallicity is -0.66 dex."
],
[
"Introduction",
"The hierarchical model of galaxies assembly predicts that a series of accretion and merging events led to the formation of the Milky Way [71], [83], [9], [11], [78].",
"Such assembly mechanisms are encoded in the stellar members of the Milky Way's halo, which comprises at least two diffuse components, inner- and outer-halo [14], [15], [16], [3], [1], [81], [80], several streams [63], [32], and numerous overdensities [5], [8].",
"Stellar members of halo streams and overdensities carry information on the merging event that brought these stars into the Galaxy.",
"Therefore, their identification is an important step to understanding galaxy formation.",
"Stars stripped from the merging Galaxy may form structures in the halo in the form of streams, shell or clouds, which can be detected in density space [39], [62], [53], [5], in phase-space [79], [88], [41] or in age space [68], [17], [13].",
"Thanks to the wide-field photometric surveys, such as Sloan Digital Sky Survey [92] and Two Micron All Sky Survey [76], many straightforward observational evidences can be found easily from density map of stars.",
"The most prominent and coherent tidal streams are from the Sagittarius dwarf galaxy [39], [53], which have been traced entirely around the Milky Way [53], [72].",
"Besides the Sgr streams, many other substructures such as the Virgo Overdensity, the Monoceros ring, the Orphan Stream, Pal 5 and GD-1 were also found from photometric surveys [63], [62], [32], [5].",
"However, it is difficult to distinguish the stream members from the Galactic field stars using only sky positions and multi-band photometry.",
"Furthermore, the analysis of simulations showed that the halo streams are distributed smoothly in space after a phase-mixing of $\\sim \\mathrm {10~Gyr}$ , but appear clumped in velocity space, especially in inner parts of the galaxy [37], [38].",
"With the development of the spectroscopic surveys, such as Sloan Digital Sky Survey SDSS [92] and the Large Sky Area Multi-Object Fibre Spectroscopic Telescope [93], it is possible to obtain 3D positions and radial velocities of numerous stars.",
"However, it was impossible to measure proper motions of distant stars ($\\mathrm {>20~kpc}$ ) with the technology of the day.",
"Many studies have indicated that the Galactic stellar halo indeed possesses detectable substructure in 4D position-velocity space.",
"[79] developed a clustering estimator named 4distance to calculate the “distance\" between two stars in four dimensional position-velocity space of $(l,b,d,rv)$ .",
"Combining with friends-of-friends (FoF) algorithm, they identified groups of stars with similar positions and radial velocities from 101 K giants observed by the Spaghetti survey [58].",
"Recently, [88] and [41] adopted 4distance to quantify substructure using much larger samples of halo stars selected from SDSS/SEGUE survey.",
"Furthermore, [41] has applied FoF to identify grouped stars in 4D position-velocity space associated with Sgr streams, Orphan stream, Cetus Polar stream and other unknown substructure.",
"However, the lack of proper motions is likely to reduce the reliability of identified stream members.",
"The second data release of $Gaia$ ($Gaia$ DR2) provides most accurate proper motions (good to $\\mathrm {0.2~mas~yr^{-1}}$ for $\\mathrm {G=17^m}$ ) and parallaxes (good to $\\mathrm {0.1~mas}$ at $\\mathrm {G=17^m}$ ) for more than 1.3 billion sources with $\\mathrm {3^m<G<21^m}$ [30] so far.",
"For majority of $Gaia$ DR2 stars, reliable distance can not be obtained by inverting the parallax, so [2] inferred the distances and their uncertainties of 1.33 billion stars using a weak distance prior that varies smoothly as a function of Galactic longitude and latitude according to a Galaxy model.",
"They pointed out that their approach can infer meaningful distances for stars with negative parallaxes and/or low parallax precision, but will underestimate the distances of distant giants because the distance prior they adopted is dominated by the nearer dwarfs in the model.",
"Therefore, $Gaia$ DR2 parallaxes do not apply to distant giants.",
"Giants of spectral type K are luminous enough ($\\mathrm {-3^m<M_r<1^m}$ ) to be observed in distant halo, and have been specifically targeted by many wide-field spectroscopic surveys to explore the outer halo of the Galaxy.",
"For example, [89] published a catalog of $\\sim $ 6000 halo K giants with distances up to $\\mathrm {80~kpc}$ drawn from the Sloan Extension for Galactic Understanding and Exploration [91].",
"Recently, the fifth data release (DR5) of LAMOST has published about 9 million spectra, containing about 13,000 halo K giants with good distance estimations (extending to distances of $\\mathrm {100~kpc}$ ; Xue et al.",
"2019 in preparation), radial velocities, sky positions and metallicities.",
"Hence, in combination with good proper motions published by $Gaia$ DR2, the sample of K giants with LAMOST spectra constitutes by far the largest set of halo stars with 3D positions, 3D velocities and metallicities.",
"This sample enables the attempt at identifying substructures in full phase space.",
"This paper is organized as follows.",
"In Section , we simply describe the selection of halo K giants and the estimate of their distances.",
"The methodology of quantifying substructure and group finding approach of friends-of-friends are represented in Section .",
"We present the results of quantifying substructure in Section and the identification of substructures in Section .",
"A brief summary is in Section 6."
],
[
"The Sample",
"LAMOST, located in Xinglong station of National Astronomical Observatories of Chinese Academy of Sciences, is a large spectroscopic survey covering -10$^\\circ <\\delta <+90^\\circ $ .",
"It can take 4000 low-resolution ($R \\sim $ 1800) optical spectra in a single exposure to the magnitude as faint as $V = 17.8^{\\rm {m}}$ .",
"Exploring the structure and evolution of the Milky Way is one of the major science goals of LAMOST, and the corresponding target selections are designed to fit the scientific motivation [93], [26], [51].",
"The stellar parameters and radial velocities can be derived by the well-calibrated LAMOST 1D pipeline, which can achieve typical uncertainties of 167 K in effective temperature $T_{\\rm {eff}}$ , $\\mathrm {0.34~dex}$ in surface gravity $\\log g$ , $\\mathrm {0.16~dex}$ in metallicity $\\mathrm {[Fe/H]}$ , and $\\mathrm {5~kms^{-1}}$ in radial velocity $rv$ [85], [84]."
],
[
"K Giants in LAMOST DR5",
"LAMOST DR5 released about 9 million spectra, of which about 5 million spectra have measurements of stellar parameters and radial velocities.",
"K giants are selected using $T_{\\rm {eff}}$ and log $g$ described in [51].",
"The distances of the K giants are determined using a Bayesian method described in [89], of which the fundamental basis is the color-magnitude diagrams (so-called fiducials) of three globular clusters and one open cluster observed by SDSS.",
"The multi-band photometry of LAMOST K giants is obtained from cross-match with Pan-STARRS1 [19] using a match radius of 1.",
"The PS1 magnitudes can be transformed to SDSS system using linear functions of $(g-i)_{P1}$ [29], which are derived through common LAMOST K giants with both PS1 and SDSS magnitudes (Xue et al.",
"2019 in preparation).",
"The extinction is corrected by subtracting the product of $E(B-V)$ from [70] and coefficients (3.303 for SDSS $g$ band and 2.285 for SDSS $r$ band) listed in Table 6 of [69] from apparent magnitudes.",
"Similar to the Bayesian method of [89], the best estimates of the distance moduli and their errors can be estimated using the mean and central $\\mathrm {68\\%}$ interval of the likelihood of the distance moduli.",
"LAMOST $\\log g$ is not accurate to discriminate between red clump stars (RC) and red giants, so we avoid assigning distances to giants below the level of the horizontal branch (HB) defined as $\\mathrm {(g-r)_0^{HB}= 0.087[Fe/H]^2 + 0.39[Fe/H] + 0.96}$ , which is derived by [89] from $\\mathrm {[Fe/H]}$ and the $(g-r)_0$ color of the giant branch at the level of HB of eight clusters.",
"After cross-match with $Gaia$ DR2 with a match radius of 1, there are $\\mathrm {39,774}$ LAMOST K giants with sky positions, distances, radial velocities, and proper motions.",
"Figure REF (upper panel) shows the line-of-sight velocity distribution along with distances of all $\\mathrm {39,774}$ K giants, on which the obvious $sin-shape$ indicates a large portion of disk stars."
],
[
"Halo selection",
"Since we focus on the Galactic halo in this work, we eliminate the K giants within $\\mathrm {5~kpc}$ above or below the Galactic disk plane ($|z|\\leqslant $ 5 kpc).",
"The right-handed Cartesian coordinate is centered at the Galactic center.",
"The $x$ -axis is positive toward the Galactic Center from the Sun, the $y$ -axis is along the rotation of the disk, and the $z$ -axis towards the North Galactic Pole.",
"The Sun's position is at (-8,0,0) kpc [66].",
"All velocities are converted to the Galactic standard of rest (GSR) frame by adopting a solar motion of (+10.0,+5.25,+7.17) km s$^{-1}$ [25] and the local standard of rest (LSR) velocity of 220 km s$^{-1}$ [45].",
"After applying the cut of $|z| > 5$ kpc, the majority of disk stars are eliminated as shown in the lower panel of Figure REF .",
"Finally, we build a sample of 13,554 halo K giants with 3D positions, 3D velocities, and metallicities.",
"The spatial distribution of the halo K giants in $x-z$ plane is shown in Figure REF , and the distributions of distances and velocities are shown in Figure REF .",
"The majority of halo K giants in our sample have Galactocentric distances in the range $\\mathrm {5-60~kpc}$ , with some stars up to $\\mathrm {120~kpc}$ .",
"The errors of velocities and distances are shown in Figure REF .",
"The typical errors are 13% in distance, 7 km s$^{-1}$ in line-of-sight velocity and 20 km s$^{-1}$ in tangential velocities.",
"The sky coverage of the halo K giants with velocity color-coded in Figure REF shows that some K giants in the region of Sgr streams have similar velocities, so next we will detect and identify the substructures in position-velocity space from LAMOST halo K giants."
],
[
"6Distance and Friends-of-friends Algorithm",
"We now start quantifying the presence of any kinematic substructure and identifying members of the substructure in 6D phase-space using LAMOST halo K giants.",
"The kinematically cold streams are not strongly phase-mixed, so the adjacent stars in stellar streams are supposed to have similar velocities.",
"Here, we follow [79] and [41] and develop a statistic that focuses on the incidence of close pairs in $(l,b,d,V_{los},V_l,V_b)$ , and then we combine the friends-of-friends algorithm to group stars that are possible in structure."
],
[
"6Distance",
"We develop 6Distance from 4distance [79] to calculate a 6D separation of $(l,b,d,V_{los},V_{l},V_{b})$ of any two stars.",
"$(l,b)$ are celestial position in the Galactic coordinate system, $d$ is distance to the Sun, $V_{los}$ is line-of-sight velocity, and $(V_{l},V_{b})$ are tangential velocities along $(l,b)$ .",
"All velocities of $(V_{los},V_{l},V_{b})$ are in GSR frame (see Section REF ).",
"6Distance between two stars $i$ and $j$ is defined as follows: $\\delta ^2_{6d}=\\omega _{\\theta }\\theta ^2_{ij}+\\omega _{\\Delta {d}}(d_i-d_j)^2+\\omega _{\\Delta {V_{l}}}(V_{l,i}-V_{l,j})^2+\\omega _{\\Delta {V_{b}}}(V_{b,i}-V_{b,j})^2+\\omega _{\\Delta {V_{\\rm {los}}}}(V_{\\rm {los},i}-V_{\\rm {los},j})^2,$ where $\\theta _{ij}$ is the great circle distance between two stars and calculated by: $\\cos {\\theta _{ij}}= \\cos {b_i}\\cos {b_j}\\cos {(l_i-l_j)}+\\sin {b_i}\\sin {b_j}.$ The five weights $\\omega _{\\theta }, \\omega _{\\Delta {d}}, w_{\\Delta V_{\\rm {los}}}, w_{\\Delta V_l},$ and $w_{\\Delta V_b}$ are used to normalize the corresponding components, and define as follows: $\\begin{array}{l}\\omega _{\\theta } = \\frac{1}{\\pi ^2},\\\\\\omega _{\\Delta {d}} = \\frac{1}{100^2} \\frac{(d_{\\rm {err}}(i)/d(i))^2+(d_{\\rm {err}}(j)/d(j))^2}{2<d_{\\rm {err}}/d>^2},\\\\\\omega _{\\Delta {V_*}} = \\frac{1}{500^2}\\frac{V^2_{\\rm {*,err}}(i)+V^2_{\\rm {*,err}}(j)}{2<V_{\\rm {*,err}}>^2},\\\\\\end{array}$ where $V_*$ stands for $V_l, V_b$ , or $V_{\\rm {los}}$ , $<...>$ denotes the average over all stars.",
"The constants in the weights are the largest angular separation ($\\pi $ ), the largest heliocentric distance separation (100 kpc), and the largest velocity separation (500 km s$^{-1}$ ), for LAMOST halo K-giant sample.",
"[79] and [88] had pointed out that 4Distance is insensitive to small changes in the weighting factors.",
"We also tried weights ($w_\\theta =\\frac{1}{<\\theta ^2>}$ , $w_{\\Delta d}=\\frac{1}{<(\\Delta d)^2>}$ , $w_{\\Delta V_*}=\\frac{1}{<(\\Delta V_*)^2>}$ ) defined by [88] and find that different weights affect little to the substructure quantification and identification."
],
[
"The diffuse halo system",
"If position-velocity substructure is present, it is expected that the distribution of $\\delta _{6d}$ for the observed sample has more close pairs than the null hypothesis of a diffuse halo system where positions and velocities are uncorrelated.",
"We construct the diffuse halo by only shuffling distances and velocities of our sample, but keeping the angular positions: $\\delta ^2_{6d_{r}}=\\omega _{\\theta }\\theta ^2_{ij}+\\omega _{\\Delta {d}}(d_{i_{r}}-d_{j_{r}})^2+\\omega _{\\Delta {V_{l}}}(V_{l,i_{r}}-V_{l,j_{r}})^2+\\omega _{\\Delta {V_{b}}}(V_{b,i_{r}}-V_{b,j_{r}})^2+\\omega _{\\Delta {V_{\\rm {los}}}}(V_{{\\rm {los}},i_{r}}-V_{{\\rm {los}},j_{r}})^2,$ where $\\omega _{\\theta }, \\omega _{\\Delta {d}}, w_{\\Delta V_{\\rm {los}}}, w_{\\Delta V_l}, w_{\\Delta V_b}$ , and the indices $(i,j)$ are exactly the same as in $\\delta _{6d}$ , but $(i_r,j_r)$ are shuffling indices.",
"The selection function of LAMOST K giants varies with line-of-sight [52].",
"However, it is a reasonable assumption that the distance of the stars in the same part of the sky are uncorrelated to the sample selection.",
"Therefore, we do not shuffle the angular positions when constructing the diffuse halo system.",
"Now, we can quantify the degree of substructure in LAMOST halo K giants by comparing the cumulative distribution of $\\delta _{6d}$ for halo K-giant sample, $N_{obs}(<\\delta _{6d})$ , with those of 100 null hypotheses of diffuse halo system $N_{null}(<\\delta _{6d})$ .",
"Figure REF shows that $N_{obs}(<\\delta _{6d})$ exceeds $N_{null}(<\\delta _{6d})$ obviously for small values of $\\delta _{6d}$ , which means there are more close pairs in halo K giants.",
"Since the null hypotheses of diffuse halo system have the same selection function with LAMOST halo K giants, more close pairs in the halo K giants is unlikely to be a result of selection function.",
"Consequently, LAMOST halo K giants have substructure indeed."
],
[
"Friends-of-Friends Algorithm",
"The quantification of substructure is just the first step, and the identification of streams is of particular importance to understand the formation of the Milky Way, such as finding the progenitors of streams, exploring the chemical properties and mass of the progenitors, and constraining the dynamics of the Milky Way.",
"FoF is a popular percolation algorithm of group finding.",
"It defines groups that contain all stars separated by 6Distance less than a given linking length.",
"[41] pointed out that FoF algorithm tends to find groups in the region of higher stellar density.",
"As shown in Figure REF , LAMOST mainly observes northern Galactic hemisphere.",
"[87] found the Galactic halo density profile traced by LAMOST halo K giants shows a single power law with index of $-4\\sim -5$ .",
"Therefore, we adopt a sky-distance-dependent linking length (i.e., we divide our sample and allocate larger linking lengths for southern Galactic hemisphere and distant sub-samples).",
"Sgr streams are the most prominent, coherent and widely studied tidal streams in the Milky Way, so its a good criterion to test our linking length.",
"We choose linking length for each part by getting enough reliable members of Sgr streams.",
"The reliability of Sgr members is evaluated by positions and velocities of Sgr stream in literature.",
"We will show below (Section REF ) that the Sgr members obtained by our linking lengths are very consistent with simulation [48] and observations [7], [47].",
"The details about the sub-samples and linking lengths will be discussed in Section .",
"Obviously, the method employed here to identify stars in each substructure has an intrinsic uncertainty due to the choice of the linking length and the working coordinates (i.e.",
"position-velocity space in this paper) itself.",
"Therefore, the stellar samples associated to streams or overdensities in this paper suffer of contamination due to the mentioned reasons.",
"However, it is not easy to quantify such contamination exactly.",
"The Sgr streams are very coherent and dense in phase space, so the linking length suitable to identify Sgr streams should be a stringent choice.",
"From the comparison with some known substructure properties in Section , we find the fraction of contamination is not high."
],
[
"Results on Quantifying Substructure in LAMOST K Giants",
"Both observations and simulations found that the Galactic halo is comprised of at least two overlapping components, an inner halo and an outer halo, with different metallcities, spatial distribution, and kinematics.",
"The inner halo is the dominant component at galactocentric distance up to $\\sim $ 15-20 kpc and for metallicity $\\mathrm {[Fe/H]>-2.0}$ dex, while the outer halo dominates the region beyond 20 kpc and at metallicity $\\mathrm {[Fe/H]<-2.0}$ dex [14], [15], [23], [42], [3], [46], [1], [35], [44], [81], [80].",
"The large sample size of LAMOST halo K giants enables us to quantify the substructure in inner halo and outer halo, as well as in different ranges of metallicity, and to test the contribution of Sgr streams."
],
[
"Substructure in Inner and Outer Halo",
"As predicted by the hierarchical galaxy formation model, substructures orbiting in outer halo are short after infall and very coherent in space, but substructures in inner halo are long after infall and spatially well-mixed [36].",
"Recent studies used main-sequence turnoff stars (MSTO), blue horizontal branch stars (BHB), and K giants to quantify the degree of substructure and found the Galactic halo significantly more structured at larger radii $r_{\\rm {gc}} > $ 20 kpc [4], [88], [20], [41], [68], [17].",
"Many cosmological simulations show a fully phase-mixed inner halo and increasing fraction of the substructure with distance [12], [59], [11], [24], [21], [65], [13].",
"To test it, we divide LAMOST halo K giants into two sub-samples - one with $\\mathrm {5 ~kpc<}$ $r_{\\mathrm {gc}}$ $\\mathrm {<20~kpc}$ and the other with $r_{\\mathrm {gc}}$ $\\mathrm {>20~kpc}$ , and compare the substructure signals in them.",
"Figure REF shows that the sub-sample beyond $\\mathrm {20~kpc}$ presents a stronger structure signal, $\\sim $ 3 times more than halo stars within $\\mathrm {20~kpc}$ at $lg(\\delta _{6d})=-1.0$ .",
"It means the outer halo is more structured than the inner halo, which is consistent with the previous findings based on observations and simulations."
],
[
"Substructure Dependence on Metallicity",
"Covering large range of metallicities makes K giants good representative tracers of Galactic stellar halo.",
"Metallicity of accreted stars can be used to infer the mass of their progenitor according to the mass-metallicity relation [49].",
"The relation tells us that if a massive dwarf galaxy is accreted, its stellar populations are likely to be metal-rich.",
"The meaningful statistics require large enough sample.",
"We divide LAMOST halo K giants into three sub-samples with comparable sizes: one with [Fe/H] $<$ -1.6 dex, one with -1.6 dex $\\leqslant $ [Fe/H] $<$ -1.2 dex, and another with [Fe/H] $\\geqslant $ -1.2 dex.",
"Figure REF shows the substructure measurements of the three sub-samples.",
"The most metal-rich sub-sample has the strongest substructure signal, with $\\sim $ 9 times more than diffuse halo at $lg(\\delta _{6d})=-1.0$ .",
"The sub-sample with intermediate metallicity has $\\sim $ 3 times more pairs than diffuse halo at $lg(\\delta _{6d})=-1.0$ , while the most metal-poor sub-sample shows the weakest substructure signal, with $\\sim $ 2.5 more pairs than diffuse halo at $lg(\\delta _{6d})=-1.0$ .",
"These results suggest that the substructure signal is increasing with metallicity."
],
[
"Contribution of Sgr Streams to the Substructure",
"To study the contribution of the Sgr Stream to the substructure strength, we test the substructure-metallicity relation in two cases, with and without the Sgr Stream stars.",
"In the non-Sgr Stream case, all the K giant stars with $|B|< 12^\\circ $ are removed following [53] and [5].",
"Figure REF shows the distribution of the relation for the stars with different metallicity ranges in the two cases, solid lines represent the results for all K giant stars and dashed lines for those out of the plane.",
"We can find a clear relation that the substructure strength is higher for the metal-richer sample in both cases, and a significant decrease of the substructure strength with $lg(\\delta _{6d})=-1.0$ between the two cases.",
"What's more, the results of the metal-richer stars decrease more than that of the metal-poorer samples, e.g.",
"the substructure strength of the most metal-rich sample decreases from 0.75 down to 0.53.",
"While the strength of the most metal-poor samples decrease from 0.39 down to 0.33.",
"The difference between the two cases indicates that the Sgr Stream significantly enhances this relation, which was also claimed by [41].",
"All results above are suggesting that LAMOST is able to provide more help for further substructure investigation in the halo, including the Sgr Stream."
],
[
"FoF Results",
"As described in Section REF , we divide our sample and allocate different linking lengths for each sub-sample.",
"Specifically, we divide our sample into 7 sub-samples according to the sky coverage and distance, as shown in Table REF .",
"Note that there are some overlapped regions between the 7 sub-samples.",
"The overlapped regions of sub-samples will produce common grouped members in the result of FoF groups.",
"We will remove the common members from distant sub-sample groups to make sure the grouped members are unduplicated.",
"By comparing our Sgr groups with the known properties of Sgr stream (e.g., distance, position, and velocities), we determine the linking length for each sub-sample.",
"The specific physical sizes of each component corresponding to the linking lengths can be found in Table REF .",
"The physical size is assuming two stars have 5 identical components of 6 phase-space, then calculating the difference component at a given linking lengths.",
"For example, if two stars have identical values of $l,b,d,V_{l},V_{b}$ , a difference of 25 km s$^{-1}$ in $V_{\\rm {los}}$ would produce a $\\delta _{6d}$ of 0.05.",
"Finally, we identify 25 groups (1517 K giants), associated to 5 known substructures: Sgr stream [39], [53], Monoceros Ring [62], Virgo Overdensity [62], Hercules-Aquila Cloud [6], and Orphan Stream [31], [6].",
"Besides, 18 groups (350 K giants) can not be linked to any known substructure, so they may relate to some unknown substructures.",
"In total, 1867 grouped stars are identified as shown in Figure REF , and the known substructures' sky distribution is shown in Figure REF .",
"The corresponding properties of known substructures and unknown groups are listed in Table and Table , respectively."
],
[
"Attributing Groups to Sgr Stream",
"Sgr stream is the most prominent stellar stream, and it has become an important tool for studying the Milky Way halo.",
"In Section , the spatial and velocity distributions have shown the exist of Sgr stream in the sample of LAMOST K giants (see Figure REF and Figure REF ).",
"In this section, we link the FoF groups to Sgr streams by comparing them with models (LM10) and observations [7], [47].",
"Comparing with the five most recent pericentric passages of LM10 model, we find 11 groups match well with LM10 model (see Figure REF ), of which 8 groups belong to Sgr leading arm (blue stars), and 3 groups belong to Sgr trailing arm (red stars).",
"Figure REF shows Sgr streams traced by K giants have larger dispersion (even offset) in distance and tangential velocities than LM10 model.",
"Larger dispersion may be caused by the errors of distances.",
"Unlike “standard candles\" (e.g., BHB with distances good to 5%, RR Lyrae stars with distances good to 3% and red clump stars with distances better than 10% ), K giants have a typical error of about 15% because their intrinsic luminosities vary by two orders of magnitude with color and depend on metallicity and age.",
"Given that tidal stripping generally eats away a satellite from the outside, so more recent pericentric passages means smaller mean internal (to the dwarf galaxy) orbital radii before they were unbound, and having higher metallicity [54], [34].",
"Figure REF shows the Sgr trailing members are located in more recent pericentic passages than Sgr leading members, and the mean [Fe/H] value of the Sgr trailing members are indeed higher than that of the Sgr leading members.",
"Besides the matched groups with LM10 model, there are 2 groups beyond distance range of LM10 model shown as the grey stars in Figure REF .",
"However, they match well with Sgr debris found by [7].",
"[47] and [7] traced the Sgr streams using red clump stars (RC), blue horizontal branch stars (BHB), main-sequence turn-off stars (MSTO), and red giants (RGB) drown from SDSS.",
"Figure REF shows that the members associated with Sgr streams match well in line-of-sight velocity with tracks found by [7] using SDSS giant stars, but locate closer than the tracks traced by BHB stars and RC stars."
],
[
"Attributing Groups to Monoceros Ring",
"Monoceros Ring is a large overdensity firstly discovered by [62], and subsequent studies have shown it is a ring-like low latitude structure and could potentially encircle the entire galaxy [90], [40], [67].",
"[90] traced the structure from $l =180^\\circ $ to $227^\\circ $ with SDSS faint turnoff stars ($(g-r)_0=0.2$ , $g_0$ =19.45).",
"They found the substructure extends 5 kpc above and below the plane of the Galaxy, and stars at southern portion is about 2 kpc farther than those of northern portion.",
"[67] used M giants from 2MASS, and they found the structure both in the north and south hemispheres and spanned at least 100$^\\circ $ .",
"[40] detected the structure from colour-magnitude diagram in many lower latitude ($|b|<50^\\circ $ ) fields of Isaac Newton Telescope Wide Field Survey.",
"Their structure from $(V-i)_0 \\sim 0.45$ , $V_0\\sim 0.9$ curved to $(V-i)_0 \\sim 1.0$ , $V_0\\sim 21.45$ in colour-magnitude diagram, which was also seen in the SDSS Monoceros fields [62].",
"[77] found the structure stretching from 100$^\\circ $ to 230$^\\circ $ in Galactic longitude, and covering from -30$^\\circ $ to 35$^\\circ $ in Galactic latitude using Pan-STARRS1 survey.",
"[50] identified the structure from SEGUE spectroscopy in northern Galactic hemisphere, and found a good match with the disrupting dwarf galaxy model by [64].",
"At present, there is little consensus on the origin of the Monoceros Ring.",
"Some studies attributed Monoceros Ring to the accretion debris from satellite [90], [55].",
"While some studies argued that it may be parts of the flare or warp of the disk [57], [56], [18], [33].",
"Rencently, [86] discovered an oscillating asymmetry in the disk in the direction of the anticenter, and associated the third oscillation line with the Monoceros Ring.",
"In this work, we identify spectroscopic members of Monoceros Ring in both northern and southern Galactic hemisphere for the first time.",
"There are four groups belonging to Monoceros Ring, of which one group is in the northern Galactic hemisphere and the other three groups are in the southern Galactic hemisphere.",
"The members of Monoceros Ring mostly locate at 5-7 kpc from Galactic disk plane and show a mean rotation of $\\sim $ 185 km s$^{-1}$ , and mean metallicity is -0.66 dex (see Figure REF ).",
"Comparing with [22], Gaia DR2 proper motions of Monoceros Ring show smaller dispersion and slighter gradients with Galactic latitude than SDSS-Gaia-DR1 shown as Figure REF .",
"We compare the Monoceros members with the simulation by [64], which modeled the Monoceros Ring as the result of a disrupted dwarf galaxy.",
"Figure REF shows the Monoceros members are consistent with the model in the northern Galactic hemisphere, but more distant and lower in Galactic latitude than the model in the southern Galactic hemisphere.",
"The values of mean rotation velocity and metallicity for the Ring may reflect the contamination from other Galactic components, the thick disk in particular.",
"Thus, the data-model inconsistency is likely caused by different origin mechanisms, or by contamination from other Galactic components."
],
[
"Attributing Groups to Virgo Overdensity",
"The substructure in Virgo constellation is very complex, and its nature is still uncertain.",
"Because of a much higher stellar density exhibited by turnoff stars from SDSS, this region has become known as the “Virgo Overdensity\" [62].",
"Virgo Overdensity is located 10$\\sim $ 20 kpc away from the Sun, over 1000 deg$^2$ [61], [43], [10], [28].",
"We find four groups located in Virgo Overdensity.",
"As shown in Figure REF , the Virgo Overdensity members has a mean metallicity of -1.31 dex, and heliocentric distance from 13 to 25 kpc."
],
[
"Attributing Groups to Hercules-Aquila Cloud",
"Hercules-Aquila Cloud was found as an overdensity using MSTO in SDSS DR5 by [6].",
"They suggested that this cloud covers a huge area of sky, centered on Galactic longitude $l \\sim 40^\\circ $ , Galactic latitude $b$ from $- 50^\\circ $ to $+ 50^\\circ $ , and line-of-sight velocity $V_{\\rm {los}} \\sim 180\\ \\rm {km\\ s^{-1}}$ .",
"Subsequently, [73] used RR Lyrae from SDSS found the Hercules-Aquila Cloud contained at least 1.6 times stellar density of the halo at heliocentric distance of 15 to 25 kpc, and its mean metallicity is similar to Galactic halo.",
"[82] found the heliocentric distance of RR Lyrae stars in Hercules-Aquila Cloud is 21.9$\\pm $ 12.1 kpc, and metallicity is -1.43$\\pm $ 0.36 dex.",
"Their study additionally presented a estimate of velocity of Hercules-Aquila Cloud.",
"In their bottom panel of Figure 17, the mean velocity of MSTO stars in Hercules-Aquila Cloud is centered around $V_{\\rm {los}}=$ 25km s$^{-1}$ .",
"[75] mapped the Hercules-Aquila Cloud using RR Lyrae from the Catalina Sky Survey [27].",
"They found this substructure is more prominent in the southern Galactic hemisphere than in the north, peaking at a heliocentric distance of 18 kpc.",
"We find three groups are asscoiated with Hercules-Aquila Could in the halo K giants.",
"As shown in Figure REF , the Hercules-Aquila Cloud members has a mean metallicity of -1.31 dex, heliocentric distance from 12 to 21 kpc, and mean line-of-sight velocity $V_{\\rm {los}}=$ 33.37 km s$^{-1}$ ."
],
[
"Attributing Groups to Orphan Stream",
"Orphan stream is a roughly $1 \\sim 2^\\circ $ wide stellar stream, and its progenitor has not been identified yet.",
"It runs from ($165^\\circ ,-17^\\circ $ ) to ($143^\\circ ,48^\\circ $ ) in equatorial coordinate [31], [6].",
"In addition, it was also traced with RR Lyrae stars by [74].",
"They found that the most distant parts of Orphan stream is $40 \\sim 50$ kpc from the Sun, mean [Fe/H] value of -2.1 dex, and line-of-sight velocity $V_{\\rm {los}} \\sim 100$ km s$^{-1}$ .",
"We find a group matches well with all these conditions.",
"As shown in Figure REF , its mean [Fe/H] is -2.02 dex, and mean line-of-sight velocity $V_{\\rm {los}}=$ 118 km s$^{-1}$ ."
],
[
"Unknown Groups",
"Besides the groups that can be attributed to known streams, there are 18 remaining groups likely relate to unknown substructure.",
"The unknown groups and their velocity-position distribution are shown in Figure REF and Table .",
"In addition, we plot the [Fe/H] distribution for groups with more than 20 members in the last two panels of Figure REF .",
"The full catalog of K giants with more than 5 members are published online, and a sample is shown in Table .",
"The stellar halo of our Milky Way are expected to be comprised largely of debris from disrupted satellite galaxies.",
"The debris may appear as coherent streams for some time, but will phase-mix until they become difficult to recognize.",
"Several prominent substructures in Galactic stellar halo have been found in the published literature, such as the famous Sgr streams.",
"Some studies have attempted to quantify the position-velocity substructure of the stellar halo using K giants and BHB stars [79], [88], [20], [41].",
"[41] even tried to identify members of substructures using SEGUE K giants.",
"However, all previous studies used only 3D positions and 1D radial velocities because of the lack of proper motions at that time.",
"Now, Gaia DR2 can provide useful proper motions for distant halo stars.",
"LAMOST combining with Gaia enables to construct a large sample of $\\mathrm {13,554}$ halo K giants with distances up to 100 kpc, radial velocities, metallicities, and proper motions.",
"This paper presents the first attempt to quantify and identify the substructure of the Milky Way’s stellar halo in 6D phase-space.",
"Based on 4Distance used in previous studies [79], [88], [20], [41], we develop 6Distance to define the distance of two stars in phase-space.",
"By comparing the number of close pairs between observed sample and the diffuse halo constructed by shuffling distances and velocities of the observed sample, we can quantify the amount of substructure in the sample.",
"We find that the substructure increases from inner halo to outer halo, and from metal-poor population to metal-rich population, in agreement with the results of [88], [41].",
"Besides quantifying substructures in stellar halo, identifying members of substructure is of particular importance to explore their the origin.",
"We combine 6Distance with FoF algorithm, and manually assign a sky-distance-dependent linking length to identify the substructures.",
"Finally, we find 43 FoF groups (1867 group members), in which 25 groups belong to 5 known substructures: Sgr stream (13 groups), Monoceros Ring (4 groups), Virgo Overdenstiy (4 groups), Hercules-Aquila Cloud (3 groups), and Orphan Stream (1 group); and 18 remaining groups are likely related to unknown substructures.",
"It is worth to point out that for the first time we identify the spectroscopic members of Monoceros Ring both in northern and southern hemispheres, which demonstrates the advantage of LAMOST.",
"The members of Sgr streams locate in distant halo, and are more metal-rich than other halo stars, we conclude that the Sgr stream dominates both trends of substructure versus metallicity and distance.",
"In addition, we analyze the kinematics and metellicities of the Monoceros Ring, Hercules-Aqulia Cloud, Virgo Overdensity, and the unknown groups with more than 20 grouped members.",
"Monoceros Ring shows more metal-rich than the typical halo stars, and its mean rotation velocity are closer to the thick disk, which may reflect the contamination from other Galactic components, the thick disk in particular.",
"This study is supported by the National Natural Science Foundation of China undergrants (NSFC) Nos.",
"11873052, 11890694, 11390371/2, 11573032 and 11773033.",
"X.-X.X.",
"thanks the “Recruitment Program of Global Youth Experts\" of China.",
"J.L.",
"acknowledges the NSFC under grants 11703019.",
"L.Z.",
"acknowledges supports from NSFC grants 11703038.",
"This project was developed in part at the 2018 Gaia-LAMOST Sprint workshop supported by the NSFC under grants 11333003 and 11390372.",
"Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope, LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences.",
"Funding for the project has been provided by the National Development and Reform Commission.",
"This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium).",
"Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.",
"Figure: Galactic longitude ll against the line-of-sight velocity V los V_{\\rm {los}} of LAMOST K giants.",
"The top panel shows the entire sample, and the signature of disk rotation (sin-shaped) is clear here.",
"In the bottom panel, after removing disk stars (|z|>5|z| > 5 kpc), the signature of disk largely disappeared.",
"Some halo substructures are visible.",
"Figure: The spatial distribution (x-zx-z plane) of 13,554\\mathrm {13,554} LAMOST halo K giants Figure: The Galactocentric distance distribution and velocity distributions along with Galactocentric distance of LAMOST halo K giants.",
"Figure: The error distributions of distances and velocities along with distances.",
"The distances have a typical error of 13%.",
"A typical error of 7 km s -1 ^{-1} in line-of-sight velocity makes it the most accurate velocity component.",
"The mean errors of the two tangential-velocity components are about 20 km s -1 ^{-1}, and can spread to ∼\\sim 100 km s -1 ^{-1}.",
"Figure: Galactic sky coverage for LAMOST halo K giants.",
"Stars are colored according to Galactocentric distance r gc r_{\\rm {gc}}, line-of-sight velocity V los V_{\\rm {los}}, and tangential velocities (V l V_{l}, V b V_{b}).",
"In the region of Sgr streams (120 ∘ <l<180 ∘ 120^\\circ < l < 180^\\circ & -60 ∘ <b<-30 ∘ -60^\\circ < b < -30^\\circ and 180 ∘ <l<200 ∘ 180^\\circ < l < 200^\\circ & 30 ∘ <b<60 ∘ 30^\\circ < b < 60^\\circ ), stars show similar distances and velocities.",
"Figure: Top panel: the close pairs distribution for 13554 halo K giants.",
"δ 6d \\delta _{6d} is the separation of 6D phase-space between any two stars.",
"The solid line is the cumulative distribution of δ 6d \\delta _{6d} for observed sample N obs (<δ 6d )N_{\\rm {obs}}(<\\delta _{6d}).",
"The dash line is the mean cumulative distribution of δ 6d \\delta _{6d} of 100 Monte Carlo representations of diffuse halo N null (<δ 6d )N_{\\rm {null}}(<\\delta _{6d}).",
"The black thick error bars are distribution enclosing 68% diffuse halo.",
"The red thin error bars enclose 95% of diffuse halo.",
"Bottom panel: the result of quantifying the halo K giants, N obs (<δ 6d )/N null (<δ 6d )N_{\\rm {obs}}(<\\delta _{6d})/N_{\\rm {null}}(<\\delta _{6d}).",
"Both panels demonstrate the halo K gaints have more close pairs in 6D phase-space than the diffuse halo system.",
"Figure: The result of quantifying the halo K giants in r gc <r_{\\rm {gc}} < 20 kpc (red dashed line) and in r gc >r_{\\rm {gc}} > 20 kpc (black solid line).",
"The substructure signal exists in both regions, while it is stronger in r gc >r_{\\rm {gc}} > 20 kpc.",
"Figure: The result of quantifying the halo K giants in different metallicity ranges.",
"The substructure signal increases with metallicity.",
"Figure: The comparison of quantifying the halo K giants before (solid lines) and after (dash lines) removing the Sgr stream (see more detals in the text).",
"The colors represent the different metallicity ranges, which are marked in the legend.s Figure: The x-zx-z plane distribution of 43 groups (1867 stars) identified from LAMOST halo K giants.",
"We combine different colors and symbols to represent different groups.",
"Dashed lines are every 20 kpc in r gc r_{\\rm {gc}}.",
"Because of the limited number of colors and symbols, some different groups are shown with same color and symbol.",
"Figure: The sky coverage of five known substructures identified from LAMOST halo K giants: Sgr Stream (red stars), Monoceros Ring (blue triangles), Virgo Overdensity (green circles), Hercules-Aquila Cloud (orange pluses), and Orphan Stream (orange diamonds).",
"The background (gray dots/filled circles) is literature sky coverage of these known substructure: Sgr leading arm and Sgr trailing arm , Monoceros , Virgo Overdensity , Hercules-Aquila Could , and Orphan stream .",
"Figure: The comparison of Sgr members identified in LAMOST halo K giants with LM10 model.",
"There are 8 FoF groups relate to Sgr leading arm (blue star symbols), 3 FoF groups attributed to Sgr trailing arm (red star symbols), and 2 FoF groups belonging to Sgr debris (grey star symbols).",
"The color dots are from LM10 model, which is color-coded by the pericentric passage (values of -1 indicate debris which is still bound at the present day, values of 0 indicate debris stripped on the most recent pericentric passage of Sgr, and values of 1 indicate debris stripped on the previous pericentric passage, see LM10 for details).",
"Comparing with LM10 model, Sgr streams traced by LAMOST K gaints are more diffuse, and locate closer.",
"The mean [Fe/H] values of Sgr leading and trailing arms are -1.24 dex and -1.12 dex, and their dispersion are 0.47 dex and 0.49 dex, which consistent with the prediction of the model that the leading arm is composed by more metal-poor stars due to the origin from the periphery of Sgr dwarf galaxy.",
"Figure: Comparison with literature Sgr streams traced by SDSS BHB and RC in r gc r_{\\rm {gc}}-Λ ˜ ⊙ \\widetilde{\\Lambda }_\\odot plane and V los V_{\\rm {los}}-Λ ˜ ⊙ \\widetilde{\\Lambda }_\\odot plane.",
"Λ ˜ ⊙ \\widetilde{\\Lambda }_\\odot is same as the definition in .",
"Blue, red, and grey star symbols represent the members of Sgr leading arm, trailing arm, and debris identified in LAMOST halo K giants.",
"Black dots with error bars are from tables 1-5 of , and the red dots with error bars are from table 2 of , which has been increased by 0.35 mag to correct for the reddening towards the progenitor.",
"The Sgr streams traced by LAMOST K giants locate closer than both the BHB stars of and the RC stars of .",
"Figure: The position and velocity distributions of Virgo Overdensity (blue dots), Hercules-Aquila Cloud (red star symbols), and Orphan Stream (green triangles) identified in LAMOST K giants, and the metallicity distributions of the Virgo Overdensity, the Hercules-Aquila Cloud.",
"There are 4 FoF groups relate to Virgo Overdensity, 3 FoF groups relate to Hercules-Aquila Cloud, and 1 FoF group relate to Orphan Stream.",
"The mean [Fe/H] values of Hercules-Aquila Cloud and Virgo Overdensity are both -1.31 dex, and their dispersion are 0.42 dex and 0.40 dex, and the metallicity distribution of Hercules-Aquila Cloud shows three peaks at -1.6 dex, -1.2 dex, and -0.6 dex.",
"Figure: The position and velocity distributions of 18 unknown groups (350 group members), and metallicity distributions of the groups with more than 20 members.",
"The detail properties of these unknown groups are listed in Table .",
"Table: Properties of Each sub-sample Table: Maximum Physical Component Size for Different Linking Length[h] Known Substructures in LAMOST Halo K Giants Table: NO_CAPTION[h] Unknown Groups in LAMOST Halo K Giants Table: NO_CAPTIONcccccccc Group Catalog: Groups with Five or More Members obsida R.A. Decl.",
"$d$ $V_{\\rm {hel}}$ pmra pmdec GroupIDb (deg) (deg) (kpc) (km s$^{-1})$ (mas yr$^{-1}$ ) (mas yr$^{-1}$ ) 77201039 15.597092 5.310736 13.6997 -215.36 0.265 -4.238 0 20603144 19.739155 -0.7779 13.9554 -173.39 -1.366 -3.4597 0 185709128 39.33012 3.758879 12.3098 -179.17 -0.346 -2.451 0 187012195 23.141436 -0.383224 13.8292 -188.84 -0.805 -3.233 0 381006238 36.810744 -1.672415 12.8698 -148.22 -0.31 -2.639 0 21110160 38.951621 1.017566 12.9395 -156.68 -0.15 -2.698 0 381014061 33.997727 -0.539124 13.6571 -174.38 -0.4502 -3.003 0 203913110 39.6415 4.362719 13.0332 -199.16 -0.194 -2.99 0 187009063 22.400078 -1.022162 14.6656 -172.4 -0.861 -3.43 0 496704119 18.590844 8.725264 14.7694 -216.16 -1.173 -3.4581 0 aUnique identifier in LAMOST.",
"b0-Sgr trailing arm; 1-Sgr leading arm; 2-Monoceros Ring; 3-Virgo Overdensity; 4-Hercules-Aquila; 5-Sgr Debris; 6-Orphan Stream; $>6$ -unknown.",
"(This table is available in its entirety in a machine-readable form in the online journal.",
"A portion is shown here for guidance regarding its form and content.)"
]
] | 1906.04373 |
[
[
"Classification of EEG Signals using Genetic Programming for Feature\n Construction"
],
[
"Abstract The analysis of electroencephalogram (EEG) waves is of critical importance for the diagnosis of sleep disorders, such as sleep apnea and insomnia, besides that, seizures, epilepsy, head injuries, dizziness, headaches and brain tumors.",
"In this context, one important task is the identification of visible structures in the EEG signal, such as sleep spindles and K-complexes.",
"The identification of these structures is usually performed by visual inspection from human experts, a process that can be error prone and susceptible to biases.",
"Therefore there is interest in developing technologies for the automated analysis of EEG.",
"In this paper, we propose a new Genetic Programming (GP) framework for feature construction and dimensionality reduction from EEG signals.",
"We use these features to automatically identify spindles and K-complexes on data from the DREAMS project.",
"Using 5 different classifiers, the set of attributes produced by GP obtained better AUC scores than those obtained from PCA or the full set of attributes.",
"Also, the results obtained from the proposed framework obtained a better balance of Specificity and Recall than other models recently proposed in the literature.",
"Analysis of the features most used by GP also suggested improvements for data acquisition protocols in future EEG examinations."
],
[
"Introduction",
"About 40% of the world's population suffers from some sleep disorder [41], [28].",
"Sleep quality directly affects the health and quality of life of the human being.",
"Poor sleep causes many people to seek out specialized clinics for an accurate diagnosis.",
"One of the most common techniques of analysis is done by observing brain activity, eye movement, muscle tension, and other body signals by polysomnography (PSG).",
"The examination consists of collecting data through a series of electrodes connected to the patient's skin and scalp during his or her usual nighttime sleep.",
"This examination allows the diagnosis of several disorders, such as obstructive sleep apnea, insomnia, narcolepsy, restless legs syndrome and bruxism.",
"It is also useful for the identification of visible waveforms like sleep spindles (SS) and K-complexes (KC) which, besides assisting in sleep staging, are related to the consolidation of memory and sensory systems.",
"Abnormalities in their forms may indicate neuropathologies or sleep disorders.",
"In patients with sleep or neurological disorders, the study of these waveforms helps in the understanding of the neurophysiological functioning and thus, allows to raise hypotheses about the problem.",
"In particular, sleep spindles have a number of theoretical and clinical implications in understanding how brain activity during sleep is affected and the development of the disorder [42].",
"Figure: Framework proposed in this work for the identification of structures in sleep EEGThe identification of waveforms on EEG signals is usually done by visual inspection by experts.",
"This is a time-consuming and tiring process, which may introduce biases and errors [34].",
"In consequence, specialists not always arrive in the same identification, as illustrated in Figure REF .",
"Because of this, there is an interest in the development of automated tools for waveform detection [42].",
"Figure: Example of Sleep Spindles and K-complexes identified from EEG data by two different experts.There are many challenges related to EEG signal analysis.",
"They have spatial and temporal co-variance, implying highly dependent samples.",
"They are also non-stationary, noisy and sensitive to external interference [19].",
"In order to describe these signals without losing information, a high number of features are necessary from the original signals, implying in high dimensionality samples.",
"One way to improve the automated classification of EEG signal structures is by using dimensionality reduction and feature construction techniques.",
"In this sense, Genetic Programming (GP) can be used to generate a function that generates a set of new, reduced features from the original ones.",
"Using GP for feature construction has two advantages: First, GP can generate non-linear combinations of features, making it more expressive than traditional feature reduction techniques.",
"Second, an analysis of the rules created by GP may allow insights about the importance of the different original features, as suggested by Ivert et al. [17].",
"Guo et al.",
"have previously proposed a GP framework for feature construction in the context of seizure detection in EEG signals using KNN [13].",
"In this paper, we build upon this work and apply it to the more difficult problem of detecting structures such as Sleep Spindles and K-complexes.",
"More specifically, we use short samples (2s vs 26s) to precisely identify the locations of the structure, we use AUC instead of Precision as the fitness function, and we explore five different classifiers instead of just KNN.",
"To test the proposed framework (Figure REF ) we perform the identification of Sleep Spindles and K-complexes on the DREAMS [9] dataset.",
"Starting from a set of 75 features per sample, the proposed GP finds a constructed set with a median of 12 features.",
"We show that the feature set found by GP achieved better AUC than using the full set of features, or even a set of 29 features selected by PCA.",
"Additionally, the proposed model achieve a better balance of Recall and Specificity when compared with other recently proposed models for the same problem.",
"Finally, and perhaps more interestingly, an analysis of the rules constructed by GP showed that we could use only one of the three EEG channels in the dataset to obtain the same quality of identification.",
"This result suggests that a simpler examination could be used, causing less discomfort for patients."
],
[
"The EEG Classification Problem",
"Electroencephalogram (EEG) is a typically noninvasive examination for the observation of electrical activity of the brain [36].",
"This information is obtained through electrodes attached to the scalp with a conductive paste.",
"Through the analysis of these data it is possible to detect diseases and psychiatric and neurological problems.",
"Usually the analysis of these signals done visually by experts (Figure REF ), which makes the process tiresome, tedious and susceptible to errors [34].",
"To assist specialists in this visual task, a number of methods of automatic processing and analysis of EEG signals has been proposed.",
"We emphasize the use of automatic methods for the study of apnea [38], epilepsy [13], drowsiness [4], sleep spindles [9], [8], [1], [21], [43], [5], K complexes [14], [31], [37], sleep stages [22] and schizophrenia [33]."
],
[
"Sleep Spindles and K-complexes",
"Sleep has two main phases: REM (Rapid Eye Movement) sleep and NREM (non-REM) sleep.",
"Occupying up to 80% of the sleep time, the NREM phase is divided into 4 stages, ranging from lightest to deepest sleep [30].",
"In particular, stage 2 of NREM sleep has as its main characteristic the appearance of specific waveforms, K-complexes and Sleep Spindles.",
"The beginning of this stage is defined by the occurrence of these signals.",
"Because of this well defined presence, they are very important for sleep staging.",
"Although Sleep Spindles mark entry into stage 2 of NREM sleep, they may also occur in stage 3 [2].",
"When a spindle occurs, the amplitude of the EEG signal increases and decreases progressively, having a minimum duration of 0.5 s with defined bandwidth between 12 and 14 Hz in the criterion of Rechtschafen and Kales [32] (some authors may consider from 11 until 16 Hz).",
"Peak-to-peak amplitude settings can also be found between 5 and 25 $\\mu V$ [9].",
"The occurrence of spindles contributes to memory consolidation, to continuous sleep [3] and in the study of sleep and neurological disorders.",
"The characteristics of the spindles change with the patient's age and sex [6].",
"The tendency is for it to occur less with advancing age [29].",
"As for gender, the phenomenon usually occurs twice more during the sleep of women, due to hormonal factors [24], [10].",
"The sleep spindles, in general, have a well-defined structure (occurrence, bandwidth, and amplitude).",
"However the advancement of the patient's age and pathologies cause inaccuracies in their shape.",
"Typically, the number of spindles decreases and their shape deteriorates [18].",
"Their shape may be distorted and are more subject to the occurrence of interference [8].",
"Patients with schizophrenia, for example, do not have normal patterns in the spindles [12].",
"Changes can also be observed due to fatal familial insomnia [27], autism and epilepsy [16].",
"This lack of standard is important for the diagnosis of neurological diseases, but it makes it more difficult to identify this phenomenon for specialists and automatic methods.",
"The K-complex is a negative acute wave immediately followed by a positive component that clearly arises in the EEG, having a minimum duration of 0.5s in the frequencies of 12 to 14 Hz [2].",
"In the identification, they can be easily confused with any waveform with high peaks [20], [11] (Figure REF ).",
"Abnormal activity of the K complex may be related to epilepsy, restless leg syndrome, and obstructive sleep apnea."
],
[
"DREAMS Data",
"We use the databases collected by the DREAMS project [9], which consist of a series of polysomnography (PSG) with expert annotations on phenomena or sleep disorders.",
"We use the sleep spindle and K-complexes datasets from this project.",
"Their purpose is to tune, train and test automatic detection algorithms.",
"The Spindles dataset consists of 30 minute stretches of the central EEG channel (extracted from full-night PSG records), independently annotated by two experts.",
"The data were acquired in a sleep laboratory of a Belgian hospital (BrainnetTM System of MEDATEC, Brussels) using a 32-channel digital polygraph.",
"It is important to highlight that all records on this dataset are from patients with various sleep pathologies: dyssonia, restless legs syndrome, insomnia, apnea syndrome or hypopnea.",
"EOG, EMG and EEG channels (channels FP1-A1, O1-A1 and C3-A1 or CZ-A1) were recorded, using the European standard data format (EDF) for storage.",
"Only EEG channels will be used in this work.",
"The sampling frequency varies between patients, having records of 200Hz, 100Hz or 50Hz of 30 minutes duration.",
"The recordings were given independently to two experts, who annotated their estimates for the locations of the sleep spindles.",
"The K-complex records were collected in the same hospital as the spindle registers, with the same equipment.",
"There are 10 polysomnographic recordings from healthy individuals.",
"Just like the previous base, we only use EEG channels.",
"The sampling frequency was 200 Hz for all patients with a 30 min duration.",
"In the same way, the excerpts were given independently to two experts.",
"To reduce bias, the experts did not have access to sleep staging of the records."
],
[
"Data preparation",
"The original EEG data was prepared for the automated identification process using the following procedure."
],
[
"Wavelets Transform",
"Wavelet transforms are mathematical tools capable of decomposing signals into several components that allow analysis at different time and frequency scales [7].",
"An input signal $x$ , passes through a low-pass filter $g$ and a high-pass filter $h$ (parallel, not sequentially), each with a cut-off frequency equal to one half of the sampling frequency of the input.",
"Then, the two generated sub-signals, that is, the output of the filters has their samples reduced in half (see Figure REF ).",
"Figure: Example of Digital Wavelet Transform in 3 levelsThis process can be repeated at several levels, causing the output of the low pass filter to be the input signal of a new pair of filters, followed by downsampling."
],
[
"Feature extraction",
"The EEG data used were sampled with different frequencies (50, 100 or 200 Hz).",
"For the application of the Digital Wavelet Transform (DWT) [23] with 5 decomposition levels, the data were resampled with an increased frequency of 256 Hz in all cases by means of interpolation through a cubic spline.",
"For each decomposition level ($D1$ to $D5$ ) in each EEG channel, the signal was separated into 2 second samples, and the following attributes were calculated for each sample: Signal amplitude average, Signal amplitude standard deviation (SD), Symmetry, Power Spectral Density (PSD), and Signal curve length.",
"Following this procedure, we obtain 900 samples (2s samples from 30 minutes of signal) with 75 real valued attributes (3 EEG channels, 5 DWT levels, and 5 attributes per level).",
"These attributes are summarized in tables REF , REF , and REF .",
"Here, the columns represent the coefficients of the DWT levels and the lines the operations performed.",
"Table: Attributes for the central EEG channel (CZ-A1 or C3-A1)Table: Attributes for the EEG channel FP1-A1Table: Attributes for the EEG channel O1-A1"
],
[
"Proposed Framework",
"Previously, Guo et al.",
"[13] proposed the use of Genetic Programming (GP) for the construction of features for EEG analysis in the classification of epileptic episodes.",
"Taking this framework as a base, we develop a framework for the identification of structures in sleep EEG, which we describe in this section.",
"There are many characteristics in the structure identification problem which differentiates it from the earlier classification work.",
"We must divide the EEG signals into multiple short samples in order to identify the position of the Spindles and K-complexes in the signal.",
"As a consequence, the data becomes highly unbalanced, complicating the training process.",
"Also, we work on three distinct EEG channels (as opposed to a single channel in the original work).",
"We tested several improvements on the original work to deal with this harder problem.",
"First, we use the Area Under the ROC Curve (AUC) of the classifiers instead of the accuracy as the fitness measure, since the AUC is more robust and discriminating [15].",
"Also, we compare several classifiers in addition to KNN, to explore the relationship between classifier choice and GP feature construction.",
"GP is widely applied in the construction and selection of features for its good performance.",
"In classification problems it is possible to evolve a tree for each problem class, selecting the best attributes and creating new features for each of them [25].",
"Even with unbalanced data, this approach with GP also gets good results [40].",
"In literature, there are also application studies in benchmarks [35] and in databases with few samples [26].",
"Finally, we publish the program of the proposed framework and experiments at our repositoryhttps://github.com/IcaroMarcelino/SleepEEG for reproducibility purposes."
],
[
"GP for Feature Construction",
"Our proposed framework uses Genetic Programming (GP) to generate the constructed set of features from the original features.",
"The GP tree is defined as follows: The input nodes are selected from the original attributes.",
"The intermediate nodes are selected from a set of arithmetic operators $\\lbrace +,-,\\times ,\\rbrace $ , as well as a set of protected operators $\\lbrace /,\\ln ,\\sqrt{\\rbrace }$ .",
"These protected operators have their definitions slightly modified to avoid errors such as division by zero, as follows: $\\text{protected division} (a, b) &=& \\left\\lbrace \\begin{array}{l l}1, & b = 0 \\\\\\frac{a}{b}, & b \\ne 0\\end{array} \\right.\\\\\\text{protected log}(a) &=& \\left\\lbrace \\begin{array}{l l}1, & b = 0 \\\\ln(|a|), & b \\ne 0\\end{array} \\right.\\\\\\text{protected square root} (a) &=& \\sqrt{|a|}$ Additionally, a special \"Feature Operator\" [13], $F$ , is used to indicate how to obtain the constructed features from the GP tree.",
"The feature operator returns the value of its input as its output, without any changes.",
"Its purpose is to mark a subtree as one of the constructed features.",
"Each $F$ operator will be the root of a subtree that expresses the function to calculate one attribute for the constructed set.",
"In this way, a GP tree containing ten nodes with the $F$ operator will generate a constructed attribute set with 10 attributes.",
"For example, the tree depicted in Figure REF shows a GP individual with two subtrees marked by the $F$ operator.",
"If we assume that the original attribute set has 26 attributes (a..z), this tree will generate a constructed attribute set with two attributes: $F_1 = a$ and $F_2 = b - 1$ .",
"The use of the $F$ operator allows a single GP tree to express multiple attribute constructing functions.",
"In this way, we avoid having to explicitly define how many attributes will be constructed beforehand, which would be necessary if each attribute were expressed by a separate tree [13].",
"Also, this allows GP trees in the population to exchange useful subtrees containing sets of constructed attributes that were successful.",
"We believe that this allows the GP to pass around the most relevant subtrees to the next generations and, with this, keep attributes and attribute subsets that facilitate classification.",
"Figure: Example of GP program marked with F operators.",
"This GP tree constructs two attributes: F 1 =aF_1 = a and F 2 =b-1F_2 = b-1To evaluate one GP tree using the structure described in the previous section, we use the following procedure.",
"First, we generate the set of constructed attributes for the GP tree using the $F$ operators.",
"Second, we train a classifier using this set of constructed attributes.",
"Finally, we use the AUC value of the classifier as the fitness value for the GP tree.",
"In this way, the product of the evolutionary process is both a set of constructed attributes, as well as a classifier trained on those attributes.",
"The parameters used for the evolutionary process in the current framework are listed in Table REF .",
"As the parameters provided good results in the initial runs, they were maintained for the following.",
"The \"Uniform mutation\" selects a random node from the individual and replaces subtree rooted in that node with a a randomly generated one.",
"Table: GP Model Parameterization"
],
[
"Classifiers",
"We compare the performance of the set of constructed attributes by GP by using five different classifiers: K Nearest Neighbors (KNN) Naive Bayes (NB) Support Vector Machines (SVM) Decision Tree (DT) Multilayer Perceptron (MLP) We perform an initial tuning procedure using the full set of 75 features to select the hyperparameters used by each classifier in the next experiments.",
"For each tuning classifier, we execute 100 runs for each parameter value tested in the following sets: KNN: $k \\in \\lbrace 3,5,7,9,11,13,15,17,19\\rbrace $ SVM: kernel $\\in \\lbrace $ Radial Basis Function (RBF), polynomial, sigmoid$\\rbrace $ MLP: activation function $\\in \\lbrace $ ReLU, logistic$\\rbrace $ , neurons in hidden layer $\\in \\lbrace 15,30,45,60,75\\rbrace $ The parameters that provide the best performance were selected.",
"KNN with $k = 5$ , SVM with RBF as kernel and MLP with a single hidden layer with 15 neurons and ReLU activation."
],
[
"Experiment",
"We perform several evaluations of the classifiers on the Sleep Spindles and K-Complexes datasets in order to analyse the performance of the propose framework.",
"The results of the classifiers are compared using the full set of 75 attributes, a reduced set of 29 attributes selected by PCA, and the set of attributes constructed by the GP framework.",
"The training dataset (used to train both the GP and the classifiers) was generated by simple random sampling 70% of the signal samples, and labelling them as positive samples (Sleep Spindles or K-complexes) if both specialists agreed on the label.",
"Additionally, because the dataset is highly unbalanced, we balance the training dataset by randomly removing samples from the majority class until both classes have the same number of signal samples.",
"The test dataset was generated by the remaining 30% of the samples, and each signal sample was labeled as positive if either specialist annotated it as positive.",
"Also, the balancing procedure is not performed on the testing data set.",
"This resulted in a slightly harder testing data set.",
"For each experiment, we repeat the training/testing procedure 10 times, and report the aggregate results of these 10 repetitions as described in the subsections below.",
"Figure: refF"
],
[
"Results",
"The first experiment was aimed at verifying the performance of the classifiers on the test dataset without the reduction of dimensionality by GP, i.e., using the 75 features.",
"The results are also useful to justify the application of feature construction.",
"The classifiers performance can be seen in Figure REF a.",
"Only MLP has good results, achieving an AUC greater than 0.7 with low SD.",
"Figure: Number of occurrences of the feature in the models generated by GP (Same models from Figure c).Figure: Number of dimensions in the models generated by GP (Same models from Figure c)Applying PCA on the data with a 95% threshold for variance, ensuring little loss of information, the initial set of 75 features can be represented with 29 attributes.",
"The performance of the trained classifiers with the feature set generated by PCA on the respective test set can be seen in Figure REF b.",
"For this problem, the PCA representation caused a performance reduction in all classifiers except NB.",
"The performance of applying the classifiers on the feature set generated by PCA can be seen in Figure REF b.",
"For this problem, the PCA representation caused a decrease in performance in all classifiers except NB.",
"Applying the GP feature reduction, the AUC of the classification increases for all classifiers except KNN (Figure REF c) and the SD reduces for all cases.",
"This method is capable to reduce the number of features from 75 to less than 29 (Figure REF ).",
"Using the same approach for training K-complexes classifiers, the models achieve high AUC scores too (Figure REF d).",
"Figure: Classifiers performance over the central EEG channel attributes for Sleep Spindles and K-complexes"
],
[
"Analysis of gender difference",
"As mentioned before, there are gender differences in sleep spindles.",
"To see if this difference affects the performance of the classifiers, the data were separated by gender.",
"Observing the Figures REF e and REF f, the classifier's performance for female patients is higher than the male patients.",
"As sleep spindles occur more often in female patients, in data there are more representative samples of the waveform (see Table REF ), which facilitates the training of more efficient classifiers."
],
[
"Constructed Features Analysis",
"In Figure REF , the frequency of occurrence of features in the models training shows that there are attributes more relevant than others in the dataset.",
"The greater occurrence of the features associated to the central EEG channel indicates that it has more important role to the identification of sleep spindles.",
"Using only this channel for training (i.e, only with the 25 first features of the dataset), the performance of all classifiers increase for sleep spindles and K-complexes identification (Figure REF ).",
"This attributes reduction contributes to a better understanding of the phenomenon, providing a more efficient approach by reducing the use of electrodes, consuming less resources and generating less discomfort to the patient.",
"Table: Comparison between the proposed model and literature models"
],
[
"Comparison to Literature Models",
"In the Table REF , the best generated model with the proposed approach (with NB classifier) was compared with the literature models which also used DREAMS data Further comparisons between sleep spindle identifiers can be seen in [21].",
"Tsanas at.",
"el [39] and Zhuang et al.",
"[44] proposed continuous wavelet transform (CWT) based approaches and the estimation of the probability of spindles occurrences.",
"Lachner-Piza et al.",
"[21] proposed a SVM approach with a feature selection method based on the label-feature and feature-feature correlations for determining the relevance and redundancy of each feature.",
"Observing the performance of the models, all obtained high specificity, indicating that the identification of samples where no spindles samples are present is reliable.",
"Moreover, there is a trade-off between sensitivity and precision.",
"In the context of applying automatic identifiers, false negatives are more unwanted than false positives.",
"That is, a highly accurate but not very sensitive classifier generates many false positives, indicating that it is not judicious.",
"In a semi-automatic application with low sensitivity, it is necessary for a specialist to inspect the markings performed by the classifier, eliminating the excess of false positives.",
"This is the case of the detector of Zhuang et al.",
"[44].",
"The detectors of Tsanas and Clifford [39] and Lachner-Piza et al.",
"[21] and the proposed model have achieved a better compromise between sensitivity (recall) and precision.",
"This implies that the identification of signal stretches as spindles is more reliable.",
"The proposed approach allows the generation of competitive models with the literature.",
"The Tsanas and Clifford model, although having a slightly higher sensitivity than the proposed model.",
"In contrast to the model of Lachner-Piza et al., our model has only minor precision, with 0.03 of difference."
],
[
"Discussion",
"The GP feature construction improves the performance of a classifier reducing the search space and generating more explicit relations between variables.",
"Observing the reduction in the number of attributes, the dimensionality of the problem is reduced by up to 7 times in most cases.",
"In addition, by analyzing the most frequent attributes, it is clear which ones are most relevant to the models.",
"With this information, it is possible to select the most important EEG channels.",
"In the case of K-complexes sleep spindles, only the central EEG channel is sufficient to perform the waveform identification.",
"The single channel approach already reduces search space by one-third.",
"Furthermore, fewer electrodes will be required for the examination, making it more comfortable for the patient, consequently, approaching the sleep in the laboratory of the daily sleep, avoiding biases."
],
[
"Conclusions",
"The use of automatic methods to identify sleep phenomena makes it possible to classify EEG signal segments with good performance, indicating whether or not a particular event occurs.",
"Excerpts of 30 minutes can have hundreds of events that you want to identify.",
"In this respect, the proposed model can be used to accelerate the process, and it is up to the expert to assign the classification.",
"The model was also useful for sleep staging, since the presence of spindles and K-complexes strongly characterize sleep stage 2.",
"It has also been shown that it is possible to significantly improve the performance of classifiers by selecting and constructing attributes.",
"In addition, the use of GP allows greater interpretability and mathematical analysis of the new attributes generated, which may help to better understand the model and the problem.",
"It is also possible to inspect the attributes generated through knowledge in the application domain.",
"The approach also does not require in-depth knowledge of the application domain.",
"In the first experiment, no assumptions about the data were performed.",
"The ease of defining the terms in which the solutions will be written, that is, the operators and the terminals, allow the creation of hybrid models with biomedical information.",
"It can also facilitate communication between specialists from different areas.",
"The automation of the selection and construction of attributes generates a dataset suitable for the desired classifier.",
"But, it is very simple to apply the attributes generated in another classifier.",
"The processing time is also reduced with the smaller number of dimensions.",
"The PSG generated signals used in sleep clinics are stored directly on computers.",
"Therefore, the application of the proposed technique can be easily applied in this context.",
"The generated models, once trained, make predictions quickly, facilitating a real-time approach.",
"Measurement of the micro-event activity on EEG signals in different populations can provide important information about abnormalities in brain signals and assist in the investigation and hypothesis assessment of observed phenomena or disturbances.",
"This underscores the importance of the study.",
"With the proposed models, the identification of the spindles or K-complexes is less costly for the specialist.",
"It may even replace its function in this task if the performance of the models is satisfactory for the requested analysis.",
"Therefore, the diagnosis can be faster and the return to the patient suffering from some disorder is more efficient.",
"Moreover, this methodology can easily be extended to other classification problems."
]
] | 1906.04403 |
[
[
"Spiral instabilities: Mechanism for recurrence"
],
[
"Abstract We argue that self-excited instabilities are the cause of spiral patterns in simulations of unperturbed stellar discs.",
"In previous papers, we have found that spiral patterns were caused by a few concurrent waves, which we claimed were modes.",
"The superposition of a few steadily rotating waves inevitably causes the appearance of the disc to change continuously, and creates the kind of shearing spiral patterns that have been widely reported.",
"Although we have found that individual modes last for relatively few rotations, spiral activity persists because fresh instabilities appear, which we suspected were excited by the changes to the disc caused by previous disturbances.",
"Here we confirm our suspicion by demonstrating that scattering at either of the Lindblad resonances seeds a new groove-type instability.",
"With this logical gap closed, our understanding of the behaviour in the simulations is almost complete.",
"We believe that our robust mechanism is a major cause of spiral patterns in the old stellar discs of galaxies, including the Milky Way where we have previously reported evidence for resonance scattering in the recently released Gaia data."
],
[
"Introduction",
"Theorists have long dreamed that the spiral patterns gracing most disc galaxies, and simulations thereof, reflect the normal modes of the disc.",
"That is, they are mildly non-linear manifestations of self-excited, linear instabilities of the stellar disc that are uniformly rotating and exponentially growing density waves.",
"Long ago, this grand program was set back by two early discoveries from normal mode analyses of apparently reasonable models of featureless discs: models where the rotation curve rose gently from the centre were dominated by bar-forming instabilities [25], [30], while models having a dense (bulge-like) centre had no modes whatsoever [61].",
"Here we begin by reviewing why we think spiral modes are in fact the most promising mechansim, despite this early setback, and add to our case by demonstrating how spiral modes recur.",
"A normal mode of any system is a self-sustaining, sinusoidal disturbance of fixed frequency and constant shape, save for a possible uniform rotation; the frequency would be complex if the mode were to grow or decay.",
"In the case of galaxy discs, the perturbed surface density of a mode is the real part of $\\delta \\Sigma (R,\\phi ,t) = A_m(R)e^{i(m\\phi - \\omega t)},$ where $m$ is the angular periodicity, $\\omega = m\\Omega _p + i\\beta $ , $\\Omega _p$ is the angular rate of rotation, usually called the pattern speed, and $\\beta $ is the growth rate.",
"The complex function $A_m(R)$ , which is independent of time, describes the radial variation of amplitude and phase of the mode.",
"Stability analysis of a system supposes small amplitude perturbations about the equilibrium state and is linearized when any terms that involve products of small amplitude terms are discarded – see [28] for a careful formulation.",
"The equilibrium is linearly unstable if any of the resulting normal modes have a positive growth rate, since its amplitude will exponentiate out of the noise until the neglected 2nd and higher order terms become no longer negligible.The word instability is sometimes used to imply a purely imaginary frequency, and a mode with a complex frequency is then described as an overstability.",
"Here we adopt the more usual convention of simply describing all frequencies with $\\beta >0$ as instabilities.",
"Normal modes can be standing wave oscillations of the system that exist between two reflecting barriers, as in organ pipes and guitar strings, which are generally described as cavity modes in galaxy discs.",
"The prime example in galaxies is the bar-forming mode, for which reflections take place at the centre and at corotation [61], [9].",
"An instability of this type is possible only if the disturbance has no inner Lindblad resonance (hereafter ILR), since linear theory [35] predicts that any disturbance that encounters an ILR will be absorbed, and therefore damped.",
"An ILR must be present for any reasonable pattern speed when the centre is dense, and therefore no small-amplitude cavity mode is possible in a featureless disc of this kind.",
"But $\\Omega _c-\\kappa /2$ has a maximum value in mass models having quasi-harmonic cores, and bar-forming instabilities avoid resonance damping by having pattern speeds that exceed this maximum.",
"Here $\\Omega _c(R)$ is the angular frequency of circular motion at radius $R$ in the disc mid-plane, and $\\kappa (R)$ is the usual frequency of small-amplitude radial oscillations about a circular orbit [9].",
"The dominant mode of several bar-unstable models has been identified in simulations, with excellent quantitative agreement of both the frequency and mode shape [49], [16].",
"Other types of cavity mode have been proposed by [36] and by [52] and are described below.",
"Galactic discs can also support another class of mode.",
"The best known examples are edge modes [61], [40] and groove modes [54], [53].",
"They are not standing waves, however, and maintain a fixed shape and frequency by other means.",
"In the case of the edge mode, a small non-axisymmetric distortion of the disc where the density decreases steeply, moves high density material out to places where the equilibrium density was lower, and conversely at other azimuthal phases.",
"On their own, such co-orbiting distortions would be neutrally stable and therefore of no interest.",
"But as [27] taught, a cool surrounding disc responds vigorously to a co-orbiting mass excess, creating a trailing wake that extends radially far into the shear flow on either side of the perturbing mass; this behaviour is a consequence of swing amplification [23], [27], [61], [8].",
"The distorted edge therefore excites a strong supporting response from the interior disc that is not balanced by the exterior response because the equilibrium density drops rapidly with radius at the edge.",
"The attraction of the interior wake on the original density excess increases its angular momentum, causing it to grow exponentially as it rotates.",
"The necessary conditions for instability were summarized by [62].",
"A groove in a disc is effectively two closely spaced edges, which however give rise to a single mode because the distortions on each edge are gravitationally coupled.",
"Note that it is a steep gradient in the angular momentum density that matters for both edge and groove modes; epicyclic blurring can mask the steepness in the actual surface density profile.",
"As for edge modes, it is the supporting response of the surrounding disc that causes the groove mode to have a substantial radial extent and to grow rapidly [53], and these authors were able to obtain good quantitative agreement between their analytic predictions and simulations.",
"Note that swing amplication, famously illustrated in the dust-to-ashes figure of [61], is not a mode both because the shape changes with time and the amplitude variation is not a simple exponential.",
"Also the wake response to an imposed co-orbiting mass clump [27] is not a mode because, to first order, it would disperse if the clump were removed, and it therefore is not self-sustaining.",
"Both are simply responses of the disc to externally imposed disturbances.",
"However, they are both very helpful concepts when trying to understand the mechanisms of self-sustaining modes."
],
[
"Noise, resonances, and heating",
"The collisionless fluid of stars invoked in these analytical treatments is an idealization that assumes the stars to be infinitely finely divided so that phase space is smooth.",
"The numbers of stars in galaxy discs is large enough that this assumption holds quite well [47], but galaxies contain mass clumps such as star clusters and giant molecular clouds, and the number of particles employed in a simulation is generally several orders of magnitude fewer than the number of stars.",
"Thus shot noise in both real galaxies and in simulations gives rise to significant inhomogeneities in the disc.",
"The spectrum of shot noise in a shearing distribution of randomly distributed gravitating masses inevitably contains leading wave components that will be strongly amplified as the shear carries them from leading to trailing.",
"This behaviour has two important consequences.",
"The first consequence of swing-amplified shot noise is that each heavy particle develops a wake [27], [8].",
"The two point correlation function of the particles becomes greater along the direction of the wake and lower in other directions, and the particle distribution is said to be polarized.",
"Since the distribution of particles is no longer perfectly random, the amplitude of all components of the noise spectrum is enhanced, causing subsequent noise-induced fluctuations to be stronger, although linear theory predicts this cycle should asymptote in a few epicycle periods to a mean steady excess over the level expected from uncorrelated noise [27], [64].",
"Second, the collective amplified response of any one component of the noise orbiting at the angular frequency $\\Omega _p$ , creates a coherent trailing wave in the disc that propagates away from corotation [59], [61] until it reaches a Lindblad resonance where it is absorbed [35].",
"For near circular orbits the Lindblad resonances occur where the Doppler shifted frequency at which the stars encounter the wave is equal to the epicyclic frequency of their small radial excursions, i.e., the radii at which $m\\left[\\Omega _p - \\Omega _c(R)\\right] = l\\kappa (R),$ with $l=\\pm 1$ at Lindblad resonances.",
"The negative sign is for the ILR, where stars overtake the wave, and the positive is for the OLR where the wave overtakes the stars, at the local epicycle frequency in both cases.",
"Wave-particle interactions at the resonance cause localized irreversible changes to the energy and angular momenta of stars.",
"Jacobi's invariant [9] implies that changes are related as $\\Delta E = \\Omega _p \\Delta L_z$ .",
"On average and to second order, particles lose $L_z$ at the ILR and gain at the OLR [34], [10], and this outward transfer of $L_z$ allows the wave to extract free energy from the galactic potential enabling the scattered particles to acquire additional random energy at both resonances.",
"The resulting depopulation of stars on near circular orbits over the narrow region of each resonance creates a “scratch” in the disc that may alter its stability properties.",
"It is important to realize that linear theory neglects this second order effect by assumption, i.e.",
"it does not allow for changes to the equilibrium state.",
"In fact, [46] found that the amplitudes of successive episodes of uncorrelated swing amplified noise rose slowly, but continuously, because the consequent scratches to the disc from each episode caused partial reflections of subsequent disturbances that allowed further amplification [52], [18].",
"This process continued until the the partial reflections became strong enough that the disc was able to support an unstable mode [46], [13], and coherent growth to large amplitude began.",
"We discuss this behaviour further in §REF .",
"Here we have used the word “scratch” to describe quite mild changes to the distribution function (hereafter DF) that can cause partial reflections of a wave propagating radially within the disc.",
"But scattering at a Lindblad resonance could also carve a similar feature that seeds a groove mode instead, and we will show below that this appears to be the more usual behaviour.",
"Note that the Lindblad resonances are closer to corotation, where $\\Omega _p = \\Omega _c(R_{\\rm CR})$ ($l=0$ in eq.",
"REF ), when $m$ is large than for waves of lower $m$ , implying that more free energy can be extracted from the potential, causing more rapid heating when the disc supports larger-scale waves.",
"The value of $m$ that is amplified most strongly [27], [61] is $m \\approx {R_{\\rm CR}\\kappa ^2 \\over 2\\pi X G\\Sigma },$ with $1 < X < 2$ in a flat rotation curve.",
"Thus the preferred $m$ varies inversely with the disc surface density $\\Sigma $ [51], [2].",
"Whatever the origin of disturbances, a sub-maximal disc will prefer higher $m$ , and therefore heat more slowly than would heavier discs.",
"Since spiral activity heats collisionless particles, or stars, an uncooled disc must become less able to support collective disturbances over time, as is well known.",
"[51] established that dynamical cooling by gas and star formation is needed to counter secular heating and to maintain spiral activity, a result that has been confirmed in many more recent simulations [43], [1], [3].",
"It nicely accounts for the observation [39] that almost all spiral patterns are seen in galaxies that contain gas and are forming stars."
],
[
"The spiral challenge",
"As already noted, a galaxy model having a dense centre and no sharp features, such as an edge or a groove, should not support any normal modes at all.",
"Since spirals are ubiquitous in disc galaxies containing a modest gas fraction, and also develop spontaneously in simulations of isolated discs, some mechanism is needed to excite them.",
"Currently, there are at least four proposed mechanisms: Following [36], [6] suggest that spirals result from a cavity type mode in a low-mass disc that is dynamically cool over most of the disc, but which also possess an inner “$Q$ barrier” to shield the ILR.",
"These authors imagine that most galaxies support a single, long-lived, mildly-unstable mode that persists for many tens of galactic rotations and becomes “quasi-steady” due to dissipative shocks in the gas, but allow that superposition of a second mode may be needed in some cases.",
"As given by eq.",
"(REF ), strong swing-amplification for $X\\sim 2$ sets a preference for multi-arm instabilities over 2-arm modes in low mass discs.",
"By considering only bi-symmetric disturbances in low-mass discs, [7] exploited the mild amplification when $X > 3$ in order to obtain slowly-growing spiral modes in their global stability analysis of many galaxy models.",
"Simulations by [45] of one of the cases presented by [7] confirmed that a single, slowly-growing mode was present when disturbance forces were restricted to $m=2$ .",
"Not surprisingly, however, he also found much more vigorous instabilities appeared when higher sectoral harmonics contributed to disturbance forces, and the contrived basic state of the disc that was designed to support the $m=2$ mode was rapidly changed by these more vigorous instabilities.",
"This evidence alone should have ruled out the theory, although [56] ignored it as he continued to advocate for their picture.",
"[63] and [64] abandoned the idea of spirals as normal modes, and advocated instead that a collection of massive clumps in the disc, each of which becomes dressed with its own wake, would create a “kaleidoscope” of shearing spiral patterns.",
"Their simulations of this process were all confined to a single shearing patch with a modest number of particles.",
"[15] conducted global simulations of a low-mass disc, embedded in a rigid halo, employing $10^8$ star particles to which they added a sprinkling of heavy particles that induced evolving multi-arm spiral patterns in the stars.",
"In a separate experiment they also tried a single perturber of mass $10^7\\;$ M$_\\odot $ , which they removed again after it had completed one orbit; the simulation continued to manifest spiral activity in response to the non-axisymmetric density distribution created by the original imposed mass, which they described as non-linear behaviour.",
"These authors suggest that “ragged” spiral activity in galaxies results from such responses to co-orbiting giant molecular clouds, massive star clusters, etc., and to the lingering disc responses should any disperse.",
"Although we do not doubt their numerical results, we remain unconvinced of their importance for spiral activity in galaxies.",
"First, the $10^7\\;$ M$_\\odot $ particle that [15] employed produced only a modest, and not very extensive wake in their low-mass disc.",
"Spirals in real galaxies generally have greater amplitude and radial extent than this, suggesting that yet more massive clumps would be needed.",
"Second, a collection of randomly placed heavy particles does not seem likely to produce a net response that is predominantly 2- or 3-armed, as are observed in the overwhelming majority of galaxies [11], [24], [65].",
"Third, clumps massive and numerous enough that their associated wakes produce large-amplitude and radially-extensive spiral patterns would scatter disc stars, and heat the disc rapidly so that the responses would fade quickly unless the disc were cooled aggressively, and the necessary cooling [63] seems rather extreme.",
"A fourth, and overriding, reason is that the idea is unnecessary, because discs readily support unstable spiral modes, as we discuss next.",
"[52] demonstrated that spiral activity in simulations resulted from superposition of a number of coherent, uniformly rotating waves.",
"They found that each wave grew and decayed, but was detectable over a period of some ten rotations at its corotation radius, and they presented substantial evidence that the waves were in fact modes.",
"The vigorous modes in their new picture differ substantially from those invoked by [6] because they work best where swing-amplication is strongest, do not last for nearly as long, and fresh instabilities develop to maintain spiral activity.",
"Note that the spiral appearance changes still more rapidly than do the modes in their simulations.",
"This is because the superposition of several modes, each having a different pattern speed and perhaps also angular periodicity, as well as time varying amplitude, causes the pattern of visible spiral arms to change radically in less than one orbit.An animation showing the time evolution of the net density when two notional patterns are superposed is at http://www.physics.rutgers.edu/$\\sim $ sellwood/spirals.html [52] also postulated a new cavity mode that relied upon partial reflection of travelling waves off an impedance variation created by a previous perturbation within the disc.",
"They suggested that ILR scattering by a past wave would have “scratched” an originally smooth distribution function (see §REF ) to create a deficiency of stars on near circular orbits at some radius.",
"A subsequent ingoing trailing wave encountering the scratch before reaching its own ILR, would find that the scratch in the otherwise smooth disc represents an abrupt change of impedance, and is therefore partially reflected into a outgoing leading wave.",
"A second reflection of the wave by swing amplification at corotation creates a cavity that will support standing waves having frequencies allowed by the usual phase closure condition.",
"They dubbed this new type of cavity mode a “mirror mode.” As for the bar mode that also includes swing amplification from leading to trailing, a reasonable reflected fraction at the partial mirror will cause a mirror mode to grow rapidly, and the instability must run its course within a modest number of pattern rotations.",
"Nevertheless, spiral activity can be maintained because other modes develop in rapid succession.",
"We present evidence in §REF below that the first real instability in the tests of the Mestel discs reported in [46] was almost certainly of this type.",
"Some mirror modes may also have been present in the simulations of [52], but we here (§) argue for a more probable recurrence cycle.",
"[20], [21], [5], [42], and others present simulations of mostly sub-maximal discs and report that spiral patterns are shearing structures, in which the density maxima wind as would material arms, or nearly so.",
"[20], [21], [31], and other authors have described spiral arm streaming motions that are consistent with expectations set out by [29]; however, such a flow pattern is required for any self-consistent spiral, irrespective of its nature or origin.",
"[5] and [37] find that spirals in their simulations behave as predicted by swing amplification theory [61].",
"[22] reported a correlation between the mean pitch angle of spiral arms in their simulations and the shear rate in the disc, which they also suggested was consistent with swing amplification theory.",
"[4] reported similar shearing and evolving spirals in the outer discs of barred simulations.",
"[32] excited regular spiral patterns that were particle wakes in the disc forced by a few heavy particles equally spaced around rings, finding additional swing amplification as the wakes lined up.",
"The earlier part of this body of work was summarized in the review by [14], and all these authors propose that shearing patterns are the fundamental character of spiral arms.",
"We discuss these findings in §REF .",
"The Gaia satellite has revealed the local phase space structure of the Milky Way in unprecedented detail [19].",
"[55] used these data to try to discriminate among the different theories just described.",
"They calculated the changes to a smooth distribution function that would be caused by a single episode of each of the last three spiral models and compared the predictions with the Gaia data.",
"They concluded that the features in action space seemed more consistent with the transient spiral mode model [52] than with either of the other two.",
"Note that [38] found evidence in the same data for scattering by the Milky Way bar, but some features remained that they could not attribute to the bar.",
"However, [26] concluded that other features in the same data were consistent with shearing spiral arms.",
"[55] also argued that the long-lived spiral mode model advocated by [6] would not cause any pronounced changes, because the ILR, where the largest changes generally occur, is shielded by the $Q$ barrier, no net change is expected at corotation, and the OLR would probably lie too far outside the solar circle to affect the local distribution.",
"Thus if their delicate mechanism for spiral generation were indeed to operate in the Milky Way, it would contribute little to the observd extensive sub-structure in phase space and would implausibly have to survive in the mild disequilibrium state of the disc revealed in the Gaia data."
],
[
"Modes or shearing patterns?",
"The shearing spiral behaviour that is apparent in almost all simulations of cool, isolated discs possesses many aspects of swing-amplification and wakes, as the above cited papers have reported.",
"We have also reported such behaviour: e.g., Fig.",
"3 of [51] indicated the time evolution of a single 3-arm spiral disturbance that apparently sheared and amplified to an open trailing pattern before decaying as it continued to wind and, in simulations of much improved numerical quality, [52] illustrated constantly changing patterns, which is qualitatively similar to the behaviour that all authors report.",
"However, we have also found [44], [45], [52] that the constantly changing appearance of spiral patterns results from the superposition of a modest number of longer lived waves [41].",
"Even though a mode (eq.",
"REF ) has a fixed pattern speed at all radii and a constant shape function, $A_m(R)$ , shearing patterns are the inevitable consequence of superposed modes.",
"All that is required to produce the appearance of a shearing transient spiral (see footnote 2) from two or more superposed patterns of fixed shape is that the modes closer to the centre have higher pattern speed, which is always true.",
"The proposition that the shearing patterns are the fundamental behaviour is not a satisfactory explanation for the origin of the spirals, because it is unable to answer a key question: how can the spiral amplitudes in simulations be largely independent of the number of particles?",
"No recent paper that argues for this interpretation addresses this issue, yet there is substantial evidence to support it: [45], [46], and [52] all reported similar final amplitudes in experiments in which the number of particles ranged over orders of magnitude.",
"As $N$ is increased, the amplitude of shot noise fluctuations must decrease as $N^{-1/2}$ , and some kind of growth mechanism is required to produce final amplitudes that are independent of $N$ .",
"Swing-amplification and/or wakes do not lead to indefinite growth, and therefore cannot account for final amplitudes that are independent of $N$ , at least in linear theory.",
"We would agree with an argument that non-linear scattering (§REF ) is important, but that plays into our case for unstable modes [52].",
"It is likely that most models adopted in simulations are unstable, because they allow some feedback through the centre and/or possess outer edges sharp enough to excite normal modes.",
"Mildly unstable modes will be seeded at low amplitude when $N$ is large and may not cause visible changes for several rotations.",
"Both [45] and [52] present cases in which the first visible features took longer and longer to appear as the number of particles was increased.Pronouncements of stability after a short period of evolution in large-$N$ models [15] are unlikely to hold in longer integrations.",
"Once the first instability has appeared, we find that activity takes off, as the particle distribution becomes more and more structured by the previous evolution; the new results presented in § below show in more detail how a recurrent cycle of modes can occur.",
"Figure: An old simulation of the half-mass Mestel disc.",
"The toppanel shows the amplitude evolution of m=2m=2 disturbances reproducedfrom the N=5×10 7 N=5\\times 10^7 case in Fig.",
"2 of .",
"The powerspectrum over the time range 100≤t≤400100\\le t \\le 400 in the middlepanel shows no coherent modes were present over this interval.",
"Thesolid curve indicates the radial variation of mΩ c m\\Omega _c while thedashed curves show mΩ c ±κm\\Omega _c \\pm \\kappa .",
"The bottom panel is forthe period 1300≤t≤16001300\\le t \\le 1600 and reveals that the more rapidamplitude growth from t=1000t=1000 is caused by more coherentinstabilities, dark horizontal streaks, that developed later in thismodel, as reported.",
"Note that the grey scaleamplitudes are logarithmic and differ in the two panels.However, it is not neccessary to start from a disc that possesses one or more global instabilities.",
"[61] claimed that the half-mass Mestel disc is linearly stable, and this case was studied in simulations by [46].",
"Fig.",
"REF illustrates the evolution of the $N = 5 \\times 10^7$ simulation from his paper; details of the model and the numerical method are given in § below.",
"The top panel reproduces the cyan curve from Fig.",
"2 of [46], which reported at each instant the greatest value of the ratio $\\Sigma _2(R)/\\Sigma _0(R)$ within the radius range $1.2 < R < 12$ , where $\\Sigma _2$ is the amplitude of the bi-symmetric disturbance density.",
"After an initial surge by a factor of a few as each particle created its own wake (§REF ), these $m=2$ features manifested slow secular growth.",
"During this phase, the amplitudes of successive swing-amplified episodes increased slowly because each created mild scratches in the previously smooth disc that enabled partial reflections.",
"The partial mirrors so created at first reflected small fractions of the incident waves, but as the disturbance amplitudes rose with each episode, the new scratches became more reflective.",
"No coherent modes were detectable during this secular growth phase, and the power spectrum, such as shown in the middle panel, was characterized by multiple uncorrelated frequencies arising from whichever noise features were strongest at the time; the corotation radius of each event determines the frequency of the wave that propagates to its ILR where it is absorbed.",
"These findings were consistent with Toomre's linear theory prediction of global stability, since the secular growth was caused by the non-linear scattering terms that his analysis neglected.",
"The largest amplitude disturbances reached an overdensity of $\\sim 2$ % by $t \\sim 1000$ , at which point the reflections became strong enough to cause true unstable modes in this originally stable disc.",
"The amplitude in the top panel of Fig.",
"REF began to rise more rapidly and the power spectrum, in the bottom panel, that has a different amplitude scale shows much stronger and more coherent waves, which are modes.",
"Figure: Fig.",
"6 of showed the shape of the mode fitted tothe data from his model 50c.",
"Here we show the logarithmic spiraltransform of that mode density.",
"Positive values of tanγ\\tan \\gamma aretrailing, negative are leading, where γ\\gamma is the angle betweenthe tangent to the spiral and the radius vector.",
"The amplitudescale is arbitrary.",
"[18] and [17] successfully developed a quasi-linear theoretical description of this behaviour.",
"[13] conducted a global mode analysis of this model at $t=1400$ , and estimated the frequency of the dominant instability to be $\\omega = 0.597 + 0.013i$ in excellent agreement with Sellwood's own estimate (his paper has factor 10 typo in the growth rate).",
"Although both analyses were of the model denoted 50c, in which the particles coordinates had been scrambled at $t=1400$ , the faster wave in the bottom panel of Fig.",
"REF confirms a mode of similar pattern speed when the evolution is uninterrupted.",
"Fig.",
"REF shows the logarithmic spiral decomposition of the mode [46] fitted to the data from his model 50c.",
"The leading/trailing bias reflects the trailing appearance of the mode, but the leading side has significant amplitude indicating that it was a mirror, or cavity, mode that operated by an inner reflection off the partial mirror to a leading wave, as [52] described.",
"[57] calculates the changes to the DF at the principal resonances caused by a transient disturbance, and attempts to show how they lead to further instabilities.",
"However, the correspondence between the modes he calculates and the groove and mirror modes that we have reported is not immediately apparent from his analysis.",
"Thus we consider that every simulation of a cool, unperturbed disc must inevitably possess, or develop, unstable modes.",
"The unbounded growth of instabilities, until second order terms become important, provides the only viable mechanism to give rise to large-amplitude spirals no matter how large a number of particles is employed.",
"With a recurrence mechanism, such as that described here (§), the simulations must support multiple unstable modes and their superposition naturally leads to the shearing transient activity that is almost universally reported.",
"Note that externally perturbed models, such as those presented by [15] and [32], may not manifest modes if the initial responses develop before any modes could grow to significant amplitude."
],
[
"Objective of this paper",
"The complicated mechanism just described would be required to account for the origin of spirals only in models that do not possess any linear instabilities.",
"The purpose of this paper is to show how a recurring cycle of instabilities can arise in models that do possess a single mild instability at the outset.",
"We argue that such an instability cycle is the origin of spiral activity in all simulations, and hopefully in real galaxies also.",
"[52] also reported that coherent waves appear and decay at increasing radii and lower frequency over time.",
"It is possible that the later instabilities are independent modes caused by uncorrelated noise fluctuations scratching the DF at larger radii.",
"However, [52] preferred the idea that the decay of one mode created conditions to seed a new instability, although they did not provide a detailed mechanism for recurrence.",
"Here we describe the recurrence mechanism in detail.",
"In fact, [54] presented a recurrence mechanism for self-gravitating spiral modes in low mass particle discs orbiting around a central mass.",
"The mechanism they proposed was that scattering at the OLR created a groove in phase space that excited a new instability.",
"We thought it unlikely that the same recurrence mechanism would work in heavy discs, for reasons that we give in §, but we here demonstrate that this expectation was wrong."
],
[
"Technique",
"Since the dynamical behaviour of fully self-consistent simulations can be very complicated, we find it fruitful to run simplified simulations that can capture the phenomena we wish to study without them being obscured by unrelated activity.",
"Once understood, it will naturally be important to show that the behaviour persists under more general conditions.",
"Accordingly, we adopt the razor-thin Mestel disc used in the studies by [66] and [61], which is characterized by a constant circular speed $V_0$ at all radii.",
"Note that since $\\Omega _c = V_0/R$ and $\\kappa = \\surd 2V_0/R$ , $\\Omega _c - \\kappa /m$ increases indefinitely towards the centre except for $m=1$ , when it is negative at all radii.",
"Thus an ILR will intervene to damp all disturbances having $\\Omega _p>0$ when $m\\ge 2$ , prohibiting possible cavity modes in a smooth disc, except for $m=1$ .",
"The axisymmetric surface density $\\Sigma _0(R) = V_0^2 / (2\\pi G R)$ would self-consistently yield the appropriate central attraction for centrifugal balance.",
"If the surface density is reduced to fraction $x$ ($0 < x \\le 1$ ), with the removed mass added to a rigid halo to maintain centrifugal balance, then eq.",
"(REF ) implies that the most vigorously amplified disturbances ($X\\simeq 1.5$ ) will have $m=1.33/x$ .",
"Since [66] had found that the full-mass disc was prone to $m=1$ cavity modes, [61] preferred a model with $x=0.5$ in order to avoid lop-sided modes.",
"Note that although swing-amplification will be near its most vigorous for $m=2$ disturbances for this choice of $x$ , bisymmetric cavity modes in a smooth disc are disallowed because an ILR would block the feedback loop.",
"Toomre employed the DF [60], [9] $f(E,L_z) = \\left\\lbrace \\begin{array}{ll}{ x F L_z^q e^{-E/\\sigma _R^2} & L_z>0 \\cr 0 & otherwise, \\cr }\\end{array}where \\right.q = V_0^2/\\sigma _R^2 - 1 and the normalization constant is\\begin{equation}F = {1 \\over G R_0(R_0V_0)^q} { (q/2 + 0.5)^{q/2+1} \\over \\pi ^{3/2}(q/2-0.5)!",
"}.\\end{equation}[\\cite {Se12} erroneously omitted the factor R_0^{-(q+1)}.",
"]Choosing q=11.44 yields a Gaussian distribution of velocities suchthat the x=0.5 disc has Q=1.5.",
"Toomre further multiplied the DFf by the double taper function\\begin{equation}T(L_z) =\\left[ 1 + \\left( {R_0V_0 \\over L_z} \\right)^\\nu \\right]^{-1}\\left[ 1 + \\left( {L_z \\over R_1V_0} \\right)^\\mu \\right]^{-1},\\end{equation}to create a central cut out centered at R_0 and an outer tapercentered at R_1, while maintaining the centripetal acceleration-V_0^2/R everywhere.",
"Setting the taper indices \\nu =4 and \\mu =5yielded an idealized, smooth disc model that Toomre claimed possessedno small amplitude unstable modes.",
"We choose R_1 = 11.5R_0, andlimit the radial extent of the disc by an energy cut-off thateliminates particles having sufficient energy to pass R=20R_0.",
"Herewe adopt units such that V_0 = R_0 = G = 1.$ The evolution of one of the simulations of this model reported by [46] was illustrated in Fig.",
"REF above.",
"The simulations in the present paper use the same code and disc model, have the same number of particles (see Table REF ), but are integrated to $t = 1000$ only and so are unaffected by the later rapid growth phase.",
"Table: Numerical parametersThe particles in our simulations are constrained to move in a plane over a 2D polar mesh at which the accelerations are calculated, and interpolated to the position of each particle.",
"In most cases, disturbance forces are restricted to a single sectoral harmonic, $m=2$ or $m=3$ .",
"Since the gravitational field is a convolution of the mass density with a Green function that is most efficiently computed by Fourier transforms, it is easy to restrict the sectoral hamonics that contribute to the field when using a polar grid.",
"A full description of our numerical procedures is given in the on-line manual [48] and the code itself is available for download.",
"Table REF gives the values of the numerical parameters for most simulations reported here, and we note when they are varied in a few cases.",
"[46] reported, and we have reconfirmed in this study, that all our results are insensitive to reasonable changes to grid resoution, time step and zones, and number of particles.",
"Changes to the softening length do affect the frequencies of the modes, but not the qualititative behaviour.",
"To set up each model we draw particles from the tapered DF (eqs.",
"REF and ) as described in the appendix of [12].",
"However, in some experiments we modified the distribution of selected particles in order to introduce a single additional feature into the DF, as set out in §REF .",
"We also restarted some simulations after “scrambling” a copy of the particle distribution.",
"By this we mean that we changed $(R,\\phi ,v_R,v_\\phi ) \\rightarrow (R,\\phi ^\\prime ,v_R,v_\\phi )$ for every particle, with $\\phi ^\\prime $ being chosen at random from a distribution that is uniform in 0 to $2\\pi $ .",
"Clearly scrambling resets the amplitudes of all non-axisymmetric disturbances back to the shot noise level of the initial disc, while preserving the radius and both velocity components (in polar coordinates), so that any features in the action distribution of the particles that had been introduced during prior evolution would be preserved.",
"This is therefore not equivalent to a fresh start with a different random seed.",
"Note that the radial action, $J_R \\equiv \\oint \\dot{R} dR/(2\\pi )$ [9], has dimensions of angular momentum, is zero for a circular orbit, and increases with the eccentricity of the orbit.",
"In an axisymmetric potential, the angular momentum, $L_z$ , is the other action for orbits confined to a plane."
],
[
"Results",
"In this section we study the effect of introducing two distinct features into the otherwise unperturbed disc, both designed to excite an initial mode of a specific type, namely: a groove mode [54], [53] and an outer edge mode [61], [40].",
"In both cases, we follow its evolution and study the further modes that are excited.",
"We show for each case that the second mode is a true instability of the modified disc by erasing all non-axisymmetric structure at the time the second mode began to grow, as just described, and demonstrating that the scrambled particle distribution possessed a very similar instability to that which developed in the continued original run.",
"Furthermore we identify the scattering feature in action space that was responsible for the second instability, as reported below in §REF .",
"Figure: The change in the densities of particles in action spacecreated by hand in order to insert a groove-like feature into theinitial model."
],
[
"Initial groove mode",
"We create a deficiency of low-$J_R$ particles over a narrow inclined range in $(L_z,J_R)$ space by giving some particles a larger $J_R$ , as shown in Fig.",
"REF .",
"The slope of this feature, $-1/2$ , approximately traces the locus of the ILR of a bisymmetric disturbance having a pattern speed $\\Omega _n = 0.1$ , although the disturbance was, in fact, purely hypothetical.",
"The ILR of a disturbance having this $\\Omega _n$ would lie at $L_z = 2.93$ for circular orbits, for which $J_R=0$ .Note that all the major resonances with any pattern speed and any angular periodicity in the self-similar Mestel disc lie on lines of slope $\\simeq -1/2$ in the space of these actions, for $J_R \\ll L_z$ .",
"We describe this feature as a groove, since it is a deficiency at low $J_R$ over a narrow range in $L_z$ .",
"Since orbits in this self-similar disc have typical epicyclic radii of $\\sigma _R/\\kappa \\simeq 0.2R_g$ about the guiding centre radius, this narrow groove in action space causes no noticeable change to the surface density profile.",
"Figure: The evolution of model G, which was seeded with the grooveshown in Fig. .",
"The red line in the top panelshows the amplitude evolution, the dotted black line indicates whatwould have happened without the groove, and explanations of thegreen and blue lines given in §§ & respectively.",
"The rapid, but small, variations in the amplitudereflect alternating constructive and destructive interference byuncorrelated episodes of swing-amplified noise, that have longerperiods at later times as this behaviour spreads to the outer partof the radial range.",
"The middle and bottom panels, for which againthe grey scales differ, present power spectra over the time ranges100≤t≤400100\\le t \\le 400 and 400≤t≤700400\\le t \\le 700 respectively, whicheach manifest a single mode.Table: Summary of the modes fitted to data from simulationsdescribed in §, including the fitted complexfrequencies, ω\\omega with β\\beta being the growth rate of themode, and the radii of the principal resonances for circular orbitsgiven by (m+l√2)/ℜ(ω)(m+l\\surd 2)/\\Re (\\omega ), although we leave uninterestingentries blank for clarity.Our, perhaps somewhat clumsy, procedure for shifting particles was motivated by our observations of resonance scattering in previous simulations.",
"We calculated the frequency distance from the resonance $\\delta \\omega = 2\\Omega _\\phi - \\Omega _R - 2\\Omega _n $ for each particle, where $\\Omega _\\phi $ and $\\Omega _R$ are respectively the angular frequencies of the guiding centre and radial oscillation for orbits of arbitrary eccentricity; note that these frequencies tend to the familiar quantities $\\Omega _\\phi \\rightarrow \\Omega _c$ and $\\Omega _R \\rightarrow \\kappa $ as the eccentricity $\\rightarrow 0$ .",
"We then select particles that have $g^2 = 1 - (\\delta \\omega / w_\\omega \\Omega _n)^2 > 0$ as candidates to be scattered.",
"The relative frequency width, $w_\\omega = 0.06$ , of this feature is deliberately rather narrow.",
"Of these, we further select only those particles whose energy of random motion $E_r = E - E_c(L_z) < g E_{r,{\\rm lim}}$ , where the groove extends to the limiting random energy $E_{r,{\\rm lim}} = 0.025$ at $g=1$ , i.e.",
"along the resonance locus.",
"The probability that these selected particles are given larger random energy decreases from $d$ to zero as $E_r$ increases from zero to this maximum, where the fractional depth of the groove $d = 0.25$ .",
"The new $E_r = E_{r,{\\rm lim}}(1+3p)$ where $p$ is the solution of $\\exp (-p^2)=s$ , with $s$ being randomly drawn from a distribution uniform in 0–1, and finally $\\Delta L_z = \\Delta E / \\Omega _n$ .",
"A simulation that employed this modified DF, model G, supported a groove instability at first.",
"The red curve in the top panel of Fig.",
"REF presents the time evolution of the ratio $\\langle \\Sigma _2(R)/\\Sigma _0(R)\\rangle $ averaged over the radius range $2<R<8$ .",
"Since this is a different measure from that presented in Fig.",
"REF , the dotted curve indicates this measure for the same model without the groove.",
"The green and blue lines are explained below.",
"Power spectra over two periods of growth are shown in the lower two panels.",
"The middle panel is for the same time period as the middle panel in Fig.",
"REF , although the grey scales differ, and the contrast in the appearance is quite striking.",
"The disturbance in model G grew slowly until $t\\sim 400$ , and the power spectrum over this time interval in the middle panel of Fig.",
"REF is dominated by a coherent disturbance of angular frequency $2\\Omega _p \\approx 0.65$ that is mostly localized between the ILR and corotation.",
"A couple of other frequencies that are probably due to swing-amplified noise are also faintly visible, and these uncorellated mild disturbances are responsible for the small, but rapid, amplitude fluctuations in the red line in the top panel.",
"We employed the mode-fitting procedure devised by [49] to the data from model G over the time interval $100 \\le t \\le 400$ and give its estimated eigenfrequency in line 1 of Table REF .",
"The shape of the fitted mode, with its principal resonances for circular orbits ($J_R=0$ ) marked by the circles, is shown in the upper panel of Fig REF , and the decomposition of the mode into logarithmic spirals is in the lower panel.",
"The mechanism for the groove instability was fully explained by [53] and outlined above in §1.1.",
"It is not a cavity-type mode, and the frequency is determined by the gradients in angular momentum density for orbits of small radial action.",
"The density changes caused by non-axisymmetric disturbances at the groove “edges” excite a supporting response from the surrounding disc, making it a large-scale spiral mode.",
"In principle, the circular angular frequency of the groove centre should be the pattern speed of the mode, but geometric factors in low-$m$ modes and finite growth rate cause the corotation radius to lie slightly farther out.",
"The shape of the mode is determined by the shear rate and surface density profile of the disc, so that in the present self-similar disc, groove modes should have closely similar shapes and pitch angles and their spatial scale will vary with the corotation radius.",
"Figure: Top panel: contours of the relative over density of the modefitted to the results from model G; distances are marked in units ofR 0 R_0 and the circles mark the radii of the principal resonances forcircular orbits, given in line 1 of Table .",
"Bottompanel: the logarithmic spiral transform of the mode density and asin Fig.",
", the amplitude scale is arbitrary.Evidence that this mode is a groove instability [54] is two-fold.",
"First, corotation (line 1 of Table REF ) lies close to, but just outside [53], the groove centre at $L_z = 2.93$ .",
"Second, the leading component in the mode transform (bottom panel of Fig.",
"REF ) is much weaker than that in Fig.",
"REF because groove modes (§REF ) do not operate by a feedback cycle of combined leading and trailing waves.",
"Our choice of a very narrow frequency width for the groove ensured a low growth rate for the mode [53].",
"We found in other simulations, not described in detail here, that wider grooves excited a mode that grew more vigorously to higher amplitude.",
"We preferred to study an instability that caused mild non-linear changes in order to make it easier to follow further activity that was a consequence of the initial mode."
],
[
"Second mode",
"After the slow growth to $t=400$ , the disturbance amplitude in model G, shown by the red line in the top panel of Fig.",
"REF , rose more rapidly over the time range $400 < t < 700$ .",
"The power spectrum over this time range in the bottom panel revealed a new coherent wave having a frequency $\\sim 0.45$ .",
"As the mode had clearly saturated by the end of this period, we fitted a mode over the time interval $200 < t < 500$ , obtaining good fits with eigenfrequencies in the range given in line 2 of Table REF .",
"In order to verify that this second mode was a true linear instability of the modified disc, we made a copy of the particle distribution at $t=400$ and evolved the scrambled (see §) particle distribution in a separate simulation, designated GR.",
"The amplitude evolution in the simulation starting from this scrambled disc is shown by the green line in the top panel of Fig.",
"REF .",
"The very first value was measured from the particles before they were scrambled, after which the amplitude rose out of the shot noise steadily to $t=700$ when the run was stopped.",
"We found that the evolution of the scrambled disc, model GR, was again dominated by an exponentially growing disturbance of fixed pattern speed.",
"The eigenfrequency fitted to the data from this run, given in line 3 of Table REF , is within the uncertainties the same as that in line 2 – the second mode found in model G. A second, milder instability was also present that had the fitted frequency given in line 4.",
"At first we thought it could be the original groove mode, although the pattern speed is distinctly higher, but we identify its progeny at the end of §REF .",
"Figure: Difference from a pristine undisturbed DF and that created bysplicing those particles in the OLR scattering feature fromFig. .",
"Note that the colour contrast hasbeen enhanced in order to show how nearly seamless the splicingprocess was.",
"These particles were employed in model GS OLR _{\\rm OLR}."
],
[
"Cause of the second mode",
"The second more vigorous mode, which was not present in the early evolution of the same model G (middle panel of Fig.",
"REF ), was caused by changes to the DF produced by the first mode.",
"Fig.",
"REF reveals the changes to the density of particles in action space between times 0 and 400.",
"The colour scale in this figure shows where the density of particles at $t=400$ is both greater and less than that at $t=0$ , and reveals several features.",
"The dashed line marks the locus of the corotation resonance, and the dotted lines the Lindblad resonances, for the fitted frequency of the first mode in this simulation.",
"These lines indicate that the most prominent feature is the result of ILR scattering and the faint pair of features at larger $L_z$ are associated with the OLR.",
"A paired deficiency and excess have appeared near the location of the original groove at $L_z \\simeq 2.93$ , reflecting the changes that must have been caused by the instability.",
"As such changes occur near corotation, marked by the dashed line, they should not cause any significant change to $J_R$ [50], which is largely true.",
"Since the groove instability is so mild, Fig.",
"REF also shows the effects of scattering by weak noise features that generally had lower frequencies than the excited mode.",
"Note the differences in the nature of the changes caused by scattering at the ILR from those at the OLR.",
"In action space, the scattering vector at a resonance has slope $\\Delta J_R/\\Delta Lz = l/m$ [50], where $m$ is the angular multiplicity of the pattern and $l=0,\\;\\mp 1$ at respectively corotation and the inner and outer Lindblad resonances.",
"Formally the slope $l/m$ is exact only as $J_R\\rightarrow 0$ but we find it is an excellent approximation over the range of $J_R$ of interest here.",
"As noted before [46], [52], [55], the scattering vector slope, $-1/2$ , is almost perfectly aligned with the ILR locus for $m=2$ waves, implying that particles stay on resonance as they gain $J_R$ , allowing large changes to build up.",
"Thus ILR scattering moves particles along the resonance locus, creating a deficiency of particles at low $J_R$ , and an excess for larger $J_R$ , just as we created artificially at the start of the first simulation (Fig.",
"REF ).",
"On the other hand, scattering vectors at the OLR have positive slope $+1/m$ , while the resonance locus again has slope $-1/2$ .",
"Thus we see particles are scattered across the resonance from a sloping deficiency to an almost parallel line where there is a slight excess.",
"It is perhaps interesting that the lines are approximately parallel, indicating that the action changes are roughly independent of the initial $J_R$ .",
"More importantly, $J_R$ is an intrinsically positive quantity, and therefore Lindblad resonance scattering always creates a deficiency at $J_R 0$ .",
"Figure: The later part of the evolution of model E having a steeperouter edge.",
"The orbit period at the corotation radius of the edgemode (R=12.8R=12.8) is ∼80\\sim 80 in these units.",
"The circles at t=600t=600mark corotation and the ILR of the edge mode, those at t=650t=650 arefor corotation and both Lindblad resonances for the second mode.The indicated colour scale shows the absolute over- or under-densityof non-axisymmetric features, which has been scaled up by a factorof 100.We suspected that one of the features in Fig.",
"REF was responsible for the new instability in model G. In order to determine which, we extracted all the particles in one of the features and spliced them into a pristine undisturbed particle distribution that we then evolved.",
"In detail, we began the splicing procedure by creating a fresh set of particles from the doubly tapered Toomre-Zang DF described in § that lacked any additional features; recall that [61] had predicted, and [46] had confirmed, that this model has no coherent, small amplitude instabilties.",
"We then extracted all the particles from the evolved distribution at $t=400$ that lay within a trapezium in action space of width $\\Delta L_z = 1$ that was defined by $|L_z + 0.5 J_R - L_{z,0}|< 0.5$ , for some value of $L_{z,0}$ .",
"We spliced the extracted particles into the undisturbed set by substituting the values of $(x,y,v_x,v_y)$ of these selected particles from the evolved set for the values in the undisturbed set of particles that lay within the same trapezium in action space.Since the central attraction, $-V_0^2/R$ , is an unchanging function throughout these experiments, we do not have to worry about possible changes to the axisymmetric potential.",
"This process required attention to two points in order to obtain a near seamless splice.",
"First, almost all the particles in the evolved model had changed their $L_z$ values during the evolution to $t=400$ , making it impossible to just replace particle $n$ from the pristine set with particle $n$ from the evolved set, even though they were both originally drawn from the same DF by the same algorithm.",
"We therefore had to compute $J^\\prime = L_z + 0.5J_R$ for every particle in both sets and rank order them to identify those in the desired range.",
"Second, the large numbers of particles in the desired range of $J^\\prime $ , typically some 4.5 million, was not precisely the same in the two sets, so that we had to skip the occasional particle, usually one in several hundred, in the larger set to ensure equal numbers that spanned the entire range.",
"Fig.",
"REF shows the difference between the spliced and undisturbed sets for $L_{z,0}=5$ .",
"It is clear that the OLR scattering feature from Fig.",
"REF has been satisfactorily isolated from all other features in action space at that time, without introducing sharp edges to the splice.",
"The time evolution of the disturbance density in a simulation started from the spliced distribution, model GS$_{\\rm OLR}$ , is indicated by the blue line in the top panel of Fig.",
"REF .",
"The mild non-axisymmetry of the spliced particles at $t=400$ , was erased by scrambling before the evolution commenced, and the disturbance density grew out of the noise, closely following the evolution from the simply scrambled disc, model GR, traced by the green line.Other modes are excited after $t\\sim 700$ , and interference between disturbances rotating at different rates can cause large temporary decreases in the measured amplitude.",
"The eigenfrequency fitted to the data from this spliced run is given in line 5 of Table REF and is, again within the uncertainties, the same as those in lines 2 and 3, the second mode in model G and the mode in the scrambled disc, model GR.",
"This evidence conclusively shows that the OLR scattering feature of the first mode in model G was responsible for creating the second instability.",
"Note that the corotation resonance for near circular orbits of the modes given in lines 2, 3, and 5 of Table REF , all lie near the OLR feature created by the first mode (Fig.",
"REF ) strongly suggesting that the second mode was also a groove instability.",
"It seems likely that the deficit of particles near $J_R 0$ was the cause.",
"We were quite surprised to find it was the OLR scattering feature that was responsible for the second instability for the reasons set out in the discussion below (§).",
"Yet our experiment with the isolated weak scattering feature of the OLR leaves no doubt that it did indeed provoke a new instability that was more vigorous than the original mode.",
"The near coincidence of the OLR for the first mode (line 1 of Table REF ) with CR for the second (lines 2, 3 and 5 of Table REF ) suggests non-linear mode coupling.",
"Resonance scattering to create the new groove is a non-linear effect, but we have strong evidence against the hypothesis of coupling at resonances between large amplitude waves that was first proposed by [58].",
"A technical reason is that their theory requires the interaction of three waves, with the third wave having rotational symmetry that is the sum or difference of the two other waves, and our simulation in which disturbance forces were restricted to $m=2$ could not support disturbances having $m=0$ or 4.",
"However, the more compelling argument is that scrambling the particles erased all non-axisymmetric features in the density distribution, which need to have significant amplitude for the mode coupling mechanism to work.",
"Instead, our simulations provide clear evidence that the second mode is a true, linear instability of the modified disc at $t=400$ .",
"We tried other experiments for which we chose other splice centres, but do not illustrate the results.",
"Choosing $L_{z,0}=3$ , model GS$_{\\rm CR}$ supported a mild groove mode with frequency given in line 6 of Table REF ; the deficiency of the original DF had been shifted to lower $L_z$ by the evolution of the first groove mode, causing the slightly higher pattern speed than that reported in line 1.",
"It is satisfying to note that this instability approximately matches that of the second mode in the scrambled model GR, reported in line 4 of the table.",
"Simulations of particle distributions that resulted from splicing with $L_{z,0}=1$ or from $L_{z,0}=2$ appeared not to support any coherent instability.",
"The strong ILR scattering feature from the first mode lies within the inner taper, and is therefore unable to excite a significant instability because the tapered surounding disc would provide only a weak supporting response of [53]."
],
[
"Initial edge mode",
"All we needed to do to excite an edge mode was to increase the value of $\\mu $ for the outer index taper (eq. ).",
"Setting $\\mu =20$ created a vigorous instability that saturated at large amplitude, strongly distorting a large fraction of the disc.",
"After some experimentation, we found that setting $\\mu =12$ , model E, yielded an outer edge mode with a moderate growth rate that enabled us to understand the subsequent evolution.",
"Note that for the sequence of runs reported in this section, in which the initial disturbances were in the outer disc where the grid becomes coarser, we used a 4 times finer grid ($404 \\times 512$ ).",
"We also restricted disturbance forces to $m=3$ , rather than $m=2$ , in order that the modes have smaller radial extent, but the other numerical parameters were unchanged.",
"Figure: The evolution of model E having a steeper outer edge.",
"Thered line in the top panel shows the amplitude evolution, while themiddle and bottom panels show power spectra over the time ranges350≤t≤650350\\le t \\le 650 and 600≤t≤1000600\\le t \\le 1000 respectively andagain the grey scales differ.The evolution of the non-axisymmetric density of model E is shown in Fig.",
"REF .",
"A 3-arm spiral grows slowly in the outer disc until $t\\sim 600$ , after which additional patterns appear closer to the disc centre.",
"The amplitude evolution of the $m=3$ disturbance density over the radius range $3<R<12$ and power spectra over two periods of growth are shown in Fig.",
"REF .",
"Figure: Top panel: contours of the relative over density of the modefitted to the simulation results of model E given in line 7 ofTable .",
"Distances are marked in units of R 0 R_0 andthe circles mark the radii of the principal resonances for circularorbits.",
"Bottom panel: the logarithmic spiral transform of the modedensity.",
"Again the amplitude scale is arbitrary."
],
[
"First mode",
"The evolution of model E to $t=600$ , is dominated (middle panel of Fig.",
"REF ) by a single, exponentially growing disturbance of frequency $3\\Omega _p 0.2$ , that extends from its ILR to OLR, with corotation perhaps near $R \\sim 13$ .",
"The mode fitting software yielded an eigenfrequency given in line 7 of Table REF and the fitted mode and its logarithmic spiral transform are presented in Fig.",
"REF .",
"It has the hallmarks of an edge mode.",
"As expected [62], $R_{\\rm CR}$ (line 7 of Table REF ) lies outside the radius of the steepest gradient, which is at $L_z=11.5$ , the centre of the outer taper.",
"Again, the logarithmic spiral transform of the mode, lower panel of Fig.",
"REF , has no significant leading component as expected for an edge mode, since the mechanism does not require feedback through leading waves (see §REF ).",
"Figure: Difference between the densities of particles in action spacefrom t=0t=0 to t=600t=600, from model E having the sharper outer edge.The dashed line shows the locus of the corotation resonance for themeasured mode frequency, line 7 of Table , and thedotted lines are the Lindblad resonances.",
"Note that the range ofabscissae differs from other figures of this type and our initialupper energy bound (§) excluded particles from thewhite region.Fig.",
"REF displays the changes to the density of particles in action space between $t=0$ and $t=600$ in model E, and the dashed and dotted lines mark the loci of the principal resonances for the measured pattern speed of the original edge mode.",
"It reveals scattering features at corotation and the ILR, but there are no significant changes at the OLR because there is very little disc left at those radii, and angular momentum transport by this edge mode is from the ILR to CR only.",
"Since the power spectra (Fig.",
"REF ) and the fitted mode shape (Fig.",
"REF ) were drawn using the relative overdensity at each radius, they exaggerate the apparent amplitude of the mode in the very outer part of the disc.",
"The very low frequency features in the bottom panel of Fig.",
"REF have a small absolute amplitude, and probably arise from shot noise due to the low particle density near the disc edge.",
"As noted above, scattering vectors in action space have slopes $=l/m$ .",
"Thus the near perfect alignment of the scattering vector with the locus of the ILR occurs only for $m=2$ disturbances, and therefore the excess and deficiency in Fig.",
"REF at the ILR are slightly offset from the resonance locus for this $m=3$ pattern, and the groove carved by the mode at $L_z=6.9$ is fractionally exterior to its ILR."
],
[
"An inwardly propagating cascade of groove modes",
"The power spectrum of the later part of this simulation, in the bottom panel of Fig.",
"REF , indicates several additional coherent waves, in addition to the continuing presence of the first mode, which had not even finished growing by $t=600$ .",
"We have fitted two modes to the data from the simulation over the period $500 < t < 780$ , recovering both the edge mode and the second mode, which has an estimated frequency given in line 8 of Table REF .",
"Corotation for this second mode, also given in line 8, is somewhat outside the groove centre carved by the original edge mode, as is usual groove mode [53].",
"The green line in Fig.",
"REF shows the amplitude evolution we obtain in a new simulation, model ER, started from the scrambled particles of model E at $t=600$ .",
"Once again we find the perturbation amplitude grows rapidly with the estimated frequency given in line 9 of Table REF , which is in reasonable agreement with the frequency of the second mode given in line 8.",
"Note the second mode outgrows the original edge mode, which is also detectable in the scrambled simulation.",
"The true uncertainties in the measured frequencies and resonance radii in this case are probably greater then the formal values given in the table because we were attempting to fit two vigorously growing modes over a short time interval.",
"The numerical evidence just presented implies that the second mode in model E is a groove mode seeded by ILR scattering by the first mode (Fig.",
"REF ).",
"Once again, the near coincidence of the ILR of mode 1 (line 7 of Table REF ) is CR of its daughter (line 8) is not evidence for non-linear mode coupling, for the same reasons as given above.",
"We obtained the daughter mode (line 9) even when we eliminated all pre-existing waves by scrambling the disc particles, and we have strong evidence that the new mode is a linear instability caused by scattering at the ILR of the first mode.",
"The power spectrum in the bottom panel of Fig.",
"REF strongly suggests a cascade of groove instabilities to ever higher frequency, with corotation of each subsequent instability lying at the ILR of the previous.",
"Since the initial outer edge mode had a low growth rate, the inner disc had to wait for a succession of destabilising grooves to be carved before each mode could develop.",
"This delay is all the more remarkable because the dynamical clock runs faster at smaller radii.",
"The long absence of coherent disturbances in the inner disc could therefore happen only if the single instability the disc possesses is at its outer edge."
],
[
"Changes of angular symmetry",
"Disturbance forces in the two sets of experiments described above were each confined to a single sectoral harmonic: $m=2$ for the initial groove mode (§REF ) and $m=3$ for the initial edge mode (§REF ).",
"Furthermore, the simulations with scrambled or spliced particle sets were also restricted to the same symmetry as in the parent simulation.",
"Here we show that the features in action space created by the first mode can also give rise to instabilities having other rotational symmetries.",
"Figure: The evolution of model ER2, the scrambled disc at t=600t=600from model E after the m=3m=3 edge mode had created the scatteringfeatures in Fig.",
", but in this casedisturbance forces were restricted to m=2m=2.",
"The top panel showsthe evolution of the m=2m=2 amplitude and the bottom panel presentsthe power spectrum over the time range 600≤t≤900600 \\le t \\le 900."
],
[
"From 3 to 2",
"The initial $m=3$ edge mode of model E presented in §REF scattered particles to produce the features shown in Fig.",
"REF by $t=600$ .",
"We have already demonstrated that the ILR scattering feature causes a new $m=3$ groove instability that we identified both in the continued run, model E, and in a separate simulation, model ER, started from a scrambled copy of the particles at $t=600$ .",
"Here we report another simulation, model ER2, with the same scrambled copy of the particles at $t=600$ , but with disturbance forces restricted to $m=2$ instead.",
"The disturbance amplitude exponentiates rapidly, as shown in the top panel of Fig.REF , and the power spectrum in the bottom panel reveals two simultaneous instabilities.",
"Our mode fitting software finds two modes, a groove instability given in line 10 of Table REF and an edge mode having a frequency given in line 11.",
"Figure: Early part of the evolution of model GU, in which disturbanceforces included 0≤m≤80 \\le m \\le 8, except for m=1m=1, terms.",
"Theindicated colour scale shows the over- or under-density ofdisturbances that are scaled up by a factor 100.As noted above (§REF ), the $m=3$ edge mode in model E was still growing at $t=600$ , and it is therefore hardly surprising to find that the edge of this disc remains steep enough to excite a new edge mode this time.",
"The new $m=2$ mode, line 11 of Table REF , has a higher growth rate than the $m=3$ edge mode because the swing amplifier is at full strength in this case, $X=2$ , whereas $X=4/3$ for $m=3$ .",
"Because the instability is more vigorous, corotation is at the larger radius, as also indicated in the Table.",
"The $m=2$ groove mode in model ER2 also has a higher growth rate, line 10, than at $m=3$ instability from the same file of particles (line 9 of Table REF ), consistent with more vigorous swing amplification, and corotation is still farther from the groove created by ILR scattering by the original $m=3$ edge mode at $R\\simeq 6.9$ , as expected for a more vigorous and larger-scale groove instability [53]."
],
[
"From 2 to 3",
"In this next case, model GS3$_{\\rm OLR}$ , we start from the spliced set of particles shown in Fig.",
"REF , but this time restrict perturbation forces to the $m=3$ sectoral harmonic.",
"Our estimated frequency of the dominant instability is given in line 12 of Table REF , and corotation is again almost exactly at the location of the groove in action space, as for the modes in lines 2, 3 and 5.",
"Notice that the growth rate is a little lower than for the corresponding $m=2$ case (line 5 of Table REF ), consistent with expectations since swing amplification is less vigorous for $m=3$ than for $m=2$ .",
"These last two experiments have demonstrated that a groove created by Lindblad resonance scattering from a pattern of one sectoral harmonic can excite an instability of a different $m$ .",
"Figure: Power spectra from model GU, §, over the timerange 0≤t≤4000 \\le t \\le 400.",
"The top panel is for m=2m=2 and thebottom panel for m=3m=3.Figure: Changes in action space of model GU (§).",
"Toppanel over the interval t=0t=0 to t=300t=300, bottom panel from t=400t=400to t=500t=500.",
"Note that the color scale has been compressed incomparison with other similar figures."
],
[
"A very slightly more general case",
"Perturbation forces in all the preceding simulations were restricted to a single sectoral harmonic, which we now relax.",
"We start a simulation, model GU, from the particle distribution used in the groove mode case, shown in Fig.",
"REF , which we know from model G will excite at least a mild $m=2$ instability.",
"However, perturbation forces in this new experiment now include $0 \\le m \\le 8$ terms, with the exception that $m=1$ is excluded to avoid possible unbalanced forces from the fixed rigid halo component.",
"Multiple patterns develop over time, as shown in Fig.",
"REF , with perhaps the first feature to appear being an $m=3$ mode.",
"Fig.",
"REF presents power spectra from model GU over the same time interval for both $m=2$ and $m=3$ , revealing a number of coherent waves with pattern speeds independent of radius.",
"A number of strong scattering features in action space are visible at $t=300$ in Fig.",
"REF (top).",
"The bottom panel shows that changes become so intricate in the later evolution that we are no longer able to isolate the effects of individual modes, but there seems little doubt that the disc is being repeatedly destabilized by scattering at resonances of successive disturbances, each of which removes particles from near-circular orbits over narrow ranges of $L_z$ .",
"The simulations of [52] manifested an apparently outwardly propagating cascade of spiral modes to ever larger radii over time.",
"Although they presented evidence of resonance scattering and argued that the changes to the DF excited subsequent modes, they did not identify the mechanism.",
"Here we have presented compelling evidence that the recurrence mechanism identified long ago by [54] for $m=4$ spirals a low-mass disc with a Keplerian rotation curve can, in fact, operate in massive discs having flat rotation curves.",
"We doubted that this would be viable for three principal reasons that, in hindsight, no longer seem particularly cogent.",
"Probably the strongest reason was that the radial extent of spiral modes, which are expected to extend from the ILR to the OLR [33], is much greater for $m=2$ waves in a disc having a flat rotation curve.",
"Both the lower $m$ value and the shape of the rotation curve cause the ratio $R_{\\rm OLR}/R_{\\rm ILR}$ , which was 1.67 for $m=4$ waves in a Kepler potential, to rise to $\\sim 5.8$ for $m=2$ patterns in a flat rotation curve.",
"If the patterns were required not to overlap, it would be hard to fit many into any disc of reasonable extent.",
"But in fact, for a cascade of outwardly propagating $m=2$ modes, successive disturbances have radii in the ratio $R_{\\rm OLR}/R_{\\rm CR} \\simeq 1.7$ .",
"Even for an inwardly propagating $m=2$ cascade initiated by an edge mode, say, the ratio is $R_{\\rm CR}/R_{\\rm ILR} \\simeq 3.4$ .",
"These smaller ratios, which also allow extensive overlap of successive modes, would be even further reduced for larger values of $m$ .",
"It is well known that $m=2$ spiral disturbances have much greater amplitude inside corotation than outside – examples include the famous dust-to-ashes figure from [61], as well as the power spectra and modes presented here, e.g.",
"the middle panel of Fig.",
"REF .",
"However, the fitted mode in that case (Fig.",
"REF ) does have a weak spiral that extends past the OLR circle.",
"This general disparity suggested that scattering at the OLR would not be as important as at the ILR.",
"Since self-excited modes in a disc must conserve angular momentum (there is no externally applied torque), the gains and losses must balance and the angular momentum emitted at the ILR must be absorbed at other resonances.",
"With the exception of edge modes, where corotation takes up almost all the angular momentum lost at the ILR (Fig.",
"REF ), the OLR must absorb a significant amount.",
"The third reason we expected that OLR scattering would be less consequential than at the ILR has already been stated.",
"The close alignment of the ILR locus and the scattering vector causes particles to stay on resonance as they are scattered allowing large changes to build up.",
"This does not happen at the OLR, where scattered particles are quickly moved off resonance and the angular momentum gain must therefore be shared among many more particles.",
"Again this is true and consistent with our results presented in Fig.",
"REF , which led us to doubt that the feeble OLR scattering seen there could excite the second mode.",
"However, it seems that the depopulation of the low-$J_R$ region, as many particles are shifted away from near-circular orbits, is the dominant source of excitation of the next mode.",
"These arguments make it clear that we were wrong to doubt the possibility of a cascade of groove instabilities, running either radially outward or inward, could occur in a massive disc having a flat rotation curve.",
"The evidence presented in § demonstrates that cascades of groove modes, with each causing changes to excite the next, arise naturally."
],
[
"Conclusions",
"In this paper, we have presented a new part of our picture that spiral patterns result from true instabilities in galaxy discs.",
"We have demonstrated that scattering at either Lindblad resonance by any one wave carves a new groove in the disc that excites a further mode.",
"We presented examples of both inwardly and outwardly propagating cascades of instabilities in simulations restricted to a single sectoral harmonic.",
"We also showed that a groove created by one wave of a certain angular symmetry can excite a mode of different angular symmetry.",
"Thus, in more general simulations that include disturbance forces from multiple sectoral harmonics, such as are inevitable when tree codes are employed, it quickly becomes very difficult to follow the causal chain from one wave to the next.",
"The mirror mode mechanism, proposed by [52], seemed to be responsible for the first true mode in models 50 of [46], as evidenced by Fig.",
"REF shown here.",
"However, we suspect it is important only in models that lack strong instabilities.",
"Once a recurrent cycle of groove modes becomes established, mild variations of impedance to cause partial reflections of travelling waves must become physically less important as scattering features in phase space increase in both the strength and number – see e.g.",
"the bottom panel of our Fig.",
"REF and Fig.",
"6 of [52], where 3D motion was allowed.",
"The recurrent cycle of groove modes reported here is then far more likely to be the main driver of evolution.",
"The additional evidence we have presented here strengthens our argument that spirals in simulations are manifestations of global instabilities.",
"The modes in our simulations are each detectable as coherent waves of fixed frequency for a number of rotations, but the visual pattern of spirals changes on a much shorter time scale.",
"The superposition of two or more modes inevitably leads to the apparently shearing spiral features that are widely reported.",
"But only unbounded growth of true instabilities in the linear regime can account for the final amplitudes being independent of the number of particles, as we have repeatedly shown in our previous work [45], [46], [52].",
"None of the many papers arguing for the fundamental nature of shearing features has so far presented any mechanism that could deliver unbounded growth.",
"Swing amplification and wakes, which are invoked in these papers, undoubtedly enhance collective features, but only by a fixed factor in linear theory [27], [61].",
"[15] describe non-linear consequences of responses to the mass clumps they introduced into their disc models, which do appear to cause more persisting non-axisymmetric structures, but it is unclear as yet whether they could account for indefinite growth.",
"The non-linear scattering at resonances, that is a crucial part of our picture, readily excites new instabilities, leading to a robust cycle of recurring modes that accounts for the kind of behaviour that is almost universally observed in simulations of disc galaxies.",
"Naturally, resonant scattering causes irreversible changes, and the consequent secular increase in random motion would, if unchecked, degrade the ability of the disc to support continuing collective instabilities, and spiral activity must fade.",
"But [51] were the first to show that allowing for dissipation and star formation in a reasonable gas fraction could allow spiral activity to continue “indefinitely,” an idea that has been supported in much subsequent work.",
"In their analysis of the local Gaia data, [55] argued that an action-space representation of local phase space revealed evidence for Lindblad resonance scattering, that was not all attributable to the bar [38].",
"This evidence suggests that the Milky Way has supported global spiral modes of the type we describe in our simulations, and we are reasonably confident that spirals in the old stellar discs of many other galaxies also result from such instabilities."
],
[
"Acknowledgements",
"We thank an anonymous referee for thoughtful comments on the paper, and also Elena D'Onghia and especially Daisuke Kawata for helpful conversations during the Gaia19 program at KITP, where this paper was advanced significantly.",
"KITP is supported in part by NSF grant PHY-1748958.",
"JAS also acknowledges the hospitality of Steward Observatory."
]
] | 1906.04191 |
[
[
"Performance of the NOF-MP2 method in hydrogen abstraction reactions"
],
[
"Abstract The recently proposed natural orbital functional second-order M{\\o}ller-Plesset (NOF-MP2) method is capableof achieving both dynamic and static correlation even for those systems with significant multiconfigurational character.",
"We test its reliability to describe the electron correlation in radical formation reactions, namely, in the homolytic X-H bond cleavage of LiH, BH, CH4, NH3, H2O and HF molecules.",
"Our results are compared with CASSCF and CASPT2 wavefunction calculations and the experimental data.",
"For a dataset of 20 organic molecules, the thermodynamics of C-H homolytic bond cleavage, in which the C-H bond is broken in the presence of different chemical environments, is presented.",
"The radical stabilization energies obtained for such general dataset are compared with the experimental data.",
"It is observed that NOF-MP2 is able to give a quantitative agreement for dissociation energies, with a performance comparable to that of the accurate CASPT2 method."
],
[
"Introduction",
"Natural orbital functional (NOF) theory [1] is being configured as an alternative formalism to both DFT and wavefunction methods, by describing the electronic structure in terms of the natural orbitals (NOs) and their occupation numbers (ONs).",
"Various functionals have been developed in the last years, a comprehensive review can be found in Refs.",
"[2], [3].",
"Recently [4], a single-reference global method for the electron correlation was introduced taking as reference the Slater determinant formed with the NOs of an approximate NOF.",
"In this approach, called natural orbital functional - second-order MÞller–Plesset (NOF-MP2) method, the total energy of an N-electron system can be attained by the expression $E=\\tilde{E}_{hf}+E^{corr}=\\tilde{E}_{hf}+E^{dyn}+E^{sta}$ where $\\tilde{E}_{hf}$ is the Hartree-Fock energy obtained with the NOs, the dynamic energy ($E^{dyn}$ ) is derived from a modified MP2 perturbation theory, while the non-dynamic energy ($E^{sta}$ ) is obtained from the static component of the employed NOF.",
"In fact, NOF theory is a particular case of the one-particle reduced density matrix (1RDM) functional theory [5], [6], [7], in which the spectral decomposition of the 1RDM is assumed.",
"In this representation, restrictions on the ONs to the range $\\left[0,1\\right]$ represent the necessary and sufficient conditions for ensemble N-representability of the 1RDM [8] under the Lowdin's normalization.",
"The exact functional in terms of the 1RDM has been an unattainable goal so far, and we really work with approximations.",
"Approximating the energy functional implies that the functional N-representability problem arises [9].",
"To date, only NOFs proposed by Piris and coworkers [10] rely on the reconstruction of the two-particle reduced density matrix (2RDM) subject to ensemble N-representability conditions.",
"The success of the NOF-MP2 method is determined by the NOs used to generate the reference.",
"The functional PNOF7s proved [11] to be the functional of choice for the method.",
"The s emphasizes that this interacting-pair model takes into account only the static correlation between pairs, and therefore avoids double counting in the regions where the dynamic correlation predominates, already in the NOF optimization.",
"Moreover, the correction $E^{dyn}$ is based on the orbital-invariant formulation of the MP2 energy [12].",
"In the present paper, we analyze the performance of NOF-MP2 in the description of X-H bond dissociations, important process in biological [13], [14] and organic chemistry [15].",
"Firstly, we evaluate the dissociation energy for the X-H bonds in LiH, BH, CH$_{4}$ , NH$_{3}$ , H$_{2}$ O and HF molecules.",
"Results are compared to our previous calculations [16], and the experimental data.",
"The proper description of the X-H homolytic bond dissociation curves is a fundamental step for the accurate characterization of the electronic structure of these important species [17], [18], [19], [20].",
"This requires the appropriate treatment of strong correlation effects since a single Slater determinant wavefunction leads to incorrect results.",
"We need to include several determinants that lead to computationally demanding methods.",
"An alternative is the density functional theory (DFT), however, it suffers from methodological problems to treat strong electron correlation or near-degeneracy effects [21], [22], [23].",
"It is worth noting that cost-effective bond dissociation energies can be obtained in the context of spin-dependent DFT, but at the price of obtaining solutions with breaking symmetry [24], [25].",
"Valence bond theory has also been used for this type of systems [26].",
"The formation of radicals by hydrogen abstraction is a fundamental step to explain the oxidation of hydrocarbons [27], [28], [29], lipid-peroxidation [14], formation of reactive oxygen species [30], Fenton chemistry [31] and DNA damage [32].",
"Due to this widespread interest on the thermodynamic stability of organic radicals, we analyze secondly the cleavage of the C-H bond in a dataset of 20 organic molecules, previously designed in our group [16].",
"As a measure of radical stability we employ the bond dissociation energy ($\\mathrm {D_{e}}$ ) which has often been used in the literature [33], [34].",
"Based on $\\mathrm {D_{e}}$ , we estimate the radical stabilization energy (RSE) for a variety of hydrogen abstraction reactions of the type: $\\mathrm {XH}+\\mathrm {Y}^{.",
"}\\rightarrow \\mathrm {X}^{.",
"}+\\mathrm {YH}$ RSE is equivalent to the difference in bond dissociation energies of XC-H and Y-H species."
],
[
"Theory",
"In this work, we address only singlet states, so we adopt the spin-restricted theory in which a single set of orbitals is used for $\\alpha $ and $\\beta $ spins.",
"We shall use PNOF7s [11], which is a NOF based on the electron-pairing approach in NOF theory [35].",
"Consider the orbital space $\\Omega $ is divided into N/2 mutually disjoint subspaces $\\Omega {}_{g}$ , so each orbital belongs only to one subspace.",
"Each subspace contains one orbital $g$ below the level N/2, and $\\mathrm {N}_{g}$ orbitals above it, which is reflected in additional sum rules for the ONs, ${\\displaystyle \\sum _{p\\in \\Omega _{g}}}n_{p}=1,\\quad g=1,2,\\ldots ,\\mathrm {N}/2$ Taking into account the spin, each subspace contains only an electron pair.",
"The Lowdin's normalization condition is automatically fulfilled, $\\begin{array}{c}{\\displaystyle 2\\sum \\limits _{p\\in \\Omega }n_{p}=2\\sum _{g=1}^{\\mathrm {N}/2}}{\\displaystyle \\sum _{p\\in \\Omega _{g}}}n_{p}=\\mathrm {N}\\end{array}$ Coupling each orbital $g$ below the N/2 level with only one orbital above it ($\\mathrm {N}_{g}=1$ ) leads to the orbital perfect-pairing approach.",
"In general, we fix $\\mathrm {N_{g}}$ to the maximum allowed value determined by the basis set used in calculations.",
"It is important to note that orbitals satisfying the pairing conditions (REF ) are not required to remain fixed throughout the orbital optimization process [36].",
"The energy of PNOF7s can be conveniently written as $\\begin{array}{c}E=\\sum \\limits _{g=1}^{\\mathrm {N}/2}E_{g}+\\sum \\limits _{f\\ne g}^{\\mathrm {N}/2}E_{fg}\\\\\\\\E_{g}=\\sum \\limits _{p\\in \\Omega _{g}}n_{p}\\left(2\\mathcal {H}_{pp}+\\mathcal {J}_{pp}\\right)+\\sum \\limits _{p,q\\in \\Omega _{g},p\\ne q}\\Pi _{qp}^{g}\\mathcal {L}_{pq}\\\\\\\\E_{fg}=\\sum \\limits _{q\\in \\Omega _{f}}\\sum \\limits _{p\\in \\Omega _{g}}\\left[n_{q}n_{p}\\left(2\\mathcal {J}_{pq}-\\mathcal {K}_{pq}\\right)+\\Pi _{qp}^{s}\\mathcal {L}_{pq}\\right]\\end{array}$ where $\\begin{array}{c}\\Pi _{qp}^{g}=\\left\\lbrace \\begin{array}{cc}-\\sqrt{n_{q}n_{p}}\\,, & p=g\\textrm { or }q=g\\\\+\\sqrt{n_{q}n_{p}}\\,, & p,q>\\mathrm {N}/2\\end{array}\\right.\\\\\\\\\\Pi _{qp}^{s}=-4n_{q}\\left(1-n_{q}\\right)n_{p}\\left(1-n_{p}\\right)\\quad \\;\\end{array}$ $\\mathcal {J}_{pq}$ , $\\mathcal {K}_{pq}$ , and $\\mathcal {L}_{pq}$ are the usual direct, exchange, and exchange-time-inversion two-electron integrals.",
"The first term of the energy in Eq.",
"(REF ) draws the system as independent N/2 electron pairs, whereas the second term contains the interactions between electrons belonging to different pairs.",
"PNOF7s provides the reference NOs to form $\\tilde{E}_{hf}$ in the NOF-MP2 method, Eq.",
"(REF ).",
"$E^{sta}$ is the sum of the static intrapair and interpair electron correlation energies: $\\begin{array}{c}E^{sta}=\\sum \\limits _{g=1}^{\\mathrm {N}/2}\\sum \\limits _{q\\ne p}\\sqrt{\\Lambda _{q}\\Lambda _{p}}\\,\\Pi _{qp}^{g}\\mathcal {\\,L}_{pq}+\\sum \\limits _{f\\ne g}^{\\mathrm {N}/2}\\sum \\limits _{p\\in \\Omega _{f}}\\sum \\limits _{q\\in \\Omega _{g}}\\Pi _{qp}^{s}\\mathcal {\\,L}_{pq}\\end{array}$ where $\\Lambda _{p}=1-\\left|1-2n_{p}\\right|$ is the amount of intra-pair static correlation in each orbital as a function of its occupancy.",
"$E^{dyn}$ is obtained from the second-order correction $E^{\\left(2\\right)}$ of the MP2 method.",
"The first-order wavefunction is a linear combination of all doubly excited configurations, and their amplitudes $T_{pq}^{fg}$ are obtained by solving the equations for the MP2 residuals [12].",
"The dynamic energy correction takes the form $E^{dyn}=\\sum \\limits _{g,f=1}^{\\mathrm {N}/2}\\sum \\limits _{p,q>N/2}^{M}\\left\\langle gf\\right|\\left.pq\\right\\rangle \\left[2T_{pq}^{gf}\\right.\\left.-T_{pq}^{fg}\\right]$ where $M$ is the number of basis functions, and $\\left\\langle gf\\right|\\left.pq\\right\\rangle $ are the matrix elements of the two-particle interaction.",
"In fact, $E^{dyn}$ is the modified $E^{\\left(2\\right)}$ in order to avoid double counting of the electron correlation.",
"It is divided into intra- and inter-pair contributions, and the amount of dynamic correlation in each orbital $p$ is defined by functions $C_{p}$ of its occupancy, namely, $\\begin{array}{c}C_{p}^{tra}={\\left\\lbrace \\begin{array}{ll}\\begin{array}{c}\\begin{array}{c}1-4h_{p}^{2}\\end{array}\\\\1-4n_{p}^{2}\\end{array} & \\begin{array}{c}p\\le \\mathrm {N}/2\\\\p>\\mathrm {N}/2\\end{array}\\end{array}\\right.",
"}\\\\\\:C_{p}^{ter}={\\left\\lbrace \\begin{array}{ll}\\begin{array}{c}\\begin{array}{c}1\\end{array}\\\\1-4h_{p}n_{p}\\end{array} & \\begin{array}{c}p\\le \\mathrm {N}/2\\\\p>\\mathrm {N}/2\\end{array}\\end{array}\\right.",
"}\\end{array}$ According to Eq.",
"(REF ), fully occupied and empty orbitals yield a maximal contribution to dynamic correlation, whereas orbitals with half occupancies contribute nothing.",
"Using these functions as the case may be (intra-pair or inter-pair), the modified off-diagonal elements of the Fock matrix ($\\tilde{\\mathcal {F}}$ ) are defined as $\\tilde{\\mathcal {F}}_{pq}={\\left\\lbrace \\begin{array}{ll}C_{p}^{tra}C_{q}^{tra}\\mathcal {F}_{pq}, & p,q\\in \\Omega _{g}\\\\C_{p}^{ter}C_{q}^{ter}\\mathcal {F}_{pq}, & otherwise\\end{array}\\right.",
"}$ as well as modified two-electron integrals: $\\widetilde{\\left\\langle pq\\right|\\left.rt\\right\\rangle }={\\left\\lbrace \\begin{array}{ll}C_{p}^{tra}C_{q}^{tra}C_{r}^{tra}C_{t}^{tra}\\left\\langle pq\\right|\\left.rt\\right\\rangle , & p,q,r,t\\in \\Omega _{g}\\\\C_{p}^{ter}C_{q}^{ter}C_{r}^{ter}C_{t}^{ter}\\left\\langle pq\\right|\\left.rt\\right\\rangle , & otherwise\\end{array}\\right.",
"}$ where the subspace index $g=1,...,\\mathrm {N}/2$ .",
"This leads to the following linear equation for the modified MP2 residuals: $\\widetilde{\\left\\langle ab\\right|\\left.ij\\right\\rangle }+\\left(\\mathcal {F}_{aa}\\right.+\\mathcal {F}_{bb}-\\mathcal {F}_{ii}-\\left.\\mathcal {F}_{jj}\\right)T_{ab}^{ij}\\:+$ ${\\displaystyle \\sum _{c\\ne a}\\mathcal {\\tilde{F}}_{ac}T_{cb}^{ij}}+{\\displaystyle \\sum _{c\\ne b}}T_{ac}^{ij}\\mathcal {\\tilde{F}}_{cb}-{\\displaystyle \\sum _{k\\ne i}}\\tilde{\\mathcal {F}}_{ik}T_{ab}^{kj}-{\\displaystyle \\sum _{k\\ne j}}T_{ab}^{ik}\\mathcal {\\tilde{F}}_{kj}=0$ where $i,j,k$ refer to the strong occupied NOs, and $a,b,c$ to weak occupied ones.",
"It should be noted that diagonal elements of the Fock matrix ($\\mathcal {F}$ ) are not modified.",
"By solving this linear system of equations the amplitudes $T_{pq}^{fg}$ are obtained, which are inserted into the Eq.",
"(REF ) to achieve $E^{dyn}$ .",
"All calculations have been carried out using the DoNOF code developed by M. Piris and coworkers.",
"The procedure is simple, showing a formal scaling of $M^{5}$ ($M$ : number of basis functions).",
"However, our implementation in the molecular basis set requires also four-index transformation of the electron repulsion integrals, which is a time-consuming step, though a parallel implementation of this part of the code has substantially improved its performance.",
"As a result, the possibility of addressing large systems opens up."
],
[
"Results and Discussion",
"Results are organized as follows.",
"First, the X-H bond dissociation energies for LiH, BH, CH$_{4}$ , NH$_{3}$ , H$_{2}$ O and HF molecules are studied using NOF-MP2.",
"Next, we analyze the performance of NOF-MP2 for describing C-H bond cleavage in a variety of 20 organic molecules.",
"Finally, radical stabilization energies are calculated based on the calculated C-H bond dissociation energies.",
"In all calculations, recall that the maximum value allowed by the basis set used is assumed for $\\mathrm {N}_{g}$ by default.",
"In NOF-MP2(3) calculations, only three orbitals ($\\mathrm {N}_{g}=3$ ) above the N/2 level in each electron pair are considered.",
"Geometries are taken from our previous publication [16], which were obtained at the M06-2X level of theory [37].",
"The dissociation limit is calculated by considering a frozen X-H distance of 5$\\textrm {Ã}$ , and optimizing the rest of internal coordinates.",
"At these geometries, single-point energies are evaluated at the NOF-MP2 level of theory.",
"The correlation-consistent valence double-$\\zeta $ (cc-pVDZ) or triple-$\\zeta $ (cc-pVTZ) basis sets developed by Dunning et al.",
"[38] are used.",
"The zero point vibrational energies (ZPVEs) were taken from the NITS Computational Chemistry Comparison and Benchmark Database (CCCBDB) [39], and corresponds to CCSD(T)/cc-pVTZ values.",
"We also provide the PNOF6 [40] and wavefunction-based calculations obtained in Ref.",
"[16].",
"For the latter, an active space was defined by the distribution of two electrons in two molecular orbitals, CASSCF(2,2) [41], [42].",
"The dynamic correlation effects were included through complete active space second-order perturbation theory calculations, CASPT2(2,2) [43].",
"MOLCAS 7.0 suite of programs [44] was used in Ref.",
"[16], for these wavefunction-based calculations."
],
[
"X-H homolytic bond cleavage ",
"X-H bond dissociation energies were calculated according to the following reaction: $\\mathrm {XH}\\rightarrow \\mathrm {^{.",
"}X}+\\mathrm {H}^{.",
"}$ with X = Li, B, CH$_{3}$ , NH$_{2}$ , OH, F. The results are presented in Figure REF and Table REF .",
"The different hydrides considered expand a wide range of dissociation energies, from 58.0 kcal/mol for LiH to 141.1 kcal/mol for FH.",
"The ordering in dissociation energies is $\\mathrm {LiH}<\\mathrm {BH}<\\mathrm {CH_{4}}<\\mathrm {NH_{3}}<\\mathrm {H_{2}O}<\\mathrm {FH}$ .",
"In general, NOF-MP2 reproduces satisfactorily these trends.",
"Figure: NOF-MP2 dissociation energies, in kcal/mol,for X-H bonds (X= Li ,B, CH 3 , NH 2 , OH ,F\\mathrm {X=Li,B,CH_{3},NH_{2},OH,F}) versus experimentalones.",
"Calculations carried out with the cc-pVTZ basis set.Let us focus our attention, for example, on the delicate case of the CH$_{4}$ /NH$_{3}$ ordering.",
"The difference in experimental dissociation energies for these two molecules is very small, only 2.9 kcal/mol with $\\mathrm {NH_{3}}$ having a higher dissociation energy.",
"NOF-MP2 is able to reproduce the correct ordering $\\mathrm {CH_{4}<NH_{3}}$ , except for NOF-MP2(3)/cc-pVDZ.",
"It should be noted that CASSCF(2,2) and PNOF6 gives the reverse order, whereas CASPT2(2,2) recovers the right trend.",
"Table: Dissociation energies, in kcal/mol, calculatedfrom single-point energies a ^{a}.",
"ZPVEs were added b ^{b} to theexperimental dissociation energies .",
"PNOF6, CASSCF(2,2)and CASPT2(2,2) results are taken from Ref.",
".It is well known that to reach the experimental values we must go to the complete basis set limit.",
"Therefore, taking into account the moderate basis sets used here, we can say that a good semi-quantitative agreement has been achieved with the experimental data by the NOF-MP2 method.",
"In general, NOF-MP2 shows an intermediate performance between the CASSCF(2,2) and CASPT2(2,2) methods, and a significant improvement with respect to the previously tested PNOF6.",
"For the six reactions considered, a mean absolute error (MAE) of 5.5 kcal/mol is obtained at NOF-MP2/cc-pVTZ level of theory.",
"NOF-MP2(3)/cc-pVTZ, leads to a higher MAE, namely 9.6 kcal/mol, but this is mainly due to LiH case.",
"For the latter, only one effective pair appears so more $\\mathrm {N}_{g}$ orbitals are needed in order to describe properly the dominant intra-pair electron correlation [10] in this system.",
"On the other hand, for $\\mathrm {CH_{4},NH_{3},H_{2}O}$ and FH, NOF-MP2(3) yield very reasonable results.",
"Therefore, we can say that $\\mathrm {N}_{g}=3$ is a good compromise for the characterization of the electron pairs, except for small systems like LiH and BH.",
"Comparing the performance of NOF-MP2 with wavefunction methods, it is clear that NOF-MP2 and NOF-MP2(3) show a better performance than CASSCF(2,2) (MAE=16.5 kcal/mol with the cc-pVTZ basis set).",
"Introduction of dynamical electron correlation at the CASPT2(2,2) level of theory, reduces the MAE to 3.4 kcal/mol, however, if we reduce the set to CH$_{4}$ , NH$_{3}$ , H$_{2}$ O and FH molecules, there is a similar performance of NOF-MP2 with respect to CASPT2(2,2) method.",
"As in our previous work [16], we have considered a dataset of 20 organic molecules to evaluate the performance of NOF-MP2 for the C-H bond dissociation energy ($\\mathrm {D_{e}^{CH}}$ ).",
"The selected set covers a wide range of $\\mathrm {D_{e}^{CH}}$ values, from 95.6 kcal/mol (H$_{2}$ CO) to 141.8 kcal/mol (C$_{2}$ H$_{2}$ ), showing the sensitivity of the C-H bond to different chemical environments.",
"We have considered functional groups with different degree of electron withdrawing/donating ability (-F, -OH, -NO$_{2}$ , -CN, -CH$_{3}$ , ...), aromaticity (-C$_{6}$ H$_{5}$ ), variety of C-X bonds (HCN, H$_{2}$ CO, CH$_{3}$ NO$_{2}$ , CH$_{3}$ CF$_{3}$ , ...), different chain lengths (CH$_{4}$ , CH$_{3}$ CH$_{3}$ , CH$_{3}$ CH$_{2}$ CH$_{3}$ ) and different C-C bond orders, single (as in CH$_{3}$ CH$_{3}$ ), double (as in C$_{2}$ H$_{4}$ ) and triple (as in C$_{2}$ H$_{2}$ ).",
"We have decided to use the cc-pVDZ basis set due to the large number of compounds to be treated.",
"Table: C-H Bond Dissociation energies, in kcal/mol, fora dataset of 20 organic molecules.",
"ZPVEs at the M062X/cc-pVTZ levelof theory were added to the experimental dissociation energies , .In case of CH 4 _{4}, this leads to a experimental D e _{e} of 112.7kcal/mol, 0.3 kcal/mol lower than the value estimated in Table .PNOF6(3), CASSCF(2,2) and CASPT2(2,2) data is taken from Ref.",
".Calculations carried out with the cc-pVDZ basis set, and consideringthe X-H distance of 5 Ã\\textrm {Ã} as the dissociation limit.Figure: C-H bond dissociation energies, in kcal/mol, forthe Table dataset of 20 organic molecules.",
"All calculationswere done with the cc-pVDZ basis set.",
"The dissociation limit distancewas taken as 5Ã\\textrm {Ã}.Figure: Radical Stabilization Energies, in kcal/mol, basedon the combination of dissociation energies of Table .All calculations were done with the cc-pVDZ basis set.",
"The mean absoluteerrors with respect to the experimental values are 5.5 kcal/mol, 4.4kcal/mol, and 4.1 kcal/mol for NOF-MP2(3), CASSCF(2,2), and CASPT2(2,2),respectively.The results can be found in Table REF and Figure REF .",
"The agreement between NOF-MP2(3) and experimental values is remarkable, with a MAE of 3.7 kcal/mol, even smaller than the MAE for the very accurate CASPT2(2,2) method, namely 5.0 kcal/mol.",
"Notice that previously tested PNOF6(3) method has a MAE of 9.0 kcal/mol, slightly better than CASSCF(2,2), 11.1 kcal/mol.",
"Thus, NOF-MP2(3) method allows for a quantitative description of these dissociation energies, with a similar degree of accuracy as CASPT2(2,2).",
"Specifically, NOF-MP2(3) is able to reproduce important trends in C-H bond energies.",
"For instance, the experimental $\\mathrm {D_{e}^{CH}}$ increases in the following order [34]: CH$_{3}$ CH$_{3}$ (109.7) < C$_{2}$ H$_{4}$ (119.3) < C$_{2}$ H$_{2}$ (141.8) .",
"NOF-MP2(3) is able to reproduce properly this trend, namely, CH$_{3}$ CH$_{3}$ (103.4) < C$_{2}$ H$_{4}$ (116.9) < C$_{2}$ H$_{2}$ (143.4).",
"The effect of aromaticity can be inferred from the comparison of these dissociation energies with that of the phenyl C-H bond.",
"C$_{6}$ H$_{6}$ , with a formal 1.5 C-C bond order, shows a high dissociation energy (120.5 kcal/mol) even slightly larger than that observed (119.3 kcal/mol) in C$_{2}$ H$_{4}$ , with a formal bond order of 2.",
"This is a clear signature of aromaticity in C$_{6}$ H$_{6}$ , partially lost upon hydrogen abstraction and radical formation.",
"NOF-MP2(3) yields larger values of bond dissociation energies for benzene than for ethene, with values of 124.3 kcal/mol and 116.9 kcal/mol, respectively.",
"In the case of the benzylic C-H bond (C$_{6}$ H$_{6}$ CH$_{2}$ -H), the effect of the aromaticity works in the opposite direction.",
"In this case, the C-H cleavage does not break the aromaticity, furthermore, the radical itself is stabilized by the aromatic character of the phenyl ring, and consequently, one obtains a much lower $\\mathrm {D_{e}^{CH}}$ than for C$_{6}$ H$_{6}$ , namely 96.1 kcal/mol versus 120.5 kcal/mol.",
"NOF-MP2(3) correctly describes this effect, $\\mathrm {D_{e}^{CH}}$ for the benzylic C-H bond (103.4 kcal/mol) is also much lower than for the phenyl C-H bond (124.3 kcal/mol) at a magnitude very similar to the experimental value.",
"It is remarkable the right description of aromatic radical stabilization by the NOF-MP2 method, since aromatic stabilization is key to describe radical stability in chemistry.",
"The chain length is also a factor influencing the C-H bond strength [33], [27], [29].",
"It is known that a larger chain stabilizes the resulting radical: observe the first 3 lines of Table REF .",
"However, NOF-MP2(3) exhibits a poorer sensitivity of radical stability towards chain-lengths with a similar $\\mathrm {D_{e}^{CH}}$ for these three molecules.",
"On the other hand, if we consider the same alkane, CH$_{3}$ CH$_{2}$ CH$_{3}$ , and measure both possibilities for hydrogen abstraction, namely, from the central -CH$_{2}$ - or from the terminal -CH$_{3}$ group, NOF-MP2(3) correctly reproduces the more favorable hydrogen abstraction from the central carbon by 1.9 kcal/mol.",
"There is also a sizable effect in hydrogen abstraction upon the inclusion of electron withdrawing groups.",
"For instance, fluorination [47] and oxidation [34] of methane tend to alter the dissociation energy of the C-H bond.",
"Regarding fluorination, a decrease of $\\mathrm {D_{e}^{CH}}$ is observed upon the inclusion of a first flour, from 112.7 kcal/mol (CH$_{4}$ ) to 108.7 kcal/mol in (CH$_{3}$ F).",
"However, upon higher degree of fluorination in the fluoromethane, $\\mathrm {D_{e}^{CH}}$ increases again, 111.8 kcal/mol in CF$_{2}$ H$_{2}$ and 113.5 kcal/mol in CF$_{3}$ H. NOF-MP2(3) yields a higher $\\mathrm {D_{e}^{CH}}$ for CH$_{3}$ F (108.5 kcal/mol) than for CH$_{4}$ (104.5 kcal/mol).",
"Nevertheless, NOF-MP2(3) describes the proper trend in increasing $\\mathrm {D_{e}^{CH}}$ with the degree of fluorination in fluoromethane, namely, CH3F (108.5 kcal/mol) < CH$_{2}$ F$_{2}$ (116.1 kcal/mol) < CHF$_{3}$ (117.6 kcal/mol).",
"With respect to the oxidation of a methyl group, NOF-MP2(3) gives the right trend.",
"For instance, in going from CH$_{3}$ OH to H$_{2}$ CO, there is an important reduction in C-H bond strength, from 103.2 kcal/mol to 95.6 kcal/mol.",
"NOF-MP2(3) yields a similar, although more discrete reduction, of $\\mathrm {D_{e}^{CH}}$ from 106.7 kcal/mol to 101.6 kcal/mol.",
"In general, we can conclude that NOF-MP2(3) represents an accurate balance between dynamical and non-dynamical electron correlation for this set of molecules, yielding $\\mathrm {D_{e}^{CH}}$ values that are of the CASPT2(2,2) quality."
],
[
"Radical Stabilization Energies",
"RSEs are defined as the energy change in the isodesmic reaction for hydrogen abstraction [48], [24], [25] of Eq.",
"(REF ).",
"Thus, the RSE for a pair X,Y is defined as $\\mathrm {RSE^{XY}}=\\mathrm {D_{e}^{XH}}-\\mathrm {D_{e}^{YH}}$ For the dataset of 21 dissociation energies of Table REF , there are 210 possible combinations of RSEs.",
"It provides with an extensive dataset for the determination of the suitability of a given method to estimate the effect of the substituents on the radical stability in organic molecules.",
"The results for NOF-MP2(3) are summarized in Fig.",
"REF , compared to the performance of wavefunction methods such as CASSCF(2,2) and CASPT2(2,2).",
"In general, there is a reasonable agreement with experimental RSEs for NOF-MP2(3) with a MAE of 5.5 kcal/mol.",
"Slightly better values are obtained for CASSCF(2,2) (4.4 kcal/mol) and CASPT2(2,2) (4.1 kcal/mol) levels of theory.",
"Another way to compare the results with respect to experimental values is to calculate the linear fit of the theoretical versus the experimental values, and determine the correlation coefficient ($r$ ).",
"In this sense, NOF-MP2(3) shows a similar performance to the CASSCF(2,2) and CAS2PT2(2,2) methods with an $r$ of 0.9314 versus a value of 0.9482 for both wavefunction methods.",
"In summary, taking into account the large number of hydrogen abstraction reactions considered, the correlation between NOF-MP2(3) and experimental data is highly satisfactory, yielding a quantitative agreement with respect to well established wavefunction methods such as CASPT2(2,2), and providing results close to chemical accuracy."
],
[
"Conclusions",
"The recently proposed parameter-free natural orbital functional second-order MÞller–Plesset (NOF-MP2) method has been applied to the description of radical formation reactions, a delicate problem in quantum chemistry.",
"The application of NOF-MP2(3) to the calculation of the C-H bond dissociation energy in a dataset of 20 organic molecules, and the estimation of the corresponding radical stabilization energies support the use of NOF-MP2(3) as a quantitative theory for the description of these important set of reactions.",
"Comparison of NOF-MP2 with experimental data reveals a similar performance of NOF-MP2 to well-established wavefunction methods such as CASPT2 for these type of problems.",
"We conclude that NOF-MP2 is capable of recovering both dynamical and non-dynamical electron correlation effects in this type of systems.",
"NOF-MP2 is a global electron correlation method for the description of radical stability, which provides results close to chemical accuracy as the widely used and well-established CASPT2 wavefunction method.",
"Financial support comes from Ministerio de EconomÃa y Competitividad (Ref.",
"CTQ2015-67608-P).",
"The authors thank for technical and human support provided by IZO-SGI SGIker of UPV/EHU and European funding (ERDF and ESF)."
]
] | 1906.04432 |
[
[
"The merits of using Ethereum MainNet as a Coordination Blockchain for\n Ethereum Private Sidechains"
],
[
"Abstract A Coordination Blockchain is a blockchain with the task of coordinating activities of multiple private blockchains.",
"This paper discusses the pros and cons of using Ethereum MainNet, the public Ethereum blockchain, as a Coordination Blockchain.",
"The requirements Ethereum MainNet needs to fulfil to perform this role are discussed within the context of Ethereum Private Sidechains, a private blockchain technology which allows many blockchains to be operated in parallel, and allows atomic crosschain transactions to execute across blockchains.",
"Ethereum MainNet is a permissionless network which aims to offer strong authenticity, integrity, and non-repudiation properties, that incentivises good behaviour using crypto economics.",
"This paper demonstrates that Ethereum MainNet does deliver these properties.",
"It then provides a comprehensive review of the features of Ethereum Private Sidechains, with a focus on the potential usage of Coordination Blockchains for these features.",
"Finally, the merits of using Ethereum MainNet as a Coordination Blockchain are assessed.",
"For Ethereum Private Sidechains, we found that Ethereum MainNet is best suited to storing long term static data that needs to be widely available, such as the Ethereum Registration Authority information.",
"However, due to Ethereum MainNet's probabilistic finality, it is not well suited to information that needs to be available and acted upon immediately, such as the Sidechain Public Keys and Atomic Crosschain Transaction state information that need to be accessible prior to the first atomic crosschain transaction being issued on a sidechain.",
"Although this paper examined the use of Ethereum MainNet as a Coordination Blockchain within reference to Ethereum Private Sidechains, the discussions and observations of the typical tasks a Coordination blockchain may be expected to perform are applicable more widely to any multi-blockchain system."
],
[
"Introduction",
"This paper analyses the advantages and disadvantages of using Ethereum MainNet as a Coordination Blockchain, by demonstrating the ways in which a Coordination Blockchain may be leveraged in a blockchain network that runs several parallel blockchains.",
"We conduct an in-depth review of the features of Ethereum Private Sidechains, which is an example of such a blockchain system, to explore their potential usage of Coordination Blockchains as an exposition of using Coordination Blockchains more generally.",
"The analysis builds on the Symposium on Distributed Ledger Technology paper Future of Blockchain [1], and other work on Ethereum Private Sidechains including: Requirements for Ethereum Private Sidechains [2], Ethereum Registration Authorities [3], Anonymous Pinning [4], and Atomic Crosschain Transactions [5].",
"Ethereum MainNet is the largest public deployment of the Ethereum platform.",
"It is a permissionless network, allowing any node to join the network.",
"It is said to offer good authenticity, integrity, and non-repudiation properties, along with an economic system to discourage transaction spamming [6], [7].",
"To date there has been no work that has analysed all of these assertions.",
"This paper remedies this deficiency by carefully analysing whether these properties are successfully delivered.",
"Sidechains are blockchains that rely on a separate blockchain, a Coordination Blockchain, for their overall utility.",
"This could be to enhance security by pinning the state of the sidechain to the Coordination Blockchain [4], for addressing information [3], or for storing data that is used across all sidechains.",
"We analyse the appropriateness of using Ethereum MainNet as a Coordination Blockchain for the various features of sidechains, using as a reference Ethereum Private Sidechains.",
"This paper is organised as follows: the Background section briefly introduces Ethereum MainNet, the platform that forms the basis for this paper.",
"We describe the concept of private blockchains and the enterprise version of Ethereum, and introduce the concept of block `finality'.",
"Next cryptanalysis of message digest and asymmetric algorithms is reviewed given classical and quantum cryptanalytical techniques.",
"The Ethereum MainNet Features section analyses whether Ethereum MainNet delivers authenticity, integrity, non-repudiation, and crypto-economic anti-spam properties.",
"The Ethereum Private Sidechains section describes the features of Ethereum Private Sidechains and their usage of Coordination Blockchains.",
"The Pros and Cons of using Ethereum MainNet as a Coordination Blockchain section analyses the advantages and disadvantages of using Ethereum MainNet as the Coordination Blockchain for each of the Ethereum Private Sidechain features.",
"Ethereum [8] is a blockchain platform that allows users to upload and execute computer programs known as Smart Contracts.",
"Ethereum Smart Contracts can be written in a variety of Turing Complete languages, the most popular being Solidity [9].",
"Source code is compiled into a bytecode representation.",
"The bytecode can then be deployed using a contract creation transaction.",
"Contracts have a special constructor function that only runs when the contract creation transaction is being processed.",
"This function is used to initialize memory and call other contract code.",
"Miners execute the bytecode inside the Ethereum Virtual Machine (EVM).",
"At present, each miner must execute all transactions for all contracts and hold the current value of all the memory associated with all of the contracts.",
"The Ethereum community is actively working on methodologies to scale the Ethereum network by sharding the blockchain [10].",
"Ethereum transactions update the state of the distributed ledger but do not return values.",
"They fall into three categories: Ether transfer, contract creation, and calling a function on a contract.",
"Ether transfer transactions move Ether from the user's account to another account.",
"Contract creation transactions put code into the distributed ledger and call the constructor of the contract code, setting the contract data's initial state.",
"Function call transactions call a function on a contract and result in updated state.",
"Contract creation and function call transactions also allow Ether to be transferred.",
"All types of transactions must be signed by a private key corresponding to an account and include a nonce value that prevents replay attacks.",
"In addition to Ethereum transactions, “View\" function calls can be executed on the Smart Contract code.",
"These View function calls return a value and do not update the state of the Smart Contract.",
"Executing code and accessing resources, such as memory, costs certain amounts of “Gas\".",
"The “Gas Cost” of executing code is closely tied to the real world cost of executing each type of instruction.",
"Miners preferentially mine transactions that are prepared to pay a higher “Gas Price”.",
"Accounts instigating transactions specify the “Gas Price\" they are prepared to pay for their transaction and specify the maximum amount of gas a transaction can use known as “Start Gas\".",
"This commits an account holder to paying up to a certain amount of Ether for the transaction.",
"Any unused gas is returned to the account holder at the end of the transaction.",
"Transactions that run out of gas prior to completion are aborted, with all of the gas being expended.",
"In the Ethereum public network, “MainNet\", all contract code and data are readable by any user of any node that connects to the network.",
"Smart Contracts on Ethereum MainNet can only perform permissioning in contract code, limiting which accounts can update the state of a contract.",
"However, there is no mechanism to limit which users can read contract code and data."
],
[
"Private Blockchains and Enterprise Ethereum",
"Private blockchains are blockchain networks that are established between nodes operated by enterprises [2].",
"Only permissioned nodes belonging to participating enterprises are allowed to join the private blockchain's peer-to-peer network and only permissioned accounts belonging to participating enterprises are allowed to submit transactions to the nodes.",
"These blockchains provide the privacy and permissioning required by enterprises [11].",
"The need for security and permissioning features over and above what is available in standard Ethereum [11] has led to a range of platforms being developed.",
"J.P. Morgan developed Quorum [12], a fork of the Golang Ethereum implementation called Geth [13].",
"ConsenSys's Protocol Engineering Group, PegaSys created Pantheon [14], an Ethereum MainNet compatible client that aims to meet the permissioning and privacy requirements of the Enterprise Ethereum Client Specification [11].",
"Hyperledger Fabric [15] is a distributed ledger platform originally created by IBM and now hosted by The Linux Foundation.",
"Similar to Quorum and Pantheon, the platform offers privacy and permissioning features.",
"Whereas Quorum offers Ethereum based private transactions, Pantheon offers private smart contracts that are private to a set of participants.",
"Hyperledger Fabric offers the ability to host one or more smart contracts on a private blockchain called a “channel\".",
"Hyperledger Fabric allows multiple channels to be operated on the one network, thus allowing for multiple sets of private contracts between different sets of participants to operate on the one network.",
"An analysis of the merits of Hyperledger Fabric and Quorum has been analysed elsewhere (see Requirements for Ethereum Private Sidechains [2])."
],
[
"Finality",
"A block is deemed final when it can no longer be changed.",
"All transactions contained within a finalised block are also deemed final.",
"Ethereum transactions are included in blocks.",
"An Ethereum MainNet miner that solves the Proof of Work cryptographic puzzle can add a block to the end of the blockchain.",
"If two or more miners solve the puzzle simultaneously, then two or more chains are created with common ancestors, and this is known as a fork [16].",
"In Bitcoin the longest chain of blocks is deemed to be the valid blockchain [17], [18].",
"In Ethereum, the fork choice is solved by means of a modified Greediest Heaviest Observed Subtree (GHOST) protocol [16] that takes into account the mining power in creating blocks that have links to the main chain, but have become stale [19].",
"These blocks are commonly referred to as uncle blocks.",
"The weight of a block relates to the number of previous blocks in the chain and uncle blocks.",
"The heaviest chain of blocks is deemed to be the valid blockchain.",
"If an Ethereum MainNet miner becomes aware of a heavier chain than it knew about, it should then only attempt to add blocks to the new chain.",
"Blocks on the old heaviest chain that are not in common with the new longest chain are deemed reordered.",
"If none of the transactions in a reordered block have been included in the blocks of the new longest chain, then the block can be included as an uncle block.",
"Otherwise, the transactions that are not included in the reordered chain need to be included in a new block.",
"There is no certainty that these transactions will be included in a new block, or that transactions in a proposed uncle block will be included in the blockchain.",
"As more blocks are added to the end of Ethereum MainNet's blockchain, the probability of a miner finding a longer blockchain and reordering the blockchain is reduced [16].",
"This is because a miner would need to repeatedly solve the Proof of Work cryptographic puzzle for each block faster than all other miners.",
"As the probability of a block being reordered is reduced, the probability of the transactions included in a block being final increases.",
"Hence, Ethereum MainNet is said to have, probabilistic finality [18].",
"Consensus algorithms such as Istanbul Fault Byzantine Tolerant (IBFT) [20] and Istanbul Fault Byzantine Tolerant version 2 (IBFT2) [21] used in consortium blockchains give instant finality, where once a transaction has been included in a block minted by a validator, it can no longer be changed."
],
[
"Pinning",
"The state of a blockchain or sidechain can be represented by the Block Hash of a block.",
"The Block Hash of a final block could be submitted to a contract on a Coordination Blockchain at regular intervals [4], as shown in Figure REF .",
"This process is know as pinning.",
"Regularly pinning sidechain state helps to protect minority sidechain participants from state reversion due to collusion by the majority of sidechain participants [4].",
"Figure: Pinning"
],
[
"Cryptanalysis",
"This section provides background material on cryptanalysis that is needed to understand the analysis of the security properties of Ethereum MainNet."
],
[
"Message Digest Algorithm Cryptanalysis",
"Message digest algorithms have three main security properties: Preimage Resistance, Second Preimage Resistance, and Collision Resistance.",
"Message digest algorithms are commonly called Cryptographic Hash algorithms, or simply Hash algorithms.",
"Given a Hash algorithm h, the three security properties can be stated as: Preimage Resistance: Given $y$ , it is difficult to determine $x$ such that $y = h(x)$ .",
"Second Preimage Resistance: Given $y$ and $x_1$ , it is difficult to determine $x_2$ such that $y = h(x_1) = h(x_2)$ and $x_1 \\ne x_2$ .",
"Collision Resistance: It is difficult to determine $x_1$ and $x_2$ such that $h(x_1) = h(x_2)$ and $x_1 \\ne x_2$ ."
],
[
"Classical Computing Cryptanalysis",
"Gordon Moore, co-founder of Intel, stated in his ”Moore'€™s Law” that the number of transistors on an integrated circuit doubles approximately every two years [22].",
"With the increased number of transistors has come a decrease in transistor size, which has resulted in decreased power consumption per transistor.",
"This has resulted in an increase in computation power, while keeping the power consumption relative static over a fifty year period.",
"This rate of increase of computation power and decrease of transistor size though slowing, is still continuing [23].",
"Additionally, new alternative approaches are being developed to deliver increased computational power [24].",
"Classical computational power can be used to break algorithms such as message digest algorithms by trying all possible combinations using a “Brute Force\" attack.",
"Complexity theory predicts how many attempts are likely to be needed to break an algorithm.",
"For message digest algorithms, using classical computing power, the complexity of breaking an algorithm's Preimage Resistance or Second Preimage Resistance property is $O(N)$ , where $N$ is the number of combinations of the digest output, whereas the complexity of breaking an algorithms Collision Resistance is $O(\\sqrt{N})$ .",
"The USA's National Institute of Standards and Technology (NIST) defines Security Strength [25] as, “A number associated with the amount of work (that is, the number of operations) that is required to break a cryptographic algorithm or system.” Security Strength and complexity are related.",
"The Security Strength of a message digest algorithm's Preimage and Second Preimage Resistance properties is $\\log _2{N}$ and the Collision Resistance Security Strength is $\\log _2{\\sqrt{N}}$ .",
"Recall that $\\log _2{N}$ corresponds to the message digest output length in bits.",
"As such, the algorithm SHA-256's Preimage and Second Preimage Security Strength is 256-bits and its Collision Resistance Security Strength is 128 bits, assuming classical computers [25].",
"In some instances, a message digest output is truncated.",
"For example in Ethereum, Keccak-256 is used to generate account numbers with the output truncated from 256-bits to 160-bits.",
"In this usage, the analysis of Security Strength remains unchanged: the complexity and hence Security Strength relates to the number of possible values of the digest output.",
"If a message digest output is truncated then the Security Strength of the overall algorithm is proportionally reduced.",
"NIST defines algorithms with Security Strengths of 80, 112, 128, 192, and 256 bits [26].",
"NIST have mandated the phasing out of 80-bit Security Strength algorithms in 2010 and, based on Moore'€™s Law, had indicated the phasing out of 112-bit Security Strength algorithms by 2030."
],
[
"Quantum Computing Cryptanalysis",
"Quantum computers are expected to allow all currently used popular asymmetric cryptographic algorithms to be defeated and are expected to reduce the Security Strength of message digest and symmetric cipher cryptographic algorithms [27].",
"Aggarwal et al.",
"[28] estimate that ECC 256-bit schemes will be able to be compromised with a Quantum computer using the Shor algorithm [29] in less than ten minutes sometime between 2027 and 2040.",
"Grover's algorithm [30] provides a speedup for database search style algorithms, such as searching for a message digest preimage or second preimage.",
"Using Grover's algorithm the complexity of message digest algorithm's Preimage or Second Preimage Resistance properties are reduced from $O(N)$ to $O(\\sqrt{N})$ .",
"This means that the Security Strength assuming a sufficiently powerful quantum computer is half that when compared to the Security Strength due to classical computing power.",
"Brassard and Tapp [31] claimed to have developed an algorithm for use with quantum computers that reduces the complexity of finding message digest collisions to $O(\\@root 3 \\of {N})$ .",
"Bernstein [32] has refuted this claim, stating that there is no real advantage provided by Brassard and Tapp's algorithm given the cost - performance analysis over classical computing power.",
"However, Aaronson and Shi [33] have determined a tight lower bound for the complexity of the collision problem as $O(\\@root 3 \\of {N})$ .",
"As such, despite Bernstein's refutation of Brassard and Tapp's algorithm, it can be conjectured that another algorithm may be found that meets the theoretical bound, that has a better cost - performance metric.",
"Despite the reduced Security Strength offered by message digest algorithms, assuming a quantum computer, they are unlikely to be a point of weakness in the near term.",
"Developing a complex quantum computer that can defeat message digest algorithms is expected to be significantly more complex than developing one to defeat ECC 256-bit [34].",
"As such, it is likely that a quantum computer that can be used to attack message digest algorithms will not be available until at least the 2030s."
],
[
"Algorithmic Weaknesses",
"Researchers search for weaknesses in algorithms.",
"These weaknesses when found can reduce the effective Security Strength offered by the algorithm.",
"For example various weakness have been found in the MD-5 message digest algorithm [35] [36].",
"It is impossible to predict if a weakness in an algorithm such as Keccak-256 will be found, and the degree to which the algorithm would be weakened with such a compromise.",
"Algorithmic weaknesses will not be considered in the analysis of Ethereum MainNet given the uncertainty as to whether such weakness will be found, when they will be found, and the impact such weaknesses might have."
],
[
"Ethereum MainNet Features",
"This section discusses in detail the features of Ethereum MainNet that are important to its usage as a Coordination Blockchain."
],
[
"Authentication",
"The International Telecommunications Union (ITU) define authentication in X.805 [37] as: ...serves to confirm the identities of communicating entities.",
"Authentication ensures the validity of the claimed identities of the entities participating in communication (e.g., person, device, service or application) and provides assurance that an entity is not attempting a masquerade or unauthorized replay of a previous communication.",
"In the context of Ethereum, this means ensuring Ethereum transactions are directly attributable to participants who operate Ethereum Accounts.",
"Ethereum transactions are signed using the private key belonging to a participant [8].",
"The public key associated with the private key can be derived from the transaction signature of any transaction signed by the private key.",
"The account number is the twenty-byte truncated Keccak-256 message digest of the public key.",
"In Ethereum, each transaction includes a nonce [8].",
"The initial nonce value for each account is zero.",
"The nonce is incremented for each successfully mined transaction.",
"Miners reject transactions with out of order or repeated nonces.",
"Doing this protects Ethereum from transaction replay attacks.",
"The nonce value is represented as a 64-bit signed number in Geth [13] and Pantheon [14].",
"Adding one to the maximum representable number would result in the largest negative number.",
"If this situation was not guarded against in the code, it would lead to unexpected results, and possibly an authentication failure.",
"However, 63-bits is large enough such that even if a single account issued every transaction on Ethereum MainNet, and could craft sufficiently small transactions and could have the gas limit increased such that they could execute 1000 transactions per second, the nonce value would not wrap around for 584 million years.",
"Ethereum private keys are 256-bits long.",
"The signature algorithm ECDSA / Keccak-256 using the secp256k1 curve is used for signing transactions.",
"The secp256k1 curve has been analysed and found to not have any weaknesses [38].",
"This signature algorithm provides 128-bits of Security Strength [26] assuming Classical Cryptanalysis.",
"The conversion of the public key to an account number using a twenty-byte truncated Keccak-256 message digest offers 160-bits of Security Strength assuming Classical Cryptanalysis, as an attacker would need to exploit the Second Preimage Resistance property of the message digest function to determine another public key which could hash to the same value as the authentic public key.",
"As such, overall the Ethereum signing mechanism provides 128-bits of Security Strength assuming Classical Cryptanalysis.",
"NIST has issued guidance that usage of algorithms offering 112-bit Security Strength assuming Classical Cryptanalysis should be phased out by 2030 [25].",
"This means that Ethereum's transaction signing technique should be secure well beyond 2030, assuming Classical Cryptanalysis, given its 128-bit Security Strength.",
"If an attacker had access to a sufficiently powerful Quantum Computer, they could determine private keys associated with the public keys.",
"The attacker could observe transactions that have been submitted and determine the public keys associated with each transaction using the standard ecrecover technique [8].",
"Once an attacker had access to a private key, they could issue arbitrary transactions using that private key.",
"Aggarwal's [28] analysis indicates that the authenticity of transactions may be able to be compromised in this way some time after 2027.",
"The Ethereum community have recognised the threat that Quantum Cryptanalysis poses to Ethereum transaction signing.",
"There are plans to roll-out “Account Security Abstraction\" changes that will authenticate transactions programmatically using user supplied code [39] [40][41].",
"This would allow for users to choose to use Quantum Cryptanalysis resistant algorithms.",
"In summary, the existing transaction authentication techniques are likely to be secure until at least 2027.",
"Prior to 2027, Ethereum is likely to be upgraded to mitigate the threat of quantum computers, thus ensuring the authenticity of transactions into the future."
],
[
"Integrity",
"ITU defines data integrity [37] as: ... ensures the correctness or accuracy of data.",
"The data is protected against unauthorized modification, deletion, creation, and replication and provides an indication of these unauthorized activities.",
"In the context of Ethereum, this means ensuring that authenticated transactions and data in the distributed ledger are stored such that they can not be modified.",
"Ethereum transactions are combined into blocks using Merkle Patricia trees [8].",
"Similarly, data in the distributed ledger is protected using Merkle Patricia trees.",
"Compromising values in the Merkle Particia trees would require breaking the Second Preimage Resistance property of Keccak-256.",
"This is unlikely to occur in foreseeable future using either Quantum or Classical Cryptanalysis techniques.",
"However, there is always the possibility that a weakness in Keccak-256 will be found."
],
[
"Non-Repudiation",
"ITU defines non-repudiation [37] as: ...provides means for preventing an individual or entity from denying having performed a particular action related to data by making available proof of various network-related actions (such as proof of obligation, intent, or commitment; proof of data origin, proof of ownership, proof of resource use).",
"It ensures the availability of evidence that can be presented to a third party and used to prove that some kind of event or action has taken place.",
"In the context of Ethereum, this means ensuring that authenticated transactions are stored such that they can not be revoked.",
"Ethereum blocks are linked together using Keccak-256 message digests.",
"Compromising this linkage would require breaking the Preimage Resistance property of Keccak-256, which is unlikely to occur in foreseeable future.",
"As discussed in Section REF , Finality, Ethereum MainNet has probabilistic finality.",
"When blocks are added to the blockchain after a block containing a transaction, the probability of a miner proposing a heavier chain that does not include the block decreases.",
"The number of blocks added after a block is known as the number of block confirmations.",
"Nakamoto [17] showed the probability of a Bitcoin block being removed after six blocks, assuming an attacker has 10% of the mining power was 0.00024.",
"A greater number of block confirmations should be observed if an attacker were assumed to have a greater percentage of the total mining power available to them, or if the user wished to have greater certainty that the block was not going to be removed.",
"In 2016, Gervais [42] determined that 37 Ethereum MainNet block confirmations were needed to offer the same level of security as six Bitcoin block confirmations, assuming Ethereum was being attacked with 30% of mining power.",
"Since 2016, the mining power devoted to Ethereum has increased considerably such that a 30% attack now seems inconceivable.",
"Major miners are unlikely to attack their own network as this would risk devaluing the cryptocurrency they are mining [43], [44].",
"The maximum hash power which can be rented in a straightforward way is 5% [45].",
"Purchasing hardware to generate 30% hash power (174TH/s [46]) would cost in excess of US$400 million [47].",
"Scaling the results of Gervais's work [42] based on the changed mining rewards of Bitcoin and Ethereum, the changed valuations, and allowing for a 10% mining power attack, indicates that eight Ethereum block confirmations corresponds to six Bitcoin confirmations.",
"Using a different methodology, Buterin [48] determined that six to twelve confirmations where required to deem a transaction final, depending on the level of risk a user was prepared to assume.",
"Based on a fourteen second target block time and assuming twelve confirmations, a block on Ethereum MainNet could be deemed final in approximately three minutes.",
"The finality time is not a precise number as the block time is randomly distributed with an average of fourteen seconds.",
"When Ethereum MainNet client vendors and miners agree to changes in the Ethereum protocol, the system is updated via changes known as Hard Forks.",
"A Hard Fork requires all Ethereum MainNet client vendors to release updated software which will activate new functionality at a certain Ethereum MainNet block number.",
"For the Spurious Dragon Hard Fork in November 2016 [49] the changes were implemented slightly differently.",
"This resulted in the Ethereum MainNet blockchain forking for some hours [50].",
"The fork is resolved once the vendors software has been corrected.",
"However, it is possible that a transaction which was part of a block accepted into the fork which was discarded was reverted and not resubmitted to the blockchain.",
"This type of forking and state reversion due to mismatched feature implementation is much less likely to occur now and in the future than it did in 2016 as Ethereum MainNet clients undergo significantly more review and testing than they did in 2016 [51], [52].",
"If an attacker could dedicate 51% of the total mining power to attacking the network, they would be able to mount a 51% Attack [53].",
"This would allow the attacker to rewrite the history of the blockchain.",
"The three largest Ethereum MainNet miners could collude to mount such as attack.",
"However, these miners are disincentivized to do such an attack as this would adversely affect confidence in Ethereum MainNet.",
"This would lead to a dramatic drop in the value of Ether [43], [44], substantially decreasing the value of their Ether and their Ethereum infrastructure investments.",
"Though the Ethereum MainNet system typically can not be modified, after a re-entrancy bug was exploited in the DAO attack [54], the system was modified to reverse the results of the attack.",
"Doing this caused some to question trust in blockchain systems and Ethereum MainNet in particular [55].",
"However, this type of irregular state change [56] to reverse the results of such an attack appear unlikely to occur again in Ethereum MainNet.",
"Despite a bug in the Parity Wallet contract that resulted in hundreds of millions of dollars of funds becoming inaccessible, proposals to alter history to restore the funds have been refused [57][58]."
],
[
"Crypto Economic Anti-Spam",
"As described in Section REF , each transaction on Ethereum MainNet costs Gas to execute, which participants pay for with Ether.",
"Ethereum MainNet currently aims to produce new blocks each 14 seconds with eight million Gas available for each block [46].",
"Each transaction has as a minimum cost, the transaction fee, that is currently 21,000 Gas.",
"Simple balance transfers between accounts just cost the transaction fee, whereas complex function calls can cost more than a million gas.",
"As the block gas limit is eight million, it means that no transaction can use more than eight million gas.",
"This translates to Ethereum MainNet supporting between four transactions per minute and twenty-seven transactions per second.",
"A typical simple transaction, adding a Pin to a pinning contract, costs 64972 Gas [4].",
"Given the eight million Gas limit, 8.8 of these transactions could execute per second.",
"Participants are disincentivized from flooding the network with transactions as each transaction has an economic cost.",
"The cost of Gas depends on the block utilisation [59].",
"Historically, the Gas price has spiked high when block utilisation has been high [60].",
"If many entities attempted to issue adding a Pin to a pinning contract transactions regularly, such that the block utilisation was high, then the cost of issuing the adding a Pin to a pinning contract transactions would increase.",
"This would incentivise the entities to find alternatives, such as reducing the frequency of submitting the transactions."
],
[
"Summary",
"Based on the analysis in this section, it can be said that Ethereum MainNet contains transactions for which the authenticity and integrity is certain.",
"Once twelve blocks have been appended to the block containing a transaction, the probability of the blockchain being reorganised such that the transaction is reverted is small.",
"As such, Ethereum MainNet offers strong non-repudiation properties.",
"Ethereum's Gas mechanism operates as an effective anti-spam tool."
],
[
"Ethereum Private Sidechains",
"Ethereum Private Sidechains are Ephemeral, On-demand, Permissioned, Private, Confidential, blockchains that allow for Atomic Crosschain Transactions.",
"They are Ephemeral in that they are created, they operate, and then they can be archived when they are no longer needed.",
"Their On-demand nature allows them to be created when needed between parties that have no prior relationship.",
"Permissioning ensures that only authorised nodes are able to join a sidechain.",
"Their design is such that to the greatest extent possible, their membership and their transactions are kept Private.",
"Confidentiality is ensured by encrypting the sidechain data when being communicated between nodes and stored on nodes.",
"Atomic Crosschain Transactions enable transactions that update state across sidechains atomically.",
"Ethereum Private Sidechains have been described in terms of their requirements [2], and aspects of their technology [3], [4], [5].",
"This paper is the first to present this technology holistically.",
"Additionally, this paper introduces the idea of pinning the final state of a sidechain prior to archiving, thus allowing the sidechain to be reinstated if needed, and introduces the idea of using Ethereum MainNet gas pricing as a mechanism for rate control of Atomic Crosschain Transactions."
],
[
"Ephemeral",
"Ethereum Private Sidechains are Ephemeral: they are created, they are used for a period, and then archived when they are no longer required.",
"This limited lifespan matches many real world requirements, such as Letters of Credit and other business deals, which have a limited lifespan.",
"The ability to archive the blockchain data in a sidechain is in contrast to existing blockchain technologies that are designed to be operational indefinitely.",
"The life span of a sidechain could vary widely.",
"For usages in which sidechains are used to deploy a contract and automatically negotiate a deal, it might only be needed for some minutes, hours or days.",
"Other usages, such as an Oracle, require a long or indefinite lifespan.",
"Indefinite lifespans can be accommodated by never archiving the sidechain.",
"While a sidechain is operational, the sidechain could be pinned to a Coordination Blockchain at regular intervals [4].",
"Regularly pinning sidechain state helps to protect minority sidechain participants from state reversion due to collusion by the majority of sidechain participants [4].",
"A key aspect of Ephemeral sidechains is the requirement to be able to restart the sidechain after archiving.",
"This can be achieved by pinning the last block of the sidechain to a Coordination Blockchain.",
"Now that the Block Hash of the last block has been securely stored in the Coordination Blockchain, the state of the sidechain can then be stored offline.",
"To restart the sidechain, the stored data is compared against the final Block Hash to confirm the correct state is being used to restart the sidechain."
],
[
"On-demand Between Parties with No Prior Relationship",
"Ethereum Private Sidechains need to be able to be deployed between parties that have no prior relationship.",
"That is, the parties need to be able to establish a sidechain without knowing each others' node IP addresses, cryptographic keys, or other information required to set-up a secure connection.",
"Establishing sidechains in this dynamic way is in contrast to existing permissioned blockchains that are largely static systems that require complex set-up.",
"For example, set-up of a Quorum [12] network requires enode addresses (IP addresses and Ethereum account numbers) for each node to be shared out of band with all other nodes.",
"Adding new nodes to the network requires this sharing and manual intervention on each node.",
"The on-demand sidechain establishment is analogous to a user of a web browser establishing a secure connection with a web server by simply entering in a URL such as https://example.com/.",
"The user does not know the IP address of the computer corresponding to example.com or the public key that can be used to verify the communications emanating from example.com.",
"However, using the domain name, some initial trust, and the Domain Name Service (DNS) and Transport Layer Security (TLS) protocols, they are able to establish a secure connection.",
"Similarly, Ethereum Private Sidechains need to be able to establish a secure sidechain using just domain names.",
"The Ethereum Registration Authorities system is a set of smart contracts that can be used to provide discoverable information to enable establishment of sidechains between organisations with no prior relationship [3].",
"A Coordination Blockchain could be used to locate the information using domain names that can be grouped according to different trust levels and different trust relationships.",
"Moreover, a Coordination Blockchain that provides organisations with a secure, decentralized, censorship-resistant mechanism for storing information that can be located using domain names and grouped according to different trust levels and different trust relationships would overcome the limitations of previous technologies that did not provide the security and censorship resistance properties that users of blockchain technologies expect."
],
[
"Permissioned",
"Ethereum Private Sidechains need to be operated by authorised nodes using authorised Ethereum accounts.",
"These requirements match those of the Enterprise Ethereum Client Specification [11].",
"The implementation of these requirements do not use a Coordination Blockchain."
],
[
"Private",
"Ethereum Private Sidechains should, to the greatest extent possible, keep their membership private from other sidechains they interact with and from any Coordination Blockchains they use to facilitate their actions."
],
[
"Confidential",
"Ethereum Private Sidechains should encrypt their blockchain and state data such that the transaction information is kept confidential, both when it is communicated between nodes on a sidechain and when it is stored in a node's local data store.",
"The implementation of this feature does not use a Coordination Blockchain."
],
[
"Atomic Crosschain Transactions",
"Ethereum Private Sidechains technology needs to enable transactions that update state across sidechains atomically [5].",
"That is, if an Atomic Crosschain Transaction is across sidechains A, B, and C, then the state updates related to the transaction are either applied on all sidechains or ignored on all sidechains.",
"A Coordination Blockchain holds a Crosschain Coordination Contract.",
"This contract is used to indicate that an Atomic Crosschain Transaction has commenced, has been committed, or should be ignored.",
"The contract acts as a common time-out reference for all sidechains and helps prevent denial of service attacks.",
"The data in the Crosschain Coordination Contract needs to be available until the last sidechain using it has been archived.",
"The Atomic Crosschain Transaction system uses threshold signatures to prove values across sidechains.",
"The public key that corresponds to the private key shares held by each of the sidechain validators is known as a Sidechain Public Key.",
"This key needs to be available to all sidechains that need to verify values coming from a sidechain.",
"As such, this value should be stored on a Coordination Blockchain.",
"The Sidechain Public Key needs to be re-generated and uploaded to the Coordination Blockchain each time a validator is added or removed from the sidechain.",
"Assuming that sidechain membership is largely static, this regeneration and upload is likely to be a rare event."
],
[
"Pros and Cons of using Ethereum MainNet as a Coordination Blockchain",
"The subsections below analyse the advantages and disadvantages of using Ethereum MainNet as the Coordination Blockchain for the operations of an Ethereum Private Sidechain.",
"The findings of the subsections are summarised in Table REF .",
"Table: Advantages and Disadvantages of using Ethereum MainNet at Coordination Blockchain for Ethereum Private Sidechains"
],
[
"Private Node Discovery - Ethereum Registration Authorities",
"The Ethereum Registration Authorities system [3] uses smart contracts on a Coordination Blockchain to enable discovery of sidechain node address and cryptographic key information, as described in Section REF .",
"As the information is used to bootstrap a sidechain, it is fundamental to the entire Ethereum Private Sidechain system that this information is authentic.",
"The data in the Ethereum Registration Authority smart contracts is largely static.",
"That is, the IP address and cryptographic key information, once set, changes rarely.",
"Given this largely static data, the economic cost of storing information on Ethereum MainNet would only be incurred rarely.",
"It is likely to cost less that US$1.00 to set-up an enterprise in the Ethereum Registration Authority system on Ethereum MainNet, based on current prices [3].",
"Sidechain users who wish to establish a sidechain need to be able to access the bootstrap information stored in Ethereum Registration Authority smart contracts for the system to be useful.",
"The information needs to be stored on a permissionless network or a permissioned network that has a black list of banned nodes.",
"Doing this allows users who have no prior relationship with the operators of the Coordination Blockchain to access the information."
],
[
"State Pinning",
"A private blockchain state pinning approach should be used to prevent state reversion as described in Section REF .",
"Posting Pins to Ethereum MainNet leverages the authenticity, integrity and non-repudiation properties of Ethereum MainNet.",
"However, submitting transactions costs money.",
"Pinning once per hour for a year would cost US$508 [4].",
"Additionally, if many sidechains pinned to Ethereum MainNet simultaneously, it would cause transaction congestion.",
"Another issue with pinning directly to Ethereum MainNet is that any disputes that occur would need to occur on Ethereum MainNet, thus making the participant list of the sidechain public.",
"Pins could be posted directly to a smart contract on Ethereum MainNet, or could be posted via a smart contract on an intermediate blockchain using a hierarchical pinning approach [4].",
"Using a hierarchical pinning approach, many private blockchains could treat another private blockchain as a Coordination Blockchain posting Pins to it.",
"This private blockchain could in turn post Pins to another private blockchain or to Ethereum MainNet.",
"This is shown diagrammatically in Figure REF .",
"Pinning to a hierarchy of Coordination Blockchains in this way means that only a small number of Pins on Ethereum MainNet could be used to secure a large number of private blockchains.",
"The cost of submitting Pins to the private blockchain could be either free or significantly less than Ethereum MainNet.",
"Figure: Hierarchical PinningA benefit of pinning directly to Ethereum MainNet, rather than via an intermediate blockchain, is that the pinned state becomes final faster.",
"That is, if a Pin is posted to a private blockchain, whose state is in turn pinned to Ethereum MainNet, then the sidechain Pin could be deemed to become final only once the private blockchain in pinned to Ethereum MainNet.",
"Posting Pins via a private blockchain significantly reduces the cost of pinning, as only one blockchain needs to submit transactions to pin its state to Ethereum MainNet, and sidechains can pin to that private blockchain.",
"Doing this reduces the number of transactions on Ethereum MainNet, thus reducing congestion, and means that the cost of submitting transactions is only incurred once for the private blockchain, rather than once for each sidechain.",
"A disadvantage of posting Pins via a private blockchain is that participants of the sidechain need to observe and be ready to challenge Pins being posted at each level of the hierarchy.",
"If sidechain state Pins are posted directly to Ethereum MainNet, then the sidechain participants only need to observe the pinning contract on Ethereum MainNet.",
"An additional benefit of pinning to a private blockchain is that the chain's permissioning could be set such that only certain nodes could view the blockchain and only certain accounts could submit transactions to the blockchain.",
"Pinning directly to Ethereum MainNet means that the organisation pinning to the contract is public.",
"If there is a dispute, then masked participants will need to unmask themselves, and thus link themselves to the sidechain and the other organisations on the sidechain.",
"If an intermediate blockchain was used, then the pinning and any disputes could happen in a more private setting."
],
[
"Final State Pinning for Archiving",
"Final State Pinning is the same as State Pinning, with the exception that rather than the pinning being on an ongoing basis, it is just to pin the final state of a sidechain prior to archiving, as described in Section REF .",
"As such, the advantages and disadvantages are similar to those described in the previous section.",
"As only one pin is posted, the concerns over having to observe pins on a private blockchain in addition to Ethereum MainNet are not significant as the observation is for a single event.",
"Similarly, concerns over cost of posting pins to Ethereum MainNet and congestion are reduced.",
"As such, the advantages are reduced to the pin becoming final sooner and the disadvantages are reduced to any dispute over the value of the pin being public."
],
[
"Sidechain Public Keys",
"As described in Section REF , the Atomic Crosschain Transactions feature needs Sidechain Public Keys to be stored on a Coordination Blockchain.",
"The Sidechain Public Keys need to be stored in a contract [5] that allows voting on new public keys, and allows masked and unmasked participants.",
"Given the participants are the same as those for the pinning scheme, it makes sense for these to be stored in the same contract as the pinning information.",
"Keeping the logic in the same contract for pinning and holding the Sidechain Public Keys is useful as it means that membership changes need to only occur in one contract.",
"However, the Sidechain Public Keys need to be visible by all sidechains that wish to verify information coming from the sidechain, whereas the pinning information need only be visible by sidechain participants and government regulators who would be appealed to in case of dispute.",
"Given the Sidechain Public Key is likely to be set once only, the economic cost of storing the key is likely to only be incurred once.",
"No analysis of the gas cost of setting a Sidechain Public Key has been undertaken yet.",
"However, given the small size of the public keys, 48 bytes, the incremental gas cost of storing the public key is likely to be in the order of 60,000 Gas, assuming the voting infrastructure has already been set-up.",
"However, if the voting infrastructure did need to be set-up, the gas cost could be much larger.",
"If a sidechain was short lived, then incurring the cost of setting up the voting infrastructure and posting the Sidechain Public Key to Ethereum MainNet could be deemed considerable.",
"However, if the sidechain was long lived, then this relative cost might not be deemed as significant.",
"A disadvantage of using Ethereum MainNet to hold Sidechain Public Keys is transactions take at least 12 blocks before they should be deemed final (see Section REF ).",
"This means that, given a target block time of fourteen seconds, users could not use the Sidechain Public Keys for Atomic Crosschain Transactions for three minutes after the transaction that posts the Sidechain Public Key is included in a block on Etheurum MainNet."
],
[
"Atomic Crosschain Transaction State",
"The Atomic Crosschain Transactions capability described in Section REF uses a Crosschain Coordination Contract to control when a crosschain transaction has started, been committed, or should be ignored.",
"This information need to be available to all validators on all sidechains involved in the crosschain transaction.",
"The information in the contract needs to be available until the last sidechain using the contract is archived.",
"Storing the Atomic Crosschain State on Ethereum MainNet means that each Atomic Crosschain Transaction costs money to execute.",
"This economic cost could be seen as an advantage, as it provides an anti-spam control external to the sidechain system.",
"However, forcing enterprises to incur a cost for each crosschain transaction is likely to be viewed as an unnecessary cost.",
"Additional issues with storing the Atomic Crosschain State on Ethereum MainNet is that this would leak the participants of a sidechain, as a transaction would need to be submitted linking the sidechain and the participant.",
"Furthermore, this would leak the rate that the participant was issuing crosschain transactions.",
"In a similar way that storing Sidechain Public Keys on Ethereum MainNet delays when the first Atomic Crosschain Transaction can be issued, as discussed in Section REF , storing Atomic Crosschain Transaction State could delay the effective start of each transaction.",
"This is because sidechain participants might want to wait for blocks that contain transactions that indicate the Atomic Crosschain Transaction start to be final prior to acting on the start indication."
],
[
"Conclusion",
"Coordination Blockchains perform various coordination tasks in private blockchain systems.",
"We used Ethereum Private Sidechains as an exposition of such a system, highlighting the features of Ethereum Private Sidechains and discussing each feature's need to leverage a Coordination Blockchain.",
"Based on the unique requirements of each feature and coordination activity, we examine whether public Ethereum MainNet would be a suitable platform for each of those tasks.",
"We found that Ethereum Registration Authority smart contracts of Ethereum Private Sidechains need to store long term data that have to be available in a permissionless blockchain.",
"Ethereum MainNet would therefore be well suited to this task, as it is a permissionless blockchain that incentivises good behaviour using crypto economics, and provides good authenticity, integrity, and non-repudiation properties.",
"Ethereum MainNet's strong security properties are also useful for State Pinning and in particular Final State Pinning, where the data needs to be stored securely for long periods of time.",
"However, pinning directly to Ethereum MainNet could lead to congestion on Ethereum MainNet, would incur high costs, and would lead to the membership of a sidechain becoming public in the case of a dispute over the value of a Pin.",
"These issues are significantly reduced by pinning via an intermediate private blockchain.",
"However, doing this introduces other issues, such as participants having to observe pinned values at multiple levels in the pinning hierarchy and the pinned values taking longer to become final.",
"Ethereum MainNet is not an appropriate location for Coordination Blockchain information that needs to be final quickly, such as Sidechain Public Keys and Atomic Crosschain Transaction State."
],
[
"Acknowledgments",
"This research has been undertaken whilst I have been employed full-time at ConsenSys and have been completing my PhD part-time at University of Queensland.",
"I acknowledge the support of my PhD supervisor Dr Marius Portmann.",
"I thank Dr Catherine Jones, Horacio Mijail Anton Quiles, David Hyland-Wood, and Sandra Johnson for reviewing this paper and providing astute feedback."
]
] | 1906.04421 |
[
[
"Topology Attack and Defense for Graph Neural Networks: An Optimization\n Perspective"
],
[
"Abstract Graph neural networks (GNNs) which apply the deep neural networks to graph data have achieved significant performance for the task of semi-supervised node classification.",
"However, only few work has addressed the adversarial robustness of GNNs.",
"In this paper, we first present a novel gradient-based attack method that facilitates the difficulty of tackling discrete graph data.",
"When comparing to current adversarial attacks on GNNs, the results show that by only perturbing a small number of edge perturbations, including addition and deletion, our optimization-based attack can lead to a noticeable decrease in classification performance.",
"Moreover, leveraging our gradient-based attack, we propose the first optimization-based adversarial training for GNNs.",
"Our method yields higher robustness against both different gradient based and greedy attack methods without sacrificing classification accuracy on original graph."
],
[
"Introduction",
"Graph structured data plays a crucial role in many AI applications.",
"It is an important and versatile representation to model a wide variety of datasets from many domains, such as molecules, social networks, or interlinked documents with citations.",
"Graph neural networks (GNNs) on graph structured data have shown outstanding results in various applications [10], [20], [22].",
"However, despite the great success on inferring from graph data, the inherent challenge of lacking adversarial robustness in deep learning models still carries over to security-related domains such as blockchain or communication networks.",
"In this paper, we aim to evaluate the robustness of GNNs from a perspective of first-order optimization adversarial attacks.",
"It is worth mentioning that first-order methods have achieved great success for generating adversarial attacks on audios or images [4], [23], [7], [24], [6].",
"However, some recent works [8], [1] suggested that conventional (first-order) continuous optimization methods do not directly apply to attacks using edge manipulations (we call topology attack) due to the discrete nature of graphs.",
"We close this gap by studying the problem of generating topology attacks via convex relaxation so that gradient-based adversarial attacks become plausible for GNNs.",
"Benchmarking on node classification tasks using GNNs, our gradient-based topology attacks outperform current state-of-the-art attacks subject to the same topology perturbation budget.",
"This demonstrates the effectiveness of our attack generation method through the lens of convex relaxation and first-order optimization.",
"Moreover, by leveraging our proposed gradient-based attack, we propose the first optimization-based adversarial training technique for GNNs, yielding significantly improved robustness against gradient-based and greedy topology attacks.",
"Our new attack generation and adversarial training methods for GNNs are built upon the theoretical foundation of spectral graph theory, first-order optimization, and robust (mini-max) optimization.",
"We summarize our main contributions as follows: We propose a general first-order attack generation framework under two attacking scenarios: a) attacking a pre-defined GNN and b) attacking a re-trainable GNN.",
"This yields two new topology attacks: projected gradient descent (PGD) topology attack and min-max topology attack.",
"Experimental results show that the proposed attacks outperform current state-of-the-art attacks.",
"With the aid of our first-order attack generation methods, we propose an adversarial training method for GNNs to improve their robustness.",
"The effectiveness of our method is shown by the considerable improvement of robustness on GNNs against both optimization-based and greedy-search-based topology attacks."
],
[
"Related Works",
"Some recent attentions have been paid to the robustness of graph neural network.",
"Both [25] and [8] studied adversarial attacks on neural networks for graph data.",
"[8] studied test-time non-targeted adversarial attacks on both graph classification and node classification.",
"Their work restricted the attacks to perform modifications on discrete structures, that is, an attacker is only allowed to add or delete edges from a graph to construct a new graph.",
"White-box, practical black-box and restricted black-box graph adversarial attack scenarios were studied.",
"Authors in [25] considered both test-time (evasion) and training-time (data poisoning) attacks on node classification task.",
"In contrast to [8], besides adding or removing edges in the graph, attackers in [25] may modify node attributes.",
"They designed adversarial attacks based on a static surrogate model and evaluated their impact by training a classifier on the data modified by the attack.",
"The resulting attack algorithm is for targeted attacks on single nodes.",
"It was shown that small perturbations on the graph structure and node features are able to achieve misclassification of a target node.",
"A data poisoning attack on unsupervised node representation learning, or node embeddings, has been proposed in [1].",
"This attack is based on perturbation theory to maximize the loss obtained from DeepWalk [17].",
"In [26], training-time attacks on GNNs were also investigated for node classification by perturbing the graph structure.",
"The authors solved a min-max problem in training-time attacks using meta-gradients and treated the graph topology as a hyper-parameter to optimize."
],
[
"Problem Statement",
"We begin by providing preliminaries on GNNs.",
"We then formalize the attack threat model of GNNs in terms of edge perturbations, which we refer as `topology attack'."
],
[
"Preliminaries on GNNs",
"It has been recently shown in [10], [20], [22] that GNN is powerful in transductive learning, e.g., node classification under graph data.",
"That is, given a single network topology with node features and a known subset of node labels, GNNs are efficient to infer the classes of unlabeled nodes.",
"Prior to defining GNN, we first introduce the following graph notations.",
"Let $\\mathcal {G} = (\\mathcal {V}, \\mathcal {E})$ denote an undirected and unweighted graph, where $\\mathcal {V}$ is the vertex (or node) set with cardinality $|\\mathcal {V}| = N$ , and $\\mathcal {E} \\in (\\mathcal {V} \\times \\mathcal {V})$ denotes the edge set with cardinality $|\\mathcal {E}| = M$ .",
"Let $\\mathbf {A}$ represent a binary adjacency matrix.",
"By definition, we have $A_{ij} = 0$ if $(i, j) \\notin \\mathcal {E}$ .",
"In a GNN, we assume that each node $i$ is associated with a feature vector $\\mathbf {x}_i \\in \\mathbb {R}^{M_0}$ and a scalar label $y_i$ .",
"The goal of GNN is to predict the class of an unlabeled node under the graph topology $\\mathbf {A}$ and the training data $\\lbrace (\\mathbf {x}_i, y_i )\\rbrace _{i=1}^{N_{\\text{train}}}$ .",
"Here GNN uses input features of all nodes but only $N_{\\text{train}} < N$ nodes with labeled classes in the training phase.",
"Formally, the $k$ th layer of a GNN model obeys the propagation rule of the generic form $\\mathbf {h}_i^{(k)} = g^{(k)} \\left( \\lbrace \\mathbf {W}^{(k-1)} \\mathbf {h}_j^{(k-1)} \\tilde{A}_{ij}, ~ \\forall j \\in \\mathcal {N}(i)\\rbrace \\right), ~ \\forall i\\in [N]$ where $\\mathbf {h}_i^{(k)} \\in \\mathbb {R}^{M_{k}}$ denotes the feature vector of node $i$ at layer $k$ , $\\mathbf {h}_i^{(0)} = \\mathbf {x}_i\\in \\mathbb {R}^{M_0}$ is the input feature vector of node $i$ , $ g^{(k)}$ is a possible composite mapping (activation) function, $\\mathbf {W}^{(k-1)} \\in \\mathbb {R}^{M_k \\times M_{k-1}}$ is the trainable weight matrix at layer $(k-1)$ , $\\tilde{ A}_{ij}$ is the $(i,j)$ th entry of $\\tilde{\\mathbf {A}}$ that denotes a linear mapping of $\\mathbf {A}$ but with the same sparsity pattern, and $\\mathcal {N}(i)$ denotes node $i$ 's neighbors together with itself, i.e., $\\mathcal {N}(i) = \\lbrace j | (i,j) \\in \\mathcal {E}, \\text{ or } j = i \\rbrace $ .",
"A special form of GNN is graph convolutional networks (GCN) [10].",
"This is a recent approach of learning on graph structures using convolution operations which is promising as an embedding methodology.",
"In GCNs, the propagation rule (REF ) becomes [10] $\\mathbf {h}_i^{(k)} = \\sigma \\left( \\sum _{j \\in \\mathcal {N}_i} \\left( \\mathbf {W}^{(k-1)} \\mathbf {h}_j^{(k-1)} \\tilde{A}_{ij} \\right) \\right),$ where $\\sigma (\\cdot )$ is the ReLU function.",
"Let $\\tilde{A}_{i,:}$ denote the $i$ th row of $\\tilde{\\mathbf {A}}$ and $\\mathbf {H}^{(k)} = \\left[ (\\mathbf {h}_1^{(k)})^\\top ; \\ldots ; (\\mathbf {h}_N^{(k)})^\\top \\right]$ , we then have the standard form of GCN, $\\mathbf {H}^{(k)} = \\sigma \\left( \\tilde{\\mathbf {A}} \\mathbf {H}^{(k-1)} ( \\mathbf {W}^{(k-1)} )^\\top \\right).$ Here $\\tilde{\\mathbf {A}}$ is given by a normalized adjacency matrix $\\tilde{\\mathbf {A}} = \\hat{\\mathbf {D}}^{-1/2} \\hat{\\mathbf {A}} \\hat{\\mathbf {D}}^{-1/2}$ , where $\\hat{\\mathbf {A}} = \\mathbf {A} + \\mathbf {I}$ , and $\\hat{\\mathbf {D}}_{ij} = 0$ if $i \\ne j$ and $\\hat{\\mathbf {D}}_{ii} = \\mathbf {1}^\\top \\hat{\\mathbf {A}}_{:,i}$ ."
],
[
"Topology Attack in Terms of Edge Perturbation",
"We introduce a Boolean symmetric matrix $\\mathbf {S} \\in \\lbrace 0,1\\rbrace ^{N \\times N}$ to encode whether or not an edge in $\\mathcal {G}$ is modified.",
"That is, the edge connecting nodes $i$ and $j$ is modified (added or removed) if and only if $S_{ij} = S_{ji}=1$ .",
"Otherwise, $S_{ij } = 0$ if $i = j $ or the edge $(i,j)$ is not perturbed.",
"Given the adjacency matrix $\\mathbf {A}$ , its supplement is given by $\\bar{\\mathbf {A}} = \\mathbf {1} \\mathbf {1}^T - \\mathbf {I} - \\mathbf {A} $ , where $\\mathbf {I}$ is an identity matrix, and $(\\mathbf {1} \\mathbf {1}^T - \\mathbf {I})$ corresponds to the fully-connected graph.",
"With the aid of edge perturbation matrix $\\mathbf {S}$ and $\\bar{\\mathbf {A}}$ , a perturbed graph topology $\\mathbf {A}^\\prime $ against $\\mathbf {A}$ is given by $\\mathbf {A}^\\prime = \\mathbf {A} + \\mathbf {C} \\circ \\mathbf {S}, ~\\mathbf {C} = \\bar{\\mathbf {A}} - \\mathbf {A},$ where $\\circ $ denotes the element-wise product.",
"In (REF ), the positive entry of $\\mathbf {C}$ denotes the edge that can be added to the graph $\\mathbf {A}$ , and the negative entry of $\\mathbf {C}$ denotes the edge that can be removed from $\\mathbf {A}$ .",
"We then formalize the concept of topology attack to GNNs: Finding minimum edge perturbations encoded by $\\mathbf {S}$ in (REF ) to mislead GNNs.",
"A more detailed attack formulation will be studied in the next section."
],
[
"Topology Attack Generation: A First-Order Optimization Perspective",
"In this section, we first define attack loss (beyond the conventional cross-entropy loss) under different attacking scenarios.",
"We then develop two efficient attack generation methods by leveraging first-order optimization.",
"We call the resulting attacks projected gradient descent (PGD) topology attack and min-max topology attack, respectively."
],
[
"Attack Loss & Attack Generation",
"Let $\\mathbf {Z} (\\mathbf {S}, \\mathbf {W}; \\mathbf {A}, \\lbrace \\mathbf {x}_i\\rbrace )$ denote the prediction probability of a GNN specified by $\\mathbf {A}^\\prime $ in (REF ) and $\\mathbf {W}$ under input features $ \\lbrace \\mathbf {x}_i\\rbrace $ .",
"Then $Z_{i,c}$ denotes the probability of assigning node $i$ to class $c$ .",
"It has been shown in existing works [9], [11] that the negative cross-entropy (CE) loss between the true labels ($ y_i$ ) and the predicted labels ($\\lbrace Z_{i,c} \\rbrace $ ) can be used as an attack loss at node $i$ , denoted by $f_i(\\mathbf {S}, \\mathbf {W};\\mathbf {A}, \\lbrace \\mathbf {x}_i\\rbrace , y_i )$ .",
"We can also propose a CW-type loss similar to Carlili-Wagner (CW) attacks for attacking image classifiers [3], $& f_i(\\mathbf {S}, \\mathbf {W}; \\mathbf {A}, \\lbrace \\mathbf {x}_i\\rbrace , y_i ) =\\max \\left\\lbrace Z_{i,y_i} - \\max _{c \\ne y_i} Z_{i,c}, - \\kappa \\right\\rbrace ,$ where $\\kappa \\ge 0$ is a confidence level of making wrong decisions.",
"To design topology attack, we seek $\\mathbf {S}$ in (REF ) to minimize the per-node attack loss (CE-type or CW-type) given a finite budge of edge perturbations.",
"We consider two threat models: a) attacking a pre-defined GNN with known $\\mathbf {W}$ ; b) attacking an interactive GNN with re-trainable $\\mathbf {W}$ .",
"In the case a) of fixed $\\mathbf {W}$ , the attack generation problem can be cast as $\\begin{array}{ll}\\displaystyle \\operatornamewithlimits{\\text{minimize}}_{\\mathbf {s} } & \\sum _{i \\in \\mathcal {V}}f_i(\\mathbf {s} ; \\mathbf {W}, \\mathbf {A}, \\lbrace \\mathbf {x}_i\\rbrace , y_i )\\\\\\operatornamewithlimits{\\text{subject to}}& \\mathbf {1}^\\top \\mathbf {s} \\le \\epsilon , ~ \\mathbf {s}\\in \\lbrace 0,1\\rbrace ^{n},\\end{array}$ where we replace the symmetric matrix variable $\\mathbf {S}$ with its vector form that consists of $n \\mathrel {\\mathop :}=N(N-1)/2$ unique perturbation variables in $\\mathbf {S}$ .",
"We recall that $f_i$ could be either a CE-type or a CW-type per-node attack loss.",
"In the case b) of re-trainable $\\mathbf {W}$ , the attack generation problem has the following min-max form $\\begin{array}{cc}\\displaystyle \\operatornamewithlimits{\\text{minimize}}_{ \\mathbf {1}^\\top \\mathbf {s} \\le \\epsilon , \\mathbf {s}\\in \\lbrace 0,1\\rbrace ^{n} } \\displaystyle \\operatornamewithlimits{\\text{maximize}}_{\\mathbf {W}} \\, \\sum _{i \\in \\mathcal {V}}f_i(\\mathbf {s} , \\mathbf {W} ; \\mathbf {A}, \\lbrace \\mathbf {x}_i\\rbrace , y_i ) ,\\end{array}$ where the inner maximization aims to constrain the attack loss by retraining $\\mathbf {W}$ so that attacking GNN is more difficult.",
"Motivated by targeted adversarial attacks against image classifiers [3], we can define targeted topology attacks that are restricted to perturb edges of targeted nodes.",
"In this case, we require to linearly constrain $\\mathbf {S}$ in (REF ) as $S_{i, \\cdot } = 0$ if $i$ is not a target node.",
"As a result, both attack formulations (REF ) and (REF ) have extra linear constraints with respect to $s$ , which can be readily handled by the optimization solver introduced later.",
"Without loss of generality, we focus on untargeted topology attacks in this paper."
],
[
"PGD Topology Attack",
"Problem (REF ) is a combinatorial optimization problem due to the presence of Boolean variables.",
"For ease of optimization, we relax $\\mathbf {s} \\in \\lbrace 0,1\\rbrace ^{n}$ to its convex hull $s \\in [ 0,1]^{n}$ and solve the resulting continuous optimization problem, $\\begin{array}{ll}\\displaystyle \\operatornamewithlimits{\\text{minimize}}_{\\mathbf {s}} & f(\\mathbf {s}) \\mathrel {\\mathop :}=\\sum _{i \\in \\mathcal {V}}f_i(\\mathbf {s} ; \\mathbf {W}, \\mathbf {A}, \\lbrace \\mathbf {x}_i\\rbrace , y_i ) \\\\\\operatornamewithlimits{\\text{subject to}}& \\mathbf {s} \\in \\mathcal {S},\\end{array}$ where $\\mathcal {S} = \\lbrace \\mathbf {s} \\, | \\, \\mathbf {1}^T s \\le \\epsilon , \\mathbf {s} \\in [0,1]^n\\rbrace $ .",
"Suppose that the solution of problem (REF ) is achievable, the remaining question is how to recover a binary solution from it.",
"Since the variable $\\mathbf {s}$ in (REF ) can be interpreted as a probabilistic vector, a randomization sampling [13] is suited for generating a near-optimal binary topology perturbation; see details in Algorithm REF .",
"Random sampling from probabilistic to binary topology perturbation [1] Input: probabilistic vector $\\mathbf {s}$ , $K$ is # of random trials $k = 1,2,\\ldots , K$ draw binary vector $\\mathbf {u}^{(k)}$ following $u^{(k)}_i = \\left\\lbrace \\begin{array}{ll}1 & \\text{with probability $s_{i}$}\\\\0 & \\text{with probability $1- s_{i}$}\\end{array} \\right.",
", \\forall i$ choose a vector $\\mathbf {s}^*$ from $\\lbrace \\mathbf {u}^{(k)} \\rbrace $ which yields the smallest attack loss $f(\\mathbf {u}^{(k)} )$ under $\\mathbf {1}^T \\mathbf {s} \\le \\epsilon $ .",
"We solve the continuous optimization problem (REF ) by projected gradient descent (PGD), $\\mathbf {s}^{(t)} = \\Pi _{\\mathcal {S}} \\left[ \\mathbf {s}^{(t-1)} - \\eta _{t} \\hat{\\mathbf {g}}_t \\right],$ where $t$ denotes the iteration index of PGD, $\\eta _t > 0$ is the learning rate at iteration $t$ , $\\hat{\\mathbf {g}}_t = \\nabla f(\\mathbf {s}^{(t-1)})$ denotes the gradient of the attack loss $f$ evaluated at $\\mathbf {s}^{(t-1)}$ , and $\\Pi _{\\mathcal {S}} (\\mathbf {a}) \\mathrel {\\mathop :}=\\operatornamewithlimits{arg\\,min}_{\\mathbf {s} \\in \\mathcal {S}} \\Vert \\mathbf {s} - \\mathbf {a} \\Vert _2^2$ is the projection operator at $\\mathbf {a}$ over the constraint set $\\mathcal {S}$ .",
"In Proposition REF , we show that the projection operation yields the closed-form solution.",
"Proposition 1 Given $\\mathcal {S} = \\lbrace \\mathbf {s} \\, | \\, \\mathbf {1}^T s \\le \\epsilon , \\mathbf {s} \\in [0,1]^n\\rbrace $ , the projection operation at the point $\\mathbf {a}$ with respect to $\\mathcal {S}$ is $\\Pi _{\\mathcal {S}} (\\mathbf {a}) = \\left\\lbrace \\begin{array}{ll}P_{[\\mathbf {0}, \\mathbf {1}]} [\\mathbf {a} - \\mu \\mathbf {1}] &\\begin{array}{l}\\text{If $\\mu > 0$ and} \\\\\\mathbf {1}^T P_{[\\mathbf {0}, \\mathbf {1}]} [\\mathbf {a} - \\mu \\mathbf {1}] = \\epsilon ,\\end{array} \\\\& \\\\P_{[\\mathbf {0}, \\mathbf {1}]} [\\mathbf {a} ] & \\text{If $\\mathbf {1}^T P_{[\\mathbf {0}, \\mathbf {1}]} [\\mathbf {a} ] \\le \\epsilon $},\\end{array}\\right.$ where $P_{[0,1]} (x ) = x$ if $x \\in [0,1]$ , 0 if $x < 0$ , and 1 if $x > 1$ .",
"Proof: We express the projection problem as $\\begin{array}{ll}\\displaystyle \\operatornamewithlimits{\\text{minimize}}_{\\mathbf {s}} & \\frac{1}{2}\\Vert \\mathbf {s} - \\mathbf {a} \\Vert _2^2 + \\mathcal {I}_{[\\mathbf {0}, \\mathbf {1}]}(\\mathbf {s})\\\\\\operatornamewithlimits{\\text{subject to}}& \\mathbf {1}^\\top \\mathbf {s} \\le \\epsilon ,\\end{array}$ where $\\mathcal {I}_{[\\mathbf {0}, \\mathbf {1}]}(\\mathbf {s}) = 0$ if $\\mathbf {s} \\in [0,1]^{n}$ , and $\\infty $ otherwise.",
"The Lagrangian function of problem (REF ) is given by $& \\frac{1}{2} \\Vert \\mathbf {s} - \\mathbf {a} \\Vert _2^2 + \\mathcal {I}_{[\\mathbf {0}, \\mathbf {1}]}(\\mathbf {s}) + \\mu (\\mathbf {1}^\\top \\mathbf {s} - \\epsilon ) \\nonumber \\\\= & \\sum _{i} \\left( \\frac{1}{2} (s_i - a_i)^2 + \\mathcal {I}_{[0,1]}(s_i) + \\mu s_i \\right) - \\mu \\epsilon ,$ where $\\mu \\ge 0$ is the dual variable.",
"The minimizer to the above Lagrangian function (with respect to the variable $\\mathbf {s}$ ) is $\\mathbf {s} = P_{[\\mathbf {0}, \\mathbf {1}]} (\\mathbf {a} - \\mu \\mathbf {1} ),$ where $P_{[\\mathbf {0}, \\mathbf {1}]}$ is taken elementwise.",
"Besides the stationary condition (REF ), other KKT conditions for solving problem (REF ) are $& \\mu (\\mathbf {1}^\\top \\mathbf {s} - \\epsilon ) = 0 , \\\\& \\mu \\ge 0, \\\\& \\mathbf {1}^\\top \\mathbf {s} \\le \\epsilon .",
"$ If $\\mu > 0$ , then the solution to problem (REF ) is given by (REF ), where the dual variable $\\mu $ is determined by (REF ) and (REF ) $\\mathbf {1}^T P_{[\\mathbf {0}, \\mathbf {1}]} [\\mathbf {a} - \\mu \\mathbf {1}] = \\epsilon ,~\\text{and}~ \\mu > 0.$ If $\\mu = 0$ , then the solution to problem (REF ) is given by (REF ) and (), $\\mathbf {s} = P_{[\\mathbf {0}, \\mathbf {1}]} (\\mathbf {a}),~\\text{and}~ \\mathbf {1}^\\top \\mathbf {s} \\le \\epsilon ,$ The proof is complete.",
"$\\square $ In the projection operation (REF ), one might need to solve the scalar equation $\\mathbf {1}^T P_{[\\mathbf {0}, \\mathbf {1}]} [\\mathbf {a} - \\mu \\mathbf {1}] = \\epsilon $ with respect to the dual variable $\\mu $ .",
"This can be accomplished by applying the bisection method [2], [12] over $\\mu \\in [\\min (\\mathbf {a}-\\mathbf {1}), \\max (\\mathbf {a}) ]$ .",
"That is because $\\mathbf {1}^T P_{[\\mathbf {0}, \\mathbf {1}]} [\\mathbf {a} - \\max (\\mathbf {a}) \\mathbf {1}] \\le \\epsilon $ and $\\mathbf {1}^T P_{[\\mathbf {0}, \\mathbf {1}]} [\\mathbf {a} - \\min (\\mathbf {a} - \\mathbf {1}) \\mathbf {1} ] \\ge \\epsilon $ , where $\\max $ and $\\min $ return the largest and smallest entry of a vector.",
"We remark that the bisection method converges in the logarithmic rate given by $\\log _2 {[ (\\max (\\mathbf {a}) -\\min (\\mathbf {a}-\\mathbf {1}) )/\\xi ]}$ for the solution of $\\xi $ -error tolerance.",
"We summarize the PGD topology attack in Algorithm REF .",
"PGD topology attack on GNN [1] Input: $\\mathbf {s}^{(0)}$ , $\\epsilon > 0$ , learning rate $\\eta _t$ , and iterations $T$ $t = 1,2,\\ldots , T$ gradient descent: $\\mathbf {a}^{(t)} = \\mathbf {s}^{(t-1)} - \\eta _{t} \\nabla f(\\mathbf {s}^{(t-1)}) $ call projection operation in (REF ) call Algorithm REF to return $\\mathbf {s}^*$ , and the resulting $\\mathbf {A}^\\prime $ in (REF )."
],
[
"Min-max Topology Attack",
"We next solve the problem of min-max attack generation in (REF ).",
"By convex relaxation on the Boolean variables, we obtain the following continuous optimization problem $\\hspace*{-10.84006pt}\\begin{array}{cc}\\displaystyle \\operatornamewithlimits{\\text{minimize}}_{ \\mathbf {s}\\in \\mathcal {S} } \\displaystyle \\operatornamewithlimits{\\text{maximize}}_{\\mathbf {W}} \\, f(\\mathbf {s}, \\mathbf {W}) = \\sum _{i \\in \\mathcal {V}}f_i(\\mathbf {s} , \\mathbf {W} ; \\mathbf {A}, \\lbrace \\mathbf {x}_i\\rbrace , y_i ) ,\\end{array}$ where $\\mathcal {S}$ has been defined in (REF ).",
"We solve problem (REF ) by first-order alternating optimization [14], [15], where the inner maximization is solved by gradient ascent, and the outer minimization is handled by PGD same as (REF ).",
"We summarize the min-max topology attack in Algorithm REF .",
"We remark that one can perform multiple maximization steps within each iteration of alternating optimization.",
"This strikes a balance between the computation efficiency and the convergence accuracy [5], [18].",
"Min-max topology attack to solve (REF ) [1] Input: given $\\mathbf {W}^{(0)}$ , $\\mathbf {s}^{(0)}$ , learning rates $\\beta _t$ and $\\eta _t$ , and iteration numbers $T$ $t = 1,2,\\ldots , T$ inner maximization over $\\mathbf {W}$ : given $\\mathbf {s}^{(t-1)}$ , obtain $\\mathbf {W}^{t} = \\mathbf {W}^{t-1} + \\beta _t \\nabla _{\\mathbf {W}} f (\\mathbf {s}^{t-1},\\mathbf {W}^{t-1})$ outer minimization over $\\mathbf {s}$ : given $\\mathbf {W}^{(t)}$ , running PGD (REF ), where $\\hat{\\mathbf {g}}_t = \\nabla _{\\mathbf {s}} f (\\mathbf {s}^{t-1},\\mathbf {W}^{t})$ call Algorithm REF to return $\\mathbf {s}^*$ , and the resulting $\\mathbf {A}^\\prime $ in (REF )."
],
[
"Robust Training for GNNs",
"With the aid of first-order attack generation methods, we now introduce our adversarial training for GNNs via robust optimization.",
"Similar formulation is also used in [16].",
"In adversarial training, we solve a min-max problem for robust optimization: $\\begin{array}{cc}\\displaystyle \\operatornamewithlimits{\\text{minimize}}_{\\mathbf {W}} \\operatornamewithlimits{\\text{maximize}}_{ \\mathbf {s} \\in \\mathcal {S}} \\, -f(\\mathbf {s}, \\mathbf {W}) ,\\end{array}$ where $ f(\\mathbf {x}, \\mathbf {W})$ denotes the attack loss specified in (REF ).",
"Following the idea of adversarial training for image classifiers in [16], we restrict the loss function $f$ as the CE-type loss.",
"This formulation tries to minimize the training loss at the presence of topology perturbations.",
"We note that problems (REF ) and (REF ) share a very similar min-max form, however, they are not equivalent since the loss $f$ is neither convex with respect to $\\mathbf {s}$ nor concave with respect to $\\mathbf {W}$ , namely, lacking saddle point property [2].",
"However, there exists connection between (REF ) and (REF ); see Proposition REF .",
"Proposition 2 Given a general attack loss function $f$ , problem (REF ) is equivalent to $\\operatornamewithlimits{\\text{maximize}}_{\\mathbf {W}} \\operatornamewithlimits{\\text{minimize}}_{ \\mathbf {s} \\in \\mathcal {S}} \\, f(\\mathbf {s}, \\mathbf {W}),$ which further yields $(\\ref {eq: min_max_sol_same}) \\le (\\ref {eq: robust_attack_cont})$ .",
"Proof: By introducing epigraph variable $p$ [2], problem (REF ) can be rewritten as $\\begin{array}{ll}\\displaystyle \\operatornamewithlimits{\\text{minimize}}_{\\mathbf {W}, p} & p \\\\\\operatornamewithlimits{\\text{subject to}}& -f(\\mathbf {s}, \\mathbf {W}) \\le p, \\forall \\mathbf {s} \\in \\mathcal {S}.\\end{array}$ By changing variable $q \\mathrel {\\mathop :}=- p$ , problem (REF ) is equivalent to $\\begin{array}{ll}\\displaystyle \\operatornamewithlimits{\\text{maximize}}_{\\mathbf {W}, q} & q \\\\\\operatornamewithlimits{\\text{subject to}}&f(\\mathbf {s}, \\mathbf {W}) \\ge q, \\forall \\mathbf {s} \\in \\mathcal {S}.\\end{array}$ By eliminating the epigraph variable $q$ , problem (REF ) becomes (REF ).",
"By max-min inequality [2], we finally obtain that $\\operatornamewithlimits{\\text{maximize}}_{\\mathbf {W}} \\operatornamewithlimits{\\text{minimize}}_{ \\mathbf {s} \\in \\mathcal {S}} \\, f(\\mathbf {s}, \\mathbf {W}) \\le \\operatornamewithlimits{\\text{minimize}}_{ \\mathbf {s}\\in \\mathcal {S} } \\operatornamewithlimits{\\text{maximize}}_{\\mathbf {W}} \\, f(\\mathbf {s}, \\mathbf {W}).$ The proof is now complete.",
"$\\square $ We summarize the robust training algorithm in Algorithm for solving problem (REF ).",
"Similar to Algorithm REF , one usually performs multiple inner minimization steps (with respect to $\\mathbf {s}$ ) within each iteration $t$ to have a solution towards minimizer during alternating optimization.",
"This improves the stability of convergence in practice [18], [16].",
"Robust training for solving problem (REF ) [1] Input: given $\\mathbf {W}^{(0)}$ , $\\mathbf {s}^{(0)}$ , learning rates $\\beta _t$ and $\\eta _t$ , and iteration numbers $T$ $t = 1,2,\\ldots , T$ inner minimization over $\\mathbf {s}$ : given $\\mathbf {W}^{(t-1)}$ , running PGD (REF ), where $\\hat{\\mathbf {g}}_t = \\nabla _{\\mathbf {s}} f (\\mathbf {s}^{t-1},\\mathbf {W}^{t-1})$ outer maximization over $\\mathbf {W}$ : given $\\mathbf {s}^{(t)}$ , obtain $\\mathbf {W}^{t} = \\mathbf {W}^{t-1} + \\beta _t \\nabla _{\\mathbf {W}} f (\\mathbf {s}^{t},\\mathbf {W}^{t-1})$ return $\\mathbf {W}^{T}$ ."
],
[
"Experiments",
"In this section, we present our experimental results for both topology attack and defense methods on a graph convolutional networks (GCN) [10].",
"We demonstrate the misclassification rate and the convergence of the proposed 4 attack methods: negative cross-entropy loss via PGD attack (CE-PGD), CW loss via PGD attack (CW-PGD), negative cross-entropy loss via min-max attack (CE-min-max), CW loss via min-max attack (CW-min-max).",
"We then show the improved robustness of GCN by leveraging our proposed robust training against topology attacks."
],
[
"Experimental Setup",
"We evaluate our methods on two well-known datasets: Cora and Citeseer [19].",
"Both datasets contain unweighted edges which can be generated as symmetric adjacency matrix $\\mathbf {A} $ and sparse bag-of-words feature vectors which can be treated the input of GCN.",
"To train the model, all node feature vectors are fed into GCN but with only 140 and 120 labeled nodes for Cora and Citeseer, respectively.",
"The number of test labeled nodes is 1000 for both datasets.",
"At each experiment, we repeat 5 times based on different splits of training/testing nodes and report mean $\\pm $ standard deviation of misclassification rate (namely, 1 $-$ prediction accuracy) on testing nodes."
],
[
"Attack Performance",
"We compare our four attack methods (CE-PGD, CW-PGD, CE-min-max, CW-min-max) with DICE (`delete edges internally, connect externally’) [21], Meta-Self attack [26] and greedy attack, a variant of Meta-Self attack without weight re-training for GCN.",
"The greedy attack is considered as a fair comparison with our CE-PGD and CW-PGD attacks, which are generated on a fixed GCN without weight re-training.",
"In min-max attacks (CE-min-max and CW-min-max), we show misclassification rates against both natural and retrained models from Algorithm REF , and compare them with the state-of-the-art Meta-Self attack.",
"For a fair comparison, we use the same performance evaluation criterion in Meta-Self, testing nodes' predicted labels (not their ground-truth label) by an independent pre-trained model that can be used during the attack.",
"In the attack problems (REF ) and (REF ), unless specified otherwise the maximum number of perturbed edges is set to be $5\\%$ of the total number of existing edges in the original graph.",
"In Algorithm REF , we set the iteration number of random sampling as $K = 20$ and choose the perturbed topology with the highest misclassification rate which also satisfies the edge perturbation constraint.",
"In Table REF , we present the misclassification rate of different attack methods against both natural and retrained model from (REF ).",
"Here we recall that the retrained model arises due to the scenario of attacking an interactive GCN with re-trainable weights (Algorithm REF ).",
"For comparison, we also show the misclassification rate of a natural model with the true topology (denoted by `clean').",
"As we can see, to attack the natural model, our proposed attacks achieve better misclassification rate than the existing methods.",
"We also observe that compared to min-max attacks (CE-min-max and CW-min-max), CE-PGD and CW-PGD yield better attacking performance since it is easier to attack a pre-defined GCN.",
"To attack the model that allows retraining, we set 20 steps of inner maximization per iteration of Algorithm REF .",
"The results show that our proposed min-max attack achieves very competitive performance compared to Meta-Self attack.",
"Note that evaluating the attack performance on the retrained model obtained from (REF ) is not quite fair since the retrained weights could be sub-optimal and induce degradation in classification.",
"Table: Misclassification rates (%\\%) under 5%5\\% perturbed edgesIn Fig.",
"REF , we present the CE-loss and the CW-loss of the proposed topology attacks against the number of iterations in Algorithm REF .",
"Here we choose $T = 200$ and $\\eta _t = 200/\\sqrt{t}$ .",
"As we can see, the method of PGD converges gracefully against iterations.",
"This verifies the effectiveness of the first-order optimization based attack generation method.",
"Figure: CE-PGD and CW-PGD attack losses on Cora and Citeseer datasets."
],
[
"Defense Performance",
"In what follows, we invoke Algorithm to generate robust GCN via adversarial training.",
"We set $T = 1000$ , $\\beta _t = 0.01$ and $\\eta _t = 200/\\sqrt{t}$ .",
"We run 20 steps for inner minimization.",
"Inspired by [16], we increase the hidden units from 16 to 32 in order to create more capacity for this more complicated classifier.",
"Initially, we set the maximum number of edges we can modify as $5\\%$ of total existing edges.",
"In Figure REF , we present convergence of our robust training.",
"As we can see, the loss drops reasonably and the $1,000$ iterations are necessary for robust training rather than normal training process which only need 200 iterations.",
"We also observe that our robust training algorithm does not harm the test accuracy when $\\epsilon = 5\\%$ , but successfully improves the robustness as the attack success rate drops from $28.0\\%$ to $22.0\\%$ in Cora dataset as shown in Table REF , After showing the effectiveness of our algorithm, we explore deeper in adversarial training on GCN.",
"We aim to show how large $\\epsilon $ we can use in robust training.",
"So we set $\\epsilon $ from $5\\%$ to $20\\%$ and apply CE-PGD attack following the same $\\epsilon $ setting.",
"The results are presented in Table REF .",
"Note that when $\\epsilon =0$ , the first row shows misclassification rates of test nodes on natural graph as the baseline for lowest misclassification rate we can obtain; the first column shows the CE-PGD attack misclassification rates of natural model as the baseline for highest misclassification rate we can obtain.",
"We can conclude that when a robust model trained under an $\\epsilon $ constraint, the model will gain robustness under this $\\epsilon $ distinctly.",
"Considering its importance to keep the original graph test performance, we suggest generating robust model under $\\epsilon = 0.1$ .",
"Moreover, please refer to Figure REF that a) our robust trained model can provide universal defense to CE-PGD, CW-PGD and Greedy attacks; b) when increasing $\\epsilon $ , the difference between both test accuracy and CE-PGD attack accuracy increases substantially, which also implies the robust model under larger $\\epsilon $ is harder to obtain.",
"Table: Misclassification rates (%\\%) of robust training (smaller is better for defense task) with at most 5%5\\% of edge perturbations.",
"𝐀\\mathbf {A} means the natural graph, 𝐀 ' \\mathbf {A}^\\prime means the generated adversarial graph under ϵ=5%\\epsilon = 5\\%.",
"𝐗/M\\mathbf {X} / M means the misclassification rate of using model MM on graph 𝐗\\mathbf {X}.Table: Misclassification rates (%\\%) of CE-PGD attack against robust training model versus (smaller is better) different ϵ\\epsilon (%\\%) on Cora dataset.",
"Hereϵ=0\\epsilon =0 in training means natural model and ϵ=0\\epsilon =0 in attack means unperturbed topology.Figure: Robust training loss on Cora and Citeseer datasets.Figure: Test accuracy of robust model (no attack),CE-PGD attack against robust model, CW-PGD attack against robust model, Greedy attack against robust model and CE-PGD attack against natural model for different ϵ\\epsilon used in robust training and test on Cora dataset."
],
[
"Conclusion",
"In this paper, we first introduce an edge perturbation based topology attack framework that overcomes the difficulty of attacking discrete graph structure data from a first-order optimization perspective.",
"Our extensive experiments show that with only a fraction of edges changed, we are able to compromise state-of-the-art graph neural networks model noticeably.",
"Additionally, we propose an adversarial training framework to improve the robustness of GNN models based on our attack methods.",
"Experiments on different datasets show that our method is able to improve the GNN model's robustness against both gradient based and greedy search based attack methods without classification performance drop on original graph.",
"We believe that this paper provides potential means for theoretical study and improvement of the robustness of deep learning models on graph data."
],
[
"Acknowledgments",
"This work is supported by Air Force Research Laboratory FA8750-18-2-0058 and the MIT-IBM Watson AI Lab."
]
] | 1906.04214 |
[
[
"Introducing the Hearthstone-AI Competition"
],
[
"Abstract The Hearthstone AI framework and competition motivates the development of artificial intelligence agents that can play collectible card games.",
"A special feature of those games is the high variety of cards, which can be chosen by the players to create their own decks.",
"In contrast to simpler card games, the value of many cards is determined by their possible synergies.",
"The vast amount of possible decks, the randomness of the game, as well as the restricted information during the player's turn offer quite a hard challenge for the development of game-playing agents.",
"This short paper introduces the competition framework and goes into more detail on the problems and challenges that need to be faced during the development process."
],
[
"Introduction",
"The development of artificial intelligence (AI) was often guided by the plethora of available real world applications.",
"The recent success of AI agents such as AlphaGo, brought game AI back into the center of attention for media and researchers alike.",
"Games pose challenging and often well-balanced problems demanding the development of new agent architectures and allowing their evaluation based on their success in playing the game.",
"Due to their competitive nature, games offer an ideal test-bed for the comparison of multiple agents under differing conditions.",
"Game based competitions and benchmarks motivated researchers to create specialized agents in games of many different genres, such as Chess [1], Go [2], Poker [3], Pac-Man [4], Starcraft [5] and many more.",
"While these focused on the development of an agent for a single game, other competitions and frameworks tried to generalize the solutions to a broader scope.",
"The Arcade Learning Environment framework (ALE) [6] as well as the General Video Game AI framework (GVGAI) [7] test an agent's success on a wide range of games.",
"All of these benchmarks pose unique demands on the agent's planning and reasoning capabilities, while there accessibility ensures a simple generation and evaluation of new agents.",
"In this paper we introduce the Hearthstone AI competition, because we believe that it is an excellent addition to the set of currently available benchmarks.",
"The focus of this competition is the development of autonomously playing agents in the context of the online collectible card game Hearthstone.",
"Interesting features of collectible card games include, but are not limited to: Partial observable state space: Critical information is hidden from the player.",
"The agent typically does not know which cards it will draw and is unaware of the opponent's deck and hand cards.",
"This is especially relevant when estimating the risk of an action, since this often depends on the current options of our opponent.",
"High complexity: Hearthstone currently features more than 2000 different cards.",
"This high amount and the number of their unique effects drastically increases the game-tree complexity.",
"Randomness: In contrast to similar deck-building card-games, card effects in Hearthstone often involve randomness.",
"This makes it particularly difficult to plan ahead, such that the agent needs to continuously adapt its strategy according to the observed result.",
"Deck-building: A deck in Hearthstone consists of 30 cards, which do not necessarily need to be unique.",
"Card synergies mean that certain combinations of cards often have a stronger effect than playing the respective cards separately.",
"Exploiting these synergies is a very complex task but allows to play cards to their full potential.",
"Dynamic Meta-Game: The odds of winning the next game depend not only on the player's skill or the current deck, but also on the rate of other decks being played.",
"The analysis of this meta-game can be crucial for creating or choosing the next deck to be played.",
"Finally, a deep understanding of the meta-game can be used to predict future enemy moves based on previously seen cards and select appropriate actions based on them.",
"In the following we will give a short introduction to Hearthstone (hearthstone) and present the competition framework (competition).",
"Competition tracks of the Hearthstone AI'19 competition are highlighted in after which an outline of future competition tracks is presented (competition-tracks).",
"More information on the competition, results of the 2018's installment and additional resources for research on Hearthstone can be found at: ....... http://www.ci.ovgu.de/Research/HearthstoneAI.html We invite all developers to participate in this exciting research topic and hope to receive many interesting submissions.",
"Figure: Elements of the Hearthstone game board: (1) weapon slot (2) hero (bottom: player, top: opponent) (3) opponent's minions, (4) player's minions, (5) hero power, (6) hand cards, (7) mana, (8) decks, (9) history"
],
[
"Hearthstone: Heroes of Warcraft",
"Hearthstone is a turn-based digital collectible card game developed and published by Blizzard Entertainment [8].",
"Players compete in one versus one duels using self-constructed decks belonging to one hero out of nine available hero-classes.",
"In those matches players try to beat their opponents by reducing their starting health from 30 to 0.",
"This can be achieved by playing cards from the hand onto the game board at the cost of mana.",
"Played cards can be used to inflict damage to the opponent's hero or to destroy cards on his side of the game board.",
"The amount of mana available to the player increases every turn (up to a maximum of 10).",
"More mana gives access to increasingly powerful cards and increases the complexity of turn while the game progresses.",
"At the beginning of each turn the player draws a new card until his deck is empty, in which case he receives a step-wise increasing amount of fatigue-damage.",
"The standard game board is shown in fig:game-board.",
"Players need to construct their own decks to play the game.",
"Those consist of 30 cards, which can be chosen out of more than 2000 currently available cards.",
"Cards and game mechanics are added in regular updates.",
"Each card bears unique effects, which the players can use to their advantage.",
"Additionally, each player chooses a hero, which gives access to a class specific pool of cards and hero power.",
"Figure: General card types: cards include (1) mana cost, (2) attack damage, (3) health/durability, (4) and special effects.Cards can be of the type minion, spell, or weapon.",
"fig:cardtypes shows one example of each card type.",
"Minion cards assist and fight on behalf of the hero.",
"They usually have an attack, health, and mana cost-value, as well as a short ability text.",
"Furthermore, minions can belong to a special minion type, which is the basis for many synergy effects.",
"Once played, they can attack the enemies side of the board in every consecutive turn to inflict damage on either the opponent's minions or hero.",
"Attacking a target also reduces the attacker's health by the target's attack value.",
"In case any minion's health drops to zero, it is removed from the board and put into its player's graveyard.",
"Spell cards can be cast at the cost of mana to activate various abilities and are discarded after use.",
"They can have a wide range of effects, e.g.",
"raising a minion's attack or inflicting damage to a random minion.",
"Secrets, which are a special kind of spells, can be played without immediately activating their effect.",
"After a trigger condition was fulfilled, the secret will be activated.",
"Once activated, the secret is removed from the board.",
"Weapon cards are directly equipped to the player's hero and enable him to attack.",
"Their durability value limits the number of attacks till the weapon breaks.",
"Only one weapon can be equipped at the same time.",
"Hearthstone decks are often created around a common theme.",
"Multiple cards that positively influence each other can create strong synergies and increase the value of each card in context of its deck.",
"For this reason, the value of a single card highly depends on the player's hand, current elements on the board, and the deck in general.",
"Common examples are minion cards of the same type, e.g.",
"”Murloc“, which give each other additional advantages, e.g.",
"an attack boost.",
"Each of these minions is comparatively weak, but their value increases when they are played together.",
"Generated decks can be categorized into three major categories: aggro, mid-range, and control.",
"Aggro decks build on purely offensive strategies, which often include a lot of minions.",
"Control decks try to win in the long run by preventing the opponent's strategy and dominating the game situation.",
"The playing style of mid-range decks is between aggro and control.",
"They try to counter early attacks to dominate the game board with high-cost minions in the middle of the game.",
"Game length and branching factor can be dependent on the player's decks in the current game.",
"Some decks try to play single high-cost cards, whereas others build on versatile combinations.",
"The complexity of each turn and the uncertainty faced during the game makes Hearthstone a challenging problem for AI research."
],
[
"Hearthstone-AI Competition Framework",
"The Hearthstone-AI competition is based on the community driven simulator Sabberstone.",
"This framework is written in C# and the competition extends the original framework by multiple helper classes which provide an agent simple means of accessing the current game state limited to variables that would have been observable to a human player.",
"More specifically, each agent needs to inherit from the AbstractAgent class.",
"The included functions InitializeAgent and FinalizeAgent can be used to load and store information at the beginning and end of each session.",
"Additionally, InitializeGame and FinalizeGame are called at the beginning and end of each simulated game.",
"These functions can be used to setup a strategy or updating it based on the games' outcome.",
"During a game each time the agent needs to choose an action its GetMove function is called.",
"The agent is given a POGame object representing the partial observation of the current game-state.",
"It contains information about the visible part of the game board, a set of remaining cards in its deck, its hand cards, and the number of cards in their opponents' hand.",
"Furthermore, the opponent's deck as well as its hand cards are replaced by dummy-cards to ensure that this information remains hidden to the agent.",
"The agent is ensured 60s of computation time to step-wise return a set of actions and concluding its turn.",
"In case the turn was not ended by the agent, the returned action will be processed irreversibly and an updated game-state will be returned to the agent while asking for its next action.",
"Actions of both players are applied until a winner can be determined or a maximum number of turns (default = 50) is exceeded.",
"In the latter the game ends with a draw.",
"The POGameHandler class controls the simulation of multiple games and reports the result of these simulations in terms of a GameStats object.",
"The number of wins, draws, and loses as well as the total and average response times per agent are tracked and reported at the end of a simulation session."
],
[
"Competition Tracks",
"During the first years of the Hearthstone-AI competition two tracks will be open for entry.",
"We plan to extend this list in the following years to give users some time to accommodate with the framework and the game itself.",
"The following two tracks will be open for submission in the Hearthstone-AI'19: Premade Deck Playing”-track In the “Premade Deck Playing”-track participants will receive a list of six decks and play out all combinations against each other.",
"Only three of the six decks will be known to the developers before the final submission.",
"Determining and using the characteristics of player’s and the opponent’s deck to the player’s advantage will help in winning the game.",
"The long-term goal of this track will be the development of an agent that is capable of playing any deck.",
"User Created Deck Playing-track The “User Created Deck Playing”-track invites all participants to create their own decks or to choose from the vast amount of decks available online.",
"Finding combinations of decks and agents that can consistently beat others will play a key role in this competition track.",
"Additionally, it gives the participants the chance to optimize the agent’s strategy to the characteristics of their chosen deck.",
"A round robin tournament will be used to determine the average win-rate of each agent and rank them accordingly.",
"In case this process becomes unfeasible due to a large number of participants, we will split submissions into multiple sub-tournaments to determine the best performing agents among them and use a round robin tournament to determine the winner of the competition.",
"Matches will be repeated multiple times to accommodate for the randomness in the card draw.",
"In order to support incremental improvements of the agents' performance, we plan to make all submissions publicly available on the competition website after the competition evaluation has been completed."
],
[
"Conclusions and Future Plans",
"Previous work on collectible card games has been scattered on different games and frameworks.",
"With this competition we want to provide a unified way to develop and compare AI approaches on multiple collectible card game related tasks.",
"While current competition tracks focus on the agents' basic game playing capabilties, we like to cover various different tasks in future installments of this competition.",
"Specifically, we plan to implement the following future tracks: Deck-building: As soon as agents are reasonably skilled in playing a given deck we want to further explore the deck building process.",
"This is currently considered a highly creative task in which many game characteristics need to be considered to build outstanding decks.",
"Draft mode deck-building: A special form of deck building is the draft mode.",
"Here, an agent is presented 3 cards at a time of which it needs to choose one to include it in its deck.",
"This process is repeated until the deck is filled.",
"Strategic planning and estimating the value of each card are important to exploit synergies and building a competitive deck.",
"Game balancing/Card generation: During the development process of a collectible card game much of the time is put into the generation of new game mechanics and cards, since these need to fit and complement the current card pool.",
"A future track will aim to explore these balancing tasks in more detail.",
"We hope that this short introduction of our competition framework motivates further research in this very interesting topic.",
"More information on how to participate in the Hearthstone-AI'19 competition, results of the 2018's installment and additional resources for research on Hearthstone can be found at: ....... http://www.ci.ovgu.de/Research/HearthstoneAI.html"
],
[
"Acknowledgement",
"We would like to thank all who contributed to the Sabberstone framework on which this competition is based on.",
"Special thanks goes to darkfriend77 and Milva who are currently organizing the framework's ongoing development process.",
"This competition would not have been possible without them.",
"[Figure: NO_CAPTION [Figure: NO_CAPTION"
]
] | 1906.04238 |
[
[
"Metrics for Learning in Topological Persistence"
],
[
"Abstract Persistent homology analysis provides means to capture the connectivity structure of data sets in various dimensions.",
"On the mathematical level, by defining a metric between the objects that persistence attaches to data sets, we can stabilize invariants characterizing these objects.",
"We outline how so called contour functions induce relevant metrics for stabilizing the rank invariant.",
"On the practical level, the stable ranks are used as fingerprints for data.",
"Different choices of contour lead to different stable ranks and the topological learning is then the question of finding the optimal contour.",
"We outline our analysis pipeline and show how it can enhance classification of physical activities data.",
"As our main application we study how stable ranks and contours provide robust descriptors of spatial patterns of atmospheric cloud fields."
],
[
"Modelling data spaces",
"Topological data analysis (TDA) and particularly its subfield persistent homology, or persistence, aim at quantifying the global connectivity structure of data sets [1], [2], [3].",
"Given a set of data points it is often possible to endow it with some reasonable notion of relation between points, e.g.",
"distance measure or correlation.",
"Study of the connectivity is facilitated by first combining points into larger entities called simplices.",
"A $k$ -simplex is a declared subset of $k+1$ related points from the data set.",
"Collection of simplices makes up a simplicial complex $C$ , namely it is a collection of certain subsets of the data.",
"Requirements are that if $\\sigma $ is a simplex in $C$ then any subset of $\\sigma $ is also a simplex in $C$ and that the intersection of two simplices is a simplex or the empty set.",
"Above we have described an abstract simplicial complex.",
"Simplices can always be realized geometrically in some $\\mathbf {R}^n$ as convex hulls of their vertices: 0-simplices as points, 1-simplices as line segments, 2-simplices as filled triangles, 3-simplices as filled tetrahedra etc.",
"Simplicial complex is hence a model of the relational structure in the data.",
"Relational structure can be modelled by a graph but graphs only consider pairwise relations between points.",
"In many cases it makes sense to use higher-dimensional connectivity instead modelled with simplices.",
"As a justification consider the example explained in Fig.",
"REF .",
"More fundamental reason is that the simpicial approach views data as spaces spanned by their points and enables the use of powerful mathematical machinery of algebraic topology for the analysis of these spaces, as will be outlined in the following section.",
"Figure: Simplicial model of social relations.",
"To model relations between k+1k+1 points it is reasonable to use kk-simplex for the purpose.",
"Here the relations between {Arya, Bran, Jon} is depicted by the 2-simplex represented by the purple triangle.",
"From the point of view of TDA the prominent feature of this data is the single loop structure, whereas a graph would see two loops (the closed path (Arya, Bran, Jon) spanning the other loop in this case).For persistence analysis we define the relation to be a function $R$ on the data with values in $\\mathbf {R}=[0,\\infty )$ , i.e.",
"$R(x,y) \\mapsto t \\in \\mathbf {R}$ for data points $x$ and $y$ .",
"Concretely we say that $k+1$ data points $x_i$ create a $k$ -simplex at scale $t$ if the points satisfy pairwise $R(x_i,x_j) \\le t.$ This construction is called the Vietoris-Rips simplicial complex at scale $t$ .",
"At fixed scale we can then study the connectivity structure.",
"As a standard example, when data is endowed with a distance measure, clustering at some fixed scale corresponds to the 0-dimensional connectivity by only looking at the connected components of the simplicial complex.",
"Simplicial complexes can also contain 1-dimensional connectivity information in the form of loops and holes (see Fig.",
"REF ), 2-dimensional information in the form of voids or cavities, etc.",
"These are collectively called topological features.",
"Persistence aims to quantify the topological features in a data set and use this information for data analysis.",
"Loop structure might signal about a recurrent dynamics of the phenomenon behind the data.",
"Various dimensional voids can mark lack of information and connectivity or insufficient data collection.",
"Finding such voids in data sets has aroused interest in different areas of data analysis community, see for example [4] and references therein.",
"As noted in [4], voids can also indicate non-allowed combinations of feature values of data vectors.",
"One immediate difficulty arises in the simplicial modelling above: what is the appropriate scale of $R$ to capture the connectivity in various dimensions of an arbitrary set of points?",
"Persistence circumvents this by forming simplicial complexes at all scales $t \\ge 0$ and capturing the evolution of topological features.",
"If a simplex is generated at scale $t$ it is then present at any subsequent scale and the simplicial complexes are connected by inclusions: $\\cdots \\subseteq C_a \\subseteq C_b \\subseteq C_c \\subseteq \\cdots $ for $\\dots \\le a \\le b \\le c \\le \\dots $ The end result of the modelling step is then a mapping called filtration, $(D,R) \\times \\mathbf {R} \\rightarrow (C_t,\\subseteq _t)_{t \\in \\mathbf {R}}$ , where $(D,R)$ denotes a data set with real-valued relation and $(C_t,\\subseteq _t)_{t \\in \\mathbf {R}}$ denotes an $\\mathbf {R}$ -parameterized sequence of simplicial complexes and inclusions."
],
[
"Algebraic fingerprinting",
"Filtration contains all the information about the relations in the data set on various scales.",
"It is therefore very complicated object for infering the global structure of data and simplification is thus necessary.",
"TDA employs tools from mathematical field of algebraic topology, essentially it uses homology of simplicial complexes which transforms the geometric information into algebraic information.",
"We will outline the algorithm for computing homology to illustrate its very implementable nature and to gain intuition on why we are interested in homology in data analysis.",
"For details into homology and its computation see [5], [6], [7].",
"For simplicity we fix the field of coefficients to be $\\mathbf {F}_2$ , the field with two elements 0 and 1.",
"Let $C$ be a simplicial complex and denote by $C_k$ its set of $k$ -simplices.",
"Concretely, $C_0$ consists of the points of the original data set.",
"1) Choose an ordering (starting from zero) on $C_0$ and use it to order elements in any simplex.",
"If $\\lbrace \\text{Arya}, \\text{Bran}, \\text{Jon}\\rbrace $ is a 2-simplex in Fig.",
"REF , fix the order in which the points are listed and denote this ordered simplex by $[\\text{Arya}, \\text{Bran}, \\text{Jon}].$ 2) For natural numbers $k$ and $0\\le i\\le k$ and a simplex $\\sigma $ in $C_k$ , define a function $d_i \\colon C_k \\rightarrow C_{k-1}$ such that $d_i(\\sigma )$ is a simplex in $C_{k-1}$ formed by removing from $\\sigma $ its $i$ -th element.",
"The ordering on $C_0$ was needed to specify the $i$ -th element in a simplex.",
"For example, $d_1([\\text{Arya}, \\text{Bran}, \\text{Jon}]) = [\\text{Arya}, \\text{Jon}].$ 3) For any natural number $k$ , let $\\Delta (C)_k$ be the vector space over $\\mathbf {F}_2$ with a base given by all simplices in $C_k$ .",
"An element $\\tau $ in $\\Delta (C)_k$ is then given by a linear combination $\\tau = \\sum _{\\sigma \\in C_k} t_\\sigma \\sigma , \\ t_\\sigma \\in \\mathbf {F}_2.$ The base for $\\Delta (C)_2$ of the simplicial complex in Fig.",
"REF would be [Arya, Bran, Jon] whereas [Arya, Bran]+[Bran, Jon]+[Arya, Jon] would be linear combination of three basis elements in $\\Delta (C)_1$ .",
"4) Define $\\partial _k \\colon \\Delta (C)_k \\rightarrow \\Delta (C)_{k-1}$ to be the linear function assigning to a base element given by a simplex $\\sigma $ in $C_k$ the linear combination $\\sum _{i=0}^{k}d_i(\\sigma )$ of $k$ -1-simplices.",
"The map $\\partial _k$ is called the boundary operator.",
"Then $\\partial _k([\\text{Arya}, \\text{Bran}, \\text{Jon}]) = [\\text{Bran}, \\text{Jon}]+[\\text{Arya}, \\text{Jon}]+[\\text{Arya}, \\text{Bran}].$ The boundary operator thus formalizes the intuition that $[\\text{Bran}, \\text{Jon}]+[\\text{Arya}, \\text{Jon}]+[\\text{Arya}, \\text{Bran}]$ forms the boundary of $[\\text{Arya}, \\text{Bran}, \\text{Jon}].$ Define $\\Delta (C)_{-1} = 0$ and $\\Delta (C)_k = 0$ for $k>m$ , where 0 denotes the zero vector space.",
"5) The boundary operators connect the various simplices of a simplicial complex together.",
"Computationally the matrices of boundary operators store the global connectivity information in their elements, with coefficient field $\\mathbf {F}_2$ these are just binary matrices.",
"Homology on degree $k$ of a simplicial complex $C$ (over coefficients $\\mathbf {F}_2$ ) is then defined as a quotient vector space: $H_k(C)= \\frac{\\text{kernel of }\\partial _k\\colon \\Delta (C)_k \\rightarrow \\Delta (C)_{k-1}}{\\text{image of }\\partial _{k+1}\\colon \\Delta (C)_{k+1}\\rightarrow \\Delta (C)_{k}}, \\ \\ \\text{for } k \\ge 0.$ As noted in step 4) above, some 1-simplices might form the boundary of a 2-simplex.",
"Some 1-simplices on the other hand might form the boundary of an actual hole in the simplicial complex as in Fig.",
"REF .",
"Similarly some $k$ -1-simplices might form the boundary of a $k$ -simplex and some might form the boundary of a $k$ -dimensional hole.",
"By its definition homology quotients out linear combinations of simplices that are boundaries and we are left with those that actually represent linearly independent $k$ -dimensional holes in the complex.",
"For $k=0$ , $H_0$ measures the number of linearly independent points that make up boundaries of 1-simplices, effectively the number of connected components.",
"Homology thus gives us exactly the global connectivity information of the relational structure of data that we seek.",
"The full complexity of a filtration is now simplified by applying homology on degree $k$ .",
"Each simplicial complex is turned into a homology vector space and the inclusion functions are turned into linear maps.",
"The result is an $\\mathbf {R}$ -parameterized sequence of vector spaces and linear maps: $\\cdots \\rightarrow H_k(C_a) \\rightarrow H_k(C_b) \\rightarrow H_k(C_c) \\rightarrow \\cdots .$ We will abbreviate $H_k(C_a)$ as $H_{k,a}$ .",
"In this parameterized sequence the dimensions of homology vector spaces encode topological information: $H_{0,t}$ effectively measuring the number of connected components, $H_{1,t}$ measuring the number of one-dimensional holes and $H_{k,t}$ those of $k$ -dimensional voids at scale $t$ .",
"This algebraic step gives a mapping $(C_t,\\subseteq _t)_{t \\in \\mathbf {R}} \\rightarrow (H_{k,t},\\rightarrow _t)_{t \\in \\mathbf {R}}.$ The obtained result is not an arbitrary $\\mathbf {R}$ -parameterized vector space.",
"The vector spaces $H_{k,t}$ are finite dimensional and there are finitely many numbers $0<t_0<\\cdots <t_n$ in $\\mathbf {R}$ such that the map $H_{k,a}\\rightarrow H_{k,b}$ may not be an isomorphism only if $a<t_i\\le b$ , for $i$ in $\\lbrace 0,\\dots ,n\\rbrace $ .",
"These considerations follow from the fact that data sets always contain only finite number of points so topological changes in the relational structure can only occur in discrete steps.",
"Such parameterized vector spaces are called tame [8].",
"An essential result in persistence theory is that any tame $\\mathbf {R}$ -parameterized vector space decomposes into interval indecomposables called bars and the collection of bars in such a decomposition is unique [9].",
"Bars are enumerated by pairs of numbers $b<d$ in $\\mathbf {R}$ .",
"The bar $[b,d)$ at scale $t$ is either a one dimensional vector space, if $b\\le t<d$ , and the zero vector space otherwise.",
"The maps between any non-zero vector spaces in a bar are isomorphisms.",
"For a bar $[b,d)$ , some topological feature is understood to have appeared in the simplicial complex at filtration value $b$ .",
"It is then present in the subsequent simplicial complexes until filtration value $d$ .",
"For example, points in the data might connect to create a 1-dimensional loop.",
"This loop persists until at some larger filtration value the points connect further to higher dimensional simplices and the loop vanishes.",
"The bar decomposition can be visualized in a stem plot on a $(b,d-b)$ -coordinate system as shown later in Fig.",
"REF .",
"The actual data analysis step in persistence pipeline is to infer information from the $\\mathbf {R}$ -parameterized sequence of homology vector spaces and linear maps obtained from the map $(D,R) \\times \\mathbf {R} \\rightarrow (C_t,\\subseteq _t)_{t \\in \\mathbf {R}} \\rightarrow (H_{k,t},\\rightarrow _t)_{t \\in \\mathbf {R}}$ constructed above.",
"To simplify notation we let $\\mathbf {R}$ -Vec denote the space of tame $\\mathbf {R}$ -parameterized sequences of vector spaces $V=\\cdots \\rightarrow V_a \\rightarrow V_b \\rightarrow V_c \\rightarrow \\cdots .$ Our framework of extracting information from objects in this space is through stabilizing a rank invariant attached to them.",
"Aim of the paper is on the practical data analysis aspects and we only outline the theoretical backgound.",
"For more details we refer to [8], [14], [15]."
],
[
"Rank invariant",
"The rank, or the dimension, is the fundamental invariant characterizing vector spaces.",
"Similarly we want to assign rank for sequences of vector spaces in $\\mathbf {R}$ -Vec.",
"Let $V$ be in $\\mathbf {R}$ -Vec.",
"Due to tameness there is a sequence $0 < t_0 < \\cdots < t_k$ in $\\mathbf {R}$ such that $V_a \\rightarrow V_b$ is not an isomorphism only if $a < t_i \\le b$ .",
"Recall that for a linear map $f \\colon X \\rightarrow Y$ its cokernel is the quotient vector space of $Y$ by the image of $f$ : $\\text{coker} f = Y / \\text{im}f.$ We then define $\\beta _0(V) =V_0 \\oplus \\text{coker}(V_0 \\rightarrow V_{t_0}) \\oplus \\text{coker}(V_{t_0} \\rightarrow V_{t_1}) \\cdots \\oplus \\text{coker}(V_{t_{k-1}} \\rightarrow V_{t_k}),$ where $V_0$ is the homology vector space in $V$ at filtration value 0.",
"Let us consider what information $\\beta _0(V)$ carries.",
"Since the maps $V_{t_i} \\rightarrow V_{t_{i+1}}$ are not isomorphisms the cokernels may not be zero.",
"The quotient by the image removes from the homology vector space $V_{t_{i+1}}$ the generators, or basis elements, which come from previous non-isomorphic homology vector space.",
"$\\beta _0$ is thus a vector space of the new homology generators that appear in the sequence of homology vector spaces.",
"In the context of filtrations of input data sets, this is a way of keeping track of how topological features created by the relational structure evolve in the simplicial complexes of the filtration.",
"For $V$ in $\\mathbf {R}$ -Vec, its rank is now defined to be a discrete invariant given by the number $\\text{rank}(V)&=\\text{dim}(\\beta _0(V))=\\\\\\text{dim}(V_0) + \\text{dim(coker}(V_0 \\rightarrow & V_{t_0})) + \\cdots + \\text{dim(coker}(V_{t_{k-1}} \\rightarrow V_{t_k})).$"
],
[
"Hierarchical stabilization and contour metrics",
"The rank defined above is not a stable invariant.",
"Effectively the number $\\text{rank}(V)$ measures the smallest number of homology generators of $V$ .",
"A small perturbation of input data can result in a number of non-essential homology generators.",
"We therefore seek to stabilize the rank invariant to deal with inherent noise in data.",
"Our approach is a general framework for stabilizing discrete invariants.",
"Let $T$ be a set of interesting objects and $I$ the attached invariant.",
"For us $T$ is of course a collection of $\\mathbf {R}$ -parameterized vector spaces associated to data sets with $\\mathbf {R}$ -valued relation and $I$ is the rank.",
"The key in converting a discrete invariant into a stable one is to choose a (pseudo)metric $d$ on $T$ .",
"Once a metric is chosen, we can define an $\\varepsilon $ -radius ball around $X \\in T$ , $B(X,\\varepsilon )= \\lbrace Y \\ |\\ d(X,Y)\\le \\varepsilon \\rbrace $ , and look at the function $\\widehat{I}_d(X)$ taking the minimum value of $I$ on balls around $X$ with increasing radii $\\varepsilon $ : $\\widehat{I}_d(X)(\\varepsilon )= \\text{min}\\lbrace I(Y)\\ |\\ Y \\in B(X,\\varepsilon )\\rbrace .$ Since we are minimizing the invariant in larger and larger balls around $X$ , the function $\\widehat{I}_d(X)$ is decreasing and piecewise constant, namely a simple function.",
"Due to being a decreasing function with non-negative values, there is some $t$ such that for all $s \\ge t$ in $\\mathbf {R}$ , $\\widehat{I}_d(X)(s) = \\widehat{I}_d(X)(t)$ .",
"The function $\\widehat{I}_d(X)$ is thus eventually constant with a limit, $\\text{lim}\\ \\widehat{I}_d(X)$ .",
"The needed metrics in the stabilization can be shown [15] to arise from so called contours.",
"Contour is function $C: \\mathbf {R} \\times \\mathbf {R} \\rightarrow \\mathbf {R}$ satisfying the following inequalities for all $v,w,\\varepsilon ,\\tau $ in $\\mathbf {R}$ : $v \\le C(v,\\varepsilon ) \\le C(w,\\tau )$ , for $v \\le w$ and $\\varepsilon \\le \\tau ,$ $C(C(v,\\varepsilon ),\\tau ) \\le C(v,\\varepsilon + \\tau )$ .",
"For example, $C(v,\\varepsilon ) = v + \\varepsilon $ , $C(v,\\varepsilon ) = v + \\varepsilon ^2$ and $C(v,\\varepsilon ) =r^\\varepsilon v$ with a positive number $r$ are all examples of contours.",
"The contour $C(v,\\varepsilon ) = v + \\varepsilon $ is called the standard contour.",
"There is a generic way of producing contours.",
"Let $f \\colon \\mathbf {R} \\rightarrow (0,\\infty )$ be a function with strictly positive values which we refer to as density.",
"Then it can be shown that the function $C(v,\\varepsilon )$ given by $C(v,\\varepsilon )=v+\\int _{y}^{y+\\varepsilon }f(x)dx,$ where for $v$ in $\\mathbf {R}$ , we have taken the unique $y$ in $\\mathbf {R}$ such that $v=\\int _{0}^{y}f(x)dx$ .",
"For more background on contours we refer to [14].",
"It is also shown in [14] how the choice of a contour leads to a pseudometric $d_C$ in $\\mathbf {R}$ -Vec.",
"The stabilization of the rank invariant with respect to the chosen contour is then defined as $\\widehat{\\text{rank}}_{C} V(\\varepsilon )=\\text{min}\\left\\lbrace \\text{rank}(W) \\ |\\ W \\in \\mathbf {R}\\textbf {-Vec}\\text{ and } d_C(V,W)\\le \\varepsilon \\right\\rbrace .$ As noted above, the stable rank function $\\widehat{\\text{rank}}_{C} V$ is decreasing and piecewise constant and from $\\mathbf {R}$ to $\\mathbf {R}$ .",
"Our approach does not conceptually rely on the bar decomposition of $V$ in $\\mathbf {R}$ -Vec.",
"Computation of the decomposition is however standard procedure in persistence analysis with various dedicated implementations [3] and when the decomposition is given, the stable rank can be computed algorithmically in a very efficient way: $\\widehat{\\text{rank}}_C V (\\varepsilon ) = |\\lbrace [b_i,d_i)\\ |\\ C(b_i,\\varepsilon ) < d_i\\rbrace |.$ The stable rank of $V$ at $\\varepsilon $ is thus the number of those bars in the decomposition that satisfy the relation between the start and end points given by the contour.",
"In practical computations the limit of $\\widehat{\\text{rank}}_C V$ is always zero, or can be set to zero.",
"By fixing some values of $\\varepsilon $ the contour $C(v,\\varepsilon )$ reduces to a single variable function and we can plot it.",
"In Fig.",
"REF this is illustrated with few values of $\\varepsilon $ in the stem plot of a bar decomposition.",
"This visualization is helpful in understanding how the contour affects the stable rank in Eq.",
"REF : the value of stable rank $\\widehat{\\text{rank}}_C V (\\varepsilon )$ at $\\varepsilon $ is the number of bars that reach over the function $C(v,\\varepsilon )$ .",
"If the function $C(v,\\varepsilon )$ has lower values it therefore makes bars relatively longer and vice versa with larger values.",
"The contour can thus be seen as controlling pointwise with respect to $b_i$ the length scale that we use to measure bars."
],
[
"Topological learning with stable ranks",
"The stable rank attached to an input data set is a topological fingerprint of the data.",
"In the actual data analysis task these fingerprints are used in, for example, classifying various data sets.",
"Recall from the construction above that the stable rank is derived by choosing a contour function $C$ which induces a metric $d_C$ needed for the stabilization in Eq.",
"REF .",
"Each choice of a contour gives a different stable rank capturing different aspects of the data.",
"The learning step in our pipeline is then to choose an appropriate contour for the analysis at hand and we explore this in Section .",
"As stable ranks are $\\mathbf {R}$ -valued functions we have various choices of metrics for comparing them.",
"In particular we have standard $L_p$ -metrics for $p \\ge 1$ : $L_p(f,g) =\\left( \\int _0^\\infty |f(t)-g(t)|^p dt \\right)^{1/p}.$ We can also define interleaving distance between functions $f$ and $g$ .",
"We first define the set of horizontal shifts of the functions satisfying the indicated inequalities: $S =\\lbrace \\varepsilon \\in \\mathbf {R} \\, | \\, f(t) \\ge g(t+\\varepsilon ) \\ \\text{and} \\ g(t) \\ge f(t+\\varepsilon ) \\ \\text{for all} \\ t \\in \\mathbf {R}\\rbrace .$ The interleaving distance $d_{\\bowtie }$ is then defined as the minimum of those shifts: $d_{\\bowtie } (f,g) ={\\left\\lbrace \\begin{array}{ll}\\text{inf}(S) &\\text{, if $S$ is non-empty,}\\\\\\infty &\\text{, otherwise.}\\end{array}\\right.",
"}$ In Section we use these constructions in demonstrating our approach with concrete data analyses.",
"We emphasize that our approach does not rely on any algebraic decomposition of persistence and is thus applicable to multiparameter persistence [16].",
"The initial theory behind our pipeline was indeed formulated for multiparameter persistence in [8] and later specialized for 1-parameter persistence in [14].",
"In the case of one parameter we obtain the convenient algorithm, Eq.",
"REF , for computing stable rank.",
"Traditional view in persistence analysis has been that long bars in the bar decomposition are of importance and smaller bars are noise.",
"This view, however, is challenged by many recent studies showing that smaller features carry important information: study of brain artery trees in [17], functional networks of [18], analysis of protein structure in [19] and the relation of observed diffraction peaks to small loops in atomic configurations of amorphous silica in [20].",
"With our pipeline we can flexibly choose different contours to learn what are in fact the essential features in the data.",
"To produce the bar decompositions we used Ripser software [22]."
],
[
"Classifying physical activities",
"We studied PAMAP2 data obtained from [10] to classify different physical activities.",
"The data consisted of seven persons performing different activities such as walking, cycling or sitting.",
"Test subjects were fitted with three Inertial Measurements Units (IMUs) and a heart rate monitor.",
"Measurements were registered every 0.1 seconds.",
"Each IMU measured 3D acceleration, 3D gyroscopic and 3D magnetometer data.",
"One data set thus consisted 28-dimensional data points indexed by 0.1 second timesteps.",
"Figure: Confusion matrices for the classification of ascending and descending stairs activities with standard contour (left) and with contour visualized in Fig.",
".Figure: Density function used for H 1 H_1 stable rank in the activities classification (left) and contour lines for few values of ε\\varepsilon (right).",
"Persistence bar stems are shown for single data sets from each (subject,activity) class.We looked at two activities which from the outset are very similar and expected to be difficult to distinguish: ascending and descending stairs.",
"For the analysis we randomly sampled without replacement 100 points from each data set, repeated 100 times.",
"For each subject we thus obtained 100 resamplings from the activity data and computed their stable ranks with respect to a chosen contour.",
"Out of these we computed the point-wise means of 40 stable ranks in $H_0$ and $H_1$ .",
"These means were used as classifiers, denoted by $\\hat{P}_{H_0}$ and $\\hat{P}_{H_1}$ .",
"Altogether we had 14 classifier pairs $(\\hat{P}_{H_0},\\hat{P}_{H_1})$ corresponding to all (subject, activity) combinations.",
"Remaining 60 stable ranks in $H_0$ and $H_1$ were used as test data and denoted by $T_{H_0}$ and $T_{H_1}$ .",
"For a test pair $(T_{H_0}, T_{H_1})$ we found $\\text{min}(L_1(\\hat{P}_{H_0},T_{H_0}) + L_1(\\hat{P}_{H_1},T_{H_1}))$ by computing $L_1$ distances between the test pair and all classifier pairs.",
"The classification is successful if the minimum is obtained with $\\hat{P}_\\bullet $ and $T_\\bullet $ belonging to the same (subject, activity) class in both $H_0$ and $H_1$ .",
"For cross validation we randomly sampled which of the stable ranks constitute classifier and which are test data for the class.",
"Result for 20-fold cross validation is shown in the confusion matrix on the left in Fig.",
"REF for the standard contour.",
"Each cell of the confusion matrix is the number of classifications in the corresponding classifier (columns) and test data (rows) pair relative to the total number of test stable ranks which was 60.",
"Correct classifications are on the diagonal.",
"Overall accuracy (mean over diagonal of the confusion matrix) with standard contour was 60%.",
"We then repeated the above cross validation process but using a different contour in computing $H_1$ stable rank.",
"Contour was obtained from the density function on the left side of Fig.",
"REF .",
"Contour lines and the bars from persistence computation are visualized on the right side of Fig.",
"REF .",
"This contour puts more weight on topological features appearing with larger filtration scales.",
"Cross-validation results are shown on the right in Fig.",
"REF .",
"Overall accuracy increased to 65%.",
"Note particularly increase in the accuracy of subject 4.",
"Also noteworthy is that ascendings mainly get confused with ascendings of different subjects and the same for descendings.",
"These (subject,activity) data thus exhibit different character and changing the contour we could make this difference more pronounced."
],
[
"Cloud pattern characterization",
"We analysed the spatial distribution of shallow cumulus clouds.",
"These clouds form in fair-weather conditions due to the convective transport of heat and moisture in the atmosphere.",
"Convection is a classic example of a pattern-forming system [12], [13].",
"Cloud formation is known to be influenced by diverse physical processes across spatial scales ranging from molecular sizes to kilometers.",
"Such spatial scales and all their physical variables cannot be explicitly resolved in numerical climate models, which calls for the development of cloud parametrization schemes.",
"Moreover, the spatial distribution of clouds influences their formation processes.",
"It is therefore important to include this distribution in parametrization schemes.",
"This problem has been studied from different perspectives, notably the influence of land surface conditions on cloud formation [23].",
"Here we describe an approach based on persistence and the use of stable ranks as descriptors of the spatial distribution of clouds.",
"See [11] for further results and references.",
"The data was produced by the Dutch Atmospheric Large-Eddy Simulation model and covered the time period between 09:00h and 18:00h during one day, saved for analysis at 15 minute intervals, with model setup similar to that in [24].",
"We simulated 10 days with different initial conditions.",
"The data consists of large amount of physical information from which cloud fields can be extracted.",
"The spatial simulation domain in $x,y,z$ coordinates is $12.8 \\times 12.8 \\times 5$ in kilometers with horizontal resolution of 50 meters and vertical resolution of 40 meters.",
"The computation domain thus consists of cells.",
"A homogeneous land surface is prescribed and the lateral boundaries are periodic.",
"The 3D cloud field from the simulation domain was then flattened in the $z$ -direction onto a 2D plane by taking the maximum liquid water content, $ql$ , values in the vertical direction.",
"The resulting cloud fields are then as visualized in Fig.",
"REF (b).",
"Figure: a) Values of the vertical wind velocity ww for a two-dimensional horizontal slice at an altitude of1.8.",
"This corresponds to cloud base height (red – w>0w > 0; blue – w<0w < 0).b) Column liquid water content qlql (i. e. the maximum liquid water value in the vertical direction).c) Point representation of the cloud field by the local maxima of qlql(only connected components formed by at least 3 cells are considered),and 1-simplices of the Vietoris-Rips filtration using the distance relation between the points, at a distance scale of 1.5.An important issue in the study of cloud formation is the quantification of spatial organization, or lack thereof, in a given cloud field.",
"While methods to study spatial distributions exist in the statistical literature for objects which can be idealized as points, it is harder to work with objects that possess a spatial extent (i.e.",
"area or volume), as clouds do.",
"This leads to the necessity of computing a point representation for a cloud before being able to assess the spatial distribution of the cloud field.",
"Here we consider three different representations: assigning to each cloud its geometric centroid, its point with maximum $ql$ value, and a set of its points chosen at random.",
"A common metric in the assessment of spatial organization is the $I_\\text{org}$ index [21], defined as follows.",
"For a two-dimensional cloud field, such as the one shown in Fig.",
"REF (b), index the connected components (the individual clouds) as $c_i$ , and compute their geometric centroids, $\\bar{c}_i$ .",
"We are interested in how the spatial distribution of the $\\bar{c}_i$ compares to what we would expect under complete spatial randomness (CSR), that is, if the centroids represent a realization of a homogeneous Poisson point process.",
"To that end, we consider the nearest-neighbor distances $d_i$ , which are defined as $d_i = \\text{min}\\lbrace d(\\bar{c}_i, x) \\ | \\ x \\in \\bar{\\mathcal {C}} \\setminus \\lbrace \\bar{c}_i \\rbrace \\rbrace $ , where $\\bar{\\mathcal {C}}$ represents the set of all centroids.",
"The cumulative distribution function (CDF) of the $d_i$ is $G_{d_i}(r) = P[d_i \\le r],$ which in the case of a Poisson point process has the analytic expression $G_{CSR}(r) = 1 - \\exp {(-\\lambda \\, \\pi \\, r^2)},$ where $\\lambda $ is the Poisson intensity parameter.",
"The value of $I_\\text{org}$ is then defined to be the area under the graph $(G_{CSR}(r), \\hat{G}(r))$ , where $\\hat{G}(r) = \\frac{\\# \\lbrace \\bar{c}_i \\in \\bar{\\mathcal {C}} \\mid d_i \\le r \\rbrace }{\\# \\lbrace \\bar{c}_i \\in \\bar{\\mathcal {C}} \\rbrace }$ is the empirical estimator of $G(r)$ .",
"If $\\hat{G}$ matches well with $G_{CSR}$ , the value of $I_\\text{org}$ will be close to 0.5.",
"A value larger than this suggests spatial clustering, while a smaller one suggests dispersion or regularity.",
"Figure: Stable rank functions obtained from 100 realizations of a homogeneous Poisson point process with λ=100\\lambda = 100.",
"Left: S 0 * S_0^*.",
"Right: S 1 * S_1^*.Let $S^*_i$ denote the stable rank of $H_i$ with respect to the standard contour (Eq.",
"REF ), normalized by its value at 0.",
"If we define the function $G_{PH}^i(r) = 1 - S_i^*(r)$ , we note that it increases monotonically towards 1.",
"In fact, since the normalized stable rank at $r$ is an indication of the relative amount of homological features that persist beyond $r$ , the function $G_{PH}^i(r)$ can be understood as the empirical CDF of homological persistence.",
"For $n$ realizations of a Poisson point process with intensity parameter $\\lambda $ , we find that their normalized stable ranks $S^*_i$ , and therefore also $G_{PH}^i$ , oscillate within a narrow band (see Fig.",
"REF ).",
"At this point we do not have an analytic expression for the stable rank functions obtained from a Poisson point process, but we can define persistent homology analogues to the $I_\\text{org}$ index via a Monte Carlo procedure by taking the area under the curves defined by $(G_{PH,CSR}^{i}(r), G_{PH}^i(r))$ .",
"In the case of a point process in the plane we would then get two values $I_{PH,0}$ and $I_{PH,1}$ .",
"We define the index as their arithmetic mean, $I_{PH} = \\frac{I_{PH,0} + I_{PH,1}}{2}.$ Figure: Density histograms of the I org I_\\text{org} index and I PH I_{PH} (Eq. )",
"for 360 distinct cloud fields.",
"A: qlql max, B: qlql max removing cloud structures with size smaller than 3 cells, C: Geometric centroids, D: Geometric centroids removing cloud structures with size smaller than 3 cells.We tested the performance of the index $I_{PH}$ defined above, and compared it to the corresponding values of $I_\\text{org}$ in the dataset consisting of 360 distinct cloud fields (36 per simulation day).",
"The values of both indices are shown in Fig.",
"REF .",
"Each panel shows the 360 values of each index for all cloud fields, computed using 4 different point representations.",
"Panel A shows the values obtained from assigning to each connected component its point with maximum $ql$ value (local maxima); panel B shows the indices obtained when using the local maxima but only of those components with size at least 3 grid cells (all smaller components are ignored).",
"Panel C shows the results of using the geometric centroid of each connected component.",
"Finally, for panel D the geometric centroids were used after discarding the smaller components.",
"These small components can be attributed to numerical imprecision in the underlying model, and hence are not physically meaningful.",
"As discussed above, if these indices have a value close to 0.5, it would indicate that the point process that they are evaluated on is close to complete spatial randomness, or a Poisson point process.",
"In the simulations used here, we have cause to expect spatially random behavior: the domain size is too small to allow for deep convection and spatial organization to happen.",
"Moreover, the lack of land surface features or patterns means there are no forcings at different spatial scales.",
"Thus the spatial distribution of physical variables is dominated by the characteristic patterns present in atmospheric turbulence, itself an essentially random process.",
"The values of the persistent homology index $I_{PH}$ strongly support this hypothesis, while $I_\\text{org}$ exhibits values in general larger than 0.5.",
"This can be attributed to the fact that it is based on nearest-neighbor distances only, whereas the stable rank functions reflect the spatial relationships of the points throughout all spatial scales.",
"This is confirmed by the fact that removing the smaller structures in the fields (those less than 3 grid cells in size) brings the values of $I_\\text{org}$ closer to 0.5 on average, whereas the average for $I_{PH}$ is barely affected.",
"This highlights the fact that, by virtue of using all the spatial information available, the persistent homology based method is inherently more robust than any nearest-neighbor method.",
"Figure: Contour 1 (left) and contour 2 (right) used in the analysis of cloud fields.",
"Stem plot is from one sampling ofa cloud field at one time step.This result has been arrived at by using the standard contour only, which implies that spatial randomness in these cloud fields is obtained when all spatial scales present in the data are given the same weight.",
"It is possible to obtain different morphological classifications of the same fields by using alternative contours, which emphasize spatial features differently at varying scales, as presented with the classification in Section REF .",
"We used standard contour and contours visualized in Fig.",
"REF .",
"These contours are referred to as contour 1, denoted $C_1$ , and contour 2, denoted $C_2$ .",
"Figure: Cloud fields which are classified into different clusters, according to the methodology described in the text.We use the H 1 H_1 stable ranks and the interleaving metric to compute the distances between them.a) and b) are classified using contour C 1 C_1, and have I org I_\\text{org} values of 0.45 and 0.53 respectively.Cloud cover is similar at 14% for both.c) and d) are classified with C 2 C_2, and have I org I_\\text{org} values of 0.65 and 0.63 respectively, and cloud coverfor both is 9.2%.To reduce the effect of sampling, 10 random samples were drawn from each of the 360 cloud fields, with sample rate 5% of cloud size.",
"To each cloud field we assign the mean stable rank of these 10 samples.",
"Stable ranks were computed in $H_1$ with respect to standard contour, contour 1 and contour 2 and normalized to give $S^*_1$ function as explained above.",
"After removing those cloud fields without $H_1$ features, we have 254 normalized stable ranks $S^*_1$ for each class of contours.",
"Distance matrices using interleaving, $L_1$ - and $L_2$ -metrics (see Section REF ) were then computed for the three different classes of stable ranks.",
"Dendrograms from the distance matrices were visually analyzed to decide on a number of clusters of stable ranks.",
"From these computations the interleaving distance gave the clearest clustering results.",
"With respect to contours, $C_1$ and $C_2$ gave better clustering than standard contour.",
"An example of diverging morphological characteristics educed from the $C_{1,2}$ clustering schemes is shown in Fig.",
"REF : (a) and (b) are representatives of two different clusters obtained by using contour $C_1$ , while (c) and (d) stem from clusters in the $C_2$ classification.",
"As expected from the definition of the contours, the classifications they induce are influenced by different spatial scales.",
"Namely, despite the fact that cloud fields a) and b) have identical cloud cover, and their $I_\\text{org}$ values are very similar, the large-scale distribution of the individual clouds is significantly different for both.",
"In similar fashion, both c) and d) are indistinguishable in terms of cloud cover and $I_\\text{org}$ , yet are distinguished by the spatial pattern of smaller structures, even if the large-scale distribution is similar in both.",
"This study of cloud fields shows that the use of stable rank functions as descriptors for spatial distributions can reveal morphological properties which other methods cannot.",
"Crucially, the possibility of changing the contour enriches the scope for determining such properties.",
"Future investigation in this direction will address questions such as: what the optimal contour is for a given problem, what these methods can reveal about the temporal evolution of cloud formation, and how the homological properties thus discovered can be related to different physical variables in the system.",
"From general data analysis point of view, particularly the optimization of contours is crucial for making our pipeline a full scale machine learning approach."
],
[
"Acknowledgments",
"We gratefully acknowledge Roel Neggers for providing the DALES simulation data.",
"JLS acknowledges support by the DFG-funded transregional research collaborative TR32 on Patterns in Soil–Vegetation–Atmosphere Systems."
]
] | 1906.04436 |
[
[
"A Graph-theoretic Method to Define any Boolean Operation on Partitions"
],
[
"Abstract The lattice operations of join and meet were defined for set partitions in the nineteenth century, but no new logical operations on partitions were defined and studied during the twentieth century.",
"Yet there is a simple and natural graph-theoretic method presented here to define any n-ary Boolean operation on partitions.",
"An equivalent closure-theoretic method is also defined.",
"In closing, the question is addressed of why it took so long for all Boolean operations to be defined for partitions."
],
[
"Introduction",
"The lattice operations of join and meet were defined on set partitions during the late nineteenth century, and the lattice of partitions on a set was used as an example of a non-distributive lattice.",
"But during the entire twentieth century, no new logical operations were defined on partitions.",
"Equivalence relations are so ubiquitous in everyday life that we often forget about their proactive existence.",
"Much is still unknown about equivalence relations.",
"Were this situation remedied, the theory of equivalence relations could initiate a chain reaction generating new insights and discoveries in many fields dependent upon it.",
"This paper springs from a simple acknowledgement: the only operations on the family of equivalence relations fully studied, understood and deployed are the binary join $\\vee $ and meet $\\wedge $ operations.",
"[3] Papers on the \"logic\" of equivalence relations [7] or partitions only involved the join and meet, and not the crucial logical operation of implication.",
"Yet, there is a general graph-theoretic methodThe method is, strictly speaking, an algorithm only when $U$ is finite.",
"by which any $n$ -ary Boolean (or truth-functional) operation $f:\\left\\lbrace T,F\\right\\rbrace ^{n}\\rightarrow \\left\\lbrace T,F\\right\\rbrace $ can be used to define the corresponding $n$ -ary operation $f:\\prod \\left( U\\right) ^{n}\\rightarrow \\prod \\left(U\\right) $ where $\\prod \\left( U\\right) $ is the set of partitions on a set $U$ .",
"A partition $\\pi =\\left\\lbrace B,B^{\\prime },...\\right\\rbrace $ on a set $U=\\left\\lbrace u,u^{\\prime },...\\right\\rbrace $ is a set of disjoint non-empty subsets $B,B^{\\prime },...$ of $U$ , called blocks, whose union is $U$ .",
"The corresponding equivalence relation, denoted $\\operatornamewithlimits{indit}\\left(\\pi \\right) $ , is the set of ordered pairs of elements of $U$ that are in the same block of $\\pi $ , and are called the indistinctions or indits of $\\pi $ , i.e., $\\operatornamewithlimits{indit}\\left( \\pi \\right) =\\left\\lbrace \\left( u,u^{\\prime }\\right) \\in U\\times U:\\exists B\\in \\pi ,u,u^{\\prime }\\in B\\right\\rbrace $ .",
"The complement $\\operatornamewithlimits{dit}\\left( \\pi \\right) =U\\times U-\\operatornamewithlimits{indit}\\left( \\pi \\right) $ is the set of distinctions or dits of $\\pi $ , i.e., ordered pairs of elements in different blocks.",
"As binary relations, the sets of distinctions or ditsets $\\operatornamewithlimits{dit}\\left( \\pi \\right) $ of some partition $\\pi $ on $U$ are called partition (or apartness) relations.",
"Given partitions $\\pi =\\left\\lbrace B,B^{\\prime },...\\right\\rbrace $ and $\\sigma =\\left\\lbrace C,C^{\\prime },...\\right\\rbrace $ on $U$ , the refinement relation is the partial order defined by: $\\sigma \\preceq \\pi $ if $\\forall B\\in \\pi ,\\exists C\\in \\sigma ,B\\subseteq C $ .",
"At the top of the refinement partial order is the discrete partition $\\mathbf {1}=\\left\\lbrace \\left\\lbrace u\\right\\rbrace :u\\in U\\right\\rbrace $ of all singletons and at the bottom is the indiscrete partition $\\mathbf {0}=\\left\\lbrace U\\right\\rbrace $ with only one block consisting of $U$ .",
"In terms of binary relations, the refinement partial order is just the inclusion partial order on ditsets, i.e., $\\sigma \\preceq \\pi $ iff $\\operatornamewithlimits{dit}\\left( \\sigma \\right) \\subseteq \\operatornamewithlimits{dit}\\left( \\pi \\right) $ .",
"It should be noted that most of the previous literature on partitions (e.g., [1]) uses the opposite partial order of `unrefinement' corresponding to the inclusion relation on equivalence relations–which reverses the definitions of the join and meet of partitions."
],
[
"The Join Operation on Partitions",
"The join $\\pi \\vee \\sigma $ of partitions $\\pi $ and $\\sigma $ (least upper bound using the refinement partial order) is the partition whose blocks are the non-empty intersections $B\\cap C$ of the blocks of $\\pi $ and $\\sigma $ (under the unrefinement ordering, it is the meet).",
"In terms of ditsets, $\\operatornamewithlimits{dit}\\left( \\pi \\vee \\sigma \\right) =\\operatornamewithlimits{dit}\\left(\\pi \\right) \\cup \\operatornamewithlimits{dit}\\left( \\sigma \\right) $ .",
"The general method for defining Boolean operations on partitions will be first illustrated with the join operation whose corresponding Boolean operation is disjunction with the truth table.",
"Table: NO_CAPTIONTruth table for disjunction.",
"Let $K\\left( U\\right) $ be the complete undirected graph on $U$ .",
"The links $u-u^{\\prime }$ corresponding to dits, i.e., $\\left( u,u^{\\prime }\\right)\\in \\operatornamewithlimits{dit}\\left( \\pi \\right) $ , of a partition are labelled with the `truth value' $T_{\\pi }$ and corresponding to indits $\\left( u,u^{\\prime }\\right) \\in \\operatornamewithlimits{indit}\\left( \\pi \\right) $ are labelled with the `truth value' $F_{\\pi }$ .",
"Given the two partitions $\\pi $ and $\\sigma $ , each link in the complete graph $K\\left( U\\right) $ is labelled with a pair of truth values.",
"The graph $G\\left( \\pi \\vee \\sigma \\right) $ of the join is obtained by putting a link $u-u^{\\prime }$ where the truth function applied to the pair of truth values on the link in $K\\left(U\\right) $ gives an $F$ .",
"Thus in the case at hand, the only links in $G\\left( \\pi \\vee \\sigma \\right) $ are for the $u-u^{\\prime }$ labelled with $F_{\\pi }$ and $F_{\\sigma }$ in $K\\left( U\\right) $ .",
"Then the partition $\\pi \\vee \\sigma $ is obtained as the connected components of its graph $G\\left(\\pi \\vee \\sigma \\right) $ .",
"Thus $u$ and $u^{\\prime }$ are in the same block (connected component of $G\\left( \\pi \\vee \\sigma \\right) $ ) if and only if the link $u-u^{\\prime }$ was labelled $F_{\\pi }$ and $F_{\\sigma }$ , i.e., $u$ and $u^{\\prime }$ were in the same block of $\\pi $ and in the same block of $\\sigma $ .",
"Thus the graph-theoretic definition of the join reproduces the set-of-blocks definition of the join defined as having its blocks the non-empty intersections of the blocks of $\\pi $ and $\\sigma $ ."
],
[
"The Meet Operation on Partitions",
"On the combined set of blocks $\\pi \\cup \\sigma $ of $\\pi $ and $\\sigma $ , define the overlap relation $B\\between C$ on two blocks if they have a non-empty intersection or overlap (see [8]).",
"The reflexive-symmetric-transitive closure of this relation is an equivalence relation, and the union of the blocks in each equivalence class gives the blocks of the meet $\\pi \\wedge \\sigma $ .",
"The corresponding truth-functional operation is conjunction with the following truth table.",
"Table: NO_CAPTIONTruth table for conjunction.",
"The same method is applied except that the links of the graph $G\\left(\\pi \\wedge \\sigma \\right) $ are the ones for which the conjunction truth table gives an $F$ when applied to the truth values on each link $u-u^{\\prime }$ .",
"Thus $G\\left( \\pi \\wedge \\sigma \\right) $ contains a link $u-u^{\\prime }$ if $\\left( u,u^{\\prime }\\right) \\in \\operatornamewithlimits{indit}\\left( \\pi \\right) $ , $\\left( u,u^{\\prime }\\right) \\in \\operatornamewithlimits{indit}\\left( \\sigma \\right)$ , or both.",
"Then the blocks of the partition $\\pi \\wedge \\sigma $ are the connected components of the graph $G\\left( \\pi \\wedge \\sigma \\right) $ .",
"The proof that the graph-theoretic definition of the meet gives the usual set-of-blocks definition of the meet boils down to showing that: $B\\in \\pi $ and $C\\in \\sigma $ are contained in the same block of the usual meet $\\pi \\wedge \\sigma $ (i.e., there is a chain of overlaps $B\\between C^{\\prime }\\between ...\\between B^{\\prime }\\between C$ connecting $B$ and $C$ ) if and only for any $u\\in B$ and $u^{\\prime }\\in C$ , $u$ and $u^{\\prime }$ are in the same connected component of $G\\left( \\pi \\wedge \\sigma \\right) $ .",
"If any two blocks $B^{\\prime }\\between C^{\\prime }$ overlap in the overlap chain, then there is an element $u^{\\prime \\prime }\\in B^{\\prime }\\cap C^{\\prime }$ such any $u\\in B^{\\prime }$ had a link $u-u^{\\prime \\prime }$ in $G\\left( \\pi \\wedge \\sigma \\right) $ and similarly any $u^{\\prime }\\in C^{\\prime }$ has a link $u^{\\prime \\prime }-u^{\\prime }$ in $G\\left( \\pi \\wedge \\sigma \\right) $ .",
"Hence the existence of an overlap chain connecting $B$ and $C$ implies that any $u\\in B$ and $u^{\\prime }\\in C$ are in the same connected component of $G\\left( \\pi \\wedge \\sigma \\right) $ .",
"Conversely, if $u\\in B$ and $u^{\\prime }\\in C$ are in the same connected component of $G\\left( \\pi \\wedge \\sigma \\right) $ , then there is some chain of links $u=u_{0}-u_{1}-...-u_{n-1}-u_{n}=u^{\\prime }$ where each link $u_{i}-u_{i+1}$ for $i=0,...,n-1$ has either $\\left( u_{i},u_{i+1}\\right) \\in \\operatornamewithlimits{indit}\\left( \\pi \\right) $ , $\\left( u_{i},u_{i+1}\\right)\\in \\operatornamewithlimits{indit}\\left( \\sigma \\right) $ , or both.",
"Every link $u_{i}-u_{i+1}$ that is in one indit set but not the other, say, $\\left(u_{i},u_{i+1}\\right) \\in \\operatornamewithlimits{indit}\\left( \\pi \\right) $ and $\\left( u_{i},u_{i+1}\\right) \\notin \\operatornamewithlimits{indit}\\left(\\sigma \\right) $ , establishes an overlap between the block of $\\pi $ containing $u_{i},u_{i+1}$ and the block of $\\sigma $ containing $u_{i}$ as well as the different block of $\\sigma $ containing $u_{i+1}$ .",
"Thus the chain of links connecting $u\\in B$ and $u^{\\prime }\\in C$ establishes a chain of overlapping blocks connecting $B$ and $C$ ."
],
[
"The Implication Operation on Partitions",
"The real beginning of the logic of partitions, as opposed to the lattice theory of partitions, was the discovery of the set-of-blocks definition of the implication operation $\\sigma \\Rightarrow \\pi $ for partitions ([5], [6]).",
"The intuitive idea is that $\\sigma \\Rightarrow \\pi $ functions like an indicator or characteristic function to indicate which blocks $B$ of $\\pi $ are contained in a block of $\\sigma $ .",
"View the discretized version of $B\\in \\pi $ , i.e., $B$ replaced by the set of singletons of the elements of $B$ , as the local version $\\mathbf {1}_{B}$ of the discrete partition $\\mathbf {1}$ , and view the block $B$ remaining whole as the local version $\\mathbf {0}_{B}$ of the indiscrete partition $\\mathbf {0}$ .",
"Then the partition implication as the inclusion indicator function is: the blocks of $\\sigma \\Rightarrow \\pi $ are for any $B\\in \\pi $ : $\\left\\lbrace \\begin{array}[c]{l}\\mathbf {1}_{B}\\text{ if }\\exists C\\in \\sigma ,B\\subseteq C\\\\\\mathbf {0}_{B}=B\\text{ otherwise.}\\end{array}\\right.",
".$ In the case of the Boolean logic of subsets, for any subsets $S,T\\subseteq U$ , the conditional $S\\supset T=S^{c}\\cup T$ has the property: $S\\supset T=U$ iff $S\\subseteq T$ , i.e., the conditional $S\\supset T$ equals the top of the lattice of subsets of $U$ iff the inclusion relation $S\\subseteq T$ holds.",
"Similarly, it is immediate that the corresponding relation holds in the partition case: $\\sigma \\Rightarrow \\pi =\\mathbf {1}$ iff $\\sigma \\preceq \\pi $ .",
"This set-of-blocks definition of the partition implication operation accounts for the important new non-lattice-theoretic properties revealed in the algebra of partitions $\\prod \\left( U\\right) $ on $U $ (defined with the join, meet, and implication as partition operations).",
"A logical formula in the language of join, meet, and implication is a subset tautology if for any non-empty universe $U$ and any subsets of $U$ substituted for the variables, the whole formula evaluates by the set-theoretic operations of join, meet, and implication (conditional) to the top $U$ .",
"Similarly, a formula in the same language is a partition tautology if for any universe $U$ with $\\left|U\\right|>1$ and for any partitions on $U$ substituted for the variables, the whole formula evaluates by the partition operations of join, meet, and implication to the top $\\mathbf {1}$ (the discrete partition).",
"All partition tautologies are subset tautologies but not vice-versa.",
"Modus ponens $\\left(\\sigma \\wedge \\left( \\sigma \\Rightarrow \\pi \\right) \\right) \\Rightarrow \\pi $ is both a subset and partition tautology but Peirce's law, $\\left( \\left(\\sigma \\Rightarrow \\pi \\right) \\Rightarrow \\sigma \\right) \\Rightarrow \\sigma $ , accumulation, $\\sigma \\Rightarrow \\left( \\pi \\Rightarrow \\left( \\sigma \\wedge \\pi \\right) \\right) $ , and distributivity, $\\left( \\left( \\pi \\vee \\sigma \\right) \\wedge \\left( \\pi \\vee \\tau \\right) \\right) \\Rightarrow \\left(\\pi \\vee \\left( \\sigma \\wedge \\tau \\right) \\right) $ , are examples of subset tautologies that are not partition tautologies.",
"The importance of the implication for partition logic is emphasized by the fact that the only partition tautologies using only the lattice operations, e.g., $\\pi \\vee \\mathbf {1}$ , correspond to general lattice-theoretic identities, i.e., $\\pi \\vee \\mathbf {1}=\\mathbf {1}$ (see [9]).",
"The graph-theoretic method automatically gives a partition operation corresponding to the Boolean conditional or implication with the truth table: Table: NO_CAPTIONTruth table for conditional and it is not trivial that the two definitions are the same.",
"It may be helpful to restate the truth table in terms of the partitions.",
"Table: NO_CAPTIONImplication truth table for partition `truth values'.",
"For the graph-theoretic definition of $\\sigma \\Rightarrow \\pi $ , we again label the links $u-u^{\\prime }$ in the complete graph $K\\left( U\\right) $ with $T_{\\pi }$ if $\\left( u,u^{\\prime }\\right) \\in \\operatornamewithlimits{dit}\\left(\\pi \\right) $ and $F_{\\pi }$ otherwise, and similarly for $\\sigma $ .",
"Then we construct the graph $G\\left( \\sigma \\Rightarrow \\pi \\right) $ by putting in a link $u-u^{\\prime }$ only in the case the link is labeled $T_{\\sigma }$ and $F_{\\pi }$ , i.e., $F_{\\sigma \\Rightarrow \\pi }$ .",
"Then the partition $\\sigma \\Rightarrow \\pi $ is the partition of connected components in the graph $G\\left( \\sigma \\Rightarrow \\pi \\right) $ .",
"To prove the graph-theoretic and set-of-blocks definitions equivalent, we might first note that if $\\left( u,u^{\\prime }\\right) \\in \\operatornamewithlimits{dit}\\left( \\pi \\right) $ , then $T_{\\pi }$ is assigned to that link in $K\\left(U\\right) $ so there is no link $u-u^{\\prime }$ in $G\\left( \\sigma \\Rightarrow \\pi \\right) $ .",
"And if $\\left( u,u^{\\prime }\\right) \\in \\operatornamewithlimits{indit}\\left( \\pi \\right) $ but also $\\left( u,u^{\\prime }\\right) \\in \\operatornamewithlimits{indit}\\left( \\sigma \\right) $ , then $T_{\\sigma \\Rightarrow \\pi }$ is assigned to the link in $K\\left( U\\right) $ so again there is no link $u-u^{\\prime }$ in $G\\left( \\sigma \\Rightarrow \\pi \\right) $ .",
"There is a link $u-u^{\\prime }$ in $G\\left( \\sigma \\Rightarrow \\pi \\right) $ in and only in the following situation where $\\left(u,u^{\\prime }\\right) \\in \\operatornamewithlimits{indit}\\left( \\pi \\right) $ and $\\left( u,u^{\\prime }\\right) \\in \\operatornamewithlimits{dit}\\left( \\sigma \\right)$ –which is exactly the situation when $B$ is not contained in any block $C$ of $\\sigma $ : Figure: NO_CAPTION Figure 1: Links $u-u^{\\prime }$ in $G\\left( \\sigma \\Rightarrow \\pi \\right) $ .",
"Then for any other element $u^{\\prime \\prime }\\in B$ so that $\\left(u,u^{\\prime \\prime }\\right) $ and $\\left( u^{\\prime },u^{\\prime \\prime }\\right)\\in \\operatornamewithlimits{indit}\\left( \\pi \\right) $ , we must have either $\\left(u,u^{\\prime \\prime }\\right) \\in \\operatornamewithlimits{dit}\\left( \\sigma \\right) $ or $\\left( u^{\\prime },u^{\\prime \\prime }\\right) \\in \\operatornamewithlimits{dit}\\left(\\sigma \\right) $ so $u^{\\prime \\prime }$ is linked in $G\\left( \\sigma \\Rightarrow \\pi \\right) $ to either $u$ or to $u^{\\prime }$ .",
"Thus all the elements of $B$ are in the same connected component of the graph $G\\left(\\sigma \\Rightarrow \\pi \\right) $ whenever $B$ is not contained in any block of $\\sigma $ .",
"If, on the other hand, $B$ is contained in some block $C$ of $\\sigma $ , then any $u\\in B$ cannot be linked to any other $u^{\\prime }$ .",
"In order to that $F_{\\pi }$ assigned to the link $u-u^{\\prime }$ , the two elements have to both belong to $B$ and thus since $B\\subseteq C$ , they both belong to $C$ so $F_{\\sigma }$ and thus $T_{\\sigma \\Rightarrow \\pi }$ is also assigned to that link.",
"Thus when $B$ is contained in a block $C\\in \\sigma $ , then any point $u\\in B$ is a disconnected component to itself in $G\\left( \\sigma \\Rightarrow \\pi \\right) $ so $B$ is discretized in the graph-theoretic construction of $\\sigma \\Rightarrow \\pi $ .",
"Thus the graph-theoretic and set-of-blocks definitions of the partition implication are equivalent.",
"Figure: NO_CAPTION Figure 2: Example of graph for partition implication Example 1 Let $U=\\left\\lbrace a,b,c,d\\right\\rbrace $ so that $K(U)=K_{4}$ is the complete graph on four points.",
"Let $\\sigma =\\left\\lbrace \\left\\lbrace a\\right\\rbrace ,\\left\\lbrace b,c,d\\right\\rbrace \\right\\rbrace $ and $\\pi =\\left\\lbrace \\left\\lbrace a,b\\right\\rbrace ,\\left\\lbrace c,d\\right\\rbrace \\right\\rbrace $ so we see immediately from the set-of-blocks definition, that the $\\pi $ -block of $\\left\\lbrace c,d\\right\\rbrace $ will be discretized while the $\\pi $ -block of $\\left\\lbrace a,b\\right\\rbrace $ will remain whole so the partition implication is $\\sigma \\Rightarrow \\pi =\\left\\lbrace \\left\\lbrace a,b\\right\\rbrace ,\\left\\lbrace c\\right\\rbrace ,\\left\\lbrace d\\right\\rbrace \\right\\rbrace $ .",
"After labelling the links in $K\\left( U\\right) $ , we see that only the $a-b$ link has the $F_{\\sigma \\Rightarrow \\pi }$ `truth value' so the graph $G\\left(\\sigma \\Rightarrow \\pi \\right) $ has only that $a-b$ link (thickened in Figure 2).",
"Then the connected components of $G\\left( \\sigma \\Rightarrow \\pi \\right) $ give the same partition implication $\\sigma \\Rightarrow \\pi =\\left\\lbrace \\left\\lbrace a,b\\right\\rbrace ,\\left\\lbrace c\\right\\rbrace ,\\left\\lbrace d\\right\\rbrace \\right\\rbrace $ .",
"The partition implication is quite rich in defining new structures in the algebra of partitions (i.e., the lattice of partitions extended with other partition operations such as the implication).",
"For instance, for a fixed partition $\\pi $ on $U$ , all the partitions of the form $\\sigma \\Rightarrow \\pi $ (for any partitions $\\sigma $ on $U$ ) form a Boolean algebra under the partition operations of implication, join, and meet, e.g., $\\left(\\sigma \\Rightarrow \\pi \\right) \\Rightarrow \\pi $ is the negation of $\\sigma \\Rightarrow \\pi $ , called the Boolean core of the upper segment $\\left[ \\pi ,\\mathbf {1}\\right] $ in the partition algebra $\\prod \\left(U\\right) $ .",
"A relation is a subset of a product, and, dually, a corelation is a partition on a coproduct.",
"Any partition $\\pi $ on $U$ can be canonically represented as a relation: $\\operatornamewithlimits{dit}\\left(\\pi \\right) \\subseteq U\\times U$ .",
"Dually any subset $S\\subseteq U$ can be canonically represented as a corelation, namely the partition $\\pi \\left(S\\right) $ on the coproduct (disjoint union) $U\\uplus U$ where the only nonsingleton blocks in $\\pi \\left( S\\right) $ are the pairs $\\left\\lbrace u,u^{\\ast }\\right\\rbrace $ of $u$ and its copy $u^{\\ast }$ for $u\\notin S$ .",
"Using this corelation construction, any powerset Boolean algebra $\\wp \\left(U\\right) $ can be canonically represented as the Boolean core of the upper segment $\\left[ \\pi ,\\mathbf {1}\\right] $ in the partition algebra $\\prod \\left( U\\uplus U\\right) $ where $\\pi =\\pi \\left( \\emptyset \\right) $ is the partition on the disjoint union $U\\uplus U$ whose blocks are all the pairs $\\left\\lbrace u,u^{\\ast }\\right\\rbrace $ for each element $u\\in U$ and its copy $u^{\\ast }$ .",
"Each partition of the form $\\sigma \\Rightarrow \\pi $ on $U\\uplus U$ is $\\pi \\left( S\\right) $ for some $S\\subseteq U$ since $\\sigma \\Rightarrow \\pi $ is essentially the characteristic function of some subset $S$ of $U$ with $\\mathbf {1}\\Rightarrow \\pi =\\pi \\left( \\emptyset \\right) $ playing the role of the empty set $\\emptyset $ and $\\pi \\Rightarrow \\pi =\\mathbf {1}_{U\\uplus U}$ playing the role of $U$ ."
],
[
"The General Graph-Theoretic Method",
"Let $f:\\left\\lbrace T,F\\right\\rbrace ^{n}\\rightarrow \\left\\lbrace T,F\\right\\rbrace $ be an $n$ -ary Boolean function and let $\\pi _{1},...,\\pi _{n}$ be $n$ partitions on $U$ .",
"In order to define the corresponding $n$ -ary partition operation $f\\left( \\pi _{1},...,\\pi _{n}\\right) $ , we again consider the complete graph $K\\left( U\\right) $ and then use each partition $\\pi _{i}$ to label each link $u-u^{\\prime }$ with $T_{\\pi _{i}}$ if $\\left( u,u^{\\prime }\\right)\\in \\operatornamewithlimits{dit}\\left( \\pi _{i}\\right) $ and $F_{\\pi _{i}}$ if $\\left(u,u^{\\prime }\\right) \\in \\operatornamewithlimits{indit}\\left( \\pi _{i}\\right) $ .",
"Then on each link we may apply $f$ to the $n$ `truth values' on the link and retain the link in $G\\left( f\\left( \\pi _{1},...,\\pi _{n}\\right) \\right) $ if the result was $F_{f\\left( \\pi _{1},...,\\pi _{n}\\right) }$ .",
"The partition $f\\left( \\pi _{1},...,\\pi _{n}\\right) $ is obtained as the connected components of the graph $G\\left( f\\left( \\pi _{1},...,\\pi _{n}\\right)\\right) $ ."
],
[
"An Equivalent Closure-theoretic Method",
"Given any subset $S\\subseteq U\\times U$ , the reflexive-symmetric-transitive (RST) closure $\\overline{S}$ is the intersection of all equivalence relations on $U$ containing $S$ .",
"The `topological' terminology of calling a subset closed if $S=\\overline{S}$ is used even though the RST closure operator is not a topological closure operator since the union of two closed sets is not necessarily closed.",
"The closed sets in $U\\times U$ are the equivalence relations (or indit sets of partitions), and their complements, the open sets, are the partition relations (or ditsets of partitions).",
"As usual, the interior operator $\\operatornamewithlimits{int}\\left( S\\right) =\\left(\\overline{S^{c}}\\right) ^{c}$ is the complement of the closure of the complement, and the open sets are the ones equalling their interiors.",
"The closure-theoretic method of defining Boolean operations on partitions will be illustrated using the symmetric difference or inequivalence operation $\\pi \\oplus \\sigma $ .",
"Every $n$ -ary Boolean operation can be defined by a truth table such as the one for symmetric difference in this case: Table: NO_CAPTIONTruth table for symmetric difference.",
"The disjunctive normal form (DNF) for the formula $P\\oplus Q$ is given by the rows where the formula evaluates as $T$ , i.e., $P\\oplus Q=\\left( P\\wedge \\lnot Q\\right) \\vee \\left( \\lnot P\\wedge Q\\right) $ , while the DNF for the negation of the formula is given by the other rows where the formula evaluates as $F$ , i.e., $\\lnot \\left( P\\oplus Q\\right) =\\left( P\\wedge Q\\right)\\vee \\left( \\lnot P\\wedge \\lnot Q\\right) $ .",
"Given two partitions $\\pi $ and $\\sigma $ on $U$ , the closure-theoretic method of obtaining the partition $\\pi \\oplus \\sigma $ is to start with the DNF for the negated Boolean formula and replace each unnegated variable by the corresponding ditset and each negated variable by the corresponding indit set–as well as replacing the disjunctions and conjunctions by the corresponding subset operations of union and intersection.",
"Applied to $\\lnot \\left( P\\oplus Q\\right) =\\left( P\\wedge Q\\right) \\vee \\left( \\lnot P\\wedge \\lnot Q\\right) $ , this procedure would yield $\\left( \\operatornamewithlimits{dit}\\left( \\pi \\right) \\cap \\operatornamewithlimits{dit}\\left( \\sigma \\right) \\right) \\cup \\left( \\operatornamewithlimits{indit}\\left(\\pi \\right) \\cap \\operatornamewithlimits{indit}\\left( \\sigma \\right) \\right) \\subseteq U\\times U$ .",
"Then the indit set of $\\pi \\oplus \\sigma $ is obtained as the RST closure: $\\operatornamewithlimits{indit}\\left( \\pi \\oplus \\sigma \\right) =\\overline{\\left(\\operatornamewithlimits{dit}\\left( \\pi \\right) \\cap \\operatornamewithlimits{dit}\\left(\\sigma \\right) \\right) \\cup \\left( \\operatornamewithlimits{indit}\\left( \\pi \\right)\\cap \\operatornamewithlimits{indit}\\left( \\sigma \\right) \\right) }$ and the partition $\\pi \\oplus \\sigma $ is the set of equivalence classes of this equivalence relation.",
"The graph-theoretic method of obtaining the partition $\\pi \\oplus \\sigma $ would label each link $u-u^{\\prime }$ in $K\\left( U\\right) $ by the two `truth values' given by $\\pi $ and $\\sigma $ , and then retain in the graph $G\\left(\\pi \\oplus \\sigma \\right) $ the links where the truth values evaluated to $F_{\\pi \\oplus \\sigma }$ , namely the ones labelled with $T_{\\pi },T_{\\sigma }$ and $F_{\\pi },F_{\\sigma }$ .",
"Then the partition $\\pi \\oplus \\sigma $ is obtained as the connected components of the graph $G\\left( \\pi \\oplus \\sigma \\right) $ .",
"To see the equivalence between the two methods, note first that the links retained in $G\\left( \\pi \\oplus \\sigma \\right) $ are precisely the pairs $\\left( u,u^{\\prime }\\right) $ in $\\left( \\operatornamewithlimits{dit}\\left(\\pi \\right) \\cap \\operatornamewithlimits{dit}\\left( \\sigma \\right) \\right) \\cup \\left(\\operatornamewithlimits{indit}\\left( \\pi \\right) \\cap \\operatornamewithlimits{indit}\\left(\\sigma \\right) \\right) $ .",
"The equivalence proof is completed by showing that taking connected components in the graph $G\\left( \\pi \\oplus \\sigma \\right) $ is equivalent to taking the RST closure of $\\left( \\operatornamewithlimits{dit}\\left(\\pi \\right) \\cap \\operatornamewithlimits{dit}\\left( \\sigma \\right) \\right) \\cup \\left(\\operatornamewithlimits{indit}\\left( \\pi \\right) \\cap \\operatornamewithlimits{indit}\\left(\\sigma \\right) \\right) $ .",
"The elements $u$ and $u^{\\prime }$ are in the same connected component of $G\\left( \\pi \\oplus \\sigma \\right) $ iff there is a chain of links $u=u_{0}-u_{1}-...-u_{n-1}-u_{n}=u^{\\prime }$ in the graph $G\\left( \\pi \\oplus \\sigma \\right) $ so each link has to be originally labelled $T_{\\pi },T_{\\sigma }$ or $F_{\\pi },F_{\\sigma }$ in the graph on $K\\left(U\\right) $ .",
"But the condition for $\\left( u,u^{\\prime }\\right) $ to be included in the RST closure $\\overline{\\left( \\operatornamewithlimits{dit}\\left(\\pi \\right) \\cap \\operatornamewithlimits{dit}\\left( \\sigma \\right) \\right) \\cup \\left(\\operatornamewithlimits{indit}\\left( \\pi \\right) \\cap \\operatornamewithlimits{indit}\\left(\\sigma \\right) \\right) }$ is that there is a chain of pairs $\\left(u,u_{1}\\right) ,\\left( u_{1},u_{2}\\right) ,...,\\left( u_{n-1},u^{\\prime }\\right) $ such that each pair is either in $\\operatornamewithlimits{dit}\\left(\\pi \\right) \\cap \\operatornamewithlimits{dit}\\left( \\sigma \\right) $ or in $\\operatornamewithlimits{indit}\\left( \\pi \\right) \\cap \\operatornamewithlimits{indit}\\left(\\sigma \\right) $ .",
"Hence the two methods give the same result.",
"The example suffices to illustrate the general closure-theoretic method and its equivalence to the graph-theoretic method of defining Boolean operations on partitions."
],
[
"Relationships between Boolean operations on partitions",
"For two subset variables, there are $2^{4}=16$ binary Boolean operations on subsets–corresponding to the sixteen ways to fill in the truth table for a binary Boolean operation.",
"Any compound Boolean function of two variables will be truth-table equivalent to one of the sixteen binary Boolean operations.",
"For instance, the Pierce's Law formula $\\left( \\left( Q\\Rightarrow P\\right)\\Rightarrow Q\\right) \\Rightarrow Q$ defines a compound binary operation that is equivalent to the constant function $T$ since it is a subset tautology.",
"Certain subsets of the sixteen binary operations suffice to define all the binary operations, e.g., $\\lnot $ and $\\vee $ .",
"Matters are rather different for the Boolean operations on partitions.",
"Using the graph-theoretic or the closure-theoretic method, partition versions of sixteen binary Boolean operations are easily defined.",
"And certain combinations of the sixteen operations suffice to define all sixteen, e.g., $\\vee $ , $\\wedge $ , $\\Rightarrow $ , and $\\oplus $ [5].",
"But when the sixteen operations are compounded, still keeping to two variables, then the resulting binary partition operations does not necessarily reduce to one of the sixteen–due to the complicated compounding of the closure operations.",
"For instance, the Pierce's Law formula $\\left( \\left(\\sigma \\Rightarrow \\pi \\right) \\Rightarrow \\sigma \\right) \\Rightarrow \\sigma $ for partitions is not equivalent to the constant function $\\mathbf {1}$ since it is not a partition tautology.",
"The topic of the total number of binary operations on partitions obtained by compounding the sixteen basic binary Boolean operations is one of many topics in partition logic that awaits future research."
],
[
"Concluding Remarks",
"In conclusion, perhaps some remarks are in order as to why it took so long to extend the Boolean operations to partitions.",
"The Boolean operations are normally associated with subsets of a set or, more specifically, with propositions.",
"Boole originally defined his logic as the logic of subsets [2] of a universe set.",
"It is then a theorem that the same set of subset tautologies is obtained as the truth-table tautologies.",
"Perhaps because “logic” has been historically associated with propositions, the texts in mathematical logic throughout the twentieth century (to the author's knowledge) ignored the Boolean logic of subsets and started with the special case of the logic of propositions and then took the truth-table characterization as the definition of a tautology.",
"By the middle of the twentieth century, category theory was defined [4] and the category-theoretic duality was established between subobjects and quotient objects, e.g., between subsets of $U$ and quotient sets (or equivalently equivalence relations or partitions) of $U$ .",
"The conceptual cost of restricting subset logic to the special case of propositional logic is that subsets have the category-theoretic dual concept of partitions while propositions have no such dual concept.",
"Hence the focus on “propositional logic” did not lead to the search for the dual logic of partitions ([5], [6]) or to the simple and natural application of Boolean operations to partitions as well as subsets–which has been our topic here."
]
] | 1906.04539 |
[
[
"Almost Optimal Semi-streaming Maximization for k-Extendible Systems"
],
[
"Abstract In this paper we consider the problem of finding a maximum weight set subject to a $k$-extendible constraint in the data stream model.",
"The only non-trivial algorithm known for this problem to date---to the best of our knowledge---is a semi-streaming $k^2(1 + \\varepsilon)$-approximation algorithm (Crouch and Stubbs, 2014), but semi-streaming $O(k)$-approximation algorithms are known for many restricted cases of this general problem.",
"In this paper, we close most of this gap by presenting a semi-streaming $O(k \\log k)$-approximation algorithm for the general problem, which is almost the best possible even in the offline setting (Feldman et al., 2017)."
],
[
"positioning arrows backgrounds calc,arrows.meta every node=[font=] [ framed, align=center, node distance=0.25cm, roundnode/.style=circle, draw=black!60, minimum width=width(\"-2L+2\"),, blanknode/.style=circle, draw=white!60, minimum width=width(\"-2L+2\"),font=, ] blanknode] (-2L) $\\vdots $ ; blanknode] (-2L+1) [right=of -2L] $\\vdots $ ; blanknode] (-2L+2) [right=of -2L+1] $\\vdots $ ; blanknode] (qdots0) [right=of -2L+2] ; roundnode, fill=green] (-L) [below=of -2L] -$\\ell $ ; roundnode, fill=orange] (-L+1) [right=of -L] -$\\ell $ +1; roundnode, fill=yellow] (-L+2) [right=of -L+1] -$\\ell $ +2; blanknode] (qdots1) [right=of -L+2] $\\cdots $ ; roundnode, fill=red] (-1) [right=of qdots1] -1; roundnode, fill=green] (0) [below=of -L] 0; roundnode, fill=orange] (1) [right=of 0] 1; roundnode, fill=yellow] (2) [right=of 1] 2; blanknode] (qdots2) [right=of 2] $\\cdots $ ; roundnode, fill=red] (L-1) [right=of qdots2] $\\ell $ -1; roundnode, fill=green] (L) [below=of 0] $\\ell $ ; roundnode, fill=orange] (L+1) [right=of L] $\\ell $ +1; roundnode, fill=yellow] (L+2) [right=of L+1] $\\ell $ +2; blanknode] (qdots3) [right=of L+2] $\\cdots $ ; roundnode, fill=red] (2L-1) [right=of qdots3] 2$\\ell $ -1; blanknode] (2L) [below=of L] $\\vdots $ ; blanknode] (2L+1) [right=of 2L] $\\vdots $ ; blanknode] (2L+2) [right=of 2L+1] $\\vdots $ ; blanknode] (qdots4) [right=of 2L+2] ; blanknode] (-L-1) [above=of -1] $\\vdots $ ; blanknode] (3L-1) [below=of 2L-1] $\\vdots $ ;"
]
] | 1906.04449 |
[
[
"Pykg2vec: A Python Library for Knowledge Graph Embedding"
],
[
"Abstract Pykg2vec is an open-source Python library for learning the representations of the entities and relations in knowledge graphs.",
"Pykg2vec's flexible and modular software architecture currently implements 16 state-of-the-art knowledge graph embedding algorithms, and is designed to easily incorporate new algorithms.",
"The goal of pykg2vec is to provide a practical and educational platform to accelerate research in knowledge graph representation learning.",
"Pykg2vec is built on top of TensorFlow and Python's multiprocessing framework and provides modules for batch generation, Bayesian hyperparameter optimization, mean rank evaluation, embedding, and result visualization.",
"Pykg2vec is released under the MIT License and is also available in the Python Package Index (PyPI).",
"The source code of pykg2vec is available at https://github.com/Sujit-O/pykg2vec."
],
[
"Introduction",
"In recent years, Knowledge Graph Embedding (KGE) methods have been applied in benchmark datasets including Wikidata ([5]), Freebase ([2]), DBpedia ([1]), and YAGO ([8]).",
"Applications of KGE methods include fact prediction, question answering, and recommender systems.",
"KGE is an active area of research and many authors have provided reference software implementations.",
"However, most of these are standalone reference implementations and therefore it is difficult and time-consuming to: (i) find the source code; (ii) adapt the source code to new datasets; (iii) correctly parameterize the models; and (iv) compare against other methods.",
"Recently, this problem has been partially addressed by libraries such as OpenKE [6] and AmpliGraph [4] that provide a framework common to several KGE methods.",
"However, these frameworks take different perspectives, make specific assumptions, and thus the resulting implementations diverge substantially from the original architectures.",
"Furthermore, these libraries often force the user to use preset hyperparameters, or make implicit use of golden hyperparameters, and thus make it tedious and time-consuming to adapt the models to new datasets.",
"This paper presents pykg2vec, a single Python library with 16 state-of-the-art KGE methods.",
"The goals of pykg2vec are to be practical and educational.",
"The practical value is achieved through: (a) proper use of GPUs and CPUs; (b) a set of tools to automate the discovery of golden hyperparameters; and (c) a set of visualization tools for the training and results of the embeddings.",
"The educational value is achieved through: (d) a modular and flexible software architecture and KGE pipeline; and (e) access to a large number of state-of-the-art KGE models."
],
[
"Knowledge Graph Embedding Methods",
"A knowledge graph contains a set of entities $\\mathbb {E}$ and relations $\\mathbb {R}$ between entities.",
"The set of facts $\\mathbb {D}^+$ in the knowledge graph are represented in the form of triples $(h, r, t)$ , where $h,t\\in \\mathbb {E}$ are referred to as the head (or subject) and the tail (or object) entities, and $r\\in \\mathbb {R}$ is referred to as the relationship (or predicate).",
"The problem of KGE is in finding a function that learns the embeddings of triples using low dimensional vectors such that it preserves structural information, $f:\\mathbb {D}^+\\rightarrow \\mathbb {R}^d$ .",
"To accomplish this, the general principle is to enforce the learning of entities and relationships to be compatible with the information in $\\mathbb {D}^+$ .",
"The representation choices include deterministic point ([3]), multivariate Gaussian distribution ([7]), or complex number ([9]).",
"Under the Open World Assumption (OWA), a set of unseen negative triplets, $\\mathbb {D}^-$ , are sampled from positive triples $\\mathbb {D}^+$ by either corrupting the head or tail entity.",
"Then, a scoring function, $f_r(h, t)$ is defined to reward the positive triples and penalize the negative triples.",
"Finally, an optimization algorithm is used to minimize or maximize the scoring function.",
"KGE methods are often evaluated in terms of their capability of predicting the missing entities in negative triples $(?, r, t)$ or $(h, r, ?",
")$ , or predicting whether an unseen fact is true or not.",
"The evaluation metrics include the rank of the answer in the predicted list (mean rank), and the ratio of answers ranked top-k in the list (hit-k ratio)."
],
[
"Software Architecture",
"The pykg2vec library is built using Python and TensorFlow.",
"TensorFlow allows the computations to be assigned on both GPU and CPU.",
"In addition to the main model training process, pykg2vec utilizes multi-processing for generating mini-batches and performing an evaluation to reduce the total execution time.",
"The various components of the library (see Figure REF ) are as follows: Figure: Pykg2vec software architecture[align=left,leftmargin=*] KG Controller: handles all the low-level parsing tasks such as finding the total unique set of entities and relations; creating ordinal encoding maps; generating training, testing and validation triples; and caching the dataset data on disk to optimize tasks that involve repetitive model testing.",
"Batch Generator: consists of multiple concurrent processes that manipulate and create mini-batches of data.",
"These mini-batches are pushed to a queue to be processed by the models implemented in TensorFlow.",
"The batch generator runs independently so that there is a low latency for feeding the data to the training module running on the GPU.",
"Core Models: consists of 16 KGE algorithms implemented as Python modules in TensorFlow.",
"Each module consists of a modular description of the inputs, outputs, loss function, and embedding operations.",
"Each model is provided with configuration files that define its hyperparameters.",
"Configuration: provides the necessary configuration to parse the datasets and also consists of the baseline hyperparameters for the KGE algorithms as presented in the original research papers.",
"Trainer and Evaluator: the Trainer module is responsible for taking an instance of the KGE model, the respective hyperparameter configuration, and input from the batch generator to train the algorithms.",
"The Evaluator module performs link prediction and provides the respective accuracy in terms of mean ranks and filtered mean ranks.",
"Visualization: plots training loss and common metrics used in KGE tasks.",
"To facilitate model analysis, it also visualizes the latent representations of entities and relations on the 2D plane using t-SNE based dimensionality reduction.",
"Bayesian Optimizer: pykg2vec uses a Bayesian hyperparameter optimizer to find a golden hyperparameter set.",
"This feature is more efficient than brute-force based approaches."
],
[
"Usage Examples",
"Pykg2vec provides users with two utilization examples ($train.py$ and $tune\\_model.py$ ) available in the pykg2vec/example folder.",
"Training is performed with the following script: $ python train.py -h # Check the manual for input arguments$ python train.py -mn transe Train TransE To apply the best setting described in the paper, the following script can be invoked.",
"$ python train.py -mn transe -ghp True # Train using the golden setting$ Some of the results plotted after training $TransE$ and $TransH$ are shown in Figure REF .",
"Figure: Visualization examples from Freebase15KTo tune the model, the following script can be invoked.",
"$ python tune_model.py -mn transe # Tune hyperparametersFound Golden Setting: # dummy tuning result{^{\\prime }L1_flag^{\\prime }: False, ^{\\prime }batch_size^{\\prime }: 256, ^{\\prime }epochs^{\\prime }: 5, ^{\\prime }hidden_size^{\\prime }: 32,^{\\prime }learning_rate^{\\prime }: 0.001, ^{\\prime }margin^{\\prime }: 0.4, ^{\\prime }opt^{\\prime }: ^{\\prime }sgd^{\\prime },^{\\prime }samp^{\\prime }: ^{\\prime }bern^{\\prime }}$"
],
[
"Conclusion",
"Pykg2vec is a Python library with extensive documentation that includes the implementations of a variety of state-of-the-art Knowledge Graph Embedding methods and modular building blocks of the embedding pipeline.",
"This library aims to help researchers and developers to quickly test algorithms against their custom knowledge base or utilize the modular blocks to adapt the library for their custom algorithms."
]
] | 1906.04239 |
[
[
"N\\'{e}el and stripe ordering from spin-orbital entanglement in\n $\\alpha$-Sr$_2$CrO$_4$"
],
[
"Abstract The rich phenomenology engendered by the coupling between the spin and orbital degrees of freedom has become appreciated as a key feature of many strongly-correlated electron systems.",
"The resulting emergent physics is particularly prominent in a number of materials, from Fe-based unconventional superconductors to transition metal oxides, including manganites and vanadates.",
"Here, we investigate the electronic ground states of $\\alpha$-Sr$_2$CrO$_4$, a compound that is a rare embodiment of the spin-1 Kugel-Khomskii model on the square lattice -- a paradigmatic platform to capture the physics of coupled magnetic and orbital electronic orders.",
"We have used resonant X-ray diffraction at the Cr-$K$ edge to reveal N\\'{e}el magnetic order at the in-plane wavevector $\\mathbf{Q}_N = (1/2, 1/2)$ below $T_N = 112$ K, as well as an additional electronic order at the 'stripe' wavevector $\\mathbf{Q}_s = (1/2, 0)$ below T$_s$ $ \\sim 50$ K. These findings are examined within the framework of the Kugel-Khomskii model by a combination of mean-field and Monte-Carlo approaches, which supports the stability of the spin N\\'{e}el phase with subsequent lower-temperature stripe orbital ordering, revealing a candidate mechanism for the experimentally observed peak at $\\mathbf{Q}_s$.",
"On the basis of these findings, we propose that $\\alpha$-Sr$_2$CrO$_4$ serves as a new platform in which to investigate multi-orbital physics and its role in the low-temperature phases of Mott insulators."
],
[
" Néel and stripe ordering from spin-orbital entanglement in $\\alpha $ -Sr$_2$ CrO$_4$ Z. H. Zhu$^1$ [email protected] W. Hu$^2$ C. A. Occhialini$^1$ J. Li$^1$ J. Pelliciari$^1$ C. S. Nelson$^3$ M. R. Norman$^4$ Q. Si$^2$ R. Comin$^1$ [email protected] $^1$ Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA $^2$ Department of Physics and Astronomy, Rice Center for Quantum Materials, Rice University, Houston, TX 77005, USA $^3$ National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, NY 11973, USA $^4$ Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA The rich phenomenology engendered by the coupling between the spin and orbital degrees of freedom has become appreciated as a key feature of many strongly-correlated electron systems.",
"The resulting emergent physics is particularly prominent in a number of materials, from Fe-based unconventional superconductors to transition metal oxides, including manganites and vanadates.",
"Here, we investigate the electronic ground states of $\\alpha $ -Sr$_2$ CrO$_4$ , a compound that is a rare embodiment of the spin-1 Kugel-Khomskii model on the square lattice – a paradigmatic platform to capture the physics of coupled magnetic and orbital electronic orders.",
"We have used resonant X-ray diffraction at the Cr-$K$ edge to reveal Néel magnetic order at the in-plane wavevector $\\mathbf {Q}_N = (1/2, 1/2)$ below $T_N = 112$ K, as well as an additional electronic order at the “stripe” wavevector $\\mathbf {Q}_s = (1/2, 0)$ below T$_s$ $ \\sim 50$ K. These findings are examined within the framework of the Kugel-Khomskii model by a combination of mean-field and Monte-Carlo approaches, which supports the stability of the spin Néel phase with subsequent lower-temperature stripe orbital ordering, revealing a candidate mechanism for the experimentally observed peak at $\\mathbf {Q}_s$ .",
"On the basis of these findings, we propose that $\\alpha $ -Sr$_2$ CrO$_4$ serves as a new platform in which to investigate multi-orbital physics and its role in the low-temperature phases of Mott insulators.",
"Multi-orbital physics plays an important role for several collective electronic phenomena in strongly-correlated systems, including high-$T_c$ superconductivity and exotic magnetism [1], [2].",
"Some model materials, such as LaMnO$_3$ [3], KCuF$_3$ [4], and YVO$_3$ [5], have enabled the exploration of the active role played by the orbital degrees of freedom in determining the symmetry of the electronic ground state.",
"In particular, the essential ingredients of unconventional superconductivity in the iron-based superconductors (FeSCs) are purported to involve multi-orbital electronic states and their interplay with magnetism [6], [7], [8], in remarkable contrast to the effective single-orbital nature of cuprate superconductors [9].",
"A few chromate perovskites ACrO$_3$ (A =Ca, Sr, and Pb) with a Cr$^{4+}$ valence state were synthesized by several groups four decades ago [10], [11], [12].",
"Recent studies on CaCrO$_3$ and SrCrO$_3$ have revealed novel physics, drawing more attention to these chromate compounds [13], [14], [15].",
"For example, both compounds are antiferromagnetic conductors, but the exact nature of the ground state has remained elusive.",
"An interesting orbital ordering transition with electronic phase coexistence has been discovered in SrCrO$_3$ [14], though it is unclear if the orbital degree of freedom is essential to carrier itinerancy within the antiferromagnetic ground state.",
"Moreover, despite the isovalency of Ba and Sr, the sister compound BaCrO$_3$ was found to be an antiferromagnetic insulator [16], [17], with theoretical studies proposing that the mechanism for the insulating state involves, again, some form of orbital ordering [18], [19].",
"A recent study of bilayered Sr$_3$ Cr$_2$ O$_7$ revealed the presence of an exotic ordered phase of orbital singlets arising from the interplay between the spin and orbital degrees of freedom [20].",
"It is thus not surprising that similar phenomena at the nexus between magnetism and orbital physics were proposed also for the single-layer ($n=1$ ) member of the Ruddlesden-Popper series of chromates, Sr$_2$ CrO$_4$ [21].",
"However, the microscopic nature of the ground state of this compound remains unknown.",
"In the present work, we focus on $\\alpha $ -Sr$_2$ CrO$_4$ , which crystallizes in the K$_2$ NiF$_4$ -type structure that makes it isostructural to La$_2$ CuO$_4$ .",
"With its rare Cr$^{4+}$ oxidation state, it realizes a unique $3d^2$ electronic configuration among layered oxides, resulting in active $t_{2g}$ orbitals with total spin $S=1$ on a square lattice.",
"Despite several studies characterizing this unique material [22], [23], [24], a clear picture of its physical properties has remained elusive, partly owing to known challenges with the synthesis of bulk single crystals.",
"Successful growth of polycrystalline samples of $\\alpha $ -Sr$_2$ CrO$_4$ by means of high-pressure ($> 5$ GPa) and high-temperature (1500 $^{\\circ }$ C) synthesis has been reported by several groups[22], [25], [26].",
"Magnetization characterization of these polycrystalline samples reveal a sharp antiferromagnetic transition at T$_N$ $\\approx 112$ K[22], which was further confirmed by muon spin rotation ($\\mu ^+$ SR) measurements [23], [24].",
"The resistivity of sintered pellets of $\\alpha $ -Sr$_2$ CrO$_4$ reveal insulating behavior [22].",
"In addition, single-crystalline samples of $\\alpha $ -Sr$_2$ CrO$_4$ have been synthesized as thin films using pulsed laser deposition (PLD), which are determined by optical conductivity to be Mott-Hubbard type insulators with a charge gap $\\Delta $ $\\approx $ 0.3 eV [27].",
"While previous studies provide evidence for an antiferromagnetic insulating ground state in $\\alpha $ -Sr$_2$ CrO$_4$ , no direct reciprocal space studies of the long-range electronic order in these compounds have been reported to date.",
"Figure: Resonant X-ray scattering measurements of α\\alpha -Sr 2 _2CrO 4 _4 at the Cr-KK edge.",
"A: unit cell of α\\alpha -Sr 2 _2CrO 4 _4, which is tetragonal with I4/mmm space group at room temperature.",
"B: Reciprocal space cartography of scattering scans in the H,K\\left( H, K \\right) plane.",
"C1-C4: individual RXS scans across the symmetry-equivalent ordering vectors ±0.5,±0.5,4.84\\left(\\pm 0.5, \\pm 0.5, 4.84 \\right) (see insets).To address this open question, we have synthesized single-crystalline films of $\\alpha $ -Sr$_2$ CrO$_4$ with the aim of resolving its magnetic ground state and to search for other ordered phases within the electronic degrees of freedom.",
"We have sought reciprocal space signatures of electronic long-range ordering using resonant x-ray scattering (RXS) measurements at the Cr-$K$ edge (with photon energy $E_{ph} \\approx 6.0$ keV).",
"At the Cr-$K$ edge, photons that are tuned to the Cr-$1s \\rightarrow 4p$ transition become resonantly sensitive to the charge, spin, and orbital degrees of freedom at the Cr site, a capability that has made RXS a valuable probe to study electronic ordering in transition metal oxides [28], [29].",
"Thin film samples of $c$ -oriented Sr$_2$ CrO$_4$ have been grown on (001) SrTiO$_3$ substrates using pulsed laser deposition (PLD) from a KrF excimer laser ($\\lambda = 248$ nm).",
"Deposition was performed under high vacuum with a laser pulse energy and repetition rate of 400 mJ and 4 Hz, respectively, with the substrate heated to 900 $^{\\circ }$ C. The laser target contained a stoichiometric mixture of SrCO$_3$ and Cr$_2$ O$_3$ prepared by a solid state reaction at 920 $^{\\circ }$ C under a constant N$_2$ flow for 24 hours.",
"The products of this reaction were ground and pressed into a pellet, then sintered under an Ar flow at 1200 $^{\\circ }$ C for 24 hours.",
"The main phase of the target was determined to be $\\beta $ -Sr$_2$ CrO$_4$ , however the final $\\sim 100$ nm film was confirmed by X-ray diffraction to be pure $\\alpha $ -Sr$_2$ CrO$_4$ , with no traces of the sister phase (see Supplementary Information, in particular Fig.",
"S1, for additional details).",
"Resonant X-ray measurements were performed at the Integrated In-Situ and Resonant X-ray Studies (ISR) beamline- 4-ID of NSLS-II at Brookhaven National Laboratory.",
"Figure: Nature and characteristics of the Néel-ordered state.",
"A: energy-integrated sample rocking scans across the resonant reflection 𝐐 N =1/2,1/2,L{\\mathbf {Q}}_{N} = \\left( 1/2, 1/2, L \\right) at various temperatures.",
"B: temperature dependence of the integrated scattering intensity around 𝐐 N {\\mathbf {Q}}_{N} (markers) and of the magnetic susceptibility χT\\chi \\left( T \\right), from (gray line).",
"C: energy dependence of the integrated RXS intensity around 𝐐 N {\\mathbf {Q}}_{N} (purple diamonds) and near-edge X-ray absorption scan (XANES, orange trace), both acquired at 20 K; the dash-dotted gray and teal lines represent, respectively, simulations of the XANES spectrum and the RXS signals from magnetic scattering using FDMNES program (see text and Supplementary Information).",
"D: azimuthal angle (ψ\\psi ) dependence of the integrated intensity (green squares) of the 𝐐 N {\\mathbf {Q}}_{N} reflection, with the blue curve representing the simulated azimuthal dependence expected for the magnetic model described in the text.Figure REF A shows the unit cell of $\\alpha $ -Sr$_2$ CrO$_4$ , which is tetragonal with space group I4/mmm at room temperature.",
"Throughout this paper, reciprocal lattice vectors (H,K,L) are scaled to reciprocal lattice units (r.l.u) for the room temperature tetragonal structure (that is, in units of $2\\pi / a$ , $2\\pi / a$ , $2\\pi / c$ , where $a = b = 3.87$ Å, $c = 12.55$ Å).",
"Resonant X-ray scattering scans were performed in the reciprocal $\\left( H, K \\right)$ plane parallel to the crystallographic CrO$_2$ layers, as illustrated in Figure REF B.",
"We have surveyed the high-symmetry directions $\\left[ H, 0 \\right]$ and $\\left[ H, H \\right]$ using incident X-rays tuned to the Cr-$K$ edge resonance E = 6.009 keV to resonantly enhance Bragg reflections of electronic nature.",
"Under these experimental conditions, we have observed four well-defined, symmetry-equivalent diffraction peaks at the Néel vectors ${\\mathbf {Q}}_{N} = \\left( \\pm 1/2, \\pm 1/2, L \\right)$ (r.l.u), with $L \\sim 4.84$ , as shown in Fig.",
"REF C1-C4.",
"These diffraction signatures reveal the presence of a $\\sqrt{2} \\times \\sqrt{2}$ electronic superlattice which is forbidden by the high-symmetry structure.",
"The disappearance of the superlattice reflection away from the Cr-$K$ resonance suggests that it results from a reordering of the electronic subspace.",
"The in-plane correlation length $\\xi _{ab}$ = a$(\\pi FWHM)^{-1}$ $\\sim $ 4000 Å($\\sim $ 1000 unit cells) is much greater than the out-of-plane correlation length $\\xi _{c}$ $\\sim $ 250 Å($\\sim $ 20 unit cells), though we note that the latter could be limited by the approximate 1000 Åthickness of the films in the $c$ direction.",
"The non-integer out-of-plane projection of the ordering vector ($L \\approx 4.84$ ) reveals an incommensurate nature of the ordering pattern along the c axis.",
"To uncover the nature of these reflections, we have studied their evolution as a function of temperature, photon energy, and azimuthal angle.",
"Figure REF A shows a series of representative rocking curves at ${\\mathbf {Q}}_{N} = \\left( 1/2, 1/2, L \\right)$ for various temperatures, with the integrated intensity extracted from Gaussian fits and plotted in Fig.",
"REF B.",
"A clear transition occurs at $T \\approx 110$ K, which coincides with the reported Néel temperature for polycrystalline samples revealed by bulk magnetometry (see susceptibility trace from Ref.",
"[22] in Fig.",
"REF B), as well as muon spin rotation measurements [23], [24].",
"In Fig.",
"REF C, we plot the integrated intensity of the rocking curve for the $ {\\mathbf {Q}}_{N} $ reflection as a function of the photon energy at $T = 20$ K, showing a clear resonant enhancement around $E = 6.009$ keV, at the main Cr-$K$ edge, measured using near edge X-ray absorption (XANES) spectrum measured in fluorescence mode.",
"Given the close correspondence of the transition temperature of this peak with the reported Néel temperature, we performed ab initio calculations using the FDMNES program to model the RXS intensity resulting from Néel-type magnetic order .",
"The RXS intensity and XANES curves are overlaid onto the experimental data points in Fig.",
"REF C, confirming the resonant enhancement of the magnetic peak at the Cr-$1s \\rightarrow 4p$ transition (for more details, see Supplementary Information).",
"To further examine the symmetry of the ordered phase, we also study the azimuthal angle dependence of the integrated intensity at $ {\\mathbf {Q}}_{N} $ .",
"RXS scans across $ {\\mathbf {Q}}_{N} $ were collected for different values of the azimuthal $\\psi $ , which corresponds to a rotation around the axis parallel to $ {\\mathbf {Q}}_{N} $ .",
"Here, an azimuthal angle $\\psi = 0$ coincides with a sample orientation that has the $\\left( 1, 1, L \\right)$ crystallographic direction lying in the scattering plane.",
"The clear 2-fold modulation of the RXS intensity as a function of the azimuthal angle, shown in Fig.",
"REF D, is consistent with a magnetic structure with moments oriented along $\\left( 1, -1, 0 \\right)$ , i.e.",
"perpendicular to the in-plane projection of the Néel ordering vector.",
"The collective information, including the resonance energy profile, azimuthal symmetry and polarization restrictions (from the incoming photons) are most consistent with magnetic scattering.",
"This interpretation is further bolstered by the close correspondence between the observed onset temperature and previous reports of the Néel transition [22], [23], [24].",
"We also note that, among the possible high-symmetry magnetic space subgroups, the one determined to be most consistent with the azimuthal symmetry is equivalent to that found in other isostructural compounds, such as La$_2$ CuO$_4$ .",
"Figure: Stripe-like electronic ordering along H,0\\left[ H, 0 \\right].",
"A: representative momentum scans across 𝐐 s =1/2,0,L {\\mathbf {Q}}_{s} = \\left( 1/2, 0, L \\right) at a photon energy E=6.009E = 6.009 keV.",
"B: integrated RXS intensity at 𝐐 s {\\mathbf {Q}}_{s} vs. temperature; the inset shows a series of RXS rocking curves across 𝐐 s {\\mathbf {Q}}_{s} for a few representative temperatures.Having identified the nature and microscopic traits of the ordered state below $T_N \\sim 110$ K, we have focused on the $\\left[ H, 0 \\right]$ reciprocal space direction, to search for additional ordering instabilities.",
"This further exploration was motivated by previous heat capacity data hinting at a second ordering transition around 140 K, whose nature has remained undisclosed [22].",
"Consistently with a dual-instability scenario, we have observed a resonant reflection at the wavevector for period-2 stripe order $ {\\mathbf {Q}}_{s} = \\left( 1/2, 0, L \\right)$ (with $L = 4.84$ here, as well) as shown in Figure REF A.",
"Similar to the reflection at the Néel vector $ {\\mathbf {Q}}_{N}$ , the diffraction peak at the stripe ordering vector also exhibits a resonant enhancement as the photon energy is tuned to $E = 6.009$ keV (see Supplementary Information).",
"The integrated intensity of the rocking curves as a function of temperature is shown in Fig.",
"REF B (RXS scans at representative temperatures are reported in the inset).",
"Surprisingly, the transition was seen to occur around $T_s \\sim 50$ K, namely at a significantly lower temperature than the proposed orbital ordering transition.",
"Here, we should note that a remnant scattering signal persists above the transition and up to the highest measured temperature (200 K), yet shows no variation around 140 K. This unusual temperature dependence has been noted in other transition metal compounds with coupled spin and orbital order parameters (OPs), in particular KCuF$_3$ , though we cannot rule out additional OPs of possibly different symmetry contributing to the remnant signal above the clear transition at $T_s$ .",
"We reconcile the puzzle of this second transition at $T_s$ by noting that the ordering temperature is close to that of the appearance of a second oscillatory component near 45 K as recorded by zero-field muon spin rotation (ZF-$\\mu ^+$ SR) measurements on polycrystalline samples [24].",
"Figure: Phase diagrams at zero and nonzero temperatures.",
"A: Thephase diagram at zero temperature for the Kugel-Khomskii model with J 1 =J 4 =1.0J_1=J_4=1.0, based on a site-factorized wavefunction approach.",
"Three stable phases appear in the phase diagram, including S Ne ´el T CAFO S_{N\\acute{e}el}T_{CAFO}, S Ne ´el T AFO S_{N\\acute{e}el}T_{AFO}, and S CAFM T AFO S_{CAFM}T_{AFO}.",
"The inset is a schematic illustration of the S Ne ´el T CAFO S_{N\\acute{e}el}T_{CAFO} ordered state.",
"The blue and red ovals represent |xz〉\\vert xz \\rangle and |yz〉\\vert yz \\rangle orbitals, respectively.",
"The black arrows denote the spin state on the |xz〉\\vert xz \\rangle or |yz〉\\vert yz \\rangle orbital.",
"S Ne ´el _{N\\acute{e}el} and S CAFM _{CAFM} stand for Néel order and stripe antiferromagnetic order, respectively.",
"T CAFO T_{CAFO} marks a stripe-like orbital order with momentum (π,0)/(0,π)(\\pi ,0)/(0,\\pi ),and T AFO _{AFO} represents Néel-like orbital order with momentum (π,π)(\\pi ,\\pi ).",
"B: The temperature-dependence of the magnetic (m S m_S) and orbital (m T m_T) order parameters from classical Monte Carlo simulations of the phase S Ne ´el T CAFO S_{N\\acute{e}el}T_{CAFO} with J 1 =J 4 =1.0,J 6 =0.4,J 8 =0.0J_1=J_4=1.0,J_6=0.4,J_8=0.0 on L=24,32,40L=24,32,40 clusters (marked by the black star in panel A).",
"The model parameters J i J_i are set as J i =t i 2 /UJ_i=t^2_i/U, where UU is the on-site interaction and t i t_i represent hopping amplitudes between orbitals up to next-nearest neighbors.",
"J 1 J_1 and J 4 J_4 are the nearest-neighbor couplings, and J 6 J_6 and J 8 J_8 the next-nearest-neighbor couplings.",
"Further details are given in the Supplementary Information.To understand the microscopic physics behind the observed scattering peaks, we examine an effective spin-1 Kugel-Khomskii model for $\\alpha $ -Sr$_2$ CrO$_4$ (see Supplementary Information for additional details).",
"In this compound, the necessarily high-spin $3d^2$ electronic configuration form an active $t_{2g}$ orbital subspace.",
"Of particular interest are results from density functional theory (DFT) calculations showing a reversed crystal field splitting compared to typical tetragonal systems with an elongated c-axis lattice parameter [21].",
"Within this scenario, two electrons occupy the $\\left\\lbrace \\vert xy \\rangle , \\vert xz \\rangle , \\vert yz \\rangle \\right\\rbrace $ $t_{2g}$ orbital manifold, with a proposed negative energy level splitting $\\Delta \\sim -0.6eV$ between the low-lying $\\vert xy \\rangle $ and $\\vert xz \\rangle $ /$ \\vert yz \\rangle $ orbitals.",
"This reversed crystal field leaves a single electron in the two-fold degenerate $\\left\\lbrace \\vert xz \\rangle , \\vert yz \\rangle \\right\\rbrace $ submanifold, setting the stage for active orbital physics as encoded in a pseudospin-1/2 degree of freedom.",
"In order to elucidate the essential physics arising from the interplay of the spin and orbital degrees of freedom in $\\alpha $ -Sr$_2$ CrO$_4$ , , , we consider the spin-1 Kugel-Khomskii Hamiltonian as the effective model: $\\small H_{KK}^{(i,j)} = -\\frac{1}{3}({\\bf S}_i\\cdot {\\bf S}_j+2)Q^{(1)}({\\bf T}_i,{\\bf T}_j)+ \\frac{1}{3}({\\bf S}_i\\cdot {\\bf S}_j-1) Q^{(2)}({\\bf T}_i,{\\bf T}_j),$ with $&&Q^{(n)}({\\bf T}_i,{\\bf T}_j)=\\nonumber \\\\&&f_{zz}^{(n)}T_{i}^{z}T_{j}^{z}+\\frac{1}{2} f_{+-}^{(n)}(T_{i}^{+}T_{j}^{-}+T_{i}^{-}T_{j}^{+})+f_{z}^{(n)}(T_{i}^{z}+T_{j}^{z})\\nonumber \\\\&&+\\frac{1}{2} f_{++}^{(n)}(T_{i}^{+}T_{j}^{+}+T_{i}^{-}T_{j}^{-})+f_{zx}^{(n)}(T_{i}^{z}T_{j}^{x}+T_{i}^{x}T_{j}^{z})\\nonumber \\\\&&+f_{x}^{(n)}(T_{i}^{x}+T_{j}^{x}) +f_{0}^{(n)},$ where ${\\bf S}_i$ and ${\\bf T}_i$ are the spin-1 and pseudospin-$1/2$ operators, respectively.",
"$f^{(n)}$ with $n=1,2$ are functions of microscopic parameters that include the hopping amplitudes between orbitals up to next-nearest neighbor and the onsite Hubbard interactions and Hund's coupling (see Supplementary Information for details).",
"We have performed calculations using a combination of mean-field and classical Monte Carlo methods.",
"As shown in Fig.",
"REF A, we find S$_{N\\acute{e}el}$ T$_{CAFO}$ , a Néel order accompanied by a stripe-like orbital order, over an extended parameter regime in the phase diagram.",
"This phase corresponds to the same ordering vector as found in RXS experiments.",
"In Fig.REF B, we show the results of classical Monte Carlo simulations for the evolution of the magnetic and orbital order parameters vs. temperature for the phase S$_{N\\acute{e}el}$ T$_{CAFO}$ .",
"The numerical results clearly show that, as temperature is lowered, the magnetic order parameter develops first, which is followed by the emergence of the stripe-like orbital order.",
"The proposed theoretical description captures the salient features of the experimental observations, including the ordering wavevectors ($\\mathbf {Q}_s$ and $\\mathbf {Q}_N$ ) and their respective transition temperatures.",
"The Néel spin order setting in at the higher-temperature transition is consistent with our RXS measurements (see Supplementary Information) as well as the magnetic measurements in polycrystalline samples [22], [24].",
"In addition, because the orbital order below the lower-temperature transition develops in a time-reversal-broken background, it is expected to develop an additional orbital moment at the same wavevector, in the presence of spin-orbit coupling.",
"Given that the involved orbitals are $\\vert xz \\rangle $ and $\\vert yz \\rangle $ , the orbital moment is likely to dominantly point along the $c$ -axis, and this appears to be the most consistent with the existing RXS azimuthal dependence (see Supplementary Information) although further spectroscopic studies will be needed in order to reach a firm conclusion.",
"Finally, while our model study has focused on the largest part of the energetics, a complete understanding will require the addition of couplings with smaller magnitude, such as the interlayer couplings.",
"In summary, our experimental and theoretical results uncover the microscopic nature of the antiferromagnetic state in $\\alpha $ -Sr$_2$ CrO$_4$ , which is here shown using resonant X-ray scattering to arise from collinear Néel-type order at ${\\mathbf {Q}}_{N} = \\left( \\pm 1/2, \\pm 1/2, L \\right)$ with the spin axis in the CrO$_2$ plane, in close analogy with the parent state of cuprate superconductors.",
"In addition to Néel order, we have detected coexisting stripe-like order setting in at a lower temperature $T_s \\sim 50$ K, with ordering vector $ {\\mathbf {Q}}_{s} = \\left( 1/2, 0, L \\right)$ .",
"A Monte Carlo study of the spin-1 Kugel-Khomskii Hamiltonian captures the observed experimental signatures of Néel and stripe phases and their ordering sequence, thereby pointing to the physical realization of this model for a system of $d^2$ ions on a square lattice.",
"Our study singles out $\\alpha $ -Sr$_2$ CrO$_4$ as a unique platform to explore the rich electronic phases in a Kugel-Khomskii-like spin-orbital system, and underscores the great scientific potential of a unique transition metal oxide, whose electronic phase diagram as a function of carrier doping is yet to be explored.",
"We are grateful to A. Chubukov, D. Puggioni, J.M.",
"Tranquada, and P.A.",
"Lee for insightful discussions.",
"Work at MIT (Z.H.Z., C.A.O., J.L., J.P., and R.C.)",
"has been supported by the Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award Number DE-SC0019126.",
"Part of this research concerning sample synthesis (Z.H.Z.)",
"has been supported by NSF through the Massachusetts Institute of Technology Materials Research Science and Engineering Center DMR – 1419807.",
"Work at Rice (W.H.",
"and Q.S.)",
"has in part been supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award No.",
"DE-SC0018197 and by the Robert A. Welch Foundation Grant No.",
"C-1411.",
"R.C.",
"acknowledges support from the Alfred P. Sloan Foundation.",
"M.N.",
"was supported by the Materials Sciences and Engineering Division, Basic Energy Sciences, Office of Science, US DOE.",
"J. P. acknowledges financial support by the Swiss National Science Foundation Early Postdoc Mobility Fellowship Project No.",
"P2FRP2_171824 and P400P2_180744.",
"This research used beamline 4-ID of the National Synchrotron Light Source II, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Brookhaven National Laboratory under Contract No.",
"DE-SC0012704.",
"D.~Vaknin, S.~K.",
"Sinha, D.~E.",
"Moncton, D.~C.",
"Johnston, J.~M.",
"Newsam, C.~R.",
"Safinya, and H.~E.",
"King~Jr., Physical Review Letters 58, 2802 (1987).",
"L.~Paolasini, R.~Caciuffo, A.~Sollier, P.~Ghigna, and M.~Altarelli, Physical Review Letters 88, 106403 (2002).",
"A.~M.",
"Oleś, G.~Khaliullin, P.~Horsch, and L.~F.",
"Feiner, Physical Review B 72, 214431 (2005), URL https://link.aps.org/doi/10.1103/PhysRevB.72.214431.",
"A.~M.",
"Oleś, P.~Horsch, and G.~Khaliullin, Physical Review B 75, 184434 (2007), URL https://link.aps.org/doi/10.1103/PhysRevB.75.184434.",
"F.~Krüger, S.~Kumar, J.~Zaanen, and J.~van~den Brink, Physical Review B 79, 054504 (2009), URL https://link.aps.org/doi/10.1103/PhysRevB.79.054504.",
"authorD.~Vaknin, authorS.~K.",
"Sinha, authorD.~E.",
"Moncton, authorD.~C.",
"Johnston, authorJ.~M.",
"Newsam, authorC.~R.",
"Safinya, and authorH.~E.",
"King~Jr., journalPhysical Review Letters volume58, pages2802 (year1987).",
"authorL.~Paolasini, authorR.~Caciuffo, authorA.~Sollier, authorP.~Ghigna, and authorM.~Altarelli, journalPhysical Review Letters volume88, pages106403 (year2002).",
"authorA.~M.",
"Oleś, authorG.~Khaliullin, authorP.~Horsch, and authorL.~F.",
"Feiner, journalPhysical Review B volume72, pages214431 (year2005), https://link.aps.org/doi/10.1103/PhysRevB.72.214431.",
"authorA.~M.",
"Oleś, authorP.~Horsch, and authorG.~Khaliullin, journalPhysical Review B volume75, pages184434 (year2007), https://link.aps.org/doi/10.1103/PhysRevB.75.184434.",
"authorF.~Krüger, authorS.~Kumar, authorJ.~Zaanen, and authorJ.~van~den Brink, journalPhysical Review B volume79, pages054504 (year2009), https://link.aps.org/doi/10.1103/PhysRevB.79.054504.",
"authorY.~Joly, authorO.~Bunǎu, authorJ.~E.",
"Lorenzo, authorR.~M.",
"Galéra, authorS.~Grenier, and authorB.~Thompson, journalJournal of Physics: Conference Series volume190, pages012007 (year2009), ISSN issn1742-6596,"
]
] | 1906.04194 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.